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Advances in Mathematical Physics
Volume 2010 (2010), Article ID 509538, 12 pages
Gauge Symmetry and Howe Duality in 4D Conformal Field Theory Models
Institute for Nuclear Research and Nuclear Energy, Tsarigradsko Chaussee 72, 1784 Sofia, Bulgaria
Received 17 September 2009; Accepted 28 October 2009
Academic Editor: Richard Kerner
Copyright © 2010 Ivan Todorov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 [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  implies the Huygens principle (3.6) and rationality of correlation functions . 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 and a commuting with its compact Lie group . The state space splits into a direct sum of irreducible modules, so that each irreducible representation (IR) of appears with a multiplicity equal to the dimension of an associated IR of . The pair 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 . 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  is a (highly structured) concept that plays a unifying role in such widely different topics as Weil's metaplectic group approach [14, 15] to 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 , among others (in physics).
Howe begins in  with a -dimensional real symplectic manifold where is spanned by symbols , called annihilation operators and is spanned by their conjugate, the creation operators satisfying the canonical commutation relations (CCR)
The commutator of two elements of the real vector space being a real number it defines a (nondegenerate, skew-symmetric) bilinear form on it which vanishes on and on separately and for which appears as the dual space to (the space of linear functionals on ). The real symplectic Lie algebra spanned by antihermitean quadratic combinations of and acts by commutators on preserving its reality and the above bilinear form. This action extends to the Fock space (unitary, irreducible) representation of the CCR. It is, however, only exponentiated to the double cover of , the metaplectic group (that is not a matrix group, i.e., has no faithful finite-dimensional representation; we can view its Fock space, called by Howe  oscillator representation as the defining one). Two subgroups and of are said to form a (reductive) dual pair if they act reductively on (that is automatic for a unitary representation like the one considered here) and each of them is the full centralizer of the other in . The oscillator representation of 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. . 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 ). It took over 20 years and the courage and dedication of Haag's (then) young collaborators, Doplicher and Roberts , to carry out this program to completion. They proved that all superselection sectors of a local QFT with a mass gap are contained in the vacuum representation of a canonically associated (graded local) field extension , and they are in a one-to-one correspondence with the unitary irreducible representations (IRs) of a compact gauge group of internal symmetries of , so that consists of the fixed points of under . The pair in 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 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 , according to the Liouville theorem, the quantum mechanical conformal group in space-time dimensions is finite (in fact, )-dimensional. It is (a covering of) the spin group Spin .(2)The representation theory of affine Kac-Moody algebras  and of the Virasoro algebra  is playing a crucial role in constructing soluble models of (rational) CFT. There are, on the other hand, no local Lie fields in higher dimensions. After an inconclusive attempt by Robinson  (criticized in ) this was proven for scalar fields by Baumann .(3)The light cone in two dimensions is the direct product of two light rays. This geometric fact is the basis of splitting variables into right- and left-movers' chiral variables. No such splitting seems to be available in higher dimensions.(4)There are chiral algebras in CFT whose local currents satisfy the axioms of vertex algebras (As a mathematical subject vertex algebras were anticipated by Frenkel and Kac  and introduced by Borcherds ; 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 CFT on a torus have elliptic correlation functions , 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 CFT does have, when properly understood, a higher-dimensional counterpart.
(1) The presence of a conformal anomaly (a nonzero Virasoro central charge ) tells us that the infinite conformal symmetry in dimension is, in fact, broken. What is actually used in 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 , infinite-dimensional Lie algebras are generated by bifields which naturally arise in the OPE of a (finite) set of (say, hermitean, scalar) local fields of dimension :
where are defined as (infinite) sums of OPE contributions of (twist two) conserved local tensor currents (and the real symmetric matrix 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 , is given by Thus and are the compact picture counterparts of “left” and “right” chiral variables (see ). The factorization of cross-ratios into chiral parts again has a higher-dimensional analogue : 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 . 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 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., ) are expressed as linear combinations of modular forms .
