Shifted Poisson structures and deformation quantization Damien Calaque, Tony Pantev, Bertrand Toën, Michel Vaquié, Gabriele Vezzosi To cite this version: Damien Calaque, Tony Pantev, Bertrand Toën, Michel Vaquié, Gabriele Vezzosi. Shifted Poisson structures and deformation quantization. Journal of topology, Oxford University Press, 2017, 10 (2), pp.483-584. 10.1112/topo.12012. hal-01253029v2 HAL Id: hal-01253029 https://hal.archives-ouvertes.fr/hal-01253029v2 Submitted on 5 Oct 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la difusion de documents entifc research documents, whether they are pub- scientifques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Shifted Poisson Structures and deformation quantization ∗ ∗ D. Calaque, T. Pantev, B. To¨en, M. Vaqui´e , G. Vezzosi Abstract This paper is a sequel to [PTVV]. We develop a general and flexible context for differential calculus in derived geometry, including the de Rham algebra and the mixed algebra of polyvector fields. We then introduce the formalism of formal derived stacks and prove formal localization and gluing results. These allow us to define shifted Poisson structures on general derived Artin stacks, and to prove that the non-degenerate Poisson structures correspond exactly to shifted symplectic forms. Shifted deformation quantization for a derived Artin stack endowed with a shifted Poisson structure is discussed in the last section. This paves the road for shifted deformation quantization of many interesting derived moduli spaces, like those studied in [PTVV] and many others. Contents Introduction 3 1 Relative differential calculus 11 1.1 Model categories setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 ∞-Categories setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3 De Rham theory in a relative setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.3.1 Cotangent complexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3.2 De Rham complexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3.3 Strict models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.4 Differential forms and polyvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4.1 Forms and closed forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4.2 Shifted polyvectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.4.3 Pn-structures and symplectic forms. . . . . . . . . . . . . . . . . . . . . . . . . 44 1.5 Mixed graded modules: Tate realization . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ∗ Partially supported by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02 1 2 Formal localization 57 2.1 Derived formal stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.2 Perfect complexes on affine formal derived stacks . . . . . . . . . . . . . . . . . . . . . 64 2.3 Differential forms and polyvectors on perfect formal derived stacks . . . . . . . . . . . 75 2.3.1 De Rham complex of perfect formal derived stacks . . . . . . . . . . . . . . . . 75 2.3.2 Shifted polyvectors over perfect formal derived stacks . . . . . . . . . . . . . . 83 2.4 Global aspects and shifted principal parts . . . . . . . . . . . . . . . . . . . . . . . . . 85 2.4.1 Families of perfect formal derived stacks . . . . . . . . . . . . . . . . . . . . . . 85 2.4.2 Shifted principal parts on a derived Artin stack. . . . . . . . . . . . . . . . . . 89 3 Shifted Poisson structures and quantization 95 3.1 Shifted Poisson structures: definition and examples . . . . . . . . . . . . . . . . . . . . 95 3.2 Non-degenerate shifted Poisson structures . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.3 Proof of Theorem 3.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.3.1 Derived stacks associated with graded dg-Lie and graded mixed complexes . . 101 3.3.2 Higher automorphisms groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.3.3 Infinitesimal theory of shifted Poisson and symplectic structures . . . . . . . . 106 3.3.4 Completion of the proof of Theorem 3.2.5 . . . . . . . . . . . . . . . . . . . . . 108 3.4 Coisotropic structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.5 Existence of quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.5.1 Deformation quantization problems . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.5.2 Solution to the deformation quantization problem . . . . . . . . . . . . . . . . 123 3.6 Examples of quantizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.6.1 Quantization formally at a point . