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are considering a space which has a topology of the type ΓD = ( S1 d × RD−d, with d (≤ D) being the number of compactified dimensions, each with the topology of a circle. This is the type of topology that emerges, for instance, in quantum field theory at finite temperature: the Matsubara formalism imposes that the time direction is compactified in a circle with length β = 1/T , where T is the temperature; its topology is then Γ4 = S 1 × R3 in the notation introduced above. Another important example involves spacetimes of dimensions D larger than four, with the “extra” or “hidden” dimensions being compactified and assumed to be very small, as in Kaluza–Klein and string theories. In any case, the topology ΓD mentioned above corresponds to a generalized Matsubara formalism, in which imaginary-time and spatial coordinates may be simultaneously compactified.

Ginzburg–Landau Theory of Phase Transitions in Compactified Spaces