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 and the leading term in its conformal partial wave expansion. The case of a theory of scalar fields of dimension 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 in terms of the three real division rings . In Section 5 we formulate the main result of [6, 7] on the Fock space representations of the Lie algebra coupled to the (dual, in the sense of Howe ) compact gauge group where is the central charge of .
3. Four-Point Functions and Conformal Partial Wave Expansions
The conformal bifields of dimension which arise in the OPE (2.2) (as sums of integrals of conserved tensor currents) satisfy the d'Alembert equation in each argument ; we will call them harmonic bifields. Their correlation functions depend on the dimension of the local scalar fields . For one is actually dealing with the theory of a free massless field. We will, therefore, assume . A basis of invariant amplitudes such that
is given by
where are the “chiral variables" (2.6)
, corresponding to single pole terms  in the 4-point correlation functions :
We have where stands for the substitution of the arguments and . Clearly, for (or , ) only the amplitudes contribute to the 4-point function (3.1). It has been demonstrated in  that the lowest angular momentum () contribution to corresponds to . The corresponding OPE of the bifield starts with a local scalar field of dimension for , with a conserved current (of ) for , with the stress energy tensor for . Indeed, the amplitude 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 have twist two.) conformal partial waves  starting with (for a derivation see [4, Appendix ])
Remark 3.1. Equations (3.2) and (3.5) provide examples of solutions of the d'Alambert equation in any of the arguments . In fact, the general conformal covariant (of dimension in each argument) such solution has the form of the right-hand side of (3.1) with
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 .
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 can only close on the 's and the unit operator for . In this case the bifields are proven, in addition, to be Huygens bilocal .
Remark 3.3. In general, irreducible positive energy representations of the (connected) conformal group are labeled by triples including the dimension and the Lorentz weight , . It turns out that for there is a spin-tensor bifield of weight whose commutator algebra does close; for there is a conformal tensor bifield of weight with this property. These bifields may be termed lefthanded. They are analogues of chiral currents; a set of bifields invariant under space reflections would also involve their righthanded counterparts (of weights and , resp.).
4. Infinite-Dimensional Lie Algebras and Real Division Rings
Our starting point is the following result of .
Proposition 4.1. The harmonic bilocal fields arising in the OPEs of a (finite) set of local hermitean scalar fields of dimension can be labeled by the elements of an unital algebra of real matrices closed under transposition, , in such a way that the following commutation relations (CR) hold: here is the free field commutator, , and where is the -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 -algebras, that is, real associative matrix algebras closed under transposition. We first observe that each such can be equipped with a Frobenius inner product
which is symmetric, positive definite, and has the property . This implies that for every right ideal its orthogonal complement is again a right ideal while its transposed is a left ideal. Therefore, is a semisimple algebra so that every module over is a direct sum of irreducible modules.
Let now be irreducible. It then follows from the Schur's lemma (whose real version  is richer but less popular than the complex one) that its commutant in coincides with one of the three real division rings (or not necessarily commutative fields): the fields of real and complex numbers and , and the noncommutative division ring 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 with this property have been extensively studied by mathematicians (for a review and references, see ); for an extension to the infinite-dimensional case, see . If is the centre of and is a closed maximal subgroup of such that is compact then is characterized by the property that is a hermitean symmetric pair. Such groups give rise to simple space-time symmetries in the sense of  (see also earlier work—in particular by Günaydin—cited there).
It was proven, first in the theory of a single scalar field (of dimension two) , and eventually for an arbitrary set of such fields , that the bilocal fields can be written as linear combinations of normal products of free massless scalar fields :
For each of the above types of Lie systems has a canonical form, namely,
where are real, are complex, and are quaternionic valued fields (corresponding to (3.2) with , , and , resp.). We will denote the associated infinite-dimensional Lie algebra by , , , or .
