An infinite number of hidden variables in hyper-Kähler metrics

Abstract

Three types of hidden variables, both independent and dependent, are shown to underlie the hyper-Kähler geometry in a complexified setting. The variables satisfy an infinite set of differential equations (‘‘hierarchy’’), just as in the case of most nonlinear integrable systems. The notion of the Plebanski key functions [J. Math. Phys. 16, 2395 (1975)] is extended to this hierarchy to give an analog of the notion of the ‘‘τ function.’’ Two examples of special solutions, which are reminiscent of several solution techniques in the theory of nonlinear integrable systems, are presented for illustration.