Skip to main content
SpringerLink
Log in

Quantum group structure in the Fock space resolutions of\(\widehat{sl}(n)\) representations

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We describe a complex of Wakimoto-type Fock space modules for the affine Kac-Moody algebra\(\widehat{sl}(n)\). The intertwining operators that build the complex are obtained from contour integrals of so-called screening operators. We show that a quantum group structure underlies the algebra of screening operators. This observation greatly facilitates the explicit determination of the intertwiners. We conjecture that the complex provides a resolution of an irreducible highest weight module in terms of Fock spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  • [AGS] Alvarez-Gaumé, L., Gomez, C., Sierra, G.: Quantum group interpretation of some conformal field theories. Phys. Lett.220B, 142–152 (1989)

    Google Scholar 

  • [BaGo] Bagger, J., Goulian, M.: Coulomb-gas representation on higher-genus surfaces. Nucl. Phys.B330, 488–508 (1990)

    Article  Google Scholar 

  • [BaRa] Barut, A. O., Raczka, R.: Theory of group representations and applications. Warszawa: Polish Scientific 1980

    Google Scholar 

  • [BBSS1] Bais, F. A., Bouwknegt, P., Schoutens, K. Surridge, M.: Extensions of the Virasoro algebra constructed from Kač-Moody algebras using higher order Casimir invariants. Nucl. Phys.B304, 348–370 (1988)

    Article  Google Scholar 

  • [BBSS2] Bais, F. A., Bouwknegt, P., Schoutens, K., Surridge, M.: Coset construction for extended Virasoro algebras. Nucl. Phys.B304, 371–391 (1988)

    Article  Google Scholar 

  • [BeOo] Bershadsky, M., Ooguri, H.: HiddenSL(n) symmetry in conformal field theory. Commun. Math. Phys.126, 49–84 (1989)

    Google Scholar 

  • [BF] Bernard, D., Felder, G.: Fock representations and BRST cohomology inSL(2) current algebra. Commun. Math. Phys.127, 145–168 (1990)

    Article  Google Scholar 

  • [BGG1] Bernstein, I. N., Gel'fand, I. M., Gel'fand, S. I.: Structure of representations generated by vectors of highest weight. Funct. An. Appl.5, 1–8 (1971)

    Article  Google Scholar 

  • [BGG2] Bernstein, I. N., Gel'fand, I. M., Gel'fand, S. I.: Differential operators on the base affine space and a study ofg-modules. Summer school of the Bolyai János Math. Soc., ed. Gel'fand, I. M., pp. 21–64, New York 1975

  • [Bi1] Bilal, A.: Bosonization ofZ N parafermions andSU(2) N Kač-Moody algebra. Phys. Lett.226B, 272–278 (1989)

    Google Scholar 

  • [Bi2] Bilal, A.: Fusion and braiding in extended conformal theories. CERN-TH.5403/89

  • [Bi3] Bilal, A.: Fusion and braiding inW-algebra extended conformal theories (II); Generalization to chiral screened vertex operators labelled by arbitrary Young tableaux. CERN-TH.5493/89

  • [BMP1] Bouwknegt, P., McCarthy, J., Pilch, K.: Free field realizations of WZNW-models; BRST complex and its quantum group structure. Phys. Lett.234B, 297–303 (1990)

    Google Scholar 

  • [BMP2] Bouwknegt, P., McCarthy, J., Pilch, K.: unpublished

  • [Bo] Bott, R.: Homogeneous vector bundles. Ann. Math.66, 203–248 (1957)

    Google Scholar 

  • [BPZ] Belavin, A. A., Polyakov, A. M., Zamolodchikov, A. B.: Infinite conformal symmetry in two dimensional quantum field theory. Nucl. Phys.B241, 333–380 (1984)

    Article  Google Scholar 

  • [DF1] Dotsenko, VI.S., Fateev, V. A.: Conformal algebra and multipoint correlation functions in 2D statistical models. Nucl. Phys.B240 [FS12], 312–348 (1984)

