2025/07/31 by Michele Schiavina, Jonas Schnitzer, Schiavina, Michele +1
#math-ph #math.DG #math.MP #math.SG
paper · pdf · doi:10.48550/arxiv.2508.00133
Consider the variational bicomplex for E the space of sections of a graded, affine bundle. Local functionals F are defined as an equivalence class of density-valued functionals, which represent Lagrangian densities. A choice of a k-symplectic local form ω on E induces a Lie[k] algebra structure on (Hamiltonian) local functionals (Fham,\⋅,⋅\ham). For any ω and any choice of a cohomological vector field Q compatible with ω, we build three explicit L_∞ algebras on a resolution of Fham, which are all L_∞ quasi-isomorphic to a dgL[k]a (Fham,dham,\⋅,⋅\ham). In particular, one of our equivalent L_∞ algebras is a dgL[k] algebra. In the case k=-1, this provides an explicit lift of the standard Batalin--Vilkovisky framework to local forms enriched by the L_∞ structure, in terms of local homotopies, which interprets the modified classical master equation as a Maurer--Cartan equation for the distinguished dgL[k]a we construct. We conjecture that the data of a lift to local forms of a BV theory contains a homotopy moment map on the cohomology of the Koszul complex of the underlying Lagrangian field theory.