Bonus symmetry in conformal field theory

Abstract

Conformal field theories typically have an enlarged symmetry over that of the chiral algebra. These enlarged symmetries simplify the analysis of a theory by linking representations that would appear independent based on considerations of the smaller symmetry of the chiral algebra. It will be shown that this bonus symmetry occurs whenever a primary field g has a fusion rule with only the identity on the r.h.s. It will be seen that the additional symmetry generated by such a field g will be reflected in the fusion rules in the modular transformation properties of the chiral characters. The way in which this enlarged symmetry may be exploited is illustrated in some simple examples. When the field g is of integer conformal dimension, g can be incorporated into an extended chiral algebra; the resulting extended, modular invariant partition function will be constructed. It will also be seen that especially strong simplifications arise when the field g with the mentioned fusion rule is of neither integer nor half-integer conformal dimension.