ap-3-10.dvi


Acta Polytechnica Vol. 50 No. 3/2010

Gause Symmetry and HoweDuality in 4D Conformal Field TheoryModels

I. Todorov

To Jiri Niederle on the occasion of his 70th birthday

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. Themutually 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.

The paper reviews joint work of B. Bakalov, N. M. Nikolov, K.-H. Rehren and the author.

1 Introduction
We review results of [32, 33, 34, 35, 36] and [2, 3] on
4Dconformalfield theory (CFT)models,which canbe
summed up as follows. The requirement of global con-
formal invariance (GCI) in compactified Minkowski
space togetherwith theWightmanaxioms [41] implies
the Huygens principle (Eq. (3.6) below) and rational-
ity of correlation functions [32]. A class of 4D GCI
quantumfield theorymodels gives rise to a (reducible)
Fock space representation of a pair consisting of an in-
finite dimensionalLie algebraLanda commutingwith
it 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 appearswith
a multiplicity equal to the dimension of an associated
IRof U. Thepair (L, U) illustratesa interconnects two
independent developments: (i) it appears as a reduc-
tive dual pair, [16, 17], 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 superselec-
tion sectors and compact gauge groups, [8, 14]. I shall
first briefly recall Howe’s and DHR’s theories; then
(in Sect. 2) I will explain how some 2D CFT tech-
niques can be extended to four space-time dimensions
(in spite of persistent doubts that this is at all possi-
ble). After these preliminaries we shall proceed with
our survey of 4D CFT models and associated infinite
dimensionalLie algebraswhich relate the two indepen-
dent developments.

1.1 Reductive dual pairs

The notion of a (reductive) dual pair was introduced
by Roger Howe in an influential preprint of the 1970’s
thatwas eventuallypublished in [17]. Itwaspreviewed
in two earlier papers of Howe, [15, 16], 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 mu-
tual commutants of von Neumann algebras ([14]), is a
(highly structured) concept that plays a unifying role
in such widely different topics as Weil’s metaplectic
group approach [44] 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) [31], among others
(in physics).
Howe begins in [16] with a 2n-dimensional real

symplecticmanifoldW = V+V′ whereV is spannedby
n symbols ai, i = 1, . . . , n, called annihilation opera-
tors andV′ is spannedby their conjugate, the creation
operators a∗i satisfying the canonical commutation re-
lations (CCR)

[ai, aj] = 0= [a
∗
i , a

∗
j], [ai, a

∗
j] = δij . (1.1)

The commutator of two elements of the real vec-
tor space W being a real number it defines a (non-
degenerate, skew-symmetric) bilinear formon itwhich
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 combi-
nations of ai and a

∗
j acts by commutators on W pre-

serving its realityandtheabovebilinear form. This ac-
tion extends to theFock space F (unitary, irreducible)
representation of the CCR. It is, however, only expo-
nentiated to the double cover of Sp(2n,R), the meta-
plectic group M p(2n) (which is not a matrix group –
i.e., has no faithful finite-dimensional representation;
we canview its Fock space, calledbyHowe [16]oscilla-
tor representationas thedefiningone). Twosubgroups
G and G′ of M p(2n) are said to forma (reductive) dual
pair if they act reductively on F (that is automatic for

54



Acta Polytechnica Vol. 50 No. 3/2010

a unitary representation like the one considered here)
and each of them is the full centralizer of the other in
M p(2n). The oscillator representation of M p(2n) dis-
plays aminimality property, [19] that keeps attracting
the attention of both physicists andmathematicians –
see e.g. [25, 12, 26].

1.2 Local observables determine a
compact gauge group

Observables (unlike charge carrying fields) are left in-
variant by (global) gauge transformations. This is, in
fact, part of the definition of a gauge symmetry or a
superselection rule as explained by Wick, Wightman
and Wigner [45]. It required the non-trivial vision of
Rudolf Haag to predict in the 1960’s that a local net
of obsevable algebras should determine the compact
gauge group that governs the structure of its super-
selection sectors (for a review and references to the
original work – see [14]). It took over 20 years and
the courage and dedication of Haag’s (then) young
collaborators, Doplicher and Roberts [8] to carry out
this program to completion. They proved that all su-
perselection sectors of a local QFT A with a mass
gap are contained in the vacuum representation of a
canonically associated (graded local) field extensionE,
and they are in a one-to-one correspondence with the
unitary irreducible representations (IR) 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) inE provides a general realizationof a dual pair
in a local quantu theory.

