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058c3896dac2a5233643dbb3193a95b28f3638a5 | subsection | 6 | 17 | Toroidal Map | Hence we distinguish two families of toroidal regular maps of type \lbrace 3,6\rbrace : \lbrace 3, 6\rbrace _{(s,0)} and \lbrace 3, 6\rbrace _{(s,s)}.
The group of symmetries of \lbrace 3, 6\rbrace _{(s,0)} and \lbrace 3, 6\rbrace _{(s,s)} are factorizations of the Coxeter group [3, 6] by(\rho _0\rho _1\rho _2)^{2s}=1\... | {
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"Maria Elisa Fernandes",
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44d08b0629f7dcab1f047cc0d316e9872a233619 | subsection | 7 | 17 | Toroidal Map | We have the following equalitiesg^{\rho _1}=g,\,g^{\rho _2}=h^{-1}\mbox{ and }h^{\rho _1}=j^{-1}.@-1.8pc{
&&&&&& &&&& &&&& &&&&&\\
&&&&&& &&&& &&&& &&&&&\\
&&*{}@{-}[rrrrrrrrrrrrrrrr]@{-}[ddddrrrr]&&&&*{}@{-}[ddddrrrr]@{-}[ddddllll]&&&&*{}@{-}[ddddllll]@{-}[ddddrrrr]&&&&*{}@{-}[ddddllll]@{-}[ddddrrrr]&&&&*{}@{-}[ddddll... | {
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f38c6e472e67df28e3e2f9708ed04444749e8cf2 | subsection | 8 | 17 | String C-groups and CPR graphs | The group of symmetries of a toroidal map is a string C-group of rank 3, the unique exception is the group of \lbrace 4,4\rbrace _{(1,0)} (that is not a polytope).
In general a string C-group of rank r is a group generated by r involutions
\rho _0, \rho _1,\ldots , \rho _{r-1} satisfying the following conditions.(\rho ... | {
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c0ef96823db0ea8d2a8ef7e7507107bf9c8c3b39 | subsection | 9 | 17 | Preliminary results | In this section we include the results that can be applied to both the toroidal regular maps of type \lbrace 4,4\rbrace and of type \lbrace 3,6\rbrace .
We start by proving a proposition on the degrees of a transitive action of a direct product of cyclic groups for later use.Proposition 3.1
Let H=C_a\times C_b be a di... | {
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} | 1808.09705 | Faithful permutation representations of toroidal regular maps | [
"Maria Elisa Fernandes",
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c9fb6b26a4469129dae732308c993ceaa570c3fc | subsection | 10 | 17 | Preliminary results | Hence G\cong T\rtimes G_1 is G_1 is the stabilizer of the identity (Section 1.7, ).Proposition 3.3
If G is the group of \lbrace 4,4\rbrace _{(s,s)} or \lbrace 3,6\rbrace _{(s,s)}, then T is intransitive.First suppose that G is the group of the map \lbrace 4,4\rbrace _{(s,s)} and that T is transitive.
Let \alpha =\rho ... | {
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} | 1808.09705 | Faithful permutation representations of toroidal regular maps | [
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676cadce0eeab385859516db29bf8c4b9d9df2bd | subsection | 11 | 17 | Preliminary results | \lbrace 4,4\rbrace _{(s,s)}) then 2n is a degree of \lbrace 4,4\rbrace _{(s,s)} (resp. \lbrace 4,4\rbrace _{(2s,0)}); and
if n is the degree of \lbrace 3,6\rbrace _{(s,0)} (resp. \lbrace 3,6\rbrace _{(s,s)}) then 3n is a degree of \lbrace 3,6\rbrace _{(s,s)} (resp. \lbrace 3,6\rbrace _{(3s,0)}); | {
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} | 1808.09705 | Faithful permutation representations of toroidal regular maps | [
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5c567ad5f8e45ebc2bd1092575071fcb084c8062 | subsection | 12 | 17 | The maps of type | In this section we consider the regular maps of type \lbrace 4,4\rbrace , determine all possible degrees for these maps and give CPR graphs for some of those degrees. | {
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} | 1808.09705 | Faithful permutation representations of toroidal regular maps | [
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d422e08db81eceefaca1bed55d5cd0e1b457f5ab | subsection | 13 | 17 | The possible degrees for the map | The groups of \lbrace 4,4\rbrace _{(s,0)} (s>2) act faithfully on the sets of vertices, faces, edges, darts and flags (as the dihedral groups \langle \rho _i, \rho _j\rangle and its subgroups are core-free, with i,j\in \lbrace 0,1,2\rbrace ). Let us consider the exceptional cases s\in \lbrace 1,2\rbrace .
The only prop... | {
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} | 1808.09705 | Faithful permutation representations of toroidal regular maps | [
"Maria Elisa Fernandes",
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ac5182e295e90e7e3b87a12d1d3e7c728664e110 | subsection | 14 | 17 | The possible degrees for the map | If x\notin T then \rho _0 \rho _0^{\rho _1}\in T, a contradiction. Thus x\in T and therefore as in (1) we conclude that x=1.
The order of H is \frac{2s^2}{ab} thus |G:H|=4ab.(3) Suppose that x\in H\cap H^{\rho _1}=\langle u^a, v^b\rangle \rtimes \langle \rho _0,\rho _2\rangle \cap \langle u^b, v^a\rangle \rtimes \langl... | {
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02753cc71c655b3df01d98c2439e84526c9003da | subsection | 15 | 17 | The possible degrees for the map | By Corollary REF and Theorem REF there are faithful permutation representations for \lbrace 4,4\rbrace _{(s,s)} for n\in \lbrace 2s^2,4ab,8ab,16ab\rbrace with s=lcm(a,b). In what follows we prove that those are the unique possibilities for n. By Proposition REF and Lemma REF the possibility to rule out is n=2ab with s=... | {
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b08e10c9bd9d45accc9fbe843e8d16482a896528 | subsection | 16 | 17 | CPR graphs of | Proposition 4.5
The following graphs are CPR graphs of \lbrace 4,4\rbrace _{(s,0)} of degree 2s (s\ge 3).
