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\section{Introduction}
The millimeter (mmWave) band in the range of $30$-$300$ GHz and massive multiple-input multiple-output (MIMO) are expected to significantly enhance the performance of future wireless communications systems \cite{6894453}. The high sampling frequency and large number of radio frequency (RF) chains, however, result in a prohibitive amount of power consumption. Among various solutions, hybrid architectures and low-resolution analog-to-digital converters (ADCs) are considered as practical candidates that achieve high energy efficiency \cite{7876856}.
To perform sparse recovery in massive MIMO systems with low-resolution ADCs, various compressed sensing (CS) algorithms were proposed \cite{8171203, 8320852, 8310593, 8515242}. The generalized approximate message passing (GAMP) \cite{8171203}, vector AMP (VAMP) \cite{8171203}, generalized expectation consistent-signal recovery (GEC-SR) \cite{8320852}, and variational Bayesian-sparse Bayesian learning (VB-SBL) \cite{8310593} algorithms are CS techniques that estimate the parameters of interest on the grid, so the off-grid error is inevitable. The atomic norm minimization (ANM) algorithm for mixed one-bit ADCs \cite{8515242}, on the other hand, estimates the channel off the grid. ANM, however, requires high-resolution ADCs to determine the regularization parameter that is essential for ANM to run.
In this paper, the Newtonized fully corrective forward greedy selection-cross validation-based (NFCFGS-CV-based) channel estimator is proposed for mmWave hybrid massive MIMO systems with low-resolution ADCs. The proposed NFCFGS algorithm is a gridless CS technique that is inspired by the FCFGS \cite{doi:10.1137/090759574} and Newtonized orthogonal matching pursuit (NOMP) \cite{7491265} algorithms. NFCFGS estimates single path off the grid at each iteration based on the previously estimated paths. To determine when to terminate without any prior knowledge on the number of paths, the CV technique is adopted, which is a model validation technique for overfitting prevention.
\textbf{Notation:} $a$, $\mathbf{a}$, and $\mathbf{A}$ denote a scalar, vector, and matrix. The element restriction of $\mathbf{a}$ to the index set $\mathcal{I}$ is $\mathbf{a}_{\mathcal{I}}$. $\mathbf{a}\succ\mathbf{b}$ represents the element-wise inequality between $\mathbf{a}$ and $\mathbf{b}$. The column-wise vectorization of $\mathbf{A}$ is $\mathrm{vec}(\mathbf{A})$. The Kronecker product of $\mathbf{A}$ and $\mathbf{B}$ is $\mathbf{A}\otimes\mathbf{B}$. $\llbracket n\rrbracket$ denotes $\llbracket n\rrbracket=\{1, \dots, n\}$.
\section{System Model}
Consider an uplink massive MIMO system with a base station and $K$ single-antenna users. The base station has $M$ antennas and $R$ RF chains where $R\leq M$. As a means to reduce the power consumption, each RF chain is equipped with a pair of $B$-bit ADCs for the real and imaginary parts. The system operates in the mmWave wideband with $D$ delay taps and sampling period $T_{s}$. In this section, the system model of the channel estimation phase is described.
The received signal $\mathbf{y}[n]\in\mathbb{C}^{R}$ at time $n$ is
\begin{align}\label{received_signal_at_n}
\mathbf{y}[n]=&\mathbf{W}[n]^{\mathrm{H}}\times\notag\\
&\left(\underbrace{\begin{bmatrix}\mathbf{H}[0]&\cdots&\mathbf{H}[D-1]\end{bmatrix}}_{=\mathbf{H}}\underbrace{\begin{bmatrix}\mathbf{s}[n]\\\vdots\\\mathbf{s}[n-D+1]\end{bmatrix}}_{=\mathbf{s}_{n}}+\bar{\mathbf{v}}[n]\right)\notag\\
=&\mathbf{W}[n]^{\mathrm{H}}\mathbf{H}\mathbf{s}_{n}+\mathbf{W}[n]^{\mathrm{H}}\bar{\mathbf{v}}[n]\notag\\
=&(\underbrace{\mathbf{s}_{n}^{\mathrm{T}}\otimes\mathbf{W}[n]^{\mathrm{H}}}_{=\bar{\mathbf{A}}[n]})\underbrace{\mathrm{vec}(\mathbf{H})}_{=\mathbf{h}}+\underbrace{\mathbf{W}[n]^{\mathrm{H}}\bar{\mathbf{v}}[n]}_{=\mathbf{v}[n]}
\end{align}
where the last line is due to $\mathrm{vec}(\mathbf{ABC})=(\mathbf{C}^{\mathrm{T}}\otimes\mathbf{A})\mathrm{vec}(\mathbf{B})$. Here, $\mathbf{H}[d]\in\mathbb{C}^{M\times K}$ is the $d$-th delay tap, $\mathbf{s}[n]\in\mathbb{C}^{K}$ is the training signal at time $n$ under the transmit power constraint $\mathbb{E}\{\mathbf{s}[n]\mathbf{s}[n]^{\mathrm{H}}\}=\rho\mathbf{I}_{K}$, $\bar{\mathbf{v}}[n]\sim\mathcal{CN}(\mathbf{0}_{M}, \mathbf{I}_{M})$ is the additive white Gaussian noise (AWGN) at time $n$, and $\mathbf{W}[n]\in\mathbb{C}^{M\times R}$ is the RF combiner at time $n$ with a network of phase shifters that satisfies $\mathbf{W}[n]^{\mathrm{H}}\mathbf{W}[n]=\mathbf{I}_{R}$. Therefore, $\mathbf{v}[n]\sim\mathcal{CN}(\mathbf{0}_{R}, \mathbf{I}_{R})$, and the signal-to-noise ratio (SNR) is $\rho$.
The channels in the mmWave band consist of a small number of paths due to the severe path loss. Therefore, the channel of the $k$-th user that corresponds to the $k$-th column of $\mathbf{H}[d]$ is \cite{6834753}
\begin{align}\label{channel}
\mathbf{h}_{k}[d]&=\sum_{\ell=1}^{L_{k}}\alpha_{k, \ell}\underbrace{p(dT_{s}-\tau_{k, \ell})\mathbf{a}(\theta_{k, \ell})}_{=\mathbf{a}_{d}(\theta_{k, \ell}, \tau_{k, \ell})}\notag\\
&=\underbrace{\begin{bmatrix}\mathbf{a}_{d}(\theta_{k, 1}, \tau_{k, 1})&\cdots&\mathbf{a}_{d}(\theta_{k, L_{k}}, \tau_{k, L_{k}})\end{bmatrix}}_{=\mathbf{F}_{k}[d]}\underbrace{\begin{bmatrix}\alpha_{k, 1}\\\vdots\\\alpha_{k, L_{k}}\end{bmatrix}}_{=\bm{\alpha}_{k}}
\end{align}
where $L_{k}$ is the number of paths, $\alpha_{k, \ell}\sim\mathcal{CN}(0, 1)$ is the $\ell$-th path gain, $\theta_{k, \ell}\sim\mathrm{Uniform}([-\pi/2, \pi/2])$ is the $\ell$-th angle-of-arrival (AoA), $\tau_{k, \ell}\sim\mathrm{Uniform}([0, (D-1)T_{s}])$ is the $\ell$-th delay, $\mathbf{a}(\theta)\in\mathbb{C}^{M}$ is the array response vector, and $p(t)$ is the pulse shaping filter.
The received signal over the channel estimation phase of length $N$ is
\begin{equation}\label{received_signal_over_N}
\mathbf{y}=\begin{bmatrix}\mathbf{y}[1]\\\vdots\\\mathbf{y}[N]\end{bmatrix}=\underbrace{\begin{bmatrix}\bar{\mathbf{A}}[1]\\\vdots\\\bar{\mathbf{A}}[N]\end{bmatrix}}_{=\bar{\mathbf{A}}}\mathbf{h}+\underbrace{\begin{bmatrix}\mathbf{v}[1]\\\vdots\\\mathbf{v}[N]\end{bmatrix}}_{=\mathbf{v}}.
\end{equation}
Then, $\mathbf{y}$ is quantized by $B$-bit ADCs as $\hat{\mathbf{y}}=\mathrm{Q}(\mathbf{y})$ where $\mathrm{Q}(\cdot)$ is the $B$-bit quantization function that operates element-wise as
\begin{equation}\label{quantization_function}
\hat{y}=\mathrm{Q}(y)\iff\begin{cases}\mathrm{Re}(\hat{y}^{\mathrm{lo}})\leq\mathrm{Re}(y)\leq\mathrm{Re}(\hat{y}^{\mathrm{up}})\\\mathrm{Im}(\hat{y}^{\mathrm{lo}})\leq\mathrm{Im}(y)\leq\mathrm{Im}(\hat{y}^{\mathrm{up}})\end{cases}.
\end{equation}
Here, $\hat{y}^{\mathrm{lo}}\in\mathbb{C}$ and $\hat{y}^{\mathrm{up}}\in\mathbb{C}$ are the lower and upper thresholds that map to $\hat{y}\in\mathbb{C}$, so the real and imaginary parts of $\hat{y}^{\mathrm{lo}}$, $\hat{y}^{\mathrm{up}}$, and $\hat{y}$ correspond to one of the $2^{B}$ quantization intervals. For notational convenience, the collection of the lower and upper thresholds that map to $\hat{\mathbf{y}}$ are denoted as $\hat{\mathbf{y}}^{\mathrm{lo}}\in\mathbb{C}^{RN}$ and $\hat{\mathbf{y}}^{\mathrm{up}}\in\mathbb{C}^{RN}$.
Now, let us rewrite $\mathbf{h}$ as
\begin{equation}
\mathbf{h}=\underbrace{\begin{bmatrix}\mathbf{F}_{1}[0]&&\\&\ddots&\\&&\mathbf{F}_{K}[0]\\&\vdots&\\\mathbf{F}_{1}[D-1]&&\\&\ddots&\\&&\mathbf{F}_{K}[D-1]\end{bmatrix}}_{=\mathbf{F}(\mathcal{P})}\underbrace{\begin{bmatrix}\bm{\alpha}_{1}\\\vdots\\\bm{\alpha}_{K}\end{bmatrix}}_{=\bm{\alpha}}
\end{equation}
using \eqref{received_signal_at_n} and \eqref{channel}. Here, $\mathcal{P}$ is the collection of all $(\theta_{k, \ell}, \tau_{k, \ell})$. Then, \eqref{received_signal_over_N} becomes
\begin{equation}\label{modified_received_signal_over_N}
\mathbf{y}=\bar{\mathbf{A}}\mathbf{F}(\mathcal{P})\bm{\alpha}+\mathbf{v}
\end{equation}
where $\mathbf{A}(\mathcal{P})=\bar{\mathbf{A}}\mathbf{F}(\mathcal{P})\in\mathbb{C}^{RN\times(L_{1}+\cdots+L_{K})}$. The goal is to estimate $(\bm{\alpha}, \mathcal{P})$ from $\hat{\mathbf{y}}$. The problem, however, is that $\mathbf{F}(\mathcal{P})$ is nonlinear with respect to $\mathcal{P}$. Furthermore, there is no prior knowledge on the number of paths, which further complicates the situation.
\section{Proposed NFCFGS-CV Algorithm}
In this section, the NFCFGS-CV-based channel estimator is proposed. The NFCFGS algorithm is a gridless CS technique that performs single path estimation at each iteration. The termination condition of NFCFGS is determined by the CV technique, which is a model validation technique that prevents overfitting.
\subsection{Proposed NFCFGS Algorithm}
The proposed NFCFGS algorithm is a gridless CS technique that combines FCFGS \cite{doi:10.1137/090759574} and the NOMP algorithm \cite{7491265}. In particular, one iteration of NFCFGS consists of single path estimation that is performed based on the previously estimated paths. To develop NFCFGS, define $\hat{\mathcal{P}}$ as the collection of all the previously estimated $(\theta_{k, \ell}, \tau_{k, \ell})$, and $\hat{\bm{\alpha}}\in\mathbb{C}^{|\hat{\mathcal{P}}|}$ as the collection of all the previously estimated $\alpha_{k, \ell}$. The goal is to estimate the path that is parametrized by $(\alpha, \theta, \tau, k)$ using $(\hat{\bm{\alpha}}, \hat{\mathcal{P}})$, while $k\in\llbracket K\rrbracket$ is unknown a priori.
To proceed, define $\mathbf{a}(\theta, \tau, k)\in\mathbb{C}^{RN}$ using $(\theta, \tau, k)$ by the same logic as $\mathbf{A}(\mathcal{P})$ is defined in \eqref{modified_received_signal_over_N} using $\mathcal{P}$. Then, the unquantized received signal becomes
\begin{equation}
\mathbf{y}=\alpha\mathbf{a}(\theta, \tau, k)+\mathbf{A}(\hat{\mathcal{P}})\hat{\bm{\alpha}}+\mathbf{v}
\end{equation}
where $\mathbf{z}=\mathbf{A}(\hat{\mathcal{P}})\hat{\bm{\alpha}}\in\mathbb{C}^{RN}$, and the likelihood function is
\begin{align}\label{likelihood_function}
&\ell_{\mathbf{z}}(\alpha, \theta, \tau, k)=\Pr\left[\hat{\mathbf{y}}\middle|\alpha, \theta, \tau, k, \hat{\bm{\alpha}}, \hat{\mathcal{P}}\right]=\notag\\
&\Pr\left[\mathrm{Re}(\hat{\mathbf{y}}^{\mathrm{lo}})\preceq\mathrm{Re}(\mathbf{y})\preceq\mathrm{Re}(\hat{\mathbf{y}}^{\mathrm{up}})\middle|\alpha, \theta, \tau, k, \hat{\bm{\alpha}}, \hat{\mathcal{P}}\right]\times\notag\\
&\Pr\left[\mathrm{Im}(\hat{\mathbf{y}}^{\mathrm{lo}})\preceq\mathrm{Im}(\mathbf{y})\preceq\mathrm{Im}(\hat{\mathbf{y}}^{\mathrm{up}})\middle|\alpha, \theta, \tau, k, \hat{\bm{\alpha}}, \hat{\mathcal{P}}\right]
\end{align}
where the last equality is due to $\eqref{quantization_function}$ and the fact that the real and imaginary parts of $\mathbf{v}\sim\mathcal{CN}(\mathbf{0}_{RN}, \mathbf{I}_{RN})$ are independent. Since
\begin{gather}
\mathrm{Re}(\hat{\mathbf{y}}^{\mathrm{lo}})\preceq\mathrm{Re}(\mathbf{y})\preceq\mathrm{Re}(\hat{\mathbf{y}}^{\mathrm{up}})\notag\\
\Updownarrow\notag\\
\mathrm{Re}(\hat{\mathbf{y}}^{\mathrm{lo}}-\mathbf{z})\preceq\mathrm{Re}(\alpha\mathbf{a}(\theta, \tau, k)+\mathbf{v})\preceq\mathrm{Re}(\hat{\mathbf{y}}^{\mathrm{up}}-\mathbf{z})\notag,
\end{gather}
the first term in the last equality of \eqref{likelihood_function} corresponds to the probability that $\mathcal{N}(\mathrm{Re}(\alpha\mathbf{a}(\theta, \tau, k)), 1/2\cdot\mathbf{I}_{RN})$ be in the box that has $\mathrm{Re}(\hat{\mathbf{y}}^{\mathrm{lo}}-\mathbf{z})$ and $\mathrm{Re}(\hat{\mathbf{y}}^{\mathrm{up}}-\mathbf{z})$ as the lower and upper boundaries. The same argument holds for the imaginary part. The two terms of $\ell_{\mathbf{z}}(\alpha, \theta, \tau, k)$ have well-known closed-form expressions that are log-concave with respect to $\alpha$ \cite{9351751, 7439790, boyd2004convex}. Therefore, $\ell_{\mathbf{z}}(\alpha, \theta, \tau, k)$ is log-concave with respect to $\alpha$. The problem, however, is that $\ell_{\mathbf{z}}(\alpha, \theta, \tau, k)$ is nonconvex with respect to $(\theta, \tau, k)$.
In analogy to FCFGS that performs support recovery by identifying the support that maximizes the derivative of the log-likelihood function with respect to the support element at $0$, NFCFGS estimates $(\theta, \tau, k)$ as follows. First, NFCFGS interprets $(\theta, \tau, k)$ and $\alpha$ as the support and support element. Then, $(\theta, \tau, k)$ is estimated by solving
\begin{alignat}{2}\label{aoa_delay_estimation}
&\underset{\theta, \tau, k}{\text{maximize}}\ &&\underbrace{|\nabla_{\alpha}\log \ell_{\mathbf{z}}(0, \theta, \tau, k)|^{2}}_{=f_{\mathbf{z}}(\theta, \tau, k)}\notag\\
&\text{subject}\ \text{to}\ &&(\theta, \tau, k)\in[-\pi/2, \pi/2]\times[0, (D-1)T_{s}]\times\llbracket K\rrbracket.
\end{alignat}
To solve \eqref{aoa_delay_estimation}, NFCFGS first maximizes $f_{\mathbf{z}}(\theta, \tau, k)$ on the grid
\begin{equation}
\Omega\subseteq[-\pi/2, \pi/2]\times[0, (D-1)T_{s}]\times\llbracket K\rrbracket
\end{equation}
that discretizes the constraint in \eqref{aoa_delay_estimation}. In practice, $[-\pi/2, \pi/2]$ and $[0, (D-1)T_{s}]$ are discretized by $\geq 2M$ and $\geq 2D$ points \cite{7491265, 7961152, 7458188}. Then, the coarse estimate $(\hat{\theta}, \hat{\tau})$ is refined off the grid using Newton's method for nonconvex optimization as \cite{murphy2012machine}
\begin{align}
(\hat{\theta}, \hat{\tau})+\begin{cases}\eta\mathbf{n}_{\mathbf{z}}(\hat{\theta}, \hat{\tau}, \hat{k})&\text{if}\ \nabla_{(\theta, \tau)}^{2}f_{\mathbf{z}}(\hat{\theta}, \hat{\tau}, \hat{k})\prec\mathbf{0}_{2\times 2}\\
\eta\mathbf{g}_{\mathbf{z}}(\hat{\theta}, \hat{\tau}, \hat{k})&\text{if}\ \nabla_{(\theta, \tau)}^{2}f_{\mathbf{z}}(\hat{\theta}, \hat{\tau}, \hat{k})\nprec\mathbf{0}_{2\times 2}\end{cases}
\end{align}
at each iteration where
\begin{align}
&\mathbf{n}_{\mathbf{z}}(\theta, \tau, k)=-\nabla_{(\theta, \tau)}^{2}f_{\mathbf{z}}(\theta, \tau, k)^{-1}\nabla_{(\theta, \tau)}f_{\mathbf{z}}(\theta, \tau, k),\\
&\mathbf{g}_{\mathbf{z}}(\theta, \tau, k)=\nabla_{(\theta, \tau)}f_{\mathbf{z}}(\theta, \tau, k)
\end{align}
are the Newton and gradient steps, and $\eta$ is the step size. The objective of the Newton refinement is to reduce the off-grid error, which is a NOMP-inspired technique. Finally, NFCFGS updates $\hat{\mathcal{P}}$ as $\hat{\mathcal{P}}\cup\{(\hat{\theta}, \hat{\tau}, \hat{k})\}$.
To update the path gains, denote the collected path gains of $\hat{\mathcal{P}}$ that are to be estimated as $\mathbf{x}\in\mathbb{C}^{|\hat{\mathcal{P}}|}$. Then, the unquantized received signal is $\mathbf{y}=\mathbf{A}(\hat{\mathcal{P}})\mathbf{x}+\mathbf{v}$, and the likelihood function is given as
\begin{align}
&\ell(\mathbf{x}, \hat{\mathcal{P}})=\Pr\left[\hat{\mathbf{y}}\middle|\mathbf{x}, \hat{\mathcal{P}}\right]=\notag\\
&\Pr\left[\mathrm{Re}(\hat{\mathbf{y}}^{\mathrm{lo}})\preceq\mathrm{Re}(\mathbf{A}(\hat{\mathcal{P}})\mathbf{x}+\mathbf{v})\preceq\mathrm{Re}(\hat{\mathbf{y}}^{\mathrm{up}})\middle|\mathbf{x}, \hat{\mathcal{P}}\right]\times\notag\\
&\Pr\left[\mathrm{Im}(\hat{\mathbf{y}}^{\mathrm{lo}})\preceq\mathrm{Im}(\mathbf{A}(\hat{\mathcal{P}})\mathbf{x}+\mathbf{v})\preceq\mathrm{Im}(\hat{\mathbf{y}}^{\mathrm{up}})\middle|\mathbf{x}, \hat{\mathcal{P}}\right].
\end{align}
Again, $\ell(\mathbf{x}, \hat{\mathcal{P}})$ has a well-known closed form expression that is log-concave with respect to $\mathbf{x}$, which can be verified using the same argument from \eqref{likelihood_function}. So, NFCFGS estimates $\mathbf{x}$ according to the maximum likelihood criterion as
\begin{equation}
\hat{\bm{\alpha}}=\argmax_{\mathbf{x}\in\mathbb{C}^{|\hat{\mathcal{P}}|}}\underbrace{\log\ell(\mathbf{x}, \hat{\mathcal{P}})}_{=g(\mathbf{x}, \hat{\mathcal{P}})},
\end{equation}
whose global optimum is attained by convex optimization, gradient descent method, for example.
The NFCFGS algorithm is outlined in Algorithm \ref{nfcfgs_cv}, and the CV technique along with $\mathcal{E}$ and $\mathcal{CV}$ in the superscripts and subscripts are explained shortly after. In Line 5, $(\theta, \tau, k)$ is coarsely estimated on the grid. Then, $(\hat{\theta}, \hat{\tau})$ is refined off the grid using Newton's method in Lines 8 and 10. After updating $\hat{\mathcal{P}}$ in Line 13, the path gains are estimated using convex optimization in Line 14. The exact forms of $f_{\mathbf{z}}(\cdot, \cdot, \cdot)$, $\mathbf{n}_{\mathbf{z}}(\cdot, \cdot, \cdot)$, $\mathbf{g}_{\mathbf{z}}(\cdot, \cdot, \cdot)$, and $g(\cdot, \cdot)$ are omitted due to the page limit, and the interested reader is referred to \cite{9351751}.
\begin{algorithm}[t]
\caption{NFCFGS-CV algorithm}\label{nfcfgs_cv}
\begin{algorithmic}[1]
\Require $\hat{\mathbf{y}}$
\Ensure $(\hat{\bm{\alpha}}, \hat{\mathcal{P}})$
\State // $g_{\mathcal{CV}}(\hat{\bm{\alpha}}, \emptyset)=-\infty$ and $\mathbf{A}(\emptyset)\hat{\bm{\alpha}}=\mathbf{0}_{RN}$ by convention
\State $\hat{\mathcal{P}}\coloneqq\emptyset$
\Do
\State $\epsilon\coloneqq g_{\mathcal{CV}}(\hat{\bm{\alpha}}, \hat{\mathcal{P}})$ and $\mathbf{z}\coloneqq\mathbf{A}(\hat{\mathcal{P}})\hat{\bm{\alpha}}$
\State $(\hat{\theta}, \hat{\tau}, \hat{k})\coloneqq\displaystyle\argmax_{(\theta, \tau, k)\in\Omega}f_{\mathbf{z}}^{\mathcal{E}}(\theta, \tau, k)$
\While {termination condition}
\If {$\nabla_{(\theta, \tau)}^{2}f_{\mathbf{z}}^{\mathcal{E}}(\hat{\theta}, \hat{\tau}, \hat{k})\prec\mathbf{0}_{2\times 2}$}
\State $(\hat{\theta}, \hat{\tau})\coloneqq(\hat{\theta}, \hat{\tau})+\eta\mathbf{n}_{\mathbf{z}}^{\mathcal{E}}(\hat{\theta}, \hat{\tau}, \hat{k})$
\Else
\State $(\hat{\theta}, \hat{\tau})\coloneqq(\hat{\theta}, \hat{\tau})+\eta\mathbf{g}_{\mathbf{z}}^{\mathcal{E}}(\hat{\theta}, \hat{\tau}, \hat{k})$
\EndIf
\EndWhile
\State $\hat{\mathcal{P}}\coloneqq\hat{\mathcal{P}}\cup\{(\hat{\theta}, \hat{\tau}, \hat{k})\}$
\State $\hat{\bm{\alpha}}\coloneqq\displaystyle\argmax_{\mathbf{x}\in\mathbb{C}^{|\hat{\mathcal{P}}|}}g_{\mathcal{E}}(\mathbf{x}, \hat{\mathcal{P}})$
\doWhile $g_{\mathcal{CV}}(\hat{\bm{\alpha}}, \hat{\mathcal{P}})>\epsilon$
\end{algorithmic}
\end{algorithm}
\subsection{Proposed CV Technique}
The optimal termination condition of NFCFGS depends on $\{L_{k}\}_{\forall k}$. In practice, however, the prior knowledge on $\{L_{k}\}_{\forall k}$ is hardly available, which calls for a more feasible termination condition. In this paper, the CV technique \cite{4301267, 5319752} is adopted as an indicator of termination, which is a model validation technique that assesses the quality of the estimate to prevent overfitting. The resulting algorithm is dubbed NFCFGS-CV.
To apply CV, $\hat{\mathbf{y}}$ is partitioned to the estimation data $\hat{\mathbf{y}}_{\mathcal{E}}\in\mathbb{C}^{|\mathcal{E}|}$ and CV data $\hat{\mathbf{y}}_{\mathcal{CV}}\in\mathbb{C}^{|\mathcal{CV}|}$ where $\mathcal{E}$ and $\mathcal{CV}$ are the index sets that form a partition of $\llbracket RN\rrbracket$. Then, channel estimation is performed based on $\hat{\mathbf{y}}_{\mathcal{E}}$, while the estimation quality is assessed based on $\hat{\mathbf{y}}_{\mathcal{CV}}$. The disjoint nature of $\hat{\mathbf{y}}_{\mathcal{E}}$ and $\hat{\mathbf{y}}_{\mathcal{CV}}$ enables CV to assess the estimation quality without any bias.
To solidify the concept of NFCFGS-CV, define the estimation functions
\begin{equation}\label{estimation_function}
\begin{alignedat}{2}
&f_{\mathbf{z}}^{\mathcal{E}}(\theta, \tau, k) &&\text{: square of the magnitude of the derivative},\\
&\mathbf{n}_{\mathbf{z}}^{\mathcal{E}}(\theta, \tau, k)&&\text{: Newton step},\\
&\mathbf{g}_{\mathbf{z}}^{\mathcal{E}}(\theta, \tau, k)&&\text{: gradient step},\\
&g_{\mathcal{E}}(\mathbf{x}, \hat{\mathcal{P}}) &&\text{: log-likelihood function},
\end{alignedat}
\end{equation}
and CV function
\begin{equation}\label{cv_function}
g_{\mathcal{CV}}(\mathbf{x}, \hat{\mathcal{P}})\text{: log-likelihood function}
\end{equation}
where the estimation (and CV) functions are obtained by deriving $f_{\mathbf{z}}(\theta, \tau, k)$, $\mathbf{n}_{\mathbf{z}}(\theta, \tau, k)$, $\mathbf{g}_{\mathbf{z}}(\theta, \tau, k)$, and $g(\mathbf{x}, \hat{\mathcal{P}})$ as in the previous subsection but under the assumption that the quantized received signal is $\hat{\mathbf{y}}_{\mathcal{E}}$ (and $\hat{\mathbf{y}}_{\mathcal{CV}}$) instead of $\hat{\mathbf{y}}$. Then, NFCFGS-CV proceeds as follows in Algorithm \ref{nfcfgs_cv}. In Lines 5, 8, 10, 13, and 14, NFCFGS performs channel estimation as discussed in the previous subsection but based on \eqref{estimation_function}. Then, Line 15 cross validates the estimation quality based on \eqref{cv_function}, which is terminated when the estimation quality starts to fall---an indication of overfitting. As a sidenote, $g_{\mathcal{CV}}(\hat{\bm{\alpha}}, \hat{\mathcal{P}})$ is in fact proportional to $-\|\hat{\mathbf{h}}-\mathbf{h}\|^{2}$, which is rigorously stated and proved in \cite{9351751}. Therefore, CV enables NFCFGS-CV to achieve the minimum squared error (SE) by terminating at the right timing.
\section{Simulation Results}
In this section, the accuracy of the NFCFGS-CV-based channel estimator is investigated based on the normalized mean SE (NMSE) $\mathbb{E}\{\|\hat{\mathbf{h}}-\mathbf{h}\|^{2}/\|\mathbf{h}\|^{2}\}$. The base station employs a half-wavelength antenna spacing uniform linear array (ULA) with uniform ADCs, whose quantization intervals vary with the SNR. The columns of the RF combiner are composed of circularly shifted ZC sequences. Likewise, the training signals are constructed as circularly shifted ZC sequences, while the raised-cosine (RC) pulse shaping filter is adopted with a roll-off factor of $0.35$. For NFCFGS-CV, $80$\% of the training signals are allocated to the estimation data, and the remaining $20$\% to the CV data.
As baselines, the GAMP \cite{8171203}, VAMP \cite{8171203}, GEC-SR \cite{8320852}, and VB-SBL \cite{8310593} algorithms are considered, which are state-of-the-art on-grid CS algorithms for low-resolution ADCs. GAMP, VAMP, and GEC-SR model the prior distribution of $\mathbf{h}$ as a Gaussian mixture, while VB-SBL uses the Student-t distribution. Unlike NFCFGS-CV, these algorithms use all the training signals to estimate the channel.
In Fig. \ref{figure_1}, the NMSEs of NFCFGS-CV and other baselines are shown for various SNRs with $B=1, 2, 3, 4$. The simulation setting is $M=32$, $R=8$, $K=4$, $D=4$, $L_{k}=2$ for all $k$, $N=1600$, and $T_{s}=1/600$ $\mu$s, which translates to $600$ MHz bandwidth. Fig. \ref{figure_1} shows the superior performance of NFCFGS-CV over other baselines at all SNRs and $B$. The performance gain is due to the gridless nature of NFCFGS-CV along with the proper termination condition that captures the minimum SE-achieving timing. Therefore, NFCFGS-CV yields an accurate channel estimate, which is critical to establishing a reliable data transmission link.
\begin{figure}[t]
\centering
\includegraphics[width=1\columnwidth]{figure_1.eps}
\caption[caption]{NMSE versus SNR for $B=1, 2, 3, 4$. In the last subplot, $B=\infty$ is shown as a reference with a cross.}\label{figure_1}
\end{figure}
\section{Conclusion}
The NFCFGS-CV-based channel estimator was proposed for mmWave hybrid massive MIMO systems with low-resolution ADCs. The channel was estimated by the NFCFGS algorithm, which is a gridless CS algorithm that performs single path estimation at each iteration. The CV technique was adopted to identify the proper termination condition when the number of paths is unknown a priori. The simulation results demonstrated that NFCFGS-CV outperforms state-of-the-art on-grid CS algorithms for low-resolution ADCs.
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2022-07-15T02:00:36",
"yymm": "2207",
"arxiv_id": "2207.06451",
"language": "en",
"url": "https://arxiv.org/abs/2207.06451"
}
|
\section{Introduction}
Realistic quantum systems inherently interact with their surroundings and can be generally modeled by the open quantum dynamics. In the weak coupling limit between the system and the environment, the dynamics would be Markovian and described by the so-called quantum Markov semigroup (QMS) or the Lindblad equation, which is a natural analog of the Fokker-Planck equation in the quantum setting \cite{breuer2002theory}. Similarly to the theory of Markov semigroups, the analysis of the mixing time
is of central importance for a QMS, and is closely related to the functional inequalities.
In this work, we are interested in a class of convex Sobolev inequalities, referred to as quantum Beckner's inequalities. We will investigate their main properties and relations with other known quantum functional inequalities, such as Poincar\'{e} and modified log-Sobolev inequalities, via both algebraic and geometric approaches.
\subsection{Classical convex Sobolev inequality} To motivate this work, we first review the results on the convex Sobolev inequalities in the classical setting.
Let $(P_t)_{t \ge 0}$ be the symmetric diffusion semigroup associated with a Markov process $(X_t)_{t \ge 0}$ on a Riemannian manifold $M$ with metric $g(\cdot,\cdot)$ that has a unique invariant measure $\pi$. We denote by $L$ the generator of $P_t$ and define the Dirichlet form $\mc{E}(f,g) := - \pi [f L g]$ for functions $f$ and $g$. Bakry and \'{E}mery in their seminal work \cite{bakry1985diffusions} showed that if there exists $\kappa > 0$ such that for $f \ge 0$,
\begin{align} \label{eq:curvature_dimension}
\Gamma_2(f) \ge \kappa \Gamma (f)\,,
\end{align}
then the convex Sobolev inequality holds:
\begin{align} \label{eq:convex_ineq}
2 \kappa \Ent_\pi^\phi(f) \le \mc{E}(\phi'(f),f)\,,
\end{align}
which is equivalent to the exponential decay of the $\phi$-entropy $\Ent_\pi^\phi(f) := \pi[\phi(f)] - \phi(\pi[f])$ and characterizes the convergence rate of the Markov process towards its invariant measure. Here $\Gamma$ and $\Gamma_2$ are carr\'{e} du champ operators \cite{bakry2014analysis}; $\phi:[0,\infty) \to \mathbb{R}$ is assumed to be
a smooth convex function such that $\phi(1) = \phi'(1) = 0$ and $1/\phi''$ is concave. In the cases: $\phi_1(s) = s(\log s - 1) + 1$ and $\phi_2(s) = s^2 - 2 s + 1$, up to some constant, \eqref{eq:convex_ineq} gives the well-known modified log-Sobolev inequality (MLSI) and the Poincar\'{e} inequality, respectively,
\begin{align} \label{ineq_a}
\alpha \big(\pi[f \log f] - \pi[f] \log \pi[f]\big) \le \mc{E}(\log f, f)\,,
\end{align}
and
\begin{align} \label{ineq_b}
\lambda \big(\pi[f^2] - \pi[f]^2 \big) \le \mc{E}(f , f)\,.
\end{align}
If we consider the interpolating family $\phi_p(s) = (s^p - s)/(p - 1) - s + 1$, $1 < p \le 2$, between $\phi_1$ and $\phi_2$, we obtain the Beckner's inequality:
\begin{equation} \label{ineq_c}
\alpha_p \big( \pi[f^p] - \pi[f]^p \big) \le p \mc{E}(f^{p - 1}, f)\,.
\end{equation}
Moreover, note from the diffusion property: $L \psi(f) = \psi'(f)L f + \psi''(f) \Gamma f$ for suitably smooth functions $\psi$ and $f$, that $\mc{E}(\log f , f) = 4 \mc{E}(f^{1/2}, f^{1/2})$ and $\mc{E}(f^q, f^{2 - q}) = (2 q - q^2) \mc{E}(f,f)$. By substituting $f = g^2$ and $f = g^q$ with $q = 2/p$ into \eqref{ineq_a} and \eqref{ineq_c}, respectively, up to constants, we have the usual log-Sobolev inequality (LSI):
\begin{equation} \label{ineq_d}
\beta \big(\pi[g^2 \log g^2] - \pi[g^2] \log \pi[g^2]\big) \le \mc{E}(g, g)\,,
\end{equation}
and the original Beckner's inequality first introduced in \cite{beckner1989generalized} for the Gaussian measure on $\mathbb{R}^n$:
\begin{equation} \label{ineq_e}
\beta_q \big( \pi[g^2] - \pi[g^q]^{2/q} \big) \le (2-q) \mc{E}(g, g)\,.
\end{equation}
The condition \eqref{eq:curvature_dimension} admits a deep geometric interpretation, and it is called the
Bakry-\'{E}mery (curvature-dimension) condition or $\Gamma_2$-criterion. To make this point clearer, let $L = \Delta_g - \nabla W \cdot \nabla$ be the generator associated with the Ornstein–Uhlenbeck process on the manifold $M$ that admits an invariant measure $d \pi = e^{- W} d \text{vol}_M$, where $W$ is the potential and $\text{vol}_M$ is the volume form on $M$. With the help of Bochner's formula, we can compute $\Gamma(f) = |\nabla f|^2$ and
$\Gamma_2(f) = |\nabla^2 f|^2 + \Ric(L)(\nabla f , \nabla f)$, where $\Ric(L)$ is the Ricci tensor for the generator $L$, defined by $\Ric(L): = \Ric_g + \nabla^2 W$ with $\Ric_g$ being the standard Ricci curvature of $M$. It is easy to prove that the condition \eqref{eq:curvature_dimension} holds if and only if the Ricci curvature of $L$ is bounded below: $\Ric(L)(\cdot,\cdot) \ge \kappa g(\cdot,\cdot)$. Otto-Villani \cite{otto2000generalization} and von Renesse-Sturm \cite{von2005transport} further observed that \eqref{eq:curvature_dimension} is also equivalent to that the relative entropy with respect to $d \pi$ is displacement $\kappa$-convex on the Wasserstein space of probability measures on $M$. Inspired by this characterization, Sturm \cite{sturm2006geometry} and Lott-Villani \cite{lott2009ricci} extended the notion of Ricci curvature
to metric measure spaces by exploiting the convexity properties of entropy functionals. See \cite{bakry2014analysis,villani2009optimal} for more details.
The above framework establishes a beautiful connection between various subjects such as partial differential equations (PDE), probability, and geometry, and has led to important research progress in these fields. The key step in the Bakry-\'{E}mery arguments \cite{bakry1985diffusions} is to estimate the second derivative of the relative entropy along the Markov semigroup, where the calculation depends on Bochner’s formula or, more abstractly, the diffusion property. Arnold et al. \cite{arnold2001convex,arnold2008large,arnold2014sharp} revisited the Bakry-\'{E}mery method in the PDE framework and characterized the long-time asymptotics for various classes of Fokker-Planck type equations based on the convex Sobolev inequalities; see also \cite{otto2001geometry,del2002best,carrillo2000asymptotic,markowich2000trend} for the applications of functional inequalities in nonlinear Fokker-Planck type equations. Among the general convex Sobolev inequalities, the Beckner's inequality is of particular interest, since it provides an interpolating family between MLSI and Poincar\'{e} inequality. The recent work \cite{gentil2021family} proved a class of weighted Beckner's inequalities and the refined ones based on the Bakry-\'{E}mery method and the curvature-dimension conditions. We also mention that \cite{arnold2007interpolation} proved the inequality \eqref{ineq_e} by the hypercontractivity and spectral estimates. In particular, Dolbeault et al. \cite{dolbeault2009new,dolbeault2012poincare} explored the gradient flow structure of the Fokker-Planck equation for general entropy functionals and proved the contraction of the associated transport distance along the Fokker-Planck flow, which gave a unified gradient flow framework for investigating the convex Sobolev inequalities \eqref{eq:convex_ineq}.
It is also desirable to extend the theory of convex Sobolev inequalities to the setting of finite Markov chains. In this case, due to the lack of chain rule, the inequalities \eqref{ineq_a} and \eqref{ineq_d}, also, \eqref{ineq_c} and \eqref{ineq_e} are not equivalent (one is stronger than the other) \cite{bobkov2006modified}. For instance, Dai Pra et al. \cite{dai2002entropy} provided an example where the MLSI \eqref{ineq_a} holds while the LSI \eqref{ineq_d} fails. In what follows, to avoid confusion between \eqref{ineq_c} and \eqref{ineq_e}, following \cite{adamczak2022modified} and \cite{chafai2004entropies} we call the inequality of the form \eqref{ineq_e} the dual Beckner's inequality.
Similarly to the diffusion case, the Bakry-\'{E}mery method and gradient flow techniques are two main approaches for the validity of \eqref{eq:convex_ineq}. J\"{u}ngel and Yue \cite{jungel2017discrete} followed Bakry-\'{E}mery's ideas and gave the conditions of $\phi$ under which \eqref{eq:convex_ineq} holds. The proof
relies on a discrete Bochner-type identity that was first introduced in \cite{boudou2006spectral,caputo2009convex}. Recently, Weber and Zacher \cite{weber2021entropy} proposed discrete analogs of the condition \eqref{eq:curvature_dimension} such that the MLSI \eqref{ineq_a} and Beckner’s inequality \eqref{ineq_c} hold. Their argument, different from \cite{jungel2017discrete}, is based on the modified $\Gamma$ and $\Gamma_2$ operators that satisfy some kind of discrete diffusion property.
We point out that a probabilistic approach, based on the Bakry-\'{E}mery method and the coupling arguments, for the discrete convex Sobolev inequalities can be found in \cite{conforti2022probabilistic} by Conforti.
The starting point of the gradient flow approach for discrete functional inequalities is \cite{maas2011gradient} where Maas defined a discrete transport distance such that the continuous time finite Markov chains can be identified as the gradient flow of the relative entropy. Following the ideas of \cite{lott2009ricci,sturm2006geometry}, Erbar and Maas \cite{erbar2012ricci} introduced the discrete Ricci curvature based on this discrete Wasserstein metric, and derived a number of functional inequalities including the discrete MLSI and the transport cost inequalities; see also \cite{erbar2018poincare}. Later, Fathi and Maas \cite{fathi2016entropic} generalized the discrete Bochner formula \cite{caputo2009convex} and developed a systematic approach for estimating the discrete Ricci curvature lower bounds. It is worth mentioning that both the discrete Bakry-\'{E}mery condition in \cite{weber2021entropy} and the discrete Ricci curvature in \cite{erbar2012ricci} enjoy the tensorization properties, and the aforementioned general results can be applied to several interesting models such as birth-death processes, random transposition models and Bernoulli-Laplace models (see related papers for details). We refer the readers to the review \cite{maas2017entropic} and the references therein for other notions of the Ricci curvature in the discrete setting and their implications on functional inequalities.
\subsection{Quantum functional inequalities}
In analogy with the classical case, quantum functional inequalities play a fundamental role in understanding the asymptotic behavior of a QMS. The study of LSI in the noncommutative setting may date back to \cite{gross1975hypercontractivity},
and its connections with the hypercontractivity were fully discussed in the seminal work \cite{olkiewicz1999hypercontractivity}.
The quantum MLSI was first introduced by Kastoryano and Temme \cite{kastoryano2013quantum} for deriving improved bounds on the mixing time of
primitive quantum Markov processes than those obtained in \cite{temme2010chi} via the Poincar\'{e} inequality. The quantum MLSI constant was investigated in detail in \cite{muller2016entropy,muller2016relative} for some special models, e.g., the depolarizing semigroup and the doubly stochastic qubit Lindbladian. Regarding the general framework for the validity of quantum MLSI, the Bakry-\'{E}mery method,
which has been successfully adapted to the discrete case, (seems) has not been explored in the literature for the quantum semigroups and needs further research. However, the notion of Ricci curvature lower bounds (geodesic convexity) in the quantum setting, pioneered by Carlen and Maas \cite{carlen2014analog,carlen2017gradient}, has shown its utility in proving the quantum MLSI and related functional inequalities with numerous applications in concrete physical models. We briefly outline the main progress in this direction below. Carlen and Maas \cite{carlen2017gradient} introduced a quantum analog of 2-Wasserstein distance such that the primitive QMS satisfying the $\sigma$-detailed balance condition (cf.\,Definition \ref{def:sidbc}) can be written as the gradient flow of the relative entropy and showed that the relative entropy is geodesically convex for the Fermi and Bose Ornstein-Uhlenbeck semigroups, which extended their previous work \cite{carlen2014analog}. Based on Carlen and Maas's results, Datta and Rouz{\'e} \cite{rouze2019concentration,datta2020relating} considered the Ricci curvature of a QMS, and obtained some quantum Sobolev and concentration inequalities, generalizing the results in the classical regime \cite{otto2001geometry,erbar2012ricci,erbar2018poincare}. Wirth and Zhang \cite{wirth2021curvature} further introduced noncommutative curvature-dimension conditions and derived some dimension-dependent quantum functional inequalities.
One of the favorable features of the classical LSI is the tensorization property, which enables obtaining the functional inequalities for the tensor product systems from those for the subsystems. However, this property is known to fail for the quantum MLSI (cf.\cite[Proposition 4.21]{brannan2022complete}). To circumvent such difficulty, Gao et al. \cite{gao2020fisher} introduced the complete modified log-Sobolev inequality (CMLSI), which is a stronger notion than the quantum MLSI,
and showed that it satisfies the desired tensorization properties. Very recently, Gao and Rouz\'{e} proved, by a two-sided estimate for the relative entropy, that the CMLSI holds for any finite dimensional $\sigma$-detailed balance QMS. We refer the interested readers to \cite{brannan2022complete,li2020graph,brannan2021complete} for
the geometrical approaches for studying the CMLSI, which is based on the noncommutative Ricci curvature and the group transference techniques.
\subsection{Main results and layout} As reviewed above, although there has been much progress on Beckner's inequalities in both diffusion and discrete cases, the results for quantum Beckner's inequalities are quite limited. We only note the recent work by Li \cite{li2020complete} where the author investigated the matrix-valued Beckner's inequalities, in terms of the Bregman relative entropy \cite{molnar2016maps}, for symmetric semigroups on a finite von Neumann algebra. In this work we try to close this gap. We consider the primitive QMS with $\sigma$-detailed balance condition for $\sigma \in \mathcal{D}_+(\mc{H})$, and define the family of quantum $p$-Beckner's inequalities and their dual version, by extending the definitions in \cite{adamczak2022modified,latala2000between,chafai2004entropies} for classical Markov semigroups; see Definition \ref{def:quantum_func} for the functional inequalities that we will mainly focus on.
It turns out that the $p$-Beckner's inequality \eqref{ineq_becp} describes the rate at which the quantum $p$-divergence $\mc{F}_{p,\sigma}$ \eqref{def:quanpdivi}, which can be viewed as the normalized noncommutative $L_p$-norm, tends to zero. The diagram in Figure \ref{fig:diagram} below summarizes the main results of Sections \ref{sec:inter_primi} and \ref{sec:positive_stab}.
\begin{figure}[h]
\begin{tikzcd}[row sep= 2.5 em , column sep= 5 em]
\eqref{ineq_becp} \arrow[Leftrightarrow]{dd}[swap]{\text{Theorem}\ \ref{thm:beck_log_sobo}} \arrow[Leftrightarrow]{rd}{\text{Theorem}\ \ref{thm:beck_poincare},\ \text{Corollary}\ \ref{coro:pitobeck} } & &
\eqref{ineq_dbecq} \arrow[Leftrightarrow]{dd}{\text{Proposition} \ \ref{dualbecktolsi}} \arrow[Rightarrow]{ll}[swap]{\text{Proposition} \ \ref{prop:beck_to_dual_beck}} \\
& \eqref{ineq_pi} \arrow[Leftrightarrow]{rd}[swap]{\text{\cite[Corollary 6]{temme2014hypercontractivity}}} & \\
\eqref{ineq_mlsi} \arrow[Leftrightarrow]{ur}{\text{\cite{gao2021complete,cao2019gradient}}} && \eqref{ineq_lsi} \arrow[Rightarrow]{ll}{\text{\cite[Proposition 13]{kastoryano2013quantum}}}
\end{tikzcd}
\caption{Chain of quantum convex Sobolev inequalities.}
\label{fig:diagram}
\end{figure}
We first note that the relations between \eqref{ineq_pi}, \eqref{ineq_mlsi}, and \eqref{ineq_lsi} have been well investigated. \cite[Corollary 6]{temme2014hypercontractivity} provided a two-sided bound for the log-Sobolev constant by the Poincar\'{e} constant; \cite{kastoryano2013quantum} showed that the modified log-Sobolev constant is bounded below by the log-Sobolev constant and above by the Poincar\'{e} constant. This also allows us to conclude that \eqref{ineq_mlsi} is equivalent to \eqref{ineq_pi} for the primitive QMS; see also \cite[Proposition I.7]{cao2019gradient} and \cite[Theorem 3.3]{gao2021complete}. Here, in Section \ref{sec:inter_primi}, we prove that the dual $q$-Beckner's inequalities \eqref{ineq_dbecq}, as an interpolating family between \eqref{ineq_lsi} and \eqref{ineq_pi}, is equivalent to \eqref{ineq_lsi} in the sense of Proposition \ref{dualbecktolsi}. We also show in Proposition \ref{prop:beck_to_dual_beck} that \eqref{ineq_dbecq} is stronger than \eqref{ineq_becp}. Propositions \ref{prop:mono_dual_beck} and \ref{prop:mono_beck} discuss the monotonicity of the dual Beckner constant $\beta_q(\mc{L})$ and the Beckner constant $\alpha_p(\mc{L})$, respectively. Then, in Section \ref{sec:positive_stab}, we investigate the quantum Beckner's inequalities in detail. Inspired by the work \cite{gao2021complete}, we show in Lemma \ref{lem:two-sided} that the quantum $p$-divergence $\mc{F}_{p,\sigma}$ can be bounded from above and below by the $\chi_{\kappa_{1/p}}^2$-divergence with $\kappa_{1/p}$ being the power difference. This key lemma enables us to derive a two-sided estimate for the Beckner constant $\alpha_p(\mc{L})$ in terms of the Poincar\'{e} constant $\lambda(\mc{L})$ in a similar way as \cite[Theorem 3.3]{gao2021complete}; see Theorem \ref{thm:beck_poincare}. We proceed to prove, by contradiction, in Theorem \ref{thm:beck_log_sobo} that for each $p \in (1,2]$, \eqref{ineq_becp} is equivalent to \eqref{ineq_mlsi}, and there holds $\alpha_p(\mc{L}) \to \alpha_1(\mc{L})$ as $p \to 1^+$, where $\alpha_1(\mc{L})$ is the modified log-Sobolev constant. This extends the result \cite[Theorem 1.1]{adamczak2022modified} in the classical setting. We also extend the main result in \cite{junge2019stability} and provide a stability estimate for the Beckner constant $\alpha_p(\mc{L})$ with respect to the invariant state $\sigma$; see Theorem \ref{thm:beck_stab}. In Section \ref{sec:app_exp}, we first show in Proposition \ref{prop:mixing} that the family of $p$-Beckner's inequalities may imply a better mixing time bound than the ones in \cite{kastoryano2013quantum,temme2010chi}. In Proposition \ref{prop:moment}, we extend \cite[Proposition 3.3]{adamczak2022modified} for the classical case to the quantum setting and obtain moment estimates under \eqref{ineq_becp}. Then we analyze the quantum Beckner constant for the depolarizing semigroup and reduce it to the computation of the classical one for a Markov chain on the two-point space; see Proposition \ref{prop:beck_depol}.
Another motivation for the current work is \cite{dolbeault2009new,dolbeault2012poincare}, where the authors provided a gradient flow approach for the classical Beckner's inequalities. To be precise,
Dolbeault et al. \cite{dolbeault2009new} defined the following class of transport distances $\mc{W}_{2,\alpha,\gamma}$ with $\alpha \in [0,1]$ \`{a} la Benamou-Brenier: for probability measures $\mu_0$ and $\mu_1$ on $\mathbb{R}^d$,
\begin{align} \label{def:distance_savare}
\mc{W}_{2,\alpha,\gamma}(\mu_0,\mu_1): = \inf\Big\{
\int_0^1\int_{\mathbb{R}^d} \rho_t^{-\alpha}|w_t|^2\ d \gamma dt\,;\ \partial_t \mu_t + \nabla \cdot {\bf \nu}_t = 0\,,\ \mu_t = \rho_t \gamma + \mu_t^\perp\,,\ \nu_t = w_t \gamma \ll \gamma
\Big\}\,,
\end{align}
where $\gamma$ is a reference Radon measure. In the case $\alpha = 1$, $\mc{W}_{2,\alpha,\gamma}(\mu_0,\mu_1)$ gives the quadratic Wasserstein distance \cite{benamou2000computational}, while when $\alpha = 0$, it is equivalent to the weighted homogeneous $\dot{H}_\gamma^{-1}$ Sobolev distance \cite{peyre2018comparison,villani2003topics}:
\begin{align*}
\norm{\mu_0 - \mu_1}_{\dot{H}_\gamma^{-1}} = \sup \Big\{\int_{\mathbb{R}^d} \xi\ d(\mu_0 - \mu_1)\,;\ \xi \in C_c^1(\mathbb{R}^d)\,,\ \int_{\mathbb{R}^d} |\nabla \xi|^2\ d \gamma \le 1 \Big\}\,.
\end{align*}
Thus $\mc{W}_{2,\alpha,\gamma}$ can be viewed as a natural interpolating family between them. Moreover, let the reference measure be $\gamma := e^{-V} \mathscr{L}^d $ with the potential $V$ being smooth and convex, where $\mathscr{L}^d$ is the Lebesgue measure on $\mathbb{R}^n$.
With such choice of $\gamma$, they showed that the gradient flow of the (Tsallis) functional, for $\alpha \in [0,1)$,
\begin{align} \label{def:class_func}
\mathscr{F}_\alpha(\mu): = \frac{1}{(2-\alpha)(1-\alpha)} \int_{\mathbb{R}^d} \rho^{2-\alpha}\ d\gamma\,,\quad \mu = \rho \gamma\,,
\end{align}
is the Fokker-Planck equation:
\begin{align*}
\partial_t \mu - \Delta \mu - \nabla \cdot (\mu \nabla V) = 0\,.
\end{align*}
In the subsequent work \cite{dolbeault2012poincare}, they further proved that $\mc{F}_\alpha$ is geodesically $\lambda$-convex under the assumption: $\nabla^2 V \ge \lambda I$ for $\lambda > 0$, which implies the classical Beckner's inequality.
It is easy to note that in the commutative setting, up to some constant, our quantum $p$-divergence $\mc{F}_{p,\sigma}$ \eqref{def:quanpdivi} is nothing else but the functional $ \mathscr{F}_\alpha$ in \eqref{def:class_func} with $p = 2 - \alpha$. In Section \ref{sec:qot_beck}, we extend the results in \cite{dolbeault2009new,dolbeault2012poincare}
to the quantum regime and provide a geometric characterization for the quantum $p$-Beckner's inequality. To do so, we first construct a Riemannian metric $g_{p,\rho}$ on the quantum states in Section \ref{sec:gradient}, so that the QMS with $\sigma$-detailed balance is the gradient flow of the quantum $p$-divergence $\mc{F}_{p,\sigma}$ with respect to $g_{p,\rho}$. Then in Section \ref{sec:quantum_distance}, we investigate the properties of the associated Riemannian distance, denoted by $W_{2,p}$ (cf.\,\eqref{def:wp1}), which can be regarded as a quantum analog of $W_{2,\alpha,\gamma}$ in \eqref{def:distance_savare}. The main result in this section is Theorem \ref{thm:main_wasser}, where we show that $(\mathcal{D}(\mc{H}),W_{2,p})$ is a complete geodesic metric space. We also prove in Proposition \ref{prop:repkernel} that similarly to the classical case,
the new class of distances $W_{2,p}$ is an interpolating family between the quantum 2-Wasserstein distance defined by Carlen and Maas \cite{carlen2017gradient} and the noncommutative $\dot{H}^{-1}$ Sobolev distance \eqref{eq:distance_1}. With these results, it is straightforward to define the entropic Ricci curvature associated with the functional $\mc{F}_{p,\sigma}$
in the spirit of \cite{sturm2006geometry,lott2009ricci,datta2020relating}. We then derive an HWI-type interpolation inequality from the Ricci curvature lower bound and show that the positive Ricci curvature can imply the quantum Beckner's inequalities. Furthermore , we prove the
following chain of quantum functional inequalities:
\begin{equation} \label{eq:chain}
\eqref{ineq_becp} \xLongrightarrow{\text{Proposition}\ \ref{propa}} \eqref{ineq:tc} \xLongrightarrow{\text{Proposition}\ \ref{propb}} \eqref{ineq_pi}\,,
\end{equation}
where \eqref{ineq:tc} is a transport cost inequality associated with $W_{2,p}$. These results are presented in Section \ref{sec:geodesic_convexity}.
We end the introduction with a list of notations used throughout this work:
\begin{enumerate}[\textbullet]
\item Let $\mc{B}(\mc{H})$ denote the space of bounded operators on a complex Hilbert space $\mc{H}$ of dimension $d < \infty$. $\mc{B}_{sa}(\mc{H})$ is the subspace of self-adjoint operators on $\mc{H}$, while $\mc{B}^{+}_{sa}(\mc{H})$ is the cone of positive semidefinite operators. For simplicity, in what follows, by $A \ge 0$ (resp., $A > 0$) we mean a positive semidefinite (resp., definite) operator.
\item We denote by $\langle\cdot,\cdot\rangle$ the Hilbert-Schmidt inner product on $\mathcal{B}(\mc{H})$, i.e., $\langle X, Y\rangle = \tr (X^*Y)$, where $X^*$ is the adjoint operator of $X$. Moreover, we write $\Phi^\dag$ for the adjoint of a superoperator $\Phi: \mc{B}(\mc{H}) \to \mc{B}(\mc{H})$ with respect to the inner product $\langle \cdot,\cdot \rangle $.
The modulus of $X \in \mathcal{B}(\mc{H})$ is defined by $|X|: = \sqrt{X^* X} $.
\item We define the Schatten $p$-norm by $\norm{X}_p = (\tr(|X|^p))^{1/p}$ for $X \in \mathcal{B}(\mc{H})$ and $p \in [1,\infty]$. In particular, $\norm{\cdot}_\infty$ is the operator norm. For a superoperator on $\mathcal{B}(\mc{H})$, the operator norm is simply denoted by $\norm{\cdot}$.
\item We denote by $\mc{D}(\mc{H}) := \{\rho \in \mc{B}_{sa}^+(\mc{H})\,;\ \tr \rho =1 \}$ the set of density operators (quantum states), and by $\mc{D}_+(\mc{H})$ the full-rank density operators.
\item The identity operator on $\mc{H}$ is written as ${\bf 1}_{\mc{H}}$ (or ${\bf 1}$, when there is no confusion). Similarly, the identity superoperator on $\mathcal{B}(\mc{H})$ is denoted by ${\rm id}_{\mc{H}}$, or simply ${\rm id}$.
\item Let $\mc{M}$ be a subset of $\mc{B}(\mc{H})$. We denote by $\mc{M}^J$ the set of vector fields over $\mc{M}$, i.e., ${\bf A} = (A_1, \cdots, A_J) \in \mc{M}^J$ for $A_j \in \mc{M}$, $1 \le j \le J$. The Hilbert-Schmidt inner product naturally extends to $\mc{M}^J$ as $\langle {\bf A} , {\bf B} \rangle = \sum_{j = 1}^J \langle A_j, B_j\rangle$.
\item We shall always write $\h{p}$ for the H\"{o}lder conjugate for $ 1 \le p \le \infty$, i.e., $1/p + 1/\h{p} = 1$.
\end{enumerate}
\section{Preliminaries}
\subsection{Quantum Markov semigroup}
Let us first recall preliminaries about the Markovian open quantum dynamics. We say that $(\mc{P}_t)_{t \ge 0}: \mc{B}(\mc{H}) \to \mc{B}(\mc{H})$ is a quantum Markov semigroup if $\mc{P}_t$ is a $C_0$-semigroup of completely positive, unital maps, whose generator $\mc{L}$ is called the Lindbladian, defined by $\mc{L}(X): = \lim_{t \to 0} t^{-1}(\mc{P}_t (X) - X)$.
A quantum channel $\Phi: \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})$ is a completely positive trace preserving (CPTP) map. Then, we see that the dual QMS $\mc{P}_t^\dag$ is a semigroup of quantum channels. The associated equation $\partial_t \rho = \mc{L}^\dag \rho$ is referred to as the Lindblad equation. In this work, we shall always consider the QMS $\mc{P}^\dag_t$ that
admits a unique full-rank invariant state $\sigma$, that is, there exists a unique $\sigma \in \mc{D}_+(\mc{H})$ such that $\mc{P}_t^\dag(\sigma) = \sigma$ for $t \ge 0$. In this case, the QMS $\mc{P}_t$ is said to be primitive; and it holds that \cite{frigerio1982long}
\begin{align} \label{eq:conver_qms}
\lim_{t \to \infty} \mc{P}_t(X) = \tr(\sigma X){\bf 1}\quad \text{for} \ X \in \mathcal{B}(\mc{H})\,.
\end{align}
For $A \in \mathcal{B}(\mc{H})$ we define the left and right multiplication operator on $\mathcal{B}(\mc{H})$ by $L_A = A X$ and $R_A = X A$, respectively. It is easy to see that $L_{f(A)} = f(L_A)$ and $R_{f(A)} = f(R_A)$ holds for $A > 0$ and
functions $f: (0,\infty) \to \mathbb{R}$. We also introduce the relative modular operator $\Delta_{\rho,\sigma} = L_\rho R_\sigma^{-1}: \mc{B}(\mc{H}) \to \mc{B}(\mc{H})$ for
$\rho,\sigma \in \mc{D}_+(\mc{H})$. When $\sigma = \rho$, we simply write it as $\Delta_\sigma$. We next introduce the quantum detailed balance condition (DBC). For this, we define a family of inner products on $\mathcal{B}(\mc{H})$: for a given $\sigma \in \mc{D}_+(\mc{H})$ and $s \in \mathbb{R}$,
\begin{align} \label{def:s-inner}
\langle X, Y\rangle_{\sigma,s} := \tr(\sigma^s X^* \sigma^{1-s} Y) = \langle X, \Delta_\sigma^{1-s} Y \sigma \rangle\,,
\end{align}
where $\langle \cdot, \cdot\rangle_{\sigma,1}$ and $\langle \cdot, \cdot \rangle_{\sigma,1/2}$ are called the GNS and KMS inner products, respectively. In particular, when $\sigma$ is the maximally mixed state ${\bf 1}/d$, all the inner products $\langle \cdot,\cdot \rangle_{\sigma,s}$ reduce to the normalized Hilbert-Schmidt inner product:
\begin{align*}
\langle X , Y\rangle_{\frac{{\bf 1}}{d}} := \frac{1}{d} \langle X , Y\rangle\,,\quad \forall X, Y \in \mathcal{B}(\mc{H})\,.
\end{align*}
\begin{definition} \label{def:sidbc}
We say that a QMS $\mc{P}_t$, or its generator $\mc{L}$, satisfies the $\sigma$-DBC for some $\sigma \in \mc{D}_+(\mc{H})$ if the operator $\mc{L}$ is self-adjoint with respect to the inner product $\langle\cdot, \cdot\rangle_{\sigma,1}$, that is,
\begin{equation*}
\langle \mc{L}(X), Y\rangle_{\sigma,1} = \langle X, \mc{L}(Y)\rangle_{\sigma,1}\,,\quad \forall X, Y \in \mc{B}(\mc{H})\,.
\end{equation*}
\end{definition}
One can readily see that $\mc{P}_t$ is symmetric, i.e., $\mc{P}_t = \mc{P}_t^\dag$, if and only if it satisfies $\sigma$-DBC for the maximally mixed state $\sigma = {\bf 1}/d$, and that if $\mc{P}_t$ satisfies $\sigma$-DBC, then $\sigma$ is an invariant state of $\mc{P}_t$. More generally, for any function $f: (0,\infty) \to (0,\infty)$ and $\sigma \in \mc{D}_+(\mc{H})$, we define the operator:
\begin{align} \label{def:operator_kernel}
J_\sigma^f := R_\sigma f(\Delta_\sigma): \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})\,,
\end{align}
and the associated inner product:
\begin{align} \label{eq:general_inner}
\langle X, Y\rangle_{\sigma,f} := \langle X, J_\sigma^f (Y)\rangle\,.
\end{align}
It is clear that $\langle X, Y\rangle_{\sigma,f}$ with $f = x^{1-s}$ gives the inner product \eqref{def:s-inner}; and the adjoint of a linear operator $\mc{K}$ on $\mathcal{B}(\mc{H})$ with respect to $\langle\cdot,\cdot\rangle_{\sigma,f}$ is given by $ ( J_\sigma^f)^{-1} \mc{K}^\dag J_\sigma^f $.
The following result from \cite[Theorem 2.9]{carlen2017gradient} relates the self-adjointness of $\mc{L}$ with respect to different inner products.
\begin{lemma} \label{lem:self_adjoint}
If a QMS $\mc{P}_t$ satisfies the $\sigma$-DBC for some state $\sigma \in \mc{D}_+(\mc{H})$, then its generator $\mc{L}$
commutes with the modular operator $\Delta_\sigma$, and it is self-adjoint with respect to $\langle\cdot, \cdot\rangle_{\sigma,f}$ for any $f:(0,\infty) \to (0,\infty)$, i.e.,
\begin{align*}
\mc{L}\Delta_\sigma = \Delta_\sigma \mc{L}\,,\quad J_\sigma^f \mc{L} = \mc{L}^\dag J_\sigma^f\,.
\end{align*}
In particular, $\mc{L}$ is self-adjoint with respect to the KMS inner product $\langle \cdot,\cdot \rangle_{\sigma,1/2}$ and there holds
\begin{align*}
\Gamma_\si \mc{L} = \mc{L}^\dag \Gamma_\si\,.
\end{align*}
\end{lemma}
The next lemma, due to Alicki \cite{alicki1976detailed}, characterizes the structure of the generator $\mc{L}$ of a QMS $\mc{P}_t$ satisfying $\sigma$-DBC; see also \cite{kossakowski1977quantum,carlen2017gradient}.
\begin{lemma}\label{lem:struc_gene}
Suppose that $\mc{P}_t$ satisfies $\sigma$-DBC for some $\sigma \in \mc{D}_+(\mc{H})$. Then it holds that
\begin{align} \label{eq:structure}
\mc{L}(X) = \sum_{j = 1}^{J} (e^{-\omega_j/2}V_j^*[X,V_j] + e^{\omega_j/2}[V_j,X]V^*_j)\,,
\end{align}
with $\omega_j \in \mathbb{R}$ and $J \le n^2 -1$. Here $V_j \in \mathcal{B}(\mc{H})$, $1\le j \le J$, are trace zero and orthogonal eigenvectors of $\Delta_\sigma$:
\begin{align} \label{eq:eigmodular}
\Delta_\sigma(V_j) = e^{-\omega_j}V_j\,,\quad \langle V_j, V_k\rangle = c_j\delta_{j,k}\,,\quad \tr(V_j) = 0\,,
\end{align}
where $c_j > 0$ are normalization constants. Moreover, for each $1 \le j \le J$, there exists $1 \le j' \le J $ such that
\begin{align} \label{eq:adjoint_index}
V_j^* = V_{j'}\,,\quad \omega_j = - \omega_{j'}\,.
\end{align}
\end{lemma}
The real numbers $\omega_j \in \mathbb{R}$ are called Bohr frequencies of the Lindbladian $\mc{L}$, which are uniquely determined by the invariant state $\sigma$, while the operators $V_j$ are only unique up to unitary transformations. In what follows, we will fix a set of $V_j$ for the representation \eqref{eq:structure} with the properties in Lemma \ref{lem:struc_gene}.
\begin{remark} \label{eq:tracial}
When the invariant state $\sigma$ is the maximally mixed state ${\bf 1}/d$, we have that
$\Delta_\sigma$ is the identity operator and hence $\omega_j = 0$ by \eqref{eq:eigmodular}, and we can take the operators $V_j$ to be self-adjoint. In this symmetric case, the QMS $\mc{P}_t$ may be regarded as a noncommutative heat semigroup, and it generator has the form:
\begin{align} \label{eq:gen_symm}
\mc{L}(X) = - \sum_j [V_j, [V_j, X]]\,.
\end{align}
\end{remark}
\begin{example} \label{exp:dep_channel}
An important example of QMS is the generalized depolarizing semigroup: for $\sigma \in \mathcal{D}_+(\mc{H})$ and $\gamma > 0$,
\begin{align} \label{def:depolt}
\mc{P}_t(X) = e^{- \gamma t} X + (1 - e^{- \gamma t}) \tr(\sigma X) {\bf 1}\,, \quad X \in \mathcal{B}(\mc{H})\,,
\end{align}
generated by
\begin{align} \label{def:depol}
\mc{L}_{depol}(X) = \gamma (\tr(\sigma X) {\bf 1} - X)\,.
\end{align}
It is easy to see that $\mc{P}_t$ is primitive and satisfies $\sigma$-DBC.
\end{example}
Lemma \ref{lem:struc_gene} actually gives a useful first-order differential structure associated with the QMS with $\sigma$-DBC. We introduce the weighting operator for a full-rank quantum state
$\sigma \in \mc{D}_+(\mc{H})$:
\begin{align*}
\Gamma_\si X = \sigma^{\frac{1}{2}}X\sigma^{\frac{1}{2}}:\ \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})\,,
\end{align*}
and the noncommutative analog of partial derivatives (associated with the generator $\mc{L}$):
\begin{align*}
\partial_j X = [V_j, X]:\ \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})\,.
\end{align*}
Then the noncommutative gradient $\nabla: \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})^J$ and divergence $ {\rm div} : \mathcal{B}(\mc{H})^J \to \mathcal{B}(\mc{H})$ can be defined as
\begin{equation*}
\nabla X = (\partial_1 X, \cdots, \partial_J X) \quad \text{for} \ X \in \mathcal{B}(\mc{H})\,,
\end{equation*}
and
\begin{equation*}
{\rm div} {\bf X} = - \sum_{j = 1}^J \partial_j^\dag X_j \quad \text{for} \ {\bf X} \in \mathcal{B}(\mc{H})^J\,,
\end{equation*}
respectively. By definition and \eqref{eq:eigmodular},
it follows that the adjoint of $\partial_j$ with respect to $\langle \cdot, \cdot \rangle_{\sigma,1/2}$ is given by \begin{align} \label{eq:adjpjkms}
\partial_{j,\sigma}^\dag B = \Gamma_\si^{-1}\partial_j^\dag \Gamma_\si B = e^{-\omega_j/2} V_j^* B - e^{\omega_j/2} B V_j^*\,.
\end{align}
With the help of $\partial^\dag_{j,\sigma}$ \eqref{eq:adjpjkms}, we can rewrite \eqref{eq:structure} as
\begin{align} \label{eq:rep_gen}
\mc{L} (X) = - \sum_{j = 1}^J \partial_{j,\sigma}^\dag \partial_j X\,,
\end{align}
and a noncommutative integration by parts formula holds:
\begin{align} \label{eq:inte_by_parts}
- \langle Y, \mc{L}(X)\rangle_{\sigma,1/2} = \sum_{j = 1}^J \langle \partial_j Y, \partial_j X \rangle_{\sigma,1/2} \quad \text{for} \ X, Y \in \mathcal{B}(\mc{H})\,.
\end{align}
The following lemma will be useful below, which generalizes \cite[Proposition 4.12]{carlen2020non}.
\begin{lemma}\label{lem:DBC}
Suppose that $\mc{P}_t$ satisfies $\sigma$-DBC, then for a continuous function $f:(0,\infty) \to (0,\infty)$, there holds
\begin{align*}
- \langle Y, \mc{L} X\rangle_{\sigma,f} = \sum_{j = 1}^J \left\langle \partial_j Y, R_{e^{-\omega_j/2} \sigma} f\left(L_{e^{\omega_j/2}\sigma} R^{-1}_{e^{-\omega_j/2} \sigma}\right) \partial_j X \right\rangle\,,\quad X, Y \in \mathcal{B}(\mc{H})\,.
\end{align*}
\end{lemma}
\begin{proof}
By Stone-Weierstrass theorem, it suffices to consider $f = x^s$. For this, we have
\begin{align*}
- \langle Y, R_\sigma \Delta_\sigma^s \mc{L} X\rangle = \sum_{j = 1}^J \langle Y, \Delta_\sigma^{s-\frac{1}{2}}\partial_j^\dag \Gamma_\si \partial_j X \rangle & = \sum_{j = 1}^J e^{(s-\frac{1}{2})\omega_j} \left\langle \partial_j Y, R_\sigma \Delta_\sigma^s \partial_j X \right\rangle \\
& = \sum_{j = 1}^J \left\langle \partial_j Y, R_{e^{-\omega_j/2}\sigma} \left(L_{e^{\omega_j/2}\sigma} R^{-1}_{e^{-\omega_j/2}\sigma}\right)^s \partial_j X \right\rangle\,,
\end{align*}
by noting $
\Delta_\sigma^s \partial_j X = e^{-s\omega_j} \partial_j \Delta_\sigma^s X
$ from \eqref{eq:eigmodular}.
\end{proof}
\subsection{Quantum entropy and Dirichlet form}
In this section, we shall introduce the notions of quantum relative entropies and Dirichlet forms, and discuss their basic properties. We start with the definition of the noncommutative weighted $L_p$ space. For $p > 0$ and $\sigma \in \mathcal{D}_+(\mc{H})$, we define the $\sigma$-weighted $p$-norm (quasi-norm if $p < 1$) on $\mathcal{B}(\mc{H})$ \cite{olkiewicz1999hypercontractivity,kastoryano2013quantum}:
\begin{align*}
\norm{X}_{p,\sigma} := \tr\Big(|\Gamma_\sigma^{1/p}(X)|^p \Big)^{1/p}\,.
\end{align*}
In particular, if $\sigma = \frac{{\bf 1}}{d}$, then $\norm{\cdot}_{p,\frac{{\bf 1}}{d}} = \frac{1}{d^{1/p}}\norm{\cdot}_p$ is the normalized Schatten $p$-norm. We also need the power operator $I_{q,p}$ for $p,q \in \mathbb{R} \backslash \{0\}$:
\begin{align*}
I_{q,p}(X):= \Gamma_{\sigma}^{-1/q}\left(|\Gamma_{\sigma}^{1/p}(X)|^{p/q} \right)\,,\quad X \in \mc{B}(\mc{H}).
\end{align*}
We summarize some important properties of $\norm{\cdot}_{p,\sigma}$ and $I_{p,q}$, see \cite[Lemmas 1,\,2]{kastoryano2013quantum} and \cite[Proposition 2.6]{gu2019interpolation}.
\begin{lemma} \label{lem:prop:norm_power}
For $p,q,r \in \mathbb{R} \backslash \{0\}$ and $X \in \mathcal{B}(\mc{H})$, it holds that
\begin{enumerate}[1.]
\item $\norm{X}_{p,\sigma} \le \norm{X}_{q,\sigma}$ for $0 < p \le q$.
\item $\norm{I_{q,p}(X)}_{q,\sigma}^q = \norm{X}_{p,\sigma}^p$ and $I_{q,r} \circ I_{r,p} = I_{q,p}$.
\item $I_{p,p}(X) = X$ for $X \ge 0$.
\end{enumerate}
\end{lemma}
We define the $p$-Dirichlet form ($p > 1$) for a QMS $\mc{P}_t$ with the generator $\mc{L}$ (our definition differs from the one in \cite{kastoryano2013quantum} by a factor of $p/2$): for any full-rank invariant state $\sigma = \mc{P}_t^\dag(\sigma)$,
\begin{align} \label{def:p_diri}
\mc{E}_{p,\mc{L}}(X) := - \frac{\h{p}p}{4} \langle I_{\h{p},p}(X),\mc{L}(X) \rangle_{\sigma,1/2}\,,\quad X \in \mathcal{B}(\mc{H})\,,
\end{align}
which, by Lemma \ref{lem:prop:norm_power}, readily gives
\begin{align*}
\mc{E}_{2,\mc{L}}(X) = - \langle X, \mc{L}(X) \rangle_{\sigma,1/2}\,, \quad X \ge 0\,.
\end{align*}
The $1$-Dirichlet form is defined as the limit $p \to 1^+$ of $\mc{E}_{p,\mc{L}}$ \cite[Proposition 8]{kastoryano2013quantum}:
\begin{align} \label{def:1diri}
\mc{E}_{1, \mc{L}}(X) := \lim_{p \to 1^+} \mc{E}_{p,\mc{L}}(X) = - \frac{1}{4} \langle \log \Gamma_\si(X) - \log \sigma, \mc{L}(X) \rangle_{\sigma,1/2}\,,\quad X \ge 0\,.
\end{align}
For the generator $\mc{L}$ satisfying $\sigma$-DBC, by the formula \eqref{eq:inte_by_parts}, we have
\begin{align} \label{eq:symm_p_diri}
\mc{E}_{p,\mc{L}}(X) & = \frac{\h{p}p}{4} \sum_{j = 1}^J \langle \partial_j I_{\h{p},p}(X), \partial_j X \rangle_{\sigma,1/2}\,.
\end{align}
Thanks to the relation \eqref{eq:eigmodular}, we can further compute, for $p,q \neq 0$,
\begin{align} \label{eq:partialpower_1}
\partial_j I_{q,p}(X) & = V_j \Gamma_\sigma^{-1/q}\left(|\Gamma_\sigma^{1/p}(X)|^{p/q} \right) - \Gamma_\sigma^{-1/q}\left(|\Gamma_\sigma^{1/p}(X)|^{p/q} \right) V_j \notag \\
& = \Gamma_\sigma^{-1/q} \left( V_j |\Gamma_\sigma^{1/p}(e^{-\omega_j/2p}X)|^{p/q} - |\Gamma_\sigma^{1/p}(e^{\omega_j/2p}X)|^{p/q} V_j \right)\,,
\end{align}
and, for $s \neq 0$,
\begin{align} \label{eq:partialpower_2}
\partial_j X = \Gamma_\sigma^{-1/s}\left(V_j \Gamma_\sigma^{1/s}(e^{-\omega_j/2s}X) - \Gamma_\sigma^{1/s}(e^{\omega_j/2s}X)V_j\right)\,.
\end{align}
Then, substituting the formulas \eqref{eq:partialpower_1} with $q=\h{p}$ and \eqref{eq:partialpower_2} with $s = p$ back into \eqref{eq:symm_p_diri}, and using Lemma \ref{lem:chain_rule}, we find the following representation of $\mc{E}_{p,\mc{L}}$:
\begin{align} \label{eq:rep_epl}
\mc{E}_{p,\mc{L}}(X)
& = \frac{p^2}{4} \sum_{j = 1}^J \left\langle \Gamma_\si^{1/p} \big(\partial_j X\big), f_p^{[1]}\left(e^{\omega_j/2p}\Gamma_\si^{1/p}(X), e^{-\omega_j/2p}\Gamma_\si^{1/p}(X)\right) \Gamma_\si^{1/p} \big(\partial_j X \big) \right\rangle\,, \quad X \ge 0\,,
\end{align}
where $f_p^{[1]}(\cdot,\cdot)$ is the double sum operator associated with the divided difference of the function:
\begin{equation} \label{def:funfp}
f_p(x): = \frac{1}{p-1} x^{p-1}\,,
\end{equation}
see \eqref{def:doutble_op_sum} and \eqref{def:divi_diff} for related definitions.
Before we proceed, we derive some integral representation formulas of $f_p^{[1]}$ for later use. We recall \cite[p.116]{bhatia2013matrix}
\begin{align*}
x^{p - 1} = \frac{\sin(p \pi)}{\pi}\int_0^\infty \frac{ s^{p-1}}{s + x}\ ds \quad \text{for}\ x > 0\,,\ 0 < p < 1\,,
\end{align*}
which yields, for $1 < p < 2$,
\begin{align} \label{eq:inte_divi_p}
f_p^{[1]}(x,y) & = \frac{1}{x - y} \int_x^y t^{p - 2}\ dt \notag \\
& = \frac{1}{x - y} \frac{\sin((p-1) \pi)}{\pi} \int_x^y \int_0^\infty \frac{s^{p-2}}{s + t}\ d s dt \notag \\
& = \frac{\sin((p - 1) \pi)}{\pi} \int_0^\infty s^{p-2} g_s^{[1]}(x,y)\ d s d t \,,
\end{align}
where $g_s(x) = \log(x + s)$ and $g_s^{[1]}$ is the associated divided differences.
By the integral form of $g_0^{[1]}$:
\begin{align*}
g_0^{[1]}(x,y) = \int_0^\infty \frac{1}{(t + x)(t + y)} dt\,,
\end{align*}
we further have, from the formula \eqref{eq:inte_divi_p},
\begin{align} \label{eq:integral_thetap}
f_p^{[1]}(x,y)
& = \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty s^{p-2} \int_0^\infty \frac{1}{(t + s + x)(t + s + y)}\ dt ds\,.
\end{align}
We next recall the comparison result for the Dirichlet forms $\mc{E}_{p,\mc{L}}$, known as the quantum Stroock-Varopoulos inequality, that was proved in \cite[Theorem 14]{beigi2020quantum}.
\begin{lemma} \label{lem:qvs}
Let $\mc{L}$ be the generator of a QMS $\mc{P}_t$ that satisfies $\sigma$-DBC for some invariant state $\sigma \in \mc{D}_+(\mc{H})$, and $\mc{E}_{p,\mc{L}}$ be defined in \eqref{def:p_diri}. Then, for $X \ge 0$, we have
\begin{equation} \label{eq:quantumsv}
\mc{E}_{p,\mc{L}}(I_{p,2}(X)) \ge \mc{E}_{q,\mc{L}}(I_{q,2}(X))\,, \quad 0 < p \le q \le 2\,.
\end{equation}
\end{lemma}
In the special case $p \ge 1$ and $q = 2$, Lemma \ref{lem:qvs} gives the strong $L_p$ regularity of the Dirichlet form \cite{kastoryano2013quantum,bardet2017estimating}, which we slightly generalize as follows. The proof follows from the basic inequality: for $p \in (1,2]$, $a,b > 0$,
\begin{align*}
(a-b)(a^{p-1} - b^{p-1}) \le \left( a^{p/2} - b^{p/2}\right)^2 \le \frac{p^2}{4(p-1)}(a-b)(a^{p-1} - b^{p-1})\,,
\end{align*}
and similar arguments in \cite[Theorem 4.1]{bardet2017estimating}. Hence we omit it here.
\begin{corollary} \label{lem:com_diri}
Under the same assumptions as in Lemma \ref{lem:qvs}, it holds that, for $p \in (1,2]$ and $X \ge 0$,
\begin{align} \label{eq:lpreg}
\mc{E}_{2, \mc{L}}(I_{2,p}(X)) \le \mc{E}_{p, \mc{L}}(X) \le \frac{p^2}{4(p-1)} \mc{E}_{2, \mc{L}}(I_{2,p}(X))\,.
\end{align}
In particular, the lower inequality in \eqref{eq:lpreg} holds for all $p \ge 1$.
\end{corollary}
We now introduce the entropy function $\Ent_{p,\sigma}(X)$, for $p \ge 1$ and $\sigma \in \mc{D}_+(\mc{H})$, as follows \cite{beigi2020quantum} (our definition differs from the one in \cite{olkiewicz1999hypercontractivity,kastoryano2013quantum} by a factor of $p$):
\begin{align*}
\Ent_{p,\sigma}(X) := \tr \left( \left(\Gamma_\si^{1/p}(X)\right)^p \left(\log \left(\Gamma_\si^{1/p}(X) \right)^p - \log \sigma \right) \right) - \norm{X}_{p,\sigma}^p \log \norm{X}_{p,\sigma}^p\,,\quad X \ge 0\,,
\end{align*}
and Umegaki's relative entropy:
\begin{equation} \label{def:relative_entropy}
D(\rho \| \sigma) = \tr(\rho \log \rho - \rho \log \sigma)\,,\quad \rho \in \mathcal{D}(\mc{H})\,.
\end{equation}
We recall from \cite[Proposition 3]{beigi2020quantum} that
\begin{align} \label{eq:differ_norm}
\frac{d}{d p} \norm{Y}_{p,\sigma} = \frac{1}{p^2} \norm{Y}_{p,\sigma}^{1-p}
\Ent_{p,\sigma}(I_{p,p}(Y))\,,\quad Y \in \mc{B}_{sa}(\mc{H})\,,
\end{align}
which, by the chain rule, implies
\cite[Theorem 2.7]{olkiewicz1999hypercontractivity}
\begin{equation} \label{eq:dev_pnorm}
\frac{d}{d p} \norm{Y}_{p,\sigma}^p = \tr \Big((\Gamma_\sigma^{1/p}(Y))^p (\log \Gamma_\sigma^{1/p}(Y) - \frac{1}{p} \log \sigma ) \Big)\,,\quad Y \ge 0\,.
\end{equation}
The above formulas relate the differential of $L_p$-norm $\norm{\cdot}_{p,\sigma}$ and the entropy function $\Ent_{p,\sigma}$. The following lemma provides some basic properties of entropy functions $\Ent_{p,\sigma}(X)$; see \cite{kastoryano2013quantum,beigi2020quantum}.
\begin{lemma} \label{lem:ent_rela}
For all $X \ge 0$, we have $\Ent_{p,\sigma}(X) \ge 0$. Moreover, for any density matrix $\rho \in \mc{D}(\mc{H})$, it holds that
\begin{align*}
\Ent_{2,\sigma}\big(\Gamma_\si^{-1/2}\left(\sqrt{\rho}\right)\big) = D(\rho || \sigma) \quad \text{and}\quad \Ent_{1,\sigma}\big(\Gamma_\si^{-1}(\rho) \big) = D(\rho || \sigma)\,.
