Abstract
We construct a family of representationsK ξ,w of the Neveu-Schwarz and Ramond algebras, which generalize the Fock representations of the Virasoro algebra. We show that the representationsK ξ,w are intertwined by a vertex operator.
The above results are used to give the proof of the conjectured formulas for the determinant of the contravariant form on the highest weight representations of the Neveu-Schwarz and Ramond algebras. Further results on the representation theory of the latter are derived from the determinant formulas.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Corwin, L., Ne'eman, Y., Sternberg, S.: Graded Lie algebras in mathematics and physics (Bose-Fermi symmetry). Rev. Mod. Phys.47, 573–603 (1975)
Feigin, B.L., Fuchs, D.B.: Funkts. Anal. Prilozh.16, 47–63 (1982)
Feingold, A., Frenkel, I.B.: Classical affine algebras. Adv. Math. (to appear)
Frenkel, I.B.: Spinor representations of affine Lie algebras. Proc. Natl. Acad. Sci. USA77, 6303–6306 (1980)
Frenkel, I.B.: Private communication to A. Meurman
Frenkel, I.B., Kac, V.G.: Invent. Math.62, 23–66 (1980)
Friedan, D.: Private communication to A. Rocha-Caridi
Friedan, D., Qiu, Z., Shenker, S.H.: Superconformal invariance in two dimensions and the tricritical Ising model. Phys. Lett.151 B, 37 (1985)
Garland, H., Lepowsky, J.: Invent. Math.34, 37–76 (1976)
Garland, H.: Publ. Math. Inst. Hautes Etud. Sci.52, (1980)
Goodman, R., Wallach, N.R.: Projective Unitary Positive-Energy Representations of Diff(S 1), preprint (1984)
Goodman, R., Wallach, N.R.: J. Reine Angew. Math.347, 69–133 (1984)
Kac, V.G.: Proceedings of the International Congress of Mathematicians, Helsinki (1978)
Kac, V.G.: Group theoretical methods in physics. In: Lecture Notes in Physics. Beiglböck, W., Böhm, A., Takasugi, E. (eds.), Vol. 94, p. 441. Berlin, Heidelberg, New York: Springer 1979
Kac, V.G.: Adv. Math.26, 8–96 (1977)
Kac, V.G.: Differential geometrical methods in mathematical physics II. Lecture Notes in Mathematics. Bleuler, K., Petri, H., Reetz, A. (eds.), Vol. 676, pp. 597–626. Berlin, Heidelberg, New York: Springer 1978
Kac, V.G., Peterson, D.: Spin and wedge representations of infinite-dimensional Lie algebras and groups. Proc. Natl. Acad. Sci. USA78, 3308–3312 (1981)
Milnor, J., Moore, J.: Ann. Math.81, 211–264 (1965)
Neveu, P., 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)
Retakh, V.S., Feigin, B.L., Usp. Mat. Nauk.37, 233–234 (1982)
Rocha-Caridi, A.: Vertex operators in mathematics and physics. Proceedings of a Conference, November 10–17, 1983. In: Mathematical Sciences Research Institute Publications. Lepowsky, J., Mandelstam, S., Singer, I.M. (eds.), Vol. 3. Berlin, Heidelberg, New York: Springer-Verlag 1984
Rocha-Caridi, A., Wallach, N.R.: Invent. Math.72, 57–75 (1983)
Rocha-Caridi, A., Wallach, N.R.: Math. Z.180, 151–177 (1982)
Segal, G.: Unitary representations of some infinite dimensional groups. Commun. Math. Phys.80, 301–342 (1981)
Shapovalov, N.N.: Funct. Anal. Appl.6, 307–312 (1972)
Thorn, C.B.: Computing the Kac determinant using dual model techniques and more about the no-ghost theorem. Nucl. Phys. B248, 551 (1984)
Author information
Authors and Affiliations
Additional information
Communicated by S.-T. Yau
Partially supported by the National Science Foundation through the Mathematical Sciences Research Institute
Partially supported by NSF grant MCS-8201260
Rights and permissions
About this article
Cite this article
Meurman, A., Rocha-Caridi, A. Highest weight representations of the Neveu-Schwarz and Ramond algebras. Commun.Math. Phys. 107, 263–294 (1986). https://doi.org/10.1007/BF01209395
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01209395