[Table: Acknowledgements] | {
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dd8ebfe3dcb3f2830414878dcaea6a02ef2b1821 | abstract | 0 | 53 | Abstract | Modular graph forms are a class of modular covariant functions which appear
in the genus-one contribution to the low-energy expansion of closed string
scattering amplitudes. Modular graph forms with holomorphic subgraphs enjoy the
simplifying property that they may be reduced to sums of products of modular
graph forms ... | {
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70ee5927a5fdb405be7d81d8ef7b4a46bf2a2808 | subsection | 1 | 53 | Introduction | In the genus-one contribution to the low-energy expansion of closed string amplitudes, a natural generalization of non-holomorphic Eisenstein series known as modular graph forms arises. A modular graph form can be understood as an assignment of a certain modular covariant functionThroughout, we will call a function f m... | {
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bc3ac141a5e77e9253fd26466cbad5a432ad624f | subsection | 2 | 53 | Introduction | The weight of this modular graph form is \sum _{r=1}^n\left({a_r-b_r \over 2}, {b_r-a_r\over 2}\right), which in particular is always integer since the sum in (REF ) vanishes by antisymmetry if \sum _{r=1}^n(a_r + b_r) is odd.An interesting special class of modular graph forms are those with holomorphic subgraphs, name... | {
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9b53aa71059d7d1daba808c3ab705cc9f5b2b646 | subsection | 3 | 53 | Introduction | This leads to the expressionQ_1(p_1,\dots ,p_n) = -\sum _{i=1}^n {1 \over p_i} - {\pi \over (n+1) \tau _2}\sum _{i=1}^n(p_i - \bar{p}_i )We finally show that the regularization guessed in this way coincides with the use of the Eisenstein summation prescription to evaluate the original sum. Since no ambiguity arises in ... | {
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edbe0c3ffed69cbbf44564ba1f8e8175c9d7ae9b | subsection | 4 | 53 | Modular graph functions and forms in physics | In addition to the massless supergravity spectrum, string theory predicts an infinite tower of massive particles with masses of order (\alpha ^{\prime })^{-1/2}. Though the direct production of such particles seems unlikely in the near or distant future, one may hope to identify this stringy spectrum indirectly through... | {
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470ef89780d0e4402a11785c6becd48871e9d721 | subsection | 5 | 53 | Modular graph functions and forms in physics | In the simplest case of m=4, it is known , , , thatc_{4,0}(\eta ) &= \pi ^{3/2}\, {\rm E}_{3/2}(\eta ) & c_{4,1}(\eta )&= 0 & c_{4,2}(\eta )&= \pi ^{5/2}\, {\rm E}_{5/2}(\eta )where E_s(\eta ) is the non-holomorphic Eisenstein series, defined as{\rm E}_s(\eta ) = \sum ^{\prime }_{(m,n) \in {\mathbb {Z}}^2}{\eta _2^s \o... | {
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a5c1dc91bf0e050381048378590c32fed6cc84b0 | subsection | 6 | 53 | Modular graph functions and forms in physics | Instead, in these cases one must generally settle for perturbative results obtained via calculation of four-graviton scattering amplitudes. For example, one may begin with the four-graviton tree-level amplitude {\cal A}_0^{(4)} , which takes the familiar form{\cal A}_0^{(4)} &= {R^4 \eta _2^2 \over s t u} {\Gamma (1-s)... | {
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19d92458d9f3b4965460e47e0f24d2206c87dae2 | subsection | 7 | 53 | Modular graph functions and forms in physics | The scalar Green's function on the torus admits the following Fourier representation,G(z|\tau ) = \sum _{p \in \Lambda }^{\prime } { \tau _2 \over \pi |p|^2 } \, e^{2 \pi i (n \alpha - m \beta )}where z=\alpha + \beta \tau with \alpha , \beta \in {\mathbb {R}}/{\mathbb {Z}}. The integers m,n parametrize the discrete mo... | {
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1004e1299853a9965e4a1f3ced4299a1ccf00325 | subsection | 8 | 53 | Modular graph functions and forms in physics | As usual, we represent a Green's function graphically by an edge in a Feynman diagram,\begin{}[baseline=-0.5ex,scale=1.7]
(1,0) -- (2.5,0) ;
(1,0) [fill=white] circle(0.05cm) ;
(2.5,0) [fill=white] circle(0.05cm) ;
(1,-0.25) node{z_i};
(2.5,-0.25) node{z_j};
\end{}
=~ G(z_i-z_j|\tau )The integration over the position o... | {
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decd286ab0de50aa1358cbdb52ee58ba80afc385 | subsection | 9 | 53 | Modular graph functions and forms in physics | In terms of the Fourier series for the Green's function (REF ), this expression is given by,\mathcal {C}_{\Gamma }(\tau ) = \sum _{p_1,\ldots ,p_w \in \Lambda }^{\prime }
\left( \prod _{r=1}^w {{\tau _2}\over {\pi |p_{r}|^2}} \right)
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2f13d496b4e5b8fb44fa77695f0a7a875a9362d1 | subsection | 10 | 53 | Holomorphic subgraph reduction of dihedral graphs | We now give a brief overview of the holomorphic subgraph reduction procedure for dihedral graphs, as introduced in . Since we will present the calculation of the trihedral holomorphic subgraph reduction formulae in detail in Section , we will refrain from providing technical details here, instead focusing on the main c... | {
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b00a518cc0de528ea6073d473d6a75b6b088173a | subsection | 11 | 53 | Holomorphic subgraph reduction of dihedral graphs | The resulting expression then has one less momentum, and thus one less edge, than the original modular graph form. This implies that modular graph forms with a holomorphic subgraph are reducible to sums of products of modular graph forms with fewer loops.A subtlety in this procedure is that by naively distributing the ... | {
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1cbfa45d6dfb1e170d642605b6b5b1e6277c0a80 | subsection | 12 | 53 | Holomorphic subgraph reduction of dihedral graphs | An important point is that the term -\frac{\pi }{2\tau _{2}}p_{0} in Q_{1}(p_{0}) and the term \frac{\pi }{\tau _{2}} in Q_{2}(p_{0}) have different modular weights than the sums on the respective left-hand sides. But when plugged into the full expression resulting from partial fraction decomposition of (REF ), these t... | {
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583cd0a5c921e2da785215ab1df120f1ce809dbf | subsection | 13 | 53 | Extension to trihedral graphs | In this section, we will generalize the holomorphic subgraph reduction procedure outlined in the previous section to trihedral modular graph forms.When the graph corresponding to a modular graph form has dihedral topology, it is sufficient to consider only two-point holomorphic subgraphs in order to arrive at a general... | {
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4751bc8e33b9cd85e1a4c60e26216b32768ba083 | subsection | 14 | 53 | Two-point holomorphic subgraph reduction | A general trihedral graph with a two-point holomorphic subgraph is depicted in the following figure:[scale=1]
[xshift=-5cm,yshift=-0.4cm]
[directed,very thick] (8.73,-1) node\bullet –node[right]\mathfrak {p}_{3} (7,0.8) node\bullet ;
[directed,very thick] (7,0.8) node\bullet –node[left]\mathfrak {p}_{1} (5.27,-1) node\... | {
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96f4a26147c10eba289a1d8ede33d3b3b4444a37 | subsection | 15 | 53 | Two-point holomorphic subgraph reduction | As in the previous case we have introduced the collective momenta \mathfrak {p}_i, defined by\mathfrak {p}_i = \sum _{n_i=1}^{R_i} p_{n_i}^{(i)}as well as the shorthand notation\prod {1 \over \mathfrak {p}^A \bar{\mathfrak {p}}^B} \equiv \left(\frac{\tau _{2}}{\pi }\right)^{\frac{1}{2}(a_{+}+a_{-})}\prod _{i=1,2,3} \pr... | {
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60be6483523089c704ee111747013373beece5e5 | subsection | 16 | 53 | Two-point holomorphic subgraph reduction | For a_{0}=a_{+}+a_{-}\ge 3, we have\,{\cal C}\!\left[\protect \begin{matrix}A_{1}\\B_{1}\protect \end{matrix}|\protect \begin{matrix}a_{+}&a_{-}&A_{2}\\0&0&B_{2}\protect \end{matrix}|\protect \begin{matrix}A_{3}\\B_{3}\protect \end{matrix}\right]=&(-1)^{a_{+}}{\tau _{2}^{\frac{1}{2}a_{0}}}{\rm G}_{a_{0}}\,{\cal C}\!\le... | {
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} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
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603440aae065da817210055d466f10e77892ab51 | subsection | 17 | 53 | Three-point holomorphic subgraph reduction | We now proceed to the main focus of this work, which is holomorphic subgraph reduction of three-point holomorphic subgraphs in trihedral modular graph forms.
