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

Abstract: The goal of this talk is to explain a concrete instance of the theory of non-semisimple TQFT in three dimensions, as discussed in the talk of Nathan Geer in this seminar on Feb. 22nd, 2023. I will describe a recent construction of a TQFT 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 describe 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.

mathematical physicsalgebraic topologycategory theoryquantum algebra

Audience: researchers in the topic

Topological Quantum Field Theory Club (IST, Lisbon)

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