Unification Theories Including Gravity and its Mathematical Structure

在这个项目中，我们研究了两个主题，即弦理论的p-adar方法和标准模型和超越标准模型的统一理论。超变换是包括引力在内最突出的统一理论。虽然已经发现了许多有趣的东西，但弦的真空结构是什么等重要问题依赖于弦理论中的非微扰计算能力。基于弦树幅值的无穷乘积公式，提出了p向弦的新的数学结构。今年，我们重新考虑了基于Bruhat-Tits树的p向弦幅度，它是对普通弦的世界单的自然类比。我们可以用这一思想来表示p进环路的幅度。剩下的问题包括澄清p-进弦理论和正规弦理论之间的关系，以及寻找定义在Bruhat-Tits树上的场论的可能应用。至于统一理论，Frampton和Okada考虑了夸克质量和味道混合的问题。在三代标准模型中，夸克区有10个自由参数。这些参数之间的任何关系都表明物理模型超出了标准模型。我们对夸克质量矩阵作了一个简单的变换，导出了小林-Maskawa矩阵的两个元素与上夸克质量、魅力夸克质量和顶夸克质量之间的新关系。一旦知道了顶夸克的质量，并对B介子衰变进行了更精确的测量，我们就可以检查所提出的关系的有效性。Frampton还与T.Yanagida一起研究了具有额外规范对称性和重费米子的模型中的强CP问题，该模型称为Sark。他们指出，在具有Peccei-Quinn对称性的Sark模型中，可以避免轻轴粒子。

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.