Abstract
New representations of affine Lie algebras are constructed using symplectic bosons of the sort that occur naturally in the BRST treatment of fermionic string theories. These representations are shown to have analogous properties to the current algebra representations in terms of free fermion fields, though they do not act in a positive space. In particular, the condition for the Sugawara construction of the Virasoro algebra to equal the free one is the existence of a superalgebra with a quadratic Casimir operator, paralleling the symmetric space theorem for fermionic field constructions. Both results are seen to be particular cases of a more general super-symmetric space theorem, which arises from considering an affinisation of the superalgebras. These algebras are realised in terms of free fermions and symplectic bosons and lead to a super-Sugawara construction of the Virasoro algebra. The conditions for this to equal a Virasoro algebra obtained from the free fields are provided by the super-symmetric space theorem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Goddard, P., Olive, D.: Int. J. Mod. Phys. A1, 303 (1986)
Sugawara, H.: A field theory of currents. Phys. Rev.170, 1659 (1968)
Goddard, P., Olive, D.: Kac-Moody algebras, conformal symmetry and critical exponents. Nucl. Phys. B257 [FS14], 226 (1985)
Goddard, P., Nahm, W., Olive, D.: Symmetric spaces, Sugawara's energy momentum tensor in two dimensions and free fermions. Phys. Lett.160 B, 111 (1985)
Frenkel, I., Kac, V.: Invent. Math.62, 23 (1980); Segal, G.: Unitary representations of some infinite dimensional groups. Commun. Math. Phys.80, 301 (1981)
Feingold, A., Frenkel, I.: Adv. Math.56, 117 (1985)
Ezawa, Z.F., Nakamura, S., Tezuka, A.: Tohoku University preprint TU/86/304
Friedan, D., Martinec, E., Shenker, S.: Conformal invariance, supersymmetry and string theory. Nucl. Phys. B271, 93 (1986)
Kac, V.G.: Adv. Math.26, 8 (1977)
Freund, P.G.O.: Introduction to supersymmetry. Cambridge: Cambridge University Press 1986
Serganova, V.V.: Funct. Anal. Appl.17, 200 (1983) [English translation]
Leites, D.A.: Itogi Nauki i Tekhniki, Ser. Sov. Prob. Mat., Noveishie Dostizheniya,25, 3 (1984) and references therein
Kac, V.: In: Geometrical and topological methods in field theory. Singapore: World Scientific 1987
Gomes, J.F.: The triviality of representations of the Virasoro algebra with vanishing central element andL 0 positive. Phys. Lett.171B, 75 (1986)
Brower, R.C.: Spectrum-generating algebra and no-ghost theorem for the dual model. Phys. Rev. D6, 1655 (1972); Goddard, P., Thorn, C.B.: Compatibility of the dual pomeron with unitarity and the absence of ghosts in the dual resonance model. Phys. Lett.40 B, 235 (1972)
Kawai, T.: Remarks on a class of BRST operators. Phys. Lett. B168, 355 (1986)
Altschuler, D.: University of Geneva preprint UGVA-DPT 1986/10-521
Author information
Authors and Affiliations
Additional information
Communicated by L. Alvarez-Gaumé
Rights and permissions
About this article
Cite this article
Goddard, P., Olive, D. & Waterson, G. Superalgebras, symplectic bosons and the Sugawara construction. Commun.Math. Phys. 112, 591–611 (1987). https://doi.org/10.1007/BF01225374
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01225374