Refined topological amplitudes in N=1 flux compactification

We study the implication of refined topological string amplitudes in the supersymmetric N=1 flux compactification. They generate higher derivative couplings among the vector multiplets and graviphoton with generically non-holomorphic moduli dependence. For a particular term, we can compute them by assuming the geometric engineering. We claim that the Dijkgraaf-Vafa large N matrix model with the beta-ensemble measure directly computes the higher derivative corrections to the supersymmetric effective action of the supersymmetric N=1$ gauge theory.Comment: 16 pages, v2: reference adde

The Omega Deformation, Branes, Integrability, and Liouville Theory

We reformulate the Omega-deformation of four-dimensional gauge theory in a way that is valid away from fixed points of the associated group action. We use this reformulation together with the theory of coisotropic A-branes to explain recent results linking the Omega-deformation to integrable Hamiltonian systems in one direction and Liouville theory of two-dimensional conformal field theory in another direction.Comment: 96 p

Refined Cigar and Omega-deformed Conifold

Antoniadis et al proposed a relation between the Omega-deformation and refined correlation functions of the topological string theory. We investigate the proposal for the deformed conifold geometry from a non-compact Gepner model approach. The topological string theory on the deformed conifold has a dual description in terms of the c=1 non-critical string theory at the self-dual radius, and the Omega-deformation yields the radius deformation. We show that the refined correlation functions computed from the twisted SL(2,R)/U(1) Kazama-Suzuki coset model at level k=1 have direct c=1 non-critical string theory interpretations. After subtracting the leading singularity to procure the 1PI effective action, we obtain the agreement with the proposal.Comment: 15 pages, v2: reference added, v3: published versio

Counting Exceptional Instantons

We show how to obtain the instanton partition function of N=2 SYM with exceptional gauge group EFG using blow-up recursion relations derived by Nakajima and Yoshioka. We compute the two instanton contribution and match it with the recent proposal for the superconformal index of rank 2 SCFTs with E6, E7 global symmetry.Comment: 16 pages, references adde

The matrix model version of AGT conjecture and CIV-DV prepotential

Recently exact formulas were provided for partition function of conformal (multi-Penner) beta-ensemble in the Dijkgraaf-Vafa phase, which, if interpreted as Dotsenko-Fateev correlator of screenings and analytically continued in the number of screening insertions, represents generic Virasoro conformal blocks. Actually these formulas describe the lowest terms of the q_a-expansion, where q_a parameterize the shape of the Penner potential, and are exact in the filling numbers N_a. At the same time, the older theory of CIV-DV prepotential, straightforwardly extended to arbitrary beta and to non-polynomial potentials, provides an alternative expansion: in powers of N_a and exact in q_a. We check that the two expansions coincide in the overlapping region, i.e. for the lowest terms of expansions in both q_a and N_a. This coincidence is somewhat non-trivial, since the two methods use different integration contours: integrals in one case are of the B-function (Euler-Selberg) type, while in the other case they are Gaussian integrals.Comment: 27 pages, 1 figur

Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions

We give a concise summary of the impressive recent development unifying a number of different fundamental subjects. The quiver Nekrasov functions (generalized hypergeometric series) form a full basis for all conformal blocks of the Virasoro algebra and are sufficient to provide the same for some (special) conformal blocks of W-algebras. They can be described in terms of Seiberg-Witten theory, with the SW differential given by the 1-point resolvent in the DV phase of the quiver (discrete or conformal) matrix model (\beta-ensemble), dS = ydz + O(\epsilon^2) = \sum_p \epsilon^{2p} \rho_\beta^{(p|1)}(z), where \epsilon and \beta are related to the LNS parameters \epsilon_1 and \epsilon_2. This provides explicit formulas for conformal blocks in terms of analytically continued contour integrals and resolves the old puzzle of the free-field description of generic conformal blocks through the Dotsenko-Fateev integrals. Most important, this completes the GKMMM description of SW theory in terms of integrability theory with the help of exact BS integrals, and provides an extended manifestation of the basic principle which states that the effective actions are the tau-functions of integrable hierarchies.Comment: 14 page

ADHM and the 4d quantum Hall effect

Yang-Mills instantons are solitonic particles in d=4+1 dimensional gauge theories. We construct and analyse the quantum Hall states that arise when these particles are restricted to the lowest Landau level. We describe the ground state wavefunctions for both Abelian and non-Abelian quantum Hall states. Although our model is purely bosonic, we show that the excitations of this 4d quantum Hall state are governed by the Nekrasov partition function of a certain five dimensional supersymmetric gauge theory with Chern-Simons term. The partition function can also be interpreted as a variant of the Hilbert series of the instanton moduli space, counting holomorphic sections rather than holomorphic functions. It is known that the Hilbert series of the instanton moduli space can be rewritten using mirror symmetry of 3d gauge theories in terms of Coulomb branch variables. We generalise this approach to include the effect of a five dimensional Chern-Simons term. We demonstrate that the resulting Coulomb branch formula coincides with the corresponding Higgs branch Molien integral which, in turn, reproduces the standard formula for the Nekrasov partition function

Exact results for N = 2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants

We provide a contour integral formula for the exact partition function of N = 2 supersymmetric U(N) gauge theories on compact toric four-manifolds by means of supersymmetric localisation. We perform the explicit evaluation of the contour integral for U(2) N = 2 17 theory on CP2 for all instanton numbers. In the zero mass case, corresponding to the N = 4 supersymmetric gauge theory, we obtain the generating function of the Euler characteristics of instanton moduli spaces in terms of mock-modular forms. In the decoupling limit of infinite mass we find that the generating function of local and surface observables computes equivariant Donaldson invariants, thus proving in this case a longstanding conjecture by N. Nekrasov. In the case of vanishing first Chern class the resulting equivariant Donaldson polynomials are new. \ua9 2016, The Author(s)

S-duality as a beta-deformed Fourier transform

An attempt is made to formulate Gaiotto's S-duality relations in an explicit quantitative form. Formally the problem is that of evaluation of the Racah coefficients for the Virasoro algebra, and we approach it with the help of the matrix model representation of the AGT-related conformal blocks and Nekrasov functions. In the Seiberg-Witten limit, this S-duality reduces to the Legendre transformation. In the simplest case, its lifting to the level of Nekrasov functions is just the Fourier transform, while corrections are related to the beta-deformation. We calculate them with the help of the matrix model approach and observe that they vanish for beta=1. Explicit evaluation of the same corrections from the U_q(sl(2)) infinite-dimensional representation formulas due to B.Ponsot and J.Teshner remains an open problem.Comment: 21 page