Heriot-Watt Mathematics Report Series
HWM06-34, 13 Sep 2006
Topological Strings, Two-Dimensional Yang-Mills Theory and Chern-Simons Theory on Torus Bundles
N Caporaso, M Cirafici, L Griguolo, S Pasquetti, D Seminara and R J Szabo
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.
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