Skip to main content

W-algebras and their representations

  • Conference paper
  • First Online:
Conformal Field Theories and Integrable Models

Part of the book series: Lecture Notes in Physics ((LNP,volume 498))

Abstract

An introduction is given to the basic concepts, construction and representation theory of W-algebras.

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

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H. Awata, M. Fukuma, Y. Matsuo, and S. Odake, Character and determinant formulae of quasifinite representation of the 1+∞ algebra, Commun. Math. Phys. 172 (1995) 377, hep-th/9405093; V. Kac and A. Radul, Representation theory of the vertex algebra W 1+∞, hep-th/9512150.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. F.A. Bais, P. Bouwknegt, K. Schoutens and M. Surridge, Extensions of the Virasoro algebra constructed from Kac-Moody algebras using higher order Casimir invariants, Nucl. Phys. B304 (1988) 348.

    Article  ADS  MathSciNet  Google Scholar 

  3. F.A. Bais, P. Bouwknegt, K. Schoutens and M. Surridge, Coset construction for extended Virasoro algebras, Nucl. Phys. B304 (1988) 371.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Z. Bajnok, Singular vectors of the WA 2 algebra, Phys. Lett. B329 (1994) 225, hep-th/9403032.

    ADS  MathSciNet  Google Scholar 

  5. Z. Bajnok, L. Palla and G. Takács, A 2 Toda theory in reduced WZNW framework and the representations of the W algebra, Nucl. Phys. B385 (1992) 329, hep-th/9206075.

    Article  ADS  Google Scholar 

  6. A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B241 (1984) 333.

    Article  ADS  MathSciNet  Google Scholar 

  7. R. Blumenhagen, M. Flohr, A. Kliem, W. Nahm, A. Recknagel, and R. Varnhagen. W-algebras with two and three generators, Nucl. Phys. B361 (1991) 255.

    Article  ADS  MathSciNet  Google Scholar 

  8. R. Blumenhagen, W. Eholzer, A. Honecker, K. Hornfeck, and R. Hübel, Unifying W-algebras, Phys. Lett. B322 (1994) 51, hep-th/9404113

    ADS  Google Scholar 

  9. R. Blumenhagen, W. Eholzer, A. Honecker, K. Hornfeck, and R. Hübel, Coset realisation of unifying W-algebras, Int. J. Mod. Phys. A10 (1995) 2367, hep-th/9406203

    ADS  Google Scholar 

  10. J. de Boer, L. Feher and A. Honecker, A class of W algebras with infinitely generated classical limit, Nucl. Phys. B420 (1993) 409, hep-th/9312049.

    Google Scholar 

  11. J. de Boer and T. Tjin, The relation between quantum W algebras and Lie algebras, Commun. Math. Phys. 160 (1994) 317, hep-th/9302006

    Article  MATH  ADS  Google Scholar 

  12. P. Bouwknegt, Extended Conformal algebras, Phys. Lett. 207B (1988) 295.

    ADS  MathSciNet  Google Scholar 

  13. P. Bouwknegt and K. Schoutens, W symmetry in conformal field theory, Phys. Rept. 223 (1993) 183.

    Article  ADS  MathSciNet  Google Scholar 

  14. P. Bowcock, Representation theory of a W algebra from generalised DS reduction, Durham preprint DTP-94-5, hep-th/9403157

    Google Scholar 

  15. P. Bowcock and G.M.T. Watts, On the classification of quantum W-algebras, Nucl. Phys. B379 (1992) 63, hep-th/9111062.

    Article  ADS  MathSciNet  Google Scholar 

  16. A. Cappelli, C. Itzykson and J.-B. Zuber, The A-D-E classification of minimal and a (1)1 conformal field theories, Commun. Math. Phys. 113 (1987) 1.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. A. Cappelli, C.A. Trugenberger and G.R. Zemba, Stable hierarchical quantum Hall fluids as W 1+∞ minimal models, Nucl. Phys. B448 (1995) 470, hep-th/9502021; W 1+∞ minimal models and the hierarchy of the quantum Hall effect, Nucl. Phys. Proc. Suppl. 45A (1996) 112.

