Symmetries and tau function of higher dimensional dispersionless integrable hierarchies

A higher dimensional analogue of the dispersionless KP hierarchy is introduced. In addition to the two-dimensional “phase space” variables (k, x) of the dispersionless KP hierarchy, this hierarchy has extra spatial dimensions compactified to a two (or any even) dimensional torus. Integrability of this hierarchy and the existence of an infinite dimensional of “additional symmetries” are ensured by an underlying twistor theoretical structure (or a nonlinear Riemann-Hilbert problem). An analogue of the tau function, whose logarithm gives the F function (“free energy” or “prepotential” in the contest of matrix models and topological conformal field theories), is constructed. The infinite dimensional symmetries can be extended to this tau (or F ) function. The extended symmetries, just like those of the dispersionless KP hierarchy, obey an anomalous commutation relations.