Supermatrix Model Research Articles

Aspects of supergroup gauge theory

In this paper, provide a survey of recent studies of supergroup gauge theory. We first discuss the supermatrix model as a zero-dimensional toy model of supergroup gauge theory and its geometric and algebraic characterization. We then focus on four-dimensional Yang–Mills theory with supergroup gauge symmetry and explore its nonperturbative properties, including instanton calculus, Seiberg–Witten geometry, Bethe/gauge correspondence, and its realization with intersecting defects.

Complex (super)-matrix models with external sources and q-ensembles of Chern–Simons and ABJ(M) type

The Langmann–Szabo–Zarembo (LSZ) matrix model is a complex matrix model with a quartic interaction and two external matrices. The model appears in the study of a scalar field theory on the non-commutative plane. We prove that the LSZ matrix model computes the probability of atypically large fluctuations in the Stieltjes–Wigert matrix model, which is a q-ensemble describing U(N) Chern–Simons theory on the three-sphere. The correspondence holds in a generalized sense: depending on the spectra of the two external matrices, the LSZ matrix model either describes probabilities of large fluctuations in the Chern–Simons partition function, in the unknot invariant or in the two-unknot invariant. We extend the result to supermatrix models, and show that a generalized LSZ supermatrix model describes the probability of atypically large fluctuations in the ABJ(M) matrix model.

Quantized Coulomb branches, monopole bubbling and wall-crossing phenomena in 3d mathcal{N} = 4 theories

To study the quantized Coulomb branch of 3d mathcal{N} = 4 unitary SQCD theories, we propose a new method to compute correlators of monopole and Casimir operators that are inserted in the mathrm{mathbb{R}}times {mathrm{mathbb{R}}}_{upepsilon}^2 Omega background. This method combines results from supersymmetric localization with inputs from the brane realisation of the correlators in type IIB string theory. The main challenge is the computation of the partition functions of certain Super-Matrix-Models (SMMs), which appear in the contribution of monopole bubbling sectors and are realised as the theory living on the D1 strings in the brane construction. We find that the non-commutativity arising in the monopole operator insertions is related to a wall-crossing phenomenon in the FI parameter space of the SMM. We illustrate our method in various examples and we provide explicit results for arbitrary correlators of non-bubbling bare monopole operators. We also discuss the realisation of the non-commutative product as a Moyal (star) product and use it to successfully test our results.

Hanany-Witten transition in quantum curves

It was known that the U(N )4 super Chern-Simons matrix model describing the worldvolume theory of D3-branes with two NS5-branes and two (1, k)5-branes in IIB brane configuration (dual to M2-branes after taking the T-duality and the M-theory lift) corresponds to the D5 quantum curve. For deformations of these two objects, on one hand the super Chern-Simons matrix model has three degrees of freedom (of relative rank defor- mations interpreted as fractional branes in brane configurations), while on the other hand the D5 curve has five degrees of freedom (characterized by point configurations of asymp- totic values). To identify the three-dimensional parameter space of brane configurations in the five-dimensional space of point configurations, we propose the necessity to cut the compact T-duality circle (or the circular quiver diagram) open, which is similar to the idea of “fixing a reference frame” or “fixing a local chart”. Since the parameter space of curves enjoys the D5 Weyl group beautifully, we are naturally led to conjecture that M2-branes are not only deformed by fractional branes but more obscure geometrical backgrounds.

Symmetry breaking in quantum curves and super Chern-Simons matrix models

It was known that quantum curves and super Chern-Simons matrix models correspond to each other. From the viewpoint of symmetry, the algebraic curve of genus one, called the del Pezzo curve, enjoys symmetry of the exceptional algebra, while the super Chern-Simons matrix model is described by the free energy of topological strings on the del Pezzo background with the symmetry broken. We study the symmetry breaking of the quantum cousin of the algebraic curve and reproduce the results in the super Chern-Simons matrix model.

Random supermatrices with an external source

In the past we have considered Gaussian random matrix ensembles in the presence of an external matrix source. The reason was that it allowed, through an appropriate tuning of the eigenvalues of the source, to obtain results on non-trivial dual models, such as Kontsevich’s Airy matrix models and generalizations. The techniques relied on explicit computations of the k-point functions for arbitrary N (the size of the matrices) and on an N-k duality. Numerous results on the intersection numbers of the moduli space of curves were obtained by this technique. In order to generalize these results to include surfaces with boundaries, we have extended these techniques to supermatrices. Again we have obtained quite remarkable explicit expressions for the k-point functions, as well as a duality. Although supermatrix models a priori lead to the same matrix models of 2d-gravity, the external source extensions considered in this article lead to new geometric results.

