Narrow Results

It is shown how a noncommutative spacetime leads to an area, mass and entropy quantization condition which allows to derive the Schwarzschild black hole entropy $\frac{A}{4G}$, the logarithmic corrections, and further corrections, from the discrete mass transitions taking place among different mass states in $D=4$. The higher-dimensional generalization of the results in $D=4$ follows. The discretization of the entropy-mass relation $S=S(M)$ leads to an entropy quantization of the form $S=S(M_n)=n$, such that one may always assign $n$ “bits” to the discrete entropy, and in doing so, make contact with quantum information. The physical applications of mass quantization, like the counting of states contributing to the black hole entropy, black hole evaporation, and the direct connection to the black holes-string correspondence [G. Horowitz and J. Polchinski, A correspondence principle for black holes and strings, Phys. Rev. D55 (1997) 6189.] via the asymptotic behavior of the number of partitions of integers, follows. To conclude, it is shown how the recent large $N$ Matrix model (fuzzy sphere) of C.-S. Chu [A matrix model proposal for QG and the QM of black holes, preprint, arXiv:2406.01466] leads to very similar results for the black hole entropy as the physical model described in this work which is based on the discrete mass transitions originating from the noncommutativity of the spacetime coordinates.

We continue our study of zero-dimensional field theories in which the fields take values in a strong homotopy Lie algebra. In the first part, we review in detail how higher Chern–Simons theories arise in the AKSZ-formalism. These theories form a universal starting point for the construction of $L_{\infty }$-algebra models. We then show how to describe superconformal field theories and how to perform dimensional reductions in this context. In the second part, we demonstrate that Nambu–Poisson and multisymplectic manifolds are closely related via their Heisenberg algebras. As a byproduct of our discussion, we find central Lie $p$-algebra extensions of $𝔰𝔬(p+2)$. Finally, we study a number of $L_{\infty }$-algebra models which are physically interesting and which exhibit quantized multisymplectic manifolds as vacuum solutions.

We review the BV formalism in the context of $0$-dimensional gauge theories. For a gauge theory $(X_0,S_0)$ with an affine configuration space $X_0$, we describe an algorithm to construct a corresponding extended theory $(\tilde{X},\tilde{S})$, obtained by introducing ghost and anti-ghost fields, with $\tilde{S}$ a solution of the classical master equation in ${{𝒪}}_{\tilde{X}}$. This construction is the first step to define the (gauge-fixed) BRST cohomology complex associated to $(\tilde{X},\tilde{S})$, which encodes many interesting information on the initial gauge theory $(X_0,S_0)$. The second part of this article is devoted to the application of this method to a matrix model endowed with a $U(2)$-gauge symmetry, explicitly determining the corresponding $\tilde{X}$ and the general solution $\tilde{S}$ of the classical master equation for the model.

Differential reformulations of field theories are often used for explicit computations. We derive a one-matrix differential formulation of two-matrix models, with the help of which it is possible to diagonalize the one- and two-matrix models using a formula by Itzykson and Zuber that allows diagonalizing differential operators with respect to matrix elements of Hermitian matrices. We detail the equivalence between the expressions obtained by diagonalizing the partition function in differential or integral formulation, which is not manifest at first glance. For one-matrix models, this requires transforming certain derivatives to variables. In the case of two-matrix models, the same computation leads to a new determinant formulation of the partition function, and we discuss potential applications to new orthogonal polynomials methods.

We study the question whether matrix models obtained in the zero volume limit of 4d Yang–Mills theories can describe large N QCD strings. The matrix model we use is a variant of the Eguchi–Kawai model in terms of Hermitian matrices, but without any twists or quenching. This model was originally proposed as a toy model of the IIB matrix model. In contrast to common expectations, we do observe the area law for Wilson loops in a significant range of scale of the loop area. Numerical simulations show that this range is stable as N increases up to 768, which strongly suggests that it persists in the large N limit. Hence the equivalence to QCD strings may hold for length scales inside a finite regime.

q-deformed fermion oscillators are used to construct q-deformed higher order Virasaro algebra in the Fock space representation. These facilitate the realization of q-deformed W_∞-algebra. The vertex operators (positively dressed) in 2D string theory are interpreted in terms of q-fermionic states.

