large-N Reduction Research Articles

Complex Langevin method on rotating matrix quantum mechanics at thermal equilibrium

Abstract Rotating systems in thermal equilibrium are ubiquitous in our world. In the context of high-energy physics, rotations would affect the phase structure of quantum chromodynamics (QCD). However, the standard Monte Carlo methods in rotating systems are problematic because the chemical potentials for the angular momenta (angular velocities) cause sign problems even for bosonic variables. In this article, we demonstrate that the complex Langevin method (CLM) may overcome this issue. We apply the CLM to the Yang–Mills (YM)-type one-dimensional matrix model (matrix quantum mechanics) that is a large-N reduction (or dimensional reduction) of the (D + 1)-dimensional U(N) pure YM theory [bosonic Banks–Fischler–Shenker–Susskind (BFSS) model]. This model shows a large-N phase transition at finite temperature, which is analogous to the confinement/deconfinement transition of the original YM theory, and our CLM predicts that the transition temperature decreases as the angular momentum chemical potential increases. In order to verify our results, we compute several quantities via the minimum sensitivity method and find good quantitative agreements. Hence, the CLM works properly in this rotating system. We also argue that our results are qualitatively consistent with a holography and the recent studies of the imaginary angular velocity in QCD. As a byproduct, we develop an analytic approximation to treat the so-called “small black hole” phase in the matrix model.

Equivalence of large-N gauge theories on a group manifold and its coset space

In 2010, Kawai, Shimasaki and one of the present authors (A.T.) showed that the large-N reduction holds on group manifolds in the sense that a large-N gauge theory on a group manifold is realized by a matrix model which is obtained by dimensionally reducing the original theory to zero dimension. In this note, generalizing the above statement, we show that a large-N gauge theory on a group manifold is equivalent to a theory which is obtained by reducing the original theory to its coset space. This is analogous to the statement of the large-N reduction on flat spaces that large-N gauge theories are independent of the volume.

Direct test of the AdS/CFT correspondence by Monte Carlo studies of $ \mathcal{N}=4 $ super Yang-Mills theory

We perform nonperturbative studies of N=4 super Yang-Mills theory by Monte Carlo simulation. In particular, we calculate the correlation functions of chiral primary operators to test the AdS/CFT correspondence. Our results agree with the predictions obtained from the AdS side that the SUSY non-renormalization property is obeyed by the three-point functions but \emph{not} by the four-point functions investigated in this paper. Instead of the lattice regularization, we use a novel regularization of the theory based on an equivalence in the large-N limit between the N=4 SU(N) theory on RxS^3 and a one-dimensional SU(N) gauge theory known as the plane-wave (BMN) matrix model. The equivalence extends the idea of large-N reduction to a curved space and, at the same time, overcomes the obstacle related to the center symmetry breaking. The adopted regularization preserves 16 SUSY, which is crucial in testing the AdS/CFT correspondence with the available computer resources. The only SUSY breaking effects, which come from the momentum cutoff $\Lambda$ in R direction, are made negligible by using sufficiently large $\Lambda$.

Exact results for perturbative partition functions of theories with SU(2|4) symmetry

In this paper, we study the theories with SU(2|4) symmetry which consist of the plane wave matrix model (PWMM), super Yang-Mills theory (SYM) on RxS^2 and SYM on RxS^3/Z_k. The last two theories can be realized as theories around particular vacua in PWMM, through the commutative limit of fuzzy sphere and Taylor's T-duality. We apply the localization method to PWMM to reduce the partition function and the expectation values of a class of supersymmetric operators to matrix integrals. By taking the commutative limit and performing the T-duality, we also obtain the matrix integrals for SYM on RxS^2 and SYM on RxS^3/Z_k. In this calculation, we ignore possible instanton effects and our matrix integrals describe the perturbative part exactly. In terms of the matrix integrals, we also provide a nonperturbative proof of the large-N reduction for circular Wilson loop operator and free energy in N=4 SYM on RxS^3.

LARGE-N REDUCTION FOR $\mathcal{N} = 2$ CHERN-SIMONS THEORIES AND LOCALIZATION

We study reduced matrix models obtained by the dimensional reduction of [Formula: see text] quiver Chern-Simons theories on S3 and show that, around a certain multiple fuzzy sphere background, the reduced models are equivalent to the original theories in the large-N limit. This is regarded as a large-N reduction on S3. By using the localization technique, we prove the large-N equivalence with respect to supersymmetric observables.

Perturbative tests for a large-N reduced model of $ \mathcal{N} = {4} $ super Yang-Mills theory

We study a non-perturbative formulation of $ \mathcal{N} = {4} $ super Yang-Mills theory (SYM) on R × S 3 in the planar limit proposed in arXiv:0807.2352. This formulation is based on the large-N reduction, and the theory can be described as a particular large-N limit of the plane wave matrix model (PWMM), which is obtained by dimensionally reducing the original theory over S 3. In this paper, we perform some tests for this proposal. We construct an operator in the PWMM that corresponds to the Wilson loop in SYM in the continuum limit and calculate the vacuum expectation value of the operator for the case of the circular contour. We find that our result indeed agrees with the well-known result first obtained by Erickson, Semenoff and Zarembo. We also compute the beta function at the 1-loop level based on this formulation and see that it is indeed vanishing.