    Article  Google Scholar 

  • [DF2] Dotsenko, Vl. S., Fateev, V. A.: Four-point correlation functions and the operator algebra in 2D conformal invariant theories with central chargec<1. Nucl. Phys.B251 [FS13], 691–734 (1985)

    Google Scholar 

  • [DM] Deligne, P., Mostow, G. D.: Monodromy of hypergeometric functions and non-lattice integral monodromy. Publ. Math. IHES63, 5–89 (1986)

    Google Scholar 

  • [Dr1] Drinfel'd, V. G.: Hopf algebras and the quantum Yang-Baxter equation. Soc. Math. Dokl.32, 254–258 (1985)

    Google Scholar 

  • [Dr2] Drinfel'd, V. G.: Quantum groups. In: Proc. of the International Congress of Math., Berkeley, 1986

  • [DQ] Distler, J., Quiu, Z. BRS cohomology and a Feigin-Fuchs representation of Kač-Moody and parafermionic theories. Cornell preprint CLNS 89/911

  • [Fe] Felder, G.: BRST approach to minimal models. Nucl. Phys.B317, 215–236 (1989); erratum. Nucl. Phys.B324, 548 (1989)

    Article  Google Scholar 

  • [FeFr1] Feigin, B. L., Frenkel, E. V.: The family of representations of affine Lie algebras. Usp. Mat. Nauk.43, 227–228 (1988); (Russ. Math. Surv.)43, 221–222 (1989)

    Google Scholar 

  • [FeFr2] Feigin, B. L., Frenkel, E. V.: Representations of affine Kač-Moody algebras and bosonization. Moscow preprint, March '89, to appear in the V. Knizhnik memorial volume

  • [FeFr3] Feigin, B. L., Frenkel, E. V.: Representations of affine Kac-Moody algebras, bosonization and resolutions. Moscow preprint, April 1989

  • [FeFr4] Feigin, B. L., Frenkel, E. V.: Affine Kac-Moody algebras and semi-infinite flag manifolds, Moscow preprint, May 1989, to appear in Commun. Math. Phys.

  • [FeFu] Feigin, B. L., Fuchs, D. B.: Representations of the Virasoro algebra. Seminar on supermanifolds no. 5, Leites, D. (ed.)

  • [FFK] Felder, G., Fröhlich, J., Keller, G.: Braid matrices and structure constants for minimal models. Commun. Math. Phys.124, 647–664 (1989)

    Article  Google Scholar 

  • [FL1] Fateev, V. A., Lukyanov, S. L.: The models of two-dimensional conformal quantum field theory withZ n symmetry. Int. J. Mod. Phys.3, 507–520 (1988)

    Article  Google Scholar 

  • [FL2] Fateev, V. A., Lukyanov, S. L.: Additional symmetries and exactly soluble models in two-dimensional field theory, I, II and III, Moscow preprints

  • [FLMS1] Frau, M., Lerda, A., McCarthy, J., Sciuto, S.: Minimal models on Riemann surfaces. Phys. Lett.228B, 205–213 (1989)

    Google Scholar 

  • [FLMS2] Frau, M., Lerda, A., McCarthy, J., Sciuto, S.: Operator formalism and free field representation for minimal models on Riemann surfaces. MIT preprint #1803. Nucl. Phys.B (to appear)

  • [FMS] Friedan, D., Martinec, E., Shenker, S.: Conformal invariance, supersymmetry and string theory. Nucl. Phys.B271, 93–165 (1986)

    Article  Google Scholar 

  • [FS1] Felder, G., Silvotti, R.: Modular covariance of minimal model correlation functions. Commun. Math. Phys.123, 1–15 (1989)

    Article  Google Scholar 

  • [FS2] Felder, G., Silvotti, R.: Free field representation of minimal models on a Riemann surface. Phys. Lett.231B, 411–416 (1989)

