Noncommutative Geometry and String Field Theory

The thesis is organized as follows: In chapter 2 we explain basic ideas of noncommutative field theory, its stringy origin and various interesting properties. We explain two of our papers, one with Bonora and Tomasiello [28] on anomalies, and another one with Bonora, Sheikh-Jabbari and Tomasiello [36] on the possibility of having SO(N) and Sp(N) gauge groups. Chapter 3 is devoted to the cubic string field theory. As there are not as many reviews as on the other subjects, we shall be slightly more detailed. We will discuss various formulations and approaches, and relations between them. vVe also explain the Sen's conjectures which triggered much of the recent developments. In chapter 4 we explain some properties of the string field algebra, the star product, the wedge states and in particular the identity state. Results here are mostly original. Chapter 5 discusses some approaches to the problem of finding the exact solution for the tachyon condensate, which we consider to be an interesting and important problem. One section of that chapter is based on our paper [86], the other parts are new results. Chapter 6 sort of merges the two noncommutativities encountered above, it adds the B field into the string field theory. It is based on our paper [91] with some updates, especially regarding the K-theory. Finally chapter 7 deals with the most mathematical aspects of noncommutative geometry and applies much of the fancy techniques to the problem of tachyon condensation on the torus. It is a simplified version of our paper with Krajewski [123].