Promoting finite to infinite symmetries: the 3+1-dimensional analogue of the Virasoro algebra and higher-spin fields

TL;DR

This paper constructs infinite-dimensional extensions of 3+1D conformal symmetry, generalizing Virasoro algebra, and explores their implications for higher-spin fields, integrable models, and non-commutative geometry.

Contribution

It introduces explicit infinite-dimensional symmetry algebras extending 3+1D conformal symmetry, enabling new approaches to higher-spin theories and integrable models.

Findings

Explicit construction of infinite-dimensional symmetry algebras

Application to higher-spin field theories in 3+1 dimensions

Proposal of a non-commutative geometric interpretation

Abstract

Infinite enlargements of finite pseudo-unitary symmetries are explicitly provided in this letter. The particular case of u(2,2)=so(4,2)+u(1) constitutes a (Virasoro-like) infinite-dimensional generalization of the 3+1-dimensional conformal symmetry, in addition to matter fields with all conformal spins. These algebras provide a new arena for integrable field models in higher dimensions; for example, Anti-de Sitter and conformal gauge theories of higher-so(4,2)-spin fields. A proposal for a non-commutative geometrical interpretation of space is also outlined.