Flat Complex Vector Bundles,the Beltrami Differential and W-Algebras

Since the appearance of the paper by Bilal et al. in 1991, it has been widely assumed that W-algebras originating from the Hamiltonian reduction of an SL(n,C)-bundle over a Riemann surface give rise to a flat connection, in which the Beltrami differential may be identified. In this Letter, it is shown that the use of the Beltrami parametrization of complex structures on a compact Riemann surface over which flat complex vector bundles are considered, allows the construction of the above mentioned flat connection. It is stressed that the modulus of the Beltrami differential is of necessity less than one, and that solutions of the so-called Beltrami equation give rise to an orientation-preserving smooth change of local complex coordinates. In particular, the latter yields a smooth equivalence between flat complex vector bundles. The role of smooth diffeomorphisms which induce equivalent complex structures is specially emphasized. Furthermore, it is shown that, while the construction given here applies to the special case of the Virasoro algebra, the extension to flat complex vector bundles of arbitrary rank does not provide generalizations of the Beltrami differential usually considered as central objects for such non-linear symmetries.