It is important to understand the full string theory spectrum, not just the supergravity states. Each factor is a group manifold. The sphere, orbifold, and torus part are standard. This problem is not fully understood but a Wakimoto construction is sufficient to study strings near the boundary.
String theory describes states in terms of vertex-operators.
A large class of string states can be constructed explicitly. A subset reproduces the supergravity chiral primaries.