Finite and Infinite Symmetries in (2+1)-Dimensional Field Theory

TL;DR

This paper explores the role of $SO(2,1)$ conformal symmetry in non-relativistic Chern--Simons theory, demonstrating how it governs solutions and leads to complete integrability of static configurations.

Contribution

It reveals the application of $SO(2,1)$ symmetry in non-relativistic Chern--Simons theory and its extension to an infinite group, enabling full integrability of static solutions.

Findings

$SO(2,1)$ symmetry controls solution structure

Infinite symmetry group renders static problem integrable

Application of symmetry concepts to Chern--Simons theory

Abstract

These days, Franco Iachello is {\it the\/} eminent practitioner applying classical and finite groups to physics. In this he is following a tradition at Yale, established by the late Feza Gursey, and succeeding Gursey in the Gibbs chair; Gursey in turn, had Pauli as a mentor. Iachello's striking achievement has been to find an actual realization of arcane supersymmetry within mundane adjacent even-odd nuclei. Thus far this is the only {\it physical\/} use of supersymmetry, and its fans surely must be surprised at the venue. Here we describe the role of $SO(2,1)$ conformal symmetry in non-relativistic Chern--Simons theory: how it acts, how it controls the nature of solutions, how it expands to an infinite group on the manifold of static solutions thereby rendering the static problem completely integrable. Since Iachello has also used the $SO(2,1)$ group in various contexts, this essay is presented to him on the occasion of his fiftieth birthday.