The graphs in question are shown in the following figure,[scale=0.9]
[xshift=-5cm,yshift=-0.4cm]
[directed,very thick,dashed] (8.73,-1) node\bullet .. controls (8... | {
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} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
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f9d17797b59e848df658b6c5ad54c639d6a5e3b5 | subsection | 18 | 53 | Three-point holomorphic subgraph reduction | The notation is as before, though we have redefined\prod {1 \over \mathfrak {p}^A \bar{\mathfrak {p}}^B} \equiv \left(\frac{\tau _{2}}{\pi }\right)^{\frac{1}{2}(a_{2}+a_{4}+a_{6})}\prod _{i=1,3,5} \prod _{n_i=1}^{R_i} \left({\tau _2 \over \pi }\right)^{{1\over 2}(a_{n_i}^{(i)}+b_{n_i}^{(i)})} {1 \over (p_{n_i}^{(i)})^{... | {
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cfa867c4f739961051297f63bf356d8d05bd1db9 | subsection | 19 | 53 | Decomposing | In order to perform the sum (REF ), we first separate out all cases in which \mathfrak {p}_{15} and \mathfrak {p}_{35} are equal to each other or to zero. In particular, there are five cases to study,&\mathfrak {p}_{15}= \mathfrak {p}_{35}=0 & \mathcal {L}_1 &= \sum ^{\prime }_{p_6} {1 \over p_6^{a_0}}
\\
&\mathfrak {p... | {
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bde9d1f173e6f56790957c936849c479382b568e | subsection | 20 | 53 | Decomposing | The first sum is trivial,\mathcal {L}_1 = {\mathcal {G}}_{a_0}To evaluate the second sum, we begin by utilizing the following partial fraction identity\frac{1}{p^{a}(q-p)^{b}}=\sum _{k=1}^{a}\binom{a+b-k-1}{a-k}\frac{1}{p^{k}q^{a+b-k}}+\sum _{k=1}^{b}\binom{a+b-k-1}{b-k}\frac{1}{q^{a+b-k}(q-p)^{k}}which allows us to re... | {
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a91aafcd9ee182995c24274db7300bd7a680735f | subsection | 21 | 53 | Decomposing | Upon applying the following identities,\sum _{k=1}^{a_1}\binom{a_1+a_2-k-1}{a_1-k}+\sum _{k=1}^{a_2}\binom{a_1+a_2-k-1}{a_2-k}&= \binom{a_1+a_2}{a_1}\\
\binom{a_{0}-3}{a_{2}+a_{4}-2}+\binom{a_{0}-3}{a_{6}-2}&=\binom{a_{0}-2}{a_{6}-1}the sum \mathcal {L}_{2} simplifies to(-1)^{a_2 + a_4}\mathcal {L}_2 =& \sum _{k=4}^{a_... | {
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73db393e4b72d18733566db156343ca9f8993b10 | subsection | 22 | 53 | Decomposing | To begin, we apply the decomposition formula (REF ) twice to obtain(-1)^{a_2 + a_4} \mathcal {L}_5 =& \sum _{k=1}^{a_6}\sum _{\ell =1}^k \binom{a_2+a_6-k-1}{a_6-k}\binom{a_4+k-\ell -1}{k-\ell } {Q_\ell (\mathfrak {p}_{15},\mathfrak {p}_{35}) \over (\mathfrak {p}_{15})^{a_2+a_6-k}(\mathfrak {p}_{35})^{a_4+k-\ell }}
\\
&... | {
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8bc5f7980f75a602ed56975c162cd71cd414706d | subsection | 23 | 53 | Evaluating | In order to evaluate (REF ), we may insert the expressions for Q_\ell (p_1,p_2) as in the case of \mathcal {L}_2. In particular, we useQ_{1}(p_1,p_2)&=-{1 \over p_1} - {1 \over p_2} - x {\pi \over \tau _2}(p_1 + p_2 - \bar{p}_1 - \bar{p}_2)
\\
Q_{2}(p_1,p_2)&=-{1 \over {p_1}^2}-{1 \over {p_2}^2} +\mathcal {\hat{G}}_2 +... | {
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0cd6c5ff26d6bc90f924b5129ce59ec82acafc75 | subsection | 24 | 53 | Evaluating | This choice will be justified in Section REF , where it arises as a special case of the general expression (REF ) for an arbitrary number of excluded momenta.Returning to the evaluation of \mathcal {L}_5, we may insert (REF ) and rewrite (REF ) as(-1)^{a_2 + a_4} \mathcal {L}_5 =& \sum _{k=1}^{a_6} \binom{a_2+a_6-k-1}{... | {
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0a64b71cd2ac66294806d31ea64ee33149f31812 | subsection | 25 | 53 | Summation over non-holomorphic momenta | With (REF ), we have completed the evaluation of the five sums \mathcal {L}_i listed in (REF ) which make up the sum \mathcal {S} in (REF ). In order to obtain our final formula for three-point holomorphic subgraph reduction of (REF ), we must now carry out the sums over the remaining momenta. | {
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64799de04875b060177d1be06dac0984edbca830 | subsection | 26 | 53 | Summation over non-holomorphic momenta | We denote the completely summed versions of the \mathcal {L}_{i} by L_{i}, such that our final answer is given by\,{\cal C}\!\left[ \begin{matrix}A_1 \, a_2 \cr B_1 \, 0 \cr \end{matrix}| \begin{matrix}A_3 \, a_4 \cr B_3 \, 0 \cr \end{matrix}| \begin{matrix}A_5 \, a_6 \cr B_5 \, 0 \cr \end{matrix} \right] = \sum _{i=1}... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
"math.NT"
] | 2,018 | en | Physics | [
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dbc37c3ce3d17e9dec9367fdcfa7d971baac1a38 | subsection | 27 | 53 | Summation over non-holomorphic momenta | To simplify the result, we introduce the following shorthand notation\,{\cal C}\!\left[ \begin{matrix} m_1 \cr n_1 \, \cr \end{matrix}| \begin{matrix}m_2 \cr n_2 \cr \end{matrix}|\,\, \right] &\equiv (-1)^{m_1+n_1+m_2+n_2} \,\,{\cal C}\!\left[ \begin{matrix}A_1 ~m_1 \, \cr B_1 ~ n_1\, \cr \end{matrix}| \begin{matrix}A_... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
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] | 2,018 | en | Physics | [
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63b38c8f974be33e1fc07a7785f62853a4f17727 | subsection | 28 | 53 | Summation over non-holomorphic momenta | \\
&\hphantom{- \sum _{\ell =1}^{a_{4}} \binom{a_4 + k -\ell -1}{a_6 - \ell } (-1)^\ell \Big \lbrace }\left.+ \sum _{m=1}^{ \ell } \binom{a_4+k -m-1}{\ell - m}(-1)^{m}(-1)^{\epsilon (a_{4}+k-m)}\,{\cal C}\!\left[\,\, \Big | \begin{matrix}a_0-m \cr 0 \, \cr \end{matrix}\Big |\begin{matrix}m \cr 0 \cr \end{matrix} \right... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
"math.NT"
] | 2,018 | en | Physics | [
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fb1274494416579bc0647d9feb87d9f03c3f5496 | subsection | 29 | 53 | Divergent modular graph forms in the reduced expression | When applying this formula one must be careful with the order in which the three blocks of the trihedral function are plugged into the formula, since an incorrect choice leads to divergent modular graph forms in the result. These divergences manifest themselves in\begin{bmatrix}
\,\,\,1&\,\,1\\
-1&\,\,1
\end{bmatrix}su... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
"math.NT"
] | 2,018 | en | Physics | [
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ad632eeef24ab83e9a5b3af9e7978e0b2b854c05 | subsection | 30 | 53 | Examples | We now offer a few examples to illustrate the utility of the three-point holomorphic subgraph reduction formula. First, consider the following trihedral modular graph form,\,{\cal C}\!\left[\protect \begin{matrix}1&2\\1&0\protect \end{matrix}|\protect \begin{matrix}1&1\\1&0\protect \end{matrix}|\protect \begin{matrix}1... | {
