Homotopies for Lagrangian field theory

Abstract

Consider the variational bicomplex for $\mathcal{E}$ the space of sections of a graded, affine bundle. Local functionals $\mathcal{F}$ are defined as an equivalence class of density-valued functionals, which represent Lagrangian densities. A choice of a $k$-symplectic local form $\omega$ on $\mathcal{E}$ induces a Lie$[k]$ algebra structure on (Hamiltonian) local functionals $({\mathcal{F}}_{\mathrm{ham}},\{\cdot ,\cdot {\}}_{\mathrm{ham}})$. For any $\omega$ and any choice of a cohomological vector field $Q$ compatible with $\omega$, we build three explicit $L_{\infty }$ algebras on a resolution of ${\mathcal{F}}_{\mathrm{ham}}$, which are all $L_{\infty }$ quasi-isomorphic to a dgL$[k]$a $({\mathcal{F}}_{\mathrm{ham}},d_{\mathrm{ham}},\{\cdot ,\cdot {\}}_{\mathrm{ham}})$. In particular, one of our equivalent $L_{\infty }$ 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 $L_{\infty }$ 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.

Improvements throughout. We sharpened our comments in section five into Conjecture 5.2.5. 33 pages + biblio