Point-like limit of the hyperelliptic Zhang-Kawazumi invariant

TL;DR

This paper investigates the boundary behavior of the Zhang-Kawazumi invariant on the moduli space of Riemann surfaces, establishing its point-like limit in hyperelliptic cases via combinatorial dual graphs.

Contribution

It proves the existence of the point-like limit of the Zhang-Kawazumi invariant for hyperelliptic Riemann surfaces and relates it to a combinatorial invariant on dual graphs.

Findings

Point-like limit exists for hyperelliptic Riemann surfaces.

Limit is given by a combinatorial analogue on dual graphs.

Connects geometric invariants with combinatorial structures.

Abstract

The behavior near the boundary in the Deligne-Mumford compactification of many functions on the moduli space of pointed Riemann surfaces can be conveniently expressed using the notion of "point-like limit" that we adopt from the string theory literature. In this note we study a function on the moduli space of Riemann surfaces that has been introduced by N. Kawazumi and S. Zhang, independently. We show that the point-like limit of the Zhang-Kawazumi invariant in a family of hyperelliptic Riemann surfaces in the direction of any hyperelliptic stable curve exists, and is given by evaluating a combinatorial analogue of the Zhang-Kawazumi invariant, also introduced by Zhang, on the dual graph of that stable curve.