Gauge theories on compact toric manifolds

TL;DR

This paper computes the supersymmetric partition function of gauge theories on compact toric manifolds using equivariant localization, revealing wall-crossing phenomena, dualities, and connections to Donaldson invariants and anomaly equations.

Contribution

It provides explicit formulas for the partition function on toric manifolds, linking residues of $ ext{C}^2$ partition functions with wall-crossing and duality symmetries, and explores gauge coupling anomalies.

Findings

Partition function is piecewise constant with wall-crossing behavior.

Residue calculations are simplified via an abstruse duality.

Reproduces known Donaldson invariants and reveals gauge coupling anomalies.

Abstract

We compute the ${\cal N}=2$ supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the K\"ahler form with jumps along the walls where the gauge symmetry gets enhanced. The partition function on such manifolds is written as a sum over the residues of a product of partition functions on $\mathbb{C}^2$. The evaluation of these residues is greatly simplified by using an "abstruse duality" that relates the residues at the poles of the one-loop and instanton parts of the $\mathbb{C}^2$ partition function. As particular cases, our formulae compute the $SU(2)$ and $SU(3)$ {\it equivariant} Donaldson invariants of $\mathbb{P}^2$ and $\mathbb{F}_n$ and in the non-equivariant limit reproduce the results obtained via wall-crossing and blow up methods in the $SU(2)$ case. Finally, we show that the $U(1)$ self-dual connections induce an anomalous dependence on the gauge coupling, which turns out to satisfy a $\mathcal{N}=2$ analog of the $\mathcal{N}=4$ holomorphic anomaly equations.