Symplectic Q-Manifolds in Derived Geometry

• Symplectic Q-manifolds are non-negatively graded manifolds with a closed, nondegenerate 2-form and a degree-1 homological vector field that encode Poisson, Courant, and higher derived structures.
• They facilitate the translation of classical geometric data into derived contexts, underpinning applications in deformation quantization and the AKSZ formalism for topological field theories.
• Their framework supports rigorous reduction theories and canonical models for Lagrangian submanifolds, ensuring integrability in graded symplectic and homotopy Poisson geometries.

A symplectic Q-manifold is a non-negatively graded manifold equipped with a closed, nondegenerate 2-form of fixed degree (the "symplectic form") and a homological vector field (Q-structure) of degree 1 that preserves the symplectic form. This structure encodes both Poisson and more generally, higher derived geometric data such as Lie algebroids, Courant algebroids, and their quantizations. Symplectic Q-manifolds are central objects in derived geometry, the theory of shifted symplectic structures, and mathematical physics, particularly in the AKSZ formalism for topological field theories.

1. Graded Symplectic Geometry and Homological Vector Fields

A graded manifold $M$ is specified by a sheaf of functions locally modeled on $C^\infty(U) \otimes S^\bullet(V)$ with generators of positive degree. A symplectic structure of degree $m$ is a closed 2-form $\omega \in \Omega^2_m(M)$ such that the map $$\omega^\flat: \mathfrak{X}^{1}_\bullet(M) \to \Omega^{1}_{\bullet+m}(M)$$, $X \mapsto \iota_X \omega$, is an isomorphism. The homological vector field $Q \in \mathfrak{X}^{1}_{1}(M)$ satisfies $Q^2 = \frac{1}{2}[Q, Q] = 0$, turning the algebra of functions into a cochain complex.

In the presence of $\omega$, functions $f \in C_k(M)$ admit Hamiltonian vector fields $X_f$ defined by $\omega^\flat(X_f) = df$, and the graded Poisson bracket

$\{f,g\} = \iota_{X_f} \iota_{X_g} \omega$

has degree $-m$. In a symplectic Q-manifold $(M, \omega, Q)$, one requires $L_Q \omega = 0$. In important cases, $Q = X_\Theta$ for a Hamiltonian $\Theta \in C_{m+1}(M)$, imposing the classical master equation $\{\Theta,\Theta\}=0$.

The standard models include degree-1 symplectic Q-manifolds $(T^*[1]M, d\theta, Q_\pi)$ encoding Poisson structures, and degree-2 models corresponding to Courant algebroids via cubic Hamiltonian functions.

2. Lagrangian Q-Submanifolds and the Graded Weinstein Theorem

A submanifold $j: L \hookrightarrow M$ is Lagrangian when $j^*\omega = 0$ and the induced map from the graded normal bundle (defined via the mapping cone of tangent complexes) to $T^*L[m-1]$ is a quasi-isomorphism. For Q-submanifolds, the vanishing ideal $I_L$ is preserved by $Q$: $Q(I_L) \subset I_L$.

In the graded setting, the (graded) Weinstein tubular neighborhood theorem ensures that near a Lagrangian Q-submanifold $L$ of an $m$-symplectic manifold $M$, there exists a symplectomorphism $\Psi: T^*[m]L \to M|_U$ identifying the zero section with $L$. This theorem is pivotal for deformation theory and for constructing canonical models of Lagrangians in derived contexts.

3. Derived Brackets, Homotopy Poisson Structures, and Reduction

The construction of symplectic Q-manifolds is intimately tied to homotopy Poisson structures, where graded manifolds are endowed with multilinear brackets $\beta_\ell: S^\ell(A[1]) \to A[1]$ (for algebra $A = C^\infty(M)$) via derived bracket formulas involving degree $2-\ell$ multivector fields $\pi_\ell$; integrability requires $[\pi, \pi]=0$ for $\pi = \sum_\ell \pi_\ell$.

Reduction theory extends Marsden–Weinstein-type results using homotopy Poisson Lie group actions equipped with multiplicative multivector fields $\varphi$ satisfying $[\varphi,\varphi]=0$; symmetry reduction of symplectic Q-manifolds leads to quotients that inherit compatible homological and symplectic structures. Applications range from cotangent-lifted Poisson reduction to generalized Courant and higher reductions (Mehta, 2010).

4. Degree-One and Degree-Two Symplectic Q-Manifolds: Poisson and Courant Geometry

Degree-One: Every degree-one symplectic Q-manifold is (noncanonically) isomorphic to $T^*[1]M$ with canonical symplectic form, and corresponds bijectively to Poisson manifolds (Vitagliano, 2014). The homological vector field $Q_\pi$ recasts both classical and graded Poisson data.

