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Towards a categorification of the Turaev--Viro TQFT

David Rose (UNC)
Tue, Jun 4 2024, 4:00pm
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For each compact, orientable surface whose connected components have non-empty boundary, we define a dg category that categorifies the Temperley--Lieb skein of the surface​. In the case when the surface is a disk, our categories are quasi-equivalent to the so-called "Bar-Natan category": the natural setting for Khovanov's celebrated categorification of the Jones polynomial. We expect that our dg categories form part of the 2-dimensional layer of a categorified version of the (stable) Turaev--Viro TQFT associated to the quantum group U_q(sl(2)). In particular, a twisted Hom-pairing on our dg categories recovers the canonical bilinear form on the state spaces of the Turaev--Viro theory. (This is joint work with M. Hogancamp and P. Wedrich.)