Topological strings, two-dimensional Yang-Mills theory and Chern-Simons theory on torus bundles

Abstract

We study the relations between two-dimensional Yang–Mills theory on the torus, topological string theory on a Calabi–Yau threefold whose local geometry is the sum of two line bundles over the torus, and Chern–Simons theory on torus bundles. The chiral partition function of the Yang–Mills gauge theory in the large $N$ limit is shown to coincide with the topological string amplitude computed by topological vertex techniques. We use Yang–Mills theory as an efficient tool for the computation of Gromov–Witten invariants and derive explicitly their relation with Hurwitz numbers of the torus. We calculate the Gopakumar–Vafa invariants, whose integrality gives a non-trivial confirmation of the conjectured non-perturbative relation between two-dimensional Yang–Mills theory and topological string theory. We also demonstrate how the gauge theory leads to a simple combinatorial solution for the Donaldson–Thomas theory of the Calabi–Yau background. We match the instanton representation of Yang–Mills theory on the torus with the non-abelian localization of Chern–Simons gauge theory on torus bundles over the circle. We also comment on how these results can be applied to the computation of exact degeneracies of $BPS$ black holes in the local Calabi–Yau background.