On the isomonodromic tau-function for the Hurwitz spaces of branched coverings of genus zero and one

TL;DR

This paper explicitly constructs the isomonodromic tau-function for Hurwitz spaces of genus zero and one coverings, linking it to Frobenius manifold structures and caustic interpretations.

Contribution

It provides explicit formulas for the tau-function on Hurwitz spaces and relates it to Frobenius manifold structures and caustic geometry.

Findings

Explicit construction of the tau-function for genus zero and one Hurwitz spaces

Reformulation of the tau-function and G-function in flat coordinates

Interpretation of the tau-function in terms of caustics and non-semisimple multiplication

Abstract

The isomonodromic tau-function for the Hurwitz spaces of branched coverings of genus zero and one are constructed explicitly. Such spaces may be equipped with the structure of a Frobenius manifold and this introduces a flat coordinate system on the manifold. The isomonodromic tau-function, and in particular the associated $G$-function, are rewritten in these coordinates and an interpretation in terms of the caustics (where the multiplication is not semisimple) is given.