Gauge theories on compact toric manifolds

We compute the N = 2 supersymmetric partition function of a gauge theory on a fourdimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the Kdhler 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 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 C-2 partition function. As particular cases, our formulae compute the SU (2) and SU (3) equivariant Donaldson invariants of P-2 and 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 N = 2 analog of the N = 4 holomorphic anomaly equations.

Automated summary

This study discusses the computation of the N = 2 supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The results show that the partition function expression has specific features that simplify the evaluation of residues through an abstruse duality. Ultimately, it is found that U(1) self-dual connections exhibit an anomalous dependence on the gauge coupling, satisfying an N = 2 analog of the N = 4 holomorphic anomaly equations.