Gauge theories on compact toric manifolds

Abstract

We compute the $\mathcal{N}=2$ 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 Kähler 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.

arXiv comment41 pages, 3 figures, discussion on SU(3) results moved to a different section, added a discussion on stability in section 2 and a mathematical appendix on this subject

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