Affine SL(2) conformal blocks from 4d gauge theories

TL;DR

This paper connects Nekrasov's instanton partition functions of 4d N=2 gauge theories with affine SL(2) conformal blocks, revealing their role as eigenfunctions of quantized Hitchin Hamiltonians in the critical limit.

Contribution

It demonstrates that Nekrasov's partition functions for SU(2) quiver gauge theories correspond to modified affine SL(2) conformal blocks, linking gauge theory and conformal field theory.

Findings

Partition functions match affine SL(2) conformal blocks.

Conformal blocks become eigenfunctions of Hitchin Hamiltonians.

Results extend to the critical limit of the theories.

Abstract

We study Nekrasov's instanton partition function of four-dimensional N=2 gauge theories in the presence of surface operators. This can be computed order by order in the instanton expansion by using results available in the mathematical literature. Focusing in the case of SU(2) quiver gauge theories, we find that the results agree with a modified version of the conformal blocks of affine SL(2) Lie algebra. These conformal blocks provide, in the critical limit, the eigenfunctions of the corresponding quantized Hitchin Hamiltonians.