Higher dimensional CFTs as 2D conformally-equivariant topological field theories

TL;DR

This paper explores a novel algebraic framework linking four-dimensional conformal field theories to two-dimensional topological field theories with $so(4,2)$ symmetry, including free and perturbative cases, and extends to non-integer dimensions.

Contribution

It introduces an algebraic construction of 4D CFT correlators via $so(4,2)$ equivariant TFT2, including free, perturbative, and non-integer dimensional theories.

Findings

Algebraic formulation of CFT4 correlators using $so(4,2)$ equivariant algebra.

Relation of crossing symmetry to algebra associativity.

Extension to non-integer dimensions via diagram algebras.

Abstract

Two and three-point functions of primary fields in four dimensional CFT have a simple space-time dependences factored out from the combinatoric structure which enumerates the fields and gives their couplings. This has led to the formulation of two dimensional topological field theories with $so(4,2)$ equivariance which are conjectured to be equivalent to higher dimensional conformal field theories. We review this CFT4/TFT2 construction in the simplest possible setting of a free scalar field, which gives an algebraic construction of the correlators in terms of an infinite dimensional $so(4,2)$ equivariant algebra with finite dimensional subspaces at fixed scaling dimension. Crossing symmetry of the CFT4 is related to associativity of the algebra. This construction is then extended to describe perturbative CFT4, by making use of deformed co-products. Motivated by the Wilson-Fisher CFT we outline the construction of U(so($d$,2)) equivariant TFT2 for non-integer $d$, in terms of diagram algebras and their representations.