A Study of the Complex Action Problem in a Simple Model for Dynamical Compactification in Superstring Theory Using the Factorization Method
The IIB matrix model proposes a mechanism for dynamically generating four dimensional space--time in string theory by spontaneous breaking of the ten dimensional rotational symmetry $\textrm{SO}(10)$. Calculations using the Gaussian expansion method (GEM) lend support to this conjecture. We study a simple $\textrm{SO}(4)$ invariant matrix model using Monte Carlo simulations and we confirm that its rotational symmetry breaks down, showing that lower dimensional configurations dominate the path integral. The model has a strong complex action problem and the calculations were made possible by the use of the factorization method on the density of states $\rho_n(x)$ of properly normalized eigenvalues $\tilde\lambda_n$ of the space--time moment of inertia tensor. We study scaling properties of the factorized terms of $\rho_n(x)$ and we find them in agreement with simple scaling arguments. These can be used in the finite size scaling extrapolation and in the study of the region of configuration space obscured by the large fluctuations of the phase. The computed values of $\tilde\lambda_n$ are in reasonable agreement with GEM calculations and a numerical method for comparing the free energy of the corresponding ansatze is proposed and tested.

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: Dynamical generation of gauge groups in the massive Yang-Mills-Chern-Simons matrix model
abstract: It has been known that the dynamics of $k$ coincident D-branes in string theory is described effectively by $\mathrm{U}(k)$ Yang-Mills theory at low energy. While these configurations appear as classical solutions in matrix models, it was not clear whether it is possible to realize the $k\ensuremath{\ne}1$ case as the true vacuum. The massive Yang-Mills\char21{}Chern-Simons matrix model has classical solutions corresponding to all the representations of the SU(2) algebra, and provides an opportunity to address the above issue on a firm ground. We investigate the phase structure of the model, and find, in particular, that there exists a parameter region where $\mathrm{O}(N)$ copies of the spin-$1/2$ representation appear as the true vacuum, thus realizing a nontrivial gauge group dynamically. Such configurations are analogous to the ones that are interpreted in the Berenstein-Maldacena-Nastase (BMN) matrix model as coinciding transverse 5-branes in M-theory.

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: 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: De Sitter metric as a self-consistent solution of the back reaction problem
abstract: The de Sitter metric is a solution of the Einstein equation even when quantum effects of matter fields are included. However between the curvature and the cosmological constant Λ changes drastically. For some values of Λ, there are two or three solutions, while there is none in some other cases. There can be even a positive curvature solution for negative Λ. These features depend on mass and spin of matter fields and the matter-curvature coupling.

title: Equilibrium Condition of Dilaton Black Holes with Electric Charge

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: Classification of an infinite series of static solutions to the vacuum Einstein equations
abstract: We propose a classification of an infinite series of the static solutions to the vacuum Einstein equation by means of their soliton number and the real and/or complex pole trajectories. We also show that in the 4-soliton solution there appears an intriguing ring solution of which the curvature invariant has a finite limit at the rings.

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