Investigation of Matrix Theory via Super Lie Algebra
This paper reports the investigation of a matrix model via super Lie algebra, following the proposal of L. Smolin. We consider the osp(1|32,R) nongauged matrix model and gl(1|32,R) gauged matrix model, especially paying attention to the supersymmetry and the relationship with IKKT model. This paper is based on the collaboration with the collaboration with S.Iso, H.Kawai and Y.Ohwashi.

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: 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: 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: 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: 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: 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

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: 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: Multi-Weyl Solutions to the Einstein Equation Coupled with Scalar Field
abstract: We study the static Einstein equation coupled with a scalar field with axial symmetry. The zero curvature equation both for the scalar field and a part of the metric in the scalar-Einstein equation enables one to obtain soliton solutions for both the scalar field and the metric. Of the infinite series of solutions for the metric and the scalar field we focus on the solution composed of 4-soliton solution of metric and 4-soliton solution of scalar field. We give a physical interpretation of the solution that two Weyls with scalar charges are exactly sustained by the tension of a string lying between the two Weyls. One of the most interesting problems of black holes is the study of bifurcation of a black hole and coalescence of multi black holes_ Before the dynamical behaviors of black holes is understood through the exact solutions of the Einstein equation, we should first understand the static and stationary solutions of multi black holes. In the preceding papers!) we have studied the vacuum soliton solutions to the static Einstein equation which includes the ring-type solutions and the multi-Weyl solution. In the 4-soliton solution or two-Weyl solution, what is peculiar in the solution is the exis­ tence of a conical singularity which may be interpreted as a "cosmic string" or strut between two Weyls. We showed that the string tension cancels the gravitational attractive force. There are two possible solutions which satisfy the asymptotic flatness: (i) The masses of two Weyls could be both positive while the string tension or the line density of the strut is negative; (ii) One of the masses of two Weyls should be negative if the string tension has the positive sign_ Either case comprises physical difficulty: negative mass or negative energy density. Classically there can be several ways for particles to balance despite of their attractive gravitational forces. The first is the introduction of strut to sustain their gravitational forces. The second is the introduction of negative mass to the system of particles. The third is the introduction of another force which can cancel the gravitational attractive force. As for the third one of the realization in the general relativity is the work by Papapetrou and Majumdar (the PM solution)_2) They introduced the coulomb force to cancel the gravitational attractive force. However the electric charge and the mass of each black hole are not independent charges but should satisfy the relation which realizes the extremal case of the Reissner-Nordstrom solution. What if the constraint between the mass and the electric charge are removed? This has not been answered so far. In the general relativity what could be the correspondence to the strut and the negative mass for balancing? This can be answered by showing the exact solutions to the Einstein equation. As far as we consider the vacuum Einstein equation we could not find physically

title: Equilibrium Condition of Dilaton Black Holes with Electric Charge