QFT on rotating boxes at finite temperature

We formulate thermal quantum field theory on a finite spatial periodic volume undergoing rotation. Traditional compactifications at finite temperature without rotations typically involve ${\mathbb{T}}^4$ as the space-time manifold within a path integral formulation and also moving frames can be accommodated by shifted boundary conditions on the same space. We show that consistent descriptions of a rotating box are possible on space-time manifolds with topology different from ${\mathbb{T}}^4$ but still flat and without boundary and we classify all possible geometries. The non-trivial topology may be implemented by rotated boundary conditions allowing for a path integral formulation. The purely imaginary angular velocity in temperature units cannot be arbitrary but several discrete values are possible. We also discuss finite volume effects in detail.