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

TL;DR

The paper develops a comprehensive framework for higher-genus modular graph functions, anchored by the Arakelov Green function and the canonical Kähler form, and analyzes their exact non-separating degeneration as the degeneration parameter $t\to\infty$. It constructs a simple generating class ${\\cal F}^{(h)}(s_{ij}|\\Omega)$ and explicitly studies the genus-two four-graviton amplitude ${\\cal B}^{(2)}(s_{ij}|\\Omega)$, deriving Laurent-polynomial asymptotics with coefficients given by genus-$h$ modular graph data and their puncture-generalizations. The authors prove four theorems (Theorems 1–4) describing how key objects such as the Arakelov Green function, Kawazumi–Zhang invariant, and string invariants degenerate, and they demonstrate that genus-two amplitudes decompose into $t$-polynomials with coefficients that are invariant under the relevant modular groups and depend on lower-genus moduli and puncture data. The results provide exact, all-orders-in-$t$ asymptotics up to exponentially suppressed terms, offering a robust blueprint for understanding higher-genus invariants and their relations to string perturbation theory and Arakelov geometry, with implications for tropical limits and supergravity matching.

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.

Figures (8)

• Figure 1: Canonical homology cycles on a compact genus-two Riemann surface $\Sigma$.
• Figure 2: The contributions to ${\cal B}_{(0,1)}$ arise from the above graph with coefficient $s_{12}$ plus the graphs obtained by all permuting the points $(1,2)$ to $(i,j)$ multiplied by the coefficient $s_{ij}$.
• Figure 3: The contributions to ${\cal B}_{(2,0)}^{(a)}$, ${\cal B}_{(2,0)}^{(b)}$, ${\cal B}_{(2,0)}^{(c)}$ arise respectively from the above graphs $(a), (b), (c)$ with coefficient $(s_{12})^2, s_{12}s_{14}$ and $s_{12}s_{34}$, plus the graphs obtained by permuting the points and multiplying by the corresponding coefficient bilinear in the $s_{ij}$.
• Figure 4: The non-separating degeneration of a genus-two surface.
• Figure 5: Fay's construction of a surface near a non-separating divisor in terms of a surface $\Sigma_h$, punctures $p_a,p_b$, and the annuli bounded by $\mathfrak{C}'_a, \mathfrak{C}"_a$ and by $\mathfrak{C}"_b, \mathfrak{C}'_b$ identified for $h=1$.