INFINITE DIMENSIONAL GRASSMANNIAN STRUCTURE OF 2-DIMENSIONAL QUANTUM-GRAVITY

We study the infinite dimensional Grassmannian structure of 2D quantum gravity coupled to minimal conformal matters, and show that there exists a large symmetry, the W1 + infinity symmetry. Using this symmetry structure, we prove that the square root of the partition function, which is a tau-function of the p-reduced KP hierarchy, satisfies the vacuum condition of the W1 + infinity algebra. We further show that this condition is reduced to the vacuum condition of the W(p) algebra when the redundant variables for the p-reduction are eliminated. This mechanism also gives a prescription for extracting the W(p) algebra from the W1 + infinity algebra.