On elementary proof of AGT relations from six dimensions

TL;DR

This paper presents an elementary proof of AGT relations from six-dimensional theories by explicitly choosing contours in integrals, reducing the relations to a symmetry identity, demonstrated with a simple example.

Contribution

It introduces a novel contour choice approach that simplifies the proof of AGT relations from 6D theories, emphasizing elementary symmetry arguments.

Findings

Contours can be explicitly chosen to simplify AGT relations

The approach reduces complex relations to elementary symmetry identities

The method applies to various cases beyond the example provided

Abstract

The actual definition of the Nekrasov functions participating in the AGT relations implies a peculiar choice of contours in the LMNS and Dotsenko-Fateev integrals. Once made explicit and applied to the original triply-deformed (6-dimensional) version of these integrals, this approach reduces the AGT relations to symmetry in q_{1,2,3}, which is just an elementary identity for an appropriate choice of the integration contour (which is, however, a little non-traditional). We illustrate this idea with the simplest example of N=(1,1) U(1) SYM in six dimensions, however, all other cases can be evidently considered in a completely similar way.