Seiberg-Witten Theory and Extended Toda Hierarchy

TL;DR

This paper constructs the quasiclassical solution to the extended Toda hierarchy related to deformed Seiberg-Witten theory, expressing derivatives via integrals and deriving Virasoro constraints, with potential nonabelian generalizations.

Contribution

It explicitly formulates the quasiclassical solution of the extended Toda hierarchy in terms of complex curves and introduces Virasoro constraints for the deformed Seiberg-Witten theory.

Findings

Explicit construction of the quasiclassical solution

Expression of derivatives through multiple integrals

Derivation of Virasoro constraints

Abstract

The quasiclassical solution to the extended Toda chain hierarchy, corresponding to the deformation of the simplest Seiberg-Witten theory by all descendants of the dual topological string model, is constructed explicitly in terms of the complex curve and generating differential. The first derivatives of prepotential or quasiclassical tau-function over the extra times, extending the Toda chain, are expressed through the multiple integrals of the Seiberg-Witten one-form. We derive the corresponding quasiclassical Virasoro constraints, discuss the functional formulation of the problem and propose generalization of the extended Toda hierarchy to the nonabelian theory.