Gauge Symmetry and Howe Duality in 4D Conformal Field Theory Models

Abstract

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.

1. Introduction

We review results of [1–7] on 4D conformal field theory (CFT) models, which can be summed up as follows. The requirement of global conformal invariance (GCI) in compactified Minkowski space together with the Wightman axioms [8] implies the Huygens principle (3.6) and rationality of correlation functions [1]. A class of 4D GCI quantum field theory models gives rise to a (reducible) Fock space representation of a pair consisting of an infinite-dimensional Lie algebra ℒ and a commuting with its compact Lie group 𝑈. The state space ℱ splits into a direct sum of irreducible ℒ×𝑈 modules, so that each irreducible representation (IR) of ℒ appears with a multiplicity equal to the dimension of an associated IR of 𝑈. The pair (ℒ,𝑈) illustrates a interconnects two independent developments: (i) it appears as a reductive dual pair [9, 10], within (a central extension of) an infinite-dimensional symplectic Lie algebra; (ii) it provides a representation theoretic realization of the Doplicher-Haag-Roberts' (DHR) theory of superselection sectors and compact gauge groups, [11, 12]. I will first briefly recall Howe's and DHR's theories, then (in Section 2) I will explain how some 2D CFT technics can be extended to four space-time dimensions (in spite of persistent doubts that this is at all possible). After these preliminaries we will proceed with our survey of 4D CFT models and associated infinite-dimensional Lie algebras which relate the two independent developments.

1.1. Reductive Dual Pairs

The notion of a (reductive) dual pair was introduced by Roger Howe in an influential preprint of the 1970s that was eventually published in [10]. It was previewed in two earlier papers of Howe [9, 13], highlightening the role of the Heisenberg group and the applications of dual pairs to physics. For Howe a dual pair, the counterpart for groups and for Lie algebras of the mutual commutants of von Neumann algebras [12] is a (highly structured) concept that plays a unifying role in such widely different topics as Weil's metaplectic group approach [14, 15] to 𝜃 functions and automorphic forms (an important chapter in number theory) and the quantum mechanical Heisenberg group along with the description of massless particles in terms of the ladder representations of 𝑈(2,2) [16], among others (in physics).

Howe begins in [9] with a 2𝑛-dimensional real symplectic manifold 𝒲=𝒱+𝒱^′ where 𝒱 is spanned by 𝑛 symbols 𝑎_𝑖,𝑖=1,…,𝑛, called annihilation operators and 𝒱^′ is spanned by their conjugate, the  creation operators  𝑎^∗_𝑖 satisfying the canonical commutation relations (CCR)

[𝑎_𝑖,𝑎_𝑗]=0=[𝑎^∗_𝑖,𝑎^∗_𝑗],[𝑎_𝑖,𝑎^∗_𝑗]=𝛿_𝑖𝑗. (1.1) The commutator of two elements of the real vector space 𝒲 being a real number it defines a (nondegenerate, skew-symmetric) bilinear form on it which vanishes on 𝒱 and on 𝒱^′ separately and for which 𝒱^′ appears as the dual space to 𝒱 (the space of linear functionals on 𝒱). The real symplectic Lie algebra 𝑠𝑝(2𝑛,ℝ) spanned by antihermitean quadratic combinations of 𝑎_𝑖 and 𝑎^∗_𝑗 acts by commutators on 𝒲 preserving its reality and the above bilinear form. This action extends to the Fock space  ℱ (unitary, irreducible) representation of the CCR. It is, however, only exponentiated to the double cover of 𝑆𝑝(2𝑛,ℝ), the metaplectic group  𝑀𝑝(2𝑛) (that is not a matrix group, i.e., has no faithful finite-dimensional representation; we can view its Fock space, called by Howe [9] oscillator representation as the defining one). Two subgroups 𝐺 and 𝐺^′ of 𝑀𝑝(2𝑛) are said to form a (reductive) dual pair if they act reductively on ℱ (that is automatic for a unitary representation like the one considered here) and each of them is the full centralizer of the other in 𝑀𝑝(2𝑛). The oscillator representation of 𝑀𝑝(2𝑛) displays a minimality property, [17, 18] that keeps attracting the attention of both physicists and mathematicians, see, for example, [19–21].

