Non Semi-Simple ${\mathfrak {sl}(2)}$ Quantum Invariants, Spin Case
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Non Semi-Simple sl(2) Quantum Invariants, Spin Case Christian Blanchet · Francesco Costantino · Nathan Geer · Bertrand Patureau-Mirand
Received: 18 May 2014 / Accepted: 5 August 2014 / Published online: 25 November 2014 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2014
Abstract Invariants of 3-manifolds from a non semi-simple category of modules over a version of quantum sl(2) were obtained by the last three authors in Costantino et al. (To appear in J. Topology. 2014). In their construction, the quantum parameter q is a root of unity of order 2r where r > 1 is odd or congruent to 2 modulo 4. In this paper, we consider the remaining cases where r is congruent to zero modulo 4 and produce invariants of 3-manifolds with colored links, equipped with generalized spin structure. For a given 3manifold M, the relevant generalized spin structures are (non canonically) parametrized by H 1 (M; C/2Z). Keywords Quantum invariants · 3-manifolds · Non semi-simple · Spin structures Mathematics Subject Classification (2010) 57M27
C. Blanchet () IMJ-PRG, UMR 7586 CNRS, Univ Paris Diderot, Sorbonne Paris Cit´e, Univ Paris Diderot, 75013 Paris, France e-mail: [email protected] F. Costantino Institut de Math´ematiques de Toulouse (IMT), Universit´e de Toulouse III Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse, France e-mail: [email protected] N. Geer Mathematics and Statistics, Utah State University, Logan, UT 84322, USA e-mail: [email protected] B. Patureau-Mirand Univ. Bretagne - Sud, UMR 6205, LMBA, 56000 Vannes, France e-mail: [email protected]
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1 Introduction New quantum invariants of 3-manifolds equipped with 1-dimensional cohomology class over C/2Z or equivalently C∗ flat connection have been constructed in [8] from a variant of quantum sl(2). This family of invariants is indexed by integers r ≥ 2, r ≡ 0 modulo 4, which give the order of the quantum parameter. The relevant representation category is non semi-simple, so that the usual modular category framework does not apply and is replaced by more general relative G-modular category. The required non degeneracy condition is not satisfied in cases r ≡ 0 module 4. In the present paper, we show that the procedure can be adapted to the remaining cases and leads to invariants of 3-manifolds with colored links, equipped with some generalized spin structure. These spin structures can be defined as certain cohomology classes on the tangent framed bundle and can be interpreted as C∗ flat connections on this framed bundle. The non semi-simple sl(2) invariants from [8] have been extended to TQFTs in [6]. For r = 2, they give a TQFT for a canonical normalization of Reidemeister torsion; in particular, they recover classification of lens spaces. In general, they give new representations of Mapping Class Groups with opened faithfulness question. They contain the Kashaev invariants and give an extended formulation of the volu
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