Due to the extensive researches in two-dimensional quantum gravity, a great deal of its mathematical and physical structures is almost completely revealed at present. However, perhaps the most important problem in this field is to construct a formalism which describes the whole theory space in a unified manner and also gives manifest relations among the operators.
As a first and important attempt to solve this problem, the authors of the present paper, by using Sato’s infinite dimensional Grassmannian theory of the KP hierarchy, study at length the global structure of the theory space of 2D quantum gravity coupled to various minimal conformal fields labeled by a pair of integers \((p,q)\). Thus, a rigorous proof of the equivalence of Douglas’ equation and Schwinger-Dyson’s equation is performed and moreover, the \(p-q\) duality at Green’s function level is completely established. Therefore, the results obtained by this analysis give a clue for constructing the universal description of the two- dimensional quantum gravity. Finally, the metamorphosis of operators under unitarity-preserving renormalization group flows is concretely worked out as an interesting application.