It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit (highly reducible) unitary positive energy representations in a Fock space. The multiplicity of their irreducible components is governed by a compact gauge group. The mutually commuting observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of local scalar fields of conformal dimension two in four space-time dimensions the associated dual pairs are constructed and classified.

We review results of [

The notion of a (reductive) dual pair was introduced by Roger Howe in an influential preprint of the 1970s that was eventually published in [

Howe begins in [

Observables (unlike charge carrying fields) are left invariant by (global) gauge transformations. This is, in fact, part of the definition of a gauge symmetry or a

A number of reasons are given why 2-dimensional conformal field theory is, in a way, exceptional so that extending its methods to higher dimensions appears to be hopeless.

The

The representation theory of affine Kac-Moody algebras [

The light cone in two dimensions is the direct product of two light rays. This geometric fact is the basis of splitting

There are chiral algebras in

Furthermore, the chiral currents in a

We will argue that each of the listed features of

(1) The presence of a conformal anomaly (a nonzero Virasoro central charge

(2) For

(3) We will exhibit a factorization of higher-dimensional intervals by using the following parametrization of the conformally compactified space-time ([

(4) It turns out that the requirement of

(5) Local GCI fields have elliptic thermal correlation functions with respect to the (differences of) conformal time variables in any even number of space-time dimensions; the corresponding energy mean values in a Gibbs (KMS) state (see, e.g., [

The rest of the paper is organized as follows. In Section

The conformal bifields

Equations (

We note that albeit each individual conformal partial wave is a transcendental function (like (

It can be deduced from the analysis of 4-point functions that the commutator algebra of a set of harmonic bifields generated by OPE of scalar fields of dimension

In general, irreducible positive energy representations of the (connected) conformal group are labeled by triples

Our starting point is the following result of [

The harmonic bilocal fields

We call the set of bilocal fields closed under the CR (

Let now

It was proven, first in the theory of a single scalar field

We note that the quaternions (represented by

In order to determine the Lie algebra corresponding to the CR (

The analysis of [

The Lie algebras

To summarize the discussion of the last section, there are three infinite-dimensional irreducible Lie algebras,

(i) In any unitary irreducible positive energy representation (UIPER) of

(ii) All UIPERs of

(iii) The ground states of equivalent UIPERs in

The

Theorem

Theorem

The infinite-dimensional Lie algebra

The author would like to thank his coauthors Bojko Bakalov, Nikolay M. Nikolov, and Karl-Henning Rehren. All results (reported in Sections