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. These framed DDFs allow for an explicit covariant formulation of the general solution of the Virasoro constraints both on-shell and off-shell which depends on a unique set of tangent space lightcone polarizations. This is possible since the frame allows us to embed the SO(D-2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$SO(D-2)$$\\end{document} lightcone physical polarizations into the SO(1,D-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$SO(1,D-1)$$\\end{document} covariant ones in the most general way. The solution to the Virasoro constraints is not in the gauge that is usually used 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. We then show that different embeddings are connected by states generated by improved Brower operators both on shell and off shell. Brower states are null on-shell and equivalent to BRST exact states. Therefore on shell one embedding is sufficient to compute all amplitudes except the ones with vanishing momentum. Off-shell a change of frame cannot be obtained by null states but only by Brower states. Improved Brower operators generate a gauge invariance associated with local frame rotations, which replaces the usual gauge invariance generated by Virasoro operators or BRST. To check the identification, we verify the matching of the expectation value of the second Casimir of the Poincaré group for some lightcone states with the corresponding covariant states built using the framed DDFs.