Deformation quantisation for unshifted symplectic structures on derived Artin stacks

We prove that every $0$-shifted symplectic structure on a derived Artin $n$-stack admits a curved $A_{\infty}$ deformation quantisation. The classical method of quantising smooth varieties via quantisations of affine space does not apply in this setting, so we develop a new approach. We construct a map from DQ algebroid quantisations of unshifted symplectic structures on a derived Artin $n$-stack to power series in de Rham cohomology, depending only on a choice of Drinfeld associator. This gives an equivalence between even power series and certain involutive quantisations, which yield anti-involutive curved $A_{\infty}$ deformations of the dg category of perfect complexes. In particular, there is a canonical quantisation associated to every symplectic structure on such a stack, which agrees for smooth varieties with the Kontsevich--Tamarkin quantisation for even associators.


Introduction
For n > 0, existence of quantisations of n-shifted Poisson structures is a formality, following from the equivalence E n+1 ≃ P n+1 of operads.Quantisations of positively shifted symplectic structures thus follow immediately from the equivalence in [Pri5, CPT + ] between symplectic and non-degenerate Poisson structures.In [Pri4], quantisation for non-degenerate (−1)-shifted Poisson structures was established, and we now consider the n = 0 case, fleshing out the details sketched in [Pri4,§4.3].
Beyond the setting of smooth Deligne-Mumford stacks, unshifted symplectic structures only arise on objects incorporating both stacky and derived structures, as nondegeneracy of the symplectic form implies that the cotangent complex must have both positive and negative terms.Examples of such symplectic derived stacks include the derived moduli stack of perfect complexes on an algebraic K3 surface, or the derived moduli stack of locally constant G-torsors on a compact oriented topological surface, for an algebraic group G equipped with a Killing form on its Lie algebra.In the latter example, the symplectic structure on the smooth locus is that of [Gol].
For Poisson structures on smooth algebraic varieties, the Kontsevich-Tamarkin quantisations rely on local formality of the Hochschild complex, so will not readily adapt to arbitrary Poisson structures on derived stacks.We show, however, that all nondegenerate Poisson structures can be quantised even when the Hochschild complex is not formal, by a similar mechanism to the quantisation of non-degenerate (−1)-shifted Poisson structures in [Pri4].
The proof in [Pri5] of the correspondence between n-shifted symplectic and nondegenerate Poisson structures relied on the existence, for all Poisson structures π, of a CDGA morphism µ(−, π) from the de Rham algebra to the algebra T π Pol(X, n) of shifted polyvectors with differential twisted by π.In [Pri4], this idea was extended to establish the existence of quantisations for (−1)-shifted symplectic structures, with µ being an A ∞ -morphism from the de Rham algebra to the ring of differential operators.
In order to adapt these constructions to 0-shifted symplectic structures, we replace polyvectors or differential operators with the Hochschild complex CC • R (X) of a derived Artin stack X, defined in terms of a resolution by stacky CDGAs (commutative bidifferential bigraded algebras).Since this has an E 2 -algebra structure, a choice w of Levi decomposition for the Grothendieck-Teichmüller group gives it a P 2 -algebra structure.Quantisations ∆ are defined as certain Maurer-Cartan elements ∆ ∈ CC • R (X) ; these give rise to curved deformations of the dg category of perfect complexes.
Each quantisation ∆ then defines a morphism µ w (−, ∆) from the de Rham complex DR(X/R) to CC • R (X) twisted by ∆.In more detail, since [∆, −] defines a derivation from O X to CC • R (X) , it determines a map Ω 1 X → CC • R (X) [1] and µ w (−, ∆) is the resulting morphism of CDGAs.This gives rise to a notion of compatibility between E 1 quantisations ∆ and generalised pre-symplectic structures (power series ω of elements of the de Rham complex): we say that ω and ∆ are w-compatible if µ w (ω, ∆) ≃ 2 ∂∆ ∂ .
Proposition 2.16 shows that every non-degenerate quantisation ∆ of a stacky CDGA A has a unique w-compatible generalised pre-symplectic structure, thus giving us a map on the space of non-degenerate 0-shifted E 1 quantisations of A.
Moreover, we have spaces QP(A, 0)/G k+1 consisting of E 1 quantisations of order k, by which we mean Maurer-Cartan elements in j≥2 (F j CC • R (A)/F j−k−1 ) j−1 , for F the good truncation filtration in the Hochschild direction.Via induction on levels of the filtration, and an analysis of the associated DGLA obstruction theory, Proposition 2.17 then shows that the resulting map QP(A, 0) nondeg → (QP(A, 0) nondeg /G 2 ) × 2 H 2 (DR(A)) underlies an equivalence.Thus quantisation reduces to a first order problem.
This first order problem is resolved by introducing a notion of self-duality.In [Pri4], self-dual quantisations were defined for line bundles L with an involution L ≃ L ∨ to the Grothendieck-Verdier dual.The analogous notion in our setting is given by considering involutive associative algebras and categories.Explicitly, when X is a smooth variety, a self-dual quantisation of O X is an associative deformation (O X , ⋆ ) of O X with a ⋆ − b = b ⋆ a.More generally, a self-dual quantisation of X over R leads to a curved A ∞ -category with R -semilinear contravariant involution, deforming the dg category of perfect complexes on X.
Restricting to self-dual quantisations ensures that the first-order obstruction vanishes, leading to Theorem 2.20, which shows that the equivalence class of self-dual quantisations of a given non-degenerate Poisson structure is parametrised by 2 H 2 (DR(A)) 2 , and in particular such quantisations always exist.Global versions of these results for derived Artin N -stacks are summarised in Theorem 3.13.The structure of the paper is as follows.
