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Topological Casimir effect in Maxwell Electrodynamics on a Compact Manifold

We study the Topological Casimir effect, in which extra vacuum energy emerges as a result of the topological features of the theory, rather than due to the conventional fluctuations of the physical propagating degrees of freedom. We compute the corresponding topological term in quantum Maxwell theory defined on a compact manifold. Numerically, the topological effect is much smaller than the conventional Casimir effect. However, we argue that the Topological Casimir Effect is highly sensitive to an external magnetic field, which may help to discriminate it from the conventional Casimir effect. It is quite amazing that the external magnetic field plays the role of the $θ$ state, similar to a $θ$ vacuum in QCD, or $θ=π$ in topological insulators.