| \section{Global dimension bounds with imperfect parallel compatibility} |
| \label{sec:global-imperfect-compatibility} |
|
|
| We now allow the unary boundary to fail parallel compatibility and quantify |
| the amount of failure required to reduce coefficient dimension. The |
| boundary is normalized to be a fiberwise isometry. Without a normalization, |
| rescaling a boundary by a small constant and its cancelling form by the |
| reciprocal constant would make an absolute compatibility norm arbitrarily |
| small without changing the cancelled curvature. |
|
|
| Let $M$ be a connected Riemannian manifold, let $(V,\nabla^V)$ be a |
| rank-$d$ flat Hermitian bundle with $d\ge2$, and let $(U,\nabla^U)$ be a rank-$r$ |
| Hermitian bundle with a unitary connection. Flatness of $\nabla^U$ is not |
| needed for the lower bound below, so that bound also applies to the more |
| restricted flat auxiliary systems. Let $Q:U\to V$ be a smooth fiberwise |
| isometry and define |
| \[ |
| P=QQ^*,\qquad |
| K=\nabla^{\operatorname{Hom}}Q |
| =\nabla^VQ-Q\nabla^U,\qquad K_\perp=(I-P)K. |
| \] |
| For a unit tangent vector $X$, use the Hilbert--Schmidt norm of |
| $K_\perp(X):U_x\to V_x$, and put |
| \[ |
| \kappa_\perp=\sup_{x,\,\lVert X\rVert=1} |
| \lVert K_\perp(X)\rVert_{\rm HS}. |
| \] |
| The same lower bounds hold for the possibly larger norm defined using |
| $K$ instead of $K_\perp$. |
|
|
| Fix $p\in M$ and finitely many piecewise smooth loops |
| $\gamma_1,\ldots,\gamma_m$ based at $p$, of lengths $L_1,\ldots,L_m$, |
| with $\sum_iL_i^2>0$. |
| Write $H_i\in U(V_p)$ for their holonomies. On the complex traceless |
| matrix space, with Hilbert--Schmidt norm, define |
| \[ |
| \mathcal L(S)=([S,H_1],\ldots,[S,H_m]),\qquad |
| s=\inf_{\substack{\operatorname{tr}S=0\\\lVert S\rVert_{\rm HS}=1}} |
| \left(\sum_i\lVert[S,H_i]\rVert_{\rm HS}^2\right)^{1/2}. |
| \] |
| If the $H_i$ act irreducibly, Schur's lemma and finite dimensionality imply |
| $s>0$. |
|
|
| \begin{theorem}[A global compatibility cost for every variable boundary] |
| \label{thm:global-compatibility-cost} |
| For every smooth isometric boundary $Q$ as above, |
| \[ |
| \boxed{\quad |
| \kappa_\perp\ge |
| \frac{s}{\sqrt{2\sum_iL_i^2}} |
| \sqrt{\frac{r(d-r)}d}.\quad} \tag{G1} |
| \] |
| In particular the bound is not restricted to constant boundaries or |
| constant coefficient subspaces. A more detailed loopwise inequality is |
| \[ |
| s^2\frac{r(d-r)}d |
| \le 2\sum_i |
| \left(\int_{\gamma_i}\lVert K_\perp(\dot\gamma_i)\rVert_{\rm HS} |
| \,dt\right)^2. \tag{G2} |
| \] |
| The right side uses any piecewise smooth parametrizations, with the |
| velocity included as displayed. |
| \end{theorem} |
|
|
| \begin{proof} |
| Metric compatibility and $Q^*Q=I$ imply that $Q^*K(X)$ is skew-Hermitian. |
| Differentiating $P=QQ^*$ therefore cancels this tangential skew-Hermitian |
| part and gives |
| \[ |
| \nabla^{\operatorname{End}}_X P |
| =K_\perp(X)Q^*+QK_\perp(X)^*,\qquad |
| \lVert\nabla_X P\rVert_{\rm HS}^2 |
| =2\lVert K_\perp(X)\rVert_{\rm HS}^2. \tag{G3} |
| \] |
| The equality follows in an orthogonal splitting |
| $V_x=\operatorname{im}Q_x\oplus(\operatorname{im}Q_x)^\perp$: |
| the two summands occupy opposite off-diagonal blocks of equal norm. |
|
|
| Parallel-transport $P$ along a loop back into the fixed fiber $V_p$. |
| The resulting curve of projectors starts at $P_p$ and ends at |
| $H_i^{-1}P_pH_i$. Unitary transport preserves Hilbert--Schmidt norm, |
