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

Chengjian Yao, Ziyi Zhou

TL;DR

The paper classifies closed $ mG_2$-structures with effective $\mathbb{T}^3$-symmetry by orbit type, revealing a precise three-way reduction: non-isotropic orbits yield a hypersymplectic 4-manifold base with a principal $\mathbb{T}^3$-bundle over it, associative orbits give a good hypersymplectic (orbifold) base and a toric reduction, and isotropic orbits produce a foliation by hypersymplectic leaves with a local multi-Hamiltonian structure. When the structures are torsion-free, these reductions sharpen to flat ($\rmG_2$-flat) in Type 1, to good hyperkähler orbifolds in Type 2, and to locally toric in Type 3, linking hypersymplectic geometry intimately with closed $\rmG_2$-manifolds under $\mathbb{T}^3$-symmetry. The work also analyzes singular orbits via multi-moment maps, culminating in a trivalent-graph description of singular strata for isotropic orbits. Overall, it extends the toric/locally toric framework to a broader setting, clarifying how hypersymplectic data capture the geometric content of closed and torsion-free $\rmG_2$-structures with $\mathbb{T}^3$-symmetry and highlighting the role of Riemannian submersions in these reductions.

Abstract

Closed $\mathrm{G}_2$-structures $\varphi$ with an effective $\mathbb{T}^3$-symmetry on connected manifolds are roughly classified into three types according to the evaluation of $\varphi$ on the principal orbits. Type 1: if there is neither associative nor isotropic orbit, then the action is free and $\varphi$ reduces to a hypersymplectic structure on the quotient manifold admitting three linearly independent closed 1-forms; in particular, it is diffeomorphic to $\mathbb{T}^4$ if the manifold is compact. Type 2: if some orbit is associative, then the action is almost-free and $\varphi$ reduces to a good hypersymplectic orbifold with cyclic isotropic groups. Type 3: 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 and complete, then the hypersymplectic manifold is flat and $\varphi$ is flat for Type 1; the good hypersymplectic orbifold is good hyperkähler orbifold for Type 2; $\varphi$ is locally toric for Type 3. As shown, hypersymplectic structures have intimate link with closed $\mathrm{G}_2$-structure with effective $\mathbb{T}^3$-symmetry.

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

TL;DR

The paper classifies closed -structures with effective -symmetry by orbit type, revealing a precise three-way reduction: non-isotropic orbits yield a hypersymplectic 4-manifold base with a principal -bundle over it, associative orbits give a good hypersymplectic (orbifold) base and a toric reduction, and isotropic orbits produce a foliation by hypersymplectic leaves with a local multi-Hamiltonian structure. When the structures are torsion-free, these reductions sharpen to flat (-flat) in Type 1, to good hyperkähler orbifolds in Type 2, and to locally toric in Type 3, linking hypersymplectic geometry intimately with closed -manifolds under -symmetry. The work also analyzes singular orbits via multi-moment maps, culminating in a trivalent-graph description of singular strata for isotropic orbits. Overall, it extends the toric/locally toric framework to a broader setting, clarifying how hypersymplectic data capture the geometric content of closed and torsion-free -structures with -symmetry and highlighting the role of Riemannian submersions in these reductions.

Abstract

Closed -structures with an effective -symmetry on connected manifolds are roughly classified into three types according to the evaluation of on the principal orbits. Type 1: if there is neither associative nor isotropic orbit, then the action is free and reduces to a hypersymplectic structure on the quotient manifold admitting three linearly independent closed 1-forms; in particular, it is diffeomorphic to if the manifold is compact. Type 2: if some orbit is associative, then the action is almost-free and reduces to a good hypersymplectic orbifold with cyclic isotropic groups. Type 3: if some orbit is isotropic, then the action is locally multi-Hamiltonian for . Moreover, the open and dense subset of principal orbits is foliated by -invariant hypersymplectic manifolds. If is torsion-free and complete, then the hypersymplectic manifold is flat and is flat for Type 1; the good hypersymplectic orbifold is good hyperkähler orbifold for Type 2; is locally toric for Type 3. As shown, hypersymplectic structures have intimate link with closed -structure with effective -symmetry.
Paper Structure (11 sections, 8 theorems, 92 equations)

This paper contains 11 sections, 8 theorems, 92 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Non-isotropic orbit
  3. Neither associative nor isotropic orbits
  4. For $\varphi$ being closed
  5. For $\varphi$ being torsion-free
  6. Associative orbits
  7. For $\varphi$ being closed
  8. For $\varphi$ being torsion-free
  9. Isotropic orbit
  10. Hypersymplectic foliation
  11. Trivalent graph

Key Result

Lemma 1.8

If $\alpha$ is closed and invariant, then $U_i\lrcorner\alpha$ is also closed and invariant.

Theorems & Definitions (23)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4: multi-Hamiltonian/locally multi-Hamiltonian
  • Definition 1.5: Toric/Locally toric $\mathrm{G}_2$-manifold
  • Example 1.6
  • Remark 1.7
  • Lemma 1.8
  • proof
  • Theorem 2.1
  • ...and 13 more