Invariants of algebraic curves and topological expansion

TL;DR

This paper introduces a new sequence of invariants for algebraic curves that connect to topological expansions in matrix models, revealing their modular properties and applications to integrals like Kontsevitch's.

Contribution

It defines novel invariants for algebraic curves that unify topological expansions across various matrix models and integrable systems.

Findings

Invariants match topological expansions of matrix models.

Invariants satisfy Hirota equations, leading to tau functions.

Application to Kontsevitch integral shows dependence only on odd times.

Abstract

For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the curve becomes singular. In addition we find that they can be used to define a formal series, which satisfies formally an Hirota equation, and we thus obtain a new way of constructing a tau function attached to an algebraic curve. These invariants are constructed in order to coincide with the topological expansion of a matrix formal integral, when the algebraic curve is chosen as the large N limit of the matrix model's spectral curve. Surprisingly, we find that the same invariants also give the topological expansion of other models, in particular the matrix model with an external field, and the so-called double scaling limit of matrix models, i.e. the (p,q) minimal models of conformal field theory. As an example to illustrate the efficiency of our method, we apply it to the Kontsevitch integral, and we give a new and extremely easy proof that Kontsevitch integral depends only on odd times, and that it is a KdV tau-function.