Instanton counting with a surface operator and the chain-saw quiver

TL;DR

This paper characterizes the moduli space of SU(N) instantons with surface operators using chain-saw quivers, enabling explicit computation of the instanton partition function and its relation to W-algebra Verma modules.

Contribution

It introduces a novel description of instanton moduli spaces with surface operators via chain-saw quivers, linking partition functions to W-algebra structures.

Findings

Partition function expressed as a sum over Young diagram fixed points.

Partition function depends on the ordering of n_I, reflecting parabolic structure.

Agreement between partition function and W-algebra Verma module norms.

Abstract

We describe the moduli space of SU(N) instantons in the presence of a general surface operator of type N=n_1+ ... +n_M in terms of the representations of the so-called chain-saw quiver, which allows us to write down the instanton partition function as a summation over the fixed point contributions labeled by Young diagrams. We find that the instanton partition function depends on the ordering of n_I which fixes a choice of the parabolic structure. This is in accord with the fact that the Verma module of the W-algebra also depends on the ordering of n_I. By explicit calculations, we check that the partition function agrees with the norm of a coherent state in the corresponding Verma module.