Conformal invariance: From Weyl to SO(2,d)

Institute of Cosmology, Relativity and Astrophysics (ICRA - CBPF) Rua Dr. Xavier Sigaud, 150, Urca, CEP 22290-180, Rio de Janeiro, RJ, Brasil

^(a) sofiane@cbpf.br

Received: 11 December 2012
Accepted: 20 January 2013

Abstract

The present work deals with two different but subtilely related kinds of conformal mappings: Weyl rescaling in d > 2 dimensional spaces and SO(2,d) transformations. We express how the difference between the two can be compensated by diffeomorphic transformations. This is well known in the framework of string theory but in the particular case of d = 2 spaces. Indeed, the Polyakov formalism describes world sheets in terms of two-dimensional conformal field theory (CFT). On the other hand, Zumino had shown that a classical four-dimensional Weyl-invariant field theory restricted to live in Minkowski space leads to an SO(2,4)-invariant field theory. We extend Zumino's result to relate Weyl and SO(2,d) symmetries in arbitrary conformally flat spaces (CFS). This allows us to assert that a classical SO(2,d)-invariant field does not distinguish, at least locally, between two different d-dimensional CFSs.