Geometry of String Theory and Its Nonperturbative Effect

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The focus of the research in these three years is the interpretation of D-brane in terms of the string field theory.In the first year (2001), I started the research by examining the relation between the projection operators of the open string star product and D-brane in the viewpoint of the boundary conformal field theory. Among other projectors, we examined identity, sliver and butterfly operators anc argued that the identity operator seems the most promising one since it reproduces the string tension correctly. At the same time, however, realized that a singularity at the midpoint of the open string prevents the rigorous treatment.In the second year (2002). I began to develop a regularized version of the open string field theory based on the simplified star product, Moyal product, which is used extensively in the noncommutative geometry. In this language, I found that the singularity appears as an anomaly of the Associativity of the star product. We can not escape from the anomaly as … More long as we work with the infinite number of components and we introduced a regularization scheme by cutting off the component field to the finite number. With this reformulation, we can make a reliable computation of all the off-shell amplitudes: At the same time, we can also make an analytic study of the tachyon vacuum. The computation of off-shell amplitude is quite satisfactory since we can write all the amplitudes in the explicit form with finite rank matrices. At the same time, we realized that the singularity from the midpoint is actually dominant to the computation of the tachyon vacuum. In this sense, we realized that open string language is not appropriate to describe the D-brane.Finally, we start to analyze the D-brane in the closed string language. It is known that D-brane is best described by the boundary state which live in the closed string sector. We show that all the known boundary state satisfy a universal idempotency relation, which has exactly the same form of the VSFT equation of motion. Less