2 How do 2D CFT methods
work in higher dimensions?

Anumber of reasons are givenwhy 2-dimensional con-
formal field theory is, in away, exceptional so that ex-
tending 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 andantianalytic conformalmappings.)
By contrast, for D > 2, according to the Liou-
ville theorem, the quantummechanical conformal
group in D space-time dimensions is finite (in

fact,
(D +1)(D +2)

2
)-dimensional: it is (a cov-

ering of) the spin group Spin(D,2).
2. The representationtheoryofaffineKac-Moodyal-
gebras [20] and of the Virasoro algebra [23] plays
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 [39] (criticized
in [28]) this was proven for scalar fields by Bau-
mann [4].

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 split-
ting seems to be available in higher dimensions.

4. There are chiral algebras in 2D CFT whose lo-
cal currents satisfy the axioms of vertex algebras1

andhave rational correlation functions. Itwasbe-
lieved for a long time that they havenophysically
interesting higher dimensional CFT analogue.

5. Furthermore, the chiral currents in a 2D CFT
on a torus have elliptic correlation functions [46],
the 1-point function of the stress energy tensor
appearing as a modular form (these can be also
interpreted as finite temperature correlation func-
tionsanda thermal energymeanvalueon theRie-
mann sphere). Again, there seemed to be no good
reason to expect higher dimensional analogues of
these attractive properties.
We shall argue that eachof the listed features of 2D

CFT does have, when properly understood, a higher
dimensional counterpart.
1. The presence of a conformal anomaly (a non-zero
Virasoro central charge c) tells us that the infinite
conformal symmetry in 1+1dimension is, in fact,
broken. What is actually used in 2D CFTare the
(conformal) operator product expansions (OPEs)
which can be derived for any D and allow to ex-
tend the notion of a primary field (for instance
with respect to the stress-energy tensor).

2. For D = 4, infinite dimensional Lie algebras are
generated by bifields Vij(x1, x2) which naturally
arise in the OPE of a (finite) set of (say, her-
mitean, scalar) localfields φi ofdimension d(> 1):

(x212)
d φi(x1)φj(x2)=

Nij + x
2
12 Vij(x1, x2)+ O((x

2
12)
2) ,

x12 = x1 − x2 , x2 =x2 − x0
2
, (2.2)

Nij = Nji ∈ R

where Vij are defined as (infinite) sums of OPE
contributionsof (twist two) conserved local tensor
currents (and the real symmetric matrix (Nij) is
positive definite). We say more on this in what
follows (reviewing results of [33, 34, 35, 36, 2, 3]).

3. We shall exhibit a factorization of higher
dimensional intervals by using the following
parametrization of the conformally compactified
space-time ([43, 42, 37, 38]):

1As a mathematical subject vertex algebras were anticipated by I. Frenkel and V. Kac [11] and introduced by R. Borcherds [5]; for
reviews and further references see e.g. [21] [10]

55



Acta Polytechnica Vol. 50 No. 3/2010

M̄ = {zα = eit uα , α =1, . . . , D; (2.3)

t, uα ∈ R ; u2 =
D∑

α=1

u2α =1} =
S

D−1 × S1
{1, −1}

.

The real interval between two points z1 = e
it1 u1,

z2 = e
it2 u2 is given by:

z212 (z
2
1 z
2
2)

−1/2 = 2(cost12 −cosα)= (2.4)
−4sint+ sint− , z12 = z1 − z2

t± = 1/2(t12 ± α) , (2.5)
u1 · u2 = cosα , t12 = t1 − t2 .

Thus t+ and t− are the compact picture coun-
terparts of “left” and “right” chiral variables (see
[38]). The factorizationof2D cross ratios into chi-
ral parts again has a higher dimensional analogue
[7]:

s :=
x212 x

2
34

x213 x
2
24
= u+ u− , (2.6)

t :=
x214 x

2
23

x213 x
2
24
=(1− u+)(1− u−) , xij = xi − xj

which yields a separation of variables in the
d’Alembert equation (cf. Remark 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 con-
formal invariance (GCI) in Minkowski space to-
getherwith the standardWightman axioms of lo-
cal commutativity and energy positivity entails
the rationalityof correlation functions in any even
number of space-time dimensions [32]. Indeed,
GCI and local commutativity of Bose fields (for
space-like separationsof thearguments) imply the
Huygens principle and, in fact, the strong (alge-
braic) locality condition

(x212)
n[φi(x1), φj(x2)]= 0 (2.7)

for n sufficiently large,

a condition only consistentwith the theory of free
fields for an even number of space time dimen-
sions. It is this Huygens locality condition which
allows the introduction of higher dimensional ver-
tex algebras [37, 38, 1].