1.2. Local Observables Determine a Compact Gauge Group

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 superselection rule as explained by Wick et al. [22]. It required the nontrivial vision of Rudolf Haag to predict in the 1960s that a local net of obsevable algebras should determine the compact gauge group that governs the structure of its superselection sectors (for a review and references to the original work see [12]). It took over 20 years and the courage and dedication of Haag's (then) young collaborators, Doplicher and Roberts [11], to carry out this program to completion. They proved that all superselection sectors of a local QFT 𝒜 with a mass gap are contained in the vacuum representation of a canonically associated (graded local) field extension ℰ, and they are in a one-to-one correspondence with the unitary irreducible representations (IRs) of a compact gauge group 𝐺 of internal symmetries of ℰ, so that 𝒜 consists of the fixed points of ℰ under 𝐺. The pair (𝒜,𝒢) in ℰ provides a general realization of a dual pair in a local quantu theory.

2. How Do 2D CFT Methods Work in Higher Dimensions?

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.

(1)The 2𝐷 conformal group is infinite-dimensional. It is the direct product of the diffeomorphism groups of the left and right (compactified) light rays. (In the euclidean picture it is the group of analytic and antianalytic conformal mappings.) By contrast, for 𝐷>2, according to the Liouville theorem, the quantum mechanical conformal group in 𝐷 space-time dimensions is finite (in fact, (𝐷+1)(𝐷+2)/2)-dimensional. It is (a covering of) the spin group Spin (𝐷,2).(2)The representation theory of affine Kac-Moody algebras [23] and of the Virasoro algebra [24] is playing a crucial role in constructing soluble 2𝐷 models of (rational) CFT. There are, on the other hand, no local Lie fields in higher dimensions. After an inconclusive attempt by Robinson [25] (criticized in [26]) this was proven for scalar fields by Baumann [27].(3)The light cone in two dimensions is the direct product of two light rays. This geometric fact is the basis of splitting 2𝐷 variables into right- and left-movers' chiral variables. No such splitting seems to be available in higher dimensions.(4)There are chiral algebras in 2𝐷 CFT whose local currents satisfy the axioms of vertex algebras (As a mathematical subject vertex algebras were anticipated by Frenkel and Kac [28] and introduced by Borcherds [29]; for reviews and further references see, e.g., [30, 31].) and have rational correlation functions. It was believed for a long time that they have no physically interesting higher-dimensional CFT analogue.(5)Furthermore, the chiral currents in a 2𝐷 CFT on a torus have elliptic correlation functions [32], the 1-point function of the stress energy tensor appearing as a modular form (these can be also interpreted as finite temperature correlation functions and a thermal energy mean value on the Riemann sphere). Again, there seemed to be no good reason to expect higher-dimensional analogues of these attractive properties.

We will argue that each of the listed features of 2𝐷 CFT does have, when properly understood, a higher-dimensional counterpart.

(1) The presence of a conformal anomaly (a nonzero Virasoro central charge 𝑐) tells us that the infinite conformal symmetry in 1+1 dimension is, in fact, broken. What is actually used in 2𝐷 CFT are the (conformal) operator product expansions (OPEs) which can be derived for any D and allow to extend the notion of a primary field (e.g., with respect to the stress-energy tensor).

(2) For 𝐷=4, infinite-dimensional Lie algebras are generated by bifields  𝑉_𝑖𝑗(𝑥₁,𝑥₂) which naturally arise in the OPE of a (finite) set of (say, hermitean, scalar) local fields 𝜙_𝑖 of dimension 𝑑(>1):

(𝑥²₁₂)^𝑑𝜙_𝑖(𝑥₁)𝜙_𝑗(𝑥₂)=𝑁_𝑖𝑗+𝑥²₁₂𝑉_𝑖𝑗(𝑥₁,𝑥₂)+𝑂((𝑥²₁₂)²𝑥),₁₂=𝑥₁−𝑥₂,𝑥²=𝑥²−𝑥⁰²,𝑁_𝑖𝑗=𝑁_𝑗𝑖∈ℝ, (2.1)(2.2) where 𝑉_𝑖𝑗 are defined as (infinite) sums of OPE contributions of (twist two) conserved local tensor currents (and the real symmetric matrix (𝑁_𝑖𝑗) is positive definite). We say more on this in what follows (reviewing results of [2–7]).

