Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories

Abstract

In this paper, we construct invariants of 3-manifolds 'à la Reshetikhin-Turaev' in the setting of non-semi-simple ribbon tensor categories. We give concrete examples of such categories that lead to a family of 3-manifold invariants indexed by the integers. We prove that this family of invariants has several notable features, including: they can be computed via a set of axioms, they distinguish homotopically equivalent manifolds that the standard Witten-Reshetikhin-Turaev invariants do not and they allow the statement of a version of the Volume Conjecture and a proof of this conjecture for an infinite class of links. n i=1 V i ⊕ W where V i is a simpleŪ -module with non-zero quantum dimension and W is aŪ -module with zero quantum dimension. By quotienting the category of finite-dimensionalŪ -modules byŪ -modules with zero quantum dimension, one obtains a modular category D. Loosely speaking, a modular category is a semi-simple ribbon category with a finite number of isomorphism classes of simple objects satisfying some axioms. Let M be a manifold obtained by surgery on L. If the ith component of L is labeled by a

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.…”

“…As a U q sl(2)-module C H kr is isomorphic to the trivial module. The modules C H kr are important tools in the work of [4,8]. We also use another notation to distinguish among these modules, those that are in the degree 0 part of C : we define for any k ∈ Z,…”

“…These basis are used in [4,8,10,22,25] to produce numerical invariant of C -colored framed trivalent graphs embedded in S 3 , and numerical 6j-symbols. The basis of Hom C (C, V α ⊗ V −α ) induce isomorphisms w α : V α → V * −α forming what is called a basic data (see [25]).…”

“…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.…”

The trace on projective representations of quantum groups

“…Recall C H odd is graded by the group D. Let X D be all the non-regular elements of D. It follows from Lemma 7.1 of [8] that C H odd is a generically D-semi-simple category with the singular locus X D .…”