Representations of the algebra of parafermion currents of spin 4/3 in two-dimensional conformal field theory. Minimal models and the tricritical Potts Z3 model

One of the main tasks of the theory of phase transitions of the second kind is the classification of all possible types of critical behavior together with calculation of the corresponding critical indices. If the hypothesis of conformal invariance of large-scale critical fluctuations is adopted, this problem reduces to the construction of all conformally invariant solutions of (Euclidean) quantum field theory. Of course, in the multidimensional case D > 2 it is hardly possible even to discuss this problem in such a general formulation. However, in the two-dimensional case there exists a method of constructing exact conformally invariant solutions based on the fact that the conformal group of two-dimensional space is infinite dimensional. In this paper, series of conformally invariant solutions is found for a two-dimensional quantum field theory possessing global symmetry with respect to the group S3 of permutations of three elements