Spectral asymptotics of Euclidean quantum gravity with diff-invariant boundary conditions

Abstract

A general method is known to exist for studying Abelian and non-Abelian gauge theories, as well as Euclidean quantum gravity, at one-loop level on manifolds with boundary. In the latter case, boundary conditions on metric perturbations h can be chosen to be completely invariant under infinitesimal diffeomorphisms, to preserve the invariance group of the theory and BRST symmetry.In the de Donder gauge, however, the resulting boundary-value problem for the Laplace type operator acting on h is known to be self-adjoint but not strongly elliptic. The latter is a technical condition ensuring that a unique smooth solution of the boundary-value problem exists, which implies, in turn, that the global heat-kernel asymptotics yielding one-loop divergences and oneloop effective action actually exists. The present paper shows that, on the Euclidean four-ball, only the scalar part of perturbative modes for quantum gravity are affected by the lack of strong ellipticity. Further evidence for lack of strong ellipticity, from an analytic point of view, is therefore obtained. Interestingly, three sectors of the scalar-perturbation problem remain elliptic, while lack of strong ellipticity is "confined" to the remaining fourth sector. The integral representation of the resulting ζ-function asymptotics is also obtained; this remains regular at the origin by virtue of a spectral identity here obtained for the first time.

Boundary effects in super-Yang–Mills theory

“…It would also be interesting to analyze the BRST symmetry of such a system using both linear and non-linear gauges. Furthermore, the BRST symmetry and gauge fixing have been studied for perturbative quantum gravity [51][52][53][54][55][56]. It is possible to generalize this work to supergravity solutions, and analyze the supersymmetry of such supergravity solutions, when there is a boundary.…”

Traces of heat operators on Riemannian foliations

“…Many researchers have studied this expansion and its generalizations and have worked to compute the coefficient functions (see [35], [3], [33], [25]), because the heat kernel is not only used to compute heat flows but is also used in many areas of geometric and topological analysis. The asymptotic expansions above (and their generalizations) have been used to study the spectrum of the Laplacian (see [3], [4], [14], [33]), the determinant of the Laplacian (see [39], [46]), conformal classes of metrics (see [40]), analytic torsion (see [44], [15]), modular forms (see [23]), index theory (see [2], [49]), stochastic analysis (see [13], [32]), gauge theory/mathematical physics (see [22], [12], [6]), and so on.…”

Quantum Cosmology From Three Different Perspectives

“…For pure gravity, one-loop quantum cosmology in the limit of small three-geometry [22] describes a vanishing probability of reaching the singularity at the origin (of the Euclidean four-ball) only with diffeomorphism-invariant boundary conditions [23,24], which are a particular case of the previous scheme. All other sets of boundary conditions lead instead to a divergent one-loop wave function [22,24,25].…”

“…Peculiar cancellations occur on the Euclidean four-ball, and the spectral (also called generalized) ζ-function remains regular at the origin [23,24], despite the lack of strong ellipticity of the boundary-value problem [21].…”