In §1 we recall the description from [Pri5] of commutative bidifferential bigraded algebras as formal completions of derived N -stacks along derived affines, together with the complex of polyvectors Pol(A, 0) on such objects, and the space P(A, 0) of Poisson than cochain complexes -this will enable us to distinguish easily between derived (chain) and stacky (cochain) structures.
Definition 1.1.A stacky CDGA is a chain cochain complex A • • equipped with a commutative product A ⊗ A → A and unit Q → A. Given a chain CDGA R, a stacky CDGA over R is then a morphism R → A of stacky CDGAs.We write DGdgCAlg(R) for the category of stacky CDGAs over R, and DG + dgCAlg(R) for the full subcategory consisting of objects A concentrated in non-negative cochain degrees.
When working with chain cochain complexes V • • , we will usually denote the chain differential by δ : V i j → V i j−1 , and the cochain differential by ∂ : . Readers interested only in DM (as opposed to Artin) stacks may ignore the stacky part of the structure and consider only chain CDGAs A • = A 0 • throughout this section.Example 1.2.We now recall an important example of a class of stacky CDGAs from [Pri5,Example 3.28].Given a Lie algebra g of finite rank acting as derivations on a derived affine scheme Y , we write O([Y /g]) for the stacky CDGA given by the Chevalley- Definition 1.3.Say that a morphism U → V of chain cochain complexes is a levelwise quasi-isomorphism if U i → V i is a quasi-isomorphism for all i ∈ Z. Say that a morphism of stacky CDGAs is a levelwise quasi-isomorphism if the underlying morphism of chain cochain complexes is so.
The following is [Pri5,Lemma 3.4]: Lemma 1.4.There is a cofibrantly generated model structure on stacky CDGAs over R in which fibrations are surjections and weak equivalences are levelwise quasiisomorphisms.
There is a denormalisation functor D from non-negatively graded CDGAs to cosimplicial algebras, with left adjoint D * as in [Pri1,Definition 4.20].Given a cosimplicial chain CDGA A, D * A is then a stacky CDGA in non-negative cochain degrees.By [Pri5,Lemma 3.5], D * is a left Quillen functor from the Reedy model structure on cosimplicial chain CDGAs to the model structure of Lemma 1.4.
Since DA is a pro-nilpotent extension of A 0 , when H <0 (A) = 0 we think of the simplicial hypersheaf RSpec DA as a stacky derived thickening of the derived affine scheme RSpec A 0 .Definition 1.5.Given a chain cochain complex V , define the cochain complex T ot with differential ∂ ± δ.
Definition 1.6.Given a stacky CDGA A and A-modules M, N in chain cochain complexes, we define internal Homs Hom A (M, N ) by , where V # # denotes the bigraded vector space underlying a chain cochain complex V .
We then define the Hom complex Ĥ om A (M, N ) by Ĥ om A (M, N ) := T ot Hom A (M, N ).
Note that there is a multiplication Ĥ om A (M, N ) ⊗ Ĥ om A (N, P ) → Ĥ om A (M, P ) (the same is not true for Tot Π Hom A (M, N ) in general).
Definition 1.7.A morphism A → B in DG + dgCAlg(R) is said to be homotopy formally étale when the map Combining [Pri5,Proposition 3.12] with [Pri3,Theorem 4.15 and Corollary 6.35], every strongly quasi-compact derived Artin N -stack over R can be resolved by a derived DM hypergroupoid (a form of homotopy formally étale cosimplicial diagram) in DG + dgCAlg(R).
Assumption 1.8.As in [Pri5,§3.3],we now assume that A ∈ DG + dgCAlg(R) has the following properties: (1) for any cofibrant replacement Ã → A in the model structure of Lemma 1.4, the morphism Ω 1 Ã/R → Ω 1 A/R is a levelwise quasi-isomorphism, (2) the A # -module (Ω 1 A/R ) # in graded chain complexes is cofibrant (i.e. it has the left lifting property with respect to all surjections of A # -modules in graded chain complexes), (3) there exists N for which the chain complexes (Ω 1 A/R ⊗ A A 0 ) i are acyclic for all i > N .
Of particular interest for us is that these conditions are satisfied when A = D * O(X) for derived Artin N -hypergroupoids X.The following is adapted from [Pri5,Definition 3.19] along the lines of [Pri4,Definition 1.3], with the introduction of a dummy variable of cohomological degree 0.
Definition 1.9.Define the complex of 0-shifted polyvector fields (or strictly speaking, multiderivations) on A by with graded-commutative multiplication (a, b) → ab following the usual conventions for symmetric powers.The Lie bracket on Ĥ om A (Ω 1 A/R , A) then extends to give a bracket (the Schouten-Nijenhuis bracket) determined by the property that it is a bi-derivation with respect to the multiplication operation.
Thus Pol(A/R, 0) has the natural structure of a P 2 -algebra, and in particular Pol(A/R, 0)[1] is a differential graded Lie algebra (DGLA) over R.
The product on polyvectors makes this a CDGA (with no need to rescale the product by ), and it inherits the filtration F from Pol.
Given π ∈ MC(F 2 Pol(A/R, 0)[1]/F p ), we define T π Pol(A/R, 0)/F p similarly.This is a CDGA because 1.3.The Hochschild complex of a stacky CDGA.Definition 1.13.For an A-module M in chain cochain complexes, we define the cohomological Hochschild complex CC • R (A, M ) over R as we would for dg algebras, but using the Hom-complexes Ĥ om.Thus CC There is also a quasi-isomorphic normalised version N c CC • R (A, M ), given by the subspaces of functions f with f (a 1 , . . ., a i−1 , 1, a i , . . ., a n ) = 0 for all i.
