Remarks on Hamiltonian structures in G2-geometry

Abstract

Abstract. In this article, we treat G 2 -geometry as a special case of multisymplectic geometry and make a number of remarks regarding Hamiltonian multivector fields and Hamiltonian differential forms on manifolds with an integrable G 2 -structure; in particular, we discuss existence and make a number of identifications of the spaces of Hamiltonian structures associated to the two multisymplectic structures associated to an integrable G 2 -structure. Along the way, we prove some results in multisymplectic geometry that are generalizations of results from symplectic geometry.

An invitation to multisymplectic geometry

“…Example 6.8 (Abelian L ∞ -algebra). In [CST13] a 2-plectic 7-manifold with no non-trivial Hamiltonian vector fields is constructed. Thus, Ω n−1 Ham (M, ω) = Ω n−1 cl (M ), l 2 = 0 and l 3 = 0.…”

Spin(7)-Manifolds and Multisymplectic Geometry

“…As the differential forms associated to exceptional holonomy manifolds naturally define multisymplectic structures, studying exceptional holonomy manifolds in this context may clarify the relationships between these spaces and symplectic manifolds. This programme to approach exceptional geometry from the perspective of multisymplectic geometry has been explored in the G 2 -holonomy context (and that of closed G 2 -Structures and co-closed G 2 -Structures) by Cho, Salur, and Todd [12]. In this work we investigate if similar results hold…”

“…The Hodge star and musical isomorphisms extend immediately to the multivector field context which allows us to prove similiar identities on an n-dimensional manifold for an ℓ-multivector field Q and k-form β [12].…”