New classical string solutions in AdS3 through null scrolls

Abstract

We introduce a new way to study null scrolls in AdS3. They are timelike surfaces generated by the evolution of a curve through a transversal lightlike geodesic flow. This new approach deals with AdS3 as a quadric in and that allows us to obtain an algorithm to construct null scrolls explicitly. We see that those surfaces are strongly related to the solutions of generalized Liouville equations. In fact, under the Virasoro constraints, we show that there exists a one-to-one correspondence between null scrolls and solutions of these equations. In particular, those with constant mean curvature are modeled by Liouville equations. That also holds for stationary null scrolls (zero mean curvature), which provide classical string solutions. As a consequence, we obtain that the classical string solutions modeled by stationary null scrolls appear, in the Pohlmeyer reduced theory, as wave solutions of a Liouville equation. Furthermore, we exploit the new approach to determine the moduli space of classical string solutions modeled by null scrolls. This space can be identified with that of parameterized timelike curves in a Lorentz plane, modulo affine parameterizations. In addition, we obtain a simple algorithm to explicitly construct those classical string solutions which can be considered as an alternative to the own Pohlmeyer reduced mechanism for classical string solutions.Dedicated to Professor Peter M Gruber, with admiration and respect.