We define increasing filtrations F on CC • R (A, M ) and CC • R (A, M ) by good truncation in the Hochschild direction, so We simply write CC Lemma 1.14.There is a natural brace algebra structure on CC • R (A) over R, compatible with the filtration F .In particular, CC • R (A)[1] is a filtered DGLA over R. On the associated graded brace algebra gr F CC • R (A), the Lie bracket and higher braces vanish, and there is a surjective quasi-isomorphism of brace algebras, where we set all the braces to be 0 on HH * .
Proof.As in [Vor,§3], there is a brace algebra structure on CC • R (A), with cup product • of cohomological degree 0 and brace operations (f, Compatibility of b with the bracket then implies that [F p , F q ] ⊂ F p+q−1 , and degree considerations also give ) is a filtered brace algebra; the bracket vanishes on gr F , as do the braces for n ≥ 2.
Since F is defined as good truncation in the Hochschild direction, Hochschild cohomology HH * is automatically a quasi-isomorphic quotient of gr F .Any operation of negative degree necessarily vanishes on this quotient, so the quotient map is a brace algebra morphism.
Lemma 1.15.There is an involutive map i : CC This involution corresponds under the HKR isomorphism to the involution of Pol(A, 0) which acts on Ĥ om A (Ω p A/R , A) as scalar multiplication by (−1) p−1 .
Proof.The first statement is proved in [Bra,§2.1],taking the trivial involution on A.
Definition 1.17.Define a decreasing filtration F on Q Pol(A, 0) by the subcomplexes This filtration is complete and Hausdorff, with For more general stacky CDGAs, the stacky and derived structures interact in a non-trivial way for quantisations, and indeed for Poisson structures.
Remark 1.19.To strengthen the analogy between this construction and [Pri4], we could replace N c CC • (A) with its quasi-isomorphic subcomplex of polydifferential operators.The filtration F is then quasi-isomorphic to the order filtration for polydifferential operators, but the latter does not interact so well with the Lie bracket.
If we wished to consider uncurved A ∞ -algebra deformations without inner automorphisms, we would have to replace CC • (A) with its sub-DGLA ker(CC • R (A) → T ot A).The analogue for [Pri4] is the kernel of the map D A → A given by evaluating at 1.As in [Pri4,Remark 1.13], this means that the E 0 analogue of a strict quantisation is a BV algebra deformation.
Example 1.20.When the stacky CDGA A is bounded in the stacky (cochain) direction, we may identify CC • R (A) with the Hochschild complex of the CDGA Tot A, as Hom(A ⊗n , A) is then also bounded in the cochain direction, and the functors Tot , T ot , Tot Π agree for such double complexes.In particular, this applies to stacky CDGAs of the form O([Y /g]) in the notation of Example 1.2.
Given a finite rank Lie algebra g acting on a smooth affine Y over R, the derived cotangent stack T * [Y /g] carries a non-degenerate Poisson structure.Explicitly, if Y = Spec B, this derived formal stack is represented by the stacky CDGA given by the Chevalley- with its natural Poisson structure as a complex of polyvectors.
A quantisation of this Poisson structure is given by the Rees algebra Following [Hin], define the Maurer-Cartan space MC(L) (a simplicial set) of a nilpotent DGLA L by MC(L where is the commutative dg algebra of de Rham polynomial forms on the n-simplex, with the t i of degree 0. Definition 1.22.We now define another decreasing filtration G on Q Pol(A, 0) by setting We then set Definition 1.23.Define the space QP(A, 0) of E 1 quantisations of A over R to be given by the simplicial set When R and A = A 0 are concentrated in non-negative homological degrees, we can interpret QP(A, 0) as a space of deformations of A as an R-linear dg category up to quasi-equivalence, and in general when A = A 0 and has bounded cohomology, [LdB] interprets QP(A, 0) as a space of deformations of A as an R-linear dg category up to derived Morita equivalence.
For stacky CDGAs, good behaviour of the functor T ot gives us a natural map CC (rarely an equivalence), so E 1 quantisations give rise to curved A ∞ deformations of the CDGA T ot A. We now give a stronger statement.
Definition 1.24.If A ∈ DG + dgCAlg(R), define the bi-dg category Per (A) as follows.Objects are A-modules M in chain cochain complexes for which M # is cofibrant as a graded chain complex over A # , M 0 is perfect over A 0 , and the map M 0 ⊗ A 0 A # → M # is a levelwise quasi-isomorphism.Morphisms are given by the chain cochain complexes Hom A (M, N ).
We then define per dg (A) to have the same objects as Per (A), and morphisms Ĥ om A (M, N ).
For every M ∈ per dg (A), we have a T ot A-module T ot M , but this need not be cofibrant or perfect.For instance, given b ∈ Z 0 Z 1 A, we may set A b to be the chain cochain complex Proposition 1.25.For A ∈ DG + dgCAlg(R), there is a natural map in the ∞category of simplicial sets from QP(A, 0) to the space of curved A ∞ deformations (per dg (A) , {m (i) } i≥0 ) of the dg category per dg (A), with i−1 | m (i) for i ≥ 3.
Proof.For any R-linear bi-dg category B, we have a Hochschild complex built from the spaces with QP(B, 0) defined analogously.Properties of T ot then give us a natural map from QP(B, 0) to QP( T ot B, 0), which is the space of curved A ∞ deformations (( T ot B) , {m (i) } i≥0 ) of the dg category T ot B with i−1 | m (i) for i ≥ 3; the Maurer-Cartan conditions ensure that i | bm (i) , so every such m does lie in the appropriate piece of the good truncation filtration.
