Unification Theories Including Gravity and its Mathematical Structure

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In this program we studied two subjects, namely the p-adic approach to the string theory and the unified theories of the standard and beyond standard models.Supersrting is the most prominent unified theory including gravity. Although many interesting things have been found, the important questions like what is the vacuum structure of string rely on the ability of non-perturbative calculation in string theory. The p-adic string was initiated to understand new mathematical structure of string in terms of an infinite product formula of the string tree amplitude.This year we reconsidered the p-adic string amplitude based on Bruhat-Tits tree which is a natural analogy to the world sheet of the ordinary string. We can formulate the p-adic loop amplitude using this idea. The remaining problems include to clarify the relationship between the p-adic and ordinarystring theories and to find possible applications of field theories defined over Bruhat-Tits tree.As for unification theories, Frampton and Okada consider the problem of quark masses and flavor mixings. In the standard model with three generations, we have 10 free parameters in the quark sector. Any relations among these parameters suggest physics beyond the standard model. We start a simple ansatz for the quark mass matrices and derive a new relation among two elements of Kobayashi-Maskawa matrix and up-, charm-and top-quark masses. Once the top quark mass is known and more precise measurements of B-meson decay are done we can check the validity of the proposed relation.With T. Yanagida, Frampton also studied the strong CP problem in a model with an extra gauge symmetry and heavy fermions called sark. They pointed out that in the sark model with Peccei-Quinn symmetry a light axion can be avoided.