Higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds

Abstract:We study the physics of multiple M5-branes compactified on a hyperbolic 3-manifold. On the one hand, it leads to the 3d-3d correspondence which maps an N = 2 superconformal field theory to a pure Chern-Simons theory on the 3-manifold. On the other hand, it leads to a warped AdS 4 geometry in M-theory holographically dual to the superconformal field theory. Combining the holographic duality and the 3d-3d correspondence, we propose a conjecture for the large N limit of the perturbative free energy of a Chern-Simons theory on hyperbolic 3-manifold. The conjecture claims that the tree, one-loop and two-loop terms all share the same N 3 scaling behavior and are proportional to the volume of the 3-manifold, while the three-loop and higher terms are suppressed at large N . Under mild assumptions, we prove the tree and one-loop parts of the conjecture. For the two-loop part, we test the conjecture numerically in a number of examples and find precise agreement. We also confirm the suppression of higher loop terms in a few examples.

show abstract

“…(3.10) and (3.17). For the one-loop part S (conj) 1 , its large N behavior can be derived using a mathematical theorem found in [45].…”

Section: Conjecture On the Large N Behavior Of Perturbative Invariantsmentioning

confidence: 99%

Holography of 3d-3d correspondence at large N

Gang

Kim

Lee

2015

J. High Energ. Phys.

View full text Add to dashboard Cite

show abstract

“…For a µ-component hyperbolic link L there are 2 µ possible lifts of the holonomy representation ρ 0 : G(L) → P SL(2, C) to an SL(2, C)representation. It is known that there is a canonical one-to-one correspondence between the set of lifts of the holonomy representation and the set of spin structures of the exterior of a link (see [17]). Among them we focus on the lift ρ 0 : G(L) → SL(2, C) such that the images of the meridians of L by ρ 0 are matrices in SL(2, C) with the trace two.…”

Section: A Generalizationmentioning

confidence: 99%

Twisted Alexander Polynomials of Hyperbolic Links

Morifuji¹,

Tran²

2017

Publ. Res. Inst. Math. Sci.

View full text Add to dashboard Cite

In this paper we apply the twisted Alexander polynomial to study the fibering and genus detecting problems for oriented links. In particular we generalize a conjecture of Dunfield, Friedl and Jackson on the torsion polynomial of hyperbolic knots to hyperbolic links, and confirm it for an infinite family of hyperbolic 2-bridge links. Moreover we consider a similar problem for parabolic representations of 2-bridge link groups.2010 Mathematics Subject Classification. Primary 57M27, Secondary 57M05, 57M25. 1 2 TAKAYUKI MORIFUJI AND ANH T. TRAN Theorem 1.1 (Theorem 4.12). For the double twist link L as in Figure 2, the twisted Alexander polynomial ∆ L,ρ0 (t) associated to ρ 0 determines the Thurston norm. Moreover L is fibered if and only if ∆ L,ρ0 (t) is monic.As is well-known, these topological properties of a 2-bridge link are detected by the reduced Alexander polynomial (see [4], [21], [22]). However there seems to be no a priori reason that the same must be true for the twisted Alexander polynomial.Since a lift of the holonomy representation of a hyperbolic link L is one of the parabolic representations (namely, it is a nonabelian SL(2, C)-representation and the traces of the images of all the meridians of L are two), it is natural to consider the following problem: For an oriented hyperbolic link L and its parabolic representation ρ : π 1 (S 3 \ L) → SL(2, C), does the twisted Alexander polynomial associated to ρ determine the Thurston norm and fiberedness of L? In this paper, we give a partial answer to this question in the case of a 2-bridge link. More precisely we show that not all parabolic representations detect the genus (in this case, the Thurston norm is equivalent to the genus) of a hyperbolic 2-bridge link.This paper is organized as follows. In Section 2, we briefly review some basic materials for the SL(2, C)-character variety of a finitely generated group and the twisted Alexander polynomial of an oriented link associated to a two-dimensional linear representation. In Section 3 we review a conjecture of Dunfield, Friedl and Jackson for hyperbolic knots and state its generalization for oriented hyperbolic links. Section 4 is devoted to the calculation of the loci of the character variety which characterize fiberedness and the genus of a wide family of 2-bridge links. This result can be regarded as a generalization of [12] and [11] which discussed the same problem in the case of knots. In the last section, we give an answer to the question on parabolic representations mentioned above.

show abstract

“…For a hyperbolic knot K and its holonomy representation ρ hol : π 1 (E K ) → SL 2 (C), it was shown in [MFP14,God] that the growth order of log |T K, σ n •ρ hol | or log |∆ K, σ n •ρ hol (1)| is the same as n 2 and the leading coefficient converges to the hyperbolic volume of E K divided by 4π. The set of SL 2 (C)-representations of π 1 (E K ) can be regarded as a deformation space of hyperbolic structures of E K .…”

Section: Introductionmentioning

confidence: 99%

Higher dimensional twisted Alexander polynomials for metabelian representations

Tran

Yamaguchi

2017

Topology and its Applications

View full text Add to dashboard Cite

We study the asymptotic behavior of the twisted Alexander polynomial for the sequence of SL n (C)-representations induced from an irreducible metabelian SL 2 (C)representation of a knot group. We give the limits of the leading coefficients in the asymptotics of the twisted Alexander polynomial and related Reidemeister torsion. The concrete computations for all genus one two-bridge knots are also presented.2010 Mathematics Subject Classification. 57M27, 57M50.

show abstract

Higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds

Cited by 32 publications

References 28 publications

Holography of 3d-3d correspondence at large N

Holography of 3d-3d correspondence at large N

Twisted Alexander Polynomials of Hyperbolic Links

Higher dimensional twisted Alexander polynomials for metabelian representations

Contact Info

Product

Resources

About