It therefore suffices to show that the map QP(Per (A), 0) → QP(A, 0) given by restriction to the object A ∈ Per (A) is a weak equivalence.By the theory of pro-nilpotent DGLAs, this will follow if CC n R (Per (A)) → CC n R (A) is a filtered quasi-isomorphism.We now observe that for any A-linear bi-dg category B with cofibrant Hombicomplexes, there is a spectral sequence When B is homotopy Cartesian in the sense that the map We then note that when B 0 # is Morita equivalent to A 0 # as a graded category, the map M → HH * ) is an isomorphism of graded modules for all A 0 # -modules M .Putting these together gives quasi-isomorphisms ) : the bi-dg category Per (A) is homotopy Cartesian because its objects are; since Per (A) 0 # is equivalent to the category of graded projective A 0 # -modules, it is Morita equivalent to A 0 # .Thus QP(Per (A), 0) → QP(A, 0) is indeed a weak equivalence.Remark 1.26.In [Pri4], we were able to consider E 0 quantisations not just of the structure sheaf O X , but also of line bundles, by constructing a G m -action on quantised polyvectors.
Similarly, the methods of this paper can be adapted to study E 1 quantisations of any A-linear bi-dg category B for which the map T ot A → gr F CC • A (B) is a quasi-isomorphism -by analogy, line bundles are A-modules for which the map T ot A → R Ĥ om A (M, M ) is a quasi-isomorphism.In particular, we can study étale G m -gerbes by establishing BG m -equivariance.One way to do this is to consider QP(Per (A), 0) as in the proof of Proposition 1.25, since Per (A) admits an action of the Picard 2-group and hence a BG m -action.
The resulting action is necessarily trivial modulo G 1 , so comes from pro-unipotent L ∞ -automorphisms of Q Pol(A, 0).Since pro-unipotent L ∞ -automorphisms are exponentials of pro-nilpotent L ∞ -derivations, we will in fact have an action of BG m ⊗ Z Q, so a notion of quantisation for (G m ⊗ Z Q)-gerbes.
1.5.The centre of a quantisation.Definition 1.27.Define the filtered tangent space to quantised polyvectors by ), define the centre of (A, ∆) by with derivation ∂ ± δ ± b + [∆, −] (necessarily square-zero by the Maurer-Cartan conditions).This has a filtration making T ∆ Q Pol(A, 0) a filtered brace algebra by Lemma 1.14.Given ∆ ∈ MC(F 2 Q Pol(A, 0)/ F p ), we define T ∆ Q Pol(A, 0)/ F p similarly -this is also a brace algebra as F p is a brace ideal.

Observe that T
Similarly to Definition 1.22, there are filtrations G on T Q Pol(A, 0), T ∆ Q Pol(A, 0) given by powers of .Since gr i G which are quasi-isomorphisms by our hypotheses on A (see Assumption 1.8).
For the filtration F of Definition 1.10, we may rewrite these maps as Since the cohomology groups of T π ∆ Pol(A, 0) are Poisson cohomology, we will refer to the cohomology groups of T ∆ Q Pol(A, 0) as quantised Poisson cohomology.
Definition 1.29.Say that an ) is a perfect complex over A 0 .Definition 1.30.Define the tangent spaces with T QP(A, 0)/G k , defined similarly.
Definition 1.31.Define the canonical tangent vector Note that this is a morphism of filtered DGLAs, so gives a map σ : for the involution i of Lemma 1.15.
Definition 1.33.Lemma 1.15 ensures that * is a morphism of DGLAs, and we define the space QP(A, 0) sd ⊂ QP(A, 0) of self-dual quantisations to be the fixed points of the involution * .This inherits cofiltrations F and G from QP(A, 0).
In particular, this means that when A = A 0 is concentrated in degree zero, elements of QP(A, 0) sd can be represented by associative algebra deformations (A ′ , ⋆ ) of A, with More generally, when R and A = A 0 are concentrated in non-negative homological degrees, elements of QP(A, 0) sd are algebroid quantisations of A equipped with a contravariant involution which is semilinear under the transformation → − .
For general stacky CDGAs A, QP(A, 0) sd gives rise via Proposition 1.25 to curved A ∞ -deformations P of the dg category per dg (A) of perfect complexes, equipped with an involution P− ≃ Popp lifting the duality functor Hom A (−, A) on per dg (A).
Remark 1.34.As in Remark 1.26, we may also consider self-duality for G m -gerbes.Since the functor sending a gerbe to its opposite is just given by the inversion map on B 2 G m , involutive gerbes are are classified by B 2 µ 2 , the homotopy fixed points of the inversion map.
However, as observed in Remark 1.26, the space of quantisations over B 2 G m is the pullback of a space over B 2 (G m ⊗ Z Q), so the space of self-dual quantisations over B 2 µ 2 is constant.This means that to every self-dual quantisation of A there correspond selfdual quantisations of all µ 2 -gerbes, and in particular of per dg (A) with duality functor RH om(−, L ) for any line bundle L .One way to make sense of this example is that even if L does not have a square root, there is necessarily an automorphism of the Hochschild complex acting as a square root of L , and thus intertwining between the respective duality functors.
Lemma 1.35.There are canonical weak equivalences Proof.This follows in much the same way as [Pri4,Lemma 4.5].Lemma 1.15 ensures that the involution * acts trivially on Pol(A, 0), since it maps The results then follow from the fibre sequences coming from obstruction theory for abelian extensions of DGLAs.

Quantisations and de Rham power series
Recall that we are fixing a chain CDGA R over Q, and a cofibrant stacky CDGA A over R. We denote the chain differentials on A and R by δ, and the cochain differential on A by ∂.
Definition 2.1.Define the de Rham complex DR(A/R) to be the product total complex of the bicomplex We define the Hodge filtration F on DR(A/R) by setting Definition 2.2.When A is a cofibrant stacky CDGA over R, recall that a 0-shifted pre-symplectic structure ω on A/R is an element Definition 2.3.Define a decreasing filtration F on DR(A/R) by Definition 2.4.Define the space of generalised 0-shifted pre-symplectic structures on A/R to be the simplicial set where we regard the cochain complex DR(A/R)[1] as a DGLA with trivial bracket.Write PreSp = GPreSp/G 1 .Also write GPreSp(A, 0)/ k := lim 0) to consist of the points whose images in PreSp(A, 0) are symplectic structures -this is a union of path-components.
