Closed $\mathrm{G}_2$-structures with $\mathbb{T}^3$-symmetry and hypersymplectic structures

TL;DR

This paper classifies closed G2-structures with T^3-symmetry, decomposing them into types based on orbit isotropy, and explores their geometric properties, including hypersymplectic structures and torsion-free cases.

Contribution

It provides a canonical decomposition of G2-structures with T^3-symmetry and classifies them into distinct types, extending known ansatzes and analyzing their geometric implications.

Findings

Classified G2-structures into isotropic and non-isotropic orbit types.

Derived a generalized Gibbons-Hawking Ansatz for torsion-free cases.

Identified conditions under which hypersymplectic orbifolds are flat or hyperkähler.

Abstract

We decompose linear $\mathrm{G}_2$-structure in canonical ways adapted to 3-dimensional subspaces, in terms of certain natural 1-forms and definite triple of 2-forms, and apply the decompositions to the study of $\mathrm{G}_2$-structure with $\mathbb{T}^3$-symmetry. Closed $\mathrm{G}_2$-structures $\varphi$ with an effective $\mathbb{T}^3$-symmetry on connected manifolds are roughly classified into two types according the orbits being non-isotropic or isotropic. Type I: if some orbit is non-isotropic, then the action is almost-free and $\varphi$ reduces to a good hypersymplectic orbifold with cyclic isotropic groups. Type II: if some orbit is isotropic, then the action is locally multi-Hamiltonian for $\varphi$. Moreover, the open and dense subset of principal orbits is foliated by $\mathbb{T}^3$-invariant hypersymplectic manifolds. If $\varphi$ is torsion-free, then for Type I, there arises another natural hypersymplectic structure, and a generalized Gibbons-Hawking Ansatz extending Madsen-Swann Ansatz is derived. For Type II, $\varphi$ is locally toric. Assuming moreover completeness and constant orbit volume, exactly three possibilities occur. Type Ia: orbits are purely non-isotropic non-associative, then the hypersymplectic 4-orbifold becomes a flat manifold. Type Ib: orbits are purely associative, then the $\mathbb{T}^3$-action is flat, and the hypersymplectic 4-orbifold becomes a hyperk\"ahler 4-orbifold. Type II: orbits are isotropic, then all orbits are principal, and $\varphi$ is flat.