Topological Strings and Nekrasov's formulas

TL;DR

Using geometric transition and the topological vertex, Eguchi and Kanno compute all-genus topological-string amplitudes on local ${\bf F}_0$ and prove exact agreement with Nekrasov's five-dimensional ${\cal N}=2$ SU(2) instanton formula parameterized by $\beta$ and $\hbar$, identifying $\beta$ with the fiber size and $\hbar$ with the string coupling. They extend the analysis to blow-ups yielding ${\bf dP}_2$ (with matter) and establish precise Nekrasov–topological-string correspondences, including four-dimensional limits that recover Seiberg-Witten prepotentials and their gravity couplings. The work also proves key Hopf-link invariant identities and provides explicit genus-zero, -one, -two, and -three GV invariants, underscoring Nekrasov's formula as encoding complete closed-string data for these local geometries. Together, the results bridge topological string theory, 5D/4D gauge theories, and GV/GW invariants, and point toward operator-method approaches for further insights.

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 βand \hbar. βis 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 βand/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.