Exact Shifted Symplectic Structures

• Exact shifted symplectic structures are advanced geometric frameworks on derived stacks, defined by closed, nondegenerate forms with global Liouville primitives.
• They link nondegenerate Poisson structures to symplectic forms through formal derivations, enabling unique deformation quantization schemes across various shifts.
• Applications include the construction of virtual cycles in moduli spaces and canonical examples like shifted cotangent stacks and moduli of perfect complexes on Calabi–Yau varieties.

Exact shifted symplectic structures arise at the intersection of derived algebraic geometry, shifted symplectic theory, and deformation quantization. They generalize the notion of exactness from classical symplectic geometry to the context of derived stacks and moduli problems, playing a central role in the categorification of invariants, deformation quantization schemes, and the paper of virtual cycles in moduli theory. Their formalism encompasses derived algebraic, analytic, and $\mathcal{C}^{\infty}$-stacks, links nondegenerate Poisson structures to symplectic forms with additional data, and underpins key functorial correspondences in modern enumerative geometry and mathematical physics.

1. Definition and Characterization

Let $X$ be a derived stack equipped with its cotangent complex $L\Omega^1_X$. The (untruncated) de Rham complex $DR(X)$ is doubly graded by form degree and weight, equipped with commuting differentials $\delta$ (internal) and $d$ (de Rham). Using the Hodge filtration $F^p DR(X)=\Omega^p\oplus\Omega^{p+1}\oplus\cdots$, the space of $n$-shifted presymplectic forms is given by elements

$\omega \in H^0 MC(F^2 DR(X)[n]).$

Explicitly, $\omega$ can be written as a sequence of closed weighted forms $(\omega_2, \omega_3, \ldots)$ with $\omega_2 \in \Omega^2_X[n]$.

A presymplectic form $\omega$ is $n$-shifted symplectic if the leading component $\omega_2$ induces an isomorphism $T_X \stackrel{\sim}{\to} L\Omega^1_X[n]$ in the derived category. Exactness requires a global $(n-1)$-form $\theta$ such that

$d\theta = \omega \qquad \text{in}\quad DR(X),$

that is, $\omega$ is exact as a cocycle in the de Rham complex. This is equivalently encoded by the Maurer–Cartan (MC) space of the mapping cone $\operatorname{cocone}(F^2 DR(X)\to DR(X))[n-1]$.

In the case of (twisted) cotangent stacks, such as $T^*[n]X$, Calaque establishes that the canonical $n$-shifted symplectic form is given by $\omega = d\lambda$, with $\lambda$ the tautological Liouville $1$-form, showing exactness in the shifted sense (Calaque, 2016).

2. Relationship with Poisson Structures and Formal Derivations

Pridham extends the correspondence between shifted symplectic and non-degenerate Poisson structures by introducing the notion of a "formal derivation" $D$ on the Poisson side: $$L = \widehat{\mathrm{Pol}}(X,n)[n+1] = \mathbb{R}\Gamma\big(X,\,\mathrm{Hom}(\mathrm{OpSym}^*(\Omega^1_X[-n-1]), \mathcal{O}_X)\big),$$ filtered by polyvector degree, equipped with the Maurer–Cartan space $MC(F^2 L)$ parameterizing $n$-shifted Poisson structures.

The grading operator $\sigma$ acts as $\sigma\pi_p = (p-1)\pi_p$, and the pair $(\pi, D)$, with $D$ a "Poisson derivation" homotoping $\sigma(\pi)$ to zero, satisfies the MC-equation

$\delta D + [\pi, D] = -\sigma(\pi).$

The space of MC elements in $\operatorname{cocone}(\sigma F^2L \to L)$, denoted $P^D(X,n)$, provides the avatar of exactness on the Poisson side: Pridham establishes a weak equivalence

$$MC\left(\operatorname{cocone}(F^2 DR(X) \to DR(X))[n+1]\right)^\mathrm{nondeg} \simeq MC\left(\operatorname{cocone}(\sigma F^2L \to L)\right)^\mathrm{nondeg},$$

thereby showing that an exact symplectic form is equivalent to a Poisson structure with a formal derivation (Pridham, 10 Nov 2025).

3. Local and Global Exactness: Darboux Theorems and Obstructions

The local Darboux lemma for shifted symplectic forms asserts that, on any sufficiently small (Zariski or étale) neighborhood in $\operatorname{Spec}A$, every nondegenerate closed $2$-form of degree $n$ is homotopic to a strict constant form: $\omega = d_{DR}\theta$ for some $\theta \in DR(A)^{n+1}(1)$. This property ensures local exactness via a splitting of the Hodge filtration. The global obstruction to exactness arises from the possibility that $\omega$ fails to be in the image of $A^1(X, n-1)$ under the de Rham differential, leading to an obstruction class in

$H^1(X, \mathbb{L}_X[n-1]).$

For example, in the moduli of perfect complexes on Calabi–Yau varieties, the obstruction can be directly calculated in Hodge cohomology (Calaque et al., 2015).

