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Single-valued multiple zeta values in genus 1 superstring amplitudes

Federico Zerbini

Abstract

We study the modular graph functions introduced by Green, Russo, Vanhove in the context of type II superstring scattering amplitudes of 4 gravitons on a torus. In particular we describe a method to algorithmically compute the coefficients in their expansion at the cusp in terms of conical sums. We perform explicit computations for 3-graviton functions, which naturally suggest to conjecture that only single-valued multiple zeta values appear.

Single-valued multiple zeta values in genus 1 superstring amplitudes

Abstract

We study the modular graph functions introduced by Green, Russo, Vanhove in the context of type II superstring scattering amplitudes of 4 gravitons on a torus. In particular we describe a method to algorithmically compute the coefficients in their expansion at the cusp in terms of conical sums. We perform explicit computations for 3-graviton functions, which naturally suggest to conjecture that only single-valued multiple zeta values appear.

Paper Structure

This paper contains 5 sections, 9 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Conical sums
  3. The 2-point case
  4. The 3-point case
  5. The 4-point case

Key Result

Theorem 1.1

For every $\underline{l}=(l_1,l_2,l_3,l_4,l_5,l_6)$ we have where for every $\mu,\nu\geq 0$ is a Laurent polynomial with coefficients $a_j^{(\mu,\nu)}$ lying in the algebra of conical sums $\mathcal{C}$, which will be defined in the next section, and $q=e^{2\pi i\tau}$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 1
  • Theorem 3.1
  • Corollary 1
  • Theorem 3.2
  • Theorem 4.1
  • ...and 3 more