MONODROMY OF SOLUTIONS OF THE KNIZHNIK-ZAMOLODCHIKOV EQUATION: SL(2)k WZNW MODEL

2004 ◽  
Vol 19 (supp02) ◽  
pp. 336-347
Author(s):  
B. PONSOT
Keyword(s):  
Field Theory ◽  
Conformal Blocks ◽  
Wznw Model ◽  
Liouville Field Theory ◽  
Zamolodchikov Equation

Three explicit and equivalent representations for the monodromy of the conformal blocks in the non compact SL(2)k WZNW model are proposed in terms of the same quantity computed in Liouville 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.

RECENT PROGRESS IN LIOUVILLE FIELD THEORY (REVIEW)

2004 ◽  
Vol 19 (supp02) ◽  
pp. 311-335 ◽  
Author(s):  
B. PONSOT
Keyword(s):  
Field Theory ◽  
Quantum Group ◽  
Recent Progress ◽  
Explicit Construction ◽  
Conformal Blocks ◽  
Crossing Symmetry ◽  
Boundary Operators ◽  
Liouville Field Theory ◽  
Theory Review

An explicit construction for the monodromy of the Liouville conformal blocks in terms of Racah-Wigner coefficients of the quantum group [Formula: see text] is proposed. As a consequence, crossing-symmetry for four point functions is analytically proven, and the expression for the correlator of three boundary operators is obtained.

N = 3 AND N = 4 SUPERCONFORMAL WZNW SIGMA MODELS IN SUPERSPACE II: THE N = 4 CASE

1992 ◽  
Vol 07 (02) ◽  
pp. 287-316 ◽  
Author(s):  
E. A. IVANOV ◽  
S. O. KRIVONOS ◽  
V. M. LEVIANT
Keyword(s):  
Sigma Models ◽  
Conformal Blocks ◽  
Wznw Model ◽  
Chiral Superfields ◽  
Zamolodchikov Equation

The issues related to the U (1) × O (4) N = 4 superconformal WZNW sigma models [with the bosonic target spaces U (1) × SU (2) and U (1) × U (1) × O (4)] are investigated in the framework of a 2D N = 4 superspace. We define the corresponding N = 4 supercurrents, both on classical and quantum levels, in terms of the basic primary N = 4 WZNW superfields and show that the generalized Sugawara form for the dimension 3/2 and 2 component currents directly follows from the constraints on the basic superfields. The N = 4 superfield analog of the Knizhnik–Zamolodchikov equation for conformal blocks in WZNW sigma models is derived. We also analyze the N = 4 WZNW superfields from the standpoint of two SU(2) N = 4 SCAs entering as subalgebras into the underlying U (1) × O (4) N = 4 SCA. We demonstrate that in the linearizing limit the U (1) × SU (2) WZNW model reproduces the recently discussed SU(2) N = 4 superconformal system of free chiral superfields.

scholarly journals Towards Feynman rules for conformal blocks

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sarah Hoback ◽  
Sarthak Parikh
Keyword(s):  
Field Theory ◽  
Effective Field Theory ◽  
Hypergeometric Functions ◽  
Power Series Expansion ◽  
Effective Field ◽  
Conformal Block ◽  
Tree Level ◽  
Conformal Blocks ◽  
Simple Set ◽  
Feynman Rules

Abstract We conjecture a simple set of “Feynman rules” for constructing n-point global conformal blocks in any channel in d spacetime dimensions, for external and exchanged scalar operators for arbitrary n and d. The vertex factors are given in terms of Lauricella hypergeometric functions of one, two or three variables, and the Feynman rules furnish an explicit power-series expansion in powers of cross-ratios. These rules are conjectured based on previously known results in the literature, which include four-, five- and six-point examples as well as the n-point comb channel blocks. We prove these rules for all previously known cases, as well as two new ones: the seven-point block in a new topology, and all even-point blocks in the “OPE channel.” The proof relies on holographic methods, notably the Feynman rules for Mellin amplitudes of tree-level AdS diagrams in a scalar effective field theory, and is easily applicable to any particular choice of a conformal block beyond those considered in this paper.

scholarly journals Higher rank FZZ-dualities

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Thomas Creutzig ◽  
Yasuaki Hikida
Keyword(s):  
Field Theory ◽  
Reduction Method ◽  
Operator Algebras ◽  
The Self ◽  
Liouville Theory ◽  
Conformal Field ◽  
Self Duality ◽  
Higher Rank ◽  
Corner Vertex ◽  
Liouville Field Theory

Abstract We examine strong/weak dualities in two dimensional conformal field theories by generalizing the Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality between Witten’s cigar model described by the $$ \mathfrak{sl}(2)/\mathfrak{u}(1) $$ sl 2 / u 1 coset and sine-Liouville theory. In a previous work, a proof of the FZZ-duality was provided by applying the reduction method from $$ \mathfrak{sl}(2) $$ sl 2 Wess-Zumino-Novikov-Witten model to Liouville field theory and the self-duality of Liouville field theory. In this paper, we work with the coset model of the type $$ \mathfrak{sl}\left(N+1\right)/\left(\mathfrak{sl}(N)\times \mathfrak{u}(1)\right) $$ sl N + 1 / sl N × u 1 and investigate the equivalence to a theory with an $$ \mathfrak{sl}\left(N+\left.1\right|N\right) $$ sl N + 1 N structure. We derive the duality explicitly for N = 2, 3 by applying recent works on the reduction method extended for $$ \mathfrak{sl}(N) $$ sl N and the self-duality of Toda field theory. Our results can be regarded as a conformal field theoretic derivation of the duality of the Gaiotto-Rapčák corner vertex operator algebras Y0,N,N+1[ψ] and YN,0,N+1[ψ−1].

TETRAHEDRA AND POLYNOMIAL EQUATIONS IN TOPOLOGICAL FIELD THEORY

1991 ◽  
Vol 06 (20) ◽  
pp. 3571-3598 ◽  
Author(s):  
NOUREDDINE CHAIR ◽  
CHUAN-JIE ZHU
Keyword(s):  
Field Theory ◽  
Topological Field Theory ◽  
Fundamental Representation ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Witten Theory ◽  
Topological Field ◽  
General Program ◽  
The One ◽  
Chern Simons

Some tetrahedra in SUk(2) Chern-Simons-Witten theory are computed. The results can be used to compute an arbitrary tetrahedron inductively by fusing with the fundamental representation. The results obtained are in agreement with those of quantum groups. By associating a (finite) topological field theory (FTFT) to every rational conformal field theory (RCFT), we show that the pentagon and hexagon equations in RCFT follow directly from some skein relations in FTFT. By generalizing the operation of surgery on links in FTFT, we also derive an explicit expression for the modular transformation matrix S(k) of the one-point conformal blocks on a torus in RCFT and the equations satisfied by S(k), in agreement with those required in RCFT. The implication of our results on the general program of classifying RCFT is also discussed.

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