Matrix models and the gravitational interaction
The superstring theory is now regarded as the most promising candidate for the unication of the Standard Model and gravity, and this eld has been rigorously investigated. However, we have seen a setback of the perturbative analysis of the superstring theory, because it has so many vacua that we have no way to determine which are the true ones. In order to remedy this problem and thus for the superstring theory to have a power to predict our real four-dimensional world, we need the nonperturbative formulation of the superstring theory. In the late 1990’s, our understanding of the nonperturbative aspects of the superstring theory have been greatly deepened. Especially, the large-N (N is the size of the matrices) reduced models have been proposed as the nonperturbative formulation of the superstring theory. One of the most promising candidates is the IIB matrix model, which is dened by the dimensional reduction of the ten-dimensional N = 1 super-Yang-Mills theory to zero dimension. It has been conjectured that this model resurrects the behavior of the string theory at the large-N limit. There have been a lot of interesting discoveries of the IIB matrix model, such as the dynamical generation of the four-dimensional spacetime and the interpretation of the dieomorphism invariance. On the other hand, there are a lot of problems to surmount, if a large-N reduced model is to be an eligible framework to unify the gravitational interaction. Firstly, it is still an enigma how we can realize the local Lorentz invariant matrix model. In addition, we need to understand how we can describe the curved spacetime more manifestly, in terms of a large-N reduced model. This thesis discusses several attempts to address these issues concerning the gravitational interaction. This thesis is based on the following works [26,38,46,60].

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: New states of gauge theories on a circle
abstract: A bstractWe study a one-dimensional large-N U(N) gauge theory on a circle as a toy model of higher dimensional Yang-Mills theories at finite temperature. To investigate the profile of the thermodynamical potential in this model, we evaluate a stochastic time evolution of several states, and find that an unstable confinement phase at high temperature does not decay to a stable deconfinement phase directly. Before it reaches the deconfinement phase, it develops to several intermediate states. These states are characterised by the expectation values of the Polyakov loop operators, which wind the temporal circle different times. We reveal that these intermediate states are the saddle point solutions of the theory, and similar solutions exist in a wide class of SU(N) and U(N) gauge theories on S1 including QCD and pure Yang-Mills theories in various dimensions. We also consider a Kaluza-Klein gravity, which is the gravity dual of the one-dimensional gauge theory on a spatial S1, and show that these solutions may be related to multi black holes localised on the S1. Then we present a connection between the stochastic time evolution of the gauge theory and the dynamical decay process of a black string though the Gregory-Laflamme instability.

title: The instability of intersecting fuzzy spheres
abstract: We discuss the classical and quantum stability of general configurations representing many fuzzy spheres in dimensionally reduced Yang-Mills-Chern-Simons models with and without supersymmetry. By performing one-loop perturbative calculations around such configurations, we find that intersecting fuzzy spheres are classically unstable in the class of models studied in this paper. We also discuss the large-N limit of the one-loop effective action as a function of the distance of fuzzy spheres. This shows, in particular, that concentric fuzzy spheres with different radii, which are identified with the 't Hooft-Polyakov monopoles, are perturbatively stable in the bosonic model and in the D = 10 supersymmetric model.

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: 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: 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: An Infinite Number of Static Soliton Solutions to the 5D Einstein-Maxwell Equations with a Dilaton Field
abstract: We study the 5D static Einstein-Maxwell equations with a dilaton field. We develop an infinite number of solutions using a soliton technique. We study the rod structure of a two-soliton solution and show that a 5D dilatonic black ring and black hole solutions are included.