Flat Complex Vector Bundles, The Beltrami Differential and WAlgebras
Abstract
Since the appearance of the paper by Bilal & al. in 1991, it has been widely assumed that Walgebras 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 parametrisation of complex structures on a compact Riemann surface over which flat complex vector bundles are considered, allows to construct 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 socalled 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 nonlinear symmetries.
 Publication:

arXiv eprints
 Pub Date:
 February 1998
 DOI:
 10.48550/arXiv.hepth/9802083
 arXiv:
 arXiv:hepth/9802083
 Bibcode:
 1998hep.th....2083L
 Keywords:

 High Energy Physics  Theory
 EPrint:
 LaTeX, 19 pages, no figure. Based on a talk given at the Luminy meeting on "Walgebras: Extended conformal symmetries", MarseilleLuminy, July 37, 1995