Shifted Symplectic Structures

Updated 13 October 2025

Shifted symplectic structures are closed 2-forms of degree n that induce quasi-isomorphisms between the tangent and cotangent complexes in derived settings.
They extend classical symplectic and Poisson geometry to derived moduli spaces, allowing the formal treatment of deformation theory, quantization, and enumerative invariants.
The framework supports local Darboux theorems and plays a key role in constructing moduli spaces for perfect complexes, mapping stacks, and Lagrangian intersections.

Shifted symplectic structures are a fundamental generalization of classical symplectic forms, developed in the context of derived algebraic geometry to capture both global and deformation-theoretic aspects of moduli spaces. They enable rigorous treatments of moduli problems, deformation quantization, intersection theory, and enumerative invariants in contexts with higher (derived, stacky, or graded) structure. The theory unifies and extends classical results in symplectic geometry, Poisson geometry, moduli theory, mathematical physics, and arithmetic geometry.

1. Foundational Definition and Context

The notion of an $n$ -shifted symplectic structure on a derived Artin $n$ -stack $X$ or on an object in higher or derived geometry is specified by a closed 2-form $\omega$ of cohomological degree $n$ ,

$\omega \in \mathcal{A}^{2,\mathrm{cl}}(X,n)$

such that the induced morphism on the (homotopy) tangent complex,

$\omega_0^\sharp : T_X \longrightarrow L_X[n]$

is a quasi-isomorphism (i.e., an equivalence in the derived category). Here $L_X$ denotes the cotangent complex of $X$ . The closedness involves additional "higher" components in the derived or graded de Rham algebra, with appropriate compatibility conditions encoding both de Rham and homological differentials.

This definition generalizes classical symplectic geometry (the case $n=0$ for smooth manifolds or schemes) and Poisson geometry (via $n=1$ and the correspondence between 1-shifted symplectic structures and exact Dirac pairs) to the setting of derived stacks, groupoids, higher categories, and even analytic or arithmetic spaces.

2. Local Normal Forms and the Shifted Darboux Theorem

For $n < 0$ , the local structure of shifted symplectic derived schemes and stacks is governed by generalized Darboux theorems. Any $k$ -shifted symplectic derived scheme (for $k<0$ ) is locally modeled on a standard cdga with coordinates and a "Darboux-form" symplectic structure, with the shift encoded in the grading and differential. In the $k = -1$ case, such schemes are Zariski-locally equivalent to a derived critical locus $\mathrm{Crit}(\Phi)$ for a function $\Phi$ (the Hamiltonian), making the classical truncation into a "d-critical locus" in the sense of Joyce (Brav et al., 2013, Ben-Bassat et al., 2013).

This equivalence allows for new construction and computation of invariants (virtual cycles, perverse sheaves, motives) and supports the extension of obstruction theories in Donaldson–Thomas theory for Calabi–Yau spaces.

3. Construction on Moduli Spaces and Mapping Stacks

Canonical shifted symplectic structures appear naturally on fundamental moduli spaces. The prime examples include:

Classifying stacks of reductive groups: These admit 2-shifted symplectic structures arising from the Killing form, connecting with moduli of principal bundles and representation theory.
Moduli of perfect complexes: The derived stack $\mathsf{RPerf}$ carries a canonical 2-shifted symplectic structure constructed using the universal Chern character and the Atiyah class, leading to a nondegenerate closed 2-form (as in (Pantev et al., 2011)).
Mapping stacks: If $F$ has an $n$ -shifted symplectic structure and $X$ is Calabi–Yau of dimension $d$ , then $\operatorname{Map}(X,F)$ carries a canonical $(n-d)$ -shifted symplectic structure. This yields a rich source of derived moduli spaces with shifted symplectic forms (e.g., moduli of perfect complexes or of local systems on oriented manifolds).
Lagrangian intersections: If $Y$ and $Z$ are Lagrangians in an $n$ -shifted symplectic stack $X$ , then their derived fiber product $Y \times_X Z$ carries a canonical $(n-1)$ -shifted symplectic structure.