Remark 4.2. We note that the quaternions (represented by real matrices) appear both in the definition of —that is, of the matrix algebra , and of its commutant , the two mutually commuting sets of imaginary quaternionic units and corresponding to the splitting of the Lie algebra of real skew-symmetric matrices into a direct sum of “a left and a right” Lie subalgebras: where are the Pauli matrices, , is the totally antisymmetric Levi-Civita tensor normalized by . We have
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 ) belong to it. The basis, say where are multi-indices, corresponds to the expansion  of a free massless scalar field in creation and annihilation operators of fixed energy states
where form a basis of homogeneous harmonic polynomials of degree in the complex 4-vector (of the parametrization (2.3) of ). The generators of the conformal Lie algebra are expressed as infinite sums in with a finite number of diagonals (cf. Appendix in ). The requirement thus restricts the topology of implying that the last (c-number) term in (4.1) gives rise to a nontrivial central extension of .
Proposition 4.3. The Lie algebras , are -parameter central extensions of appropriate completions of the following inductive limits of matrix algebras: In the free field realization (4.4) the suitably normalized central charge coincides with the positive integer .
5. Fock Space Representation of the Dual Pair
To summarize the discussion of the last section, there are three infinite-dimensional irreducible Lie algebras, that are generated in a theory of GCI scalar fields of dimension and correspond to the three real division rings (Proposition 4.3). For an integer central charge they admit a free field realization of type (4.3) and a Fock space representation with (compact) gauge group :
It is remarkable that this result holds in general.
(i) In any unitary irreducible positive energy representation (UIPER) of the central charge is a positive integer.
(ii) All UIPERs of are realized (with multiplicities) in the Fock space of free hermitean massless scalar fields.
(iii) The ground states of equivalent UIPERs in form irreducible representations of the gauge group (5.1). This establishes a one-to-one correspondence between UIPERs of occurring in the Fock space and the irreducible representations of .
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 the representation theory of the resulting infinite-dimensional Lie algebra is then essentially equivalent to that of the vertex algebra studied in  (see the introduction in  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  and Haag  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 ) 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 and ].)
The infinite-dimensional Lie algebra and the compact gauge group 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 of the series of Lie superalgebras studied in ).
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.
- N. M. Nikolov and I. T. Todorov, “Rationality of conformally invariant local correlation functions on compactified Minkowski space,” Communications in Mathematical Physics, vol. 218, no. 2, pp. 417–436, 2001.
- N. M. Nikolov, Ya. S. Stanev, and I. T. Todorov, “Four-dimensional conformal field theory models with rational correlation functions,” Journal of Physics A, vol. 35, no. 12, pp. 2985–3007, 2002.
- N. M. Nikolov, Ya. S. Stanev, and I. T. Todorov, “Globally conformal invariant gauge field theory with rational correlation functions,” Nuclear Physics B, vol. 670, no. 3, pp. 373–400, 2003.
- N. M. Nikolov, K.-H. Rehren, and I. T. Todorov, “Partial wave expansion and Wightman positivity in conformal field theory,” Nuclear Physics B, vol. 722, no. 3, pp. 266–296, 2005.
- N. M. Nikolov, K.-H. Rehren, and I. Todorov, “Harmonic bilocal fields generated by globally conformal invariant scalar fields,” Communications in Mathematical Physics, vol. 279, no. 1, pp. 225–250, 2008.
- B. Bakalov, N. M. Nikolov, K.-H. Rehren, and I. Todorov, “Unitary positive-energy representations of scalar bilocal quantum fields,” Communications in Mathematical Physics, vol. 271, no. 1, pp. 223–246, 2007.
- B. Bakalov, N. M. Nikolov, K.-H. Rehren, and I. Todorov, “Infinite-dimensional Lie algebras in 4D conformal quantum field theory,” Journal of Physics A, vol. 41, no. 19, Article ID 194002, 12 pages, 2008.
