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

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.


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 infinitedimensional 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

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 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 .

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.

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.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 stressenergy tensor .4, infinite-dimensional Lie algebras are generated by bifields V ij x 1 , x 2 which naturally arise in the OPE of a finite set of say, hermitean, scalar local fields φ i of dimension d > 1 : 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 : The real interval between two points z 1 e it 1 u 1 , z 2 e it 2 u 2 is given by 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 : 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 spacetime 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 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 higherdimensional 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.

Four-Point Functions and Conformal Partial Wave Expansions
The conformal bifields V x 1 , x 2 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 where u ± are the "chiral variables" 2.6

3.4
We have w ν2 P 34 w ν1 P w ν1 where P ij stands for the substitution of the arguments x i and x j .Clearly, for x 1 x 2 or s 0, t 1 only the amplitudes f 0i contribute to the 4point function 3.1 .It has been demonstrated in 4 that the lowest angular momentum contribution to f νi corresponds to 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 s, t 39 starting with for a derivation see 4, Appendix B 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 .

Infinite-Dimensional Lie Algebras and Real Division Rings
Our starting point is the following result of 5 .
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 which is symmetric, positive definite, and has the property This implies that for every right ideal I ⊂ M its orthogonal complement is again a right ideal while its transposed t I 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 infinitedimensional 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 infinitedimensional 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 G ünaydin-cited there .

Advances in Mathematical Physics
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 : 4.3 For each of the above types of Lie systems V M has a canonical form, namely, 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 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: where σ k are the Pauli matrices, iσ 2 , ε μναβ is the totally antisymmetric Levi-Civita tensor normalized by ε 0123 1.We have

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 where h μ z , μ 1, . . ., 1 2 form a basis of homogeneous harmonic polynomials of degree 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:

4.8
In the free field realization 4.4 the suitably normalized central charge coincides with the positive integer N.

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 : It is remarkable that this result holds in general.ii All UIPERs of L F are realized (with multiplicities) in the Fock space F of Ndim R F 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 .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 .

5 Remark 3 . 1 .
μ, μ; 2μ; u .3.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

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 → t M, in such a way that the following commutation relations (CR) hold:

Theorem 5 . 1 .
i In any unitary irreducible positive energy representation (UIPER) of L F the central charge N is a positive integer.

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.