Homotopies for Lagrangian field theory
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909791082774528 |
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| author | Schiavina, Michele Schnitzer, Jonas |
| author_facet | Schiavina, Michele Schnitzer, Jonas |
| contents | 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 $ω$ on $\mathcal{E}$ induces a Lie$[k]$ algebra structure on (Hamiltonian) local functionals $(\mathcal{F}_{\mathrm{ham}},\{\cdot,\cdot\}_{\mathrm{ham}})$. For any $ω$ and any choice of a cohomological vector field $Q$ compatible with $ω$, 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 further provide a multisymplectic interpretation of the resulting data. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_00133 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homotopies for Lagrangian field theory Schiavina, Michele Schnitzer, Jonas Mathematical Physics Differential Geometry Symplectic Geometry 81T70, 17B55, 16E45 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 $ω$ on $\mathcal{E}$ induces a Lie$[k]$ algebra structure on (Hamiltonian) local functionals $(\mathcal{F}_{\mathrm{ham}},\{\cdot,\cdot\}_{\mathrm{ham}})$. For any $ω$ and any choice of a cohomological vector field $Q$ compatible with $ω$, 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 further provide a multisymplectic interpretation of the resulting data. |
| title | Homotopies for Lagrangian field theory |
| topic | Mathematical Physics Differential Geometry Symplectic Geometry 81T70, 17B55, 16E45 |
| url | https://arxiv.org/abs/2508.00133 |