Park introduces the notion of "locked" forms and "locked" exactness, constructing a fibered model of the exact/locked/closed forms. Here, exactness is detected via the presence of a Liouville primitive in the locked forms, forming a crucial ingredient in the structure theory for shifted symplectic fibrations (Park, 27 Jun 2024).

4. Canonical Examples and Symplectic Pushforwards

Exact shifted symplectic forms arise canonically in various geometric settings:

• Shifted Cotangent Stacks: For any derived Artin stack $X$, $T^*[n]X$ carries a canonical exact $n$-shifted symplectic structure with Liouville primitive, as established by Calaque (Calaque, 2016).
• Twisted Cotangent Bundles and Symplectic Pushforwards: Park shows that twisted cotangent bundles $T^*_{B,a}[d]$ provide models for exact shifted symplectic pushforwards, constructed from locked $(d+1)$-forms $a$ as the symplectic pushforward $p_*(U,\omega_{U,a})$ (Park, 27 Jun 2024).
• Moduli of Perfect Complexes: The moduli space $\operatorname{Perf}(X/B,\operatorname{ch})$ of perfect complexes on a Calabi–Yau $4$-fold family carries a natural $-2$-shifted symplectic structure. The exact locus is characterized by the vanishing of the $(0,4)$-piece in the Hodge decomposition of the second Chern character.

These examples illustrate both the local-to-global issues surrounding exactness and foundational constructions in derived moduli problems.

5. Virtual Cycles, Lagrangians, and Enumerative Applications

In the context of $(-2)$-shifted symplectic fibrations, the exact locus—where the symplectic form admits a global Liouville primitive—becomes the geometric stage for defining virtual Lagrangian cycles. Park defines, for such exact loci $Z_B(w)$ in the base $B$, a canonical bivariant class

$$[M/B]_{\mathrm{lag}}: A_*(Z_B(w)) \longrightarrow A_{*+\frac12 \dim(TM/B)}(M)$$

characterized by functorial properties under Lagrangian correspondences, pushforwards, and base changes. This construction produces unique, deformation-invariant cycles that underpin virtual invariants in Donaldson–Thomas theory for Calabi–Yau 4-folds, with the invariants locally constant on the exact locus $w=0$ (Park, 27 Jun 2024).

Exact shifted symplectic structures also allow the description of Lagrangian branes inside higher critical loci, providing a bridge between derived symplectic geometry and categorified enumerative theories.

6. Deformation Quantization and Self-Dual Quantizations

Pridham establishes that for nondegenerate shifted Poisson structures with formal derivation, there exists a unique self-dual deformation quantization wherever quantization is defined. In the $0$-shifted analytic or $\mathcal{C}^\infty$ setting, this generalizes Fedosov’s parametrization, showing independence from the associator choice due to the elimination of $\hbar^2$-derivation ambiguities.

For higher positive shifts, deformation quantization relies on the formality of $E_{n+1}$-Hochschild complexes and associated Drinfeld associators, with exactness ensuring uniqueness. In the negative ($n=-1$) shift, where all symplectic structures are canonically exact, the formalism recovers sheaves of twisted $BD_0$-algebras with $\hbar_2$-derivation. On the critical locus of a function, the quantized module $E$ exhibits a sesquilinear pairing and a canonical $\hbar_2$-connection; passing to D-modules and de Rham complexes recovers the perverse sheaf of vanishing cycles with monodromy operator, in line with the work of Behrend–Bryan–Davison–Joyce–Song (Pridham, 10 Nov 2025).

7. Summary Table of Properties

Context	Exactness Criterion	Consequence
Shifted cotangent stack $T^*[n]X$	$\omega = d\lambda$, $\lambda$ Liouville	Canonical exact $n$-shifted symplectic structure
Moduli of complexes on CY4	$(\operatorname{ch}_2)_{(0,4)}=0$	Exactness locus for virtual Lagrangian cycle construction
Poisson side	$\delta D + [\pi, D] = -\sigma(\pi)$	Corresponds to exact shifted symplectic via MC equivalence
Deformation quantization	Exactness eliminates $\hbar^2$ ambiguity	Unique self-dual quantization, associator independence

Exact shifted symplectic structures thus provide the framework for both concrete geometric constructions and advanced invariants in derived and enumerative geometry, serving as the technical bridge between shifted Poisson geometry, quantization theory, and the construction of virtual cycles in moduli spaces.