Novel phase structure of twisted O(N) φ4 model on MD−1⊗S1

We study the O(N) φ4 model compactified on MD−1⊗S1, which allows to impose twisted boundary conditions for the S1-direction. The O(N) symmetry can be broken to H explicitly by the boundary conditions and further broken to I spontaneously by vacuum expectation values of the fields. The symmetries H and I are completely classified and the model turns out to have unexpectedly a rich phase structure. The unbroken symmetry I is shown to depend on not only the boundary conditions but also the radius of S1, and the symmetry breaking patterns are found to be unconventional. The spontaneous breakdown of the translational invariance is also discussed.