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"Jan E. Gerken",
"Justin Kaidi"
] | [
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26f2c7b0af6cd1eb3a15ba7ee89f96ad559b6f0d | subsection | 31 | 53 | Examples | To avoid this, we instead consider the equivalent expression\,{\cal C}\!\left[\protect \begin{matrix}1&1\\1&0\protect \end{matrix}|\protect \begin{matrix}1&2\\1&0\protect \end{matrix}|\protect \begin{matrix}1\\0\protect \end{matrix}\right]=\sum ^{\prime }_{p_{i}\in \Lambda }\left(\frac{\tau _{2}}{\pi }\right)^{4}\frac{... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
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] | 2,018 | en | Physics | [
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171dcbb69ea71e649f396b36f12666c816c2d381 | subsection | 32 | 53 | Examples | This is because these can be reduced to dihedral holomorphic subgraph reduction without introducing new divergent sums by doing a more careful partial fraction decomposition hand-tailored to these specific examples. | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
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58c557a95f1dbd85167bfe529d1987d4f95706a9 | subsection | 33 | 53 | Examples | A calculation along those lines is outlined in Appendix .Finally, at third order in \alpha ^{\prime } in the heterotic calculation, a more complex example arises which decomposes into dihedral graphs and lower-loop trihedral graphs,\,{\cal C}\!\left[ \begin{matrix} 2 & 1 \\ 1& 0\end{matrix} | \begin{matrix} 1 & 1 \\ 1 ... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
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aae614359f26606e8ddfe6ec92df15e43b14f5a1 | subsection | 34 | 53 | Definition of | As we have seen in the previous section, our derivation of holomorphic subgraph reduction formulae relies on the regularization of sums of the form\sum ^{\prime }_{p\ne p_1,\dots , p_n} {1 \over p}We have already noted above that the appropriate regularization scheme for the case of n=2 is (REF ) with x= {1\over 3}. In... | {
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} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
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0ae928ae0e17416fead0dd1211e973b5a2c433cc | subsection | 35 | 53 | Definition of | (6,2);
[directed, very thick] (6,2) ..controls (4.7,1) ..(4.27,-1);
[directed, very thick,dashed] (4.27,-1) ..controls (6,-1.6) .. | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
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"math.NT"
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ff525a2eaea298ffc3821ab0d4917ff2d7432799 | subsection | 36 | 53 | Definition of | (7.73,-1);
[ directed, very thick,dashed] (6,2) node{\bullet } -- (6,0) node{\bullet };
[ directed, very thick,dashed] (4.27,-1) node{\bullet } -- (6,0) node{\bullet };
[ directed, very thick] (7.73,-1) node{\bullet } -- (6,0) node{\bullet };
(7.6,1) node{p_2};
(4.4,1) node{p_3};
(6,-1.2) node{p_1};
(6.3,1.1) node{p_5}... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
"math.NT"
] | 2,018 | en | Physics | [
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fc1664908a54f17c153a43d1822565458253c4dc | subsection | 37 | 53 | Definition of | \\
&\hphantom{\sum ^{\prime }_{{p_1 \ne p_3, p_4 \\p_{1}\ne p_3 + p_4 }} \Big [}\left. +\frac{2p_{3}^{2}-4p_{3}p_{4}+p_{4}^{2}}{p_{3}^{2}p_{4}^{3}(p_{3}-p_{4})^{2}}\,\frac{1}{p_{1}-p_{4}} - {2p_3^2 + 4 p_3 p_4 + p_4^2 \over p_3^2 p_4^3 (p_3 + p_4)^2}\,{1 \over p_1} - {1 \over p_3 p_4^{2} (p_3 + p_4)}\,{1 \over p_1^2} \... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
"math.NT"
] | 2,018 | en | Physics | [
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c3183539169863e3b4ab24126011253084fc8951 | subsection | 38 | 53 | Definition of | For example, we could instead have begun with\,{\cal C}\!\left[\begin{matrix} 2 \\ 0\end{matrix} \bigg | \begin{matrix} 2 \\ 0\end{matrix} \bigg | \begin{matrix} 1 \\ 1\end{matrix} \bigg | \begin{matrix} 1 \\ 1\end{matrix} \bigg | \begin{matrix} 2 \\ 0\end{matrix} \bigg | \begin{matrix} 2 \\ 0\end{matrix}\right] = \sum... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
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86f5fc4b820f4a44e84ad364f93c55ebcec0cfd4 | subsection | 39 | 53 | Definition of general | The general strategy is now clear. The appearance of Q_1(p_1,\dots , p_n) in holomorphic subgraph reduction of modular graph forms always comes from the decomposition of sums of the form\sum ^{\prime }_{p \ne p_{1}, \dots , p_{n}}{1 \over p^{a_0} (p - p_1)^{a_1} \dots (p-p_n)^{a_n}}for some external momenta p_i and cor... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
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] | 2,018 | en | Physics | [
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0e955fd66be47696d07fcac7a213b5d52bbd647a | subsection | 40 | 53 | Definition of general | The validity of this interchange of derivatives and sums follows by uniform convergence.\sum ^{\prime }_{p \ne p_{1}, \dots , p_{n}} {1 \over p^{a_0} (p - p_1) \dots (p-p_n)}For any a_{0},n\ge 1 we may now use a partial fraction decomposition to re-expressThis can be proven by induction over n either directly, or alter... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
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3736cfebcba3c8d39f7b7d238ec31a68ad35aec0 | subsection | 41 | 53 | Definition of general | In particular, we may work with the Eisenstein summation prescription, denoted by \hspace{-5.0pt}\operatornamewithlimits{\hspace{5.0pt}\sum \raisebox {-0.5em}{\mbox{[}0.2em]{\scriptstyle \mathrm {E}}}\hspace{3.00003pt}} and defined in (REF ) of Appendix REF , and then distribute the sums in (REF ), yielding\sum _{i=1}^... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
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5b2c55beb9c4ebbd63c107df478e756deb259151 | subsection | 42 | 53 | Definition of general | The correct replacements for this matching are\sum _{p\ne p_{1},\dots ,p_n}^{\prime } \frac{1}{p}&\longrightarrow Q_{1}(p_{1},\dots ,p_n)\\
\sum _{p\ne p_{1},\dots ,p_n}^{\prime } \frac{1}{p_{i}-p}&\longrightarrow Q_{1}(p_{i},\underbrace{p_{i}-p_{1},\dots ,p_{i}-p_{n}}_{\text{omit $p_{i}-p_{i}$}})with Q_1(p_1, \dots , ... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
