DETERMINANT FORMULAS FOR THE FREE FIELD REPRESENTATIONS OF THE VIRASORO AND KAC-MOODY ALGEBRAS

The determinant formulas are given for the Feigin-Fuchs modules over the Virasoro algebra and for the Wakimoto modules over the affine Kac-Moody algebras. These formulas imply that some of the modules are isomorphic to the Verma modules or their duals. In particular, this is the case for all Feigin-Fuchs modules over the Virasoro algebra with central charge c greater-than-or-equal-to 25. Seiberg's condition is a simple criterion as to whether such a module is isomorphic to a dual Verma module.