\end{align*}
\end{lemma}
We also recall the sandwiched R\'{e}nyi relative entropy
introduced in \cite{muller2013quantum,wilde2014strong}: for $p \in (0,1) \cup (1,\infty)$,
\begin{align} \label{def:sandwi}
D_p(\rho \| \sigma) :=
\h{p}\log\big(\norm{\Gamma_\sigma^{-1}(\rho)}_{p,\sigma}\big), \quad \rho \in \mathcal{D}(\mc{H})\,,
\end{align}
and
\begin{align} \label{def:maxentropy}
D(\rho\|\sigma) = \lim_{p \to 1} D_p(\rho\|\sigma)\,, \quad D_{\infty}(\rho \| \sigma) := \log \inf\{c > 0\,;\ \rho \le c \sigma
\} = \lim_{p \to \infty} D_p(\rho \| \sigma)\,,
\end{align}
where $ D(\rho\|\sigma)$ is the relative entropy \eqref{def:relative_entropy}, and $D_\infty(\rho\|\sigma)$ is the max-relative entropy \cite{datta2009min,muller2013quantum}. It has been proved in \cite{frank2013monotonicity,beigi2013sandwiched} that $D_p$ satisfies the data processing inequality for $p \in (\frac{1}{2},1)\cup (1,\infty)$.
The operator $\Gamma_\sigma^{-1}(\rho)$ can be viewed as the relative density of a quantum state $\rho \in \mathcal{D}(\mc{H})$ with respect to the full-rank reference state $\sigma \in \mathcal{D}_+(\mc{H})$. For convenience, we shall call operators $X \ge 0$ with $\norm{X}_{1,\sigma} = 1$ relative densities with respect to $\sigma$. In this work, we mainly consider a modified sandwiched R\'{e}nyi entropy, referred to as \emph{quantum $p$-divergence},
\begin{align} \label{def:quanpdivi}
\mc{F}_{p,\sigma}(\rho) := \frac{1}{p(p-1)} \big(\norm{\Gamma_\sigma^{-1}(\rho)}_{p,\sigma}^p - 1 \big)\,,\quad \rho \in \mathcal{D}(\mc{H})\,,
\end{align}
where $\sigma \in \mathcal{D}_+(\mc{H})$ and $p \in (0,1) \cup (1,\infty)$. By definition, it follows that
\begin{equation} \label{eq:rela_sandpdivi}
\mc{F}_{p,\sigma}(\rho) = \frac{1}{p(p-1)} \big(e^{(p-1)D_p(\rho\|\sigma)} - 1 \big)\,,
\end{equation}
and thus, by \eqref{def:maxentropy},
\begin{align*}
\lim_{p \to 1} \mc{F}_{p,\sigma}(\rho) = D(\rho\| \sigma)\,.
\end{align*}
We recall that the variance of $X \in \mc{B}_{sa}(\mc{H})$ is defined by
\begin{align*}
\Var_\sigma(X): = \norm{X - \tr(\sigma X)}_{2,\sigma}^2 = \norm{X}_{2,\sigma}^2 - \norm{X}_{1,\sigma}^2\,.
\end{align*}
Clearly, when $p = 2$, $\mc{F}_{p,\sigma}(\rho)$ reduces to the variance of the relative density of $\rho$, up to a constant factor,
\begin{align*}
\mc{F}_{p,\sigma}(\rho) = \frac{1}{2} \Var_{\sigma}(X)\,, \quad X = \Gamma_\si^{-1} (\rho)\,.
\end{align*}
According to \eqref{eq:rela_sandpdivi}, many properties of $D_p(\rho\|\sigma)$ can be directly translated to $\mc{F}_{p,\sigma}(\rho)$ as follows.
\begin{lemma} \label{lem:prop_divi}
For any $\rho \in \mathcal{D}(\mc{H}), \sigma \in \mathcal{D}_+(\mc{H})$, we have
\begin{enumerate}[1.]
\item $\mc{F}_{p,\sigma}(\rho) \ge 0$, and $ \mc{F}_{p,\sigma}(\rho) = 0$ if and only if $\rho = \sigma$.
\item $\mc{F}_{p,\sigma}(\rho)$ is jointly convex with respect to $(\rho,\sigma)$.
\item The data processing inequality holds for $ \mc{F}_{p,\sigma}(\rho)$, $p \in (\frac{1}{2},1)\cup (1,\infty)$,
\begin{align} \label{eq:dpi_divi}
\mc{F}_{p,\Phi(\sigma)}(\Phi(\rho)) \le \mc{F}_{p,\sigma}(\rho)\,,
\end{align}
where $\Phi$ is a quantum channel.
\end{enumerate}
\end{lemma}
We finally recall the Araki–Lieb–Thirring (ALT) inequality \cite{lieb1997inequalities,araki1990inequality}, which is also useful in the sequel.
\begin{lemma}
For any $A \ge 0$, $B \ge 0$, and $q \ge 0$, it holds that
\begin{align} \label{eq_alt_2}
\tr( (B^r A^r B^r)^q) \le \tr( (BAB)^{rq})\,,\quad 0\le r \le 1\,.
\end{align}
\end{lemma}
\section{Quantum interpolation functional inequalities}
Recall that we limit our discussion to the primitive QMS $\mc{P}_t$ satisfying $\sigma$-DBC for some $\sigma \in \mathcal{D}_+(\mc{H})$. In this section, we introduce two new families of quantum functional inequalities: the quantum $p$-Beckner's inequality and the quantum dual $q$-Beckner's inequality, which interpolate the quantum Sobolev-type inequalities and the Poincar\'{e} inequality. To motivate the definition of quantum Beckner's inequalities, we consider the convergence of a QMS in terms of the quantum $p$-divergence $\mc{F}_{p,\sigma}$. By the limit \eqref{eq:conver_qms} and the contractivity \eqref{eq:dpi_divi} of $\mc{F}_{p,\sigma}$ under CPTP maps, we have that $\mc{F}_{p,\sigma}$ decreases along the dynamic $\rho_t = \mc{P}_t^\dag(\rho)$ and there holds $\mc{F}_{p,\sigma}(\rho_t) \to 0$ as $t \to \infty$. Recalling that the convergence of the Umegaki’s relative entropy $D(\rho \| \sigma)$ is equivalent to the quantum modified log-Sobolev inequality \cite{muller2016entropy,carlen2017gradient}, we will see that a similar result holds for $\mc{F}_{p,\sigma}(\rho)$, namely, the quantum Beckner's inequality characterizes the decay rate of $\mc{F}_{p,\sigma}(\rho_t)$. It is convenient to consider the dynamics of the relative density $X = \Gamma_\si^{-1}(\rho)$, equivalently, the QMS in the
Heisenberg picture. From definition and Lemma \ref{lem:self_adjoint}, we have
\begin{align} \label{eq:evo_density}
X_t := \Gamma_\si^{-1}(\rho_t) = \Gamma_\si^{-1} e^{t \mc{L}^\dag} \Gamma_\si (X) = \mc{P}_t(X)\,.
\end{align}
Then we compute the time derivative of $\mc{F}_{p,\sigma}(\rho_t)$ as
\begin{align} \label{eq:ep_divi}
\frac{d}{d t} \mc{F}_{p,\sigma}(\rho_t) & = \frac{1}{p-1} \langle \Gamma_\si^{-1/\h{p}}((\Gamma_\si^{-1/\h{p}}(\rho_t))^{p-1}), \mc{L}^\dag(\rho_t)\rangle \notag \\
& = \frac{1}{p-1} \langle \Gamma_\si^{-1/\h{p}}(\Gamma_\si^{1/p}(X_t))^{p-1}), \Gamma_\si\mc{L}(X_t)\rangle \notag \\
& = - \frac{4}{p^2} \mc{E}_{p,\mc{L}}(X_t) \le 0\,.
\end{align}
Similarly to the case of $D(\rho \| \sigma)$ \cite{spohn1978entropy}, we can define the entropy production of the semigroup $\mc{P}_t$ associated with the $p$-divergence $\mc{F}_{p,\sigma}$ by the quantity $- \frac{4}{p^2} \mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho))$. As a corollary of \eqref{eq:ep_divi}, we have the nonnegativity of the $p$-Dirichlet form: $\mc{E}_{p,\mc{L}}(X) \ge 0$ for any $X \ge 0$. Moreover, again by \eqref{eq:ep_divi}, a simple use of Gr\"{o}nwall’s inequality concludes that the exponential decay of $\mc{F}_{p,\sigma}$:
\begin{align} \label{eq:exp_divi}
\mc{F}_{p,\sigma}(\rho_t) \le e^{- \frac{4 \alpha_p}{p}t} \mc{F}_{p,\sigma}(\rho)\,, \quad \forall\rho \in \mathcal{D}(\mc{H})\,,
\end{align}
for some $\alpha_p > 0$, is equivalent to the functional inequality:
\begin{equation} \label{eq_beck}
\alpha_p \mc{F}_{p,\sigma}(\rho) \le p^{-1} \mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho))\,, \quad \forall\rho \in \mathcal{D}(\mc{H})\,,
\end{equation}
which we refer to as the quantum Beckner's inequality. By analogy with the classical case \eqref{ineq_e}, the quantum dual Beckner's inequality can also be easily defined; see \eqref{ineq_dbecq} below.
\subsection{Interpolation between Sobolev and Poincar\'{e} inequalities} \label{sec:inter_primi}
Let us now formally define the quantum functional inequalities in the Heisenberg picture that will be investigated in detail
in the following sections.
\begin{definition} \label{def:quantum_func} For a primitive QMS $\mc{P}_t$ with generator $\mc{L}$ satisfying $\sigma$-DBC, let $\mc{E}_{p,\mc{L}}$ be the $p$-Dirichlet form \eqref{def:p_diri}. Then we say that the QMS $\mc{P}_t$ satisfies:
\begin{enumerate}[1.]
\item the Poincar\'{e} inequality if there exists a constant $\lambda > 0$ such that for all $X \in \mc{B}(\mc{H})$,
\begin{equation} \label{ineq_pi}
\lambda \norm{X - \tr(\sigma X) {\bf 1}}^2_{\sigma,f} \le - \langle X, \mc{L} X \rangle_{\sigma,f}\,, \tag{PI}
\end{equation}
where $f:(0,\infty) \to (0,\infty)$ and the norm $\norm{\cdot}_{\sigma,f}$ is defined by the inner product \eqref{eq:general_inner}.
\item the modified log-Sobolev inequality (MLSI) if there exists $\alpha_1 > 0$ such that for all $X \ge 0$,
\begin{equation} \label{ineq_mlsi}
\alpha_1 \Ent_{1,\sigma}(X) \le \mc{E}_{1,\mc{L}}(X)\,. \tag{mLSI}
\end{equation}
\item the $p$-Beckner's inequality with $p \in (1,2]$ if there exists $\alpha_p > 0$ such that for all $X \ge 0$,
\begin{align} \label{ineq_becp}
\alpha_p (\norm{X}^p_{p,\sigma} - \norm{X}_{1,\sigma}^p) \le (p-1) \mc{E}_{p,\mc{L}}(X)\,. \tag{Bec-$p$}
\end{align}
\item the log-Sobolev inequality (LSI) if there exists $\beta > 0$ such that for all $Y \ge 0$,
\begin{align} \label{ineq_lsi}
\beta \Ent_{2,\sigma}(Y) \le \mc{E}_{2,\mc{L}}(Y)\,. \tag{LSI}
\end{align}
\item the dual $q$-Beckner's inequality with $q \in [1,2)$ if there exists $\beta_q > 0$ such that for all $Y \ge 0$,
\begin{align} \label{ineq_dbecq}
\beta_q \Var_{q,\sigma}(Y)\le (2-q) \mc{E}_{2,\mc{L}}(Y)\,, \tag{Bec'-$q$}
\end{align}
where $\Var_{q,\sigma}(Y)$ is the $q$-variance: $\Var_{q,\sigma}(Y): = \norm{Y}_{2,\sigma}^2 - \norm{Y}_{q,\sigma}^2$.
\end{enumerate}
\end{definition}
The above functional inequalities can be easily reformulated in the Schr\"{o}dinger picture by inserting $X = \Gamma_\si^{-1}(\rho)$. For instance, from the definition of $\mc{F}_{p,\sigma}(\rho)$ \eqref{def:quanpdivi}, \eqref{ineq_becp} is clearly equivalent to \eqref{eq_beck}, while, by Lemma \ref{lem:ent_rela}, \eqref{ineq_mlsi} with $X = \Gamma_\si^{-1}(\rho)$ gives the familiar one in terms of quantum states \cite{rouze2019concentration}:
\begin{equation*}
\alpha_1 D(\rho \| \sigma) \le - \frac{1}{4} \tr \left(\mc{L}^\dag(\rho) (\log \rho - \log \sigma) \right)\,, \quad \forall \rho \in \mathcal{D}(\mc{H})\,.
\end{equation*}
For ease of exposition, we call the optimal constant in \eqref{ineq_becp}
the quantum Beckner constant, denoted by $\alpha_p(\mc{L})$. Similar
notions apply to other functional inequalities defined above. In particular, the Poincar\'{e} constant $\lambda(\mc{L})$ is nothing but the spectral gap of $\mc{L}$, which is independent of the choice of $f$. Indeed, note from Lemma \ref{lem:self_adjoint} and Lemma \ref{lem:DBC} that $\mc{L}$ is self-adjoint with respect to $\langle\cdot,\cdot\rangle_{\sigma,f}$, and $\ker({\mc{L}}) = \ker(\nabla) = {\rm span}\{ {\bf 1}_{\mc{H}} \}$ holds, which also equals to the commutant of operators $(V_j)_{j \in J}$. Then, by the min-max theorem, the Poincar\'{e} constant
\begin{equation} \label{def:spect}
\lambda(\mc{L}) := \inf_{X \in \mathcal{B}(\mc{H}), X \neq {\bf 1}} \frac{- \langle X, \mc{L} X \rangle_{\sigma,f}}{\norm{X - \tr(\sigma X){\bf 1}}^2_{\sigma,f}}
\end{equation}
characterizes the smallest non-zero eigenvalue of $- \mc{L}$ (i.e., the spectral gap). It is worth pointing out that since the Lindbladian $\mc{L}$ is Hermitian-preserving, the infimum in \eqref{def:spect} can be equivalently taken
over $X$ in $\mc{B}_{sa}(\mc{H})$.
In this section, we discuss some properties of the optimal constants for the functional inequalities in Definition \ref{def:quantum_func} and the relations among them. The basic relations between \eqref{ineq_mlsi}, \eqref{ineq_lsi}, and \eqref{ineq_pi} have been investigated and understood in \cite[Proposition 13 and Theorem 16]{kastoryano2013quantum}, which we state below for completeness and future use.
\begin{lemma} \label{lem:aux_func}
For a primitive QMS $\mc{P}_t$ satisfying $\sigma$-DBC, it holds that
\begin{align*}
2 \beta(\mc{L}) \le 2 \alpha_1(\mc{L}) \le \lambda (\mc{L})\,.
\end{align*}
\end{lemma}
We first consider the properties of quantum dual $q$-Beckner's inequalities \eqref{ineq_dbecq}. It is clear from definition that when $q = 1$, the inequality \eqref{ineq_dbecq} reduces to the Poincar\'{e} inequality \eqref{ineq_pi} with $
f = x^{1/2}$. On the other hand, it is easy to see that in the limit $q \to 2$, \eqref{ineq_dbecq} gives the quantum LSI \eqref{ineq_lsi}. In this sense, \eqref{ineq_dbecq} can be considered as an interpolating family between the quantum LSI and the Poincar\'{e} inequality. Indeed, we have Proposition \ref{dualbecktolsi} below. The proof is based on the following monotonicity lemma.
\begin{lemma}\label{lem:monotone}
The function $ \frac{\Var_{q,\sigma}(Y)}{1/q - 1/2}$ is monotone increasing for $q \in [1,2)$.
\end{lemma}
\begin{proof}
We recall the interpolation of the noncommutative $L_p$ space
\cite[Corollary 3]{beigi2013sandwiched}: for $1\le p_0 < p_1 \le \infty$, and $1/p_\theta = (1-\theta)/p_0 + \theta/p_1$ with $\theta \in [0,1]$,
$$
\norm{Y}_{p_\theta,\sigma} \le \norm{Y}_{p_0,\sigma}^\theta \norm{Y}_{p_1,\sigma}^{1-\theta}\,, \quad \forall\, Y \in \mathcal{B}(\mc{H})\,.
$$
We immediately see that the function $\log \norm{Y}_{1/t,\sigma}$ is a convex function for $t \in [0,1]$, which implies that
\begin{equation*}
\varphi(t) := \exp(2 \log \norm{Y}_{1/t,\sigma}) =\norm{Y}^2_{1/t,\sigma}
\end{equation*}
is also convex. Therefore, we have that the function
\begin{align*}
\frac{\Var_{q,\sigma}(Y)}{1/q - 1/2} = \frac{\varphi(1/2) - \varphi(1/q)}{1/q - 1/2}
\end{align*}
is increasing in $q$.
\end{proof}
\begin{proposition} \label{dualbecktolsi}
Let $\mc{P}_t = e^{t\mc{L}}$ be a primitive QMS. If \eqref{ineq_dbecq} holds for every $q \in [1,2)$ with $\limsup_{q \to 2^-}\beta_q > 0$, then \eqref{ineq_lsi} holds with constant $\beta \ge \limsup_{q \to 2^-} \beta_q/2$. Conversely, if \eqref{ineq_lsi} holds with $\beta > 0$, then \eqref{ineq_dbecq} holds for every $q \in [1,2)$ with $\beta_q \ge q \beta$.
\end{proposition}
\begin{proof}
Suppose that \eqref{ineq_dbecq} holds with $\limsup_{q \to 2^-}\beta_q > 0$. By the formula \eqref{eq:differ_norm}, we have
\begin{align} \label{auxeq_1:limit}
\frac{1}{2}\Ent_{2,\sigma}(Y) = \lim_{q \to 2^-}\frac{\norm{Y}_{2,\sigma}^2 - \norm{Y}_{q,\sigma}^2}{2-q} \quad \text{for any}\ Y \ge 0\,.
\end{align}
Then taking the upper limit as $q \to 2^-$ in \eqref{ineq_dbecq}, we find
\begin{align*}
\frac{1}{2} \bigl(\limsup_{q \to 2^-} \beta_q\bigr) \Ent_{2,\sigma}(Y) \le \mc{E}_{2,\mc{L}}(Y)\,.
\end{align*}
Thus, by definition, \eqref{ineq_lsi} holds with $\beta \ge \limsup_{q \to 2^-} \beta_q/2$.
For the reverse direction, by Lemma \ref{lem:monotone},
it follows, from \eqref{auxeq_1:limit} and the assumption that \eqref{ineq_lsi} holds, that
\begin{align*}
\frac{\Var_{q,\sigma}(Y)}{1/q - 1/2} \le \limsup_{q \to 2^-} \frac{\Var_{q,\sigma}(Y)}{1/q - 1/2} = 2 \Ent_{2,\sigma}(Y) \le 2\beta^{-1} \mc{E}_{2,\mc{L}}(Y)\,,
\end{align*}
that is, \eqref{ineq_dbecq} holds with $\beta_q \ge q \beta$.
\end{proof}
The quantum dual Beckner constant $\beta_q(\mc{L})$ has the following monotonicity property, which implies that if \eqref{ineq_dbecq} holds for some $q \in [1,2)$, then it holds for all $q \in [1,2)$.
\begin{proposition} \label{prop:mono_dual_beck}
Let $\mc{P}_t = e^{t\mc{L}}$ be a primitive QMS.
For the optimal dual Beckner constant $\beta_q(\mc{L})$ in \eqref{ineq_dbecq}, it holds that $\beta_q(\mc{L})/(2-q)$ is increasing and $\beta_q(\mc{L})/q$ is decreasing for $q \in [1,2)$.
\end{proposition}
\begin{proof}
For the first claim, by the ordering of $\norm{\cdot}_{p,\sigma}$ in Lemma \ref{lem:prop:norm_power}, we have, for $ 1 \le q \le q' < 2$ and $Y \ge 0$,
\begin{equation*}
\beta_q(\norm{Y}^2_{2,\sigma} - \norm{Y}^2_{q',\sigma}) \le \beta_q(\norm{Y}^2_{2,\sigma} - \norm{Y}^2_{q,\sigma}) \le \frac{2-q}{2-q'} (2 - q') \mc{E}_{2,\mc{L}}(Y)\,,
\end{equation*}
that is, $\beta_{q'}/(2-q') \ge \beta_q/(2-q)$. The second claim is a direct consequence of Lemma \ref{lem:monotone}. Indeed, due to the monotonicity, we have
\begin{equation*}
2q \beta_{q'} \frac{\Var_{q,\sigma}(Y)}{2 - q} \le 2q' \beta_{q'} \frac{\Var_{q',\sigma}(Y)}{2-q'} \le 2q' \mc{E}_{2,\mc{L}}(Y)\,,
\end{equation*}
which clearly shows $\beta_q/q \ge \beta_{q'}/q'$.
\end{proof}
We also prove an analog result for the quantum Beckner constant $\alpha_p(\mc{L})$.
\begin{proposition} \label{prop:mono_beck}
Let $\mc{P}_t = e^{t\mc{L}}$ be a primitive QMS with $\sigma$-DBC.
If the Beckner's inequality \eqref{ineq_becp} holds for some $p' \in (1,2]$ with $\alpha_{p'} > 0$, then for any $1 < p \le p'$, the inequality \eqref{ineq_becp} holds with constant $\alpha_p$ satisfying
\begin{align} \label{eq:monoconst}
\frac{p}{p-1}\alpha_p \ge \frac{p'}{p'-1}\alpha_{p'}.
\end{align}
Equivalently, $\frac{p}{p-1} \alpha_p(\mc{L})$, as a function of $p \in (1,2]$, is nonincreasing.
\end{proposition}
\begin{proof}
It suffices to prove the inequality \eqref{eq:monoconst}.
For this, by the quantum Stroock-Varopoulos inequality in \eqref{eq:quantumsv} and \eqref{ineq_becp} for $p'$, we have
\begin{align} \label{auxeq:mono}
\mc{E}_{p,\mc{L}}(X) \ge \mc{E}_{p',\mc{L}}(I_{p',p}(X)) \ge \alpha_{p'} (p'-1)^{-1} \left(\norm{I_{p',p}(X)}_{p',\sigma}^{p'} - \norm{I_{p',p}(X)}_{1,\sigma}^{p'}\right)\,.
\end{align}
Note from Lemma \ref{lem:prop:norm_power} that $\norm{I_{p',p}(X)}_{p',\sigma}^{p'} = \norm{X}_{p,\sigma}^p$. By ALT inequality \eqref{eq_alt_2}, we find
\begin{align*}
\norm{I_{p',p}(X)}_{1,\sigma} & = \tr\left( \sigma^{\frac{1}{2\widehat{p'}}} \left(\sigma^{\frac{1}{2p}} X \sigma^{\frac{1}{2p}}\right)^{\frac{p}{p'}}\sigma^{\frac{1}{2\widehat{p'}}}\right) \le \norm{X}^{\frac{p}{p'}}_{\frac{p}{p'},\sigma}\,,
\end{align*}
since
\begin{equation*}
\frac{p}{p'} < 1\,,\quad \frac{1}{2\widehat{p'}} \frac{p'}{p} + \frac{1}{2p} = \frac{p'}{2p}\,.
\end{equation*}
Therefore, by \eqref{auxeq:mono} and Lemma \ref{lem:prop:norm_power}, it follows that
\begin{equation*}
\mc{E}_{p,\mc{L}}(X) \ge \alpha_{p'} (p'-1)^{-1} \left(\norm{X}_{p,\sigma}^p - \norm{X}^p_{\frac{p}{p'},\sigma}\right) \ge \alpha_{p'} (p'-1)^{-1} \left(\norm{X}_{p,\sigma}^p - \norm{X}^p_{1,\sigma}\right)\,.
\end{equation*}
The proof is complete by definition of Beckner's inequality.
\end{proof}
We finally relate the $p$-Beckner's inequality and the dual $q$-Beckner's inequality with $q = 2/p$.
\begin{proposition} \label{prop:beck_to_dual_beck}
Let $\mc{P}_t = e^{t\mc{L}}$ be a primitive QMS with $\sigma$-DBC.
Let $p \in (1,2]$ and $q = 2/p \in [1,2)$. If \eqref{ineq_dbecq} holds with $\beta_q$, then \eqref{ineq_becp} holds with $\alpha_p \ge p \beta_q/2$.
\end{proposition}
\begin{proof}
We substitute $Y = I_{2,p}(X)$ for $X \ge 0$ in \eqref{ineq_dbecq} and find
\begin{equation} \label{auxeq_dualbeck_to_beck}
\norm{I_{2,p}(X)}^2_{2,\sigma} - \norm{I_{2,p}(X)}_{2/p,\sigma}^{2} = \norm{X}^p_{p,\sigma} - \norm{I_{2,p}(X)}_{2/p,\sigma}^{2} \le \beta_q^{-1}(2-q) \mc{E}_{2,\mc{L}}(I_{2,p}(X))\,.
\end{equation}
By ALT inequality \eqref{eq_alt_2} and Lemma \ref{lem:prop:norm_power}, we have
\begin{align} \label{auxeq_dualbeck_to_beck_2}
\norm{I_{2,p}(X)}_{2/p,\sigma}^{2/p} & = \tr \big(\sigma^{\frac{p}{4} - \frac{1}{4}} \big(\sigma^{\frac{1}{2p}} X \sigma^{\frac{1}{2p}} \big)^{\frac{p}{2}}\sigma^{\frac{p}{4} - \frac{1}{4}}\big)^{\frac{2}{p}} \notag \\
& \le \tr \bigl((\Gamma_\sigma(X))^{\frac{p}{2}}\bigr)^{\frac{2}{p}} \le \norm{X}_{1,\sigma}.
\end{align}
Then, by $L_p$ regularity in Corollary \ref{lem:com_diri}, it follows from \eqref{auxeq_dualbeck_to_beck} and \eqref{auxeq_dualbeck_to_beck_2} that
\begin{align*}
\norm{X}^p_{p,\sigma} - \norm{X}_{1,\sigma}^p \le \beta_q^{-1}(2-q) \mc{E}_{p,\mc{L}}(X)\,,
\end{align*}
which gives $\alpha_p \ge p \beta_q/2$.
\end{proof}
\subsection{Quantum Beckner constant} \label{sec:positive_stab}
In this section, we focus on the properties of the quantum Beckner constant $\alpha_p(\mc{L})$. We first provide a two-sided bound for $\alpha_p(\mc{L})$ in terms of the Poincar\'{e} constant $\lambda(\mc{L})$, which will yield the uniform positivity of $\alpha_p(\mc{L})$ (i.e., $\inf_{p \in (1,2]}\alpha_p(\mc{L}) > 0$). Then, we investigate the relations between the quantum MLSI \eqref{ineq_mlsi} and the $p$-Beckner’s inequality \eqref{ineq_becp}. Finally, we will extend the quantum Holley-Stroock’s argument from \cite{junge2019stability} and give the stability estimates for the Beckner constant $\alpha_p(\mc{L})$ with respect to the invariant state $\sigma$.
\subsubsection{Uniform positivity of $\alpha_p(\mc{L})$}
Let us start with the following proposition generalizing \cite[Proposition 2.8]{adamczak2022modified} in the classical setting to the quantum regime, which shows that the quantum Poincar\'{e} inequality \eqref{ineq_pi} implies $p$-Beckner's inequality \eqref{ineq_becp} for $p \in (1,2]$.
\begin{proposition} \label{prop:pb}
Let $\mc{P}_t = e^{t\mc{L}}$ be a primitive QMS with $\sigma$-DBC.
If the Poincar\'{e} inequality \eqref{ineq_pi} holds with constant $\lambda$, then, for all $p \in (1,2]$, the Beckner's inequality \eqref{ineq_becp} holds with constant $\alpha_p$ satisfying
\begin{equation} \label{eq:poin_to_beck}
\alpha_p \ge \lambda (p-1)\,.
\end{equation}
In particular, if \eqref{ineq_mlsi} holds with constant $\alpha_1$, then $\alpha_p \ge 2 \alpha_1(p-1)$.
\end{proposition}
\begin{proof}
We first claim that there holds
\begin{equation} \label{auxeq:convex_bec}
\norm{X}_{p,\sigma}^p - \norm{X}_{1,\sigma}^p \le \langle I_{\h{p},p}(X), X - \tr(\sigma X) {\bf 1} \rangle_{\sigma,1/2}\,,\quad \forall X \ge 0\,.
\end{equation}
Indeed, a direct computation gives
\begin{align} \label{auxeq:ipp}
\left\langle I_{\h{p},p}(X), X - \tr(\sigma X) {\bf 1} \right\rangle_{\sigma,1/2} = \norm{X}_{p,\sigma}^p - \norm{X}_{1,\sigma} \tr(\sigma^{1/p}(\Gamma_\si^{1/p}X)^{p-1})\,.
\end{align}
Then, by ALT inequality \eqref{eq_alt_2} and the ordering of $\norm{\cdot}_{p,\sigma}$ in Lemma \ref{lem:prop:norm_power}, we have
\begin{align*}
\tr(\Gamma_\si^{1/p}(\Gamma_\si^{1/p}X)^{p-1}) & \le \tr((\Gamma_\si^{1/(p-1)}X )^{p-1}) = \norm{X}_{p-1,\sigma}^{p-1} \le \norm{X}_{1,\sigma}^{p-1}\,,
\end{align*}
which, along with \eqref{auxeq:ipp}, implies the desired inequality \eqref{auxeq:convex_bec}.
One can readily note that up to some constant, the right hand term in \eqref{auxeq:convex_bec} is the $p$-Dirichlet form $\mc{E}_{p,\mc{L}_{depol}}$ defined by \eqref{def:p_diri}
associated with the generator $\mc{L}_{depol}$ with $\gamma = 1$; see Example \ref{exp:dep_channel}. We proceed by using Corollary \ref{lem:com_diri} and find
\begin{align} \label{auxeq:diri_depo}
\mc{E}_{p, \mc{L}_{depol}}(X) \le \frac{p^2}{4(p-1)} \Var_{2,\sigma}(I_{2,p}(X)) \le \frac{1}{\lambda}\frac{p^2}{4(p-1)} \mc{E}_{2,\mc{L}}(I_{2,p}(X)) \le \frac{1}{\lambda}\frac{p^2}{4(p-1)} \mc{E}_{p, \mc{L}}(X)\,,
\end{align}
where we have also used the assumption that \eqref{ineq_pi} holds, and the observation:
\begin{equation*}
\Var_{2,\sigma}(X) = \langle X - \tr(\sigma X) {\bf 1}, X - \tr(\sigma X) {\bf 1} \rangle_{\sigma,1/2} = \mc{E}_{2,\mc{L}_{depol}}(X)\,.
\end{equation*}
Then it follows from \eqref{auxeq:convex_bec} and \eqref{auxeq:diri_depo} that
\begin{align*}
\lambda (\norm{X}_{p,\sigma}^p - \norm{X}_{1,\sigma}^p) \le \mc{E}_{p, \mc{L}}(X)\,,
\end{align*}
that is, $\alpha_p \ge \lambda (p-1)$ holds. The proof is completed by Lemma \ref{lem:aux_func}.
\end{proof}
Unfortunately, the lower bound for the Beckner constant $\alpha_p(\mc{L})$ provided in \eqref{eq:poin_to_beck}
vanishes as $p \to 1^+$, and the proof techniques used in the classical case \cite[Proposition 2.9 and Theorem 2.1]{adamczak2022modified} to deal with this limiting regime cannot be easily extended to the noncommutative setting.
In the following, we establish a uniform lower bound for $\alpha_p(\mc{L})$ by adopting the analysis framework recently proposed in \cite{gao2021complete} for the complete modified log-Sobolev inequality. We first extend the key estimates in \cite[Lemma 2.1 and Lemma 2.2]{gao2021complete} to our case.
\begin{lemma}\label{lem:mono_norm}
Let operators $X_i, Y_i > 0$, $ i = 1,2$, satisfy $X_i \le c Y_i$ for some $c > 0$. It holds that
\begin{align} \label{eq:mono_norm_two}
\langle A, f_p^{[1]}(Y_1, Y_2) A\rangle \le c^{2-p} \langle A , f_p^{[1]}( X_1, X_2) A \rangle\,, \quad \forall A \in \mathcal{B}(\mc{H})\,,
\end{align}
where $f_p^{[1]}$ is the divided difference of the function $f_p$ defined in \eqref{def:funfp}.
\end{lemma}
\begin{proof}
The proof follows from the integral form \eqref{eq:integral_thetap} of $f_p^{[1]}$:
\begin{align} \label{eq:auxest_mono}
\langle A, f_p^{[1]}(X_1, X_2) A\rangle
& = \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty s^{p-2} \int_0^\infty \tr \left( A^* \frac{1}{t + s + X_1} A \frac{1}{t + s + X_2} \right) dt ds \notag \\
& \ge \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty s^{p-2} \int_0^\infty \tr \left(A^* \frac{1}{t + s + c Y_1} A \frac{1}{t + s + c Y_2} \right) dt ds \notag \\
& = \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty s^{p-2} \frac{1}{c^2} \int_0^\infty \tr \left(A^* \frac{1}{\frac{t + s}{c} + Y_1} A \frac{1}{\frac{t + s}{c} + Y_2} \right) dt ds \notag \\
& = \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty (cs)^{p-2} \int_0^\infty \tr \left(A^* \frac{1}{t + s + Y_1} A \frac{1}{t + s + Y_2} \right) dt ds \\
& = c^{p-2} \langle A, f_p^{[1]}(Y_1, Y_2) A \rangle\,,\notag
\end{align}
where in the first inequality we have used the fact that $t^{-1}$ is operator monotone decreasing; in the third line we have used the change of variable $r \to r/s$ and $t \to t/s$.
\end{proof}
For our subsequent discussion, several useful observations are in order. First, we define the function
\begin{align*}
\varphi_p(x) = \frac{1}{p-1}\frac{x - x^{1/p}}{x^{1/p} - 1}\,.
\end{align*}
Then the kernel $J_\sigma^{\varphi_p} = \varphi_p(\Delta_\sigma) R_\sigma $ for the inner product $\langle \cdot, \cdot\rangle_{\sigma,\varphi_p}$ (cf.\,\eqref{def:operator_kernel} and \eqref{eq:general_inner}) can be reformulated, in terms of the double sum operator associated with $f_p^{[1]}$, as follows:
\begin{align} \label{eq:kernel_iden}
\varphi_p(\Delta_\sigma) R_\sigma & = \frac{1}{p-1} \frac{\Delta_{\sigma} - \Delta_\sigma^{1/p}}{\Delta_{\sigma}^{1/p} - 1} R_\sigma \notag \\
& = \frac{1}{p-1} \frac{L_{\sigma}^{(p-1)/p} - R^{(p-1)/p}_\sigma}{L_{\sigma}^{1/p} - R^{1/p}_\sigma}L_\sigma^{1/p}R_\sigma^{1/p} \notag \\ & = \Gamma_\si^{1/p} f_p^{[1]}\left(\sigma^{1/p},\sigma^{1/p}\right)\Gamma_\si^{1/p}\,.
\end{align}
Second, it turns out that the inner product $\langle \cdot , \cdot \rangle_{\sigma,\varphi_p}$ is also related to the $\chi^2$-divergence associated with the power difference $\kappa_\alpha$ (cf.\,Definition \ref{def:quantumchi} and \eqref{def:pdiff}). Indeed, for $\rho \in \mathcal{D}(\mc{H})$ with the relative density $X = \Gamma_\si^{-1}(\rho)$, we can directly compute
\begin{equation*}
\norm{X - {\bf 1}}_{\sigma,\varphi_p}^2 = \langle \rho - \sigma, \Gamma_\si^{-1}\varphi_p(\Delta_\sigma)R_\sigma \Gamma_\si^{-1}(\rho - \sigma)\rangle = \langle \rho-\sigma, \Omega_\sigma^{x^{-1}\varphi_p}(\rho - \sigma)\rangle\,,
\end{equation*}
by noting
\begin{align} \label{eq:kernel_iden_2}
\Gamma_\si^{-1}\varphi_p(\Delta_\sigma)R_\sigma \Gamma_\si^{-1} = \Delta_\sigma^{-1}\varphi_p(\Delta_\sigma)R_\sigma^{-1} = \Omega_\sigma^{x^{-1}\varphi_p}\,,
\end{align}
see \eqref{def:omega} for the definition of the operator $\Omega_\sigma^{x^{-1}\varphi_p}$. Then we find, by setting $\alpha = 1/p$ and \eqref{def:pdiff},
\begin{align} \label{auxeq_power_diff}
x^{-1}\varphi_{1/\alpha}(x) = \frac{1}{p-1}\frac{1-x^{(1-p)/p}}{x^{1/p}-1} = \frac{\alpha}{\alpha-1} \frac{x^{\alpha-1} - 1}{x^{\alpha} - 1} = \kappa_\alpha\,.
\end{align}
With the notation defined above, we can conclude
\begin{align} \label{eq:chi_heisen}
\norm{X - {\bf 1}}_{\sigma,\varphi_p}^2 = \chi^2_{\kappa_\alpha}(\rho,\sigma)\,.
\end{align}
Third, by Lemma \ref{lem:DBC}, it holds that
\begin{align} \label{eq:rep_dbcl}
- \left\langle X, \mc{L} X \right\rangle_{\sigma,\varphi_p}
& = \left\langle \partial_j X, R_{e^{-\omega_j/2} \sigma} \varphi_p\left(L_{e^{\omega_j/2}\sigma} R^{-1}_{e^{-\omega_j/2} \sigma}\right) \partial_j X \right\rangle
\notag \\
& = \left\langle \Gamma_\si^{1/p} \big(\partial_j X\big), f_p^{[1]}\left(e^{\omega_j/2p}\sigma^{1/p}, e^{-\omega_j/2p}\sigma^{1/p}\right) \Gamma_\si^{1/p}\big(\partial_j X\big) \right\rangle\,,\quad X \in \mathcal{B}(\mc{H})\,,
\end{align}
where the second equality follows from the same calculation as in \eqref{eq:kernel_iden}.
We are now ready to give the following crucial lemma that provides a two-sided estimate of the quantum $p$-divergence $\mc{F}_{p,\sigma}(\rho)$ in terms of the quantum $\chi^2$-divergence associated with the power difference $\kappa_{1/p}$.
\begin{lemma} \label{lem:two-sided}
For $\rho \in \mathcal{D}_+(\mc{H})$ satisfying $\rho \le c \sigma$ for some $c > 0$, it holds that
\begin{equation} \label{eq:two_sided_dense}
k_p(c) \chi_{\kappa_{1/p}}^2(\rho,\sigma) \le \mc{F}_{p,\sigma}(\rho) \le p^{-1} \chi_{\kappa_{1/p}}^2(\rho,\sigma)\,,
\end{equation}
where the constant $k_p(c)$ is given by
\begin{align*}
k_p(c) = \frac{c^p - 1 - p(c-1)}{p(c-1)^2 (p-1)}.
\end{align*}
In particular, the upper bound estimate in \eqref{eq:two_sided_dense} holds for any $\rho \in \mathcal{D}_+(\mc{H})$.
\end{lemma}
\begin{proof}
Recalling the relation \eqref{eq:chi_heisen}, we will prove the inequality \eqref{eq:two_sided_dense} in the Heisenberg picture:
\begin{align} \label{eq:two_sided_est}
k_p(c) \norm{X - {\bf 1}}^2_{\sigma,\varphi_p} \le \frac{1}{p(p-1)}\left(\norm{X}_{p,\sigma}^p - 1\right) \le p^{-1} \norm{X - {\bf 1}}^2_{\sigma,\varphi_p},
\end{align}
for $X > 0$ satisfying $\tr(\sigma X) = 1$ and $X \le c {\bf 1}$.
We define $X_t = (1 - t) {\bf 1} + t X$, $t \in [0,1]$, and consider the function:
\begin{equation*}
\varphi(t) := \frac{1}{p(p-1)}\left(\norm{X_t}_{p,\sigma}^p - 1 \right).
\end{equation*}
It is easy to compute the derivatives:
\begin{align*}
\varphi'(t) = \frac{1}{p-1}\tr \left( \left( \Gamma_\si^{1/p}(X_t)\right)^{p-1} \Gamma_\si^{1/p}\left(X-{\bf 1}\right) \right)\,,
\end{align*}
and
\begin{align} \label{auxeq:2nd_der}
\varphi''(t) = \left\langle \Gamma_\si^{1/p}(X-{\bf 1}), f_p^{[1]}\left(\Gamma_\si^{1/p}(X_t),\Gamma_\si^{1/p}(X_t) \right) \Gamma_\si^{1/p}(X-{\bf 1}) \right\rangle.
\end{align}
By assumption $X \le c {\bf 1}$, we have
\begin{align} \label{aauxeq}
(1 - t) {\bf 1} \le X_t \le (1 + (c - 1) t) {\bf 1}\,,
\end{align}
and hence $(1-t) \sigma^{1/p}\le \Gamma_\si^{1/p}(X_t) \le (1 + (c - 1) t) \sigma^{1/p}$.
Then applying Lemma \ref{lem:mono_norm} to \eqref{auxeq:2nd_der} gives
\begin{align*}
(1 + (c-1)t)^{p-2} \left \langle X -{\bf 1}, J_\sigma^{\varphi_p} (X -{\bf 1}) \right\rangle \le \varphi''(t) \le (1 - t)^{p-2} \left \langle X -{\bf 1}, J_\sigma^{\varphi_p} (X -{\bf 1}) \right\rangle,
\end{align*}
where we have used \eqref{aauxeq} and the observation \eqref{eq:kernel_iden}.
Note from $\tr (\sigma X) = 1$ that $\varphi'(0)= 0$. It follows that
\begin{align*}
\varphi(1) - \varphi(0) = \int_0^1 \int_0^x \varphi''(t) dt dx & \le \int_0^1 \int_0^x (1 - t)^{p-2} dt dx \left \langle X -{\bf 1}, J_\sigma^{\varphi_p} (X -{\bf 1}) \right\rangle \\
& \le p^{-1} \norm{X - {\bf 1}}^2_{\sigma,\varphi_p}\,.
\end{align*}
Similarly, for the lower bound, we have
\begin{align*}
\varphi(1) - \varphi(0) & \ge \int_0^1 \int_0^x (1 + (c - 1) t)^{p-2} dt dx \left \langle X -{\bf 1}, J_\sigma^{\varphi_p} (X -{\bf 1}) \right\rangle \\
& \ge \frac{c^p - 1 - p(c-1)}{p(c-1)^2 (p-1)} \norm{X - {\bf 1}}^2_{\sigma,\varphi_p}.
\end{align*}
The proof is complete by noting $\varphi(0) = 0$.
\end{proof}
With the help of above lemmas, we can extend \cite[Theorem 3.3]{gao2021complete} to uniformly bound the Beckner constant $\alpha_p(\mc{L})$. For this, we define a constant for the invariant state $\sigma$ of $\mc{P}_t^\dag$:
\begin{align} \label{def:constcsi}
C(\sigma):= & \inf\{c > 0\,;\ \rho \le c \sigma
\ \text{for all}\ \rho \in \mc{D}(\mc{H})
\}\,.
\end{align}
It is easy to note that $C(\sigma)$ is closely related to the max-relative entropy $D_{\infty}$ in \eqref{def:maxentropy} and can be explicitly represented:
\begin{align*}
C(\sigma) = \sup_{\rho \in \mc{D}(\mc{H})} \exp\left(D_{\infty}(\rho \| \sigma) \right) = \sigma_{\min}^{-1}\,,
\end{align*}
where $\sigma_{\min}$ is the smallest eigenvalue of $\sigma$. Indeed, \cite[Lemma 2.1]{muller2018sandwiched} shows that for $\sigma \in \mathcal{D}_+(\mc{H})$ and $p \in [1,\infty]$, the sandwiched R\'{e}nyi relative entropy $D_p$ satisfies
\begin{align} \label{eq:suprhodp}
\sup_{\rho \in \mathcal{D}(\mc{H})} D_p(\rho \| \sigma) = \log \sigma_{\min}^{-1}\,.
\end{align}
We remark here that $\sigma_{\min}$ has been estimated in several interesting cases; see \cite[Remark 1]{kastoryano2013quantum} and \cite[Section 7]{muller2018sandwiched}.
\begin{theorem} \label{thm:beck_poincare}
Let $\mc{P}_t = e^{t \mc{L}}$ be a primitive QMS satisfying $\sigma$-DBC. Then the Beckner's inequality \eqref{ineq_becp} holds for all $p \in (1,2]$ with constant $\alpha_p(\mc{L})$ satisfying the estimate:
\begin{align} \label{eq:beck_poin}
\frac{p^2 \sigma_{\min}^{2-p}}{4} \lambda(\mc{L}) \le \alpha_p(\mc{L}) \le \frac{p}{2}\lambda(\mc{L})\,,
\end{align}
where $\lambda(\mc{L})$ is the Poincar\'{e} constant.
\end{theorem}
\begin{proof}
We first show the right inequality in \eqref{eq:beck_poin}, that is, \eqref{ineq_becp} implies \eqref{ineq_pi}, by the standard linearization argument.
We consider $Z = {\bf 1} + \varepsilon X$ for an arbitrary $X \in \mc{B}_{sa}(\mc{H})$ with $\tr(\sigma X) = 0$, where $\varepsilon > 0$ is small enough such that $Z > 0$. By a direct expansion with respect to $\varepsilon$, we find
\begin{equation} \label{eq:asy_divi}
\norm{Z}_{p,\sigma}^p = 1 + \varepsilon p \tr(\sigma X) + \frac{\varepsilon^2}{2}p(p-1)\norm{X}_{\sigma,\varphi_p}^2 + O(\varepsilon^3)\,,
\end{equation}
and
\begin{align*}
\norm{Z}_{1,\sigma}^p = \big(1 + \varepsilon \tr(\sigma X)\big)^p = 1 + \varepsilon p \tr(\sigma X) + \frac{\varepsilon^2}{2}p(p-1)\tr(\sigma X)^2 + O(\varepsilon^3)\,.
\end{align*}
We also compute
\begin{align*}
\Gamma_\si I_{\h{p},p}(Z) = \sigma + \varepsilon (p-1) \varphi_p(\Delta_\sigma)R_\sigma(X) + O(\varepsilon^2)\,,
\end{align*}
and $\mc{L}(Z) = \varepsilon \mc{L}(X)$, which yields
\begin{align} \label{eq:asy_diri}
\mc{E}_{p,\mc{L}}(Z) = - \varepsilon^2(p-1)\frac{\h{p}p}{4} \langle X, \mc{L} X\rangle_{\sigma,\varphi_p} + O(\varepsilon^3)\,.
\end{align}
Hence, by applying \eqref{ineq_becp} to $Z$, we have
\begin{align} \label{midauxeq}
\alpha_p \frac{\varepsilon^2}{2}p(p-1) \left(\norm{X}_{\sigma,\varphi_p}^2 - \tr(\sigma X)^2\right) + O(\varepsilon^3)
\le - \varepsilon^2(p-1)^2\frac{\h{p}p}{4} \langle X, \mc{L} X\rangle_{\sigma,\varphi_p} + O(\varepsilon^3)\,.
\end{align}
Due to $\tr(\sigma X) = 0$, it readily follows that $$2 \alpha_p \norm{X}_{\sigma,\varphi_p}^2 \le - p \langle X, \mc{L} X \rangle_{\sigma,\varphi_p}\,,$$ by dividing both sides of \eqref{midauxeq} by $\varepsilon^2$ and letting $\varepsilon \to 0$,
which gives the desired estimate: $2 \alpha_p \le p \lambda(\mc{L})$.
We next show the left inequality in \eqref{eq:beck_poin}.
It suffices to consider relative densities $X \ge 0$ with $\norm{X}_{1,\sigma} = 1$, for which there holds $X \le C(\sigma) {\bf 1} $, by definition \eqref{def:constcsi} of the constant $C(\sigma)$. Using the upper estimate in \eqref{eq:two_sided_est} and the positivity of the spectral gap of $\mc{L}$, we have
\begin{align} \label{auxeqq_1}
\frac{1}{p-1}(\norm{X}_{p,\sigma}^p - 1) \le \norm{X - {\bf 1}}_{\sigma,\varphi_p}^2 \le - \lambda(\mc{L})^{-1}
\langle X, \mc{L} X \rangle_{\sigma,\varphi_p}\,.
\end{align}
Then, by the formulas \eqref{eq:rep_epl}, \eqref{eq:rep_dbcl}, and Lemma \ref{lem:mono_norm}, it follows that
\begin{align} \label{auxeqq_2}
- \left\langle X, \mc{L} X \right\rangle_{\sigma,\varphi_p}
& = \left\langle \Gamma_\si^{1/p} \big(\partial_j X \big), f_p^{[1]}\left(e^{\omega_j/2p}\sigma^{1/p}, e^{-\omega_j/2p}\sigma^{1/p}\right) \Gamma_\si^{1/p} \big(\partial_j X\big) \right\rangle \notag \\
& \le C(\sigma)^{2-p} \left\langle \Gamma_\si^{1/p} \big(\partial_j X\big), f_p^{[1]}\left(e^{\omega_j/2p} \Gamma_\si^{1/p}(X), e^{-\omega_j/2p} \Gamma_\si^{1/p}(X) \right) \Gamma_\si^{1/p}\big(\partial_j X\big) \right\rangle \notag \\
& \le 4 p^{-2} C(\sigma)^{2-p} \mc{E}_{p,\mc{L}}(X)\,.
\end{align}
Therefore, by \eqref{auxeqq_1} and \eqref{auxeqq_2}, we obtain
\begin{equation*}
\frac{1}{p-1}(\norm{X}_{p,\sigma}^p - 1) \le \frac{4 C(\sigma)^{2-p}}{p^2 \lambda(\mc{L})}
\mc{E}_{p,\mc{L}}(X)\,. \qedhere
\end{equation*}
\end{proof}
For a specific invariant state $\sigma$, the lower bound $\alpha_p \ge \lambda (p-1)$ in \eqref{eq:poin_to_beck} may be better than the one in \eqref{eq:beck_poin} for some values of $p \in (1,2]$. Combining these two estimates, we obtain a sharper lower bound for $\alpha_p$.
\begin{corollary} \label{coro:pitobeck}
Let $\mc{P}_t = e^{t \mc{L}}$ be a primitive QMS satisfying $\sigma$-DBC. Then the Beckner constant $\alpha_p(\mc{L})$ satisfies
\begin{equation} \label{eq:lower_est}
\alpha_p(\mc{L}) \ge \max\Big\{ \lambda(\mc{L}) (p-1)\,, \frac{p^2 \sigma_{\min}^{2-p}}{4} \lambda(\mc{L}) \Big\}\,.
\end{equation}
The lower bound \eqref{eq:lower_est} still holds if we replace $\lambda(\mc{L})$ by $2 \alpha_1(\mc{L})$.
\end{corollary}
\subsubsection{Connections with $\alpha_1(\mc{L})$}
We next discuss the relations between the quantum $p$-Beckner's inequalities \eqref{ineq_becp} and the quantum MLSI \eqref{ineq_mlsi}.
First, by dividing both sides of \eqref{ineq_becp} by a factor $(p-1)$, and then taking the right limit $p \to 1^+$ and using formulas \eqref{def:1diri} and \eqref{eq:dev_pnorm}, we obtain the following lemma.
\begin{lemma} \label{lem:mb}
Let $\mc{P}_t = e^{t \mc{L}}$ be a primitive QMS.
If \eqref{ineq_becp} holds for all $p \in (1,2]$ with $\limsup_{p \to 1^+} \alpha_p > 0$, then \eqref{ineq_mlsi} holds with
\begin{equation} \label{eq:beckmsli_one}
\alpha_1 \ge \limsup_{p \to 1^+} \alpha_p\,.
\end{equation}
\end{lemma}
\begin{theorem} \label{thm:beck_log_sobo}
Let $\mc{P}_t = e^{t \mc{L}}$ be a primitive QMS with $\sigma$-DBC. Then for every $p \in (1,2]$, the Beckner's inequality \eqref{ineq_becp} holds
if and only if the modified log-Sobolev inequality \eqref{ineq_mlsi} holds. Moreover, there holds
\begin{equation} \label{eq:opt_becktoopt_msli}
\alpha_1(\mc{L}) = \lim_{p \to 1^+} \alpha_p({\mc{L}})\,.
\end{equation}
\end{theorem}
The proof of \eqref{eq:opt_becktoopt_msli} depends on the following lemma that extends the result \cite[Theorem 6.5]{bobkov2006modified} for the discrete modified log-Sobolev inequality.
\begin{lemma} \label{lem:extremal_func}
Let $\mc{P}_t = e^{t \mc{L}}$ be a primitive QMS.
If $\alpha_p(\mc{L}) < p \lambda(\mc{L})/2$ holds for the quantum Beckner constant, then the following infimum is attained:
\begin{align} \label{auxeq:defoptap}
\alpha_{p}(\mc{L}) = \inf_{\substack{X\ge 0, X \neq {\bf 1} \\ \norm{X}_{1,\sigma} = 1}} \frac{ (p-1)\mc{E}_{p,\mc{L}}(X)}{\norm{X}_{p,\sigma}^p- 1}\,.
\end{align}
\end{lemma}
\begin{proof}
By definition, there exists a sequence of $X_n \in \{X \ge 0\,;\ \norm{X}_{1,\sigma} = 1\,,\, X \neq {\bf 1}\}$ such that
\begin{align} \label{eq:assp}
V_p(X_n): = \frac{ (p-1)\mc{E}_{p,\mc{L}}(X_n)}{\norm{X_n}_{p,\sigma}^p- 1} \to \alpha_p(\mc{L})\,,\quad \text{as}\ n \to \infty\,.
\end{align}
Since the set $\{X \ge 0\,;\ \norm{X}_{1,\sigma} = 1\}$ is compact, without loss of generality, we assume that $X_n \to X $ as $n \to \infty$ for some $X \ge 0$ with $\norm{X}_{1,\sigma} = 1$. Suppose that $X = {\bf 1}$. Then we can write $X_n = {\bf 1} + Y_n$ with $Y_n \to 0$ and $\tr(\sigma Y_n) = 0$. Recalling the asymptotic expansions \eqref{eq:asy_divi} and \eqref{eq:asy_diri}, we obtain, by \eqref{ineq_pi},
\begin{align*}
\liminf_{n \to \infty} V_p(X_n) = \liminf_{n \to \infty} - \frac{p}{2} \frac{\langle Y_n, \mc{L} Y_n \rangle_{\sigma,\varphi_p} + O(\norm{Y_n}^3_{1,\sigma})}{\norm{Y_n}^2_{\sigma,\varphi_p} + O(\norm{Y_n}^3_{1,\sigma})} \ge \frac{p}{2}\lambda(\mc{L}) > \alpha_p(\mc{L})\,,
\end{align*}
which contradicts \eqref{eq:assp}. Thus, the operator $X$ is in the desired set $\{X \ge 0\,;\ \norm{X}_{1,\sigma} = 1\,,\, X \neq {\bf 1}\}$ and the infimum in \eqref{auxeq:defoptap} is attained.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:beck_log_sobo}]
The first statement follows from Theorem \ref{thm:beck_poincare} and \cite[ Theorem 3.3]{gao2021complete}.
In order to establish \eqref{eq:opt_becktoopt_msli}, by Lemma \ref{lem:mb}, it suffices to prove
\begin{align} \label{auxeq:infmb}
\liminf_{p \to 1^+} \alpha_p (\mc{L}) \ge \alpha_1 (\mc{L})\,.
\end{align}
We shall prove it by contradiction. If \eqref{auxeq:infmb} does not hold, there is a sequence $p_n \to 1^+$ as $n \to \infty$ such that
\begin{align*}
\lim_{n \to \infty} \alpha_{p_n}(\mc{L}) \le \alpha_1(\mc{L}) - \varepsilon\,,
\end{align*}
for some small enough $\varepsilon > 0$, that is, for any $k > 0$, there exists $N$ such that for $n \ge N$,
\begin{align} \label{auxeq:seq}
\alpha_{p_n}(\mc{L}) \le \alpha_1(\mc{L}) - \varepsilon + \frac{1}{k}\,.
\end{align}
Suppose that $\alpha_{p_n}(\mc{L}) = {p_n}\lambda(\mc{L})/2$ holds for infinitely many $n$. It follows from \eqref{auxeq:seq} that $\lambda(\mc{L})/2 \le \alpha_1(\mc{L}) - \varepsilon$, which is a contradiction with Lemma \ref{lem:aux_func}. Hence, without loss of generality, we assume that $\alpha_{p_n}(\mc{L}) < {p_n}\lambda(\mc{L})/2$ for all $n$. Then, Lemma \ref{lem:extremal_func} gives the existence of the minimizer $X_n$ associated with $\alpha_{p_n}(\mc{L})$. By compactness, we further assume that $X_n$ converges to some $X \ge 0$ with $\norm{X}_{1,\sigma} = 1$. If $X = {\bf 1}$, similarly to the proof of Lemma \ref{lem:extremal_func} above,
we write $X_n = {\bf 1} + Y_n$ and find
\begin{align*}
\alpha_1(\mc{L}) - \varepsilon + \frac{1}{k} \ge \liminf_{n \to \infty} \alpha_{p_n}(\mc{L}) = \liminf_{n \to \infty} - \frac{p_n}{2} \frac{\langle Y_n, \mc{L} Y_n \rangle_{\sigma,\varphi_{p_n}} + O(\norm{Y_n}_{1,\sigma}^3)}{\norm{Y_n}^2_{\sigma,\varphi_{p_n}} + O(\norm{Y_n}_{1,\sigma}^3)} \ge \frac{1}{2}\lambda(\mc{L})\,,
\end{align*}
which again contradicts with Lemma \ref{lem:aux_func}, by letting $k \to \infty$. If $X \neq {\bf 1}$, by definition \eqref{ineq_becp} and \eqref{auxeq:seq}, we have
\begin{align*}
\mc{E}_{p_n,\mc{L}}(X_{n}) \le \Big(\alpha_1(\mc{L}) - \varepsilon + \frac{1}{k}\Big) \frac{\norm{X_{n}}_{p_n,\sigma}^{p_n}- 1}{p_n-1}\,,\quad \text{for}\ n \ \text{large enough}\,.
\end{align*}
It implies that, by letting $n \to \infty$ and $k \to \infty$, and using \eqref{eq:dev_pnorm} with an elementary analysis,
\begin{align*}
\mc{E}_{1,\mc{L}}(X) \le \big(\alpha_1(\mc{L}) - \varepsilon\big) \Ent_{1,\sigma}(X)\,.
\end{align*}
It clearly contradicts the optimality of $\alpha_1(\mc{L})$. The proof is complete.
\end{proof}
\subsubsection{Stability of $\alpha_p(\mc{L})$} We proceed to investigate the stability of the quantum Beckner constant $\alpha_p(\mc{L})$ with respect to the invariant state. We will compare the constants $\alpha_p(\mc{L})$
for the following two generators $\mc{L}_\sigma$ and $\mc{L}_{\sigma'}$ that satisfy the detailed balance conditions with respect to two different but commuting full-rank states $\sigma$ and $\sigma'$:
\begin{align} \label{eq:gen_1}
\mc{L}_\sigma(X) = \sum_{j = 1}^{J} \big(e^{-\omega_j/2}V_j^*[X,V_j] + e^{\omega_j/2}[V_j,X]V^*_j\big)\,,
\end{align}
and
\begin{align} \label{eq:gen_2}
\mc{L}_{\sigma'}(X) = \sum_{j = 1}^{J} \big(e^{-\nu_j/2}V_j^*[X,V_j] + e^{\nu_j/2}[V_j,X]V^*_j\big)\,,
\end{align}
where $e^{-\omega_j}$ and $e^{-\nu_j}$ are the eigenvalues of $\Delta_\sigma$ and $\Delta_{\sigma'}$, respectively. For ease of exposition, we assume that the states $\sigma$ and $\sigma'$ admit the spectral decompositions:
\begin{align} \label{auxeq_specdecomp}
\sigma = \sum_{k = 1}^{d} \sigma_k v_k\,,\quad \sigma' = \sum_{k = 1}^{d} \sigma'_k v_k\,,
\end{align}
respectively.
\begin{theorem} \label{thm:beck_stab}
Let $\mc{L}_\sigma$ and $\mc{L}_{\sigma'}$ be the generators of two primitive QMS satisfying $\sigma$-DBC and $\sigma'$-DBC, given in \eqref{eq:gen_1} and \eqref{eq:gen_2}, respectively.
Then it holds that
\begin{align*}
\frac{\Lambda_{\min}}{\Lambda_{\max}} \min_{j} e^{-|\omega_j-\nu_j|(2-p)/2p} \alpha_p(\mc{L}_{\sigma'}) \le \alpha_p(\mc{L}_\sigma)\,,
\end{align*}
where constants $\Lambda_{\min}$ and $\Lambda_{\max}$ are defined as
\begin{align} \label{eq:const_state}
\Lambda_{\min} = \min_k \frac{\sigma_k}{\sigma'_k}\,, \quad \text{and} \quad \Lambda_{\max} = \max_k \frac{\sigma_k}{\sigma'_k}\,.
\end{align}
\end{theorem}
\begin{proof}
Let $X := \Gamma_{\sigma'}^{-1}(\rho) \ge 0$ for $\rho \in \mathcal{D}(\mc{H})$. We first establish a comparison result for the quantum $p$--divergence $\mc{F}_{p,\sigma}(\rho)$. We define the map
\begin{align*}
\Phi(A): = \mm
\Lambda_{\max}^{-1} \Gamma_\si \Gamma_{\sigma'}^{-1} (A) & 0 \\ 0 &
\tr(A) - \Lambda_{\max}^{-1}\norm{\Gamma_{\sigma'}^{-1}(A)}_{1,\sigma}
\nn : \ \mathcal{B}(\mc{H}) \to \mc{B}(\mc{H}\oplus \mathbb{C})\,.
\end{align*}
It is easy to check from \eqref{eq:const_state} that $\tr \big( \Lambda_{\max}^{-1} \Gamma_\si \Gamma_{\sigma'}^{-1} (A) \big) \le \tr (A) $ for any $A \in \mathcal{B}(\mc{H})$. Hence $\Phi$ is completely positive and trace preserving by Kraus representation theorem. Then by the data processing inequality \eqref{eq:dpi_divi}, we have
\begin{align} \label{eq:dpi}
\mc{F}_{p, \Phi(\sigma')}\left(\Phi(\rho)\right) \le \mc{F}_{p,\sigma'}\left(\rho\right) = \frac{1}{p(p-1)} \left( \norm{X}_{p,\sigma'}^p - 1 \right).
\end{align}
We now compute, by definition,
\begin{align} \label{eq:auxdp_2}
\mc{F}_{p, \Phi(\sigma')}\left(\Phi(\rho)\right) = \frac{1}{p(p-1)} \left(
\Lambda^{-1}_{\max} \norm{X}_{p,\sigma}^p + \big(1 - \Lambda^{-1}_{\max} \norm{X}_{1,\sigma}\big)^p \left(1 - \Lambda^{-1}_{\max}\right)^{1-p} - 1
\right).
\end{align}
Note from \eqref{eq:const_state} that $\Lambda_{\max} \ge 1 \ge \Lambda_{\min}$ and $\Lambda^{-1}_{\max} \norm{X}_{1,\sigma} \le 1$.
Then the convexity of $x^p$, $1 < p \le 2$, gives
\begin{align*}
&(1 - \Lambda^{-1}_{\max}) \left(\frac{1 - \Lambda^{-1}_{\max} \norm{X}_{1,\sigma}}{1 - \Lambda^{-1}_{\max}} \right)^p + \Lambda^{-1}_{\max} \left(\frac{ \Lambda^{-1}_{\max} \norm{X}_{1,\sigma}}{\Lambda^{-1}_{\max}} \right)^p \\
= & \big(1 - \Lambda^{-1}_{\max} \norm{X}_{1,\sigma}\big)^p \big(1 - \Lambda^{-1}_{\max}\big)^{1-p} + \Lambda^{-1}_{\max} \norm{X}_{1,\sigma}^p \ge 1\,,
\end{align*}
which, by \eqref{eq:auxdp_2}, implies
\begin{align} \label{auxeq_stab}
\mc{F}_{p, \Phi(\sigma') }\left(\Phi(\rho) \right) \ge \frac{1}{p(p-1)} \left(
\Lambda^{-1}_{\max} \norm{X}_{p,\sigma}^p - \Lambda^{-1}_{\max} \norm{X}^p_{1,\sigma}
\right)\,.
\end{align}
Combining \eqref{eq:dpi} and \eqref{auxeq_stab}, we can find
\begin{equation} \label{eq:sta_main_est_1}
\frac{1}{p(p-1)} \left(
\norm{X}_{p,\sigma}^p - \norm{X}^p_{1,\sigma}
\right) \le \frac{\Lambda_{\max}}{p(p-1)} \left( \norm{X}_{p,\sigma'}^p - 1 \right)\,.
\end{equation}
We next give the comparison result for $\mc{E}_{p,\mc{L}}$. We recall \eqref{eq:rep_epl}, and, by Lemma \ref{lem:mono_norm}, obtain
\begin{align} \label{est:p-diri}
\mc{E}_{p,\mc{L}_\sigma}(X) \ge \left(\inf_{j} e^{-|\omega_j - \nu_j|(2-p)/2p}\right) \frac{p^2}{4} \sum_{j = 1}^J \left\langle \Gamma_\si^{1/p}(\partial_j X), f_p^{[1]}\left(e^{\nu_j/2p}\Gamma_\si^{1/p}(X), e^{-\nu_j/2p}\Gamma_\si^{1/p}(X)\right) \Gamma_\si^{1/p} (\partial_j X) \right\rangle,
\end{align}
since
\begin{equation*}
e^{\pm \omega_j/2p} \Gamma_\si^{1/p}(X) \le (\max_j e^{|\omega_j - \nu_j|/2p}) e^{\pm \nu_j/2p} \Gamma_\si^{1/p}(X)\,.
\end{equation*}
By the integral representation \eqref{eq:integral_thetap} of $f_p^{[1]}$, we can estimate
\small
\begin{align} \label{auxeqq_stabbdiri}
& \Gamma_\si^{1/p} f_p^{[1]}\left(e^{\nu_j/2p}\Gamma_\si^{1/p}(X), e^{-\nu_j/2p}\Gamma_\si^{1/p} (X)\right) \Gamma_\si^{1/p} \\ = & \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty s^{p-2}
\Gamma_\si^{1/p} g_0^{[1]}\left(s + e^{\nu_j/2p}\Gamma_\si^{1/p}(X), s + e^{-\nu_j/2p}\Gamma_\si^{1/p} (X)\right) \Gamma_\si^{1/p}
\ ds \notag \\
\ge & \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty s^{p-2} \Gamma_\si^{1/p} g_0^{[1]}\left(s \Lambda_{\min}^{-1/p} \sigma^{1/p}(\sigma')^{-1/p} + e^{\nu_j/2p}\Gamma_\si^{1/p}(X), s \Lambda_{\min}^{-1/p} \sigma^{1/p}(\sigma')^{-1/p} + e^{-\nu_j/2p}\Gamma_\si^{1/p} (X)\right) \Gamma_\si^{1/p}\ ds\,, \notag
\end{align}
\normalsize
where we used the following observation from \eqref{auxeq_specdecomp} and \eqref{eq:const_state}:
\begin{equation} \label{eq:simple_ob}
\Lambda_{\max}^{-1/p} \sigma^{1/p} (\sigma')^{-1/p} \le {\bf 1} \le \Lambda_{\min}^{-1/p} \sigma^{1/p} (\sigma')^{-1/p}\,,
\end{equation}
and the operator monotonicity of $t^{-1}$. The observation \eqref{eq:simple_ob} also implies that $\Lambda_{\min}^{1/p} \Gamma_\si^{-1/p}\Gamma_{\sigma'}^{1/p}$ is completely positive and trace non-increasing. Then, by \cite[Proposition 3.6]{junge2019stability}, it follows that
\small
\begin{multline*}
\Gamma_\si^{1/p} g_0^{[1]}\left(s \Lambda_{\min}^{-1/p} \sigma^{1/p}(\sigma')^{-1/p} + e^{\nu_j/2p}\Gamma_\si^{1/p}(X), s \Lambda_{\min}^{-1/p} \sigma^{1/p}(\sigma')^{-1/p} + e^{-\nu_j/2p}\Gamma_\si^{1/p} (X)\right) \Gamma_\si^{1/p} \\
\ge \Lambda_{\min}^{2/p} \Gamma_{\sigma'}^{1/p} g_0^{[1]}\left(s + \Lambda_{\min}^{1/p} e^{\nu_j/2p} \Gamma_{\sigma'}^{1/p}(X), s + \Lambda_{\min}^{1/p} e^{-\nu_j/2p}\Gamma_{\sigma'}^{1/p} (X)\right) \Gamma_{\sigma'}^{1/p}\,.
\end{multline*}
\normalsize
Therefore, by \eqref{eq:integral_thetap} and \eqref{auxeqq_stabbdiri}, we have
\begin{align*}
& \Gamma_\si^{1/p} f_p^{[1]}\left(e^{\nu_j/2p}\Gamma_\si^{1/p}(X), e^{-\nu_j/2p}\Gamma_\si^{1/p} (X)\right) \Gamma_\si^{1/p} \\
\ge & \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty s^{p-2} \Lambda_{\min}^{2/p} \Gamma_{\sigma'}^{1/p} g_0^{[1]}\left(s + \Lambda_{\min}^{1/p} e^{\nu_j/2p} \Gamma_{\sigma'}^{1/p}(X), s + \Lambda_{\min}^{1/p} e^{-\nu_j/2p}\Gamma_{\sigma'}^{1/p} (X)\right) \Gamma_{\sigma'}^{1/p}
\ ds \\
\ge & \Lambda_{\min} \Gamma_{\sigma'}^{1/p} f_p^{[1]}\left(e^{\nu_j/2p}\Gamma_{\sigma'}^{1/p}(X), e^{-\nu_j/2p}\Gamma_{\sigma'}^{1/p} (X)\right) \Gamma_{\sigma'}^{1/p}\,.
\end{align*}
Combining the above estimate with \eqref{est:p-diri} and recalling \eqref{eq:rep_epl}, we readily have
\begin{align} \label{eq:sta_main_est_2}
\mc{E}_{p,\mc{L}_\sigma}(X) \ge \left(\inf_{j} e^{-|\omega_j - \nu_j|(2-p)/2p}\right) \Lambda_{\min} \mc{E}_{p,\mc{L}_{\sigma'}}(X)\,.
\end{align}
The proof is completed by the following simple estimate, with the help of \eqref{eq:sta_main_est_1} and \eqref{eq:sta_main_est_2},
\begin{align*}
\frac{1}{p-1} \left(
\norm{X}_{p,\sigma}^p - \norm{X}_{1,\sigma}
\right) & \le \frac{\Lambda_{\max}}{\alpha_p(\mc{L}_{\sigma'})} \mc{E}_{p,\mc{L}_{\sigma'}}(X) \\
& \le \frac{\Lambda_{\max}}{\alpha_p(\mc{L}_{\sigma'})} \left(\inf_{j} e^{-|\omega_j-\nu_j|(2-p)/2p}\right)^{-1} \Lambda_{\min}^{-1} \mc{E}_{p,\mc{L}_\sigma}(X)\,. \qedhere
\end{align*}
\end{proof}
\subsection{Applications and examples} \label{sec:app_exp}
This section is devoted to the applications of the $p$-Beckner's inequalities. We first analyze the Beckner constant for the depolarizing semigroup. We then derive an improved bound on the mixing time of quantum Markov dynamics in terms of the Beckner constants $\alpha_p(\mc{L})$. For the symmetric semigroups, by borrowing the techniques from \cite{adamczak2022modified,junge2015noncommutative}, we obtain the moment estimates from Beckner's inequalities \eqref{ineq_becp}, which further allows us to derive concentration inequalities.
\subsubsection{Beckner constant for depolarizing semigroups} In general, it is challenging to explicitly compute or estimate the optimal constant for the functional inequalities, even in the classical setting. In this section, we consider the quantum Beckner constant $\alpha_p$ for the simplest QMS: the depolarizing semigroup \eqref{def:depolt} with $\gamma = 1$ and $\sigma = {\bf 1}/d$, and show that in this case the computation of $\alpha_p$ is equivalent to the classical one for a random walk. We mention that the explicit values of the log-Sobolev constant $\beta$ and the modified log-Sobolev constant $\alpha_1$ for $\mc{L}_{depol}$ with a general invariant state $\sigma \in \mathcal{D}_+(\mc{H})$ have been obtained in \cite{beigi2020quantum} and \cite{muller2016relative}, respectively.
\begin{proposition} \label{prop:beck_depol}
Let $\mc{L}_{depol}(X) = \tr(X/d){\bf 1} - X$ be the Lindbladian of the depolarizing semigroup. Then we have
\begin{align} \label{eq:const_beck}
\alpha_p(\mc{L}_{depol}) = \inf_{\substack{\theta x + (1 - \theta) y = 1 \\ x,y \ge 0,\ \theta \in \{\frac{1}{d},\ldots, 1 - \frac{1}{d}\} }} \frac{p^2}{4} \frac{ (\theta x^p + (1-\theta)y^p) - (\theta x^{p-1} + (1-\theta)y^{p-1})}{(\theta x^p + (1-\theta)y^p) - 1}\,.
\end{align}
\end{proposition}
\begin{proof}
We first compute from definition that
\begin{equation} \label{auxeq:expbeck}
\alpha_{p}(\mc{L}_{depol}) = \inf_{X \ge 0} \frac{(p-1)\mc{E}_{p,\mc{L}_{depol}}(X)}{\norm{X}_{p,\frac{{\bf 1}}{d}}^p - \norm{X}_{1,\frac{{\bf 1}}{d}}^p} = \frac{p^2}{4} \inf_{X \ge 0} \frac{\norm{X}_{p,\frac{{\bf 1}}{d}}^p- \norm{X}_{p-1,\frac{{\bf 1}}{d}}^{p-1}\norm{X}_{1,\frac{{\bf 1}}{d}} }{\norm{X}_{p,\frac{{\bf 1}}{d}}^p - \norm{X}_{1,\frac{{\bf 1}}{d}}^p}\,.
\end{equation}
We only consider $p \in (1,2)$, since the case $p = 2$, corresponding to the spectral gap, is trivial. Let $\mu_i \ge 0$, $1 \le i \le d$, be the eigenvalues of $X \ge 0$. It is easy to reformulate \eqref{auxeq:expbeck} as
\begin{align*}
\alpha_{p}(\mc{L}_{depol}) = \frac{p^2}{4} \inf_{\mu_i \ge 0} \frac{ \sum_i \mu_i^p - d^{-1} \big(\sum_{i} \mu_i^{p-1}\big)\big(\sum_i \mu_i\big)}{\sum_i \mu_i^p - d^{-(p-1)}\big(\sum_i \mu_i\big)^p }\,,
\end{align*}
which is equivalent to, for any $\mu_i \ge 0$,
\begin{equation} \label{auxeq:expbeck_2}
F(\mu_1,\ldots, \mu_d): = \sum_i \mu_i^p - d^{-1} \big(\sum_{i} \mu_i^{p-1}\big)\big(\sum_i \mu_i\big) - \frac{4 \alpha_{p}(\mc{L}_{depol}) }{p^2} \Big(\sum_i \mu_i^p - d^{-(p-1)}\big(\sum_i \mu_i\big)^p \Big) \ge 0\,.
\end{equation}
Suppose that $(r_i)_{i =1}^d$ achieves the equality in \eqref{auxeq:expbeck_2}. We claim that all $r_i$ are strictly positive. If not, without loss of generality, we assume $r_1 = 0$, $r_2 > 0$, and
$\sum_{i = 2}^{d} r_i = 1$. Then, from \eqref{auxeq:expbeck_2}, for small enough $\varepsilon > 0$, we have
\begin{align*}
F(\varepsilon, r_2 - \varepsilon, r_3, \ldots, r_d) = &\sum_{i = 3} r_i^p - d^{-1} \big(\sum_{i = 3}^d r_i^{p-1}\big) - C_p \Big(\sum_{i = 3}^d r_i^p - d^{-(p-1)} \Big) \\
& - d^{-1} \big( (r_2 - \varepsilon) ^{p-1} + \varepsilon^{p-1} \big) - (C_p - 1) \big( (r_2 - \varepsilon)^p + \varepsilon^p \big) \\
= & - d^{-1} \varepsilon^{p-1} + O(\varepsilon) < 0\,,
\end{align*}
where $C_p := 4 \alpha_p/p^2$.
This fact contradicts the assumption that $(r_i)_{i}$ saturates the equality, so the claim holds.
Now, by $r_i > 0$ for all $i$, we have $\nabla F (r_1,\ldots,r_d) = 0$, which, by a direct computation, gives the following equations in the variables $r_i$:
\begin{equation} \label{auxeq:expbeck_3}
\big(\frac{p^2}{4} - \alpha_p \big)r_i^{p-1} - \frac{p(p-1)}{4 d}\norm{X}_1 r_i^{p-2}
= \frac{p}{4d}\norm{X}_{p-1}^{p-1} - \alpha_p d^{1-p}\norm{X}_1^{p-1}.
\end{equation}
When $\norm{X}_1$ and $\norm{X}_{p-1}$ are fixed, the above equation clearly
has at most two solutions, denoted by $a$ and $b$, which means that $r_i$ takes the value either $a$ or $b$. Let $n$ be the number of $r_i$ equal to $a$. Then, it follows that
\begin{align*}
\alpha_{p}(\mc{L}_{depol}) & = \inf_{\substack{n a + (d - n) b = 1 \\ a,b \ge 0,\ n \in \{1,\ldots, d-1\} }} \frac{p^2}{4} \frac{ (n a^p + (d - n)b^p) - d^{-1}(n a^{p-1} + (d-n)b^{p-1})}{(n a^p + (d-n)b^p) - d^{-(p-1)}} \\
& = \inf_{\substack{\theta x + (1 - \theta) y = 1 \\x,y \ge 0,\ \theta \in \{\frac{1}{d},\ldots, 1-\frac{1}{d}\} }} \frac{p^2}{4} \frac{ (\theta x^p + (1-\theta)y^p) - (\theta x^{p-1} + (1-\theta)y^{p-1})}{(\theta x^p + (1-\theta)y^p) - 1}\,,
\end{align*}
by setting $\theta = n/d$, $x = d a$, and $y = d b$.
\end{proof}
To connect the expression \eqref{eq:const_beck} with the classical Beckner constant, we consider a Markov chain on $\{0,1\}$ with the transition matrix:
$$
P = \mm \theta & 1- \theta \\ \theta & 1- \theta \nn\,,
$$
which has the invariant measure $\pi(0) = \theta$, $\pi(1) = 1-\theta$. The Beckner constant for this chain is give by \cite[(4.1)]{bobkov2006modified}
\begin{align} \label{eq:rep_beck}
\alpha_{p,\theta} & = \inf_{f \ge 0} \frac{p}{2} \frac{- \langle f^{p-1}, (P - I) f \rangle_{\pi}}{\pi(f^p) - \pi(f)^p } \notag \\
& = \inf_{\substack{\theta x + (1 - \theta) y = 1 \\x,y \ge 0 }} \frac{p}{2} \frac{ (\theta x^p + (1-\theta)y^p) - (\theta x^{p-1} + (1-\theta)y^{p-1})}{(\theta x^p + (1-\theta)y^p) - 1}\,.
\end{align}
Then we can see $\alpha_{p}(\mc{L}_{depol}) = \inf \big\{ \alpha_{p,\theta}\,;\ \theta \in \{\frac{1}{d},\ldots, 1-\frac{1}{d}\}\big\}$. However, although the representation \eqref{eq:rep_beck} is simple, numerical techniques seem still to be necessary to find the explicit values of $ \alpha_{p,\theta}$ and $\alpha_{p}(\mc{L}_{depol})$.
\subsubsection{Mixing times}
In this section, we analyze the mixing time of QMS from quantum $p$-Beckner's inequalities and show that the improved bounds on the mixing time may be obtained.
For a primitive QMS $\mc{P}_t = e^{t\mc{L}}$, we define the $l_1$ mixing time for $\varepsilon > 0$ by
\begin{align} \label{def:l1mix}
t_1(\varepsilon) = \inf\{t > 0\,;\ \big\lVert\mc{P}^\dag_t(\rho) - \sigma\big\rVert_1 \le \varepsilon \quad \text{for all} \ \rho \in \mathcal{D}(\mc{H}) \}\,.
\end{align}
To bound $t_1(\varepsilon)$, we need the following lemma
extending \cite[Theorem 4.1]{li2020complete} for the symmetric QMS, which characterizes the convergence of QMS in terms of $\sigma$-weighted $p$-norm.
\begin{lemma} \label{lem:conver_p_norm}
Let $\mc{P}_t$ be a primitive QMS satisfying $\sigma$-DBC. Then it holds that
\begin{align*}
\norm{\mc{P}_t(X) - \tr(\sigma X) {\bf 1}}_{p,\sigma} \le e^{- \frac{2 \alpha_p(\mc{L})}{p}t} \norm{X}_{p,\sigma}^{1-p/2} \sqrt{\frac{2}{p(p-1)} \left( \norm{X}_{p,\sigma}^p - \norm{X}_{1,\sigma}^{p} \right)}\,, \quad X \ge 0\,,
\end{align*}
where $\alpha_p (\mc{L}) > 0$ is the quantum Beckner constant associated with $\mc{P}_t$.
\end{lemma}
\begin{proof}
We define, for $X, Y \in \mc{B}_{sa}(\mc{H})$,
\begin{equation*}
G_{X,Y}(s) := \norm{X + s Y}_{p,\sigma}^p - \frac{p(p-1)}{2} s^2 \norm{X + s Y}_{p,\sigma}^{p-2} \norm{Y}_{p,\sigma}^2.
\end{equation*}
Similarly to \cite[Theorem 4.1]{li2020complete}, by results in \cite{ricard2016noncommutative}, it is easy to prove that $G_{X,Y}''(0) \ge 0$ for any self-adjoint $X, Y$, which implies that $G_{X,Y}(s)$ is a convex function on $\mathbb{R}$. We now consider $A := \tr(\sigma X) {\bf 1} $ and $B := X_t - \tr(\sigma X) {\bf 1}$ with $X_t = \mc{P}_t(X)$ for $X \ge 0$. Then, by Lemma \ref{lem:prop:norm_power}, we have, for any $s \in \mathbb{R}$,
\begin{equation*}
\norm{A + s B}_{p,\sigma}^p \ge \norm{A + s B}_{1,\sigma}^p \ge \tr(\sigma(A + s B))^p = \tr(\sigma X)^p = \norm{A}_{p,\sigma}^p\,.
\end{equation*}
It then follows from definition that $G_{A,B}'(0) \ge 0$, which, along with the convexity of $G_{A,B}(s)$, yields $G'_{A,B}(s) \ge 0$ for any $s \ge 0$. Hence, we have $G_{A,B}(1) \ge G_{A,B}(0) $, that is (recalling the definitions of $A, B$),
\begin{equation} \label{auxeq:decay}
\norm{X_t}_{p,\sigma}^p - \frac{p(p-1)}{2} \norm{X_t}_{p,\sigma}^{p-2} \norm{X_t - \tr(\sigma X) {\bf 1}}_{p,\sigma}^2 \ge \norm{X}_{1,\sigma}^p\,.
\end{equation}
By \eqref{ineq_becp} and Gr\"{o}nwall’s inequality, it holds that
\begin{align*}
\norm{X_t}_{p,\sigma}^p - \norm{X_t}_{1,\sigma}^{p} \le e^{- \frac{4 \alpha_p}{p}t} \left( \norm{X}_{p,\sigma}^p - \norm{X}_{1,\sigma}^{p} \right)\,,\quad X \ge 0\,,
\end{align*}
which, along with \eqref{auxeq:decay}, implies
\begin{align*}
\norm{X_t - \tr(\sigma X) {\bf 1}}_{p,\sigma}^2 \le \frac{2}{p(p-1)} e^{- \frac{4 \alpha_p}{p}t} \norm{X}_{p,\sigma}^{2-p}\left( \norm{X}_{p,\sigma}^p - \norm{X}_{1,\sigma}^{p} \right)\,.