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08c62bcb074397d9facb349297eaf454c98e6387 | subsection | 43 | 53 | Definition of general | Since we are interested only in the terms of abnormal modular weight, we may discard all terms in the sum over \ell for which a_0 - \ell + 1 >2. | {
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} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
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34b68440112036504d8dc67bb7aa2bbe332e2617 | subsection | 44 | 53 | Definition of general | For a_{0}\ge 2 we insert (REF ) into (REF ) and keep only the terms -{\pi \over (n+1)\tau _2}\sum _{\ell =1}^np_\ell in Q_{1} and \frac{\pi }{\tau _{2}} in Q_{2}, to obtainFor a_{0}=1, n\ge 2, one finds F(p_{1},\dots ,p_{n};1)=0 and the second term in (REF ) is absent.{\tau _2 \over \pi } \sum ^{\prime }_{p \ne p_{1}, ... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
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13ea74d3d416c8931a97c50e084d94cb4cb59718 | subsection | 45 | 53 | Summary | In this work, we have extended the results of to obtain holomorphic subgraph reduction formulae for trihedral modular graph forms. The two-point holomorphic subgraph reduction formula was given in (REF ), and is a simple generalization of the two-point formula for dihedral modular graph forms. The three-point holomorph... | {
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"openalex_id": "https://openalex.org/W2963674950",
"raw": "E. D'Hoker and M. B. Green, “Identities between Modular Graph Forms,” J. Number Theor. 189, 25 (2018) [arXiv:1603.00839 [hep-th]].",
"s... | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
"math.NT"
] | 2,018 | en | Physics | [
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7ab3d54ca45e9045d1a979efb2a4c363fd818e12 | subsection | 46 | 53 | Trihedral holomorphic subgraph reduction without | In this section we outline derivations of the decompositions (REF ) and (REF ) which do not involve divergent sums which must be regularized. This will serve as a check for the consistency of our regularization procedure. | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
"math.NT"
] | 2,018 | en | Physics | [
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d2df209e2fe7eb4afbbe15592ce736d9348945f3 | subsection | 47 | 53 | Trihedral holomorphic subgraph reduction without | Note that derivations of this sort must be found on a case-by-case basis, and do not admit a nice systematization like that studied in the main text.First, consider the sum\left(\frac{\pi }{\tau _{2}}\right)^{4}\,{\cal C}\!\left[\protect \begin{matrix}1&1\\1&0\protect \end{matrix}|\protect \begin{matrix}1&2\\1&0\protec... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
"math.NT"
] | 2,018 | en | Physics | [
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daa835624895608063914a4b9e25fa11df778f64 | subsection | 48 | 53 | Trihedral holomorphic subgraph reduction without | The second term is dihedral and the last term factorizes completely. The third term can be shown to be a dihedral modular graph form by relabeling p_{1}\rightarrow p_{1}-p_{3} and then p_{3}\rightarrow -p_{3},\sum _{{p_{1},p_{2},p_{3}\\p_{2},p_{3}\ne 0\\p_{1}+p_{2}\ne 0\\p_{1}+p_{3}\ne 0}}\frac{1}{|p_{2}|^{2}|p_{3}|^{2... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
"math.NT"
] | 2,018 | en | Physics | [
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43a96fcf8c603636235cd211a87547f4ee203529 | subsection | 49 | 53 | Trihedral holomorphic subgraph reduction without | Note that in this derivation, all the sums appearing in every step were absolutely convergent and no regularization was needed.As a second example, consider\left(\frac{\pi }{\tau _{2}}\right)^{4}\,{\cal C}\!\left[\protect \begin{matrix}2\\0\protect \end{matrix}|\protect \begin{matrix}1&1\\0&1\protect \end{matrix}|\prot... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
"math.NT"
] | 2,018 | en | Physics | [
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1268ae341d34ddef5b2e40d138c5f5c3e81931f0 | subsection | 50 | 53 | Eisenstein summation of simple sums | In this Appendix, we apply the Eisenstein summation prescription to sums which are needed to evaluate the expression (REF ). The Eisenstein summation prescription is defined as follows,\hspace{-5.0pt}\operatornamewithlimits{\hspace{5.0pt}\sum \raisebox {-0.5em}{\mbox{[}0.2em]{\scriptstyle \mathrm {E}}}\hspace{3.00003pt... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
"math.NT"
] | 2,018 | en | Physics | [
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72f9538bda252276b874c613232d831c43a135d2 | subsection | 51 | 53 | Eisenstein summation of simple sums | Again using vanishing of the sum over 1/m, as well as (REF ), we have\hspace{-5.0pt}\operatornamewithlimits{\hspace{5.0pt}\sum \raisebox {-0.5em}{\mbox{[}0.2em]{\scriptstyle \mathrm {E}}}\hspace{3.00003pt}}_{p\notin P}^{\prime }\frac{1}{p_{i}-p}=-\frac{1}{p_{i}}-\sum _{{p\in P\\p\ne p_{i}}}\frac{1}{p_{i}-p}-i\pi \lim _... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
"math.NT"
] | 2,018 | en | Physics | [
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c2fac7673e1b1703d93cb590b84dfd37c3f2b873 | subsection | 52 | 53 | Body | We now obtain the expression for Q_{1}(p_1,\dots ,p_n) given in (REF ) of the main text by Eisenstein summing a certain linear combination of shifted sums. We start by definingQ_1 (p_1, \dots , p_n) \equiv {1\over 2}\left[\hspace{5.0pt}\hspace{-5.0pt}\operatornamewithlimits{\hspace{5.0pt}\sum \raisebox {-0.5em}{\mbox{[... | {
"cite_spans": []
} | 10.1007/JHEP01(2019)131 | 1809.05122 | Holomorphic subgraph reduction of higher-point modular graph forms | [
"Jan E. Gerken",
"Justin Kaidi"
] | [
"hep-th",
"math.NT"
] | 2,018 | en | Physics | [
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2cc402c1d1a71424b2802eefab4ac8dbf30c2e83 | abstract | 0 | 20 | Abstract | A noncommutative polynomial is stable if it is nonsingular on all tuples of
matrices whose imaginary parts are positive definite. In this paper a
characterization of stable polynomials is given in terms of strongly stable
linear matrix pencils, i.e., pencils of the form $H+iP_0+P_1x_1+\cdots+P_dx_d$,
where $H$ is hermi... | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
"math.RA",
"math.FA"
] | 2,018 | en | Mathematics | [
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d61d88929d0d037d0666611096773cc7253db133 | subsection | 1 | 20 | Introduction | A multivariate polynomial f\in x_1,\dots ,x_d] is stable if
