We study the continuum Schwinger-Dyson equations for nonperturbative two-dimensional quantum gravity coupled to various matter fields. The continuum Schwinger-Dyson equations for the one-matrix model are explicitly derived and turn out to be a formal Virasoro condition on the square root of the partition function, which is conjectured to be the τ function of the KdV hierarchy. Furthermore, we argue that general multi-matrix models are related to the W algebras and suitable reductions of KP hierarchy and its generalizations.

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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 W 1 + O0 symmetry. Using this symmetry structure, we prove that the square root of the partition function, which is a τ function of the p-reduced KP hierarchy, satisfies the vacuum condition of the W 1 + ao 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 W 1 + 00 algebra.

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