Presymplectic AKSZ Framework

• The presymplectic AKSZ framework is a generalized sigma-model that unifies the BV formalism, gauge PDEs, and frame-like Lagrangians by incorporating degenerate presymplectic structures.
• It is built on a graded bundle architecture from T[1]X to a target with a degree-1 BRST vector field and closed 2-form, ensuring consistency through presymplectic master equations and symplectic reduction.
• Practical applications include Einstein gravity, supergravity, higher-spin theories, and Yang–Mills, offering a covariant geometric foundation for local gauge field theories and deformation analysis.

The presymplectic AKSZ framework is a generalization of the Alexandrov–Kontsevich–Schwartz–Zaboronsky (AKSZ) sigma-model construction that systematically unifies the Batalin–Vilkovisky (BV) formalism, gauge PDEs, frame-like Lagrangians, and the geometry of local gauge field theories with local degrees of freedom. It extends the AKSZ method to encompass targets equipped with degenerate (presymplectic) structures, enabling treatments of fields with dynamical content, non-topological interactions, and systematic inclusion of antifields and ghosts in a finite-dimensional and geometrical setting. The framework encompasses gravity, higher-spin theory, supergravity, Yang–Mills, and more, and is now foundational for the covariant geometric approach to local gauge systems.

1. Fundamental Data and Geometric Structures

A presymplectic AKSZ sigma-model is constructed from the following data:

• Source: The shifted tangent bundle $T[1]X$ of an $n$-dimensional spacetime $X$, equipped with the de Rham differential $d_X = \theta^\mu \partial / \partial x^\mu$, where $\theta^\mu$ are odd coordinates of degree $+1$ (Grigoriev et al., 2020, Grigoriev, 2022).
• Target: A graded fiber bundle $\pi \colon E \to T[1]X$ carrying a degree-$1$ vector field $Q$ (the BRST or homological structure), and a closed 2-form $\omega$ of total degree $n{-}1$, possibly degenerate (presymplectic) (Alkalaev et al., 2013, Dneprov et al., 2024).
• Compatibility conditions (“presymplectic master equations”):
  • $d\omega = 0$
  • $L_Q\omega \in I$ (vertical; $I$ is the ideal generated by forms pulled back from $T[1]X$).
  • $i_Q\omega + dL \in I$ for a Hamiltonian $L$ of degree $n$ (the “covariant Hamiltonian”).
  • For weak models: $i_Q i_Q \omega = 0$ (ensures closure after symplectic reduction) (Frias et al., 2024, Grigoriev, 2022).

An AKSZ sigma-model is then defined on the mapping space from the source to the target: fields are supermaps $\sigma\colon T[1]X \to E$. In the presence of degeneracies, a symplectic reduction yields minimal models corresponding to conventional BV (field, ghost, antifield) content (Dneprov et al., 2024, Alkalaev et al., 2013, Grigoriev, 2022).

2. From BV Action to Gauge PDEs and Jet Bundles

The presymplectic AKSZ formalism encodes both the action and the full-scale BV data (including the antibracket and BRST symmetry) as geometric and cohomological structures:

• BV master action:

$$S_{\mathrm{BV}}[\sigma] = \int_{T[1]X} \left( \sigma^*(\chi)(d_X) + \sigma^* L \right)$$

where $\chi$ is a presymplectic potential ($\omega = d\chi$). This construction encompasses all ghosts and antifields as components of the supermap $\sigma$ (Alkalaev et al., 2013, Grigoriev et al., 2020).

• Mapping space and jet prolongation: The total BRST differential becomes $s = d_X + Q$, and the construction naturally lives on the super-jet bundle $SJ^\infty(E)$, where horizontal (spacetime) and vertical (field) gradings are treated systematically (Grigoriev, 2022).
• Equation descent and boundaries: The framework is equipped for universal descent to lower-dimensional submanifolds (e.g., boundaries), providing a built-in link to the BFV (boundary) formalism, the covariant Hamiltonian formalism, and the multisymplectic approach (Canepa et al., 2020, Grigoriev, 2022).

3. Construction of Minimal and Regular Models

To control auxiliary fields, contractible pairs, and degeneracy, the formalism incorporates:

• Field–antifield content as symplectic quotient: The kernel of $\omega$ (including pure gauge degrees and contractible ghosts) is quotiented out; the action and the symplectic structure are transferred to a minimal model on a finite-dimensional vector bundle (Dneprov et al., 2024, Grigoriev, 2022).
• Uniqueness and algebraic minimality: In the formal (algebraic) setup the minimal presymplectic AKSZ model is unique up to isomorphism and corresponds to the minimal $L_\infty$ model of the full BRST algebra (Dneprov et al., 2024).
• Partial Lagrangians: In non-variational cases (where $\omega$ does not come from a Lagrangian), the action is “partial”: its critical locus contains the original equation manifold but may encompass additional solutions (Grigoriev, 2016).