(3) We will exhibit a factorization of higher-dimensional intervals by using the following parametrization of the conformally compactified space-time ([33–36]): 𝑀={𝑧_𝛼=𝑒^𝑖𝑡𝑢_𝛼,𝛼=1,…,𝐷;𝑡,𝑢_𝛼∈ℝ;𝑢²=^𝐷_𝛼=1𝑢²_𝛼𝕊=1}=^𝐷−1×𝕊¹{1,−1}. (2.3) The real interval between two points 𝑧₁=𝑒^𝑖𝑡1𝑢₁, 𝑧₂=𝑒^𝑖𝑡2𝑢₂ is given by 𝑧²₁₂(𝑧²₁𝑧²₂)^−1/2=2(cos𝑡₁₂−cos𝛼)=−4sin𝑡₊sin𝑡₋,𝑧₁₂=𝑧₁−𝑧₂,𝑡_±=12(𝑡₁₂±𝛼),𝑢₁⋅𝑢₂=cos𝛼,𝑡₁₂=𝑡₁−𝑡₂. (2.4)(2.5) Thus 𝑡₊ and 𝑡₋ are the compact picture counterparts of “left” and “right” chiral variables (see [36]). The factorization of 2𝐷 cross-ratios into chiral parts again has a higher-dimensional analogue [37]: 𝑥𝑠∶=²¹²𝑥²₃₄𝑥²₁₃𝑥²₂₄=𝑢₊𝑢₋𝑥,𝑡∶=²¹⁴𝑥²₂₃𝑥²₁₃𝑥²₂₄=(1−𝑢₊)(1−𝑢₋),𝑥_𝑖𝑗=𝑥_𝑖−𝑥_𝑗, (2.6) which yields a separation of variables in the d'Alembert equation (cf. equation (2.1)). One should, in fact, be able to derive the factorization (2.6) from (2.4).

(4) It turns out that the requirement of global conformal invariance (GCI) in Minkowski space together with the standard Wightman axioms of local commutativity and energy positivity entails the rationality of correlation functions in any even number of space-time dimensions [1]. Indeed, GCI and local commutativity of Bose fields (for space-like separations of the arguments) imply the Huygens principle and, in fact, the strong (algebraic) locality condition

(𝑥²₁₂)^𝑛[𝜙_𝑖(𝑥₁),𝜙_𝑗(𝑥₂)]=0for𝑛sufficientlylarge, (2.7) a condition only consistent with the theory of free fields for an even number of space time dimensions. It is this Huygens locality condition which allows the introduction of higher-dimensional vertex algebras [35, 36, 38].

(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., [12]) are expressed as linear combinations of modular forms [36].

The rest of the paper is organized as follows. In Section 3 we reproduce the general form of the 4-point function of the bifield 𝑉 and the leading term in its conformal partial wave expansion. The case of a theory of scalar fields of dimension 𝑑=2 is singled out, in which the bifields (and the unit operator) close a commutator algebra. In Section 4 we classify the arising infinite-dimensional Lie algebras ℒ in terms of the three real division rings 𝔽=ℝ,ℂ,ℍ. In Section 5 we formulate the main result of [6, 7] on the Fock space representations of the Lie algebra ℒ(𝔽) coupled to the (dual, in the sense of Howe [9]) compact gauge group 𝑈(𝑁,𝔽), where 𝑁 is the central charge of ℒ.

