$U_q(\mathfrak{gl}(1 \vert 1))$ and $U(1 \vert 1)$ Chern--Simons theory

Abstract: Chern--Simons theory, as introduced by Witten, is a three dimensional quantum gauge theory associated to a compact simple Lie group and a level. The mathematical model of this theory as a topological quantum field theory was introduced by Reshetikhin and Turaev and is at the core of modern quantum topology. The goal of this talk is to explain a non-semisimple modification of the construction of Reshetikhin and Turaev which realizes Chern--Simons theory with gauge supergroup $U(1 \vert 1)$, as studied in the physics literature by Rozansky--Saleur and Mikhaylov. In particular, I'll motivate and explain various relative modular structures on the category of representations of the quantum group of $\mathfrak{gl}(1 \vert 1)$ which should be seen as non-semisimple analogues of modular tensor categories associated to the quantum representation theory of a simple Lie algebra at a root of unity. Based on joint work with Nathan Geer.

HEP - theorymathematical physicsalgebraic geometryalgebraic topologyquantum algebrarepresentation theoryquantum physics

Audience: researchers in the topic

PIMS Geometry / Algebra / Physics (GAP) Seminar