Closed forms and multi-moment maps

Abstract

We extend the notion of multi-moment map to geometries defined by closed forms of arbitrary degree. We give fundamental existence and uniqueness results and discuss a number of essential examples, including geometries related to special holonomy. For forms of degree four, multi-moment maps are guaranteed to exist and are unique when the symmetry group is (3,4)-trivial, meaning that the group is connected and the third and fourth Lie algebra Betti numbers vanish. We give a structural description of some classes of (3,4)-trivial algebras and provide a number of examples.Comment: 36 page

Existence and uniqueness of weak homotopy moment maps

“…Also notice that the n-th component of a weak moment map is precisely the moment map introduced by Madsen and Swann in [6] and [7].…”

“…Notice that for the case n = k, it is always true that H 0 (g, Ω n−k cl ) = 0 since any-non zero constant function is closed. Hence Theorem 5.6 gives a generalization of Theorems 3.5 and 3.14 of [6] and [7] respectively. Moreover, by taking n = k = 1, we see that we are obtaining a generalization from symplectic geometry.…”

“…Weak moment maps generalize the moment maps of Madsen and Swann in [6] and [7], and were also used to give a multisymplectic version of Noether's theorem in [5]. In this paper, we study the existence and uniqueness of weak homotopy moment maps and show that the theory is a generalization from symplectic geometry.…”

Momentum Sections in Hamiltonian Mechanics and Sigma Models

“…It is important to compare other generalizations of a moment map theory to a multisymplectic manifold such as Madsen-Swann's multimoment map on the n-th Lie kernel [20,21], a homotopy moment map [11], and a weak moment map [13].…”

Noether's theorem in multisymplectic geometry

“…Instead, we need to restrict our attention to multivector fields in the Lie kernel, which we defined in Definition 2.8. We quickly recall this definition and some terminology and notation introduced in [12].…”