Derived Stacks in Symplectic Geometry

Calaque, Damien

doi:10.1017/9781108854399.007

Cited by 9 publications

(20 citation statements)

References 45 publications

(46 reference statements)

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“…Our major geometric motivation actually lies in the construction of lagrangian (rather than just symplectic) structures, and their connections with critical loci or Calabi-Yau geometry. To be more specific, inspired by classical symplectic geometry [32], we are interested in deformations of conormal stacks that appear as relative critical loci (see [7,1]). In certain cases, these relative critical loci can be obtained as moduli of representations of relative versions of Ginzburg's dg-algebras.…”

Section: Introductionmentioning

confidence: 99%

Relative critical loci and quiver moduli

Bozec¹,

Calaque²,

Scherotzke³

2020

Preprint

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In this paper we identify the cotangent to the derived stack of representations of a quiver Q with the derived moduli stack of modules over the Ginzburg dg-algebra associated with Q. More generally, we extend this result to finite type dg-categories, to a relative setting as well, and to deformations of these. It allows us to recover and generalize some results of Yeung, and leads us to the discovery of seemingly new lagrangian subvarieties in the Hilbert scheme of points in the plane.

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Section: Introductionmentioning

confidence: 99%

Relative critical loci and quiver moduli

Bozec¹,

Calaque²,

Scherotzke³

2020

Preprint

Self Cite

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“…More precisely, we view our construction as a derived intersection of two Lagrangians in a 1-shifted symplectic stack and obtain a -shifted symplectic structure from [PTVV13, Theorem 2.9]. There are similar statements in the literature for Marsden–Weinstein–Meyer reduction [Cal20, § 2.1.2] and quasi-Hamiltonian reduction [Saf16]. We refer the reader to [PTVV13, Get14, Cal20, PS20] for the relevant background on shifted symplectic geometry.…”

Section: Shifted Symplectic Interpretationmentioning

confidence: 90%

“…Given an algebraic symplectic groupoid , recall that the quotient stack inherits a -shifted symplectic structure (see e.g. [Cal20, § 1.2.3], [Get14], and [Saf21, Proposition 3.31]). Recall also that a Hamiltonian action of on a smooth symplectic variety with moment map gives rise to a Lagrangian structure on the map (see [Cal20, Example 1.31]).…”

Section: Shifted Symplectic Interpretationmentioning

confidence: 99%

Symplectic reduction along a submanifold

Crooks¹,

Mayrand²

2022

Compositio Math.

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We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex algebraic varieties, and has an interpretation in terms of derived stacks in shifted symplectic geometry. It also encompasses Marsden–Weinstein–Meyer reduction, Mikami–Weinstein reduction, the pre-images of Poisson transversals under moment maps, symplectic cutting, symplectic implosion, and the Ginzburg–Kazhdan construction of Moore–Tachikawa varieties in topological quantum field theory. A key feature of our construction is a concrete and systematic association of a Hamiltonian $G$ -space $\mathfrak {M}_{G, S}$ to each pair $(G,S)$ , where $G$ is any Lie group and $S\subseteq \mathrm {Lie}(G)^{*}$ is any submanifold satisfying certain non-degeneracy conditions. The spaces $\mathfrak {M}_{G, S}$ satisfy a universal property for symplectic reduction which generalizes that of the universal imploded cross-section. Although these Hamiltonian $G$ -spaces are explicit and natural from a Lie-theoretic perspective, some of them appear to be new.

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“…One take-away is that being closed is structure/data as opposed to a property of a form. [Cal,PTVV13] Let X be a (derived) stack with cotangent complex L X . We will assume that L X is dualizable and the dual will be the tangent complex T X .…”

Section: Mapping Stacks and Geometric Structuresmentioning

confidence: 99%

Quantizing Derived Mapping Stacks

Grady

2020

Preprint

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In this review we discuss several topological and geometric invariants obtained by quantizing σmodels. More precisely, we don't quantize the entire mapping stack of fields, but rather only the substack of low energy fields. The theory restricted to this substack can be presented Lie theoretically and the problem is reduced to perturbative gauge theory. Throughout, we make extensive use of derived symplectic geometry and the BV formalism of Costello and Gwilliam. Finally, we frame the AJ Conjecture in knot theory as a question of quantizing character stacks. Contents 1. Introduction 1.1. Mapping stacks and L ∞ -models 1.2. σ-Models 2. Mapping Stacks and Geometric Structures 2.1. Derived symplectic geometry 2.2. The L ∞ -model 2.3. An extended example: the derived loop space 3. The Costello-Gwilliam Approach to BV Quantization 3.1. BV theory 3.2. BV theory and volume forms: integration via homology 3.3. Observable theory 3.4. RG flow 4. Examples 4.1. BF and Chern-Simons type theories 4.2. Yang-Mills theories 5. Boundary Conditions/Theories 5.1. Derived lagrangians 5.2. BV with boundary 5.3. CS/WZW correspondence 5.4. A new proof of the Bursztyn-Dolgushev-Waldmann Theorem 6. Character Varieties: A View of the AJ Conjecture 6.1. Character varieties and their shifted symplectic structure 6.2. The quantum nature of Skein modules 6.3. The AJ conjecture

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Derived Stacks in Symplectic Geometry

Cited by 9 publications

References 45 publications

Relative critical loci and quiver moduli

Relative critical loci and quiver moduli

Symplectic reduction along a submanifold

Quantizing Derived Mapping Stacks

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