3. Four-Point Functions and Conformal Partial Wave Expansions

The conformal bifields 𝑉(𝑥₁,𝑥₂) of dimension (1,1) which arise in the OPE (2.2) (as sums of integrals of conserved tensor currents) satisfy the d'Alembert equation in each argument [3]; we will call them harmonic bifields. Their correlation functions depend on the dimension 𝑑 of the local scalar fields 𝜙. For 𝑑=1 one is actually dealing with the theory of a free massless field. We will, therefore, assume 𝑑>1. A basis {𝑓_𝜈𝑖,𝜈=0,1,…,𝑑−2,𝑖=1,2} of invariant amplitudes 𝐹(𝑠,𝑡) such that

⟨0∣𝑉₁(𝑥₁,𝑥₂)𝑉₂(𝑥₃,𝑥₄1)∣0⟩=𝜌₁₃𝜌₂₄𝜌𝐹(𝑠,𝑡),_𝑖𝑗=𝑥²_𝑖𝑗+𝑖0𝑥⁰_𝑖𝑗,𝑥²=𝐱²−(𝑥⁰)² (3.1) is given by

(𝑢₊−𝑢₋)𝑓_𝜈1𝑢(𝑠,𝑡)=^+𝜈+1(1−𝑢₊)^𝜈+1−𝑢₋^𝜈+1(1−𝑢₋)^𝜈+1,(𝑢₊−𝑢₋)𝑓_𝜈2(𝑠,𝑡)=(−1)^𝜈(𝑢₊^𝜈+1−𝑢₋^𝜈+1),𝜈=0,1,…,𝑑−2, (3.2) where 𝑢_± are the “chiral variables" (2.6)

𝑓₀₁=1𝑡,𝑓₀₂=1;𝑓₁₁=1−𝑠−𝑡𝑡²,𝑓₁₂𝑓=𝑡−𝑠−1₂₁=(1−𝑡)²−𝑠(2−𝑡)+𝑠²𝑡³,𝑓_𝜈21(𝑠,𝑡)=𝑡𝑓_𝜈1(𝑠𝑡,1𝑡), (3.3)𝑓_𝜈,𝑖, 𝑖=1,2, corresponding to single pole terms [5] in the 4-point correlation functions 𝑤_𝜈𝑖(𝑥₁,…,𝑥₄)=𝑓_𝜈𝑖(𝑠,𝑡)/𝜌₁₃𝜌₂₄:

𝑤₀₁=1𝜌₁₄𝜌₂₃,𝑤₀₂=1𝜌₁₃𝜌₂₄,𝑤₁₁=𝜌₁₃𝜌₂₄−𝜌₁₄𝜌₂₃−𝜌₁₂𝜌₃₄𝜌²₁₄𝜌²₂₃,𝑤₁₂=𝜌₁₄𝜌₂₃−𝜌₁₃𝜌₂₄−𝜌₁₂𝜌₃₄𝜌²₁₃𝜌²₂₄,𝑤₂₁=(𝜌₁₃𝜌₂₄−𝜌₁₄𝜌₂₃)²−𝜌₁₂𝜌₃₄(2𝜌₁₃𝜌₂₄−𝜌₁₄𝜌₂₃)+𝜌²₁₂𝜌²₃₄𝜌³₁₄𝜌³₂₃,𝑤₂₂=(𝜌₁₄𝜌₂₃−𝜌₁₃𝜌₂₄)²−𝜌₁₂𝜌₃₄(2𝜌₁₄𝜌₂₃−𝜌₁₃𝜌₂₄)+𝜌²₁₂𝜌²₃₄𝜌³₁₃𝜌³₂₄. (3.4) We have 𝑤_𝜈2=𝑃₃₄𝑤_𝜈1(=𝑃₁₂𝑤_𝜈1) where 𝑃_𝑖𝑗 stands for the substitution of the arguments 𝑥_𝑖 and 𝑥_𝑗. Clearly, for 𝑥₁=𝑥₂ (or 𝑠=0, 𝑡=1) only the amplitudes 𝑓_0𝑖 contribute to the 4-point function (3.1). It has been demonstrated in [4] that the lowest angular momentum (ℓ) contribution to 𝑓_𝜈𝑖 corresponds to ℓ=𝜈. The corresponding OPE of the bifield 𝑉 starts with a local scalar field 𝜙 of dimension 𝑑=2 for 𝜈=0, with a conserved current 𝑗_𝜇 (of 𝑑=3) for 𝜈=1, with the stress energy tensor 𝑇_𝜆𝜇 for 𝜈=2. Indeed, the amplitude 𝑓_𝜈1 admits an expansion in twist two (the twist of a symmetric traceless tensor is defined as the difference between its dimension and its rank. All conserved symmetric tensors in 4𝐷 have twist two.) conformal partial waves  𝛽_ℓ(𝑠,𝑡) [39] starting with (for a derivation see [4, Appendix B])