\end{align*}
The proof is complete by taking the square root of the above inequality.
\end{proof}
\begin{proposition} \label{prop:mixing}
Under the same assumption as in Lemma \ref{lem:conver_p_norm}, it holds that
\begin{align} \label{eq:mixing}
t_1(\varepsilon) \le \inf_{p \in (1,2]} h(p,\sigma_{\min},\varepsilon)\,,
\end{align}
where
\begin{align*}
h(p,\sigma_{\min},\varepsilon) := \frac{p}{2\alpha_p(\mc{L})} \log \bigg( \varepsilon^{-1} \sqrt{\frac{2}{p(p-1)} \left( \sigma_{\min}^{\frac{2}{p}-2} - \sigma_{\min}^{ p + \frac{2}{p} -3} \right)} \bigg)\,.
\end{align*}
\end{proposition}
\begin{proof}
Let $X_t$ be the relative density of $\rho_t = e^{t \mc{L}^\dag}(\rho)$; see \eqref{eq:evo_density}. We write
\begin{align*}
\big\lVert\mc{P}^\dag_t(\rho) - \sigma\big\rVert_1 = \big\lVert \Gamma_\si (\Gamma_\si^{-1} \mc{P}^\dag_t(\rho) - {\bf 1})\big\rVert_1 = \norm{X_t - {\bf 1}}_{1,\sigma}\,.
\end{align*}
By Lemma \ref{lem:conver_p_norm}, we have
\begin{align} \label{eq:sup_pnorm}
\sup_{\rho \in \mathcal{D}(\mc{H})} \big\lVert\mc{P}^\dag_t(\rho) - \sigma\big\rVert_1 \le e^{- \frac{2 \alpha_p(\mc{L})}{p}t} \sup_{X \ge 0,\, \tr(\sigma X) = 1} \sqrt{\frac{2}{p(p-1)} \left( \norm{X}_{p,\sigma}^2 - \norm{X}_{p,\sigma}^{2-p} \right)}\,.
\end{align}
Recalling the formula \eqref{eq:suprhodp}, by definition of $D_p(\rho \| \sigma) $, there holds
\begin{align*}
\sup_{X \ge 1,\, \tr(\sigma X) = 1} \norm{X}_{p,\sigma} = \sigma_{\min}^{\frac{1}{p}-1}\,.
\end{align*}
Also note that the function $x^2 - x^{2 - p}$ with $p \in (1,2]$ is increasing for $x \ge 1$. Then it follows from \eqref{eq:sup_pnorm} that
\begin{align*}
\sup_{\rho \in \mathcal{D}(\mc{H})} \big\lVert\mc{P}^\dag_t(\rho) - \sigma\big\rVert_1 \le e^{- \frac{2 \alpha_p(\mc{L})}{p}t} \sqrt{\frac{2}{p(p-1)} \left( \sigma_{\min}^{\frac{2}{p}-2} - \sigma_{\min}^{ p + \frac{2}{p} -3} \right)}\,.
\end{align*}
By definition \eqref{def:l1mix} and a direct computation, we obtain the estimate \eqref{eq:mixing} for $t_1(\varepsilon)$.
\end{proof}
By elementary calculus, we find
\begin{align*}
h(2,\sigma_{\min},\varepsilon) = \frac{1}{\lambda(\mc{L})} \log \bigg(\varepsilon^{-1} \sqrt{ \sigma_{\min}^{-1} - 1 } \bigg)\,,
\end{align*}
and, as $p \to 1^+$,
\begin{align*}
h(p, \sigma_{\min}, \varepsilon) \to \frac{1}{2\alpha_1(\mc{L})} \log \bigg(\varepsilon^{-1} \sqrt{ 2 \log \big(\sigma_{\min}^{-1}\big)} \bigg)\,,
\end{align*}
which are nothing but the mixing time bounds obtained from the decays of the variance and the relative entropy, respectively \cite{kastoryano2013quantum,temme2010chi}. It means that $\inf_{p \in (1,2]} h(p,\sigma_{\min},\varepsilon)$ in \eqref{eq:mixing} can be indeed an improved bound for the mixing time in some cases.
\subsubsection{Moment estimates and concentration inequalities} In this section, we consider the primitive symmetric QMS $\mc{P}_t = \mc{P}_t^\dag$; see Remark \ref{eq:tracial}. We will derive the moment estimates from quantum $p$-Beckner's inequalities, by extending the arguments of \cite[Proposition 3.3]{adamczak2022modified} for classical Markov semigroups to the quantum setting. This helps us to obtain concentration inequalities in a similar manner as \cite{junge2015noncommutative}. We first recall the carr\'{e} du champ operator (gradient form) associated with $\mc{P}_t = e^{t \mc{L}}$ \cite{junge2015noncommutative,wirth2021curvature}:
\begin{align} \label{def:gamma}
\Gamma(X,Y) = \frac{1}{2}(\mc{L}(X^*Y) - X^* (\mc{L}Y) - (\mc{L}X)^*Y)\quad \text{for}\ X,Y \in \mc{B}(\mc{H})\,.
\end{align}
As usual, we write $\Gamma(X)$ for $\Gamma(X,X)$. By the self-adjointness $\mc{L}^\dag = \mc{L}$ and $\mc{L}({\bf 1}) = 0$, a direct computation gives the relation between the $\Gamma$ operator and the Dirichlet form $\mc{E}_{2,\mc{L}}(X,Y) := -\langle X, \mc{L} Y \rangle_{\frac{{\bf 1}}{d}} $:
\begin{align} \label{eq:rela_gamma}
\mc{E}_{2,\mc{L}}(X,Y) = \big\langle \frac{{\bf 1}}{d}, \Gamma(X,Y)\big\rangle\,.
\end{align}
Before we state our main result on moment estimates, we need the following useful lemma.
\begin{lemma}\label{lem:diri_convex}
For any differentiable convex and increasing function $\varphi: [0,\infty) \to \mathbb{R}$ and $c \in \mathbb{R}$, it holds that
\begin{align} \label{eq:diri_convex}
\mc{E}_{2,\mc{L}}(\varphi(|X + c|), |X + c|) \le 2 \langle\varphi'(|X + c|), \Gamma(X) \rangle_{\frac{{\bf 1}}{d}} \le 2 \norm{\varphi'(|X+c|)}_{p,\frac{{\bf 1}}{d}} \norm{\Gamma(X)}_{\h{p},\frac{{\bf 1}}{d}}\,,
\end{align}
for $X \in \mc{B}_{sa}(\mc{H})$, where $p \ge 1$.
\end{lemma}
\begin{proof}
Assume that $X \in \mc{B}_{sa}(\mc{H})$ has the spectral decomposition $X = \sum \lambda_i E_i$, where $E_i$ are the eigen-projections associated with the eigenvalues $\lambda_i$. Note from the convexity of $\varphi$ that for any $x,y \ge 0$,
\begin{align} \label{auxeq:convex_fun}
\frac{\varphi(x) - \varphi(y)}{x - y} \le \max \{\varphi'(x), \varphi'(y)\} \le \varphi'(x) + \varphi'(y)\,,
\end{align}
since $\varphi'(x) \ge 0$ holds by the monotonicity of $\varphi$. Recalling the formula \eqref{eq:inte_by_parts}, by the eigen-decomposition of $X$ and the inequality \eqref{auxeq:convex_fun}, we have
\begin{align} \label{auxeq_momen_1}
\mc{E}_{2,\mc{L}}(\varphi(|X+c|), |X + c|) & = \sum_{j \in J} \big\langle \partial_j \varphi(|X+c|), \partial_j |X + c| \big\rangle_{\frac{{\bf 1}}{d}} \notag \\
& = \frac{1}{d} \sum_{j \in J} \sum_{i,k} (\varphi(|\lambda_i+c|) - \varphi(|\lambda_k + c|))(|\lambda_i + c| - |\lambda_k+c|) \tr (E_k V_j E_i V_j) \notag \\
& \le \frac{1}{d} \sum_{j \in J} \sum_{i,k} (\varphi'(|\lambda_i+c|) + \varphi'(|\lambda_k + c|))(|\lambda_i + c| - |\lambda_k+c|)^2 \tr (E_k V_j E_i V_j)\,.
\end{align}
Similarly, by definition \eqref{def:gamma} of $\Gamma(X)$, we also compute
\begin{align} \label{auxeq_momen_2}
2 \big\langle\varphi'(|X+c|), \Gamma(X) \big\rangle_{\frac{{\bf 1}}{d}} = & \frac{1}{d} \sum_{j \in J} \sum_{i,k} \big\{(\varphi'(|\lambda_i+c|) - \varphi'(|\lambda_k + c|))(\lambda^2_i - \lambda_k^2) \notag \\
& \qquad \qquad - 2(\lambda_i\varphi'(|\lambda_i + c|) - \lambda_k \varphi'(|\lambda_k + c|))(\lambda_i - \lambda_k) \big\} \tr (E_k V_j E_i V_j) \notag \\
= & \frac{1}{d} \sum_{j \in J} \sum_{i,k} (\varphi'(|\lambda_i+c|) + \varphi'(|\lambda_k + c|))(\lambda_i - \lambda_k)^2 \tr (E_k V_j E_i V_j)\,.
\end{align}
Since there holds $\big||\lambda_i + c| - |\lambda_k+c| \big| \le |\lambda_i - \lambda_k|$, by \eqref{auxeq_momen_1} and \eqref{auxeq_momen_2}, and using H\"{o}lder's inequality, we obtained the desired estimate \eqref{eq:diri_convex}.
\end{proof}
\begin{proposition} \label{prop:moment}
Suppose that the quantum $p$-Beckner's inequality \eqref{ineq_becp} holds
for all $p \in (1,2]$ with $\alpha_p \ge a(p-1)^s$ for some $a > 0$ and $s \ge 0$. Then we have, for $X \in \mc{B}_{sa}(\mc{H})$ and $r \ge 2$,
\begin{align} \label{eq:moment}
\norm{X - \norm{X}_{1,\frac{{\bf 1}}{d}}}_{r,\frac{{\bf 1}}{d}}^2 \le \frac{r^{s + 1}\kappa(s)}{a}\norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}}\,,
\end{align}
where $\kappa(s) := (1 - e^{-(s+1)/2})^{-1}$.
\end{proposition}
\begin{proof}
We shall prove by induction that for all positive integers $k$ and $r \in (k,k+1]$, there holds
\begin{align} \label{eq:ineq_induc}
\norm{X - \norm{X}_{1,\frac{{\bf 1}}{d}}}_{r,\frac{{\bf 1}}{d}}^2 \le c_r \norm{\Gamma(X)}_{\max\{\frac{r}{2},1\},\frac{{\bf 1}}{d}}\,, \quad \forall X \in \mc{B}_{sa}(\mc{H})\,,
\end{align}
where
\begin{align} \label{def:constant_moment}
c_r := \frac{r^{s + 1}\kappa_r(s)}{a} \,, \quad \kappa_r(s) := \Big( 1 - \big( \frac{r-1}{r} \big)^{(s+1)r/2}\Big)^{-1}\,.
\end{align}
The desired estimate \eqref{eq:moment} is a direct consequence of \eqref{eq:ineq_induc}, since $\kappa_r$ increases in $r$ and $\kappa_r(s) \to \kappa(s)$ as $r \to \infty$. We first note that by Theorem \ref{thm:beck_poincare}, \eqref{ineq_pi} holds with $\lambda(\mc{L}) \ge a$. For $k = 1$ and $r \in (1,2]$, by \eqref{ineq_pi} and \eqref{eq:rela_gamma}, we have
\begin{align*}
\norm{X - \norm{X}_{1,\frac{{\bf 1}}{d}}}_{r,\frac{{\bf 1}}{d}}^2 \le \norm{X - \norm{X}_{1,\frac{{\bf 1}}{d}}}_{2,\frac{{\bf 1}}{d}}^2 \le \frac{1}{\lambda(\mc{L})} \mc{E}_{2,\mc{L}}(X) \le c_r \norm{\Gamma(X)}_{1,\frac{{\bf 1}}{d}}\,,
\end{align*}
since $c_r$ is increasing in $r$, which gives $c_r \ge c_1 = 1/a \ge 1/\lambda(\mc{L})$. Suppose that \eqref{eq:ineq_induc} holds for all integers smaller than some $k > 1$. Now we consider $r \in (k,k+1]$. We define $Y = | X - \norm{X}_{1,\frac{{\bf 1}}{d}}|$, and then estimate by using \eqref{ineq_becp},
\begin{align} \label{auxeqpr_1}
\alpha_{\h{r}}\big(\norm{Y^{r-1}}_{\h{r},\frac{{\bf 1}}{d}}^{\h{r}} - \norm{Y^{r-1}}_{1,\frac{{\bf 1}}{d}}^{\h{r}}\big) \le (\h{r} - 1)\mc{E}_{\h{r},\mc{L}}(Y^{r-1})\,.
\end{align}
By Lemma \ref{lem:diri_convex} with $\varphi(x) = x^{r - 1}$, $c = - \norm{X}_{1,\frac{{\bf 1}}{d}}$, and $p = r/(r-2)$, we have
\begin{align} \label{auxeqpr_2}
(\h{r} - 1)\mc{E}_{\h{r},\mc{L}}(Y) = - \frac{\h{r}^2}{4}\langle Y^{r-1}, \mc{L}Y \rangle_{\frac{{\bf 1}}{d}} & \le \frac{\h{r}^2}{2}(r - 1)\norm{Y^{r-2}}_{p,\frac{{\bf 1}}{d}} \norm{\Gamma(X)}_{\h{p},\frac{{\bf 1}}{d}} \notag \\
& = \frac{1}{2} \frac{r^2}{r-1} \norm{Y}_{r,\frac{{\bf 1}}{d}}^{r-2} \norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}}\,.
\end{align}
We write $l_r = \norm{Y}_{r,\frac{{\bf 1}}{d}}$ and note $\alpha_{\h{r}} \ge a(\h{r} - 1)^s = a (r - 1)^{-s}$. Then combining estimates \eqref{auxeqpr_1} and \eqref{auxeqpr_2} gives
\begin{align} \label{auxeqpr_3}
l_r^r - l_{r-1}^{r} \le \frac{r^2 (r-1)^{s-1}}{2a} l_r^{r-2} \norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}} \le \frac{r^{s+1}}{a} l_r^{r-2} \norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}}\,,
\end{align}
since $ r/(r-1)\le 2$. Applying the assumption \eqref{eq:ineq_induc} to bound $l_{r-1}$ and by \eqref{auxeqpr_3}, we obtain
\begin{align} \label{auxeqpr_4}
l_r^r \le \big(c_{r-1}\norm{\Gamma(X)}_{\max\{\frac{r-1}{2},1\},\frac{{\bf 1}}{d}}\big)^{\frac{r}{2}} + \frac{r^{s+1} }{a} l_r^{r-2} \norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}}\,.
\end{align}
Note from \eqref{def:constant_moment} that
\begin{align} \label{auxeqpr_5}
\frac{c_{r-1} \norm{\Gamma(X)}_{\max\{\frac{r-1}{2},1\},\frac{{\bf 1}}{d}}}{c_r \norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}}} \le \frac{c_{r-1}}{c_r} \le \frac{(r-1)^{s+1}\kappa_{r-1}(s)}{r^{s+1}\kappa_r(s)}\,.
\end{align}
By dividing both sides of \eqref{auxeqpr_4} by $(c_r\norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}})^{r/2}$ and using \eqref{auxeqpr_5}, it follows that
\begin{align} \label{auxeqpr_6}
\Big( \frac{l_r^2}{c_r\norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}}}\Big)^{\frac{r}{2}} \le \Big( \frac{r-1}{r} \Big)^{\frac{(s+1)r}{2}} + \frac{1}{\kappa_r} \Big( \frac{l_r^2}{c_r \norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}}} \Big)^{\frac{r-2}{2}}\,.
\end{align}
To complete the proof, we need the fact from \cite[Proposition 3.1]{adamczak2022modified} that the function $h(x) = (1-1/r)^{(1+s)r/2} + \kappa_r^{-1} x^{1-2/r} - x$ is strictly concave on $[0,\infty)$ and satisfies $h(0) > 0$ and $h(1) = 0$, which means that $h(x) \ge 0$ implies $x \le 1$. This fact, along with \eqref{auxeqpr_6}, readily implies
\begin{equation*}
\norm{X - \norm{X}_{1,\frac{{\bf 1}}{d}}}_{r,\frac{{\bf 1}}{d}}^2 \le c_r \norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}}\,. \qedhere
\end{equation*}
\end{proof}
\begin{remark}
In the case of $s = 0$, Proposition \ref{prop:moment} shows that if \eqref{ineq_becp} holds with $\inf_{p\in(1,2]}\alpha_p \ge a$, then we have
\begin{align*}
\norm{X - \norm{X}_{1,\frac{{\bf 1}}{d}}}^2_{r,\frac{{\bf 1}}{d}} \le \frac{r\kappa}{a } \norm{\Gamma(X)}_{\infty}\,,\quad \forall X \in \mc{B}_{sa}(\mc{H})\,,
\end{align*}
where $\kappa := (1 - e^{-1/2})^{-1}$, since $\norm{\Gamma(X)}_{\frac{r}{2},\frac{{\bf 1}}{d}} \le \norm{\Gamma(X)}_{\infty}$. A similar result was obtained by Junge and Zeng \cite{junge2015noncommutative} for non-primitive symmetric QMS under the Bakry-\'{E}mery curvature-dimension condition ($\Gamma_2$-criterion):
\begin{align}\label{def:becdc}
\Gamma_2(X,X) \ge \alpha \Gamma(X,X)\,, \quad \forall X \in \mathcal{B}(\mc{H})\,, \tag{BE$(\alpha,\infty)$}
\end{align}
for some $\alpha > 0$, where $\Gamma_2$ is the iterated carr\'{e} du champ operator: $\Gamma_2(X,Y): = - \frac{1}{2}(\Gamma(X,\mc{L} Y) + \Gamma(\mc{L}X,Y) - \mc{L}\Gamma(X,Y))$. To be precise, they used the noncommutative martingale methods and proved that
under some necessary regularity condition, if \ref{def:becdc} holds, then we have
\begin{align*}
\norm{X - E(X)}_r^2 \le \frac{8r}{\alpha} \norm{\Gamma(X)}_{\infty}\,,\quad \forall X \in \mc{B}_{sa}(\mc{H})\,,
\end{align*}
where $E$ is the conditional expectation to the fixed point algebra $\{X\,;\ \mc{P}_t(X)= X\ \text{for}\ t \ge 0\}$. It is currently unknown whether or not the quantum Beckner's inequality can be implied from $\Gamma_2$-criterion \ref{def:becdc}. Hence, our result is complementary with theirs.
\end{remark}
Similarly to \cite[Corollary 4.13]{junge2015noncommutative}, we next show that the moment estimates \eqref{eq:moment} with $s= 0$ can imply a noncommutative exponential integrability and an associated Gaussian concentration inequality. The idea of the proof is borrowed from the commutative case \cite{ben2008poincare} by Efraim and Lust-Piquard.
\begin{corollary} \label{cor:concern}
Suppose that the quantum $p$-Beckner's inequality \eqref{ineq_becp} holds
with $\inf_{p\in(1.2]}\alpha_p \ge a$ for some $a > 0$.
Then it holds that
\begin{equation} \label{eq:ex_int}
\norm{\exp\big(\big|X- \norm{X}_{1,\frac{{\bf 1}}{d}}\big|\big)}_{1,\frac{{\bf 1}}{d}} \le 2 \exp\bigg( \frac{e\kappa \norm{\Gamma(X)}_{\infty}}{2 a} \bigg)\,,
\end{equation}
and, for any $t > 0$,
\begin{align} \label{eq:concentration}
\frac{1}{d}\tr\Big(1_{[t,\infty)}\big(\big|X- \norm{X}_{1,\frac{{\bf 1}}{d}}\big|\big)\Big) \le 2 \exp \left( -
\frac{a t^2}{2 e \kappa \norm{\Gamma(X)}_{\infty}}
\right)\,,
\end{align}
where $\kappa = (1 - e^{-1/2})^{-1}$.
\end{corollary}
\begin{proof}
Note that $\Gamma(X) = \Gamma\big(X - \norm{X}_{1,\frac{{\bf 1}}{d}}\big)$. Without loss of generality, we assume $\norm{X}_{1,\frac{{\bf 1}}{d}} = 0$. By functional calculus and
Proposition \ref{prop:moment}, we obtain
\begin{align} \label{auxeq:ex_int}
\frac{1}{2d} \tr (e^{|X|}) &\le \frac{1}{d}\tr(\cosh X) = 1 + \sum_{j = 1}^\infty \frac{1}{(2j)!}\norm{X}^{2j}_{2j, \frac{{\bf 1}}{d}} \notag \\
& \le 1 + \sum_{j = 1}^\infty \frac{1}{(2j)!} \frac{(2j)^{j}\kappa^j}{a^j}\norm{\Gamma(X)}_{j,\frac{{\bf 1}}{d}}^j \notag \\
& \le 1 + \sum_{j = 1}^\infty \frac{j^j}{j! (2 j -1)!!} \bigg( \frac{\kappa \norm{\Gamma(X)}_{j,\frac{{\bf 1}}{d}} }{a} \bigg)^j \notag \\
& \le 1 + \sum_{j = 1}^\infty \frac{1}{j!} \bigg( \frac{e\kappa \norm{\Gamma(X)}_{j,\frac{{\bf 1}}{d}}}{2a} \bigg)^j\,,
\end{align}
where in the last inequality we have used $\frac{j^j}{(2j-1)!!} \le \left( \frac{e}{2} \right)^j$ for all $j \in \mathbb{N}$ from Stirling's formula. Then the inequality \eqref{eq:ex_int} follows from \eqref{auxeq:ex_int} and again $\norm{\Gamma(X)}_{j,\frac{{\bf 1}}{d}} \le \norm{\Gamma(X)}_\infty$. For the concentration inequality \eqref{eq:concentration}, by \eqref{eq:ex_int} and Chebyshev inequality, we have, for any $\lambda > 0$,
\begin{align*}
\frac{1}{d}\tr\big(1_{[t,\infty)}(|X|)\big) \le e^{-\lambda t} \frac{1}{d} \tr \big( e^{\lambda |X|} \big) \le 2 \exp\bigg( \frac{e\kappa \norm{\Gamma(X)}_{\infty}}{2a} \lambda^2 - t \lambda \bigg)\,.
\end{align*}
Then, choosing $\lambda$ to minimize the right-hand side of the above estimate gives the desired \eqref{eq:concentration}.
\end{proof}
\begin{remark}
Bobkov and G\"{o}tze \cite{bobkov1999exponential} established the exponential integrability in the commutative case from a variant of log-Sobolev inequality. Moreover, they showed that the exponential integrability implies the transportation cost inequality of order $1$. A noncommutative analog of this result can be found in \cite[Corollary 4.19]{junge2015noncommutative}. Also note that Rouz\'{e} and Datta \cite{rouze2019concentration} showed a similar Gaussian concentration inequality from the transportation cost inequalities that can be implied by MLSI. Corollary \ref{cor:concern} above provides another approach from the quantum MLSI to the concentration inequality, in view of Corollary \ref{coro:pitobeck}.
\end{remark}
\section{Generalized quantum optimal transport} \label{sec:qot_beck}
\subsection{Gradient flow of quantum \texorpdfstring{$p$} --divergence} \label{sec:gradient}
This section aims to identify the dual QMS $\mc{P}_t^\dag = e^{t \mc{L}^\dag}$ with $\sigma$-DBC as the gradient flow of the quantum $p$--divergence $\mc{F}_{p,\sigma}$, for any $p \in (1,2]$, with respect to the Riemannian metrics $g_{p,\mc{L}}$ constructed in \eqref{def:metric_tensor} below. The arguments follow closely with those in \cite{carlen2017gradient} (see also \cite{otto2001geometry} for the classical case).
For our purpose, we first compute the functional derivative $\delta_\rho \mc{F}_{p,\sigma}(\rho)$ of $\mc{F}_{p,\sigma}$ by
\begin{align*}
\frac{d}{d t}\Big|_{t = 0} \mc{F}_{p,\sigma}(\rho_t) = \langle \delta_\rho \mc{F}_{p,\sigma}(\rho), \dot{\rho}\rangle\,,
\end{align*}
for any smooth curve $\rho_t: (-\varepsilon,\varepsilon) \to \mc{D}(\mc{H})$, $\varepsilon > 0$, satisfying $\rho_0 = \rho$. Similarly to the computation for \eqref{eq:ep_divi}, we find
\begin{equation} \label{eq:deriv_func}
\delta_\rho \mc{F}_{p,\sigma}(\rho) = \frac{1}{p-1} \Gamma_\si^{-1/\h{p}}\Big(\left(\Gamma_\si^{-1/\h{p}}(\rho)\right)^{p-1}\Big)\,, \quad \forall \rho \in \mc{D}(\mc{H})\,.
\end{equation}
Then, letting $Y := \Gamma_\si^{-1/\h{p}}(\rho)$, by \eqref{eq:partialpower_2} and Lemma \ref{lem:chain_rule},
we derive, for each $1 \le j \le J$,
\begin{align} \label{eq:deriv_entropy}
\partial_j \delta_\rho \mc{F}_{p,\sigma}(\rho) & = \frac{1}{p-1} \partial_j\Gamma_\si^{-1/\h{p}}\left(Y^{p-1}\right) \notag \\ & = \frac{1}{p-1}\Gamma_\si^{-1/\h{p}}\left(V_j \left(e^{-\omega_j/2p} Y\right)^{p-1} - \left(e^{\omega_j/2p} Y\right)^{p-1} V_j\right) \notag \\
& = \Gamma_\si^{-1/\h{p}} \left(f_p^{[1]}\left(e^{\omega_j/2p} Y, e^{-\omega_j/2p} Y \right) \left(V_j \left(e^{-\omega_j/2p} Y \right) - \left(e^{\omega_j/2p} Y\right) V_j \right)\right),
\end{align}
where $f_p$ is given in \eqref{def:funfp}.
Again by \eqref{eq:partialpower_2} and noting $\partial_{j,\sigma} = \Gamma_\si \partial_j \Gamma_\si^{-1}$ from \eqref{eq:adjpjkms}, it follows that
\begin{align} \label{eq:deriv_entropy_aux}
V_j \left(e^{-\omega_j/2p} Y\right) - \left(e^{\omega_j/2p} Y\right) V_j &= \Gamma_\si^{1/p} \left( \partial_j \Gamma_\si^{-1/p} (Y) \right) \notag \\ & = \Gamma_\si^{1/p} \left( \partial_j \Gamma_\si^{-1} (\rho) \right) = \Gamma_\si^{-1/\h{p}} (\partial_{j,\sigma}\rho)\,.
\end{align}
Combining \eqref{eq:deriv_entropy} and \eqref{eq:deriv_entropy_aux} readily gives
\begin{align} \label{eq:double_deriv}
\partial_j \delta_\rho \mc{F}_{p,\sigma}(\rho) = \Gamma_\si^{-1/\h{p}} \left(f_p^{[1]}\left(e^{\omega_j/2p} \Gamma_\si^{-1/\h{p}}(\rho), e^{-\omega_j/2p} \Gamma_\si^{-1/\h{p}}(\rho)\right)\left(\Gamma_\si^{-1/\h{p}} (\partial_{j,\sigma} \rho)\right)\right).
\end{align}
We next define the Riemannian structure, associated with $\mc{F}_{p,\sigma}(\rho)$, on the matrix manifold $\mc{D}_+(\mc{H})$, i.e., a family of inner products on the tangent space $T_\rho = \mc{B}^0_{sa}(\mc{H}) := \{X \in \mc{B}_{sa}(\mc{H})\,;\ \tr(X) = 0\}$ that depends smoothly on $\rho$.
Motivated by the formula \eqref{eq:double_deriv}, we first define the operator $ [\rho]_{p,\omega}: \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})$ for $\rho \in \mc{D}_+(\mc{H})$ and $\omega \in \mathbb{R}$ by
\begin{align} \label{def:kernel_multiplication}
[\rho]_{p,\omega} = \Gamma_\si^{1/\h{p}} \circ
\theta_p\left(e^{\omega/2p} \Gamma_\si^{-1/\h{p}}(\rho), e^{-\omega/2p} \Gamma_\si^{-1/\h{p}}(\rho)\right) \circ
\Gamma_\si^{1/\h{p}}\,,
\end{align}
where
\begin{align} \label{def:theta_p}
\theta_p(x,y) := \big(f_p^{[1]}(x,y)\big)^{-1}
= \left\{
\begin{aligned}
& (p-1) \frac{x - y}{x^{p-1} - y^{p-1}}\,, \quad & x \neq y\,, \\
& x^{2-p} \,,\quad & x = y\,.
\end{aligned}
\right.
\end{align}
It is clear that when $\sigma = {\bf 1}/d$ and $\omega = 0$, we have $[\rho]_{p,\omega} A = d^{1-p} \rho^{2-p} A = \Gamma_\si( (\Gamma_\si^{-1}(\rho))^{2-p}) A$ for any $A$ that commutes with $\rho$. Therefore, $ [\rho]_{p,\omega}$ can be regarded as
a noncommutative analog of the multiplication by the relative density $(\Gamma_\si^{-1}\rho)^{2-p}$ with respect to the reference state $\sigma$. Noting $\Gamma_\si^{-1/\h{p}}(\rho) > 0$, by Lemma \ref{lem:double_inner} the operator $[\rho]_{p,\omega}$ is
evidently invertible. Then, we immediately obtain from \eqref{eq:double_deriv} that
\begin{align} \label{auxeq_deriva}
\partial_j \delta_\rho \mc{F}_{p,\sigma}(\rho) = [\rho]_{p,\omega_j}^{-1}\partial_{j,\sigma}\rho\,.
\end{align}
With the help of $[\rho]_{p,\omega}$, we now introduce the operator:
\begin{align} \label{def:dprho}
\mf{D}_{p,\rho}(A) := \sum_{j = 1}^J \partial_j^\dag \left([\rho]_{p,\omega_j} \partial_j A \right):\ \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})\,,
\end{align}
which, as we shall see soon, is crucial for defining the desired Riemannian metrics on $\mc{D}_+(\mc{H})$. The following result summarizes the main properties of $[\rho]_{p,\omega}$ and $\mf{D}_{p,\rho}$ and can be proved in the same manner as \cite[Lemma 5.8]{carlen2017gradient} and \cite[Lemma 7.3]{carlen2020non}. Hence we omit its proof.
\begin{lemma} \label{lem:basic_krho}
Let $\rho \in \mc{D}_+(\mc{H})$ and $\omega \in \mathbb{R}$. It holds that
\begin{enumerate}[1.]
\item $\langle \cdot, [\rho]_{p,\omega}(\cdot)\rangle$ gives an inner product on $\mathcal{B}(\mc{H})$. Moreover, both $[\rho]_{p,\omega}$ and $[\rho]_{p,\omega}^{-1}$ are $C^\infty$ on $\mc{D}_+(\mc{H})$.
\item $\mf{D}_{p,\rho}$ is a positive semidefinite operator on $\mathcal{B}(\mc{H})$ and preserves self-adjointness. Moreover, we have
\begin{align*}
\ran(\mf{D}_{p,\rho}) = \ran(\mc{L}^\dag) = \ran({\rm div})\,,\quad \ker(\mf{D}_{p,\rho}) = \ker(\mc{L})=\ker(\nabla)\,.
\end{align*}
\end{enumerate}
\end{lemma}
Recalling $\ker(\mc{L}) = {\rm span}\{{\bf 1}_{\mc{H}}\}$ for a primitive QMS $\mc{P}_t = e^{t \mc{L}}$, a direct consequence of the above lemma is that $\mf{D}_{p,\rho}$ for $\rho \in \mc{D}_+(\mc{H})$ is a positive definite operator on $\mc{B}_{sa}^0(\mc{H}) = (\ker(\mf{D}_{p,\rho}))^\perp$ that depends $C^\infty$ on $\rho$.
These facts allow us to define the following class of Riemannian metrics on $\mc{D}_+(\mc{H})$.
\begin{definition}\label{def:metric}
For each $\rho \in \mc{D}_{+}(\mc{H})$, we define the metric tensor $g_{p,\rho}$ on the tangent space $T_\rho = \mc{B}_{sa}^0(\mc{H})$ by
\begin{align} \label{def:metric_tensor}
g_{p,\rho}(\nu_1, \nu_2) := \langle \mf{D}_{p,\rho}^{-1} (\nu_1), \nu_2 \rangle \,,\quad \nu_1, \nu_2 \in T_\rho\,.
\end{align}
\end{definition}
We introduce $U_i := \mf{D}_{p,\rho}^{-1} (\nu_i) \in \mc{B}_{sa}^0(\mc{H})$ and define the inner product on $\mc{B}(\mc{H})^J$:
\begin{align*}
\langle {\bf A}, {\bf B}\rangle_{p,\rho} := \sum_{j = 1}^J \langle A_j, [\rho]_{p,\omega_j} B_j\rangle \quad \text{for} \ {\bf A}, {\bf B} \in \mc{B}(\mc{H})^J\,.
\end{align*}
Then, by \eqref{def:dprho}, the metric tensor $g_{p,\rho}$ can be rewritten as
\begin{align} \label{def:gp2}
g_{p,\rho}(\nu_1, \nu_2) = \sum_{j = 1}^{J} \langle \partial_j U_1, [\rho]_{p,\omega_j} \partial_j U_2\rangle = \langle \nabla U_1, \nabla U_2\rangle_{p,\rho}\,.
\end{align}
We are now ready to conclude Proposition \ref{prop:gradient_flow} below. We first recall that the Riemannian gradient of $\mc{F}_{p,\sigma}$
with respect to the metric $g_{p,\rho}$, denoted by
$\grad \mc{F}_{p,\sigma}(\rho)$,
is determined by the relation:
\begin{align*}
g_{p,\rho}({\rm grad}\mc{F}_{p,\sigma}(\rho), \nu) = \langle \delta_\rho \mc{F}_{p,\sigma}(\rho), \nu\rangle \,, \quad \forall \nu \in T_\rho\,.
\end{align*}
Then, by Definition \ref{def:metric} and formulas \eqref{auxeq_deriva} and \eqref{def:dprho}, it follows that
\begin{align} \label{eq:riemannian_grad}
{\rm grad} \mc{F}_{p,\sigma}(\rho) = \mf{D}_{p,\rho} (\delta_\rho \mc{F}_{p,\rho}(\rho)) = - \mc{L}^\dag(\rho)\,,
\end{align}
and the following result holds.
\begin{proposition}\label{prop:gradient_flow}
The dual primitive QMS, $\rho_t = e^{t \mc{L}^\dag}(\rho)$ for $\rho \in \mathcal{D}_+(\mc{H})$, satisfying $\sigma$-DBC is the gradient flow of the quantum $p$--divergence $\mc{F}_{p,\sigma}$ with respect to the Riemannian metric $g_{p,\rho}$ in \eqref{def:metric_tensor}:
\begin{align*}
\partial_t{\rho}_t = - {\rm grad} \mc{F}_{p,\sigma}(\rho_t) = \mc{L}^\dag(\rho_t)\,.
\end{align*}
\end{proposition}
An immediate consequence of Proposition \ref{prop:gradient_flow} above is the decrease of the entropy functional $\mc{F}_{p,\sigma}(\rho)$ along the quantum dynamic $\rho_t = \mc{P}_t^\dag(\rho)$:
\begin{align} \label{eq:grad_entropy}
\frac{d}{d t} \mc{F}_{p,\sigma}(\rho_t) & = g_{p,\rho_t}({\rm grad} \mc{F}_{p,\sigma}(\rho_t),\dot{\rho}_t) = - g_{p,\rho_t}(\mc{L}^\dag(\rho_t), \mc{L}^\dag(\rho_t)) \le 0\,.
\end{align}
We shall see later in Section \ref{sec:geodesic_convexity} that the decay rate of $\mc{F}_{p,\sigma}(\rho_t)$, characterized by the quantum Beckner's inequality (cf.\,\eqref{eq:exp_divi} and \eqref{eq_beck}), naturally connects with the geodesic convexity of the functional $\mc{F}_{p,\sigma}$.
Inspired by the result by Dietert \cite{dietert2015characterisation}, that if a Markov chain with finite states can be formulated as the gradient flow of the relative entropy, then it must be time-reversible, and its quantum analogs \cite{carlen2020non,cao2019gradient}, we consider the necessary condition for $\mc{P}_t^\dag$ being the gradient flow of $\mc{F}_{p,\sigma}$ for some Riemannian metric. Before we proceed, we give the following lemma that will be useful in the sequel.
\begin{lemma} \label{lem:key_rela}
For $\sigma \in \mathcal{D}_+(\mc{H})$, it holds that
\begin{equation} \label{auxeqq_rela_00}
[\sigma]_{p,0} = J_\sigma^{\kappa^{-1}_{1/p}} = R_\sigma \kappa_{1/p}^{-1}(\Delta_\sigma)\,,
\end{equation}
and
\begin{align} \label{auxeqq_rela}
[\sigma]_{p,\omega_j} \circ \partial_j \circ [\sigma]_{p,0}^{-1} = \Gamma_\si \circ \partial_j \circ \Gamma_\si^{-1}\,,
\end{align}
where $[\sigma]_{p,0}$ and $[\sigma]_{p,\omega}$ are defined as in \eqref{def:kernel_multiplication}; the function $\kappa_{1/p}$ is the power difference \eqref{def:pdiff}.
\end{lemma}
\begin{proof}
For the first identity \eqref{auxeqq_rela_00}, note from \eqref{eq:kernel_iden}, \eqref{eq:kernel_iden_2}, \eqref{auxeq_power_diff}, and definition \eqref{def:kernel_multiplication} that
\begin{equation*}
[\sigma]_{p,0} = (\Omega_{\sigma}^{x^{-1}\varphi_p})^{-1} = R_\sigma \kappa_{1/p}^{-1}(\Delta_\sigma)\,.
\end{equation*}
With the help of the representation \eqref{eq:rep_gen} and the operator $[\cdot]_{p,\omega}$ defined in \eqref{def:kernel_multiplication}, we observe that \eqref{eq:rep_dbcl} can be reformulated as
\begin{align} \label{auxeqq_rela_0}
- \langle X, \mc{L} X \rangle_{\sigma,\varphi_p} = \sum_{j = J} \langle \Gamma_\si X, [\sigma]_{p,0}^{-1} \partial_j^\dag \Gamma_\si \partial_j X\rangle = \sum_{j = J} \langle \Gamma_\si \partial_j X, [\sigma]_{p,\omega_j}^{-1} \Gamma_\si \partial_j X\rangle\,,
\end{align}
which is equivalent to the second identity \eqref{auxeqq_rela}.
\end{proof}
\begin{proposition}
If the dual primitive QMS $\mc{P}_t^\dag = e^{t \mc{L}^\dag}$ with $\sigma$-DBC is the gradient flow of the functional $\mc{F}_{p,\sigma}(\rho)$ with respect to some smooth Riemannian metric $g_\rho$, then $\mc{L}$ is self-adjoint with respect to the inner product:
\begin{equation} \label{def:nece_inner}
\langle X , Y \rangle_{[\sigma]_{p,0}}: = \langle X, [\sigma]_{p,0} Y \rangle\,,\quad X, Y \in \mathcal{B}(\mc{H})\,,
\end{equation}
where $[\sigma]_{p,0}$ is defined by \eqref{def:kernel_multiplication}.
\end{proposition}
\begin{proof}
Note that any Riemannian metric $g_\rho$ on $\mathcal{D}_+(\mc{H})$ can be written as
\begin{align*}
g_\rho(\nu_1,\nu_2) = \langle \mf{D}_\rho^{-1} \nu_1, \nu_2 \rangle\,,\quad \nu_1,\nu_2 \in T_\rho\,,
\end{align*}
where $\mf{D}_\rho$ is a positive definite operator from the cotangent space $T^*_\rho$ to the tangent space $T_\rho$ at $\rho$ (both $T_\rho$ and $T^*_\rho$ can be identified with $\mc{B}^0_{sa}(\mc{H})$). The dual QMS $\mc{P}_t^\dag$ is the gradient flow of $\mc{F}_{p,\sigma}(\rho)$ with respect to $g_\rho$ means that
\begin{align*}
\mc{L}^\dag \rho = - \mf{D}_{\rho}(\delta_\rho \mc{F}_{p,\sigma}(\rho))\,.
\end{align*}
Substituting $\rho = \sigma + \varepsilon X$ for $X \in \mc{B}^0_{sa}(\mc{H})$ into the above formula and differentiating it with respect to $\varepsilon$ gives
\begin{align*}
\mc{L}^\dag X = - \mf{D}_{\rho} \Gamma_\si^{-1/\h{p}}f_p^{[1]}(\sigma^{1/p},\sigma^{1/p})\Gamma_\si^{-1/\h{p}}(X) = - \mf{D}_{\rho}[\sigma]^{-1}_{p,0} X\,,
\end{align*}
by \eqref{eq:deriv_func} and \eqref{eq:chain_curve}. Hence, it follows that, for $X, Y \in \mc{B}_{sa}^0(\mc{H})$,
\begin{align*}
\langle \mc{L}Y, X\rangle_{[\sigma]_{p,0}} = - \langle Y, \mf{D}_\rho X \rangle = - \langle \mf{D}_\rho Y, X \rangle = \langle Y, \mc{L} X \rangle_{[\sigma]_{p,0}}\,,
\end{align*}
which implies that $\mc{L}$ is self-adjoint on $\mc{B}(\mc{H})$ with respect to $\langle \cdot,\cdot \rangle_{[\sigma]_{p,0}}$, since $\mc{L}$ and $[\sigma]_{p,0}$ are Hermitian-preserving and satisfy
$\mc{L}({\bf 1}) = 0$ and $[\sigma]_{p,0}{\bf 1} = \sigma$.
\end{proof}
In the limiting case $p \to 1^{+}$, corresponding to the gradient flow of
the Umegaki’s relative entropy, the inner product $ \langle \cdot , \cdot \rangle_{[\sigma]_{p,0}}$ \eqref{def:nece_inner} reduces to the BKM inner product:
\begin{equation*}
\langle X, Y \rangle_{{\rm BKM}}: = \int_0^1 \tr(\sigma^{1-t} X^* \sigma^t Y)\ dt\,,
\end{equation*}
and hence our result generalizes the one \cite[Theorem 2.9]{carlen2020non} by Carlen and Maas. In the case of $p = 2$, where $\mc{F}_{p,\sigma} $ is the variance (up to a constant factor), $ \langle \cdot , \cdot \rangle_{[\sigma]_{p,0}}$ becomes the familiar KMS inner product $\langle \cdot,\cdot \rangle_{\sigma,1/2}$. The characterizations of, and the relations between the quantum Markov semigroups that are self-adjoint with respect to different inner products have been discussed extensively in \cite{carlen2020non,amorim2021complete,benoist2021deviation}. For our case, noting the formula \eqref{auxeqq_rela_00}, it follows from Lemma \ref{lem:self_adjoint} that the QMS satisfying $\sigma$-DBC is also self-adjoint with respect to the inner product $ \langle \cdot , \cdot \rangle_{[\sigma]_{p,0}}$. But the reverse direction is still not well understood and needs further investigations.
\subsection{Generalized quantum transport distances} \label{sec:quantum_distance} The Riemannian distance $W_{2,p}$ induced by the metric $g_{p,\rho}$ in \eqref{def:metric_tensor} can be defined as: for $\rho_0, \rho_1 \in \mathcal{D}_+(\mc{H})$,
\begin{align}
W_{2,p}(\rho_0,\rho_1)^2 =\ & \inf \left\{
\int_0^1 g_{p, \gamma(s)}\left(\dot{\gamma}(s),\dot{\gamma}(s) \right) ds\,;\ \gamma(0) = \rho_0\,, \gamma(1) = \rho_1
\right\} \label{def:wp1} \\
\overset{\eqref{def:gp2}}{=} &
\inf \left\{
\int_0^1 \norm{\nabla U(s)}^2_{p,\gamma(s)} ds \,;\ \dot{\gamma}(s) = \mf{D}_{p,\rho}U(s)\ \text{with}\ \gamma(0) = \rho_0\,, \gamma(1) = \rho_1
\right\}, \label{def:wp2}
\end{align}
where the infimum is taken over the smooth $(C^\infty)$ curves $\gamma(s):[0,1] \to \mathcal{D}_+(\mc{H})$. In this section, we will investigate the basic properties of the distance function $W_{2,p}$. By the standard reparameterization techniques (cf.\cite[Lemma 1.1.4]{ambrosio2005gradient} or \cite[Theorem 5.4]{dolbeault2009new}),
we have that $W_{2,p}$ equals to the minimum arc length:
\begin{align} \label{def:wp3}
W_{2,p}(\rho_0,\rho_1) = \inf \left\{
\int_0^1 g_{p, \gamma(s)}\left(\dot{\gamma}(s),\dot{\gamma}(s) \right)^{1/2} ds\,;\ \gamma(0) = \rho_0\,, \gamma(1) = \rho_1
\right\},
\end{align}
where the infimum is taken over the smooth curves of constant speed (i.e., $g_{p, \gamma(s)}\left(\dot{\gamma}(s),\dot{\gamma}(s) \right)^{1/2}$ is constant). Then it follows from the expression \eqref{def:wp3} that the Riemannian manifold $\mc{D}_+(\mc{H})$ equipped with the distance $W_{2,p}$ is a metric space. Moreover, it turns out that $W_{2,p}$ can be extended continuously to the boundary of $\mc{D}_+(\rho)$.
\begin{lemma} \label{lem:exten_bry}
The metric $W_{2,p}$ on $\mc{D}_+(\mc{H})$ extends continuously to $\mathcal{D}(\mc{H})$.
\end{lemma}
Since the proof of Lemma \ref{lem:exten_bry} is similar to that of \cite[Proposition 9.2]{carlen2020non}, we sketch its main steps in Appendix \ref{app:proof} for completeness. In analogy with the classical 2-Wasserstein distance, it is helpful to introduce the quantum moment variable ${\bf B} = ([\rho]_{p,\omega_1}\partial_1 U, \ldots, [\rho]_{p,\omega_J}\partial_J U )$ for $U \in \mc{B}_{sa}^0$ and then reformulate \eqref{def:wp2} as a convex optimization problem:
\begin{align} \label{def:wp4}
W_{2,p}(\rho_0,\rho_1)^2 = \inf \Big\{
\int_0^1 \norm{{\bf B}}^2_{-1,p,\gamma(s)} ds \,;\ &\dot{\gamma}(s) + {\rm div} {\bf B} (s) = 0\ \text{with}\ \gamma(0) = \rho_0\,, \gamma(1) = \rho_1, \notag \\
& \gamma(s) \in C\left([0,1]; \mathcal{D}(\mc{H})\right),\ \text{and}\ {\bf B} (s)\in L^1\left([0,1]; \mathcal{B}(\mc{H})^J\right)
\Big\}\,,
\end{align}
where the continuity equation $\dot{\gamma}(s) + {\rm div} {\bf B}(s) = 0$ is understood in the sense of distributions, and
$\norm{\cdot}_{-1,p,\rho}$ is the norm from
the inner product $\langle\cdot,\cdot\rangle_{-1,p,\rho}$ on $\mc{B}(\mc{H})^J$ defined by
\begin{align*}
\langle {\bf A}, {\bf B}\rangle_{-1,p,\rho} := \sum_{j = 1}^J \langle A_j, [\rho]_{p,\omega_j}^{-1} B_j\rangle\,.
\end{align*}
Indeed, by approximation techniques as in \cite[Proposition 1]{wirth2022dual} and a mollification argument from \cite[Lemma 2.9]{erbar2012ricci}, we can show that the infimum in the above representation \eqref{def:wp4} of $W_{2,p}$ can be equivalently taken over smooth curves $\gamma \in C^\infty([0,1];\mathcal{D}_+(\mc{H}))$. Then, the equivalence between the formulations \eqref{def:wp2} and \eqref{def:wp4} follows from the same arguments as in \cite[Lemma 9.1]{carlen2020non}, while the convexity of the optimization problem \eqref{def:wp4} is a simple consequence of the following lemma and \cite[Theorem 9.7]{carlen2020non}.
\begin{lemma}\label{lem:joint_convexity}
$\langle X, [\rho]_{p,\omega}^{-1}X \rangle$ is jointly convex in $(\rho,X) \in \mc{D}_+(\mc{H}) \times \mc{B}(\mc{H})$.
\end{lemma}
\begin{proof}
Note from \eqref{def:kernel_multiplication} that $$\langle X, [\rho]_{p,\omega}^{-1} X\rangle = \langle \Gamma_\si^{-1/\h{p}}(X), f_p^{[1]}(e^{\omega/2p} \Gamma_\si^{-1/\h{p}}(\rho), e^{-\omega/2p} \Gamma_\si^{-1/\h{p}}(\rho))(\Gamma_\si^{-1/\h{p}}(X))\rangle\,.$$ Then the statement follows from \cite[Lemma 2.3]{zhang2020optimal}, which shows the joint convexity of $\langle Y, f_p^{[1]}(A,B) Y \rangle$ for $Y \in \mathcal{B}(\mc{H})$ and full-rank $A,B \in \mc{B}_{sa}^+(\mc{H})$.
\end{proof}
We summarize the above discussion as follows.
\begin{proposition} \label{prop:dist_convexity}
$W_{2,p}^2$ has an equivalent convex optimization formulation \eqref{def:wp4}. More precisely, let $\rho_0^i, \rho_1^i\in \mc{D}(\mc{H})$ for $i = 0,1$, and set $\rho_0^s := (1-s)\rho_0^0 + s\rho_0^1$ and $\rho_1^s := (1-s)\rho_1^0 + s\rho_1^1$ for $s \in [0,1]$. Then there holds
\begin{equation*}
W_{2,p}(\rho^s_0,\rho^s_1)^2 \le (1-s) W_{2,p}(\rho^0_0,\rho^0_1)^2 + s W_{2,p}(\rho^1_0,\rho^1_1)^2\,.
\end{equation*}
\end{proposition}
The main result of this section is Theorem \ref{thm:main_wasser} below.
\begin{theorem}\label{thm:main_wasser}
$(\mc{D}(\mc{H}), W_{2,p})$ is a complete geodesic space. In particular, for any $\rho_0, \rho_1 \in \mathcal{D}(\mc{H})$, the minimizer to \eqref{def:wp4} exists and gives the minimizing geodesic $(\gamma_*(s))_{s \in [0,1]}$ connecting $\rho_0$ and $\rho_1$ and satisfying
\begin{equation} \label{eq:geo_mini}
W_{2,p}(\gamma_*(s),\gamma_*(t)) = |s - t| W_{2,p}(\rho_0,\rho_1)\,, \quad \forall s,t \in [0,1]\,.
\end{equation}
\end{theorem}
The completeness of the metric space $(\mc{D}(\mc{H}), W_{2,p})$
needs the following lemma, which is of interest itself. We postpone both proofs of Theorem \ref{thm:main_wasser} and Lemma \ref{lem:lower_bound} to Appendix \ref{app:proof} for ease of reading.
\begin{lemma} \label{lem:lower_bound}
There exists a constant $C > 0$, independent of $p \in (1,2]$, such that for any $\rho_0, \rho_1 \in \mathcal{D}(\mc{H})$,
\begin{align*}
\norm{\rho_1 - \rho_0}_1 \le C W_{2,p}(\rho_0,\rho_1)\,.
\end{align*}
\end{lemma}
\begin{remark}
It is straightforward to generalize the arguments in \cite{wirth2022dual} to derive the dual formulation for the distance $W_{2,p}$ in terms of noncommutative Hamilton-Jacobi-Bellman-type equations. Actually, a formal calculation gives
\begin{align*}
\frac{1}{2} W_{2,p}^2(\rho_0,\rho_1) = \sup\Big\{\tr(A(1)\rho_1 - A(0)\rho_0)\,;\ \tr(\dot{A}(t)\rho) + \frac{1}{2}\norm{\nabla A(t)}^2_{p,\rho} \le 0\,,\ \forall \rho \in \mathcal{D}(\mc{H}) \Big\}\,.
\end{align*}
The above dual formula allows us to fit our distance $W_{2,p}$ into the general framework of noncommutative transportation metrics proposed in \cite{gao2021ricci}, and to further discuss the coarse Ricci curvature of quantum channels with respect to $W_{2,p}$. The detailed investigation on these issues would be interesting but beyond the scope of this work.
\end{remark}
We next derive the geodesic equations for the Riemannian manifold $(\mathcal{D}_+(\mc{H}), g_{p,\rho})$. Instead of regarding the geodesic equation as the Euler-Lagrange equation for the minimization problem \eqref{def:wp2} as in \cite{datta2020relating,carlen2020non}, we interpret it as the Hamiltonian flow of the Hamiltonian associated with the metric $g_{p,\rho}$:
$$
H(\rho,U) := \frac{1}{2} g^{-1}_{p,\rho}(U,U) = \frac{1}{2}\langle \mf{D}_{p,\rho}U, U\rangle \quad \text{for}\ \rho \in \mathcal{D}_+(\mc{H})\ \text{and}\ U \in \mc{B}_{sa}^0(\mc{H})\,,
$$
where $g^{-1}_{p,\rho}$ is the inverse of the metric tensor. Then we can immediately write the geodesic equations:
\begin{align} \label{def:geo_eq}
& \dot{\rho} = \delta_{U} H(\rho,U)\,,\quad \dot{U} = -\delta_\rho H(\rho,U)\,.
\end{align}
By definition, it is clear that $\delta_{U} H(\rho,U) = \mf{D}_{p,\rho}U$. To find $\delta_\rho H$, by Lemma \ref{lem:high_order}, we have
\begin{align} \label{eq:func_dev}
\langle \delta_\rho H(\rho, U), A\rangle = & \lim_{\varepsilon \to 0} \frac{1}{2} \left\langle \mf{D}_{p, \rho + \varepsilon A} U, U\right\rangle \notag \\
= & \lim_{\varepsilon \to 0} \frac{1}{2} \left\langle \partial_j U, [\rho+\varepsilon A]_{p,\omega_j} \partial_j U \right\rangle \notag \\
= & \frac{1}{2} \sum_{j = 1}^J\left\langle \Gamma_\si^{1/\h{p}} (\partial_j U), (\delta_1 \theta_p)\big((l_j(\rho),l_j(\rho)), r_j(\rho)\big)\big(l_j(A), \Gamma_\si^{1/\h{p}}(\partial_j U)\big)\right\rangle \notag
\\
& + \frac{1}{2} \sum_{j=1}^J \left\langle \Gamma_\si^{1/\h{p}} (\partial_j U), (\delta_2 \theta_p)\big(l_j(\rho), ( r_j(\rho), r_j(\rho))\big)\big(\Gamma_\si^{1/\h{p}}(\partial_j U), r_j(A)\big)\right\rangle\,,
\end{align}
where for any $X \in \mathcal{B}(\mc{H})$,
\begin{align*}
l_j(X): = e^{\omega_j/2p} \Gamma_\si^{-1/\h{p}}(X)\,,\quad r_j(X): = e^{-\omega_j/2p} \Gamma_\si^{-1/\h{p}}(X)\,.
\end{align*}
We also note
\begin{align} \label{eq:observation}
& \left\langle \Gamma_\si^{1/\h{p}} (\partial_{j'} U), (\delta_1 \theta_p)\big((l_{j'}(\rho),l_{j'}(\rho)), r_{j'}(\rho)\big)\big(l_{j'}(A), \Gamma_\si^{1/\h{p}}(\partial_{j'} U)\big)\right\rangle \notag \\
= & \left\langle \Gamma_\si^{1/\h{p}} ((\partial_{j} U)^*), (\delta_1 \theta_p)\big((r_j(\rho),r_j(\rho)), l_j(\rho)\big)\big(r_j(A), \Gamma_\si^{1/\h{p}}((\partial_{j} U)^*)\big)\right\rangle \notag \\
= & \left\langle \Gamma_\si^{1/\h{p}} (\partial_{j} U), (\delta_2 \theta_p)\big(l_j(\rho), (r_j(\rho),r_j(\rho))\big)\big(\Gamma_\si^{1/\h{p}}(\partial_{j} U), r_j(A)\big)\right\rangle \,,
\end{align}
where the first equality is from
the relations $l_j(\rho) = r_{j'}(\rho)$, $r_j(\rho) = l_{j'}(\rho)$, and $(\partial_j U)^* = - \partial_{j'}U$ by
\eqref{eq:adjoint_index}, and the second equality is by the following formula from definitions \eqref{def_1:divi_diff} and \eqref{def_2:multiple}:
\begin{equation*}
\langle X^*, (\delta_1 f)((A,A),B)(Y,X^*) \rangle = \langle X, (\delta_2 f)(B,(A,A))(X,Y)\rangle \,,
\end{equation*}
for any symmetric $f$: $f(s,t) = f(t,s)$, $X \in \mathcal{B}(\mc{H})$, $Y \in \mc{B}_{sa}(\mc{H})$, and commuting matrices $A,B \in \mc{B}_{sa}(\mc{H})$. The identity \eqref{eq:observation} implies that the two sums in \eqref{eq:func_dev} are actually equal. Therefore, we obtain from \eqref{def:geo_eq} and \eqref{eq:func_dev} the following proposition, where the local existence and uniqueness of geodesics follows from the standard theory of ODE.
\begin{proposition}
On the Riemannian manifold $(\mathcal{D}_+(\mc{H}), g_{p,\rho})$, the unique constant speed geodesic $(\rho(t))_{t \in (-\varepsilon,\varepsilon)}$, $\varepsilon > 0$, with initial data: $\rho(0) = \rho_0 \in \mathcal{D}_+(\mc{H})$ and $\dot{\rho}(0) = \nu_0 \in \mc{B}_{sa}^0(\mc{H})$, satisfies the equation:
\begin{equation} \label{eq:geod_equa}
\begin{split}
& \dot{\rho} = \mf{D}_{p,\rho}U \,,\\
& \langle \dot{U}, A\rangle = - \sum_{j = 1}^J\left\langle \partial_j U, \mc{K}_{\rho,A}^{(i),j} \left[ \partial_j U \right] \right \rangle\,, \quad \forall A \in \mc{B}^0_{sa}(\mc{H})\,,
\end{split}
\end{equation}
with $\rho(0) = \rho_0$ and $U(0) = \mf{D}_{p,\rho_0}^{-1} \nu_0$,
where $i = 1$ or $2$, and $\mc{K}_{\rho,A}^{(i),j}$ is defined by, for $\rho \in \mathcal{D}(\mc{H})$ and $A \in \mc{B}_{sa}^0(\mc{H})$,
\begin{equation} \label{def:kgeode}
\begin{split}
&\mc{K}_{\rho,A}^{(1),j}[\cdot] = \Gamma_\si^{1/\h{p}} \circ
(\delta_1 \theta_p)\big((l_j(\rho),l_j(\rho)), r_j(\rho)\big)\big[l_j(A), \Gamma_\si^{1/\h{p}}(\cdot) \big]: \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})\,,
\\ & \mc{K}_{\rho,A}^{(2),j}[\cdot] = \Gamma_\si^{1/\h{p}} \circ (\delta_2 \theta_p)\big(l_j(\rho), ( r_j(\rho), r_j(\rho))\big)\big[\Gamma_\si^{1/\h{p}}(\cdot), r_j(A)\big]: \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})\,.
\end{split}
\end{equation}
\end{proposition}
We end this section with the following proposition that the Riemannian metrics $g_{p,\rho}$ with $1 < p\le 2$ serve as an interpolating family between the metric defined by Carlen and Mass in \cite{carlen2017gradient} and the one induced from the KMS inner product $\langle \cdot, \cdot\rangle_{\sigma,1/2}$. It can be easily proved by an elementary analysis with
the fact, from the analytic perturbation theory \cite{rellich1969perturbation}, that the eigenvalues and eigenfunctions of $\Gamma_\si^{\scriptscriptstyle -1/\h{p}}(\rho)$ are differentiable with respect to $p$.
\begin{proposition}\label{prop:repkernel}
Suppose that $\Gamma_\si^{-1/\h{p}}(\rho)$ has the eigen-decomposition:
$
\Gamma_\si^{-1/\h{p}}(\rho) = \sum_{j = 1}^{d} \lambda_{j,p} E_{j,p}.
$ Then, we have
\begin{align} \label{eq:repre_rpw}
[\rho]_{p,\omega}A = \sum_{i,k = 1}^{d}
\theta_p\left(e^{\omega/2p}\lambda_{k,p}, e^{-\omega/2p}\lambda_{i,p} \right)
\Gamma_\si^{1/\h{p}} \left(E_{k,p} \Gamma_\si^{1/\h{p}}(A) E_{i,p}\right)\,, \quad A \in \mathcal{B}(\mc{H})\,,
\end{align}
where $\theta_p$ is given in \eqref{def:theta_p}. Moreover, $[\rho]_{p,\omega}$ is continuous in $p$, and it holds that when $p = 2$,
\begin{align*}
[\rho]_{2,\omega}A = \Gamma_\si(A)\,,
\end{align*}
and when $p \to 1^+$,
\begin{align} \label{def:metricpeq1}
[\rho]_{p,\omega}A \to [\rho]_{\omega}A := \sum_{i,k = 1}^{d}
\theta_{log}\left(e^{\omega/2}\lambda_k, e^{-\omega/2}\lambda_i \right)
E_k A E_i\,,
\end{align}
where $\theta_{log}$ is the logarithmic mean:
\begin{align*}
\theta_{log}(x,y): = \left\{
\begin{aligned}
& \frac{x - y}{\log x - \log y} \,, \quad & x \neq y\,, \\
& x \,,\quad & x = y\,,
\end{aligned}
\right.
\end{align*}
and $\lambda_j$ are eigenvalues of $\rho$ with $E_j$ being the associated rank-one eigen-projections: $\rho = \sum_{j = 1}^{d} \lambda_j E_j$.
\end{proposition}
Recalling the integral formula: $\frac{x - y}{\log x - \log y} = \int_0^1 x^{1-s} y^s \, ds $, we can write the operator $[\rho]_\omega$ in \eqref{def:metricpeq1} as
\begin{align*}
[\rho]_\omega(A) = \int_0^1 e^{\omega(1/2-s)} \rho^{1-s} A \rho^{s}\, ds\,,\quad A \in \mathcal{B}(\mc{H})\,,
\end{align*}
which is nothing but the noncommutative multiplication by $\rho$ involved in the definition of the quantum quadratic Wasserstein distance $W_2$ \cite{carlen2014analog,carlen2017gradient}. It follows that in the limiting case $p \to 1^+$, our transport distance $W_{2,p}$ reduces to the one $W_2$ introduced by Carlen and Maas. When $p = 2$, the representation \eqref{def:wp4} of $W_{2,p}$ gives
\begin{align} \label{eq:distance_1}
W_{2,2}(\rho_0,\rho_1)^2 = \inf \Big\{
\int_0^1 \sum_{j = 1}^J \langle B_j, \Gamma_\si^{-1} B_j \rangle \, ds \,;\ &\dot{\gamma}(s) + {\rm div} {\bf B} (s) = 0\ \text{with}\ \gamma(0) = \rho_0\,, \gamma(1) = \rho_1
\Big\}\,,
\end{align}
which has a similar form as the classical distance $\mathscr{W}_{2,\alpha,\gamma}$ in \eqref{def:distance_savare} with $\alpha = 0$, and thus can be regarded as a quantum analog of dual $\dot{H}^{-1}$ Sobolev distance. These facts allow us to (at least formally) conclude that our new family of quantum distances $W_{2,p}$ interpolates the noncommutative $2$-Wasserstein and the $\dot{H}^{-1}$ Sobolev ones. One may further expect a stronger result that for $\rho_0,\rho_1 \in \mathcal{D}(\mc{H})$, $W_{2,p}(\rho_0,\rho_1)$ is continuous in $p \in (1,2]$ and $W_{2,p}(\rho_0,\rho_1) \to W_2(\rho_0,\rho_1)$ as $p \to 1^+$, but this task seems to be challenging, and we leave it for future investigation.
\subsection{Ricci curvature and functional inequalities} \label{sec:geodesic_convexity}
In this section, we will investigate the entropic Ricci curvature lower bound in terms of the quantum $p$-divergence $\mc{F}_{p,\sigma}(\rho)$. Thanks to the gradient flow structure obtained in Section \ref{sec:gradient}, we are allowed to derive some new functional inequalities from the Ricci curvature lower bound: a quantum HWI-type inequality and a transportation cost inequality, and connect them with the quantum Beckner's inequality \eqref{ineq_becp}, in the spirit of Otto and Villani \cite{otto2000generalization}.
Let us first introduce the Ricci curvature in our setting. We follow the terminology of \cite{erbar2012ricci,datta2020relating}, and say that a primitive QMS $\mc{P}_t$ with $\sigma$-DBC has the Ricci curvature lower bound $\kappa \in \mathbb{R}$ associated with $\mc{F}_{p,\sigma}$ if
\begin{equation} \label{eq:ricci_p}
\frac{d^2}{ds^2}\Big|_{s = 0} \mc{F}_{p,\sigma}(\gamma(s)) \ge \kappa g_{p,\gamma(0)}(\dot{\gamma}(0),\dot{\gamma}(0))\,,
\end{equation}
where $\gamma(s)$, $s \in (-\varepsilon,\varepsilon)$, is a geodesic satisfying $\gamma(0) = \rho \in \mathcal{D}_+(\mc{H})$. We compute the second derivative of $\mc{F}_{p,\sigma}$ along the constant-speed geodesic $\gamma(s)$. For this, noting the Riemannian gradient \eqref{eq:riemannian_grad} of $\mc{F}_{p,\sigma}$, by definition, we have
\begin{align} \label{eq:first_dev_f}
\frac{d}{ds} \mc{F}_{p,\sigma}(\gamma(s)) &= - g_{p,\gamma(s)} (\dot{\gamma}(s), \mc{L}^\dag(\gamma(s))) = - \langle U(s), \mc{L}^\dag(\gamma(s))\rangle\,,
\end{align}
where $(\gamma,U)$ is the unique solution to \eqref{eq:geod_equa}. Then, differentiating \eqref{eq:first_dev_f} again with respect to $s$, we obtain
\begin{align} \label{eq:second_dev_f}
\frac{d^2}{ds^2}\Big|_{s = 0} \mc{F}_{p,\sigma}(\gamma(s)) & = - \langle \dot{U}(s), \mc{L}^\dag(\gamma(s))\rangle - \langle U(s), \mc{L}^\dag(\dot{\gamma}(s))\rangle \Big|_{s = 0} \notag \\
& = \sum_{j = 1}^J \big\langle \partial_j U(0), \mc{K}_{\rho,\mc{L}^\dag(\gamma(0))}^{(i),j} \left[ \partial_j U(0) \right] \big \rangle - \left\langle U(0), \mc{L}^\dag(\mf{D}_{p,\gamma(0)} U(0)) \right \rangle\,.
\end{align}
Recall that the Riemannian Hessian of $\mc{F}_{p,\sigma}$ at $\rho \in \mathcal{D}_+(\mc{H})$ is defined by, for $U \in \mc{B}_{sa}^0(\mc{H})$,
\begin{align*}
\hess \mc{F}_{p,\sigma}(\rho)[U,U] := \frac{d^2}{ds^2}\Big|_{s = 0} \mc{F}_{p,\sigma}(\gamma(s))\,,
\end{align*}
where $(\gamma(s),U(s))$ satisfies the geodesic equation \eqref{eq:geod_equa} with initial conditions $\gamma(0) = \rho$, $U(0) = U$, and $\dot{\gamma}(0) = \mf{D}_{p,\rho}U$. We readily conclude from the formula \eqref{eq:second_dev_f} that
\begin{align} \label{eq:hess_of_f}
\hess \mc{F}_{p,\sigma}(\rho)[U,U] = \sum_{j = 1}^J \big\langle \partial_j U, \mc{K}_{\rho,\mc{L}^\dag(\rho)}^{(i),j} \left[ \partial_j U \right] \big \rangle - \left\langle U, \mc{L}^\dag(\mf{D}_{p,\rho} U) \right \rangle\,, \quad \forall U \in \mc{B}_{sa}^0(\mc{H})\,.
\end{align}
Hence, it follows from definition that \eqref{eq:ricci_p} is equivalent to
\begin{align} \label{eq:ricci_hess}
\hess\mc{F}_{p,\sigma}(\rho)[U,U] \ge \kappa \langle U, \mf{D}_{p,\rho} U\rangle\,, \tag{${\rm Ric}_p(\mc{L}) \ge \kappa$}
\end{align}
for $\rho \in \mathcal{D}_+(\mc{H})$ and $U \in \mc{B}^0_{sa}(\mc{H})$. The next proposition provides several equivalent characterizations of \eqref{eq:ricci_hess}, in terms of the $\kappa$-geodesic convexity of $\mc{F}_{p,\sigma}(\rho)$, the gradient estimates, and the evolution variational inequality. In what follows, we will use the notation:
\begin{align} \label{def:upper_deriv}
\frac{d^+}{d t} f(t) = \limsup_{h \to 0^+} \frac{f(t+h) - f(t)}{h}\,.
\end{align}
\begin{proposition} \label{prop:geodesic_convex}
Let $\mc{P}_t$ be a primitive QMS satisfying $\sigma$-DBC with $\sigma \in \mathcal{D}_+(\mc{H})$. For $\kappa \in \mathbb{R}$, \eqref{eq:ricci_hess} is equivalent to the following statements:
\begin{enumerate}[(i)]
\item $\mc{F}_{p,\sigma}(\rho)$ is geodesically $\kappa$-convex on $(\mc{D}(\mc{H}), W_{2,p})$, that is, for any constant-speed geodesic $(\gamma(s))_{s \in [0,1]} \subset \mathcal{D}(\mc{H})$,
\begin{align} \label{eq:geo_conv}
\mc{F}_{p,\sigma}(\gamma(s)) \le (1-s) \mc{F}_{p,\sigma}(\gamma(0)) + s \mc{F}_{p,\sigma}(\gamma(1)) - \frac{\kappa}{2}s(1-s) W_{2,p}(\gamma(0),\gamma(1))^2\,.
\end{align}
\item For any $\rho_0, \rho_1 \in \mathcal{D}(\mc{H})$, the following evolution
variational inequality (EVI) holds: $\forall t \ge 0$,
\begin{align} \label{eq:evi}
\frac{1}{2}\frac{d^+}{d t} W_{2,p}(\mc{P}_t^\dag \rho_0 ,\rho_1)^2 + \frac{\kappa}{2} W_{2,p}(\mc{P}_t^\dag \rho_0 ,\rho_1)^2 \le \mc{F}_{p,\sigma}(\rho_1) - \mc{F}_{p,\sigma}(\mc{P}_t^\dag \rho_0)\,.
\end{align}
\item The following gradient estimate holds: for $\rho \in \mathcal{D}_+(\mc{H})$ and $U \in \mc{B}_{sa}^0(\mc{H})$, $\forall t > 0$,
\begin{align*}
\norm{\nabla \mc{P}_t(U)}^2_{p,\rho} \le e^{-2 \kappa t} \norm{\nabla U}^2_{p, \mc{P}_t^\dag\rho}\,.
\end{align*}
\item The following contraction of the transport distance $W_{2,p}$ along the gradient flow holds:
\begin{equation*}
W_{2,p}(\mc{P}_t^\dag \rho_0, \mc{P}_t^\dag \rho_1) \le e^{-\kappa t} W_{2,p}(\rho_0,\rho_1) \quad \forall \rho_0,\rho_1 \in \mathcal{D}_+(\mc{H}) \,.
\end{equation*}
\end{enumerate}
\end{proposition}
\begin{proof}
The equivalence: \eqref{eq:ricci_hess} $\Longleftrightarrow (i) \Longleftrightarrow (ii)$ can be proved in the same manner as \cite[Theorem 3]{datta2020relating} with the gradient flow techniques from \cite{otto2005eulerian,daneri2008eulerian,dolbeault2012poincare}; see also \cite[Theorem 10.2]{carlen2020non}. \eqref{eq:ricci_hess} $\Longleftrightarrow (iii)$ follows from a similar semigroup interpolation argument as in the proof of \cite[Theorem 10.4]{carlen2020non}, while the proof of \eqref{eq:ricci_hess} $\Longleftrightarrow (iv)$ can be easily modified from those of
Proposition 3.1 and (2.12) of \cite{daneri2008eulerian}.
\end{proof}
With the notion of Ricci curvature, we next prove some interesting implications between functional inequalities, following the arguments of Otto and Villani \cite{otto2000generalization} (see also \cite{erbar2012ricci,datta2020relating}).
We start with an HWI-type inequality, which relates the generalized quantum transport distance $W_{2,p}$, the quantum $p$-divergence $\mc{F}_{p,\sigma}$, and the entropy production ($p$-Dirichlet form) $\mc{E}_{p,\mc{L}}$. The following lemma will be used later on.
\begin{lemma} \label{lem:deriva_wasser}
Let $\rho, \w{\rho} \in \mathcal{D}_+(\mc{H})$ and $\rho_t = \mc{P}_t^\dag \rho$. Then there holds, for $t \ge 0$,
\begin{align*}
\frac{d^+}{d t} W_{2,p}(\rho_t,\w{\rho}) \le \frac{2}{p} \sqrt{\mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho_t))}\,.
\end{align*}
\end{lemma}
\begin{proof}
By definition \eqref{def:upper_deriv} and the triangle inequality, we have
\begin{align*}
\frac{d^+}{dt}W_{2,p}(\rho_t,\w{\rho}) = \limsup_{s \to 0^+} \frac{1}{s}\big(W_{2,p}(\rho_{t + s},\w{\rho}) - W_{2,p}(\rho_t,\w{\rho})\big) \le \limsup_{s \to 0^+} \frac{1}{s}W_{2,p}(\rho_t,\rho_{t+s})\,.
\end{align*}
The expression \eqref{def:wp3} with a time scaling gives
\begin{align*}
W_{2,p}(\rho_t,\rho_{t+s}) \le \int_t^{t+s} \sqrt{g_{p,\gamma(\tau)}(\dot{\gamma}(\tau),\dot{\gamma}(\tau))} \ d\tau\,,
\end{align*}
for any smooth curve $\gamma$ in $\mathcal{D}_+(\mc{H})$ such that $\gamma(t) = \rho_t$ and $\gamma(t+s) = \rho_{t+s}$. Note from \eqref{eq:ep_divi} and \eqref{eq:grad_entropy} that
\begin{align*}
g_{p,\gamma(\tau)}(\dot{\gamma}(\tau),\dot{\gamma}(\tau)) = \frac{4}{p^2} \mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\gamma(\tau)))\,.
\end{align*}
Therefore, we can find
\begin{equation*}
\frac{d^+}{dt}W_{2,p}(\rho_t,\w{\rho}) \le \limsup_{s \to 0^+} \frac{1}{s} \int_t^{t+s} \frac{2}{p} \sqrt{\mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\gamma(\tau)))} \ d\tau = \frac{2}{p} \sqrt{\mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho_t))}\,. \qedhere
\end{equation*}
\end{proof}
\begin{theorem} \label{them:riccitohwi}
If \eqref{eq:ricci_hess} holds for some $\kappa \in \mathbb{R}$, then the following HWI-type inequality holds:
\begin{equation} \label{eq:HWI}
\mc{F}_{p,\sigma}(\rho) \le \frac{2}{p} W_{2,p}(\rho, \sigma) \sqrt{\mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho))}- \frac{\kappa}{2} W_{2,p}(\rho, \sigma)^2\,, \quad \text{for all}\ \rho \in \mathcal{D}(\mc{H})\,.
\end{equation}
\end{theorem}
\begin{proof}
It suffices to consider $\rho \in \mathcal{D}_+(\mc{H})$, since $\mathcal{D}_+(\mc{H})$ is dense in $\mathcal{D}(\mc{H})$ and the inequality \eqref{eq:HWI} is continuous with respect to $\rho$. Letting $\rho_t = \mc{P}_t^\dag \rho$ for $\rho \in \mathcal{D}_+(\mc{H})$, we derive from Lemma \ref{lem:deriva_wasser} that
\begin{align*}
- \frac{1}{2} \frac{d^+}{d t}\Big|_{t =0} W_{2,p}(\rho_t, \sigma)^2 = &\liminf_{t \to 0^+} \frac{1}{2t} \big(W_{2,p}(\rho, \sigma)^2 - W_{2,p}(\rho_t, \sigma)^2\big) \\
\le & \limsup_{t \to 0^+} \frac{1}{2t} \big( W_{2,p}(\rho, \rho_t)^2 + 2 W_{2,p}(\rho, \rho_t) W_{2,p}(\rho_t,\sigma) \big) \\
\le & \frac{2}{p} \sqrt{\mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho))} W_{2,p}(\rho,\sigma)\,.
\end{align*}
Then, by above estimate, recalling the EVI \eqref{eq:evi} in Proposition \ref{prop:geodesic_convex} with $\rho_0 = \rho$ and $\rho_1 = \sigma$, we obtain
\begin{align*}
\mc{F}_{p,\sigma}(\rho) & \le - \frac{1}{2}\frac{d^+}{d t}\Big|_{t= 0} W_{2,p}(\rho_t, \sigma)^2 - \frac{\kappa}{2} W_{2,p}(\rho, \sigma)^2 \\
& \le \frac{2}{p} W_{2,p}(\rho,\sigma) \sqrt{\mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho))} - \frac{\kappa}{2} W_{2,p}(\rho, \sigma)^2 \,. \qedhere
\end{align*}
\end{proof}
As a direct consequence of Theorem \ref{them:riccitohwi} above, in the case of positive Ricci curvature lower bound, we can obtain the quantum Beckner's inequality.
\begin{corollary} \label{coro:poricci_beck}
If \eqref{eq:ricci_hess} holds for some $\kappa > 0$, then the quantum $p$-Beckner's inequality \eqref{eq_beck} holds with constant $\alpha_p \ge (\kappa p)/2 $.
\end{corollary}
\begin{proof}
By Theorem \ref{them:riccitohwi} and Young's inequality, we have
\begin{align*}
\mc{F}_{p,\sigma}(\rho) \le \frac{2}{p} \Big( \frac{1}{2 C} W_{2,p}(\rho, \sigma)^2 + \frac{C}{2} \mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho)) \Big) - \frac{\kappa}{2} W_{2,p}(\rho, \sigma)^2\,.
\end{align*}
Letting $C = 2/(p\kappa)$, by definition \eqref{eq_beck}, we complete the proof.
\end{proof}
Another implication of positive Ricci curvature is the finite diameter of the metric space $(\mathcal{D}(\mc{H}), W_{2,p})$, which can be viewed as a noncommutative Bonnet–Myers theorem.
\begin{corollary}
If \eqref{eq:ricci_hess} holds for some $\kappa > 0$, then it holds that
\begin{equation} \label{est:diameter}
\sup_{\rho_0, \rho_1\in \mc{D}(\mc{H})} W_{2,p}(\rho_0,\rho_1)^2 \le \frac{8}{\kappa p (p-1)} (\sigma_{\min}^{1-p} - 1)\,,
\end{equation}
where $\sigma_{\min}$ is the minimal eigenvalue of the invariant state $\sigma \in \mathcal{D}_+(\mc{H})$.
\end{corollary}
\begin{proof}
Note that the geodesic convexity \eqref{eq:geo_conv} gives
\begin{align} \label{auxest:diameter}
\frac{\kappa}{8} W_{2,p}(\rho_0,\rho_1)^2 \le \frac{1}{2} \mc{F}_{p,\sigma}(\rho_0) + \frac{1}{2} \mc{F}_{p,\sigma}(\rho_1) \,.
\end{align}
Then the estimate \eqref{est:diameter} follows from \eqref{eq:rela_sandpdivi}, \eqref{eq:suprhodp}, and \eqref{auxest:diameter}.
\end{proof}
We say that a primitive QMS with $\sigma$-DBC satisfies a transport cost inequality associated with the distance $W_{2,p}$ with constant $c > 0$ if for all $\rho \in \mathcal{D}(\mc{H})$,
\begin{equation} \label{ineq:tc}
W_{2,p}(\rho,\sigma) \le \sqrt{c \mc{F}_{p,\sigma}(\rho)}\,. \tag{TCp}
\end{equation}
We will show the chain of quantum functional inequalities \eqref{eq:chain}.
\begin{proposition} \label{propa}
Suppose that the quantum $p$-Beckner's inequality \eqref{eq_beck} holds for some $p \in (1,2]$. Then the transport cost inequality \eqref{ineq:tc} holds with constant $c \ge p/\alpha_p$.
\end{proposition}
\begin{proof}
Again, it suffices to consider $\rho \in \mathcal{D}_+(\mc{H})$. Let $\rho_t = \mc{P}_t^\dag \rho$ and define the function
\begin{align*}
h(t) := W_{2,p}(\rho_t,\rho) + \sqrt{c \mc{F}_{p,\sigma}(\rho_t)}\,, \quad t \ge 0\,.
\end{align*}
Clearly, $h(t)$ satisfies that $h(0)= \sqrt{c \mc{F}_{p,\sigma}(\rho)}$ and $h(t) \to W_{2,p}(\sigma,\rho)$ as $t \to \infty$ by \eqref{eq:conver_qms}. We now claim that when $c \ge p/\alpha_p$, $\frac{d^+}{d t} h(t) \le 0$ holds for $t \ge 0$, which completes the proof. By Lemma \ref{lem:deriva_wasser} and \eqref{eq:ep_divi}, when $\rho_t \neq \sigma$, we compute
\begin{align*}
\frac{d^+}{d t} h(t) &\le \frac{2}{p} \sqrt{\mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho_t))} - \frac{2\sqrt{c}}{p^2 \sqrt{\mc{F}_{p,\sigma}(\rho_t)}} \mc{E}_{p,\mc{L}}\big(\Gamma_\si^{-1}(\rho_t)\big) \\
& = \frac{2}{p} \sqrt{\mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho_t))} \Big(1 - \frac{\sqrt{c}}{p \sqrt{\mc{F}_{p,\sigma}(\rho_t)}} \sqrt{\mc{E}_{p,\mc{L}}(\Gamma_\si^{-1}(\rho_t))}\big) \Big) \le 0\,,
\end{align*}
where the last inequality follows from $c \ge p/\alpha_p$ and the Beckner's inequality \eqref{eq_beck}. If $\rho_{t_0} = \sigma$ for some $t_0$, then $\rho_t = \sigma$ for $t \ge t_0$ and hence $\frac{d^+}{d t} h(t) = 0$ for $t \ge t_0$.
\end{proof}
\begin{proposition} \label{propb}
If the transport cost inequality \eqref{ineq:tc} holds with constant $c$, then the Poincar\'{e} inequality \eqref{ineq_pi} holds with $f = \varphi_p$ and constant $\lambda \ge 2/c$.
\end{proposition}
\begin{proof}
We consider $X \in \mc{B}_{sa}(\mc{H})$ with $\tr(\sigma X) = 0$, and define $\rho_\varepsilon = \Gamma_\si({\bf 1} + \varepsilon X)$. Recall Theorem \ref{thm:main_wasser} and let
$(\gamma_\varepsilon, {\bf B}_\varepsilon)$ be the minimizer to \eqref{def:wp4} for $W_{2,p}(\rho_\varepsilon,\sigma)$. Then, note from \eqref{eq:chi_heisen} that
\begin{align} \label{auxeqq_tctopi}
\norm{X}_{\sigma,\varphi_p}^2 & = \frac{1}{\varepsilon^2} \Big\langle \Omega_\sigma^{\kappa_{1/p}}(\rho_\varepsilon - \sigma), \int_0^1 - {\rm div} {\bf B}_\varepsilon(s) ds \Big\rangle \notag \\
& \le \frac{1}{\varepsilon^2} \Big( \int_0^1 \norm{\nabla \Omega_\sigma^{\kappa_{1/p}}(\rho_\varepsilon - \sigma)}^2_{p,\gamma_\varepsilon(s)} ds \Big)^{1/2} \Big( \int_0^1 \norm{{\bf B}_\varepsilon(s)}^2_{-1,p,\gamma_\varepsilon(s)} ds \Big)^{1/2} \notag \\
& \le \frac{1}{\varepsilon^2} \Big( \int_0^1 \norm{\nabla \Omega_\sigma^{\kappa_{1/p}}(\rho_\varepsilon - \sigma)}^2_{p,\gamma_\varepsilon(s)} ds \Big)^{1/2} W_{2,p}(\rho_\varepsilon,\sigma)\,,
\end{align}
by using the continuity equation in the first line, and Cauchy's inequality in the second line. Applying \eqref{ineq:tc} with the expansion \eqref{eq:asy_divi}, by definition \eqref{def:quanpdivi} of $\mc{F}_{p,\sigma}(\rho)$ and the relation \eqref{eq:chi_heisen},
we find
\begin{align} \label{auxeq:wasserexp}
\frac{1}{\varepsilon} W_{2,p}(\rho_\varepsilon,\sigma) \le \sqrt{ \frac{c}{2} \norm{X}_{\sigma,\varphi_p}^2 + O(\varepsilon)}\,.
