Generalized Toda Theory from Six Dimensions and the Conifold

TL;DR

This paper explores a geometric extension of the AGT correspondence involving six-dimensional defects and generalized conifolds, broadening the understanding of boundary modes and their relation to Toda theory.

Contribution

It introduces a geometric framework for a generalized AGT correspondence using six-dimensional defects and generalized conifolds, extending previous derivations.

Findings

Generalized conifolds model the extended AGT setup.

The ordinary conifold clarifies features of the original AGT derivation.

Boundary modes lead to Toda theory on a Riemann surface.

Abstract

Recently, a physical derivation of the Alday-Gaiotto-Tachikawa correspondence has been put forward. A crucial role is played by the complex Chern-Simons theory arising in the 3d-3d correspondence, whose boundary modes lead to Toda theory on a Riemann surface. We explore several features of this derivation and subsequently argue that it can be extended to a generalization of the AGT correspondence. The latter involves codimension two defects in six dimensions that wrap the Riemann surface. We use a purely geometrical description of these defects and find that the generalized AGT setup can be modeled in a pole region using generalized conifolds. Furthermore, we argue that the ordinary conifold clarifies several features of the derivation of the original AGT correspondence.