5. Local GCI fields have elliptic thermal correla-
tion functions with respect to the (differences of)
conformal time variables in any even number of
space-time dimensions; the corresponding energy
meanvalues in aGibbs (KMS) state (see e.g. [14])
are expressed as linear combinations of modular
forms [38].
The rest of the paper is organized as follows. In

Sect. 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 the-
ory of scalar fields of dimension d = 2 is singled out,
in which the bifields (and the unit operator) close a
commutator algebra. In Sect. 4 we classify the aris-
ing infinite dimensional Lie algebras L in terms of the
three real division rings F = R, C, H. In Sect. 5 we
formulate the main result of [2] and [3] on the Fock
space representations of the Lie algebra L(F) coupled
to the (dual, in the sense ofHowe [16]) 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 eachargument [34]; we shall call them har-
monic 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 mass-
less field. We shall, therefore, assume d > 1. A basis
{fνi, ν = 0,1, . . . , d −2, i = 1,2} of invariant ampli-
tudes F(s, t) such that

〈0 | V1(x1, x2)V2(x3, x4) | 0〉 =
1

ρ13 ρ24
F(s, t) ,

ρij = x
2
ij + i0x

0
ij , x

2 = x2 − (x0)2 (3.1)
is given by

(u+ − u−)fν1(s, t) =
uν+1+

(1− u+)ν+1
−

uν+1−
(1− u−)ν+1

,

(u+ − u−)fν2(s, t) = (−1)ν(uν+1+ − u
ν+1
− ) , (3.2)

ν = 0,1, . . . , d −2 ,

where u± are the “chiral variables” (2.6);

f01 =
1
t

, f02 =1;

f11 =
1− s − t

t2
, f12 = t − s −1;

f21 =
(1− t)2 − s(2− t)+ s2

t3
, (3.3)

fν2(s, t) =
1
t

fν1

(
s

t
,
1
t

)
fν,i, i = 1,2 corresponding to single pole terms [36]
in the 4-point correlation functions wνi(x1, . . . , x4) =
fνi(s, t)/ρ13 ρ24:

w01 =
1

ρ14 ρ23
,

w02 =
1

ρ13 ρ24
;

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Acta Polytechnica Vol. 50 No. 3/2010

w11 =
ρ13 ρ24 − ρ14 ρ23 − ρ12 ρ34

ρ214 ρ
2
23

,

w12 =
ρ14 ρ23 − ρ13 ρ24 − ρ12 ρ34

ρ213 ρ
2
24

;

w21 =
(ρ13 ρ24 − ρ14 ρ23)2

ρ314 ρ
3
23

−

ρ12 ρ34 (2ρ13 ρ24 − ρ14 ρ23)+ ρ212 ρ234
ρ314 ρ

3
23

,

w22 =
(ρ14 ρ23 − ρ13 ρ24)2

ρ313 ρ
3
24

− (3.4)

ρ12 ρ34 (2ρ14 ρ23 − ρ13 ρ24)+ ρ212 ρ234
ρ313 ρ

3
24

.

We have wν2 = P34wν1(= P12wν1) where Pij stands
for the substitution of the arguments xi and xj.
Clearly, for x1 = x2 (or s = 0, t = 1) only the am-
plitudes f0i contribute to the 4-point function (3.1).
It has been demonstrated in [35] that the lowest an-
gular momentum (�) contribution to fνi 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 jμ (of d = 3) for
ν =1; with the stress energy tensor Tλμ for ν =2. In-
deed, the amplitude fν1 admits an expansion in twist
two2 conformal partial waves β�(s, t) [6] starting with
(for a derivation see [35], Appendix B)

βν(s, t) =
Gν+1(u+)− Gν+1(u−)

u+ − u−
, (3.5)

Gμ(u) = u
μF(μ, μ;2μ;u) .

Remark 3.1 Eqs. (3.2) (3.5) provide examples of solu-
tions of the d’Alambert equation in any of the argu-
ments xi, i = 1,2,3,4. In fact, the general conformal
covariant (of dimension 1 in each argument) such so-
lution has the formof 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 con-
formal partial wave is a transcendental function (like
(3.5)) the sum of all such twist two contributions is
the rational function fν1(s, t).

