Hiroshi Itoyama

The Quiver Matrix Model and 2d-4d Conformal Connection

We review the quiver matrix model (the ITEP model) in the light of the recent progress on 2d-4d connection of conformal field theories, in particular, on the relation between Toda field theories and a class of quiver superconformal gauge theories. On the basis of the CFT representation of the beta deformation of the model, a quantum spectral curve is introduced as >=0 at finite N and for beta neq 1. The planar loop equation in the large N limit follows with the aid of W_n constraints. Residue analysis is provided both for the curve of the matrix model with the multi-log potential and for the Seiberg-Witten curve in the case of SU(n) with 2n flavors, leading to the matching of the mass parameters. The isomorphism of the two curves is made manifest.

Method of generating q-expansion coefficients for conformal block and N=2 Nekrasov function by β-deformed matrix model

Abstract We observe that, at β -deformed matrix models for the four-point conformal block, the point q = 0 is the point where the three-Penner type model becomes a pair of decoupled two-Penner type models and where, in the planar limit, (an array of), two-cut eigenvalue distribution(s) coalesce into (that of) one-cut one(s). We treat the Dotsenko–Fateev multiple integral, with their paths under the recent discussion, as perturbed double-Selberg matrix model (at q = 0 , it becomes a pair of Selberg integrals) to construct two kinds of generating functions for the q -expansion coefficients and compute some. A formula associated with the Jack polynomial is noted. The second Nekrasov coefficient for SU ( 2 ) with N f = 4 is derived. A pair of Young diagrams appears naturally. The finite N loop equation at q = 0 as well as its planar limit is solved exactly, providing a useful tool to evaluate the coefficients as those of the resolvents. The planar free energy in the q -expansion is computed to the lowest non-trivial order. A free field representation of the Nekrasov function is given.

The Dijkgraaf–Vafa prepotential in the context of general Seiberg–Witten theory

Abstract We consider the prepotential of Dijkgraaf and Vafa (DV) as one more (and in fact, singular) example of the Seiberg–Witten (SW) prepotentials and discuss its properties from this perspective. Most attention is devoted to the issue of complete system of moduli, which should include not only the sizes of the cuts (in matrix model interpretation), but also their positions, i.e., the number of moduli should be almost doubled, as compared to the DV consideration. We introduce the notion of regularized DV system (not necessarily related to matrix model) and discuss the WDVV equations. These definitely hold before regularization is lifted, but an adequate limiting procedure, preserving all ingredients of the SW theory, remains to be found.

Calculating gluino-condensate prepotential

We discuss the derivation of the CIV-DV prepotential for arbitrary power n + 1 of the original superpotential in the N = 1 SUSY YM theory (for arbitrary number n of cuts in the solution of the planar matrix model in the Dijkgraaf-Vafa interpretation). The goal is to hunt for structures, not so much for exact formulas, which are necessarily complicated, before the right language is found to represent them. Some entities, reminiscent of representation theory, clearly emerge, but a lot of work remains to be done to identify the relevant ones. As a practical application, we obtain a cubic (first non-perturbative) contribution to the prepotential for any n.

GLUINO-CONDENSATE (CIV-DV) PREPOTENTIAL FROM ITS WHITHAM–TIME DERIVATIVES

We describe an expedient way to derive the CIV-DV prepotential in power series expansion in S_i. This is based on integrations of equations for its derivatives with respect to additional (Whitham) moduli T_m. For illustrative purposes, we calculate explicitly the leading terms of the expansion and explicitly check some components of the WDVV equations to the leading order. Extension to any higher order is simple and algorithmic.

Experiments with the WDVV equations for the gluino-condensate prepotential: the cubic (two-cut) case

Abstract We demonstrate by explicit calculation that the first two terms in the CIV-DV prepotential for the two-cut case satisfy the generalized WDVV equations, just as in all other known examples of hyper-elliptic Seiberg–Witten models. The WDVV equations are non-trivial in this situation, provided the set of moduli is extended as compared to the Dijkgraaf–Vafa suggestion and includes also moduli, associated with the positions of the cuts (not only with their lengths). Expression for the extra modulus dictated by WDVV equation, however, appears different from a naive expectation implied by the Whitham theory. Moreover, for every value of the “quantum-deformation parameter” 1/ g 3 , we actually find an entire one-parameter family of solutions to the WDVV equations, of which the conventional prepotential is just a single point.

Partial breaking of N=2 supersymmetry and of gauge symmetry in the U(N) gauge model

Abstract We explore vacua of the U ( N ) gauge model with N = 2 supersymmetry recently constructed in hep-th/0409060 . In addition to the vacuum previously found with unbroken U ( N ) gauge symmetry in which N = 2 supersymmetry is partially broken to N = 1 , we find cases in which the gauge symmetry is broken to a product gauge group ∏ i = 1 n U ( N i ) . The N = 1 vacua are selected by the requirement of a positive definite Kahler metric. We obtain the masses of the supermultiplets appearing on the N = 1 vacua.

Supersymmetric U(N) Gauge Model and Partial Breaking of N = 2 Supersymmetry

We briefly review a construction of N=2 supersymmetric U(N) gauge model in which rigid N=2 supersymmetry is spontaneously broken to N=1. This model generalizes the abelian model considered by Antoniadis, Patouche and Taylor. We discuss the conditions on the vacua of the model with partial supersymmetry breaking.

Boundary ring: A way to construct approximate NG solutions with polygon boundary conditions I. Zn-symmetric configurations

We describe an algebro-geometric construction of polygon-bounded minimal surfaces in ADS5, based on consideration of what we call the ”boundary ring” of polynomials. The first non-trivial example of the Nambu-Goto (NG) solutions for Z6-symmetric hexagon is considered in some detail. Solutions are represented as power series, of which only the first terms are evaluated. The NG equations leave a number of free parameters (a free function). Boundary conditions, which fix the free parameters, are imposed on truncated series. It is still unclear if explicit analytic formulas can be found in this way, but even approximate solutions, obtained by truncation of power series, can be sufficient to investigate the Alday-Maldacena – BDS/BHT version of the string/gauge duality.

Whitham prepotential and superpotential

N=2 supersymmetric U(N) Yang–Mills theory softly broken to N=1 by the superpotential of the adjoint scalar fields is discussed from the viewpoint of the Whitham deformation theory for prepotential. With proper identification of the superpotential we derive the matrix model curve from the condition that the mixed second derivatives of the Whitham prepotential have a nontrivial kernel.