Topological Gauge Theories on Local Spaces and Black Hole Entropy Countings

TL;DR

This paper develops a method to reduce cohomological gauge theories on local spaces to their base manifolds using equivariant localization, with applications to black hole entropy and Gromov-Witten invariants.

Contribution

It provides a rigorous localization framework for gauge theories on local spaces, connecting to black hole entropy and enumerative invariants in string theory.

Findings

Proved reduction of gauge theories via path integral localization.

Matched four-dimensional gauge theory results with boundary contributions.

Studied abelian gauge theories on Calabi-Yau surfaces related to Donaldson-Thomas invariants.

Abstract

We study cohomological gauge theories on total spaces of holomorphic line bundles over complex manifolds and obtain their reduction to the base manifold by U(1) equivariant localization of the path integral. We exemplify this general mechanism by proving via exact path integral localization a reduction for local curves conjectured in hep-th/0411280, relevant to the calculation of black hole entropy/Gromov-Witten invariants. Agreement with the four-dimensional gauge theory is recovered by taking into account in the latter non-trivial contributions coming from one-loop fluctuations determinants at the boundary of the total space. We also study a class of abelian gauge theories on Calabi-Yau local surfaces, describing the quantum foam for the A-model, relevant to the calculation of Donaldson-Thomas invariants.