Symmetries and tau function of higher dimensional dispersionless integrable hierarchies

Abstract

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.

50 pages, (Changes: a few references are added; numbering of formulas are slightly modified)

References in corpus (4)

• SDiff(2) Toda equation -- hierarchy, $\tau$ function, and symmetries
• One-Dimensional Fermions as Two-Dimensional Droplets Via Chern-Simons Theory
• Classical Fermi Fluid and Geometric Action for $c=1$
• Dressing operator approach to Moyal algebraic deformation of selfdual gravity

Cited by in corpus (3)

• A Geometry for Multidimensional Integrable Systems
• Dispersionless (3+1)-dimensional integrable hierarchies
• Dispersionless limit of the noncommutative potential KP hierarchy and solutions of the pseudodual chiral model in 2+1 dimensions