We study the properties of matrix models with soft confining potentials. Their precise mathematical characterization is that their weight function is not determined by its moments. We mainly rely on simple considerations based on orthogonal polynomials and the moment problem. In addition, some of these models are equivalent, by a simple mapping, to matrix models that appear in Chern–Simons theory. The models can be solved with q deformed orthogonal polynomials (Stieltjes–Wigert polynomials), and the deformation parameter turns out to be the usual q parameter in Chern–Simons theory. In this way, we give a matrix model computation of the Chern–Simons partition function on S³ and show that there are infinitely many matrix models with this partition function.

A study of the one-loop dilatation operator in the scalar sector of SYM is presented. The dilatation operator is analyzed from the point of view of Hamiltonian matrix models. A Lie algebra underlying operator mixing in the planar large-N limit is presented, and its role in understanding the integrability of the planar dilatation operator is emphasized. A classical limit of the dilatation operator is obtained by considering a contraction of this Lie algebra, leading to a new way of constructing classical limits for quantum spin chains. An infinite tower of local conserved charges is constructed in this classical limit purely within the context of the matrix model. The deformation of these charges and their relation to the charges of the spin chain is also elaborated upon.

Mass deformations of supersymmetric Yang–Mills theories in three spacetime dimensions are considered. The gluons of the theories are made massive by the inclusion of a nonlocal gauge and Poincaré invariant mass term due to Alexanian and Nair, while the matter fields are given standard Gaussian mass-terms. It is shown that the dimensional reduction of such mass-deformed gauge theories defined on R³ or R × T² produces matrix quantum mechanics with massive spectra. In particular, all known massive matrix quantum mechanical models obtained by the deformations of dimensional reductions of minimal super Yang–Mills theories in diverse dimensions are shown also to arise from the dimensional reductions of appropriate massive Yang–Mills theories in three spacetime dimensions. Explicit formulas for the gauge theory actions are provided.

We consider SU(N) Yang–Mills theory on the space ℝ × S³ with Minkowski signature (-+++). The condition of SO(4)-invariance imposed on gauge fields yields a bosonic matrix model which is a consistent truncation of the plane wave matrix model. For matrices parametrized by a scalar ϕ, the Yang–Mills equations are reduced to the equation of a particle moving in the double-well potential. The classical solution is a bounce, i.e. a particle which begins at the saddle point ϕ = 0 of the potential, bounces off the potential wall and returns to ϕ = 0. The gauge field tensor components parametrized by ϕ are smooth and for finite time, both electric and magnetic fields are nonvanishing. The energy density of this non-Abelian dyon configuration does not depend on coordinates of ℝ × S³ and the total energy is proportional to the inverse radius of S³. We also describe similar bounce dyon solutions in SU(N) Yang–Mills theory on the space ℝ × S² with signature (-++). Their energy is proportional to the square of the inverse radius of S². From the viewpoint of Yang–Mills theory on ℝ^1,1 × S² these solutions describe non-Abelian (dyonic) flux tubes extended along the x³-axis.

In Ref. 1 a matrix model representation was found for the simplest Hurwitz partition function, which has Lambert curve ϕe^-ϕ = ψ as a classical equation of motion. We demonstrate that Fourier–Laplace transform in the logarithm of external field Ψ converts it into a more sophisticated form, recently suggested in Ref. 2.

In this short note we construct a simple cut-and-join operator representation for Kontsevich–Witten tau-function that is the partition function of the two-dimensional topological gravity. Our derivation is based on the Virasoro constraints. Possible applications of the obtained representation are discussed.

