Symplectic Stack X(dL)

• Symplectic Stack X(dL) is a derived moduli stack parameterizing flat G-bundles on closed d-manifolds, featuring a canonical (2-d)-shifted symplectic structure.
• Its (2-d)-shifted symplectic structure, constructed via PTVV transgression, ensures non-degeneracy and permits local Darboux coordinate descriptions.
• The framework bridges gauge theory, representation theory, and topology, enabling applications in moduli space analysis, Hamiltonian reduction, and intersection theory.

Symplectic Stack $X(dL)$

A symplectic stack $X(dL)$ is a derived moduli stack that parameterizes flat $G$-bundles (or local systems) on a closed oriented $d$-manifold $L$, equipped with a canonical $(2-d)$-shifted symplectic structure in the sense of derived algebraic geometry. The construction, formalism, and consequences of this structure provide a unifying framework for moduli spaces arising in gauge theory, representation theory, symplectic topology, and mathematical physics, especially in the context of derived mapping stacks and shifted symplectic structures (Pantev et al., 2011, Calaque, 2018).

1. Definition and Construction

Let $L$ be a closed, oriented, smooth $d$-manifold, and $G$ a compact or reductive Lie group. The symplectic stack $X(dL)$ is defined as the derived mapping stack from the de Rham stack $\mathrm{d}L$ of $L$ to the classifying stack $BG$ of $G$:

$X(dL) := \mathrm{Map}(\mathrm{d}L, BG)$

Here, $\mathrm{d}L$ is the stack whose ring of functions is the de Rham complex $(\Omega^\bullet(L), d_{\mathrm{dR}})$, reflecting the derived geometry of $L$. Points of $X(dL)$ correspond to flat $G$-bundles on $L$; more generally, $X(dL)$ is the derived moduli stack of $G$-local systems over $L$, or families of such over a base commutative dg algebra (Calaque, 2018).

2. Shifted Symplectic Structure

The central feature of $X(dL)$ is its canonical $(2-d)$-shifted symplectic structure, constructed via the foundational result of Pantev–Toën–Vaquie–Vezzosi (PTVV) (Pantev et al., 2011). The classifying stack $BG$ possesses a canonical 2-shifted symplectic form $\omega_{BG}$ determined by a non-degenerate, $G$-invariant symmetric bilinear form $c \in S^2(\mathfrak{g}^*)^G$ on the Lie algebra $\mathfrak{g}$.

The PTVV transgression principle states that if $E$ is a $d$-oriented derived stack and $X$ is an $n$-shifted symplectic derived stack, the mapping stack $\mathrm{Map}(E, X)$ inherits an $(n-d)$-shifted symplectic form by:

$$\omega_{\mathrm{Map}(E,X)} := \int_{[E]} \mathrm{ev}^*(\omega_X)$$

For $E = \mathrm{d}L$, $X = BG$, $n=2$, the resulting form on $X(dL)$ is:

$$\omega_{X(dL)} = \int_{[L]} \mathrm{ev}^*(\omega_{BG}) \in \Gamma(X(dL), \wedge^2 \mathbb{L}_{X(dL)}[2-d])$$

This form is closed and non-degenerate; at a point corresponding to a flat bundle $P$, the induced map of tangent complexes

$$\omega_{X(dL)}^\sharp : T_{X(dL)} \to \mathbb{L}_{X(dL)}[2-d]$$

is a quasi-isomorphism, ensuring the genuine shifted symplectic property (Calaque, 2018, Pantev et al., 2011).

3. Local Structure and Darboux Theorem

The local model for shifted symplectic stacks, including $X(dL)$, is governed by the shifted version of the Darboux theorem (Ben-Bassat et al., 2013). This guarantees that any $k$-shifted symplectic derived Artin stack (for $k<0$), near each point, admits an atlas with explicit coordinates in which the shifted 2-form has a universal standard ("Darboux") expression. For $k=2-d$, appropriate coordinates and generators for the structure sheaf and cotangent complex yield explicit formulas for the symplectic form, verifying locality and facilitating further geometrical and physical computations.

4. Classical Moduli, Hamiltonian Reduction, and Intersection Theory

On the underived locus and in classical topology, $X(dL)$ specializes to many familiar objects:

• For $d=2$, $X(dL)$ recovers the moduli space of flat $G$-connections or character variety, with the classical Atiyah–Bott–Goldman symplectic structure arising from the shifted symplectic form.
• $X(dL)$ can be described as the derived Hamiltonian reduction of the infinite-dimensional affine space of all connections $\mathrm{Conn}_G(L)$, with the curvature map furnishing the Lagrangian structure essential to the reduction (Pantev et al., 2011, Calaque, 2018).

In the general framework, if two derived stacks $X, Y$ map Lagrangianly into an $n$-shifted symplectic stack $(F, \omega)$, their fiber product $X \times^h_F Y$ is naturally $(n-1)$-shifted symplectic, expressing the derived intersection theory at the heart of Lagrangian correspondences and field-theoretic boundary conditions (Pantev et al., 2011).

5. Examples and Special Cases

A variety of rich examples and applications arise:

• For a Riemann surface ($d=2$), the stack $X(dL)$ is 0-shifted symplectic and models the character variety $\mathrm{Hom}(\pi_1(L), G)//G$ (Pantev et al., 2011).
• For a 3-manifold, $X(dL)$ has a $(-1)$-shifted symplectic structure and is central in Chern–Simons theory and topological field theory.
• In the toric quasifold or irrational stacky geometry context, $X(dL)$ exhibits noncommutative and stacky phenomena, such as irrational moment polytopes and non-Hausdorff groupoids (Hoffman et al., 2018).
• In the setting of noncommutative cluster Lagrangians and microlocal sheaf theory, $X(dL)$ is realized as a moduli stack of dg-sheaves with prescribed microlocal support, admitting an explicit cluster algebra structure and $K_2$-Lagrangian substack description (Goncharov et al., 12 Jan 2026).

6. Deformation Theory and Moduli of Symplectic Structures

The derived moduli stack $\mathrm{Symp}(X, n)$ parametrizes $n$-shifted symplectic structures on a fixed derived stack $X$. Under finiteness and orientability conditions, $\mathrm{Symp}(X, n)$ carries a canonical shifted quadratic form of degree $n+2$, generalizing classical results and enabling derived deformation theory of symplectic forms on stacks such as $X(dL)$ (Bach et al., 2017).

7. Hamiltonian Symplectic Stacks and Stacky Symplectic Reduction

In differentiable and étale settings, symplectic stacks such as $X(dL)$ fit into a broader class of 0-symplectic stacks, admitting Hamiltonian group stack actions, stacky moment maps, and satisfying stack-theoretic versions of the Kirwan convexity, Meyer-Marsden-Weinstein symplectic reduction, and Duistermaat-Heckman theorems (Hoffman et al., 2018). The theory extends to non-rational and non-Hausdorff situations, capturing new geometric phenomena beyond the classical field.

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References:

• Pantev, Toën, Vaquié, Vezzosi, "Shifted Symplectic Structures" (Pantev et al., 2011)
• Calaque, "Derived stacks in symplectic geometry" (Calaque, 2018)
• Bach, Melani, "The derived moduli stack of shifted symplectic structures" (Bach et al., 2017)
• Joyce et al., "A 'Darboux Theorem' for shifted symplectic structures..." (Ben-Bassat et al., 2013)
• Bottman et al., "Stacky Hamiltonian actions and symplectic reduction" (Hoffman et al., 2018)
• Goncharov, Kontsevich, "Non-commutative cluster Lagrangians" (Goncharov et al., 12 Jan 2026)