N=2 gauge theories on toric singularities, blow-up formulae and W-algebrae

TL;DR

This paper computes Nekrasov partition functions for N=2 gauge theories on toric singularities, explores their modular properties, and connects them to W-algebra representations and conformal field theories via the AGT correspondence.

Contribution

It introduces blow-up formulae for gauge theories on toric singularities and links these to W-algebra representations and 2D CFTs, extending the AGT correspondence.

Findings

Derived explicit Nekrasov partition functions on toric singularities.

Established the modular properties and central charges of associated 2D CFTs.

Provided evidence for the correspondence with parafermionic theories.

Abstract

We compute the Nekrasov partition function of gauge theories on the (resolved) toric singularities C^2/\Gamma in terms of blow-up formulae. We discuss the expansion of the partition function in the \epsilon_1,\epsilon_2 \to 0 limit along with its modular properties and how to derive them from the M-theory perspective. On the two-dimensional conformal field theory side, our results can be interpreted in terms of representations of the direct sum of Heisenberg plus W_N-algebrae with suitable central charges, which can be computed from the fan of the resolved toric variety.We provide a check of this correspondence by computing the central charge of the two-dimensional theory from the anomaly polynomial of M5-brane theory. Upon using the AGT correspondence our results provide a candidate for the conformal blocks and three-point functions of a class of the two-dimensional CFTs which includes parafermionic theories.