The Witten-Reshetikhin-Turaev invariants of finite order mapping tori II

Andersen, Jørgen Ellegaard; Himpel, Benjamin

doi:10.4171/qt/33

Cited by 24 publications

(19 citation statements)

References 63 publications

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“…where Stab Z 2 (a) is the stabiliser of a under the action of the Weyl group: a → −a. The fact that (3.6) is satisfied with all these ingredients (3.10)-(3.13) is a classic result of [65]; see also [60,[66][67][68][69] for recent work on WRT invariants of mapping tori. Aiming to understand the precise definition/characterization of almost abelian flat connections on general 3-manifolds, in the rest of this section we extend the notion of the two sets (3.2) and (3.3) to plumbed 3-manifolds with b 1 > 0 and 0-surgeries on some knots.…”

Section: Jhep09(2020)152mentioning

confidence: 99%

“…14 This expression for Z (±) b can be obtained in several different ways, all of which give the same result, originate on the 3-manifold side of the 3d-3d correspondence, and will be discussed in some details below. First, it can be deduced directly from the structure of WRT invariants for genus-1 mapping tori [65][66][67][68][69]. Secondly, it can be obtained via a general formula (3.19) for plumbings with loops which, in turn, can be derived by extending the arguments in [54], as we explain below.…”

Section: Jhep09(2020)152mentioning

confidence: 99%

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3d-3d correspondence for mapping tori

Chun

Gukov

Park

et al. 2020

J. High Energ. Phys.

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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.

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Section: Jhep09(2020)152mentioning

confidence: 99%

Section: Jhep09(2020)152mentioning

confidence: 99%

3d-3d correspondence for mapping tori

Chun

Gukov

Park

et al. 2020

J. High Energ. Phys.

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“…Remark 8.1. In [11] we use this formula to get an expression for the contribution from a smooth component M(Σ f ) c to the leading order term of the perturbation expansion of Z (k)…”

Section: By Thementioning

confidence: 99%

The Witten–Reshetikhin–Turaev invariants of finite order mapping tori I

Andersen

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We formulate the Asymptotic Expansion Conjecture for the Witten-Reshetikhin-Turaev quantum invariants of closed oriented three manifolds. For finite order mapping tori, we study these quantum invariants via the geometric gauge theory approach to the corresponding quantum representations and prove, using a version of the Lefschetz-Riemann-Roch Theorem due to Baum, Fulton, MacPherson & Quart, that the quantum invariants can be expressed as a sum over the components of the moduli space of flat connections on the mapping torus. Moreover, we show that the term corresponding to a component is a polynomial in the level k, weighted by a complex phase, which is k times the Chern-Simons invariant corresponding to the component. We express the coefficients of these polynomials in terms of cohomological pairings on the fixed point set of the moduli space of flat connections on the surface. We explicitly describe the fixed point set in terms of moduli spaces of the quotient orbifold Riemann surface and for the smooth components we express the aforementioned coefficients in terms of the known generators of the cohomology ring. We provide an explicit formula in terms of the Seifert invariants of the mapping torus for the contributions from each of the smooth components. We further establish that the Asymptotic Expansion Conjecture and the Growth Rate Conjecture for these finite order mapping tori.

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“…Through Witten's path integral motivation for the WRT-TQFTs, it is expected that they should admit asymptotic expansions in the level k, and this is known as the asymptotic expansion conjecture. For a summary of results, see [3,5,33,34,55] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic expansions of the Witten–Reshetikhin–Turaev invariants of mapping tori I

Andersen

Petersen

2018

Trans. Amer. Math. Soc.

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In this paper we engage in a general study of the asymptotic expansion of the Witten-Reshetikhin-Turaev invariants of mapping tori of surface mapping class group elements. We use the geometric construction of the Witten-Reshetikhin-Turaev TQFT via the geometric quantization of moduli spaces of flat connections on surfaces. We identify assumptions on the mapping class group elements that allow us to provide a full asymptotic expansion. In particular, we show that our results apply to all pseudo-Anosov mapping classes on a punctured torus and show by example that our assumptions on the mapping class group elements are strictly weaker than hitherto successfully considered assumptions in this context. The proof of our main theorem relies on our new results regarding asymptotic expansions of oscillatory integrals, which allows us to go significantly beyond the standard transversely cut out assumption on the fixed point set. This makes use of Picard-Lefschetz theory for Laplace integrals.

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The Witten-Reshetikhin-Turaev invariants of finite order mapping tori II

Cited by 24 publications

References 63 publications

3d-3d correspondence for mapping tori

3d-3d correspondence for mapping tori

The Witten–Reshetikhin–Turaev invariants of finite order mapping tori I

Asymptotic expansions of the Witten–Reshetikhin–Turaev invariants of mapping tori I

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