DETERMINANTS OF LAPLACIANS IN REAL LINE BUNDLES OVER HYPERBOLIC MANIFOLDS CONNECTED WITH QUANTUM GEOMETRY OF MEMBRANES

The possibility is discussed of generalizing the Polyakov approach to strings on membranes and the connection of such a generalization with Thurston's classification of three-dimensional geometries. The important ingredients for computing a membrane path integral are the determinants of scalar Laplacians acting in real line bundles over three-dimensional closed manifolds. In the closed bosonic membrane case, such determinants are evaluated for a class of closed 3-manifolds of the H3/G{cyrillic} form with a discrete subgroup of isometries G{cyrillic} of the three-dimensional Lobachevsky space H3 and they are expressed in terms of the Selberg zeta function. Some further possible implications of the results obtained are also discussed. © 1990 Kluwer Academic Publishers.