String Theory Journal Club

The low-energy theory of 4d N=2 supersymmetric gauge theories is encoded in the Seiberg-Witten (SW) curve which is naturally related to an integrable system. For SU(2) theories, in presence of Omega-background this integrable system is given by Painlevé equations. In this talk we we will show that the Nekrasov partition functions of 4d SU(2) susy gauge theories on the blowup of spacetime are tau functions which solve the Painlevé equations in Hirota bilinear form. These solutions can be expressed as expansions in terms of the moduli of the quantum SW curve and the construction can be applied also to non-Lagrangian theories. Furthermore, these solutions have manifest modular properties which provide a natural non-perturbative completion of the corresponding topological string partition function and directly lead to the BCOV holomorphic anomaly equations of the topological string.