THE MULTIVARIABLE ALEXANDER POLYNOMIAL AND MODERN KNOT THEORY

1992 ◽  
Vol 06 (11n12) ◽  
pp. 1857-1869
Author(s):  
H. SALEUR
Keyword(s):  
Quantum Groups ◽  
Knot Theory ◽  
Alexander Polynomial ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Link Invariant ◽  
Topological Quantum ◽  
Topological Quantum Field Theories

This note is a summary of several recent works (by the author and collaborators) that study the Conway-Alexander link invariant in the light of quantum groups and topological quantum field theories. Their purpose is to understand connections between “modern” knot theory and more classical topological concepts.

TOPOLOGICAL QUANTUM FIELD THEORIES ASSOCIATED TO FINITE GROUPS AND CROSSED G-SETS

1992 ◽  
Vol 01 (01) ◽  
pp. 1-20 ◽  
Author(s):  
DAVID N. YETTER
Keyword(s):  
Gauge Group ◽  
Quantum Groups ◽  
Finite Groups ◽  
Gauge Theories ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Link Invariants ◽  
Topological Quantum ◽  
Topological Quantum Field Theories

Using methods suggested by the work of Turaev and Viro [11, 12], we provide a detailed construction of topological quantum field theories associated to finite crossed G-sets. Our construction of theories associated to finite groups fills in some details implicit in Dijkgraaf and Witten's [3] discussion of topological gauge theories with finite gauge group, while the theories associated to finite crossed G-sets simultaneously extend Dijkgraaf and Witten's [3] results to 3-manifolds equipped with links and Freyd and Yetter's [5] construction of link invariants from crossed G-sets from links in the 3-sphere to links in arbitrary 3-manifolds. Topological interpretations of the manifold and link invariants associated to these TQFT's are provided. We conclude discussion of our results as a toy model for QFT and of their relation to quantum groups.

scholarly journals A general state-sum construction of 2-dimensional topological quantum field theories with defects

2017 ◽  
Vol 26 (04) ◽  
pp. 1750014
Author(s):  
Gathoni Kamau-Devers ◽  
Gail Jardine ◽  
David Yetter
Keyword(s):  
Knot Theory ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Two Dimensional ◽  
General State ◽  
Topological Quantum ◽  
Witten Theory ◽  
Pachner Moves ◽  
Topological Quantum Field Theories

We derive a general state sum construction for 2D topological quantum field theories (TQFTs) with source defects on oriented curves, extending the state-sum construction from special symmetric Frobenius algebra for 2D TQFTs without defects (cf. Lauda and Pfeiffer [State-sum construction of two-dimensional open-closed topological quantum field theories, J. Knot Theory Ramifications 16 (2007) 1121–1163, doi: 10.1142/S0218216507005725]). From the extended Pachner moves (Crane and Yetter [Moves on filtered PL manifolds and stratified PL spaces, arXiv:1404.3142 ]), we derive equations that we subsequently translate into string diagrams so that we can easily observe their properties. As in Dougherty, Park and Yetter [On 2-dimensional Dijkgraaf–Witten theory with defects, to appear in J. Knots Theory Ramifications], we require that triangulations be flaglike, meaning that each simplex of the triangulation is either disjoint from the defect curve, or intersects it in a closed face, and that the extended Pachner moves preserve flaglikeness.

GENERAL COVARIANCE, TOPOLOGICAL QUANTUM FIELD THEORIES AND FRACTIONAL STATISTICS

1992 ◽  
Vol 07 (02) ◽  
pp. 209-234 ◽  
Author(s):  
J. GAMBOA
Keyword(s):  
Hamiltonian Formalism ◽  
Field Theories ◽  
General Covariance ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Fractional Statistics ◽  
Multiply Connected ◽  
Topological Quantum ◽  
Gauge Fixing ◽  
Topological Quantum Field Theories

Topological quantum field theories and fractional statistics are both defined in multiply connected manifolds. We study the relationship between both theories in 2 + 1 dimensions and we show that, due to the multiply-connected character of the manifold, the propagator for any quantum (field) theory always contains a first order pole that can be identified with a physical excitation with fractional spin. The article starts by reviewing the definition of general covariance in the Hamiltonian formalism, the gauge-fixing problem and the quantization following the lines of Batalin, Fradkin and Vilkovisky. The BRST–BFV quantization is reviewed in order to understand the topological approach proposed here.

TWO-DIMENSIONAL TOPOLOGICAL QUANTUM FIELD THEORIES AND FROBENIUS ALGEBRAS

1996 ◽  
Vol 05 (05) ◽  
pp. 569-587 ◽  
Author(s):  
LOWELL ABRAMS
Keyword(s):  
Field Theories ◽  
Algebra Structure ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Two Dimensional ◽  
Direct Sums ◽  
Frobenius Algebras ◽  
Topological Quantum ◽  
Simple Demonstration ◽  
Topological Quantum Field Theories

We characterize Frobenius algebras A as algebras having a comultiplication which is a map of A-modules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either “annihilator algebras” — algebras whose socle is a principal ideal — or field extensions. The relationship between two-dimensional topological quantum field theories and Frobenius algebras is then formulated as an equivalence of categories. The proof hinges on our new characterization of Frobenius algebras. These results together provide a classification of the indecomposable two-dimensional topological quantum field theories.

scholarly journals THE RELATIONS BETWEEN GENERALIZED FIELDS AND SUPERFIELDS FORMALISMS OF THE BATALIN–VILKOVISKY METHOD OF QUANTIZATION

2004 ◽  
Vol 19 (14) ◽  
pp. 2339-2353 ◽  
Author(s):  
ÖMER F. DAYI
Keyword(s):  
General Solution ◽  
Master Equation ◽  
Gauge Theories ◽  
Relativistic Particle ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Yang Mills ◽  
Topological Quantum ◽  
Topological Quantum Field Theories

A general solution of the Batalin–Vilkovisky master equation was formulated in terms of generalized fields. Recently, a superfields approach of obtaining solutions of the Batalin–Vilkovisky master equation is also established. Superfields formalism is usually applied to topological quantum field theories. However, generalized fields method is suitable to find solutions of the Batalin–Vilkovisky master equation either for topological quantum field theories or the usual gauge theories like Yang–Mills theory. We show that by truncating some components of superfields with appropriate actions, generalized fields formalism of the usual gauge theories result. We demonstrate that for some topological quantum field theories and the relativistic particle both of the methods possess the same field contents and yield similar results. Inspired by the observed relations, we give the solution of the BV master equation for on-shell N=1 supersymmetric Yang–Mills theory utilizing superfields.

EQUIVARIANT LOCALIZATION TECHNIQUES IN TOPOLOGICAL QUANTUM FIELD THEORIES

2010 ◽  
Vol 07 (02) ◽  
pp. 247-266
Author(s):  
ADRIAN VAJIAC
Keyword(s):  
Field Theories ◽  
Configuration Spaces ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Direct Study ◽  
Topological Quantum ◽  
Four Manifolds ◽  
Topological Quantum Field Theories

This paper proposes a different approach to derive Witten's formula relating Donaldson and Seiberg–Witten invariants for four-manifolds, via equivariant localization techniques. Our approach proposes a direct study on the Donaldson–Witten and Seiberg–Witten configuration spaces, not making use the theory of non-abelian monopoles.

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