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.6.2 Quantization of BG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Appendix A 133 Appendix B 135 2 Introduction This work is a sequel of [PTVV]. We introduce the notion of a shifted Poisson structure on a general derived Artin stack, study its relation to the shifted symplectic structures from [PTVV], and construct a deformation quantization of it. As a consequence, we construct a deformation quantization of any derived Artin stack endowed with an n-shifted symplectic structure, as soon as n ≠ 0. In particular we quantize many derived moduli spaces studied in [PTVV]. In a nutshell the results of this work are summarized as follows. Main results A 1. There exists a meaningful notion of n-shifted Poisson structures on derived Artin stacks locally of finite presentation, which recovers the usual notion of Poisson structures on smooth schemes when n = 0. 2. For a given derived Artin stack X, the space of n-shifted symplectic structures on X is naturally equivalent to the space of non-degenerate n-shifted Poisson structures on X. 3. Let X be any derived Artin stack locally of finite presentation endowed with an n-shifted Poisson structure π. For n ≠ 0 there exists a canonical deformation quantization of X along π, realized as an E|n|-monoidal ∞-category Perf(X, π), which is a deformation of the symmetric monoidal ∞-category Perf(X) of perfect complexes on X. As a corollary of these, we obtain the existence of deformation quantization of most derived moduli stacks studied in [PTVV], e.g. of the derived moduli of G-bundles on smooth and proper Calabi- Yau schemes, or the derived moduli of G-local systems on compact oriented topological manifolds. The existence of these deformation quantizations is a completely new result, which is a far reaching generalization of the construction of deformation quantization of character varieties heavily studied in topology, and provides a new world of quantized moduli spaces to explore in the future. The above items are not easy to achieve. Some ideas of what n-shifted Poisson structures should be have been available in the literature for a while (see [Me, To2, To3]), but up until now no general definition of n-shifted Poisson structures on derived Artin stacks existed outside of the rather restrictive case of Deligne-Mumford stacks. The fact that Artin stacks have affine covers only up to smooth submersions is an important technical obstacle which we have to deal with already when we define shifted Poisson structures in this general setting. Indeed, in contrast to differential forms, polyvectors do not pull-back along smooth morphisms, so the well understood definition in the affine setting (see [Me, To2]) can not be transplanted to an Artin stack without additional effort, and such a transplant requires a new idea. A different complication lies in the fact that the comparison between non- degenerate shifted Poisson structures and their symplectic counterparts requires keeping track of non- trivial homotopy coherences even in the case of an affine derived scheme. One reason for this is that non-degeneracy is only defined up to quasi-isomorphism, and so converting a symplectic structure into 3 a Poisson structure by dualization can not be performed easily. Finally, the existence of deformation quantization requires the construction of a deformation of the globally defined ∞-category of perfect complexes on a derived Artin stack. These ∞-categories are of global nature, and their deformations are not easily understood in terms of local data. In order to overcome the above mentioned technical challenges we introduce a new point of view on derived Artin stacks by developing tools and ideas from formal geometry in the derived setting. This new approach is one of the technical hearts of the paper, and we believe it will be an important general tool in derived geometry, even outside the applications to shifted Poisson and symplectic structures discussed in this work. The key new idea here is to understand a given derived Artin stack X by ̂ means of its various formal completions Xx, at all of its points x in a coherent fashion. For a smooth algebraic variety, this idea has been already used with great success in the setting of deformation quantization (see for instance [Fe, Ko1, Bez-Ka]), but the extension we propose here in the setting of ̂ derived Artin stacks is new. By [Lu2], the geometry of a given formal completion Xx is controlled by a dg-Lie algebra, and our approach, in a way, rephrases many problems concerning derived Artin stacks in terms of dg-Lie algebras. In this work we explain how shifted symplectic and Poisson structures, as well as ∞-categories of perfect complexes, can be expressed in those terms. Having this formalism at our disposal is what makes our Main statement A accessible. The formalism essentially allows us to reduce the problem to statements concerning dg-Lie algebras over general base rings and their Chevalley complexes. The general formal geometry results we prove on the way are of independent interest and will be useful for many other questions related to derived Artin stacks. Let us now discuss the mathematical content of the paper in more detail. To start with, let us ex- plain the