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 nonperturbative 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 nonabelian 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.

62 pages, 5 figures

References in corpus (36)

• N=2 Extremal Black Holes
• Two Dimensional Gauge Theories Revisited
• Black Hole Attractors and the Topological String
• Macroscopic Entropy of $N=2$ Extremal Black Holes
• D-Branes and Topological Field Theories
• Chern-Simons theory, matrix integrals, and perturbative three-manifold invariants
• Quantum Foam and Topological Strings
• Microscopic Entropy of N=2 Extremal Black Holes
• Hodge integrals and degenerate contributions
• Quantum Calabi-Yau and Classical Crystals
• Two Dimensional Yang-Mills, Black Holes and Topological Strings
• Baby Universes in String Theory
• Two dimensional lattice gauge theory based on a quantum group
• Chern-Simons Theory on S^1-Bundles: Abelianisation and q-deformed Yang-Mills Theory
• Relative Lefschetz Action and BPS State Counting
• M-theory and a Topological String Duality
• Topological closed-string interpretation of Chern-Simons theory
• Phase transitions, double-scaling limit, and topological strings
• Phase transitions in q-deformed 2d Yang-Mills theory and topological strings
• Topological strings and large N phase transitions II: Chiral expansion of q-deformed Yang-Mills theory
• BPS Microstates and the Open Topological String Wave Function
• The String Partition Function for QCD on the Torus
• Chern-Simons-Witten Theory as a Topological Fermi Liquid
• Chern-Simons Theory, 2d Yang-Mills, and Lie Algebra Wanderers
• The formulae of Kontsevich and Verlinde from the perspective of the Drinfeld double
• Chern-Simons Theory in Lens Spaces from 2d Yang-Mills on the Cylinder
• Deviations from the Area Law for Supersymmetric Black Holes
• A DK Phase Transition in q-Deformed Yang-Mills on S^2 and Topological Strings
• Black-holes, topological strings and large N phase transitions
• A Note on Knot Invariants and q-Deformed 2d Yang-Mills
• The instanton contributions to Yang-Mills theory on the torus: localization, Wilson loops and the perturbative expansion
• Crystals and intersecting branes
• Conformal Field Theory Techniques in Large N Yang-Mills Theory
• On Exponential corrections to the 1/N expansion in two-dimensional Yang Mills theory
• Non-Abelian Localization For Chern-Simons Theory
• Instantons, Fluxons and Open Gauge String Theory

Cited by in corpus (4)

• Holomorphic maps and the complete 1/N expansion of 2D SU(N) Yang-Mills
• The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories
• Topological Gauge Theories on Local Spaces and Black Hole Entropy Countings
• Equivariant and supersymmetric localization in QFT