Higher genus modular graph functions, string invariants, and their exact asymptotics

TL;DR

This paper extends the theory of modular graph functions to higher genus surfaces, analyzing their asymptotic behavior under degenerations, with applications to string amplitude computations and new exact results.

Contribution

It develops a general method for analyzing higher genus modular graph functions' asymptotics, revealing their Laurent polynomial structure in degeneration limits.

Findings

Modular graph functions degenerate to Laurent polynomials in parameter t.

Coefficients are generalized modular graph functions on lower genus surfaces.

Results are exact to all orders in t, with exponentially suppressed corrections.

Abstract

The concept and the construction of modular graph functions are generalized from genus-one to higher genus surfaces. The integrand of the four-graviton superstring amplitude at genus-two provides a generating function for a special class of such functions. A general method is developed for analyzing the behavior of modular graph functions under non-separating degenerations in terms of a natural real parameter $t$. For arbitrary genus, the Arakelov Green function and the Kawazumi-Zhang invariant degenerate to a Laurent polynomial in $t$ of degree $(1,1)$ in the limit $t\to\infty$. For genus two, each coefficient of the low energy expansion of the string amplitude degenerates to a Laurent polynomial of degree $(w,w)$ in $t$, where $w+2$ is the degree of homogeneity in the kinematic invariants. These results are exact to all orders in $t$, up to exponentially suppressed corrections. The non-separating degeneration of a general class of modular graph functions at arbitrary genus is sketched and similarly results in a Laurent polynomial in $t$ of bounded degree. The coefficients in the Laurent polynomial are generalized modular graph functions for a punctured Riemann surface of lower genus.