Derived algebraic geometry with a view to quantisation II: symplectic and Poisson structures. J.P.Pridham

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1 Derived algebraic geometry with a view to quantisation II: symplectic and Poisson structures J.P.Pridham 1 / 24

2 de Rham complexes A pa, δq a CDGA over R. Kähler differentials Ω 1 A{R (a complex). Exterior powers Ω p A{R. de Rham differential d : Ω p A{R Ñ Ωp`1 A{R. de Rham complex DRpA{Rq p ź p Ω p A{Rr ps, d δq. Hodge filtration F p DRpAq ś Ω ěp A. 2 / 24

3 Derived de Rham cohomology Derived de Rham LDRpA{Rq : DRpÃ{Rq, for Ã Ñ A a cofibrant («quasi-free) resolution. LF p DRpA{Rq : F p DRpÃ{Rq. Also write LΩ p paq : Ω p pãq. Pieces of cyclic homology (Feigin Tsygan, HKR). 3 / 24

4 n-shifted pre-symplectic structures ω P Z n`2 LF 2 DRpA{Rq [ PTVV]. Explicitly, ω ř pě2 ω p, with δω 2 0, dω p δω p`1. Morphisms given by chain homotopies. Sheafify for global, so for hgpd X, ω P ś i F 2 LDRpOpX i q{rq n`2 i. Symplectic if non-degenerate: ω 7 2 : plω1 Xq _ ÝÑ LΩ 1 Xrns. 4 / 24

5 n-shifted polyvectors x PolpA{R, nq: RHom A pcosymm A plω 1 A{Rrn ` 1sq, Aq, ( «{ Symm A pt A{R rn ` 1sq). Filtration F p x PolpA{R, nq: RHom A pcosymm ěp A plω1 A{Rrn ` 1sq, Aq. Commutative product F p F q Ă F p`q. Schouten Nijenhuis Lie bracket on xpolpa{r, nqrn ` 1s, rf p, F q s Ă F p`q 1. 5 / 24

6 n-poisson structures (affine case) π ř pě2 π p P F 2 x PolpA{R, nq n`2, δπ ` 1 rπ, πs 0 (Maurer Cartan). 2 Morphisms from Thom Sullivan homotopies give space PpA{Rq of n-poisson structures. Non-degenerate if π 7 2 : LΩ1 Arns ÝÑ plω 1 Aq _. 6 / 24

7 n-poisson algebras P n`1 -algebra is CDGA with compatible Lie bracket of degree n. π P PpAq gives L 8 -structure on Arns. n-poisson structures ðñ homotopy P n`1 -algebra structures on A [Melani]. 7 / 24

8 n-poisson structures on DM stacks X a derived DM stack, resolved by hgpd ˇX (simplicial derived affine). PpX {R, nq the space of cosimplicial homotopy P n`1 -algebra structures on cosimplicial CDGA Op ˇX q. Invariance under trivial DM hgpds ùñ well-defined up to coherent homotopy (contractible space of choices). Similarly 8-functorial for étale maps. 8 / 24

9 What about Artin stacks? Don t have smooth functoriality. So resolve X by more exotic objects. Example Y acted on by G, Lie algebra g. Chart for ry {Gs will be ry {gs. OprY {gsq Chevalley Eilenberg complex OpY q B ÝÑ OpY qbg _ B ÝÑ OpY qbλ 2 g _ B ÝÑ... bigraded CDGA. 9 / 24

10 Stacky CDGAs commutative algebras in double complexes: A 0,0 B A 1,0 B A 2,0 B δ A 0, 1. δ δ B A 1, 1 δ B. δ B A 2, 1 δ B. B δ δ B... Take weak equivalences to have quasi-isomorphic columns. stacky ÝÝÝÑ derived 10 / 24

11 Functor D from cosimplicial CDGAs to stacky CDGAs. D OprY {Gsq OpY {gq. Given derived Artin hgpd ˇX, can form cosimplicial stacky CDGA D Op ˇX q: D Op ˇX q D Op ˇX 1 q D Op ˇX 2 q, a DM hgpd in stacky derived affines! Example: when ˇX BrY {Gs, this is ry {gs ry ˆ G{g 2 s ry ˆ G 2 {g 3 s 11 / 24

12 Structures on Artin stacks Internal Hom in double complexes. Semi-infinite total complex behaves Tot ˆ pv q i p à V n,i n ź V n,i n, δ Bq. nă0 ně0 Hence define ˆ Hom ˆ Tot Hom, x Pol,... Poisson structures PpAq ÝÑ Pp ˆ Tot Aq, for stacky CDGA A. 12 / 24

