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

On the property of Cachazo-Intriligator-Vafa prepotential at the extremum of the superpotential

We consider CIV-DV prepotential F for N=1 SU(n) SYM theory at the extremum of the effective superpotential and prove the relation $2F-S dF/dS = - 2 u_2 Lambda^2n /(n^2-1)$Comment: LaTeX, 10 pages; v2: some misprints corrected; v3: submitted to Phys.Rev.

On Effective Superpotentials and Compactification to Three Dimensions

We study four dimensional N=2 SO/SP supersymmetric gauge theory on R^3\times S^1 deformed by a tree level superpotential. We will show that the exact superpotential can be obtained by making use of the Lax matrix of the corresponding integrable model which is the periodic Toda lattice. The connection between vacua of SO(2N) and SO(2kN-2k+2) can also be seen in this framework. Similar analysis can also be applied for SO(2N+1) and SP(2N).Comment: 18 pages, latex file, v2: typos corrected, refs adde

\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

N=1 G_2 SYM theory and Compactification to Three Dimensions

We study four dimensional N=2 G_2 supersymmetric gauge theory on R^3\times S^1 deformed by a tree level superpotential. We will show that the exact superpotential can be obtained by making use of the Lax matrix of the corresponding integrable model which is the periodic Toda lattice based on the dual of the affine G_2 Lie algebra. At extrema of the superpotential the Seiberg-Witten curve typically factorizes, and we study the algebraic equations underlying this factorization. For U(N) theories the factorization was closely related to the geometrical engineering of such gauge theories and to matrix model descriptions, but here we will find that the geometrical interpretation is more mysterious. Along the way we give a method to compute the gauge theory resolvent and a suitable set of one-forms on the Seiberg-Witten curve. We will also find evidence that the low-energy dynamics of G_2 gauge theories can effectively be described in terms of an auxiliary hyperelliptic curve.Comment: 27 pages, late

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

Note on Matrix Model with Massless Flavors

In this note, following the work of Seiberg in hep-th/0211234 for the conjecture between the field theory and matrix model in the case with massive fundamental flavors, we generalize it to the case with massless fundamental flavors. We show that with a little modifications, the analysis given by Seiberg can be used directly to the case of massless flavors. Furthermore, this new method explains the insertion of delta functions in the matrix model given by Demasure and Janik in hep-th/0211082.Comment: 10 pages. Type fixed. Remarks adde

Conformal SO(2,4) Transformations for the Helical AdS String Solution

By applying the conformal SO(2,4) transformations to the folded rotating string configuration with two spins given by a certain limit from the helical string solution in AdS_3 x S^1, we construct new string solutions whose energy-spin relations are characterized by the boost parameter. When two SO(2,4) transformations are performed with two boost parameters suitably chosen, the straight folded rotating string solution with one spin in AdS_3 is transformed in the long string limit into the long spiky string solution whose expression is given from the helical string solution in AdS_3 by making a limit that the modulus parameter becomes unity.Comment: 16 pages, LaTex, no figure

Branched Matrix Models and the Scales of Supersymmetric Gauge Theories

In the framework of the matrix model/gauge theory correspondence, we consider supersymmetric U(N) gauge theory with $U(1)^N$ symmetry breaking pattern. Due to the presence of the Veneziano--Yankielowicz effective superpotential, in order to satisfy the $F$--term condition $\sum_iS_i=0$, we are forced to introduce additional terms in the free energy of the corresponding matrix model with respect to the usual formulation. This leads to a matrix model formulation with a cubic potential which is free of parameters and displays a branched structure. In this way we naturally solve the usual problem of the identification between dimensionful and dimensionless quantities. Furthermore, we need not introduce the $\N=1$ scale by hand in the matrix model. These facts are related to remarkable coincidences which arise at the critical point and lead to a branched bare coupling constant. The latter plays the role of the $\N=1$ and $\N=2$ scale tuning parameter. We then show that a suitable rescaling leads to the correct identification of the $\N=2$ variables. Finally, by means of the the mentioned coincidences, we provide a direct expression for the $\N=2$ prepotential, including the gravitational corrections, in terms of the free energy. This suggests that the matrix model provides a triangulation of the istanton moduli space.Comment: 1+18 pages, harvmac. Added discussion on the CSW relative shifts of theta vacua and the odd phases at the critical point. References added and typos correcte