Symplectic maps and hyperK\"ahler moment map geometry

TL;DR

This paper establishes a novel link between symplectic diffeomorphisms of a 4-torus and hyperK"ahler moment map geometry, introducing a new flow and laying groundwork for polyhedral symplectic topology.

Contribution

It introduces a correspondence between symplectic diffeomorphisms and hyperK"ahler moment map zeros, and develops a new flow and polyhedral adaptation with potential topological applications.

Findings

Established a correspondence between symplectic diffeomorphisms and hyperK"ahler moment map zeros.

Proposed a new modified moment map flow for symplectic geometry.

Proved a Duistermaat type theorem in the polyhedral setting.

Abstract

We obtain a correspondence between the group of symplectic diffeomorphisms of a 4-dimensional real torus and the vanishing locus of a certain hyperK\"ahler moment map. This observation gives rise to a new flow, called the modified moment map flow. The construction can be adapted to the polyhedral setting, for which we prove a Duistermaat type theorem. This paper lays out the ground work for some effective polyhedral symplectic geometry and for a potential Morse-Bott theory, with applications to the topology of the space of symplectic maps of the 4-torus.