Quantum coherent string states inAdS3and theSL(2,R)Wess-Zumino-Witten-Novikov model

Abstract

In this paper we make the connection between semiclassical string quantization and exact conformal field theory quantization of strings in 2+1 anti--de Sitter spacetime. More precisely, considering the Weiss-Zumino-Witten-Novikov model corresponding to $\mathrm{SL}(2,R)$ and its covering group, we construct quantum coherent string states, which generalize the ordinary coherent states of quantum mechanics, and show that in the classical limit they correspond to oscillating circular strings. After quantization, the spectrum is found to consist of two parts: A continuous spectrum of low mass states (partly tachyonic) satisfying the standard spin-level condition necessary for unitarity $|j|<k/2,$ and a discrete spectrum of high mass states with asymptotic behavior ${m}^{2}{\ensuremath{\alpha}}^{\ensuremath{'}}\ensuremath{\propto}{N}^{2}$ $(N$ positive integer). The quantization condition for the high mass states arises from the condition of finite positive norm of the coherent string states, and the result agrees with our previous results obtained using semiclassical quantization. In the $\stackrel{\ensuremath{\rightarrow}}{k}\ensuremath{\infty}$ limit, all the usual properties of coherent or quasiclassical states are recovered. It should be stressed that we consider the circular strings only for simplicity and clarity, and that our construction can easily be used for other string configurations too. We also compare our results with those obtained in the recent paper by Maldacena and Ooguri, hep-th/0001053.