scholarly journals Quantum field theory and the Jones polynomial

1989 ◽  
Vol 121 (3) ◽  
pp. 351-399 ◽  
Author(s):  
Edward Witten
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Jones Polynomial ◽  
Quantum Field

scholarly journals q-nonabelianization for line defects

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Andrew Neitzke ◽  
Fei Yan
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Surface Defects ◽  
Jones Polynomial ◽  
Quantum Field ◽  
Superconformal Field Theory ◽  
Line Defects ◽  
Bps States ◽  
Holomorphic Covering ◽  
Fold Cover

Abstract We consider the q-nonabelianization map, which maps links L in a 3-manifold M to combinations of links $$ \tilde{L} $$ L ˜ in a branched N -fold cover $$ \tilde{M} $$ M ˜ . In quantum field theory terms, q-nonabelianization is the UV-IR map relating two different sorts of defect: in the UV we have the six-dimensional (2, 0) superconformal field theory of type $$ \mathfrak{gl} $$ gl (N ) on M × ℝ2,1, and we consider surface defects placed on L × {x4 = x5 = 0}; in the IR we have the (2, 0) theory of type gl (1) on $$ \tilde{M} $$ M ˜ × ℝ2,1, and put the defects on $$ \tilde{L} $$ L ˜ × {x4 = x5 = 0}. In the case M = ℝ3, q-nonabelianization computes the Jones polynomial of a link, or its analogue associated to the group U(N ). In the case M = C × ℝ, when the projection of L to C is a simple non-contractible loop, q-nonabelianization computes the protected spin character for framed BPS states in 4d $$ \mathcal{N} $$ N = 2 theories of class S. In the case N = 2 and M = C × ℝ, we give a concrete construction of the q-nonabelianization map. The construction uses the data of the WKB foliations associated to a holomorphic covering $$ \tilde{C}\to C $$ C ˜ → C .

scholarly journals IS THE JONES POLYNOMIAL OF A KNOT REALLY A POLYNOMIAL?

2006 ◽  
Vol 15 (08) ◽  
pp. 983-1000 ◽  
Author(s):  
STAVROS GAROUFALIDIS ◽  
THANG TQ LÊ
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Analytic Function ◽  
Jones Polynomial ◽  
Laurent Polynomial ◽  
Topological Quantum Field Theory ◽  
Quantum Field ◽  
Integer Coefficients ◽  
Topological Quantum ◽  
Proper Generalization

The Jones polynomial of a knot in 3-space is a Laurent polynomial in q, with integer coefficients. Many people have pondered why this is so, and what a proper generalization of the Jones polynomial for knots in other closed 3-manifolds is. Our paper centers around this question. After reviewing several existing definitions of the Jones polynomial, we argue that the Jones polynomial is really an analytic function, in the sense of Habiro. Using this, we extend the holonomicity properties of the colored Jones function of a knot in 3-space to the case of a knot in an integer homology sphere, and we formulate an analogue of the AJ Conjecture. Our main tools are various integrality properties of topological quantum field theory invariants of links in 3-manifolds, manifested in Habiro's work on the colored Jones function.

A Review of Conformal Field Theory in 2D

2014 ◽  
Vol 6 (2) ◽  
pp. 1079-1105
Author(s):  
Rahul Nigam
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Conformal Field Theory ◽  
Statistical Mechanics ◽  
Critical Behavior ◽  
Verma Module ◽  
Virasoro Algebra ◽  
Quantum Field ◽  
Conformal Field ◽  
Insight Into

In this review we study the elementary structure of Conformal Field Theory in which is a recipe for further studies of critical behavior of various systems in statistical mechanics and quantum field theory. We briefly review CFT in dimensions which plays a prominent role for example in the well-known duality AdS/CFT in string theory where the CFT lives on the AdS boundary. We also describe the mapping of the theory from the cylinder to a complex plane which will help us gain an insight into the process of radial quantization and radial ordering. Finally we will develop the representation of the Virasoro algebra which is the well-known "Verma module".  

Sign in / Sign up

Export Citation Format

Share Document