Conformal Field Theory with Periodic Time

It is shown that time-ordered correlation functions of a unitary CFT${}_2$ in 2D Minkowski space admit a single-valued, conformally-invariant extension to the Lorentzian signature torus provided that the $S^1\times S^1$ spatial and temporal radii are equal. The result extends to Lorentzian CFT${}_D$ on equal-radii $S^{D-1}\times S^1$ under the assumption that branch cuts occur only when a pair of operator insertions are null separated.