Conformal field theories with infinitely many conservation laws

TL;DR

This paper explores conformal field theories in even-dimensional space-times that possess infinitely many conserved currents, proposing a generalization of 2D chiral algebras to higher dimensions and analyzing their structure and implications.

Contribution

It extends the understanding of higher-dimensional CFTs with infinite conservation laws, connecting them to free field constructions and generalizing concepts from 2D chiral algebras.

Findings

Infinite conserved tensor currents exist in certain higher-dimensional CFTs.

Free massless scalar fields generate these currents under specific gauge symmetries.

The structure of such theories in 4D remains an open question.

Abstract

Globally conformal invariant quantum field theories in a D-dimensional space-time (D even) have rational correlation functions and admit an infinite number of conserved (symmetric traceless) tensor currents. In a theory of a scalar field of dimension D-2 they were demonstrated to be generated by bilocal normal products of free massless scalar fields with an O(N), U(N), or Sp(2N) (global) gauge symmetry [BNRT]. Recently, conformal field theories "with higher spin symmetry" were considered for D=3 in [MZ] where a similar result was obtained (exploiting earlier study of CFT correlators). We suggest that the proper generalization of the notion of a 2D chiral algebra to arbitrary (even or odd) dimension is precisely a CFT with an infinite series of conserved currents. We shall recast and complement (part of) the argument of Maldacena and Zhiboedov into the framework of our earlier work. We extend to D=4 the auxiliary Weyl-spinor formalism developed in [GPY] for D=3. The free field construction only follows for D>3 under additional assumptions about the operator product algebra. In particular, the problem of whether a rational CFT in 4D Minkowski space is necessarily trivial remains open.