  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.

High Energy Physics - Theory;Representation Theory