Journal ArticleDOI
Invariants of 3-manifolds via link polynomials and quantum groups
Reads0
Chats0
TLDR
In this paper, the authors construct topological invariants of compact oriented 3-manifolds and of framed links in such manifolds, where the terms of the sequence are equale to the values of the Jones polynomial of the link in the corresponding roots of 1.Abstract:
The aim of this paper is to construct new topological invariants of compact oriented 3-manifolds and of framed links in such manifolds. Our invariant of (a link in) a closed oriented 3-manifold is a sequence of complex numbers parametrized by complex roots of 1. For a framed link in S 3 the terms of the sequence are equale to the values of the (suitably parametrized) Jones polynomial of the link in the corresponding roots of 1. In the case of manifolds with boundary our invariant is a (sequence of) finite dimensional complex linear operators. This produces from each root of unity q a 3-dimensional topological quantum field theoryread more
Citations
More filters
Journal ArticleDOI
Anyons in an exactly solved model and beyond
TL;DR: In this article, a spin-1/2 system on a honeycomb lattice is studied, where the interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength.
Journal ArticleDOI
State sum invariants of 3 manifolds and quantum 6j symbols
Vladimir Turaev,Oleg Viro +1 more
TL;DR: In this article, a new approach to construct quantum invariants of 3-manifolds is presented, based on the so-called quantum 6j-symbols associated with the quantized universal enveloping algebra U,&(C) where CJ is a complex root of 1 of a certain degree z > 2.
Journal ArticleDOI
Quantum affine algebras and holonomic difference equations
Igor Frenkel,N. Yu. Reshetikhin +1 more
TL;DR: In this paper, the authors derived new holonomicq-difference equations for the matrix coefficients of the products of intertwining operators for quantum affine algebra representations of levelk.
Journal ArticleDOI
Higher dimensional algebra and topological quantum field theory
John C. Baez,James Dolan +1 more
TL;DR: In this paper, it was shown that the k-fold suspension of a weak n-category stabilizes for k≥n+2, and its relation to stable homotopy theory was discussed.
Book ChapterDOI
Feynman Diagrams and Low-Dimensional Topology
TL;DR: In this paper, the authors describe a program relating Feynman diagrams, topology of manifolds, homotopical algebra, non-commutative geometry and several kinds of topological physics.
References
More filters
Journal ArticleDOI
Quantum field theory and the Jones polynomial
TL;DR: In this paper, it was shown that 2+1 dimensional quantum Yang-Mills theory with an action consisting purely of the Chern-Simons term is exactly soluble and gave a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms.
Book
Infinite Dimensional Lie Algebras
TL;DR: The invariant bilinear form and the generalized casimir operator are integral representations of Kac-Moody algebras and the weyl group as mentioned in this paper, as well as a classification of generalized cartan matrices.
Journal ArticleDOI
A q -difference analogue of U(g) and the Yang-Baxter equation
TL;DR: Aq-difference analogue of the universal enveloping algebra U(g) of a simple Lie algebra g is introduced in this article, and its structure and representations are studied in the simplest case g=sl(2).
Journal ArticleDOI
A polynomial invariant for knots via von Neumann algebras
TL;DR: In this paper, it was shown that (6, n) and (c, ra) represent the same closed braid (up to link isotopy) if and only if they are equivalent for the equivalence relation generated by Markov moves of types 1 and 2 on the disjoint union of the braid groups.
Book ChapterDOI
Quantization of Lie Groups and Lie Algebras
TL;DR: The quantum inverse scattering method for solving nonlinear equations of evolution, the quantum theory of magnets, the method of commuting transfer-matrices in classical statistical mechanics, and factorizable scattering theory as discussed by the authors emerged as a natural development of the various directions in mathematical physics.