Abstract
We clarify the notion of the DS — generalized Drinfeld-Sokolov — reduction approach to classicalW-algebras. We first strengthen an earlier theorem which showed that ansl(2) embeddingL ⊏G can be associated to every DS reduction. We then use the fact that aW-algebra must have a quasi-primary basis to derive severe restrictions on the possible reductions corresponding to a givesl(2) embedding. In the known DS reductions found to data, for which theW-algebras are denoted byW G L -algebras and are called canonical, the quasi-primary basis corresponds to the highest weights of thesl(2). Here we find some examples of noncanonical DS reductions leading toW-algebras which are direct products ofW G L -algebras and “free field” algebras with conformal weights Δ∈{0, 1/2, 1}. We also show that if the conformal weights of the generators of aW-algebra obtained from DS reduction are nonnegative Δ≥0 (which is the case for all DS reductions known to date), then the Δ≥3/2 subsectors of the weights are necessarily the same as in the correspondingW G L -algebra. These results are consistent with an earlier result by Bowcock and Watts on the spectra ofW-algebras derived by different means. We are led to the conjecture that, up to free fields, the set ofW-algebras with nonnegative spectra Δ>-0 that may be obtained from DS reduction is exhausted by the canonical ones.
Similar content being viewed by others
References
Zamolodchikov, A.B.: Infinite additional symmetries in 2-dimensional conformal quantum field theory. Theor. Math. Phys.,65, 1205–1213 (1985)
Lukyanov, S.L., Fateev, V.A.: Additional symmetries and exactly soluble models in two-dimensional conformal field theory. Sov. Sci. Rev. A. Phys.15, 1–116 (1990)
Bouwknegt, P., Schoutens, K.:W-symmetry in conformal field theory. Phys. Rep.223, 183–276 (1993)
Blumenhagen, R., Flohr, M., Kliem, A., Nahm, W., Recknagel, A., Varnhagen, R.:W-algebras with two and three generators. Nucl. Phys.B361, 255–289 (1991);
Eholzer, W., Honecker, A., Hübel, R.: How complete is the classification ofW-symmetries Phys. Lett.308B, 42–50 (1993)
Kausch, H.G., Watts, G.M.T.: A Study ofW-algebras using Jacobi identities. Nucl. Phys.B354, 740–768 (1991)
Goddard, P., Kent, A., Olive, D.: Virasoro algebras and coset space models. Phys. Lett.152B, 88–92 (1985)
Bais, F.A., Bouwknegt, P., Schoutens, K., Surridge, M.: Extensions of the Virasoro algebra constructed from Kac-Moody algebras by using higher order Casimir invariants. Nucl. Phys.B304, 348–370 (1988); Coset constructions for extended Virasoro algebras. Nucl. Phys.B304, 371–391 (1988)
Bowcock, P., Goddard, P.: Coset constructions and extended conformal algebras. Nucl. Phys.B305, 685–709 (1988)
Bouwknegt, P.: Extended conformal algebras from Kac-Moody algebras. In: Infinite dimensional Lie algebras and Lie groups. Advanced Series in Math. Phys.7, Kac, V.G. (ed.), Singapore: World Scientific 1989
Watts, G.M.T.:W-algebras and coset models. Phys. Lett.245B, 65–71 (1990)
Drinfeld, V.G., Sokolov, V.V.: Lie algebras and equations of Korteweg-de Vries type. J. Sov. Math.30, 1975–2036 (1984)
Fateev, V.A., Lukyanov, S.L.: The models of two dimensional conformal quantum field theory withZ n symmetry. Int. J. Mod. Phys.A3, 507–520 (1988)
Yamagishi, K.: The KP hierarchy and extended Virasoro algebras. Phys. Lett.205B, 466–470 (1988);
Mathieu, P.: Extended classical conformal algebras and the second Hamiltonian structure of Lax equations. Phys. Lett.208B, 101–106 (1988);
Bakas, I.: The Hamiltonian structure of the spin-4 operator algebra. Phys. Lett.213B, 313–318 (1988)
Balog, J., Fehér, L., Forgács, P., O'Raifeartaigh, L., Wipf, A.: Toda theory andW-algebra from a gauged WZNW point of view. Ann. Phys. (N. Y.)203, 76–136 (1990)
Bais, F.A., Tjin, T., van Driel, P.: Covariantly coupled chiral algebras. Nucl. Phys.B357, 632–654 (1991)
