Exact Disk Solution of the Einstein Equation
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

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: 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.

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: 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: 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: A practical solution to the sign problem in a matrix model for dynamical compactification
abstract: The matrix model formulation of superstring theory offers the possibility to understand the appearance of 4d space-time from 10d as a consequence of spontaneous breaking of the SO(10) symmetry. Monte Carlo studies of this issue is technically difficult due to the so-called sign problem. We present a practical solution to this problem generalizing the factorization method proposed originally by two of the authors (K.N.A. and J.N.). Explicit Monte Carlo calculations and large-N extrapolations are performed in a simpler matrix model with similar properties, and reproduce quantitative results obtained previously by the Gaussian expansion method. Our results also confirm that the spontaneous symmetry breaking indeed occurs due to the phase of the fermion determinant, which vanishes for collapsed configurations. We clarify various generic features of this approach, which would be useful in applying it to other statistical systems with the sign problem.

title: Perturbative versus nonperturbative dynamics of the fuzzy S2 × S2
abstract: We study a matrix model with a cubic term, which incorporates both the fuzzy S2 × S2 and the fuzzy S2 as classical solutions. Both of the solutions decay into the vacuum of the pure Yang-Mills model (even in the large-N limit) when the coefficient of the cubic term is smaller than a critical value, but the large-N behavior of the critical point is different for the two solutions. The results above the critical point are nicely reproduced by the all order calculations in perturbation theory. By comparing the free energy, we find that the true vacuum is given either by the fuzzy S2 or by the ``pure Yang-Mills vacuum'' depending on the coupling constant. In Monte Carlo simulation we do observe a decay of the fuzzy S2 × S2 into the fuzzy S2 at moderate N, but the decay probability seems to be suppressed at large N. The above results, together with our previous results for the fuzzy CP2, reveal certain universality in the large-N dynamics of four-dimensional fuzzy manifolds realized in a matrix model with a cubic term.

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.