\end{align}
A direct computation with \eqref{eq:kernel_iden_2} and Lemma \ref{lem:key_rela} gives
\begin{align} \label{newlabel_aux}
\partial_j \Omega_\sigma^{\kappa_{1/p}}(\rho_\varepsilon - \sigma) & = \varepsilon \partial_j \Gamma_\si^{-1}\varphi_p(\Delta_\sigma) R_\sigma \Gamma_\si^{-1} \Gamma_\si X \notag \\
& = \varepsilon \partial_j [\sigma]_{p,0}^{-1} \Gamma_\si X \notag\\
& = \varepsilon [\sigma]_{p,\omega_j}^{-1} \Gamma_\si \partial_j X\,.
\end{align}
By the proof of Lemma \ref{lem:exten_bry}, $\norm{\rho_\varepsilon -\sigma}_1 \to 0$ as $\varepsilon \to 0$ implies $W_{2,p}(\rho_\varepsilon,\sigma) \to 0$, which further yields $W_{2,p}(\gamma_\varepsilon(t),\sigma) = |1 - t| W_{2,p}(\rho_\varepsilon,\sigma) \to 0$, as $\varepsilon \to 0$, for all $t \in (0,1)$. Moreover, using Lemma \ref{lem:lower_bound}, we have $\norm{\gamma_\varepsilon(t)-\sigma}_1 \to 0$, as $\varepsilon \to 0$, for $t \in (0,1)$.
It follows from the dominated convergence theorem that
\begin{align*}
\frac{1}{\varepsilon^2} \int_0^1 \norm{\nabla \Omega_\sigma^{\kappa_{1/p}}(\rho_\varepsilon - \sigma)}^2_{p,\gamma_\varepsilon(s)} ds & \overset{\eqref{newlabel_aux}}{=} \sum_{j = 1}^J \int_0^1 \langle [\sigma]_{p,\omega_j}^{-1} \Gamma_\si \partial_j X, [\gamma_\varepsilon(s)]_{p,\omega_j} [\sigma]_{p,\omega_j}^{-1} \Gamma_\si \partial_j X \rangle \\
& \to \sum_{j = 1}^J \int_0^1 \langle \Gamma_\si \partial_j X, [\sigma]_{p,\omega_j}^{-1} \Gamma_\si \partial_j X \rangle \overset{\eqref{auxeqq_rela_0}}{=} - \langle X, \mc{L} X \rangle_{\sigma,\varphi_p}\,, \quad \text{as}\ \varepsilon \to 0\,.
\end{align*}
Combining the above formula with \eqref{auxeqq_tctopi} and \eqref{auxeq:wasserexp}, we conclude
\begin{equation*}
\norm{X}_{\sigma,\varphi_p}^2 \le - \frac{c}{2} \langle X, \mc{L} X \rangle_{\sigma,\varphi_p}\,. \qedhere
\end{equation*}
\end{proof}
We have seen that the entropic Ricci curvature lower bound \eqref{eq:ricci_hess} can imply a sequence of quantum functional inequalities. A natural and important following-up question is how to estimate the lower bound $\kappa$ for the Ricci curvature. Following closely the arguments in \cite[Theorem 10.6]{carlen2020non}, we can explicitly estimate the Ricci curvature lower bound for the depolarizing semigroup by definition \eqref{eq:ricci_hess}.
\begin{proposition} \label{prop:ricci_depol}
Let $\mc{L}_{depol}$ be the generator \eqref{def:depol} for the depolarizing semigroup with $\gamma > 0$ and the invariant state $\sigma = {\bf 1}/d$. Then the Ricci curvature of $\mc{L}_{depol}$ is bounded below by $\gamma p/2$.
\end{proposition}
\begin{proof}
Note from the definition of $\mc{L}_{depol}$ that $\partial_j \mc{L}_{depol} = - \gamma \partial_j$.
Recalling \eqref{eq:hess_of_f}, and we compute
\begin{align*}
-\langle \mc{L}_{depol} U, \mf{D}_{p,\sigma} U \rangle = \gamma d^{1-p} \sum_{j = 1}^J \langle \partial_j U, \theta_p(\rho,\rho) \partial_j U\rangle\,.
\end{align*}
By definition \eqref{def:kgeode}, we can also calculate
\begin{align*}
\sum_{j = 1}^J \big\langle \partial_j U, \mc{K}_{\rho,\mc{L}_{depol}^\dag(\rho)}^{(1),j} \left[ \partial_j U \right] \big \rangle & = \gamma d^{1-p} \sum_{j =1}^J \big\langle \partial_j U,
(\delta_1 \theta_p)\big((\rho,\rho), \rho\big)\big[ \frac{{\bf 1}}{d} - \rho, \partial_j U \big] \big\rangle \\
& = \gamma d^{1-p} \sum_{j =1}^J \big\langle \partial_j U, L_{\frac{{\bf 1}}{d} - \rho} (\partial_1 \theta_p)(\rho,\rho)[\partial_j U] \big\rangle\,,
\end{align*}
since $\mc{L}^\dag_{depol} = \gamma\big(\frac{{\bf 1}}{d} - \rho \big)$. Similarly, we have
\begin{align*}
\sum_{j = 1}^J \big\langle \partial_j U, \mc{K}_{\rho,\mc{L}_{depol}^\dag(\rho)}^{(2),j} \left[ \partial_j U \right] \big \rangle
= \gamma d^{1-p} \sum_{j =1}^J \big\langle \partial_j U, R_{\frac{{\bf 1}}{d} - \rho} (\partial_2 \theta_p)(\rho,\rho)[\partial_j U] \big\rangle\,.
\end{align*}
It follows that
\begin{align*}
\hess \mc{F}_{p,\sigma}(\rho)[U,U] &= \frac{1}{2} \sum_{j = 1}^J \Big\langle \partial_j U, \big(\mc{K}_{\rho,\mc{L}_{depol}^\dag(\rho)}^{(1),j} + \mc{K}_{\rho,\mc{L}_{depol}^\dag(\rho)}^{(1),j}\big) \left[ \partial_j U \right] \Big \rangle - \left\langle U, \mc{L}_{depol}^\dag(\mf{D}_{p,\rho} U) \right \rangle \\
& = \gamma d^{1-p} \sum_{j =1}^J \Big\langle \partial_j U, \big(\frac{1}{2d}\partial_1 \theta_p + \frac{1}{2d}\partial_2 \theta_p + \frac{p}{2}\theta_p \big) (\rho,\rho)[\partial_j U] \Big\rangle \\
& \ge d^{1-p} \frac{p\gamma }{2} \sum_{j =1}^J \Big\langle \partial_j U, \theta_p (\rho,\rho)[\partial_j U] \Big\rangle = \frac{p\gamma}{2} \langle U, \mf{D}_{p,\rho} U\rangle\,,
\end{align*}
where the second line is from $ x \partial_x \theta_p(x,y) + y \partial_y \theta_p(x,y) = (2-p) \theta_p(x,y)$; the third inequality follows from $ \partial_x \theta_p(x,y) + \partial_y \theta_p(x,y) \ge 0$ by the concavity of $x^{p-1}$.
\end{proof}
However, similarly to the case of estimating the functional inequality constant, there are very few examples where the explicit expressions of the Ricci curvature lower bounds can be obtained. To avoid the complicated computation and estimation based on the definition \eqref{eq:ricci_hess}, Carlen and Maas \cite{carlen2017gradient} consider the following intertwining property of the quantum Markov
semigroup: for some $\kappa \in \mathbb{R}$ and all $j$,
\begin{align} \label{eq:intertwin}
\partial_j \mc{P}_t = e^{- \kappa t} \mc{P}_t \partial_j\,,
\end{align}
which can be verified for many interesting cases, e.g., Fermi and Bose Ornstein-Uhlenbeck semigroups \cite[Section 6]{carlen2017gradient}. They showed that under the condition \eqref{eq:intertwin}, the Ricci curvature of the QMS $\mc{P}_t$ associated with the Umegaki's relative entropy is bounded from below by $\kappa$. The key step in their argument is the monotonicity of the action functional:
\begin{align} \label{eq:monotone}
\langle \mc{P}^\dag_t A, [\mc{P}^\dag_t\rho]_{\omega}^{-1} \mc{P}^\dag_t A\rangle \le \langle A, [ \rho]_{\omega}^{-1} A\rangle\,,
\end{align}
where $\mc{P}_t$ is the primitive QMS satisfying $\sigma$-DBC. To extend their approach to our case, we need to show that a similar monotonicity result as \eqref{eq:monotone} holds for $ \langle A, [\rho]_{p,\omega}^{-1} A\rangle$. It is a nontrivial task, since $\langle A, [\rho]_{p,\omega}^{-1} A\rangle$ is not $1$-homogeneous so that the contractivity of $\langle A, [\rho]_{p,\omega}^{-1} A\rangle$ under $\mc{P}_t^{\scriptscriptstyle \dag}$ can not be implied from its joint convexity \cite{lesniewski1999monotone}. For the symmetric QMS, $\mc{P}_t = \mc{P}_t^{\scriptscriptstyle \dag}$ is the unital quantum channel for each $t$. Note that \cite[Theorem 5.1]{zhang2021some} has shown that
\begin{equation} \label{auxeq_p-nomo}
\langle \Phi(A), [\Phi(\rho)]_{p,0}^{-1} \Phi(A)\rangle \le \langle A, [ \rho]_{p,0}^{-1} A\rangle\,, \quad \forall \rho \in \mathcal{D}_+(\mc{H})\,, A \in \mathcal{B}(\mc{H})\,,
\end{equation}
holds for any unital quantum channel $\Phi$. Thus, in this case, it is straightforward to conclude as in \cite[Theorem 10.9]{carlen2020non} that if the primitive symmetric QMS $\mc{P}_t$ satisfies
the property \eqref{eq:intertwin}, then its Ricci curvature associated with $\mc{F}_{p,\sigma}$ has a lower bound $\kappa$. It is known \cite{carlen2017gradient,carlen2020non} that the infinite temperature Fermi Ornstein-Uhlenbeck semigroup is symmetric and satisfies \eqref{eq:intertwin} with $\kappa = 1$, which readily gives that it has Ricci curvature lower bound $1$ and then the quantum Beckner's inequality holds with $\alpha_p \ge p/2$ by Corollary \ref{coro:poricci_beck}. But it seems not easy to extend the monotonicity result \eqref{auxeq_p-nomo} beyond the symmetric regime, namely, to show the monotonicity $\langle A, [\rho]_{p,\omega}^{-1} A\rangle$ under quantum channels with $\sigma$-DBC. We choose to investigate it in the future.
\section{Conclusions and discussion}
In this work, we have introduced the families of quantum Beckner's inequalities \eqref{ineq_becp} and \eqref{ineq_dbecq} that interpolate between the Sobolev-type and Poincar\'{e} inequalities. The basic properties of Beckner's inequalities, e.g., the monotonicity, the uniform
positivity, and the stability of the optimal constants, have been investigated in detail. We have also discussed their relations with other known quantum functional inequalities and applied Beckner's inequalities \eqref{ineq_becp} to estimating the mixing time and deriving the moment estimates. Furthermore, we have provided a quantum optimal transport framework for analyzing the Beckner's inequalities. In doing so, we have defined a new class of quantum transport distances $W_{2,p}$ such that the QMS with $\sigma$-DBC is the gradient flow of the quantum $p$-divergence. The main properties of the metric space $(\mathcal{D}(\mc{H}),W_{2,p})$ have been analyzed.
We have then introduced the associated entropic Ricci curvature and showed that it could yield a number of implications between \eqref{ineq_becp}, \eqref{ineq_pi}, an HWI-type inequality, and a transport cost inequality.
We briefly discuss some generalizations and applications of our results and methods. First, as mentioned in the introduction, the tensorization property for quantum functional inequalities is much more subtle than the classical case. Some tensorization-type results have been obtained for the quantum MLSI and LSI in the case of the depolarizing semigroups, e.g., \cite[Lemma 25]{kastoryano2013quantum} and \cite[Theorems 19 and 21]{beigi2020quantum}. By Lemma \ref{lem:aux_func} and Corollary \ref{coro:pitobeck}, these results can also provide dimension-independent lower bounds for the quantum Beckner constants (in certain scenarios). Second, thanks to Lemma \ref{lem:two-sided}, we can estimate the strong data processing inequality constant for the quantum $p$-divergence $\mc{F}_{p,\sigma}$ in terms of the $\chi^2_{\kappa_{1/p}}$ contraction coefficient, in the sense of \cite[Theorem 4.1]{gao2021complete}. We can also discuss the approximate tensorization property of $\mc{F}_{p,\sigma}$ similarly to \cite[Theorem 5.1]{gao2021complete} and the stability of the data processing inequality for the divergence $\mc{F}_{p,\sigma}$ similarly to \cite[Proposition 5.1]{junge2019stability}.
Third, in view of \cite{wirth2021curvature}, it is straightforward to consider the curvature-dimension conditions for quantum systems and investigate the finite-dimension version of the quantum Beckner's inequality. The details and refinements of these results would be worth being reported elsewhere.
We conclude with some interesting open questions. First, although we have obtained several generic results for the quantum Beckner constant, it is also desirable to establish more quantitative estimates for some concrete, physically meaningful models, for instance, the quantum spin system \cite{capel2020modified} and the quantum Markov semigroups constructed from the classical ones via the group transference \cite{gao2020fisher}. Second, the Bakry-\'{E}mery method has been well explored in the diffusion and discrete cases for proving Beckner's inequalities \cite{jungel2017discrete,weber2021entropy,gentil2021family}. However, the quantum extension of such an approach is still unknown. We are also curious about whether there exists some connections between Beckner's inequalities and the hypercontractivity of QMS, considering the existing works \cite{beckner1989generalized,arnold2007interpolation} and
the natural relations between the Beckner's inequality and the noncommutative $p$-norm $\norm{\cdot}_{p,\sigma}$. Third, in this work we have only considered the quantum Beckner's inequality for the primitive QMS. It would be important and useful to extend it to the non-primitive setting. The study of non-primitive quantum functional inequalities is initiated by Bardet \cite{bardet2017estimating} and has become an active research topic in recent years. It is known from \cite{frigerio1982long,gao2021complete} that for a general QMS, we have $E_\mc{F} \mc{P}_t = \mc{P}_t E_\mc{F} = E_\mc{F}$ and
\begin{align*}
\lim_{t \to \infty} \mc{P}_t (X - E_{\mc{F}}(X)) = \lim_{t \to \infty} \mc{P}_t X - E_{\mc{F}}(X) = 0\,,
\end{align*}
where $E_\mc{F}$ is the conditional expectation to the fixed-point algebra $\mc{F} := \left\{X \in \mathcal{B}(\mc{H});\ \forall t \ge 0,\, \mc{P}_t(X) = X \right\}$. One may attempt to define the non-primitive Beckner's inequality as
\begin{equation*}
\alpha_p \Big(\norm{X}^p_{p,\sigma} - \norm{E_\mc{F}(X)}_{p,\sigma}^p \Big) \le (p-1) \mc{E}_{p,\mc{L}}(X)\,, \quad \forall X \ge 0\,,
\end{equation*}
where $\sigma \in \mc{F}$ is a reference invariant state.
However, to make it well-defined, we need to show that the optimal constant $\alpha_p(\mc{L})$ is independent of the choice of $\sigma$, or we consider a particular state $\sigma_{\tr} = E_{\mc{F}}^\dag(\frac{{\bf 1}}{d})$ as in \cite{bardet2017estimating}. Alternatively, we may define the inequality by \eqref{eq_beck} with $\sigma = E_\mc{F}^\dag(\rho)$, i.e., consider the divergence $\mc{F}_{p,E_\mc{F}^\dag(\rho)}(\rho)$.
It is unclear which one is the most appropriate definition. It would be also interesting to further consider the complete Beckner's inequality and its properties in the spirit of \cite{gao2020fisher,brannan2021complete,brannan2022complete,gao2021complete}.
\begin{appendix}
\section{Some additional preliminaries}
\subsection{Quantum \texorpdfstring{$\chi^2$}--divergence} \label{app:quantum_diver}
In this appendix, we briefly recall the quantum $\chi^2$-divergences introduced in \cite{temme2010chi}. We consider the following class of functions:
\begin{align*}
\mc{K} = \{\kappa: (0,\infty) \to (0,\infty)\,;\ \kappa \ \text{is operator convex},\ x\kappa(x) = \kappa(x^{-1}),\ \kappa(1) = 1\}.
\end{align*}
By \cite[Proposition 2.1]{hiai2016contraction}, for $\kappa \in \mc{K}$, there exists a unique probability measure $m$ on $[0,1]$ such that
\begin{align} \label{eq:inte_rep}
\kappa(x) = \int_0^1 \left(\frac{1}{x + s} + \frac{1}{sx + 1}\right) \frac{1 + s}{2} dm\,,
\end{align}
which implies
\begin{align} \label{eq:compare_function}
\frac{2}{1 + x} \le \kappa(x) \le \frac{1+x}{2 x}\,,\quad \forall \kappa \in \mc{K}\,.
\end{align}
We now define the operator $\Omega_\sigma^\kappa$ for $\kappa \in \mc{K}$ and $\sigma > 0$:
\begin{align} \label{def:omega}
\Omega_\sigma^\kappa = R_\sigma^{-1} \kappa(\Delta_\sigma):\ \mathcal{B}(\mc{H}) \to \mathcal{B}(\mc{H})\,.
\end{align}
Recalling $J_\sigma^f$ in \eqref{def:operator_kernel}, clearly there holds
\begin{align} \label{aux_rela}
(\Omega_\sigma^\kappa)^{-1} = J_\sigma^{1/\kappa}\,.
\end{align}
\begin{definition} \label{def:quantumchi}
The quantum $\chi_\kappa^2$-divergence for $\rho, \sigma \in \mathcal{D}(\mc{H})$ is defined by
\begin{align*}
\chi_{\kappa}^2(\rho,\sigma) = \langle \rho - \sigma, \Omega_\sigma^\kappa(\rho-\sigma)\rangle\,,
\end{align*}
when ${\rm supp}(\rho) \subset {\rm supp}(\sigma)$; otherwise $\chi_\kappa^2(\rho,\sigma) = \infty$.
\end{definition}
For the purposes of this work, the following family of power difference means is of particular interest \cite{hiai1999means}:
\begin{align}\label{def:pdiff}
\kappa_\alpha = \frac{\alpha}{\alpha-1} \frac{x^{\alpha - 1} - 1}{x^{\alpha} - 1}\,, \quad \alpha \in [-1,2]\,.
\end{align}
In fact, the kernel function of the operator $J_\sigma^{1/k}$ in \eqref{aux_rela} is given by
\begin{align*}
M_\alpha = y \kappa_\alpha^{-1}(x/y) = \frac{\alpha - 1}{\alpha}\frac{x^\alpha - y^\alpha}{x^{\alpha - 1} - y^{\alpha - 1}}\,,
\end{align*}
which is the so-called A-L-G interpolation mean since $M_\alpha$, for $\alpha = -1$, $\alpha = 1/2$, $\alpha = 1$, and $\alpha = 2$, gives the harmonic mean, the geometric mean, the logarithmic mean, and the arithmetic mean, respectively.
\subsection{Noncommutative calculus} \label{app:NCC}
In this appendix, following \cite{carlen2020non,bardet2017estimating}, we
review some fundamentals about noncommutative calculus associated with the derivations $\partial_j$. Let operators $A,B \in \mc{B}_{sa}(\mc{H})$ admit the spectral decompositions:
\begin{align*}
A = \sum_{i = 1}^{d}\lambda_i A_i\,,\quad B = \sum_{k=1}^{d} \mu_k B_k\,,
\end{align*}
where $\lambda_i$ and $\mu_k$ are eigenvalues of $A$ and $B$, respectively; $A_i$ and $B_i$ are the associated rank-one spectral projections. For a function $f \in C(I \times I)$ with $I$ being a compact interval containing the spectra of $A$ and $B$, we define the Schur multiplier (double sum operator) by \cite{birman2003double,potapov2011operator,de2004differentiation}
\begin{align} \label{def:doutble_op_sum}
f(A,B) = \sum_{i,k = 1}^{d} f(\lambda_i,\mu_k) L_{A_i}R_{B_k}\,,
\end{align}
where $C(I \times I)$ is the Banach space of complex-valued continuous functions on $I \times I$. It was observed in \cite{bardet2017estimating} that given $A,B \in \mc{B}_{sa}(\mc{H})$, $f(A,B)$ is $*$-representation between $C(I \times I)$ and $\mc{B}(\mc{B}(\mc{H}))$. Indeed, we have the following lemma from \cite[Lemma 4.1]{bardet2017estimating} and \cite[Lemma 6.6]{carlen2020non}.
\begin{lemma} \label{lem:double_inner}
Let $A,B \in \mc{B}_{sa}(\mc{H})$ and the compact interval $I$ contain the spectra of $A$ and $B$. It holds that
\begin{enumerate}[1.]
\item $f(A,B)g(A,B) = (fg)(A,B)$ for $f,g \in C(I \times I)$.
\item If $f \in C(I \times I)$ is non-negative, then $f(A,B)$ is a positive semidefinite operator on $B(\mc{H})$ with respect to the inner product $\langle \cdot, \cdot \rangle$. It follows that if $f$ is strictly positive, the sequilinear form $\langle \cdot, f(A,B)(\cdot)\rangle$ defines an inner product on $\mc{B}(\mc{H})$.
\end{enumerate}
\end{lemma}
In this work, we mainly consider the case where $f$ is the divided difference of some differentiable function $\varphi$ on $I$:
\begin{align} \label{def:divi_diff}
\varphi^{[1]}(\lambda,\mu) = \left\{
\begin{aligned}
& \frac{\varphi(\lambda)-\varphi(\mu)}{\lambda-\mu}\,, \quad & \lambda \neq \mu\,, \\
& \varphi'(\lambda)\,,\quad & \lambda = \mu\,,
\end{aligned}
\right.
\end{align}
which is closely related to the chain rule for the derivation $\partial_j$ (cf.\cite[Lemma 4.2]{bardet2017estimating} and \cite[Proposition 6.2]{carlen2020non}).
\begin{lemma} \label{lem:chain_rule}
Under the same assumption as in Lemma \ref{lem:double_inner}, for a continuously differentiable function $f \in C(I)$ such that $f(0) = 0$, we have, for any $V \in \mathcal{B}(\mc{H})$,
\begin{align} \label{eq:chain_rule}
V f(B) - f(A) V = f^{[1]}(A,B)(V B - A V)\,.
\end{align}
Moreover, in the case of $A = B$, the identity \eqref{eq:chain_rule} holds
for any $f: I \to \mathbb{C}$ (that may not be differentiable).
\end{lemma}
It then follows from the above lemma that for a differentiable curve $A(t): (a,b) \to \mc{B}(\mc{H})$, we have
\begin{align}\label{eq:chain_curve}
& \frac{d}{d t} f(A(t)) = f^{[1]}(A(t),A(t))(A'(t))\,.
\end{align}
We also need a multiple operator version of \eqref{eq:chain_curve}. We recall that for a differentiable function $\varphi: \mathbb{R}^n \to \mathbb{C}$, the partial divided difference $\delta_j \varphi: \mathbb{R}^{n+1} \to \mathbb{C} $ with respect to the variable
$x_j$ is defined by
\begin{align} \label{def_1:divi_diff}
(\delta_{j} \varphi)(x_1,\cdots,x_{j-1},(\lambda,\mu),x_{j+1},\cdots,x_n)= \left(\varphi(x_1,\cdots,x_{j-1},\cdot,x_{j+1},\cdots,x_n)\right)^{[1]}(\lambda,\mu).
\end{align}
Let $A^{(k)}$, $k = 1,\ldots,n$, be self-adjoint operators with the spectral decompositions: $A_i = \sum_{i} \lambda_{i}^{(k)}A_{i}^{(k)}$, where $\lambda_i^{(k)}$ are eigenvalues and $A_{i}^{(k)}$ are the associated rank-one spectral projections. For a function $\varphi: I \times \cdots \times I \to \mathbb{C}$ with the interval $I$ containing the spectra of $A^{(i)}$,
the multiple operator sum is defined as:
\begin{align} \label{def_2:multiple}
\varphi(A_1,\cdots,A_n) = \sum_{i_1,\cdots,i_n = 1}^{d} \varphi(\lambda^{(1)}_{i_1},\cdots,\lambda^{(n)}_{i_n})A_{i_1}^{(1)}\otimes\cdots \otimes A_{i_n}^{(n)}.
\end{align}
The following chain rule from \cite[Proposition 6.8]{carlen2020non} shall be useful in the expression of the geodesic equations for the generalized quantum transport distance.
\begin{lemma} \label{lem:high_order}
Let the curves $A_t, B_t: (a,b) \to \mc{B}_{sa}(\mc{H})$ be differentiable, and let $\varphi: I \times I \to \mathbb{C}$ be differentiable with $I$ containing the spectra of $A_t$ and $B_t$ for all $t \in (a,b)$. Then there holds
\begin{align*}
\partial_t \varphi(A_t,B_t)(\cdot) = (\delta_1 \varphi) ((A_t,A_t),B_t)[\partial_t A_t, \cdot] + (\delta_2 \varphi)(A_t,(B_t,B_t))[\cdot, \partial_t B_t].
\end{align*}
\end{lemma}
\section{Additional Proofs} \label{app:proof}
\begin{proof}[Proof of Lemma \ref{lem:exten_bry}]
Consider $\rho_0,\rho_1 \in \mathcal{D}(\mc{H})$ and let $\{\rho_0^n\}, \{\rho_1^n\}$ be any sequences in $\mathcal{D}_+(\mc{H})$ such that $\norm{\rho_i^n - \rho_i}_2 \to 0$ as $n \to \infty$ for $i = 0,1$. It suffices to show that $W_{2,p}(\rho^n_0,\rho^n_1)$ is a Cauchy sequence. For this, by the triangle inequality, we only need to show $W_{2,p}(\rho_i^n, \rho_i^m) \to 0$ as $n, m \to \infty$ for $i = 0,1$. For $\varepsilon \in (0,1)$,
we define $\w{\rho}_0 = (1- \varepsilon) \rho_0 + \varepsilon {\bf 1}$, and the linear interpolation $\gamma_n(s):= (1 - s) \rho_0^n + s \w{\rho}_0$ which satisfies $\gamma_n(s) \ge s \varepsilon {\bf 1}$. Then it is easy to see
\begin{align*}
e^{\pm \omega_j/2p}\Gamma_\si^{-1/\h{p}}(\gamma_n(s)) \ge \inf_j \{e^{-|\omega_j|/2p}\} s \varepsilon \sigma^{-1/\h{p}}\,,
\end{align*}
by which, Lemma \ref{lem:mono_norm} implies that $[\gamma_n(s)]_{p,\omega}^{-1} \le C (s \varepsilon)^{p-2} {\rm id}_{\mc{H}}$ holds for some constant $C > 0$, and hence that
\begin{align*}
\mf{D}_{p,\gamma_n(s)}^{-1} \le C (s \varepsilon)^{p-2} \norm{(-{\rm div} \circ \nabla(\cdot))^{-1}} {\rm id}_{\mc{H}}\,.
\end{align*}
Recalling the expression \eqref{def:wp3}, we obtain
\begin{align} \label{auxeq:bry}
W_{2,p}(\rho_0^n, \w{\rho}_0) &\le \int_0^1 \left\langle \dot{\gamma}_n(s), \mf{D}_{p,\gamma_n(s)}^{-1} \dot{\gamma}_n(s) \right\rangle^{1/2}\, ds \notag \\
& \le C (s \varepsilon)^{p-2} \norm{(-{\rm div} \circ \nabla(\cdot))^{-1}} \int_0^1 \norm{ \dot{\gamma}_n(s)}_2\, ds\,.
\end{align}
Note the estimate:
\begin{align*}
\norm{ \dot{\gamma}_n(s)}_2 = \norm{\w{\rho}_0 - \rho^n_0}_2 \le \varepsilon \norm{\rho_0 - {\bf 1}}_2 + \norm{\rho_0 - \rho_0^n}_2\,,
\end{align*}
which, by \eqref{auxeq:bry}, gives
\begin{align*}
\limsup_{n \to \infty} W_{2,p}(\rho_0^n, \w{\rho}_0) \le C (s \varepsilon)^{p-1} \norm{(-{\rm div} \circ \nabla(\cdot))^{-1}} \norm{\rho_0 - {\bf 1}}_2\,.
\end{align*}
Since $p \in (1,2]$ and $\varepsilon$ is arbitrary, the above estimate implies $ \lim_{n,m \to \infty} W_{2,p}(\rho_0^n, \rho_0^m) = 0$, by the triangle inequality.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:main_wasser}]
We will first show the existence of the minimizer of \eqref{def:wp4} by a direct method. Let $\left\{(\gamma^{(n)},{\bf B}^{(n)})\right\}$ be the minimizing sequence such that
$
\sup_n \int_0^1 \big\lVert{\bf B}^{(n)}\big\rVert^2_{-1,p,\gamma^{(n)}}\ ds < + \infty\,.
$
We claim that there exists constant $C_j$, depending on $\sigma \in \mathcal{D}_+(\mc{H})$ and $\omega_j \in \mathbb{R}$, such that
\begin{align} \label{claim_one}
\langle X, [\rho]_{p,\omega_j} X\rangle \le C_j \norm{X}_2^2\,,\quad \forall \rho \in \mathcal{D}(\mc{H})\,.
\end{align}
To show this, by the representation \eqref{eq:repre_rpw} of $[\rho]_{p,\omega}$ with notations from Proposition \ref{prop:repkernel}, we first have
\begin{align*}
\langle X, [\rho]_{p,\omega} X \rangle \le (p-1) \bigg( \sup_{i,k} \theta_p\left(e^{\omega/2p}\lambda_{k,p}, e^{-\omega/2p}\lambda_{i,p}\right)
\bigg) \norm{\Gamma_\si^{1/\h{p}}(X)}_2^2\,.
\end{align*}
We also note that there exists a closed interval $I$ containing $e^{\pm \omega/2p} \lambda_{k,p}$ for all $\rho \in \mathcal{D}(\mc{H})$, and that the function $\theta_p(x,y)$ is bounded for $x,y \in I$. Then the claim \eqref{claim_one} readily follows. Since ${\bf B}^{(n)} \in L^1\left([0,1],\mathcal{B}(\mc{H})^J\right)$, we
can consider the $\mathcal{B}(\mc{H})^J$-valued measure $\mathsf{B}^{(n)}(ds) := {\bf B}^{(n)}(s) ds$. Then, for every Borel set $E \subset [0,1]$, we estimate
\begin{align} \label{eq:estimate_minimizing}
|\mathsf{B}^{(n)}|(E) & \le \int_E \bigg(\sum_{j = 1}^J \norm{B_j^{(n)}(s)}_2^2 \bigg)^{1/2} ds \notag \\
& = \int_E \bigg(\sum_{j = 1}^J \left\langle [\gamma^{(n)}]_{p,\omega_j}^{-1/2} B_j^{(n)}, [\gamma^{(n)}]_{p,\omega_j} [\gamma^{(n)}]_{p,\omega_j}^{-1/2} B_j^{(n)} \right\rangle \bigg)^{1/2} ds \notag \\
& \le C \mathscr{L}(E)^{1/2} \bigg( \int_E \sum_{j = 1}^J \left\langle B_j^{(n)}, [\gamma^{(n)}]^{-1}_{p,\omega_j} B_j^{(n)} \right\rangle ds \bigg)^{1/2} \notag \\
& \le C \mathscr{L}(E)^{1/2} \bigg(\int_0^1 \big\lVert{\bf B}^{(n)}\big\rVert^2_{-1,p,\gamma^{(n)}} ds \bigg)^{1/2} < +\infty\,,
\end{align}
for $C := \max_j\{C_j^{1/2}\}$, where $\mathscr{L}$ is the Lebesgue measure on $\mathbb{R}$ and the third line is from H\"{o}lder's inequality and the estimate \eqref{claim_one}.
It follows that the total variations of the measures $\mathsf{B}^{(n)}$ are uniformly bounded in $n$. Hence there exists a subsequence, still denoted by $\mathsf{B}^{(n)}$, converging weakly* to a $\mathcal{B}(\mc{H})^J$-valued measure $\mathsf{B}_*$. Then, by \eqref{eq:estimate_minimizing}, we can obtain
\begin{equation*}
|\mathsf{B}_*|(E) \le \liminf_{n \to \infty} |\mathsf{B}^{(n)}|(E) \le C \mathscr{L}(E)^{1/2}\,,
\end{equation*}
for some constant $C$, which implies that $\mathsf{B}_* \ll \mathscr{L}$ and hence the R-N derivative ${\bf B}_*: = \frac{d \mathsf{B}_*}{d \mathscr{L}}$ exists. We next prove that $\gamma^{(n)}(s)$ converges pointwise to some $\gamma(s): [0,1] \to \mathcal{D}(\mc{H})$. For this, we note
\begin{equation*}
\gamma^{(n)}(t) - \gamma^{(n)}(0) = - \int_0^t {\rm div} {\bf B}^{(n)}(s)\ ds.
\end{equation*}
Then by the weak* convergence of $\mathsf{B}^{(n)}$ and $\gamma^{(n)}(0) = \rho_0$, we have the pointwise convergence of $\gamma^{(n)}(s)$ with the limit denoted by $\gamma_*(s)$.
Moreover, it is easy to check that $\gamma_*(s) \in C([0,1],\mathcal{D}(\mc{H})) $ and ${\bf B}_* \in L^1([0,1],\mathcal{B}(\mc{H})^J)$ satisfy the continuity equation. By dominated convergence theorem, we also have $\mc{B}(\mathcal{B}(\mc{H}))$-valued measure $[\gamma^{(n)}(s)]_{p,\omega_j} ds$ weakly* converge to $[\gamma_*(s)]_{p,\omega_j} ds$. Finally, noting the integral representation for $[\rho]^{-1}_{p,\omega}$ from \eqref{eq:inte_divi_p} and \eqref{def:kernel_multiplication}:
\begin{align*}
[\rho]^{-1}_{p,\omega}(\cdot)
& =
\frac{\sin((p-1) \pi)}{\pi} \Gamma_\si^{-1/\h{p}} \int_0^\infty s^{p - 2} g_s^{[1]} \left(e^{\omega/2p} \Gamma_\si^{-1/\h{p}}(\rho), e^{-\omega/2p} \Gamma_\si^{-1/\h{p}}(\rho) \right) \Gamma_\si^{-1/\h{p}}(\cdot)\ ds\,,
\end{align*}
we have
\small
\begin{align} \label{auxeq_final}
& \liminf_{n \to \infty} \int_0^1 \big\lVert{\bf B}^{(n)}\big\rVert^2_{-1,p,\gamma^{(n)}}\ ds
\notag \\
\ge & \frac{\sin((p-1)\pi)}{\pi} \sum_{j=1}^J \int_0^\infty t^{p-2} \liminf_{n \to \infty} \int_0^1 \left\langle \Gamma_\si^{-1/\h{p}}\big(B_j^{(n)}(s)\big) , g_t^{[1]} \left(e^{\omega/2p} \Gamma_\si^{-1/\h{p}}\big(\gamma^{(n)}(s)\big), e^{-\omega/2p} \Gamma_\si^{-1/\h{p}}\big(\gamma^{(n)}(s)\big) \right) \Gamma_\si^{-1/\h{p}}\big(B_j^{(n)}(s)\big) \right\rangle ds dt \notag \\
\ge & \frac{\sin((p-1)\pi)}{\pi} \sum_{j=1}^J \int_0^\infty t^{p-2} \int_0^1 \left\langle \Gamma_\si^{-1/\h{p}}\big(B_{*,j}(s)\big) , g_t^{[1]} \left(e^{\omega/2p} \Gamma_\si^{-1/\h{p}}\big(\gamma_*(s)\big), e^{-\omega/2p} \Gamma_\si^{-1/\h{p}}\big(\gamma_*(s)\big) \right) \Gamma_\si^{-1/\h{p}}\big(B_{*,j}(s)\big) \right\rangle ds dt \notag \\
\ge & \int_0^1 \big\lVert{\bf B}_*\big\rVert^2_{-1,p,\gamma_*} ds\,,
\end{align}
\normalsize
where in the first inequality we have used Fatou's lemma, and in the second inequality we have used Theorem 3.4.3 of \cite{buttazzo1989semicontinuity} on the lower-semicontinuity of integral functionals. The estimate \eqref{auxeq_final} directly implies that $(\gamma_*, {\bf B}_*)$ is a minimizer to \eqref{def:wp4}.
Recalling formulations \eqref{def:wp1} and \eqref{def:wp3}, by Jensen's inequality, we find
\begin{align*}
W_{2,p}(\rho_0,\rho_1) = \Big(\int_0^1 g_{p, \gamma_*(s)}\left(\dot{\gamma}_*(s),\dot{\gamma}_*(s) \right) ds \Big)^{1/2} = \int_0^1 g_{p, \gamma_*(s)}\left(\dot{\gamma}_*(s),\dot{\gamma}_*(s) \right)^{1/2} ds\,.
\end{align*}
It follows that $W_{2,p}(\rho_0,\rho_1)^2 = g_{p, \gamma_*(s)}\left(\dot{\gamma}_*(s),\dot{\gamma}_*(s) \right)$ for almost everywhere $s \in [0,1]$. Then, by definition of $W_{2,p}$ and a time scaling, it is easy to check that the property \eqref{eq:geo_mini} holds. Therefore, we have proved that $(\mc{D}(\mc{H}), W_{2,p})$ is a geodesic space. The completeness of the metric space $(\mc{D}(\mc{H}), W_{2,p})$ is a simple consequence of Lemma \ref{lem:lower_bound}. Indeed, let $\{\rho_n\}$ be a Cauchy sequence such that $W_{2,p}(\rho_n,\rho_m) \to 0$ as $n,m \to \infty$. Then, by Lemma \ref{lem:lower_bound}, it is also Cauchy in the complete metric space $(\mathcal{D}(\mc{H}),\norm{\cdot}_1)$, which implies that there exists $\rho_\infty \in \mathcal{D}(\mc{H})$ such that $\norm{\rho_n - \rho_\infty}_1 \to 0$ as $n \to \infty$. Similarly to the proof of Lemma \ref{lem:exten_bry}, by $\norm{\rho_n - \rho_\infty}_1 \to 0$, we can conclude $W_{2,p}(\rho_n,\rho_\infty) \to 0$ as $n \to \infty$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:lower_bound}]
For any $\delta > 0$ and $\rho_0, \rho_1 \in \mathcal{D}_+(\mc{H})$, by Theorem \ref{thm:main_wasser}, there exists the curve $(\gamma(s),{\bf B}(s))$, $s \in [0,1]$, satisfying $\dot{\gamma}(s) + {\rm div} {\bf B}(s) = 0$ with $\gamma(0) = \rho_0$ and $\gamma(1) = \rho_1$, such that
\begin{align*}
\int_0^1 \norm{{\bf B}(s)}^2_{-1, p, \gamma(s)} ds \le W_{2,p}(\rho_0,\rho_1)^2 + \delta\,.
\end{align*}
It then follows that, by Cauchy's inequality,
\begin{align} \label{auxeq_bound1}
\tr(X(\rho_1 - \rho_0)) = \tr \left(X \int_0^1 \dot{\gamma}(s) ds \right) \notag
& = \int_0^1 \langle \nabla X, {\bf B}(s) \rangle ds \notag \\
&\le \left(\int_0^1 \norm{\nabla X}_{p,\gamma(s)}^2 d s \right)^{1/2} \left( \int_0^1 \norm{{\bf B}(s) }_{-1,p,\gamma(s)}^2 d s \right)^{1/2} \notag \\
&\le \left(\int_0^1 \sum_j \langle \partial_j X, [\gamma(s)]_{p,\omega_j} \partial_j X \rangle d s \right)^{1/2} \left( W_{2,p}(\rho_0,\rho_1)^2 + \delta \right)^{1/2}.
\end{align}
To deal with the term $\langle \partial_j X, [\gamma(s)]_{p,\omega_j} \partial_j X \rangle$, we next estimate the kernel operator $[\rho]_{p,\omega}$ for $\rho \in \mc{D}_+$ and $\omega \in \mathbb{R}$. For this, recalling \eqref{eq:repre_rpw}:
\begin{align} \label{eq:kernel2}
[\rho]_{p,\omega}(\cdot) = (p-1)\sum_{i,k}\frac{e^{\omega/2p}\lambda_{k} - e^{-\omega/2p}\lambda_{i}}{\left(e^{\omega/2p}\lambda_{k}\right)^{p-1} - \left(e^{-\omega/2p}\lambda_{i}\right)^{p-1}} \Gamma_\si^{1/\h{p}} \left(E_{k} \Gamma_\si^{1/\h{p}}(\cdot) E_{i}\right),
\end{align}
where $\lambda_{i}$ and $E_{i}$ are eigenvalues and the associated eigen-projections of $\Gamma_\si^{-1/\h{p}}(\rho)$, respectively (we omit the subscript $p$ of $\lambda_i$ and $E_i$ for simplicity).
By the integral presentation \eqref{eq:inte_divi_p}, we can estimate
\begin{align} \label{eq:ker_estimate}
&\frac{1}{p-1} \frac{\left(e^{\omega/2p}\lambda_{k}\right)^{p-1} - \left( e^{-\omega/2p}\lambda_{i}\right)^{p-1}}{e^{\omega/2p}\lambda_{k} - e^{-\omega/2p}\lambda_{i}}
\notag \\
= & \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty s^{p-2}
\frac{\log (s + e^{\omega/2p} \lambda_k) - \log (s + e^{-\omega/2p}\lambda_i) }{e^{\omega/2p} \lambda_k - e^{-\omega/2p}\lambda_i}
\ ds \notag\\
\ge & \frac{\sin((p-1) \pi)}{\pi} \int_0^\infty s^{p-2} \frac{2}{2s + e^{\omega/2p} \lambda_k + e^{-\omega/2p}\lambda_i}\ ds \notag \\
= & \frac{\sin((p-1) \pi)}{\pi} \left(e^{\omega/2p}\lambda_k + e^{-\omega/2p} \lambda_i\right)^{p-2} \int_0^\infty s^{p-2} \frac{2}{1+ 2 s}\ ds\,,
\end{align}
where the third line is from the elementary inequality:
\begin{equation*}
\frac{x - y}{\log x - \log y} \le \frac{x + y}{2}\quad \text{for all}\ x, y > 0\,,
\end{equation*}
and the last line is by the change of variable. We define the constant:
\begin{equation}\label{def:cons_cp}
C_p := \frac{\sin((p - 1) \pi)}{\pi} \int_0^\infty s^{p-2} \frac{2}{1+ 2 s} \ d t\,.
\end{equation}
By \eqref{eq:kernel2} and \eqref{eq:ker_estimate}, there holds
\begin{align*}
\left\langle \partial_j X, [\rho]_{p,\omega}\partial_j X \right\rangle \le & C_p^{-1} \sum_{i,k} \left(e^{\omega/2p}\lambda_k + e^{-\omega/2p} \lambda_i\right)^{2-p} \left\langle \Gamma_\si^{1/\h{p}} (\partial_j X), \left(E_{k} \Gamma_\si^{1/\h{p}}(\partial_j X) E_{i}\right) \right\rangle \\
\le & C_p^{-1} \sum_{i,k} \left(\left(e^{\omega/2p}\lambda_k \right)^{2-p} + \left(e^{-\omega/2p} \lambda_i\right)^{2-p}\right) \left\langle \Gamma_\si^{1/\h{p}} (\partial_j X), \left(E_{k} \Gamma_\si^{1/\h{p}}(\partial_j X) E_{i}\right) \right\rangle \\
= & C_p^{-1} \left\langle \Gamma_\si^{1/\h{p}} (\partial_j X), \left(L_{\left(e^{\omega/2p}\Gamma_\si^{-1/\h{p}}(\rho)\right)^{2-p}} + R_{\left(e^{-\omega/2p} \Gamma_\si^{-1/\h{p}}(\rho) \right)^{2-p}}\right) \Gamma_\si^{1/\h{p}}(\partial_j X) \right\rangle \\
\le & C_p^{-1} \left\langle \Gamma_\si^{1/\h{p}} (\partial_j X), \left(L_{\left(e^{\omega/2p}\sigma^{-1/\h{p}}\right)^{2-p}} + R_{\left(e^{-\omega/2p} \sigma^{-1/\h{p}} \right)^{2-p}}\right) \Gamma_\si^{1/\h{p}}(\partial_j X) \right\rangle \\
\le & C_p^{-1} \left( e^{(2-p)\omega/2p} + e^{(p-2)\omega/2p} \right) \tr \left( \sigma^{(p-2)/\h{p}} \right) \norm{\Gamma_\si^{1/\h{p}}(\partial_j X)}_{\infty}^2 \\
\le & 4 C_p^{-1} \left(e^{(2-p)\omega/2p} + e^{(p-2)\omega/2p} \right) \tr \left( \sigma^{(p-2)/\h{p}} \right) \norm{\sigma}_\infty^{2/\h{p}}\norm{V_j}^2_\infty \norm{X}^2_\infty\,,
\end{align*}
where the second inequality is by $(x + y)^p \le x^p + y^p$ for $p \in (0,1)$ and $x, y > 0$; the third inequality is by $\rho \le {\bf 1}$ for all $\rho \in \mathcal{D}(\mc{H})$ and the operator monotonicity of $t^p$ for $0 \le p \le 1$; the fourth inequality is by
H\"{o}lder's inequality. Then we arrive at, by \eqref{auxeq_bound1},
\begin{align} \label{eq:est_lower}
\norm{\rho_1 - \rho_0}_1 & \le \left( 4 C_p^{-1} \tr \left( \sigma^{(p-2)/\h{p}} \right) \norm{\sigma}_\infty^{2/\h{p}} \sum_j \left(e^{(2-p)\omega_j/2p} + e^{(p-2)\omega_j/2p} \right) \norm{V_j}_\infty^2 \right)^{1/2} W_{2,p}(\rho_0,\rho_1)\,.
\end{align}
We finally prove the uniform boundedness of the prefactor in \eqref{eq:est_lower} for $p \in (1,2]$. It suffices to consider the constant $C_p$ \eqref{def:cons_cp}. By elementary calculation, we derive
\begin{align*}
\frac{2}{3}\left( \frac{1}{p - 1} + \frac{1}{2 - p} \right) \le \int_0^\infty \frac{2 s^{p-2} }{1+ 2 s} d s &= \int_0^1 \frac{2 s^{p-2}}{1+ 2 s} d s + \int_1^\infty \frac{2 s^{p-2}}{1+ 2 s} d s \le \frac{2}{p - 1} + \frac{1}{2 - p}\,,
\end{align*}
which immediately gives the following estimates:
\begin{align*}
\frac{2}{3} + O(p - 1) \le C_p \le 2 + O(p-1) \quad \text{as} \ p \to 1^+\,,
\end{align*}
and
\begin{align*}
\frac{2}{3} + O(p - 1) \le C_p \le 1 + O(p-2) \quad \text{as} \ p \to 2\,.
\end{align*}
The proof is complete.
\end{proof}
\end{appendix}
|
{
"timestamp": "2022-07-27T02:16:50",
"yymm": "2207",
"arxiv_id": "2207.06422",
"language": "en",
"url": "https://arxiv.org/abs/2207.06422"
}
|
\section{Introduction}
A complete understanding of symmetries in gauge theories, including gravity, is one of the key ingredients in the description of the quantum properties of these theories. Since even in the same community the word 'symmetry' can have various meanings, let us clarify its meaning for us. A global symmetry of a theory expresses that there exists a conserved charge of motion, i.e., a quantity conserved dynamically. Classically, this is often seen at the level of the classical action for the theory, for instance a $U(1)$ global symmetry for a complex scalar field. From a quantum standpoint, one introduces a charge operator that acts on the Hilbert space of the theory, and conserved quantities are then interpreted as quantum numbers labeling states.
Gauge symmetries on the other hand express mere redundancy of the description of a physical system. They express an underlying local reparameterization of the variables of the system, that leaves the latter invariant. The prime example for us is gravitational theories, where diffeomorphisms are local gauge symmetries. These gauge symmetries reduce the independent degrees of freedom of a given problem, that define the physical field space of the theory.
As a result, the space of all fields is only a presymplectic manifold, by which one means that the gauge symmetries correspond to zero modes of a presymplectic form.
Global and gauge symmetries are at the core of Noether theorems \cite{Noether1918}: the first states that a global symmetry implies the existence of a conserved current, while the second states that conserved currents for gauge symmetries are weakly vanishing, that is, they vanish on the equations of motion. Being conserved and weakly vanishing, a current associated to a gauge symmetry is at most a total derivative. Subtleties arise in the presence of boundaries, where this last statement opens the door to the possibility of having a non-trivial gauge symmetry current. This is the scenario we explore in the context of diffeomorphism invariant theories.
In the presence of a boundary --- or, more generally, a subregion of interest --- part of the gauge symmetries acquires a non-trivial action on the field space approaching the boundary. Among bulk gauge symmetries, one must then distinguish between those that are still trivial from those that are no longer trivial. The former are still pure gauge symmetries while the latter are termed asymptotic symmetries. They are not global symmetries of the full bulk field space, but nonetheless they are distinguished from the gauge symmetries that remain pure gauge even in the presence of a boundary thanks to their non-vanishing Noether charge, which is now a codimension-$2$ integral from the bulk perspective, as first observed in the seminal work of Regge and Teitelboim \cite{Regge:1974zd}. These codimension-$2$ charges are called surface charges or, more recently, corner charges. The term corner refers to any codimension-$2$ surface on which these charges may have support. Given that such charges are some of the most robust observables of a gauge theory, we regard corners as the building blocks of the \hlt{corner proposal} \cite{Donnelly:2016auv, Speranza:2017gxd,Geiller:2017whh,Freidel:2020xyx,Freidel:2020svx,Freidel:2020ayo,Donnelly:2020xgu,Ciambelli:2021vnn,Freidel:2021cjp,Ciambelli:2021nmv}, which is the starting point of this manuscript. We will return to it presently.
One of the main features of gauge theories is that they possess a holographic feature: since an asymptotic symmetry gives rise to corner charges, one can reinterpret it as a global symmetry of a theory supported on the codimension-$1$ boundary. This result is perfectly in line with the AdS/CFT correspondence \cite{Maldacena:1997re,Witten:1998qj}, where the bulk asymptotic symmetries become the global conformal symmetries of the boundary field theory, as also precursorily observed in AdS$_3$ by Brown and Henneaux \cite{Brown:1986nw}. It is by now more and more clear that gauge theories have in general this holographic nature, although we should emphasize that we do not imply that in any gauge theory the full bulk solution can be reconstructed from the boundary theory, but rather that the asymptotic field space and symmetries can always be reinterpreted in terms of a (not necessarily universal) boundary theory. Recent progress in this direction notably includes flat holography \cite{Arcioni:2003xx,Arcioni:2003td,deBoer:2003vf,Dappiaggi:2005ci,Barnich:2010eb}, celestial holography \cite{Strominger:2013jfa,Kapec:2016jld,Cheung:2016iub,Pasterski:2016qvg,Pasterski:2017kqt,Donnay:2020guq,Pate:2019lpp,Fotopoulos:2019vac,Donnay:2021wrk,Pasterski:2021raf}, Carrollian holography \cite{Bagchi:2010zz,Hartong:2015xda,Ciambelli:2018wre, Ciambelli:2018ojf,Campoleoni:2018ltl, Figueroa-OFarrill:2021sxz,Donnay:2022aba}, but also finite distance boundaries, such as null hypersurfaces \cite{Hopfmuller:2016scf,Hopfmuller:2018fni,Chandrasekaran:2018aop,Ciambelli:2019lap,Speranza:2019hkr,Adami:2021nnf,Chandrasekaran:2021hxc} and in particular black hole horizons \cite{Donnay:2015abr,Donnay:2016ejv,Donnay:2019jiz,Grumiller:2019fmp,Carlip:2019dbu}. In \cite{Ciambelli:2021vnn}, we found that the subset of diffeomorphisms in gravitational theories that can contribute to asymptotic symmetries is universal. Due to the special role played by embeddings we called this algebra the maximal embedding algebra; we will take this opportunity to rebrand it as the \hlt{universal corner symmetry} (UCS), to emphasize its key role beyond classical embeddings. Indeed, we believe that this algebra should be taken as one of the fundamental ingredients of gravitational theories, and that it could lead us to an understanding of quantum aspects of gravity.
The UCS is not realized fully in the vicinity of a single corner. Instead, we observe that only complementary subalgebras inside it are supported at different corners. The main distinction comes from finite- versus infinite distance corners, from a bulk spacetime viewpoint. Finite distance corners realize a subalgebra called the \hlt{extended corner symmetry} (ECS) \cite{Donnelly:2016auv, Speranza:2017gxd,Freidel:2020xyx,Freidel:2020svx,Freidel:2020ayo,Donnelly:2020xgu,Ciambelli:2021vnn}, in which everything but Weyl is in principle sourced. This happens because the bulk metric approaches smoothly the corner, without poles in the normal directions, so the leading geometric structures are uncharged under Weyl. In contrast, the algebra on asymptotic corners, i.e., corners at infinite metric distance, includes Weyl transformations, and a different subalgebra of the UCS is realized, which we name the \hlt{asymptotic corner symmetry} (ACS). The introduction of the ACS is a new result of this paper, and pertains both to asymptotically flat spacetimes, where BMSW and subalgebras are realized\cite{Sachs:1962wk,doi:10.1098/rspa.1962.0161,doi:10.1098/rspa.1962.0206,Barnich:2010eb,Campiglia:2014yka,Compere:2018ylh,Campiglia:2020qvc,Flanagan:2019vbl,Freidel:2021fxf}, but also asymptotically AdS spacetimes, that have received tremendous attention in recent years with attempts to enlarge the asymptotic symmetry algebras \cite{Troessaert:2013fma,Grumiller:2016pqb,Ciambelli:2019bzz,Compere:2019bua,Alessio:2020ioh,Fiorucci:2020xto,Ciambelli:2020eba,Geiller:2022vto}. One of the main results of this paper is to clarify this universal structure and discuss how the ECS and ACS can be simultaneously realized inside the UCS. By focussing on this symmetry structure, this then allows for an interpretation of a corner purely group-theoretically, without necessarily a reference to a classical spacetime in which it might be embedded. This is one of the first goals of the corner proposal and suggests that a novel quantum theory is not far away.
With this in mind then, we focus attention on these symmetries of gravitational theories.
Especially in the field of asymptotic symmetries, most discussions follow a bulk-to-boundary (or bulk-to-corner) perspective, where the bulk and its classical dynamics are the starting point and the endpoint is a full characterization of the classical field space at a designated boundary. Given our interest in quantum aspects of gravity, we instead adopt a corner-to-bulk viewpoint, asking how this universal symmetry instructs us about gravity. We are not here looking for a holographic corner (or boundary) theory, but rather a full understanding of gravity itself from its `atomic' constituents, the corners.
We have used classical gravity to identify the universal corner symmetry, but here we will start over with the UCS in hand but not necessarily an associated classical geometry. Our analysis will be presented in two steps.
The first is algebraic, and consists of understanding the representation theory of the UCS. To do so, we will apply the method of coadjoint orbits,\footnote{The coadjoint orbit method is becoming more and more utilized in our community \cite{Witten:1987ty,Balog:1997zz,Barnich:2014zoa,Barnich:2014kra,Barnich:2015uva,Donnelly:2020xgu,Barnich:2021dta,Lahlali:2021nrf,Bergshoeff:2022eog,Riello:2022din} (see also \cite{Oblak:2016eij} and references therein).} developed by Kirillov in \cite{Kirillov_1962,
Kirillov1976ElementsOT,Kirillov_Merits,
Kirillov1990,kirillov2004lectures}, and further discussed in \cite{Kostant_2006,ginz,Duistermaat:1982aa,ALEKSEEV1989719,wildberger_1990,Brylinski_1994}, to the full universal corner symmetry.
To this end, the Kirillov-Kostant-Souriau (KKS) symplectic two form \cite{souriau1970structure,Kostant_2006,
Kirillov1976ElementsOT,Kostant2009} is of central relevance, and we will find that the ECS and ACS emerge as important substructures of the orbits of the UCS. A key result is that the UCS does not possess Casimirs except those extracted from the corner reparameterization invariance, whereas the ACS and ECS do. Locally, at a point on the corner, the analysis drastically simplifies, such that useful information can be extracted. The ECS and ACS at a point turns out to be ideals inside the UCS, and a full characterization of the tangent space to a generic point of the dual algebra is possible.
The second step identifies the proper geometric framework to interpret the UCS orbit analysis. This involves the introduction of an Atiyah Lie algebroid, a concept that was originally described in \cite{Atiyah:1957,Atiyah:1979iu} (see also the extensive works of Mackenzie, \cite{mackenzie_1987,mackenzie_2005}), over the corner. Atiyah Lie algebroids have been advocated to be the mathematical foundation of gauge theories \cite{Lazzarini_2012,Fournel:2012uv,Jordan:2014uza,Carow-Watamura:2016lob,Kotov:2016lpx,Attard:2019pvw}. Here though we emphasize that the base manifold is not a spacetime, but instead a corner, such that the value of a local section of the algebroid is given precisely by a generator of the full UCS.
We recently explored in \cite{Ciambelli:2021ujl} the theory of algebroids, which finds a natural application in this context. We interpret the coadjoint analysis as a characterization of the dual Lie algebroid, where all the algebraic results derived locally can be imported in this more general setup.
The algebroid comes equipped with the adjoint representation, while other representations are realized by introducing associated bundles to the algebroid. In particular, we identify a certain rank-$2$ affine associated bundle that we implicate in what we interpret in terms of a reconstruction of a classical bulk geometry. That is, this affine bundle can be endowed with fibre coordinates that may be identifed with the two normal directions in such a classical geometry, and we show that derivations on the affine bundle naturally encode the local bulk metric data that are involved in the non-zero Noether charges.
This is thus the bulk classical representation. While it seems compelling that such classical geometry can emerge, we believe that the construction will be even more powerful in more general quantum gravitational contexts.
This is the beginning of our elaboration of the corner proposal.
Indeed, once the correct geometric structure underlying corners and UCS is introduced, the next item to explore is the relationship between a classical field space and the various orbits in the dual algebroid. This is made explicit through the moment map, described in general in \cite{Kostant_2006,souriau1970structure}.
The moment map enters naturally in our algebroid, and it is strictly related to the construction of the associated affine bundle. There are two moment maps relevant for us: the moment map for finite distance corners into the dual of the ECS, and for asymptotic corners into the dual of the ACS.
We discuss how to realize these moment maps inside the dual of the full UCS, such that it captures both images of these moment maps in a unified way. The associated affine bundle is crucial, because it gives the representation of the UCS that is realized in classical gravity around corners.
Although this paper is not about the covariant phase space, it is important that we have recently found \cite{Ciambelli:2021nmv,Freidel:2021dxw}
a formalism where all diffeomorphisms are on an equal footing, and associated Noether charges are always integrable. Indeed, as a consequence of the original works of Wald et al. \cite{Lee:1990nz,Wald:1993nt,Iyer:1994ys,Wald:1999wa} (see also \cite{Compere:2019qed,Harlow:2019yfa}), and of Barnich and Brandt \cite{Barnich:2001jy}, surface charges are not in general integrable. The new formalism introduces embedding fields as a refinement in gravity of the general theory of edge modes explored in \cite{Freidel:2015gpa,Donnelly:2016auv,Speranza:2017gxd,Geiller:2017whh,Freidel:2020xyx,Freidel:2020svx,Freidel:2020ayo,Donnelly:2020xgu,Freidel:2021cjp}. While an active area of research \cite{Speranza:2022lxr,Carrozza:2022xut,Kabel:2022efn,Goeller:2022rsx}, our extended phase space considerably facilitates the construction of the moment map for finite distance corners, allowing to map the full field space to the dual algebra, and not only the part without symplectic flux.
The paper is organized as follows. We first review in Section \ref{sec2} the main results outlined in \cite{Ciambelli:2021vnn}, and start the process of disentangling them from the classical bulk spacetime. In Section \ref{sec3} we focus on the group-theoretical aspects of the UCS at single points on the corners. In this simpler setup, we exploit in Subsection \ref{sec3.1} the coadjoint orbit method to invert the KKS 2-form on generic points and thus have a full understanding of the algebra structure. We derive a general formalism that we use to find the KKS 2-form intrinsically for the ECS \ref{sec3.2} and ACS \ref{sec3.3}, at a point on the corner. We conclude this section discussing how to embed the two intrinsic analysis inside the full algebra. An intrinsic analysis of the various orbits of UCS at a single point is offered in Appendix \ref{appA}. Section \ref{sec4} is then devoted to the geometric properties of corners. The reparametrization symmetry of the corner is introduced in Subsection \ref{sec4.1}, where we pass from the algebraic viewpoint to the theory of algebroids, which is described in Subsection \ref{sec4.2}. The study of associated bundles allows us to identify the bulk spacetime representation. Eventually, we discuss how the concept of moment map is naturally encompassed in this geometric structure in Subsection \ref{sec4.3}, and explicitly construct the moment map for finite distance and asymptotic corners in
Subsection \ref{sec4.4}. We conclude with a recap and future perspectives of this work in Section \ref{sec5}.
\section{Universal Corner Symmetry}\label{sec2}
In this section, we review previous works on corner symmetries with the goal of setting up notation and nomenclature. One of the goals of this paper is to emphasize the importance \cite{Ciambelli:2021vnn} of the group $Diff(S)\ltimes GL(2,\mathbb{R})\ltimes \mathbb{R}^2$. As such, we will refer to the algebra associated to this group as the universal corner symmetry (UCS). We will often have occasion to refer to the group $GL(2,\mathbb{R})\ltimes \mathbb{R}^2$ and we will generally call this group $H$, and its Lie algebra $\mathfrak{h}$. We will also be interested in certain subalgebras (actually, ideals) of $\mathfrak{h}$, which we will introduce shortly. First let us recall the defining representation of the UCS.
In \cite{Ciambelli:2021vnn}, given a classical spacetime manifold $M$, we identified the UCS as a maximal closed subalgebra of $\mathfrak{diff}(M)$ associated with a codimension-2 embedded (or immersed) subspace $\phi:S\to M$. This was obtained by introducing local coordinates on $M$ near the subspace $y^M=(u^a,x^i)$, with $a=0,1$ and $i=1,...,n$, with $d=n+2$. Introducing intrinsic local coordinates $\sigma^\alpha$ ($\alpha=1,...,n$) on $S$, the embedding map may be described by $\phi : (u^a(\sigma),x^i(\sigma))$. We refer to the special case $\phi_0 : (u^a(\sigma)=0,x^i(\sigma)=\delta^i_\alpha\sigma^\alpha)$ as the trivial embedding, in which the bulk coordinates are adapted to those of $S$.
An arbitrary vector field on $M$ can be written as $\un\xi=\xi^i(u,x)\un\partial_i+\xi^a(u,x)\un\partial_a$ in these coordinates. In the case of the trivial embedding,\footnote{Here we refer to the trivial embedding for convenience. In \cite{Ciambelli:2021vnn}, we also emphasized the importance of embeddings that are infinitesimally close to the trivial embedding. In such a case, the expansion in powers of $u^a$ still makes sense. } we can expand its component functions near $u^a=0$,
\begin{eqnarray}\label{exp1}
\xi^b(u,x)&=&\sum_{n=0} \frac{1}{n!}u^{a_1}...u^{a_n} \xi_{(n)}^b{}_{a_1...a_n}(x)=\xi_{(0)}^b(x)+\xi_{(1)}{}^b{}_{a_1}(x)u^{a_1}+{1\over 2}\xi_{(2)}{}^b{}_{a_1a_2}(x)u^{a_1}u^{a_2}+\dots
\\
\xi^i(u,x)&=&\sum_{n=0} \frac{1}{n!}u^{a_1}...u^{a_n} \xi_{(n)}^i{}_{a_1...a_n}(x)=\xi_{(0)}^i(x)+\xi_{(1)}^i{}_{a_1}(x)u^{a_1}+{1\over 2}\xi_{(2)}^i{}_{a_1a_2}(x)u^{a_1}u^{a_2}+\dots\label{exp2}
\end{eqnarray}
The UCS is obtained by restricting to vector fields for which these expansions are truncated at first order for $\xi^b$ and zeroth order for $\xi^i$,
\begin{eqnarray}\label{subalgebragens}
\un{\xi}=\xi_{(0)}^k(x)\un\partial_k +\Big(\xi_{(0)}^a(x)+u^b\xi_{(1)}{}^a{}_b(x)\Big)\un\partial_a,
\end{eqnarray}
and we use a hat to denote $\un{\hat\xi}_{(0)}=\xi_{(0)}^i(x)\un\partial_i$.
Indeed, these vector fields close under the Lie bracket. That is, they
satisfy the closed algebra
\begin{eqnarray}\label{LieMsub}
\big[\un{\xi}_1,\un{\xi}_2\big]
&=&\nonumber
\big[\un{\hat\xi}_{(0)1},\un{\hat\xi}_{(0)2}\big]^j\un\partial_j
\\&&\nonumber
+\Big[
\un{\hat\xi}_{(0)1}(\xi_{(0)2}^b)
-\un{\hat\xi}_{(0)2}(\xi_{(0)1}^b)
-\xi_{(1)1}{}^b{}_a\xi_{(0)2}^a
+\xi_{(1)2}{}^b{}_a\xi_{(0)1}^a
\Big]\un\partial_b
\\&&
+u^c
\Big[-\big[\xi_{(1)1},\xi_{(1)2}\big]^b{}_c
+\un{\hat\xi}_{(0)1}(\xi_{(1)2}{}^b{}_c)
-\un{\hat\xi}_{(0)2}(\xi_{(1)1}{}^b{}_c)
\Big]\un\partial_b.
\end{eqnarray}
The term closure refers here to the $u^a$ expansion. The aforementioned algebra is the maximal finitely generated truncation of $\mathfrak{diff}(M)$, in the sense that if the vector fields have certain powers of $u^a$ only, then the Lie bracket of two such fields has, at best, the same powers of $u^a$ appearing in its expansion.
As discussed in \cite{Ciambelli:2021vnn}, one then observes that the generators are valued in the Lie algebra of the group
\begin{eqnarray}\label{maximo}
\underbrace{Diff(S)}_\text{$\xi^j_{(0)}$}\ltimes\Big( \underbrace{GL(2,\mathbb{R})}_\text{$\xi_{(1)}{}^a{}_b$}\ltimes \underbrace{\mathbb{R}^{2}}_\text{$\xi_{(0)}^a$}\Big).
\end{eqnarray}
The semi-direct product structure is identified by associating terms in the Lie bracket as
\begin{eqnarray}\label{LieMsubexpl}
\big[\un{\xi}_1,\un{\xi}_2\big]
&=&
\underbrace{\big[\un{\hat\xi}_{(0)1},\un{\hat\xi}_{(0)2}\big]^j\un\partial_j}_\text{$\mathfrak{diff}(S)$}
\nonumber\\&&
+\Big[
\underbrace{\un{\hat\xi}_{(0)1}(\xi_{(0)2}^b)
-\un{\hat\xi}_{(0)2}(\xi_{(0)1}^b)}_\text{$\mathfrak{diff}(S)$ acts on $\mathbb{R}^2$}
+\underbrace{\xi_{(1)2}{}^b{}_a\xi_{(0)1}^a
-\xi_{(1)1}{}^b{}_a\xi_{(0)2}^a}_\text{$\mathfrak{gl}(2,\mathbb{R})$ acts on $\mathbb{R}^2$}
\Big]\un\partial_b
\nonumber\\&&
+u^c
\Big[-\underbrace{\big[\xi_{(1)1},\xi_{(1)2}\big]^b{}_c}_\text{$\mathfrak{gl}(2,\mathbb{R})$}
+\underbrace{\un{\hat\xi}_{(0)1}(\xi_{(1)2}{}^b{}_c)
-\un{\hat\xi}_{(0)2}(\xi_{(1)1}{}^b{}_c)}_\text{$\mathfrak{diff}(S)$ acts on $\mathfrak{gl}(2,\mathbb{R})$}
\Big]\un\partial_b.
\end{eqnarray}
The algebra $\mathfrak{diff}(S)\loplus \big( \mathfrak{gl}(2,\mathbb{R})) \loplus \mathbb{R}^2\big)$ of the group $Diff(S)\ltimes \big(GL(2,\mathbb{R})\ltimes \mathbb{R}^{2}\big)$ is the UCS (in \cite{Ciambelli:2021vnn}, this was referred to as the maximal embedding algebra). The importance of this algebra lies in its universality; in particular, we have not referred to any dynamical fields (such as a metric) or to any particular spacetime geometry. It has been identified with very few assumptions.
In \cite{Ciambelli:2021vnn} (see also \cite{Speranza:2017gxd,Freidel:2020xyx,Freidel:2020svx,Freidel:2020ayo,Donnelly:2020xgu,Freidel:2021cjp}), it was also found that corners support (a subalgebra of) the UCS. By this we mean that only a subset of all of the diffeomorphism Noether charge densities\footnote{Here, the reader should recall that in the case of gauge symmetries, a Noether current is an exact form determined by an $n$-form ($n=d-2$) on $M$, which given a codimension-$2$ embedding $\phi:S\to M$ can be pulled back to a top form on $S$ and integrated to obtain the corresponding charge.} can pull back to non-zero values on a corner. Thus corner symmetries are important from this symmetry perspective as well; in particular most of $\mathfrak{diff}(M)$ is always pure gauge, with a subset realized as the residual global symmetry associated with the presence of a given corner. For example, the Noether charges of the classical Einstein-Hilbert theory at finite distance corners are non-zero only for $Diff(S)\ltimes H_s$, where $H_s=SL(2,\mathbb{R})\ltimes \mathbb{R}^2$. The associated algebra is known as the extended corner symmetry (ECS); we now believe that this is the maximal symmetry supported by Noether charges of any diffeomorphism-invariant theory on generic 'finite distance' corners. Nevertheless, we will see in this paper that it is the UCS which should be regarded as the primary symmetry of diffeomorphism invariant theories. Indeed, it is the UCS that allows a unified treatment of finite distance corners and 'asymptotic' corners. The latter support another important subalgebra of the UCS, the Lie algebra of the group
\begin{eqnarray}
Diff(S)\ltimes H_w,\qquad H_w := W\ltimes \mathbb{R}^2,
\end{eqnarray}
which we refer to as the asymptotic corner symmetry (ACS). The group $Diff(S)\ltimes H_w$ in fact contains BMSW, recently shown to be the general asymptotic algebra \cite{Freidel:2021fxf} in asymptotically-flat geometries. Here by $W$ we mean the group $\mathbb{R}\subset GL(2,\mathbb{R})$ consisting of the trace element of $GL(2,\mathbb{R})$.
Therefore the ECS and ACS are both contained in the UCS. The latter is not itself realized in the vicinity of a single corner, but it contains both finite distance and asymptotic corner algebras as subalgebras. At a given point on the corner $S$, we can reduce our attention to the group $H$. Then, the algebras of the two subgroups $H_w=H/SL(2,\mathbb{R})$ and $H_s=H/W$, $\mathfrak{h}_w$ and $\mathfrak{h}_s$ respectively, are ideals inside $\mathfrak{h}$. We believe that this is an important feature. As we will see in the next section, it is this property that allows at each point on the dual space $\mathfrak{h}^*$ to identify $\mathfrak{h}_w$ or $\mathfrak{h}_s$ directions uniquely. This is why we should regard the UCS as the fundamental algebra, whose representation theory will dictate both finite distance and asymptotic distance physics.
Before we continue to the coadjoint orbit analysis, we set up some notation. We introduce a basis for the Lie algebra $\mathfrak{h}$ as $\left(\un t^a{}_b,\un t_a\right)$, with $a,b=0,1$. Then any element $\un\mu\in\mathfrak{h}$ can be written
\begin{eqnarray}
\un\mu=\theta^a{}_b\un t^b{}_a+ b^a\un t_a,
\end{eqnarray}
and thus we can regard $\un\mu\leftrightarrow (\theta^a{}_b,b^a)$. The Lie bracket on $\mathfrak{h}$ is given by (see Section \ref{sec3} for further details)
\begin{eqnarray}
[\un\mu,\un\nu]=
-[\theta,\theta']^b{}_c\un t^c{}_b
-(\theta^a{}_bb'{}^b-\theta'{}^a{}_bb^b)\un t_a,
\end{eqnarray}
where $\un\nu\leftrightarrow (\theta'{}^a{}_b,b'{}^a)$.
The defining representation $\rho:\mathfrak{h}\to T\mathbb{R}^2$ is of the form
\begin{eqnarray}\label{defrep}
\rho(\un t^a{}_b)=u^a\un\partial_b,\qquad
\rho(\un t_b)=\un\partial_b.
\end{eqnarray}
If we denote the coordinates by $u^a=(u,\rho)$, then the $\mathfrak{gl}(2,\mathbb{R})=\mathfrak{w} \oplus \mathfrak{sl}(2,\mathbb{R})$ generators can be written
\begin{eqnarray}
\un W=\tfrac12 (u\un\partial_u+\rho\un\partial_\rho),\qquad \un L_3=\tfrac12 (u\un\partial_u-\rho\un\partial_\rho),\qquad \un L_+=u\un\partial_\rho,\qquad \un L_-=\rho\un\partial_u.
\end{eqnarray}
So $\un W$ is the trace of $\rho (\un t^a{}_b)$ while the $\mathfrak{sl}(2,\mathbb{R})$ generators are the three traceless elements. The Lie brackets are
\begin{eqnarray}
[\un W,\un L_3]=0=[\un W,\un L_\pm],\qquad
[\un L_3,\un L_\pm]=\pm \un L_\pm,\quad
[\un L_+,\un L_-]= 2\un L_3.
\end{eqnarray}
We denote the translations by
\begin{eqnarray}
\un T_-=\un\partial_u,\qquad \un T_+=\un\partial_\rho,
\end{eqnarray}
and we then have
\begin{eqnarray}
\left[\un W,\un T_\pm\right]=-\tfrac12\un T_\pm,\qquad
\left[\un L_3,\un T_\pm\right]=\pm\tfrac12 \un T_\pm,\qquad
\left[\un T_+,\un T_-\right]= \un 0,
\\
\left[\un L_+,\un T_+\right]=\un 0=\left[\un L_-,\un T_-\right],\quad
\left[\un L_+,\un T_-\right]=-T_+,\qquad \left[\un L_-,\un T_+\right]=-T_-,
\end{eqnarray}
which is the algebra $(\mathfrak{w}\oplus \mathfrak{sl}(2,\mathbb{R})) \loplus \mathbb{R}^2$. The algebra $\big(\mathfrak{diff}(S)\loplus (\mathfrak{w}\oplus \mathfrak{sl}(2,\mathbb{R}))\big) \loplus \mathbb{R}^2$ is then obtained by taking vector fields with coefficients that are arbitrary functions of the corner coordinates.
For finite distance corners in $M$, as found in \cite{Ciambelli:2021vnn,Freidel:2020xyx} and previously discussed in \cite{Speranza:2017gxd,Donnelly:2020xgu}, the most general algebra dynamically realized\footnote{These statements apply to any diffeomorphism-invariant theory, \cite{Speranza:2017gxd}.} is the aforementioned extended corner symmetry $(\mathfrak{diff}(S)\loplus \mathfrak{sl}(2,\mathbb{R})) \loplus \mathbb{R}^2$, obtained from the differential representation by dropping $\un W$. On the other hand, for asymptotic corners, it has been found in \cite{Freidel:2021fxf} that the most general algebra dynamically realized in asymptotically flat spacetimes is the Weyl BMS algebra (BMSW), given by the $(\mathfrak{diff}(S)\loplus \mathfrak{w}) \loplus \mathbb{R}$ subalgebra of the ACS, obtained by excluding the $\mathfrak{sl}(2,\mathbb{R})$ and $\un T_+$ generators. This algebra has numerous physically relevant subalgebras; the generalized BMS algebra $\mathfrak{diff}(S)\loplus \mathbb{R}$ found in \cite{Campiglia:2014yka,Compere:2018ylh}, the extended BMS algebra found in \cite{Barnich:2010eb}, where $\mathfrak{diff}(S)$ is reduced to the locally well-defined conformal Killing vectors on $S$, and finally the original BMS algebra found in \cite{doi:10.1098/rspa.1962.0161,doi:10.1098/rspa.1962.0206}, where $\mathfrak{diff}(S)$ is reduced to the globally well-defined conformal Killing vectors on $S$.
We emphasize that the ACS algebra and ECS algebra are not nested into one another. Rather, they are distinct quotients, as mentioned above, of the UCS in which complementary subalgebras of $\mathfrak{gl}(2,\mathbb{R})$ are retained. While it was so far unclear why only the charges of ECS (ACS) have support on finite distance (asymptotic) corners in diffeomorphism-invariant theories, we will find an explanation of this fact below by studying the coadjoint orbits of the UCS: the tangent space to the orbit passing through a generic point can be split into ECS and ACS parts universally by enforcing the constancy of Casimirs along the orbit. Which Casimir to enforce is dictated by which subalgebra one wants to restrict to, and it is this at the end the physical input on the system.
Ultimately, we are interested then in the representation theory of the UCS and its subalgebras. For this reason, we will not make further use of the defining differential representation given above, but work more abstractly. In fact, we will abandon an interpretation of the algebras as realized by vector fields in a classical spacetime. Instead we will eventually be led to a geometrically well-defined interpretation in terms of certain bundles over the space $S$. Of central importance is an Atiyah Lie algebroid over $S$ with structure group $H$, and the representation theory of the UCS can be explored by considering bundles associated to the Lie algebroid. In particular, we will introduce a rank-2 affine bundle over $S$ whose total space may be thought of as a local picture of a corner in a spacetime manifold. There are specific moment maps that involve the introduction of certain geometric structures on the affine bundle and map these to the ECS or ACS orbits.
\section{Coadjoint Orbit Method}\label{sec3}
In this section, we study the coadjoint orbits of the UCS, and its various subcases previously introduced. While mostly mathematical, this analysis has important repercussions in understanding gravity and its constituents, that is, the codimension-2 corners. While $\mathfrak{diff}(S)$ is an important part of the UCS, it turns out to be instructive to first consider the coadjoint orbits of $\mathfrak{h}$. Thus we will split our analysis into two parts: we begin by focussing our attention on $\mathfrak{h}$ and its ideals only, and later add back $\mathfrak{diff}(S)$ consistently.
\subsection{The Algebra $\mathfrak{h}$}\label{sec3.1}
Given the Lie group $H$, its Lie algebra $\mathfrak{h}$ is a vector space. One can consider the dual vector space $\mathfrak{h}^*$, defined as
\begin{eqnarray}
m\in \mathfrak{h}^*,\quad \un\mu\in \mathfrak{h},\quad m: \mathfrak{h}\to \mathbb{R}
,\qquad m(\un\mu)\in \mathbb{R}.
\end{eqnarray}
While the Lie algebra bracket on $\mathfrak{h}$ defines the adjoint action of $\mathfrak{h}$ on itself,
\begin{eqnarray}
ad_{\un\mu}\un\nu := [\un\mu,\un\nu],
\end{eqnarray}
there is a corresponding coadjoint action of $\mathfrak{h}$ on $\mathfrak{h}^*$, with the defining property
\begin{eqnarray}\label{coad}
(ad_{\un\mu}^*m)(\un\nu)=-m(ad_{\un\mu}\un \nu).
\end{eqnarray}
From this we can define the map $ad^*_{\un\mu} : \mathfrak{h}^*\to \mathfrak{h}^*$.
These are all the essential ingredients one needs to study coadjoint orbits.
Coadjoint orbits are usually introduced at the group level
\begin{eqnarray}
O_m=\{g\in H \ \vert \ Ad^*_{g}m\},
\end{eqnarray}
with $Ad^*$ the group coadjoint action, but it is sufficient for our purposes to use the local algebra version
\begin{eqnarray}
{\cal O}_m=\{\un\mu\in \mathfrak{h} \ \vert \ ad^*_{\un\mu}m\},
\end{eqnarray}
which is more suitable for the forthcoming discussion that also includes $\mathfrak{diff}(S)$. Indeed one can interpret the $ad^*$ action as exploring the tangent space to an orbit at the given point $m\in\mathfrak{h}^*$. Not all algebra elements act on $m$ in a non-trivial way. Those that do not define the stabilizer subalgebra
\begin{eqnarray}
{\cal S}_m=\{\un\mu \in \mathfrak{h} \ \vert \ ad^*_{\un\mu}m=m\},
\end{eqnarray}
with associated stabilizer (or isotropy) group $S_m$. An important result is the isomorphism
\begin{eqnarray}
O_m\simeq H/S_m.
\end{eqnarray}
Another important result, and the key reason why we focus on coadjoint orbits, is that, in very simple instances, classifying them gives information about the representations of the algebra \cite{Kirillov1976ElementsOT}. There is no known direct correspondence in general (we indeed refer to it as the orbit {\it method}), and it has limitations, but with such a complicated object as the UCS, we believe it is one of the best paths towards an understanding of the algebra.
Crucially, one can perform a unified treatment of orbits and stabilizers using the KKS symplectic $2$-form \cite{souriau1970structure,
Kostant_2006}:\footnote{In an abuse of notation, we call KKS 2-form both the contracted and non-contracted objects, $\Omega_m(\un\mu,\un\nu)$ and $\Omega_m(.,.)$. The meaning and distinction will always be clear from context.}
\begin{eqnarray}\label{KKS2f}
\Omega_m: {\cal O}_m\otimes {\cal O}_m\to \mathbb{R}, \qquad \Omega_m(\un\mu,\un\nu):=\Omega_m(ad^*_{\un\mu}m,ad^*_{\un\nu}m)= m([\un\mu,\un\nu]).
\end{eqnarray}
The notation $\Omega_m(\un\mu,\un\nu)$ is merely a shortform that we use for convenience.
By definition, the KKS $2$-form is non-degenerate and thus invertible when restricted to the orbits. This is due to the property of $\mathfrak{h}^*$ being a Poisson manifold, partitioned by symplectic leaves, on which the KKS form becomes non-degenerate. The analysis of the various orbits and different points in $\mathfrak{h}^*$ reduces then to a full characterization of $\Omega$.
For the specific algebra $\mathfrak{h}$, this is a tractable problem, as it can be thought of as a simple extension of the familiar analysis of the Poincar\'e group (in $1+1$ dimensions). We denote a basis of $\mathfrak{h}$ as $\left(\un t^a{}_b,\un t_a\right)$ with $a,b=0,1$. The Lie brackets on $\mathfrak{h}$ are of the form
\begin{eqnarray}\label{Alg}
\left[\un t^a{}_b,\un t^c{}_d\right]
=
\delta^c{}_b\un t^a{}_d -\delta^a{}_d\un t^c{}_b,
\qquad
\left[\un t^a{}_b,\un t_c\right]=-\pi^a{}_c{}^d{}_b\un t_d,
\qquad
\left[\un t_b,\un t_c\right]=\un 0.
\end{eqnarray}
The values of the coefficients $\pi^a{}_d{}^c{}_b$ that determine the structure constants depend on the precise choice of group, because given such a choice, the $\un t^a{}_b$ may satisfy some conditions.\footnote{The notation is set up to apply generally to $GL(k,\mathbb{R})\ltimes \mathbb{R}^k$ where $a,b=0,...,k-1$ (which appears in the context of codimension-$k$ embeddings), but we restrict attention here to $k=2$. The notation also applies to subgroups of $GL(2,\mathbb{R})$, for which we modify $\pi^a{}_b{}^c{}_d$ accordingly. We will see examples of this in forthcoming subsections.} For the case studied in this subsection, $GL(2,\mathbb{R})$, there are no such conditions, and we have
$\pi^a{}_c{}^d{}_b=\delta^a{}_c\delta^d{}_b$.
