Toric hypersymplectic quotients

Abstract:

We study the hypersymplectic spaces obtained as quotients of flat hypersymplectic space $\mathbb {R}^{4d}$ by the action of a compact Abelian group. These $4n$-dimensional quotients carry a multi-Hamilitonian action of an $n$-torus. The image of the hypersymplectic moment map for this torus action may be described by a configuration of solid cones in $\mathbb {R}^{3n}$. We give precise conditions for smoothness and non-degeneracy of such quotients and show how some properties of the quotient geometry and topology are constrained by the combinatorics of the cone configurations. Examples are studied, including non-trivial structures on $\mathbb {R}^{4n}$ and metrics on complements of hypersurfaces in compact manifolds.