Hank Chen | Applied Mathematics, University of Waterloo
Categorification of Classical and Quantum Dynamical Systems
Quantum groups provide the symmetry structure to discuss quantum field theories. Their quasitriangular structure encodes physically the consistency of three-particle scattering. Moreover, quantum groups also play a central role in three-dimensional topological field theories (such as Chern-Simons theory or BF theory), which has many applications to various areas of physics, such as 3d quantum gravity and (2+1)d topological phases like the Kitaev toric code. Now it to study higher-dimensional topological field theories, the symmetry structures are in general “categorified” --- namely new layers of structures emerge; this categorification procedure has been seen in various areas of physics already, mainly in application to 4-dimensional models. This leads us to begin our program of categorifying quantum groups to obtain a notion of “quantum 2-gorups”. In this research, we tackle this problem in a systematic manner, starting at the semiclassical level of the Drinfel’d double. I explain the structural coherences leading to an appropriate notion of a 2-Drinfel’d double, and construct higher-gauge topological field theories that enjoy such a categorified symmetry. Applications of such a structure are briefly outlined, of note is a notion of “2-graded integrability” suitable for 2-dimensional Integrable lattice systems. We now aim to pin down the quantum 2-group reconstructed from the semiclassical 2-Drinfel’d double, and also examine its associated categorified Fourier duality.