Topological strings and Nekrasov's formulas

We apply the method of geometric transition and compute all genus topologicalclosed string amplitudes compactified on local {\bf F}_0 by making use of theChern-Simons gauge theory. We find an exact agreement of the results of ourcomputation 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 sizeof the fiber of {\bf F}_0 and \hbar corresponds to the string couplingconstant. Thus Nekrasov's formula encodes all the information of topological stringamplitudes on local {\bf F}_0 including the number of holomorphic curves atarbitrary genus. By taking suitable limits \beta and/or \hbar \to 0 onerecovers the four-dimensional Seiberg-Witten theory and also its coupling toexternal graviphoton fields. We also compute topological string amplitude for the local 2nd del Pezzosurface and check the consistency with Nekrasov's formula of SU(2) gauge theorywith a matter field in the vector representation.Comment: 38 pages, 4 figures, (v2) References added, Appendices expanded, (v3) Typos corrected, References adde