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

TL;DR

This paper introduces a new class of non-semi-simple TQFTs based on nilpotent quantum sl(2) representations at roots of unity, extending link invariants like Kashaev's and providing more sensitive mapping class group representations.

Contribution

It constructs a novel non-semi-simple TQFT framework that generalizes previous invariants, including Reidemeister torsion and Kashaev's invariants, using the universal construction without modular categories.

Findings

Provides a super-TQFT for even roots of unity.

Re-proves Lens space classification via non-semi-simple invariants.

Shows mapping class group representations are more sensitive, with infinite order Torelli group actions.

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 (arXiv:1202.3553) 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 (arXiv:1202.3553). 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.