Partner symmetries of the complex Monge-Ampere equation yield hyper-Kahler metrics without continuous symmetries

TL;DR

This paper develops a method to construct hyper-Kähler metrics without continuous symmetries by extending the Lax pair for the complex Monge-Ampère equation, leading to new non-invariant solutions.

Contribution

It introduces an extended Lax pair framework that allows deriving hyper-Kähler metrics without Killing vectors through a novel use of partner symmetries and Legendre transformations.

Findings

Constructed hyper-Kähler metrics with no continuous symmetries.

Developed a linear system for a real potential from the extended Lax pair.

Enabled generation of non-invariant solutions to the Legendre-transformed Monge-Ampère equation.

Abstract

We extend the Mason-Newman Lax pair for the elliptic complex Monge-Amp\`ere equation so that this equation itself emerges as an algebraic consequence. We regard the function in the extended Lax equations as a complex potential. We identify the real and imaginary parts of the potential, which we call partner symmetries, with the translational and dilatational symmetry characteristics respectively. Then we choose the dilatational symmetry characteristic as the new unknown replacing the K\"ahler potential which directly leads to a Legendre transformation and to a set of linear equations satisfied by a single real potential. This enables us to construct non-invariant solutions of the Legendre transform of the complex Monge-Amp\`ere equation and obtain hyper-K\"ahler metrics with anti-self-dual Riemann curvature 2-form that admit no Killing vectors.