Knizhnik-Zamolodchikov equations and spectral flow in AdS3 string theory
Sylvain Ribault
TL;DR
The paper tackles the challenge of extending the Knizhnik–Zamolodchikov framework to AdS3 string theory, where spectral flow enlarges the operator content. It derives generalized KZ-type equations for both spectral-flow preserving and violating correlators, and uses Sklyanin’s separation of variables to connect these equations to BPZ equations in Liouville theory. A central result is a precise mapping between H3+ (Euclidean AdS3) correlators and Liouville theory correlators with degenerate insertions, including explicit relations for structure constants, OPEs, and conformal blocks. This Liouville-based reformulation provides a practical computational toolkit for AdS3 string correlators and offers a path toward understanding fusing matrices and parafermionic data in this noncompact setting, with potential implications for discrete/state content via Wick rotations.
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.