Generalized Conformal Representations of Orthogonal Lie Algebras

TL;DR

This paper generalizes conformal representations of orthogonal Lie algebras to tensor spaces of irreducible modules, involving a hidden central transformation, with conditions for irreducibility, potentially impacting higher-dimensional conformal field theory.

Contribution

It introduces a new non-homogeneous conformal representation of $o(n+2,{C})$ on tensor spaces, extending classical conformal transformations with a central transformation involved.

Findings

Established a condition for irreducibility of the generalized representation

Connected the representation to higher-dimensional conformal field theory

Utilized Pieri's formulas, invariant operators, and Kostant's identities

Abstract

The conformal transformations with respect to the metric defining $o(n,\mbb{C})$ give rise to a nonhomogeneous polynomial representation of $o(n+2,\mbb{C})$. Using Shen's technique of mixed product, we generalize the above representation to a non-homogenous representation of $o(n+2,\mbb{C})$ on the tensor space of any finite-dimensional irreducible $o(n,\mbb{C})$-module with the polynomial space, where a hidden central transformation is involved. Moreover, we find a condition on the constant value taken by the central transformation such that the generalized conformal representation is irreducible. In our approach, Pieri's formulas, invariant operators and the idea of Kostant's characteristic identities play key roles. The result could be useful in understanding higher-dimensional conformal field theory with the constant value taken by the central transformation as the central charge. Our representations virtually provide natural extensions of the conformal transformations on a Riemannian manifold to its vector bundles.