    Article  ADS  MathSciNet  Google Scholar 

  18. E.B. Dynkin, Transl. Am. Math. Soc. Series 2, 6 (1957) 112.

    Google Scholar 

  19. P. van Driel and K. de Vos, The Kazhdan-Lusztig conjecture for W-algebras, Bonn preprint BONN-TH-95-14, hep-th/9508020.

    Google Scholar 

  20. V.G. Drinfel'd and V.V. Sokolov, Lie algebras and equations of Korteweg-de Vries type, J. Sov. Math. 30 (1985) 1975.

    Article  MATH  Google Scholar 

  21. W. Eholzer, M. Flohr, A. Honecker, R. Hübel, W. Nahm and R. Varnhagen, Representations of W-algebras with two-generators and new rational models, Nucl. Phys. B383 (1992) 249.

    Article  ADS  Google Scholar 

  22. W. Eholzer, A. Honecker and R. Hübel, How complete is the classification of W-symmetries?, Phys. Lett. B308 (1993) 42, hep-th/9302124.

    ADS  Google Scholar 

  23. V.A. Fateev and S.L. Luk'yanov, The models of two-dimensional conformal quantum field theory with Z n symmetry, Int. J. Mod. Phys. A3 (1988) 507.

    ADS  MathSciNet  Google Scholar 

  24. V.A. Fateev and S.L. Luk'yanov, Additional symmetries and exactly-soluble models in two-dimensional conformal field theory, Sov. Sci. Rev. A15 (1990) 1.

    Google Scholar 

  25. V.A. Fateev and A.B. Zamolodchikov, Conformal quantum field theory models in two dimensions having Z 3 symmetry, Nucl. Phys. B280 [FS18] (1987) 644.

    Article  ADS  MathSciNet  Google Scholar 

  26. L. Feher, L. O'Raifeartaigh, P. Ruelle, I. Tsutsui and A. Wipf, On Hamiltonian reductions of the Wess-Zumino-Novikov-Witten theories, Phys. Rep. 222 (1992) 1.

    Article  ADS  MathSciNet  Google Scholar 

  27. L. Feher, L O'Raifeartaigh and I. Tsutsui, The vacuum preserving Lie algebra of a classical W algebra, Phys. Lett. B316 (1993) 275, hep-th/9307190

    ADS  MathSciNet  Google Scholar 

  28. B. Feigin and E. Frenkel, Quantization of the Drinfeld-Sokolov reduction, Phys. Lett. B246 (1990) 75.

    ADS  MathSciNet  Google Scholar 

  29. B.L. Feigin and D.B. Fuchs, On the cohomology of some nilpotent subalgebras of Kac-Moody and the Virasoro algebras, J. Geom. Phys. 5 (1988) 209.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  30. J.M. Figueroa-O'Farrill and S. Schrans, The spin 6 extended conformal algebra, Phys. Lett. B245 (1990) 471.

    ADS  MathSciNet  Google Scholar 

  31. E.V. Frenkel, V. Kac and M. Wakimoto, Characters and fusion rules for W algebras via quantized Drinfeld-Sokolov reductions, Commun. Math. Phys. 147 (1992) 295.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  32. P. Furlan, A.C. Ganchev, R. Paunov and V.B. Petkova, Solutions of the Knizhnik-Zamolodchikov equation with rational isospins and the reduction to the minimal models, Nucl. Phys. B394 (1993) 665, hep-th/9201080.

    Article  ADS  MathSciNet  Google Scholar 

  33. P. Furlan, A.C. Ganchev and V.B. Petkova, Singular vectors of W algebras via DS reduction of A (1)2 Nucl. Phys. B431 (1994) 622, hep-th/9403075.

    Article  ADS  MathSciNet  Google Scholar 

  34. M.R. Gaberdiel, Fusion rules of chiral algebras, Nucl. Phys. B417 (1994) 130, hep-th/9309105; Fusion in conformal field theory as the tensor product of the symmetry algebra, Int. J. Mod. Phys. A9 (1994) 4619, hep-th/9307183.

