--- title: Symplectic Stack X(dL) url: https://www.emergentmind.com/topics/symplectic-stack-x-dl type: topic --- # Symplectic Stack X(dL) 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 [1111.3209][1802.09643]. ## 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 [1802.09643]. ## 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) [1111.3209]. 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 [1802.09643][1111.3209]. ## 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 [1312.0090]. 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 [1111.3209][1802.09643]. 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 [1111.3209]. ## 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 \(\mathrm{Hom}(\pi_1(L), G)//G\) [1111.3209]. - 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 [1808.01003]. - 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 [2601.07538]. ## 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 \(X(dL)\) [1706.08369]. ## 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 [1808.01003]. The theory extends to non-rational and non-Hausdorff situations, capturing new geometric phenomena beyond the classical realm. --- **References:** - Pantev, Toën, Vaquié, Vezzosi, "Shifted Symplectic Structures" [1111.3209] - Calaque, "Derived stacks in symplectic geometry" [1802.09643] - Bach, Melani, "The derived moduli stack of shifted symplectic structures" [1706.08369] - Joyce et al., "A 'Darboux Theorem' for shifted symplectic structures..." [1312.0090] - Bottman et al., "Stacky Hamiltonian actions and symplectic reduction" [1808.01003] - Goncharov, Kontsevich, "Non-commutative cluster Lagrangians" [2601.07538] Source: https://www.emergentmind.com/topics/symplectic-stack-x-dl