𝛽_𝜈𝐺(𝑠,𝑡)=^𝜈+1(𝑢₊)−𝐺_𝜈+1(𝑢₋)𝑢₊−𝑢₋,𝐺_𝜇(𝑢)=𝑢^𝜇𝐹(𝜇,𝜇;2𝜇;𝑢). (3.5)

Remark 3.1. Equations (3.2) and (3.5) provide examples of solutions of the d'Alambert equation in any of the arguments 𝑥_𝑖,𝑖=1,2,3,4. In fact, the general conformal covariant (of dimension 1 in each argument) such solution has the form of the right-hand side of (3.1) with 𝐹(𝑠,𝑡)=𝑓(𝑢₊)−𝑓(𝑢₋)𝑢₊−𝑢₋. (3.6)

Remark 3.2. We note that albeit each individual conformal partial wave is a transcendental function (like (3.5)) the sum of all such twist two contributions is the rational function 𝑓_𝜈1(𝑠,𝑡).

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 𝑑 can only close on the 𝑉's and the unit operator for 𝑑=2. In this case the bifields 𝑉 are proven, in addition, to be Huygens bilocal [5].

Remark 3.3. In general, irreducible positive energy representations of the (connected) conformal group are labeled by triples (𝑑;𝑗₁,𝑗₂) including the dimension 𝑑 and the Lorentz weight (𝑗₁,𝑗₂)(2𝑗_𝑖∈ℕ), [40]. It turns out that for 𝑑=3 there is a spin-tensor bifield of weight ((3/2;1/2,0),(3/2;0,1/2)) whose commutator algebra does close; for 𝑑=4 there is a conformal tensor bifield of weight ((2;1,0),(2;0,1)) with this property. These bifields may be termed lefthanded. They are analogues of chiral 2𝐷 currents; a set of bifields invariant under space reflections would also involve their righthanded counterparts (of weights ((3/2;0,1/2),(3/2;1/2,0)), and ((2;0,1),(2;1,0)), resp.).

4. Infinite-Dimensional Lie Algebras and Real Division Rings

Our starting point is the following result of [5].

Proposition 4.1. The harmonic bilocal fields 𝑉 arising in the OPEs of a (finite) set of local hermitean scalar fields of dimension 𝑑=2 can be labeled by the elements 𝑀 of an unital algebra ℳ⊂𝑀𝑎𝑡(𝐿,ℝ) of real matrices closed under transposition, 𝑀→^𝑡𝑀, in such a way that the following commutation relations (CR) hold: [𝑉_𝑀1(𝑥₁,𝑥₂),𝑉_𝑀2(𝑥₃,𝑥₄)]=Δ₁₃𝑉_{𝑡𝑀1𝑀2}(𝑥₂,𝑥₄)+Δ₂₄𝑉_{𝑀1𝑡𝑀2}(𝑥₁,𝑥₃)+Δ₂₃𝑉_𝑀1𝑀2(𝑥₁,𝑥₄)+Δ₁₄𝑉_𝑀2𝑀1(𝑥₃,𝑥₂)+tr(𝑀₁𝑀₂)Δ_12,34+tr(^𝑡𝑀₁𝑀₂)Δ_12,43, (4.1) here Δ_𝑖𝑗 is the free field commutator, Δ_𝑖𝑗∶=Δ⁺_𝑖𝑗−Δ⁺_𝑗𝑖, and Δ_12,𝑖𝑗=Δ⁺_1𝑖Δ⁺_2𝑗−Δ⁺_𝑖1Δ⁺_𝑗2, where Δ⁺_𝑖𝑗=Δ⁺(𝑥_𝑖−𝑥_𝑗) is the 2-point Wightman function of a free massless scalar field.