Remarks 2.5.Note that Definition 2.4 is not the obvious analogue of the definition of generalised (−1)-shifted pre-symplectic structures from [Pri4, Definition 1.29], which used the convolution (G * F ) 2 = F 2 + G 1 in place of F 2 for reasons specific to negatively shifted structures.The only difference lies in the linear term, which is where the correspondence between generalised symplectic structures and non-degenerate quantisations breaks down anyway -replacing F 2 with (G * F ) 2 would not significantly affect the main results of either paper, nor would eliminating the linear term altogether.
Also note that GPreSp(A, 0) is canonically weakly equivalent to the Dold-Kan denormalisation of the good truncation complex τ ≤0 ( F 2 DR(A/R) [2]) (and similarly for the various quotients we consider), but the description in terms of MC will simplify comparisons.In particular, we have

Formality.
Definition 2.6.Write GT for the Grothendieck-Teichmüller group.This is an affine group scheme over Q, with reductive quotient G m .Denote the pro-unipotent radical ker(GT → G m ) by GT 1 .
Write Levi GT for the space of Levi decompositions of GT, i.e. sections of GT → G m .By the general theory of pro-algebraic groups in characteristic 0, the space Levi GT is an affine scheme over Q equipped with the structure of a trivial GT 1 -torsor via the adjoint action.
For any λ ∈ k × , a Levi decomposition w ∈ Levi GT (k) gives a λ-Drinfeld associator w(λ) ∈ GT(k).When λ is not a root of unity, this map from Levi GT (k) to the set of λ-Drinfeld associators is an isomorphism, since both sets are GT 1 (k)-torsors.
As explained succinctly in [Pet], formality of the Q-linear E 2 operad is a consequence of the observation that the Grothendieck-Teichmüller group is a pro-unipotent extension of G m .Since GT acts on E 2 , any Levi decomposition w : G m → GT gives a weight decomposition (i.e. a G m -action) of E 2 which splits the good truncation filtration, so gives an equivalence between E 2 and P 2 respecting the natural map from the Lie operad.
Definition 2.7.Given a Levi decomposition w ∈ Levi GT (Q), we denote by p w the resulting ∞-functor from E 2 -algebras to P 2 -algebras over Q, which respects the underlying L ∞ -algebras.
As in [Vor], brace algebras are naturally E 2 -algebras, so CC • R (A) has an E 2 -algebra structure.Moreover, the equivalence between E 2 and P 2 necessarily respects the good truncation filtrations, and the filtered complex (CC • R (A), F ) is an algebra with respect to the brace operad filtered by good truncation.This yields a filtered P 2 -algebra ). Definition 2.8.For any of the definitions from §1, we add the subscript w to indicate that we are replacing (CC • R (A), F ) with (p w CC • R (A), F ) in the construction.Since these DGLAs are quasi-isomorphic and MC preserves weak equivalences, in particular we have canonical weak equivalences QP w (A, 0) ≃ QP(A, 0).Properties of the filtration F then ensure that the complexes T ∆ Q Pol w (A, 0) are filtered P 2 -algebras.
Remark 2.9.Rather than just choosing w ∈ Levi GT (Q), a more natural approach might be to consider the simplicial set RLevi GT (R) of all Levi decompositions over R.This would lead to a space QP Levi (A, 0) over RLevi GT (R) with fibre QP w (A, 0) over w and a canonical weak equivalence QP Levi (A, 0) ≃ RLevi GT (R) × QP(A, 0).

Compatible quantisations.
We will now develop the notion of compatibility between a generalised pre-symplectic structure and an E 1 quantisation, generalising the notion of compatibility between 0-shifted pre-symplectic and Poisson structures from [Pri5].The following definitions are adapted from [Pri5,Definition 1.16].
Definition 2.10.Given a stacky CDGA B over A and a derivation ∆ ∈ MC( Ĥ om to be the morphism of graded A-algebras given on generators Ω 1 A/R by setting µ(adf, ∆) := a∆(f ), and then applying T ot (noting that Tot Π Ω p A = T ot Ω p A ).The proof of [Pri5,Lemma 1.17] ensures that this becomes a chain map (and hence an R-CDGA morphism) by applying Definition 2.10 to the stacky CDGAs and the derivation [∆, −], then taking the limit over all k.Observe that this map preserves the filtration F .
Definition 2.12.We say that a generalised pre-symplectic structure ω and an E 1 quantisation ∆ are w-compatible (or a w-compatible pair) if where σ = −∂ −1 is the canonical tangent vector of Definition 1.31.
Definition 2.13.Given a simplicial set Z, an abelian group object A in simplicial sets over Z, a space X over Z and a morphism s : X → A over Z, define the homotopy vanishing locus of s over Z to be the homotopy limit of the diagram Definition 2.14.Define the space QComp w (A, 0) of w-compatible quantised 0-shifted pairs to be the homotopy vanishing locus of We define a cofiltration on this space by setting QComp w (A, 0)/G j to be the homotopy vanishing locus of When j = 1, note that this recovers the notion of compatible 0-shifted pairs from [Pri5,§3.3.3].

The equivalences.
Proposition 2.16.For any Levi decomposition w of GT, the canonical map QComp w (A, 0) nondeg → QP w (A, 0) nondeg ≃ QP(A, 0) nondeg is a weak equivalence.In particular, there is a morphism in the homotopy category of simplicial sets.
Proof.We adapt the proof of [Pri5,Proposition 1.26].For any ∆ ∈ QP w (A, 0), the homotopy fibre of QComp w (A, 0) nondeg over ∆ is just the homotopy fibre of The map µ w (−, ∆) : on the associated gradeds gr k G gr p F .We therefore have a quasi-isomorphism of bifiltered complexes, so we have isomorphisms on homotopy groups: Proposition 2.17.For any Levi decomposition w of GT, the maps coming from Proposition 2.16 are weak equivalences for all j ≥ 2.