Fehér, L., O'Raifeartaigh, L., Ruelle, P., Tsutsui, I., Wipf, A.: Generalized Toda theories andW-algebras associated with integral gradings. Ann. Phys. (N. Y.)213, 1–20 (1992)
Frappat, L., Ragoucy, E., Sorba, P.:W-algebras and superalgebras from constrained WZW models: A group theoretical classification. Commun. Math. Phys.157, 499–548 (1993)
Bershadsky, M., Ooguri, H.: HiddenSL(n) symmetry in conformal field theories. Commun. Math. Phys.126, 49–83 (1989)
Figueroa-O'Farrill, J.M.: On the homological construction of Casimir algebras. Nucl. Phys.B343, 450–466 (1990)
Feigin, B.L., Frenkel, E.: Quantization of the Drinfeld-Sokolov reduction. Phys. Lett.246B, 75–81 (1990)
Frenkel, E., Kac, V.G., Wakimoto, M.: Characters and fusion rules forW-algebras via quantized Drinfeld-Sokolov reduction. Commun. Math. Phys.147, 295–328 (1992)
de Boer, J., Tjin, T.: The relation between quantumW algebras and Lie algebras. Commun. Math. Phys.160, 317–332 (1994)
Sevrin, A., Troost, W.: Extensions of the Virasoro algebra and gauged WZW models. Phys. Lett.315B, 304–310 (1993)
Fehér, L., O'Raifeartaigh, L., Ruelle, P., Tsutsui, I., Wipf, A.: On Hamiltonian reductions of the Wess-Zumino-Novikov-Witten theories. Phys. Rep.222, 1–64 (1992)
Bilal, A., Gervais, J.-L.: Systematic approach to conformal systems with extended Virasoro symmetries. Phys. Lett.206B, 412–420 (1988); Extendedc=∞ conformal systems from classsical Todal field theories. Nucl. Phys.B314, 646–686 (1989); Systematic construction of conformal theories with higher spin Virasoro symmetries. Nucl. Phys.B318, 579–630 (1989)
Saveliev, M.: On some connections and extensions ofW-algebras. Mod. Phys. Lett.A5, 2223–2229 (1990)
Mansfield, P., Spence, B.: Toda theory, the geometry ofW-algebras and minimal models. Nucl. Phys.B362, 294–328 (1991)
Dynkin, E.B.: Semisimple subalgebras of semisimple Lie algebras. Am. Math. Soc. Transl.6 [2], 111–244 (1957)
Polyakov, A.M.: Gauge transformations and diffeomorphisms. Int. J. Mod. Phys.A5, 833–842 (1990)
Bershadsky, M.: Conformal field theories via Hamiltonian reduction. Commun. Math. Phys.139, 71–82 (1991)
Fehér, L., O'Raifeartaigh, L., Ruelle, P., Tsutsui, I.: Rational versus polynomial character ofW l n -algebras. Phys. Lett.283B, 243–251 (1992)
Bowcock, P., Watts, G.M.T.: On the classification of quantumW-algebras. Nucl. Phys.379B, 63–96 (1992)
Goddard, P., Schwimmer, A.: Factoring out free fermions and superconformal algebras. Phys. Lett.214B, 209–214 (1988)
Fehér, L., O'Raifeartaigh, L., Tsutsui, I.: The vacuum preserving Lie algebra of a classicalW-algebra. Phys. Lett.316B, 275–281 (1993)
Beukers, F.: Differential Galois theory. In: From number theory to physics. Waldschmidt, M., Moussa, P., Lucke, J.-M., Itzykson, C. (eds.), Berlin, Heidelberg, New York: Springer 1992
Deckmyn, A., Thielemans, K.: Factoring out free fields. Preprint KUL-TF-93/26 (1993), hepth/9306129
Delduc, F., Frappat, L., Ragoucy, E., Sorba, P., Toppan, F.: RationalW-algebras from composite operators. Phys. Lett.318B, 457 (1993)
Di Francesco, P., Itzykson, C., Zuber, J.-H.: ClassicalW-algebras. Commun. Math. Phys.140, 543–567 (1991)
Bonora, L., Xiong, C.S.: Covariantsl 2 decomposition of thesl n Drinfeld-Sokolov equations and theW n -algebras. Int. J. Mod. Phys.A7, 1507–1525 (1992)
Author information
Authors and Affiliations
Additional information
Communicated by R. H. Dijkgraaf
Rights and permissions
About this article
Cite this article
Fehér, L., O'Raifeartaigh, L., Ruelle, P. et al. On the completeness of the set of classicalW-algebras obtained from DS reductions-algebras obtained from DS reductions. Commun.Math. Phys. 162, 399–431 (1994). https://doi.org/10.1007/BF02102024
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02102024