We call the set of bilocal fields closed under the CR (4.1) a Lie system. The types of Lie systems are determined by the corresponding 𝑡-algebras, that is, real associative matrix algebras ℳ closed under transposition. We first observe that each such ℳ can be equipped with a Frobenius inner product

⟨𝑀₁,𝑀₂⟩=tr(^𝑡𝑀₁𝑀₂)=_𝑖𝑗(𝑀₁)_𝑖𝑗(𝑀₂)_𝑖𝑗, (4.2) which is symmetric, positive definite, and has the property ⟨𝑀₁𝑀₂,𝑀₃⟩=⟨𝑀₁,𝑀₃^𝑡𝑀₂⟩. This implies that for every right ideal ℐ⊂ℳ its orthogonal complement is again a right ideal while its transposed ^𝑡ℐ is a left ideal. Therefore, ℳ is a semisimple algebra so that every module over ℳ is a direct sum of irreducible modules.

Let now ℳ be irreducible. It then follows from the Schur's lemma (whose real version [41] is richer but less popular than the complex one) that its commutant ℳ^′ in Mat(𝐿,ℝ) coincides with one of the three real division rings (or not necessarily commutative fields): the fields of real and complex numbers ℝ and ℂ, and the noncommutative division ring ℍ of quaternions. In each case the Lie algebra of bilocal fields is a central extension of an infinite-dimensional Lie algebra that admits a discrete series of highest weight representations. Finite dimensional simple Lie groups 𝐺 with this property have been extensively studied by mathematicians (for a review and references, see [42]); for an extension to the infinite-dimensional case, see [43]. If 𝑍 is the centre of 𝐺 and 𝐾 is a closed maximal subgroup of 𝐺 such that 𝐾/𝑍 is compact then 𝐺 is characterized by the property that (𝐺,𝐾) is a hermitean symmetric pair. Such groups give rise to simple space-time symmetries in the sense of [44] (see also earlier work—in particular by Günaydin—cited there).

It was proven, first in the theory of a single scalar field 𝜙 (of dimension two) [2], and eventually for an arbitrary set of such fields [5], that the bilocal fields 𝑉_𝑀 can be written as linear combinations of normal products of free massless scalar fields 𝜑_𝑖(𝑥):

𝑉_𝑀(𝑥₁,𝑥₂)=^𝐿_𝑖,𝑗=1𝑀^𝑖𝑗∶𝜑_𝑖(𝑥₁)𝜑_𝑗(𝑥₂). (4.3) For each of the above types of Lie systems 𝑉_𝑀 has a canonical form, namely,

ℝ∶𝑉(𝑥₁,𝑥₂)=^𝑁_𝑖=1∶𝜑_𝑖(𝑥₁)𝜑_𝑖(𝑥₂),ℂ∶𝑊(𝑥₁,𝑥₂)=^𝑁_𝑗=1∶𝜑^∗_𝑗(𝑥₁)𝜑_𝑗(𝑥₂),ℍ∶𝑌(𝑥₁,𝑥₂)=^𝑁_𝑚=1∶𝜑⁺_𝑚(𝑥₁)𝜑_𝑚(𝑥₂), (4.4) where 𝜑_𝑖 are real, 𝜑_𝑗 are complex, and 𝜑_𝑚 are quaternionic valued fields (corresponding to (3.2) with 𝐿=𝑁, 2𝑁, and 4𝑁, resp.). We will denote the associated infinite-dimensional Lie algebra by ℒ(𝔽), 𝔽=ℝ, ℂ, or ℍ.

