\beta-deformed matrix model and Nekrasov partition function

We study Penner type matrix models in relation with the Nekrasov partition function of four dimensional \mathcal{N}=2, SU(2) supersymmetric gauge theories with N_F=2,3 and 4. By evaluating the resolvent using the loop equation for general \beta, we explicitly construct the first half-genus correction to the free energy and demonstrate the result coincides with the corresponding Nekrasov partition function with general \Omega-background, including higher instanton contributions after modifying the relation of the Coulomb branch parameter with the filling fraction. Our approach complements the proof using the Selberg integrals directly which is useful to find the contribution in the series of instanton numbers for a given deformation parameter.Comment: 25 page

Penner Type Matrix Model and Seiberg-Witten Theory

We discuss the Penner type matrix model recently proposed by Dijkgraaf and Vafa for a possible explanation of the relation between four-dimensional gauge theory and Liouville theory by making use of the connection of the matrix model to two-dimensional CFT. We first consider the relation of gauge couplings defined in UV and IR regimes of N_f = 4, N = 2 supersymmetric gauge theory being related as $q_{{\rm UV}}={\vartheta_2(q_{{\rm IR}})^4/\vartheta_3(q_{{\rm IR}})^4}$. We then use this relation to discuss the action of modular transformation on the matrix model and determine its spectral curve. We also discuss the decoupling of massive flavors from the N_f = 4 matrix model and derive matrix models describing asymptotically free N = 2 gauge theories. We find that the Penner type matrix theory reproduces correctly the standard results of N = 2 supersymmetric gauge theories.Comment: 22 pages; v2: references added, typos corrected; v3: a version to appear in JHE

Uniformization, Calogero-Moser/Heun duality and Sutherland/bubbling pants

Inspired by the work of Alday, Gaiotto and Tachikawa (AGT), we saw the revival of Poincar{\'{e}}'s uniformization problem and Fuchsian equations obtained thereof. Three distinguished aspects are possessed by Fuchsian equations. First, they are available via imposing a classical Liouville limit on level-two null-vector conditions. Second, they fall into some A_1-type integrable systems. Third, the stress-tensor present there (in terms of the Q-form) manifests itself as a kind of one-dimensional "curve". Thereby, a contact with the recently proposed Nekrasov-Shatashvili limit was soon made on the one hand, whilst the seemingly mysterious derivation of Seiberg-Witten prepotentials from integrable models become resolved on the other hand. Moreover, AGT conjecture can just be regarded as a quantum version of the previous Poincar{\'{e}}'s approach. Equipped with these observations, we examined relations between spheric and toric (classical) conformal blocks via Calogero-Moser/Heun duality. Besides, as Sutherland model is also obtainable from Calogero-Moser by pinching tori at one point, we tried to understand its eigenstates from the viewpoint of toric diagrams with possibly many surface operators (toric branes) inserted. A picture called "bubbling pants" then emerged and reproduced well-known results of the non-critical self-dual c=1 string theory under a "blown-down" limit.Comment: 17 pages, 4 figures; v2: corrections and references added; v3: Section 2.4.1 newly added thanks to JHEP referee advice. That classical four-point spheric conformal blocks reproducing known SW prepotentials is demonstrated via more examples, to appear in JHEP; v4: TexStyle changed onl

Matrix models for irregular conformal blocks and Argyres-Douglas theories

As regular conformal blocks describe the N=2 superconformal gauge theories in four dimensions, irregular conformal blocks are expected to reproduce the instanton partition functions of the Argyres-Douglas theories. In this paper, we construct matrix models which reproduce the irregular conformal conformal blocks of the Liouville theory on sphere, by taking a colliding limit of the Penner-type matrix models. The resulting matrix models have not only logarithmic terms but also rational terms in the potential. We also discuss their relation to the Argyres-Douglas type theories.Comment: 42 pages, 2 figures; v2: minor corrections; v3: minor corrections, added the second paragraph in section

