  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.