Remark 4.2. We note that the quaternions (represented by 4×4 real matrices) appear both in the definition of 𝑌—that is, of the matrix algebra ℳ, and of its commutant ℳ^′, the two mutually commuting sets of imaginary quaternionic units ℓ_𝑖 and 𝑟_𝑗 corresponding to the splitting of the Lie algebra 𝑠𝑜(4) of real skew-symmetric 4×4 matrices into a direct sum of “a left and a right” 𝑠𝑜(3) Lie subalgebras: ℓ₁=𝜎₃⊗𝜖,ℓ₂=𝜖⊗𝟏,ℓ₃=ℓ₁ℓ₂=𝜎₁⊗𝜖,(ℓ_𝑗)_𝛼𝛽=𝛿_𝛼0𝛿_𝑗𝛽−𝛿_𝛼𝑗𝛿_0𝛽−𝜀_{0𝑗𝛼𝛽}𝑟,𝛼,𝛽=0,1,2,3,𝑗=1,2,3,₁=𝜖⊗𝜎₃,𝑟₂=𝟏⊗𝜖,𝑟₃=𝑟₁𝑟₂=𝜖⊗𝜎₁, (4.5) where 𝜎_𝑘 are the Pauli matrices, 𝜖=𝑖𝜎₂, 𝜀_{𝜇𝜈𝛼𝛽} is the totally antisymmetric Levi-Civita tensor normalized by 𝜀₀₁₂₃=1. We have 𝑌(𝑥₁,𝑥₂)=𝑉₀(𝑥₁,𝑥₂)𝟏+𝑉₁(𝑥₁,𝑥₂)ℓ₁+𝑉₂(𝑥₁,𝑥₂)ℓ₂+𝑉₃(𝑥₁,𝑥₂)ℓ₃=𝑌(𝑥₂,𝑥₁)⁺(ℓ⁺_𝑖=−ℓ_𝑖,[ℓ_𝑖,𝑟_𝑗𝑉]=0),_𝜅(𝑥₁,𝑥₂)=^𝑁_𝑚=1∶𝜑^𝛼_𝑚(𝑥₁)(ℓ_𝜅)_𝛼𝛽𝜑^𝛽_𝑚(𝑥₂),ℓ₀=𝟏. (4.6)

In order to determine the Lie algebra corresponding to the CR (4.1) in each of the three cases (4.5) we choose a discrete basis and specify the topology of the resulting infinite matrix algebra in such a way that the generators of the conformal Lie algebra (most importantly, the conformal Hamiltonian 𝐻) belong to it. The basis, say (𝑋_𝑚𝑛) where 𝑚,𝑛 are multi-indices, corresponds to the expansion [34] of a free massless scalar field 𝜑 in creation and annihilation operators of fixed energy states

𝜑(𝑧)=^∞_ℓ=0^(ℓ+1)2_𝜇=1((𝑧²)^−ℓ−1𝜑_ℓ+1,𝜇+𝜑_{−ℓ−1,𝜇})ℎ_ℓ𝜇(𝑧), (4.7) where (ℎ_ℓ𝜇(𝑧),𝜇=1,…,(ℓ+1)²) form a basis of homogeneous harmonic polynomials of degree ℓ in the complex 4-vector 𝑧 (of the parametrization (2.3) of 𝑀). The generators of the conformal Lie algebra 𝑠𝑢(2,2) are expressed as infinite sums in 𝑋_𝑚𝑛 with a finite number of diagonals (cf. Appendix B in [6]). The requirement 𝑠𝑢(2,2)⊂ℒ thus restricts the topology of ℒ implying that the last (c-number) term in (4.1) gives rise to a nontrivial central extension of ℒ.

The analysis of [6, 7] yields the following.

Proposition 4.3. The Lie algebras ℒ(𝔽), 𝔽=ℝ,ℂ,ℍ are 1-parameter central extensions of appropriate completions of the following inductive limits of matrix algebras: ℝ∶𝑠𝑝(∞,ℝ)=lim_𝑛→∞𝑠𝑝(2𝑛,ℝ),ℂ∶𝑢(∞,∞)=lim_𝑛→∞𝑢(𝑛,𝑛),ℍ∶𝑠𝑜^∗(4∞)=lim_𝑛→∞𝑠𝑜^∗(4𝑛). (4.8) In the free field realization (4.4) the suitably normalized central charge coincides with the positive integer 𝑁.

