Symmetry behavior in curved spacetime: Finite-size effect and dimensional reduction

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Abstract
We discuss the effect of certain aspects of curvature and topology on symmetry breaking in curved spacetime with the aim of understanding phase transitions in the early Universe. We show that for spacetimes with some compact spatial dimensions where the invariant operator of the scalar fluctuation field has a discrete spectrum, the most important contribution to the infrared behavior comes from its zero mode (or band). The decoupling of higher modes gives rise to dimensional reduction in the infrared domain. We introduce the notion of effective infrared dimension and explain how it can be useful for physically understanding the symmetry behavior in spacetimes of different topology. We also introduce an eigenvalue analysis to study dimensional reduction in both direct-product spaces and spaces which can approximate product spaces under extreme deformations. We illustrate this method by analyzing the symmetry behavior of the Taub universe in the small- and large-anisotropy limits. These geometric effects in curved space are of the same nature as finite-size effects in condensed matter and surface physics. We use the two-particle-irreducible formalism and the large-N approximation to derive the higher-loop corrections. We give the results of the effective mass and the effective potential for various systems and spacetimes with compact dimensions. The ideas, techniques, and results developed here are useful for the study of finite-size effects in field theory, cosmology, and condensed-matter physics.