Higher Genus Modular Graph Functions, String Invariants, and their Exact Asymptotics

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.