Integrating simple genus two string invariants over moduli space

TL;DR

This paper investigates genus two string invariants integrated over moduli space, demonstrating that certain integrals reduce to boundary terms and ultimately vanish, revealing new insights into string theory invariants.

Contribution

It introduces methods to reduce complex genus two string invariants to boundary integrals, showing their vanishing and advancing understanding of string moduli space integrals.

Findings

Certain genus two string invariants' integrals reduce to boundary terms.

All considered integrals over moduli space vanish.

Boundary terms are fully determined by the Kawazumi--Zhang invariant.

Abstract

We consider an Sp(4,Z) invariant expression involving two factors of the Kawazumi--Zhang (KZ) invariant each of which is a modular graph with one link, and four derivatives on the moduli space of genus two Riemann surfaces. Manipulating it, we show that the integral over moduli space of a linear combination of a modular graph with two links and the square of the KZ invariant reduces to a boundary integral. We also consider an Sp(4,Z) invariant expression involving three factors of the KZ invariant and six derivatives on moduli space, from which we deduce that the integral over moduli space of a modular graph with three links reduces to a boundary integral. In both cases, the boundary term is completely determined by the KZ invariant. We show that both the integrals vanish.