WKB expansion of Yang-Yang generating function and Bergman tau-function

TL;DR

This paper explores the symplectic structure of monodromy maps for second order equations on Riemann surfaces, linking WKB analysis, the Yang-Yang function, and the Bergman tau-function to deepen understanding of their geometric and analytical properties.

Contribution

It establishes a connection between the WKB expansion of the Yang-Yang generating function and the Bergman tau-function, revealing new insights into the symplectic geometry of monodromy maps.

Findings

Poisson bracket implies Goldman Poisson structure

Leading WKB term linked to Bergman tau-function

Monodromy symplectomorphism characterized by new geometric relations

Abstract

We study the symplectic properties of the monodromy map of second order equations on a Riemann surface whose potential is meromorphic with double poles. We show that the Poisson bracket defined in terms of periods of meromorphic quadratic differential implies the Goldman Poisson structure on the monodromy manifoldThese results are applied to the WKB analysis of the equation. It is shown that the leading term in the WKB expansion of the generating function of the monodromy symplectomorphism (the "Yang-Yang function" of Nekrasov, Rosly and Shatashvili) is determined by the Bergman tau-function on the moduli space of meromorphic quadratic differentials.