Vacuum structure of twisted scalar field theories on M D − 1 ⊗ S 1

Scalar field theories on ${M}^{D\ensuremath{-}1}\ensuremath{\bigotimes}{S}^{1},$ which allow us to impose twisted boundary conditions for the ${S}^{1}$ direction, are studied in detail, and several novel features overlooked so far are revealed. One of the characteristic features is the appearance of critical radii of the circle ${S}^{1},$ at which some symmetries are broken or restored. A phase transition can occur at the classical level or can be caused by quantum effects. Radiative corrections can restore broken symmetries or can break symmetries for a small radius R. A surprising feature is that the translational invariance for the ${S}^{1}$ direction can spontaneously be broken. A particular class of coordinate-dependent vacuum configurations is clarified and the $O(N)$ ${\ensuremath{\varphi}}^{4}$ model on ${M}^{D\ensuremath{-}1}\ensuremath{\bigotimes}{S}^{1}$ is extensively studied, as an illustrative example.