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Single-valued hyperlogarithms, correlation functions and closed string amplitudes

Pierre Vanhove, Federico Zerbini

TL;DR

This work develops an analytic, single-valued perspective on genus-zero closed-string amplitudes, unifying open- and closed-string data through the single-valued projection framework. It achieves a constructive holomorphic factorisation of conformal correlators into Aomoto–Gel'fand blocks, from which the KLT relations naturally emerge as specialisations, linking closed-string integrals to products of open-string data. Locally, it proves that the asymptotic coefficients in the α′-expansion are single-valued multiple zeta values by building a theory of integrating single-valued hyperlogarithms and showing that these coefficients live in the sv-MZV ring. The results provide an intuitive, technically explicit bridge between conformal-field-theory correlators, hyperlogarithms, and the arithmetic of string amplitudes, with potential extensions to higher genus. The approach also yields algorithmic avenues to compute α′-expansions and clarifies the appearance of single-valued periods in closed-string physics, reinforcing the role of single-valued periods in connecting open and closed string theories.

Abstract

We give new proofs of a global and a local property of the integrals which compute closed string theory amplitudes at genus zero. Both kinds of properties are related to the newborn theory of single-valued periods, and our proofs provide an intuitive understanding of this relation. The global property, known in physics as the KLT formula, is a factorisation of the closed string integrals into products of pairs of open string integrals. We deduce it by identifying closed string integrals with special values of single-valued correlation functions in two dimensional conformal field theory, and by obtaining their conformal block decomposition. The local property is of number theoretical nature. We write the asymptotic expansion coefficients as multiple integrals over the complex plane of special functions known as single-valued hyperlogarithms. We develop a theory of integration of single-valued hyperlogarithms, and we use it to demonstrate that the asymptotic expansion coefficients belong to the ring of single-valued multiple zeta values.

Single-valued hyperlogarithms, correlation functions and closed string amplitudes

TL;DR

This work develops an analytic, single-valued perspective on genus-zero closed-string amplitudes, unifying open- and closed-string data through the single-valued projection framework. It achieves a constructive holomorphic factorisation of conformal correlators into Aomoto–Gel'fand blocks, from which the KLT relations naturally emerge as specialisations, linking closed-string integrals to products of open-string data. Locally, it proves that the asymptotic coefficients in the α′-expansion are single-valued multiple zeta values by building a theory of integrating single-valued hyperlogarithms and showing that these coefficients live in the sv-MZV ring. The results provide an intuitive, technically explicit bridge between conformal-field-theory correlators, hyperlogarithms, and the arithmetic of string amplitudes, with potential extensions to higher genus. The approach also yields algorithmic avenues to compute α′-expansions and clarifies the appearance of single-valued periods in closed-string physics, reinforcing the role of single-valued periods in connecting open and closed string theories.

Abstract

We give new proofs of a global and a local property of the integrals which compute closed string theory amplitudes at genus zero. Both kinds of properties are related to the newborn theory of single-valued periods, and our proofs provide an intuitive understanding of this relation. The global property, known in physics as the KLT formula, is a factorisation of the closed string integrals into products of pairs of open string integrals. We deduce it by identifying closed string integrals with special values of single-valued correlation functions in two dimensional conformal field theory, and by obtaining their conformal block decomposition. The local property is of number theoretical nature. We write the asymptotic expansion coefficients as multiple integrals over the complex plane of special functions known as single-valued hyperlogarithms. We develop a theory of integration of single-valued hyperlogarithms, and we use it to demonstrate that the asymptotic expansion coefficients belong to the ring of single-valued multiple zeta values.

Paper Structure

This paper contains 39 sections, 27 theorems, 193 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Content
  3. Closed string amplitudes and correlation functions
  4. Closed string amplitudes and single-valued hyperlogarithms
  5. Organisation of the paper
  6. Hyperlogarithms
  7. Definition and first properties
  8. Rings of hyperlogarithms
  9. Special values of hyperlogarithms
  10. Generating series and monodromy
  11. Single-valued hyperlogarithms
  12. Special values of single-valued hyperlogarithms
  13. Correlation functions and closed string amplitudes
  14. Single-valuedness of CFT correlators
  15. Closed string theory amplitudes building blocks
  16. ...and 24 more sections

Key Result

Theorem 1.1

Let $N\in\mathbb{N}$. The $(N+1)!$ Aomoto-Gel'fand hypergeometric functions $I_{(\rho,\sigma)}(\pmb a,\pmb b,\pmb c;\pmb d;\eta):=F_{\Delta_{(\rho,\sigma)}(\eta)}(\pmb a,\pmb b,\pmb c;\pmb d;\eta)$ which are associated, for permutations $\rho\in\mathfrak{S}_r$, $\sigma\in\mathfrak{S}_s$ with $r+s=N$ form a basis the vector space of ($N$-dimensional) integrals AGgendef1varIntro over the field

Figures (3)

  • Figure 4.1: Integration contour of Lemma \ref{['lemmaproofbasis']}.
  • Figure 4.2: Two contours leading to eqs. \ref{['eq:twocontours']}.
  • Figure 5.1: The contour of integration $C^-$ for the variable $v^-+i\epsilon\delta$ depending on the position of $v^+$ on the real axis.

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Brown, BrownNote
  • Proposition 2.2: Brown, BrownNote
  • Theorem 2.3: Brown, BrownNote
  • Proposition 2.4
  • proof
  • Definition 2.5: Single-valued multiple zeta values Brown:2013gia
  • Proposition 2.6
  • ...and 38 more