Boundary Conditions, Symmetries, and Bootstrap in 2D Conformal Field Theory

This dissertation contains two parts of study on the Friedan states and on the conformal bootstrap with symmetries. The Friedan states are potential physical boundary states of the free boson. However, they exhibited a continuous spectrum in the open string sector, in contrast to more standard examples. The explicit expressions for the density of states of the Friedan states is obtained. Some pathologies and possible contradictions of these states are explored. The g functions of the Friedan states are shown to be infinite, suggesting an infinite number of degrees of freedom in the theory. The modular conformal bootstrap using the moment curve approach and focused on two-dimensional conformal field theory with c 1 is discussed. A refinement is made to incorporate conserved currents. Specializing to theories with symmetries, the anyon partition functions, which transform as a vector under modular transformations, are introduced. Applying the moment curve bootstrap to these leads to bounds on the minimum conformal weights in different sectors. In the analytical approach, the correlated bounds between different anyon sectors are obtained, with the abelian symmetries, Z₂ and Z₃, and with the non-abelian symmetry, S₃. The explicit bounds on the free boson theories with the presence of the symmetries are then calculated and compared with the bootstrap bounds. Finally, some numerical bootstrap results with Z₂ and Z₃ symmetries are presented.