Topological Casimir energy for a general class of Clifford–Klein space–times

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Abstract
Using zeta regularization we compute the vacuum energy for free massless scalar fields on ultrastatic space–times R×(Γ\X), where X is an arbitrary noncompact irreducible rank 1 symmetric space and Γ is a cocompact torsion free subgroup of isometries of X. The spaces X include hyperbolic manifolds on which previous authors have focused. Specifically, using a general trace formula, we extend the work of Bytsenko, Goncharov, Zerbini (and others), where X=SO1(m,1)/SO(m), to the other classical rank 1 symmetric spaces X=SU(m,1)/U(m), SP(m,1)/(SP(m)×SP(1)), and the exceptional space X=F4(−20)/Spin(9). We find in general that the trivial unitary character of Γ always induces a negative topological component of the energy.