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Topological gauge theories on local spaces and black hole entropy countings. (English) Zbl 1153.83352

Summary: 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 arXiv: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 fluctuation 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.

MSC:

83C57 Black holes
81T13 Yang-Mills and other gauge theories in quantum field theory
81T45 Topological field theories in quantum mechanics