Chern-Simons terms in field and string theories

In this thesis we study effect of Chern-Simons terms in field and string theory. In the introductory section we formulate the problems and state our the results. The first chapter is about three dimensional abelian spin Chern-Simons theories. We propose a simple classification of quantum spin Chern-Simons theories with the gauge group T = U(1)N. While the classical spin Chern-Simons theories are classified by an integral lattice the quantum theories are classified differently. Two quantum theories are equivalent if they have the same invariants on 3-manifolds with spin structure, or equivalently if they lead to equivalent representations of the modular group. We prove the quantum theory is completely determined by three invariants which can be constructed from the data in the classical action. We comment on implications for the classification of the fractional quantum Hall fluids. In particular, we derived the most general formula for Hall conductivity with denominators pn and 2n which can be obtained from abelian Chern-Simons theory.; The second chapter is about the singleton sector of type IIB supergravity. This singlet sector is a free field sector with Chern-Simons like interactions. We give a simple derivation of the conformal blocks of the singleton sector of compactifications of IIB string theory on spacetimes of the form X5 x Y5 with Y5 compact, while X5 has as conformal boundary an arbitrary 4-manifold M4. We retain the second-derivative terms in the action for the B, C fields and thus the analysis is not purely topological. The unit-normalized conformal blocks agree exactly with the quantum partition function of the U(1) gauge theory on the conformal boundary. We reproduce the action of the magnetic translation group and the SL(2, Z )S-duality group obtained from the purely topological analysis of Witten. An interesting subtlety in the normalization of the IIB Chern-Simons phase is noted.; The third chapter is about open cubic bosonic string field theory. It happens that the classical action of this theory is the Chern-Simons action on some infinite dimensional algebra A . In order for this action to be well defined the algebra A must associative. However in the literature there was no analytic prove of its associativity, and moreover there are claims that the associativity is violated by anomalies. In the chapter 3 we give an analytic prove of the associativity.