Abstract
We consider 3-point and 4-point correlation functions in a conformal field theory with a W-algebra symmetry. Whereas in a theory with only Virasoro symmetry the three-point functions of descendant fields are uniquely determined by the three-point function of the corresponding primary fields this is not the case for a theory withW 3 algebra symmetry. The generic 3-point functions of W-descendant fields have a countable degree of arbitrariness. We find, however, that if one of the fields belongs to a representation with null states that this has implications for the 3-point functions. In particular, if one of the representations is doubly degenerate, then the 3-point function is determined up to an overall constant. We extend our analysis to 4-point functions and find that if two of the W-primary fields are doubly degenerate then the intermediate channels are limited to a finite set and that the corresponding chiral blocks are determined up to an overall constant. This corresponds to the existence of a linear differential equation for the chiral blocks with two completely degenerate fields as has been found in the work of Bajnok et al.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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; Z. Bajnok,C 2 Toda theory in reduced WZNW framework, Inst. Theor. Phys., Univ. Budapest, preprint ITP-Budapest-501 (1993).
V. G. Kac, Contravariant form for infinite-dimensional Lie algebras and superalgebras,in: Proc. Int. Congress Math. 1978, Helsinki.
a. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Nucl. Phys.B241 (1984) 333.
B. L. Feigin and D. B. Fuchs,Representations of the Virasoro algebra, Repts. Dept. Math. Stockholm Univ. (1986), published in ‘Representations of infinite-dimensional Lie groups and Lie algebras’, eds. A. Vershik and D. Zhelobenko, Gordon and Breach (1989).
B. L. Feigin and D. B. Fuchs, Functional Analysis and its Applications16 (1982) 114.
B. L. Feigin and D. B. Fuchs, J. Geom. Phys.5 (1988) 209.
R. P. Langlands, Commun. Math. Phys.124 (1989) 261.
M. Bauer, P. di Francesco, C. Itzykson, and J.-B. Zuber, Phys. Lett. B 260 (1991) 323. Nucl. Phys.B362 (1991) 515.
A. Kent, Phys. Lett.278B (1992) 443.
A. B. Zamolodchikov, Theor. Math. Phys.65 (1985) 347.
G. M. T. Watts, Nucl. Phys.B326 (1989) 648; erratum Nucl. Phys.B336 (1989) 720.
V. A. Fateev and A. B. Zamolodchikov, Nucl. Phys.B280 [FS18] (1987) 644.
V. A. Fateev and S. L. Luk'yanov, Int. J. Mod. Phys.A3 (1988) 507; Sov. Sci. Rev.A15 (1990) 1.
F. A. Bais, P. Bouwknegt, K. Schoutens, and M. Surridge, Nucl. Phys.B304 (1988) 348; Nucl. Phys.B304 (1988) 371.
I. N. Bernstein, I. M. Gelfand, and S. I. Gelfand, Differential operators based on affine space and a study ofg-modules,in: Proc. 1971 Summer School in Math., Janos Bolyai Math. Soc. Budapest, ed. I. M. Gelfand, pp. 21–64.
P. Bouwknegt, J. McCarthy, and K. Pilch,Semi-infinite cohomology of W-algebras, University of Southern California, preprint USC-93/11 (1993);On the BRST structure of W 3 gravity coupled to c=2 matter, University of Southern California, preprint USC-93/14 (1993), To appear in Proc. of AMS Special Session on Geometry and Physics, Los Angeles, Nov. 1992.
P. Bowcock and G. M. T. Watts, Phys. Lett.297B (1992) 282.
E. V. Frenkel, V. Kac, and M. Wakimoto, Commun. Math. Phys.147 (1992) 195.
Additional information
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98. No. 3, pp. 500–508, March, 1994
Rights and permissions
About this article
Cite this article
Bowcock, P., Watts, G.M.T. Null vectors, 3-point and 4-point functions in conformal field theory. Theor Math Phys 98, 350–356 (1994). https://doi.org/10.1007/BF01102212
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01102212