We will denote general elements $\mathfrak{h}\ni\un\mu=\theta^a{}_b\un t^b{}_a+b^a\un t_a$, $\un\nu=\theta'{}^a{}_b\un t^b{}_a+b'{}^a\un t_a$, etc. Thus we have
\begin{eqnarray}\label{genalg}
ad_{\un\mu}\un\nu=[\un\mu,\un\nu]=
-[\theta,\theta']^b{}_c\un t^c{}_b
-(\theta^a{}_bb'{}^c-\theta'{}^a{}_bb^c)\pi^b{}_c{}^d{}_a\un t_d.
\end{eqnarray}
The dual vector space $\mathfrak{h}^*$ can be endowed with the dual basis $\{t^a{}_b,t^a\}$ satisfying
\begin{eqnarray}
t^a{}_b(\un t^c{}_d)=\pi^a{}_d{}^c{}_b
,\qquad
t^a{}_b(\un t_c)=0,\qquad
t^a(\un t^b{}_c)=0,\qquad
t^a(\un t_b)=\delta^a{}_b.
\end{eqnarray}
We denote a general element of $\mathfrak{h}^*$ as $m=J^a{}_bt^b{}_a+P_at^a$, and so we have
\begin{eqnarray}
m(\un\mu)
=
\theta^b{}_aJ^a{}_b+b^aP_a.
\end{eqnarray}
One can regard $(J^a{}_b,P_a)$ as the coordinates of the point $m$ in this basis.
The coadjoint action \eqref{coad} of $\mathfrak{h}$ on $\mathfrak{h}^*$ satisfies
\begin{eqnarray}
(ad^*_{\un\mu}m)(\un\nu)=-m(ad_{\un\mu}\un\nu)
=J^a{}_b [\theta,\theta']^b{}_a
+ P_a(\theta^a{}_bb'{}^b-\theta'{}^a{}_bb^b),
\end{eqnarray}
from which it is then a simple matter to deduce
\begin{eqnarray}\label{adactionhdual}
ad^*_{\un\mu}m=
\Big([J,\theta]^a{}_b - b^aP_b\Big) t^b{}_a
+P_b\theta^b{}_ct^c.
\end{eqnarray}
Given the explicit expression of $m$,
one can interpret the $ad^*$ action as a transformation of its components,
\begin{eqnarray}\label{dJdP}
\delta_{\un\mu} J^a{}_b
= [J,\theta]^a{}_b -b^aP_b,
\qquad
\delta_{\un\mu} P_a = P_b\theta^b{}_a.
\end{eqnarray}
Here $\delta_{\un\mu} J^a{}_b$ and $\delta_{\un\mu} P_a$ should be regarded as components of tangent vectors to the orbit at the point $m$.
A relevant intermediate step in the calculation of the symplectic form is that we can invert these relations to obtain
\begin{eqnarray}\label{inversions}
b^a&=&-\delta_{\un\mu} J^a{}_b J^b{}_c \kappa^{cd}P_d-J^a{}_b \delta_{\un\mu} J^b{}_c \kappa^{cd}P_d+{\cal J} \delta_{\un\mu} J^a{}_b\kappa^{bc}P_c, \\
\theta^a{}_b&=&-\delta_{\un\mu} J^a{}_c \kappa^{cd} P_d P_b+\big(P_e\delta_{\un\mu} J^e{}_c \kappa^{cd} P_d\big) \delta^a{}_b+\delta_{\un\mu} P_c \kappa^{cd}P_d J^a{}_b
-P_eJ^e{}_d\kappa^{cd}\delta_{\un\mu} P_c \delta^a{}_b.
\label{inversionsb}
\end{eqnarray}
In this expression, we have introduced the trace ${\cal J}=J^a{}_b\delta^b{}_a$, and we have defined the $\mathfrak{h}$-invariant quantity
\begin{eqnarray}\label{C3}
\kappa^{ab}={\varepsilon^{ab}\over C_3}, \qquad C_3:= P_aJ^a{}_b\varepsilon^{bc}P_c.
\end{eqnarray}
The symbol $\varepsilon$ is the $2x2$ Levi-Civita symbol with conventions $\varepsilon^{01}=-1$, and we note that $\varepsilon^{ab}$ and $C_3$ are $\mathfrak{h}_s$ invariants.
We regard $C_3$ as an element of the enveloping algebra of $\mathfrak{h}^*$. The facts that $b^a$ depends only on $\delta_{\un\mu}J^a{}_b$ while $\theta^a{}_b$ depends on both $\delta_{\un\mu}J^a{}_b$ and $\delta_{\un\mu}P_a$ are a result of the semi-direct sum structure of the algebra. The algebra $\mathfrak{h}$ has dimension $6$. That the inversion \eqref{inversions} and \eqref{inversionsb} are possible is testament to the fact that the orbit at a generic point $m$ can be $6$-dimensional. On the other hand,
the inversion formulas fail if $C_3$ vanishes, indicating that the special points that satisfy this condition have lower dimensional orbits. This is the sense in which $\mathfrak{h}^*$ fails to be a symplectic manifold, as there are a series of non-generic points for which the orbits are 4-,2- or 0-dimensional. We refer to such orbits as singular orbits, and explore them in Appendix \ref{appA}. Clearly $C_3$ plays a central role here, and we will return to it presently.
As a final step, we write the KKS symplectic form
\begin{eqnarray}\label{KKS}
\Omega^{(\mathfrak{h})}_m(\un\mu,\un\nu)
=
-J^a{}_b[\theta,\theta']^b{}_a -P_a(\theta^a{}_cb'{}^c-\theta'{}^a{}_cb^c) ,
\end{eqnarray}
which is a non-degenerate 2-form on the orbit.
Assuming $C_3\neq 0$, we can now use the inversion formulas \eqref{inversions} and \eqref{inversionsb} to rewrite this in terms of $\delta_{\un\mu} J^a{}_b$ and $\delta_{\un\mu} P_a$. It is convenient to first separate the trace and traceless pieces,
\begin{eqnarray}
J^a{}_b=\bar J^a{}_b+\tfrac12 {\cal J}\delta^a{}_b,\qquad
\theta^a{}_b=\bar\theta^a{}_b+\tfrac12 w\delta^a{}_b,\qquad w=\theta^a{}_b\delta^b{}_a,
\end{eqnarray}
and we then obtain
\begin{eqnarray}
\Omega^{(\mathfrak{h})}_m(\un\mu,\un\nu)=\delta_{\un\mu}P_a\big(\delta_{\un\nu}\bar J^a{}_b\bar J^b{}_c+\bar J^a{}_b \delta_{\un\nu}\bar J^b{}_c\big)\kappa^{cd}P_d
-\tfrac{1}{2}(P_a\delta_{\un\mu}\bar J^a{}_b)\kappa^{bc}(P_d\delta_{\un\nu}\bar J^d{}_c)
+\tfrac12\delta_{\un\mu}\log C_3\delta_{\un\nu}{\cal J}
-(\un\mu \leftrightarrow \un\nu).\label{KKSdJdP}
\end{eqnarray}
To derive this, we have also used the identity
\begin{eqnarray}
\delta_{\un\mu} \log C_3=w,
\end{eqnarray}
which follows from \eqref{inversionsb}.
We note that the Pfaffian of $\Omega^{(\mathfrak{h})}_m$ is
\begin{eqnarray} \label{pfaf}
Pf[\Omega^{(\mathfrak{h})}_m]=-\frac{1}{C_3},
\end{eqnarray}
which is further evidence that the KKS form is non-degenerate iff $C_3\neq 0$. In the formula \eqref{KKSdJdP}, there are terms involving only the traceless part of $J^a{}_b$ (and thus associated with $\mathfrak{h}_s$). The last term instead indicates that the trace ${\cal J}$ is canonically conjugate to $\log C_3$.
Now if we denote $Z^A=(J^a{}_b,P_a)$ and write the symplectic 2-form as \[\Omega_m(\un\mu,\un\nu)= \Omega_{AB}(Z)(\delta_{\un\mu}Z^A\delta_{\un\nu}Z^B-\delta_{\un\nu}Z^A\delta_{\un\mu}Z^B),\] then by inverting the matrix $\Omega_{AB}(Z)$, we obtain Poisson brackets as usual $\{Z^A,Z^B\}=\Omega^{AB}(Z)$. In this case, we find simply the Poisson bracket representation of the full $\mathfrak{h}$ algebra, as long as $C_3\neq 0$.
Let us recap our findings so far. The central result is that \eqref{KKS} at the generic point $m$ can be inverted to then obtain the KKS symplectic form on $6$-dimensional orbits, giving the Poisson bracket representation of $\mathfrak{h}$. For such orbits, there is no isotropy group and no Casimirs. This is achievable only if $C_3\neq 0$, indicating that $\mathfrak{h}^*$ is locally symplectic, but there may be lower dimensional singular orbits if $C_3=0$, as we show in Appendix \ref{appA}. There is a completely different structure arising in the case $\mathfrak{h}_s^*$ and $\mathfrak{h}_w^*$, where we will find that generic orbits have always non-trivial isotropy groups. As already discussed previously, one can single out $\mathfrak{h}_s$ and $\mathfrak{h}_w$ directions at generic $\mathfrak{h}^*$ points. We will show this after presenting an intrinsic analysis of $\mathfrak{h}_s$ and $\mathfrak{h}_w$ coadjoint orbits.
\subsection{The Algebra $\mathfrak{h}_s$}\label{sec3.2}
We begin with the intrinsic derivation of the orbits of $\mathfrak{h}_s$. Our general analysis is ready-made for such a scenario. Indeed, it suffices to consider \eqref{Alg},
\begin{eqnarray}
\left[\un t^a{}_b,\un t^c{}_d\right]
=
\delta^c{}_b\un t^a{}_d -\delta^a{}_d\un t^c{}_b,
\qquad
\left[\un t^a{}_b,\un t_c\right]=-\pi^a{}_c{}^d{}_b\un t_d,
\qquad
\left[\un t_b,\un t_c\right]=\un 0,
\end{eqnarray}
with the choice $\pi^a{}_c{}^d{}_b=\delta^a{}_c\delta^d{}_b-\tfrac12 \delta^a{}_b\delta^d{}_c$. Then one obtains
\begin{eqnarray}
m(\un\mu)
=
\theta^b{}_aJ^a{}_b-\tfrac12 {\cal J}w+b^aP_a=\bar\theta^b{}_aJ^a{}_b+b^aP_a,
\end{eqnarray}
where we recall that $\bar\theta^a{}_b=\theta^a{}_b-\tfrac12 w\delta^a{}_b$ is the traceless part of $\theta^a{}_b$. We see that the trace $w$ decouples, indicating that we are appropriately describing the algebra $\mathfrak{h}_s$. We also see that consistently only the traceless parts of $J^a{}_b$ appear here (since only those coordinatize $\mathfrak{h}_s^*$), and so we will write it as $\bar J^a{}_b$. The KKS form then reads
\begin{eqnarray}
\Omega^{(\mathfrak{h}_s)}_m(\un\mu,\un\nu)
&=&
-J^a{}_b[\theta,\theta']^b{}_a -P_a(\theta^a{}_cb'{}^c-\theta'{}^a{}_cb^c)+\tfrac12 P_a(w b'{}^a-w' b^a)\nonumber\\
&=&
-\bar J^a{}_b[\bar\theta,\bar\theta']^b{}_a -P_a(\bar\theta^a{}_cb'{}^c-\bar\theta'{}^a{}_cb^c).\label{intKKS}
\end{eqnarray}
Again, we see that only the traceless quantities contribute to the $\mathfrak{h}_s$ symplectic form.
Working through a similar analysis as in the last subsection leads to the following expressions for the variations corresponding to the $ad^*$ action of $\mathfrak{h}_s$ on $\mathfrak{h}_s^*$,
\begin{eqnarray}\label{var2}
\delta_{\un\mu}\bar J^a{}_b
= [\bar J,\bar \theta]^a{}_b -b^aP_b+\tfrac12 b^cP_c\delta^a{}_b,
\qquad
\delta_{\un\mu} P_a = P_b\bar\theta^b{}_a.
\end{eqnarray}
One observes that $\delta^b{}_a \delta_{\un\mu}J^a{}_b=0$, and therefore there are only $3$ variations inside $\delta_{\un\mu}J^a{}_b$, which are the traceless parts, $\delta_{\un\mu}\bar J^a{}_b$.
While the algebra has dimension $5$, the tangent space at a generic point in $\mathfrak{h}_s^*$ is only $4$-dimensional (consistent with such orbits being symplectic). This can be seen by attempting to invert \eqref{var2}. While there are five equations depending on the five parameters $(\bar\theta^a{}_b,b^a)$, they are not all independent. Indeed one finds that the variations satisfy a relation independent of $(\bar\theta^a{}_b,b^a)$,
\begin{eqnarray}\label{P0v}
2(\bar J^0{}_1P_0-\bar J_3P_1)\delta_{\un\mu}P_0=2(P_1\bar J^1{}_0+\bar J_3 P_0)\delta_{\un\mu}P_1-P_0^2\delta_{\un\mu}\bar J^0{}_1+2P_0P_1\delta_{\un\mu}\bar J_3+P_1^2\delta_{\un\mu}\bar J^1{}_0,
\end{eqnarray}
where for brevity we have introduced
$\bar J_3=\bar J^0{}_0=-\bar J^1{}_1$.
This result is easy to understand: since we are considering here $\mathfrak{h}_s$, we have that $C_3$ is a Casimir. Indeed, regarding $\delta_{\un\mu}$ as a derivation, we readily compute
\begin{eqnarray}
\delta_{\un\mu}C_3&=&\delta_{\un\mu}(P_a\bar J^a{}_b\varepsilon^{bc}P_c)\nonumber\\
&=&2\delta_{\un\mu}P_a\,\bar J^a{}_b\varepsilon^{bc}P_c
+P_a\delta_{\un\mu}\bar J^a{}_b\varepsilon^{bc}P_c\nonumber
\\
&=&2\delta_{\un\mu}P_0(\bar J^0{}_1P_0-\bar J_3P_1)
-2\delta_{\un\mu}P_1(\bar J_3 P_0+\bar J^1{}_0P_1)
+P_0^2\delta_{\un\mu}\bar J^0{}_1
-2P_0P_1\delta_{\un\mu}\bar J_3
-P_1^2\delta_{\un\mu}\bar J^1{}_0.
\end{eqnarray}
The vanishing of the variation of $C_3$ is thus equivalent to the relation \eqref{P0v}: since $C_3$ is a Casimir, it is constant along an $\mathfrak{h}_s$ orbit.
Given that, we can obtain the KKS form on such an orbit by using the above relation to eliminate one of the variations. This cannot be done globally, and there is in general some freedom of choice. As an example, suppose we are at a point $m\in\mathfrak{h}_s^*$ where $\bar J^0{}_1P_0-\bar J_3P_1\neq 0$. We could then choose to eliminate $\delta_{\un\mu}P_0$ by solving \eqref{P0v} for it as
\begin{eqnarray}\label{P0v2}
\delta_{\un\mu}P_0=\frac{2(P_1\bar J^1{}_0+\bar J_3 P_0)\delta_{\un\mu}P_1-P_0^2\delta_{\un\mu}\bar J^0{}_1+2P_0P_1\delta_{\un\mu}\bar J_3+P_1^2\delta_{\un\mu}\bar J^1{}_0}{2(\bar J^0{}_1P_0-\bar J_3P_1)}.
\end{eqnarray}
If we then define
\begin{eqnarray}
J_C^2:=\bar J^a{}_b\bar J^b{}_a=2(\bar J_3^2+\bar J^1{}_0\bar J^0{}_1),
\end{eqnarray}
we obtain from \eqref{intKKS} the non-degenerate KKS form on the $4$-dimensional orbits
\begin{eqnarray}\label{KKSintS}
\Omega^{(\mathfrak{h}_s)}_m(\un\mu,\un\nu)
=\frac{2P_0\delta_{\un\mu}\bar J_3\delta_{\un\nu}\bar J^0{}_1+P_1\delta_{\un\mu}\bar J^1{}_0\delta_{\un\nu}\bar J^0{}_1+\delta_{\un\mu}P_1\delta_{\un\nu}J_C^2}{2 (\bar J^0{}_1P_0-\bar J_3P_1)}-(\un\mu \leftrightarrow \un\nu).
\end{eqnarray}
The quantity $J_C^2$ is of course an $\mathfrak{sl}(2,\mathbb{R})$ Casimir; it is not on the other hand an $\mathfrak{h}_s$ Casimir and consequently varies along the orbit. Inverting the components of this symplectic form gives (basis ordering $(P_1,\bar J_3,\bar J^0{}_1,\bar J^1{}_0)$)
\begin{eqnarray}
\Omega_{AB}^{(\mathfrak{h}_s)}=\begin{pmatrix}0&\tfrac12 P_1&0&-P_0\cr-\tfrac12 P_1&0&-\bar J^0{}_1&\bar J^1{}_0\cr 0&\bar J^0{}_1&0&-2\bar J_3\cr P_0&-\bar J^1{}_0&2\bar J_3&0\end{pmatrix},
\end{eqnarray}
and thus gives rise to the Poisson bracket realization of $\mathfrak{sl}(2,\mathbb{R})$ along with the appropriate brackets of $P_1$ with $\bar J^a{}_b$. We reiterate that although we are not here using $\delta P_0$ as a basis form, $P_0$ does vary along the orbit. Thus the full $\mathfrak{h}_s$ algebra is encoded in the Poisson brackets if we recall that $\delta C_3$ is normal to the orbits.
To recap, the intrinsic $\mathfrak{h}_s$ analysis has revealed that generic orbits are $4$-dimensional, and thus there is a constraint among the $5$ variations. We have seen that this is a result of the existence of an $\mathfrak{h}_s$ cubic Casimir, $C_3$, which can then be used to express one variation in terms of the others. Solving this for $\delta_{\un\mu}P_0$, one obtains exactly \eqref{P0v}. Enforcing the Casimir $C_3$ to be constant is indeed what we will do in order to see $\mathfrak{h}_s$ directions directly inside the $6$-dimensional tangent space of $\mathfrak{h}$ orbits in Subsection \ref{sec3.4}. First, we will in the next subsection discuss the intrinsic analysis for $\mathfrak{h}_w$ orbits.
\subsection{The Algebra $\mathfrak{h}_w$}\label{sec3.3}
A similar intrinsic analysis can be performed to find generic $\mathfrak{h}_w$ orbits. Here again our setup is ready to be applied. We use \eqref{Alg}, i.e.,
\begin{eqnarray}
\left[\un t^a{}_b,\un t^c{}_d\right]
=
\delta^c{}_b\un t^a{}_d -\delta^a{}_d\un t^c{}_b,
\qquad
\left[\un t^a{}_b,\un t_c\right]=-\pi^a{}_c{}^d{}_b\un t_d,
\qquad
\left[\un t_b,\un t_c\right]=\un 0,
\end{eqnarray}
with now $\pi^a{}_c{}^d{}_b=\tfrac12 \delta^a{}_b\delta^d{}_c$. This choice results in only the trace of the generators $\un t^a{}_b$ appearing. It immediately follows that
\begin{eqnarray}
m(\un\mu)
=\tfrac12 {\cal J}w+b^aP_a,
\end{eqnarray}
and the KKS form then reads
\begin{eqnarray}
\Omega^{(\mathfrak{h}_w)}_m(\un\mu,\un\nu)
=-\tfrac12 P_a(w b'{}^a-w' b^a).
\end{eqnarray}
Again, we see that the traceless $\bar\theta^a{}_b$ does not contribute to the intrinsic $\mathfrak{h}_w$ analysis, with only the trace $w$ appearing.
Similar to previous discussions, we read off the variations
\begin{eqnarray}\label{var3hw}
\delta_{\un\mu} {\cal J}
= -b^cP_c,
\qquad
\delta_{\un\mu} P_a = \tfrac12 w P_a,
\end{eqnarray}
corresponding to the $ad^*$ action of $\mathfrak{h}_w$ on $\mathfrak{h}_w^*$.
Here the algebra has dimension $3$, but the tangent space to an orbit at a generic point in $\mathfrak{h}_w^*$ will only be $2$-dimensional. Indeed, we see from the second two equations in \eqref{var3hw} that $w$ may be eliminated giving an equation amongst the variations. At all points in $\mathfrak{h}_w$ for which $P_0\neq 0$, we can write this equation as
\begin{eqnarray}\label{solveCas1}
\delta_{\un\mu}P_1=P_1 {\delta_{\un\mu}P_0\over P_0}.
\end{eqnarray}
Using this to eliminate $\delta_{\un\mu}P_1$ allows us to extract the non-degenerate KKS form on the $2$-dimensional orbits,
\begin{eqnarray}\label{KKSintW}
\Omega^{(\mathfrak{h}_w)}_m(\un\mu,\un\nu)={\delta_{\un\mu}P_0\delta_{\un\nu}{\cal J}-\delta_{\un\mu}{\cal J}\delta_{\un\nu}P_0\over P_0},
\end{eqnarray}
which is valid on the domain $P_0\neq 0$. We see that ${\cal J}$ and $\log P_0$ are Darboux coordinates on these 2-dimensional orbits. Note that eq. \eqref{solveCas1} can be interpreted as the vanishing of an $\mathfrak{h}_w$ Casimir, which is of the form
\begin{eqnarray}
C_1={P_1\over P_0}.
\end{eqnarray}
Indeed, since the $P_a$ simply rescale with the same weight under the Weyl transformation, their ratio is invariant, and thus each orbit has a constant value of $C_1$.
In the following subsection, we will see how to realize $\mathfrak{h}_s$ and $\mathfrak{h}_w$ directions directly inside the $6$-dimensional tangent space on $\mathfrak{h}^*$ by making use of the facts gleaned from the intrinsic analyses.
\subsection{Orbits and Ideals of $\mathfrak{h}$}\label{sec3.4}
The algebra $\mathfrak{h}$ contains three ideals: an Abelian one, $\mathfrak{h}_0\equiv \mathbb{R}^2$, and the two non-Abelian ideals, $\mathfrak{h}_s$ and $\mathfrak{h}_w$, whose orbits we considered above. It is not a coincidence that the two non-Abelian ideals are exactly the algebra of finite distance and asymptotic corners. Indeed, the property of being ideals of $\mathfrak{h}$ implies that one can reach them from a quotient of the original group $H$:
\begin{eqnarray}
H_0=H/GL(2,\mathbb{R}), \qquad H_w=H/SL(2,\mathbb{R}),\qquad H_s=H/W.
\end{eqnarray}
In this section, we will discuss how the $6$-dimensional tangent space to a generic point in $\mathfrak{h}^*$ can be regarded as containing complementary subspaces that can be associated to the orbits of $\mathfrak{h}_w^*$ and $\mathfrak{h}_s^*$, respectively.
We have of course an action of $\mathfrak{h}$ on $\mathfrak{h}^*$. We can ask how we might reduce to an action of the ideals $\mathfrak{h}_s$ and $\mathfrak{h}_w$ in a geometrically meaningful way. We can make progress in this direction by noting that the two ideals have Casimirs which are local functions on $\mathfrak{h}^*$. In the case of $\mathfrak{h}_s$, there is one such Casimir $C_3$ that we have seen in the above discussions, while for $\mathfrak{h}_w$ there are three Casimirs that we will introduce momentarily.
Let us first discuss the case of $\mathfrak{h}_s\subset\mathfrak{h}$. Formally, we can introduce the distribution $ker(\delta C_3)$, which we regard as a locally integrable distribution in $T\mathfrak{h}^*$; identifying $T_m\mathfrak{h}^*$ with $\mathfrak{h}$, this consists of all vectors $\un\mu$ that satisfy $\delta_{\un\mu}C_3=0$. Since
we know that $\delta_{\un\mu}C_3=wC_3$, we see that $\delta_{\un\mu}C_3$ vanishes when $w=0$, and so we can identify $ker(\delta C_3)\sim \mathfrak{h}_s$.
This defines a $5$-dimensional subspace of $T_m\mathfrak{h}^*$. We know from eq. \eqref{KKSdJdP} that on this subspace, the KKS form is singular and reduces to that of the intrinsic $\mathfrak{h}_s$ analysis, eq. \eqref{KKSintS}. This indicates that a $4$-dimensional subspace of $ker(\delta C_3)$ can be associated with an $\mathfrak{h}_s$ orbit. However, if we examine the action of $\mathfrak{h}_s$ on $\mathfrak{h}^*$, we find from \eqref{dJdP}
\begin{eqnarray}
\delta_{\un\mu}\bar J^a{}_b
= [\bar J,\bar\theta]^a{}_b -b^aP_b+\tfrac12 b^c P_c \delta^a{}_b ,
\qquad
\delta_{\un\mu} P_a = P_b\bar\theta^b{}_a,
\qquad
\delta_{\un\mu}{\cal J}
= -b^c P_c.
\end{eqnarray}
In this expression, we recover the intrinsic result \eqref{var2}. Moreover, whereas the trace ${\cal J}$ decouples from the KKS form when pulled back to $ker(\delta C_3)$, it is not invariant under $\mathfrak{h}_s$. This indicates that the orbit is not just a constant-${\cal J}$ slice through $\mathfrak{h}^*$.
Similarly, one can identify $\mathfrak{h}_w\subset \mathfrak{h}$ by constructing the $3$-dimensional distribution $ker(\delta C_1,\delta C_2,\delta C_2')$, where
\begin{eqnarray}
C_1:={P_1\over P_0},\qquad C_2:= C_1^{-1}J^0{}_1-J^0{}_0,\qquad C_2':= C_1J^1{}_0-J^1{}_1.
\end{eqnarray}
These three functions have been chosen such that $ker(\delta C_1,\delta C_2,\delta C_2')$ consists of those vectors $\un\mu$ for which $\bar\theta^a{}_b=0$. Indeed, calling $\theta_3={1\over 2}(\theta^0{}_0-\theta^1{}_1)$, one finds
\begin{eqnarray}
\delta_{\un\mu}\begin{pmatrix}C_1\cr C_2\cr C_2'\end{pmatrix}
&=& \begin{pmatrix}1&-C_1^2&-2C_1\cr
(C_2'-C_2)/C_1&0&0\cr 0&-(C_2'-C_2)C_1&0\end{pmatrix}
\begin{pmatrix}\theta^0{}_1\cr\theta^1{}_0\cr \theta_3\end{pmatrix}.
\end{eqnarray}
Since the determinant of the matrix appearing here is given by $2(C_2'-C_2)^2C_1$, generically the left-hand side vanishes only for $\bar\theta^a{}_b=0$, as required. Thus we conclude that $ker(\delta C_1,\delta C_2,\delta C_2')\sim \mathfrak{h}_w$, and $C_1,C_2,C_2'$ are Casimirs of $\mathfrak{h}_w$. In terms of these variables, the KKS form on $\mathfrak{h}^*$ can be written
\begin{eqnarray}
\Omega^{(\mathfrak{h})}_m(\un\mu,
\un\nu)=
-\delta_{\mu}\log C_1 \delta_{\nu} J_3
+\delta_{\mu} C_2\delta_{\nu}\log (C_2'-C_2)
+\tfrac12\delta_{\mu}\log (C_1P_0^2)\delta_{\nu}{\cal J}-(\un\mu\leftrightarrow\un\nu).
\end{eqnarray}
Thus we immediately see that this pulls back to $\delta\log P_0\wedge \delta {\cal J}$, which coincides with the KKS form on $\mathfrak{h}_w^*$. Similarly to what happens for $\mathfrak{h}_s$, while $J_3$ decouples from the KKS form, it is not invariant under $\mathfrak{h}_w$. Indeed, from \eqref{dJdP} reduced to $\mathfrak{h}_w$, we have
\begin{eqnarray}
\delta_{\un\mu} J_3
= -\tfrac12(b^0-b^1C_1)P_0,
\qquad
\delta_{\un\mu} {\cal J}
= -(b^0+b^1C_1)P_0,
\qquad
\delta_{\un\mu}\log P_0 = \tfrac12 w.
\end{eqnarray}
Finally, we note that the reduction to $\mathfrak{h}_0$ can be performed by considering the distribution corresponding to setting $\bar\theta^a{}_b=0$ and $w=0$; this however yields
\begin{eqnarray}
\Omega^{(\mathfrak{h})}_m(\un\mu,\un\nu)=0,
\end{eqnarray}
and the isotropy algebra coincides with the full algebra itself.
So we have seen that the three ideals of $\mathfrak{h}$ give rise to $4$-, $2$-, and $0$-dimensional immersions of their orbits at generic points. On the other hand, the generic orbits of $\mathfrak{h}$ itself are $6$-dimensional, and indeed there are no Casimirs. Casimir operators, and related quantum numbers, are at the core of the quantum representation theory, and we plan to address quantum features of this analysis in future publications.
We can now continue the discussion of the beginning of this subsection. We have shown that the $6$ directions $\{\delta_{\un\mu}P_a,\delta_{\un\mu}J^a{}_b\}$ are all independent at a generic point in $\mathfrak{h}^*$. Furthermore, we have seen that the orbits of the ideals in $\mathfrak{h}$ can be regarded as immersed in $\mathfrak{h}^*$.
The algebra $\mathfrak{h}$, and more generally the full UCS, serves as an organizing principle for the orbits of $\mathfrak{h}_s$ and $\mathfrak{h}_w$, which we have argued correspond to the physically relevant finite distance and asymptotic corners, respectively. More precisely, we should lift this analysis to the ECS and ACS, by including as well the semi-direct product with $\mathfrak{diff}(S)$.
A full accounting of the coadjoint orbits of UCS, ECS and ACS is beyond the scope of this paper. However, we believe that the simplified discussion given above of $\mathfrak{h},\mathfrak{h}_s$ and $\mathfrak{h}_w$ is helpful for organization. In the next section, we will add back in $\mathfrak{diff}(S)$ and thus discuss the full UCS. We will find that this can be usefully organized in terms of certain Lie algebroids over $S$.
Importantly, our analysis will suggest that there may be a useful semi-classical construction of spacetime geometries making use of these algebroid structures that have support on corners. To make such a contact with classical physics, we will build moment maps pertaining to orbits of the ECS and ACS.
\section{UCS and Its Algebroid Interpretation}\label{sec4}
Restoring the $\mathfrak{diff}(S)$ part of the UCS has dramatic effects on the coadjoint analysis. After discussing them, we will show how algebroids offer the natural playground to geometrically describe the UCS over a corner. Associated bundles to the algebroid give various representations of the UCS, including what we identify as the classical spacetime representation. We then reach the right point to introduce moment maps, and discuss how the dual of the UCS algebroid contains the image of both the ECS and ACS moment maps.
\subsection{Reintroduction of $\mathfrak{diff}(S)$}\label{sec4.1}
We would like now to reintroduce the $\mathfrak{diff}(S)$ part of the algebra. Given that it acts on everything to its right, but it is not acted upon, due to the semi-direct structure, the analysis carried so far goes through, with however the important modification that elements on $\mathfrak{h}$ are now valued on $S$. In particular, the UCS, ECS, and ACS are obtained from $\mathfrak{h}$, $\mathfrak{h}_s$, and $\mathfrak{h}_w$ by adding $\mathfrak{diff}(S)$ and making the latter algebras local, respectively. The UCS is the biggest, and the ECS and ACS are subalgebras inside it. The ECS and ACS are not however ideals, because $\mathfrak{diff}(S)$ acts non-trivially on the complementary elements of the UCS in these two reductions. This has important repercussions on the KKS form, that we detail below.
In this context, the dual space UCS$^*$ is coordinatized by a one-form $\alpha=\alpha_\beta(\sigma)d\sigma^\beta$ on $S$ together with an element of $\mathfrak{h}^*$ with values in $S$, that is, calling $M\in$ UCS$^*$,
\begin{eqnarray}
M=\alpha_\beta(\sigma) d\sigma^\beta+J^a{}_b(\sigma) t^b{}_a+ P_a(\sigma) t^a.
\end{eqnarray}
Similarly, an element $\un\mathfrak{X}\in $UCS is given by a vector field on $S$ together with an element of $\mathfrak{h}$ valued on $S$,
\begin{eqnarray}
\un\mathfrak{X}=\xi^\beta(\sigma)\un\partial_\beta+\theta^a{}_b(\sigma)\un t^b{}_a+ b^a(\sigma)\un t_a.
\end{eqnarray}
One can then act with $M$ on $\un\mathfrak{X}$ producing a function on $S$ which evaluates to
\begin{eqnarray}
M(\un\mathfrak{X})=i_{\un\xi}\alpha+\theta^a{}_bJ^b{}_a+b^aP_a,
\end{eqnarray}
where the explicit $\sigma$ dependence is from now on implicitly assumed, and we have introduced the interior product $i_{\un\xi}$ on $S$.
The invariant pairing $\langle \ , \rangle:$ UCS$^*\otimes$ UCS $\to \mathbb{R}$ is then introduced as
\begin{eqnarray}\label{ucspair}
\langle M,\un\mathfrak{X}\rangle=\int_S vol_S\, M(\un\mathfrak{X}),
\end{eqnarray}
where $vol_S$ is a volume form on the corner $S$. Once a field space is introduced, $M(\un\mathfrak{X})$ will be the image in UCS$^*$ of a local charge aspect via a moment map. On the other hand, the invariant pairing is the image under the moment map of an integrated charge on the corner. We will discuss this further in Subsection \ref{sec4.3}.
Now, the adjoint action of the Lie algebra UCS on itself is given by
\begin{eqnarray}
ad_{\un\mathfrak{X}}\un\mathfrak{Y}:= [\un\mathfrak{X},\un\mathfrak{Y}].
\end{eqnarray}
Correspondingly, the coadjoint action is defined through the pairing given above,
\begin{eqnarray}\label{adad}
\langle ad^*_{\un\mathfrak{X}}M,\un\mathfrak{Y}\rangle =- \langle M, ad_{\un\mathfrak{X}}\un\mathfrak{Y}\rangle.
\end{eqnarray}
Thanks to this pairing, we extend the coadjoint computations on $\mathfrak{h}^*$ to UCS$^*$, with some important limitations. Indeed, as we will discuss further in the subsequent section, we will interpret a generator of the UCS geometrically as a section of an algebroid rather than an algebra. While in the latter we can descend to a group analysis, one should generalize certain results to groupoids in order to do so here. We expect this to be a promising avenue of research, but would go beyond the scope of this paper.
Given $M$ in UCS$^*$, we can define from \eqref{adad} the map $ad^*_{\un\mathfrak{X}} :$ UCS$^*\to$ UCS$^*$. Elements of the UCS whose coadjoint action is trivial defines the stabilizer subalgebroid
\begin{eqnarray}
{\cal S}_M=\{\un\mathfrak{X} \in \text{UCS} \ \vert \ ad^*_{\un\mathfrak{X}}M=M\}.
\end{eqnarray}
Contrarily, one can define the UCS coadjoint orbit algebroid of $M$ as
\begin{eqnarray}
{\cal O}_M=\{\un\mathfrak{X}\in \text{UCS} \ \vert \ ad^*_{\un\mathfrak{X}}M\neq M \}.
\end{eqnarray}
The integral lines of these elements of the UCS then define the coadjoint orbits. Although we are here talking about the local tangent plane at the point in UCS$^*$, and confine our attention to the local action, we will colloquially refer to ${\cal O}_M$ as the coadjoint orbit.
As in the case of $\mathfrak{h}^*$, one can here perform a unified treatment of orbits and stabilizers using the KKS symplectic $2$-form:
\begin{eqnarray}\label{ucskks2}
\Omega_M: {\cal O}_M\otimes {\cal O}_M\to \mathbb{R}, \qquad \Omega_M(\un\mathfrak{X},\un\mathfrak{Y})=\langle M, [\un\mathfrak{X},\un\mathfrak{Y}]\rangle.
\end{eqnarray}
By definition, the KKS $2$-form is non-degenerate on the orbits. The analysis of the various orbits and different points in UCS$^*$ reduces then to a full characterization of $\Omega$.
There is already a hint that the underlying structure is an algebroid at the algebraic level. Indeed, one observes that the basis $\{\un\partial_\beta,\un t^a{}_b,\un t_a\}$ of the UCS satisfies\footnote{As we will discuss further below, this holds in a local trivialization, so only in a local patch.}
\begin{eqnarray}\label{comm}
\left[\un\partial_\beta,\un \partial_\gamma\right]=\left[\un\partial_\beta,\un t^a{}_b\right]=\left[\un\partial_\beta,\un t_a\right]=0,\quad \left[\un t^a{}_b,\un t^c{}_d\right]
=
\delta^c{}_b\un t^a{}_d -\delta^a{}_d\un t^c{}_b,
\quad
\left[\un t^a{}_b,\un t_c\right]=-\pi^a{}_c{}^d{}_b\un t_d,
\quad
\left[\un t_b,\un t_c\right]=\un 0,
\end{eqnarray}
which tells us that the non-trivial $\mathfrak{diff}(S)$ action comes entirely from the explicit $\sigma$ dependence of the components $\alpha_\beta$, $\theta^a{}_b$, and $b^a$. From this perspective, $\mathfrak{diff}(S)$ plays a special role, and indeed we will associate it to the reparameterization invariance of the base, in the algebroid picture below.
Using \eqref{comm}, the Lie bracket of two elements $\un\mathfrak{X}=\xi^\beta(\sigma)\un\partial_\beta+\theta^a{}_b(\sigma)\un t^b{}_a+ b^a(\sigma)\un t_a$,
and $\un\mathfrak{Y}=\zeta^\beta(\sigma)\un\partial_\beta+\theta'^a{}_b(\sigma)\un t^b{}_a+ b'^a(\sigma)\un t_a$ of the UCS is given by
\begin{eqnarray}
ad_{\un\mathfrak{X}}\un\mathfrak{Y}=[\un\mathfrak{X},\un\mathfrak{Y}]=
[\un\xi,\un\zeta]^\beta\un\partial_\beta+\big(\un\xi(\theta'^b{}_a)-\un\zeta(\theta^b{}_a)-[\theta,\theta']^b{}_a\big)\un t^a{}_b+\big(\un\xi(b'^a)-\un\zeta(b^a)-\theta^a{}_bb'{}^b-\theta'{}^a{}_bb^b\big)\un t_a,
\end{eqnarray}
which, upon applying the defining representation \eqref{defrep}, is by construction equal to \eqref{LieMsubexpl}. From this we evaluate
\begin{eqnarray}\label{varAlg}
\langle ad^*_{\un\mathfrak{X}}M,\un\mathfrak{Y}\rangle=-\langle M,ad_{\un\mathfrak{X}}\un\mathfrak{Y}\rangle
&=&\int_S vol_S\Big(-i_{[\un\xi,\un\zeta]}\alpha+J^a{}_b\big(-\un\xi(\theta'^b{}_a)+\un\zeta(\theta^b{}_a)+ [\theta,\theta']^b{}_a
\big)\nonumber\\
&&+ P_a(-\un\xi(b'^a)+\un\zeta(b^a)+\theta^a{}_bb'{}^b-\theta'{}^a{}_bb^b)\Big).
\end{eqnarray}
We would like to obtain the variations $\{\delta_{\un\mathfrak{X}}\alpha,\delta_{\un\mathfrak{X}}J^a{}_b,\delta_{\un\mathfrak{X}}P_a\}$ from this expression. Special attention should be devoted to $vol_S$, because it transforms under $\delta_{\un\mathfrak{X}}$ non-trivially. So in order to read the variations one should compare \eqref{varAlg} with\footnote{Perhaps a better notation here would be to write \[ \langle *M,\un\mathfrak{Y}\rangle=\int_S *M(\un\mathfrak{Y}),\] and then interpret the variation as $\langle \delta_{\un\mathfrak{X}}*M,\un\mathfrak{X}\rangle$. That is, the integral involves the top form dual to $M(\un\mathfrak{Y})$ and we are varying that. In \cite{Donnelly:2020xgu}, a similar analysis was performed, but densities were introduced rather than tensors, to incorporate the $vol_S$ contributions.}
\begin{eqnarray}
\delta_{\un\mathfrak{X}}\langle M,\un\mathfrak{Y}\rangle
&=&\int_S \Big(\zeta^\beta\delta_{\un\mathfrak{X}}(\alpha_\beta vol_S)+\theta'^a{}_b\delta_{\un\mathfrak{X}}(J^b{}_a vol_S)+b'^a\delta_{\un\mathfrak{X}}(P_a vol_S)\Big),
\end{eqnarray}
where $\zeta^\beta\delta_{\un\mathfrak{X}}(\alpha_\beta vol_S)\equiv (i_{\un\zeta}\delta_{\un\mathfrak{X}}\alpha)vol_S+(i_{\un\zeta}\alpha)\delta_{\un\mathfrak{X}}vol_S$.
Using $\delta_{\un\mathfrak{X}}vol_S=\nabla_\beta\xi^\beta vol_S$, and assuming $S$ has no boundary, we find the variations
\begin{eqnarray}\label{UCSdeltaalpha}
\delta_{\un\mathfrak{X}}\alpha&=& {\cal L}_{\un\xi}\alpha+J^a{}_bd\theta^b{}_a+P_a db^a,\\
\delta_{\un\mathfrak{X}}J^a{}_b&=&\un\xi(J^a{}_b)+[J,\theta]^a{}_b -b^aP_b,
\label{UCSdeltaJ}\\
\delta_{\un\mathfrak{X}}P_a&=&\un\xi(P_a)+P_b\theta^b{}_a.\label{UCSdeltaP}
\end{eqnarray}
In this expression, we introduced the exterior derivative $d$ on $S$ and the associated Lie bracket acting on forms as ${\cal L}=id+di$. In addition to the assumption that $S$ is without boundary, we have also assumed smoothness, necessarily required for the manipulations performed here.
In our previous discussion of the coadjoint orbits of $\mathfrak{h}$, we were able to convert the KKS form from a function of the transformation parameters to a function of the variations of the covector space (see for example eq. \eqref{KKS} versus \eqref{KKSdJdP}). In the present case (including diffeomorphisms) such a process is much more challenging as it would inevitably be non-local on $S$. In the following subsection, we will reinterpret the current construction in terms of a certain Atiyah Lie algebroid. It is perhaps then the case that one could manage the conversion via the introduction of corresponding groupoids, but this is beyond the scope of the present paper. Nevertheless, we take this as a strong indication that the proper local geometric structure on $S$ is an algebroid over $S$. As we describe in the next subsection, this will allow for a completely geometric account of the representation theory of the ECS and ACS.
Indeed, instead of attempting the aformentioned conversion process, we continue by considering the restriction of the full KKS form
\begin{eqnarray}\label{KKSUCS}
\Omega(\un\mathfrak{X},\un\mathfrak{Y})=\int_S vol_S\Big(i_{[\un\xi,\un\zeta]}\alpha+J^a{}_b\big(\un\xi(\theta'^b{}_a)-\un\zeta(\theta^b{}_a)- [\theta,\theta']^b{}_a
\big)+ P_a(\un\xi(b'^a)-\un\zeta(b^a)-\theta^a{}_bb'{}^b+\theta'{}^a{}_bb^b)\Big)
\end{eqnarray}
to the ECS and ACS subalgebras. By such restrictions, we mean the analogues of the discussion in Section \ref{sec3}. These are restrictions on the $(\alpha,\theta,b)$ in eqs. (\ref{UCSdeltaalpha}--\ref{UCSdeltaP}). For ECS, we obtain
\begin{eqnarray}\label{ECSdeltaalpha}
\delta_{\un\mathfrak{X}_s}\alpha&=& {\cal L}_{\un\xi}\alpha+\bar J^a{}_bd\bar\theta^b{}_a+P_a db^a,\\
\delta_{\un\mathfrak{X}_s}\bar J^a{}_b&=&\un\xi(\bar J^a{}_b)+[\bar J,\bar\theta]^a{}_b -b^aP_b+\tfrac12 b\cdot P \delta^a{}_b,
\label{ECSdeltaJ}\\
\delta_{\un\mathfrak{X}_s}P_a&=&\un\xi(P_a)+P_b\bar\theta^b{}_a,\label{ECSdeltaP}
\end{eqnarray}
while for ACS we have
\begin{eqnarray}\label{ACSdeltaalpha}
\delta_{\un\mathfrak{X}_w}\alpha&=& {\cal L}_{\un\xi}\alpha+\tfrac12{\cal J}dw+P_a db^a,\\
\delta_{\un\mathfrak{X}_w}{\cal J}&=&\un\xi({\cal J}) -b^a P_a,
\label{ACSdeltaJ}\\
\delta_{\un\mathfrak{X}_w}P_a&=&\un\xi(P_a)+\tfrac12 wP_b.\label{ACSdeltaP}
\end{eqnarray}
We would like to stress the importance of these equations. They show how any field variation dictated by the ECS and the ACS are realized at points of UCS$^*$. This is the analogous of the discussion for $\mathfrak{h}$ and its two sub-algebras $\mathfrak{h}_s$ and $\mathfrak{h}_w$ in the previous section. We will now show how these results are accommodated into the general theory of Atiyah Lie algebroids.
\subsection{Lie Algebroids}\label{sec4.2}
In the previous section we have introduced the invariant pairing
\begin{eqnarray}\label{pair}
\langle M,\un\mathfrak{X}\rangle=\int_S vol_S\, M(\un\mathfrak{X}),
\end{eqnarray}
where
\begin{eqnarray}\label{localmXM}
\un\mathfrak{X}=\xi^\alpha\un\partial_\alpha+\theta^a{}_b\un t^b{}_a+b^a\un t_a,\qquad
M(\un\mathfrak{X})=i_{\un\xi}\alpha+\theta^a{}_bJ^b{}_a+b^aP_a.
\end{eqnarray}
We have also seen how introducing $\mathfrak{diff}(S)$ complicates the algebraic analysis, and calls for a deeper geometric understanding of the UCS.
In this section, we note that $\un\mathfrak{X}$ can be interpreted as a section of an Atiyah Lie algebroid \cite{Atiyah:1957,Atiyah:1979iu} over $S$, associated to the group $H=GL(2,\mathbb{R})\ltimes \mathbb{R}^2$, in terms of which eqs. \eqref{localmXM} can be interpreted as formulas valid within a local trivialization. The construction is as follows (details can be found in our recent exploratory paper \cite{Ciambelli:2021ujl}, from which most conventions are taken). We first introduce the principal $H$-bundle $\pi:P_c\to S$. The corresponding Atiyah Lie algebroid is obtained as the quotient of $TP_c$ by the right action of $H$, $A_c=TP_c/H$. This is a vector bundle of rank $n+6$ over $S$ ($n=\dim S$) that is equipped with an anchor map $\rho:A_c\to TS$ whose kernel is the vertical sub-bundle of $A_c$ isomorphic to the adjoint bundle $L_c=P_c\times_{Ad_H} \mathfrak{h}$. That is, there is a short exact sequence
\begin{eqnarray}\label{shortExactSeq}
\begin{tikzcd}
0
\arrow{r}
&
L_c
\arrow{r}{j}
&
A_c
\arrow{r}{\rho}
&
TS
\arrow{r}
&
0
\end{tikzcd}.
\end{eqnarray}
Locally, we can think of $L_c$ as having fibres $\mathfrak{h}$, and thus the value of a section of $L_c$ at a point $p\in U\subset S$ is an element of the Lie algebra $\un\mu_p\in\mathfrak{h}$. A local trivialization of $A_c$ on $U\subset S$ is a map $\tau:A_c|_U\to TU\times L_c|_U$, and so we can always think of a section $\un\mathfrak{X}$ of $A_c$ as given locally by a vector on $S$ together with an element of the Lie algebra, that is
\begin{eqnarray}
\tau(\un\mathfrak{X})=\xi^\alpha\un\partial_\alpha+\mu^A\un t_A,
\end{eqnarray}
which explicitly associates a section of $A_c$ with an element of the UCS, that is $\un\mathfrak{X}_p \to (\un\xi_p,\un\mu_p)$. Here, we reiterate that $\sigma^\alpha, (\alpha=1,\dots,n)$ are local coordinates on $U$ and we introduced $\un t_A, (A=1,\dots,6)$, a local frame for $L_c$.
Each of the vector bundles described above is equipped with a skew bracket: $L_c$ has the Lie algebra bracket $[\cdot,\cdot]_L$, $TS$ the Lie bracket of vector fields $[\cdot,\cdot]$, while the bracket $[\cdot,\cdot]_A$ on $A_c$ is such that $\rho$ and $j$ are morphisms, that is, given two (local) sections $\un\mathfrak{X}$ and $\un\mathfrak{Y}$ of $A_c$ and two sections $\un\mu,\un\nu$ of $L_c$, we have
\begin{eqnarray}
\rho([\un\mathfrak{X},\un\mathfrak{Y}]_A)=[\rho(\un\mathfrak{X}),\rho(\un\mathfrak{Y})],\qquad
j([\un\mu,\un\nu]_L)=[j(\un\mu),j(\un\nu)]_A.
\end{eqnarray}
A connection on the algebroid $A_c$ is given by a map $\sigma:TS\to A_c$ (not to be confused with the local coordinates $\sigma^\alpha$ on $S$) and $\omega:A_c\to L_c$ with $\omega\circ\sigma=0$. The curvature of the connection is a measure of the failure of $\sigma$ and $-\omega$ to be morphisms of the brackets; we again refer to \cite{Ciambelli:2021ujl} for more details.
The map $\sigma$ can be interpreted as an Ehresmann connection, which provides a lift of a vector field in $TS$ to a section of the horizontal sub-bundle $H_c$ of $A_c$, such that $A_c=H_c\oplus V_c$. In a local trivialization, we can write a `split' basis of sections of $H_c$, denoted $\un E_{\un\alpha}$ with $({\un \alpha}=1,\dots,n)$, and of sections of $V_c$, denoted $\un E_{\un A}$ with $({\un A}=1,\dots,6)$, as
\begin{eqnarray}\label{trivbasisA}
\tau(\un E_{\un\alpha})=\tau(\sigma)^\alpha{}_{\un\alpha} (\un\partial_\alpha+ a^A_\alpha(\sigma) \un t_A),\qquad
\tau(\un E_{\un A})=\tau(\sigma)^A{}_{\un A} \un t_A,
\end{eqnarray}
respectively. Note that here we are using a short form notation $\un t_A$, but in the case at hand, we could also use the notation $(\un t^a{}_b,\un t_a)$. Similarly, we have written the connection coefficients here as $a^A_{\alpha}$ but if we again regard the $\un t_A$ as the pair $(\un t^a{}_b,\un t_a)$, we have a pair of gauge fields $(a_{\alpha}^{(0)}{}^a,a_{\alpha}^{(1)}{}^a{}_b)$. These form an $H$-connection, in the sense that on an overlap $U_i\cap U_j$, we have
\begin{eqnarray}
(a_{i,\alpha}^{(0)}{}^a,a_{i,\alpha}^{(1)}{}^a{}_b)=J^\beta{}_\alpha
(R^{-1})^a{}_c (a_{j,\beta}^{(0)}{}^c+\partial_\beta b^c-a_{j,\beta}^{(1)}{}^c{}_db^d,
-\partial_\beta R^c{}_b+a_{j,\beta}^{(1)}{}^c{}_d R^d{}_b).
\end{eqnarray}
Furthermore, what we previously called $M\in$ UCS$^*$ can now be interpreted as a section of the dual bundle $A_c^*$. As usual, a section of the dual bundle, $M\in\Gamma(A_c^*)$ is defined as a map $M:A_c\to C^\infty(S)$. Thus,
we can import all the results of Subsection \ref{sec4.1}, and re-interpret them on this Atiyah Lie algebroid.
The basis $\{E^{\un\alpha},E^{\un A}\}$ for $A_c^*$ that is dual to \eqref{trivbasisA} is given in a local trivialization by\footnote{In Ref. \cite{Ciambelli:2021ujl}, we denoted the inverse map $\tau^*:A_c^*|_U\to T^*U\oplus L_c^*|_U$. To avoid confusion with the algebroid pullback, we now refer to this map as $\bar\tau$ and denote its matrix elements by $\tau^{-1}$.
Also as described in \cite{Ciambelli:2021ujl}, we enforce $\tau\circ j = Id_{L|_U}$. }
\begin{eqnarray}\label{trivbasisAdual}
\bar\tau(E^{\un\alpha})=(\tau^{-1}(\sigma))^{\un\alpha}{}_\alpha d\sigma^\alpha,\qquad
\bar\tau(E^{\un A})=(\tau^{-1}(\sigma))^{\un A}{}_A(t^A-a^A_\alpha(\sigma)d\sigma^\alpha).
\end{eqnarray}
The coadjoint orbit analysis discussed previously can now be formulated more geometrically in terms of $A_c^*$, rather than UCS$^*$. The purpose of the orbit analysis is to better understand representations.
The algebroid geometric structure is well-suited to do so, for any representation of $\mathfrak{h}$ corresponds to an associated bundle $E\to S$ whose fibres correspond to the representation space. Furthermore, one can construct a notion of differentiation of sections of $E$ by establishing a morphism between $A_c$ and another Lie algebroid, the algebroid of derivations of sections of $E$, called $Der(E)$, such that
\begin{eqnarray}\label{DerEshortExactSeqGen}
\begin{tikzcd}
&
L_c
\arrow{r}{j}
\arrow{dd}{v_E}
&
A_c
\arrow{dr}{\rho}
\arrow{dd}{\phi_E}
&
&
\\
0
\arrow{ur}
\arrow{dr}
&&&
TS
\arrow{r}
&
0
\\
&
End(E)
\arrow{r}{j_E}
&
Der(E)
\arrow{ur}{\rho_E}
&
&
\end{tikzcd}.
\end{eqnarray}
For $\psi\in\Gamma(E)$ a local section of $E$, one then has $\phi_E(\un\mathfrak{X})(\psi)=\hat{d}\psi(\un\mathfrak{X})$, with $\hat{d}\psi\in\Gamma(A_c^*\times E)$. This can be extended to sections of $\wedge^kA_c^*\times E$, with $\hat{d}:\wedge^kA_c^*\times E\to \wedge^{k+1}A_c^*\times E$. Given a connection on $A_c$, there is an induced connection on $Der(E)$ such that the horizontal part of $\hat{d}\psi$ can be interpreted as a covariant derivative, $\hat{d}\psi(\un\mathfrak{X}_H)=\nabla^E_{\rho(\un\mathfrak{X})}\psi$. In the diagram above, the $End(E)$ can be thought of as giving rise to a matrix representation of $\mathfrak{h}$.
Let us explore possible representations in our current context.
In most applications, the associated bundles are taken to be vector bundles, whose transition functions are linear maps.
In the case at hand, where $H=GL(2,\mathbb{R})\ltimes \mathbb{R}^2$, there is an important associated bundle $\pi_{{\cal B}}:{{\cal B}}\to S$ which is a rank-2 {\it affine} bundle. This bundle is said to be modeled on a rank-2 vector bundle (with fibre $V_{{\cal B}}$), with transition functions given by affine maps. That is for $U_i,U_j\subset S$, a local section $\psi\in\Gamma({{\cal B}})$ satisfies (a,b=1,2)
\begin{eqnarray}\label{transfnB}
(\sigma_i^\alpha,\psi^a_i)=(\sigma_i^\alpha(\sigma_j),R_{ij}{}^a{}_b\psi^b_j+b_{ij}{}^a),
\end{eqnarray}
on $U_i\cap U_j$,
where we are writing the components of the section with respect to a basis for $V_{{\cal B}}$. Thus the transition functions are determined by $(R_{ij}{}^a{}_b, b_{ij}{}^a)$, which indeed correspond to an element of $H$. Infinitesimally, we have
\begin{eqnarray}\label{transfnBinf}
(\sigma_i^\alpha,\psi^a_i(\sigma_i))=(\sigma_j^\alpha-\xi^\alpha(\sigma_j),{\cal L}_{\un\xi}\psi^a_j(\sigma_j)+\theta^a{}_b(\sigma_j)\psi^b_j(\sigma_j)+b^a(\sigma_j)),
\end{eqnarray}
so we see that the infinitesimal transformation of $\psi^a$ is given by a local element of the UCS. The affine bundle might be referred to as the fundamental representation.
Note that if we had called the section $u\in\Gamma({{\cal B}})$, we could regard the components $u^a$ as local fibre coordinates, and then \eqref{transfnB} appear as diffeomorphisms of the total space of ${{\cal B}}$, restricted to lie in $H$. We will argue later that there is a `semi-classical' correspondence between the bundle ${{\cal B}}$ and a spacetime $M$, near the corner, in the sense that the restricted diffeomorphisms are precisely those that have non-zero charges in a classical theory. This is the reason why we focus on this particular representation in the rest of this subsection.
The corresponding structure for the dual bundles is given by
\begin{eqnarray}\label{DerEshortExactSeqGen}
\begin{tikzcd}
&
L_c^*
\arrow{dl}
&
A_c^*
\arrow{l}{j^*}
&
&
\\
0
&&&
T^*S
\arrow{dl}{\rho_{{\cal B}}^*}
\arrow{ul}{\rho^*}
&
0
\arrow{l}
\\
&
End({{\cal B}})^*
\arrow{uu}{v_{{\cal B}}^*}
\arrow{ul}
&
Der({{\cal B}})^*
\arrow{l}{j_{{\cal B}}^*}
\arrow{uu}{\phi_{{\cal B}}^*}
&
&
\end{tikzcd}.\label{derBdiagram}
\end{eqnarray}
Consider a section $M$ of $A_c^*$. In a local split basis, we can write $M=M_{\un \alpha}E^{\un\alpha}+M_{\un A}E^{\un A}$. Given the above discussion, in a local trivialization, one then has
\begin{eqnarray}\label{tauM}
\bar\tau (M)&=&M_{\un A}(\tau^{-1})^{\un A}{}_A(t^A-a^A_\alpha d\sigma^\alpha)+M_{\un\alpha} (\tau^{-1})^{\un\alpha}{}_\alpha d\sigma^\alpha\nonumber\\
&\equiv &
M_{ A}(t^A-a^A_\alpha d\sigma^\alpha)+M_{\alpha} d\sigma^\alpha.\nonumber\\
&=&M^b{}_at^a{}_b+M_at^a
+(M_{\alpha}-a^{(0)}_\alpha{}^aM_a-a^{(1)}_\alpha{}^a{}_bM^b{}_a) d\sigma^\alpha.
\end{eqnarray}
The first three terms in the last line of this expression relate to $GL(2,\mathbb{R})$, $\mathbb{R}^2$ and $Diff(S)$ respectively and they coincide in form with the Noether charge aspects found on a finite distance corner in a classical spacetime \cite{Ciambelli:2021vnn}, we will expand on this in the next subsection.
The last term in \eqref{tauM} deserves some comment. If we compare this result to that found in the Einstein-Hilbert theory on a classical spacetime \cite{Ciambelli:2021vnn}, in the latter case, only the third term, involving $a^{(1)}$, appeared. This can be explained first by the fact that the embedding condition implied that $a^{(0)}$ pulls back to zero on $S$, while the $Diff(S)$ that appeared there was actually associated with changes of coordinates for the embedding $\phi(S)$ as opposed to changes of the intrinsic coordinates on $S$ (which would be pure gauge from the point of view of the bulk theory defined on the spacetime manifold).
We can then re-interpret these restrictions on the section of $A^*_c$ as a sort of 'gauge fixing' on the various fundamental fields.
We would like to rewrite \eqref{tauM} in terms of the associated bundle ${{\cal B}}$. Denoting the corresponding section of $Der({{\cal B}})^*$ by $M_{{\cal B}}$, we have $\phi_{{\cal B}}^*(M_{{\cal B}})=M$.
Now, given the local trivialization $\bar\tau$ of $A_c^*$, there is a corresponding trivialization $\bar\tau_{{\cal B}}:Der({{\cal B}})^*|_U\to T^*U\oplus End({{\cal B}})^*|_U$, satisfying $\bar\tau\circ\phi_{{\cal B}}^*=(1\otimes v_{{\cal B}}^*)\circ\bar\tau_{{\cal B}}$.
We can take a basis for $Der({{\cal B}})$ as $(\un F_{\un{\alpha}},\un F_{\un{A}})=(\phi_{{\cal B}}(\un E_{\un\alpha}),\phi_{{\cal B}}(\un E_{\un A}))$,
and the dual basis for $Der({{\cal B}})^*$ is then $(F^{\un\alpha},F^{\un A})$. We thus obtain
\begin{eqnarray}
\bar\tau_{{\cal B}}(F^{\un\alpha})=(\tau^{-1}(\sigma))^{\un\alpha}{}_\alpha d\sigma^\alpha,\qquad
\bar\tau_{{\cal B}}(F^{\un A})=(\tau^{-1}(\sigma))^{\un A}{}_A(v^A-a^A_\alpha(\sigma)d\sigma^\alpha),
\end{eqnarray}
where the upper triangular matrices $v^A$ are a basis for $End({{\cal B}})^*$, such that $v_{{\cal B}}^*(v^A)=t^A$, with $v^A(v_{{\cal B}}(\un t_C))=\delta^A{}_C$. The latter can be interpreted as a matrix trace.
Then, we can eventually write \eqref{tauM} from the point of view of ${{\cal B}}$ as
\begin{eqnarray}
\bar\tau_{{\cal B}} (M_{{\cal B}})&=&
M_{{\cal B}}{}_{\un\alpha}(\tau^{-1}(\sigma))^{\un\alpha}{}_\alpha d\sigma^\alpha
+M_{{\cal B}}{}_{\un A}(\tau^{-1}(\sigma))^{\un A}{}_A(v^A-a^A_\alpha(\sigma)d\sigma^\alpha)\nonumber\\
&\equiv&
M_{{\cal B}}{}_{\alpha} d\sigma^\alpha
+M_{{\cal B}}{}_{A}(v^A-a^A_\alpha(\sigma)d\sigma^\alpha)
\nonumber\\
&=&M_{{\cal B}}{}^b{}_av^a{}_b+M_{{\cal B}}{}_av^a
+(M_{{\cal B}}{}_{\alpha}-a^{(0)}_\alpha{}^aM_{{\cal B}}{}_a-a^{(1)}_\alpha{}^a{}_bM_{{\cal B}}{}^b{}_a) d\sigma^\alpha.\label{tauM1}
\end{eqnarray}
This result is the starting point to make contact with the classical spacetime representation. Indeed, if we start with $A_c^*$ alone, there is no notion of normal coordinates and classical spacetime reconstruction. it is only via the affine bundle ${{\cal B}}$ that these quantities appear. In our endeavor to understand gravity better, ${{\cal B}}$ is the basic ingredient to probe classical aspects and representations of the UCS.
Since this subsection contains various new perspectives on the corner proposal, we would like to offer a brief summary before continuing. The rewriting of the UCS in terms of an Atiyah Lie algebroid over $S$ sheds light on its geometric aspects, and allows the inclusion of $\mathfrak{diff}(S)$ in a natural way, as the reparameterization invariance of the base manifold. Clearly, the issue faced in the previous section in trying to invert the KKS 2-form still persists, and a full classification of the UCS representations is for now out of reach. Nonetheless, we now have a different way to appreciate certain representations of the UCS, via the construction of associated bundles to the algebroid. We have in particular identified one representation, on an affine bundle modeled on a rank-$2$ vector bundle, in which $2$ 'normal' directions naturally appear on the fibres. We have moreover seen that the components of a section $M$ of $A^*_c$, or the corresponding section $M_B$ of $Der({{\cal B}})^*$, have the same structure as the fields that enter into the non-zero charges of a finite distance corner on a classical gravitational spacetime. This will become more precise in the next section when we discuss moment maps, which relate classical on-shell field spaces to specific orbits inside $A_c^*$. So there are two main results in this subsection. The first is the construction of $A^*_c$, which contains the image of both finite distance and asymptotic corners moment maps. The second is the identification of ${{\cal B}}$ as the first step in the reconstruction of a classical spacetime, if the starting point is $A^*_c$. In the corner proposal, we in fact stipulate that corners are the atomic constituents of gravity. We advocate that this is the correct path toward a better understanding of gravity. In particular, other representations of $A^*_c$ might be relevant, and could instruct us about quantum properties of the theory.
\subsection{Moment Map}\label{sec4.3}
So far, we have extrapolated the UCS from classical gravity and then worked only intrinsically to the corner. We now connect the classical field space to the dual algebroid $A^*_c$ via the moment map. Before doing so in specific examples, we offer the general analysis.
Calling $\chi$ the set of physical classical fields of a dynamical theory, the moment map is the map $\mu:\chi\to A^*_c$ that links these fields to a point in $A^*_c$. Then, the symmetry algebra acts on the tangent space at the point and infinitesimally moves it on the orbit via the coadjoint action \eqref{adad}. The orbit is a symplectic manifold whose non-degenerate $2$-form is precisely the KKS $2$-form \eqref{ucskks2}.
The moment map applies only to symplectomorphisms, and therefore, it is properly defined only for integrable Noether charges. That is, calling $H_{\un\mathfrak{X}}$ the Noether charge of a given dynamical theory with symplectic $2$-form $\Omega$, we have
\begin{eqnarray}\label{int}
\delta H_{\un\mathfrak{X}}=-I_{V_{\un\mathfrak{X}}}\Omega,
\end{eqnarray}
where $V_{\un\mathfrak{X}}$ is the symplectomorphism associated to the symmetry generator $\un\mathfrak{X}$ and $I$ and $\delta$ are the interior product and exterior derivative on $\chi$, respectively. Our extended phase space of gravity proposed in \cite{Ciambelli:2021vnn,Ciambelli:2021nmv} (see also \cite{Freidel:2021dxw}) is such that all gravitational diffeomorphism surface charges are integrable, and thus the whole field space can be mapped to the dual algebroid $A_c^*$. We consider this a very important improvement, whose consequences are yet to be fully explored. We will return to this discussion momentarily.
In practice, the identification of the moment map is performed via the pairing introduced in \eqref{ucspair}. Using the notation $\mu(\chi)=M$, one has that
\begin{eqnarray}\label{mupair}
\langle\mu(\chi),\un \mathfrak{X}\rangle=\int_S vol_S\, \mu(\chi)(\un\mathfrak{X})\equiv \mu(H_{\un\mathfrak{X}}),
\end{eqnarray}
where in the last equality we introduced a compact, yet slightly abusive, notation. Then, using this notation and requiring that the moment map relates the symplectic $2$-form $\Omega$ on $\chi$ to the KKS $2$-form $\Omega_{\mu(\chi)}$ at the point $\mu(\chi)\in A^*_c$, that is,\footnote{In this equation and henceforth, we are using the notation $\delta_{\un\mathfrak{X}}$ to denote the field space variation along $V_{\un\mathfrak{X}}$, that is, the field space Lie derivative $I_{V_{\un\mathfrak{X}}}\delta+\delta I_{V_{\un\mathfrak{X}}}$. This should not be confused with our previous use of the symbol in Section \ref{sec4.1}. }
\begin{eqnarray}
\Omega_{\mu(\chi)}(\un\mathfrak{X},\un\mathfrak{Y})=-\langle ad^*_{\un\mathfrak{X}}(\mu(\chi)),\un\mathfrak{Y})\rangle
= \mu(H_{[\un\mathfrak{X},\un\mathfrak{Y}]_A})
=-\mu(\delta_{\un\mathfrak{X}}H_{\un\mathfrak{Y}})=\mu(I_{V_{\un\mathfrak{X}}}I_{V_{\un\mathfrak{Y}}}\Omega),
\end{eqnarray}
one obtains that $\mu$ is compatible with the field space variation $\delta_{\un\mathfrak{X}}$ on $\chi$:
\begin{eqnarray}
\mu\circ \delta_{\un\mathfrak{X}}=ad^*_{\un\mathfrak{X}}\circ \mu.
\end{eqnarray}
This means that the charge algebra is realized on the orbit in $A^*_c$ via the coadjoint action.
This discussion has followed the typical path from a classical theory to the abstract dual algebra and, indeed, there was no need to introduce ${{\cal B}}$. The theme of this work is to make one further step and, once $A^*_c$ is understood, take it as the starting point and study its representations. If we knew little or nothing about the classical field space (which is the state of the art concerning quantum representations and observables), the first step would have been to introduce an associated bundle on which one can identify two fibre coordinates that play the role of the 'normal directions' in the vicinity of a corner in a classical spacetime. Then, the algebra acting at points in $A^*_c$ is mapped to derivations of this bundle, which can be packaged and reinterpreted as probing the normal directions. This is exactly why ${{\cal B}}$ is a fundamental ingredient when thinking of a classical field space from the intrinsic algebroid viewpoint. It is this corner-to-bulk shift of paradigm that allowed us to appreciate the ECS and ACS as sub-algebras of the UCS. Although a complete reconstruction picture of a classical spacetime is still under investigation, we have at least set the stage here to do so. This is also a framework where quantum features of gravity seem within reach, as they could be revealed studying other representations and associated bundles of the algebroid $A_c^*$, without any interference from classical inputs, such as dynamics and, even more importantly, the notion of a metric and spacetime.
\subsection{ECS and ACS Moment Maps}\label{sec4.4}
In this concluding subsection, we explicitly construct the moment maps of finite distance and asymptotic corners, since we have introduced all the important ingredients. The common thread of this manuscript is how to unify the treatment of ECS and ACS inside the UCS, the universal symmetry algebra. This applies also to the moment maps: while the intrinsic moment maps for finite distance and asymptotic corners link their solution spaces respectively to points on the dual algebras ECS$^*$ and ACS$^*$, we have seen how to directly interpret them on the dual algebra UCS$^*$.
\paragraph{Finite-distance corners:} Following the conventions established in \cite{Ciambelli:2021vnn}, we begin with the bulk metric parameterization of an arbitrary spacetime adapted to a corner
\begin{eqnarray}\label{randers}
g=
h_{ab}(u,x)n^a\otimes n^b+\gamma_{ij}(u,x)dx^i\otimes dx^j,
\end{eqnarray}
and expand the metric constituents order by order as in (\ref{exp1},\ref{exp2}). This parameterization is well-suited and preferred, since the metric constituents transform as affine tensors under the full UCS (see \cite{Ciambelli:2021vnn} for more details). The Noether charges are given by
\begin{eqnarray}
H_{\un\mathfrak{X}}
=\int_S vol_S\ \Big(\xi_{(1)}{}^a{}_bN^b{}_a+\xi_{(0)}^jb_j+\xi_{(0)}^ap_a\Big)\label{fullcharge},
\end{eqnarray}
with\footnote{Note that the first and last equations can be rewritten as
\begin{eqnarray}
N^a{}_b=-\frac{1}{\sqrt{-\det h^{(0)}}}\varepsilon^{ac}h^{(0)}_{cb},\qquad
p_a=-\frac{1}{\sqrt{-\det h^{(0)}}}\tfrac12\varepsilon^{bc}(h^{(1)}_{ab,c}-h^{(1)}_{ac,b}).
\end{eqnarray}
}
\begin{eqnarray}\label{N}
N^b{}_a =
\sqrt{-\det h^{(0)}}
h_{(0)}^{bc}\varepsilon_{ca},\qquad
b_j = -N^b{}_aa^{(1)}_j{}^a{}_b,\qquad
p_d = \tfrac12 N^{a}{}_{c}h_{(0)}^{cb}(h^{(1)}_{db,a}-h^{(1)}_{da,b}).
\end{eqnarray}
Clearly the classical result is of the same form as written above, in eq. \eqref{ucspair} for example. We note however that $N^a{}_b$ satisfies two conditions: first, it is traceless, which should be regarded as the reduction of the symmetry to ECS for such finite distance corners. Second, it satisfies $\tfrac12tr\, N^2=1$ (equivalently, $\det N=-1$); this should be regarded as a special property of the point on the ECS orbit that we happen to find ourselves. This statement can be understood in two ways: first, $tr\, N^2$ is not a Casimir, and so would vary along an orbit; second, and equivalently, we have made a choice by writing the metric in the form \eqref{randers} -- if it were modified by $g\to g/\Omega(u,x)^2$ then the net effect on the charges is to take $N^a{}_b\to N^a{}_b/\Omega^{(0)}(x)^n$. Note that the assumption that $\Omega^{(0)}(x)\neq 0$ is equivalent to the statement that the corner is at finite distance.
Now let us go back to the algebroid construction. For a finite distance corner, given that the symmetry reduces to ECS$^*$, we have an invariant tensor $\varepsilon^{ab}$, and using it, we can convert a section of $End({{\cal B}}_s)^*$ to a symmetric bilinear form
\begin{eqnarray}
\varepsilon: End({{\cal B}}_s)^*\to S^2{{\cal B}}_s.
\end{eqnarray}
Here by ${{\cal B}}_s$, we mean the affine bundle associated to the G-structure $A_s$ whose structure group is reduced to $H_s$ (operationally, this just means that $M^a{}_b$ is traceless).
In particular, a section of $S^2{{\cal B}}_s$ is a pair $(h_{(0)}^{ab},h_{(1)}^a)$ where $h_{(0)}$ is symmetric, and we identify\footnote{As a section of $S^2{{\cal B}}$, the pair $(h_{(0)}^{ab},h_{(1)}^a)$ can be organized into a matrix $\begin{pmatrix} h_{(0)}^{ab} & h_{(1)}^a\cr h_{(1)}^b&1\end{pmatrix}$ and transform under $\mathfrak{h}_s$ as
\begin{eqnarray}
\delta_{\un\mu}h_{(0)}^{ab}=
\bar\theta^a{}_ch_{(0)}^{cb}+\bar\theta^b{}_dh_{(0)}^{ad}+b^ah_{(1)}^{b}+h_{(1)}^{a}b^b,\qquad
\delta_{\un\mu}h_{(1)}^a=
\bar\theta^a{}_ch_{(1)}^{c}+b^a.
\end{eqnarray}
}
\begin{eqnarray}\label{N}
h_{(0)}^{ab}:=M^a{}_c\varepsilon^{cb},\qquad
h_{(1)}^{a}:=M_b\varepsilon^{ba}.
\end{eqnarray}
By inverting $h_{(0)}$, and using it to lower indices, we can equivalently regard this data as a section $(h^{(0)}_{ab},h^{(1)}_a)$ of $S^2{{\cal B}}_s^*$, and we identify
\begin{eqnarray}
h^{(1)}_a=\tfrac12\varepsilon^{bc}(h^{(1)}_{ab,c}-h^{(1)}_{ac,b}).
\end{eqnarray}
We note then that this data, along with the connection on the affine bundle and a metric on $S$, is equivalent to the data contained in a bulk metric near a corner, in the sense that these are the components of the bulk metric that contribute to the Noether charges for a finite distance corner. To recap then, we have seen that the affine bundle ${{\cal B}}$ may be regarded as a local model for a bulk spacetime, while a section of $Der({{\cal B}})^*$ can be interpreted as giving rise to a local metric on that bulk spacetime.
We can interpret \eqref{N} as essentially a moment map: we regard the field space as
$\chi=\{h^{(0)}_{ab},a^{(1)}_j{}^a{}_b,h^{(1)}_{a[b,c]}\}$, and the moment map as given by
\begin{eqnarray}
\mu_{s}(N^a{}_b)=J^a{}_b\qquad \mu_{s}(p_a)=P_a\qquad \mu_{s}(b_i)=\alpha_i.
\end{eqnarray}
More precisely, this may be regarded as a ``pre-moment map" that maps onto ECS$^*$, but not obviously to the non-degenerate orbits.
However, of the five $N^a{}_b,p_a$, only four will be independent on an orbit.\footnote{Indeed one finds that evaluating the charges on a finite distance corner in any particular classical geometry, only a subset of the $\mathfrak{h}_s$ charges are independent, which we now understand as this restriction to the $\mathfrak{h}_s$-orbit.} To demonstrate this, we can construct the Casimir
\begin{eqnarray}
{\cal C}_3=\int_S vol_S\ p_aN^a{}_b\varepsilon^{bc}p_c,
\end{eqnarray}
and show that
\begin{eqnarray}
ad^*_{\un\mathfrak{X}}{\cal C}_3=0.
\end{eqnarray}
From a classical bulk perspective, this result would be interpreted as a constraint equation.
\paragraph{Asymptotic corners:} The corresponding analysis for the asymptotic corner symmetry is more involved on the classical side because of the need for holographic renormalization, and some details will depend on precisely which asymptotic setting (e.g., asymptotically flat or AdS) is studied. Nevertheless, we expect that a moment map relates the renormalized charges to the expressions involving ACS$^*$. At a point on $S$ the ACS reduces to $\mathfrak{h}_w$. The latter is a three-dimensional ideal of $\mathfrak{h}$, whose orbits are at best two-dimensional, because of the existence of Casimirs.
The pairing \eqref{pair} for ACS is given by
\begin{eqnarray}\label{acspair}
\langle M,\un\mathfrak{X}\rangle=\int_S vol_S\, (i_{\un\xi}\alpha+\tfrac12 {\cal J}w+b^aP_a).
\end{eqnarray}
We stress that the ACS pertains to both asymptotically flat and AdS solution spaces, and so \eqref{acspair} is the starting point to construct both moment maps.
We begin by discussing asymptotically flat spacetimes, using results from \cite{Freidel:2021fxf}. The setup is the following. We work in $4$ spacetime dimensions, the normal coordinates to the corner are $(u,r)$ while coordinates on the corner are $x^A$, and the asymptotic boundary is located at $r\to \infty$. The leading order of the vector field generating asymptotic symmetries is
\begin{eqnarray}\label{BMSW}
\un\xi=T(x)\un\partial_u+ Y^A(x)\un\partial_A+W(x)(u\un\partial_u-r\un\partial_r),
\end{eqnarray}
and generates the so-called Weyl BMS symmetry, \cite{Freidel:2021fxf}. To compare with our setting, where the normal coordinates are expanded around the corner as $u^a\to 0$, we need to perform the change of coordinate $r={1\over \rho}$, such that the conformal boundary is at $\rho\to 0$. Then, eq. \eqref{BMSW} becomes
\begin{eqnarray}\label{BMSWrho}
\un\xi=T(x)\un\partial_u+ Y^A(x)\un\partial_A+W(x)(u\un\partial_u+\rho\un\partial_\rho).
\end{eqnarray}
Comparing with the defining representation introduced in Section \ref{sec2}, this vector field is included in the ACS, with the normal translation along $\un\partial_\rho$ turned off, i.e., $b^\rho=0$. Using the conventions of the present paper, the other parameters are identified as $T(x)=b^u(\sigma)$, $W(x)=2w(\sigma)$, and $Y^A(x)=\xi^\alpha(\sigma)\delta^A_\alpha$.
The renormalized charges $Q_R$ (referred to as $H_{\un\mathfrak{X}}$ in Section \ref{sec4.3}) in $4$-dimensional Einstein gravity with vanishing cosmological constant, evaluated at the corner on the asymptotic boundary at $u=0$, were found in \cite{Freidel:2021fxf} to be given by
\begin{eqnarray}
Q^R=\int_S vol_S \left(T (M-\tfrac12 \bar D_A U^A)+4 W \bar \beta+Y^A(\bar P_A+2\partial_A \bar\beta)\right),
\end{eqnarray}
where $M,U^A,\bar\beta$, and $\bar P_A$ are pieces of the bulk metric in Bondi gauge, and $\bar D_A$ is the corner covariant derivative.
The asymptotic solution space is thus parameterized by $\chi=\{M,U^A,\bar\beta,\bar P_A\}$ and, comparing with \eqref{acspair}, we identify the moment map:
\begin{eqnarray}
\mu_w(M-\tfrac12 \bar D_AU^A)=P_u,\qquad \mu_w(16\bar\beta)={\cal J},\qquad \mu_w(\bar P_A+2\partial_A\bar\beta)=\delta_A^\beta \alpha_\beta,
\end{eqnarray}
where in the last expression we converted the corner coordinates $x^A$ into $\sigma^\beta$. Contrary to the finite distance discussion, this is a moment map, that is, it maps directly into the non-degenerate orbits. Indeed, the solution space is here fully on-shell of the bulk equations of motion. From the point of view of $Der(B)$, we observe that the ACS collapses the $4$-dimensional non-affine part of the symmetries to the Weyl singlet. This implies that only the conformal factor of the full boundary metric matters in the solution space. However, in order to understand this better, one should perform an analysis of asymptotic corners with an adapted parameterization (not a gauge) and expansion, as done for finite distance corners in \eqref{randers}. Such an investigation would be certainly interesting to pursue.
We would like now to turn our attention to asymptotically AdS spacetimes. Our starting point is \cite{Alessio:2020ioh}, so the context is $3$-dimensional Einstein gravity with negative cosmological constant. Working in the Fefferman-Graham gauge with $\rho$ radial direction (conformal boundary at $\rho=0$) and $x^a=(t,\phi)$ boundary coordinates, the leading order of the residual vector field is
\begin{eqnarray}
\un\xi=\big(\tfrac12 \bar D_aY^a-\omega(x)\big)\rho\un\partial_\rho+Y^a(x)\un\partial_a,
\end{eqnarray}
where $\bar D_a$ is the boundary covariant derivative.
This vector field generates a priori a bigger algebra than the ACS, but, once expanded around the asymptotic corner at $(t,\rho)=0$, only $\omega^{(0)}(\phi),Y^t_{(0)}(\phi)$, and $Y^\phi_{(0)}(\phi)$ contribute to the charges, and generate Weyl, supertranslations, and diffeomorphisms of the $1$-dimensional corner, respectively. Thus, the physical asymptotic symmetry algebra around a corner is just $(\mathfrak{diff}(S)\loplus \mathbb{R})\loplus \mathbb{R}$, which is again included in the ACS, as advertised.
The renormalized charges, as computed in \cite{Alessio:2020ioh} in the chiral splitting and expanded around $t=0$, are given by\footnote{Conventions when comparing with \cite{Alessio:2020ioh}: $\Xi_t={\Xi_{++}+\Xi_{--}\over \ell}$, $\Xi_\phi=\Xi_{++}-\Xi_{--}$.}
\begin{eqnarray}
Q_{(\omega,Y)}={\ell\over 8\pi G}\int_0^{2\pi}d\phi \Big(Y^t_{(0)}(\phi)\Xi_{t}+Y^\phi_{(0)}(\phi)\Xi_{\phi}-2\ell\omega^{(0)}(\phi)\partial_t\varphi\Big),
\end{eqnarray}
where $\ell$ is the AdS radius, $G$ is the Newton constant, $\Xi_t$ and $\Xi_\phi$ are the bulk metric field composing the boundary stress tensor, and $\varphi$ is the conformal factor of the boundary metric. These three last quantities compose thus the sourced solution space of the bulk theory, $\chi=\{\Xi_t,\Xi_\phi,\varphi\}$. Then, comparing with \eqref{acspair}, we can identify the moment map:
\begin{eqnarray}
\mu_w^{\text{AdS}}(e^{-2\varphi}\Xi_t)=P_u, \qquad \mu_w^{\text{AdS}}(-2\ell e^{-2\varphi}\partial_t \varphi)={\cal J}, \qquad \mu_w^{\text{AdS}}(e^{-2\varphi}\Xi_\phi)=\alpha_\phi.
\end{eqnarray}
This final result concludes this section. It would be interesting to explore higher dimensions, using the existing literature \cite{Fiorucci:2020xto}, but utilizing the more suitable Weyl-Fefferman-Graham gauge introduced in \cite{Ciambelli:2019bzz}. We leave this analysis for future investigations.
All the results derived previously in this manuscript, on how to simultaneously realize the ECS and ACS orbits inside the UCS, apply here and, in particular, this demonstrates that finite distance and asymptotic corner physics can be described as specific reductions of the UCS. While this analysis has taken a step further in the understanding and unification of the treatment of symmetries of gravity, there are still many open questions that we discuss in the next section.
\section{Outlook}\label{sec5}
In this paper, we have started to explore a new avenue of investigation in which a minimal amount of symmetry information has been extracted that we expect to hold in the quantum regime. The identification of the universal symmetry algebra at corners is, in this regard, a lamppost. We thus studied its representations using the orbit method. We have shown that the classical gravitational phase space both at finite distance and asymptotic corners is sent through the moment map to the dual algebra of the UCS, proving that the latter contains and describes both simultaneously. We have furthermore geometrically realized the UCS as the symmetry group of an Atiyah Lie algebroid associated to a $GL(2,\mathbb{R})\ltimes \mathbb{R}^2$-principal bundle over the corner. This provides the correct arena to study representations of the UCS, each of which corresponds to a specific bundle associated to the Lie algebroid. We have identified the representation giving rise to the asymptotic phase space of a classical spacetime near a corner.
