Exact results for ${\cal N}=2$ supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants

TL;DR

This paper derives an exact contour integral formula for ${ m N}=2$ supersymmetric gauge theories on compact toric four-manifolds, evaluates it explicitly for $U(2)$ on $P^2$, and connects it to Donaldson invariants and mock modular forms.

Contribution

It provides the first explicit contour integral formula for ${ m N}=2$ theories on compact toric manifolds and links supersymmetric localization to Donaldson invariants.

Findings

Explicit contour integral formula for ${ m N}=2$ partition functions.

Connection between supersymmetric gauge theories and Donaldson invariants.

Generation of new equivariant Donaldson polynomials for specific cases.

Abstract

We provide a contour integral formula for the exact partition function of ${\cal N}=2$ supersymmetric $U(N)$ gauge theories on compact toric four-manifolds by means of supersymmetric localisation. We perform the explicit evaluation of the contour integral for $U(2)$ ${\cal N}=2$ theory on $\mathbb{P}^2$ for all instanton numbers. In the zero mass case, corresponding to the ${\cal N}=4$ supersymmetric gauge theory, we obtain the generating function of the Euler characteristics of instanton moduli spaces in terms of mock-modular forms. In the decoupling limit of infinite mass we find that the generating function of local and surface observables computes equivariant Donaldson invariants, thus proving in this case a long-standing conjecture by N. Nekrasov. In the case of vanishing first Chern class the resulting equivariant Donaldson polynomials are new.