Perturbative 4D conformal field theories and representation theory of diagram algebras

Abstract: The correlators of free four dimensional conformal field theories (CFT4) have been shown to be given by amplitudes in two-dimensional $so(4,2)$ equivariant topological field theories (TFT2), by using a vertex operator formalism for the correlators. We show that this can be extended to perturbative interacting conformal field theories, using two representation theoretic constructions. A co-product deformation for the conformal algebra accommodates the equivariant construction of composite operators in the presence of non-additive anomalous dimensions. Explicit expressions for the co-product deformation are given within a sector of $ \mathcal{N} =4 $ SYM and for the Wilson-Fischer fixed point near four dimensions. The extension of conformal equivariance beyond integer dimensions (relevant for the Wilson-Fischer fixed point) leads to the definition of an associative diagram algebra $ {\bf U}{*} $, abstracted from $ Uso(d)$ in the limit of large integer $d$, which admits extension of $ Uso(d)$ representation theory to general real (or complex) $d$. The algebra is related, via oscillator realisations, to $so(d)$ equivariant maps and Brauer category diagrams. Tensor representations are constructed where the diagram algebra acts on tensor products of a fundamental diagram representation. A similar diagrammatic algebra ${\bf U}{\star ,2}$, related to a general $d$ extension for $ Uso(d,2)$ is defined, and some of its lowest weight representations relevant to the Wilson-Fischer fixed point are described.