Homotopy momentum sections on multisymplectic manifolds

Abstract

We introduce a notion of a homotopy momentum section on a Lie algebroid over a pre-multisymplectic manifold. A homotopy momentum section is a generalization of the momentum map with a Lie group action and the momentum section on a pre-symplectic manifold, and is also regarded as a generalization of the homotopy momentum map on a multisymplectic manifold. We show that a gauged nonlinear sigma model with Wess-Zumino term with Lie algebroid gauging has the homotopy momentum section structure.

Compatible E-Differential Forms on Lie Algebroids Over (Pre-)Multisymplectic Manifolds

“…Moreover we claim that the condition is unified with the twisted Poisson structure, and generalized to higher algebroids and multisymplectic settings. A generalization of the momentum section to a (pre)-multisymplectic manifold called a homotopy momentum section has be proposed in [20]. Our condition for an E-n-form in this paper is consistent with the definition of the homotopy momentum section.…”

“…for e, e 1 , e 2 ∈ g. Here −, − is the pairing of g * and g. As explained in Example 2.3, the Lie group action induces an action Lie algebroid structure on a trivial bundle E = M × g. The anchor map is induced from the map of the Lie algebra action as ρ : M × g → T M. The momentum map is regarded as a section of the dual trivial bundle µ ∈ Γ(M × g * ). Under the condition (20), Equation ( 21) is changed to [5] E dµ(e 1 , e 2 ) = −ι 2 ρ ω(e 1 , e 2 ), (22) i.e., E dµ = −ι 2 ρ ω. Here E d is a Lie algebroid differential with respect to the action Lie algebroid.…”

“…. , e m+1 ), (28) for e i ∈ g, We can rewrite n equations ( 24)-( 27) to equivalent n equations [20],…”

“…Example 4.7 (Homotopy momentum sections) [20] A homotopy momentum map on a pre-multisymplectic manifold is generalized to a homotopy momentum section.…”

“…We claim that the above structure is interesting and worth to analyze, since it has many examples already listed above. Generalizations of momentum maps such as momentum sections on a Lie algebroid, homotopy momentum maps [16] and homotopy momentum sections [20] on multisymplectic manifolds are examples of our geometric structure.…”