Abstract
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 H 3/Г form with a discrete subgroup of isometries Г of the three-dimensional Lobachevsky space H 3 and they are expressed in terms of the Selberg zeta function. Some further possible implications of the results obtained are also discussed.
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Goncharov, Y.P. Determinants of Laplacians in real line bundles over hyperbolic manifolds connected with quantum geometry of membranes. Lett Math Phys 19, 73–81 (1990). https://doi.org/10.1007/BF00402263
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DOI: https://doi.org/10.1007/BF00402263