Nonperturbative studies of fuzzy spheres in a matrix model with the Chern-Simons term
Fuzzy spheres appear as classical solutions in a matrix model obtained via dimensional reduction of 3-dimensional Yang-Mills theory with the Chern-Simons term. Well-defined perturbative expansion around these solutions can be formulated even for finite matrix size, and in the case of k coincident fuzzy spheres it gives rise to a regularized U(k) gauge theory on a noncommutative geometry. Here we study the matrix model nonperturbatively by Monte Carlo simulation. The system undergoes a first order phase transition as we change the coefficient (α) of the Chern-Simons term. In the small α phase, the large-N properties of the system are qualitatively the same as in the pure Yang-Mills model (α = 0), whereas in the large α phase a single fuzzy sphere emerges dynamically. Various `multi fuzzy spheres' are observed as meta-stable states, and we argue in particular that the k coincident fuzzy spheres cannot be realized as the true vacuum in this model even in the large-N limit. We also perform one-loop calculations of various observables for arbitrary k including k = 1. Comparison with our Monte Carlo data suggests that higher order corrections are suppressed in the large-N limit.

title: Towards an Effective Importance Sampling in Monte Carlo Simulations of a System with a Complex Action
abstract: The sign problem is a notorious problem, which occurs in Monte Carlo simulations of a system with a partition function whose integrand is not positive. One way to simulate such a system is to use the factorization method where one enforces sampling in the part of the configuration space which gives important contribution to the partition function. This is accomplished by using constraints on some observables chosen appropriately and minimizing the free energy associated with their joint distribution functions. These observables are maximally correlated with the complex phase. Observables not in this set essentially decouple from the phase and can be calculated without the sign problem in the corresponding "microcanonical" ensemble. These ideas are applied on a simple matrix model with very strong sign problem and the results are found to be consistent with analytic calculations using the Gaussian Expansion Method.

title: Monte Carlo studies of the GWW phase transition in large-N gauge theories
abstract: In the study of the small ten-dimensional Schwarzschild black hole, the black hole to string transition is an important problem. In [L. Alvarez-Gaume, P. Basu, M. Marino, S.R. Wadia,  (2006) 647 ], a possible identification is made between the Gross–Witten–Wadia (GWW) type third-order large- N phase transition in the boundary gauge theory and the string-black hole transition in the bulk. In this Letter, we exhibit the existence of the GWW transition by Monte Carlo simulation in the zero mode bosonic action of the finite-temperature N = 4 SYM theory on S 3 . Exhibiting this transition in the truncated but highly non-trivial gauge theory implies that in the vicinity of the critical temperature T c , the system goes critical, and the fluctuations give rise to universal formulas derived in [L. Alvarez-Gaume, P. Basu, M. Marino, S.R. Wadia,  (2006) 647, ]. We also discuss the issue of SO ( 6 ) R -symmetry breaking.

title: Monte Carlo simulations of a supersymmetric matrix model of dynamical compactification in non perturbative string theory
abstract: The IKKT or IIB matrix model has been postulated to be a non perturbative definition of superstring theory. It has the attractive feature that spacetime is dynamically generated, which makes possible the scenario of dynamical compactification of extra dimensions, which in the Euclidean model manifests by spontaneously breaking the SO(10) rotational invariance (SSB). In this work we study using Monte Carlo simulations the 6 dimensional version of the Euclidean IIB matrix model. Simulations are found to be plagued by a strong complex action problem and the factorization method is used for effective sampling and computing expectation values of the extent of spacetime in various dimensions. Our results are consistent with calculations using the Gaussian Expansion method which predict SSB to SO(3) symmetric vacua, a finite universal extent of the compactified dimensions and finite spacetime volume.

title: Monte Carlo studies of the spontaneous rotational symmetry breaking in dimensionally reduced super Yang-Mills models
abstract: A bstractIt has long been speculated that the spontaneous symmetry breaking (SSB) of SO(D) occurs in matrix models obtained by dimensionally reducing super Yang-Mills theory in D = 6, 10 dimensions. In particular, the D = 10 case corresponds to the IIB matrix model, which was proposed as a nonperturbative formulation of superstring theory, and the SSB may correspond to the dynamical generation of four-dimensional space-time. Recently, it has been shown by using the Gaussian expansion method that the SSB indeed occurs for D = 6 and D = 10, and interesting nature of the SSB common to both cases has been suggested. Here we study the same issue from first principles by a Monte Carlo method in the D = 6 case. In spite of a severe complex-action problem, the factorization method enables us to obtain various quantities associated with the SSB, which turn out to be consistent with the previous results obtained by the Gaussian expansion method. This also demonstrates the usefulness of the factorization method as a general approach to systems with the complex-action problem or the sign problem.

title: Matrix model with manifest general coordinate invariance
abstract: We present a formulation of a matrix model which manifestly possesses the general coordinate invariance when we identify the large N matrices with differential operators. In order to build a matrix model which has the local Lorentz invariance, we investigate how the so (9,1) Lorentz symmetry and the u ( N ) gauge symmetry are mixed together. We first analyze the bosonic part of the model, and we find that the Einstein gravity is reproduced in the classical low-energy limit. We then present a proposal to build a matrix model which has N =2 SUSY and reduces to the type IIB supergravity in the classical low-energy limit.

