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String theory amplitudes and partial fractions

Hjalmar Rosengren

TL;DR

The paper provides rigorous proofs and generalizations of the symmetric and asymmetric partial fraction expansions for string amplitudes originally found by Saha and Sinha. It develops a general framework for symmetric rational functions in $r$ variables, reduces multi-variable partial fractions to one-variable problems, and then applies these results to open- and closed-string amplitudes, yielding explicit infinite-series representations that converge under precise conditions. Key contributions include new parameterized symmetric/asymmetric expansions, limit-transition techniques from finite to infinite series, and connections to known constants such as $\pi$ via specialized parameter choices. The work enhances the mathematical understanding of string amplitudes and offers versatile expansions that may aid both theoretical analyses and computational evaluations in string theory.

Abstract

We give rigorous proofs and generalizations of partial fraction expansions for string amplitudes that were recently discovered by Saha and Sinha.

String theory amplitudes and partial fractions

TL;DR

The paper provides rigorous proofs and generalizations of the symmetric and asymmetric partial fraction expansions for string amplitudes originally found by Saha and Sinha. It develops a general framework for symmetric rational functions in variables, reduces multi-variable partial fractions to one-variable problems, and then applies these results to open- and closed-string amplitudes, yielding explicit infinite-series representations that converge under precise conditions. Key contributions include new parameterized symmetric/asymmetric expansions, limit-transition techniques from finite to infinite series, and connections to known constants such as via specialized parameter choices. The work enhances the mathematical understanding of string amplitudes and offers versatile expansions that may aid both theoretical analyses and computational evaluations in string theory.

Abstract

We give rigorous proofs and generalizations of partial fraction expansions for string amplitudes that were recently discovered by Saha and Sinha.
Paper Structure (5 sections, 5 theorems, 91 equations)

This paper contains 5 sections, 5 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Partial fraction expansions for symmetric rational functions
  3. Expansions of open string amplitudes
  4. Asymmetric expansions of closed string amplitudes
  5. Symmetric expansions of closed string amplitudes

Key Result

Proposition 2.1

Let $P$ be a symmetric polynomial of $r$ variables, which is of degree at most $n+1$ in each variable. Let $Q_0,\dots,Q_n$ be symmetric polynomials that are of degree exactly $1$ in each variable. Then, where $\mathbf t_k$ denotes the vector defined up to permutations by

Theorems & Definitions (6)

  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Corollary 3.1
  • Corollary 4.1
  • Corollary 5.1