4. Presymplectic AKSZ in Gravity, Supergravity, and Higher-Spin Theories

Presymplectic AKSZ models have yielded significant advances in the geometric formulation of gravitational and higher-spin theories:

• Einstein gravity: The Cartan–Weyl frame-like action and its full BV complex are encoded as a presymplectic AKSZ sigma model with target $\mathfrak{g}[1]$, where $\mathfrak{g}$ is the (A)dS or Poincaré algebra; the presymplectic structure is constructed from $\epsilon_{abcd}\,\xi^d\,d\xi^a\wedge d\rho^{bc}$, and the action matches the standard frame Lagrangian on shell (Grigoriev et al., 2020, Alkalaev et al., 2013).
• $N=1$, $D=4$ supergravity: The target is the Chevalley–Eilenberg complex of the super-Poincaré algebra, with an invariant presymplectic degree-3 form, and the resulting master action packages all ghosts, antifields, and the classical action, while manifestly realizing both spacetime, group-manifold (rheonomic), and superspace formulations as different choices of source (Grigoriev et al., 6 Mar 2025).
• Conformal gravity and higher-spin fields: Minimal (frame-like) models yield compatible presymplectic structures on the reduced total BRST complexes, producing new first-order frame-like actions and their BV extensions, with the off-shell gauge symmetry algebra fully encoded at the geometric level (Dneprov et al., 2022, Sharapov et al., 2021).
• Bigravity: The ghost-free massive bigravity theory admits a presymplectic AKSZ realization, with the target modeled as a quasi-regular submanifold of a product of two shifted Poincaré algebras and algebraic constraints encoding the Einstein–Hilbert and interaction sectors; the Deser–van Nieuwenhuizen constraints emerge as supergeometric relations in the model (Grigoriev et al., 2024).

5. Deformation Theory, Interactions, and Consistency Conditions

Consistent deformations—local, strictly kinetic-preserving interactions—are naturally analyzed within this framework:

• Homological deformation classification: Interactions correspond to deformations of the Hamiltonian $L$ and BRST vector field $Q$, while holding the presymplectic form $\omega$ fixed, with obstructions residing in the BRST cohomology $H^\bullet(Q_0, A)$, where $A$ is the algebra of Hamiltonian functions modulo the kernel of $\omega$ (Frias et al., 2024).
• Chern–Simons and Yang–Mills: The standard non-Abelian interactions are constructed from cocycles in this cohomology, with higher-order Maurer–Cartan equations describing compatibility and associativity, paralleling but not identical to traditional $L_\infty$ deformation theory (Frias et al., 2024, Sharapov et al., 2021).
• Minimal model spectrum: The same formalism recovers all known consistent cubic and higher-order couplings in both topological and non-topological gauge systems, including higher-spin gravity (Sharapov et al., 2021).

6. Extensions: Backgrounds, Homogeneous Spaces, and Generalized Symmetries

The presymplectic AKSZ approach has been extended to treat bundles over nontrivial base gauge PDEs (“background fields”) and to model generalized symmetries:

• Background fields: The formalism generalizes to bundles over a base $B$ which is itself a gauge PDE (e.g., gravity as a background for matter fields); the total $Q$ combines both field and background differentials, and compatibility must be checked modulo new vertical ideals (Dneprov et al., 15 May 2025).
• Homogeneous spaces and symmetry gauging: Homogeneous presymplectic gauge PDEs provide uniform models of fields with explicit $G$-symmetry, using bundles modeled on $G/H$ with appropriate Chevalley–Eilenberg differentials (Dneprov et al., 15 May 2025).
• Higher-form symmetries: Internal 1-form symmetries (e.g., in abelian gauge theory) and their gauging are captured using the presymplectic structure and background bundle framework, unifying the treatment of traditional and generalized symmetries (Dneprov et al., 15 May 2025).

7. Examples, Applications, and Outlook

Key instances of the presymplectic AKSZ framework include:

Theory	Target Structure	Key Features
Einstein gravity	Poincaré / (A)dS $\mathfrak{g}[1]$	Frame-like Cartan–Weyl action, BV master action, boundary BFV structure
$N=1$, $D=4$ supergravity	Super-Poincaré CE complex	Unified spacetime, group-manifold, and superspace formulations; BV master action
Higher-spin gravity	Extended HS algebra, $\mathrm{gl}(\mathcal{A})$	AdS/CFT-compatible presymplectic cocycles, unique couplings, Fradkin–Vasiliev interactions
Bigravity	Product of $\mathfrak{g}[1]$	Algebraic supergeometric realization of DN constraints; proper BV extension
Yang–Mills, Chern–Simons	$$\mathfrak{g}[1],\ \mathfrak{g}[1] \oplus \Lambda^2\mathfrak{g}$$	Homological construction of standard gauge interactions as cocycles
Maxwell, $p$-form fields	Simpler degree models	Covariant first-order actions, BF/BFV matching, higher-form symmetry gauging

The broad unifying feature is that local gauge theories with dynamical degrees of freedom and their BV formulations are encoded as geometrical data: a graded bundle $E \to T[1]X$, a homological vector field $Q$, and a regular compatible (possibly degenerate) closed 2-form $\omega$ of degree $n-1$. Interactions, symmetries, and consistent truncations are systematically analyzed within this algebraic–geometric structure.

The presymplectic AKSZ construction forms the supergeometric foundation for modern Lagrangian, Hamiltonian, and BV-BFV approaches to local gauge field theory, and is playing a central role in the development of the geometry of PDEs, boundary dynamics, quantum field theory, and applications to mathematical physics (Grigoriev, 2022, Dneprov et al., 2024, Dneprov et al., 2022, Grigoriev et al., 6 Mar 2025, Dneprov et al., 15 May 2025, Alkalaev et al., 2013, Sharapov et al., 2021, Frias et al., 2024).