Narrow Results

In this paper, we construct the super Virasoro algebra with an arbitrary conformal dimension $\mathrm{\Delta }$ from the generalized ${ℛ}(p,q)$-deformed quantum algebra and investigate the ${ℛ}(p,q)$-deformed super Virasoro algebra with the particular conformal dimension $\mathrm{\Delta }=1.$ Furthermore, we perform the ${ℛ}(p,q)$-conformal Virasoro $n$-algebra, the ${ℛ}(p,q)$-conformal super Virasoro $n$-algebra ($n$-even) and discuss a toy model for the ${ℛ}(p,q)$-conformal Virasoro constraints and ${ℛ}(p,q)$-conformal super Virasoro constraints. Besides, we generalized the notion of the ${ℛ}(p,q)$-elliptic Hermitian matrix model with an arbitrary conformal dimension $\mathrm{\Delta }$. Finally, we deduce relevant particular cases generated by quantum algebras known in the literature.

Thoma's theorem states that a group algebra $C^{\ast }(\mathrm{\Gamma })$ is of type I if and only if $\mathrm{\Gamma }$ is virtually abelian. We discuss here some similar questions for the quantum groups, our main result stating that, under suitable virtually abelianity conditions on a discrete quantum group $\mathrm{\Gamma }$, we have a stationary model of type $\pi :C^{\ast }(\mathrm{\Gamma })\to M_F(C(L))$, with $F$ being a finite quantum group, and with $L$ being a compact group. We discuss then some refinements of these results in the quantum permutation group case, $\widehat{\mathrm{\Gamma }}\subset S_N^{+}$, by restricting the attention to the matrix models which are quasi-flat, in the sense that the images of the standard coordinates, known to be projections, have rank $\le 1$.

We study matrix models related to SO/Sp gauge theories with flavors. We give the effective superpotentials for gauge theories with arbitrary tree level superpotential up to first instanton level. For quartic tree level superpotential, we obtained exact one-cut solution. We also derive Seiberg–Witten curve for these gauge theories from matrix model argument.

We describe an integrable model, related to the Gaudin magnet, and its relation to the matrix model of Brézin, Itzykson, Parisi and Zuber. Relation is based on Bethe ansatz for the integrable model and its interpretation using orthogonal polynomials and saddle point approximation. Large-N limit of the matrix model corresponds to the thermodynamic limit of the integrable system. In this limit (functional) Bethe ansatz is the same as the generating function for correlators of the matrix models.

Solutions of matrix quantum mechanics have been shown to describe time dependent backgrounds in the holographically dual two-dimensional closed string theory. We review some recent work dealing with nontrivial spacetimes which arise in this fashion and discuss aspects of physical phenomena in them. These include spacetimes with spacelike boundaries.

We construct a two-dimensional (2D) causal dynamical triangulation (CDT) model from a matrix model which represents the loop gas model of closed string. The target-space index is reinterpreted as time or geodesic distance. We apply stochastic quantization method to the model to obtain the Generalized CDT (GCDT), which has additional interaction of creating baby universe. If we take a specific scaling in continuum limit, we realize an extended GCDT model characterized by the noncritical string field theory.

Gribov's observation that global gauge fixing is impossible has led to suggestions that there may be a deep connection between gauge fixing and confinement. We find an unexpected relation between the topological nontriviality of the gauge bundle and colored states in SU(N) Yang–Mills theory, and show that such states are necessarily impure. We approximate QCD by a rectangular matrix model that captures the essential topological features of the gauge bundle, and demonstrate the impure nature of colored states explicitly. Our matrix model also allows the inclusion of the QCD θ-term, as well as to perform explicit computations of low-lying glueball masses. This mass spectrum is gapped.

Since an impure state cannot evolve to a pure one by a unitary transformation, our result shows that the solution to the confinement problem in pure QCD is fundamentally quantum information-theoretic.

We have calculated the interaction between two fuzzy spheres in three dimensions. It depends on the distance r between the spheres and the radii ρ₁, ρ₂. There is no force between the spheres when they are at infinite distance. We have also studied the interaction for the r = 0 case. We find that an attractive force exists between two fuzzy sphere surfaces for small and large distances. We have also studied an extension of this model in the ten-dimensional case.

We study a disk amplitude which has a complicated heterogeneous matter configuration on the boundary in a system of the (3,4) conformal matter coupled to two-dimensional gravity. It is analyzed using the two-matrix chain model in the large N limit. We show that the disk amplitude calculated by Schwinger–Dyson equations can be completely reproduced through purely geometrical consideration. From this result, we speculate that all heterogeneous loop amplitudes can be derived from the geometrical consideration and the consistency among relevant amplitudes.

