3d-3d correspondence for mapping tori

One of the main challenges in 3d-3d correspondence is that no existent approach offers a complete description of 3d $$ \mathcal{N} $$ N = 2 SCFT T [M3] — or, rather, a “collection of SCFTs” as we refer to it in the paper — for all types of 3-manifolds that include, for example, a 3-torus, Brieskorn spheres, and hyperbolic surgeries on knots. The goal of this paper is to overcome this challenge by a more systematic study of 3d-3d correspondence that, first of all, does not rely heavily on any geometric structure on M3 and, secondly, is not limited to a particular supersymmetric partition function of T [M3]. In particular, we propose to describe such “collection of SCFTs” in terms of 3d $$ \mathcal{N} $$ N = 2 gauge theories with “non-linear matter” fields valued in complex group manifolds. As a result, we are able to recover familiar 3-manifold invariants, such as Turaev torsion and WRT invariants, from twisted indices and half-indices of T [M3], and propose new tools to compute more recent q-series invariants $$ \hat{Z} $$ Z ̂ (M3) in the case of manifolds with b1> 0. Although we use genus-1 mapping tori as our “case study,” many results and techniques readily apply to more general 3-manifolds, as we illustrate throughout the paper.

LENS SPACE SURGERIES ALONG CERTAIN 2-COMPONENT LINKS RELATED WITH PARK’S RATIONAL BLOW DOWN, AND REIDEMEISTER-TURAEV TORSION

We study lens space surgeries along two different families of 2-component links, denoted by${A}_{m, n} $and${B}_{p, q} $, related with the rational homology$4$-ball used in J. Park’s (generalized) rational blow down. We determine which coefficient$r$of the knotted component of the link yields a lens space by Dehn surgery. The link${A}_{m, n} $yields a lens space only by the known surgery with$r= mn$and unexpectedly with$r= 7$for$(m, n)= (2, 3)$. On the other hand,${B}_{p, q} $yields a lens space by infinitely many$r$. Our main tool for the proof are the Reidemeister-Turaev torsions, that is, Reidemeister torsions with combinatorial Euler structures. Our results can be extended to the links whose Alexander polynomials are same as those of${A}_{m, n} $and${B}_{p, q} $.

SOME FINITENESS PROPERTIES FOR THE REIDEMEISTER–TURAEV TORSION OF THREE-MANIFOLDS

We prove for the Reidemeister–Turaev torsion of closed oriented three-manifolds some finiteness properties in the sense of Goussarov and Habiro, that is, with respect to some cut-and-paste operations which preserve the homology type of the manifolds. In general, those properties require the manifolds to come equipped with an Euler structure and a homological parametrization.