Stokes phenomena and quantum integrability in non-critical string/M theory

We study Stokes phenomena of the k×k isomonodromy systems with an arbitrary Poincaré index r, especially which correspond to the fractional-superstring (or parafermionic-string) multi-critical points (p-hat,q-hat)=(1,r-1) in the k-cut two-matrix models. Investigation of this system is important for the purpose of figuring out the non-critical version of M theory which was proposed to be the strong-coupling dual of fractional superstring theory as a two-matrix model with an infinite number of cuts. Surprisingly the multi-cut boundary-condition recursion equations have a universal form among the various multi-cut critical points, and this enables us to show explicit solutions of Stokes multipliers in quite wide classes of (k,r). Although these critical points almost break the intrinsic Zk symmetry of the multi-cut two-matrix models, this feature makes manifest a connection between the multi-cut boundary-condition recursion equations and the structures of quantum integrable systems. In particular, it is uncovered that the Stokes multipliers satisfy multiple Hirota equations (i.e. multiple T-systems). Therefore our result provides a large extension of the ODE/IM correspondence to the general isomonodromy ODE systems endowed with the multi-cut boundary conditions. We also comment about a possibility that N=2 QFT of Cecotti-Vafa would be "topological series" in non-critical M theory equipped with a single quantum integrability.