Monodromies of CFT correlates on the Lorentzian Cylinder

TL;DR

This paper explores the complex multi-sheet structure of Lorentzian CFT correlators on the cylinder, revealing their spiral lightcone structure and classifying the sheets and causal configurations involved.

Contribution

It demonstrates the infinite sheet access of Lorentzian correlators on the cylinder and provides a complete classification of these sheets and causal configurations.

Findings

Correlators on the Lorentzian cylinder access an infinite number of sheets.

Classification of sheets and causal configurations in Lorentzian CFT correlators.

Physical interpretation of the infinite sheets of correlators.

Abstract

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.