The power of vortices – dual perspectives on exotic quantum phases

The analysis of topological excitations in interacting many-body systems often provides important insights into quantum phases and phase transitions. Seminal examples are the (thermal) BKT transition as well as the (quantum) superfluid-Mott insulator transition, which are naturally described in terms of vortices. More recently, such a ‘dual’ formulation has proven extremely illuminating in the study of exotic phases of matter that host fractional excitations, e.g., in topological phases and quantum magnets. In my talk, I will review the dual description of conventional phases of matter and explain how exotic, fractionalized phases are captured within this approach. I will then generalize these dualities to fermionic systems, and discuss the implications for the half-filled Landau level and strongly interacting surfaces of topological insulators. Finally, I will describe how two-dimensional Dirac fermions (and their symmetries) can be mapped onto interacting bosons.