4. Equivalences, Functoriality, and Higher Groupoids

The theory extends to higher and stacky contexts using Lie $n$ -groupoids and differentiable formal stacks (see (Cueca et al., 10 Oct 2025, Cueca et al., 2021)). In this setting:

Shifted symplectic forms on Lie $n$ -groupoids are encoded by collections of forms $\omega_\bullet = \sum_{i=0}^m \omega_i$ on the simplex levels, satisfying a total closedness condition $D\omega_\bullet = 0$ .
The non-degeneracy is formulated via the infinity–multiplicative (IM) pairing, requiring the induced map from the tangent to shifted cotangent complex to be a quasi-isomorphism.
Morita invariance: These structures descend under Morita equivalence, so shifted symplectic forms are objects inherent to the corresponding higher stack.

Correspondences and functoriality appear through the symmetric monoidal functor from the span category of perfect derived prestacks to the category of Lagrangian correspondences, preserving shifted symplectic and Lagrangian data under compositions (see (Calaque et al., 11 Jul 2024)).

5. Key Constructions: Cotangent Bundles, Groupoids, Zero Loci

Shifted Cotangent Stacks: If $X$ is a derived stack (with perfect cotangent complex), then its $n$ -shifted cotangent bundle $T^*[n]X$ admits a canonical $n$ -shifted symplectic structure constructed from the universal 1-form and its de Rham differential (Calaque, 2016, Calaque et al., 11 Jul 2024).
Shifted Symplectic Groupoids: These structures generalize symplectic groupoids in Poisson geometry to the derived, shifted setting, with the groupoid object equipped with multiplicative, closed shifted forms. The quotient stack inherits an $(n+1)$ -shifted symplectic structure (Calaque et al., 11 Jul 2024).
Symplectic Zero Loci and Twisted Cotangent Bundles: Symplectic zero loci—homotopy fibers of isotropic sections of vector bundles—carry natural $(-2)$ -shifted symplectic structures. Twisted cotangent bundles, as symplectic pushforwards, produce new local models and underpin the local structure theorems of symplectic fibrations (Park, 27 Jun 2024).

6. Deformation Theory, Virtual Cycles, and Enumerative Geometry Applications

A central theme is the use of shifted symplectic structures in defining and understanding enumerative invariants:

Deformation to the normal cone: For Lagrangian morphisms, the deformation space carries a relative (twisted) shifted Lagrangian structure; at the special fiber, one recovers the shifted conormal bundle—a construction vital for virtual fundamental class arguments (Calaque et al., 11 Jul 2024, Park, 27 Jun 2024).
Virtual Lagrangian cycles in Donaldson–Thomas theory: For $(-2)$ -shifted symplectic fibrations (e.g., moduli of perfect complexes on Calabi–Yau 4-folds), a unique functorial bivariant class $[M/B]_\mathrm{Lag}$ is associated, invariant under deformation and controlling counts of stable objects (Park, 27 Jun 2024).
Relation to perverse sheaves and motives: In the $(-1)$ -shifted (d-critical) setting, categorical and motivic extensions of Donaldson–Thomas theory are constructed, utilizing gluing of local vanishing cycle sheaves and motives (Ben-Bassat et al., 2013).

7. Poisson Duality, Quantization, and New Geometric Contexts

Poisson structures: There is an equivalence between nondegenerate (weighted) shifted symplectic and shifted Poisson structures, structurally encoded via derived Schouten–Nijenhuis brackets and DGLAs (Pym et al., 2016, Pridham, 2022).
Quantization: Shifted symplectic forms control the deformation quantization of moduli spaces, providing quantizations as solutions to quantum master equations or in the homotopy BV formalism (Safronov, 2020, Pridham, 2018).
Generalized complex and holomorphic symplectic geometry: Generalized complex structures are reformulated as holomorphic symplectic stacks, with coisotropic intersections inheriting shifted Poisson structures, extending the reach of the theory into new geometric domains (Qin, 22 Jul 2024).

The theory of shifted symplectic structures connects derived, stacky, and graded symplectic geometry, integrating methods from deformation theory, moduli spaces, Poisson and quantization theory, and mathematical physics. Its applications span from the structure of moduli of sheaves and local systems, Donaldson–Thomas invariants, and categorified enumerative geometry, to the geometric foundations of field theory and arithmetic geometry. The generality, functoriality, and compatibility of shifted symplectic forms with modern derived geometric machinery make them an indispensable tool in contemporary algebraic and differential geometry.