scholarly journals Conformal group theory of tensor structures

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Ilija Burić ◽  
Volker Schomerus ◽  
Mikhail Isachenkov
Keyword(s):  
Correlation Functions ◽  
Conformal Group ◽  
Wave Functions ◽  
Dimensional Euclidean Space ◽  
Theoretic Approach ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Local Operators ◽  
Indispensable Tool ◽  
Systematic Derivation

Abstract The decomposition of correlation functions into conformal blocks is an indispensable tool in conformal field theory. For spinning correlators, non-trivial tensor structures are needed to mediate between the conformal blocks, which are functions of cross ratios only, and the correlation functions that depend on insertion points in the d-dimensional Euclidean space. Here we develop an entirely group theoretic approach to tensor structures, based on the Cartan decomposition of the conformal group. It provides us with a new universal formula for tensor structures and thereby a systematic derivation of crossing equations. Our approach applies to a ‘gauge’ in which the conformal blocks are wave functions of Calogero-Sutherland models rather than solutions of the more standard Casimir equations. Through this ab initio construction of tensor structures we complete the Calogero-Sutherland approach to conformal correlators, at least for four-point functions of local operators in non-supersymmetric models. An extension to defects and superconformal symmetry is possible.

NONARCHIMEDEAN CONFORMAL FIELD THEORIES

1989 ◽  
Vol 04 (18) ◽  
pp. 4877-4908 ◽  
Author(s):  
EZER MELZER
Keyword(s):  
Correlation Functions ◽  
Central Charge ◽  
Conformal Symmetry ◽  
Conformal Group ◽  
Field Theories ◽  
Dimensional Euclidean Space ◽  
Conformal Field Theories ◽  
Linear Transformations ◽  
Associative Algebras ◽  
Conformal Field

We present a general formalism for conformal field theories defined on a non-Archimedean field. Such theories are defined by complex-valued correlation functions of fields of a [Formula: see text]-adic variable. Conformal invariance is imposed by requiring the correlation functions to be unchanged under fractional linear transformations, the latter forming the full analogue of the conformal group in two-dimensional, euclidean space-time. All fields in the theory can be taken to be "primary", under the "non-Archimedean conformal group". The conformal symmetry fixes completely the form of all correlation functions, once we are given the weight-spectrum of the theory and the OPE coefficients (which must be the structure constants of certain commutative, associative algebras). We explicitly construct non-Archimedean CFT's having the same weight spectrum as that of Archimedean models of central charge c < 1. The OPE coefficients of these "local" Archimedean and non-Archimedean models are related by adelic formulae.

scholarly journals On 2d CFTs that interpolate between minimal models

2019 ◽  
Vol 6 (6) ◽  
Author(s):  
Sylvain Ribault
Keyword(s):  
Correlation Functions ◽  
Central Charge ◽  
Field Theories ◽  
Two Dimensional ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Exactly Solvable ◽  
Structure Constants

We investigate exactly solvable two-dimensional conformal field theories that exist at generic values of the central charge, and that interpolate between A-series or D-series minimal models. When the central charge becomes rational, correlation functions of these CFTs may tend to correlation functions of minimal models, or diverge, or have finite limits which can be logarithmic. These results are based on analytic relations between four-point structure constants and residues of conformal blocks.

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 THE FREE FIELD REPRESENTATION OF SU(3) CONFORMAL FIELD THEORY

1993 ◽  
Vol 08 (23) ◽  
pp. 4031-4053
Author(s):  
HOVIK D. TOOMASSIAN
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Correlation Functions ◽  
Free Field ◽  
Field Representation ◽  
Conformal Field ◽  
Point Correlation

The structure of the free field representation and some four-point correlation functions of the SU(3) conformal field theory are considered.

scholarly journals Geometrical four-point functions in the two-dimensional critical Q-state Potts model: the interchiral conformal bootstrap

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Yifei He ◽  
Jesper Lykke Jacobsen ◽  
Hubert Saleur
Keyword(s):  
Potts Model ◽  
Correlation Functions ◽  
Liouville Theory ◽  
Analytical Determination ◽  
Conformal Weight ◽  
Two Dimensional ◽  
Conformal Blocks ◽  
Conformal Bootstrap ◽  
Point Correlation

