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 F0 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 N = 2 SU(2) gauge theory with two parameters β and ℏ. β is related to the size of the fiber of F0 and ℏ corresponds to the string coupling constant. Thus Nekrasov's formula encodes all the information of topological string amplitudes on local F0 including the number of holomorphic curves at arbitrary genus. By taking suitable limits β and/or ℏ → 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. © SISSA/ISAS 2003.