Dijkgraaf-Vafa Matrix Model Research Articles

Genus-one correction to asymptotically free Seiberg-Witten prepotential from Dijkgraaf-Vafa matrix model

We find perfect agreements on the genus-one correction to the prepotential of SU(2) Seiberg-Witten theory with N_f=2, 3 between field theoretical and Dijkgraaf-Vafa-Penner type matrix model results.

THE MICROSCOPIC APPROACH TO $\mathcal{N} = 1$ SUPER YANG-MILLS THEORIES

We give a brief account of the recent progresses in super Yang-Mills theories based in particular on the application of Nekrasov's instanton technology to the case of [Formula: see text] supersymmetry. We have developed a first-principle formalism from which any chiral observable in the theory can be computed, including in strongly coupled confining vacua. The correlators are first expressed in terms of some external variables as sums over colored partitions. The external variables are then fixed to their physical values by extremizing the microscopic quantum superpotential. Remarquably, the results can be shown to coincide with the Dijkgraaf-Vafa matrix model approach, which uses a totally different mathematical framework. These results clarify many important properties of [Formula: see text] theories, related in particular to generalized Konishi anomaly equations and to Veneziano-Yankielowicz terms in the glueball superpotentials. The proof of the equivalence between the formalisms based on colored partitions and on matrices is also a proof of the open/closed string duality in the chiral sector of the theories.

Extended 𝒩 = 1 super Yang-Mills theory

We solve a generalization of ordinary N=1 super Yang-Mills theory with gauge group U(N) and an adjoint chiral multiplet X for which we turn on both an arbitrary tree-level superpotential term \int d^{2}\theta Tr W(X) and an arbitrary field-dependent gauge kinetic term \int d^{2}\theta Tr V(X)W^{\alpha}W_{\alpha}. When W=0, the model reduces to the extended Seiberg-Witten theory recently studied by Marshakov and Nekrasov. We use two different points of view: a ''macroscopic'' approach, using generalized anomaly equations, the Dijkgraaf-Vafa matrix model and the glueball superpotential; and the recently proposed ''microscopic'' approach, using Nekrasov's sum over colored partitions and the quantum microscopic superpotential. The two formalisms are based on completely different sets of variables and statistical ensembles. Yet it is shown that they yield precisely the same gauge theory correlators. This beautiful mathematical equivalence is a facet of the open/closed string duality. A full microscopic derivation of the non-perturbative N=1 gauge dynamics follows.

Glueball operators and the microscopic approach to 𝒩 = 1 gauge theories

We explain how to generalize Nekrasov's microscopic approach to = 2 gauge theories to the = 1 case, focusing on the typical example of the U(N) theory with one adjoint chiral multiplet X and an arbitrary polynomial tree-level superpotential Tr W(X). We provide a detailed analysis of the generalized glueball operators and a non-perturbative discussion of the Dijkgraaf-Vafa matrix model and of the generalized Konishi anomaly equations. We compute in particular the non-trivial quantum corrections to the Virasoro operators and algebra that generate these equations. We have performed explicit calculations up to two instantons, that involve the next-to-leading order corrections in Nekrasov's Ω-background.

LargeNgauge theories and topological cigars

We analyze the conjectured duality between a class of double-scaling limits of a one-matrix model and the topological twist of non-critical superstring backgrounds that contain the N=2 Kazama-Suzuki SL(2)/U(1) supercoset model. The untwisted backgrounds are holographically dual to double-scaled Little String Theories in four dimensions and to the large N double-scaling limit of certain supersymmetric gauge theories. The matrix model in question is the auxiliary Dijkgraaf-Vafa matrix model that encodes the F-terms of the above supersymmetric gauge theories. We evaluate matrix model loop correlators with the goal of extracting information on the spectrum of operators in the dual non-critical bosonic string. The twisted coset at level one, the topological cigar, is known to be equivalent to the c=1 non-critical string at self-dual radius and to the topological theory on a deformed conifold. The spectrum and wavefunctions of the operators that can be deduced from the matrix model double-scaling limit are consistent with these expectations.

