Affiliation: College of Arts and Sciences, Department of Mathematics
Classical invariants for representations of one Lie group can often be related to invariants of some other Lie group. Physics suggests that the right objects to consider for these questions are certain refinements of classical invariants known as conformal blocks. Conformal blocks appear in algebraic geometry as spaces of global sections of line bundles on moduli stacks of parabolic bundles on a smooth curve. Rank-level duality connects a conformal block associated to one Lie algebra to a conformal block for a different Lie algebra. In this dissertation we discuss a general approach to rank-level duality questions. The main result of the dissertation is a rank-level duality for so(2r+1) conformal blocks on the pointed projective line which was suggested by T. Nakanishi and A. Tsuchiya. As an application of the general techniques developed in the thesis, we prove new symplectic rank-level dualities.