Topological Strings and Nekrasov's formulas

Abstract

We apply the method of geometric transition and compute all genus topological closed string amplitudes compactified on local {\bf F}_0 by making use of the Chern-Simons gauge theory. We find an exact agreement of the results of our computation with the formula proposed recently by Nekrasov for {\cal N}=2 SU(2) gauge theory with two parameters \beta and \hbar. \beta is related to the size of the fiber of {\bf F}_0 and \hbar corresponds to the string coupling constant. Thus Nekrasov's formula encodes all the information of topological string amplitudes on local {\bf F}_0 including the number of holomorphic curves at arbitrary genus. By taking suitable limits \beta and/or \hbar \to 0 one recovers the four-dimensional Seiberg-Witten theory and also its coupling to external graviphoton fields. We also compute topological string amplitude for the local 2nd del Pezzo surface and check the consistency with Nekrasov's formula of SU(2) gauge theory with a matter field in the vector representation.