Arithmetic and Geometric Structures in Physics Seminar
The mathematical notion of moonshine relates the theory of sporadic simple groups with that of distinguished modular objects. The first example, 'Monstrous Moonshine', was clarified in the context of two dimensional conformal field theory in the 90's. In 2010, interest in moonshine in the physics community was reinvigorated when Eguchi et. al. observed representations of the finite group M24 appearing in the elliptic genus of nonlinear sigma models on K3. In 2013, Cheng, Duncan, and Harvey provided a uniform construction of 23 new examples of moonshine, called 'Umbral moonshine', of which M24 moonshine is a special case. This talk will survey old and new developments in Monstrous, Mathieu, and Umbral moonshines, with particular emphasis on their appearance in conformal field theory and string theory. Along the way, we will introduce the special classes of automorphic and mock modular forms that appear in these moonshines and highlight their physical relevance.