Conformal Blocks in 2-Dimensional Conformal Field Theory and Montesinos Knots

1990 ◽  
Vol 13 (1) ◽  
pp. 25-29
Author(s):  
M Zaganescu
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Conformal Blocks ◽  
Conformal Field

ANYONS AND GAUSSIAN CONFORMAL FIELD THEORIES

1991 ◽  
Vol 06 (04) ◽  
pp. 289-294 ◽  
Author(s):  
DILEEP P. JATKAR ◽  
SUMATHI RAO
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Mass Parameter ◽  
Simons Theory ◽  
Conformal Field Theories ◽  
Conformal Blocks ◽  
Conformal Field ◽  
A Charge ◽  
Chern Simons ◽  
Chern Simons Theory

We identify the spin of the anyons with the holomorphic dimension of the primary fields of a Gaussian conformal field theory. The angular momentum addition rules for anyons go over to the fusion rules for the primary fields and the r↔1/2r duality of the Gaussian CFT is reproduced by a charge-flux duality of the anyons. For a U(1) Chern-Simons theory with topological mass parameter k=2n, N-anyon states on the torus have 2n components, which correspond to the 2n conformal blocks of an N-point function in the Gaussian conformal field theory.

RESHETIKHIN-TURAEV AND CRANE-KOHNO-KONTSEVICH 3-MANIFOLD INVARIANTS COINCIDE

1993 ◽  
Vol 02 (01) ◽  
pp. 65-95 ◽  
Author(s):  
SERGEY PIUNIKHIN
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Explicit Formula ◽  
Quantum Groups ◽  
Jones Polynomial ◽  
Matrix Elements ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Wznw Model ◽  
The Matrix

The coincidence of two different presentations of Witten 3-manifold invariants is proved. One of them, invented by Reshetikhin and Turaev, is based on the surgery presentation a of 3-manifold and the representation theory of quantum groups; another one, invented by Kohno and Crane and, in slightly different language by Kontsevich, is based on a Heegaard decomposition of a 3-manifold and representations of the Teichmuller group, arising in conformal field theory. The explicit formula for the matrix elements of generators of the Teichmuller group in the space of conformal blocks in the SU(2) k, WZNW-model is given,using the Jones polynomial of certain links.

scholarly journals Defect conformal blocks from Appell functions

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Ilija Burić ◽  
Volker Schomerus
Keyword(s):  
Field Theory ◽  
Correlation Function ◽  
Conformal Field Theory ◽  
Hypergeometric Functions ◽  
Conformal Symmetry ◽  
Scalar Fields ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Additional Defect ◽  
Appell Functions

Abstract We develop a group theoretical formalism to study correlation functions in defect conformal field theory, with multiple insertions of bulk and defect fields. This formalism is applied to construct the defect conformal blocks for three-point functions of scalar fields. Starting from a configuration with one bulk and one defect field, for which the correlation function is determined by conformal symmetry, we explore two possibilities, adding either one additional defect or bulk field. In both cases it is possible to express the blocks in terms of classical hypergeometric functions, though the case of two bulk and one defect field requires Appell’s function F4.

scholarly journals SEMICLASSICAL BLOCKS AND CORRELATORS IN RATIONAL AND IRRATIONAL CONFORMAL FIELD THEORY

1996 ◽  
Vol 11 (27) ◽  
pp. 4837-4896 ◽  
Author(s):  
M. B. HALPERN ◽  
N. A. OBERS
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Simple Class ◽  
High Level

The generalized Knizhnik–Zamolodchikov equations of irrational conformal field theory provide a uniform description of rational and irrational conformal field theory. Starting from the known high-level solution of these equations, we first construct the high-level conformal blocks and correlators of all the affine-Sugawara and coset constructions on simple g. Using intuition gained from these cases, we then identify a simple class of irrational processes whose high-level blocks and correlators we are also able to construct.

scholarly journals DUALITY IN osp(1|2) CONFORMAL FIELD THEORY AND LINK INVARIANTS

1998 ◽  
Vol 13 (17) ◽  
pp. 2931-2978 ◽  
Author(s):  
I. P. ENNES ◽  
A. V. RAMALLO ◽  
J. M. SANCHEZ DE SANTOS ◽  
P. RAMADEVI
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Simons Theory ◽  
Conformal Blocks ◽  
Link Invariants ◽  
Conformal Field ◽  
Free Field Realization ◽  
Chern Simons ◽  
Knots And Links ◽  
Chern Simons Theory

We study the crossing symmetry of the conformal blocks of the conformal field theory based on the affine Lie superalgebra osp(1|2). Within the framework of a free field realization of the osp(1|2) current algebra, the fusion and braiding matrices of the model are determined. These results are related in a simple way to those corresponding to the su(2) algebra by means of a suitable identification of parameters. In order to obtain the link invariants corresponding to the osp(1|2) conformal field theory, we analyze the corresponding topological Chern–Simons theory. In a first approach we quantize the Chern–Simons theory on the torus and, as a result, we get the action of the Wilson line operators on the supercharacters of the affine osp(1|2). From this result we get a simple expression relating the osp(1|2) polynomials for torus knots and links to those corresponding to the su(2) algebra. Further, this relation is verified for arbitrary knots and links by quantizing the Chern–Simons theory on the punctured two-sphere.

scholarly journals The geometry of Casimir W-algebras

2018 ◽  
Vol 5 (5) ◽  
Author(s):  
Raphaël Belliard ◽  
Bertrand Eynard ◽  
Sylvain Ribault
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Lie Algebra ◽  
Differential System ◽  
Correlation Functions ◽  
Riemann Sphere ◽  
Fuchsian System ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Affine Lie Algebra

Let \mathfrak{g}𝔤 be a simply laced Lie algebra, \widehat{\mathfrak{g}}_1𝔤̂1 the corresponding affine Lie algebra at level one, and \mathcal{W}(\mathfrak{g})𝒲(𝔤) the corresponding Casimir W-algebra. We consider \mathcal{W}(\mathfrak{g})𝒲(𝔤)-symmetric conformal field theory on the Riemann sphere. To a number of \mathcal{W}(\mathfrak{g})𝒲(𝔤)-primary fields, we associate a Fuchsian differential system. We compute correlation functions of \widehat{\mathfrak{g}}_1𝔤̂1-currents in terms of solutions of that system, and construct the bundle where these objects live. We argue that cycles on that bundle correspond to parameters of the conformal blocks of the W-algebra, equivalently to moduli of the Fuchsian system.

scholarly journals Gaudin models and multipoint cnformal blocks: general theory

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Ilija Burić ◽  
Sylvain Lacroix ◽  
Jeremy A. Mann ◽  
Lorenzo Quintavalle ◽  
Volker Schomerus
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Differential Operators ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Commuting Differential Operators ◽  
Higher Dimensional ◽  
Important Open Problem ◽  
Complete Set ◽  
Gaudin Models

Abstract The construction of conformal blocks for the analysis of multipoint correlation functions with N > 4 local field insertions is an important open problem in higher dimensional conformal field theory. This is the first in a series of papers in which we address this challenge, following and extending our short announcement in [1]. According to Dolan and Osborn, conformal blocks can be determined from the set of differential eigenvalue equations that they satisfy. We construct a complete set of commuting differential operators that characterize multipoint conformal blocks for any number N of points in any dimension and for any choice of OPE channel through the relation with Gaudin integrable models we uncovered in [1]. For 5-point conformal blocks, there exist five such operators which are worked out smoothly in the dimension d.

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