title: Cosmological solutions of the Einstein equation with the backreaction effect of quantized conformally invariant fields

title: Equilibrium Condition in the Axisymmetric N-Reissner-Nordstrom Solution
abstract: We solve the source-free Einstein-Maxwell equations in a static and axisymmetric spacetime by using the inverse scattering method that is one of the soliton techniques to solve the gravitational field equation. We obtain an exact solution that describes n Reissner-Nordstrom black holes located along the symmetry axis. We define the mass and charge of each black hole and study the condition for static equilibrium of black holes. We show that the equilibrium is realized only when all the black holes are extremal and the spacetime becomes Majumdar-Papapetrou type. Ever since the soliton techniques were first applied to solving the gravitational field equation we have been able to study the structure of many-body system in general relativity using exact solutions. Although the obtained solutions are static or stationary it may be important to study them as a first step toward considering a dynamical process such as black hole collision. The simplest example of many-body system is the axisymmetric n-Schwarzschild solution, which is obtained as a 2n­ soliton solution to the vacuum Einstein equation.ll This solution includes conical singularities between Schwarzschild black holes along the symmetry axis. These conical singularities, which we call struts, are necessary to uphold the gravitational attractive force between the black holes. Static balance without struts could be realized by introducing negative masses but naked singularities appear. Another way of static balance is to introduce some fields with repulsive force. Scalar fields act attractively and also break event horizons of black holes, which brings about naked singularities again. 2 > Repulsive Coulomb force in electromagnetic field can balance attractive gravitational one and leads the system to static equilib­ rium. The solution given by Majumdar 3 > and Papapetrou 4 > is known as such a solution. This solution is an exact solution of the source-free Einstein-Maxwell equations in a conformastatic spacetime and describes a system of extremal black holes.

title: An Infinite Number of Static Soliton Solutions to the 5D Einstein-Maxwell Equations
abstract: The soliton technique is applied to the 5D static Einstein-Maxwell equations, and an infinite number of solutions are explicitly obtained. We study the rod structure of 2-soliton solutions and we show that the 5D Reissner-Nordstrom solution and the 5D Majumdar-Papapetrou solution are included as the 2-soliton solutions.

title: Exact Disk Solution of the Einstein Equation
abstract: The 4-soliton solution with complex pole trajectories to the vacuum Einstein equation is studied. The solution describes a static and axisymmetric gravitational field with asymptotic flatness, which is produced by a thin disk. We obtain the energy-momentum tensor of the disk to reveal its properties. The soliton technique 1 ) to solve the gravitational field equation gave us a new viewpoint on the exact solutions of the vacuum Einstein equation. There is an infinite series of solutions which are characterized by the soliton number n. In the stationary case with axial symmetry the series includes the Kerr-NUT solution as a 2-soliton solution and the Tomimatsu-Sato solution as a certain limit of 4-soliton solution. In the static case 2 ),3) there is an infinite series of the Weyl solutions, which is interpreted as a nonlinear superposition of multi-Weyl solutions. Two of the present authors 3 ),4) showed that the soliton technique gives not only these series of solutions but a series of ring-type solutions by adopting complex pole trajectories in the static case. The first non-trivial example of this series is the Weyl solution expressed in the oblate spheroidal coordinates,5) which has a ring-type curvature singularity. In our classification this solution is interpreted as the 2-soliton solution with non-complex-conjugate pole trajectories. The next example is the 4-soliton solution in which there are two types. One is just a nonlinear superposition of the above 2-soliton solutions, which has two ring-type singularities with different radii. The other, which is more important, is the 4-soliton solution constructed from two pairs of complex conjugate pole trajectories. This was referred to as the two­ ring solution, because the curvature invariants have finite limits at two rings although the metric coefficients diverge there. This fact contrasts with the fact that the curvature invariants diverge at the ring in the 2-soliton case. The purpose of this paper is to study the detailed structure of the two-ring solution. It will be shown that the solution should be called a disk solution rather than a ring solution. The properties of the disk will be also clarified by calculating the energy-momentum tensor. In the next section we give the solution in an explicit form and investigate its mathematical structure. From this we find that the plane containing the rings is separated into a regular region inside the inner ring and a region with the discontinuity of curvature outside the inner ring. We shall call this outer region a "disk" because of its non-zero energy density as will be.discussed in § 3. The disk is further divided into two parts by the outer ring. In § 3 we obtain the

title: A Metric Representation of Belavin's Yang-Mills Pseudoparticle Solution and Its Relation to Some Isotropic Universe
abstract: In order to investigate a physical connection between the quantum tunneling processes (vacuum-to vacuum transition or thermal radiation) of Yang-Mills 11 •*' and gravitational" pseudoparticles, T omimatsu"'<l compared several (especially, topological) properties of the Belavin and Schwarzschild solutions with each other in terms of their Euclidean metrics. For that purpose, he transformed the Belavin solution into a metric form on the basis of Carmeli's'l formalism which describes the general theory of relativity in the form of an SL (2, C) gauge field. He showed that, of the two important topological quantities P (the Pontryagin number) and X (the Euler characteristic), P= 0 and ;: '-2 for the rewritten Belavin solution, similar to the result that P= 0 and !: ~, 2 for the Schwarzschild solution. To assure the idea that the origin of radiation with a thermal nature from both pseudoparticles is due to the fact that their x's are non-vanishing, he"b' performed a path-integral approach to the quantization of a massive scalar field in the gravitational field corresponding to Belavin' s solution. The resulting energy spectrum was an incomplete thermal one because of its non -uniform temper a ture and non -zero