Calculations in Boundary Conformal Field Theory

In recent years, it has become apparent that conformal field theories with boundary interactions in 1 + 1 dimensions can be used to answer a variety of interesting questions in particle physics and condensed matter physics. In this thesis, I review the basic technology of boundary conformal field theory (BCFT) and use it to find exact solutions to two different BCFT models. The first model may be applied to either string theory or dissipative quantum mechanics. In the context of string theory, it describes an open bosonic string coupled to a uniform background gauge field and a periodic background tachyon potential. In the case of dissipative quantum mechanics, the model describes (in a not-so-obvious way) the quantum dynamics of a charged particle in N dimensions subject to a uniform magnetic field, periodic potential, and dissipation of the Caldeira-Leggett type. I find that there is a special family of magnetic fields and periodic potentials which lead to a conformally invariant theory, as well as exact solutions for the boundary state, correlation functions, and partition function. These special cases are constructed using the root lattice of any simply-laced Lie algebra (i.e., SU(N), SO(2N), E_6, E_7, and E_8).. The second model is a generalization of Polchinski's 1 + 1 dimensional version of the Callan -Rubakov effect. It consists of N massless Dirac fermions on the half line interacting with a monopole at the boundary. The interaction is summarized by imposing linear boundary conditions on the fermion currents, which leads to non-trivial boundary scattering for the fermions when N > 2. I compute the exact boundary state, partition function, and correlation functions. I also find that the monopole has a hidden discrete degree of freedom for N > 2 which must be taken into account in order to have a unitary boundary S-matrix.