Topological Gauge Theories on Local Spaces and Black Hole Entropy Countings

We study cohomological gauge theories on total spaces of holomorphic line bundles over complex manifolds and obtain their reduction to the base manifold by U(1) equivariant localization of the path integral. We exemplify this general mechanism by proving via exact path integral localization a reduction for local curves conjectured in hep-th/0411280, relevant to the calculation of black hole entropy/Gromov-Witten invariants. Agreement with the four-dimensional gauge theory is recovered by taking into account in the latter non-trivial contributions coming from one-loop fluctuations determinants at the boundary of the total space. We also study a class of abelian gauge theories on Calabi-Yau local surfaces, describing the quantum foam for the A-model, relevant to the calculation of Donaldson-Thomas invariants.Comment: 17 page

Quantum curves and q-deformed Painlev\ue9 equations

We propose that the grand canonical topological string partition functions satisfy finite-difference equations in the closed string moduli. In the case of genus one mirror curve, these are conjectured to be the q-difference Painlev\ue9 equations as in Sakai\u2019s classification. More precisely, we propose that the tau functions of q-Painlev\ue9 equations are related to the grand canonical topological string partition functions on the corresponding geometry. In the toric cases, we use topological string/spectral theory duality to give a Fredholm determinant representation for the above tau functions in terms of the underlying quantum mirror curve. As a consequence, the zeroes of the tau functions compute the exact spectrum of the associated quantum integrable systems. We provide details of this construction for the local P1 7 P1 case, which is related to q-difference Painlev\ue9 with affine A1 symmetry, to SU(2) Super Yang\u2013Mills in five dimensions and to relativistic Toda system. \ua9 2019, Springer Nature B.V

BPS Quivers of Five-Dimensional SCFTs, Topological Strings and q-Painlevé Equations

We study the discrete flows generated by the symmetry group of the BPS quivers for Calabi–Yau geometries describing five-dimensional superconformal quantum field theories on a circle. These flows naturally describe the BPS particle spectrum of such theories and at the same time generate bilinear equations of q-difference type which, in the rank one case, are q-Painlevé equations. The solutions of these equations are shown to be given by grand canonical topological string partition functions which we identify with τ-functions of the cluster algebra associated to the quiver. We exemplify our construction in the case corresponding to five-dimensional SU(2) pure super Yang–Mills and Nf= 2 on a circle

Counting Yang-Mills Instantons by Surface Operator Renormalization Group Flow

We show that the nonperturbative dynamics of N=2 super-Yang-Mills theories in a self-dual ω background and with arbitrary simple gauge group is fully determined by studying renormalization group equations of vacuum expectation values of surface operators generating one-form symmetries. The corresponding system of equations is a nonautonomous Toda chain, the time being the renormalization group scale. We obtain new recurrence relations which provide a systematic algorithm computing multi-instanton corrections from the tree-level one-loop prepotential as the asymptotic boundary condition of the renormalization group equations. We exemplify by computing the E6 and G2 cases up to two instantons

Exact solution of Kerr black hole perturbations via CFT2 and instanton counting: Greybody factor, quasinormal modes, and Love numbers

We give explicit expressions for the finite frequency greybody factor, quasinormal modes, and Love numbers of Kerr black holes by computing the exact connection coefficients of the radial and angular parts of the Teukolsky equation. This is obtained by solving the connection problem of the confluent Heun equation in terms of the explicit expression of irregular Virasoro conformal blocks as sums over partitions via the Alday, Gaiotto, and Tachikawa correspondence. In the relevant approximation limits our results are in agreement with existing literature. The method we use can be extended to solve the linearized Einstein equation in other interesting gravitational backgrounds

N=2 gauge theories on unoriented/open four-manifolds and their AGT counterparts

We compute the exact path integral of N = 2 supersymmetric gauge theories with general gauge group on RP4 and a Z(2)-quotient of the hemi-S-4. By specializing to SU(2) superconformal quivers, we show that these, together with hemi-S-4 partition functions, compute Liouville correlators on unoriented/open Riemann surfaces. We perform explicit checks for Riemann surfaces obtained as Z(2) quotients of the sphere and the torus. We also discuss the coupled 3d-4d systems associated to Liouville amplitudes with boundary punctures

Circular quiver gauge theories, isomonodromic deformations and WN fermions on the torus

We study the relation between class S theories on punctured tori and isomonodromic deformations of flat SL(N) connections on the two-dimensional torus with punctures. Turning on the self-dual Ω -background corresponds to a deautonomization of the Seiberg–Witten integrable system which implies a specific time dependence in its Hamiltonians. We show that the corresponding τ-function is proportional to the dual gauge theory partition function, the proportionality factor being a nontrivial function of the solution of the deautonomized Seiberg–Witten integrable system. This is obtained by mapping the isomonodromic deformation problem to WN free fermion correlators on the torus

M2-branes and q-Painlevé equations

In this paper we investigate a novel connection between the effective theory of M2-branes on (C-2/Z(2)xC(2)/Z(2))/Z(k) and the q-deformed Painleve equations, by proposing that the grand canonical partition function of the corresponding four-nodes circular quiver N = 4 Chern-Simons matter theory solves the q-Painleve VI equation. We analyse how this describes the moduli space of the topological string on local dP(5) and, via geometric engineering, five dimensional N-f = 4 SU(2) N = 1 gauge theory on a circle. The results we find extend the known relation between ABJM theory, q-Painleve III3, and topological strings on local P-1 x P-1. From the mathematical viewpoint the quiver Chern-Simons theory provides a conjectural Fredholm determinant realisation of the q-Painleve VI tau-function. We provide evidence for this proposal by analytic and numerical checks and discuss in detail the successive decoupling limits down to N-f = 0, corresponding to q-Painleve III3

Gauge theories on compact toric manifolds

We compute the N= 2 supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the Kähler form with jumps along the walls where the gauge symmetry gets enhanced. The partition function on such manifolds is written as a sum over the residues of a product of partition functions on C2. The evaluation of these residues is greatly simplified by using an “abstruse duality” that relates the residues at the poles of the one-loop and instanton parts of the C2 partition function. As particular cases, our formulae compute the SU(2) and SU(3) equivariant Donaldson invariants of P2 and Fn and in the non-equivariant limit reproduce the results obtained via wall-crossing and blow up methods in the SU(2) case. Finally, we show that the U(1) self-dual connections induce an anomalous dependence on the gauge coupling, which turns out to satisfy a N= 2 analog of the N= 4 holomorphic anomaly equations

The stringy instanton partition function

We 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 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