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

Abstract

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 [12] including the Kashaev invariant of links. Here the modular category framework does not apply and we use the "universal construction". Our TQFT provides a monoidal functor from a category of surfaces and their cobordisms into the category of graded finite dimensional vector spaces and their degree 0-morphisms and depends on the choice of a root of unity of order 2r. 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 [12]. 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.

Some remarks on the unrolled quantum group of sl(2)

“…Finally, the category U q sl(2)-mod is not semi-simple nor braided and has an infinite number of non-isomorphic simple modules. However, one can easily modify U q sl(2) and obtain a braided category of highest weight modules which has been used to construct invariants of links [24], of 3-manifolds [8] and TQFTs [4]. The aim of this paper is to give an overview of the algebraic results related to this modified quantization and prove a few straightforward results.…”

“…We also study the decomposition of the tensor product of certain indecomposable modules (see Section 8). These results are used in [4] in an essential way to build a TQFT for 3-manifolds equipped with a cohomology class. The category U H q sl(2)-mod contains some indecomposable non-projective modules that are not studied in this paper (see for example their use in [9]).…”

“…This category has been used to construct quantum link and 3-manifold invariants in several papers [2,4,8,10,22,24,25]. These 3-manifold invariants have powerful new properties, including asymptotic behavior related to the Volume Conjecture and novel quantum representation of mapping class groups (see [4,8]). The existence of these properties is directly related to the unique representation theory discussed in this paper.…”

“…Understanding a deeper connection between the representation theory of this paper and CFT deserves some attention. For example, it would be interesting to compare the CFT representations of SL(2, Z) (see [17]) and more generally mapping class group representations (see [19]) with those obtained from U H q sl (2) in the TQFT of [4].…”

The trace on projective representations of quantum groups

“…In particular, when = 4 and g = sl(2) in [5] it is shown that the closed 3-manifold invariant of [8] associated to the category C H sl(2) are a canonical normalization of Reidemeister torsion defined by Turaev which gives rise to a Topological Quantum Field Theory (TQFT). It would be interesting to see what properties the analogous topological invariants have for other Lie algebras at similar level.…”