Jacobi-Trudi Identity in Super Chern-Simons Matrix Model

It was proved by Macdonald that the Giambelli identity holds if we define the Schur functions using the Jacobi-Trudi identity. Previously for the super Chern-Simons matrix model (the spherical one-point function of the superconformal Chern-Simons theory describing the worldvolume of the M2-branes) the Giambelli identity was proved from a shifted version of it. With the same shifted Giambelli identity we can further prove the Jacobi-Trudi identity, which strongly suggests an integrable structure for this matrix model.

Giambelli identity in super Chern-Simons matrix model

A classical identity due to Giambelli in representation theory states that the character in any representation is expressed as a determinant whose components are characters in the hook representation constructed from all the combinations of the arm and leg lengths of the original representation. We prove that, in a general fractional-brane background, the identity persists in taking, for each character, the matrix integration of the super Chern-Simons matrix model in the grand canonical ensemble.

Toward $$\mathrm {U}(N|M)$$ U ( N | M ) knot invariant from ABJM theory

We study U(N|M) character expectation value with the supermatrix Chern-Simons theory, known as the ABJM matrix model, with emphasis on its connection to the knot invariant. This average just gives the half BPS circular Wilson loop expectation value in ABJM theory, which shall correspond to the unknot invariant. We derive the determinantal formula, which gives U(N|M) character expectation values in terms of U(1|1) averages for a particular type of character representations. This means that the U(1|1) character expectation value is a building block for all the U(N|M) averages, and in particular, by an appropriate limit, for the U(N) invariants. In addition to the original model, we introduce another supermatrix model obtained through the symplectic transform, which is motivated by the torus knot Chern-Simons matrix model. We obtain the Rosso-Jones-type formula and the spectral curve for this case.

On supermatrix models, Poisson geometry, and noncommutative supersymmetric gauge theories

We construct a new supermatrix model which represents a manifestly supersymmetric noncommutative regularisation of the UOSp(2|1) supersymmetric Schwinger model on the supersphere. Our construction is much simpler than those already existing in the literature and it was found by using Poisson geometry in a substantial way.

Duality and integrability of a supermatrix model with an external source

We study the Hermitian supermatrix model involving an external source. We derive the determinantal formula for the supermatrix partition function, and also for the expectation value of the characteristic polynomial ratio, which yields the duality between the characteristic polynomial and the external source with an arbitrary matrix potential function. We also show that the supermatrix integral satisfies the one and two dimensional Toda lattice equations as well as the ordinary matrix model.

From Weak to Strong Coupling in ABJM Theory

The partition function of N=6 supersymmetric Chern-Simons-matter theory (known as ABJM theory) on S^3, as well as certain Wilson loop observables, are captured by a zero dimensional super-matrix model. This super-matrix model is closely related to a matrix model describing topological Chern-Simons theory on a lens space. We explore further these recent observations and extract more exact results in ABJM theory from the matrix model. In particular we calculate the planar free energy, which matches at strong coupling the classical IIA supergravity action on AdS_4 x CP^3 and gives the correct N^{3/2} scaling for the number of degrees of freedom of the M2 brane theory. Furthermore we find contributions coming from world-sheet instanton corrections in CP^3. We also calculate non-planar corrections, both to the free energy and to the Wilson loop expectation values. This matrix model appears also in the study of topological strings on a toric Calabi-Yau manifold, and an intriguing connection arises between the space of couplings of the planar ABJM theory and the moduli space of this Calabi-Yau. In particular it suggests that, in addition to the usual perturbative and strong coupling (AdS) expansions, a third natural expansion locus is the line where one of the two 't Hooft couplings vanishes and the other is finite. This is the conifold locus of the Calabi-Yau, and leads to an expansion around topological Chern-Simons theory. We present some explicit results for the partition function and Wilson loop observables around this locus.

Quantized Nambu-Poisson manifolds in a 3-Lie algebra reduced model

We consider dimensional reduction of the Bagger-Lambert-Gustavsson theory to a zero-dimensional 3-Lie algebra model and construct various stable solutions corresponding to quantized Nambu-Poisson manifolds. A recently proposed Higgs mechanism reduces this model to the IKKT matrix model. We find that in the strong coupling limit, our solutions correspond to ordinary noncommutative spaces arising as stable solutions in the IKKT model with D-brane backgrounds. In particular, this happens for S 3, $ {\mathbb{R}^3} $ and five-dimensional Neveu-Schwarz Hpp-waves. We expand our model around these backgrounds and find effective noncommutative field theories with complicated interactions involving higher-derivative terms. We also describe the relation of our reduced model to a cubic supermatrix model based on an $ \mathfrak{o}\mathfrak{s}\mathfrak{p}\left( {1\left| {32} \right.} \right) $ supersymmetry algebra.

