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.
Highlights
Spin, such as e.g. currents or the stress tensor, is clearly one of the most pressing issues that was addressed early on in [9,10,11,12,13], see [14, 15] and references therein for more recent developments
In this work we employed conformal group theory to explicitly embed the theory of Calogero-Sutherland models and their wave functions into conformal field theory
In the case relevant for four-point functions of spinning fields, Calogero-Sutherland wave functions depend on two variables ui, i = 1, 2, which can be considered as coordinates on a particular 2-dimensional abelian subgroup of the conformal group, as described in eq (3.1)
Summary
Our main goal is to reinterpret four-point correlation functions of local primary fields in a conformal field theory in terms of functions on the conformal group which. In a parityinvariant conformal field theory, a primary field which transforms in the representation π of the orthogonal group O(d) is acted on by I as ΦI (x) = IΦ(x)I−1 =. With the help of these reflections, we can define new maps Ii = Isei = seiI which are in the identity component Spin(1, d + 1) of the conformal group. This concludes our short detour on conformal inversions and their action on primary fields in a general conformal field theory. The Ward identities take a similar form
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