    Google Scholar 

  • [FZ1] Fateev, V. A., Zamolodchikov, A. B.: Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in Z N -symmetric statistical systems. Sov. Phys. JETP62, 380–399 (1985)

    Google Scholar 

  • [FZ2] Fateev, V. A.: Zamolodchikov, A. B.: Conformal quantum field theory models in two dimensions having Z3 symmetry. Nucl. Phys.B280 [FS18], 644–660 (1987)

    Article  Google Scholar 

  • [Ge] Gepner, D.: New conformal field theories associated with Lie algebras and their partition functions. Nucl. Phys.B290 [FS20], 10–24 (1987)

    Article  Google Scholar 

  • [GeWi] Gepner, D., Witten, E.: String theory on group manifolds. Nucl. Phys.B278, 493–549 (1986)

    Article  Google Scholar 

  • [Gi] Ginsparg, P.: Applied conformal field theory. HUTP-88/A054

  • [GL] Garland, H., Lepowsky, J.: Lie algebra homology and the Macdonald-Kac formula. Inv. Math.34, 37–76 (1976)

    Article  Google Scholar 

  • [GMM] Gerasimov, A., Marshakov, A., Morozow, A.: Free field representation of parafermions and related coset models. Nucl. Phys.B328, 664–676 (1989)

    Article  Google Scholar 

  • [GMMOS] Gerasimov, A., Marshakov, A., Morozow, A., Olshanetsky, M., Shatashvili, S.: Wess-Zumino-Witten models as a theory of free fields. Moscow ITEP preprint, March '89

  • [It] Ito, K.:N=2 super coulomb gas formalism. UT-Komaba 89-13

  • [ItKa] Ito, K., Kazama, Y.: Feigin-Fuchs representation ofA (1)n affine Lie algebra and the associated parafermionic algebra. UT-Komaba 89-22

  • [Ja] Jantzen, J. C.: Moduln mit einem höchsten Gewicht. Lecture Notes in Mathematics, Vol.750. Berlin, Heidelberg, New York: Springer 1979

    Google Scholar 

  • [Ji] Jimbo, M.: Aq-difference analogue ofU(g) and the Yang-Baxter equation. Lett. Math. Phys.10, 63–69 (1985)

    Article  Google Scholar 

  • [Ka] Kac, V. G.: Infinite dimensional Lie algebas. Cambridge, Cambridge University Press 1985

    Google Scholar 

  • [KaKa] Kac, V. G., Kazhdan, D. A.: Structure of representations with highest weight of infinite dimensional Lie algebras. Adv. Math.34, 97–108 (1979)

    Article  Google Scholar 

  • [Ke] Kempf, G.: Cousin-Grothendieck complex of induced representations. Adv. Math.29, 310–396 (1978)

    Article  Google Scholar 

  • [KV] Kashiwara, M., Vergne, M.: Remarque sur la covariance de certains operateurs differentiels. Lecture Notes in Mathematics Vol.587, pp. 119–137. Berlin, Heidelberg, New York: Springer 1976

    Google Scholar 

  • [Kol] Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. Math.74, 329–387 (1961)

    Google Scholar 

  • [Ko2] Kostant, B.: Verma modules and the existence of quasi-invariant differential operators. In: Non-commutative harmonic analysis. Lecture Notes in Mathematics, Vol.466, 101–128. Berlin, Heidelberg, New York: Springer 1974

    Google Scholar 

  • [Lu1] Lusztig, G.: Quantum deformations of certain simple modules over enveloping algebras. Adv. Math.70, 237–249 (1988)

    Article  Google Scholar 

  • [Lu2] Lusztig, G.: Modular representations and quantum groups. Contemp. Math.82, 59–77 (1989)

    Google Scholar 

  • [Lu3] Lusztig, G.: Finite dimensional Hopf algebra arising from quantum groups. MIT preprint

  • [MR] Moore, G., Reshetikhin, N.: A comment on quantum group symmetry in conformal field theory. Nucl. Phys.B328, 557–574 (1989)

