Abstract
A natural supersymmetric extension\((\widehat{dG})_\kappa\) is defined of the current (= affine Kac-Moody Lie) algebra\(\widehat{dG}\); it corresponds to a superconformal and chiral invariant 2-dimensional quantum field theory (QFT), and hence appears as an ingredient in superstring models. All unitary irreducible positive energy representations of\((\widehat{dG})_\kappa\) are constructed. They extend to unitary representations of the semidirect sumS κ(G) of\((\widehat{dG})_\kappa\) with the superconformal algebra of Neveu-Schwarz, for\(\kappa = \frac{1}{2}\), or of Ramond, for κ=0.
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References
Witten, E.: Non-abelian bosonization in two dimensions. Commun. Math. Phys.92, 455–472 (1984)
Knizhnik, V. G., Zamolodchikov, A. B.: Current algebra and Wess-Zumino model in two dimensions. Nucl. Phys.B247, 83–103 (1984)
Todorov, I. T.: Current algebra approach to conformal invariant two-dimensional models. Phys. Lett.153B, 77–81 (1985); Infinite Lie algebras in 2-dimensional conformal field theory, ISAS Trieste lecture notes 2/85/E.P.
Kac, V. G.: Contravariant form for infinite dimensional Lie algebras and superalgebras. Lecture Notes in Physics,94 Heidelberg, New York, Berlin: Springer, 1979 pp. 441–445; Frenkel, I. B., Kac, V. G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math.62, 23–66 (1980); Frenkel, I. B.: Two constructions of affine Lie algebra representations and Boson-Fermion correspondence in quantum field theory. J. Funct. Anal.44, 259–327 (1981)
Kac, V. G.: Infinite dimensional Lie algebras: An introduction, Boston: Birkhäuser 1983
Kac, V. G., Peterson, D. H.: Spin and wedge representations of infinite dimensional Lie algebras and groups. Proc. Natl. Acad. Sci. USA78, 3308–3312 (1981)
Kac, V. G., Peterson, D. H.: Infinite dimensional Lie algebras, theta functions and modular forms. Adv. Math.53, 125–264 (1984); Goodman, R., Wallach, N. R.: Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle. J. Reine Angew. Math.347, 69–133 (1984), Erratum, ibid.352, 220 (1984)
Goddard, P., Olive, D.: Kac-Moody algebras, conformal symmetry and critical exponents, Nucl. Phys.B257 [FS 14], 226–240 (1985)
Goddard, P., Kent, A., Olive, D.: Virasoro algebras and coset space models. Phys. Lett.152B, 88–93 (1985)
Green, M. B., Schwarz, J. H.: Anomaly cancellation in supersymmetricD = 10 gauge theory and superstring theory. Phys. Lett.149B, 117–122 (1984); Infinity cancellations in SO(32) superstring theory. Phys. Lett.151B, 21–25 (1984)
Witten, E.: Some properties of O(32) superstrings. Phys. Lett.149B, 351–356 (1984)
Gross, D. J., Harvey, J. A., Martinec, E., Rohm, R.: Heterotic string. Phys. Rev. Lett.54, 502–505 (1985); Heterotic string theory I. The free heterotic string. Princeton: Princeton University Press, preprint (1985)
Eichenherr, H.: Minimal operator algebras in superconformal quantum field theory. Phys. Lett.151B, 26–30 (1985); Bershadsky, M. A., Knizhnik, V. G., Teitelman, M. G.: Superconformal symmetry in two dimensions. Phys. Lett.151B, 31–36 (1985)
Friedan, D., Qiu, Z. Shenker, S.: Superconformal invariance in two dimensions and the tricritical Ising model. Phys. Lett.151B, 37–43 (1985)
Belavin, A. A., Polyakov, A. M., Zamolodchikov, A. B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys.B241, 333–380 (1984); Infinite conformal symmetry of critical fluctuations in two dimensions. J. Stat. Phys.34, 763–774 (1984)
Neveu, A., Schwarz, J. H.: Factorizable dual model of pions. Nucl. Phys.B31, 86–112 (1971)
Ramond, P.: Dual theory for free fermions. Phys. Rev.D3, 2415–2418 (1971)
Ferrara, S., Gatto, R., Grillo, A.: Conformal algebra in two-space time dimensions and the Thirring model. Nuovo Cim.12A, 959–968 (1972); Mansuri, F., Nambu, Y.: Gauge conditions in dual resonance models. Phys. Lett.39B, 375–378 (1972); Fubini, S., Hanson, A., Jackiw, R.: New approach to field theory. Phys. Rev.D7, 1732–1760 (1973); Lüscher, M., Mack, G.: The energy momentum tensor of critical quantum field theory in 1 + 1 dimensions (Hamburg 1975) (unpublished)
Sugawara, H.: A field theory of currents. Phys. Rev.170, 1659–1662 (1968); Sommerfield, C.: Currents as dynamical variables. Phys. Rev.176, 2019–2025 (1968); Coleman, S., Gross, D., Jackiw, R.: Fermion avatars of the Sugawara model. Phys. Rev.180, 1359–1366 (1969); Bardakci, K., Halpern, M.: New dual quark models. Phys. Rev.D3, 2493–2506 (1971)
Kac, V. G.: Infinite dimensional Lie algebras and Dedekind's η-function. Funkts. Anal. Prilozh.8, 77–78 (1974) [Engl. transl: Funct. Anal. Appl.8, 68–70 (1974)]; Garland, H.: The arithmetic theory of loop algebras. J. Algebra53, 480–551 (1978); Kac, V. G., Peterson, D. H.: Unitary structure in representations of infinite-dimensional groups and a convexity theorem. Invent. Math.76, 1–14 (1984)
Kac, V. G.: Some problems on infinite-dimensional Lie algebras and their representations. In: Lecture Notes in Mathematics. Vol.933, pp. 117–126, Heidelberg, New York, Berlin: Springer 1982
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Communicated by A. Jaffe
On leave of absence from the Institute for Nuclear Research and Nuclear Energy of the Bulgarian Academy of Sciences, BG-1184 Sofia, Bulgaria
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Kac, V.G., Todorov, I.T. Superconformal current algebras and their unitary representations. Commun.Math. Phys. 102, 337–347 (1985). https://doi.org/10.1007/BF01229384
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DOI: https://doi.org/10.1007/BF01229384