17 research outputs found
Spectral asymmetry and Riemannian geometry. III
In Parts I and II of this paper ((4),(5)) we studied the 'spectral asymmetry' of certain elliptic self-adjoint operators arising in Riemannian geometry. More precisely, for any elliptic self-adjoint operator A on a compact manifold we defined ηA(s)=Σλ+0signλ|λ|-8, where λ runs over the eigenvalues of A. For the particular operators of interest in Riemannian geometry we showed that ηA(s) had an analytic continuation to the whole complex s-plane, with simple poles, and that s=0 was not a pole. The real number ηA(0), which is a measure of 'spectral asymmetry', was studied in detail particularly in relation to representations of the fundamental group
Electromagnetic Casimir piston in higher dimensional spacetimes
We consider the Casimir effect of the electromagnetic field in a higher
dimensional spacetime of the form , where is the
4-dimensional Minkowski spacetime and is an -dimensional
compact manifold. The Casimir force acting on a planar piston that can move
freely inside a closed cylinder with the same cross section is investigated.
Different combinations of perfectly conducting boundary conditions and
infinitely permeable boundary conditions are imposed on the cylinder and the
piston. It is verified that if the piston and the cylinder have the same
boundary conditions, the piston is always going to be pulled towards the closer
end of the cylinder. However, if the piston and the cylinder have different
boundary conditions, the piston is always going to be pushed to the middle of
the cylinder. By taking the limit where one end of the cylinder tends to
infinity, one obtains the Casimir force acting between two parallel plates
inside an infinitely long cylinder. The asymptotic behavior of this Casimir
force in the high temperature regime and the low temperature regime are
investigated for the case where the cross section of the cylinder in is
large. It is found that if the separation between the plates is much smaller
than the size of , the leading term of the Casimir force is the
same as the Casimir force on a pair of large parallel plates in the
-dimensional Minkowski spacetime. However, if the size of
is much smaller than the separation between the plates, the leading term of the
Casimir force is times the Casimir force on a pair of large parallel
plates in the 4-dimensional Minkowski spacetime, where is the first Betti
number of . In the limit the manifold vanishes, one
does not obtain the Casimir force in the 4-dimensional Minkowski spacetime if
is nonzero.Comment: 22 pages, 4 figure
From simplicial Chern-Simons theory to the shadow invariant II
This is the second of a series of papers in which we introduce and study a
rigorous "simplicial" realization of the non-Abelian Chern-Simons path integral
for manifolds M of the form M = Sigma x S1 and arbitrary simply-connected
compact structure groups G. More precisely, we introduce, for general links L
in M, a rigorous simplicial version WLO_{rig}(L) of the corresponding Wilson
loop observable WLO(L) in the so-called "torus gauge" by Blau and Thompson
(Nucl. Phys. B408(2):345-390, 1993). For a simple class of links L we then
evaluate WLO_{rig}(L) explicitly in a non-perturbative way, finding agreement
with Turaev's shadow invariant |L|.Comment: 53 pages, 1 figure. Some minor changes and corrections have been mad
p-form spectra and Casimir energies on spherical tesselations
Casimir energies on space-times having the fundamental domains of
semi-regular spherical tesselations of the three-sphere as their spatial
sections are computed for scalar and Maxwell fields. The spectral theory of
p-forms on the fundamental domains is also developed and degeneracy generating
functions computed. Absolute and relative boundary conditions are encountered
naturally. Some aspects of the heat-kernel expansion are explored. The
expansion is shown to terminate with the constant term which is computed to be
1/2 on all tesselations for a coexact 1-form and shown to be so by topological
arguments. Some practical points concerning generalised Bernoulli numbers are
given.Comment: 43 pages. v.ii. Puzzle eliminated, references added and typos
corrected. v.iii. topological arguments included, references adde