Monodromies of CFT correlates on the Lorentzian Cylinder

While correlators of a CFT are single valued in Euclidean Space, they are multi valued -- and have a complicated sheet structure -- in Lorentzian space. Correlators on $R^{1,1}$ are well known to access a finite number of these sheets. In this paper we demonstrate the spiral nature of lightcones on $S^1 \times$ time allows time ordered correlators of a $CFT_2$ on this spacetime -- the Lorentzian cylinder -- to access an infinite number of sheets of the correlator. We present a complete classification, both of the sheets accessed as well as of the various distinct causal configurations that lie on a particular sheet. Our construction provides a physical interpretation for an infinite number of sheets of the correlator, while, however, leaving a larger infinity of these sheets uninterpreted.