general strategy and the general philosophy developed all along this manuscript. For a given derived Artin stack X, locally of finite presentation over a base commutative ring k of characteristic 0, we consider the corresponding de Rham stack XDR of [Si1, Si2]. As an ∞-functor on commutative dg-algebras, XDR sends A to X(Ared), the Ared-points of X (where Ared is defined to be the reduced ordinary commutative ring π0(A)red). The natural projection π : X −→ XDR realizes X as a family ̂ of formal stacks over XDR: the fiber of π at a given point x ∈ XDR, is the formal completion Xx of X at x. By [Lu2] this formal completion is determined by a dg-Lie algebra lx. However, the dg-Lie algebra lx itself does not exist globally as a sheaf of dg-Lie algebras over XDR, simply because its underlying complex is TX[−1], the shifted tangent complex of X, which in general does not have a flat connection and thus does not descend to XDR. However, the Chevalley complex of lx, viewed as a graded mixed commutative dg-algebra (cdga for short) can be constructed as a global object BX over XDR. To be more precise we construct BX as the derived de Rham complex of the natural inclusion Xred −→ X, suitably sheafified over XDR. One of the key insights of this work is the following result, expressing global geometric objects on X as sheafified notions on XDR related to BX. Main results B With the notation above: 4 1. The ∞-category Perf(X), of perfect complexes on X, is naturally equivalent, as a symmetric monoidal ∞-category, to the ∞-category of perfect sheaves of graded mixed BX-dg-modules on XDR: Perf Perf(X) ≃ BX −Mod ϵ−dggr . 2. There is an equivalence between the space of n-shifted symplectic structures on X, and the space of closed and non-degenerate 2-forms on the sheaf of graded mixed cdgas BX. The results above state that the geometry of X is largely recovered from XDR together with the sheaf of graded mixed cdgas BX, and that the assignment X →↦ (XDR, BX) behaves in a faithful manner from the perspective of derived algebraic geometry. In the last part of the paper, we take advantage of this in order to define the deformation quantization problem for objects belonging to general categories over k. In particular, we study and quantize shifted Poisson structures on X, by considering compatible brackets on the sheaf BX. Finally, we give details for three relevant quantizations and compare them to the existing literature. The procedure of replacing X with (XDR, BX) is crucial for derived Artins stacks because it essentially reduces statements and notions to the case of a sheaf of graded mixed cdgas. As graded mixed cdgas can also be understood as cdgas endowed with an action of a derived group stack, this further reduces statements to the case of (possibly unbounded) cdgas, and thus to an affine situation. Description of the paper. In the first section, we start with a very general and flexible context for (relative) differential int int calculus. We introduce the internal cotangent complex L and internal de Rham complex DR (A) A associated with a commutative algebra A in a good enough symmetric monoidal stable k-linear ∞- category M (see Section 1.1 and Section 1.2 for the exact assumptions we put on M). The internal int de Rham complex DR (A) is defined as a graded mixed commutative algebra in M. Next we recall p,cl from [PTVV] and extend to our general context the spaces A (A, n) of (closed) p-forms of degree n on A, as well as of the space Symp(A, n) of n-shifted symplectic forms on A. We finally introduce int (see also [PTVV, Me, To2, To3]) the object Pol (A, n) of internal n-shifted polyvectors on A, which int is a graded n-shifted Poisson algebra in M. In particular, Pol (A, n)[n] is a graded Lie algebra object in M. We recall from [Me] that the space Pois(A, n) of graded n-shifted Poisson structures on int A is equivalent to the mapping space from 1(2)[−1] to Pol (A, n + 1)[n + 1] in the ∞-category of graded Lie algebras in M, and we thus obtain a reasonable definition of non-degeneracy for graded n-shifted Poisson structures. Here 1(2)[−1] denotes the looping of the monoidal unit of M sitting in pure weight 2 (for the grading). We finally show that 5 int Corollary 1.5.5 If L is a dualizable A-module in M, then there is natural morphism A nd Pois (A, n) −→ Symp(A, n) nd from the space Pois (A, n) of non-degenerate n-shifted Poisson structures on A to the space Symp(A, n) of n-shifted symplectic structures on A. We end the first part of the paper by a discussion of what happens when M is chosen to be the gr ∞-category ϵ− (k −mod) of graded mixed complexes, which will be our main case of study in order to deal with the sheaf BX on XDR mentioned above. We then describe two lax symmetric monoidal t gr functors | − |, | − | : ϵ −M → M, called standard realization and Tate realization. We can apply the Tate