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

TL;DR

This paper explores the relationship between Weyl rescaling and SO(2,d) conformal transformations, extending known results from 2D and 4D to arbitrary conformally flat spaces, highlighting their local equivalence.

Contribution

It generalizes Zumino's relation between Weyl and SO(2,d) symmetries from specific cases to arbitrary conformally flat spaces, showing their local equivalence.

Findings

Weyl rescaling and SO(2,d) transformations can be compensated by diffeomorphisms.

Classical SO(2,d)-invariant fields are locally indistinguishable across different conformally flat spaces.

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. On the other hand, B. 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.