Advances in Mathematical Physics
Volume 2010 (2010), Article ID 107857, 70 pages
Review Article

Instantons, Topological Strings, and Enumerative Geometry

1Department of Mathematics, Heriot-Watt University, Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, UK
2Maxwell Institute for Mathematical Sciences, Edinburgh, UK

Received 21 December 2009; Accepted 12 March 2010

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.