f(\alpha )\ne 0 whenever \operatorname{Im}\alpha _j>0 for all j=1,\dots ,d. Stable polynomials and their variations, such as Hurwitz and Schur polynomials, originated in control theory , , , . However, recent years saw a renewed interest in stable polynomials ... | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
"math.RA",
"math.FA"
] | 2,018 | en | Mathematics | [
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0a2eb036a2e5605a6bd9b11b314a5a3628f18859 | subsection | 2 | 20 | Main results | Let {x}=(x_1,\dots ,x_d) be freely noncommuting variables. In our noncommutative setting, the positive orthant in d is replaced by the set of all tuples of matrices whose imaginary part is positive definite, which we call the matricial positive orthant and denote \mathbb {H}^d. Then we say that a linear matrix pencil L... | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
"math.RA",
"math.FA"
] | 2,018 | en | Mathematics | [
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671c4e4690d69c845fce7e9f71f51a46ce210dc2 | subsection | 3 | 20 | Stable pencils | In this section we completely characterize stable linear matrix pencils, i.e., rectangular pencils that have full rank on the matricial positive orthant. We prove that every such pencil is equivalent to a lower block triangular pencil whose diagonal blocks are stable for obvious reasons (and thus called purely stable p... | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
"math.RA",
"math.FA"
] | 2,018 | en | Mathematics | [
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e4a6b9af271089fca330e94f06a49ebc25c7a0de | subsection | 4 | 20 | Stable pencils | If v\in \ker L(X), then by Lemma REF and positive semidefiniteness we havev &\in \ker \left(H\otimes I+\sum _{j>0} P_j\otimes \operatorname{Re}X_j \right)
\cap \ker \left(P_0\otimes I+\sum _{j>0} P_j\otimes \operatorname{Im}X_j\right) \\
&= \ker (P_0\otimes I)\cap \left(\bigcap _{j>0}\ker (P_j\otimes \operatorname{Im}X... | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
"math.RA",
"math.FA"
] | 2,018 | en | Mathematics | [
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dc0dbe59fbd3b12870a22d8d85404d1979554e38 | subsection | 5 | 20 | Stable pencils | From the \mathbb {R}-linear system \operatorname{Im}(DA_1)=0 in D we deduce thatD=\begin{pmatrix}\alpha _1+i\beta & \alpha _2+2i\beta \\ \alpha _3 & 2\alpha _3+\alpha _1-i\beta \end{pmatrix},\qquad \alpha _j,\beta \in \mathbb {R}.Furthermore \det \operatorname{Im}(DA_0)=-\frac{1}{4}(\alpha _2+\alpha _3)^2, so \operator... | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
"math.RA",
"math.FA"
] | 2,018 | en | Mathematics | [
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b06a2ecdc72db678c643d7d705c89f5b624a7c72 | subsection | 6 | 20 | Notation | We start by introducing the basic terminology used throughout the paper, including purely stable pencils. | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
"math.RA",
"math.FA"
] | 2,018 | en | Mathematics | [
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69c6f04136f585dd06c3a71a82cdd7044d6416bc | subsection | 7 | 20 | Linear matrix pencils | For d\in \mathbb {N} let {x}=(x_1,\dots ,x_d) be a tuple of freely noncommuting variables and let \mathop {<}\!{x}\!\mathop {>} be the free -algebra generated by x. If A0,...,Ad, then
L=A_0+A_1x_1+\cdots +A_dx_d\in {\delta \times \varepsilon }\otimes _{\mathop {<}\!{x}\!\mathop {>}=\mathop {<}\!{x}\!\mathop {>}\!{}^{\d... | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
"math.RA",
"math.FA"
] | 2,018 | en | Mathematics | [
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b0811b328963fa2568152db6debeac8e6f821268 | subsection | 8 | 20 | Main theorem | In this subsection we apply a truncated Gelfand-Naimark-Segal (GNS) construction to prove that every stable pencil is S-stable; see Theorem REF . We start with some preliminary notation.By {x}^*=(x_1^*,\dots ,x_d^*) we denote the formal adjoints of variables x_j and endow the free algebra \mathop {<}\!{x},{x}^*\!\matho... | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
"math.RA",
"math.FA"
] | 2,018 | en | Mathematics | [
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2cb343976891a5444fea802f5dab0926a0f4ec8e | subsection | 9 | 20 | Main theorem | Then \operatorname{Re}((D+E)A_0)\succeq 0 and \operatorname{Re}((D+E)A_j)=0, -\operatorname{Im}((D+E)A_j)\succeq 0 for j>0. For \tilde{D}=i(D+E) we thus have \operatorname{Im}(\tilde{D}A_0)\succeq 0 and \operatorname{Im}(\tilde{D}A_j)=0, \operatorname{Re}(\tilde{D}A_j)\succeq 0 for j>0. By the assumption (REF ) we have... | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
"math.RA",
"math.FA"
] | 2,018 | en | Mathematics | [
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12ab4c9448c433d49a6870881b4ed0db78f5f4bf | subsection | 10 | 20 | Main theorem | Furthermore, the map\operatorname{M}_{\varepsilon }(\rightarrow \operatorname{End}(\mathcal {V}_0)\cong \operatorname{M}_{\varepsilon }(=\operatorname{M}_{\varepsilon }(\otimes \operatorname{M}_{\varepsilon }(given by a\mapsto \ell _a is a unital *-embedding of *-algebras. By a *-version of the Skolem-Noether theorem t... | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
"math.RA",
"math.FA"
] | 2,018 | en | Mathematics | [
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70c843f6dbb1f2b02f46bc24a586d27441717541 | subsection | 11 | 20 | Main theorem | Hence every solution D of (REF ) satisfies DL=0, so L(X) does not have full rank for some X\in \mathbb {H}^d\cap d by Lemma REF .Now assume the statement holds for all \varepsilon ^{\prime }<\varepsilon and that L is not S-stable. By composing the coefficients of L on the left with the projection onto \sum _j \operator... | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
"math.RA",
"math.FA"
] | 2,018 | en | Mathematics | [
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18db8fee216febec4e0f89775088dddbcc8d5c89 | subsection | 12 | 20 | An algorithm | The proof of Theorem REF can be used to devise an algorithm for testing whether a pencil is stable by solving a sequence of semidefinite programs (SDPs) , .Let L=A_0+\sum _{j>0}A_jx_j be of size \delta \times \varepsilon with \delta \ge \varepsilon .Solve the following feasibility SDP for D\in {\varepsilon \times \delt... | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
"math.RA",
"math.FA"
] | 2,018 | en | Mathematics | [
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61665622fee8c37085ad2b82c3fc06f6dabacffa | subsection | 13 | 20 | Hermitian coefficients | Classically, one is interested in symmetric or hermitian determinantal representations (REF ) of real polynomials. However, the constant term of a purely stable pencil is in general not hermitian. This can be amended for a particular class of pencils. We say that L=H_0+\sum _{j>0}H_jx_j is a hermitian pencil if H_j\in ... | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
