Classical integrability in the BTZ black hole

TL;DR

This paper demonstrates that classical string dynamics in the BTZ black hole background are integrable by constructing the monodromy matrix and deriving integral equations, linking solutions to a twisted SL(2, R) spin chain model.

Contribution

It establishes classical integrability of string propagation in BTZ black hole backgrounds and connects the solution space to a continuum limit of Bethe equations.

Findings

Classical solutions characterized by density functions in the complex plane.

Explicit solutions for geodesics, winding strings, BMN, and magnon-like states.

Dispersion relations derived for specific string configurations.

Abstract

Using the fact the BTZ black hole is a quotient of AdS_3 we show that classical string propagation in the BTZ background is integrable. We construct the flat connection and its monodromy matrix which generates the non-local charges. From examining the general behaviour of the eigen values of the monodromy matrix we determine the set of integral equations which constrain them. These equations imply that each classical solution is characterized by a density function in the complex plane. For classical solutions which correspond to geodesics and winding strings we solve for the eigen values of the monodromy matrix explicitly and show that geodesics correspond to zero density in the complex plane. We solve the integral equations for BMN and magnon like solutions and obtain their dispersion relation. Finally we show that the set of integral equations which constrain the eigen values of the monodromy matrix can be identified with the continuum limit of the Bethe equations of a twisted SL(2, R) spin chain at one loop.