Modular Symmetry in Non-Compact Conformal Field Theory

Conformal field theory (CFT) is a quantum field theory with conformal symmetry. It has found applications in statistical mechanics, condensed matter and string theory. In the last two decades, 2 dim. supersymmetric CFTs has provided an intimate connection between 2 dim. supersymmetric gauge theories, compact manifolds, and modular forms. We revisit this three-way connection in the case of 2 dim. non-compact CFTs. These CFTs have both discrete and continuous spectrum. The presence of continuum states in the spectrum is a source of many subtleties and often invalidates basic conclusions that apply for compact CFTs which have only discrete states in their spectrum. In the study of non-compact supersymmetric CFTs, we uncovered a remarkable relation between certain non-compact manifolds and mock modular forms. Mock modular forms are number-theoretic objects which were first discovered by Ramanujan almost 100 years ago but understood only 10 years back. In my talk, I will present the constructions of 2 dim. supersymmetric gauge theories which lead to these non-compact CFTs and discuss their modular properties.