Mañas Baena, Manuel EnriqueGuha, Partha2023-06-202023-06-201994-030010-361610.1007/BF02099775https://hdl.handle.net/20.500.14352/59705©Springer. One of us MM like to thank Prof. Francisco Guil for initial collaboration and PG like to thank Prof. Pierre van Moerbeke for enlighting discussion. We are also grateful to Dr. Sacha Sardo Infirri for stimulating discussion.The periodic flag manifold (in the Sato Grassmannian context) description of the modified Korteweg-de Vries hierarchy is used to analyse the translational and scaling self-similar solutions of this hierarchy. These solutions are characterized by the string equations appearing in the double scaling limit of the symmetric unitary matrix model with boundary terms. The moduli space is a double covering of the moduli space in the Sato Grassmannian for the corresponding self-similar solutions of the Korteweg-de Vries hierarchy, i.e. of stable 2D quantum gravity. The potential modified Korteweg-de Vries hierarchy, which can be described in terms of a line bundle over the periodic flag manifold, and its self-similar solutions corresponds to the symmetric unitary matrix model. Now, the moduli space is in one-to-one correspondence with a subset of codimension one of the moduli space in the Sato Grassmannian corresponding to self-similar solutions of the Korteweg-de Vries hierarchy.engjournal articlehttp://dx.doi.org/10.1007/BF02099775http://link.springer.comhttp://arxiv.org/abs/hep-th/9307016open access51-732-Dimensional quantum-gravityGeometryFísica-Modelos matemáticosFísica matemática