Counting conifolds and Dijkgraaf–Vafa matrix models for three matrices

We study superpotential perturbations of q deformed N = 4 Yang–Mills for q a root of unity. This is a special case whose geometry is associated to an orbifold with three lines of codimension two singularities meeting at the origin. We perform field theory perturbations that leave only co-dimension three singularities of conifold type in the geometry. We show that there are two “fractional brane” solutions of the F-term equations for each singularity in the deformed geometry, and that the number of complex deformations of that geometry also matches the number of singularities. This proves that for this case there are no local or non-local obstructions to deformation. We also show that the associated Dijkgraaf–Vafa matrix model has a solvable sector, and that the loop equations in this sector encode the full deformed geometry of the theory.

Double scaling limits in gauge theories and matrix models

We show that $\N=1$ gauge theories with an adjoint chiral multiplet admit a wide class of large-N double-scaling limits where $N$ is taken to infinity in a way coordinated with a tuning of the bare superpotential. The tuning is such that the theory is near an Argyres-Douglas-type singularity where a set of non-local dibaryons becomes massless in conjunction with a set of confining strings becoming tensionless. The doubly-scaled theory consists of two decoupled sectors, one whose spectrum and interactions follow the usual large-N scaling whilst the other has light states of fixed mass in the large-N limit which subvert the usual large-N scaling and lead to an interacting theory in the limit. $F$-term properties of this interacting sector can be calculated using a Dijkgraaf-Vafa matrix model and in this context the double-scaling limit is precisely the kind investigated in the "old matrix model'' to describe two-dimensional gravity coupled to $c<1$ conformal field theories. In particular, the old matrix model double-scaling limit describes a sector of a gauge theory with a mass gap and light meson-like composite states, the approximate Goldstone boson of superconformal invariance, with a mass which is fixed in the double-scaling limit. Consequently, the gravitational $F$-terms in these cases satisfy the string equation of the KdV hierarchy.

Nonperturbative Superpotentials and Compactification to Three Dimensions

We consider four-dimensional N=2 supersymmetric gauge theories with gauge group U(N) on R^3 x S^1, in the presence of a classical superpotential. The low-energy quantum superpotential is obtained by simply replacing the adjoint scalar superfield in the classical superpotential by the Lax matrix of the integrable system that underlies the 4d field theory. We verify in a number of examples that the vacuum structure obtained in this way matches precisely that in 4d, although the degrees of freedom that appear are quite distinct. Several features of 4d field theories, such as the possibility of lifting vacua from U(N) to U(tN), become particularly simple in this framework. It turns out that supersymmetric vacua give rise to a reduction of the integrable system which contains information about the field theory but also about the Dijkgraaf-Vafa matrix model. The relation between the matrix model and the quantum superpotential on R^3 x S^1 appears to involve a novel kind of mirror symmetry.

On Exact Superpotentials, Free Energies and Matrix Models

We discuss exact results for the full nonperturbative effective superpotentials of four dimensional $\mathcal{N}=1$ supersymmetric U(N) gauge theories with additional chiral superfield in the adjoint representation and the free energies of the related zero dimensional bosonic matrix models with polynomial potentials in the planar limit using the Dijkgraaf-Vafa matrix model prescription and integrating in and out. The exact effective superpotentials are produced including the leading Veneziano-Yankielowicz term directly from the matrix models. We also discuss how to use integrating in and out as a tool to do random matrix integrals in the large $N$ limit.

Five-dimensional gauge theories and quantum mechanical matrix models

We show how the Dijkgraaf-Vafa matrix model proposal can be extended to describe five-dimensional gauge theories compactified on a circle to four dimensions. This involves solving a certain quantum mechanical matrix model. We do this for the lift of the N=1* theory to five dimensions. We show that the resulting expression for the superpotential in the confining vacuum is identical with the elliptic superpotential approach based on Nekrasov's five-dimensional generalization of Seiberg-Witten theory involving the relativistic elliptic Calogero-Moser, or Ruijsenaars-Schneider, integrable system.

On exact superpotentials in confining vacua

We consider the N=1 super-Yang–Mills theory with gauge group U( N) or SU( N) and one adjoint Higgs field with an arbitrary polynomial superpotential. We provide a purely field theoretic derivation of the exact effective superpotential W( S) for the glueball superfield S=− W aα W a α /(32 Nπ 2) in the confining vacua. We show that the result matches with the Dijkgraaf–Vafa matrix model proposal. The proof brings to light a deep relationship between non-renormalization theorems first discussed by Intriligator, Leigh and Seiberg, and the fact that W( S) is given by a sum over planar diagrams.