Instanton Counting and Chern-Simons Theory

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 × 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 × IP 1 proposed recently by Nekrasov.We also obtain the partition functions for local F 1 and F 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.