Abstract Based on the spectrum identified in our earlier work [1], we numerically solve the bootstrap to determine four-point correlation functions of the geometrical connectivities in the Q-state Potts model. Crucial in our approach is the existence of “interchiral conformal blocks”, which arise from the degeneracy of fields with conformal weight hr,1, with r ∈ ℕ*, and are related to the underlying presence of the “interchiral algebra” introduced in [2]. We also find evidence for the existence of “renormalized” recursions, replacing those that follow from the degeneracy of the field $$ {\Phi}_{12}^D $$ Φ 12 D in Liouville theory, and obtain the first few such recursions in closed form. This hints at the possibility of the full analytical determination of correlation functions in this model.

scholarly journals Dispersion formulas in QFTs, CFTs and holography

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
David Meltzer
Keyword(s):  
Fourier Transform ◽  
Correlation Functions ◽  
Momentum Space ◽  
Conformal Blocks ◽  
Dispersion Formula ◽  
Space Dispersion

Abstract We study momentum space dispersion formulas in general QFTs and their applications for CFT correlation functions. We show, using two independent methods, that QFT dispersion formulas can be written in terms of causal commutators. The first derivation uses analyticity properties of retarded correlators in momentum space. The second derivation uses the largest time equation and the defining properties of the time-ordered product. At four points we show that the momentum space QFT dispersion formula depends on the same causal double-commutators as the CFT dispersion formula. At n-points, the QFT dispersion formula depends on a sum of nested advanced commutators. For CFT four-point functions, we show that the momentum space dispersion formula is equivalent to the CFT dispersion formula, up to possible semi-local terms. We also show that the Polyakov-Regge expansions associated to the momentum space and CFT dispersion formulas are related by a Fourier transform. In the process, we prove that the momentum space conformal blocks of the causal double-commutator are equal to cut Witten diagrams. Finally, by combining the momentum space dispersion formulas with the AdS Cutkosky rules, we find a complete, bulk unitarity method for AdS/CFT correlators in momentum space.

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.

scholarly journals Two point functions in defect CFTs

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Christopher P. Herzog ◽  
Abhay Shrestha
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Correlation Functions ◽  
Equations Of Motion ◽  
Physical Space ◽  
Arbitrary Spin ◽  
Momentum Tensor ◽  
Maxwell Theory ◽  
Conformal Field ◽  
Point Correlation

Abstract This paper is designed to be a practical tool for constructing and investigating two-point correlation functions in defect conformal field theory, directly in physical space, between any two bulk primaries or between a bulk primary and a defect primary, with arbitrary spin. Although geometrically elegant and ultimately a more powerful approach, the embedding space formalism gets rather cumbersome when dealing with mixed symmetry tensors, especially in the projection to physical space. The results in this paper provide an alternative method for studying two-point correlation functions for a generic d-dimensional conformal field theory with a flat p-dimensional defect and d − p = q co-dimensions. We tabulate some examples of correlation functions involving a conserved current, an energy momentum tensor and a Maxwell field strength, while analysing the constraints arising from conservation and the equations of motion. A method for obtaining bulk-to-defect correlators is also explained. Some explicit examples are considered: free scalar theory on ℝp× (ℝq/ℤ2) and a free four dimensional Maxwell theory on a wedge.

scholarly journals Instability of complex CFTs with operators in the principal series

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Dario Benedetti
Keyword(s):  
Principal Series ◽  
Conformal Group ◽  
Point Of View ◽  
Field Theories ◽  
Operator Product ◽  
Quantum Field Theories ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Primary Operator ◽  
Tensor Models

Abstract We prove the instability of d-dimensional conformal field theories (CFTs) having in the operator-product expansion of two fundamental fields a primary operator of scaling dimension h = $$ \frac{d}{2} $$ d 2 + i r, with non-vanishing r ∈ ℝ. From an AdS/CFT point of view, this corresponds to a well-known tachyonic instability, associated to a violation of the Breitenlohner-Freedman bound in AdSd+1; we derive it here directly for generic d-dimensional CFTs that can be obtained as limits of multiscalar quantum field theories, by applying the harmonic analysis for the Euclidean conformal group to perturbations of the conformal solution in the two-particle irreducible (2PI) effective action. Some explicit examples are discussed, such as melonic tensor models and the biscalar fishnet model.

Sign in / Sign up

Export Citation Format

Share Document