realization to all of the previous internal constructions and get in particular the notions of Tate n-shifted symplectic form and non-degenerate Tate n-shifted Poisson structure. We prove that, int as before, these are equivalent as soon as L is a dualizable A-module in M. A One of the main difficulties in dealing with n-shifted polyvectors (and thus with n-shifted Poisson structures) is that they do not have sufficiently good functoriality properties. Therefore, in contrast with the situation with forms and closed forms, there is no straightforward and easy global definition of n-shifted polyvectors and n-shifted Poisson structures. Our strategy is to use ideas from formal geometry and define an n-shifted Poisson structure on a derived stack X as a flat family of n-shifted Poisson structures on the family of all formal neighborhoods of points in X. The main goal of the second part of the paper is to make sense of the previous sentence for general enough derived stacks, i.e. for locally almost finitely presented derived Artin stacks over k. In order to achieve this, we develop a very general theory of derived formal localization that will be certainly very useful in other applications of derived geometry, as well. We therefore start the second part by introducing various notions of formal derived stacks: formal derived stack, affine formal derived stack, good formal derived stack over A, and perfect formal derived stack over A. It is important to note that if X is a derived Artin stack, then ̂ • the formal completion Xf : X ×X DR FDR along any map f : F → X is a formal derived stack. ̂ • the formal completion Xx along a point x : Spec(A) → X is an affine formal derived stack. • each fiber X ×X DR Spec(A) of X → XDR is a good formal derived stack over A, which is moreover perfect if X is locally of finite presentation. Our main result here is the following 6 Theorem 2.2.2 There exists an ∞-functor D from affine formal derived stacks to commutative alge- gr bras in M = ϵ − (k −mod) , together with a conservative ∞-functor φX : QCoh(X) → D(X) −modM, which becomes fully faithful when restricted to perfect modules. perf Therefore, Perf(X) is identified with a full sub-∞-category D(X)−mod M of D(X)−modM that we explicitly determine. We then prove that the de Rham theories of X and of D(X) are equivalent for a perfect algebraisable formal derived stack over A. Namely we show that: ( ) ( ) t DR D(X)/D(SpecA) ≃ DR D(X)/D(SpecA) ≃ DR(X/SpecA) gr as commutative algebras in ϵ−(A−mod) . We finally extend the above to the case of families X → Y of algebraisable perfect formal derived stacks, i.e. families for which every fiber XA := X×Y SpecA → SpecA is an algebraisable perfect formal derived stack. We get an equivalence of symmetric monoidal perf ∞-categories φX : Perf(X) ≃ D X/Y −modM as well as equivalences: ( ) ( ( ) ( ) t Γ Y,DR D X/Y /D(Y ) ≃ Γ Y,DR DX/Y /DY ) ≃ DR(X/Y ) of commutative algebras in M In particular, whenever Y = XDR, we get a description of the de Rham (graded mixed) algebra DR(X) ≃ DR(X/XDR) by means of the global sections of the relative Tate de Rham (graded mixed) algebra BX := D X/XDR over DXDR. Informally speaking, we prove that a (closed) form on X is a float family of (closed) forms on the family of all formal completions of X at various points. The above justifies the definitions of shifted polyvector fields and shifted Poisson structures that we introduce in the third part of the paper. Namely, the n-shifted Poisson algebra Pol(X/Y, n) of n-shifted polyvector fields on a family of algebraisable perfect formal derived stacks X → Y is defined to be ( ) t Γ Y,Pol (D X/Y /DX, n) The space of n-shifted Poisson structures Pois(X/Y, n) is then defined as the mapping space from k(2)[−1] to Pol(X/Y, n + 1)[n + 1] in the ∞-category of graded Lie algebras in M. Following the affine case treated in the first part (see also [Me]), we again prove that this is equivalent to the space 1 of DY -linear n-shifted Poisson algebra structures on D X/Y . We then prove the following 1 Recently J. Pridham proved this comparison theorem for derived Deligne-Mumford stacks by a different approach [Pri]. 