Abstract:
It is widely believed that via the Seiberg-Witten map, the linearly realized BPS equation in the non-commutative space is related to the non-linearly realized BPS equation in the commutative space in the zero slope limit. We show that the relation also holds without taking the zero slope limit as is expected from the arguments of the BPS equation for the non-Abelian Born-Infeld theory. This is regarded as an evidence for the relation between the two BPS equations. As a byproduct of our analysis, the non-linear instanton equation is solved exactly.

Abstract:
We solve the non-linear monopole equation of the Born-Infeld theory to all orders in the NS 2-form and give physical implications of the result. The solution is constructed by extending the earlier idea of rotating the brane configuration of the Dirac monopole in the target space. After establishing the non-linear monopole, we explore the non-commutative monopole by the Seiberg-Witten map.

Abstract:
It was suggested by Sugimoto that there is a new supersymmetry breaking mechanism by an orientifold plane which is oppositely charged as the usual one. Here we prove the trace formula for this system to show that the supersymmetry is broken not explicitly but spontaneously. We also discuss the possibility of interpreting the orientifold plane as an intrinsic object of the superstring theory.

Abstract:
Recently, it was shown by Jevicki, Kazama and Yoneya (JKY) that the Super-Yang-Mills theories (SYM) in $D\leq 4$ can reproduce the conformal symmetry of the near-horizon geometry of the D-brane solutions. However, the eikonal approximation they used is not sufficient for analyzing the conformal symmetry in SYM. We carry out the 1-loop calculation beyond the eikonal approximation for D=1 SYM (i.e., Matrix model) to confirm the claim of JKY.

Abstract:
We generalize the background gauge in the Matrix model to propose a new gauge which is useful for discussing the conformal symmetry. In this gauge, the special conformal transformation (SCT) as the isometry of the near-horizon geometry of the D-particle solution is directly reproduced with the correct coefficient as the quantum correction to the SCT in the Matrix model. We also present a general argument for the relation between the gauge choice and the field redefinition in the Matrix model.

Abstract:
We present a new Fermi gas formalism for the ABJ matrix model. This formulation identifies the effect of the fractional M2-brane in the ABJ matrix model as that of a composite Wilson loop operator in the corresponding ABJM matrix model. Using this formalism, we study the phase part of the ABJ partition function numerically and find a simple expression for it. We further compute a few exact values of the partition function at some coupling constants. Fitting these exact values against the expected form of the grand potential, we can determine the grand potential with exact coefficients. The results at various coupling constants enable us to conjecture an explicit form of the grand potential for general coupling constants. The part of the conjectured grand potential from the perturbative sum, worldsheet instantons and bound states is regarded as a natural generalization of that in the ABJM matrix model, though the membrane instanton part contains a new contribution.

Abstract:
We reexamine the oscillator construction of the D25-brane solution and the tachyon fluctuation mode of vacuum string field theory given previously. Both the classical solution and the tachyon mode are found to violate infinitesimally their determining equations in the level cut-off regularization. We study the effects of these violations on physical quantities such as the tachyon mass and the ratio of the energy density of the solution relative to the D25-brane tension. We discuss a possible way to resolve the problem of reproducing the expected value of one for the ratio.

Abstract:
We construct a 1/4 BPS soliton solution in N=4 non-commutative super Yang-Mills theory to the first order in the non-commutativity parameter \theta_{ij}. We then solve the non-commutative eigenvalue equations for the scalar fields. The Callan-Maldacena interpretation of the eigenvalues precisely reproduces the expected string junction picture: the string junction is tilted against the D3-branes with angle \theta_{ij}.

Abstract:
We analyze how lower-dimensional bosonic D-branes further decay, using the boundary string field theory. Especially we find that the effective tachyon potential of the lower-dimensional D-brane has the same profile as that of D25-brane.

Abstract:
We reveal a novel mathematical structure in physical observables, the mass of tachyon fluctuation mode and the energy density, associated with a classical solution of vacuum string field theory constructed previously [hep-th/0108150]. We find that they are expressed in terms of quantities which apparently vanish identically due to twist even-odd degeneracy of eigenvalues of a Neumann coefficient matrix defining the three-string interactions. However, they can give non-vanishing values because of the breakdown of the degeneracy at the edge of the eigenvalue distribution. We also present a general prescription of correctly simplifying the expressions of these observables. Numerical calculation of the energy density following our prescription indicates that the present classical solution represents the configuration of two D25-branes.