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.

The lattice definition of a two-dimensional topological field theory (TFT) is given generically, and the exact solution is obtained explicitly. In particular, the set of all lattice topological field theories is shown to be in one-to-one correspondence with the set of all associative algebras R, and the physical Hilbert space is identified with the center Z(R) of the associative algebra R. Perturbations of TFT's are also considered in this approach, showing that the form of topological perturbations is automatically determined, and that all TFT's are obtained from one TFT by such perturbations. Several examples are presented, including twisted N = 2 minimal topological matter and the case where R is a group ring. *

The type IIA/IIB effective actions compactified on T d are known to be invariant under the T -duality group SO(d, d; Z), although the invariance of the R-R sector cannot be seen so directly. Inspired by a work of Brace, Morariu and Zumino, we introduce new potentials, which are mixtures of R-R potentials and the NS-NS 2-form, in order to make the invariant structure of R-R sector more transparent. We give a simple proof that if these new potentials transform as a Majorana-Weyl spinor of SO (d, d; Z), the effective actions are indeed invariant under the T -duality group. The argument is made in such a way that it can apply to Kaluza-Klein forms of arbitrary degree. We also demonstrate that these new fields simplify all the expressions, including the Chern-Simons term. * )

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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