Shifted Symplectic Geometry

• Shifted symplectic geometry is a framework in derived algebraic geometry that generalizes classical symplectic structures by incorporating closed, nondegenerate 2-forms in arbitrary cohomological degrees.
• Its local models, using shifted Darboux charts and twisted cotangent bundles, provide explicit computational tools for handling singularities and complex moduli problems.
• The theory underpins key applications in quantization, Donaldson–Thomas theory, and TQFTs, bridging traditional geometric insights with modern derived and higher-categorical methods.

Shifted symplectic geometry is a framework within derived algebraic geometry and higher differential geometry that generalizes classical symplectic structures to objects (schemes, stacks, or moduli spaces) governed by differential graded or ∞-category theory. Instead of working solely with non-degenerate closed 2-forms in degree zero, shifted symplectic geometry studies closed, non-degenerate 2-forms living in arbitrary integer cohomological degrees—called “shifts”—on derived schemes, stacks, and related objects. Introduced by Pantev, Toën, Vaquié, and Vezzosi, the theory enables rigorous formulation and manipulation of geometric structures on spaces with singularities, moduli of complexes, mapping stacks, and intersections, with deep connections to quantization, Donaldson–Thomas theory, higher categories, and field theory.

1. Foundations: Definitions and Local Structure

A $k$-shifted symplectic structure on a derived stack $X$ is specified by a closed 2-form $\omega$ of cohomological degree $k$,

$\omega \in \mathcal{A}^2(X, k),$

that is nondegenerate in the sense that it induces a quasi-isomorphism

$\omega^\sharp: \mathbb{T}_X \to \mathbb{L}_X[k],$

where $\mathbb{T}_X$ and $\mathbb{L}_X$ are the tangent and cotangent complexes of $X$, respectively. This generalizes the classical case ($k=0$) to the derived and stacky setting (Pantev et al., 2011).

To make this structure concrete and computationally tractable, shifted symplectic geometry provides explicit local models—derived analogues of Darboux’s theorem. For $k<0$, a $k$-shifted symplectic derived scheme or stack admits, in the Zariski or étale topology, local presentations as “twisted” shifted cotangent bundles. This means that locally,

• The algebra of functions is a commutative differential graded algebra (cdga) of “standard form” built from a classical algebra by adjoining generators in negative degrees.
• The symplectic form is written in standard coordinates $(x_i, y_i)$ as

$\omega^0 = \sum_{i} d_{dR} x_i \wedge d_{dR} y_i,$

i.e., a shifted Darboux form.

• The differential on the cdga is given by the shifted Poisson bracket with a Hamiltonian $H$ of degree $k+1$, satisfying the classical master equation $\{H, H\}=0$ (Brav et al., 2013).

When $k=-1$, such a local model can always be taken to be the derived critical locus $\operatorname{Crit}(f)$ of a function $f$ on a smooth scheme, linking the theory to critical points, obstruction theories, and virtual cycles.

2. Lagrangians, Intersections, and Neighbourhood Theorems

A morphism $f: L \to X$ from a derived stack $L$ to a $k$-shifted symplectic stack $X$ can carry a Lagrangian structure, which in the derived context means the isotropic structure is (homotopically) maximally nondegenerate relative to $\omega$. The shifted Lagrangian Neighbourhood Theorem provides explicit local models for pairs $(X, L)$:

• $X$ is locally modeled by a shifted Darboux chart as above.
• The Lagrangian $L$ is presented as a twisted shifted conormal bundle, encoded by a submersion $A^\bullet \to B^\bullet$ of standard form cdgas (for the Lagrangian) and compatible explicit isotropic data (Joyce et al., 2015).

This local trivialization of both the shifted symplectic structure and the Lagrangian embedding is crucial for understanding intersections: If $L_1, L_2$ are Lagrangians in an $n$-shifted symplectic stack, their derived fiber product $L_1 \times_X^h L_2$ is canonically $(n-1)$-shifted symplectic. This mechanism yields strong structural theorems for moduli of sheaves, mapping spaces, and intersections, including the original virtual fundamental cycle construction and categorification in Donaldson–Thomas theory (Pantev et al., 2011, Brav et al., 2013, Grataloup, 2023).

3. Mapping Stacks and Transgression

A major source of examples comes from mapping stacks. If $Y$ is an $n$-shifted symplectic stack and $X$ is a $d$-dimensional Calabi–Yau space or a $d$-oriented derived stack, then the derived mapping stack $\operatorname{Map}(X, Y)$ acquires a canonical $(n-d)$-shifted symplectic form. The construction uses an integration (transgression) map, typically expressed via the evaluation map and the trace (Poincaré duality) on $X$:

$$\int_{[X]}: \mathcal{A}^p(Y, n) \to \mathcal{A}^p(\operatorname{Map}(X,Y), n-d).$$

This framework not only produces canonical 2-shifted symplectic forms on classifying stacks and moduli of complexes, but also underpins applications to moduli of local systems, flat bundles, and branes. The corresponding derived intersection theory ensures the “closedness” and quasi-isomorphism (nondegeneracy) conditions descend to mapping stacks (Pantev et al., 2011, Katzarkov et al., 2017, Spaide, 2016).

