Ẑ invariants at rational τ

$$ \hat{Z} $$ Z ̂ -invariants, which can reconstruct the analytic continuation of the SU(2) Chern-Simons partition functions via Borel resummation, were discovered by GPV and have been conjectured to be a new homological invariant of 3-manifolds which can shed light onto the superconformal and topologically twisted index of 3d $$ \mathcal{N} $$ N = 2 theories proposed by GPPV. In particular, the resurgent analysis of $$ \hat{Z} $$ Z ̂ has been fruitful in discovering analytic properties of the WRT invariants. The resurgent analysis of these $$ \hat{Z} $$ Z ̂ -invariants has been performed for the cases of Σ(2, 3, 5), Σ(2, 3, 7) by GMP, Σ(2, 5, 7) by Chun, and, more recently, some additional Seifert manifolds by Chung and Kucharski, independently. In this paper, we extend and generalize the resurgent analysis of $$ \hat{Z} $$ Z ̂ on a family of Brieskorn homology spheres Σ(2, 3, 6n + 5) where n ∈ ℤ+ and 6n + 5 is a prime. By deriving $$ \hat{Z} $$ Z ̂ for Σ(2, 3, 6n + 5) according to GPPV and Hikami, we provide a formula where one can quickly compute the non-perturbative contributions to the full analytic continuation of SU(2) Chern-Simons partition function.

Section: Jhep02(2021)008supporting

confidence: 89%

“…. , 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.…”

Section: Jhep02(2021)008mentioning

confidence: 84%

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

2021

“…written here in terms of the standard false theta-functions, p (q) at roots of unity, not just the radial limit of it, we were able to reproduce WRT invariant for this manifold from Z. As in [74,75], it would be interesting to study more carefully (and compare) the behavior near q = e 2πiτ , with different rational values of τ ∈ Q.…”

Section: Example: a Tadpole Diagrammentioning

confidence: 78%

3d-3d correspondence for mapping tori

Chun

Gukov

Park

et al. 2020

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.

“…Note added. While preparing the manuscript, we found that [15] appeared, which overlaps with parts of section 2 of this paper.…”

Section: Jhep02(2021)083mentioning

confidence: 68%

BPS invariants for 3-manifolds at rational level K

Chung

2021

We consider the Witten-Reshetikhin-Turaev invariants or Chern-Simons partition functions at or around roots of unity $$ q={e}^{\frac{2\pi i}{K}} $$ q = e 2 πi K with a rational level K = $$ \frac{r}{s} $$ r s where r and s are coprime integers. From the exact expression for the G = SU(2) Witten-Reshetikhin-Turaev invariants of the Seifert manifolds at a rational level obtained by Lawrence and Rozansky, we provide an expected form of the structure of the Witten-Reshetikhin-Turaev invariants in terms of the homological blocks at a rational level. Also, we discuss the asymptotic expansion of knot invariants around roots of unity where we take a limit different from the limit in the standard volume conjecture.