Deformed SW curve and the null vector decoupling equation in Toda field theory

TL;DR

This paper demonstrates how the deformed Seiberg-Witten curve relates to a differential equation in Toda field theory, providing a new proof of the AGT correspondence and extending it to include degenerate fields in conformal blocks.

Contribution

It establishes a direct mapping between the deformed SW curve and a differential equation in Toda CFT with degenerate fields, extending the AGT correspondence.

Findings

Mapped deformed SW curve to a differential equation in Toda CFT

Provided an independent proof of AGT in Nekrasov-Shatashvili limit

Extended AGT to include secondary degenerate fields in conformal blocks

Abstract

It is shown that the deformed Seiberg-Witten curve equation after Fourier transform is mapped into a differential equation for the AGT dual 2d CFT cnformal block containing an extra completely degenerate field. We carefully match parameters in two sides of duality thus providing not only a simple independent prove of the AGT correspondence in Nekrasov-Shatashvili limit, but also an extension of AGT to the case when a secondary field is included in the CFT conformal block. Implications of our results in the study of monodromy problems for a large class of $n$'th order Fuchsian differential equations are discussed.