Topological symmetry restoration

We examine spontaneously broken $N$-component $\lambda {\phi }^4$ theory at finite temperature on a static manifold whose equal-time hypersurfaces are homogeneous but may be topologically nontrivial. The alterations in the infrared structure of the field theory caused by the nontrivial topology can induce a transition from the ordered to the disordered phase, even at zero temperature. Results derived include a general expression for the zero-temperature one-loop effective potential on a topologically trivial homogeneous curved manifold and a calculation of the free energy of the self-interacting scalar field at finite temperature on a static universe whose spatial section is $\frac{S^3}{\Gamma }$, where $\Gamma$ is a discrete group.