While our results fit perfectly the finite distance corner analysis, the asymptotic corner should be studied more carefully already at the classical level. There are two main reasons for this. The first is that an analysis similar to \cite{Ciambelli:2021vnn} has yet to be done. The latter is a complicated task, due to the second reason, which is holographic renormalization. While tremendous effort has been focussed in recent years on this topic, a full understanding of this procedure for asymptotically flat spacetimes, in particular concerning the consequences it has on the coadjoint orbit analysis is yet to be explored. We plan to return to this in the near future.
In addition, we plan to study the classification of UCS representations, with two main objectives. The first one is to store all the classical information of a spacetime into the dual algebra UCS$^*$, and concretely retrieve various known gravitational solutions as a set of discrete corners with certain field content. We expect this to be possible thanks to the extended phase space introduced in \cite{Ciambelli:2021nmv,Freidel:2021dxw}, whose fundamental role has been already appreciated in the present manuscript. In this regard, it would be rewarding to appreciate the properties of edge modes in a concrete setup, such as corners sitting on a black hole horizon.
The second, more far-reaching, objective is to understand how certain representations are better suited for a quantum description of the gravitational field content. We expect that the construction of a quantum theory along these lines will lead to unitarity and locality. The presence of the BRST symmetry on Atiyah Lie algebroids \cite{Ciambelli:2021ujl} suggests that some quantum features of gravity are already present in our description, and it is worth investigating. Other questions are expected to find a natural answer in this direction. For instance, the role of gravitational electric-magnetic duality at corners, or S-matrix scattering properties, are questions on our agenda.
Finally, our recent works, starting with \cite{Ciambelli:2021ujl} and including the present one, have all in common the end goal of formulating the correct gravitational framework to understand features of quantum gravity, such as entanglement entropy between subregions, non-factorizability, and the information conundrum. This is clearly a long road yet to pave, but we believe that the direction taken here is the correct one, for it completely disentangles classical features, like a metric or dynamics, from more fundamental properties, such as symmetries and their representations.
\paragraph{Acknowledgments}
We thank G. Barnich, L. Freidel and M. Klinger for discussions. We would also like to thank the participants of the first and second online Corners Meeting for stimulating discussions. LC acknowledges \'Ecole Polytechnique and Nordita for invitations to present this work. RGL thanks the Universit\'e Libre de Bruxelles for hospitality where a portion of this work was completed, as well as the opportunity to present the work at the Belgian joint seminars.
The research of LC was partially supported by a Marina Solvay
Fellowship, by the ERC Advanced Grant ``High-Spin-Grav" and by
FNRS-Belgium (convention FRFC PDR T.1025.14 and convention IISN 4.4503.15).
The work of RGL was supported by the U.S. Department of Energy under contract DE-SC0015655.
|
{
"timestamp": "2022-07-15T02:00:30",
"yymm": "2207",
"arxiv_id": "2207.06441",
"language": "en",
"url": "https://arxiv.org/abs/2207.06441"
}
|
\section{Introduction}\label{sec:intro}
We say that a sequence of nonincreasing positive integers $\lambda = (\lambda_1, \hdots, \lambda_m)$ is a \textit{partition} of a positive integer $n$ if $\lambda_1 + \cdots + \lambda_m = n$. The \textit{Young diagram} of the partition $\lambda = (\lambda_1,\ldots,\lambda_m)$ is a left-justified array of square cells with rows of length $\lambda_1, \hdots, \lambda_m$, and the \textit{hook length} of any cell in the Young diagram is the sum of the number of cells to the right of it in its row, the number of cells below it in its column, and one (to account for the cell itself).
\begin{exstar} Here we give the Young diagram for the partition $\lambda = (6,4,3,1)$ of $n = 14$, where each cell is labelled with its corresponding hook length.
\begin{figure}[H]
\centering
\ytableausetup{centertableaux}
\begin{ytableau}
9 & 7 & 6 & 4 & 2 & 1 \\
6 & 4 & 3 & 1 \\
4 & 2 & 1 \\
1
\end{ytableau}
\label{fig:young_diagram_example}
\end{figure}
\end{exstar}
Partition hook lengths have many applications in combinatorics, number theory, and representation theory. For instance, consider the famous correspondence between complex finite-dimensional irreducible representations of the symmetric group $S_n$ and the Young diagrams of the partitions of $n$. For any prime $p$, the Young diagrams associated with the set of $p$-core partitions, that is, partitions with no hook lengths divisible by $p$, correspond precisely to those irreducible representations that remain irreducible upon reduction modulo $p$ \cite[Chapter 6]{JK84}. Moreover, the number of standard Young tableaux of the partition $\lambda$, a quantity of interest in algebraic combinatorics, is given by the Frame-Robinson-Thrall hook length formula \cite{FRT54}
\[
d_{\lambda} = \frac{n!}{\prod_{h \in \mathcal{H}(\lambda)}h},
\]
where $\mathcal{H}(\lambda)$ denotes the multiset of hook lengths associated to a partition $\lambda$. The value $d_{\lambda}$ is also the dimension of the irreducible representation of $S_n$ corresponding to $\lambda$.
In addition, hook lengths arise naturally in mathematical physics and the study of modular forms. For instance, Nekrasov--Okounkov \cite{NO06} derived the identity
\[
\sum_\lambda q^{\lambda} \prod_{h \in \mathcal{H}(\lambda)} \left( 1 -\frac{z}{h^2} \right) = \prod_{n=1}^\infty (1-q^n)^{z-1}
\] in their study of the Seiberg-Witten theory, which expresses the powers of the Euler product in terms of the hook lengths of partitions. Han \cite[Theorem 1.3]{Han10} derived the following generalization of the Nekrasov--Okounkov identity: \begin{align}\label{eqn:han_generalization} \sum_{\lambda } x^{|\lambda|}\prod_{h \in \mathcal{H}_t(\lambda)} \left(y- \frac{tyz}{h^2}\right) = \prod_{k \geq 1} \frac{(1-x^{tk})^t}{(1-(yx^t)^k)^{t-z}(1-x^k)}. \end{align} Here $\mathcal{H}_t(\lambda)$ denotes the multiset of hook lengths of $\lambda$ divisible by $t$.
In view of the importance of $\mathcal{H}_{t}(
\lambda)$, we study the statistics of the sequence of random variables $\{Y_t(n)\}$ for a fixed integer $t \geq 2$, where the values of $Y_t(n)$ are the number of hook lengths divisible by $t$ in the partitions of $n$.
For the sake of convenience, we define the polynomial
\begin{align}
P_t(n,x) := \sum_{\lambda \vdash n} x^{\#\mathcal{H}_t(\lambda)} = \sum_{m=0}^\infty p_t(m,n)x^m
\end{align} for any positive integers $t$ and $n$, where $p_t(m,n)$ denotes the number of partitions of $n$ with $m$ hooks of length divisible by $t$. Recently, Griffin--Ono--Tsai \cite{GOT22} showed for the case $t \geq 4$ that the cumulative distribution functions of $Y_t(n)$ converge pointwise to a shifted Gamma distribution with shape $k(t) := (t - 1)/2$ and scale $\theta(t) := \sqrt{2/(t - 1)}$ by estimating the polynomial $P_t(n,x)$ at specific values of $x$. To be precise, they proved that for $t \geq 4$, the random variables $\{Y_t(n)\}$ satisfy
\[
\frac{n\pi}{\sqrt{3(t-1)n}} - \frac{t\pi}{\sqrt{3(t-1)n}}Y_t(n) \sim X_{k(t), \theta(t)},\]
where $X_{k(t), \theta(t)}$ is a random variable satisfying the Gamma distribution with shape $k(t)$ and scale $\theta(t)$ that we defined earlier.
Ono asked for a resolution to the analogous question for the cases $t = 2$ and $3$. The methods in \cite{GOT22} do not apply to the cases $t=2$ and $3$ because they rely on the existence of moment generating functions for Gamma distributions with shape parameter $k(t) > 1$. Moreover, Griffin--Ono--Tsai observed that for $t=2$ and $3$, the support of $Y_t(n)$ is very sparse. For example, consider the distribution of nonzero coefficients in the polynomial
\begin{align*}
P_2(100, x) = \sum_{\lambda \vdash 100} x^{\#\mathcal{H}_2(\lambda)} &= 752 x^{11} + 8470 x^{17} + 1046705 x^{32} +3157789 x^{36} \\&\quad+ 31551450 x^{45} +
51124970 x^{47} + 103679156 x^{50},
\end{align*} also depicted in Figure \ref{fig:t2_sparse}.
Due to the frequency of zeros, the probability mass functions do not appear to converge to a continuous probability density function, which further distinguishes these cases from the case of $t \geq 4$ (see, for instance, Figure \ref{fig:conjecturet11} in Section \ref{sec:conj}).
\begin{figure}[h!]
\centering
\includegraphics[scale=0.8]{lots_of_zeros_100.png}
\caption{Coefficients of the polynomial $P_2(100,x)$. Observe the sparse support (indicated in blue) and the frequency of zeros (indicated in red).}
\label{fig:t2_sparse}
\end{figure}
Motivated by the result for $t \geq 4$, we apply a change of variables and define the probability mass functions $\widetilde{f}_{t;n}(x)$ corresponding to
\begin{align}\widetilde{Y}_t(n) := \frac{n\pi}{\sqrt{3(t-1)n}} - \frac{t\pi}{\sqrt{3(t-1)n}}Y_t(n)\end{align}
for any positive integer $n$.
Similarly, let $g_t(x)$ denote the probability density function for the random variable $X_{k(t), \theta(t)}$. Our first goal is to compare the behavior of the \textit{scaled} mass functions (to account for the change of variables) \begin{align}f_{t;n}(x) := \frac{\sqrt{3(t-1)n}}{t\pi} \; \widetilde{f}_{t;n}(x)\end{align} with $g_t(x)$ in the cases $t = 2$ and $3$. Moreover, for the sake of brevity, we define for each positive integer $n$ \[\mathcal{S}_{t;n} := \left\{x \in \mathbb{R} \ \bigg|\ \frac{n}{t} - \frac{\sqrt{3(t-1)n}}{t\pi}x \in \{0, 1,\ldots,\lfloor n/t\rfloor\}\right\},\] the set of $x \in \mathbb{R}$ where the function $f_{t;n}$ is defined.
We begin by providing an explicit description of the supports of $Y_t(n)$ and show that almost all of the coefficients of $P_t(n,x)$ are zero for $t = 2$ and $3$. To do so, we derive a formula for the coefficients $p_t(m,n)$ related to the number of $t$-colored partitions of $m$. In the case $t = 2$, we relate the vanishing of these coefficients to the count of triangular numbers at most $n/2$, and in the case $t = 3$, we obtain our results by studying the multiplicative properties of the Fourier coefficients of a cusp form with complex multiplication. Using our explicit description, we show that, in some sense, the sparse support of the mass functions $f_{t;n}$ implies that they cannot converge to $g_t(x)$ for $t = 2,3$.
\begin{thm}\label{intro_thm:support_t2}
If $t=2$, then the following are true:
\begin{itemize}
\item[(a)] We have $p_2(m,n) \neq 0$ if and only if $n - 2m$ is a triangular number.
\item[(b)] As $n \to \infty$, \[\frac{\#\{0 \leq m \leq \deg P_2(n,x)\ |\ p_2(m,n) \neq 0\}}{\deg P_{2}(n,x)} = O(n^{-1/2}).\]
\item[(c)] If there exists $n_0$ such that $x \in \mathcal{S}_{2;n_0}$, then there exists an infinite sequence $\{n_j\}_j$ such that $x \in \mathcal{S}_{2;n_j}$ for all $j$, and the sequence $f_{2,n_j}(x)$ does not converge to $g_2(x)$ as $j \to \infty$.
\end{itemize}
\end{thm}
\begin{thm}\label{intro_thm:support_t3}
If $t=3$, then the following are true:
\begin{itemize}
\item[(a)] We have $p_3(m,n) \neq 0$ if and only if $\operatorname{ord}_r(3(n-3m)+1) \equiv 0 \pmod{2}$ for every prime $r \equiv 2 \pmod{3}$.
\item[(b)] As $n \to \infty$, \[\frac{\#\{0 \leq m \leq \deg P_3(n,x)\ |\ p_3(m,n) \neq 0\}}{\deg P_{3}(n,x)} = O\left(\frac{1}{\sqrt{\log(n)}}\right).\]
\item[(c)] If there exists $n_0$ such that $x \in \mathcal{S}_{3;n_0}$, then there exists an infinite sequence $\{n_j\}_j$ such that $x \in \mathcal{S}_{3; n_j}$ for all $j$, and the sequence $f_{3,n_j}(x)$ does not converge to $g_3(x)$ as $j \to \infty$.
\end{itemize}
\end{thm}
Note that, since the points at which the functions $f_{t;n}$ are defined vary with $n$, it is not meaningful to discuss pointwise convergence to $g_t$ in the usual sense. Instead, our results in Theorems \ref{intro_thm:support_t2}(c) and \ref{intro_thm:support_t3}(c) provide an alternative, allowing us to describe the failure of $f_{t;n}$ to converge pointwise to $g_t$ for specific subsequences of $n$ where this notion makes sense.
\begin{xrem} Theorems \ref{intro_thm:support_t2}(b) and \ref{intro_thm:support_t3}(b) imply that, for both $t = 2$ and $3$, the coefficients of $P_t(n,x)$ are zero on a subset with asymptotic density $1$. Namely,
\[
\lim_{n \to \infty} \frac{\#\{ 0 \leq m \leq \deg P_t(n,x) \mid p_t(m,n) = 0\}}{\deg P_t(n,x)} = 1
\]
for both $t = 2$ and $3.$
\end{xrem}
\begin{exstar}
The behavior described in the previous remark is illustrated in Figure \ref{fig:tableofnondiffablepoints}, which contains statistics on the proportion of nonzero coefficients of $P_2(n, x)$ and $P_3(n, x)$ with respect to the degrees of the polynomials up to $n = 5000$. Observe that the proportion of nonzero coefficients decreases faster for $t = 2$ than for $t = 3$, reflecting the bounds $O(n^{-1/2})$ for the former and $O(\log(n)^{-1/2})$ for the latter. Figure \ref{fig:t3_support_sparse} plots $Y_3(n)$ for $n = 100, 500, 1000$, and $2000$.
\end{exstar}
\begin{center}
\begin{figure}[h!]
{\tabulinesep=1.2mm
\begin{tabu}{ |c|c|c| }
\hline
$n$ & $\dfrac{\#\{p_2(m,n) \neq 0\}}{\deg P_2(n,x)}$ & $\dfrac{\#\{p_3(m,n) \neq 0\}}{\deg P_3(n,x)}$ \\
\hline
\hline
100 & 0.14000\textcolor{white}{...} & 0.63636... \\
\hline
500 & 0.06400\textcolor{white}{...} & 0.50602... \\
\hline
1000 & 0.04600\textcolor{white}{...} & 0.47147...\\
\hline
1500 & 0.03600\textcolor{white}{...} & 0.46200\textcolor{white}{...} \\
\hline
2000 & 0.03100\textcolor{white}{...} & 0.45496... \\
\hline
2500 & 0.02800\textcolor{white}{...} & 0.44658... \\
\hline
3000 & 0.02600\textcolor{white}{...} & 0.44400\textcolor{white}{...} \\
\hline
3500 & 0.02400\textcolor{white}{...} & 0.43825... \\
\hline
4000 & 0.02250\textcolor{white}{...} & 0.43661... \\
\hline
4500 & 0.02088... & 0.43200\textcolor{white}{...} \\
\hline
5000 & 0.02000\textcolor{white}{...} & 0.43097... \\
\hline
\end{tabu}}
\caption{Table depicting the proportion of nonzero coefficients of $P_t(n,x)$ for $t = 2,3$ across various values of $n$.}
\label{fig:tableofnondiffablepoints}
\end{figure}
\end{center}
\begin{figure}[h!]
\begin{minipage}{.5\linewidth}
\centering
\subfloat[Plot of $Y_3(100)$ \label{subfig:y3100}]{\includegraphics[width=3in]{lots_of_zeros_3_100.png}}
\end{minipage}%
\begin{minipage}{.5\linewidth}
\centering
\subfloat[Plot of $Y_3(500)$.\label{subfig:y3500}]{\includegraphics[width=3in]{lots_of_zeros_3_500.png}}
\end{minipage}%
\par\medskip
\centering
\begin{minipage}{.5\linewidth}
\centering
\subfloat[Plot of $Y_3(1000)$.\label{subfig:y31000}]{\includegraphics[width=3in]{lots_of_zeros_3_1000.png}}
\end{minipage}%
\begin{minipage}{.5\linewidth}
\centering
\subfloat[Plot of $Y_3(2000)$. \label{subfig:y32000}]{\includegraphics[width=3in]{lots_of_zeros_3_2000.png}}
\end{minipage}
\caption{Observe that the support of $Y_3(n)$ becomes sparser as $n$ grows larger, exemplifying the behavior outlined in Theorem \ref{intro_thm:support_t3}(b).}
\label{fig:t3_support_sparse}
\end{figure}
On the other hand, Figure \ref{fig:support_looks_smooth} suggests that the points in the support of $f_{2;n}(x)$ do in fact lie on a continuous curve, and Figure \ref{fig:t3_support} indicates that the points in the support of $f_{3;n}(x)$ approximate multiple continuous curves. Thus, a natural question to ask is to what extent $f_{t;n}$ converges to $g_t$ when we restrict the domain to the support.
As we will see in Subsection \ref{sec:even_hl}, for $t=2$ the phenomenon depicted in Figure \ref{fig:support_looks_smooth} originates from the fact that the values of $f_{2;n}(x)$ are related to counts of two-colored partitions. Similarly, we discuss in Subsection \ref{sec:three_hl} how the relation between the values of $f_{3;n}(x)$ with three-colored partitions. We handle the case of $t=2$ in Theorem \ref{intro_thm:pmf_no_converge} and $t=3$ in Theorem \ref{intro_thm:pmf_converge_t3}. In particular, for $t=2$ we explicitly derive a formula for the apparent continuous curve for each $n$, but also show that these functions do not converge pointwise to $g_2(x)$.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.75]{5000_smooth_curve_better.png}
\caption{The values of $f_{2;5000}(x)$ in its support appear to approximate a continuous curve.}
\label{fig:support_looks_smooth}
\end{figure}
\newpage
\begin{thm}\label{intro_thm:pmf_no_converge}
Assuming the notation above, the following are true:
\begin{itemize}
\item[(a)] For positive integers $n$, let \[
h_{2;n}(x) := \frac{3^{1/4}n^{3/2} \exp \left(\pi \sqrt{\frac{2}{3}} \left( \sqrt{n - \frac{\sqrt{3n}}{\pi}x} - \sqrt{n}\right) \right)}{2 \pi\left( \frac{n}{2} - \frac{\sqrt{3n}}{2\pi}x \right)^{5/4}}.
\] For all $x$ such that $f_{2;n}(x) \neq 0$, we have that $f_{2;n}(x) \sim h_{2;n}(x)(1+\beta_2(x,n))$ as $n \to \infty$, where $\beta_2(x,n)$ is a constant satisfying $\beta_2(x) = O_\varepsilon((n - x\sqrt{n})^{-1/4+\varepsilon})$ for any $\varepsilon > 0$.
\item[(b)] For all $x \in \mathbb{R}$ and for any subsequence $\{n_j\}_{j=1}^\infty$, the sequence $h_{2;n_j}(x)$ does not converge to $g_2(x)$ as $j \to \infty$.
\end{itemize}
\end{thm}
\begin{figure}[h!]
\centering
\includegraphics[width=3.5in]{best_5000_smooth_curve_with_approximation.png}
\caption{The values of $f_{2;5000}(x)$ in its support (in red) along with the continuous approximation $h_{2;5000}(x)$ (in black) derived in the proof of Theorem \ref{intro_thm:pmf_no_converge}.}
\label{fig:support_with_curve}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=4in]{better_convergence_to_zero.png}
\caption{The approximations $h_{2;n}(x)$, depicted here for various values of $n$, do not converge pointwise to $g_2(x)$, depicted here in black, as $n \to \infty$, exemplifying Theorem \ref{intro_thm:pmf_no_converge}.}
\label{fig:approximations_dont_converge}
\end{figure}
In the case of $t=3$, we prove that the nonzero points of $f_{3;n}(x)$ are asymptotic to integer multiples of a continuous function $h_{3;n}(x)$. We find an explicit formula for these continuous functions and show that they converge to scalar multiples of $g_3$. This result contrasts slightly with the $t=2$ case, but still falls short of outright convergence.
\newpage
\begin{thm}\label{intro_thm:pmf_converge_t3}
Assuming the notation above, the following are true:
\begin{itemize}
\item[(a)] For positive integers $n$, let \[
h_{3;n}(x) := \frac{3\sqrt{3}n^{3/2}\exp\left(\pi\sqrt{\frac{2}{3}}\left(\sqrt{n-\frac{\sqrt{6n}}{\pi}x} - \sqrt{n}\right)\right)}{2\pi\left(n - \frac{\sqrt{6n}}{\pi}x\right)^{3/2}}.
\]
For all $x$ such that $f_{3;n}(x) \neq 0$, there exists a positive integer $\alpha(x,n)$ such that \[f_{3;n}(x) \sim \alpha(x, n)h_{3;n}(x)(1+\beta_3(x))\] as $n \to \infty$, where $\beta_3(x,n)$ is a constant satisfying $\beta_3(x,n) = O_\varepsilon((n - x\sqrt{n})^{-1/4+\varepsilon})$ for any $\varepsilon > 0$.
\item[(b)] For all $x \in \mathbb{R}$, we have \[h_{3;n}(x) \to \dfrac{3\sqrt{3}}{2\pi}g_3(x)\] as $n \to \infty$.
\end{itemize}
\end{thm}
\begin{figure}[h!]
\centering
\includegraphics[width=4in]{t3_n10000_smooth_curve.png}
\caption{The values of $f_{3;10000}$ in its support appear to approximate several continuous curves.}
\label{fig:t3_support}
\end{figure}
\begin{xrem}
In Proposition \ref{prop:explicit_coefficients_t_3}, we will show that the coefficients $p_3(m, n)$ can be written in terms of the character sum $c(k) = \sum_{d | 3k + 1} \left(\frac{d}{3}\right)$ and the number of $3$-colored partitions of $m$. The former is the source of the integer-valued scalar $\alpha(x, n)$ in Theorem \ref{intro_thm:pmf_converge_t3}.
\end{xrem}
\begin{figure}[h!]
\centering
\includegraphics[width=4in]{better_t3_n10000_smooth_curve_with_approximation.png}
\caption{The values of $f_{2;10000}(x)$ in its support (in red) along with the continuous approximations $h_{3;10000}(x)$, $2h_{3;10000}(x)$, and $4h_{3,10000}(x)$ (in black) derived in the proof of Theorem \ref{intro_thm:pmf_converge_t3}.}
\label{fig:t3_support_with_curve}
\end{figure}
Theorems \ref{intro_thm:support_t2}, \ref{intro_thm:support_t3}, and \ref{intro_thm:pmf_no_converge} illustrate how the probability mass functions of the random variables $Y_{t}(n)$ fail to converge, in various senses, to the probability density functions consistent with the work of \cite{GOT22}, while Theorem \ref{intro_thm:pmf_converge_t3} gives convergence to scalar multiples of the probability density function of the Gamma distribution at best. In contrast, we show that the cumulative distribution functions of $Y_t(n)$ \textit{do} in fact converge pointwise to the expected shifted Gamma distributions. Interestingly, the failure of the nonzero values of $f_{2;n}$ to converge to $g_2$ (Theorem \ref{intro_thm:pmf_no_converge}) suggests that, at the very least in the case $t = 2$, the spacing between points in the support plays a role in the convergence of cumulative distribution functions. By considering the pointwise convergence of characteristic functions in place of moment generating functions, our approach to Theorem \ref{intro_thm:convergence_in_distribution} avoids the obstacles that Griffin--Ono--Tsai encountered in their work.
Fix $t \geq 2$ and let $k(t) = \frac{t-1}{2}, \theta(t) = \sqrt{\frac{2}{t-1}}$. We show that the sequence of random variables \[\frac{n\pi}{\sqrt{3(t-1)n}} - \frac{t\pi}{\sqrt{3(t-1)n}}Y_t(n)\] converges to the random variable $X_{k(t),\theta(t)}$ in distribution.
\begin{thm}\label{intro_thm:convergence_in_distribution}
Assume the notation above. If $t \geq 2$, then the following are true:
\begin{itemize}
\item[(a)] The sequence $Y_t(n)$ satisfies
\begin{align*}
Y_t(n) \sim \frac{n}{t} - \frac{\sqrt{3(t-1)n}}{\pi t} \cdot X_{k(t), \theta(t)},
\end{align*}
and has mean $\mu_t(n) \sim \frac{n}{t} - \frac{(t-1)\sqrt{6n}}{2 \pi t}$, mode $\mathrm{mo}_t(n) \sim \frac{n}{t} - \frac{(t-3)\sqrt{6n}}{ 2 \pi t}$, and variance ${\sigma_t^2(n) \sim \frac{3(t-1)n}{\pi^2t^2}}.$
\item[(b)] If we let $\xi_{t,n}(x) := \mu_t(n) + \sigma_t(n)x$ and $F_t(k,n)$ denote the cumulative distribution function of $Y_t(n)$, then \[\lim_{n \to \infty} F_t(\xi_{t,n}(x),n) = \frac{\gamma\left(\frac{t-1}{2},\sqrt{\frac{t-1}{2}}x + \frac{t-1}{2}\right)}{\Gamma\left(\frac{t-1}{2}\right)},\] where $\gamma(s,t)$ is the lower incomplete gamma function.
\end{itemize}
\end{thm}
Theorem \ref{intro_thm:convergence_in_distribution} extends the results of \cite[Theorem 1.2]{GOT22} to hold for $t = 2, 3$. Thus, our theorem completes the characterization of the limiting distribution of $\{Y_t(n)\}$ for all values of $t$.
\begin{figure}[h!]
\centering
\includegraphics[width=4in]{cdf_2_2500.png}
\caption{Plot of the cumulative distribution function of $Y_2(2500)$ (in orange) and the cumulative distribution function of the shifted random variable specified in Theorem \ref{intro_thm:convergence_in_distribution}(a) with $t = 2, n = 1000$ (in dashed purple).}
\label{fig:cdf_2_2500}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=4in]{cdf_3_2500.png}
\caption{Plot of the cumulative distribution function of $Y_3(2500)$ (in orange) and the cumulative distribution function of the random variable specified in Theorem \ref{intro_thm:convergence_in_distribution}(a) with $t = 3, n = 2500$ (in dashed purple).}
\label{fig:cdf_3_1000}
\end{figure}
The structure of the article is as follows. In Section \ref{sec:pmfs}, we explicitly describe the supports of the probability mass functions $Y_{t}(n)$ for $t = 2, 3$ in Theorems \ref{intro_thm:support_t2} and \ref{intro_thm:support_t3}. Moreover, we show the failure of $f_{2;n}$ to converge to $g_2(x)$ in contrast with the convergence (up to a scalar multiple) of $f_{3;n}(x)$ to $g_3(x)$, thereby proving Theorems \ref{intro_thm:pmf_no_converge} and \ref{intro_thm:pmf_converge_t3}. Finally, in Section \ref{sec:cdfs}, we prove Theorem \ref{intro_thm:convergence_in_distribution} to show that the cumulative distribution functions of $Y_t(n)$ for $t = 2, 3$ still converge to the desired shifted Gamma distribution.
\subsection*{Acknowledgements} The authors would like to thank Wei-Lun Tsai for supervising this project and Ken Ono for his generous support. The authors were participants in the 2022 UVA REU in Number Theory. They are grateful for the support of grants from the National Science Foundation
(DMS-2002265, DMS-2055118, DMS-2147273), the National Security Agency (H98230-22-1-0020), and the Templeton World Charity Foundation.
\section{Probability Mass Functions}\label{sec:pmfs}
In this section, we study the scaled mass functions $f_{t;n}$ corresponding to $\{Y_t(n)\}$ for $t = 2, 3$. Recall from Section \ref{sec:intro} that $p_t(m, n)$ denotes the number of partitions of $n$ with $m$ hooks of length divisible by $t$. For each $t \geq 1$, Han \cite{Han10} derived the following multivariate generating function for the quantities $p_t(m,n)$: \begin{align} \label{generating_function}
G_t(x; q) =\prod_{n = 1}^\infty \frac{(1 - q^{tn})^t}{(1 - (xq^t)^n)^t(1 - q^n)} =\sum_{n=0}^\infty \sum_{m=0}^\infty p_t(m, n)x^m q^n.
\end{align} Moreover, recall the notation \[P_t(n;x) := \sum_{m=0}^\infty p_t(m,n)x^m.\] Observe that $P_t(n;x)$ is a polynomial in $x$; we will show that $\deg P_t(n;x) = \lfloor n/t\rfloor$ for each $t = 2,3$. For any positive integers $n$ and $k$, we can describe the probability mass functions of $Y_t(n)$ via \begin{align}\label{eqn:probability}\mathbb{P}(Y_t(n) = k) = \frac{p_t(k,n)}{p(n)}.\end{align}
Before we proceed, we will need to define, for a fixed nonnegative integer $t$, the $t$-colored partitions of any positive integer $n$. These $t$-colored partitions will play an important role in our characterization of the nonzero values of $p_t(m,n)$ in the cases $t = 2$ and $3$.
\begin{defn}
Fix a nonnegative integer $t$ and a positive integer $n$. A $t$-colored partition of $n$ is a multiset of $t$ (possibly empty) sequences of nonincreasing positive integers such that sum of all integers across all sequences is equal to $n$.
\end{defn}
Observe that the number of $t$-colored partitions of some positive integer $n$ is given by the coefficient of $y^n$ after formally expanding the product
\begin{align}\label{eqn:tcolored_gf} \prod_{m=1}^\infty \frac{1}{(1-y^m)^t}.\end{align}
We begin in Section \ref{sec:even_hl} with a characterization of the support of $Y_t(n)$ for $t=2,3$ and estimate how quickly it vanishes as $n$ grows. As we have seen in Figure \ref{fig:support_looks_smooth}, the nonzero values of the function $f_{2; n}(x)$ appear to approximate a continuous curve -- we determine this curve $h_{2;n}(x)$ in Section \ref{sec:even_hl} using asymptotic formulas of Meinardus and Hardy-Ramanujan. However, we show that even this continuous approximation will not converge pointwise to the probability density function of the Gamma distribution. In Section \ref{sec:three_hl} we determine the support of $f_{3; n}(x)$ and find an asymptotic upper bound on its decay as $n \to \infty$. In contrast to the case of $t = 2$, we show that $f_{2; n}(x)$ approximate multiple continuous curves which converge to scalar multiples of $g_3(x)$.
\subsection{Even Hook Lengths}\label{sec:even_hl}
In this subsection, we begin with an explicit description of the coefficients $p_2(m,n)$ in terms of the number of two-colored partitions of $m$ and prove Theorem \ref{intro_thm:support_t2}. Then, for all $x \in \mathbb{R}$ such that $f_{2; n}(x) \neq 0$, we estimate $f_{2; n}(x)$ with a continuous function $h_{2;n}(x)$ and use the explicit formula for $h_{2;n}(x)$ to prove Theorem \ref{intro_thm:pmf_no_converge}.
\begin{prop}\label{prop:explicit_coefficients_t_2}
When $t = 2$, the coefficients of $P_2(n;x)$ satisfy \[p_2(m,n) = \begin{cases} a(m) & \text{if $n - 2m$ is a triangular number} \\ 0 & \text{otherwise,}\end{cases}\] where $a(k)$ denotes the number of $2$-colored partitions of $k$.
\end{prop}
\begin{proof}
By a simple observation, we have \[\prod_{m=1}^\infty \frac{1}{(1-(xq^2)^m)^2} = \sum_{k=0}^\infty a(k)x^kq^{2k},\] where $a(k)$ is the number of $2$-colored partitions of $k$. On the other hand, a classical identity of Jacobi gives that \[\prod_{m=1}^\infty \frac{(1-q^{2m})^2}{(1-q^m)} = \sum_{\ell=1}^\infty q^{\ell(\ell+1)/2}.\] Alternatively, one can obtain the identity above by realizing that the left-hand side is the generating function for the $2$-core partitions, and noting that the $2$-core partitions are precisely the partitions of the form $(\ell,\ell-1,\ldots,2,1)$ for positive integers $\ell$. The desired result follows immediately upon studying the coefficient of $x^mq^n$ in (\ref{generating_function}): \[G_2(x;q) = \prod_{m=1}^\infty \frac{(1-q^{2m})^2}{(1-(xq^2)^m)^2(1-q^m)} = \left(\sum_{k=0}^\infty a(k)x^kq^{2k}\right)\left(\sum_{\ell=1}^\infty q^{\ell(\ell+1)/2}\right). \qedhere\]
\end{proof}
\begin{cor}\label{cor:deg_p_2}
The polynomial $P_2(n,x)$ has degree $\lfloor n/2 \rfloor$.
\end{cor}
\begin{proof}
By Proposition \ref{prop:explicit_coefficients_t_2}, we see that $\deg P_2(n,x)\leq\lfloor n/2\rfloor$. Note that $n - 2\lfloor n/2 \rfloor = 0$ or $1$, which are both triangular numbers, thus $p_2(\lfloor n/2 \rfloor,n) = a(\lfloor n/2 \rfloor) \neq 0$.
\end{proof}
As depicted in Figure \ref{fig:t2_sparse}, the distribution of $Y_2(100)$ is populated with zeros for $x \in \{0, 1, \dots, 50\}$ (i.e., the support is sparse). We provide asymptotics on the proportion of nonzero coefficients of $P_2(n;x)$ to show that in fact the density of the zeros approaches $1$ in the limit of large $n$.
\begin{cor}\label{cor:vanishing_coefficients_t_2}
As $n \to \infty$, we have \[\frac{\#\{0 \leq m \leq \deg P_2(n,x)\ |\ p_2(m,n) \neq 0\}}{\deg P_2(n,x)} = O(n^{-1/2}).\] In particular, \[\lim_{n \to \infty} \frac{ \#\{0 \leq m \leq \deg P_2(n,x)\ |\ p_2(m,n) = 0\}}{\deg P_2(n,x)} = 1.\]
\end{cor}
\begin{proof}
In light of Proposition \ref{prop:explicit_coefficients_t_2}, we see that \begin{align*}
\#\{p_2(m,n) \neq 0\} = \#\{0 \leq m \leq \deg P_2(n,x)\ |\ n-2m \text{ is a triangular number}\}.
\end{align*} However, the right-hand side is at most \[\#\{0 \leq k \leq n \ |\ k \text{ is a triangular number}\} = O(n^{1/2}).\] The result follows since Corollary \ref{cor:deg_p_2} gives us that $\deg P_2(n,x) = \lfloor n/2 \rfloor \asymp n$.
\end{proof}
Combining the previous results, we can now prove Theorem \ref{intro_thm:support_t2}.
\begin{proof}[Proof of Theorem \ref{intro_thm:support_t2}]
Observe that (a) follows from Proposition \ref{prop:explicit_coefficients_t_2} and (b) follows from Corollary \ref{cor:vanishing_coefficients_t_2}, so it suffices to prove (c).
Recall that for any positive integer $n$, we defined \[\mathcal{S}_{2;n} := \left\{x \in \mathbb{R}\ \bigg|\ \frac{n}{2} - \frac{\sqrt{3n}}{2\pi}x \in \{0,1,\ldots,\lfloor n/2\rfloor\}\right\}.\]
Fix $x$ and $n_0$ such that the scaled mass function $f_{2;n_0}$ is defined at $x$. Moreover, for the sake of convenience, we assume that $n_0/2 \in \mathbb{Z}$ and $3n_0$ is square-free (the proof is very similar without these assumptions, albeit more complicated from a notational perspective). Under these assumptions, we see that there must exist some $q \in \mathbb{Z}$ such that \[x = \frac{2q\pi}{\sqrt{3n_0}}.\] Now, for each positive integer $j$, define $n_j := j^2n_0$, so that \[\frac{n_j}{2} - \frac{\sqrt{3n_j}}{2\pi}x = \frac{j^2n_0}{2} - qj \in \{0,1,\ldots, n_j/2\}.\] Note that the nonnegativity of this quantity follows from the fact that $n_0/2 \geq q$ by assumption. Thus, we have produced an infinite sequence $\{n_j\}_{j=1}^\infty$ such that $x \in \mathcal{S}_{2;n_j}$ for all $j$. Now, we claim that there are infinitely many values of $j$ such that $f_{2;n_j}(x) = 0$.
Proposition \ref{prop:explicit_coefficients_t_2} shows us that $f_{2;n_j}(x) = 0$ whenever \[n_j - 2\left(\frac{n_j}{2} - \frac{\sqrt{3n_j}}{2\pi}x\right) = 2qj\] is not a triangular number, and indeed, for a fixed integer $q$, there are infinitely many choices of $j$ such that $2qj$ is not a triangular number. Thus, we have shown that $f_{2;n_j}(x) = 0$ for infinitely many values of $j$. Since $g_2(x) \neq 0$ for $x > 0$, we deduce that $f_{2;n_j}(x)$ cannot converge to $g_2(x)$ as $j \to \infty$.
\end{proof}
Recall from Figure \ref{fig:support_looks_smooth} that the nonzero values of $f_{2;n}(x)$ appear to approximate a continuous function. We will show using Proposition \ref{prop:explicit_coefficients_t_2} that these values are indeed asymptotic to a continuous function.
\begin{lem}\label{lem:meinardus}
For fixed $m$, we have \[\frac{p_2(m,n)}{p(n)} \sim \frac{n\exp\left(\pi\sqrt{\frac{2}{3}}(\sqrt{2m}-\sqrt{n})\right)}{3^{1/4}m^{5/4}}(1+\widetilde{\beta}_2(m)),\] as $n \to \infty$ through suitable values of $n$ (i.e. such that $n -2m$ is a triangular number), where $\widetilde{\beta}_2(m)$ is a constant satisfying $\widetilde{\beta}_2(m) = O_\varepsilon(m^{-1/4+\varepsilon})$ for any $\varepsilon > 0$.
\end{lem}
\begin{proof}
A result of Meinardus \cite[Theorem 6.2]{And84} gives us that \[a(m) =
\frac{3^{1/4}}{12 n^{5/4}}\exp\left( 2 \pi \sqrt{\frac{n}{3}} \right) (1 + \widetilde{\beta}_2(m)). \]
Moreover, the famous Hardy-Ramanujan asymptotic formula \cite{HR18} gives us \[p(n) \sim \frac{1}{4n\sqrt{3}}\exp\left(\pi\sqrt{\frac{2n}{3}}\right)\] as $n \to \infty$, whence the desired result follows after using $p_2(m,n) = a(m)$ from Proposition \ref{prop:explicit_coefficients_t_2} and simplifying.
\end{proof}
Despite the apparent continuity of $f_{2;n}(x)$ as $n \to \infty$ suggested by Lemma \ref{lem:meinardus}, we show below in the proof of Theorem \ref{intro_thm:pmf_no_converge} that the continuous approximations $h_{2;n}(x)$ still fail to converge to $g_2(x)$.
\begin{proof}[Proof of Theorem \ref{intro_thm:pmf_no_converge}] Observe that
\begin{align*}
f_{2;n}(x) = \frac{\sqrt{3n}}{2\pi} \, \frac{p_2\left(\frac{n}{2} - \frac{\sqrt{3n}}{2\pi}x, n\right)}{p(n)}.
\end{align*}
As a quick remark, Proposition \ref{prop:explicit_coefficients_t_2} tells us that $f_{2;n}(x) \neq 0$ if and only if $\frac{\sqrt{3n}}{\pi}x$ is a triangular number. In line with our result from Lemma \ref{lem:meinardus}, the choice
\[
h_{2;n}(x) := \frac{3^{1/4}n^{3/2} \exp \left(\pi \sqrt{\frac{2}{3}} \left( \sqrt{n - \frac{\sqrt{3n}}{\pi}x} - \sqrt{n}\right) \right)}{2 \pi\left( \frac{n}{2} - \frac{\sqrt{3n}}{2\pi}x \right)^{5/4}}.
\] satisfies the conditions in the theorem statement. On the other hand, for any fixed $x \in \mathbb{R}$,
\[
\exp \left(\pi \sqrt{\frac{2}{3}} \left( \sqrt{n - \frac{\sqrt{3n}}{\pi}x} - \sqrt{n}\right) \right) = \Omega(1)
\]
as $n \to \infty$, so it follows that $h_{2;n}(x) \to \infty$ as $n \to \infty$, concluding the proof.
\end{proof}
\subsection{Hook Lengths Divisible by Three}\label{sec:three_hl} In this subsection, we find an explicit formula for the coefficients of the polynomial $P_3(n;x)$ and prove Theorems \ref{intro_thm:support_t3} and \ref{intro_thm:pmf_converge_t3}.
\begin{prop}\label{prop:explicit_coefficients_t_3}
In the case $t = 3$, the coefficients are given by \[p_3(m,n) = b(m)c(n-3m)\] where $b(k)$ denotes the number of $3$-colored partitions of $k$, \[c(k) := \sum_{d |3k+1} \left(\frac{d}{3}\right),\] and $\left(\frac{a}{p}\right)$ denotes the Legendre symbol.
\end{prop}
\begin{proof}
Similar to the proof of Proposition \ref{prop:explicit_coefficients_t_2}, first notice that \[\prod_{m=1}^\infty \frac{1}{(1-(xq^3)^m)^3} = \sum_{k=0}^\infty b(k)x^kq^{3k},\] where $b(k)$ is the number of $3$-colored partitions of $n$. On the other hand, Han--Ono \cite{HO11} showed that \[\prod_{m=1}^\infty \frac{(1-q^{3m})^3}{1-q^m} = \sum_{k=0}^\infty c(k)q^k,\] where $c(k)$ is defined as in the statement of the proposition. We obtain the proposition by computing the coefficient of $x^mq^n$ in (\ref{generating_function}): \[G_3(x;q) = \prod_{m=1}^\infty \frac{(1-q^{3m})^3}{(1-(xq^3)^m)^3(1-q^m)} = \left(\sum_{k=0}^\infty b(k)x^kq^{3k}\right)\left(\sum_{k=0}^\infty c(k)q^k\right). \qedhere\]
\end{proof}
The properties of the coefficients $c(k)$ were studied in depth in \cite{HO11}. Their results allow us to explicitly characterize the nonzero coefficients and compute the degree of $P_3(n;x)$.
\begin{cor}\label{cor:zero_coeffs_t_3}
We have $p_3(m,n) = 0$ if and only if $c(n-3m) = 0$. In particular, $p_3(m,n) = 0$ if and only if there exists a prime $r \equiv 2\pmod{3}$ such that $\operatorname{ord}_r(3(n-3m)+1)$ is odd.
\end{cor}
\begin{proof}
This corollary is an immediate consequence of \cite[Theorem 1.1]{HO11} and the fact that $b(k) \neq 0$ for any nonnegative integer $k$.
\end{proof}
\begin{cor} \label{cor:deg_P_3}
The polynomial $P_3(n, x)$ has degree $\lfloor n/3 \rfloor$.
\end{cor}
\begin{proof}
Observe that $n - 3\lfloor n/3 \rfloor$ is either $0$, $1$, or $2$, and $c(0) = 1$, $c(1) = 1$, $c(2) = 2$.
\end{proof}
\begin{cor}
If $3(n - 3m)+1$ is prime, then $p_3(m,n) = 2b(m)$.
\end{cor}
\begin{proof}
If $3(n - 3m)+1$ is prime, then \[c(n-3m) = \left(\frac{1}{3}\right) + \left(\frac{3(n-3m)+1}{3}\right) = 2. \qedhere\]
\end{proof}
We now seek an asymptotic upper bound on the proportion of nonzero coefficients in $P_3(n;x)$ as $n \to \infty$. Before proceeding, we establish some notation from \cite{HO11}. Consider the series \[D(q) := \prod_{m=1}^\infty (1-q^m)^8 := \sum_{k=0}^\infty d(k)q^k.\] To further study the coefficients $d(k)$, Han--Ono \cite{HO11} noted that the \textit{renormalized series} \[\mathcal{D}(q) := \sum_{n = 1}^\infty d^*(n) q^n := \sum_{n = 0}^\infty d(n)q^{3n + 1}\] arises as the $q$-expansion of a modular form belonging to $S_4(\Gamma_0(9))$, the space of weight $4$ cusp forms on $\Gamma_0(9)$. They used the fact that $\dim S_4(\Gamma_0(9)) = 1$ to show that $\mathcal{D}(q)$ is in fact a normalized Hecke eigenform, and thus deduce general multiplicative properties of its Fourier coefficients $d(k)$. Moreover, by relating the renormalized series \[C(q) := \sum_{k=0}^\infty c(k)q^{3k+1}\] the norm form on the ring of integers of the imaginary quadratic field $\mathbb{Q}(\sqrt{-3})$, they derived the formula for $c(k)$ in the statement of Proposition \ref{prop:explicit_coefficients_t_3} and in turn showed that $c(k) = 0$ if and only $d(k) = 0$ for all nonnegative integers $k$. Thus, in light of Corollary \ref{cor:zero_coeffs_t_3}, we can quantify the rate at which the support of $Y_3(n)$ vanishes by exploiting the multiplicative properties of $d(k)$.
Recall that a set $E \subset \mathbb{Z}^{+}$ is \textit{multiplicative} if for relatively prime integers $n_1, n_2 \in \mathbb{Z}^{+}$, we have $n_1 n_2 \in E$ if and only if $n_1 \in E$ or $n_2 \in E$. We use the aforementioned properties of $d(k)$ in tandem with \cite[Theorem 2.4(a)]{Ser75} on the sizes of complements of multiplicative sets to estimate the proportion of nonzero coefficients of $P_3(n,x)$. In the same spirit as Corollary \ref{cor:vanishing_coefficients_t_2}, the following result allows us to say that ``almost every" coefficient of $P_{3}(m,n)$ is equal to zero.
\begin{cor}\label{cor:vanishing_coefficients_t_3}
As $n \to \infty$, \[\frac{\#\{0 \leq m \leq \deg P_3(n, x) \ | \ p_3(m,n) \neq 0\}}{\deg P_3(n,x)} = O\left(\frac{1}{\sqrt{\log(n)}}\right).\]
In particular, \[\lim_{n \to \infty} \frac{\#\{0 \leq m \leq \deg P_3(n, x) \ | \ p_3(m,n) = 0\}}{\deg P_3(n,x)} = 1.\]
\end{cor}
\begin{proof}
First consider the set $E := \{m \in \mathbb{Z}^{+} \ | \ d^*(m) = 0\},$ which consists of the integers $m$ where the renormalized coefficients $d^*(m)$ vanish. Moreover, consider the set $P$ consisting of all primes congruent to $2 \pmod{3}$. By \cite[Corollary 2.2]{HO11}, the set $E$ is multiplicative, and $P$ is precisely the set of primes in $E$. Due to a classical result of Dirichlet, the set $P$ has natural density $1/2$ in the natural numbers. Then, \cite[Theorem 2.4]{Ser75} tells us that for any $z > 0$, \[\overline{E}(z) := \# \{0 \leq \ell \leq z \ | \ \ell \not\in E \} = \#\{0 \leq \ell \leq z \ |\ d^*(\ell) \neq 0\} = O\left(\frac{z}{\sqrt{\log(z)}}\right).\] Thus, using the fact that $c(k) = 0$ if and only if $d(k) = 0$ (see \cite[Theorem 1.1]{HO11}), we deduce that
\begin{align*}
\# \{0 \leq m \leq \lfloor n/3 \rfloor \ | \ p_3(m, n) \neq 0 \} & \leq \# \{0 \leq k \leq n \ | \ c(k) \neq 0 \} \\[5pt] & = \# \{0 \leq k \leq n \ | \ d(k) \neq 0 \} \\[5pt]
& = \# \{0 \leq k \leq n \ | \ d^*(3k + 1) \neq 0 \} \\[5pt]
& \leq \# \{0 \leq l \leq 3n + 1 \ | \ d^*(l) \neq 0 \} \\[5pt]
& = O\left(\frac{n}{\sqrt{\log(n)}}\right).
\end{align*}
The desired results follow after dividing by $\deg P_3(n,x) = \lfloor n/3\rfloor \asymp n$. \qedhere
\end{proof}
\begin{proof}[Proof of Theorem \ref{intro_thm:support_t3}]
Combining the results in this subsection completes the proof of Theorem \ref{intro_thm:support_t3}. In particular, note that Theorem \ref{intro_thm:support_t3}(a) follows from Proposition \ref{cor:zero_coeffs_t_3}, while Theorem \ref{intro_thm:support_t3}(b) follows from Corollary \ref{cor:vanishing_coefficients_t_3}. Finally, Theorem \ref{intro_thm:support_t3}(c) follows from a similar argument as in the proof of Theorem \ref{intro_thm:support_t2}(c) in Section \ref{sec:even_hl}.
\end{proof}
In preparation for the proof of Theorem \ref{intro_thm:pmf_converge_t3}, we show that the values of $f_{3; n}(x)$ restricted to its support approximate multiple continuous curves using similar techniques as in Lemma \ref{lem:meinardus}.
\begin{lem}\label{lem:meinardus_t3}
We have \[\frac{p_3(m, n)}{p(n)} \sim \frac{n\sqrt{3} \exp\left(\pi \sqrt{\frac{2}{3}} (\sqrt{3m} - \sqrt{n})\right)}{2^{3/2}m^{3/2}} \widetilde{\alpha}(m,n)(1 + \widetilde{\beta}_3(m))\] as $n \to \infty$ for all $m$ such that $f_{3; n}(m) \neq 0$, where $\widetilde{\beta}_3(m) = O_\varepsilon(m^{-1/4 + \varepsilon})$ for any $\varepsilon > 0$.
\end{lem}
\begin{proof}
Recall from Proposition \ref{prop:explicit_coefficients_t_3} that $p_3(m,n) = b(m)c(n-3m)$, and set $\widetilde{\alpha}(m,n) := c(n-3m)$. A result of Meinardus \cite[Theorem 6.2]{And84} gives us that \[b(m) = \frac{1}{2^{7/2}m^{3/2}} \exp (\pi \sqrt{2m})(1 + \widetilde{\beta}_3(m)).\] As in the proof of Lemma \ref{lem:meinardus}, the desired result follows after applying the Hardy-Ramanujan asymptotic formula \cite{HR18} for $p(n)$ and simplifying.
\end{proof}
\begin{xrem}
We make a few comments on the values of the integers $\alpha(m,n) = c(n - 3m)$. First, for any integer $k$, note that the character sum \[c(k) = \sum_{d|3k+1} \left(\frac{d}{3}\right)\] is odd if and only if $k$ is a perfect square, as the Legendre symbol is completely multiplicative. As the perfect squares have natural density zero in the integers, it follows that $100\%$ of the scalars $\alpha(m,n)$ are even numbers in the limit of large $n$. Moreover, if we factor $k = r_1^2r_2,$ where $r_1$ is divisible only by primes equivalent to $2 \pmod{3}$ and $r_2$ is divisible only by primes equivalent to $1 \pmod{3}$, then $c(k)$ is precisely the number of divisors of $r_2$. In light of this fact, one can study the frequency with which $\alpha(m,n) = c(n-3m) = \ell$ for any positive integer $\ell$ by applying results on the density of the prime numbers.
\end{xrem}
In the case $t = 3$, we have so far shown that the nonzero points of $f_{3;n}(m)$ lie on integer multiples of the continuous function $h_{3;n}(m)$. When $t = 2$, Theorem \ref{intro_thm:pmf_no_converge} tells us that the continuous approximations $h_{2; n}(m)$ fail to converge to $g_2(m)$ as $n \to \infty$. In contrast, we now prove that $h_{3; n}(m)$ in fact converges to a scalar multiple of the density function $g_3(m)$.
\begin{proof}[Proof of Theorem \ref{intro_thm:pmf_converge_t3}]
Observe that
\[
f_{3;n}(x) = \frac{\sqrt{6n}}{3\pi} \, \frac{p_3\left( \frac{n}{3} - \frac{\sqrt{6n}}{3\pi}x, n \right)}{p(n)}.
\]
Corollary \ref{cor:zero_coeffs_t_3} tells us that $f_{3;n}(x) \neq 0$ if and only if $c\left(\frac{\sqrt{6n}}{\pi}x\right) = 0$. By Lemma \ref{lem:meinardus_t3}, the choice of
\[
h_{3;n}(x) := \frac{3\sqrt{3}n^{3/2}\exp\left(\pi\sqrt{\frac{2}{3}}\left(\sqrt{n-\frac{\sqrt{6n}}{\pi}x} - \sqrt{n}\right)\right)}{2\pi\left(n - \frac{\sqrt{6n}}{\pi}x\right)^{3/2}}
\]
and $\alpha(x,n) := \widetilde{\alpha}\left(\frac{n}{3} - \frac{\sqrt{6n}}{3\pi}x, n\right)$ satisfies the conditions of the theorem.
For statement (b), recall that $g_3(x) = e^{-x}$. For all $x \in \mathbb{R}$, applying the Taylor series expansion (in $x$) of \[n^{1/2} \left(1 - \frac{\sqrt{6}}{\sqrt{n} \pi}x\right)^{1/2}\] gives us
\begin{align*}
\lim_{n \to \infty} h_{3; n}(x) = \lim_{n \to \infty} \frac{3\sqrt{3}n^{3/2}\exp\left(\pi\sqrt{\frac{2}{3}}\left(\sqrt{n-\frac{\sqrt{6n}}{\pi}x} - \sqrt{n}\right)\right)}{2\pi\left(n - \frac{\sqrt{6n}}{\pi}x\right)^{3/2}} = \frac{3\sqrt{3}}{2 \pi} g_3(x),
\end{align*} and the desired result follows.
\end{proof}
\section{Convergence in Distribution}\label{sec:cdfs}
In this section, we show that the cumulative distribution function of $\{Y_t(n)\}$ converges to a shifted Gamma distribution for $t = 2, 3$, thereby fully resolving the question of limiting distributions of $\{Y_t(n)\}$ for all $t \geq 2$. We proceed by using the generating function (\ref{generating_function}) to estimate values of the polynomials $P_t(n,x)$ using the saddle point method. We will then write the characteristic functions of the random variables $Y_t(n)$ in terms of these polynomials and use these estimates to prove the desired pointwise convergence of the characteristic functions.
The following proposition is our main tool for obtaining the estimates necessary to prove Theorem \ref{intro_thm:convergence_in_distribution}. Griffin--Ono--Tsai proved a similar result the case where $\alpha$ takes on strictly real values \cite[Proposition 2.2]{GOT22}, but we show that the result can be extended to the case where $\alpha$ takes on purely imaginary values. The proof of \cite[Proposition 2.2]{GOT22} can be readily adapted to prove our modified proposition, so our treatment of its proof will be rather terse.
\begin{prop} \label{prop:formula_P(n, x)}
Fix $t = 2, 3$ and let $\alpha \in i\mathbb{R}$ be purely imaginary. Then, the following asymptotic estimate holds in the limit of large $n$: \[P_t\left(n,x_n\right) = \frac{1}{2^{7/4}3^{1/4}n}\sqrt{\frac{1}{\sqrt{6}} + \frac{\alpha}{\pi t}}\left(\frac{\pi t}{\pi t + \alpha\sqrt{6}}\right)^{t/2} \exp\left(\pi\sqrt{n}\left(\sqrt{\frac{2}{3}}+\frac{\alpha}{\pi t}\right)\right) \cdot (1 + O_\alpha(n^{-1/7})),\] where $x_n := \exp\left(\alpha/\sqrt{n}\right)$ for each positive integer $n$.
\end{prop}
\begin{proof}
Applying the Cauchy residue theorem to (\ref{generating_function}) gives us \[P_t(n,x_n) = \frac{1}{2\pi} \int_{-\pi}^\pi (ze^{i\tau})^{-n}G_t(x_n;ze^{i\tau})\ d\tau = \frac{1}{2\pi} \int_{-\pi}^{\pi} \exp(g_t(x_n;ze^{i\tau}))\ d\tau,\] where \begin{align*}g_t(x_n;w) &:= \mathrm{Log}(w^{-n}G_t(x_n;w))\\[10pt] &= -n\log(w) + \sum_{m=1}^\infty t\log(1-w^{tm}) - \sum_{m=1}^\infty t\log(1-(x_nw^t)^m) - \sum_{m=1}^\infty \log(1-w^m),\end{align*} provided $0 < |w| < 1$. To apply the saddle point method, we seek $\kappa\in \mathbb{C}$ with $0 < |\kappa| < 1$ such that \[g_t(x_n,\kappa) \neq 0, \quad \frac{d}{dw}\bigg|_{w=\kappa}[g_t(x_n;w)] = 0,\quad \frac{d^2}{dw^2}\bigg|_{w=\kappa}[g_t(x_n;w)]\neq 0.\] We set the derivative of $g_t(x_n;w)$ with respect to $w$ equal to zero and follow a line of reasoning identical to that in the proof of \cite[Proposition 2.2]{GOT22}. In particular,
\begin{align*}
\frac{d}{dw}g(x_n,w) = -\frac{m}{w} - \sum_{j=1}^\infty \frac{t^2j}{1 - w^{tj}}w^{tj-1} + \sum_{j=1}^\infty \frac{t^2j x_n^j}{1 - (x_nw^{j})^j} w^{tj-1} + \sum_{j=1}^\infty \frac{jw^{j-1}}{1 - w^{j}}
\end{align*}
gives
\begin{align*}
n = \sum_{j=1}^\infty \frac{t^2j}{w^{-tj} - 1} + \sum_{j=1}^\infty \frac{t^2j x_n^j}{w^{-tj} - x_n^j} + \sum_{j=1}^\infty \frac{j}{w^{-j} -1}.
\end{align*}
By the definition of $x_n$ and
\begin{align}
\label{2.8analogue} \sum_{j=1}^\infty \frac{j}{e^{j \alpha} - 1} = \frac{\pi^2}{6\alpha^2} - \frac{1}{2\alpha} + O(1),
\end{align}
we obtain the saddle point $\kappa = e^{-\rho_n}$, where \[\rho_n := \left(\frac{\pi}{\sqrt{6}} + \frac{\alpha}{t}\right)n^{-1/2} + O_\alpha(n^{-1}).\] In particular, notice that (for sufficiently large $n$) \[|\kappa| = |e^{-\rho_n}| = |e^{-\operatorname{Re}(\rho_n)}| < 1.\]
With this value of $\kappa$, we see that (again following the arguments in \cite[Proposition 2.2]{GOT22}) \begin{align*}
g_t(x_n,\kappa) &= n\rho_n + \sum_{m=1}^\infty t\log(1-e^{-tm\rho_n}) - t\sum_{m=1}^\infty \log(1-x_n^me^{-tm\rho_n}) - \sum_{m=1}^\infty \log(1-e^{-m\rho_n}) \\[10pt] &=
n\rho_n + \sum_{m=1}^\infty t\log(1-e^{-tm\rho_n}) - t\sum_{m=1}^\infty \log(1-e^{-m(t\rho_n - \alpha/\sqrt{n})}) - \sum_{m=1}^\infty \log(1-e^{-m\rho_n}).
\end{align*}
By applying Euler-Maclaurin summation in the style of \cite[Lemma 2.3]{GOT22}, we have \[\sum_{m=1}^\infty \log(1-e^{-j\gamma}) = -\frac{\pi^2}{6\gamma} - \frac{1}{2}\mathrm{log}\left(\frac{\gamma}{2\pi}\right) + O(\gamma)\] for any $\gamma \in \mathbb{C}$ with positive real part. Using this equality, we obtain the same estimate for $g_t(x_n,\kappa)$ as in the proof of \cite[Proposition 2.2]{GOT22}. In the same way as we have outlined so far, the rest of the proof follows \textit{mutatis mutandis} to the proof of \cite[Proposition 2.2]{GOT22}, using the facts that $\rho_n$ has positive real part and multiplication by $x_n$ does not affect absolute value.
\end{proof}
With this proposition in place, we can now prove the convergence in distribution of $\{Y_t(n)\}$ to a shifted Gamma distribution. Due to the following continuity theorem by L\'evy, it suffices to prove pointwise convergence of characteristic functions.
\begin{thm}[L\'evy] \cite[Section 18.1]{Wil91}
Let $(X_n)$ be a sequence of random variables with characteristic functions given by $\varphi_n(r):= \mathbb{E}[e^{iX_nr}]$ and let $X$ be a random variable with characteristic function $\varphi(r) := \mathbb{E}[e^{iXr}]$. Suppose that
\[
\varphi(r) = \lim_{n \to \infty} \varphi_n(r)
\]
for all $r \in \mathbb{R}$ and that $\varphi$ is continuous at $r = 0$. Then $X_n$ converges to $X$ in distribution.
\end{thm}
The following proof of Theorem \ref{intro_thm:convergence_in_distribution} is similar to the proof of \cite[Theorem 1.2]{GOT22} in the case of $t \geq 4$. The methods of Griffin--Ono--Tsai do not apply to the cases $t = 2, 3$ since the moment generating functions of the corresponding Gamma distributions do not exist, but replacing moment generating functions with characteristic functions allows us to extend the result to all $t \geq 2$. Recall the notation $k(t) = \frac{t-1}{2}$ and $\theta(t) = \sqrt{\frac{2}{t-1}}$ from Section \ref{sec:intro}.
\begin{proof}[Proof of Theorem \ref{intro_thm:convergence_in_distribution}]
The characteristic function of the Gamma distribution $X_{k, \theta}$ is given by $$\varphi(X_{k, \theta}, r) =\left(\frac{1}{1 - ir\theta}\right)^k,$$ where $X_{k, \theta}$ has mean $\mu_{k, \theta} = k \theta$, mode $\text{mo}_{k, \theta} = (k - 1)\theta$ and variance $\sigma_{k, \theta}^2 = k \theta^2$. For $a, b \in \mathbb{R}$, the shifted Gamma distribution $a X_{k, \theta} + b$ has characteristic function $$\varphi(a X_{k, \theta} + b, r) = \frac{e^{br}}{(1 - i \theta a r)^k},$$ with mean $a k \theta + b$, mode $a(k - 1)\theta + b$ and variance $a^2 k \theta^2$. We proceed by comparing $Y_t(n)$ with $a X_{k, \theta} + b$, where $k = \frac{t - 1}{2}, \theta = \sqrt{\frac{2}{t - 1}}, a = -1$, and $b = \sqrt{2(t - 1)}/2.$
By (\ref{eqn:probability}), the characteristic functions of $Y_t(n)$ are given by $$\varphi(Y_t(n), r) = \frac{1}{p(n)} \sum_{m = 0}^\infty p_t(m; n) e^{\frac{i(m - \mu_t(n))r}{\sigma_t(n)}}$$ for each $n \geq 1$. Due to L\'evy's continuity theorem and the above remarks, it suffices to show $$\lim_{n\to \infty} \varphi(Y_t(n); r) = \frac{e^{br}}{(1 - i \theta a r)^k}$$ for any $r \in \mathbb{R}$. We first evaluate $P_t(n; x)$ at $x = e^{\frac{ir}{\sigma_t(n)}}$ to obtain $$\varphi(Y_t(n), r) = \frac{P_t(n; e^{\frac{ir}{\sigma_t(n)}})}{p(n)}e^{-i\frac{\mu_t(n)}{\sigma_t(n)}r}.$$ By Proposition \ref{prop:formula_P(n, x)} with $\alpha = i\pi t r / \sqrt{3(t - 1)}$ and $x_n = e^{\frac{\alpha}{\sqrt{n}}}$, we find that
\begin{align*}
\varphi(Y_t(n), r) & = \frac{(2^{\frac74}3^{\frac14}n)^{-1}\sqrt{\frac{1}{\sqrt{6}} + \frac{ir}{\sqrt{3(t - 1)}}} \cdot (1 + \sqrt{\frac{2}{t - 1}}ir)^{-\frac{t}{2}} \cdot (1 + O_r(n^{-\frac17}))}{(4 \sqrt{3}n)^{-1} \cdot (1 + O(n^{-\frac17}))} \cdot e^{\frac{n}{t \sigma_t(n)}ir - \frac{\mu_t(n)}{\sigma_t(n)}ir} \\
& = \frac{e^{\frac{\sqrt{2(t - 1)}}{2}ir}}{(1 + \sqrt{\frac{2}{t - 1}}ir)^{\frac{t - 1}{2}}} \cdot (1 + O_r(n^{-\frac17})).
\end{align*}
Taking the limit as $n \to \infty$, L\'evy's continuity theorem gives us \[Y_t(n) \sim \sigma_t(n) (a X_{k, \theta} + b) + \mu_t(n).\] We also obtain claim (2) since the random variable $X_{k, \theta}$ has cumulative distribution function $F_{k, \theta}(x) = \gamma(k, \frac{x}{\theta})/\Gamma(k)$ \cite[II.2]{Fel71}, where $\gamma(s, t)$ is the lower incomplete Gamma function.
\end{proof}
\section{Conclusion}\label{sec:conj}
Theorems \ref{intro_thm:support_t2}(c) and \ref{intro_thm:support_t3}(c) describe respectively how the many vanishing terms in the distributions of $Y_t(n)$ prevent the convergence of the scaled mass functions $f_{t;n}$ for $t = 2,3$ to $g_t(x)$, for various notions of convergence. In the case of $t \geq 4$, however, none of the coefficients of $P_t(n;x)$ vanish \cite{granville_ono_1996}, even in the limit of large $n$. While the work of \cite{GOT22} establishes convergence of the cumulative distribution functions of $\{Y_t(n)\}$ to a shifted Gamma distribution for $t \geq 4,$ the corresponding probability mass functions have yet to be studied. Thus, comparison with Theorems \ref{intro_thm:support_t2}(c), \ref{intro_thm:support_t3}(c) leads to the natural question of whether the functions $f_{t;n}(x)$ converge to $g_t$ for $t \geq 4$. Indeed, there is experimental evidence in support of the positive -- see, for instance, Figure \ref{fig:conjecturet11} for $t = 11$.
\begin{figure}[h!]
\centering
\includegraphics[width=4in]{conjecturet11_5000.png}
\caption{Plot of the scaled mass function $f_{11;5000}(x)$ (in red) with the probability density function $g_{11}(x)$.}
\label{fig:conjecturet11}
\end{figure}
\begin{conj} For $t \geq 4$, we have $f_{t;n}(x)\to g_t(x)$ as $n \to \infty$ for all $x > 0$.
\end{conj}
\nocite{*}
|
{
"timestamp": "2022-07-15T02:01:20",
"yymm": "2207",
"arxiv_id": "2207.06486",
"language": "en",
"url": "https://arxiv.org/abs/2207.06486"
}
|
\section{Introduction}\label{sec:intro}
In his original work, Einstein formulated the general theory of relativity in terms of the metric tensor as the fundamental field variable of the gravitational field, which describes gravity by the curvature of its Levi-Civita connection. Numerous modified gravity theories depart from this formulation, either keeping the metric as the only fundamental field variable and modifying its dynamics through a modified action, or by adding further fundamental field which couple non-minimally to the curvature~\cite{CANTATA:2021ktz}. However, there exist also other classes of gravity theories, in which the curvature of the Levi-Civita connection plays a less prominent role, and another, independent connection is introduced as a fundamental field variable next to the metric. Unlike the Levi-Civita connection, this connection is assumed to have vanishing curvature, but instead one allows for non-vanishing torsion or nonmetricity, or both. Gravity theories of this type are known as \emph{teleparallel} gravity theories.
In fact, already Einstein studied the possibility to describe gravity in terms of the torsion of a flat, metric-compatible connection instead of curvature~\cite{Einstein:1928}, in an attempt to unify gravity and electromagnetism. While this attempt was not successful, it gave rise to a new class of gravity theories, now known as metric teleparallel gravity theories~\cite{Aldrovandi:2013wha,Bahamonde:2021gfp}, in which gravity is mediated by torsion instead of curvature. Only much later another class of gravity theories was introduced, which attributes gravity to the nonmetricity of a flat, torsion-free (i.e., symmetric) connection, and is hence known as symmetric teleparallel gravity~\cite{Nester:1998mp}. Finally, allowing for both torsion and nonmetricity leads to the realm of general teleparallel gravity~\cite{BeltranJimenez:2019odq}. It is worth mentioning that these theories are embedded in the much wider and well-studied framework of metric-affine gravity theories~\cite{Hehl:1994ue,Blagojevic:2002du}, for the metric-compatible case also in the framework of Poincaré gauge theories~\cite{Hehl:1976kj,Blagojevic:2013xpa}. However, a full account of this relationship and the historic development and studies of teleparallel gravity theories would by far exceed the scope of this chapter.
Despite the long-standing history of teleparallel gravity theories and the studies of their fundamental properties and underlying structure for several decades, a renewed and growing interest in teleparallel modifications and extension of general relativity and their phenomenology has arisen only recently with the growing number of unexplained observations and tensions in cosmology. Numerous theories have been constructed as possible candidates to explain the early and late accelerating phases of the universe, known as inflation and dark energy eras, to resolve the question of singularities and the information paradox of black holes, and to provide alternative pathways towards a quantization of gravity and a unification with other fundamental forces. The phenomenology of these theories greatly differs and depends on their choice of dynamical fields and action, so that a full account would, again, exceed the scope of this chapter, and we must limit ourselves to a more general discussion of the class of teleparallel gravity theories, and leave specific theories and their phenomenological properties for further reading.
The aim of this chapter is to provide a practical introduction to teleparallel gravity. In section~\ref{sec:teleact} we give a simplified summary of the general structure and underlying mathematical foundations of teleparallel gravity theories in their three flavors - general, symmetric and metric. In particular, we discuss the fundamental fields in these theories, the general form of the action and the field equations. This practical introduction continues in section~\ref{sec:telephys}, where we explain how to formulate physical principles and perform common calculations necessary to solve the gravitational field equations of teleparallel gravity theories. We discuss how the invariance of the action under diffeomorphisms leads to the conservation of the matter currents, and show how to construct teleparallel geometries with spacetime symmetries and their perturbations, which can be used to solve the field equations of a given theory of gravity and thus study its phenomenology. Finally, section~\ref{sec:theories} gives an overview of the most commonly studied classes of teleparallel gravity theories and their field equations, and briefly summarizes their common properties.
There are many interesting aspects of teleparallel gravity which cannot be covered in this chapter, as they would by far exceed its scope and its aim towards performing practical calculations. In particular, we do not discuss the role of gauge symmetries in teleparallel gravity, which allow its interpretation as a gauge theory of the translation group. In relation to this, we do not discuss its formulation in terms of a tetrad. Throughout the chapter, we use only the tensor notation, which is more widespread in relating gravity to observations, and avoid the use of differential form language, which is often more concise and thus preferred by theorists, but less common in practical calculations of phenomenology. Further, we cannot cover fundamental questions such as the number of degrees of freedom of these theories, which is studied in their Hamiltonian formulation, and hints towards theoretical issues known under the term strong coupling. The interested reader is encouraged to follow the references provided in this chapter for a more detailed account of these mathematical foundations, their applications and possible issues.
We use the convention that spacetime indices are labeled with lowercase Greek letters and take the values \((0,1,2,3)\), as well as the metric signature \((-1,+1,+1,+1)\).
\section{Dynamical fields, action and field equations}\label{sec:teleact}
In this introductory section we give an overview of the dynamical fields and their properties, the general structure of the action, and the variational methods used to obtain their field equations. Here we focus on three different flavors of teleparallel theories: general teleparallel theories, in which both torsion and nonmetricity are allowed to be non-vanishing, are discussed in section~\ref{ssec:genteleact}; we then restrict the theories to symmetric teleparallel gravity by imposing vanishing torsion in section~\ref{sec:symteleact}, and to metric teleparallel gravity by imposing vanishing nonmetricity in section~\ref{sec:metteleact}.
\subsection{General teleparallel gravity}\label{ssec:genteleact}
We start our discussion of teleparallel gravity theories from the viewpoint of metric-affine gravity, in which next to the metric \(g_{\mu\nu}\) a connection with coefficients \(\Gamma^{\mu}{}_{\nu\rho}\) is introduced as a fundamental field on the spacetime manifold \(M\), which is independent of the Levi-Civita connection. To distinguish these two connections, we write the latter, and all derived quantities such as the covariant derivative and the curvature tensor, with an empty circle on top, i.e.,
\begin{equation}\label{eq:levicivita}
\lc{\Gamma}^{\mu}{}_{\nu\rho} = \frac{1}{2}g^{\mu\sigma}\left(\partial_{\nu}g_{\sigma\rho} + \partial_{\rho}g_{\nu\sigma} - \partial_{\sigma}g_{\nu\rho}\right)\,.
\end{equation}
Given another, independent connection, their difference can always be written in the form
\begin{equation}\label{eq:conndec}
\Gamma^{\mu}{}_{\nu\rho} - \lc{\Gamma}^{\mu}{}_{\nu\rho} = M^{\mu}{}_{\nu\rho} = K^{\mu}{}_{\nu\rho} + L^{\mu}{}_{\nu\rho}\,.
\end{equation}
Here, \(M^{\mu}{}_{\nu\rho}\) is called the \emph{distortion}: it is the difference between two connection coefficients, and hence a tensor field. If one of these two connections is the Levi-Civita connection of a metric, the distortion decomposes further into the \emph{contortion} \(K^{\mu}{}_{\nu\rho}\) and the \emph{disformation} \(L^{\mu}{}_{\nu\rho}\), which can be obtained as follows. First, define the \emph{torsion}
\begin{equation}\label{eq:torsion}
T^{\mu}{}_{\nu\rho} = \Gamma^{\mu}{}_{\rho\nu} - \Gamma^{\mu}{}_{\nu\rho}\,,
\end{equation}
as well as the \emph{nonmetricity}
\begin{equation}\label{eq:nonmetricity}
Q_{\mu\nu\rho} = \nabla_{\mu}g_{\nu\rho} = \partial_{\mu}g_{\nu\rho} - \Gamma^{\sigma}{}_{\nu\mu}g_{\sigma\rho} - \Gamma^{\sigma}{}_{\rho\mu}g_{\nu\sigma}\,.
\end{equation}
These are, again, tensor fields. Using the metric to raise and lower indices, one then obtains the contortion
\begin{equation}\label{eq:contortion}
K^{\mu}{}_{\nu\rho} = \frac{1}{2}\left(T_{\nu}{}^{\mu}{}_{\rho} + T_{\rho}{}^{\mu}{}_{\nu} - T^{\mu}{}_{\nu\rho}\right)\,,
\end{equation}
as well as the disformation
\begin{equation}\label{eq:disformation}
L^{\mu}{}_{\nu\rho} = \frac{1}{2}\left(Q^{\mu}{}_{\nu\rho} - Q_{\nu}{}^{\mu}{}_{\rho} - Q_{\rho}{}^{\mu}{}_{\nu}\right)\,.
\end{equation}
Hence, in the presence of a metric, an independent connection can always uniquely be specified in terms of its torsion and nonmetricity, which determine its deviation from the Levi-Civita connection.
The dynamical fields then enter the action of the theory, which is of the general form
\begin{equation}
S[g, \Gamma, \psi] = S_{\text{g}}[g, \Gamma] + S_{\text{m}}[g, \Gamma, \psi]\,,
\end{equation}
where the gravitational part \(S_{\text{g}}\) of the action depends only on the metric and the connection, while the matter part \(S_{\text{m}}\) also depends on some set of matter fields \(\psi^I\), whose components we do not specify further and simply label them with an index \(I\). By variation with respect to these matter fields, the matter action determines the matter field equations, which govern the dynamics of the matter fields in a given gravitational field background. In general, this background depends both on the metric \(g_{\mu\nu}\) and the connection \(\Gamma^{\mu}{}_{\nu\rho}\). Further, varying the matter action with respect to the metric and the connection gives rise to the \emph{energy-momentum} \(\Theta^{\mu\nu}\) and \emph{hypermomentum} \(H_{\mu}{}^{\nu\rho}\)\ defined by the variation~\cite{Hehl:1994ue}
\begin{equation}\label{eq:metricmatactvar}
\delta S_{\text{m}} = \int_M\left(\frac{1}{2}\Theta^{\mu\nu}\delta g_{\mu\nu} + H_{\mu}{}^{\nu\rho}\delta\Gamma^{\mu}{}_{\nu\rho} + \Psi_I\delta\psi^I\right)\sqrt{-g}\mathrm{d}^4x\,,
\end{equation}
where \(\Psi_I = 0\) are the matter field equations. The specific form of \(\Theta^{\mu\nu}\) and \(H_{\mu}{}^{\nu\rho}\) depends on the type of matter under consideration and its coupling to the background geometry. These terms will act as the source of the gravitational field equations. To obtain the latter, one writes the variation of the gravitational part of the action in the similar form
\begin{equation}\label{eq:metricgravactvar}
\delta S_{\text{g}} = -\int_M\left(\frac{1}{2}W^{\mu\nu}\delta g_{\mu\nu} + Y_{\mu}{}^{\nu\rho}\delta\Gamma^{\mu}{}_{\nu\rho}\right)\sqrt{-g}\mathrm{d}^4x\,,
\end{equation}
where any necessary integration by parts has been carried out in order to eliminate derivatives acting on the variations. This variation defines two further tensor fields, which we denote \(W^{\mu\nu}\) and \(Y_{\mu}{}^{\nu\rho}\), and which will enter as the dynamical part of the gravitational field equations.
The action and variation given above constitute the general form for a metric-affine theory of gravity. In teleparallel gravity, however, the connection is further restricted to have vanishing curvature,
\begin{equation}\label{eq:curvature}
R^{\mu}{}_{\nu\rho\sigma} = \partial_{\rho}\Gamma^{\mu}{}_{\nu\sigma} - \partial_{\sigma}\Gamma^{\mu}{}_{\nu\rho} + \Gamma^{\mu}{}_{\tau\rho}\Gamma^{\tau}{}_{\nu\sigma} - \Gamma^{\mu}{}_{\tau\sigma}\Gamma^{\tau}{}_{\nu\rho} \equiv 0\,.
\end{equation}
Note that this condition involves both the connection coefficients and their derivatives. In the context of Lagrange theory, such type of condition constitutes a nonholonomic constraint. Different possibilities exist to implement this constraint~\cite{Hohmann:2021fpr}. One possibility is to add another term of the form
\begin{equation}
S_{\text{r}} = \int_M\tilde{r}_{\mu}{}^{\nu\rho\sigma}R^{\mu}{}_{\nu\rho\sigma}\mathrm{d}^4x\,,
\end{equation}
where the tensor density \(\tilde{r}_{\mu}{}^{\nu\rho\sigma}\) acts as a Lagrange multiplier, and can be taken to be antisymmetric in its last two indices, \(\tilde{r}_{\mu}{}^{\nu\rho\sigma} = \tilde{r}_{\mu}{}^{\nu[\rho\sigma]}\), since the contraction of its symmetric part with the antisymmetric indices of the curvature tensor vanishes and thus does not contribute to the action. Variation with respect to \(\tilde{r}_{\mu}{}^{\nu\rho\sigma}\) then yields the constraint equation \(R^{\mu}{}_{\nu\rho\sigma} = 0\). In order to derive the variation with respect to the connection coefficients, note that the variation of the curvature can be expressed as
\begin{equation}\label{eq:metaffvarflat}
\delta R^{\mu}{}_{\nu\rho\sigma} = \nabla_{\rho}\delta\Gamma^{\mu}{}_{\nu\sigma} - \nabla_{\sigma}\delta\Gamma^{\mu}{}_{\nu\rho} + T^{\tau}{}_{\rho\sigma}\delta\Gamma^{\mu}{}_{\nu\tau}\,.
\end{equation}
With the help of this expression, as well as performing integration by parts, one obtains the variation of the Lagrange multiplier term \(S_{\text{r}}\) in the action with respect to the connection as
\begin{equation}
\begin{split}
\delta_{\Gamma}S_{\text{r}} &= \int_M\tilde{r}_{\mu}{}^{\nu\rho\sigma}\left(\nabla_{\rho}\delta\Gamma^{\mu}{}_{\nu\sigma} - \nabla_{\sigma}\delta\Gamma^{\mu}{}_{\nu\rho} + T^{\tau}{}_{\rho\sigma}\delta\Gamma^{\mu}{}_{\nu\tau}\right)\mathrm{d}^4x\\
&= \int_M\left(T^{\sigma}{}_{\sigma\rho}\tilde{r}_{\mu}{}^{\nu\rho\tau} - T^{\rho}{}_{\rho\sigma}\tilde{r}_{\mu}{}^{\nu\tau\sigma} + T^{\tau}{}_{\rho\sigma}\tilde{r}_{\mu}{}^{\nu\rho\sigma} - \nabla_{\rho}\tilde{r}_{\mu}{}^{\nu\rho\tau} + \nabla_{\sigma}\tilde{r}_{\mu}{}^{\nu\tau\sigma}\right)\delta\Gamma^{\mu}{}_{\nu\tau}\mathrm{d}^4x\,.
\end{split}
\end{equation}
Combining all terms, one finds that the gravitational field equations are given by the metric field equation
\begin{equation}\label{eq:genmetfield}
W_{\mu\nu} = \Theta_{\mu\nu}\,,
\end{equation}
as well as the connection field equation
\begin{equation}\label{eq:genconnfieldlag}
\tilde{Y}_{\mu}{}^{\nu\tau} = \tilde{H}_{\mu}{}^{\nu\tau} + T^{\sigma}{}_{\sigma\rho}\tilde{r}_{\mu}{}^{\nu\rho\tau} - T^{\rho}{}_{\rho\sigma}\tilde{r}_{\mu}{}^{\nu\tau\sigma} + T^{\tau}{}_{\rho\sigma}\tilde{r}_{\mu}{}^{\nu\rho\sigma} - \nabla_{\rho}\tilde{r}_{\mu}{}^{\nu\rho\tau} + \nabla_{\sigma}\tilde{r}_{\mu}{}^{\nu\tau\sigma}\,,
\end{equation}
where it is convenient to define the tensor densities
\begin{equation}\label{eq:connvardens}
\tilde{Y}_{\mu}{}^{\nu\tau} = Y_{\mu}{}^{\nu\tau}\sqrt{-g}\,, \quad
\tilde{H}_{\mu}{}^{\nu\tau} = H_{\mu}{}^{\nu\tau}\sqrt{-g}\,.
\end{equation}
Note that the connection still contains the undetermined Lagrange multiplier \(\tilde{r}_{\mu}{}^{\nu\rho\sigma}\). However, the latter can be eliminated using the following procedure. First, we calculate the divergence
\begin{multline}\label{eq:divconnfield}
\nabla_{\tau}\tilde{Y}_{\mu}{}^{\nu\tau} = \nabla_{\tau}\tilde{H}_{\mu}{}^{\nu\tau} + \nabla_{\tau}\left(T^{\sigma}{}_{\sigma\rho}\tilde{r}_{\mu}{}^{\nu\rho\tau} - T^{\rho}{}_{\rho\sigma}\tilde{r}_{\mu}{}^{\nu\tau\sigma} + T^{\tau}{}_{\rho\sigma}\tilde{r}_{\mu}{}^{\nu\rho\sigma}\right)\\
- \nabla_{\tau}\nabla_{\rho}\tilde{r}_{\mu}{}^{\nu\rho\tau} + \nabla_{\tau}\nabla_{\sigma}\tilde{r}_{\mu}{}^{\nu\tau\sigma}\,.
\end{multline}
The last two terms can be simplified by realizing that the Lagrange multiplier \(\tilde{r}_{\mu}{}^{\nu\rho\sigma}\) is antisymmetric in its last two indices, so that one can apply the commutator of covariant derivatives given by
\begin{multline}
2\nabla_{[\rho}\nabla_{\sigma]}\tilde{r}_{\mu}{}^{\nu\rho\sigma} = -T^{\tau}{}_{\rho\sigma}\nabla_{\tau}\tilde{r}_{\mu}{}^{\nu\rho\sigma}\\
- R^{\tau}{}_{\mu\rho\sigma}\tilde{r}_{\tau}{}^{\nu\rho\sigma} + R^{\nu}{}_{\tau\rho\sigma}\tilde{r}_{\mu}{}^{\tau\rho\sigma} + R^{\rho}{}_{\tau\rho\sigma}\tilde{r}_{\mu}{}^{\nu\tau\sigma} + R^{\sigma}{}_{\tau\rho\sigma}\tilde{r}_{\mu}{}^{\nu\rho\tau} - R^{\tau}{}_{\tau\rho\sigma}\tilde{r}_{\mu}{}^{\nu\rho\sigma}\,.
