Multisymplectic AKSZ sigma models

TL;DR

This work extends the AKSZ sigma‑model framework to multisymplectic settings by equipping a target $Q$‑manifold with a form $Ω$ of arbitrary degree satisfying $(d+L_Q)Ω=0$, enabling higher‑derivative, gauge‑invariant actions formulated via a Chern–Weil map. The construction yields a broad class of theories, including higher‑dimensional Chern–Simons, MMSW gravity, self‑dual gravity and its higher‑spin extensions, as well as twistor and Sparling formulations, all recast as multisymplectic AKSZ models. By developing the Chern–Weil morphism for $Q$‑bundles and deriving explicit equations of motion and gauge symmetries, the paper provides a cohesive, geometry‑driven framework to reformulate diverse gauge theories with local degrees of freedom in a finite‑dimensional setting. The results illuminate connections between multisymplectic geometry, PDE geometry, and BV/BV‑like formalisms, offering new avenues for applying multisymplectic ideas to higher‑spin gravity and related field theories. Overall, the multisymplectic AKSZ approach consolidates several known theories under a unified, gauge‑invariant, higher‑derivative formalism with potential for further generalisation to nontrivial $Q$‑bundles and PDE‑based extensions.

Abstract

The Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) construction encodes all the data of a topological sigma-model in the finite-dimensional symplectic $Q$-manifold. Relaxing the nondegeneracy condition i.e. considering a presymplectic form instead, extends the construction to non-topological models. The gauge-invariant action functional of (presymplectic) AKSZ sigma model is written in terms of space-time differential forms and can be seen as a covariant multidimensional analogue of the usual 1st order Hamiltonian action. In this work, we show that the AKSZ construction has a natural generalisation where the target space $Q$-manifold is equipped with a form of arbitrary degree $Ω$ (possibly inhomogeneous) which is $(\mathrm{d}+L_Q)$-closed. This data defines a higher-derivative generalisation of the AKSZ action which is still invariant under the natural gauge transformations determined by $Q$ and which is efficiently formulated in terms of a version of Chern-Weil map introduced by Kotov and Strobl. It turns out that a variety of interesting gauge theories, including higher-dimensional Chern-Simons theory, MacDowell-Mansouri-Stelle-West action and self-dual gravity as well as its higher spin extension, can be concisely reformulated as such multisymplectic AKSZ models. We also present a version of the construction in the setup of PDE geometry and demonstrate that the counterpart of the multisymplectic AKSZ action is precisely the standard multisymplectic formulation, where the Chern-Weil map corresponds to the usual pullback map.