Simple Space-Time Symmetries: Generalizing Conformal Field Theory

TL;DR

This paper classifies simple space-time symmetry groups that generalize conformal groups, identifying their structure, properties, and the conditions under which they admit positive energy representations and time reflection automorphisms.

Contribution

It provides a comprehensive classification of simple Lie groups acting as space-time symmetries with properties similar to conformal groups, extending the framework beyond classical conformal symmetry.

Findings

Allowed groups include universal coverings of SU(m,m), SO(2,D), Sp(l,R), SO*(4n), and E_7(-25).

These groups correspond to fractional linear transformations of Euclidean Jordan algebras.

Only SO(2,D) groups have a time reflection automorphism; others have intrinsic chiral structures.

Abstract

We study simple space-time symmetry groups G which act on a space-time manifold M=G/H which admits a G-invariant global causal structure. We classify pairs (G,M) which share the following additional properties of conformal field theory: 1) The stability subgroup H of a point in M is the identity component of a parabolic subgroup of G, implying factorization H=MAN, where M generalizes Lorentz transformations, A dilatations, and N special conformal transformations. 2) special conformal transformations in N act trivially on tangent vectors to the space-time manifold M. The allowed simple Lie groups G are the universal coverings of SU(m,m), SO(2,D), Sp(l,R), SO*(4n) and E_7(-25) and H are particular maximal parabolic subgroups. They coincide with the groups of fractional linear transformations of Euklidean Jordan algebras whose use as generalizations of Minkowski space time was advocated by Gunaydin. All these groups G admit positive energy representations. It will also be shown that the classical conformal groups SO(2,D) are the only allowed groups which possess a time reflection automorphism; in all other cases space-time has an intrinsic chiral structure.