Non-semi-simple TQFTs, Reidemeister torsion and Kashaev's invariants

Abstract

We construct and study a new family of TQFTs based on nilpotent highest weight representations of quantum sl(2) at a root of unity indexed by generic complex numbers. This extends to cobordisms the non-semi-simple invariants defined in [9] including the Kashaev invariant of links. Our case is not covered by the modular category framework and so we give a new example of application of the so-called “universal construction”. For each root of unity of order 2r where r≥2 is not divisible by 4, our TQFT provides a monoidal functor from a category of surfaces and their cobordisms into the category of graded finite dimensional vector spaces. The functor is always symmetric monoidal but for even values of r the braiding on GrVect has to be the super-symmetric one, thus our TQFT may be considered as a super-TQFT. In the special case r=2 our construction yields a TQFT for a canonical normalization of Reidemeister torsion and we re-prove the classification of Lens spaces via the non-semi-simple quantum invariants defined in [9]. We prove that the representations of mapping class groups and Torelli groups resulting from our constructions are potentially more sensitive than those obtained from the standard Reshetikhin–Turaev functors; in particular we prove that the action of the bounding pairs generators of the Torelli group has always infinite order.