String Theory on AdS_3 as Discrete Light-Cone Liouville Theory

We investigate (super) string theory on $Ad{S}_3$ background based on an approach of free field realization. We demonstrate that this string theory can be reformulated as a string theory defined on a linear dilaton background along the transverse direction (Liouville mode'') and compactified onto $S^1$ along a {\em light-like} direction. Under this reformulation we analyze the physical spectrum as that of a free field system, and discuss the consequences when we turn on the Liouville potential. The discrete light-cone momentum in our framework is naturally interpreted as the winding number’’ of the long string configuration and is identified with the spectral flow parameter that is introduced in the recent work by Maldacena and Ooguri \cite{MO}. Moreover we show that there exist infinite number of the on-shell chiral primary states possessing the different light-cone momenta and the spectral flow consistently acts on the space of chiral primaries. We observe that they are also chiral primaries in the sense of space-time (or the conformal theory of long string) and the spectrum of space-time $U(1{)}_R$ charge is consistent with the expectation from the $Ad{S}_3\mathrm{/}CF{T}_2$-duality. We also clarify the correspondence between our framework and the symmetric orbifold theory of multiple long string system \cite{HS2}.