Multi-Weyl Solutions to the Einstein Equation Coupled with Scalar Field
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: The Renormalized Energy-Momentum Tensor In a Robertson -Walker Universe
abstract: An expectation value of the energy·momentum tensor for a massive scalar field propagating in a Robertson-\\;'alker universe is calculated by taking account of both effects caused by the particle creation and the vacuum polarization. A special case of the Chitre-Hartle universe which enables us to perform an explicit calculation is also treated.

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: Axially symmetric solution of the Einstein equation with solitons in extra dimensions
abstract: We give the solution to the higher dimensional Einstein equation with axial symmetry by using the inverse scattering method. The solution consists of two solitons in the external dimensions and any number of solitons in the internal dimensions.

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: 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.

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: 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: Supermatrix Models
abstract: We investigate several matrix models based on super Lie algebras, osp(1|32, R), u(1|16, 16) and gl(1|32, R). They are natural generalizations of IIB matrix model and were first proposed by Smolin . In particular, we study the supersymmetry structures of these models and discuss possible reductions to IIB matrix model. We also point out that diffeomorphism invariance is hidden in gauge theories on noncommutative space which are derived from matrix models. This symmetry is independent of the global SO(9, 1) invariance in IIB matrix model and we report our trial to extend the global Lorentz invariance to local symmetry by introducing u(1|16, 16) or gl(1|32, R) super Lie algebras. ∗ e-mail address : azuma@gauge.scphys.kyoto-u.ac.jp † e-mail address : satoshi.iso@kek.jp ‡ e-mail address : hkawai@gauge.scphys.kyoto-u.ac.jp § e-mail address : yuhi@gauge.scphys.kyoto-u.ac.jp

title: Matrix models and the gravitational interaction
abstract: 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: Curved-space classical solutions of a massive supermatrix model
abstract: We investigate here a supermatrix model with a mass term and a cubic interaction. It is based on the super Lie algebra osp(1|32,R), which could play a role in the construction of the eleven-dimensional M-theory. This model contains a massive version of the IIB matrix model, where some fields have a tachyonic mass term. Therefore, the trivial vacuum of this theory is unstable. However, this model possesses several classical solutions where these fields build noncommutative curved spaces and these solutions are shown to be energetically more favorable than the trivial vacuum. In particular, we describe in details two cases, the SO(3)×SO(3)×SO(3) (three fuzzy 2-spheres) and the SO(9) (fuzzy 8-sphere) classical backgrounds.