| so integrating (G3) gives |
| \[ |
| \lVert[H_i,P_p]\rVert_{\rm HS} |
| =\lVert H_i^{-1}P_pH_i-P_p\rVert_{\rm HS} |
| \le\sqrt2\int_{\gamma_i} |
| \lVert K_\perp(\dot\gamma_i)\rVert_{\rm HS}\,dt. |
| \] |
| Since $P_p$ is a rank-$r$ orthogonal projector, |
| \[ |
| \left\lVert P_p-\frac rd I\right\rVert_{\rm HS}^2 |
| =r-\frac{r^2}d=\frac{r(d-r)}d. |
| \] |
| The scalar matrix commutes with each holonomy. Apply the definition of |
| $s$ to this traceless matrix, sum the squared loop bounds, and obtain |
| (G2). Each integral is at most $L_i\kappa_\perp$, giving (G1). |
| \end{proof} |
|
|
| \begin{corollary}[A positive compatibility tolerance still forces full rank] |
| Suppose the holonomies above act irreducibly and a prescribed curvature |
| has a value of norm one at $p$. If a cancellation $QG$ has curvature |
| error less than one at that value and |
| \[ |
| \kappa_\perp< |
| \frac{s}{\sqrt{2\sum_iL_i^2}}\sqrt{\frac{d-1}d}, \tag{G4} |
| \] |
| then $r=d$. Thus the all-or-nothing obstruction persists for a |
| nonzero compatibility budget. |
| \end{corollary} |
| \begin{proof} |
| Curvature error less than one excludes $r=0$. For every integer |
| $1\le r\le d-1$ one has $r(d-r)\ge d-1$, so (G1) excludes every |
| remaining proper rank. The full bundle with its identity boundary |
| attains zero compatibility and curvature error. |
| \end{proof} |
|
|
| \subsection{Matching-order transition on a fixed closed five-manifold} |
|
|
| The compatibility scale in (G4) is attained in order by an explicit |
| family with globally fixed potential and curvature. This strengthening |
| uses two disjoint transport handles, so it does not require identifying |
| different flat bundles only in one local chart. |
|
|
| \begin{theorem}[The rank-reduction threshold has order $|\epsilon|$] |
| \label{thm:global-linear-compatibility-scale} |
| For every fixed $d\ge2$, there are a fixed closed connected oriented |
| five-manifold $M$, a fixed trivial Hermitian bundle $V=M\times\mathbb C^d$, |
| fixed smooth forms $B,F$ with $\lVert F\rVert_\infty=1$, and flat |
| unitary connections $\nabla^V_\epsilon$ converging smoothly to the trivial |
| connection, with |
| \[ |
| d_{\nabla^V_\epsilon}B=F. |
| \] |
| There are constants $c_d,C_d,\epsilon_d>0$ such that, for |
| $0<|\epsilon|<\epsilon_d$ and every fixed curvature tolerance |
| $0\le\tau<1$, the following statements hold. |
| \begin{enumerate} |
| \item Every smooth isometric boundary from any auxiliary Hermitian |
| bundle of rank strictly between zero and $d$ that attains curvature |
| error $\lVert F-QG\rVert_{L^\infty}\le\tau$ has |
| $\lVert\nabla^{\operatorname{Hom}}Q\rVert_\infty |
| \ge c_d|\epsilon|$. |
| \item A flat unitary auxiliary line, an isometric boundary, and a |
| covariantly closed scalar five-form cancel $F$ exactly, with |
| compatibility error at most $C_d|\epsilon|$. |
| \end{enumerate} |
| Consequently the compatibility budget necessary and sufficient, up to |
| fixed positive factors, for any rank reduction is $|\epsilon|$. |
| The constants depend on $d$ and the fixed geometry; no dimension-uniform |
| claim is made. |
| \end{theorem} |
|
|
| \begin{proof} |
| Let |
| $M=(S^1\times S^4)\mathbin{\#}(S^1\times S^4)$. |
| There are smooth closed one-forms $\eta_1,\eta_2$ with disjoint supports, |
| one in each punctured summand, with period one around the respective |
| circle generators and vanishing on a common coordinate ball $D$. |
