Varying the non-semisimple Crane–Yetter theory over the character stack

Abstract: Associated to a certain subquotient of the category of representations of the small quantum group at a root of unity is an invertible 4d TQFT known as Crane–Yetter: it is the anomaly theory of the 3d theory called Witten–Reshetikhin–Turaev. In fact, the non-semisimplified representation category is invertible in the Morita theory of braided tensor categories: under the cobordism hypothesis this defines a non-semisimple invertible TQFT. Such an invertible theory assigns to a closed 3-manifold a 1-dimensional vector space. In this talk, we define a relative TQFT which can be seen as varying non-semisimple Crane-Yetter over the character stack: it assigns to a closed 3-manifold $M$ a line bundle on its character stack $\mathrm{Ch}_G(M)$. We construct this theory by analysing invertibility of a 1-morphism in the Morita theory of symmetric tensor categories, coming from representations of Lusztig's quantum group at a root of unity regarded as a bimodule for $\mathrm{Rep}(G)$ using the quantum Frobenius map. In the talk we will describe this 1-morphism and analyse its invertibility and the consequences of this in detail.

mathematical physicsalgebraic topologycategory theoryquantum algebra

Audience: researchers in the topic

Topological Quantum Field Theory Club (IST, Lisbon)

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