SDiff(2) KP hierarchy
This paper deals with the dispersionless KP hierarchy from the point of view of quasi-classical limit. Its Lax formalism, W-infinity symmetries and general solutions are shown to be reproduced from their counterparts in the KP hierarchy in the limit of hbar->0. Free fermions and bosonized vertex operators play a key role in the description of W-infinity symmetries and general solutions, which is technically very similar to a recent free fermion formalism of c=1 matrix models.
We study the relation between topological string theory and singularity theory using the partition function of A_N-1 topological string defined by matrix integral of Kontsevich type. Genus expansion of the free energy is considered, and the genus g=0 contribution is shown to be described by a special solution of N-reduced dispersionless KP system. We show a universal correspondences between the time variables of dispersionless KP hierarchy and the flat coordinates associated with versal deformations of simple singularities of type A. We also study the behavior of topological matter theory on the sphere in a topological gravity background, to clarify the role of the topological string in the singularity theory. Finally we make some comment on gravitational phase transition.
The equations of Löwner type can be derived in two very different contexts: one of them is complex analysis and the theory of parametric conformal maps and the other one is the theory of integrable systems. In this paper we compare the both approaches. After recalling the derivation of Löwner equations based on complex analysis we review one- and multi-variable reductions of dispersionless integrable hierarchies (dKP, dBKP, dToda, and dDKP). The one-variable reductions are described by solutions of different versions of Löwner equation: chordal (rational) for dKP, quadrant for dBKP, radial (trigonometric) for dToda and elliptic for DKP. We also discuss multi-variable reductions which are given by a system of Löwner equations supplemented by a system of partial differential equations of hydrodynamic type. The solvability of the hydrodynamic type system can be proved by means of the generalized hodograph method.