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# Advances in Theoretical and Mathematical Physics

## Volume 7 (2003)

### Number 3

### Instanton Counting and Chern-Simons Theory

Pages: 457 – 497

DOI: http://dx.doi.org/10.4310/ATMP.2003.v7.n3.a4

#### Authors

#### Abstract

The instanton partition function of *N* = 2, *D* = 4, *SU*(2) gauge theory is obtained by taking the field theory limit of the topological open string partition function, given by a Chern-Simons theory, of a CY3-fold. The CY3-fold on the open string side is obtained by geometric transition from local *IP*^{1} X *IP*^{1} which is used in the geometric engineering of the *SU*(2) theory. The partition function obtained from the Chern-Simons theory agrees with the closed topological string partition function of local *IP*^{1} X *IP*^{1} proposed recently by Nekrasov. We also obtain the partition functions for local *IF*^{1} and *IF*^{2} CY3-folds and show that the topological string amplitudes of all three local Hirzebruch surfaces give rise to the same field theory limit. It is shown that a generalization of the topological closed string partition function whose field theory limit is the generalization of the instanton partition function, proposed by Nekrasov, can be determined easily from the Chern-Simons theory.