Soliton dynamics and dualities in superstring theory

弦理论被认为是最有希望统一自然界所有四种力的候选者。据推测，所有一致的弦理论都可以从一个叫做m理论的理论中推导出来。众所周知，m理论面临着量子化等诸多理论挑战。在这个研究中，我澄清了m理论的一些方面。它可以分为(1)洛伦兹对称恢复的检验(2)开膜作为矩阵模型的量化(3)从m理论的角度解释弦结(4)将通常通过开弦边界定义的非交换时空扩展到开膜情况。在(1)中，我们只进行了经典计算，量子化微积分在未来是可取的。在(4)中，给出了与数学物理的许多分支相关的保体积微分同态的量子表示。由弦理论定义的时空几何与爱因斯坦引力中的普通黎曼几何有很大的不同。在某种极限下，已知几何被简化为Connes提出的非交换几何。一旦时空变成非交换态，就会出现一种新的孤子构型，这种构型在交换理论中是不存在的。在本研究中，我提出这样的孤子电荷可以解释为c代数的k同调。通过应用这一建议，我考虑了定义在非交换环面上的孤子构型，并指出存在一种奇怪的激发，这在现阶段很难从物理上理解。如果在这样的非交换空间上定义开弦，我们需要引入矩阵弦理论。我们为这样的理论提供了一个基础。

String theory is regarded as the most promising candidate to unify all the four forces of the nature. It has been conjectured that all consistent string theory can be derived from a single theory that is called M-theory. It is known that M-theory has a lot of theoretical challenges such as the quantization. In this research, I clarified some aspects of M-theory. It may be classified as (1) examination of the restoration of Lorentz symmetry (2) Quantization of the open membrane as the matrix model (3) Interpretation of string junction from M-theoretical viewpoint (4) extension of the noncommutative space-time which is usually defined through the boundary of the open string to the open membrane situation. In (1), we have carried out only the classical computation and quantized calculus is desirable in the future. In (4), it gives a quantum representation of the volume preserving diffeomorphism, which is related to the many branches of the mathematical physics.The space-time geometry, which is defined by the string theory, is critically different from the ordinary Riemannian geometry in Einstein gravity. In some limit, it is known that the geometry is simplified to the noncommutative geometry that Connes proposed. Once the space-time becomes noncommutative, there appears a new kind of Soliton configuration, which does not appear in the commutative theory. In this research, I proposed that such Soliton charges might be interpreted as the K-homology of the C-algebra. By applying this proposal, I considered the Soliton configuration defined on the noncommutative torus and indicated that there appears a strange excitation, which is difficult to understand physically at this stage. If open string is defined on such a noncommutative space, we need to introduce the matrix string theory. We give a foundation to such a theory.