It canbededuced fromtheanalysis of 4-point func-
tions that the commutator algebraof a set of harmonic
bifields generated byOPEof 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 addi-
tion, to be Huygens bilocal [36].

Remark 3.3 Ingeneral, irreducible positive energy rep-
resentations of the (connected) conformal group are
labeled by triples (d;j1, j2) including the dimension
d and the Lorentz weight (j1, j2)(2ji ∈ N), [29]. It

turns out that for d = 3 there is a spin-tensor bi-
field of weight ((3/2;1/2,0),(3/2;0,1/2))whose com-
mutator algebra does close; for d = 4 there is a con-
formal tensor bifield of weight ((2;1,0),(2;0,1)) with
this property. These bifields may be termed left-
handed: they are analogues of chiral 2D currents; a
set of bifields invariant under space reflections would
also involve their righthanded counterparts (ofweights
((3/2;0,1/2),(3/2;1/2,0)) and ((2;0,1),(2;1,0)), re-
spectively).

4 Infinite dimensional Lie
algebras and real division
rings

Our starting point is the following result of [36].

Proposition 4.1. The harmonic bilocal fields V aris-
ing 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:

[VM1(x1, x2), VM2(x3, x4)] =

Δ13VtM1M2(x2, x4)+Δ24VM1 tM2(x1, x3)+

Δ23VM1M2(x1, x4)+Δ14VM2M1(x3, x2)+

tr(M1M2)Δ12,34 + tr(
tM1M2)Δ12,43 ; (4.1)

here Δij is the free field commutator, Δij := Δ
+
ij −

Δ+ji, and Δ12,ij = Δ
+
1i Δ

+
2j − Δ

+
i1Δ

+
j2 where Δ

+
ij =

Δ+(xi − xj) is the 2-point Wightman function of a
free massless scalar field.

Wecall the set of bilocal fields closedunder theCR
(4.1) aLie system. The types of Lie systems are deter-
mined by the corresponding t-algebras – i.e., real asso-
ciative 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 im-
plies 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 alge-
bra 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 [27] is richer

2The 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.

57



Acta Polytechnica Vol. 50 No. 3/2010

but less popular than the complex one) that its com-
mutant M′ in M at(L, R) coincides with one of the
three real division rings (or not necessarily commu-
tative 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 representations3.
It was proven, first in the theory of a single scalar

field φ (of dimension two) [33], and eventually for an
arbitrary set of such fields [36], that the bilocal fields
VM can be written as linear combinations of normal
products of free massless scalar fields ϕi(x):

VM(x1, x2)=
L∑

i,j=1

M ij : ϕi(x1)ϕj(x2) : . (4.3)

For each of the above types of Lie systems VM has a
canonical form, namely

R : V (x1, x2) =
N∑

i=1

: ϕi(x1)ϕi(x2) : ,

C : W(x1, x2) =
N∑

j=1

: ϕ∗j(x1)ϕj(x2) : ,

H : Y (x1, x2) =
N∑

m=1

: ϕ+m(x1)ϕm(x2) : (4.4)

where ϕi are real, ϕj are complex, and ϕm are quater-
nionic valued fields (corresponding to (3.2) with L =
N,2N, and 4N, respectively). We shall denote the as-
sociated infinite dimensional Lie algebra by L(F), F =
R, C, or H.

Remark 4.1 Wenote that thequaternions (represented
by 4×4 realmatrices) appear both in the definition of
Y – i.e., of the matrix algebra M, and of its commu-
tant M′, the two mutually commuting sets of imagi-
nary quaternionic units �i and rj corresponding to the
splitting of the Lie algebra so(4) of real skew- sym-
metric 4×4 matrices into a direct sum of “a left and
a right” so(3) Lie subalgebras:

�1 = σ3 ⊗ � , �2 = � ⊗1 , �3 = �1�2 = σ1 ⊗ � ,
(�j)αβ = δα0δjβ − δαj δ0β − ε0jαβ ,
α, β =0,1,2,3 , j =1,2,3 ;

r1 = � ⊗ σ3 , r2 =1⊗ � , r3 = r1r2 = � ⊗ σ1 (4.5)

where σk are the Pauli matrices, � = iσ2, εμναβ is

the totally antisymmetric Levi-Civita tensor normal-
ized by ε0123 =1. We have

Y (x1, x2) = V0(x1, x2)1+ V1(x1, x2)�1 +

V2(x1, x2)�2 + V3(x1, x2)�3 =

Y (x2, x1)
+

(�+i = −�i , [�i, rj] = 0);

Vκ(x1, x2) =
N∑

m=1

: ϕαm(x1)(�κ)αβ ϕ
β
m(x2) :, (4.6)

�0 =1 .

