2019 Thematic Program (IV)
Shifted symplectic and Poisson structures
October , 2020
W303 School of Mathematics, Sichuan University
The aim of this lecture series is to introduce shifted symplectic structures, and give several examples of these. We will connect these new notions to more classical concepts of symplectic geometry and geometric representation theory. We may use as prerequisites the basics of derived algebraic geometry, from the first two lecture of Bertrand Toen’s mini-course. If time permits, we will also talk about shifted Poisson structures. We will try to follow the following plan, that may nevertheless be subject to changes:
• Lecture 1.
Shifted symplectic geometry I: homotopical symplectic linear algebra, de Rham graded mixed complex, (closed) forms on a derived stack, examples (smooth scheme, BG, Perf).
• Lecture 2.
Shifted symplectic geometry II: shifted symplectic and lagrangian structures on derived stacks, hamiltonian reduction in the derived setting, examples (shifted cotangent stacks, derived critical loci, BG, Perf). Optional: shifted symplectic reduction of derived critical loci.
• Lecture 3.
The AKSZ-PTVV construction I: d-oriented stacks, trangression of (closed) forms, examples (various derived moduli stacks). Optional: a noncommutative version.
• Lecture 4.
The AKSZ-PTVV construction II: relative orientations, category of lagrangian correspondences, shifted symplectic groupoids, oriented co-groupoids. Optional: the AKSZPTVV construction as a fully extended TFT.
• Lecture 5.
Shifted Poisson structures on derived schemes: dg Lie algebra of polyvector fields, definition and examples of shifted Poisson structures on (affine) derived schemes, Melani’s theorem.
• Lecture 6.
Shifted Poisson structures on derived stacks: formal geometry in the derived setting, shifted Poisson structures on derived stacks, examples.