Quantum discrete isomonodromy system from the viewpoint of quantum Teichmueller space

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The aim is to construct the quantum discretized version of the monodromy preserbing deformation of the Fuchsian equation as the system on the discretized Teichmueller space, so that one can recognize the Painleve VI system as well as the Garnier system as included into the picture, aiming that the construction will give the understanding of the symmetry structure as well as the viewpoint to the solvable lattice models from the theory of Riemann surfaces.For this aim we have succeeded in the rank two case the construction of the quantum discrete version of the isomonodromy system using the periodically reduced system of the nonautonomous discrete quantum Toda field equation. The autonomous system has been studied by Kashaev and Reshetikhin, and our result is in good coincidence with our previous result using the Weyl group action approach.