On the quantum invariants for the spherical Seifert manifolds
type: Item
instance of: scholarly article
title: On the quantum invariants for the spherical Seifert manifolds
zbMATH Open document ID: 1147.58023
author: Kazuhiro Hikami
published in: Communications in Mathematical Physics
publication date: 2007-09-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/math-ph/0504082
review text: In this paper some invariants (Witten-Reshetkhin-Turaev (WRT) quantum invariants) are studied for Seifert manifolds \(S^3/\Gamma\), where \(\Gamma\) is a finite subgroup of SU(2). Quantum invariants are invariants introduced by Witten and Atiyah in the so called topological quantum field theory in the years 1989 and 1990 and further developed by many other mathematicians (Kontsevich, Quinn, Reshetikhin, Turaev, Vassiliev, \dots). (The Chern-Simons field theory and the Wess-Zumino-Witten model can be considered as the physical back-ground of quantum invariants.) A particular attention was put on \(3\)-manifolds, where quantum invariants have been related to so called ribbon Hopf algebras and Hopf diagrams. The author, in this paper, shows that WRT quantum invariants for \(S^3/\Gamma\) can be written in terms of the Eichler integral of modular forms with half-integral weight, and gives an exact asymptotic expansion of the invariants by using the nearly property of the Eichler integral. Furthermore, a relation with polyhedral equations associated to \(\Gamma\) is proved. The paper, after a detailed introduction, splits in five more sections. 2. Preliminaries. 3. Witten-Reshetkhin-Turaev invariant. 4. WRT invariant for the spherical Seifert manifolds. 5. Relationship with the Platonic solids. 6. Conclusions and discussions. Remark. ``Quantum invariants'' were also considered in the geometric theory of PDE's, super PDE's and quantum PDE's by the reviewer of this paper. In particular, these are interpreted as evaluation of ``conservation laws'' on admissible integral Cauchy manifolds. Then, such invariants are related to ``Hopf algebras of PDE's''. Furthermore, more recently, such invariants are related also to subgroups of crystallographic groups. (See works published starting from 1990.)
reviewed by: Agostino Prástaro
Mathematics Subject Classification ID: 58J28
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 11F23
Mathematics Subject Classification ID: 81T45
zbMATH DE Number: 5189594
zbMATH Keywords: quantum invariants
zbMATH Keywords: spherical Seifert manifolds
MaRDI profile type: Publication
OpenAlex ID: W2095489530
arXiv ID: math-ph/0504082
cites work: \(q\)-series identities and values of certain \(L\)-functions
cites work: Q3998730
cites work: Topological methods to compute Chern-Simons invariants
cites work: The A-D-E classification of minimal and \(A_ 1^{(1)}\) conformal invariant theories
cites work: Q3695591
cites work: Q4299405
cites work: The Klein quartic in number theory
cites work: Instanton Homology of Seifert Fibred Homology Three Spheres
cites work: Conformal field theory
cites work: Reidemeister torsion, spectral sequences, and Brieskorn spheres.
cites work: Computer calculation of Witten's 3-manifold invariant
cites work: Casson's invariant of Seifert homology 3-spheres
cites work: Three-manifolds invariants and their relation with the fundamental group
cites work: \(q\)-series and \(L\)-functions related to half-derivatives of the Andrews-Gordon identity
cites work: Volume Conjecture and Asymptotic Expansion ofq-Series
cites work: Quantum invariant for torus link and modular forms
cites work: ON THE QUANTUM INVARIANT FOR THE BRIESKORN HOMOLOGY SPHERES
cites work: Q4673256
cites work: Quantum invariants, modular forms, and lattice points II
cites work: Torus knot and minimal model
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Modular forms of rational weights and modular varieties
cites work: Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation
cites work: Q4040877
cites work: On Modular forms of Weight (6n + 1)/5 Satisfying a Certain Differential Equation
cites work: A link invariant from quantum dilogarithm
cites work: The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl\((2,\mathbb{C})\)
cites work: Chern-Simons invariants of 3-manifolds and representation spaces of knot groups
cites work: Q3235831
cites work: Q4695272
cites work: Asymptotic expansions of Witten–Reshetikhin–Turaev invariants for some simple 3-manifolds
cites work: Q5285044
cites work: Witten-Reshetikhin-Turaev invariants of Seifert manifolds
cites work: Modular forms and quantum invariants of \(3\)-manifolds
cites work: Classical tessellations and three-manifolds
cites work: The colored Jones polynomials and the simplicial volume of a knot
cites work: SU(n)-CHERN–SIMONS INVARIANTS OF SEIFERT FIBERED 3-MANIFOLDS
cites work: Q4449110
cites work: Q5663963
cites work: Invariants of 3-manifolds via link polynomials and quantum groups
cites work: A large \(k\) asymptotics of Witten's invariant of Seifert manifolds
cites work: Q3416832
cites work: A contribution of the trivial connection to the Jones polynomial and Witten's invariant of 3d manifolds. I
cites work: Residue formulas for the large \(k\) asymptotics of Witten's invariants of Seifert manifolds. The case of \(SU(2)\)
cites work: Q3663422
cites work: Fusion rules and modular transformations in 2D conformal field theory
cites work: Quantum field theory and the Jones polynomial
cites work: The Absolute Value of The Chern-Simons-Witten Invariants of Lens Spaces
cites work: Q4284459
cites work: Vassiliev invariants and a strange identity related to the Dedekind eta-function
DOI: 10.1007/S00220-006-0094-1
description: scientific article