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Integral TQFTs from Quantum Doubles, Gauge Equivariance, and Applications
by
Thomas Kerler
The Ohio State University
Coauthors: Qi Chen (WSSU)
We review TQFT constructions extending the Hennings invariant for 3-manifolds which start from a quasitriangular Hopf algebra H. We clarify how gauge equivalent Hopf algebras give rise to naturally isormophic TQFTs. We will also review past results in integral TQFTs in the traditional WRT approach and their relevance.
We will then show, more specifically, that if H=D(A) is the double of a Hopf algebra over a Dedekind domain R, and A is projective/free as an R-module we obtain a TQFT on puctured surfaces into the category of projective/free R-modules (with equivariant H-action). For closed surfaces projectivity survives.
We also show the double of the Borel algebra of quantum sl_2 is gauge equivalent to the product of quantum sl_2 itself and a cyclic group algebra. We infer the respective factorization of TQFT's and invariants. Combining our integrality result and the relation between sl_2 Hennings and WRT invariants established by Chen, Kuppum, and Srinivasan this yields an independent new proof of integrality of the WRT invariant.
Date received: May 11, 2010
Copyright © 2010 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbaf-15.