Modularity in Quantum Systems

Coordinators: Sergei Gukov, Sarah Harrison, Jeffrey Harvey, and Ken Ono

KITP and the program coordinators will be delivering remote talk sessions to this program's participants. 

Symmetry is important for understanding physics at all scales: from phases of many-body systems to cosmological solutions to the nature and interactions of subatomic particles. Furthermore, understanding symmetry often leads to deep connections between theoretical physics and mathematics. This workshop will focus on objects which have symmetry with respect to the modular group--the group of two by two integer matrices with determinant one. Such objects include special functions in number theory known as automorphic forms, and objects in representation theory known as modular tensor categories which moreover have connections to geometry, topology, and several areas of theoretical physics. A major goal of this workshop is to bring together experts in quantum field theory, string theory, and the mathematical fields of number theory, geometry, topology, and representation theory to explore these emerging connections, learn new things from each other, and forge new collaborations.

The program will focus on the following specific connections. 1) “Moonshine”--a connection between sporadic finite groups and (mock) modular forms, with emerging relations to enumerative geometry and string theory compactifications. 2) Vertex operator algebras (VOA) which arise in BPS sectors of supersymmetric gauge theories, and connections to topology of 3- and 4-manifolds. 3) Connections between mock modular forms, supersymmetric black hole degeneracies in string theory, and the quantum geometry of Calabi-Yau manifolds (so-called Gromov-Witten/Donaldson-Thomas theory.) 4) Classification of topological phases in condensed matter systems using the framework of modular tensor categories and more general connections between VOA and representations of quantum groups (Kazhdan-Lusztig correspondence.) 5) Connections between automorphic forms and the structure of scattering amplitudes in string theory. 6) Modularity in 2d conformal field theory and connections to 3d quantum gravity.