[Paper Review] Elliptic multiple zeta values, modular graph functions and genus 1 superstring scattering amplitudes

Federico Zerbini|arXiv (Cornell University)|Apr 21, 2018

Advanced Mathematical Identities44 references16 citations

TL;DR

This paper introduces and studies elliptic multiple zeta values and modular graph functions—non-holomorphic and holomorphic analogues of multiple zeta values—showcasing their role in genus one superstring scattering amplitudes. It derives new asymptotic expansions for these functions, enabling explicit computations and revealing structural parallels between genus zero and genus one amplitudes.

ABSTRACT

In this PhD thesis we study holomorphic and non-holomorphic elliptic analogues of multiple zeta values, namely elliptic multiple zeta values and modular graph functions. Both classes of functions have been discovered very recently, and are involved in the computation of genus one superstring amplitudes. In particular, we obtain new results on the asymptotic expansion of these functions that allow us to perform explicit computations and point out analogies between genus zero and genus one amplitudes.

Motivation & Objective

• To investigate holomorphic and non-holomorphic elliptic analogues of multiple zeta values, specifically elliptic multiple zeta values and modular graph functions.
• To understand the role of these functions in the computation of genus one superstring scattering amplitudes.
• To derive new asymptotic expansions for these functions to enable explicit computations in string amplitude evaluations.
• To identify structural analogies between genus zero and genus one superstring amplitudes through the lens of these special functions.
• To establish a framework connecting number theory (elliptic MZVs) with string theory (modular graph functions) via asymptotic analysis.

Proposed method

• Introduces elliptic multiple zeta values as holomorphic modular graph functions with elliptic modular properties.
• Analyzes modular graph functions as non-holomorphic generalizations of multiple zeta values, defined via lattice sums and Eisenstein series.
• Applies asymptotic expansion techniques to modular graph functions in the limit of degenerating torus moduli.
• Uses conical sum computations and symbolic manipulation tools (e.g., HyperInt) to evaluate special function values.
• Relies on advanced number theory and modular forms to analyze the structure and behavior of these functions.
• Establishes connections between the asymptotic behavior of these functions and the structure of string amplitudes at genus one.

Experimental results

Research questions

• RQ1 How do elliptic multiple zeta values generalize classical multiple zeta values in the context of genus one string amplitudes?
• RQ2 What are the asymptotic behaviors of modular graph functions in the degeneration limit of the torus?
• RQ3 In what ways do the structures of genus one superstring amplitudes mirror those of genus zero amplitudes?
• RQ4 How can asymptotic expansions of modular graph functions be computed explicitly to facilitate amplitude evaluations?
• RQ5 What is the precise relationship between elliptic multiple zeta values and modular graph functions in string theory?

Key findings

• New asymptotic expansions for modular graph functions are derived, enabling explicit computation of genus one superstring scattering amplitudes.
• The paper establishes a structural analogy between genus zero and genus one superstring amplitudes through the behavior of these special functions.
• Elliptic multiple zeta values are shown to be natural holomorphic counterparts of modular graph functions, enriching the modular structure of string amplitudes.
• The asymptotic analysis reveals that modular graph functions exhibit a decomposition into multiple zeta values and their elliptic generalizations in the degeneration limit.
• Computational tools such as HyperInt are effectively used to evaluate conical sums, supporting the analytical results.
• The work provides a systematic framework to relate number-theoretic objects (elliptic MZVs) to physical quantities (string amplitudes) at genus one.