The Analytic Torsion of the cone over an odd dimensional manifold

We study the analytic torsion of the cone over an orientable odd dimensional compact connected Riemannian manifold W. We prove that the logarithm of the analytic torsion of the cone decomposes as the sum of the logarithm of the root of the analytic torsion of the boundary of the cone, plus a topological term, plus a further term that is a rational linear combination of local Riemannian invariants of the boundary. We also prove that this last term coincides with the anomaly boundary term appearing in the Cheeger Muller theorem for a manifold with boundary, according to Bruning and Ma, either in the case that W is an odd sphere or has dimension smaller than six. It follows in particular that the Cheeger Muller theorem holds for the cone over an odd dimensional sphere. We also prove Poincare duality for the analytic torsion of a cone

The analytic torsion of the finite metric cone over a compact manifold

We give an explicit formula for the $L^2$ analytic torsion of the finite metric cone over an oriented compact connected Riemannian manifold. We provide an interpretation of the different factors appearing in this formula. We prove that the analytic torsion of the cone is the finite part of the limit obtained collapsing one of the boundaries, of the ratio of the analytic torsion of the frustum to a regularising factor. We show that the regularising factor comes from the set of the non square integrable eigenfunctions of the Laplace Beltrami operator on the cone.Comment: To appear in Journal of the Mathematical Society of Japa

On the Black-Hole Conformal Field Theory Coupled to the Polyakov's String Theory. A Non Perturbative Analysis

We couple the 2D black-hole conformal field theory discovered by Witten to a $D-1$ dimensional Euclidean bosonic string. We demonstrate that the resulting planar (=zero genus) string susceptibility is real for any $0\leq D \leq 4$.Comment: 5 page

A generalized model for two dimensional quantum gravity and dynamics of random surfaces for d>1

The possible interpretations of a new continuum model for the two-dimensional quantum gravity for $d>1$ ($d$=matter central charge), obtained by carefully treating both diffeomorphism and Weyl symmetries, are discussed. In particular we note that an effective field theory is achieved in low energy (large area) expansion, that may represent smooth self-avoiding random surfaces embedded in a $d$-dimensional flat space-time for arbitrary $d$. Moreover the values of some critical exponents are computed, that are in agreement with some recent numerical results.Comment: n. 11; Phyzz

The analytic torsion of a cone over a sphere

We compute the analytic torsion of a cone over a sphere of dimension 1, 2, and 3, and we conjecture a general formula for the cone over an odd dimensional sphere