\end{multline}
Also using the vanishing curvature~\eqref{eq:curvature}, the only remaining term is given by
\begin{equation}
2\nabla_{[\rho}\nabla_{\sigma]}\tilde{r}_{\mu}{}^{\nu\rho\sigma} = -T^{\tau}{}_{\rho\sigma}\nabla_{\tau}\tilde{r}_{\mu}{}^{\nu\rho\sigma}\,.
\end{equation}
Further, one can use the antisymmetry of the Lagrange multiplier to write
\begin{multline}
\nabla_{\tau}\left(T^{\sigma}{}_{\sigma\rho}\tilde{r}_{\mu}{}^{\nu\rho\tau} - T^{\rho}{}_{\rho\sigma}\tilde{r}_{\mu}{}^{\nu\tau\sigma} + T^{\tau}{}_{\rho\sigma}\tilde{r}_{\mu}{}^{\nu\rho\sigma}\right) = 3\nabla_{[\tau}\left(T^{\tau}{}_{\rho\sigma]}\tilde{r}_{\mu}{}^{\nu\rho\sigma}\right).
\end{multline}
The derivative of the torsion tensor can be rewritten by making use of the curvature-free Bianchi identity
\begin{equation}
\nabla_{[\nu}T^{\mu}{}_{\rho\sigma]} + T^{\mu}{}_{\tau[\nu}T^{\tau}{}_{\rho\sigma]} = R^{\mu}{}_{[\nu\rho\sigma]} = 0\,,
\end{equation}
from which after contraction follows
\begin{equation}
3\nabla_{[\tau}T^{\tau}{}_{\rho\sigma]} = -3T^{\tau}{}_{\omega[\tau}T^{\omega}{}_{\rho\sigma]} = T^{\tau}{}_{\tau\omega}T^{\omega}{}_{\rho\sigma}\,.
\end{equation}
By combining all terms, one finds that the divergence~\eqref{eq:divconnfield} of the connection field equation~\eqref{eq:genconnfieldlag} reads
\begin{equation}
\begin{split}
\nabla_{\tau}\tilde{Y}_{\mu}{}^{\nu\tau} &= \nabla_{\tau}\tilde{H}_{\mu}{}^{\nu\tau} + T^{\tau}{}_{\tau\omega}T^{\omega}{}_{\rho\sigma}\tilde{r}_{\mu}{}^{\nu\rho\sigma} + 3T^{\tau}{}_{[\rho\sigma}\nabla_{\tau]}\tilde{r}_{\mu}{}^{\nu\rho\sigma} - T^{\tau}{}_{\rho\sigma}\nabla_{\tau}\tilde{r}_{\mu}{}^{\nu\rho\sigma}\\
&= \nabla_{\tau}\tilde{H}_{\mu}{}^{\nu\tau} + T^{\tau}{}_{\tau\omega}T^{\omega}{}_{\rho\sigma}\tilde{r}_{\mu}{}^{\nu\rho\sigma} + 2T^{\tau}{}_{\tau[\rho}\nabla_{\sigma]}\tilde{r}_{\mu}{}^{\nu\rho\sigma}\,.
\end{split}
\end{equation}
Similarly, contracting the field equation~\eqref{eq:genconnfieldlag} with the trace of the torsion tensor, one obtains
\begin{equation}
T^{\omega}{}_{\omega\tau}\tilde{Y}_{\mu}{}^{\nu\tau} = T^{\omega}{}_{\omega\tau}\tilde{H}_{\mu}{}^{\nu\tau} + T^{\omega}{}_{\omega\tau}T^{\tau}{}_{\rho\sigma}\tilde{r}_{\mu}{}^{\nu\rho\sigma} + 2T^{\omega}{}_{\omega[\rho}\nabla_{\sigma]}\tilde{r}_{\mu}{}^{\nu\rho\sigma}\,.
\end{equation}
Subtracting these two equations, the Lagrange multiplier terms cancel, and one obtains the connection field equations
\begin{equation}\label{eq:genconnfielddens}
\nabla_{\tau}\tilde{Y}_{\mu}{}^{\nu\tau} - T^{\omega}{}_{\omega\tau}\tilde{Y}_{\mu}{}^{\nu\tau} = \nabla_{\tau}\tilde{H}_{\mu}{}^{\nu\tau} - T^{\omega}{}_{\omega\tau}\tilde{H}_{\mu}{}^{\nu\tau}\,.
\end{equation}
This equation can also be rewritten by eliminating the density factors using
\begin{equation}\label{eq:covderdens}
\nabla_{\mu}\sqrt{-g} = \frac{1}{2}g^{\nu\rho}\nabla_{\mu}g_{\nu\rho}\sqrt{-g} = \frac{1}{2}Q_{\mu\nu}{}^{\nu}\sqrt{-g} = M^{\nu}{}_{\nu\mu}\sqrt{-g}\,,
\end{equation}
where the last expression follows from rewriting the covariant derivative of the metric in terms of its (vanishing) covariant derivative with respect to the Levi-Civita connection using the decomposition~\eqref{eq:conndec}. The latter can also be used to write the torsion as
\begin{equation}
T^{\mu}{}_{\nu\rho} = M^{\mu}{}_{\rho\nu} - M^{\mu}{}_{\nu\rho}\,.
\end{equation}
Using these relations and the definition~\eqref{eq:connvardens} of the densities \(\tilde{Y}_{\mu}{}^{\nu\rho}\) and \(\tilde{H}_{\mu}{}^{\nu\rho}\), the field equations become
\begin{equation}\label{eq:genconnfield}
\nabla_{\tau}Y_{\mu}{}^{\nu\tau} - M^{\omega}{}_{\tau\omega}Y_{\mu}{}^{\nu\tau} = \nabla_{\tau}H_{\mu}{}^{\nu\tau} - M^{\omega}{}_{\tau\omega}H_{\mu}{}^{\nu\tau}\,.
\end{equation}
The equations~\eqref{eq:genmetfield} and~\eqref{eq:genconnfield} constitute the field equations for the dynamical fields in teleparallel gravity.
Besides the method of Lagrange multipliers, the teleparallel field equations can also be obtained by using the method of restricted variation. Using this method, no Lagrange multiplier is introduced, the constraint equation~\eqref{eq:curvature} of vanishing curvature is imposed to restrict the connection \(\Gamma^{\mu}{}_{\nu\rho}\), and the variation \(\delta\Gamma^{\mu}{}_{\nu\rho}\) is restricted in order to preserve this constraint. Using the expression~\eqref{eq:metaffvarflat}, one finds that the variation of the connection must be of the form
\begin{equation}\label{eq:flatvar}
\delta\Gamma^{\mu}{}_{\nu\rho} = \nabla_{\rho}\xi^{\mu}{}_{\nu}
\end{equation}
for a tensor field \(\xi^{\mu}{}_{\nu}\). Indeed, for the curvature perturbation one then finds
\begin{equation}
\delta R^{\mu}{}_{\nu\rho\sigma} = \nabla_{\rho}\nabla_{\sigma}\xi^{\mu}{}_{\nu} - \nabla_{\sigma}\nabla_{\rho}\xi^{\mu}{}_{\nu} + T^{\tau}{}_{\rho\sigma}\nabla_{\tau}\xi^{\mu}{}_{\nu} = 0\,,
\end{equation}
using the formula for the commutator of covariant derivatives in the absence of curvature. It follows that the variation of the action takes the form
\begin{equation}
\begin{split}
\delta_{\Gamma}S &= \int_M\left(\tilde{H}_{\mu}{}^{\nu\rho} - \tilde{Y}_{\mu}{}^{\nu\rho}\right)\nabla_{\rho}\xi^{\mu}{}_{\nu}\mathrm{d}^4x\\
&= \int_M\left(T^{\sigma}{}_{\sigma\rho}\tilde{H}_{\mu}{}^{\nu\rho} - \nabla_{\rho}\tilde{H}_{\mu}{}^{\nu\rho} - T^{\sigma}{}_{\sigma\rho}\tilde{Y}_{\mu}{}^{\nu\rho} + \nabla_{\rho}\tilde{Y}_{\mu}{}^{\nu\rho}\right)\xi^{\mu}{}_{\nu}\mathrm{d}^4x\,,
\end{split}
\end{equation}
where the second line follows from integration by parts. Hence, one finds the same connection field equation~\eqref{eq:genconnfielddens}.
\subsection{Symmetric teleparallel gravity}\label{sec:symteleact}
The class of teleparallel gravity theories discussed in the previous section, in which the affine connection \(\Gamma^{\mu}{}_{\nu\rho}\) is restricted only by the flatness condition~\eqref{eq:curvature}, is also known as \emph{general} teleparallel gravity, and is the youngest among the different classes of teleparallel gravity theories. Two other classes of teleparallel gravity theories can be obtained by demanding that either the torsion~\eqref{eq:torsion} or the nonmetricity~\eqref{eq:nonmetricity} vanishes. We will start with the former condition, which yields the class of \emph{symmetric} teleparallel gravity theories, which refers to the fact that the coefficients of a torsion-free connection are symmetric in their lower two indices. In order to implement the condition of vanishing torsion, one may proceed in full analogy to the flatness condition in the previous section, by adding another Lagrange multiplier term
\begin{equation}\label{eq:lagmultors}
S_{\text{t}} = \int_M\tilde{t}_{\mu}{}^{\nu\rho}T^{\mu}{}_{\nu\rho}\mathrm{d}^4x\,,
\end{equation}
where variation with respect to the tensor density \(\tilde{t}_{\mu}{}^{\nu\rho}\) leads to the constraint equation \(T^{\mu}{}_{\nu\rho} = 0\). In order to derive the field equations, one then proceeds as in the previous section, by varying the full action and eliminating the Lagrange multipliers from the resulting field equations. This calculation is rather lengthy, but straightforward, and so we will not show it here. Instead, we will follow the alternative procedure of restricted variation of the action, by considering only variations \(\delta\Gamma^{\mu}{}_{\nu\rho}\) which maintain the vanishing curvature and torsion of the connection. We can use the fact that the flatness is maintained by the variation~\eqref{eq:flatvar}, and further restrict the form of \(\xi^{\mu}{}_{\nu}\). It turns out that this is achieved by setting \(\xi^{\mu}{}_{\nu} = \nabla_{\nu}\zeta^{\mu}\) for some vector field \(\zeta^{\mu}\), and thus
\begin{equation}\label{eq:symtelevar}
\delta\Gamma^{\mu}{}_{\nu\rho} = \nabla_{\rho}\nabla_{\nu}\zeta^{\mu}\,.
\end{equation}
Using the fact that covariant derivatives commute in the absence of curvature and torsion, one now immediately sees
\begin{equation}
\delta T^{\mu}{}_{\nu\rho} = \delta\Gamma^{\mu}{}_{\rho\nu} - \delta\Gamma^{\mu}{}_{\nu\rho} = \nabla_{\nu}\nabla_{\rho}\zeta^{\mu} - \nabla_{\rho}\nabla_{\nu}\zeta^{\mu} = 0\,.
\end{equation}
The variation of the action with respect to the connection is then simply given by
\begin{equation}
\begin{split}
\delta_{\Gamma}S &= \int_M\left(\tilde{H}_{\mu}{}^{\nu\rho} - \tilde{Y}_{\mu}{}^{\nu\rho}\right)\nabla_{\rho}\nabla_{\nu}\zeta^{\mu}\mathrm{d}^4x\\
&= -\int_M\nabla_{\rho}\left(\tilde{H}_{\mu}{}^{\nu\rho} - \tilde{Y}_{\mu}{}^{\nu\rho}\right)\nabla_{\nu}\zeta^{\mu}\mathrm{d}^4x\\
&= \int_M\nabla_{\nu}\nabla_{\rho}\left(\tilde{H}_{\mu}{}^{\nu\rho} - \tilde{Y}_{\mu}{}^{\nu\rho}\right)\zeta^{\mu}\mathrm{d}^4x\,,
\end{split}
\end{equation}
where integration by parts simplifies due to the vanishing torsion. The connection field equation thus becomes
\begin{equation}\label{eq:symconnfield}
\nabla_{\nu}\nabla_{\rho}\tilde{Y}_{\mu}{}^{\nu\rho} = \nabla_{\nu}\nabla_{\rho}\tilde{H}_{\mu}{}^{\nu\rho}\,.
\end{equation}
Together with the metric field equation~\eqref{eq:genmetfield}, it constitutes the field equations of symmetric teleparallel gravity.
\subsection{Metric teleparallel gravity}\label{sec:metteleact}
We finally come to the remaining class of theories, which are defined by imposing the condition of vanishing nonmetricity, so that the connection becomes metric-compatible. This class of theories is therefore known as \emph{metric} teleparallel gravity, or simply as teleparallel gravity, since it was conceived first among the three different classes we discuss here. To derive its field equations, one can also in this case either introduce a Lagrange multiplier
\begin{equation}\label{eq:lagmulnonmet}
S_{\text{q}} = \int_M\tilde{q}^{\mu\nu\rho}Q_{\mu\nu\rho}\mathrm{d}^4x\,,
\end{equation}
and vary with respect to the tensor density \(\tilde{q}^{\mu\nu\rho}\) to obtain \(Q_{\mu\nu\rho} = 0\), or find a suitable restriction on the connection variation. Here we will follow once again the latter approach. From the definition~\eqref{eq:nonmetricity} of the nonmetricity, one obtains its variation
\begin{equation}
\delta Q_{\mu\nu\rho} = \nabla_{\mu}\delta g_{\nu\rho} - g_{\sigma\rho}\delta\Gamma^{\sigma}{}_{\nu\mu} - g_{\nu\sigma}\delta\Gamma^{\sigma}{}_{\rho\mu} = \nabla_{\mu}(\delta g_{\nu\rho} - 2\xi_{(\nu\rho)})\,,
\end{equation}
provided that the variation of the connection is chosen to implement the flatness condition~\eqref{eq:flatvar}. Here we also used the metric compatibility of the connection to commute lowering an index with the covariant derivative. It turns out that the condition of vanishing nonmetricity imposes a relation
\begin{equation}
\delta g_{\mu\nu} = 2\xi_{(\mu\nu)}
\end{equation}
between the variations of the metric and the connection. Since both are now expressed in terms of the tensor field \(\xi_{\mu\nu}\), the field equations follow from the total variation
\begin{equation}
\begin{split}
\delta S &= \int_M\left(\Theta^{\mu\nu}\xi_{(\mu\nu)} + H^{\mu\nu\rho}\nabla_{\rho}\xi_{\mu\nu} - W^{\mu\nu}\xi_{(\mu\nu)} - Y^{\mu\nu\rho}\nabla_{\rho}\xi_{\mu\nu}\right)\sqrt{-g}\mathrm{d}^4x\\
&= \int_M\left(\Theta^{(\mu\nu)} - \nabla_{\rho}H^{\mu\nu\rho} + H^{\mu\nu\rho}T^{\tau}{}_{\tau\rho} - W^{(\mu\nu)} + \nabla_{\rho}Y^{\mu\nu\rho} - Y^{\mu\nu\rho}T^{\tau}{}_{\tau\rho}\right)\xi_{\mu\nu}\sqrt{-g}\mathrm{d}^4x\,,
\end{split}
\end{equation}
after performing integration by parts, and using the metric compatibility of the connection to obtain \(\nabla_{\mu}\sqrt{-g} = 0\). Keeping in mind that \(W^{\mu\nu}\) and \(\Theta^{\mu\nu}\) are defined by the variation of the action with respect to the metric, and thus symmetric by definition, one obtains the field equation
\begin{equation}\label{eq:metallfield}
W^{\mu\nu} - \nabla_{\rho}Y^{\mu\nu\rho} + Y^{\mu\nu\rho}T^{\tau}{}_{\tau\rho} = \Theta^{\mu\nu} - \nabla_{\rho}H^{\mu\nu\rho} + H^{\mu\nu\rho}T^{\tau}{}_{\tau\rho}\,.
\end{equation}
This single field equation therefore conveys the dynamics in metric teleparallel gravity.
\section{Physical aspects and formalisms in teleparallel geometry}\label{sec:telephys}
To be able to make contact with phenomenology and observations, it is necessary to discuss a few general physical principles in the framework of teleparallel gravity. The first principle, which we discuss in section~\ref{ssec:enmomhypcons}, is the conservation of the matter currents, which are energy-momentum and hypermomentum, which follows from the invariance of the action under diffeomorphisms. We then continue with spacetime symmetries in section~\ref{ssec:symmetry}, which can be used to obtain solutions of teleparallel gravity theories, such as black holes, whose phenomenology can subsequently be studied. In particular, we focus on the case of homogeneous and isotropic teleparallel spacetimes, and derive the dynamical variables which appear in teleparallel cosmology. Finally, we discuss the theory of perturbations of teleparallel geometries in section~\ref{ssec:pert}. These form the basis of testing teleparallel gravity theories using gravitational waves and high-precision post-Newtonian observations.
\subsection{Energy-momentum-hypermomentum conservation}\label{ssec:enmomhypcons}
In order to be independent of the choice of coordinates, the different components \(S_{\text{g}}\) and \(S_{\text{m}}\) of the action discussed in the previous sections are demanded to be independently invariant under diffeomorphisms. Note that an infinitesimal diffeomorphism generated by a vector field \(X^{\mu}\) changes the metric by
\begin{equation}\label{eq:mettrans}
\delta_Xg_{\mu\nu} = (\mathcal{L}_{X}g)_{\mu\nu} = X^{\rho}\partial_{\rho}g_{\mu\nu} + \partial_{\mu}X^{\rho}g_{\rho\nu} + \partial_{\nu}X^{\rho}g_{\mu\rho} = 2\lc{\nabla}_{(\mu}X_{\nu)}\,,
\end{equation}
while the connection is changed by
\begin{equation}\label{eq:conntrans}
\begin{split}
\delta_{X}\Gamma^{\mu}{}_{\nu\rho} &= (\mathcal{L}_{X}\Gamma)^{\mu}{}_{\nu\rho}\\
&= X^{\sigma}\partial_{\sigma}\Gamma^{\mu}{}_{\nu\rho} - \partial_{\sigma}X^{\mu}\Gamma^{\sigma}{}_{\nu\rho} + \partial_{\nu}X^{\sigma}\Gamma^{\mu}{}_{\sigma\rho} + \partial_{\rho}X^{\sigma}\Gamma^{\mu}{}_{\nu\sigma} + \partial_{\nu}\partial_{\rho}X^{\mu}\\
&= \nabla_{\rho}\nabla_{\nu}X^{\mu} - X^{\sigma}R^{\mu}{}_{\nu\rho\sigma} - \nabla_{\rho}(X^{\sigma}T^{\mu}{}_{\nu\sigma})\,.
\end{split}
\end{equation}
Note that both expressions are tensor fields, despite the fact that the connection coefficients are not tensor fields. Their variation, however, being an infinitesimal difference between connection coefficients, is a tensor field. In the teleparallel case, the curvature tensor vanishes. Using these formulas, it is now easy to calculate the change of the gravitational part \(S_{\text{g}}\) of the action, which reads
\begin{equation}
\begin{split}
\delta_XS_{\text{g}} &= -\int_M\left(\frac{1}{2}\sqrt{-g}W^{\mu\nu}\delta_Xg_{\mu\nu} + \tilde{Y}_{\mu}{}^{\nu\rho}\delta_X\Gamma^{\mu}{}_{\nu\rho}\right)\mathrm{d}^4x\\
&= -\int_M\left\{\sqrt{-g}W^{\mu\nu}\lc{\nabla}_{\mu}X_{\nu} + \tilde{Y}_{\mu}{}^{\nu\rho}\left[\nabla_{\rho}\nabla_{\nu}X^{\mu} - \nabla_{\rho}(X^{\sigma}T^{\mu}{}_{\nu\sigma})\right]\right\}\mathrm{d}^4x\\
&= \int_M\Big[\sqrt{-g}\lc{\nabla}_{\nu}W_{\mu}{}^{\nu} + T^{\sigma}{}_{\mu\nu}(\nabla_{\rho}\tilde{Y}_{\sigma}{}^{\nu\rho} - T^{\tau}{}_{\tau\rho}\tilde{Y}_{\sigma}{}^{\nu\rho})\\
&\phantom{=}- \nabla_{\nu}(\nabla_{\rho}\tilde{Y}_{\mu}{}^{\nu\rho} - T^{\tau}{}_{\tau\rho}\tilde{Y}_{\mu}{}^{\nu\rho}) + T^{\omega}{}_{\omega\nu}(\nabla_{\rho}\tilde{Y}_{\mu}{}^{\nu\rho} - T^{\tau}{}_{\tau\rho}\tilde{Y}_{\mu}{}^{\nu\rho})\Big]X^{\mu}\mathrm{d}^4x\,.
\end{split}
\end{equation}
Assuming that the gravitational part \(S_{\text{g}}\) of the action is invariant under diffeomorphisms, this variation must vanish identically for arbitrary vector fields \(X^{\mu}\). Hence, it follows that the terms \(W^{\mu\nu}\) and \(\tilde{Y}_{\mu}{}^{\nu\rho}\) obtained from the variation of the action satisfy
\begin{multline}\label{eq:gravconsdens}
\sqrt{-g}\lc{\nabla}_{\nu}W_{\mu}{}^{\nu} + T^{\sigma}{}_{\mu\nu}(\nabla_{\rho}\tilde{Y}_{\sigma}{}^{\nu\rho} - T^{\tau}{}_{\tau\rho}\tilde{Y}_{\sigma}{}^{\nu\rho})\\
- \nabla_{\nu}(\nabla_{\rho}\tilde{Y}_{\mu}{}^{\nu\rho} - T^{\tau}{}_{\tau\rho}\tilde{Y}_{\mu}{}^{\nu\rho}) + T^{\sigma}{}_{\sigma\nu}(\nabla_{\rho}\tilde{Y}_{\mu}{}^{\nu\rho} - T^{\tau}{}_{\tau\rho}\tilde{Y}_{\mu}{}^{\nu\rho}) = 0\,.
\end{multline}
Alternatively, one can also write this relation without density factors, and finds
\begin{multline}\label{eq:gravcons}
\lc{\nabla}_{\nu}W_{\mu}{}^{\nu} + T^{\sigma}{}_{\mu\nu}(\nabla_{\rho}Y_{\sigma}{}^{\nu\rho} - M^{\tau}{}_{\rho\tau}Y_{\sigma}{}^{\nu\rho})\\
- \nabla_{\nu}(\nabla_{\rho}Y_{\mu}{}^{\nu\rho} - M^{\tau}{}_{\rho\tau}Y_{\mu}{}^{\nu\rho}) + M^{\sigma}{}_{\nu\sigma}(\nabla_{\rho}Y_{\mu}{}^{\nu\rho} - M^{\tau}{}_{\rho\tau}Y_{\mu}{}^{\nu\rho}) = 0\,.
\end{multline}
This equation is derived from a purely geometric property of the gravitational part of the action, and so it is a geometric identity, i.e., it holds for any field configuration of the metric \(g_{\mu\nu}\) and the connection \(\Gamma^{\mu}{}_{\nu\rho}\), independently of whether these satisfy the gravitational field equations or not. Such a relation is therefore also said to hold \emph{off-shell}. This is to be contrasted with the variation of the matter action \(S_{\text{m}}\), which reads
\begin{equation}
\begin{split}
\delta_XS_{\text{g}} &= \int_M\left(\frac{1}{2}\sqrt{-g}\Theta^{\mu\nu}\delta_Xg_{\mu\nu} + \tilde{H}_{\mu}{}^{\nu\rho}\delta_X\Gamma^{\mu}{}_{\nu\rho} + \tilde{\Psi}_I\delta_X\psi^I\right)\mathrm{d}^4x\\
&= \int_M\left\{\sqrt{-g}\Theta^{\mu\nu}\lc{\nabla}_{\mu}X_{\nu} + \tilde{H}_{\mu}{}^{\nu\rho}\left[\nabla_{\rho}\nabla_{\nu}X^{\mu} - \nabla_{\rho}(X^{\sigma}T^{\mu}{}_{\nu\sigma})\right] + \tilde{\Psi}_I\mathcal{L}_X\psi^I\right\}\mathrm{d}^4x\,.
\end{split}
\end{equation}
Here, \(\tilde{\Psi}_I = 0\) (or equivalently \(\Psi_I = 0\), without using densities) are the matter field equations. If these are satisfied, and only then, demanding that the matter action is invariant under diffeomorphisms generated by an arbitrary vector field \(X^{\mu}\) leads to the energy-momentum-hypermomentum conservation law
\begin{multline}\label{eq:enmomhypconsdens}
\sqrt{-g}\lc{\nabla}_{\nu}\Theta_{\mu}{}^{\nu} + T^{\sigma}{}_{\mu\nu}(\nabla_{\rho}\tilde{H}_{\sigma}{}^{\nu\rho} - T^{\tau}{}_{\tau\rho}\tilde{H}_{\sigma}{}^{\nu\rho})\\
- \nabla_{\nu}(\nabla_{\rho}\tilde{H}_{\mu}{}^{\nu\rho} - T^{\tau}{}_{\tau\rho}\tilde{H}_{\mu}{}^{\nu\rho}) + T^{\sigma}{}_{\sigma\nu}(\nabla_{\rho}\tilde{H}_{\mu}{}^{\nu\rho} - T^{\tau}{}_{\tau\rho}\tilde{H}_{\mu}{}^{\nu\rho}) = 0\,,
\end{multline}
or, again in the version without densities,
\begin{multline}\label{eq:enmomhypcons}
\lc{\nabla}_{\nu}\Theta_{\mu}{}^{\nu} + T^{\sigma}{}_{\mu\nu}(\nabla_{\rho}H_{\sigma}{}^{\nu\rho} - M^{\tau}{}_{\rho\tau}H_{\sigma}{}^{\nu\rho})\\
- \nabla_{\nu}(\nabla_{\rho}H_{\mu}{}^{\nu\rho} - M^{\tau}{}_{\rho\tau}H_{\mu}{}^{\nu\rho}) + M^{\sigma}{}_{\nu\sigma}(\nabla_{\rho}H_{\mu}{}^{\nu\rho} - M^{\tau}{}_{\rho\tau}H_{\mu}{}^{\nu\rho}) = 0\,.
\end{multline}
Since this relation does not hold for arbitrary field configurations of the gravitational and matter field, but only for those which satisfy the matter field equations \(\Psi_I = 0\), it is said to hold \emph{on-shell}. Note that we have not made any assumptions on the properties of the connection except for vanishing curvature. In particular, we have not imposed vanishing torsion or nonmetricity. It follows that the geometric identity and energy-momentum-hypermomentum law given above hold for all three classes of teleparallel gravity theories (but their expressions will simplify in the symmetric and metric cases, as we will see below). Finally, we remark that in the case of vanishing hypermomentum, i.e., for matter which couples only to the metric and not to the connection, which is most commonly considered in the context of teleparallel gravity, the conservation law reduces to
\begin{equation}
\lc{\nabla}_{\nu}\Theta_{\mu}{}^{\nu} = 0\,,
\end{equation}
which is the well-known energy-momentum conservation.
As an alternative to imposing the matter field equations, the conservation law~\eqref{eq:enmomhypcons} can also be derived from the geometric identity~\eqref{eq:gravcons}, by imposing the gravitational field equations. This is most straightforward for the general teleparallel gravity class, whose gravitational field equations are~\eqref{eq:genmetfield} and~\eqref{eq:genconnfield}. One easily sees that the terms appearing in the identity~\eqref{eq:gravcons} are exactly the left-hand sides of the gravitational field equations. Replacing them with the respective right-hand sides, one obtains the energy-momentum-hypermomentum conservation law~\eqref{eq:enmomhypcons}. Of course, the same holds true also if one uses the tensor density version of these equations.
A similar derivation can also be used in the case of symmetric teleparallel gravity, where one assumes vanishing torsion, \(T^{\mu}{}_{\nu\rho} = 0\). In this case, it is most convenient to start from the density version~\eqref{eq:gravconsdens}, which simplifies to become
\begin{equation}\label{eq:symgravcons}
\sqrt{-g}\lc{\nabla}_{\nu}W_{\mu}{}^{\nu} - \nabla_{\nu}\nabla_{\rho}\tilde{Y}_{\mu}{}^{\nu\rho} = 0\,.
\end{equation}
Using the metric field equation~\eqref{eq:genmetfield} and the connection field equation~\eqref{eq:symconnfield}, one thus immediately obtains the conservation law
\begin{equation}\label{eq:symenmomhypcons}
\sqrt{-g}\lc{\nabla}_{\nu}\Theta_{\mu}{}^{\nu} - \nabla_{\nu}\nabla_{\rho}\tilde{H}_{\mu}{}^{\nu\rho} = 0\,,
\end{equation}
which agrees with the general form~\eqref{eq:enmomhypconsdens} in the absence of torsion. What is most remarkable in the case of symmetric teleparallel gravity is the fact that one can also proceed in a different order: by imposing the matter field equations \(\Psi_I = 0\), from which follows the conservation law~\eqref{eq:symenmomhypcons}, further imposing the metric field equation~\eqref{eq:genmetfield}, and using the identity~\eqref{eq:symgravcons}, one obtains the connection field equation~\eqref{eq:symconnfield}. In other words, any field configuration of the matter and gravitational fields, which satisfies the matter and metric field equations, automatically satisfies also the connection field equation. For this reason, one often omits the latter when it comes to solving the field equations.
Finally, we study the energy-momentum-hypermomentum conservation also in the metric teleparallel setting. In this case, one can omit the density factors in the geometric identity~\eqref{eq:gravconsdens}, since the connection is metric-compatible, so that it becomes
\begin{multline}\label{eq:metgravcons}
\lc{\nabla}_{\nu}W_{\mu}{}^{\nu} + T^{\sigma}{}_{\mu\nu}(\nabla_{\rho}Y_{\sigma}{}^{\nu\rho} - T^{\tau}{}_{\tau\rho}Y_{\sigma}{}^{\nu\rho})\\
- \nabla_{\nu}(\nabla_{\rho}Y_{\mu}{}^{\nu\rho} - T^{\tau}{}_{\tau\rho}Y_{\mu}{}^{\nu\rho}) + T^{\sigma}{}_{\sigma\nu}(\nabla_{\rho}Y_{\mu}{}^{\nu\rho} - T^{\tau}{}_{\tau\rho}Y_{\mu}{}^{\nu\rho}) = 0\,.
\end{multline}
Further, we impose the metric teleparallel gravity field equation~\eqref{eq:metallfield}, which we will write in the form
\begin{equation}\label{eq:transmetfield}
W^{\mu\nu} - \Theta^{\mu\nu} = A^{\mu\nu}\,,
\end{equation}
where we have defined the abbreviation
\begin{equation}\label{eq:metfieldabbr}
A^{\mu\nu} = \nabla_{\rho}Y^{\mu\nu\rho} - Y^{\mu\nu\rho}T^{\tau}{}_{\tau\rho} - \nabla_{\rho}H^{\mu\nu\rho} + H^{\mu\nu\rho}T^{\tau}{}_{\tau\rho}\,.
\end{equation}
Note that the left hand side of the field equation~\eqref{eq:transmetfield} is symmetric by definition. Hence, when the equation holds, also the right hand side must be symmetric, and thus \(A^{[\mu\nu]} = 0\). We then take the Levi-Civita covariant derivative of this equation, which reads
\begin{equation}\label{eq:divmetfield}
\lc{\nabla}_{\nu}W^{\mu\nu} - \lc{\nabla}_{\nu}\Theta^{\mu\nu} = \lc{\nabla}_{\nu}A^{\mu\nu}\,.
\end{equation}
On the right-hand side, we can use the relation
\begin{equation}\label{eq:covdevmetfield}
\begin{split}
\lc{\nabla}_{\nu}A^{\mu\nu} &= \nabla_{\nu}A^{\mu\nu} - K^{\mu}{}_{\rho\nu}A^{\rho\nu} - K^{\nu}{}_{\rho\nu}A^{\mu\rho}\\
&= \nabla_{\nu}A^{\mu\nu} - \frac{1}{2}\left[(T_{\rho}{}^{\mu}{}_{\nu} + T_{\nu}{}^{\mu}{}_{\rho} - T^{\mu}{}_{\rho\nu})A^{\rho\nu} - (T_{\rho}{}^{\nu}{}_{\nu} + T_{\nu}{}^{\nu}{}_{\rho} - T^{\nu}{}_{\rho\nu})A^{\mu\rho}\right]\\
&= \nabla_{\nu}A^{\mu\nu} - T_{\rho}{}^{\mu}{}_{\nu}A^{\rho\nu} - T^{\nu}{}_{\nu\rho}A^{\mu\rho}\,,
\end{split}
\end{equation}
where we have used the symmetry \(A^{[\mu\nu]} = 0\) to obtain the last line. Now combining the geometric identity~\eqref{eq:metgravcons}, the divergence~\eqref{eq:divmetfield} of the gravitational field equation and the result~\eqref{eq:covdevmetfield}, one finally arrives at
\begin{multline}\label{eq:metenmomhypcons}
\lc{\nabla}_{\nu}\Theta_{\mu}{}^{\nu} + T^{\sigma}{}_{\mu\nu}(\nabla_{\rho}H_{\sigma}{}^{\nu\rho} - T^{\tau}{}_{\tau\rho}H_{\sigma}{}^{\nu\rho})\\
- \nabla_{\nu}(\nabla_{\rho}H_{\mu}{}^{\nu\rho} - T^{\tau}{}_{\tau\rho}H_{\mu}{}^{\nu\rho}) + T^{\sigma}{}_{\sigma\nu}(\nabla_{\rho}H_{\mu}{}^{\nu\rho} - T^{\tau}{}_{\tau\rho}H_{\mu}{}^{\nu\rho}) = 0\,,
\end{multline}
which agrees with~\eqref{eq:enmomhypcons} in the case of vanishing nonmetricity.
\subsection{Spacetime symmetries and cosmology}\label{ssec:symmetry}
In the previous section we have made use of the transformation laws~\eqref{eq:mettrans} of the metric and~\eqref{eq:conntrans} of the connection under infinitesimal diffeomorphisms generated by a vector field \(X^{\mu}\). The same transformation laws also find application in the discussion of symmetric spacetimes, i.e., teleparallel geometries, which are invariant under the action of particular vector fields, \(\delta_Xg_{\mu\nu} = 0\) and \(\delta_X\Gamma^{\mu}{}_{\nu\rho} = 0\)~\cite{Hohmann:2015pva,Hohmann:2019nat}. The choice of these vector fields depends on the physical situation under consideration. A few common examples can be expressed most conveniently in spherical coordinates \((t, r, \varphi, \vartheta)\): a \emph{stationary} spacetime is invariant under the (timelike) vector field \(\partial_t\); \emph{spherical} symmetry is conveyed by the three rotation generators
\begin{equation}
\sin\varphi\partial_{\vartheta} + \frac{\cos\varphi}{\tan\vartheta}\partial_{\varphi}\,, \quad
-\cos\varphi\partial_{\vartheta} + \frac{\sin\varphi}{\tan\vartheta}\partial_{\varphi}\,, \quad
-\partial_{\varphi}\,;
\end{equation}
finally, \emph{cosmological} symmetry comprises of invariance under both rotations as given above and translations, defined by the vector fields
\begin{subequations}
\begin{align}
& \chi\sin\vartheta\cos\varphi\partial_r + \frac{\chi}{r}\cos\vartheta\cos\varphi\partial_{\vartheta} - \frac{\chi\sin\varphi}{r\sin\vartheta}\partial_{\varphi}\,,\\
& \chi\sin\vartheta\sin\varphi\partial_r + \frac{\chi}{r}\cos\vartheta\sin\varphi\partial_{\vartheta} + \frac{\chi\cos\varphi}{r\sin\vartheta}\partial_{\varphi}\,,\\
& \chi\cos\vartheta\partial_r - \frac{\chi}{r}\sin\vartheta\partial_{\vartheta}\,,
\end{align}
\end{subequations}
where we used the abbreviation \(\chi = \sqrt{1 - (ur)^2}\), and \(u\) can be any real or imaginary number, so that the sign of \(u^2 \in \mathbb{R}\) determines the curvature of the spatial hypersurfaces of constant time \(t\). For \(u^2 > 0\), their spatial curvature is positive, while \(u^2 < 0\) corresponds to negative spatial curvature. Finally, \(u^2 = 0\) is the spatially flat case.
Symmetric spacetimes are often considered as potential solutions to the field equations of a given theory, since they are completely characterized by fewer functions than there are components of the dynamical fields, and these functions depend on a smaller number of coordinates, hence leading to a simple ansatz for solving the field equations. As a simple and physically well motivated example, we show this for the case of cosmological symmetry in the teleparallel geometry. It is well known that the most general metric which is homogeneous and isotropic is the Friedmann-Lemaître-Robertson-Walker metric
\begin{equation}
g_{\mu\nu} = -n_{\mu}n_{\nu} + h_{\mu\nu}\,,
\end{equation}
where the hypersurface conormal
\begin{equation}
n_{\mu}\mathrm{d} x^{\mu} = -N\mathrm{d} t
\end{equation}
and spatial metric
\begin{equation}
h_{\mu\nu}\mathrm{d} x^{\mu} \otimes \mathrm{d} x^{\nu} = A^2\left[\frac{\mathrm{d} r \otimes \mathrm{d} r}{\chi^2} + r^2(\mathrm{d}\vartheta \otimes \mathrm{d}\vartheta + \sin^2\vartheta\mathrm{d}\varphi \otimes \mathrm{d}\varphi)\right]
\end{equation}
are fully determined by two functions of time, known as the lapse function \(N = N(t)\) and scale factor \(A = A(t)\). Using this metric, we can apply the decomposition~\eqref{eq:conndec} of the affine connection, and we find that the most general homogeneous and isotropic connection is characterized through its torsion and nonmetricity
\begin{subequations}
\begin{align}
T^{\mu}{}_{\nu\rho} &= \frac{2}{A}(\mathcal{T}_1h^{\mu}_{[\nu}n_{\rho]} + \mathcal{T}_2n_{\sigma}\varepsilon^{\sigma\mu}{}_{\nu\rho})\,,\\
Q_{\rho\mu\nu} &= \frac{2}{A}(\mathcal{Q}_1n_{\rho}n_{\mu}n_{\nu} + 2\mathcal{Q}_2n_{\rho}h_{\mu\nu} + 2\mathcal{Q}_3h_{\rho(\mu}n_{\nu)})\,,
\end{align}
\end{subequations}
by five further functions \(\mathcal{T}_1, \mathcal{T}_2, \mathcal{Q}_1, \mathcal{Q}_2, \mathcal{Q}_3\) of time, and \(\varepsilon_{\mu\nu\rho\sigma}\) is the totally antisymmetric tensor normalized such that
\begin{equation}
\varepsilon_{0123} = \sqrt{-g} = \frac{NA^3r^2\sin\vartheta}{\chi}\,.
\end{equation}
Note that in general the curvature of this connection does not vanish, and so one must impose additional constraints on the aforementioned functions. Before discussing these constraints, it is most convenient to introduce the conformal time derivative
\begin{equation}
F' = \frac{A}{N}\frac{\mathrm{d} F}{\mathrm{d} t}
\end{equation}
acting on any time-dependent scalar function \(F = F(t)\), as well as the conformal Hubble parameter
\begin{equation}
\mathcal{H} = \frac{A'}{A} = \frac{1}{N}\frac{\mathrm{d} A}{\mathrm{d} t}\,.
\end{equation}
With the help of these definitions, the conditions on the parameter functions under which the curvature tensor vanishes become
\begin{subequations}
\begin{align}
\mathcal{T}_2(\mathcal{H} - \mathcal{T}_1 + \mathcal{Q}_2) &= 0\,,\label{eq:cosmocurv1}\\
\mathcal{T}_2(\mathcal{H} - \mathcal{T}_1 + \mathcal{Q}_2 - \mathcal{Q}_3) &= 0\,,\label{eq:cosmocurv2}\\
(\mathcal{H} - \mathcal{T}_1 + \mathcal{Q}_2)(\mathcal{H} - \mathcal{T}_1 + \mathcal{Q}_2 - \mathcal{Q}_3) - \mathcal{T}_2^2 + u^2 &= 0\,,\label{eq:cosmocurv3}\\
(\mathcal{Q}_1 + \mathcal{Q}_2)(\mathcal{H} - \mathcal{T}_1 + \mathcal{Q}_2) + (\mathcal{H} - \mathcal{T}_1 + \mathcal{Q}_2)' &= 0\,,\label{eq:cosmocurv4}\\
(\mathcal{Q}_1 + \mathcal{Q}_2)(\mathcal{H} - \mathcal{T}_1 + \mathcal{Q}_2 - \mathcal{Q}_3) - (\mathcal{H} - \mathcal{T}_1 + \mathcal{Q}_2 - \mathcal{Q}_3)' &= 0\,,\label{eq:cosmocurv5}\\
\mathcal{T}_2' &= 0\,.\label{eq:cosmocurv6}
\end{align}
\end{subequations}
Note that \(u\) appears in only one of these equations; nevertheless, it plays an important role for the solutions of this system, as we will show now. For this purpose, consider first the case \(u^2 \neq 0\). In this case the condition~\eqref{eq:cosmocurv3} implies
\begin{equation}
(\mathcal{H} - \mathcal{T}_1 + \mathcal{Q}_2)(\mathcal{H} - \mathcal{T}_1 + \mathcal{Q}_2 - \mathcal{Q}_3) \neq \mathcal{T}_2^2\,.
\end{equation}
From the two conditions~\eqref{eq:cosmocurv1} and~\eqref{eq:cosmocurv2} then further follows that either the right hand side vanishes, or both factors on the left hand side vanish. We first consider this latter case. The condition~\eqref{eq:cosmocurv3} then requires \(\mathcal{T}_2 = \pm u\), and we find that all remaining equations are solved by
\begin{equation}\label{eq:cosmogaxi}
\mathcal{T}_2 = \pm u\,, \quad
\mathcal{T}_1 - \mathcal{Q}_2 = \mathcal{H}\,, \quad
\mathcal{Q}_3 = 0\,.
\end{equation}
Also we see that we must demand \(u\) to be real in order to obtain a real value of the connection coefficients; hence, this solution is valid only for positive spatial curvature \(u^2 > 0\). Alternatively, the first two equations~\eqref{eq:cosmocurv1} and~\eqref{eq:cosmocurv2} can also be solved by setting \(\mathcal{T}_2 = 0\). From the remaining equations then follows the solution
\begin{equation}\label{eq:cosmogvec}
\mathcal{T}_2 = 0\,, \quad
(\mathcal{H} - \mathcal{T}_1 + \mathcal{Q}_2)(\mathcal{H} - \mathcal{T}_1 + \mathcal{Q}_2 - \mathcal{Q}_3) = -u^2\,, \quad
\mathcal{Q}_1 + \mathcal{Q}_2 = -\frac{\mathcal{H}' - \mathcal{T}_1' + \mathcal{Q}_2'}{\mathcal{H} - \mathcal{T}_1 + \mathcal{Q}_2}\,.
\end{equation}
Now we see that both signs of \(u^2\) are allowed. These are the only two possibilities to solve the first three equations, and so we may turn our attention to the case \(u = 0\). In this case the third equation~\eqref{eq:cosmocurv3} mandates
\begin{equation}
(\mathcal{H} - \mathcal{T}_1 + \mathcal{Q}_2)(\mathcal{H} - \mathcal{T}_1 + \mathcal{Q}_2 - \mathcal{Q}_3) = \mathcal{T}_2^2\,,
\end{equation}
and we see that both sides of this equation must vanish in order to satisfy the conditions~\eqref{eq:cosmocurv1} and~\eqref{eq:cosmocurv2}, so that all solutions will have \(\mathcal{T}_2 = 0\). For the left hand side, we are free to choose at most one of the two factors to be non-vanishing. This leads to the three possible solutions
\begin{equation}\label{eq:cosmogcoin}
\mathcal{T}_2 = 0\,, \quad
\mathcal{T}_1 - \mathcal{Q}_2 = \mathcal{H}\,, \quad
\mathcal{Q}_3 = 0
\end{equation}
if both factors vanish,
\begin{equation}\label{eq:cosmogconf}
\mathcal{T}_2 = 0\,, \quad
\mathcal{T}_1 - \mathcal{Q}_2 + \mathcal{Q}_3 = \mathcal{H}\,, \quad
\mathcal{Q}_1 + \mathcal{Q}_2 = -\frac{\mathcal{Q}_3'}{\mathcal{Q}_3}
\end{equation}
if only the second factor vanishes, as well as
\begin{equation}\label{eq:cosmogpara}
\mathcal{T}_2 = 0\,, \quad
\mathcal{T}_1 - \mathcal{Q}_2 = \mathcal{H}\,, \quad
\mathcal{Q}_1 + \mathcal{Q}_2 = \frac{\mathcal{Q}_3'}{\mathcal{Q}_3}
\end{equation}
if only the first factor vanishes. These are the only possible homogeneous and isotropic teleparallel geometries. Note that for each solution one has three conditions on the five parameter functions, so that two of them can be freely chosen to parametrize the solution, and must be determined alongside the scale factor \(A\) and lapse \(N\) by solving the field equations of a given teleparallel gravity theory.
In the discussion above we have assumed a general teleparallel geometry, for which both torsion and nonmetricity are allowed to be non-vanishing. From the solutions we have found one can now easily deduce the symmetric and metric teleparallel geometries. We start with the former, by imposing the additional condition \(\mathcal{T}_1 = \mathcal{T}_2 = 0\). One immediately sees that this condition is not compatible with the first solution~\eqref{eq:cosmogaxi}, which explicitly demands \(\mathcal{T}_2 \neq 0\), and so this solution cannot be restricted to symmetric teleparallel gravity. This is different for the remaining solutions. From the solution~\eqref{eq:cosmogvec}, one obtains the spatially curved case
\begin{equation}\label{eq:cosmosvec}
(\mathcal{H} + \mathcal{Q}_2)(\mathcal{H} + \mathcal{Q}_2 - \mathcal{Q}_3) = - u^2\,, \quad
\mathcal{Q}_1 + \mathcal{Q}_2 = -\frac{\mathcal{H}' + \mathcal{Q}_2'}{\mathcal{H} + \mathcal{Q}_2}\,,
\end{equation}
while the three spatially flat solutions become
\begin{subequations}
\begin{align}
\mathcal{Q}_2 &= -\mathcal{H}\,, &
\mathcal{Q}_3 &= 0\,, \\
\mathcal{Q}_2 - \mathcal{Q}_3 &= -\mathcal{H}\,, &
\mathcal{Q}_1 + \mathcal{Q}_2 &= -\frac{\mathcal{Q}_3'}{\mathcal{Q}_3}\,,\\
\mathcal{Q}_2 &= -\mathcal{H}\,, &
\mathcal{Q}_1 + \mathcal{Q}_2 &= \frac{\mathcal{Q}_3'}{\mathcal{Q}_3}\,.
\end{align}
\end{subequations}
For each of these solutions one has two conditions on the three scalar functions \(\mathcal{Q}_{1,2,3}\) which parametrize the nonmetricity, so that one of them remains undetermined by the symmetry condition, and is left to be determined by the gravitational field equations~\cite{DAmbrosio:2021pnd,Hohmann:2021ast}.
In a similar fashion, one can also restrict the general teleparallel cosmologies to the metric teleparallel geometry, by imposing the conditions \(\mathcal{Q}_1 = \mathcal{Q}_2 = \mathcal{Q}_3 = 0\) of vanishing nonmetricity. For the first solution~\eqref{eq:cosmogaxi}, this leads to
\begin{equation}\label{eq:cosmomaxi}
\mathcal{T}_2 = \pm u\,, \quad
\mathcal{T}_1 = \mathcal{H}\,,
\end{equation}
while the second solution~\eqref{eq:cosmogvec} becomes
\begin{equation}\label{eq:cosmomvec}
\mathcal{T}_2 = 0\,, \quad
\mathcal{T}_1 = \mathcal{H} \pm iu\,.
\end{equation}
For the latter, we have explicitly solved the appearing quadratic equation. In this case we see that \(u\) must be imaginary in order to obtain a real torsion, and so we are restricted to the case \(u^2 < 0\) of negative spatial curvature. Finally, the three solutions for \(u = 0\) reduce to the common case
\begin{equation}\label{eq:cosmomflat}
\mathcal{T}_2 = 0\,, \quad
\mathcal{T}_1 = \mathcal{H}\,.
\end{equation}
In all three cases, the two free functions in the torsion scalar are fixed by the conditions of cosmological symmetry and vanishing curvature, so that the field equations are fully expressed in terms of the scale factor \(A\) and the lapse \(N\)~\cite{Hohmann:2020zre}.
\subsection{Perturbation theory}\label{ssec:pert}
Besides the use of symmetric spacetimes as shown in the previous section, another common approach to simplify the (in general non-linear) field equations of a given teleparallel gravity theory is to start from a known, usually highly symmetric solution of the field equations, given by a metric \(\bar{g}_{\mu\nu}\) and a flat affine connection with coefficients \(\bar{\Gamma}^{\mu}{}_{\nu\rho}\), and perform a perturbative expansion of the dynamical fields and their governing field equations around this solution. For this purpose, one conventionally introduces a perturbation parameter \(\epsilon\) on which the solution will depend, and which can be related, for example, to the gravitational constant for a weak-field approximation, or the inverse speed of light for a low-velocity approximation. The full solution \(g_{\mu\nu}(\epsilon)\) and \(\Gamma^{\mu}{}_{\nu\rho}(\epsilon)\), is then expanded in a Taylor series
\begin{equation}
g_{\mu\nu} = \sum_{k = 0}^{\infty}\frac{\epsilon^k}{k!}\left.\frac{\mathrm{d}^k}{\mathrm{d}\epsilon^k}g_{\mu\nu}\right|_{\epsilon = 0}\,, \quad
\Gamma^{\mu}{}_{\nu\rho} = \sum_{k = 0}^{\infty}\frac{\epsilon^k}{k!}\left.\frac{\mathrm{d}^k}{\mathrm{d}\epsilon^k}\Gamma^{\mu}{}_{\nu\rho}\right|_{\epsilon = 0}
\end{equation}
around the background solution \(\bar{g}_{\mu\nu} = g_{\mu\nu}(0)\) and \(\bar{\Gamma}^{\mu}{}_{\nu\rho} = \Gamma^{\mu}{}_{\nu\rho}(0)\). Different conventions are abundant for the terms in this Taylor expansion, either for the coefficients
\begin{equation}
\delta^kg_{\mu\nu} = \left.\frac{\mathrm{d}^k}{\mathrm{d}\epsilon^k}g_{\mu\nu}\right|_{\epsilon = 0}\,, \quad
\delta^k\Gamma^{\mu}{}_{\nu\rho} = \left.\frac{\mathrm{d}^k}{\mathrm{d}\epsilon^k}\Gamma^{\mu}{}_{\nu\rho}\right|_{\epsilon = 0}\,,
\end{equation}
or for the full terms
\begin{equation}
\order{g}{k}_{\mu\nu} = \frac{\epsilon^k}{k!}\left.\frac{\mathrm{d}^k}{\mathrm{d}\epsilon^k}g_{\mu\nu}\right|_{\epsilon = 0}\,, \quad
\order{\Gamma}{k}^{\mu}{}_{\nu\rho} = \frac{\epsilon^k}{k!}\left.\frac{\mathrm{d}^k}{\mathrm{d}\epsilon^k}\Gamma^{\mu}{}_{\nu\rho}\right|_{\epsilon = 0}\,.
\end{equation}
In the following, we will make use of the latter, as it turns out to be shorter for the examples we consider. It follows from the fact that the metric \(g_{\mu\nu}\) is a symmetric tensor field that the same property holds also for all terms \(\order{g}{k}_{\mu\nu}\) in its perturbative expansion. For the connection coefficients, one similarly concludes from the fact that both \(\Gamma^{\mu}{}_{\nu\rho}\) and \(\bar{\Gamma}^{\mu}{}_{\nu\rho}\) are connection coefficients that the remaining terms \(\order{\Gamma}{k}^{\mu}{}_{\nu\rho}\) with \(k > 0\) are tensor fields. In order to determine these terms, one performs a similar Taylor expansion of the gravitational field equations in the perturbation parameter \(\epsilon\). It is the main virtue of this expansion that at each perturbation order \(k\), the corresponding terms of the field equations comprise a linear equation for the field terms \(\order{g}{k}_{\mu\nu}\) and \(\order{\Gamma}{k}^{\mu}{}_{\nu\rho}\) at the same order, which contain the lower order terms as a source; hence, they can be solved subsequently for increasing orders, where the previously found solutions for lower orders are used in each further order to be solved.
In the case of teleparallel gravity, it is important to keep in mind that next to the gravitational field equations also the constraint~\eqref{eq:curvature} of vanishing curvature must be satisfied at any perturbation order. In the symmetric and metric teleparallel classes of theories, also either the torsion~\eqref{eq:torsion} or nonmetricity~\eqref{eq:nonmetricity} must vanish at each order. While it is possible to simply consider these constraints as additional equations which must be solved next to the field equations at each order, one may also pose the question whether it is possible to find a general perturbative solution to these constraints, which is independent of the gravity theory under consideration, and which can then be inserted into the perturbed field equations of any specific gravity theory. To obtain this solution, one needs to perform a perturbative expansion of the corresponding constraint equations. We start by showing this procedure for the flatness constraint~\eqref{eq:curvature}. At the zeroth order, this simply becomes the vanishing of the curvature
\begin{equation}
0 = \bar{R}^{\mu}{}_{\nu\rho\sigma} = \partial_{\rho}\bar{\Gamma}^{\mu}{}_{\nu\sigma} - \partial_{\sigma}\bar{\Gamma}^{\mu}{}_{\nu\rho} + \bar{\Gamma}^{\mu}{}_{\tau\rho}\bar{\Gamma}^{\tau}{}_{\nu\sigma} - \bar{\Gamma}^{\mu}{}_{\tau\sigma}\bar{\Gamma}^{\tau}{}_{\nu\rho}
\end{equation}
for the background connection \(\bar{\Gamma}^{\mu}{}_{\nu\rho}\), which we assume to be satisfied from now on. For the first-order perturbation of the curvature, one finds the condition
\begin{equation}
0 = \order{R}{1}^{\mu}{}_{\nu\rho\sigma} = \bar{\nabla}_{\rho}\order{\Gamma}{1}^{\mu}{}_{\nu\sigma} - \bar{\nabla}_{\sigma}\order{\Gamma}{1}^{\mu}{}_{\nu\rho} + \bar{T}^{\tau}{}_{\rho\sigma}\order{\Gamma}{1}^{\mu}{}_{\nu\tau}\,,
\end{equation}
where all quantities which are calculated with respect to the background connection are denoted with a bar. Now it is helpful to recall that the commutator of covariant derivatives is given by
\begin{equation}
\bar{\nabla}_{\rho}\bar{\nabla}_{\sigma}\lambda^{\mu}{}_{\nu} - \bar{\nabla}_{\sigma}\bar{\nabla}_{\rho}\lambda^{\mu}{}_{\nu} = \bar{R}^{\mu}{}_{\tau\rho\nu}\lambda^{\tau}{}_{\nu} - \bar{R}^{\tau}{}_{\nu\rho\nu}\lambda^{\mu}{}_{\tau} - \bar{T}^{\tau}{}_{\rho\sigma}\bar{\nabla}_{\tau}\lambda^{\mu}{}_{\nu}
\end{equation}
for a tensor field \(\lambda^{\mu}{}_{\nu}\), where the two curvature terms on the right hand side vanish. Hence, we can solve the flatness condition at the first perturbation order by setting
\begin{equation}\label{eq:genpert1}
\order{\Gamma}{1}^{\mu}{}_{\nu\rho} = \bar{\nabla}_{\rho}\order{\lambda}{1}^{\mu}{}_{\nu}
\end{equation}
with an arbitrary first-order tensor field \(\order{\lambda}{1}^{\mu}{}_{\nu}\). To illustrate the further procedure, we calculate the second order curvature perturbation
\begin{equation}
0 = \order{R}{2}^{\mu}{}_{\nu\rho\sigma} = \bar{\nabla}_{\rho}\order{\Gamma}{2}^{\mu}{}_{\nu\sigma} - \bar{\nabla}_{\sigma}\order{\Gamma}{2}^{\mu}{}_{\nu\rho} + \bar{T}^{\tau}{}_{\rho\sigma}\order{\Gamma}{2}^{\mu}{}_{\nu\tau} + 2\order{\Gamma}{1}^{\mu}{}_{\tau[\rho}\order{\Gamma}{1}^{\tau}{}_{|\nu|\sigma]}\,,
\end{equation}
where we now also need to take into account the first-order connection perturbation. A naive ansatz \(\bar{\nabla}_{\rho}\order{\lambda}{2}^{\mu}{}_{\nu}\) for \(\order{\Gamma}{2}^{\mu}{}_{\nu\rho}\) is therefore not sufficient. In order to cancel the term arising from the first-order connection perturbation, one also needs to include terms which are quadratic in \(\order{\lambda}{1}^{\mu}{}_{\nu}\), and must contain one derivative. One finds that a possible solution is given by
\begin{equation}\label{eq:genpert2}
\order{\Gamma}{2}^{\mu}{}_{\nu\rho} = \bar{\nabla}_{\rho}\order{\lambda}{2}^{\mu}{}_{\nu} - \order{\lambda}{1}^{\mu}{}_{\tau}\bar{\nabla}_{\rho}\order{\lambda}{1}^{\tau}{}_{\nu}\,.
\end{equation}
Note that this solution is not unique. Alternatively, one could choose, for example,
\begin{equation}
\order{\Gamma}{2}^{\mu}{}_{\nu\rho} = \bar{\nabla}_{\rho}\order{\lambda}{2}^{\mu}{}_{\nu} + \bar{\nabla}_{\rho}\order{\lambda}{1}^{\mu}{}_{\tau}\order{\lambda}{1}^{\tau}{}_{\nu}\,.
\end{equation}
This becomes clear by realizing that these solutions differ only by a term \(\bar{\nabla}_{\rho}(\order{\lambda}{1}^{\mu}{}_{\tau}\order{\lambda}{1}^{\tau}{}_{\nu})\), which can be absorbed by a redefinition of \(\order{\lambda}{2}^{\mu}{}_{\nu}\). By a similar procedure, one can also subsequently solve the flatness condition at any higher perturbation order.
For the symmetric and metric teleparallel cases, one can make use of the already determined solution of the perturbative flatness condition, and further restrict the perturbation tensor fields \(\order{\lambda}{k}^{\mu}{}_{\nu}\) in order to achieve a connection with vanishing torsion or nonmetricity at any perturbation. We start with the former, which means that the background connection coefficients \(\bar{\Gamma}^{\mu}{}_{\nu\rho}\) as well as the perturbations \(\order{\Gamma}{k}^{\mu}{}_{\nu\rho}\) must be symmetric in their lower two indices. To obtain this property, one can make use of the result that the flat connection perturbations can always be parametrized in the form
\begin{equation}
\order{\Gamma}{k}^{\mu}{}_{\nu\rho} = \bar{\nabla}_{\rho}\order{\lambda}{k}^{\mu}{}_{\nu} + \sum_{j = 1}^{k - 1}\order{\Lambda}{j,k}^{\mu}{}_{\tau}\order{\Gamma}{j}^{\tau}{}_{\nu\rho}\,,
\end{equation}
where \(\order{\Lambda}{j,k}^{\mu}{}_{\tau}\) is determined by solving the flatness condition at the $k$'th perturbation order. Indeed, we have seen this form explicitly for the first order~\eqref{eq:genpert1}, as well as the second order~\eqref{eq:genpert2}, where for the latter \(\order{\Lambda}{1,2}^{\mu}{}_{\tau} = -\order{\lambda}{1}^{\mu}{}_{\tau}\). Using this parametrization, it follows that once we have solved the condition of vanishing torsion up to the perturbation order \(k - 1\), the condition for the $k$'th order simply becomes
\begin{equation}
\bar{\nabla}_{[\rho}\order{\lambda}{k}^{\mu}{}_{\nu]} = 0\,.
\end{equation}
Further using the fact that the covariant derivatives with respect to the background connection commute in the absence of curvature and torsion, we can thus write the solution as
\begin{equation}\label{eq:sympert}
\order{\lambda}{k}^{\mu}{}_{\nu} = \bar{\nabla}_{\nu}\order{\zeta}{k}^{\mu}\,,
\end{equation}
where we introduced the perturbation parameters \(\order{\zeta}{k}^{\mu}\). Note, however, that also in this case numerous other parametrizations of the connection coefficients can be found. A parametrization which turns out particularly convenient for practical calculations arises from the fact that two flat, torsion-free connections, are locally related by a diffeomorphism. It follows that also the perturbed connection \(\Gamma^{\mu}{}_{\nu\rho}\), which depends on the perturbation parameter \(\epsilon\), and the background \(\bar{\Gamma}^{\mu}{}_{\nu\rho}\), are locally related by a family \(\Phi_{\epsilon}\) of diffeomorphisms parametrized by \(\epsilon\), such that
\begin{equation}
\Gamma^{\mu}{}_{\nu\rho}(\epsilon) = \Phi_{\epsilon}^*\bar{\Gamma}^{\mu}{}_{\nu\rho}\,.
\end{equation}
Performing a Taylor expansion in order to obtain the perturbation terms \(\order{\Gamma}{k}^{\mu}{}_{\nu\rho}\), we see that on the right hand side \(\bar{\Gamma}^{\mu}{}_{\nu\rho}\) remains fixed, and we need to expand the diffeomorphism \(\Phi_{\epsilon}\) into a corresponding Taylor series. It turns out that such an expansion gives rise to a series of vector fields \(\order{\xi}{k}^{\mu}\) for \(k > 0\), in terms of which the Taylor expansion reads~\cite{Bruni:1996im,Sonego:1997np,Bruni:1999et}
\begin{equation}
\order{\Gamma}{k}^{\mu}{}_{\nu\rho} = \sum_{l_1 + 2l_2 + \ldots = k}\frac{1}{l_1!l_2! \cdots}\left(\mathcal{L}_{\order{\xi}{1}}^{l_1} \cdots \mathcal{L}_{\order{\xi}{j}}^{l_j} \cdots \bar{\Gamma}\right)^{\mu}{}_{\nu\rho}\,.
\end{equation}
It is instructive to calculate the lower order terms explicitly, using the formula~\eqref{eq:conntrans} with vanishing curvature and torsion. For the background, the expansion trivially reduces to
\begin{equation}
\order{\Gamma}{0}^{\mu}{}_{\nu\rho} = \bar{\Gamma}^{\mu}{}_{\nu\rho}\,.
\end{equation}
For the first order, only one term appears on the right-hand side, which reads
\begin{equation}\label{eq:sympert1}
\order{\Gamma}{1}^{\mu}{}_{\nu\rho} = \left(\mathcal{L}_{\order{\xi}{1}}\bar{\Gamma}\right)^{\mu}{}_{\nu\rho} = \bar{\nabla}_{\nu}\bar{\nabla}_{\rho}\order{\xi}{1}^{\mu}\,,
\end{equation}
while for the second order one finds the two terms
\begin{equation}
\begin{split}\label{eq:sympert2}
\order{\Gamma}{2}^{\mu}{}_{\nu\rho} &= \left(\mathcal{L}_{\order{\xi}{2}}\bar{\Gamma}\right)^{\mu}{}_{\nu\rho} + \frac{1}{2}\left(\mathcal{L}_{\order{\xi}{1}}\mathcal{L}_{\order{\xi}{1}}\bar{\Gamma}\right)^{\mu}{}_{\nu\rho}\\
&= \bar{\nabla}_{\nu}\bar{\nabla}_{\rho}\order{\xi}{2}^{\mu} + \bar{\nabla}_{(\nu}\order{\xi}{1}^{\sigma}\bar{\nabla}_{\rho)}\bar{\nabla}_{\sigma}\order{\xi}{1}^{\mu} - \frac{1}{2}\bar{\nabla}_{\nu}\bar{\nabla}_{\rho}\order{\xi}{1}^{\sigma}\bar{\nabla}_{\sigma}\order{\xi}{1}^{\mu} + \frac{1}{2}\order{\xi}{1}^{\sigma}\bar{\nabla}_{\nu}\bar{\nabla}_{\rho}\bar{\nabla}_{\sigma}\order{\xi}{1}^{\mu}\,.
\end{split}
\end{equation}
It is also helpful to compare these formulas with the perturbations~\eqref{eq:genpert1} and~\eqref{eq:genpert2}, together with the substitution~\eqref{eq:sympert}. For the first perturbation order, the perturbation~\eqref{eq:genpert1} becomes
\begin{equation}
\order{\Gamma}{1}^{\mu}{}_{\nu\rho} = \bar{\nabla}_{\nu}\bar{\nabla}_{\rho}\order{\zeta}{1}^{\mu}\,,
\end{equation}
which agrees with~\eqref{eq:sympert1} for \(\order{\zeta}{1}^{\mu} = \order{\xi}{1}^{\mu}\). At the second order, the perturbation~\eqref{eq:genpert2} becomes
\begin{equation}
\order{\Gamma}{2}^{\mu}{}_{\nu\rho} = \bar{\nabla}_{\nu}\bar{\nabla}_{\rho}\order{\zeta}{2}^{\mu} - \bar{\nabla}_{\sigma}\order{\zeta}{1}^{\mu}\bar{\nabla}_{\nu}\bar{\nabla}_{\rho}\order{\zeta}{1}^{\sigma}\,,
\end{equation}
which agrees with the result~\eqref{eq:sympert2} for
\begin{equation}
\order{\zeta}{2}^{\mu} = \order{\xi}{2}^{\mu} + \frac{1}{2}\order{\xi}{1}^{\sigma}\bar{\nabla}_{\sigma}\order{\xi}{1}^{\mu}\,.
\end{equation}
Also one easily checks that the curvature and torsion vanish at any perturbation order. This type of perturbative expansion is used, for example, to determine the propagation of gravitational waves~\cite{Hohmann:2018wxu} and the post-Newtonian limit~\cite{Flathmann:2020zyj,Hohmann:2021rmp}.
Finally, we take a look at the form of the perturbations in the metric teleparallel case, which means that the nonmetricity must vanish at all perturbation orders. This holds in particular for the background,
\begin{equation}
0 = \bar{Q}_{\mu\nu\rho} = \bar{\nabla}_{\mu}\bar{g}_{\nu\rho}\,,
\end{equation}
and so raising and lowering indices of the perturbations with the metric commutes with the covariant derivative, which greatly simplifies the calculations. We make use of this fact for calculating the conditions on the connection perturbation \(\order{\lambda}{k}^{\mu}{}_{\nu}\) which we need to satisfy in order to obtain vanishing nonmetricity. At the linear order, the perturbation of the nonmetricity is given by
\begin{equation}
\order{Q}{1}_{\mu\nu\rho} = \bar{\nabla}_{\mu}\left(\order{g}{1}_{\nu\rho} - 2\order{\lambda}{1}_{(\nu\rho)}\right)\,,
\end{equation}
and so it vanishes if we fix the symmetric part of the connection perturbation by the condition
\begin{equation}
2\order{\lambda}{1}_{(\mu\nu)} = \order{g}{1}_{\mu\nu}\,.
\end{equation}
One could naively conclude that the same formula holds identically also for higher orders, replacing the first perturbation order with an arbitrary order \(k\). However, this is not the case, as follows from a perturbative expansion of the nonmetricity~\eqref{eq:nonmetricity}, whose higher than linear orders contain products of the lower order perturbations of the metric and the connection. For example, for the second order we have
\begin{equation}
\begin{split}
\order{Q}{2}_{\mu\nu\rho} &= \bar{\nabla}_{\mu}\left(\order{g}{2}_{\nu\rho} - 2\order{\lambda}{2}_{(\nu\rho)}\right) - 2\nabla_{\mu}\order{\lambda}{1}^{\sigma}{}_{(\nu}\left(\order{g}{1}_{\rho)\sigma} - \order{\lambda}{1}_{\rho)\sigma}\right)\\
&= \bar{\nabla}_{\mu}\left(\order{g}{2}_{\nu\rho} - 2\order{\lambda}{2}_{(\nu\rho)} - \bar{g}_{\sigma\omega}\order{\lambda}{1}^{\sigma}{}_{\nu}\order{\lambda}{1}^{\omega}{}_{\rho}\right)\,,
\end{split}
\end{equation}
after substituting \(\order{g}{1}_{\nu\rho}\) from the first order result, so that we can easily read off the condition for the second order perturbation. Following the same procedure also for higher order perturbations, one arrives at the general formula
\begin{equation}
2\order{\lambda}{k}_{(\mu\nu)} = \order{g}{k}_{\mu\nu} - \bar{g}_{\rho\sigma}\sum_{j = 1}^{k - 1}\order{\lambda}{j}^{\rho}{}_{\mu}\order{\lambda}{k - j}^{\sigma}{}_{\nu}\,.
\end{equation}
We see that the condition of vanishing nonmetricity links the symmetric part of the connection perturbation to the perturbation of the metric, and so the latter can always be expressed in terms of the former, leaving \(\order{\lambda}{k}_{\mu\nu}\) as the only independent perturbation variable. Like in the case of symmetric teleparallel gravity, this perturbative expansion is used for the calculation of gravitational waves~\cite{Hohmann:2018jso} and the post-Newtonian limit~\cite{Ualikhanova:2019ygl}, but also in the cosmological perturbation theory around flat~\cite{Golovnev:2018wbh} and general~\cite{Bahamonde:2022ohm} cosmological backgrounds.
\section{Teleparallel gravity theories}\label{sec:theories}
In this final section, we discuss a few selected classes of teleparallel gravity theories, their actions and field equations. These theories constitute modifications of general relativity, which depart from a reformulation of the Einstein-Hilbert action in terms of teleparallel geometries, known as the teleparallel equivalent of general relativity, which we discuss in section~\ref{ssec:tegr}. A simple modification is then obtained by replacing the Lagrangian of these theories by a free function thereof, as we show in section~\ref{ssec:fg}. Another modification arises by considering the most general action which is quadratic in torsion and nonmetricity, and which we show in section~\ref{ssec:quadlag}. Moreover, modified theories can be obtained by considering a scalar field as another dynamical variable in addition to the metric and the flat connection; we discuss theories of this type in section~\ref{ssec:scalartele}, and see how a particular subclass of them is connected to a previously discussed class of theories in section~\ref{ssec:stfg}.
\subsection{The teleparallel equivalents of general relativity}\label{ssec:tegr}
In the previous sections we have discussed the general form of the action and field equations for teleparallel gravity theories, but we have not yet considered any particular theories. As a starting point for the construction of modified teleparallel gravity theories, we now pose the question how the well-known general relativity action and field equations can be cast into the teleparallel framework. The crucial observation to answer this question is the fact that the decomposition~\eqref{eq:conndec} of the independent connection with respect to the Levi-Civita connection of the metric induces a related decomposition of the curvature given by
\begin{equation}
R^{\mu}{}_{\nu\rho\sigma} = \lc{R}^{\mu}{}_{\nu\rho\sigma} + \lc{\nabla}_{\rho}M^{\mu}{}_{\nu\sigma} - \lc{\nabla}_{\sigma}M^{\mu}{}_{\nu\rho} + M^{\mu}{}_{\tau\rho}M^{\tau}{}_{\nu\sigma} - M^{\mu}{}_{\tau\sigma}M^{\tau}{}_{\nu\rho}\,.
\end{equation}
Keeping in mind that the curvature~\eqref{eq:curvature} of the teleparallel connection is imposed to vanish, one can solve for the curvature tensor of the Levi-Civita connection, and finds
\begin{equation}\label{eq:curvdec}
\lc{R}^{\mu}{}_{\nu\rho\sigma} = -\lc{\nabla}_{\rho}M^{\mu}{}_{\nu\sigma} + \lc{\nabla}_{\sigma}M^{\mu}{}_{\nu\rho} - M^{\mu}{}_{\tau\rho}M^{\tau}{}_{\nu\sigma} + M^{\mu}{}_{\tau\sigma}M^{\tau}{}_{\nu\rho}\,.
\end{equation}
This allows us to replace the Ricci scalar \(\lc{R}\) in the Einstein-Hilbert action
\begin{equation}\label{eq:einsteinhilbert}
S_{\text{g}} = \frac{1}{2\kappa^2}\int_M\lc{R}\sqrt{-g}\mathrm{d}^4x
\end{equation}
by
\begin{equation}\label{eq:riccisplit}
\lc{R} = -G + B\,,
\end{equation}
where we defined the terms
\begin{equation}\label{eq:gtegrbndterms}
G = 2M^{\mu}{}_{\tau[\mu}M^{\tau\nu}{}_{\nu]}\,, \quad
B = 2\lc{\nabla}_{\mu}M^{[\nu\mu]}{}_{\nu}\,.
\end{equation}
One can see that \(B\) becomes a boundary term in the action, and therefore does not contribute to the field equations. Omitting this term from the action, one thus obtains~\cite{BeltranJimenez:2019odq}
\begin{equation}\label{eq:gtegraction}
S_{\text{g}} = -\frac{1}{2\kappa^2}\int_MG\sqrt{-g}\mathrm{d}^4x\,.
\end{equation}
This is the action of the \emph{general teleparallel equivalent of general relativity} (GTEGR). To study the nature of this equivalence, we calculate the gravitational field equations. Note that the variation of the distortion tensor is given by
\begin{equation}
\delta M^{\mu}{}_{\nu\rho} = \delta\Gamma^{\mu}{}_{\nu\rho} - \delta\lc{\Gamma}^{\mu}{}_{\nu\rho} = \delta\Gamma^{\mu}{}_{\nu\rho} - \frac{1}{2}g^{\mu\sigma}\left(\lc{\nabla}_{\nu}\delta g_{\sigma\rho} + \lc{\nabla}_{\rho}\delta g_{\nu\sigma} - \lc{\nabla}_{\sigma}\delta g_{\nu\rho}\right)\,,
\end{equation}
and so the variation of the gravity scalar becomes
\begin{equation}
\delta G = U^{\mu\nu}\delta g_{\mu\nu} + V^{\rho\mu\nu}\lc{\nabla}_{\rho}\delta g_{\mu\nu} + Z_{\mu}{}^{\nu\rho}\delta\Gamma^{\mu}{}_{\nu\rho}\,,
\end{equation}
where we have introduced the abbreviations
\begin{subequations}\label{eq:gvarabbrev}
\begin{align}
U^{\mu\nu} &= M^{\rho\sigma(\mu}M_{\sigma}{}^{\nu)}{}_{\rho} - M^{\rho(\mu\nu)}M^{\sigma}{}_{\rho\sigma}\\
V^{\rho\mu\nu} &= M^{\rho(\mu\nu)} - M^{\sigma(\mu}{}_{\sigma}g^{\nu)\rho} - M^{[\rho\sigma]}{}_{\sigma}g^{\mu\nu}\\
Z_{\mu}{}^{\nu\rho} &= M^{\nu\sigma}{}_{\sigma}\delta_{\mu}^{\rho} + M^{\sigma}{}_{\mu\sigma}g^{\nu\rho} - M^{\nu\rho}{}_{\mu} - M^{\rho}{}_{\mu}{}^{\nu}\,.
\end{align}
\end{subequations}
This allows us to calculate the variation of the action~\eqref{eq:gtegraction} and perform integration by parts in order to eliminate the derivatives acting on the metric perturbation \(\delta g_{\mu\nu}\). The resulting variation then takes the form~\eqref{eq:metricgravactvar} with
\begin{equation}\label{eq:gtegrmetvar}
\begin{split}
W_{\mu\nu} &= \frac{1}{\kappa^2}\left(U_{\mu\nu} - \lc{\nabla}_{\rho}V^{\rho}{}_{\mu\nu} + \frac{1}{2}Gg_{\mu\nu}\right)\\
&= \frac{1}{\kappa^2}\bigg[\lc{\nabla}_{(\mu}M^{\rho}{}_{\nu)\rho} - \lc{\nabla}_{\rho}M^{\rho}{}_{(\mu\nu)} + M^{\rho}{}_{\sigma(\mu}M^{\sigma}{}_{\nu)\rho} - M^{\rho}{}_{\sigma\rho}M^{\sigma}{}_{(\mu\nu)}\\
&\phantom{=}- \frac{1}{2}\left(\lc{\nabla}_{\rho}M^{\sigma\rho}{}_{\sigma} - \lc{\nabla}_{\rho}M^{\rho\sigma}{}_{\sigma} + M^{\rho\sigma\omega}M_{\omega\rho\sigma} - M^{\rho}{}_{\omega\rho}M^{\omega\sigma}{}_{\sigma}\right)g_{\mu\nu}\bigg]
\end{split}
\end{equation}
and
\begin{equation}\label{eq:gtegrconnvar}
Y_{\mu}{}^{\nu\rho} = \frac{1}{2\kappa^2}Z_{\mu}{}^{\nu\rho} = \frac{1}{2\kappa^2}(M^{\nu\sigma}{}_{\sigma}\delta_{\mu}^{\rho} + M^{\sigma}{}_{\mu\sigma}g^{\nu\rho} - M^{\nu\rho}{}_{\mu} - M^{\rho}{}_{\mu}{}^{\nu})\,.
\end{equation}
By comparing with the relation~\eqref{eq:curvdec}, one finds that the first equation can be rewritten as
\begin{equation}
W_{\mu\nu} = \frac{1}{\kappa^2}\left(\lc{R}_{\mu\nu} - \frac{1}{2}\lc{R}g_{\mu\nu}\right)\,,
\end{equation}
and so the metric equation resembles Einstein's equation
\begin{equation}\label{eq:einstein}
\lc{R}_{\mu\nu} - \frac{1}{2}\lc{R}g_{\mu\nu} = \kappa^2\Theta_{\mu\nu}\,.
\end{equation}
We also need to consider the connection field equation~\eqref{eq:genconnfield}, which becomes
\begin{multline}
\frac{1}{\kappa^2}\left(\lc{\nabla}_{[\mu}M^{\nu\rho}{}_{\rho]} + \lc{\nabla}^{[\nu}M_{\rho\mu}{}^{\rho]} + M^{\nu}{}_{\rho[\mu}M^{\rho\sigma}{}_{\sigma]} + M_{\rho}{}^{\sigma[\nu}M_{\sigma\mu}{}^{\rho]}\right)\\
= \nabla_{\tau}H_{\mu}{}^{\nu\tau} - M^{\omega}{}_{\tau\omega}H_{\mu}{}^{\nu\tau}\,.
\end{multline}
Once again making use of the relation~\eqref{eq:curvdec}, the term in brackets becomes
\begin{equation}
\lc{R}^{\nu\rho}{}_{\mu\rho} + \lc{R}_{\rho\mu}{}^{\nu\rho} = \lc{R}^{\nu}{}_{\mu} - \lc{R}_{\mu}{}^{\nu} = 0\,,
\end{equation}
which vanishes, since the Ricci tensor of the Levi-Civita connection is symmetric. Hence, one is left with the equation
\begin{equation}\label{eq:gtegrhypmom}
\nabla_{\tau}H_{\mu}{}^{\nu\tau} - M^{\omega}{}_{\tau\omega}H_{\mu}{}^{\nu\tau} = 0
\end{equation}
for the hypermomentum, which must be satisfied for any matter which is compatible with the gravitational action~\eqref{eq:gtegraction}. Note that the connection does not appear anywhere on the gravitational side of the field equations, due to the fact that it enters into the action only through a total derivative term. For consistency, one conventionally assumes that it does not couple to the matter fields, so that the hypermomentum vanishes, and so the constraint~\eqref{eq:gtegrhypmom} is satisfied identically. The only non-trivial field equation is then Einstein's equation~\eqref{eq:einstein}, and so the field equations of GTEGR are equivalent to those of general relativity, hence justifying the name teleparallel equivalent.
Since the connection only has a spurious appearance in the action~\eqref{eq:gtegraction}, one may expect that it will not enter the field equations also in the symmetric and metric classes of teleparallel gravity theories. This is not obvious from the Lagrange multiplier approach of deriving the field equations, since the Lagrange multiplier terms~\eqref{eq:lagmultors} and~\eqref{eq:lagmulnonmet} are not total derivatives, and so the connection enters the field equations obtained by variation with respect to the Lagrange multipliers. Nevertheless, keeping in mind that this approach yields the same field equations as the approach of restricted variation, and that in the latter the variation of the connection appears only through a total derivative in the action, one may still expect to obtain an equivalent of general relativity. This is particularly easy to see in the case of symmetric teleparallel gravity, since its metric field equation~\eqref{eq:genmetfield} takes the same form as in the case of general teleparallel gravity, and hence once again resembles the Einstein equations~\eqref{eq:einstein}, irrespective of the constraint \(T^{\mu}{}_{\nu\rho} = 0\) imposed on the connection. For the remaining field equation~\eqref{eq:symconnfield}, it is helpful to recall that for the variation~\eqref{eq:gtegrconnvar} the left hand side of the field equation~\eqref{eq:genconnfield} vanishes identically, and hence does the left hand side of the equivalent field equation~\eqref{eq:genconnfielddens}. In the absence of torsion, the torsion term vanishes, and one is left with
\begin{equation}
\nabla_{\rho}\tilde{Y}_{\mu}{}^{\nu\rho} = 0\,.
\end{equation}
Hence, also the left hand side of the symmetric teleparallel field equation~\eqref{eq:symconnfield} vanishes identically, leaving only the hypermomentum constraint
\begin{equation}
\nabla_{\nu}\nabla_{\rho}\tilde{H}_{\mu}{}^{\nu\rho} = 0\,,
\end{equation}
which can be satisfied by demanding vanishing hypermomentum.
A similar argument holds in the case of metric teleparallel gravity. Once again, one can make use of the fact that the left hand side of the field equation~\eqref{eq:genconnfielddens} vanishes identically for the variation~\eqref{eq:gtegrconnvar}. In the absence of nonmetricity, the covariant derivative~\eqref{eq:covderdens} of the density factor \(\sqrt{-g}\) vanishes, and so this factor can be canceled from the equations. One is then left with the equation
\begin{equation}
\nabla_{\tau}Y_{\mu}{}^{\nu\tau} - T^{\omega}{}_{\omega\tau}Y_{\mu}{}^{\nu\tau} = 0\,.
\end{equation}
Using this result, the metric teleparallel field equation~\eqref{eq:metallfield} reduces to
\begin{equation}
W^{\mu\nu} = \Theta^{\mu\nu} - \nabla_{\rho}H^{\mu\nu\rho} + H^{\mu\nu\rho}T^{\tau}{}_{\tau\rho}\,.
\end{equation}
Demanding once again vanishing hypermomentum, one therefore obtains Einstein's equation~\eqref{eq:einstein} also in this case.
In order to gain more insight into the underlying structure of the different teleparallel equivalents of general relativity, it is helpful to decompose the gravity scalar \(G\) and the boundary term \(B\) into the individual contributions from the torsion and the nonmetricity. Using the connection decomposition~\eqref{eq:conndec}, the contortion~\eqref{eq:contortion} and the disformation~\eqref{eq:disformation}, one finds
\begin{multline}\label{eq:gtegrscalar}
G = \frac{1}{4}Q^{\mu\nu\rho}Q_{\mu\nu\rho} - \frac{1}{2}Q^{\mu\nu\rho}Q_{\rho\mu\nu} - \frac{1}{4}Q^{\rho\mu}{}_{\mu}Q_{\rho\nu}{}^{\nu} + \frac{1}{2}Q^{\mu}{}_{\mu\rho}Q^{\rho\nu}{}_{\nu}\\
+ \frac{1}{4}T^{\mu\nu\rho}T_{\mu\nu\rho} + \frac{1}{2}T^{\mu\nu\rho}T_{\rho\nu\mu} - T^{\mu}{}_{\mu\rho}T_{\nu}{}^{\nu\rho} + T^{\mu\nu\rho}Q_{\nu\rho\mu} - T^{\mu}{}_{\rho\mu}Q_{\rho\nu}{}^{\nu} + T^{\mu}{}_{\rho\mu}Q^{\nu}{}_{\nu\rho}\,,
\end{multline}
as well as
\begin{equation}
B = \lc{\nabla}_{\mu}(2T_{\nu}{}^{\nu\mu} + Q_{\nu}{}^{\nu\mu} - Q^{\mu\nu}{}_{\nu})\,.
