Conformal algebra: R-matrix and star-triangle relation

Abstract

The main purpose of this paper is the construction of the R-operator which acts in the tensor product of two infinite-dimensional representations of the conformal algebra and solves Yang-Baxter equation. We build the R-operator as a product of more elementary operators S1, S2 and S3. Operators S1 and S3 are identified with intertwining operators of two irreducible representations of the conformal algebra and the operator S2 is obtained from the intertwining operators S1 and S3 by a certain duality transformation. There are star-triangle relations for the basic building blocks S1, S2 and S3 which produce all other relations for the general R-operators. In the case of the conformal algebra of n-dimensional Euclidean space we construct the R-operator for the scalar (spin part is equal to zero) representations and prove that the star-triangle relation is a well known star-triangle relation for propagators of scalar fields. In the special case of the conformal algebra of the 4-dimensional Euclidean space, the R-operator is obtained for more general class of infinitedimensional (differential) representations with nontrivial spin parts. As a result, for the case of the 4-dimensional Euclidean space, we generalize the scalar star-triangle relation to the most general star-triangle relation for the propagators of particles with arbitrary spins. *

$${{SO(d,1)}}$$ S O ( d , 1 ) -Invariant Yang–Baxter Operators and the dS/CFT Correspondence

“…In the case of the flat model, our expression for R θ then coincides, up to a phase, with an expression derived previously [CDI13] (see also [DKM01]) for the case of the principal series representation. These authors also proved the Yang-Baxter equation (R4"), and a version of the idempotency relation (R5").…”

“…Some parts of the following proof are similar to the one in [CDI13] -in particular, the verification of the Yang-Baxter equation (R4") via a startriangle (triality) relation.…”

Yangian bootstrap for conformal Feynman integrals

“…see e.g. [14], with Γ x = Γ(x) denoting the Gamma function. At four points, the function φ in (5) becomes nontrivial due to the presence of two non-trivial conformal invariants z andz defined by zz =…”

Generalized fishnets and exact four-point correlators in chiral CFT4

“…2) under the condition α + β + γ = 5, and being. Such formulas are actually particular reductions of a more general one derived in [50]. This generalization involves two propagators in the representation of traceless symmetric tensors of integer rank S, namely…”