Infinite dimensional Lie algebras in 4D conformal quantum field theory

TL;DR

This paper explores how global conformal invariance in four-dimensional quantum field theory leads to infinite-dimensional Lie algebras, revealing new algebraic structures and their associated gauge groups.

Contribution

It introduces a novel connection between GCI scalar fields and infinite-dimensional Lie algebras, extending algebraic techniques from 2D to 4D conformal quantum field theory.

Findings

Infinite-dimensional Lie algebras arise from GCI scalar fields of dimension two.

The associated gauge groups are O(N), U(N), and U(N,H)=Sp(2N).

The Lie algebras are central extensions of sp(infty,R), u(infty,infty), and so*(4 infty).

Abstract

The concept of global conformal invariance (GCI) opens the way of applying algebraic techniques, developed in the context of 2-dimensional chiral conformal field theory, to a higher (even) dimensional space-time. In particular, a system of GCI scalar fields of conformal dimension two gives rise to a Lie algebra of harmonic bilocal fields, V_m(x,y), where the m span a finite dimensional real matrix algebra M closed under transposition. The associative algebra M is irreducible iff its commutant M' coincides with one of the three real division rings. The Lie algebra of (the modes of) the bilocal fields is in each case an infinite dimensional Lie algebra: a central extension of sp(infty,R) corresponding to the field R of reals, of u(infty,infty) associated to the field C of complex numbers, and of so*(4 infty) related to the algebra H of quaternions. They give rise to quantum field theory models with superselection sectors governed by the (global) gauge groups O(N), U(N), and U(N,H)=Sp(2N), respectively.