- R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That, Princeton Landmarks in Physics, Princeton University Press, Princeton, NJ, USA, 2000.
- R. Howe, “Dual pairs in physics: harmonic oscillators, photons, electrons, and singletons,” in Applications of Group Theory in Physics and Mathematical Physics, M. Flato, P. Sally, and G. Zuckerman, Eds., vol. 21 of Lectures in Applied Mathematics, pp. 179–207, American Mathematical Society, Providence, RI, USA, 1985.
- R. Howe, “Transcending classical invariant theory,” Journal of the American Mathematical Society, vol. 2, no. 3, pp. 535–552, 1989.
- S. Doplicher and J. E. Roberts, “Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics,” Communications in Mathematical Physics, vol. 131, no. 1, pp. 51–107, 1990.
- R. Haag, Local Quantum Physics: Fields, Particles, Algebras, Texts and Monographs in Physics, Springer, Berlin, Germany, 1992.
- R. Howe, “On the role of the Heisenberg group in harmonic analysis,” Bulletin of the American Mathematical Society, vol. 3, no. 2, pp. 821–843, 1980.
- A. Weil, “Sur certains groupes d'opérateurs unitaires,” Acta Mathematica, vol. 111, no. 1, pp. 143–211, 1964.
- A. Weil, “Sur la formule de Siegel dans la théorie des groupes classiques,” Acta Mathematica, vol. 113, no. 1, pp. 1–87, 1965.
- G. Mack and I. T. Todorov, “Irreducibility of the ladder representations of when restricted to the Poincaré subgroup,” Journal of Mathematical Physics, vol. 10, pp. 2078–2085, 1969.
- A. Joseph, “Minimal realizations and spectrum generating algebras,” Communications in Mathematical Physics, vol. 36, pp. 325–338, 1974.
- A. Joseph, “The minimal orbit in a simple Lie algebra and its associated maximal ideal,” Annales Scientifiques de l'École Normale Supérieure Série 4, vol. 9, no. 1, pp. 1–29, 1976.
- D. Kazhdan, B. Pioline, and A. Waldron, “Minimal representations, spherical vectors and exceptional theta series,” Communications in Mathematical Physics, vol. 226, no. 1, pp. 1–40, 2002.
- M. Günaydin and O. Pavlyk, “A unified approach to the minimal unitary realizations of noncompact groups and supergroups,” Journal of High Energy Physics, vol. 2006, no. 9, article 050, 2006.
- T. Kobayashi and G. Mano, “The Schrödinger model for the minimal representation of the indefinite orthogonal group ,” to appear in Memoirs of the American Mathematical Society.
- G. C. Wick, A. S. Wightman, and E. P. Wigner, “The intrinsic parity of elementary particles,” Physical Review, vol. 88, pp. 101–105, 1952.
- V. G. Kac, Infinite-Dimensional Lie Algebras, Cambridge University Press, Cambridge, UK, 3rd edition, 1990.
- V. G. Kac and A. K. Raina, Bombay Lectures on Highest Weight Representations of Infinite-Dimensional Lie Algebras, vol. 2 of Advanced Series in Mathematical Physics, World Scientific, Teaneck, NJ, USA, 1987.
- D. W. Robinson, “On a soluble model of relativistic field theory,” Physics Letters B, vol. 9, pp. 189–190, 1964.
- J. H. Lowenstein, “The existence of scalar Lie fields,” Communications in Mathematical Physics, vol. 6, pp. 49–60, 1967.
- K. Baumann, “There are no scalar Lie fields in three or more dimensional space-time,” Communications in Mathematical Physics, vol. 47, no. 1, pp. 69–74, 1976.
- I. B. Frenkel and V. G. Kac, “Basic representations of affine Lie algebras and dual resonance models,” Inventiones Mathematicae, vol. 62, no. 1, pp. 23–66, 1980.