Proof.The proof of [Pri4,Proposition 1.40] generalises to this setting.We have a commutative diagram of fibre sequences, with N (ω, π, j) the cocone of the map given by combining Here ν(ω, π) is the tangent map of µ(ω, −) at π, given by µ(ω, π As in [Pri4,Lemma 1.39], on the associated graded piece As this is an isomorphism for all j ≥ 2, the map N (ω, π, j) → F 2−j DR(A/R) j is quasi-isomorphism, which inductively gives the required weak equivalences from the fibre sequences above.
Remark 2.18.Taking the limit over all j, Proposition 2.17 gives an equivalence in particular, this means that there is a canonical map (QP(A, 0) nondeg /G 2 ) → QP(A, 0) nondeg , dependent on w, corresponding to the distinguished point 0 ∈ MC( 2 DR(A/R) ).
Thus to quantise a non-degenerate 0-shifted Poisson structure π = j≥2 π j (or equivalently, by [Pri5,Corollary 1.38], a 0-shifted symplectic structure), it suffices to lift the power series j≥2 π j j−1 to a Maurer-Cartan element of j≥2 (F j CC • R (A)/F j+2 ) j−1 .Even if π is degenerate, a variant of Proposition 2.17 still holds.Because π ♯ • ω ♯ is homotopy idempotent, the map gr p F ν(ω, π) has eigenvalues in the interval [0, p], so we just replace (1 − j) with an operator having eigenvalues in the interval [1 Since this is still a quasi-isomorphism for j > 1, we have giving a sufficient first-order criterion for degenerate quantisations to exist.
Remark 2.19.As in Remark 2.9, we could consider the space RLevi GT (R) of R-linear Levi decompositions, and the proof of Proposition 2.17 then gives equivalences 2.4.1.Self-duality.
Theorem 2.20.For any Levi decomposition w of GT, there is a canonical weak equivalence QP(A, 0) nondeg,sd ≃ P(A, 0 In particular, w gives a canonical choice of self-dual quantisation for any non-degenerate 0-shifted Poisson structure on A.
Remark 2.21.The proof of Theorem 2.20 shows that for a self-dual quantisation of a non-degenerate 0-shifted Poisson structure, the w-compatible generalised symplectic structure is determined by its even coefficients.This raises the question of whether the odd coefficients must be homotopic to 0, as happens in the (−1)-shifted case by [Pri4,Remark 4.6].The answer depends on the choice of w, as follows.
The involution i from Lemma 1.15 is not just a DGLA automorphism.If we write into an involutive brace algebra.The opposite brace algebra B opp is most easily understood in terms of the associated B ∞ -algebra, which is a bialgebra structure on the tensor coalgebra T (B[1]): to form B opp , we just take the opposite comultiplication on T (B[1]).
We can define an involution of the E 2 operad similarly, which takes an embedding [1, k] × I 2 → I 2 of k little squares in a big square, and reverses the order of the labels [1, k] with appropriate signs.This involution comes from an element t ∈ GT which maps to −1 ∈ G m ; in other words, t is a (−1)-Drinfeld associator.It gives a notion of opposite E 2 -algebra, with (−) t : C Levi decompositions w of GT with w(−1) = t form a torsor Levi t GT for the subgroup (GT 1 ) t of t-invariants in GT 1 .(To see that Levi t GT is non-empty, first pick any Levi decomposition w 0 , and write w 0 (−1) = tu for u ∈ GT 1 .Since t and w 0 (−1) are both of order 2, we have u = ad t (u −1 ), so u ) For any such w ∈ Levi t GT (Q), the ∞-functor p w sends opposite E 2 -algebras to opposite P 2 -algebras, defined by reversing the sign of the Lie bracket.This gives µ w (ω, ∆) t = µ w (ω, −∆ t ), so ω( ) is compatible with ∆ if and only if ω(− ) is compatible with ∆ * , implying that the odd coefficients of ω must be homotopic to 0 when ∆ is non-degenerate and self-dual.
For a more explicit description of the generalised symplectic structure ω corresponding to a non-degenerate self-dual quantisation ∆, observe that we then have an isomorphism and that [ω] must be the inverse image of [ 2 ∂∆ ∂ ].Remark 2.22.Similarly to Remarks 2.9 and 2.19, we could consider the space RLevi t GT (R) of R-linear Levi decompositions with w(−1) = t, and the proof of Theorem 2.20 then combines with Remark 2.21 to give a canonical weak equivalence ). over RLevi t GT (R).2.5.Comparison with Kontsevich-Tamarkin quantisations.In [Kon2], Kontsevich showed that for a smooth algebraic variety X over a field k, and a choice w ∈ Levi GT (k), every Poisson structure π lifts to an algebroid quantisation of X.We now investigate how this quantisation relates to our quantisations above when π is nondegenerate and X = Spec A affine; by descent, this comparison will extend to the global quantisations of the next section.In Tamarkin's reinterpretation [Tam], the main intermediate step, as described in [Kon1] or [VdB,Theorem 1.1], is the existence of a canonical quasi-isomorphism of filtered P 2,∞ -algebras, lifting the HKR isomorphism, and depending only on a choice of formality quasi-isomorphism for k t 1 , . . ., t d .This immediately gives us a k -linear P 2,∞ -algebra quasi-isomorphism sending the filtration { F p } p on the left to { i F p−i i } p .
In particular, we have a section φ of the projection Q Pol w (A, 0) → Pol(A, 0), and the quantisation of [Kon2] is the induced map φ w : P(A, 0) → QP(A, 0).
