Extended Conformal Symmetry

TL;DR

This paper introduces an extended conformal symmetry framework that incorporates conformal weights into group structures, leading to a new algebraic formulation with applications in gauge theories.

Contribution

It presents a novel discrete, non-central conformal extension of groups with dilatations, including a faithful vector representation and a closed algebra containing Heisenberg and Virasoro structures.

Findings

Extended conformal group has a scale-invariant scalar product.

The algebra includes infinite Heisenberg and Virasoro algebras.

Gauging the extended symmetry automatically incorporates conformal weights.

Abstract

We show that the grading of fields by conformal weight, when built into the initial group symmetry, provides a discrete, non-central conformal extension of any group containing dilatations. We find a faithful vector representation of the extended conformal group and show that it has a scale-invariant scalar product and satisfies a closed commutator algebra. The commutator algebra contains the infinite Heisenberg and Virasoro algebras. In contrast to the classical treatment of scale invariance, covariant derivatives and gauge transformations automatically incorporate the correct conformal weights when the extended symmetry is gauged.