Fermionic construction of N = 1,2 superconformal algebras

In connection to string theory and two-dimensional critical phenomena, a great deal of attention has been paid to the conformally and superconformally invariant two-dimensional field theories. Unitary representations of the conformal and superconformal algebras give constraints on possible values of critical indices. For N = 0,1,2 Kac-determinant formurae have been obtained and possible values of the central charge c, highest weight and U(1) charge and corresponding characters for unitary irreducible representation have been derived. Unitary representations of the (super)conformal algebra are realized by several methods (free fermions, free bosons, free fermions and bosons, ZN-currents and a free boson....). It is known that fermionic representation of (super)conformal algebra is relevant to construct some models which are difficult to realized by bosons or boson-fermion models by projection. In the case of N = 2, representations with C/3 ≥ 1 are also important in their relevance to the compactification of superstring. In this report we derive conditions for fermionic construction of N = 1,2 superconformal algebras and obtain some solutions for these conditions. (author)