| For example, on each circle choose a smooth function supported in a |
| proper interval with integral one and take $f(t)dt$ on the product with |
| $S^4$. Remove the gluing ball away from that support. The form vanishes |
| near the gluing sphere and extends by zero across the connected sum. |
| The two supports are disjoint, so $\eta_1\wedge\eta_2=0$. |
|
|
| Put |
| \[ |
| A=\operatorname{diag}(1,\ldots,d),\qquad |
| J=(1)_{i,j=1}^d,\qquad |
| v=d^{-1/2}(1,\ldots,1)^T, |
| \] |
| and define |
| \[ |
| \nabla^V_\epsilon=d+i\epsilon(A\eta_1+J\eta_2). |
| \] |
| The matrices $A,J$ are Hermitian. The connection is flat because |
| $d\eta_i=0$ and every wedge product $\eta_i\wedge\eta_j$ vanishes. |
| Its two generating holonomies, based in $D$, are |
| \[ |
| H_1=e^{-i\epsilon A},\qquad |
| H_2=e^{-i\epsilon J} |
| =I+\frac{e^{-id\epsilon}-1}{d}J. |
| \] |
| For $0<|\epsilon|<2\pi/d$, the first has simple spectrum and the |
| second has every off-diagonal entry nonzero. A common commuting |
| orthogonal projector must first be diagonal and then scalar, hence |
| zero or the identity. Thus these holonomies act irreducibly. |
|
|
| Let $\beta$ be an ordinary four-form compactly supported in $D$, with |
| $\lVert d\beta\rVert_\infty=1$. Set $B=v\beta$ and $F=v\,d\beta$. |
| Both are fixed, and the connection term vanishes on their support. |
| Hence $d_{\nabla^V_\epsilon}B=F$ for every parameter. Choose the |
| base point $p\in D$ where the curvature norm is one and fix generating |
| loops of finite lengths $L_1,L_2$, independently of $\epsilon$. |
|
|
| On traceless matrices define |
| \[ |
| \mathcal L_0(S)=([S,A],[S,J]),\qquad |
| s_0=\sigma_{\min}(\mathcal L_0)>0. |
| \] |
| Its kernel is zero by the same diagonal-then-scalar argument. |
| The unitary exponential remainder estimate yields |
| \[ |
| \lVert\mathcal L_\epsilon+i\epsilon\mathcal L_0\rVert |
| \le\sqrt2\,d^2\epsilon^2. |
| \] |
| It follows that $s_\epsilon\ge s_0|\epsilon|/2$ for sufficiently |
| small $|\epsilon|$. Apply (G1), using |
| $r(d-r)\ge d-1$, to obtain the first assertion with |
| \[ |
| c_d=\frac{s_0}{2\sqrt{2(L_1^2+L_2^2)}} |
| \sqrt{\frac{d-1}d}. |
| \] |
| This lower bound allows arbitrary smooth isometric boundaries and |
| even arbitrary unitary auxiliary connections. |
|
|
| For attainment in order, take the trivial complex line with flat |
| connection |
| \[ |
| \nabla^U_\epsilon |
| =d+i\epsilon\left(\frac{d+1}{2}\eta_1+d\eta_2\right), |
| \qquad Q(1)=v,\qquad G=d\beta. |
| \] |
| Then $QG=F$ exactly, and $G$ is covariantly closed by top degree. |
| Since $Jv=dv$, |
| \[ |
| \nabla^{\operatorname{Hom}}Q |
| =i\epsilon\left(A-\frac{d+1}{2}I\right)v\eta_1, |
| \] |
| whose uniform norm is |
| \[ |
| |\epsilon|\sqrt{\frac{d^2-1}{12}}\, |
| \lVert\eta_1\rVert_\infty. |
| \] |
| This proves the second assertion and the matching order. The boundary |
| is generally not parallel, so the resulting small-error datum is not |
| an exact homological completion; its compatibility defect is the |
| specific error being optimized. |
| \end{proof} |
|
|
| The result gives a sharp order transition, not an exact formula for every |
| intermediate budget. It also distinguishes two limits: for each fixed |
| nonzero $\epsilon$ there is a positive compatibility threshold below |
| which dimension $d$ is forced, while that threshold itself tends to zero |
| as the holonomy tends to the identity. |
|
|
