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 F_0 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 F_0 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 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.Comment: 39 pages, references added, typos corrected, cosmetic change

The Vertex on a Strip

We demonstrate that for a broad class of local Calabi-Yau geometries built around a string of IP^1's - those whose toric diagrams are given by triangulations of a strip - we can derive simple rules, based on the topological vertex, for obtaining expressions for the topological string partition function in which the sums over Young tableaux have been performed. By allowing non-trivial tableaux on the external legs of the corresponding web diagrams, these strips can be used as building blocks for more general geometries. As applications of our result, we study the behavior of topological string amplitudes under flops, as well as check Nekrasov's conjecture in its most general form.Comment: 26 pages, 12 figures; v2: minor corrections, version to appear in ATM

Impact of trade openness on output growth for Pakistan: an empirical investigation

In this paper we analyze the impact of trade liberalization policy on GDP growth of Pakistan for the period ranging from 1972 to 2002. We found that there is long run negative relationship between trade growth and GDP growth. When we separate the total trade volume in export and import we find insignificant positive relationship between GDP and export and import. Both the models showed positive and significant relationship between GDP and investment.Trade Openness; Liberalization