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Identities among higher genus modular graph tensors
High Energy Physics - Theory · Submitted 2 Oct 2020
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim An infinite family of algebraic identities between one-loop and tree-level modular graph tensors are proven for arbitrary genus and arbitrary tensorial rank.
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The abstract, sentence by sentence
Higher genus modular graph tensors map Feynman graphs to functions on the Torelli space of genus-$h$ compact Riemann surfaces which transform as tensors under the modular group $Sp(2h , \mathbb Z)$, thereby generalizing a construction of Kawazumi. An infinite family of algebraic identities between one-loop and tree-level modular graph tensors are proven for arbitrary genus and arbitrary tensorial rank. We also derive a family of identities that apply to modular graph tensors of higher loop order.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Higher genus modular graph tensors map Feynman graphs to functions on the Torelli space of genus-$h$ compact Riemann surfaces which transform as tensors under the modular group $Sp(2h , \mathbb Z)$, thereby generalizing a construction of Kawazumi. | background 0.89 | 0.58 | 0.07 | asserted 1.00 |
| s1 | An infinite family of algebraic identities between one-loop and tree-level modular graph tensors are proven for arbitrary genus and arbitrary tensorial rank. | result 0.96 | 0.95 | 0.77 | asserted 1.00 |
| s2 | We also derive a family of identities that apply to modular graph tensors of higher loop order. | result 0.86 | 0.96 | 0.14 | asserted 1.00 |
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The paper
Original abstract
Higher genus modular graph tensors map Feynman graphs to functions on the Torelli space of genus-$h$ compact Riemann surfaces which transform as tensors under the modular group $Sp(2h , \mathbb Z)$, thereby generalizing a construction of Kawazumi. An infinite family of algebraic identities between one-loop and tree-level modular graph tensors are proven for arbitrary genus and arbitrary tensorial rank. We also derive a family of identities that apply to modular graph tensors of higher loop order.
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References
How the literature uses it
18 arXiv papers cite this in 35 sentences
Papers citing it · 29 sentences without a confident label
Builds on it 1 paper
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Here we shall extend this library of modular tensors by including convolutions that involve not only Arakelov Green functions (as was the case in [ref] ) but also derivatives on Arakelov Green functions and holomorphic Abelian differentials.
Constructing polylogarithms on higher-genus Riemann surfaces · Convolution of Arakelov Green functions and modular tensors · 0.99
Cites it as background 5 papers
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The modular transformation $M$ acts on the period matrix by $Ω→(A Ω+B) (C Ω+D)^-1$ and on the Abelian differentials by its non-linear $GL(h,)$ representation, [equation] Modular tensors $$ of arbitrary rank were defined in [ref] (see also [ref] ) to transform as follows, ^I_1 ⋯I_r _J_1 ⋯J_s →Q^I_1_I_1' ⋯Q^I_r_I_r' ^I_1' ⋯I_r' _J_1' ⋯J_s' R^J'_1 _J_1 ⋯R^J'_s _J_s .
Cyclic products of higher-genus Szeg\"o kernels, modular tensors and polylogarithms · Modular tensors · 0.91
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Analogous analysis has also been done at genus two [ref] based on the low momentum expansion of the two loop four and five graviton amplitudes [ref] , though much less has been understood compared to the one loop graphs.
Relations between elliptic modular graphs · Introduction · 0.80
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This has led to a detailed analysis of their properties [ref] revealing a rich underlying structure.
Integrating simple genus two string invariants over moduli space · Introduction · 0.92
What it cites · 49 sentences without a confident label
Builds on 2 papers
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We shall then summarize the interchange lemma proven in [ref] , and prove its generalization to higher rank tensors.
Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality · The interchange lemmas · 0.93
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This would amount to a higher-genus generalization of modular graph forms [ref] that appear in the moduli derivatives of modular graph functions at genus one.
Identities between Modular Graph Forms · Conclusion and outlook · 0.80
Uses the method or tool of 5 papers
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This identity was proven with the help of interchange Lemma D.1 in [ref] , which prescribes a certain interchange rule for derivatives on higher genus Arakelov Green functions similar to the interchange formula resulting from translation invariance at genus one.
Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality · Introduction · 0.83
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Using Siegel's formula for the volume of moduli space and an inhomogeneous Laplace eigenvalue equation for the Kawazumi-Zhang invariant derived in [ref] (see also [ref] ), the integrations over moduli space could be performed analytically.
Matching the $D^6 {\cal R}^4$ interaction at two-loops · Introduction · 0.90
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Using Siegel's formula for the volume of moduli space and an inhomogeneous Laplace eigenvalue equation for the Kawazumi-Zhang invariant derived in [ref] (see also [ref] ), the integrations over moduli space could be performed analytically.
Johnson's homomorphisms and the Arakelov-Green function · Introduction · 0.90
Compares with 7 papers
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It will be shown in section [ref] that this identity is identical to the one proven for genus two in appendix D of [ref] .
Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality · Weight 2 · 0.86
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The resulting coefficients were found to match the predictions of space-time supersymmetry [ref] and S-duality [ref] in Type IIB string theory [ref] and [ref] .
Matching the $D^6 {\cal R}^4$ interaction at two-loops · Introduction · 0.84
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The resulting coefficients were found to match the predictions of space-time supersymmetry [ref] and S-duality [ref] in Type IIB string theory [ref] and [ref] .
Two-loop superstrings and S-duality · Introduction · 0.84
Cites as background 13 papers
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A more general analysis of genus-one modular graph functions was initiated in [ref] where they were found to satisfy identities that extend the well-known relations between multiple zeta-values to modular functions.
On the modular structure of the genus-one Type II superstring low energy expansion · Introduction · 0.93
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A more general analysis of genus-one modular graph functions was initiated in [ref] where they were found to satisfy identities that extend the well-known relations between multiple zeta-values to modular functions.
Single-valued multiple zeta values in genus 1 superstring amplitudes · Introduction · 0.93
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A more general analysis of genus-one modular graph functions was initiated in [ref] where they were found to satisfy identities that extend the well-known relations between multiple zeta-values to modular functions.
Modular Graph Functions · Introduction · 0.93
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