[Paper Review] Genus-zero and genus-one string amplitudes and special multiple zeta values

Don Zagier, Federico Zerbini|arXiv (Cornell University)|Jun 28, 2019

Advanced Mathematical Identities25 references4 citations

TL;DR

This paper establishes a deep connection between genus-zero and genus-one string amplitudes in perturbative string theory by showing that the genus-one contribution to 2-point amplitudes is related to the genus-zero contribution to 4-point amplitudes through special linear combinations of multiple zeta values. The key result is a novel identity linking the generating series of genus-one modular graph functions to the closed-string Virasoro amplitude, proving that their coefficients lie in the rational span of special multiple zeta values and confirming a conjecture via Brown’s single-valued projection.

ABSTRACT

In this paper we show that in perturbative string theory the genus-one contribution to formal 2-point amplitudes can be related to the genus-zero contribution to 4-point amplitudes. This is achieved by studying special linear combinations of multiple zeta values that appear as coefficients of the amplitudes. We also exploit our results to relate closed strings to open strings at genus one using Brown's single-valued projection, proving a conjecture of Broedel, Schlotterer and the second author.

Motivation & Objective

• To establish a mathematical bridge between genus-zero and genus-one string amplitudes in perturbative string theory.
• To prove that the coefficients of genus-one modular graph functions (Dℓ(τ)) are rational linear combinations of special multiple zeta values Z(k,r).
• To relate closed-string amplitudes at genus one to open-string amplitudes using Brown’s single-valued projection, confirming a conjecture from the literature.

Proposed method

• The authors define special multiple zeta values Z(k,r) as weighted sums over compositions of integers into 1s and 2s, with coefficients depending on the number of 1s.
• They show that Taylor coefficients of the closed-string Virasoro amplitude V^(cl)(s,t,u) lie in the rational span of Z(k,r), using Euler’s formula for log Γ(1+x).
• A generating series W^(cl)(X,Y) is constructed from the coefficients of the Laurent polynomial dℓ(Y), which describes the genus-one amplitude.
• The key identity W^(cl)(X,Y) = V^(cl)(2X, -X-Y, Y-X) / [X(X+Y)(Y-X)] is derived, linking the genus-one amplitude to the genus-zero amplitude.
• The proof uses the metric-independence of Green’s functions on Riemann surfaces and their exponentiated forms H, which are invariant under changes of metric.
• The single-valued projection of multiple zeta values is applied to relate open and closed string amplitudes at genus one, confirming a conjecture.

Experimental results

Research questions

• RQ1 How are the coefficients of genus-one modular graph functions Dℓ(τ) related to multiple zeta values in the context of string amplitudes?
• RQ2 Can the genus-one contribution to 2-point amplitudes be expressed in terms of genus-zero 4-point amplitudes via special combinations of multiple zeta values?
• RQ3 Does Brown’s single-valued projection provide a consistent map between open and closed string amplitudes at genus one, as conjectured?
• RQ4 Are the coefficients of the Laurent polynomial dℓ(Y) in the asymptotic expansion of Dℓ(τ) rational linear combinations of Z(k,r) values?
• RQ5 What is the precise algebraic structure of the vector space spanned by the Taylor coefficients of the closed-string amplitude V^(cl)(s,t,u)?

Key findings

• The coefficients of the Laurent polynomial dℓ(Y) in the asymptotic expansion of the genus-one modular graph function Dℓ(τ) are shown to be rational linear combinations of the special multiple zeta values Z(k,r).
• The generating series W^(cl)(X,Y) for the coefficients γn,k of dℓ(Y) satisfies the identity W^(cl)(X,Y) = V^(cl)(2X, -X-Y, Y-X) / [X(X+Y)(Y-X)], linking genus-one amplitudes to genus-zero amplitudes.
• The Taylor coefficients e_{p,q} of the closed-string amplitude V^(cl)(s,t,u) lie in the rational vector space spanned by the Z(k,r) values, with both spaces having dimension O(w) for weight w.
• The paper confirms a conjecture that the genus-one closed-string amplitude can be related to the genus-zero open-string amplitude via Brown’s single-valued projection, establishing a direct link between open and closed strings at genus one.
• The special multiple zeta values Z(k,r) are shown to be the same ones appearing in the leading terms of non-holomorphic modular functions studied by Green, Russo, and Vanhove, providing a physical and number-theoretic consistency check.
• The vector space E_w spanned by the coefficients e_{p,q} of V^(cl) and the vector space D_w spanned by Z(k,r) are shown to be subspaces of the space of weight-w polynomials in odd zeta values, with both having dimension O(w).