Spectral Analysis and Zeta Determinant on the Deformed Spheres

Abstract

We consider a class of singular Riemannian manifolds, the deformed spheres S ). An application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions, and in particular ζ(0, ∆ S N k ) and). We give explicit formulas for the zeta regularized determinant in the low dimensional cases, N = 2, 3, thus generalizing a result of Dowker [25], and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter k.

Lowest Landau level on a cone and zeta determinants

“…Here we collect formulas for the spectral zeta function on the conical surfaces, derived in Refs. [28,29,30]. In order to compare the results with Eqns.…”

“…The value of the zeta function at zero can be read off from Ref. [28,Thm. 4.15] ζ(0, ∆ g ) = −1 + a 6 + 1 6a .…”

Metrics of Constant Positive Curvature with Conical Singularities, Hurwitz Spaces, and Determinants of Laplacians

“…with zero mode excluded) ζ-regularized determinant. The constant C can be found by using the result [24]: one has to consider a sphere with two antipodal singularities of conical angle 4π and compare formula (1.1) with the one given in [24]). Our main result generalizes (1.1) to the case of compact Riemann surfaces X of arbitrary genus and arbitrary meromorphic functions f : X → CP 1 (for simplicity we consider only functions f with simple critical values, the modifications required to consider the general case are insignificant and of no interest, and the result is essentially the same).…”

Zeta determinants on cones and products

“…Let C ν M be the metric cone over M , namely the space [0, 1] × M with metric g = (dx) 2 + x 2 ν 2 g M , on (0, 1] × M , and where ν is a positive constant [5]. Particular instances of this setting have been studied in [1,2,10] (m-ball), (cone over a circle) [19], and (deformed spheres) [23]. The zeta function on the cone C ν M , is defined by the series…”