5. Fock Space Representation of the Dual Pair ℒ(𝔽)×𝑈(𝑁,𝔽)

To summarize the discussion of the last section, there are three infinite-dimensional irreducible Lie algebras, ℒ(𝔽), that are generated in a theory of GCI scalar fields of dimension 𝑑=2 and correspond to the three real division rings 𝔽 (Proposition 4.3). For an integer central charge 𝑁 they admit a free field realization of type (4.3) and a Fock space representation with (compact) gauge group 𝑈(𝑁,𝔽):

𝑈(𝑁,ℝ)=𝑂(𝑁),𝑈(𝑁,ℂ)=𝑈(𝑁),𝑈(𝑁,ℍ)=𝑆𝑝(2𝑁)(=𝑈𝑆𝑝(2𝑁)). (5.1) It is remarkable that this result holds in general.

Theorem 5.1. (i) In any unitary irreducible positive energy representation (UIPER) of ℒ(𝔽) the central charge 𝑁 is a positive integer.
(ii) All UIPERs of ℒ(𝔽) are realized (with multiplicities) in the Fock space ℱ of 𝑁𝑑𝑖𝑚_ℝ𝔽 free hermitean massless scalar fields.
(iii) The ground states of equivalent UIPERs in ℱ form irreducible representations of the gauge group 𝑈(𝑁,𝔽) (5.1). This establishes a one-to-one correspondence between UIPERs of ℒ(𝔽) occurring in the Fock space and the irreducible representations of 𝑈(𝑁,𝔽).

The proof of this theorem for 𝔽=ℝ, ℂ is given in [6] (the proof of (i) is already contained in [2]); the proof for 𝔽=ℍ is given in [7].

Remark 5.2. Theorem 5.1 is also valid—and its proof becomes technically simpler—for a 2-dimensional chiral theory (in which the local fields are functions of a single complex variable). For 𝔽=ℂ the representation theory of the resulting infinite-dimensional Lie algebra 𝑢(∞,∞) is then essentially equivalent to that of the vertex algebra 𝑊_1+∞ studied in [45] (see the introduction in [6] for a more precise comparison).

Theorem 5.1 provides a link between two parallel developments, one in the study of the highest weight modules of reductive Lie groups (and of related dual pairs—see Section 1.1) [42, 43, 46, 47] (and [9, 10]), the other in the work of Doplicher and Roberts [11] and Haag [12] on the theory of (global) gauge groups and superselection sectors—see Section 1.2. (They both originate—in the paper of Irving Segal and Rudolf Haag, resp.—at the same Lille 1957 conference on mathematical problems in quantum field theory.) Albeit the settings are not equivalent the results match. The observable algebra (in our case, the commutator algebra generated by the set of bilocal fields 𝑉_𝑀) determines the (compact) gauge group and the structure of the superselection sectors of the theory. (For a more careful comparison between the two approaches, see [6, Sections 1 and 4].)

The infinite-dimensional Lie algebra ℒ(𝔽) and the compact gauge group 𝑈(𝑁,𝔽) appear as a rather special (limit-) case of a dual pair in the sense of Howe [9, 10]. It would be interesting to explore whether other (inequivalent) pairs would appear in the study of commutator algebras of (spin)tensor bifields (discussed in Remark 3.3) and of their supersymmetric extension (e.g., a limit as 𝑚,𝑛→∞ of the series of Lie superalgebras 𝑜𝑠𝑝(4𝑚^∗∣2𝑛) studied in [48]).

Acknowledgments

The author would like to thank his coauthors Bojko Bakalov, Nikolay M. Nikolov, and Karl-Henning Rehren. All results (reported in Sections 3–5) of this paper have been obtained in collaboration with them. He also acknowledges a partial support from the Bulgarian National Council for Scientific Research under Contracts Ph-1406 and DO-02-257.