- R. E. Borcherds, “Vertex algebras, Kac-Moody algebras, and the Monster,” Proceedings of the National Academy of Sciences of the United States of America, vol. 83, no. 10, pp. 3068–3071, 1986.
- V. Kac, Vertex Algebras for Beginners, vol. 10 of University Lecture Series, American Mathematical Society, Providence, RI, USA, 2nd edition, 1998.
- E. Frenkel and D. Ben-Zvi, Vertex Algebras and Algebraic Curves, vol. 88 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 2nd edition, 2004.
- Y. Zhu, “Modular invariance of characters of vertex operator algebras,” Journal of the American Mathematical Society, vol. 9, no. 1, pp. 237–302, 1996.
- A. Uhlmann, “The closure of Minkowski space,” Acta Physica Polonica, vol. 24, pp. 295–296, 1963.
- I. T. Todorov, “Infinite-dimensional Lie algebras in conformal QFT models,” in Conformal Groups and Related Symmetries: Physical Results and Mathematical Background, A. O. Barut and H.-D. Doebner, Eds., vol. 261 of Lecture Notes in Physics, pp. 387–443, Springer, Berlin, Germany, 1986.
- N. M. Nikolov, “Vertex algebras in higher dimensions and globally conformal invariant quantum field theory,” Communications in Mathematical Physics, vol. 253, no. 2, pp. 283–322, 2005.
- N. M. Nikolov and I. T. Todorov, “Elliptic thermal correlation functions and modular forms in a globally conformal invariant QFT,” Reviews in Mathematical Physics, vol. 17, no. 6, pp. 613–667, 2005.
- F. A. Dolan and H. Osborn, “Conformal four point functions and the operator product expansion,” Nuclear Physics B, vol. 599, no. 1-2, pp. 459–496, 2001.
- B. Bakalov and N. M. Nikolov, “Jacobi identity for vertex algebras in higher dimensions,” Journal of Mathematical Physics, vol. 47, no. 5, Article ID 053505, 30 pages, 2006.
- V. K. Dobrev, G. Mack, V. B. Petkova, S. G. Petrova, and I. T. Todorov, Harmonic Analysis on the n-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory, vol. 63 of Lecture Notes in Physics, Springer, Berlin, Germany, 1977.
- G. Mack, “All unitary ray representations of the conformal group with positive energy,” Communications in Mathematical Physics, vol. 55, no. 1, pp. 1–28, 1977.
- S. Lang, Algebra, vol. 211 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 3rd edition, 2002.
- T. Enright, R. Howe, and N. Wallach, “A classification of unitary highest weight modules,” in Representation Theory of Reductive Groups, vol. 40 of Progress in Mathematics, pp. 97–143, Birkhäuser, Boston, Mass, USA, 1983.
- M. U. Schmidt, “Lowest weight representations of some infinite-dimensional groups on Fock spaces,” Acta Applicandae Mathematicae, vol. 18, no. 1, pp. 59–84, 1990.
- G. Mack and M. de Riese, “Simple space-time symmetries: generalizing conformal field theory,” Journal of Mathematical Physics, vol. 48, no. 5, Article ID 052304, 21 pages, 2007.
- V. Kac and A. Radul, “Representation theory of the vertex algebra ,” Transformation Groups, vol. 1, no. 1-2, pp. 41–70, 1996.
- M. Kashiwara and M. Vergne, “On the Segal-Shale-Weil representations and harmonic polynomials,” Inventiones Mathematicae, vol. 44, no. 1, pp. 1–47, 1978.
- H. P. Jakobsen, “The last possible place of unitarity for certain highest weight modules,” Mathematische Annalen, vol. 256, no. 4, pp. 439–447, 1981.
- M. Günaydin and R. J. Scalise, “Unitary lowest weight representations of the noncompact supergroup ,” Journal of Mathematical Physics, vol. 32, no. 3, pp. 599–606, 1991.