The purpose of this paper is to use the idea in J. Geom. Phys.42, 54 (2002) to compute the topological charges for a (finite) sequence of noncommutative line bundles over the fuzzy sphere. Central to this task is to construct projective modules associated with sequence of the irreducible sub-representations of the tensor product of two different irreps of SU(2). The topological charges corresponding to such fuzzy line bundles are fractional and different from each other. However, in the commutative limit, those tend to Chern numbers of a sequence of the complex line bundles over two-sphere.

We formulate -fold supersymmetry in quantum mechanical matrix models. As an example, we construct general two-by-two Hermitian matrix two-fold supersymmetric quantum mechanical systems. We find that there are two inequivalent such systems, both of which are characterized by two arbitrary scalar functions, and one of which does not reduce to the scalar system. The obtained systems are all weakly quasi-solvable.

In the case of an invertible coordinate commutator matrix θ_ij, we derive a general instanton solution of the noncommutative gauge theories on d = 2n planes given in terms of n oscillators.

We give an introduction to the recently-established connection between supersymmetric gauge theories and matrix models. We begin by reviewing previous material that is required in order to follow the latest developments. This includes the superfield formulation of gauge theories, holomorphy, the chiral ring, the Konishi anomaly and the large N limit. We then present both the diagrammatic proof of the connection and the one based on the anomaly. Our discussion is entirely field theoretical and self contained.

Even though matrix model partition functions do not exhaust the entire set of τ-functions relevant for string theory, they seem to be elementary building blocks for many others and they seem to properly capture the fundamental symplicial nature of quantum gravity and string theory. We propose to consider matrix model partition functions as new special functions. Here we restrict our consideration to the finite-size Hermitian 1-matrix model and concentrate mostly on its phase/branch structure arising when the partition function is considered as a D-module. We discuss the role of the CIV–DV prepotential (as generating a possible basis in the linear space of solutions to the Virasoro constraints, but with a lack of understanding of why and how this basis is distinguished).

We present an analysis of the Yangian symmetries of various bosonic sectors of the dilatation operator of SYM. The analysis is presented from the point of view of Hamiltonian matrix models. In the various SU(n) sectors, we give a modified presentation of the Yangian generators, which are conserved on states of any size. A careful analysis of the Yangian invariance of the full SO(6) sector of the scalars is also presented in this paper. We also study the Yangian invariance beyond first order perturbation theory. Following this, we derive the continuum limits of the various matrix models and reproduce the sigma model actions for fast moving strings reported in some papers. We motivate the constructions of continuum sigma models (corresponding to both the SU(n) and SO(n) sectors) as variational approximations to the matrix model Hamiltonians. These sigma models retain the semiclassical counterparts of the original Yangian symmetries of the dilatation operator. The semiclassical Yangian symmetries of the sigma models are worked out in detail. The zero curvature representation of the equations of motion and the construction of the transfer matrix for the SO(n) sigma model obtained as the continuum limit of the one loop bosonic dilatation operator is carried out, and the similar constructions for the SU(n) case are also discussed.

Recent analytical and numerical solutions of above systems are reviewed. Discussed results include: a) exact construction of the supersymmetric vacua in two space-time dimensions, and b) precise numerical calculations of the coexisting, continuous and discrete, spectra in the four-dimensional system, together with the identification of dynamical supermultiplets and SUSY vacua. New construction of the gluinoless SO(9) singlet state, which is vastly different from the empty state, in the ten-dimensional model is also briefly summarized.

Using the large-N conformal invariant collective field formulation, of the dual matrix model, a strong-weak coupling duality is displayed in terms of the generators of the conformal group. In the large-N limit, the real-symmetric matrix model is dual to the quaternionic-real matrix model and the hermitian matrix model is self-dual. We construct the master Hamiltonian for dual real-symmetric and quaternionic-real matrices and discuss a connection with the hermitian matrix model.