Abstract:We elucidate the relation between Painlevé equations and four-dimensional rank one N = 2 theories by identifying the connection associated to Painlevé isomonodromic problems with the oper limit of the flat connection of the Hitchin system associated to gauge theories and by studying the corresponding renormalisation group flow. Based on this correspondence we provide long-distance expansions at various canonical rays for all Painlevé τ -functions in terms of magnetic and dyonic Nekrasov partition functions for N = 2 SQCD and Argyres-Douglas theories at self-dual Omega background ǫ 1 + ǫ 2 = 0, or equivalently in terms of c = 1 irregular conformal blocks.

We find N = 1 Lagrangian gauge theories that flow to generalized ArgyresDouglas theories with N = 2 supersymmetry. We find that certain SU quiver gauge theories flow to generalized Argyres-Douglas theories of type (A k−1 , A mk−1 ) and (I m,km , S). We also find quiver gauge theories of SO/Sp gauge groups flowing to the (

We show that the exact partition function of U(N ) six-dimensional gauge theory with eight supercharges on C 2 × S 2 provides the quantization of the integrable system of hydrodynamic type known as gl(N ) periodic Intermediate Long Wave (ILW). We characterize this system as the hydrodynamic limit of elliptic Calogero-Moser integrable system. We compute the Bethe equations from the effective gauged linear sigma model on S 2 with target space the ADHM instanton moduli space, whose mirror computes the Yang-Yang function of gl(N ) ILW. The quantum Hamiltonians are given by the local chiral ring observables of the six-dimensional gauge theory. As particular cases, these provide the gl(N ) Benjamin-Ono and Korteweg-de Vries quantum Hamiltonians. In the four dimensional limit, we identify the local chiral ring observables with the conserved charges of Heisenberg plus W N algebrae, thus providing a gauge theoretical proof of AGT correspondence.

In this paper we identify the problem of equivariant vortex counting in a (2, 2) supersymmetric two dimensional quiver gauged linear sigma model with that of computing the equivariant Gromov-Witten invariants of the GIT quotient target space determined by the quiver. We provide new contour integral formulae for the I and J -functions encoding the equivariant quantum cohomology of the target space. Its chamber structure is shown to be encoded in the analytical properties of the integrand. This is explained both via general arguments and by checking several key cases. We show how several results in equivariant Gromov-Witten theory follow just by deforming the integration contour. In particular we apply our formalism to compute Gromov-Witten invariants of the C 3 /Z n orbifold, of the Uhlembeck (partial) compactification of the moduli space of instantons on C 2 and of A n and D n singularities both in the orbifold and resolved phases. Moreover, we analyse dualities of quantum cohomology rings of holomorphic vector bundles over Grassmannians, which are relevant to BPS Wilson loop algebrae.arXiv:1307.5997v2 [hep-th]