Degree-Two: Courant algebroids over $M$ (with symmetric pairing $\langle \cdot,\cdot \rangle$, anchor $\rho$, and bracket) are equivalent to degree-2 symplectic Q-manifolds $(T^*[2]M \oplus E[1], \omega, Q_\Theta)$, with the cubic Hamiltonian $\Theta$ encoding the Dorfman bracket and other structure. Quantization is achieved via the Weyl map $\mathcal{WQ}_\hbar$ from functions on the graded manifold to differential operators on spinor bundles; skew-symmetric Dirac generating operators $D$ produce scalar invariants $D^2$ generalizing classical Cartan forms (Grützmann et al., 2014).

5. Higher Lie Groupoids, Shifted Symplectic Structures, and Morita Invariance

Symplectic Q-manifolds provide the infinitesimal models ("differentiation") of higher Lie groupoids with shifted symplectic structures; e.g., Lie 1-groupoids yield degree-1 symplectic Q-manifolds, Lie 2-groupoids yield degree-2 objects corresponding to Courant algebroids. Shifted symplectic forms $\omega_\bullet$, defined over the simplicial levels, are Morita invariant (i.e. intrinsic to the differentiable stack) (Cueca et al., 10 Oct 2025). Examples include:

• The classifying stack $BG$ for a Lie group $G$ with a 2-shifted symplectic form, linking to the Atiyah–Bott construction.
• Groupoid models for group-valued moment maps.
• Double symplectic and Courant groupoids integrating derived Poisson and Courant structures.

These constructions underpin the AKSZ formalism: mapping spaces $\operatorname{Maps}(T[1]N, M)$ between source supermanifolds and symplectic Q-manifolds serve as targets for topological field theories with the classical master equation governing their BV actions.

6. Analytic, Topological, and Representation-Theoretic Aspects

Cohomological tools such as Tseng–Yau’s symplectic Bott–Chern and Aeppli groups (Tardini et al., 2016), as well as elliptic complexes (e.g., trace-free and coupled BGG complexes (Eastwood et al., 2017)), generalize harmonic theory and characterize invariants (e.g., the Hard Lefschetz condition). These invariants facilitate the paper of global properties and the deformation theory of symplectic Q-manifolds, including non-Kähler examples (Bogomolov–Guan manifolds and their period maps, Fujiki relations, and Jordan automorphism groups (Kurnosov et al., 2019, Bogomolov et al., 2020)).

Quaternionic and generalized complex structures, as constructed in the product of symplectic manifolds (Pantilie, 2011, Jiménez-Pérez, 2019), provide further geometric frameworks illuminating the relationship between symplectic, Poisson, and generalized geometries. These lead to twistor spaces and generating function techniques.

In representation theory and combinatorics, intermediate symplectic Q-functions (Yanagida, 2022) interpolate between Schur's Q-functions and Okada's symplectic Q-functions, offering tableau-sum and Pfaffian formulas with applications to symplectic Q-manifold invariants.

7. Applications in Mathematical Physics and Emerging Directions

Symplectic Q-manifolds serve as the targets for AKSZ-type topological quantum field theories. They underlie BV formalism, deformation quantization (as in the Poisson sigma model recasting Kontsevich's star product), and moment map reduction in derived geometry. New developments such as $q$-cosymplectic geometry (Leok et al., 7 Sep 2025, Leok et al., 20 Sep 2025) generalize time-dependent and multi-time structures, providing integrable systems with multiple commuting Reeb fields, Marsden–Weinstein-type reduction, and Lie integrability for multi-time Hamiltonian systems (e.g., the extended FitzHugh–Nagumo model).

Further advances include the paper of scattering symplectic manifolds as symplectic structures on Lie algebroids, glueing constructions via weak fillings, and normal form theorems relating degenerate Poisson structures to graded symplectic geometry (Alboresi, 2017).

Summary Table: Canonical Structures in Symplectic Q-Manifolds

Degree	Model	Encoded Data
1	$T^*[1]M$, $Q_\pi$	Poisson manifold
2	$T^*[2]M \oplus E[1]$, $Q_\Theta$	Courant algebroid
$m$	$T^*[m]M$, general Hamiltonians	Derived higher structures

Symplectic Q-manifolds unify graded symplectic geometry, homotopical Poisson and Lie structures, and their quantizations. Their morphisms, reductions, and canonical neighborhoods are central to modern developments in geometry and mathematical physics, with broad implications for representation theory, deformation quantization, and the theory of differentiable stacks.