Toward a canonical formalism of non-perturbative two-dimensional gravity

On the basis of the previously proposed action principle describing the theory space of 2D gravity in less than one-dimension, we develop a systematic canonical formalism for studying the properties of the string equation in the phase space of the cosmological constant and its canonical conjugate, the puncture operator. The string equation is written in a manifestly invariant form under the group of regular canonical transformations in the phase space of generalized coordinate and momentum. As a consequence, the geometrical origin of the generalized Virasoro condition on the partition function (or more precisely, the -function) is understood to be the symmetry under the regular area-preserving diffeomorphisms (w₁₊ symmetry) in the deformed phase space. The deformed canonical formalism can be regarded as a quantization of a classical canonical formalism describing the sphere limit of the theory.