Framed DDF operators and the general solution to Virasoro constraints

Abstract

We define the framed DDF operators by introducing the concept of local frames in the usual formulation of DDF operators. In doing so it is possible to completely decouple the DDF operators from the associated tachyon and show that they are good zero-dimensional conformal operators. This allows for an explicit formulation of the general solution of the Virasoro constraints both on-shell and off-shell. We then make precise the realization of the intuitive idea that DDF operators can be used to embed light-cone states in the covariant formulation. This embedding is not unique, but depends on a coset. This coset is the little group of the embedding of the light-cone and is associated with a frame. The frame allows us to embed the $SO(D-2)$ light-cone physical polarizations into the $SO(1,D-1)$ covariant ones in the most general way. The solution to the Virasoro constraints is not in the gauge that is usually used. This happens since the states obtained from DDF operators are generically the sum of terms which are partially transverse due to the presence of a projector but not traceless and terms which are partially traceless but not transverse. To check the identification, we verify the matching of the expectation value of the second Casimir of the Poincar'e group for some light-cone states with the corresponding covariant states built using the framed DDFs.