  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 $\theta$ state, similar to a
$\theta$ vacuum in QCD, or $\theta=\pi$ in topological insulators.
