Homotopy reduction of multisymplectic structures in Lagrangian field theory

Abstract

While symplectic geometry is the geometric framework of classical mechanics, the geometry of classical field theories is governed by multisymplectic structures. In multisymplectic geometry, the Poisson algebra of Hamiltonian functions is replaced by the $L_\infty$-algebra of Hamiltonian forms introduced by Rogers in 2012. The corresponding notion of homotopy momentum maps as morphisms of $L_\infty$-algebras is due to Callies, Fr\'egier, Rogers, and Zambon in 2016. We develop a method of homotopy reduction for local homotopy momentum maps in Lagrangian field theory using these homotopy algebraic structures.