In order to determine the Lie algebra correspond-
ing 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 conformalLie algebra (most
importantly, the conformal Hamiltonian H) belong to
it. The basis, say (Xmn) where m, n are multiindices,
corresponds to the expansion [42] of a free massless
scalar field ϕ in creation and annihilation operators of
fixed energy states

ϕ(z)=
∞∑

�=0

(�+1)2∑
μ=1

((z2)−�−1ϕ�+1,μ + ϕ−�−1,μ)h�μ(z), (4.7)

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
M̄). The generators of the conformal Lie algebra
su(2,2) are expressed as infinite sums in Xmn with
a finite number of diagonals (cf. Appendix B to [2]).
The requirement su(2,2)⊂ L thus restricts the topol-
ogy of L implying that the last (c-number) term in
(4.1) gives rise to a non-trivial central extension of L.
The analysis of [2], [3] yields the following

Proposition 4.2 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 normal-
ized central charge coincides with the positive inte-
ger N.

3Finite dimensional simple Lie groups G with this property have been extensively studied by mathematicians (for a review and
references – see [9]); for an extension to the infinite dimensional case – see [40]. 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 [30] (see also earlier work – in particular by Günaydin – cited there).

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Acta Polytechnica Vol. 50 No. 3/2010

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 aregenerated in a theoryofGCI scalarfields
of dimension d = 2 and correspond to the three real
division rings F (Proposition 4.2). For an integer cen-
tral charge N they admit a free field realizationof 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) , (5.1)

U(N, H) = Sp(2N)(= U Sp(2N)) .

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 multi-
plicities) in the Fock space F of N dimRF free her-
mitean 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 cor-
respondence 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
[2] (the proof of (i) is already contained in [33]); the
proof for F = H is given in [3].

Remark 5.1 Theorem 5.1 is also valid – and its proof
becomes technically simpler – for a 2-dimensional chi-
ral theory (in which the local fields are functions of
a single complex variable). For F = C the represen-
tation theory of the resulting infinite dimensional Lie
algebra u(∞, ∞) is then essentially equivalent to that
of the vertex algebra W1+∞ studied in [22] (see the
introduction to [2] for a more precise comparison).

Theorem 5.1 provides a link between two paral-
lel developments, one in the study of highest weight
modules of reductive Lie groups (and of related dual
pairs – see Sect. 1.1) [24, 18, 9, 40] (and [16, 17]), the
other in thework ofHaag-Doplicher-Roberts [14, 8] on
the theory of (global) gauge groups and superselection
sectors – see Sect. 1.2. (They both originate – in the
talks of IrvingSegalandRudolfHaag, respectively–at
the same Lille 1957 conference onmathematical prob-
lems in quantum field theory). Albeit the settings are
not equivalent the resultsmatch. The observable alge-
bra (in our case, the commutator algebra generatedby
the set of bilocal fields VM) 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 Sections 1 and 4
of [2].)

The infinite dimensional Lie algebra L(F) and the
compact gauge group U(N, F) appear as a rather spe-
cial (limit-) case of a dual pair in the sense of Howe
[16], [17]. It would be interesting to explore whether
other (inequivalent) pairs would appear in the study
of commutator algebras of (spin)tensor bifields (dis-
cussed 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 [13]).

Acknowledgement

It is a pleasure to thank my coauthorsBojkoBakalov,
Nikolay M. Nikolov and Karl-Henning Rehren: all re-
sults (reported in Sects. 3–5) of this paper have been
obtained in collaborationwith them. I thank Cestmir
Burdik for invitingme to talk at themeeting “Selected
Topics inMathematical andParticlePhysics”,Prague,
5–7 May 2009, dedicated to the 70th birthday of Jiri
Niederle. I also acknowledge a partial support from
the BulgarianNationalCouncil for ScientificResearch
under contracts Ph-1406 and DO-02-257.

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Ivan Todorov
E-mail: todorov@inrne.bas.bg
Institute for Nuclear Research and Nuclear Energy,
TsarigradskoChaussee 72
BG-1784 Sofia, Bulgaria

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