\end{equation}
If either torsion or nonmetricity vanish, these expressions simplify. In particular, the gravity scalar~\eqref{eq:gtegrscalar} reduces to the nonmetricity scalar
\begin{equation}\label{eq:stegrscalar}
\begin{split}
Q &= \frac{1}{2}Q_{\rho\mu\nu}P^{\rho\mu\nu}\\
&= \frac{1}{4}Q^{\mu\nu\rho}Q_{\mu\nu\rho} - \frac{1}{2}Q^{\mu\nu\rho}Q_{\rho\mu\nu} - \frac{1}{4}Q^{\rho\mu}{}_{\mu}Q_{\rho\nu}{}^{\nu} + \frac{1}{2}Q^{\mu}{}_{\mu\rho}Q^{\rho\nu}{}_{\nu}
\end{split}
\end{equation}
or the torsion scalar
\begin{equation}\label{eq:mtegrscalar}
\begin{split}
T &= \frac{1}{2}T^{\rho}{}_{\mu\nu}S_{\rho}{}^{\mu\nu}\\
&= \frac{1}{4}T^{\mu\nu\rho}T_{\mu\nu\rho} + \frac{1}{2}T^{\mu\nu\rho}T_{\rho\nu\mu} - T^{\mu}{}_{\mu\rho}T_{\nu}{}^{\nu\rho}\,,
\end{split}
\end{equation}
respectively, where we have introduced the nonmetricity conjugate
\begin{equation}\label{eq:nonmetconj}
P^{\rho\mu\nu} = L^{\rho\mu\nu} - \frac{1}{2}g^{\mu\nu}(Q^{\rho\sigma}{}_{\sigma} - Q_{\sigma}{}^{\sigma\rho}) + \frac{1}{2}g^{\rho(\mu}Q^{\nu)\sigma}{}_{\sigma}
\end{equation}
and the superpotential
\begin{equation}\label{eq:suppot}
S_{\rho}{}^{\mu\nu} = K^{\mu\nu}{}_{\rho} - \delta_{\rho}^{\mu}T_{\sigma}{}^{\sigma\nu} + \delta_{\rho}^{\nu}T_{\sigma}{}^{\sigma\mu}\,.
\end{equation}
In terms of these scalars, the action of the \emph{symmetric teleparallel equivalent of general relativity} (STEGR) becomes~\cite{Nester:1998mp}
\begin{equation}\label{eq:stegraction}
S_{\text{g}} = -\frac{1}{2\kappa^2}\int_MQ\sqrt{-g}\mathrm{d}^4x\,,
\end{equation}
while for the \emph{metric teleparallel equivalent of general relativity} (MTEGR\footnote{In the literature, the abbreviation TEGR is more common, since it was developed prior to the other equivalent theories.}) one has~\cite{Maluf:2013gaa}
\begin{equation}\label{eq:mtegraction}
S_{\text{g}} = -\frac{1}{2\kappa^2}\int_MT\sqrt{-g}\mathrm{d}^4x\,.
\end{equation}
Within their respective class of teleparallel gravity theories, these actions yield the same metric field equation as general relativity, and are thus common starting points for the construction of modified gravity theories, as we will see in the following sections.
\subsection{The $f(G)$ classes of modified theories}\label{ssec:fg}
After discussing in the previous section a number of teleparallel gravity theories, whose metric field equation reproduces Einstein's field equation of general relativity for matter without hypermomentum, we now turn our focus towards modifications of these gravity theories. For the Einstein-Hilbert action~\eqref{eq:einsteinhilbert}, a well-known and thoroughly studied class of gravity theories is obtained by replacing the Ricci scalar \(\lc{R}\) by \(f(\lc{R})\), where \(f\) is an arbitrary real function of one variable, which is chosen such that the phenomenology of the resulting theory matches with observations, e.g., in cosmology. The same procedure can also be applied to the teleparallel equivalent theories~\cite{Boehmer:2021aji}. Starting with the GTEGR action~\eqref{eq:gtegraction}, one thus obtains the action
\begin{equation}\label{eq:fgaction}
S_{\text{g}} = -\frac{1}{2\kappa^2}\int_Mf(G)\sqrt{-g}\mathrm{d}^4x\,.
\end{equation}
In order to derive the field equations, one proceeds as shown in the previous section, by variation of the action and integration by parts, so that the gravitational part \(S_{\text{g}}\) takes the form~\eqref{eq:metricgravactvar}, with
\begin{equation}\label{eq:fgmetvar}
W_{\mu\nu} = \frac{1}{\kappa^2}\left[f'U_{\mu\nu} - \lc{\nabla}_{\rho}(f'V^{\rho}{}_{\mu\nu}) + \frac{1}{2}fg_{\mu\nu}\right]
\end{equation}
and
\begin{equation}\label{eq:fgconnvar}
Y_{\mu}{}^{\nu\rho} = \frac{1}{2\kappa^2}f'Z_{\mu}{}^{\nu\rho}\,.
\end{equation}
where we wrote \(f, f', \ldots\) as a shorthand for \(f(G), f'(G), \ldots\), and used the abbreviations~\eqref{eq:gvarabbrev} we introduced for the variation of the gravity scalar \(G\). Hence, it follows that the gravitational field equations are given by the metric equation
\begin{equation}\label{eq:fgmetfield}
f'U_{\mu\nu} - \lc{\nabla}_{\rho}(f'V^{\rho}{}_{\mu\nu}) + \frac{1}{2}fg_{\mu\nu} = \kappa^2\Theta_{\mu\nu}
\end{equation}
and the connection equation
\begin{equation}\label{eq:fgconnfield}
\nabla_{\rho}(f'Z_{\mu}{}^{\nu\rho}) - f'M^{\omega}{}_{\rho\omega}Z_{\mu}{}^{\nu\rho} = 2\kappa^2(\nabla_{\rho}H_{\mu}{}^{\nu\rho} - M^{\omega}{}_{\rho\omega}H_{\mu}{}^{\nu\rho})\,.
\end{equation}
These equations can be written more explicitly as follows. First, recall that
\begin{equation}
U_{\mu\nu} - \lc{\nabla}_{\rho}(V^{\rho}{}_{\mu\nu}) + \frac{1}{2}Gg_{\mu\nu} = \lc{R}_{\mu\nu} - \frac{1}{2}\lc{R}g_{\mu\nu}
\end{equation}
is the left hand side of the GTEGR field equation. Using this fact, the metric field equation~\eqref{eq:fgmetfield} becomes
\begin{equation}
f'\left(\lc{R}_{\mu\nu} - \frac{1}{2}\lc{R}g_{\mu\nu}\right) - V^{\rho}{}_{\mu\nu}\lc{\nabla}_{\rho}f' + \frac{1}{2}(f - f'G)g_{\mu\nu} = \kappa^2\Theta_{\mu\nu}\,.
\end{equation}
Finally, substituting \(V^{\rho}{}_{\mu\nu}\) using the variation~\eqref{eq:gvarabbrev}, one obtains
\begin{multline}\label{eq:fgmetfield2}
f'\left(\lc{R}_{\mu\nu} - \frac{1}{2}\lc{R}g_{\mu\nu}\right) - M^{\rho}{}_{(\mu\nu)}\lc{\nabla}_{\rho}f' + \lc{\nabla}_{(\mu}f'M^{\sigma}{}_{\nu)\sigma} + M^{[\rho\sigma]}{}_{\sigma}g_{\mu\nu}\lc{\nabla}_{\rho}f'\\
+ \frac{1}{2}(f - f'G)g_{\mu\nu} = \kappa^2\Theta_{\mu\nu}\,.
\end{multline}
Similarly, one can use the fact that the left hand side of the GTEGR connection equation vanishes, and hence
\begin{equation}
\nabla_{\rho}(Z_{\mu}{}^{\nu\rho}) - M^{\omega}{}_{\rho\omega}Z_{\mu}{}^{\nu\rho} = 0\,,
\end{equation}
to write the connection field equation as
\begin{equation}
Z_{\mu}{}^{\nu\rho}\nabla_{\rho}f' = 2\kappa^2(\nabla_{\rho}H_{\mu}{}^{\nu\rho} - M^{\omega}{}_{\rho\omega}H_{\mu}{}^{\nu\rho})\,,
\end{equation}
and substituting the variation~\eqref{eq:gvarabbrev},
\begin{multline}\label{eq:fgconnfield2}
M^{\nu\rho}{}_{\rho}\nabla_{\mu}f' + M^{\sigma}{}_{\mu\sigma}g^{\nu\rho}\nabla_{\rho}f' - M^{\nu\rho}{}_{\mu}\nabla_{\rho}f' - M^{\rho}{}_{\mu}{}^{\nu}\nabla_{\rho}f'\\
= 2\kappa^2(\nabla_{\rho}H_{\mu}{}^{\nu\rho} - M^{\omega}{}_{\rho\omega}H_{\mu}{}^{\nu\rho})\,.
\end{multline}
The most important difference which distinguishes the \(f(G)\) class of theories from GTEGR is the fact that for \(f'' \neq 0\) the connection contribution to the action is no longer a total derivative, and so the connection remains as a dynamical field in the field equations, which now also contain a non-trivial connection field equation. It follows in particular that these field equations are not equivalent to those of \(f(\lc{R})\) gravity, since the latter has the metric as its only dynamical field, and its field equations are of fourth derivative order\footnote{They can be reduced to second order by introducing an auxiliary scalar field.}. In contrast, the field equations of \(f(G)\) gravity are of second derivative order.
In analogy to the GTEGR action~\eqref{eq:gtegraction}, which is based on the general teleparallel geometry containing both torsion and nonmetricity, also the teleparallel equivalent theories based on more restricted geometries can be generalized by introducing a free function into their respective actions~\eqref{eq:stegraction} and~\eqref{eq:mtegraction}. Equivalently, one can take the action~\eqref{eq:fgaction} and impose the vanishing torsion or nonmetricity either by introducing Lagrange multipliers or by imposing the constraint alongside a restricted variation. It turns out that the resulting field equations can be simplified, as we will see in the following. We start with the symmetric teleparallel gravity action~\cite{BeltranJimenez:2017tkd}
\begin{equation}\label{eq:fqaction}
S_{\text{g}} = -\frac{1}{2\kappa^2}\int_Mf(Q)\sqrt{-g}\mathrm{d}^4x\,.
\end{equation}
The variation of this action is still given by the expressions~\eqref{eq:fgmetvar} and~\eqref{eq:fgconnvar}, but these simplify due to the vanishing torsion, and can be expressed in terms of the nonmetricity. Also the field equation simplify, which one can see as follows, starting with the connection equation. From the general form~\eqref{eq:symconnfield} follows that only the symmetric part \(\tilde{Y}_{\mu}{}^{(\nu\rho)}\) contributes, since the covariant derivatives commute in the absence of curvature and torsion. Using~\eqref{eq:fgconnvar}, this means that only the symmetric part \(Z_{\mu}{}^{(\nu\rho)}\) contributes, which is given by
\begin{equation}\label{eq:symconnvarrel}
Z_{\mu}{}^{(\nu\rho)} = Q_{\mu}{}^{\nu\rho} - \frac{1}{2}g^{\nu\rho}Q_{\mu\sigma}{}^{\sigma} + \frac{1}{2}\delta^{(\nu}_{\mu}Q^{\rho)\sigma}{}_{\sigma} - Q_{\sigma}{}^{\sigma(\nu}\delta^{\rho)}_{\mu} = -2P^{(\nu\rho)}{}_{\mu}\,.
\end{equation}
The connection equation therefore becomes
\begin{equation}\label{eq:fqconnfield}
-\nabla_{\nu}\nabla_{\rho}(f'\tilde{P}^{\nu\rho}{}_{\mu}) = \kappa^2\nabla_{\nu}\nabla_{\rho}\tilde{H}_{\mu}{}^{\nu\rho}\,,
\end{equation}
where \(\tilde{P}^{\nu\rho}{}_{\mu} = \sqrt{-g}P^{\nu\rho}{}_{\mu}\), and we omitted the symmetrization brackets around the indices, which are redundant due to the contraction with the commuting derivatives. Similarly, we can also simplify the metric field equation, which takes the same form~\eqref{eq:fgmetfield}, and can equivalently be written as
\begin{equation}
f'U^{\mu}{}_{\nu} - \frac{\lc{\nabla}_{\rho}(\sqrt{-g}f'V^{\rho\mu}{}_{\nu})}{\sqrt{-g}} + \frac{1}{2}f\delta^{\mu}_{\nu} = \kappa^2\Theta^{\mu}{}_{\nu}\,,
\end{equation}
using the fact that the Levi-Civita connection is metric compatible, so that we can raise and lower indices and introduce the density factor \(\sqrt{-g}\) inside the derivative. Changing this covariant derivative to the independent connection, one has
\begin{multline}
\lc{\nabla}_{\rho}(\sqrt{-g}f'V^{\rho\mu}{}_{\nu}) = \nabla_{\rho}(\sqrt{-g}f'V^{\rho\mu}{}_{\nu})\\
+ \sqrt{-g}f'(L^{\sigma}{}_{\sigma\rho}V^{\rho\mu}{}_{\nu} - L^{\rho}{}_{\sigma\rho}V^{\sigma\mu}{}_{\nu} - L^{\mu}{}_{\sigma\rho}V^{\rho\sigma}{}_{\nu} + L^{\sigma}{}_{\nu\rho}V^{\rho\mu}{}_{\sigma})\,.
\end{multline}
Calculating the variation terms
\begin{equation}
U^{\mu\nu} = \frac{1}{4}Q^{\mu\rho\sigma}Q^{\nu}{}_{\rho\sigma} + \frac{1}{4}(Q^{\rho\mu\nu} - Q^{(\mu\nu)\rho})Q_{\rho\sigma}{}^{\sigma} - Q^{\rho\sigma\mu}Q_{[\rho\sigma]}{}^{\nu}
\end{equation}
and
\begin{equation}
V^{\rho\mu\nu} = \frac{1}{2}Q^{\rho\mu\nu} - Q^{(\mu\nu)\rho} - \frac{1}{2}g^{\mu\nu}(Q^{\rho\sigma}{}_{\sigma} - Q_{\sigma}{}^{\sigma\rho}) + \frac{1}{2}g^{\rho(\mu}Q^{\nu)\sigma}{}_{\sigma} = P^{\rho\mu\nu}\,,
\end{equation}
one finds that they combine into
\begin{multline}
U^{\mu}{}_{\nu} - L^{\sigma}{}_{\sigma\rho}V^{\rho\mu}{}_{\nu} + L^{\rho}{}_{\sigma\rho}V^{\sigma\mu}{}_{\nu} + L^{\mu}{}_{\sigma\rho}V^{\rho\sigma}{}_{\nu} - L^{\sigma}{}_{\nu\rho}V^{\rho\mu}{}_{\sigma}\\
= \frac{1}{2}(Q^{\mu[\rho}{}_{\rho}Q_{\nu\sigma}{}^{\sigma]} - Q_{(\nu}{}^{\mu\rho}Q_{\rho)\sigma}{}^{\rho} + Q^{\rho\mu\sigma}Q_{\nu\rho\sigma}) = -\frac{1}{2}P^{\mu\rho\sigma}Q_{\nu\rho\sigma}\,.
\end{multline}
Combining these results, the metric field equation finally becomes
\begin{equation}\label{eq:fqmetfield}
-\frac{\nabla_{\rho}(\sqrt{-g}f'P^{\rho\mu}{}_{\nu})}{\sqrt{-g}} - \frac{f'}{2}P^{\mu\rho\sigma}Q_{\nu\rho\sigma} + \frac{1}{2}f\delta^{\mu}_{\nu} = \kappa^2\Theta^{\mu}{}_{\nu}\,.
\end{equation}
Note, however, that this form changes if one raises or lowers indices, which appear also under the metric-incompatible covariant derivative \(\nabla_{\rho}\).
Finally, we also take a closer look at the metric teleparallel case, by imposing vanishing nonmetricity. Under this restriction, the general action~\eqref{eq:fgaction} becomes~\cite{Bengochea:2008gz,Linder:2010py,Krssak:2015oua}
\begin{equation}\label{eq:ftaction}
S_{\text{g}} = -\frac{1}{2\kappa^2}\int_Mf(T)\sqrt{-g}\mathrm{d}^4x\,.
\end{equation}
In this case we need to consider only the single field equation~\eqref{eq:metallfield}, whose left hand side now takes the form
\begin{multline}
W^{\mu\nu} - \nabla_{\rho}Y^{\mu\nu\rho} + Y^{\mu\nu\rho}T^{\tau}{}_{\tau\rho}\\
= \frac{1}{\kappa^2}\left[f'U^{\mu\nu} - \lc{\nabla}_{\rho}(f'V^{\rho\mu\nu}) + \frac{1}{2}fg^{\mu\nu} - \frac{1}{2}\nabla_{\rho}(f'Z^{\mu\nu\rho}) + \frac{1}{2}f'Z^{\mu\nu\rho}T^{\tau}{}_{\tau\rho}\right]\,.
\end{multline}
using the variation expressions~\eqref{eq:fgmetvar} and~\eqref{eq:fgconnvar}. In order to simplify this expressions, we transform the covariant derivative with respect to the independent connection to that of the Levi-Civita connection, and find
\begin{equation}
\nabla_{\rho}(f'Z^{\mu\nu\rho}) - f'Z^{\mu\nu\rho}T^{\tau}{}_{\tau\rho} = \lc{\nabla}_{\rho}(f'Z^{\mu\nu\rho}) + f'(K^{\mu}{}_{\sigma\rho}Z^{\sigma\nu\rho} + K^{\nu}{}_{\sigma\rho}Z^{\mu\sigma\rho})\,,
\end{equation}
where the trace of the torsion tensor cancels with a trace of the contortion tensor. Now we can combine the two covariant derivatives, and evaluate
\begin{equation}
V^{\rho\mu\nu} + \frac{1}{2}Z^{\mu\nu\rho} = 2T_{\sigma}{}^{\sigma[\rho}g^{\nu]\mu} + T^{[\nu\rho]\mu} - \frac{1}{2}T^{\mu\nu\rho} = -S^{\mu\nu\rho}\,.
\end{equation}
We are left with the terms
\begin{equation}
U^{\mu\nu} - \frac{1}{2}(K^{\mu}{}_{\sigma\rho}Z^{\sigma\nu\rho} + K^{\nu}{}_{\sigma\rho}Z^{\mu\sigma\rho}) = 2K^{\nu\rho[\sigma}K_{\rho\sigma}{}^{\mu]} = S^{\rho\sigma\nu}K_{\rho\sigma}{}^{\mu}\,.
\end{equation}
Combining all terms and lowering indices, which commutes with all covariant derivatives since these are now metric-compatible, we can write the field equation as
\begin{equation}\label{eq:ftallfield}
\lc{\nabla}_{\rho}(f'S_{\mu\nu}{}^{\rho}) + f'S^{\rho\sigma}{}_{\nu}K_{\rho\sigma\mu} + \frac{1}{2}fg_{\mu\nu} = \kappa^2(\Theta_{\mu\nu} - \nabla_{\rho}H_{\mu\nu}{}^{\rho} + H_{\mu\nu}{}^{\rho}T^{\tau}{}_{\tau\rho})\,.
\end{equation}
Also this equation can be brought into various other forms by using the identities which hold for the contortion and the torsion.
\subsection{The general quadratic Lagrangians}\label{ssec:quadlag}
The GTEGR action~\eqref{eq:gtegraction} has the appealing property that the gravity scalar~\eqref{eq:gtegrbndterms}, unlike the Ricci scalar, is quadratic in first order derivatives of the dynamical fields, and hence more reminiscent of the kinetic energy of a gauge field. This invites for another class of modified teleparallel gravity theories, by considering an action which is an arbitrary linear combination of all possible scalars which can be obtained by contracting the product of the distortion tensor \(M^{\mu}{}_{\nu\rho}\) with itself. One easily checks that there are 11 possible terms: five terms arise from contracting \(M^{\mu}{}_{\nu\rho}\) with a second copy carrying the same indices in an arbitrary permutation, and six terms arising from contracting two arbitrary traces of the distortion tensor with each other, where in both cases terms which are distinguished only by the order of the factors are counted only once, since they are identical. This gives rise to the generalized gravity scalar~\cite{BeltranJimenez:2019odq}
\begin{equation}\label{eq:gngrscalar}
\begin{split}
\mathcal{G} &= M^{\mu\nu\rho}(k_1M_{\mu\nu\rho} + k_2M_{\nu\rho\mu} + k_3M_{\mu\rho\nu} + k_4M_{\rho\nu\mu} + k_5M_{\nu\mu\rho})\\
&\phantom{=}+ k_6M_{\rho\mu}{}^{\mu}M^{\rho\nu}{}_{\nu} + k_7M_{\mu\rho}{}^{\mu}M^{\nu\rho}{}_{\nu} + k_8M^{\mu}{}_{\mu\rho}M_{\nu}{}^{\nu\rho}\\
&\phantom{=}+ k_9M_{\mu\rho}{}^{\mu}M_{\nu}{}^{\nu\rho} + k_{10}M^{\mu}{}_{\mu\rho}M^{\rho\nu}{}_{\nu} + k_{11}M_{\rho\mu}{}^{\mu}M^{\nu\rho}{}_{\nu}
\end{split}
\end{equation}
with arbitrary constants \(k_1, \ldots, k_{11}\). Equivalently, one could also start from the expression~\eqref{eq:gtegrscalar}, and consider the most general scalar which is quadratic in the torsion and nonmetricity tensors. Again one finds 11 possible terms, so that their most general linear combination is of the form
\begin{equation}\label{eq:gngrtq}
\begin{split}
\mathcal{G} &= a_1T^{\mu\nu\rho}T_{\mu\nu\rho} + a_2T^{\mu\nu\rho}T_{\rho\nu\mu} + a_3T^{\mu}{}_{\mu\rho}T_{\nu}{}^{\nu\rho}\\
&\phantom{=}- b_1Q^{\mu\nu\rho}T_{\rho\nu\mu} - b_2Q^{\rho\mu}{}_{\mu}T^{\nu}{}_{\nu\rho} - b_3Q_{\mu}{}^{\mu\rho}T^{\nu}{}_{\nu\rho}\\
&\phantom{=}+ c_1Q^{\mu\nu\rho}Q_{\mu\nu\rho} + c_2Q^{\mu\nu\rho}Q_{\rho\mu\nu} + c_3Q^{\rho\mu}{}_{\mu}Q_{\rho\nu}{}^{\nu} + c_4Q^{\mu}{}_{\mu\rho}Q_{\nu}{}^{\nu\rho} + c_5Q^{\mu}{}_{\mu\rho}Q^{\rho\nu}{}_{\nu}\,,
\end{split}
\end{equation}
where we introduced the arbitrary constants \(a_1, \ldots, a_3, b_1, \ldots, b_3, c_1, \ldots, c_5\). Demanding that both expressions agree, one easily checks that these two sets of constants are related to each other by
\begin{gather}
k_1 = 2a_1 - b_1 + 2c_1\,, \quad
k_2 = -2a_2 + b_1 + 2c_2\,, \quad
k_9 = -2a_3 + 2b_2 - b_3 + 2c_5\,,\nonumber\\
k_4 = a_2 + c_2\,, \quad
k_5 = a_2 - b_1 + 2c_1\,, \quad
k_6 = c_4\,, \quad
k_7 = a_3 + b_3 + c_4\,,\\
k_8 = a_3 - 2b_2 + 4c_3\,, \quad
k_3 = -2a_1 + b_1 + c_2\,, \quad
k_{10} = -b_3 + 2c_5\,, \quad
k_{11} = b_3 + 2c_4\,.\nonumber
\end{gather}
Further, choosing the values of these constants to be
\begin{equation}\label{eq:gtegrvalues}
k_{11} = -k_2 = 1\,, \quad k_1 = k_3 = k_4 = k_5 = k_6 = k_7 = k_8 = k_9 = k_{10} = 0\,,
\end{equation}
one finds that the scalar \(\mathcal{G}\) reduces to \(G\). Hence, one may expect that the class of modified gravity theories defined by the action
\begin{equation}\label{eq:gngraction}
S_{\text{g}} = -\frac{1}{2\kappa^2}\int_M\mathcal{G}\sqrt{-g}\mathrm{d}^4x
\end{equation}
has a well-defined limit towards GTEGR, which is achieved if the constant parameters in the action take the aforementioned values. In order to derive the field equations, one can proceed in full analogy to the GTEGR field equations we discussed before. First, it is helpful to calculate the variation of the scalar~\eqref{eq:gngrscalar}, and write it in the form
\begin{equation}\label{eq:gngrvardef}
\delta\mathcal{G} = \mathcal{U}^{\mu\nu}\delta g_{\mu\nu} + \mathcal{V}^{\rho\mu\nu}\lc{\nabla}_{\rho}\delta g_{\mu\nu} + \mathcal{Z}_{\mu}{}^{\nu\rho}\delta\Gamma^{\mu}{}_{\nu\rho}\,.
\end{equation}
Here we have made use of the abbreviations
\begin{subequations}\label{eq:gngrvarabbrev}
\begin{multline}
\mathcal{U}^{\mu\nu} = k_1(M^{\mu\rho\sigma}M^{\nu}{}_{\rho\sigma} - M^{\rho\mu}{}_{\sigma}M_{\rho}{}^{\nu\sigma} - M_{\rho\sigma}{}^{\mu}M^{\rho\sigma\mu}) - k_2M_{\rho}{}^{\sigma(\mu}M_{\sigma}{}^{\nu)\rho}\\
+ k_3(M^{\mu\rho\sigma}M^{\nu}{}_{\sigma\rho} - 2M^{\rho\sigma(\mu}M_{\rho}{}^{\nu)\sigma}) - k_4M^{\rho\mu}{}_{\sigma}M^{\sigma\nu}{}_{\rho} - k_5M^{\rho\sigma\mu}M_{\sigma\rho}{}^{\nu}\\
+ k_6M^{\mu\rho}{}_{\rho}M^{\nu\sigma}{}_{\sigma} - k_7M^{\rho\mu}{}_{\rho}M^{\sigma\nu}{}_{\sigma} - k_8M_{\rho}{}^{\rho\mu}M_{\sigma}{}^{\sigma\nu} - k_9M_{\rho}{}^{\rho(\mu}M_{\sigma}{}^{\nu)\sigma}\\
- (2k_6M_{\rho\sigma}{}^{\sigma} + k_{11}M_{\sigma\rho}{}^{\sigma} + k_{10}M^{\sigma}{}_{\sigma\rho})M^{\rho(\mu\nu)}\,,
\end{multline}
as well as
\begin{multline}
\mathcal{V}^{\rho\mu\nu} = -2k_6g^{\rho(\mu}M^{\nu)\sigma}{}_{\sigma} - k_{11}g^{\rho(\mu}M_{\sigma}{}^{\nu)\sigma} - k_{10}M_{\sigma}{}^{\sigma(\mu}g^{\nu)\rho}\\
+ \frac{1}{2}g^{\mu\nu}\big[(2k_6 - k_{10} - k_{11})M^{\rho\sigma}{}_{\sigma} + (k_{11} - 2k_7 - k_9)M^{\sigma\rho}{}_{\sigma} + (k_{10} - 2k_8 - k_9)M_{\sigma}{}^{\sigma\rho}\big]\\
+ (k_4 - k_5 - k_1 - k_3)M^{(\mu\nu)\rho} + (k_5 - k_4 - k_1 - k_3)M^{(\mu|\rho|\nu)} + (k_1 - k_2 + k_3 - k_4 - k_5)M^{\rho(\mu\nu)}
\end{multline}
and
\begin{multline}
\mathcal{Z}_{\mu}{}^{\nu\rho} = 2k_1M_{\mu}{}^{\nu\rho} + k_2(M^{\nu\rho}{}_{\mu} + M^{\rho}{}_{\mu}{}^{\nu}) + 2k_3M_{\mu}{}^{\rho\nu} + 2k_4M^{\rho\nu}{}_{\mu} + 2k_5M^{\nu}{}_{\mu}{}^{\rho}\\
+ 2k_6M_{\mu\sigma}{}^{\sigma}g^{\nu\rho} + 2k_7M^{\sigma\nu}{}_{\sigma}\delta_{\mu}^{\rho} + 2k_8M_{\sigma}{}^{\sigma\rho}\delta_{\mu}^{\nu} + k_9(M_{\sigma}{}^{\rho\sigma}\delta_{\mu}^{\nu} + M_{\sigma}{}^{\sigma\nu}\delta_{\mu}^{\rho})\\
+ k_{10}(M^{\rho\sigma}{}_{\sigma}\delta_{\mu}^{\nu} + M^{\sigma}{}_{\sigma\mu}g^{\nu\rho}) + k_{11}(M^{\nu\sigma}{}_{\sigma}\delta_{\mu}^{\rho} + M^{\sigma}{}_{\mu\sigma}g^{\nu\rho})\,.
\end{multline}
\end{subequations}
By inserting the variation~\eqref{eq:gngrvardef} into the variation of the action~\eqref{eq:gngraction} and integration by parts, one obtains the form~\eqref{eq:metricgravactvar}, with
\begin{equation}\label{eq:gngrmetvar}
W_{\mu\nu} = \frac{1}{\kappa^2}\left(\mathcal{U}_{\mu\nu} - \lc{\nabla}_{\rho}\mathcal{V}^{\rho}{}_{\mu\nu} + \frac{1}{2}\mathcal{G}g_{\mu\nu}\right)
\end{equation}
and
\begin{equation}\label{eq:gngrconnvar}
Y_{\mu}{}^{\nu\rho} = \frac{1}{2\kappa^2}\mathcal{Z}_{\mu}{}^{\nu\rho}\,.
\end{equation}
Hence, by comparing with the corresponding GTEGR expressions~\eqref{eq:gtegrmetvar} and~\eqref{eq:gtegrconnvar}, we see that these have the same form, and one simply replaces the terms derived by variation of \(G\) with those obtained from \(\mathcal{G}\) in its place. One therefore finds the metric field equation
\begin{equation}\label{eq:gngrmetfield}
\mathcal{U}_{\mu\nu} - \lc{\nabla}_{\rho}(\mathcal{V}^{\rho}{}_{\mu\nu}) + \frac{1}{2}\mathcal{G}g_{\mu\nu} = \kappa^2\Theta_{\mu\nu}\,,
\end{equation}
as well as the connection field equation
\begin{equation}\label{eq:gngrconnfield}
\nabla_{\tau}\mathcal{Z}_{\mu}{}^{\nu\tau} - M^{\omega}{}_{\tau\omega}\mathcal{Z}_{\mu}{}^{\nu\tau} = 2\kappa^2(\nabla_{\tau}H_{\mu}{}^{\nu\tau} - M^{\omega}{}_{\tau\omega}H_{\mu}{}^{\nu\tau})\,,
\end{equation}
with the abbreviations~\eqref{eq:gngrvarabbrev}.
A more comprehensible set of field equations is obtained for the more restricted geometries, in which we impose either vanishing torsion or vanishing nonmetricity. This can most easily be seen from the expression~\eqref{eq:gngrtq}, which shows that numerous terms vanish identically in either of these two cases. We first consider the symmetric teleparallel case of vanishing torsion. In this case, \(\mathcal{G}\) reduces to the generalized nonmetricity scalar~\cite{BeltranJimenez:2017tkd}
\begin{equation}\label{eq:sngrscalar}
\begin{split}
\mathcal{Q} &= \frac{1}{2}Q_{\rho\mu\nu}\mathcal{P}^{\rho\mu\nu}\\
&= c_1Q^{\mu\nu\rho}Q_{\mu\nu\rho} + c_2Q^{\mu\nu\rho}Q_{\rho\mu\nu} + c_3Q^{\rho\mu}{}_{\mu}Q_{\rho\nu}{}^{\nu} + c_4Q^{\mu}{}_{\mu\rho}Q_{\nu}{}^{\nu\rho} + c_5Q^{\mu}{}_{\mu\rho}Q^{\rho\nu}{}_{\nu}\,,
\end{split}
\end{equation}
and only the five constant parameters \(c_1, \ldots, c_5\) remain present in the action. In place of the nonmetricity conjugate~\eqref{eq:nonmetconj} we now have the generalized expression
\begin{multline}\label{eq:ngrnonmetconj}
\mathcal{P}^{\rho\mu\nu} = 2c_1Q^{\rho\mu\nu} + 2c_2Q^{(\mu\nu)\rho} + 2c_3g^{\mu\nu}Q^{\rho\sigma}{}_{\sigma}\\
+ 2c_4Q_{\sigma}{}^{\sigma(\mu}g^{\nu)\rho} + c_5(g^{\mu\nu}Q_{\sigma}{}^{\sigma\rho} + g^{\rho(\mu}Q^{\nu)\sigma}{}_{\sigma})\,.
\end{multline}
For the corresponding class of gravity theories depending on these parameters, whose action reads
\begin{equation}\label{eq:sngraction}
S_{\text{g}} = -\frac{1}{2\kappa^2}\int_M\mathcal{Q}\sqrt{-g}\mathrm{d}^4x\,,
\end{equation}
the term ``Newer General Relativity'' has been coined. Its field equations can be obtained in great analogy to the other symmetric teleparallel gravity theories we have encountered before. First, we derive the connection field equation~\eqref{eq:symconnfield}, and use the fact that only the symmetric part \(Y_{\mu}{}^{(\nu\rho)}\) contributes. Using the variation~\eqref{eq:gngrconnvar}, we thus calculate
\begin{multline}
\mathcal{Z}_{\mu}{}^{(\nu\rho)} = -2c_2Q_{\mu}{}^{\nu\rho} - 2(2c_1 + c_2)Q^{(\nu\rho)}{}_{\mu} - 2g^{\nu\rho}(2c_4Q^{\sigma}{}_{\sigma\mu} + c_5Q_{\mu\sigma}{}^{\sigma})\\
- 4(c_4 + c_5)Q_{\sigma}{}^{\sigma(\nu}\delta^{\rho)}_{\mu} - 2(4c_3 + c_5)\delta^{(\nu}_{\mu}Q^{\rho)\sigma}{}_{\sigma} = -2\mathcal{P}^{(\nu\rho)}{}_{\mu}\,,
\end{multline}
which generalizes the similar relation~\eqref{eq:symconnvarrel}. Hence, we find that the connection field equation can be written in the simple form
\begin{equation}
-\nabla_{\nu}\nabla_{\rho}(\tilde{\mathcal{P}}^{\nu\rho}{}_{\mu}) = \kappa^2\nabla_{\nu}\nabla_{\rho}\tilde{H}_{\mu}{}^{\nu\rho}\,,
\end{equation}
using the tensor density \(\tilde{\mathcal{P}}^{\nu\rho}{}_{\mu}\) built from the generalized nonmetricity conjugate~\eqref{eq:ngrnonmetconj}. We then proceed with the metric equation, which still takes the general form~\eqref{eq:gngrmetfield} also in the symmetric teleparallel case, but can be simplified as follows. Raising one index and introducing a density factor, it can equivalently be written as
\begin{equation}
\mathcal{U}^{\mu}{}_{\nu} - \frac{\lc{\nabla}_{\rho}(\sqrt{-g}\mathcal{V}^{\rho\mu}{}_{\nu})}{\sqrt{-g}} + \frac{1}{2}\mathcal{G}\delta^{\mu}_{\nu} = \kappa^2\Theta^{\mu}{}_{\nu}\,.
\end{equation}
The covariant derivative with respect to the Levi-Civita connection can be transformed to the independent connection, by using the relation
\begin{multline}
\lc{\nabla}_{\rho}(\sqrt{-g}\mathcal{V}^{\rho\mu}{}_{\nu}) = \nabla_{\rho}(\sqrt{-g}\mathcal{V}^{\rho\mu}{}_{\nu})\\
+ \sqrt{-g}(L^{\sigma}{}_{\sigma\rho}\mathcal{V}^{\rho\mu}{}_{\nu} - L^{\rho}{}_{\sigma\rho}\mathcal{V}^{\sigma\mu}{}_{\nu} - L^{\mu}{}_{\sigma\rho}\mathcal{V}^{\rho\sigma}{}_{\nu} + L^{\sigma}{}_{\nu\rho}\mathcal{V}^{\rho\mu}{}_{\sigma})\,.
\end{multline}
To proceed further, we need the terms
\begin{multline}
\mathcal{U}^{\mu\nu} = (2c_1Q^{\rho\sigma\mu} + c_2Q^{\sigma\rho\mu})Q_{\rho\sigma}{}^{\nu} - (2c_1 + c_2)Q^{\rho\sigma(\mu}Q^{\nu)}{}_{\rho\sigma} - (c_1 + c_2)Q^{\mu\rho\sigma}Q^{\nu}{}_{\rho\sigma}\\
- c_4Q_{\rho}{}^{\rho(\mu}Q^{\nu)\sigma}{}_{\sigma} + 2c_4Q_{\rho}{}^{\rho\mu}Q_{\sigma}{}^{\sigma\nu} - \left(c_3 + \frac{c_5}{2}\right)Q^{\mu\rho}{}_{\rho}Q^{\nu\sigma}{}_{\sigma}\\
+ (2c_4Q^{\sigma}{}_{\sigma\rho} + c_5Q_{\rho\sigma}{}^{\sigma})\left(\frac{1}{2}Q^{\rho\mu\nu} - Q^{(\mu\nu)\rho}\right)
\end{multline}
and
\begin{multline}
\mathcal{V}^{\rho\mu\nu} = 2c_1Q^{\rho\mu\nu} + 2c_2Q^{(\mu\nu)\rho} + 2c_3g^{\mu\nu}Q^{\rho\sigma}{}_{\sigma}\\
+ 2c_4Q_{\sigma}{}^{\sigma(\mu}g^{\nu)\rho} + c_5(g^{\mu\nu}Q_{\sigma}{}^{\sigma\rho} + g^{\rho(\mu}Q^{\nu)\sigma}{}_{\sigma})\,,
\end{multline}
which are obtained from the more general expressions~\eqref{eq:gngrvarabbrev} by imposing vanishing torsion. A tedious, but straightforward calculation shows that the resulting terms can be combined to yield
\begin{multline}
\mathcal{U}^{\mu}{}_{\nu} - L^{\sigma}{}_{\sigma\rho}\mathcal{V}^{\rho\mu}{}_{\nu} + L^{\rho}{}_{\sigma\rho}\mathcal{V}^{\sigma\mu}{}_{\nu} + L^{\mu}{}_{\sigma\rho}\mathcal{V}^{\rho\sigma}{}_{\nu} - L^{\sigma}{}_{\nu\rho}\mathcal{V}^{\rho\mu}{}_{\sigma}\\
= -(c_1Q^{\mu\rho\sigma} + c_2Q^{\rho\sigma\mu})Q_{\nu\rho\sigma} - c_3Q^{\mu\rho}{}_{\rho}Q_{\nu\sigma}{}^{\sigma} - c_4Q^{\rho}{}_{\rho\sigma}Q_{\nu}{}^{\mu\sigma} - c_5Q_{(\nu}{}^{\mu\rho}Q_{\rho)\sigma}{}^{\sigma}\\
= -\frac{1}{2}\mathcal{P}^{\mu\rho\sigma}Q_{\nu\rho\sigma}\,.
\end{multline}
This finally yields the metric field equation
\begin{equation}
-\frac{\nabla_{\rho}(\sqrt{-g}\mathcal{P}^{\rho\mu}{}_{\nu})}{\sqrt{-g}} - \frac{1}{2}\mathcal{P}^{\mu\rho\sigma}Q_{\nu\rho\sigma} + \frac{1}{2}\mathcal{Q}\delta^{\mu}_{\nu} = \kappa^2\Theta^{\mu}{}_{\nu}
\end{equation}
for the Newer General Relativity class of gravity theories, where we now also used the relation \(\mathcal{G} = \mathcal{Q}\) in the absence of torsion. Note that a special case is obtained when the parameters take the values~\eqref{eq:gtegrvalues}, for which we have
\begin{equation}
c_1 = \frac{1}{4}\,, \quad
c_2 = -\frac{1}{2}\,, \quad
c_3 = -\frac{1}{4}\,, \quad
c_4 = 0\,, \quad
c_5 = \frac{1}{2}\,.
\end{equation}
In this case we find \(\mathcal{Q} = Q\) and \(\mathcal{P}^{\mu\nu\rho} = P^{\mu\nu\rho}\), so that the theory reduces to STEGR.
Finally, also in the metric teleparallel geometry we can find a general class of gravity theories, whose action is now quadratic in the torsion tensor. By imposing vanishing nonmetricity, the scalar~\eqref{eq:gngrscalar} becomes the generalized torsion scalar~\cite{Hayashi:1979qx}
\begin{equation}\label{eq:mngrscalar}
\begin{split}
\mathcal{T} &= \frac{1}{2}T^{\rho}{}_{\mu\nu}\mathcal{S}_{\rho}{}^{\mu\nu}\\
&= a_1T^{\mu\nu\rho}T_{\mu\nu\rho} + a_2T^{\mu\nu\rho}T_{\rho\nu\mu} + a_3T^{\mu}{}_{\mu\rho}T_{\nu}{}^{\nu\rho}\,,
\end{split}
\end{equation}
where the generalized superpotential is now given by
\begin{equation}\label{eq:ngrsuppot}
\mathcal{S}_{\rho}{}^{\mu\nu} = 2a_1T_{\rho}{}^{\mu\nu} + 2a_2T^{[\nu\mu]}{}_{\rho} + 2a_3T_{\sigma}{}^{\sigma[\nu}\delta^{\mu]}_{\rho}\,.
\end{equation}
The resulting class of gravity theories, which is now defined by the action
\begin{equation}\label{eq:mngraction}
S_{\text{g}} = -\frac{1}{2\kappa^2}\int_M\mathcal{T}\sqrt{-g}\mathrm{d}^4x\,,
\end{equation}
is known as ``New General Relativity''\footnote{This term is also, more commonly, used for a particular subclass of theories, in which \(2a_1 + a_2 = 0\) and \(a_3 = -1\), so that there is only one free parameter besides the gravitational constant \(\kappa\)~\cite{Hayashi:1979qx}.}. In this case, the left hand side of the field equations~\eqref{eq:metallfield} becomes
\begin{multline}
W^{\mu\nu} - \nabla_{\rho}Y^{\mu\nu\rho} + Y^{\mu\nu\rho}T^{\tau}{}_{\tau\rho}\\
= \frac{1}{\kappa^2}\left[\mathcal{U}^{\mu\nu} - \lc{\nabla}_{\rho}\mathcal{V}^{\rho\mu\nu} + \frac{1}{2}\mathcal{G}g^{\mu\nu} - \frac{1}{2}\nabla_{\rho}\mathcal{Z}^{\mu\nu\rho} + \frac{1}{2}\mathcal{Z}^{\mu\nu\rho}T^{\tau}{}_{\tau\rho}\right]\,.
\end{multline}
with the help of the formulas~\eqref{eq:gngrmetvar} and~\eqref{eq:gngrconnvar}. In order to combine the two derivative terms, we convert the covariant derivative \(\nabla_{\rho}\) with respect to the independent connection to a Levi-Civita covariant derivative \(\lc{\nabla}_{\rho}\), using the relation
\begin{equation}
\nabla_{\rho}\mathcal{Z}^{\mu\nu\rho} - \mathcal{Z}^{\mu\nu\rho}T^{\tau}{}_{\tau\rho} = \lc{\nabla}_{\rho}\mathcal{Z}^{\mu\nu\rho} + K^{\mu}{}_{\sigma\rho}\mathcal{Z}^{\sigma\nu\rho} + K^{\nu}{}_{\sigma\rho}\mathcal{Z}^{\mu\sigma\rho}\,.
\end{equation}
Now the two terms under the derivative combine into
\begin{equation}
\mathcal{V}^{\rho\mu\nu} + \frac{1}{2}\mathcal{Z}^{\mu\nu\rho} = -2a_1T^{\mu\nu\rho} - 2a_2T^{[\rho\nu]\mu} - 2a_3T_{\sigma}{}^{\sigma[\rho}g^{\nu]\mu} = -\mathcal{S}^{\mu\nu\rho}\,.
\end{equation}
The remaining terms take, once again, a very simple form, which is given by
\begin{multline}
\mathcal{U}^{\mu\nu} - \frac{1}{2}(K^{\mu}{}_{\sigma\rho}\mathcal{Z}^{\sigma\nu\rho} + K^{\nu}{}_{\sigma\rho}\mathcal{Z}^{\mu\sigma\rho})\\
= [(a_2 - 2a_1)K^{\rho\sigma\nu} + (3a_2 - 2a_1)K^{\nu\rho\sigma}]K_{\rho\sigma}{}^{\mu} + a_3K_{\rho\sigma}{}^{\sigma}K^{\nu\rho\mu} = \mathcal{S}^{\rho\sigma\nu}K_{\rho\sigma}{}^{\mu}\,.
\end{multline}
Hence, the full field equations of New General Relativity become
\begin{equation}
\lc{\nabla}_{\rho}(\mathcal{S}_{\mu\nu}{}^{\rho}) + \mathcal{S}^{\rho\sigma}{}_{\nu}K_{\rho\sigma\mu} + \frac{1}{2}\mathcal{G}g_{\mu\nu} = \kappa^2(\Theta_{\mu\nu} - \nabla_{\rho}H_{\mu\nu}{}^{\rho} + H_{\mu\nu}{}^{\rho}T^{\tau}{}_{\tau\rho})\,.
\end{equation}
Also for this class of theories a special case is obtained by choosing the parameter values~\eqref{eq:gtegrvalues}, which now implies
\begin{equation}
a_1 = \frac{1}{4}\,, \quad
a_2 = \frac{1}{2}\,, \quad
a_3 = -1\,.
\end{equation}
In this case, the theory reduces to MTEGR, with \(\mathcal{T} = T\) and \(\mathcal{S}_{\rho}{}^{\mu\nu} = S_{\rho}{}^{\mu\nu}\).
\subsection{Scalar-teleparallel theories}\label{ssec:scalartele}
While the classes of modified teleparallel gravity theories we considered so far were constructed purely from the metric and the flat affine connection as fundamental fields, we now consider a class of theories in which in addition a scalar field is introduced as a fundamental field variable. Also this class of theories can be motivated by analogy with a scalar-tensor modification of the Einstein-Hilbert action~\eqref{eq:einsteinhilbert} of general relativity, which takes the general form
\begin{equation}\label{eq:stcaction}
S_{\text{g}} = \frac{1}{2\kappa^2}\int_M\left[\mathcal{A}(\phi)\lc{R} - \mathcal{B}(\phi)g^{\mu\nu}\lc{\nabla}_{\mu}\phi\lc{\nabla}_{\nu}\phi - 2\kappa^2\mathcal{V}(\phi)\right]\sqrt{-g}\mathrm{d}^4x\,,
\end{equation}
where \(\mathcal{A}, \mathcal{B}, \mathcal{V}\) are free functions of the scalar field \(\phi\). Here we work in the so-called Jordan frame, which means that we assume no direct coupling between the scalar field and any matter fields. Recalling that the Ricci scalar \(\lc{R}\) can be written in the form~\eqref{eq:riccisplit}, one may expect that replacing \(\lc{R}\) by \(-G + B\) one obtains a teleparallel equivalent of the scalar-curvature theory, while using only \(-G\) instead leads to an inequivalent scalar-teleparallel theory, since the omitted term is not a boundary term due to the non-minimal coupling term \(\mathcal{A}(\phi)\). One can cover both cases by considering the action
\begin{equation}
S_{\text{g}} = \frac{1}{2\kappa^2}\int_M\left[-\mathcal{A}(\phi)G - \mathcal{B}(\phi)g^{\mu\nu}\lc{\nabla}_{\mu}\phi\lc{\nabla}_{\nu}\phi - \hat{\mathcal{C}}(\phi)B - 2\kappa^2\mathcal{V}(\phi)\right]\sqrt{-g}\mathrm{d}^4x\,,
\end{equation}
where we introduced another free function \(\hat{\mathcal{C}}\) of the scalar field. Keeping in mind that \(B\) is a boundary term, i.e., a total divergence, we see that the field equations do not change if we add an arbitrary constant to \(\hat{\mathcal{C}}\). To resolve this ambiguity, we can use integration by parts,
\begin{equation}
2\hat{\mathcal{C}}\lc{\nabla}_{\mu}M^{[\nu\mu]}{}_{\nu} = \lc{\nabla}_{\mu}(\hat{\mathcal{C}}M^{[\nu\mu]}{}_{\nu}) - \hat{\mathcal{C}}'M^{[\nu\mu]}{}_{\nu}\lc{\nabla}_{\mu}\phi\,,
\end{equation}
and omit the boundary term. Defining a new parameter function \(\mathcal{C} = \hat{\mathcal{C}}'\), we then have
\begin{multline}\label{eq:stgaction}
S_{\text{g}} = \frac{1}{2\kappa^2}\int_M\Big[-\mathcal{A}(\phi)G - \mathcal{B}(\phi)g^{\mu\nu}\lc{\nabla}_{\mu}\phi\lc{\nabla}_{\nu}\phi\\
+ 2\mathcal{C}(\phi)M^{[\nu\mu]}{}_{\nu}\lc{\nabla}_{\mu}\phi - 2\kappa^2\mathcal{V}(\phi)\Big]\sqrt{-g}\mathrm{d}^4x\,.
\end{multline}
Note that for \(\mathcal{A}' + \mathcal{C} = 0\), the action becomes equivalent to the scalar-curvature action~\eqref{eq:stcaction}. To derive the field equations for this generalized class of theories, we proceed by varying the action as with the previous examples. Due to the presence of an additional fundamental field, also the variation~\eqref{eq:metricgravactvar} is enhanced by an additional term, and becomes
\begin{equation}
\delta S_{\text{g}} = -\int_M\left(\frac{1}{2}W^{\mu\nu}\delta g_{\mu\nu} + Y_{\mu}{}^{\nu\rho}\delta\Gamma^{\mu}{}_{\nu\rho} + \Phi\delta\phi\right)\sqrt{-g}\mathrm{d}^4x\,,
\end{equation}
after eliminating derivatives of the variations using integration by parts. Varying the action~\eqref{eq:stgaction}, we find the terms
\begin{subequations}\label{eq:stgvar}
\begin{align}
W_{\mu\nu} &= \frac{1}{\kappa^2}\bigg\{\mathcal{A}\lc{R}_{\mu\nu} - \frac{\mathcal{A}}{2}\lc{R}g_{\mu\nu} + \mathcal{C}\lc{\nabla}_{\mu}\lc{\nabla}_{\nu}\phi - (\mathcal{B} - \mathcal{C}')\lc{\nabla}_{\mu}\phi\lc{\nabla}_{\nu}\phi\nonumber\\
&\phantom{=}+ (\mathcal{A}' + \mathcal{C})\left(\lc{\nabla}_{(\mu}\phi M^{\rho}{}_{\nu)\rho} - M^{\rho}{}_{(\mu\nu)}\lc{\nabla}_{\rho}\phi + M^{[\rho\sigma]}{}_{\sigma}\lc{\nabla}_{\rho}\phi g_{\mu\nu}\right)\nonumber\\
&\phantom{=}+ \left[\left(\frac{\mathcal{B}}{2} - \mathcal{C}'\right)\lc{\nabla}_{\rho}\phi\lc{\nabla}^{\rho}\phi - \mathcal{C}\lc{\nabla}_{\rho}\lc{\nabla}^{\rho}\phi + \kappa^2\mathcal{V}\right]g_{\mu\nu}\bigg\}\,,\\
Y_{\mu}{}^{\nu\rho} &= \frac{1}{2\kappa^2}\left[\mathcal{A}(g^{\nu\rho}M^{\sigma}{}_{\mu\sigma} + \delta_{\mu}^{\rho}M^{\nu\sigma}{}_{\sigma} - M^{\nu\rho}{}_{\mu} - M^{\rho}{}_{\nu}{}^{\mu}) + \mathcal{C}(g^{\nu\rho}\lc{\nabla}_{\mu}\phi - \delta_{\mu}^{\rho}\lc{\nabla}^{\nu}\phi)\right]\,,\\
\Phi &= \frac{1}{2\kappa^2}\left[-2\mathcal{B}\lc{\nabla}_{\mu}\lc{\nabla}^{\mu}\phi - \mathcal{B}'\lc{\nabla}_{\mu}\phi\lc{\nabla}^{\mu}\phi + \mathcal{C}B + \mathcal{A}'G\right] + \mathcal{V}\,,
\end{align}
\end{subequations}
where we have made use of the relations~\eqref{eq:curvdec} and~\eqref{eq:gtegrbndterms}, and from now on we omit the argument \(\phi\) of the parameter functions for brevity. We can then read off the field equations and study their properties. We start with the metric field equation~\eqref{eq:genmetfield}, which reads
\begin{multline}\label{eq:stgmetfield}
\mathcal{A}\lc{R}_{\mu\nu} - \frac{\mathcal{A}}{2}\lc{R}g_{\mu\nu} + \mathcal{C}\lc{\nabla}_{\mu}\lc{\nabla}_{\nu}\phi + (\mathcal{A}' + \mathcal{C})\left(\lc{\nabla}_{(\mu}\phi M^{\rho}{}_{\nu)\rho} - M^{\rho}{}_{(\mu\nu)}\lc{\nabla}_{\rho}\phi + M^{[\rho\sigma]}{}_{\sigma}\lc{\nabla}_{\rho}\phi g_{\mu\nu}\right)\\
- (\mathcal{B} - \mathcal{C}')\lc{\nabla}_{\mu}\phi\lc{\nabla}_{\nu}\phi + \left[\left(\frac{\mathcal{B}}{2} - \mathcal{C}'\right)\lc{\nabla}_{\rho}\phi\lc{\nabla}^{\rho}\phi - \mathcal{C}\lc{\nabla}_{\rho}\lc{\nabla}^{\rho}\phi + \kappa^2\mathcal{V}\right]g_{\mu\nu} = \kappa^2\Theta_{\mu\nu}\,.
\end{multline}
It is most remarkable that in the case \(\mathcal{A}' + \mathcal{C} = 0\) the only term containing the flat, affine connection vanishes from these field equations, and one finds that they indeed resemble the field equations of scalar-curvature gravity in this case. To check whether this property holds also for the connection field equation~\eqref{eq:genconnfield}, we calculate
\begin{equation}
\nabla_{\tau}Y_{\mu}{}^{\nu\tau} - M^{\omega}{}_{\tau\omega}Y_{\mu}{}^{\nu\tau} = \frac{\mathcal{A}' + \mathcal{C}}{2\kappa^2}\left[M^{\nu\rho}{}_{\rho}\lc{\nabla}_{\mu}\phi + M^{\rho}{}_{\mu\rho}\lc{\nabla}^{\nu}\phi - (M^{\nu\rho}{}_{\mu} + M^{\rho}{}_{\mu}{}^{\nu})\lc{\nabla}_{\rho}\phi\right]\,,
\end{equation}
where any terms involving the covariant derivative of the distortion \(M^{\mu}{}_{\nu\rho}\) cancel as a consequence of the flatness of the connection. We see that this expression becomes trivial for \(\mathcal{A}' + \mathcal{C} = 0\). In that case, the connection field equation
\begin{multline}\label{eq:stgconnfield}
(\mathcal{A}' + \mathcal{C})\left[M^{\nu\rho}{}_{\rho}\lc{\nabla}_{\mu}\phi + M^{\rho}{}_{\mu\rho}\lc{\nabla}^{\nu}\phi - (M^{\nu\rho}{}_{\mu} + M^{\rho}{}_{\mu}{}^{\nu})\lc{\nabla}_{\rho}\phi\right]\\
= 2\kappa^2(\nabla_{\tau}H_{\mu}{}^{\nu\tau} - M^{\omega}{}_{\tau\omega}H_{\mu}{}^{\nu\tau})
\end{multline}
becomes a constraint for the hypermomentum. Finally, we study the scalar field equation
\begin{equation}
-2\mathcal{B}\lc{\nabla}_{\mu}\lc{\nabla}^{\mu}\phi - \mathcal{B}'\lc{\nabla}_{\mu}\phi\lc{\nabla}^{\mu}\phi + \mathcal{C}B + \mathcal{A}'G + 2\kappa^2\mathcal{V}' = 0\,.
\end{equation}
Here the right hand side vanishes, since we do not consider any direct coupling between the scalar field and matter. Note that if \(\mathcal{C} = -\mathcal{A}'\), i.e., in the case of the scalar-curvature equivalent, the two terms \(\mathcal{C}B + \mathcal{A}'G\) combine to \(-\mathcal{A}'\lc{R}\), and the equation becomes independent of the teleparallel connection, as one would expect, and as we have seen for the remaining field equations. Further, one finds that the scalar field equation contains second order derivatives of both the scalar field and the metric, where the latter enter through the boundary term. In order to eliminate these metric derivatives from the equation, it is common to apply a ``debraiding'' procedure by adding a suitable multiple of the trace of the matter field equation. The latter reads
\begin{multline}
-\mathcal{A}\lc{R} - 3\mathcal{C}\lc{\nabla}_{\mu}\lc{\nabla}^{\mu}\phi + 2(\mathcal{A}' + \mathcal{C})M^{[\mu\nu]}{}_{\nu}\lc{\nabla}_{\mu}\phi\\
+ (\mathcal{B} - 3\mathcal{C}')\lc{\nabla}_{\mu}\phi\lc{\nabla}^{\mu}\phi + 4\kappa^2\mathcal{V} = \kappa^2\Theta_{\mu}{}^{\mu}\,.
\end{multline}
Hence, calculating the linear combination
\begin{multline}
\mathcal{C}W_{\mu}{}^{\mu} + 2\mathcal{A}\Phi = \frac{1}{\kappa^2}\bigg[-(2\mathcal{A}\mathcal{B} + 3\mathcal{C}^2)\lc{\nabla}_{\mu}\lc{\nabla}^{\mu}\phi + (\mathcal{B}\mathcal{C} - 3\mathcal{C}\mathcal{C}' - \mathcal{A}\mathcal{B}')\lc{\nabla}_{\mu}\phi\lc{\nabla}^{\mu}\phi\\
+ (\mathcal{A}' + \mathcal{C})\left(\mathcal{A}G + 2\mathcal{C}M^{[\mu\nu]}{}_{\nu}\lc{\nabla}_{\mu}\phi\right)\bigg] + 2\mathcal{A}\mathcal{V}' + 4\mathcal{C}\mathcal{V}\,,
\end{multline}
we find that the debraided scalar field equation
\begin{multline}\label{eq:genscaldebfield}
-(2\mathcal{A}\mathcal{B} + 3\mathcal{C}^2)\lc{\nabla}_{\mu}\lc{\nabla}^{\mu}\phi + (\mathcal{B}\mathcal{C} - 3\mathcal{C}\mathcal{C}' - \mathcal{A}\mathcal{B}')\lc{\nabla}_{\mu}\phi\lc{\nabla}^{\mu}\phi\\
+ (\mathcal{A}' + \mathcal{C})\left(\mathcal{A}G + 2\mathcal{C}M^{[\mu\nu]}{}_{\nu}\lc{\nabla}_{\mu}\phi\right) + 2\kappa^2(\mathcal{A}\mathcal{V}' + 2\mathcal{C}\mathcal{V}) = \kappa^2\mathcal{C}\Theta_{\mu}{}^{\mu}
\end{multline}
does not contain any derivatives of the independent connection, and has only first order derivatives of the metric tensor, which enter through the distortion and the Christoffel symbols contained in the covariant derivative. Also here we see that the teleparallel connection does not contribute to the field equation for \(\mathcal{A}' + \mathcal{C} = 0\). Further, one finds that the trace of the energy-momentum tensor acts as the matter source for the scalar field.
It is now easy to study how the field equations change if we consider the symmetric or metric teleparallel geometries instead of the general teleparallel geometry we have used to construct the scalar-teleparallel gravity theory discussed above. We start with the former, which yields a class of scalar-nonmetricity theories of gravity, whose action is given by~\cite{Jarv:2018bgs,Runkla:2018xrv}
\begin{multline}\label{eq:stqaction}
S_{\text{g}} = \frac{1}{2\kappa^2}\int_M\Big[-\mathcal{A}(\phi)Q - \mathcal{B}(\phi)g^{\mu\nu}\lc{\nabla}_{\mu}\phi\lc{\nabla}_{\nu}\phi\\
+ \mathcal{C}(\phi)(Q_{\nu}{}^{\nu\mu} - Q^{\mu\nu}{}_{\nu})\lc{\nabla}_{\mu}\phi - 2\kappa^2\mathcal{V}(\phi)\Big]\sqrt{-g}\mathrm{d}^4x\,.
\end{multline}
For the metric field equations, which retain the general form~\eqref{eq:genmetfield}, we see that the only change compared to the general teleparallel case arises from those terms which involve the teleparallel affine connection. These terms greatly simplify and become
\begin{equation}
\lc{\nabla}_{(\mu}\phi M^{\rho}{}_{\nu)\rho} - M^{\rho}{}_{(\mu\nu)}\lc{\nabla}_{\rho}\phi + M^{[\rho\sigma]}{}_{\sigma}\lc{\nabla}_{\rho}\phi g_{\mu\nu} = -P^{\rho}{}_{\mu\nu}\lc{\nabla}_{\rho}\phi\,,
\end{equation}
using the nonmetricity conjugate~\eqref{eq:nonmetconj}. The metric field equations therefore read
\begin{multline}\label{eq:stqmetfield}
\mathcal{A}\lc{R}_{\mu\nu} - \frac{\mathcal{A}}{2}\lc{R}g_{\mu\nu} + \mathcal{C}\lc{\nabla}_{\mu}\lc{\nabla}_{\nu}\phi - (\mathcal{B} - \mathcal{C}')\lc{\nabla}_{\mu}\phi\lc{\nabla}_{\nu}\phi - (\mathcal{A}' + \mathcal{C})P^{\rho}{}_{\mu\nu}\lc{\nabla}_{\rho}\phi\\
+ \left[\left(\frac{\mathcal{B}}{2} - \mathcal{C}'\right)\lc{\nabla}_{\rho}\phi\lc{\nabla}^{\rho}\phi - \mathcal{C}\lc{\nabla}_{\rho}\lc{\nabla}^{\rho}\phi + \kappa^2\mathcal{V}\right]g_{\mu\nu} = \kappa^2\Theta_{\mu\nu}\,.
\end{multline}
We then continue with the connection equation, which now takes the form~\eqref{eq:symconnfield}. Here we can make use of several simplifications we have employed before. First, using the variation~\eqref{eq:gvarabbrev} of the gravity scalar \(G\), we write the variation~\eqref{eq:stgvar} as
\begin{equation}
\begin{split}
Y_{\mu}{}^{\nu\rho} &= \frac{1}{2\kappa^2}\left[\mathcal{A}Z_{\mu}{}^{\nu\rho} + \mathcal{C}\left(g^{\nu\rho}\lc{\nabla}_{\mu}\phi - \delta_{\mu}^{\rho}\lc{\nabla}^{\nu}\phi\right)\right]\\
&= \frac{1}{2\kappa^2}\left[\mathcal{A}Z_{\mu}{}^{\nu\rho} + \mathcal{C}\left(g^{\nu\rho}\nabla_{\mu}\phi - \delta_{\mu}^{\rho}g^{\nu\sigma}\nabla_{\sigma}\phi\right)\right]\,,
\end{split}
\end{equation}
where we used the fact that any covariant derivative acts equally on the scalar field \(\phi\). Next, we introduce a density factor \(\sqrt{-g}\) and take a covariant derivative, to calculate
\begin{equation}
\begin{split}
\nabla_{\rho}\tilde{Y}_{\mu}{}^{\nu\rho} &= \frac{1}{2\kappa^2}\nabla_{\rho}\left[\mathcal{A}\tilde{Z}_{\mu}{}^{\nu\rho} + \sqrt{-g}\mathcal{C}\left(g^{\nu\rho}\nabla_{\mu}\phi - \delta_{\mu}^{\rho}g^{\nu\sigma}\nabla_{\sigma}\phi\right)\right]\\
&= \frac{\sqrt{-g}}{2\kappa^2}\bigg[\mathcal{A}'Z_{\mu}{}^{\nu\rho}\nabla_{\rho}\phi + \left(\frac{1}{2}Q_{\rho\tau}{}^{\tau}\mathcal{C} + \mathcal{C}'\nabla_{\rho}\phi\right)\left(g^{\nu\rho}\nabla_{\mu}\phi - \delta_{\mu}^{\rho}g^{\nu\sigma}\nabla_{\sigma}\phi\right)\\
&\phantom{=}+ \mathcal{C}\left(g^{\nu\rho}\nabla_{\rho}\nabla_{\mu}\phi - g^{\nu\sigma}\nabla_{\mu}\nabla_{\sigma}\phi - Q_{\rho}{}^{\rho\nu}\nabla_{\mu}\phi + Q_{\mu}{}^{\nu\sigma}\nabla_{\sigma}\phi\right)\bigg]\\
&= \frac{1}{2\kappa^2}(\mathcal{A}' + \mathcal{C})\tilde{Z}_{\mu}{}^{\nu\rho}\nabla_{\rho}\phi\\
&= \frac{1}{2\kappa^2}\nabla_{\rho}[(\mathcal{A} + \hat{\mathcal{C}})\tilde{Z}_{\mu}{}^{\nu\rho}]\,,
\end{split}
\end{equation}
where we used the identity \(\nabla_{\rho}\tilde{Z}_{\mu}{}^{\nu\rho} = 0\) we found in deriving the STEGR field equations, and the fact that numerous terms involving the scalar field cancel, while the remaining terms combine to a very compact form. Here \(\hat{\mathcal{C}}\) is defined by \(\hat{\mathcal{C}}' = \mathcal{C}\) only up to an irrelevant constant. To obtain the connection field equations, we apply another covariant derivative, and use the relation~\eqref{eq:symconnvarrel} to finally obtain
\begin{equation}
\nabla_{\nu}\nabla_{\rho}\tilde{Y}_{\mu}{}^{\nu\rho} = \frac{1}{2\kappa^2}\nabla_{\nu}\nabla_{\rho}[(\mathcal{A} + \hat{\mathcal{C}})\tilde{Z}_{\mu}{}^{\nu\rho}] = -\frac{1}{\kappa^2}\nabla_{\nu}\nabla_{\rho}[(\mathcal{A} + \hat{\mathcal{C}})\tilde{P}^{\nu\rho}{}_{\mu}]\,.
\end{equation}
Hence, we see that the left hand side of the connection field equations
\begin{equation}\label{eq:stqconnfield}
-\nabla_{\nu}\nabla_{\rho}[(\mathcal{A} + \hat{\mathcal{C}})\tilde{P}^{\nu\rho}{}_{\mu}] = \kappa^2\nabla_{\nu}\nabla_{\rho}\tilde{H}_{\mu}{}^{\nu\rho}
\end{equation}
vanishes identically for \(\mathcal{A}' + \mathcal{C} = 0\). At last, we come to the scalar field equation, which we consider in its debraided form~\eqref{eq:genscaldebfield}. Imposing vanishing torsion, the only affected term is given by
\begin{equation}
\mathcal{A}G + 2\mathcal{C}M^{[\mu\nu]}{}_{\nu}\lc{\nabla}_{\mu}\phi = \mathcal{A}Q + 2\mathcal{C}Q^{[\mu\nu]}{}_{\nu}\lc{\nabla}_{\mu}\phi\,,
\end{equation}
and so the scalar field equation undergoes the trivial change to become
\begin{multline}
-(2\mathcal{A}\mathcal{B} + 3\mathcal{C}^2)\lc{\nabla}_{\mu}\lc{\nabla}^{\mu}\phi + (\mathcal{B}\mathcal{C} - 3\mathcal{C}\mathcal{C}' - \mathcal{A}\mathcal{B}')\lc{\nabla}_{\mu}\phi\lc{\nabla}^{\mu}\phi\\
+ (\mathcal{A}' + \mathcal{C})\left(\mathcal{A}Q + 2\mathcal{C}Q^{[\mu\nu]}{}_{\nu}\lc{\nabla}_{\mu}\phi\right) + 2\kappa^2(\mathcal{A}\mathcal{V}' + 2\mathcal{C}\mathcal{V}) = \kappa^2\mathcal{C}\Theta_{\mu}{}^{\mu}\,.
\end{multline}
This completes the field equations for the scalar-nonmetricity class of gravity theories.
We finally also take a brief look at the metric teleparallel case, and study the field equations of a class of scalar-torsion theories defined by the action~\cite{Geng:2011aj,Hohmann:2018rwf,Hohmann:2018ijr}
\begin{multline}\label{eq:sttaction}
S_{\text{g}} = \frac{1}{2\kappa^2}\int_M\Big[-\mathcal{A}(\phi)T - \mathcal{B}(\phi)g^{\mu\nu}\lc{\nabla}_{\mu}\phi\lc{\nabla}_{\nu}\phi\\
+ 2\mathcal{C}(\phi)T_{\nu}{}^{\nu\mu}\lc{\nabla}_{\mu}\phi - 2\kappa^2\mathcal{V}(\phi)\Big]\sqrt{-g}\mathrm{d}^4x\,,
\end{multline}
which directly follows from the action~\eqref{eq:stgaction} by imposing vanishing nonmetricity. Recall that under this condition the single field equation obtained by simultaneous variation of the metric and connection is given by~\eqref{eq:metallfield}. Using the variation~\eqref{eq:stgvar}, these field equations become
\begin{multline}\label{eq:sttallfield}
\mathcal{A}\lc{R}_{\mu\nu} - \frac{\mathcal{A}}{2}\lc{R}g_{\mu\nu} + \mathcal{C}\lc{\nabla}_{\mu}\lc{\nabla}_{\nu}\phi - (\mathcal{B} - \mathcal{C}')\lc{\nabla}_{\mu}\phi\lc{\nabla}_{\nu}\phi + (\mathcal{A}' + \mathcal{C})S_{\mu\nu}{}^{\rho}\lc{\nabla}_{\rho}\phi\\
+ \left[\left(\frac{\mathcal{B}}{2} - \mathcal{C}'\right)\lc{\nabla}_{\rho}\phi\lc{\nabla}^{\rho}\phi - \mathcal{C}\lc{\nabla}_{\rho}\lc{\nabla}^{\rho}\phi + \kappa^2\mathcal{V}\right]g_{\mu\nu} = \kappa^2(\Theta^{\mu\nu} - \nabla_{\rho}H^{\mu\nu\rho} + H^{\mu\nu\rho}T^{\tau}{}_{\tau\rho})\,.
\end{multline}
These equations are supplemented by the scalar field equation, which follows from the general teleparallel equation~\eqref{eq:genscaldebfield} by using
\begin{equation}
\mathcal{A}G + 2\mathcal{C}M^{[\mu\nu]}{}_{\nu}\lc{\nabla}_{\mu}\phi = \mathcal{A}T - 2\mathcal{C}T_{\nu}{}^{\nu\mu}\lc{\nabla}_{\mu}\phi\,,
\end{equation}
in the absence of nonmetricity. Hence, the (debraided) scalar field equation takes the form
\begin{multline}
-(2\mathcal{A}\mathcal{B} + 3\mathcal{C}^2)\lc{\nabla}_{\mu}\lc{\nabla}^{\mu}\phi + (\mathcal{B}\mathcal{C} - 3\mathcal{C}\mathcal{C}' - \mathcal{A}\mathcal{B}')\lc{\nabla}_{\mu}\phi\lc{\nabla}^{\mu}\phi\\
+ (\mathcal{A}' + \mathcal{C})\left(\mathcal{A}T - 2\mathcal{C}T_{\nu}{}^{\nu\mu}\lc{\nabla}_{\mu}\phi\right) + 2\kappa^2(\mathcal{A}\mathcal{V}' + 2\mathcal{C}\mathcal{V}) = \kappa^2\mathcal{C}\Theta_{\mu}{}^{\mu}
\end{multline}
in the metric teleparallel gravity setting.
\subsection{Scalar-teleparallel representation of $f(G)$ theories}\label{ssec:stfg}
Among the general classes of scalar-teleparallel theories of gravity discussed in the previous section there is a particular subclass of theories, defined by a suitable choice of the parameter functions \(\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{V}\), whose field equations turn out to be equivalent to those of the \(f(G)\) class of theories. Note that for a given function \(f\), the choice of the parameter functions in the scalar-teleparallel representation is not unique, and different choices are connected by redefinitions of the scalar field. For the general teleparallel geometry, a straightforward procedure is to start from the action~\eqref{eq:fgaction}, and to rewrite it, similarly to the \(f(\lc{R})\) class of theories~\cite{Sotiriou:2008rp}, in the form
\begin{equation}\label{eq:stge2action}
S_{\text{g}} = -\frac{1}{2\kappa^2}\int_M[f(\phi) - \psi(\phi - G)]\sqrt{-g}\mathrm{d}^4x\,,
\end{equation}
thereby introducing two scalar fields \(\psi\) and \(\phi\). Here \(\psi\) is a Lagrange multiplier, and imposes the constraint
\begin{equation}\label{eq:stgcons1}
\phi = G
\end{equation}
for the scalar field \(\phi\). Variation with respect to the latter yields another constraint
\begin{equation}\label{eq:stgcons2}
\psi = f'(\phi)\,,
\end{equation}
which can then be used to solve for the scalar field \(\psi\). The remaining field equations are the metric field equation
\begin{multline}
\psi\left(\lc{R}_{\mu\nu} - \frac{1}{2}\lc{R}g_{\mu\nu}\right) + \left(\lc{\nabla}_{(\mu}\psi M^{\rho}{}_{\nu)\rho} - M^{\rho}{}_{(\mu\nu)}\lc{\nabla}_{\rho}\psi + M^{[\rho\sigma]}{}_{\sigma}\lc{\nabla}_{\rho}\psi g_{\mu\nu}\right)\\
+ \frac{1}{2}[f(\phi) - \phi\psi]g_{\mu\nu} = \kappa^2\Theta_{\mu\nu}\,,
\end{multline}
as well as the connection field equation
\begin{equation}
M^{\nu\rho}{}_{\rho}\lc{\nabla}_{\mu}\psi + M^{\rho}{}_{\mu\rho}\lc{\nabla}^{\nu}\psi - (M^{\nu\rho}{}_{\mu} + M^{\rho}{}_{\mu}{}^{\nu})\lc{\nabla}_{\rho}\psi = 2\kappa^2(\nabla_{\tau}H_{\mu}{}^{\nu\tau} - M^{\omega}{}_{\tau\omega}H_{\mu}{}^{\nu\tau})\,.
\end{equation}
Together with the constraints~\eqref{eq:stgcons1} and~\eqref{eq:stgcons2}, one finds that these reproduce the $f(G)$ field equations~\eqref{eq:fgmetfield2} and~\eqref{eq:fgconnfield2}.
Instead of keeping two scalar fields, one can take one further step and substitute the constraint~\eqref{eq:stgcons2} in the action~\eqref{eq:stge2action}, which then becomes
\begin{equation}\label{eq:stgeaction}
S_{\text{g}} = -\frac{1}{2\kappa^2}\int_M[f(\phi) - f'(\phi)(\phi - G)]\sqrt{-g}\mathrm{d}^4x\,.
\end{equation}
Note that this does not change the metric and connection field equations. Variation with respect to the scalar field now yields the field equation
\begin{equation}
(G - \phi)f'' = 0\,,
\end{equation}
which resembles the constraint~\eqref{eq:stgcons1} for \(f'' \neq 0\). By comparison with the general scalar-teleparallel action~\eqref{eq:stgaction}, one reads off the relations
\begin{equation}\label{eq:stgeparfun}
\mathcal{A}(\phi) = f'(\phi)\,, \quad
\mathcal{B}(\phi) = 0\,, \quad
\mathcal{C}(\phi) = 0\,, \quad
\mathcal{V}(\phi) = \frac{f(\phi) - \phi f'(\phi)}{2\kappa^2}\,.
\end{equation}
Alternatively, if the constraint~\eqref{eq:stgcons2} is invertible, one may also solve it for \(\phi\) instead, which yields a different parametrization. The resulting action then takes the form
\begin{equation}\label{eq:stgeaction2}
S_{\text{g}} = -\frac{1}{2\kappa^2}\int_M[\psi G - 2\kappa^2\mathcal{U}(\psi)]\sqrt{-g}\mathrm{d}^4x\,,
\end{equation}
where \(\mathcal{U}\) is implicitly defined by
\begin{equation}
\mathcal{U}(\psi) = \mathcal{V}(\phi)\,.
\end{equation}
To obtain a more explicit relation, one may differentiate with respect to \(\phi\) on both sides, which yields
\begin{equation}
f''(\phi)\mathcal{U}'(\psi) = -\frac{\phi f''(\phi)}{2\kappa^2}\,.
\end{equation}
This shows that \(f(\phi)\) and \(\mathcal{U}(\psi)\) are related by a Legendre transformation. In this case the scalar field equation becomes
\begin{equation}
G = -2\kappa^2\mathcal{U}'(\psi)\,,
\end{equation}
once again reproducing the constraint~\eqref{eq:stgcons1}, up to a change of parametrization.
It is easy to check that for the values~\eqref{eq:stgeparfun} of the parameter functions (and hence also for the equivalent parametrization via \(\psi\)) indeed yield a class of theories whose field equations reproduce those of the \(f(G)\), \(f(Q)\)~\cite{Jarv:2018bgs} and \(f(T)\)~\cite{Yang:2010ji} classes of gravity theories, if suitable restrictions are imposed on the torsion or nonmetricity of the connection. Substituting the values~\eqref{eq:stgeparfun} and the constraint~\eqref{eq:stgcons1} into the metric field equation~\eqref{eq:stgmetfield} yields
\begin{multline}
f'\lc{R}_{\mu\nu} - \frac{f'}{2}\lc{R}g_{\mu\nu} + f''\left(\lc{\nabla}_{(\mu}G M^{\rho}{}_{\nu)\rho} - M^{\rho}{}_{(\mu\nu)}\lc{\nabla}_{\rho}G + M^{[\rho\sigma]}{}_{\sigma}\lc{\nabla}_{\rho}G g_{\mu\nu}\right)\\
+ \frac{1}{2}(f - f'G)g_{\mu\nu} = \kappa^2\Theta_{\mu\nu}\,,
\end{multline}
which, using \(f''\lc{\nabla}_{\mu}G = \lc{\nabla}_{\mu}f'\), reproduces the field equation~\eqref{eq:fgmetfield2}. The same relation is used to show that the connection field equation~\eqref{eq:stgconnfield}, which becomes
\begin{multline}
f''\left[M^{\nu\rho}{}_{\rho}\lc{\nabla}_{\mu}G + M^{\rho}{}_{\mu\rho}\lc{\nabla}^{\nu}G - (M^{\nu\rho}{}_{\mu} + M^{\rho}{}_{\mu}{}^{\nu})\lc{\nabla}_{\rho}G\right]\\
= 2\kappa^2(\nabla_{\tau}H_{\mu}{}^{\nu\tau} - M^{\omega}{}_{\tau\omega}H_{\mu}{}^{\nu\tau})\,,
\end{multline}
resembles the connection field equation~\eqref{eq:fgconnfield2}. We then continue with the symmetric teleparallel case. Here, the connection equation~\eqref{eq:stqconnfield} becomes
\begin{equation}
-\nabla_{\nu}\nabla_{\rho}(f'\tilde{P}^{\nu\rho}{}_{\mu}) = \kappa^2\nabla_{\nu}\nabla_{\rho}\tilde{H}_{\mu}{}^{\nu\rho}\,,
\end{equation}
which is obviously identical to the corresponding equation~\eqref{eq:fqconnfield}. The metric field equation~\eqref{eq:stqmetfield} takes the form
\begin{equation}
f'\lc{R}_{\mu\nu} - \frac{f'}{2}\lc{R}g_{\mu\nu} - f''P^{\rho}{}_{\mu\nu}\lc{\nabla}_{\rho}Q + \frac{1}{2}(f - f'Q)g_{\mu\nu} = \kappa^2\Theta_{\mu\nu}\,.
\end{equation}
To bring this to the familiar form, one raises one index, and uses the fact that the left hand side of the STEGR field equation satisfies
\begin{equation}
-\frac{\nabla_{\rho}(\sqrt{-g}P^{\rho\mu}{}_{\nu})}{\sqrt{-g}} - \frac{1}{2}P^{\mu\rho\sigma}Q_{\nu\rho\sigma} + \frac{1}{2}Q\delta^{\mu}_{\nu} = R^{\mu}{}_{\nu} - \frac{1}{2}R\delta^{\mu}_{\nu}\,.
\end{equation}
This can be used to replace the Einstein tensor, so that the scalar-nonmetricity field equation becomes
\begin{equation}
-\frac{f'\nabla_{\rho}(\sqrt{-g}P^{\rho\mu}{}_{\nu})}{\sqrt{-g}} - \frac{f'}{2}P^{\mu\rho\sigma}Q_{\nu\rho\sigma} - P^{\rho\mu}{}_{\nu}\lc{\nabla}_{\rho}f' + \frac{1}{2}f\delta^{\mu}_{\nu} = \kappa^2\Theta^{\mu}{}_{\nu}\,.
\end{equation}
Observe that the two derivative terms can be combined into a single term, which yields the field equation~\eqref{eq:fqmetfield}. A similar procedure can be applied to the scalar-torsion case, whose field equations~\eqref{eq:sttallfield} now read
\begin{multline}
f'\lc{R}_{\mu\nu} - \frac{f'}{2}\lc{R}g_{\mu\nu} + f''S_{\mu\nu}{}^{\rho}\lc{\nabla}_{\rho}T + \frac{1}{2}(f - f'T)g_{\mu\nu}\\
= \kappa^2(\Theta^{\mu\nu} - \nabla_{\rho}H^{\mu\nu\rho} + H^{\mu\nu\rho}T^{\tau}{}_{\tau\rho})\,.
\end{multline}
Here one uses the left hand side of the MTEGR field equation, which can be written as
\begin{equation}
\lc{\nabla}_{\rho}(S_{\mu\nu}{}^{\rho}) + S^{\rho\sigma}{}_{\nu}K_{\rho\sigma\mu} + \frac{1}{2}Tg_{\mu\nu} = \lc{R}_{\mu\nu} - \frac{1}{2}\lc{R}g_{\mu\nu}\,.
\end{equation}
Using this relation to replace the Einstein tensor, one finds
\begin{multline}
f'\lc{\nabla}_{\rho}(S_{\mu\nu}{}^{\rho}) + f'S^{\rho\sigma}{}_{\nu}K_{\rho\sigma\mu} + S_{\mu\nu}{}^{\rho}\lc{\nabla}_{\rho}f' + \frac{1}{2}fg_{\mu\nu}\\
= \kappa^2(\Theta^{\mu\nu} - \nabla_{\rho}H^{\mu\nu\rho} + H^{\mu\nu\rho}T^{\tau}{}_{\tau\rho})\,.
\end{multline}
Once again the two derivative terms can be combined, and one has the field equation~\eqref{eq:ftallfield}.
\section{Outlook and open questions}\label{sec:outlook}
Teleparallel gravity theories are an active field of research and many questions are yet unanswered at the time of writing of this chapter. One of the most prominent open questions is known as the ``strong coupling problem''~\cite{BeltranJimenez:2019nns,Golovnev:2020zpv}. It refers to the fact that both the Hamiltonian analysis and higher order perturbation theory predict the presence of additional degrees of freedom compared to general relativity in several classes of teleparallel gravity, which are not manifest as propagating modes in the linear perturbation theory. Such modes are called strongly coupled, and their presence hints towards possible instabilities and a lack of predictability, which potentially renders the perturbation theory around such background solutions invalid. Among the most common approaches to clarify the nature and severity of these issues is the Hamiltonian analysis and the study of constraints.
Besides fundamental questions, also the phenomenology of teleparallel gravity theories leaves numerous possibilities for further studies, which can potentially lead to new experimental tests. Active fields at the time of writing this chapter include the study of cosmology using the method of dynamical systems, cosmological perturbations, black holes and other exotic compact objects, as well as their shadows and their perturbations, which are closely related to the emission of gravitational waves. Hence, it is reasonable to expect numerous future developments in this field.
\begin{acknowledgement}
The author thanks Claus Lämmerzahl and Christian Pfeifer for the kind invitation to contribute this book chapter. He acknowledges the full financial support of the Estonian Ministry for Education and Science through the Personal Research Funding Grant PRG356, as well as the European Regional Development Fund through the Center of Excellence TK133 ``The Dark Side of the Universe''.
\end{acknowledgement}
\bibliographystyle{spphys}
|
{
"timestamp": "2022-07-15T02:00:20",
"yymm": "2207",
"arxiv_id": "2207.06438",
"language": "en",
"url": "https://arxiv.org/abs/2207.06438"
}
|
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