7 Theorem 3.2.4 The subspace of non-degenerate elements in Pois(X, n) := Pois(X/XDR, n) is equiv- alent to Symp(X, n) for any derived Artin stack that is locally of finite presentation. We conclude the third Section by defining the deformation quantization problem for n-shifted 1 Poisson structures, whenever n ≥ −1. For every such n, we consider a Gm-equivariant A k-family of k-dg-operads BDn+1 such that its 0-fiber is the Poisson operad Pn+1 and its generic fiber is the k-dg-operad En+1 of chains of the little (n + 1)-disk topological operad. The general deformation quantization problem can then be stated as follows: Deformation Quantization Problem. Given a Pn+1-algebra stucture on an object, does it exist a family of BDn+1-algebra structures such that its 0-fiber is the original Pn+1-algebra structure? Let X be a derived Artin stack locally of finite presentation over k, and equipped with an n-shifted Poisson structure. Using the formality of En+1 for n ≥ 1, we can solve the deformation quantization problem for the DX DR-linear Pn+1-algebra structure on BX. This gives us, in particular, a Gm- equivariant 1-parameter family of DX DR-linear En+1-algebra structures on BX. Passing to perfect modules we get a 1-parameter deformation of Perf(X) as an En-monoidal ∞-category, which we call the n-quantization of X. We work out three important examples in some details: • the case of an n-shifted symplectic structure on the formal neighborhood of a k-point in X: we recover Markarian’s Weyl n-algebra from [Mar]. 3 g • the case of those 1-shifted Poisson structure on BG that are given by elements in ∧ (g) : we ob- fd tain a deformation, as a monoidal k-linear category, of the category Rep (g) of finite dimensional representations of g. 2 g • the case of 2-shifted Poisson structures on BG, given by elements in Sym (g) : we obtain a fd deformation of Rep (g) as a braided monoidal category. Finally, Appendices A and B contains some technical results used in Sections 1 and 3, respectively. Further directions and future works. In order to finish this introduction, let us mention that the present work does not treat some important questions related to quantization, which we hope to address in the future. For instance, we introduce a general notion of coisotropic structures for maps with an n-shifted Poisson target, analogous to the notion of Lagrangian structures from [PTVV]. However, the definition itself requires a certain additivity theorem, whose proof has been announced recently by N. Rozenblyum but is not available yet. Also, we did not address the question of comparing Lagrangian structure and co-isotropic structures that would be a relative version of our comparison between shifted symplectic and non-degenerate Poisson structures. Neither did we address the question 8 of quantization of coisotropic structures. In a different direction, our deformation quantizations are only constructed under the restriction n ≠ 0. The case n = 0 is presently being investigated, but at the moment is still open. In the same spirit, when n = −1 and n = −2, deformation quantization admits an interpretation different from our construction (see for example [To3, Section 6.2]). We believe that our present formal geometry approach can also be applied to these two specific cases. Acknowledgments. We are thankful to D. Kaledin for suggesting to us several years ago that formal geometry should give a flexible enough setting for dealing with shifted polyvectors and Poisson structures. We would also like to thank K. Costello and O. Gwilliam for their explanations about the Darboux lemma [Co-Gwi, Lemma 11.2.0.1] in the setting of minimal L∞-algebras, which can be found in a disguised form in the proof of our Lemma 3.3.11. We are grateful to M. Harris, D. Joyce, M. Kontsevich, V. Melani, M. Porta, P. Safronov, D. Tamarkin, and N. Rozenblyum for illuminating conversations on the subject of this paper. V. Melani and M. Porta’s questions were very useful in order to clarify some obscure points in a previous version of the paper. It is a pleasure to thank once again C. Simpson, for having brought to us all along these years the importance of the de Rham stack XDR, which is a central object of the present work. Damien Calaque acknoweldges the support of Institut Universitaire de France and ANR SAT. During the preparation of this work Tony Pantev was partially supported by NSF research grant DMS- 1302242. Bertrand To¨en and Michel Vaqui´e have been partially supported by ANR-11-LABX-0040- CIMI within the program ANR-11-IDEX-0002-02. Gabriele Vezzosi is a member of the GNSAGA- INDAM group (Italy) and of PRIN-Geometria delle varieta´ algebriche (Italy). In addition Gabriele Vezzosi would like to thank the IAS (Princeton) and the IHES (Bures-sur-Yvette), where part of this work was carried out, for providing excellent working conditions. Notation. • Throughout this paper k will denote a Noetherian commutative Q-algebra. • We will use (∞, 1)-categories ([Lu1]) as our model for ∞-categories. They will be simply called ∞-categories. • For a model category N, we will denote by L(N) the ∞-category defined as the homotopy coherent nerve of the Dwyer-Kan localization of N along its weak equivalences. • The ∞-category T := L(sSets) will be called the ∞-category of spaces. • All symmetric monoidal categories we use will be symmetric monoidal (bi)closed categories. • cdga will denote the ∞-category of non-positively graded differential graded k-algebras (with k −i differential increasing the degree). For A ∈ cdga k, we will write πi A = H (A) for any i ≥ 0. 9