3d-3d correspondence for mapping tori

Abstract

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.

Resurgent analysis of SU(2) Chern-Simons partition function on Brieskorn spheres Σ(2, 3, 6n + 5)

“…In this paper, the main goal is to extend the resurgent analysis of the analytically continued SU(2) Chern-Simons partition function defined on a family of Brieskorn spheres Σ(2, 3, 6n + 5) where n ∈ Z + and 6n + 5 is prime. We provide supporting evidence to the claims made in [2] on theẐ-invariants introduced in [3] that has already been initiated in various contexts [4,6,7,[25][26][27]. 4 Similar to the discussion in section 3.4 and 3.5 in [2] and [6], We will start with, in section 2, the general partition function for our interested Brieskorn spheres Σ(2, 3, 6n + 5) from the generating function introduced in [3] along with special cases of [5,27].…”

“…We provide supporting evidence to the claims made in [2] on theẐ-invariants introduced in [3] that has already been initiated in various contexts [4,6,7,[25][26][27]. 4 Similar to the discussion in section 3.4 and 3.5 in [2] and [6], We will start with, in section 2, the general partition function for our interested Brieskorn spheres Σ(2, 3, 6n + 5) from the generating function introduced in [3] along with special cases of [5,27]. Then, using the resurgence properties of mock modular forms to analyze the homological blocks [2], we will provide a general expression of the Borel transformed Chern-Simons partition function that reveals the structure of poles in the Borel plane for Σ(2, 3, 6n + 5).…”

“…. , v 9 }, and edges (1, 2), (2, 3), (1,4), (1,5), (5,6), (6, 7), (7,8), (8,9) along with det M = −1 and CokerM = {0} which remains valid for all types of Brieskorn homology spheres. Now, we first conjecture the following from the ingredients gained from the linking matrix and the generating function eq.…”

“…At last, 3 The label α can be understood as giving rise to physical boundary conditions that come to realize BPS particles of T [M3] in M2-branes ending on a pair of M5-branes which realize the A1 6d N = (2, 0) theory [3]. 4 A brief discussion on the resurgent analysis ofẐ on general Brieskorn spheres was presented in section 2.6 of [27].…”

Higher Rank Z^ and F_K

“…Furthermore we conjecture that our 0-surgery formula in [7] holds for higher rank as well: Conjecture 3.6 (higher rank 0-surgery formula). Let K ⊂ S 3 be a knot.…”