Non semi-simple sl(2) quantum invariants, spin case

TL;DR

This paper extends the construction of quantum invariants of 3-manifolds from non semi-simple categories of modules over quantum sl(2), specifically addressing cases where the quantum parameter's order is divisible by four, and incorporates generalized spin structures.

Contribution

It introduces new 3-manifold invariants for cases where the quantum parameter's order is divisible by four, expanding previous work to include generalized spin structures.

Findings

Constructed invariants for $r$ divisible by 4

Parametrized spin structures by $H^1(M; \\mathbb{C}/2\\mathbb{Z})$

Extended the class of quantum invariants to new cases

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 [arXiv:1404.7289]. 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 $3$-manifold $M$, the relevant generalized spin structures are (non canonically) parametrized by $H^1(M;\mathbb C/2\mathbb Z)$.