Contour Integral for the Partition Function of $\mathcal{N}=2$ Topologically Twisted on $\mathbb{CP}^2$ and Physical Fluxes

TL;DR

This paper computes a contour integral for the partition function of an $ =2$ topologically twisted theory on $ ext{CP}^2$, revealing a simplified flux dependence and new equivariant invariants related to Donaldson invariants.

Contribution

It introduces a novel contour integral approach that reduces flux summation and uncovers new equivariant invariants of $ ext{CP}^2$ linked to gauge bundle stability.

Findings

Partition function depends on a single physical flux.

Contour captures more poles, refining previous flux sum approaches.

New equivariant invariants generalize Donaldson invariants.

Abstract

We compute the contour integral for the partition function of an $\mathcal{N}=2$ $SU(2)$ topologically twisted theory on $\mathbb{CP}^2$, dimensionally reducing from an $\mathcal{N}=1$ theory on $S^5$. Earlier works presented the partition function as a sum over three equivariant fluxes, one for each toric divisor of $\mathbb{CP}^2$. Our result depends only on a single physical flux, assigned to the non-trivial two-cycle of the manifold. The reduced summation over fluxes is compensated by a contour of integration, arising from a different solution of the BPS equations, which captures more poles in each topological sector. As our observable involves a position-dependent Yang-Mills coupling, we compute new equivariant invariants of $\mathbb{CP}^2$, which reduce to Donaldson invariants in the non-equivariant limit. Stability conditions of gauge bundles over $\mathbb{CP}^2$ appear intrinsically via the dimensional reduction.