Functions In Conformal Field Theory Research Articles

Geometry and Classificatin of Solutions of the Classical Dynamical Yang-Baxter Equation

The classical Yang-Baxter equation (CYBE) is an algebraic equation central in the theory of integrable systems. Its solutions were classified by Belavin and Drinfeld. Quantization of CYBE led to the theory of quantum groups. A geometric interpretation of CDYB was given by Drinfeld and gave rise to the theory of Poisson-Lie groups. The classical dynamical Yang-Baxter equation (CDYBE) is an important differential equation analagous to CYBE and introduced by Felder as the consistency condition for the Knizhnik-Zamolodchikov-Bernard equations for correlation functions in conformal field theory on tori. Quantization of CDYBE allowed Felder to introduce an interesting elliptic analog of quantum groups. It becomes clear that numerous important notions and results connected with CYBE have dynamical analogs. In this paper we classify solutions to CDYBE and give geometric interpretation to CDYBE. The classification and interpretation are remarkably analogous to the Belavin-Drinfeld picture.

Null vectors, 3-point and 4-point functions in conformal field theory

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 descendants fields are uniquely determined by the three point function of the corresponding primary fields this is not the case for a theory with $W_3$ 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.

Irreducible automorphic functions and then-point functions of conformal field theory

Irreducible automorphic functions for a compact Riemann surface of arbitrary genus are used to expand two- and three-point functions of conformal quantum field theory. The divergence of this expansion at coinciding arguments is studied in particular.