The Witten–Reshetikhin–Turaev invariant for links in finite order mapping tori I

Andersen, Jørgen Ellegaard; Himpel, Benjamin; Jørgensen, Søren Fuglede; Martens, Johan; McLellan, Brendan

doi:10.1016/j.aim.2016.08.042

Cited by 10 publications

(14 citation statements)

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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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“…Thus we have identified a pseudo-Anosov homeomorphism ϕ whose fixed point set is cut out transversely, whereas ϕ 2 satisfy the conditions of Theorem 1.2 but whose fixed point set it not cut out transversely. The general construction of a Hitchin connection in [2] applies to the case of M l , for all l ∈ (−2, 2), with its Chern-Simons line bundle constructed in [6] and its family of complex structures parametrized by Teichmüller space T of Σ 1 1 constructed in the works of Daskalopoulos and Wentworth and Mehta and Seshadri [36,61]. The mapping class group Γ (Σ 1 1 ,vP ) acts on this setup and the same argument as in the co-prime case shows that this is in fact a real analytic action.…”

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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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“…Remark 5. 6 We could equally well state our main result below in terms of WLO disc rig (L) rather than WLO rig (L). For stylistic reasons we prefer WLO rig (L).…”

Section: Definition Of Wlo Disc Rig (L) and Wlo Rig (L)mentioning

confidence: 64%

“…Finally, in Sec. 6 we present our main result, Theorem 6.4 (which will be proven in [43]), before we conclude the main part of the paper with a short outlook in Sec. 7.…”

Section: Introductionmentioning

confidence: 93%

From simplicial Chern-Simons theory to the shadow invariant I

Hahn

2015

Journal of Mathematical Physics

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This is the first of a series of papers in which we introduce and study a rigorous "simplicial" realization of the non-Abelian Chern-Simons path integral for manifolds M of the form M = Σ × S 1 and arbitrary simply-connected compact structure groups G. More precisely, we will introduce, for general links L in M , a rigorous simplicial version WLO rig (L) of the corresponding Wilson loop observable WLO(L) in the so-called "torus gauge" by Blau and Thompson (Nucl. Phys. B408(2): 1993). For a simple class of links L we then evaluate WLO rig (L) explicitly in a non-perturbative way, finding agreement with Turaev's shadow invariant |L|. AMS subject classifications: 57M27, 81T08, 81T452 Chern-Simons theory on M = Σ × S 1 in the torus gauge Chern-Simons theoryLet us fix a simply-connected compact Lie group 2 G with Lie algebra g.For every smooth manifold M , every real vector space V and every n ∈ N 0 we will denote by Ω n (M, V ) the space of V -valued n-forms on M and we setBy G M we will denote the "gauge group" C ∞ (M, G). We will usually write A instead of A M = Ω 1 (M, g) and G instead of G M .In the following we will restrict ourselves to the special case where M is an oriented closed 3-manifold. Moreover, we will consider only the special case where G is simple (cf. Remark 2.2 1 or, rather, Abelian BF3-models, cf. the beginning of Sec. 5 below 2 cf. part A of the Appendix for concrete formulas in the special case G = SU (2) 3 More precisely, the normalization is chosen such that α,α = 2 for every short real corootα w.r.t. any fixed Cartan subalgebra of g. Observe that after making the identification t ∼ = t * which is induced by ·, · we have α, α = 2 for every long root α. Thus the normalization here coincides with the one in [68]. This normalization guarantees that the exponential exp(iSCS) is "gauge invariant", i.e. invariant under the standard G-operation on A 4 i.e. a smooth embedding S 1 → M 5 observe that if G is simple then every Ad-invariant scalar product on g is proportional to the Killing form ad(B(σ))) = Ad(exp(B(σ))) and that k is an Ad |T -invariant subspace of g 13 the notation [Σ, G/T ] is motivated by the fact that C ∞ (Σ, G/T )/GΣ coincides with the set of homotopy classes of maps Σ → G/T , cf. Proposition 3.2 in [40] 14 observe that for b ∈ t the valueḡbḡ −1 will not depend on the special choice of g

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The Witten–Reshetikhin–Turaev invariant for links in finite order mapping tori I

Cited by 10 publications

References 65 publications

3d-3d correspondence for mapping tori

3d-3d correspondence for mapping tori

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

From simplicial Chern-Simons theory to the shadow invariant I

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