We formulate the planar "large N limit" of matrix models with a continuously infinite number of matrices directly in terms of U(N) invariant variables. Noncommutative probability theory is found to be a good language to describe this formulation. The change of variables from matrix elements to invariants induces an extra term in the Hamiltonian, which is crucial in determining the ground state. We find that this collective potential has a natural meaning in terms of noncommutative probability theory: it is the "free Fisher information" discovered by Voiculescu. This formulation allows us to find a variational principle for the classical theory described by such large N limits. We then use the variational principle to study models more complex than the one describing the quantum mechanics of a single Hermitian matrix (i.e. go beyond the so-called D = 1 barrier). We carry out approximate variational calculations for a few models and find excellent agreement with known results where such comparisons are possible. We also discover a lower bound for the ground state by using the noncommutative analog of the Cramer–Rao inequality.

Recently we have introduced a matrix model depending on two coupling constants g² and λ, which contains the fuzzy sphere as a background; to obtain the classical limit g² must depend on N in a precise way. In this paper we show how to obtain the classical solitons of the N → ∞ limit imposing the development ; as a consequence at finite N one obtains a noncommutative version of the solitons for the fuzzy sphere.

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 ∂ℱ/∂T_m 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.

We introduce a matrix model for noncommutative gravity, based on the gauge group U(2)⊗U(2). The vierbein is encoded in a matrix Y_μ, having values in the coset space U(4)/(U(2)⊗U(2)), while the spin connection is encoded in a matrix X_μ, having values in U(2)⊗U(2). We show how to recover the Einstein equations from the θ→0 limit of the matrix model equations of motion. We stress the necessity of a metric tensor, which is a covariant representation of the gauge group in order to set up a consistent second order formalism. We finally define noncommutative gravitational instantons as generated by U(2)⊗U(2) valued quasi-unitary operators acting on the background of the matrix model. Some of these solutions have naturally self-dual or anti-self-dual spin connections.

We propose an action for gravity on a fuzzy sphere, based on a matrix model. We find striking similarities with an analogous model of two-dimensional gravity on a noncommutative plane, i.e. the solution space of both models is spanned by pure U(2) gauge transformations acting on the background solution of the matrix model, and there exist deformations of the classical diffeomorphisms which preserve the two-dimensional noncommutative gravity actions.

We find that the IIA matrix models defined on the non-compact C³/Z₆, C²/Z₂ and C²/Z₄ orbifolds preserve supersymmetry where the fermions are on-mass-shell Majorana–Weyl fermions. In these examples supersymmetry is preserved both in the orbifolded space and in the non-orbifolded space at the same time. The matrix model on C³/Z₆ orbifold has the same supersymmetry as the case of C³/Z₃ orbifold which was pointed out previously. On the other hand the matrix models on C²/Z₂ and C²/Z₄ orbifold have a half of the supersymmetry. We further find that the matrix model on C²/Z₂ orbifold with a parity-like identification preserves supersymmetry.

We investigate through the Morita equivalence the problem of mapping, odd dimensional noncommutative lattice gauge theories onto suitable matrix models. We specialize our analysis to three-dimensional noncommutative QED (NCQED) and scalar QED (NCsQED), for which we explicitly build the corresponding matrix model.

We review recent developments (up to January 2004) of the Liouville field theory and its matrix model dual. This review consists of two parts. In part I, we review the bosonic Liouville theory. After briefly reviewing the necessary background, we discuss the bulk structure constants (the DOZZ formula) and the boundary states (the FZZT brane and the ZZ brane). Various applications are also presented. In part II, we review the supersymmetric extension of the Liouville theory. We first discuss the bulk structure constants and the branes as in the bosonic Liouville theory, and then we present the matrix dual descriptions with some applications. This review also includes some original material such as the derivation of the conjectured dual action for the Liouville theory from other known dualities.

We propose a model for random forces in a turbulent incompressible fluid by balancing the energy gain from fluctuations against dissipation by viscosity. This leads to a more singular covariance distribution for the random forces than is ordinarily allowed. We then propose regularization of the fluid system by matrix models. A formula for entropy of a two dimensional fluid is derived and then a vorticity profile of a hurricane that maximizes entropy. A regularization of three dimensional incompressible fluid flow using quantum groups is also proposed.

We extend the large N reduction and the compactification in matrix models to those on manifolds represented as fiber bundles. We also discuss application of our results to the AdS/CFT correspondence and topological field theories.

Motivated by supersymmetry breaking in matrix model formulations of superstrings, we present some concrete models, in which the supersymmetry is preserved for any finite N, but gets broken at infinite N, where N is the rank of matrix variables. The models are defined as supersymmetric field theories coupled to some matrix models, and in the induced action obtained after integrating out the matrices, supersymmetry is spontaneously broken only when N is infinity.