Structure of Dubrovin-Zhang free energy functions and universal identities

TL;DR

This paper establishes a structural relationship between higher genus free energy functions of Dubrovin-Zhang hierarchies and the Witten-Kontsevich free energy, introducing universal identities applicable across these hierarchies.

Contribution

It provides a structural theorem linking Dubrovin-Zhang and Witten-Kontsevich free energies and constructs universal identities valid for all Dubrovin-Zhang hierarchies at any genus.

Findings

Proved a structural theorem relating higher genus free energies.

Constructed universal identities for Dubrovin-Zhang hierarchies.

Developed techniques to derive identities without moduli space geometry.

Abstract

We prove a structural theorem relating the higher genera free energy functions of the Dubrovin-Zhang hierarchies to the Witten-Kontsevich free energy function of the Korteweg-de Vries hierarchy. As an important application, for any given genus $g\geq 1$, we construct a set of universal identities valid for the free energy functions of any Dubrovin-Zhang hierarchy. In particular, we present some techniques that can be used to derive universal identities without relying on the geometry of the moduli space of stable curves of higher genus.