In the non-degenerate setting ω = π −1 , and in order to compare φ w (π) with the quantisations of Theorem 2.20, we need to know whether it is self-dual.Tamarkin's approach to quantisation, as described in [Kon1], relies on showing that the equivalence class of P 2 -algebra deformations of Pol(k[t 1 , . . ., t d ], 0) invariant under affine transformations is trivial.The same will necessarily be true for the equivalence class of involutive P 2 -algebra deformations, replacing the deformation complex of [Kon1,§3.4]with its subspace of odd weight.If w(−1) = t, the formality isomorphism of [VdB,Theorem 1.1] then gives φ : Pol(A, 0) 2 ≃ Q Pol w (A, 0) sd , so we have shown the following: Proposition 2.23.For a smooth algebra A over a field k ⊃ Q and for w ∈ Levi t GT (k), the Kontsevich-Tamarkin quantisation φ w (π) of any Poisson structure π on A is selfdual.When π is non-degenerate, the quantisation φ w (π) corresponds under Theorem 2.20 to the constant de Rham power series π −1 .
The explicit quantisation formulae of [Kon3] satisfy a ⋆ − b = b ⋆ a, so are self-dual and are thus covered by the proposition.
Remark 2.24.If we wish to extend Theorem 2.20 to give existence of quantisations for degenerate Poisson structures on more general stacky CDGAs A, then we need an alternative to [Kon2], and we now sketch a possible approach which we intend to investigate in future work.Instead of looking at quantisations of k t 1 , . . ., t d , we can rigidify the problem by observing that p w CC • k (A) is an involutive filtered deformation of the P 2 -algebra Pol(A, 0) whenever w(−1) = t.
Since the Maurer-Cartan functor is defined on curved L ∞ -algebras, it suffices to consider curved P 2 -algebra deformations.Regarding Pol(A, 0) as some sort of stacky CDGA with its canonical 1-shifted non-degenerate Poisson structure ̟ will yield a DGLA L := T ̟ Pol(Pol(A, 0), 1) [2].The gradings on Pol(−, n) which set m-vectors to have weight m give rise to a total grading on L, for which ̟ is of total weight 2 − 1 = 1 since it is a bivector which acts with weight −1 on Pol(A, 0).The Lie bracket on L has weight −1, and the DGLA governing filtered deformations of Pol(A, 0) will be the sub-DGLA of L consisting of elements with non-positive weight.Involutive deformations will be governed by the sub-DGLA of odd weights.
The methods of [Pri5] should then show that the complex L is G m -equivariantly quasi-isomorphic to DR(Pol(A, 0)) [2], with the subcomplex of non-positive weight quasiisomorphic to DR(A) [2], which has weight 0. Thus the subcomplex of odd weights would be trivial, so the ∞-groupoid of deformations of Pol(A, 0) as an involutive filtered curved P 2 -algebra should be contractible.This would mean that p w CC • k (A) ≃ Pol(A, 0), establishing the remaining case of [Toë,Conjecture 5.3] and giving curved quantisations for all 0-shifted Poisson structures.

Quantisation for derived stacks
As in [Pri4,§3], in order to pass from stacky CDGAs to derived Artin stacks, we will exploit étale functoriality using Segal spaces.
with the filtration F k CC • R (A, M ) defined similarly, for the Hochschild complexes of Definition 1.13.
We then write CC • R (A) := CC • R (A, A), which inherits the structure of a brace algebra from each CC • R (A(i), A(i)).For f : i → j a morphism in I, observe that the HKR maps ) are quasi-isomorphisms whenever A(i) is cofibrant in the model structure of Lemma 1.4.Also note that if u : I → J is a morphism of small categories and A is a functor from J to DG + dgCAlg(R) with B = A • u, then we have a natural map CC B).In order to ensure that CC • R (A, M ) has the correct homological properties, we now consider categories of the form which is cofibrant and fibrant for the injective model structure (i.e. each A(i) is cofibrant in the model structure of Lemma 1.4 and the maps A(i) → A(i + 1) are surjective), then gr , from which this result follows immediately via the HKR isomorphism.
The constructions in §1 now all carry over verbatim, generalising from cofibrant stacky CDGAs to [m]-diagrams of stacky CDGAs which are cofibrant and fibrant for the injective model structure.In order to identify QP/G 1 with P, and for notions such as non-degeneracy to make sense, we have to assume that for our fibrant cofibrant [m]diagram A of stacky CDGAs, each A(j) satisfies Assumption 1.8, so there exists N for which the chain complexes (Ω 1 A(j)/R ⊗ A(j) A(j) 0 ) i are acyclic for all i > N .
(1) the matching maps are fibrations for all m ≥ 0; (2) the partial matching maps are smooth surjections for all m ≥ 1 and k, and are weak equivalences for all m > N and all k.A morphism X → Y in sDG + Aff R is a trivial DG Artin (resp.DM) N -hypergroupoid if and only if the matching maps are surjective smooth fibrations for all m, and are weak equivalences for all m ≥ n.
The following is [Pri3,Theorem 4.15 and Corollary 6.35], as spelt out in [Pri2, Theorem 5.11]: Theorem 3.6.The ∞-category of strongly quasi-compact N -geometric derived Artin stacks over R is given by localising the category of DG Artin N -hypergroupoids over R at the class of trivial relative DG Artin N -hypergroupoids.
Given a DG Artin N -hypergroupoid X, we denote the associated N -geometric derived Artin stack by X ♯ .
