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Shifted cotangent stacks are shifted symplectic
Damien Calaque1
1 IMAG, Univ Montpellier, CNRS, Institut Universitaire de France, Montpellier, France
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 28 (2019) no. 1, pp. 67-90.

We prove that shifted cotangent stacks carry a canonical shifted symplectic structure. We also prove that shifted conormal stacks carry a canonical Lagrangian structure. These results were believed to be true, but no written proof was available in the Artin case.

On démontre que les champs cotangents décalés sont canoniquement munis d’une structure symplectique décalée. On démontre également que les champs conormaux décalés sont munis d’une structure Lagrangienne canonique. Ces résultats étaient attendus mais aucune démonstration n’était disponible dans le cas des champs d’Artin.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1593

Damien Calaque 1

1 IMAG, Univ Montpellier, CNRS, Institut Universitaire de France, Montpellier, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Damien Calaque. Shifted cotangent stacks are shifted symplectic. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 28 (2019) no. 1, pp. 67-90. doi : 10.5802/afst.1593. https://afst.centre-mersenne.org/articles/10.5802/afst.1593/

[1] Damien Calaque Three lectures on derived symplectic geometry and topological field theories, Poisson 2012: Poisson Geometry in Mathematics and Physics (Indagationes Mathematicæ), Volume 25, Elsevier, 2014, pp. 926-947 | MR | Zbl

[2] Damien Calaque Lagrangian structures on mapping stacks and semi-classical TFTs, Stacks and Categories in Geometry, Topology, and Algebra (Tony Pantev, ed.) (Contemporary Mathematics), Volume 643, American Mathematical Society, 2015 | DOI | MR | Zbl

[3] Damien Calaque; Tony Pantev; Bertrand Toën; Michel Vaquié; Gabriele Vezzosi Shifted Poisson Structures and Deformation Quantization, J. Topol., Volume 10 (2017) no. 2, pp. 483-584 | DOI | MR | Zbl

[4] Dennis Gaitsgory; Nick Rozenblyum A study in derived algebraic geometry. Vol. II. Deformations, Lie theory and formal geometry, Mathematical Surveys and Monographs, 221, American Mathematical Society, Providence, RI, 2017 no. 2 | MR | Zbl

[5] Rune Haugseng Iterated spans and classical topological field theories, Math. Z., Volume 289 (2018) no. 3-4, pp. 1427-1488 | DOI | MR | Zbl

[6] Valerio Melani; Pavel Safronov Derived coisotropic structures II: stacks and quantization, Selecta Math. (N.S.), Volume 24 (2018) no. 4, pp. 3119-3173 | DOI | MR | Zbl

[7] Tony Pantev; Bertrand Toën; Michel Vaquié; Gabriele Vezzosi Shifted Symplectic Structures, Publications mathématiques de l’IHÉS, Volume 117 (2013) no. 1, pp. 271-328 | DOI | MR | Zbl

[8] J. P. Pridham Shifted Poisson and symplectic structures on derived N-stacks, J. Topol., Volume 10 (2017) no. 1, pp. 178-210 | DOI | MR | Zbl

[9] P. Safronov Quasi-Hamiltonian reduction via classical Chern–Simons theory, Adv. Math., Volume 287 (2016), pp. 733-773 | DOI | MR | Zbl

[10] Bertrand Toën Champs affines, Sel. Math., New Ser., Volume 12 (2006) no. 1, pp. 39-135 | DOI | MR | Zbl

[11] Bertrand Toën; Gabriele Vezzosi Homotopical algebraic geometry. II. Geometric stacks and applications, Mem. Amer. Math. Soc., Volume 193 (2008) no. 902, 224 pages | MR | Zbl

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