Generalized matrix models and AGT correspondence at all genera

We study generalized matrix models corresponding to n-point Virasoro conformal blocks on Riemann surfaces with arbitrary genus g. Upon AGT correspondence, these describe four dimensional N=2 SU(2)^{n+3g-3} gauge theories with generalized quiver diagrams. We obtain the generalized matrix models from the perturbative evaluation of the Liouville correlation functions and verify the consistency of the description with respect to degenerations of the Riemann surface. Moreover, we derive the Seiberg-Witten curve for the N=2 gauge theory as the spectral curve of the generalized matrix model, thus providing a check of AGT correspondence at all genera.Comment: 19 pages; v2: version to appear in JHE

Surface Operator, Bubbling Calabi-Yau and AGT Relation

Surface operators in N=2 four-dimensional gauge theories are interesting half-BPS objects. These operators inherit the connection of gauge theory with the Liouville conformal field theory, which was discovered by Alday, Gaiotto and Tachikawa. Moreover it has been proposed that toric branes in the A-model topological strings lead to surface operators via the geometric engineering. We analyze the surface operators by making good use of topological string theory. Starting from this point of view, we propose that the wave-function behavior of the topological open string amplitudes geometrically engineers the surface operator partition functions and the Gaiotto curves of corresponding gauge theories. We then study a peculiar feature that the surface operator corresponds to the insertion of the degenerate fields in the conformal field theory side. We show that this aspect can be realized as the geometric transition in topological string theory, and the insertion of a surface operator leads to the bubbling of the toric Calabi-Yau geometry.Comment: 36 pages, 14 figures. v2: minor changes and typos correcte

On "Dotsenko-Fateev" representation of the toric conformal blocks

We demonstrate that the recent ansatz of arXiv:1009.5553, inspired by the original remark due to R.Dijkgraaf and C.Vafa, reproduces the toric conformal blocks in the same sense that the spherical blocks are given by the integral representation of arXiv:1001.0563 with a peculiar choice of open integration contours for screening insertions. In other words, we provide some evidence that the toric conformal blocks are reproduced by appropriate beta-ensembles not only in the large-N limit, but also at finite N. The check is explicitly performed at the first two levels for the 1-point toric functions. Generalizations to higher genera are briefly discussed.Comment: 10 page

Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals

The AGT relation identifies the Nekrasov functions for various N=2 SUSY gauge theories with the 2d conformal blocks, which possess explicit Dotsenko-Fateev matrix model (beta-ensemble) representations the latter being polylinear combinations of Selberg integrals. The "pure gauge" limit of these matrix models is, however, a non-trivial multiscaling large-N limit, which requires a separate investigation. We show that in this pure gauge limit the Selberg integrals turn into averages in a Brezin-Gross-Witten (BGW) model. Thus, the Nekrasov function for pure SU(2) theory acquires a form very much reminiscent of the AMM decomposition formula for some model X into a pair of the BGW models. At the same time, X, which still has to be found, is the pure gauge limit of the elliptic Selberg integral. Presumably, it is again a BGW model, only in the Dijkgraaf-Vafa double cut phase.Comment: 21 page

Non-Perturbative Topological Strings And Conformal Blocks

We give a non-perturbative completion of a class of closed topological string theories in terms of building blocks of dual open strings. In the specific case where the open string is given by a matrix model these blocks correspond to a choice of integration contour. We then apply this definition to the AGT setup where the dual matrix model has logarithmic potential and is conjecturally equivalent to Liouville conformal field theory. By studying the natural contours of these matrix integrals and their monodromy properties, we propose a precise map between topological string blocks and Liouville conformal blocks. Remarkably, this description makes use of the light-cone diagrams of closed string field theory, where the critical points of the matrix potential correspond to string interaction points.Comment: 36 page

Deformed planar topological open string amplitudes on Seiberg-Witten curve

We study refined B-model via the beta ensemble of matrix models. Especially, for four dimensional N=2 SU(2) supersymmetric gauge theories with N_f=0,1 and 2 fundamental flavors, we discuss the correspondence between deformed disk amplitudes on each Seiberg-Witten curve and the Nekrasov-Shatashvili limit of the corresponding irregular one point block of a degenerate operator via the AGT correspondence. We also discuss the relation between deformed annulus amplitudes and the irregular two point block of the degenerate operator, and check a desired agreement for N_f=0 and 1 cases.Comment: 27 pages. v2: minor changes and references added. v3: minor corrections and one reference adde