There is a denormalisation functor D from non-negatively graded CDGAs to cosimplicial algebras, with left adjoint D * as in [Pri1,Definition 4.20].Given a cosimplicial chain CDGA A, D * A is then a stacky CDGA, with (D * A) i j = 0 for i < 0. 3.3.Global quantisations.The following is [Pri5,Corollary 3.13], showing that a DG Artin N -hypergroupoid X can be recovered from the stacky CDGAs D * O(X ∆ j ): Lemma 3.7.For any simplicial presheaf F on DGAff(R) and any Reedy fibrant simplicial derived affine X, there is a canonical weak equivalence Lemma 3.7 and [Pri5, Proposition 3.18] ensure that if a morphism X → Y of DG Artin N -hypergroupoids becomes an equivalence on hypersheafifying, then D * O(Y ) → D * O(X) is formally étale in the sense of Lemma 3.4.In particular this means that the maps ∂ ) can be thought of as a DM hypergroupoid in stacky CDGAs, and we may make the following definition: Definition 3.8.Given a DG Artin N -hypergroupoid X over R and any of the constructions F based on QP, write for RF as in §3.1.
The proof of [Pri5,Proposition 3.29] shows that if Y → X is a trivial DG Artin hypergroupoid, then the morphism F (X) → F (Y ) is an equivalence for any of the constructions F = P, Comp, PreSp.Thus the following is well-defined: Definition 3.9.Given a strongly quasi-compact DG Artin N -stack X over R, define the spaces QP(X, 0), QComp w (X, 0), GSp(X, 0) to be the spaces QP(X, 0), Comp w (X, 0), GSp(X, 0) for any DG Artin N -hypergroupoid X with X ♯ ≃ X.
Examples 3.10.Examples of derived stacks X with canonical 0-shifted symplectic structures (elements of GSp(X, 0)/G 1 ) include the derived moduli stack RPerf S of perfect complexes on an algebraic K3 surface S, or the derived moduli stack RLoc G (Σ) = map(Σ, BG) of locally constant G-torsors on a compact oriented topological surface Σ, for an algebraic group G equipped with a Killing form on its Lie algebra.These both follow from [PTVV, §3.1], with the symplectic form in the latter case coming from the 2-shifted symplectic structure in H 4 (F 2 DR(BG)), via the composition Proposition 3.11.For any strongly quasi-compact DG Artin N -stack X over R, there is a natural map from QP(X, 0) to the space of curved A ∞ deformations (per dg (X) , {m (i) } i≥0 ) of the dg category per dg (X) of perfect O X -complexes.
This restricts to a map from QP(X, 0) sd to the space of involutive curved A ∞ deformations of per dg (X).
As in Remark 2.21, the power series w-compatible with a quantisation, although determined by its even coefficients, might have odd coefficients unless w(−1) = t.The reasoning of Remark 2.22 gives a canonical equivalence RLevi t GT (R)×QP(X, 0) nondeg,sd ≃ RLevi t GT (R)×P(X, 0) nondeg ×MC( 2 DR(X/R) 2 [1]) over RLevi t GT (R) which does send a quantisation to its family of compatible de Rham power series.
As in Remark 1.34, self-dual quantisations of O X also give rise to self-dual quantisations of all involutive G m -gerbes, and in particular of the Picard algebroid with involution given by RH om O X (−, L ) for any line bundle L .
Finally, applying étale descent to Proposition 2.23 shows that for a smooth DM stack X, the Kontsevich-Tamarkin quantisation φ w (π) of any Poisson structure π on X is self-dual whenever w ∈ Levi t GT (k).When π is non-degenerate, the quantisation φ w (π) then corresponds under Theorem 3.13 to the constant de Rham power series π −1 .
Remark 3.15 (Positively shifted symplectic structures).All of the definitions of this paper adapt to n-shifted symplectic structures for n > 0, by working with E n+1 -Hochschild cohomology.Formality of E n+1 gives canonical P n+2 -algebra quasi-isomorphisms so existence of quantisations is automatic.
However, the methods of this paper adapt to yield slightly more information for positively shifted symplectic structures, since they give equivalences QP w (X, n) nondeg,sd ≃ P(X, n) nondeg × MC( 2 DR(X/R) 2 [n + 1]) for any w ∈ Levi t GT .This provides a complete parametrisation of non-degenerate selfdual E n+1 quantisations in terms of de Rham power series, with the choice of Levi decomposition only altering the parametrisation.
of g with coefficients in the chain g-module O(Y ).When the action of g on Y is induced by the action of an affine group scheme G with Lie algebra g, the stacky CDGA can recover the relative de Rham stack [Y /g] of Y over [Y /G], and there is a formally étale simplicial resolution of [Y /G] in terms of the functors [Y × G n /g n+1 ].
i i F i D Tot O([Y /g]) of the order filtration F on the ring of differential operators, i.e. the -adically complete sub-DGA of D Tot O([Y /g]) generated by O([Y /g]) and first order differential operators divisible by .This quantisation satisfies b ⋆ a = (−1) deg a deg b a ⋆ − b, so will be included in the parametrisation of Theorem 2.20.Definition 1.21.Given a DGLA L, define the the Maurer-Cartan set by MC(L) :

3. 1 .
Quantised polyvectors for diagrams.Definition 3.1.Given a small category I, an I-diagram A in DG + dgCAlg(R), and an A-module M in I-diagrams of chain cochain complexes, define the filtered Hochschild cochain complex CC • R (A, M ) to be the equaliser of the obvious diagram i∈I of pullback along Σ × RLoc G (Σ) → BG with Poincaré duality.When Σ is the 2-sphere, the Killing form gives an equivalence RLoc G (Σ) ≃ T * BG, and for any derived Artin stack Y, [Cal] gives a 0-shifted symplectic structure on the derived cotangent stack T * Y. Example 1.20 generalises to give canonical quantisations in QP(T * Y, 0), defined in terms of differential operators.For explicit hypergroupoid resolutions of T * BG, the stacky CDGAs LD * O((T * BG) ∆ j ) featuring in our definition of Poisson structures are just given by O(T * [G j /g j+1 ]) in the notation of Example 1.20.