01519nas a2200145 4500008004100000245015300041210006900194260001300263520096300276100002001239700002301259700002401282700001601306856005101322 2014 en d00aSix-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics0 aSixdimensional supersymmetric gauge theories quantum cohomology bSpringer3 aWe show that the exact partition function of U(N) six-dimensional gauge
theory with eight supercharges on C^2 x 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.1 aBonelli, Giulio1 aSciarappa, Antonio1 aTanzini, Alessandro1 aVasko, Petr uhttp://urania.sissa.it/xmlui/handle/1963/3454601412nas a2200145 4500008004100000245004500041210004100086260001300127520099200140100002001132700002301152700002401175700001601199856005101215 2014 en d00aThe stringy instanton partition function0 astringy instanton partition function bSpringer3 aWe perform an exact computation of the gauged linear sigma model associated
to a D1-D5 brane system on a resolved A_1 singularity. This is accomplished via
supersymmetric localization on the blown-up two-sphere. We show that in the
blow-down limit C^2/Z_2 the partition function reduces to the Nekrasov
partition function evaluating the equivariant volume of the instanton moduli
space. For finite radius we obtain a tower of world-sheet instanton
corrections, that we identify with the equivariant Gromov-Witten invariants of
the ADHM moduli space. We show that these corrections can be encoded in a
deformation of the Seiberg-Witten prepotential. From the mathematical
viewpoint, the D1-D5 system under study displays a twofold nature: the
D1-branes viewpoint captures the equivariant quantum cohomology of the ADHM
instanton moduli space in the Givental formalism, and the D5-branes viewpoint
is related to higher rank equivariant Donaldson-Thomas invariants of P^1 x C^2.1 aBonelli, Giulio1 aSciarappa, Antonio1 aTanzini, Alessandro1 aVasko, Petr uhttp://urania.sissa.it/xmlui/handle/1963/3458901601nas a2200145 4500008004100000245008900041210007100130260001300201520110700214100002001321700002301341700002401364700001601388856005101404 2014 en d00aVortex Partition Functions, Wall Crossing and Equivariant Gromov–Witten Invariants0 aVortex Partition Functions Wall Crossing and Equivariant Gromov– bSpringer3 aIn 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 C3/Zn orbifold, of the Uhlembeck (partial) compactification of the moduli space of instantons on C2, and of An and Dn 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.1 aBonelli, Giulio1 aSciarappa, Antonio1 aTanzini, Alessandro1 aVasko, Petr uhttp://urania.sissa.it/xmlui/handle/1963/34652