Minimal representations and reductive dual pairs in conformal field theory

TL;DR

This paper explores minimal representations and reductive dual pairs in conformal field theory, clarifying their mathematical foundations and highlighting their implicit use in physics for understanding gauge symmetries.

Contribution

It provides a pedagogical introduction to minimal representations and dual pairs, connecting mathematical theory with their application in conformal field theory and gauge symmetry.

Findings

Clarified the concept of minimal representations in Lie groups.

Explained reductive dual pairs and their role in gauge symmetry.

Highlighted the implicit use of these concepts in physics without formal naming.

Abstract

A minimal representation of a simple non-compact Lie group is obtained by ``quantizing'' the minimal nilpotent coadjoint orbit of its Lie algebra. It provides context for Roger Howe's notion of a reductive dual pair encountered recently in the description of global gauge symmetry of a (4-dimensional) conformal observable algebra. We give a pedagogical introduction to these notions and point out that physicists have been using both minimal representations and dual pairs without naming them and hence stand a chance to understand their theory and to profit from it.