Construction of Fuzzy Spaces and Their Applications to Matrix Models
Abstract
Quantization of spacetime by means of finite dimensional matrices is the basic idea of fuzzy spaces. There remains an issue of quantizing time, however, the idea is simple and it provides an interesting interplay of various ideas in mathematics and physics. Shedding some light on such an interplay is the main theme of this dissertation. The dissertation roughly separates into two parts. In the first part, we consider a mathematical aspect of fuzzy spaces, namely, their construction. We begin with a review of the construction of fuzzy complex projective spaces CPk (k=1,2,...) in relation to geometric quantization. We then present the construction of fuzzy S4, utilizing the fact that CP3 is an S2 bundle over S4. This method is also applicable to the case of fuzzy S8. In the second part, we consider applications of fuzzy spaces to physics. We first consider gravitational theories on fuzzy spaces, anticipating that they may offer a novel way of regularizing spacetime dynamics. Particularly we obtain actions for gravity on fuzzy S2 and fuzzy CP2. We also discuss application to M(atrix) theory. Introducing extra potentials, we show that the theory has new brane solutions whose transverse directions are described by fuzzy S4 and fuzzy CP3. The extra potentials can be interpreted as fuzzy versions of differential forms or fluxes, which enable us to discuss compactification models of M(atrix) theory. Compactification down to fuzzy S4 is discussed and a physically interesting matrix model in four-dimensions is proposed.