Fractional-superstring amplitudes, multi-cut matrix models and non-critical M theory

Multi-cut two-matrix models are studied in the Zk symmetry breaking k-cut (pˆ,qˆ) critical points which should correspond to (pˆ,qˆ) minimal k-fractional superstring theory. FZZT-brane or macroscopic loop amplitudes are obtained in all of these critical points and found to have two kinds of solutions in general. Each of these solutions is expressed by hyperbolic cosine or sine functions with proper phase shifts. The algebraic geometries and ZZ-brane disk amplitudes (instanton actions) of these solutions are also studied. In particular, our results suggest that minimal ∞-fractional superstring theory can be viewed as a mother theory which includes all the minimal k-fractional superstring theories (k=1,2,…) as its perturbative vacua in the weak-coupling string Landscape. Our results also indicate that, in the strong coupling regime of this fractional superstring theory, there is a three-dimensional theory which would be understood as the non-critical version of M theory in the sense proposed by P. Hořava and C.A. Keeler.