Conformal embeddings and hidden symmetries of free fermion theories

Abstract

It is shown that the conformal field theory of free complex fermions transforming in the (anti)symmetric tensor representation R of u( N) is characterized by a nontrivial operator algebra. However, this theory can be naturally embedded into the theory of free Dirac fermions in the defining representation of u(dim R); the latter theory corresponds to an operator algebra which is trivial in the sense that e.g. φ×φ + = 1 . We argue that these observations can be generalized to any system of free fermions possessing an energy-momentum tensor of the Sugawara form. As a consequence, there exist no further cases where a free fermion theory is quantum equivalent to a WZW theory, besides the ones already well known (i.e. those based on the current algebras so (N) and u ̂ (N) with unit central charge).