    Article  ADS  MathSciNet  Google Scholar 

  35. M.R. Gaberdiel and H.G. Kausch, A rational logarithmic conformal field theory, Phys. Lett. B386 (1996) 131, hep-th/9606050; Indecomposable fusion products, Nucl. Phys. B477 (1996) 293, hep-th/9604026.

    ADS  MathSciNet  Google Scholar 

  36. T. Gannon, The classification of affine su(3) modular invariant partition functions, Commun. Math. Phys 161 (1994) 233, hep-th/9212060.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  37. T. Gannon and M.A. Walton, On the classification of diagonal coset modular invariants, Commun. Math. Phys 173 (1995) 175, hep-th/9407055

    Article  MATH  ADS  MathSciNet  Google Scholar 

  38. P. Goddard, Meromorphic Conformal Field Theory, in: Infinite Dimensional Lie Algebras and Lie Groups, ed. V. G. Kac, World Scientific, 1989, CIRM-Luminy July conference on Infinite dimensional Lie algebras and Lie Groups, Marseilles 1988.

    Google Scholar 

  39. K. Hornfeck, W-algebras with set of primary fields of dimensions (3,4,5) and (3,4,5,6), Nucl. Phys. B407 (1993) 237, hep-th/9212104.

    Article  ADS  MathSciNet  Google Scholar 

  40. K. Hornfeck, Classification of structure constants for W-algebras from highest weights, Nucl. Phys. B411 (1994) 307, hep-th/9307170.

    Article  ADS  MathSciNet  Google Scholar 

  41. K. Hornfeck, W-algebras of negative rank, Phys. Lett. B343 (1995) 94, hep-th/9410013.

    ADS  MathSciNet  Google Scholar 

  42. Y.-Z. Huang and J. Lepowsky, On the D-module and formal variable approaches to vertex algebras, in: ‘Topics in Geometry: In Memory of Joseph D'Atri', ed. S. Gindikin, Progress in Nonlinear Differential Equations, Vol. 20, Birkhauser, Boston, (1996) 175, q-alg/9603020.

    Google Scholar 

  43. H.G. Kausch, Extended conformal algebras generated by a multiplet of primary fields, Phys. Lett. B259 (1991) 448.

    ADS  MathSciNet  Google Scholar 

  44. H.G. Kausch and G.M.T. Watts, A study of W-algebras using Jacobi identities, Nucl. Phys. B354 (1991) 740.

    Article  ADS  MathSciNet  Google Scholar 

  45. W. Nahm, Chiral algebras of two-dimensional chiral field theories and their normal ordered products, in ‘Recent developments in conformal field theories’ S. Randjbar-Daemi et al. eds, (World Scientific 1990) 81.

    Google Scholar 

  46. W. Nahm, Quasi-rational fusion products, Int. J. Mod. Phys. B8 (1994) 3693, hep-th/9402039.

    ADS  MathSciNet  Google Scholar 

  47. E. Verlinde, Fusion rules and modular transformations in 2d conformal field theory, Nucl. Phys. B300 [FS22] (1988) 360.

    Article  ADS  MathSciNet  Google Scholar 

  48. G.M.T. Watts, Fusion in the W 3 algebra, Commun. Math. Phys. 171 (1995) 87, hep-th/9403163.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  49. A.B. Zamolodchikov, Infinite additional symmetries in two-dimensional conformal quantum field theory, Theor. Mat. Fiz. 65 (1985) 347.

    MathSciNet  Google Scholar 

  50. Y.-C. Zhu, Vertex operator algebras, elliptic functions and modular forms, Ph.D. thesis, Yale University, 1990

    Google Scholar 

  51. I.B. Frenkel and Y.-C. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992), 123.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Zalán Horváth László Palla

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag

About this paper

Cite this paper

Watts, G.M.T. (1997). W-algebras and their representations. In: Horváth, Z., Palla, L. (eds) Conformal Field Theories and Integrable Models. Lecture Notes in Physics, vol 498. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0105278

Download citation

  • DOI: https://doi.org/10.1007/BFb0105278

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63618-2

  • Online ISBN: 978-3-540-69613-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics