Classical integrability in the BTZ black hole

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.

45 pages, Reference added, typos corrected, discussion on geodesics improved to include all geodesics

References in corpus (4)

• Chaos Rules out Integrability of Strings in AdS_5 x T^{1,1}
• Infinite spin limit of semiclassical string states
• S-matrix for magnons in the D1-D5 system
• Integrability Lost

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• Higher spin fermions in the BTZ black hole
• Probing analytical and numerical integrability: The curious case of $(Ad{S}_5\times S^5{)}_{\eta }$
• Generating string solutions in BTZ