arXiv subjects
Publications and source records attributed to J. S. Dowker.
The specific form of the constant term in the asymptotic expansion of the heat-kernel on an axially-symmetric space with a codimension two fixed-point set of conical singularities is used to determine the associated conformal change of the effective action in four dimensions. Another derivation of the relevant coefficient is presented.
A functorial derivation is presented of a heat-kernel expansion coefficient on a manifold with a singular fixed point set of codimension two. The existence of an extrinsic curvature term is pointed out.
The recently discussed notion of geometric entropy is shown to be related to earlier calculations of thermal effects in Rindler space. The evaluation is extended to de Sitter space and to a two-dimensional black hole.
We calculate the Riemann curvature tensor and sectional curvature for the Lie group of volume-preserving diffeomorphisms of the Klein bottle and projective plane. In particular, we investigate the sign of the sectional curvature, and find a possible disagreement with a theorem of Lukatskii. We suggest an amendment to this theorem.
The scalar functional determinants on sectors of the two-dimensional disc and spherical cap are determined for arbitrary angles (rational factors of $π$). The wholesphere and hemisphere expressions are also given, in low dimensions, for both the ordinary and the conformal Laplacian. Comparisons are made with other work.
The standard formula for the change in the effective action under a conformal transformation is extended to the case when the boundary is piecewise smooth. We then find the functional determinants of the scalar Laplacian on regions of the plane obtained by stereographic projection of the fundamental domains on an orbifolded two-sphere. Examples treated are the sector of a disk and a circular crescent. The effective action on a spherical cap is also determined.
A technical point regarding the invariance of Polyakov's nonlocal form of the effective action under uniform rescalings is briefly addressed.
A derivation of the Cesàro-Fedorov relation from the Selberg trace formula on an orbifolded 2-sphere is elaborated and extended to higher dimensions using the known heat-kernel coefficients for manifolds with piecewise-linear boundaries. Several results are obtained that relate the coefficients, $b_i$, in the Shephard-Todd polynomial to the geometry of the fundamental domain. For the 3-sphere we show that $b_4$ is given by the ratio of the volume of the fundamental tetrahedron to its Schläfli reciprocal.
The effective action on an orbifolded sphere is computed for minimally coupled scalar fields. The results are presented in terms of derivatives of Barnes zeta-functions and it is shown how these may be evaluated. Numerical values are shown. An analytical, heat-kernel derivation of the Cesàro-Fedorov formula for the number of symmetry planes of a regular solid is also presented.
The vacuum energy is calculated for a free, conformally-coupled scalar field on the orbifold space-time \R$\times §^2/Γ$ where $Γ$ is a finite subgroup of O(3) acting with fixed points. The energy vanishes when $Γ$ is composed of pure rotations but not otherwise. It is shown on general grounds that the same conclusion holds for all even-dimensional factored spheres and the vacuum energies are given as generalised Bernoulli functions (i.e. Todd polynomials). The relevant $ζ$- functions are analysed in some detail and several identities are incidentally derived. The general discussion is presented in terms of finite reflection groups.
Quantum field theory is discussed in Möbius corner kaleidoscopes using the method of images. The vacuum average of the stress-energy tensor of a free field is derived and is shown to be a simple sum of straight cosmic string expressions, the strings running along the edges of the corners. It does not seem possible to set up a spin-half theory easily.
A finite-dimensional su($N$) Lie algebra equation is discussed that in the infinite $N$ limit (giving the area preserving diffeomorphism group) tends to the two-dimensional, inviscid vorticity equation on the torus. The equation is numerically integrated, for various values of $N$, and the time evolution of an (interpolated) stream function is compared with that obtained from a simple mode truncation of the continuum equation. The time averaged vorticity moments and correlation functions are compared with canonical ensemble averages.