New Discrete States in Two-Dimensional Supergravity

Two-dimensional string theory is known to contain the set of discrete states that are the SU(2) multiplets generated by the lowering operator of the SU(2) current algebra. Their structure constants are defined by the area preserving diffeomorphisms in two dimensions. In this paper we show that the interaction of d = 2 superstrings with the superconformal β − γ ghosts enlarges the actual algebra of the dimension 1 currents and hence the new ghost-dependent discrete states appear. Generally, these states are the SU(N) multiplets, if the algebra includes the currents of ghost numbers n : −N ≤ n ≤ N − 2, not related by picture-changing. We compute the structure constants of these ghost-dependent discrete states for N = 3 and express them in terms of SU(3) Clebsch-Gordan coefficients, relating this operator algebra to the volume preserving diffeomorphisms in d = 3. For general N , the operator algebra is conjectured to be isomorphic to SDiff(N). This points at possible holographic relations between two-dimensional superstrings and field theories in higher dimensions. PACS:04.50.+ h;11.25.Mj. February 2006 † [email protected] Introduction The spectrum of physical states of non-critical one-dimensional string theory (or , equivalently, of critical string theory in two dimensions) is known to contain the “discrete states” of non-standard b-c ghost numbers 0 and 2 (while the standard unintegrated vertex operators always carry the ghost number 1) [1], [2],[3],[4],[5] These states appear at the special (integer or half integer) values of the momentum p but are absent for generic p 1 The appearance of these states is closely related to the SU(2) symmetry, emerging when the theory is compactified on a circle with the self-dual radius [2], [3] The SU(2) symmetry is generated by 3 currents of conformal dimension 1: