AKSZ Formulation in Field Theories

• AKSZ formulation is a geometric framework that uses graded manifolds and QP-structures to systematically construct BV master actions for field theories.
• It extends topological sigma models to include generalized gauge theories, presymplectic extensions, and higher-spin models in a unified manner.
• The method embeds source and target geometries into mapping spaces, streamlining quantization and cohomological deformation of gauge symmetries.

The AKSZ formulation, named after Alexandrov, Kontsevich, Schwarz, and Zaboronsky, is a geometric framework for constructing a wide class of topological field theories using graded symplectic and supergeometric structures. Originally designed to provide a systematic Batalin–Vilkovisky (BV) construction of topological sigma models, it has evolved to encompass generalized gauge theories, higher and derived structures, non-topological theories via presymplectic and frame-like extensions, and even models with backgrounds, higher-form symmetries, and interactions. Central to the AKSZ approach is the idea of identifying the configuration space of a field theory with the mapping space between a source graded manifold (often encoding the spacetime de Rham complex) and a target QP-manifold (i.e. a graded symplectic manifold equipped with a compatible homological vector field), from which the BV master action and observables are constructed by transgression of the symplectic and Hamiltonian data.

1. Core Formalism: Graded Geometry, QP-Manifolds, and the AKSZ Prescription

The AKSZ construction is built upon the following data:

• A source graded manifold $T[1]X$, typically the shifted tangent bundle of a manifold $X$ with the de Rham differential (serving as a cohomological vector field).
• A target QP-manifold $(M, \omega, Q)$, where $M$ is a graded manifold, $\omega$ is a graded symplectic form of degree $n$, and $Q$ is a Hamiltonian cohomological vector field (i.e., $Q^2 = 0$, generated by a Hamiltonian function $\Theta$ satisfying $\{\Theta, \Theta\} = 0$).
• The mapping space $\mathcal{F} = \mathrm{Map}(T[1]X, M)$, forming the space of (super)fields.

The AKSZ BV action on $\mathcal{F}$ is defined as: $$S[\Phi] = \int_{T[1]X} \left( \langle \Phi^*\vartheta, d_{T[1]X} \Phi \rangle + \Phi^*\Theta \right)$$ where $\vartheta$ is the Liouville 1-form for $\omega$. The master equation $(S, S) = 0$ is satisfied automatically by virtue of the QP-structure. This construction naturally accommodates the entire gauge structure within the theory, encoding both the fields and all ghost, antifield, and symmetry data as graded components of the supermap $\Phi$.

Notable examples include the Poisson sigma model (n = 1), the Courant sigma model (n = 2), and higher BF or Chern–Simons theories for larger $n$ (Ikeda, 2012).

2. Algebraic and Homological Underpinnings

A fundamental aspect of the AKSZ framework is its encoding of the master equation and gauge symmetries in homological algebraic terms. The QP-manifold structure generalizes Lie algebraic and algebroid data (e.g., Lie algebroids, Courant algebroids, higher Lie n-algebroids) to the graded setting. The homological vector field reflects the gauge transformations and their higher analogues, while the symplectic form on the mapping space yields the odd Poisson (BV) bracket required for consistent quantization.

Vector fields on mapping spaces play a decisive role: the master action $S$ generates a Hamiltonian vector field corresponding to the "difference" of the source and target homological fields (Voronov, 2012). The converse result—namely, that the existence of a gradient vector field on the mapping space determines the AKSZ data—highlights the geometric inevitability of the AKSZ-type construction in the context of graded symplectic BV theory.

3. Generalizations: Presymplectic, Frame-like, and Higher-Spin AKSZ Models

For non-topological and locally nontrivial field theories, the conventional AKSZ requirement of a nondegenerate symplectic structure on the target is relaxed. In the presymplectic AKSZ framework, the target is a graded (super)manifold with a closed but possibly degenerate two-form $\omega$ and a compatible homological vector field $Q$ (Grigoriev et al., 2020, Alkalaev et al., 2013). This generalization accommodates systems with local degrees of freedom (such as gravity, Yang-Mills, or higher-spin fields), which are inaccessible to the standard AKSZ models with finite-dimensional, nondegenerate targets.

• For Einstein gravity, the target is the shifted Poincare or (A)dS algebra, the presymplectic potential and Hamiltonian are built from the Lie algebra data, and the full Cartan–Weyl Lagrangian (and its complete BV extension) is encoded supergeometrically. The quotient by the kernel of $\omega$ recovers the usual BV phase space, and the corresponding Hamiltonian and Lagrangian formulations are unified (Grigoriev et al., 2020, Alkalaev et al., 2013).
• For higher-spin gravity, the presymplectic AKSZ formulation leverages cyclic and Hochschild cohomology of the underlying higher-spin algebra to classify admissible presymplectic forms, yielding a two-parameter family of models capturing all expected couplings and symmetry currents (Sharapov et al., 2021).

In this context, frame-like and multi-frame models, parameterized systems, and gauge theories on homogeneous spaces are all captured as instances of presymplectic AKSZ-type sigma models.

4. AKSZ Construction with Backgrounds and Homogeneous Structures

The flexibility of the AKSZ construction extends to systems with background fields or curved geometries by representing the field theory data as a tower of gauge PDE (partial differential equation) bundles over spacetime (Dneprov et al., 15 May 2025). The presymplectic gauge PDE formalism encodes theories whose base space is itself a gauge PDE (e.g., describing background geometry, connections, or other fields), leading to a bundle-of-bundles structure. This is crucial when handling background-dependent scenarios like linearizations, gauge fields over nontrivial backgrounds, or deformations such as Fedosov quantization.

Homogeneous presymplectic gauge PDEs precisely capture gauge systems on spaces with large symmetries—examples include Fronsdal-type higher-spin fields on (A)dS or conformal spaces. Moreover, higher-form symmetries and their gauging emerge as natural extensions simply by enlarging the vertical symmetry algebra, with conserved currents arising from vertical symmetry vector fields on the gauge PDE.

5. Examples and Canonical Applications

The AKSZ framework provides a unifying structure for a large spectrum of field theories:

• Poisson sigma model (PSM): Target $T^*[1]M$ with symplectic structure $\omega$ and Hamiltonian $\Theta$ encoding the Poisson structure; quantization of the PSM yields Kontsevich's deformation quantization (Ikeda, 2012).
• Courant sigma model: The target encodes a Courant algebroid, and the action reproduces the topological structure of 3D models, incorporating H-fluxes (Marotta et al., 2021).
• Higher (semistrict) gauge theory: 3D and 4D semistrict higher BF and Chern–Simons theories are constructed using target Lie 2-algebra data, superfields, and appropriate automorphism 2-group structures (Zucchini, 2011).
• Classical mechanics as AKSZ: Classical Hamiltonian dynamics may be reformulated as a 1D AKSZ sigma model with an extended (super)phase space; gauge fixing then recovers the Koopman–von Neumann and GRT path integral formulations (Basile et al., 26 Apr 2025).
• AKSZ with reduction: By encoding target reduction data (e.g., momentum maps corresponding to Lie group symmetries) via BFV extensions, one can systematically construct "gauged" or reduced models, establishing the commutativity of quantization and reduction in the AKSZ setting (Bonechi et al., 2012).
• General relativity and supergravity: Presymplectic AKSZ formalism captures both the Einstein–Hilbert and Cartan–Weyl (frame) formulations, including compatible BV–BFV structures required for quantization with boundaries (Canepa et al., 2020), and has been systematically elaborated for $N=1, D=4$ supergravity (Grigoriev et al., 6 Mar 2025).
• Poly-symplectic geometry: The AKSZ formalism extends to poly–Poisson and poly–symplectic target spaces, generalizing the Poisson sigma model and yielding theories with vector-bundle valued symplectic forms and associated higher currents (Contreras et al., 2019).
• AKSZ in derived algebraic geometry: Functorial extensions yield families of extended TQFTs valued in higher categories of symplectic derived stacks, transferring symplectic and duality structures via integration modeled on Poincaré–Lefschetz/​Verdier duality (Calaque et al., 2021).

6. Deformation Theory, Interactions, and Consistent Couplings

A key strength of the AKSZ construction is its compatibility with cohomological deformation theory. Consistent interactions and deformations correspond to simultaneous deformations of the homological vector field $Q$ and the covariant Hamiltonian (BV Lagrangian) $L$, subject to compatibility with the (pre)symplectic structure. The deformation theory is Maurer–Cartan–like but "doubled," with intricate recursion relations and potential cohomological obstructions appearing at higher orders (Frias et al., 28 Dec 2024).

For presymplectic AKSZ models, the recent literature develops methods for classifying and extending interactions without deforming $\omega$, explicitly rederiving Chern–Simons and Yang–Mills interactions from first principles in the supergeometric context. This technique systematically identifies allowed cubic vertices, checks Jacobi identities, and tracks higher-order consistency.

7. Outlook and Impact

The AKSZ formulation remains central to the intersection of mathematical and theoretical physics, providing a rigorous and flexible language for the BV quantization of gauge and topological field theories and their generalizations. Its reach now includes:

• Locally nontrivial physics (e.g., gravity, supergravity, higher-spins) via presymplectic generalizations.
• Geometric quantization, deformation quantization, and connections to formality theorems.
• Systematic handling of backgrounds, higher-form symmetries, and symmetry gauging.
• Encapsulation of algebroid, group, and higher-categorical data as geometric, supermanifold structures.
• Full compatibility with homological algebraic techniques—spectral sequences, cohomology, and reduction procedures.

Major directions of ongoing research involve further functorial enhancements (as in derived algebraic geometry or higher categories), treatments of explicit quantum anomalies via global or differential master equations (Moshayedi, 2020), and expansive applications in generalized and doubled geometry for string/M-theory compactifications (Marotta et al., 2021).

The presymplectic and gauge PDE-based approaches, together with a robust deformation theory, now render the AKSZ framework adaptable to virtually any contemporary context in gauge and topological field theory, as well as to intricate geometric and algebraic structures beyond classical BV theory.