13 Symplectic versus Poisson Classical case: 2-form ω is symplectic iff inverse π is Poisson. Standard proof uses Darboux theorem (cotangent bundle). Shifted Darboux theorems [B-BBBJ], [BG] give local comparison for shifted structures. 13 / 24

14 A more direct approach ω 2 homotopy inverse to π 2. Higher components?? Look to generalise π 7 ω 7 π 7 π 7. Then globalise (via hypergroupoids). 14 / 24

15 The canonical tangent vector Tangent space T π P of Poisson structures at π: α P F 2 x PolpA{R, nq n`2, δα ` rπ, αs 0 (i.e. π ` αɛ Poisson). Differentiating G m -action gives σpπq : ÿ pě2pp 1qπ p P T π P. 15 / 24

16 Compatibility (the key) Contraction µp, πq from de Rham to Poisson cohomology (cf. [K-S M]). Derivative ν. Relates de Rham / Schouten Nijenhuis: rπ, µpω, πqs µpdω, πq`νpω, π; 1 rπ, πsq. 2 µpω, πq P T π P for ω pre-symplectic, π Poisson. 16 / 24

17 In detail When φ adf 1 ^... ^ df p, µpφ, πq arπ, f 1 s... rπ, f p, s, νpφ, π; bq ÿ i arπ, f 1 s... rb, f i s... rπ, f p s. Thus µpω 2, π 2 q 7 π 7 2 ω7 2 π7 2. µp, πq a qu-iso for π non-degenerate. 17 / 24

18 The affine equivalence Weak equivalences of 8-groupoids (i.e. topological spaces): tsymplectic structures ωu tcompatible pairs µpω, πq» σpπqu tnon-degenerate Poisson structures πu Thus π ÞÑ µp, πq 1 σpπq (defined up to coherent homotopy). 18 / 24

19 Governing DGLAs For A a stacky CDGA: Symplectic: F 2 LDRpA{Rqrn ` 1s (abelian). Poisson: F 2 x PolpA{R, nqrn ` 1s. T P: F 2 x PolpA{R, nqrn ` 1srɛs. Compatible pairs a homotopy limit. Equivalence via obstruction theory. 19 / 24

20 Obstructions gr p F Ñ F 2 x PolpAq{F p`1 Ñ F 2 x PolpAq{F p central extension of DGLAs. Maurer Cartan obstruction map PpA{R, nq{f p Ñ MCpgr p Frn ` 2sq, fibre PpA{R, nq{f p`1. Similar obstructions for symplectic structures, compatible pairs. Graded pieces equivalent via powers of plω 1 X q_» LΩ 1 X rns. 20 / 24

21 Derived Artin N-stacks Artin hypergroupoid resolution ˇX Ñ X always exists. Thus can look at structures on cosimplicial stacky CDGA D Op ˇX q. Affine equivalences étale functorial (up to coherent homotopy). Hence on X, symplectic Ø non-degenerate Poisson. 21 / 24

22 References Tony Pantev, Bertrand Toën, Michel Vaquié, and Gabriele Vezzosi. Shifted symplectic structures. Publ. Math. Inst. Hautes Études Sci., 117: , J. P. Pridham. Shifted Poisson and symplectic structures on derived N-stacks. arxiv: v3 [math.ag], D. Calaque, T. Pantev, B. Toën, M. Vaquié, and G. Vezzosi. Shifted Poisson structures and deformation quantization. arxiv: v3 [math.ag], / 24

23 Valerio Melani. Poisson bivectors and Poisson brackets on affine derived stacks. arxiv: v2 [math.ag], O. Ben-Bassat, C. Brav, V. Bussi, and D. Joyce. A Darboux theorem for shifted symplectic structures on derived Artin stacks, with applications. Geom. Topol., to appear. arxiv: [math.ag]. E. Bouaziz and I. Grojnowski. A d-shifted Darboux theorem. arxiv: v1 [math.ag], Yvette Kosmann-Schwarzbach and Franco Magri. Poisson Nijenhuis structures. Ann. Inst. H. Poincaré Phys. Théor., 53(1):35 81, / 24

24 J. P. Pridham. Unifying derived deformation theories. Adv. Math., 224(3): , arxiv: v6 [math.ag]. 24 / 24