Perturbative tests for a large-N reduced model of $ \mathcal{N} = {4} $ super Yang-Mills theory

We study a non-perturbative formulation of N=4 super Yang-Mills theory (SYM) on RxS^3 proposed in arXiv:0807.2352. This formulation is based on the large-N reduction, and the theory can be described as a particular large-N limit of the plane wave matrix model (PWMM), which is obtained by dimensionally reducing the original theory over S^3. In this paper, we perform some tests for this proposal. We construct an operator in the PWMM that corresponds to the Wilson loop in SYM in the continuum limit and calculate the vacuum expectation value of the operator for the case of the circular contour. We find that our result indeed agrees with the well-known result first obtained by Erickson, Semenoff and Zarembo. We also compute the beta function at the 1-loop level based on this formulation and see that it is indeed vanishing.

Large-N reduction of [formula omitted] Yang–Mills theory with massive adjoint overlap fermions

We study four-dimensional large-NSU(N) Yang–Mills theory coupled to adjoint overlap fermions on a single site lattice. Lattice simulations along with perturbation theory show that the bare quark mass has to be taken to zero as one takes the continuum limit in order to be in the physically relevant center-symmetric phase. But, it seems that it is possible to take the continuum limit with any renormalized quark mass and still be in the center-symmetric physics. We have also conducted a study of the correlations between Polyakov loop operators in different directions and obtained the range for the Wilson mass parameter that enters the overlap Dirac operator.

Testing a novel large-Nreduction for 𝒩 = 4 super Yang-Mills theory onR×S3

Recently a novel large-N reduction has been proposed as a maximally supersymmetric regularization of = 4 super Yang-Mills theory on R × S3 in the planar limit. This proposal, if it works, will enable us to study the theory non-perturbatively on a computer, and hence to test the AdS/CFT correspondence analogously to the recent works on the D0-brane system. We provide a nontrivial check of this proposal by performing explicit calculations in the large-N reduced model, which is nothing but the so-called plane wave matrix model, around a particular stable vacuum corresponding to R × S3. At finite temperature and at weak coupling, we reproduce precisely the deconfinement phase transition in the = 4 super Yang-Mills theory on R × S3. This phase transition is considered to continue to the strongly coupled regime, where it corresponds to the Hawking-Page transition on the AdS side. We also perform calculations around other stable vacua, and reproduce the phase transition in super Yang-Mills theory on the corresponding curved space-times such as R × S3/Zq and R × S2.

Applying the Wang-Landau algorithm to lattice gauge theory

We implement the Wang-Landau algorithm in the context of SU(N) lattice gauge theories. We study the quenched, reduced version of the lattice theory and calculate its density of states for N=20, 30, 40, 50. We introduce a variant of the original algorithm in which the weight function used in the update does not asymptote to a fixed function, but rather continues to have small fluctuations that enhance tunneling. We formulate a method to evaluate the errors in the density of states, and use the result to calculate the dependence of the average action density and the specific heat on the 't Hooft coupling {lambda}. This allows us to locate the coupling {lambda}{sub t} at which a strongly first-order transition occurs in the system. For N=20 and 30 we compare our results with those obtained using Ferrenberg-Swendsen multihistogram reweighting and find agreement with errors of 0.2% or less. Extrapolating our results to N={infinity}, we find ({lambda}{sub t}){sup -1}=0.3148(2). We remark on the significance of this result for the validity of quenched large-N reduction of SU(N) lattice gauge theories.

Large-N reduction in the continuum three-dimensional Yang-Mills theory.

Numerical and theoretical evidence leads us to propose the following: Three-dimensional Euclidean Yang-Mills theory in the planar limit undergoes a phase transition on a torus of side l=l(c). For l>l(c) the planar limit is l independent, as expected of a noninteracting string theory. We expect the situation in four dimensions to be similar.

Dijkgraaf-Vafa theory as large-N reduction

Dijkgraaf-Vafa theory as large-N reduction

LARGE-N REDUCTION, MASTER FIELD AND LOOP EQUATIONS IN KAZAKOV-MIGDAL MODEL

I study the large-N reduction á la Eguchi-Kawai in the Kazakov-Migdal lattice gauge model. I show that both quenching and twisting prescriptions lead to the coordinate-independent master field. I discuss properties of loop averages in reduced as well as unreduced models and demonstrate those coincide in the large mass expansion. I derive loop equations for the Kazakov-Migdal model at large-N and show they are reduced for the quadratic potential to a closed set of two equations. I find an exact strong coupling solution of these equations for any D and extend the result to a more general interacting potential.