Symmetries and tau function of higher dimensional dispersionless integrable hierarchies

TL;DR

This paper introduces a higher-dimensional dispersionless integrable hierarchy with additional spatial dimensions, explores its symmetries and tau function, and connects it to twistor theory and nonlinear Riemann-Hilbert problems.

Contribution

It extends the dispersionless KP hierarchy to higher dimensions with a novel tau function and symmetry structure, grounded in twistor theory.

Findings

Hierarchy is integrable with infinite symmetries.

Constructed tau function relates to free energy in matrix models.

Extended symmetries obey anomalous commutation relations.

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.