  Multi-cut two-matrix models are studied in the Z_k symmetry breaking k-cut
(\hat p,\hat q) critical points which should correspond to (\hat p,\hat q)
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
\infty-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. Horava and C. A.
Keeler.