4. Structure Theorems for Moduli and Critical Loci

The geometry of derived moduli stacks—such as the derived stack $\mathrm{RPerf}$ of perfect complexes or the classifying stack $BG$—is governed by canonical shifted symplectic structures. In particular, on $\mathrm{RPerf}$, the 2-shifted symplectic structure is constructed via the Chern character $\mathrm{Ch}_2$ of the universal perfect complex:

$$\mathrm{Ch}_2(\mathcal{E}) = \operatorname{Tr}(\text{mult}) \in H^2(\mathrm{RPerf}, \mathcal{A}^2_{L_{\mathrm{RPerf}}}),$$

where the trace of multiplication on endomorphisms realizes the canonical pairing on the (shifted) tangent complex $A[1]$ (Pantev et al., 2011). This nondegenerate 2-form both reflects and lifts classical invariants in moduli spaces and recovers symmetric obstruction theories on classical truncations.

The Darboux theorem for shifted symplectic derived Artin stacks (Brav et al., 2013, Ben-Bassat et al., 2013) shows that in the $k<0$ case, these moduli spaces, even when stacky, admit local charts in which the symplectic structure and the moduli problem become explicitly computable. Applications include:

• Realization of derived moduli stacks as (derived) critical loci, bringing to bear tools from singularity theory, perverse sheaves, and motivic vanishing cycles.
• The construction of d-critical structures on truncations, perverse sheaves of vanishing cycles, and associated motives (Ben-Bassat et al., 2013).

5. Shifted Symplectic Groupoids, Lie Algebroids, and Poisson Geometry

The theory extends naturally to Lie groupoid and Lie algebroid settings, where it provides higher-categorical and stacky generalizations of integration of Poisson and Dirac structures. An $n$-shifted symplectic Lie algebroid on $X$ is an $L_\infty$-algebroid $L$ such that the stack $[X / L]$ carries a shifted symplectic structure, with nondegeneracy of the corresponding higher 2-form (Pym et al., 2016). In degrees $n=0,1,2$, this classification recovers classical structures:

• 0-shifted: Lie algebroids with closed two-form transverse structures (foliations with symplectic leaves).
• 1-shifted: exact Dirac structures and Courant algebroids (twisted by 3-forms).
• 2-shifted: twisted Courant algebroids, with the twisting class arising from gerbes or codimension-two cycles.

In the groupoid case, one defines $n$-shifted symplectic Lie $k$-groupoids and establishes Morita invariance, showing that different presentations of the same stack yield canonically equivalent shifted symplectic structures. This is crucial for relating explicit finite/infinite-dimensional models for classifying stacks (such as $BG$) and their associated TQFTs (Cueca et al., 2021).

Symplectic reduction, moment map theory, and intersection theory are all recast in this context as homotopy pullbacks of Lagrangians, ensuring that reduction always results in the correct derived and shifted symplectic structure on the quotient or moduli space (Grataloup, 2023).

6. Applications: Quantization, Donaldson–Thomas Theory, and Beyond

Shifted symplectic geometry enables the formulation of deformation quantization, geometric quantization, and TQFTs in derived and higher-categorical frameworks:

• Quantization: Prequantizations (via gerbes and connections) and their global existence are ensured by descent properties, allowing extension from local computations to global data on derived stacks (Safronov, 2020).
• Donaldson–Thomas Theory: The –1 and –2-shifted symplectic structures on moduli of sheaves or perfect complexes underlie virtual fundamental cycles, categorified and motivic invariants, and the functorial construction of Lagrangian correspondences required for wall-crossing and enumerative invariants (Pantev et al., 2011, Ben-Bassat et al., 2013, Park, 27 Jun 2024, Kiem et al., 28 Apr 2025).
• Categorical and Higher Structures: The definition of higher Fukaya categories associated to $(n-1)$-shifted stacks, constructions of TQFTs via 1-shifted Lagrangian correspondences (e.g., the Moore–Tachikawa conjecture), and shifted AKSZ–type field theories all arise naturally in this framework (Crooks et al., 5 Sep 2024, Pascaleff et al., 26 Mar 2025).

The recent development of shifted contact geometry, equivalences for pushforwards along base change, and the paper of shifted Poisson and coisotropic structures further indicate the breadth of applications, allowing one to handle derived (coisotropic) intersections, reductions, and quantizations in both classical and modern contexts (Maglio et al., 2023, Calaque et al., 11 Jul 2024, Qin, 22 Jul 2024).

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In summary, shifted symplectic geometry situates classical symplectic and Poisson geometry within derived and higher-categorical settings, providing canonical structures on moduli and mapping stacks, local normal forms, and a toolkit for quantization, categorification, and enumerative invariants. The formalism captures and unifies classical constructions while extending them to singular, stacky, and noncommutative geometries, reflecting both the foundational and computational strengths of the theory.