"math.RA",
"math.FA"
] | 2,018 | en | Mathematics | [
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a74e4a84b97baa2e6d8dd23c8309d9740597df48 | subsection | 14 | 20 | Hermitian coefficients | Now if v_1 and v_2 are eigenvectors for D^*, then\langle \operatorname{Im}(DH_0) (v_1\pm v_2),(v_1\pm v_2)\rangle \ge 0since \operatorname{Im}(DH_0)\succeq 0, which together with (REF ) implies\langle \operatorname{Im}(DH_0) v_1,v_2\rangle +\langle \operatorname{Im}(DH_0) v_2,v_1\rangle =0.Now let v\in d be arbitrary. ... | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
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] | 2,018 | en | Mathematics | [
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5dfdf3d23a8908d65028ded4c16732df719e7be3 | subsection | 15 | 20 | Hurwitz and Schur stability | In control theory, there are also other stability notions, such as Hurwitz and Schur stability, that can be related to Definition REF . In this subsection we describe how to apply Theorem REF and the algorithm from Subsection REF to test other versions of noncommutative stability.We say that L is Hurwitz stable if L(X)... | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
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edd0b077db781705d0ad91d315a9456e106a20aa | subsection | 16 | 20 | Hurwitz and Schur stability | If L(X) has full rank for every X\in \mathbb {D}^d of size d\cdot \min \lbrace \delta ,\varepsilon \rbrace , then L(X) has full rank for every X\in \mathbb {D}^d.Remark 2.13 Via realization theory (see Subsection REF below), Schur stable pencils are closely related to noncommutative rational functions that are regular ... | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
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] | 2,018 | en | Mathematics | [
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5cff3378762fbf78b0a1a6c515ac9006eef12988 | subsection | 17 | 20 | Stability of noncommutative polynomials | We are now ready to apply the preceding results to noncommutative polynomials and rational functions. First we characterize noncommutative rational functions whose domains contain the matricial positive orthant \mathbb {H}^d (Theorem REF ). Next we show that every stable noncommutative polynomial admits a determinantal... | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
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6e83ec10d7ea1d46fbde01e042a76f5d4c5ff7a0 | subsection | 18 | 20 | Noncommutative rational functions | After a short introduction of the free skew field and required realization theory, we describe noncommutative rational functions defined on the matricial positive orthant. | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
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bf084ceb6b56f0d7ef784ce96318046d28039a17 | subsection | 19 | 20 | Free skew field | We give a condensed introduction of noncommutative rational functions using matrix evaluations of formal rational expressions following . Originally they were defined ring-theoretically , .Noncommutative rational expressions are syntactically valid combinations of complex numbers, variables {x}, arithmetic operations +... | {
"cite_spans": []
} | 1807.05645 | Stable noncommutative polynomials and their determinantal
representations | [
"Jurij Volčič"
] | [
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d16d939a6557ab0f867dee49e95d9e2de8dc35e6 | abstract | 0 | 5 | Abstract | Evidence in probabilistic reasoning may be 'hard' or 'soft', that is, it may
be of yes/no form, or it may involve a strength of belief, in the unit interval
[0, 1]. Reasoning with soft, [0, 1]-valued evidence is important in many
situations but may lead to different, confusing interpretations. This paper
intends to bri... | {
"cite_spans": []
} | 1807.05609 | The Mathematics of Changing one's Mind, via Jeffrey's or via Pearl's
update rule | [
"Bart Jacobs"
] | [
"cs.AI"
] | 2,018 | en | Computer Science | [
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9bf0fed9cdb6a89b06378408e1e3f4f199d80a8b | subsection | 1 | 5 | Introduction | Logical statements in a probabilistic setting are usually interpreted
as events, that is, as subsets E\subseteq \Omega of an
underlying sample space \Omega of possible worlds, or equivalently
as characteristic functions \Omega \rightarrow \lbrace 0,1\rbrace . One
typically computes the probability \footnotesize \mathrm... | {
"cite_spans": []
} | 1807.05609 | The Mathematics of Changing one's Mind, via Jeffrey's or via Pearl's
update rule | [
"Bart Jacobs"
] | [
"cs.AI"
] | 2,018 | en | Computer Science | [
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d2169bf21aa8a4efdfa5b51a431b40074f066918 | subsection | 2 | 5 | Introduction | This approach is described
operationally: extend a Bayesian network with an auxiliary node, so
that soft evidence can be emulated in terms of hard evidence on this
additional node, and so that the usual inference methods can be
applied. We shall see that extending a Bayesian network with such a
node corresponds to usin... | {
"cite_spans": []
} | 1807.05609 | The Mathematics of Changing one's Mind, via Jeffrey's or via Pearl's
update rule | [
"Bart Jacobs"
] | [
"cs.AI"
] | 2,018 | en | Computer Science | [
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557ea471c363c23970bcca7bc7007341bf3d38d2 | subsection | 3 | 5 | Introduction | It is
thus not a method that can be used in general.Some more technical background: within the compositional programming
language perspective, a Bayesian network is a (directed acyclic) graph
in the Kleisli category of the distribution monad \mathcal {D} — or the
Giry monad \mathcal {G} for continuous probability theor... | {
"cite_spans": []
} | 1807.05609 | The Mathematics of Changing one's Mind, via Jeffrey's or via Pearl's
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"Bart Jacobs"
] | [
"cs.AI"
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7b43d5cbf0e21386e0f1fcaa7d3ee6ac1a9f316b | subsection | 4 | 5 | A simple illustration | Consider a simple Bayesian network involving a test for a disease, as
on the left in Figure . There is an a priori
disease probability of 1\%. The test has a sensitivity as given by
the table on the lower-left in the figure: in presence of the disease,
written as d, the likelihood of a positive test outcome is 90\%;
in... | {
