Extrinsic Hermitian Geometry of Functional Determinants for Vector Subbundles and the Drinfeld--Sokolov Ghost System

In this paper, a novel method is presented for the study of the dependence of the functional determinant of the Laplace operator associated to a subbundle $F$ of a hermitian holomorphic vector bundle $E$ over a Riemann surface $\Sigma$ on the hermitian structure $(h,H)$ of $E$. The generalized Weyl anomaly of the effective action is computed and found to be expressible in terms of a suitable generalization of the Liouville and Donaldson actions. The general techniques worked out are then applied to the study of a specific model, the Drinfeld--Sokolov (DS) ghost system arising in $W$--gravity. The expression of generalized Weyl anomaly of the DS ghost effective action is found. It is shown that, by a specific choice of the fiber metric $H_h$ depending on the base metric $h$, the effective action reduces into that of a conformal field theory. Its central charge is computed and found to agree with that obtained by the methods of hamiltonian reduction and conformal field theory. The DS holomorphic gauge group and the DS moduli space are defined and their dimensions are computed.