    Article  Google Scholar 

  • [MS] Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys.123, 177–254 (1989)

    Article  Google Scholar 

  • [Ne1] Nemeschansky, D.: Feigin-Fuchs representation of\(\widehat{su}(2)_k\) Kac-Moody algebra. Phys. Lett.224B, 121–124 (1989)

    Google Scholar 

  • [Ne2] Nemeschansky, D.: Feigin-Fuchs representation of string functions. Preprint USC-89-012

  • [No] Novikov, S. P.: The Hamiltonian formalism and a many-valued analogue of Morse theory. Usp. Mat. Nauk.37, 3–49 (1982); (Russ. Math. Surv.37, 1–56 (1982)

    Google Scholar 

  • [Pe] Perelomov, A. M.: Generalized coherent states and their applications. Berlin, Heidelberg, New York: Springer 1986

    Google Scholar 

  • [RCW] Rocha-Caridi, A., Wallach, N. R.: Projective modules over graded Lie algebras I. Math. Z.180, 151–177 (1982)

    Google Scholar 

  • [Ro] Rosso, M.: Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra. Commun. Math. Phys.117, 581–593 (1988)

    Article  Google Scholar 

  • [Sm] Smit, D.-J.: Quantum groups and algebraic geometry in conformal field theory. PhD-thesis, University of Utrecht, 1989

  • [TK1] Tsuchiya, A., Kanie, Y.: Fock space representations of the Virasoro algebra-Intertwining operators-, Publ. RIMS, Kyoto Univ.22, 259–327 (1986)

    Google Scholar 

  • [TK2] Tsuchiya, A., Kanie, Y.: Vertex operators in conformal field theory onP 1 and monodromy representations of braid group. Adv. Stud. Pure Math.16, 297–372 (1988)

    Google Scholar 

  • [Ve] Verma, D. N.: Structure of certain induced representations of complex semi simple Lie algebras. Bull. Am. Math. Soc.74, 160–166 (1968)

    Google Scholar 

  • [Wa] Wakimoto, M.: Fock representations of the affine Lie algebraA (1)1 . Commun. Math. Phys.104, 605–609 (1986)

    Article  Google Scholar 

  • [Wi1] Witten, E.: Non-abelian bosonization in two dimensions. Commun. Math. Phys.92, 455–472 (1984)

    Article  Google Scholar 

  • [Wi2] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351–399 (1989)

    Article  Google Scholar 

  • [WZ] Wess, J., Zumino, B.: Consequences of anomalous Ward identities. Phys. Lett.37B, 95–98 (1971)

    Google Scholar 

  • [Za1] Zamolodchikov, A. B.: Infinite additional symmetries in two dimensional conformal quantum field theory. Theor. Math. Phys.65, 1205–1213 (1986)

    Article  Google Scholar 

  • [Za2] Zamolodchikov, A. B.: Montreal lectures, unpublished

  • [Ze] Zhelobenko, D. B.: Compact Lie groups and their representations. Providence, RI: American Mathematical Society 1973

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Theoretical Physics and Department of Physics, Massachusetts Institute of Technlogy, 02139, Cambridge, MA, USA

    Peter Bouwknegt

  2. Department of Physics, Brandeis University, 02254, Waltham, MA, USA

    Jim McCarthy

  3. Department of Physics, University of Scuthern California, 90089-0484, Los Angeles, CA, USA

    Krzysztof Pilch

Authors
  1. Peter Bouwknegt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jim McCarthy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Krzysztof Pilch
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by N. Reshetikhin

Supported by the U.S. Department of Energy under Contract #DE-AC02-76ER03069.

Supported by the NSF Grant #PHY-88-04561

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bouwknegt, P., McCarthy, J. & Pilch, K. Quantum group structure in the Fock space resolutions of\(\widehat{sl}(n)\) representations. Commun.Math. Phys. 131, 125–155 (1990). https://doi.org/10.1007/BF02097682

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02097682

Keywords

Search

Navigation