"cite_spans": []
} | 1807.05609 | The Mathematics of Changing one's Mind, via Jeffrey's or via Pearl's
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"Bart Jacobs"
] | [
"cs.AI"
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1bcaacde5d76af4c823757bddc8b180e2a76fd08 | abstract | 0 | 36 | Abstract | In 4d $\mathcal{N}=1$ superconformal field theories (SCFTs) the R-symmetry
current, the stress-energy tensor, and the supersymmetry currents are grouped
into a single object, the Ferrara-Zumino multiplet. In this work we study the
most general form of three-point functions involving two Ferrara-Zumino
multiplets and a ... | {
"cite_spans": []
} | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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1ad8a1fb65e2a795d248aa24b60361c7319309e7 | subsection | 1 | 36 | Body | CERN-TH-2018-096R-current three-point functionsin 4d \mathcal {N}=1 superconformal theoriesAndrea Manenti,{\!}^a Andreas Stergiou,{\!}^b
and Alessandro Vichi{}^a{}^aInstitute of Physics, École Polytechnique
Fédérale de Lausanne (EPFL),Rte de la Sorge, BSP 728, CH-1015 Lausanne, Switzerland{}^bTheoretical Physics Depart... | {
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f7e07f211fbcf30ebdbeb0080b742529d1ac53a4 | subsection | 2 | 36 | Body | This
will provide a new way to explore the space of SCFTs, and hopefully shed
more light on the “minimal” 4d \mathcal {N}=1 SCFT studied with bootstrap
techniques in , , and
attempted to be identified by analytical means in , .Unlike the case of extended supersymmetry in 4d, in \mathcal {N}=1 SCFTs
the supermultiplet c... | {
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fbed4e29fb0257df6b8a919820b6c36501bc62c0 | subsection | 3 | 36 | Body | This typically happens automatically
after the Ward identities for conservation at the first two points have
been solved, i.e. the solution for the independent three-point function
coefficients involves explicit factors of \Delta -\Delta _u, where
\Delta _u is the dimension at the unitarity bound. While in some of our
... | {
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650a1e9e06eb0f40c563bf84f0c8a952a16fa860 | subsection | 4 | 36 | Body | The three-point function is non-zero in three cases:
J(X1) J(X2) O+k,(X3)= K,,k
{.
[JJOnonsusy]
where the prefactor is
K,,k
=J3-2X12+-8+12
kX13--12 kX23--12
k ,[JJOnonsusyPrefactor]
and we have defined the tensor structuresThe quantities appearing in and (REF )
are combinations of the 6d coordinates X_i and spinors S_... | {
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"Andrea Manenti",
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9e56803a9c351efe16ad2620d41fbbeca64462de | subsection | 5 | 36 | Body | As a consequence,
\lambda ^{(2)} vanishes as well and there is only one degree of freedom. A
second exception is for k=2, \ell =0; in this case permutation symmetry
and current conservation sets the three-point function to zero, expect for
the special case \Delta =2, when \lambda ^{(3)}=\lambda ^{(4)}, while all
the re... | {
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d7178f81b8d2f1618266fbd5d50bc8737008f807 | subsection | 6 | 36 | Body | The condition implies
the following conservation and irreducibility conditions:
J= T = T= T[] = S= S= S= S =0 .
[conservation]General propertiesIn this section we study the most general form of the three-point function
of two Ferrara–Zumino multiplets and a third general superconformal
multiplet \mathcal {O}_{\gamma ... | {
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b0dc9b6a710f86d7abed54675603cd4047844bba | subsection | 7 | 36 | Body | In superspace, however, we would expect the
correlator to be non-vanishing whenever there is a
non-zero three-point function between a component of \mathcal {O} and any
pairs of the fields appearing in the expansion . For instance,
when only \theta _{1,2}={\bar{\theta }}_{1,2}=0, we could get a non-supersymmetric
three... | {
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"raw": "M. Berkooz, R. Yacoby & A. Zait, “Bounds on \\mathcal {N} = 1 superconformal theories with global symmetries”, JHEP 1408, 008 (2014), arXiv:1402.6068 [hep-th], [Erratum: JHEP 1501, 132 (2015)]",
... | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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... |
d6fb91165903e14a3c899a1a5e3fee5c72b87d61 | subsection | 8 | 36 | Body | The arbitrariness of the three-point function is now entirely
contained in the tensor t, which is only a function of the coordinates
X,\Theta ,{\hspace{0.83328pt}\overline{\hspace{-0.83328pt}\Theta \hspace{-1.111pt}}\hspace{1.111pt}}{}, while the prefactor takes care of reproducing the
correct covariance properties at ... | {
"cite_spans": []
} | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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61e11ee5a89acbc5d3e04a09bb827c3b8f4ecfe7 | subsection | 9 | 36 | Body | The choice made here is less convenient for writing down the structures explicitly.[Shortcuts]
In addition, the homogeneity property can now we
written as
t(1,2,3,1,2,3,X,,) =
23(2q+q-9)23(q+2q-9)
j t(i,i,X,,) ,
[homogeneityEta]
while the symmetry property for the first two points reads
t(1,2,3,1,2,3,X,X)=
t(2,1,3,2... | {
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{
"arxiv_id": "",
"doi": "10.1007/jhep02(2018)096",
"end": 1883,
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"raw": "P. Kravchuk & D. Simmons-Duffin, “Counting Conformal Correlators”, JHEP 1802, 096 (2018), arXiv:1612.08987 [hep-th]",
"source_ref_id": "... | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
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] | 2,018 | en | Physics | [
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2ca06f4c406308fe745e2b4bc88c2a49735c2e07 | subsection | 10 | 36 | Body | The third part is instead built with exactly one \Theta and one
{\hspace{0.83328pt}\overline{\hspace{-0.83328pt}\Theta \hspace{-1.111pt}}\hspace{1.111pt}}{}.
In order to enumerate the structures in the first part we can simply follow
a standard approach for non-supersymmetric CFTs.
One possible way is to choose a confo... | {
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{
"arxiv_id": "",
"doi": "10.1007/bf01609130",
"end": 437,
"openalex_id": "https://openalex.org/W2009996835",
"raw": "G. Mack, “Convergence of Operator Product Expansions on the Vacuum in Conformal Invariant Quantum Field Theory”, Commun. Math. Phys. 53, 155 (1977)"... | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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