Knizhnik-Zamolodchikov equations and spectral flow in AdS3 string theory

TL;DR

This paper extends the Knizhnik-Zamolodchikov equations to include spectral flowed fields in AdS3 string theory, revealing their relation to Liouville theory correlators and the effects of spectral flow violations.

Contribution

It generalizes KZ equations for spectral flowed fields in AdS3 string theory and connects these correlators to Liouville theory with degenerate insertions.

Findings

KZ equations are equivalent when spectral flow is preserved or violated by one unit.

Linear combinations of KZ equations apply when spectral flow is violated by two or more units.

Correlators relate to Liouville theory with degenerate fields, with each spectral flow violation removing one degenerate field.

Abstract

I generalize the Knizhnik-Zamolodchikov equations to correlators of spectral flowed fields in AdS3 string theory. If spectral flow is preserved or violated by one unit, the resulting equations are equivalent to the KZ equations. If spectral flow is violated by two units or more, only some linear combinations of the KZ equations hold, but extra equations appear. Then I explicitly show how these correlators and the associated conformal blocks are related to Liouville theory correlators and conformal blocks with degenerate field insertions, where each unit of spectral flow violation removes one degenerate field. A similar relation to Liouville theory holds for noncompact parafermions.