This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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.

We study q-series-valued invariants of 3-manifolds that depend on the choice of a root system G. This is a natural generalization of the earlier works by Gukov-Pei-Putrov-Vafa [arXiv:1701.06567] and Gukov-Manolescu [arXiv:1904.06057] where they focused on G = SU(2) case. Although a full mathematical definition for these "invariants" is lacking yet, we defineẐ G for negative definite plumbed 3-manifolds and F G K for torus knot complements. As in the G = SU(2) case by Gukov and Manolescu, there is a surgery formula relating F G K toẐ G of a Dehn surgery on the knot K. Furthermore, specializing to symmetric representations, F G K satisfies a recurrence relation given by the quantum A-polynomial for symmetric representations, which hints that there might be HOMFLY-PT analogues of these 3-manifold invariants.

No abstract

S. Gukov and C. Manolescu conjectured that the Melvin-Morton-Rozansky expansion of the colored Jones polynomials can be re-summed into a two-variable series F K (x, q), which is the knot complement version of the 3-manifold invariant Ẑ whose existence was predicted earlier by S. Gukov, P. Putrov and C. Vafa. In this paper we use an inverted version of the R-matrix state sum to prove this conjecture for a big class of links that includes all homogeneous braid links as well as all fibered knots up to 10 crossings. We also study an inverted version of Habiro's cyclotomic series that leads to a new perspective on F K and discovery of some regularized surgery formulas relating F K with Ẑ. These regularized surgery formulas are then used to deduce expressions of Ẑ for some plumbed 3-manifolds in terms of indefinite theta functions.