Multi-moment maps

Abstract

We introduce a notion of moment map adapted to actions of Lie groups that preserve a closed three-form. We show existence of our multi-moment maps in many circumstances, including mild topological assumptions on the underlying manifold. Such maps are also shown to exist for all groups whose second and third Lie algebra Betti numbers are zero. We show that these form a special class of solvable Lie groups and provide a structural characterisation. We provide many examples of multi-moment maps for different geometries and use them to describe manifolds with holonomy contained in G_2 preserved by a two-torus symmetry in terms of tri-symplectic geometry of four-manifolds.

27 pages

References in corpus (1)

• Toric G_2 and Spin(7) holonomy spaces from gravitational instantons and other examples

Cited by in corpus (21)

• Closed forms and multi-moment maps
• An invitation to multisymplectic geometry
• Hypersymplectic 4-manifolds, the $G_2$-Laplacian flow and extension assuming bounded scalar curvature
• Reduction of polysymplectic manifolds
• Reduction of multisymplectic manifolds
• Noether's Theorem in Multisymplectic Geometry
• Existence and Uniqueness of Weak Homotopy Moment Maps
• Momentum sections in Hamiltonian mechanics and sigma models
• Remarks on multisymplectic reduction
• Spin(7)-manifolds with three-torus symmetry
• Bryant-Salamon ${\mathrm{G}}_2$ manifolds and coassociative fibrations
• More insights into symmetries in multisymplectic field theories
• Toric geometry of $G_2$-manifolds
• Nearly Kähler six-manifolds with two-torus symmetry
• Homotopy momentum sections on multisymplectic manifolds
• Geometric structures associated with a simple Cartan 3-form
• Exact $G_2$-structures on unimodular Lie algebras
• Poisson-Poincaré reduction for Field Theories
• Remarks on Hamiltonian Structures in G_2-Geometry
• Cayley fibrations in the Bryant-Salamon $Spin(7)$ manifold
• Hamiltonian Reduction in Affine Principal Bundles