Integrable hierarchies, Hurwitz numbers and a branch point field in critical topologically massive gravity

TL;DR

This paper explores the integrable structures of a partition function in critical topologically massive gravity, revealing its connections to Hurwitz numbers, KP tau functions, and branch point fields, thus linking gravity, algebraic geometry, and integrable systems.

Contribution

It demonstrates that the partition function acts as a generating function for Hurwitz numbers and is a KP tau function, providing new insights into the integrable structure of the theory.

Findings

Partition function is a generating function of Hurwitz numbers.

Partition function is a KP tau function, indicating solitonic properties.

Logarithmic fields correspond to branch point fields in the orbifold target space.

Abstract

We discuss integrable aspects of the logarithmic contribution of the partition function of cosmological critical topologically massive gravity. On one hand, written in terms of Bell polynomials which describe the statistics of set partitions, the partition function of the logarithmic fields is a generating function of the potential Burgers hierarchy. On the other hand, the polynomial variables are solutions of the Kadomtsev-Petviashvili equation, and the partition function is a KP $\tau$ function, making more precise the solitonic nature of the logarithmic fields being counted. We show that the partition function is a generating function of Hurwitz numbers, and derive its expression. The fact that the partition function is the generating function of branched coverings gives insight on the orbifold target space. We show that the logarithmic field $\psi^{new}_{\mu \nu}$ can be regarded as a branch point field associated to the branch point $\mu l =1$.