Supermatrix models, loop equations, and duality

We study integrals over Hermitian supermatrices of arbitrary size p + q, which are parametrized by an external field X and a source Y of respective sizes m + n and p + q. We show that these integrals exhibit a simple topological expansion in powers of a formal parameter ℏ, which can be identified with 1/(p − q). The loop equation and the associated spectral curve are also obtained. The solutions to the loop equation are given in terms of the symplectic invariants introduced by Eynard and Orantin [Commun. Number Theory Phys. 1, 347 (2007)]. The symmetry property of the latter objects allows us to prove a duality that relates supermatrix models in which the role of X and Y are interchanged.

A supermatrix model for $$ \mathcal{N} $$ = 6 super Chern-Simons-matter theory

We construct the Wilson loop operator of N=6 super Chern-Simons-matter which is invariant under half of the supercharges of the theory and is dual to the simplest macroscopic open string in AdS_4 x CP^3. The Wilson loop couples, in addition to the gauge and scalar fields of the theory, also to the fermions in the bi-fundamental representation of the U(N) x U(M) gauge group. These ingredients are naturally combined into a superconnection whose holonomy gives the Wilson loop, which can be defined for any representation of the supergroup U(N|M). Explicit expressions for loops supported along an infinite straight line and along a circle are presented. Using the localization calculation of Kapustin et al. we show that the circular loop is computed by a supermatrix model and discuss the connection to pure Chern-Simons theory with supergroup U(N|M).

Superbosonization Formula and its Application to Random Matrix Theory

Starting from Gaussian random matrix models we derive a new supermatrix field theory model. In contrast to the conventional non-linear sigma models, the new model is applicable for any range of correlations of the elements of the random matrices. We clarify the domain of integration for the supermatrices, and give a demonstration of how the model works by calculating the density of states for an ensemble of almost diagonal matrices. It is also shown how one can reduce the supermatrix model to the conventional sigma model.

Superfield formulation of 4D, Script N = 1 massless higher spin gauge field theory and supermatrix model

We study the relation between a supermatrix model and the free 4D, = 1 supersymmetric field theory of a massless supermultiplet with spins (3, 5/2). In order to do this, we construct a superfield formulation of the theory. We show that solutions of the equations of motion for the supermultiplet (3, 5/2) satisfy the equations of motion of a supermatrix model.

Supermatrix models and multi ZZ-brane partition functions in minimal superstring theories

We study (p,q)=(2,4k) minimal superstrings within the minimal superstring field theory constructed in hep-th/0611045. We explicitly give a solution to the W_{1+\infty} constraints by using charged D-instanton operators, and show that the (m,n)-instanton sector with m positive-charged and n negative-charged ZZ-branes is described by an (m+n)\times (m+n) supermatrix model. We argue that the supermatrix model can be regarded as an open string field theory on the multi ZZ-brane system.

Super Yang-Mills Theory from a Supermatrix Model

It is known that Yang-Mills theories on non-commutative space can be derived from large-N reduced models. Gauge fields in non-commutative Yang-Mills theories can be described as fluctuations of matrices expanded about an appropriate classical solution of the reduced models. We investigate a generalization of this procedure in superfield formalism. We show that we can construct a supermatrix model such that D=4 $\N=1$ super Yang-Mills theory can be derived from it. In addition, we can couple matter supermatrices to this supermatrix model and also construct models corresponding to $\N=2$ and $\N=4$ super Yang-Mills theories. In these investigations, we need to introduce a new non-anti-commutative superspace, and we investigate the definition of field theories on this space.

Curved Superspaces and Local Supersymmetry in Supermatrix Model

In a previous paper, we introduced a new interpretation of matrix models, in which any d-dimensional curved space can be realized in terms of d matrices, and the diffeomorphism and the local Lorentz symmetries are included in the ordinary unitary symmetry of the matrix model. Furthermore, we showed that the Einstein equation is naturally obtained, if we employ the standard form of the action, S=-tr([A_a,A_b][A^a,A^b])+.... In this paper, we extend this formalism to include supergravity. We show that the supercovariant derivatives on any d-dimensional curved space can be expressed in terms of d supermatrices, and the local supersymmetry can be regarded as a part of the superunitary symmetry. We further show that the Einstein and Rarita-Schwinger equations are compatible with the supermatrix generalization of the standard action.