Academic Editor: M. N. Hounkonnou
Copyright © 2010 Richard J. Szabo. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We review and elaborate on certain aspects of the connections between instanton counting in
maximally supersymmetric gauge theories and the computation of enumerative invariants of
smooth varieties. We study in detail three instances of gauge theories in six, four, and two
dimensions which naturally arise in the context of topological string theory on certain noncompact
threefolds. We describe how the instanton counting in these gauge theories is related
to the computation of the entropy of supersymmetric black holes and how these results are
related to wall-crossing properties of enumerative invariants such as Donaldson-Thomas and
Gromov-Witten invariants. Some features of moduli spaces of torsion-free sheaves and the
computation of their Euler characteristics are also elucidated.