Advances in Theoretical and Mathematical Physics
Volume 12 (2008)
Topological strings, two-dimensional Yang-Mills theory and Chern-Simons theory on torus bundles
Pages: 981 – 1058
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.