QUANTISATION OF DERIVED LAGRANGIANS

J.P.PRIDHAM

Abstract. Given a 0-shifted symplectic structure on a derived Artin N-stack Y, we investigate quantisations of line bundles ℒ on derived Lagrangians X. These take the form of curved A_{∞} deformations of 𝒪_{Y} mapping to the ring of differential operators on ℒ. For each choice of formality isomorphism, we construct a map from the space of non-degenerate quantisations to power series in the de Rham cohomology cone. When ℒ is a square root of the dualising line bundle, this gives an equivalence between even power series and involutive quantisations, ensuring that the latter always exist, and giving rise to a likely candidate for the Fukaya category of algebraic Lagrangians envisaged by Behrend and Fantechi. We sketch a generalisation to Lagrangians on higher n-shifted structures.

Introduction

In [Pri4], the existence of quantisations for 0-shifted symplectic structures on derived Artin N-stacks Y was established, which in the Deligne–Mumford setting take the form of curved A_{∞} deformations of the étale structure sheaf 𝒪_{Y}. For general derived Artin Nstacks, the quantisation is formulated in terms of a site of stacky CDGAs (commutative bidifferential bigraded algebras), and leads to a deformation of the ∞-category of perfect complexes on Y. Likewise, in [Pri3], quantisations of (−1)-shifted symplectic structures on X were established, in the form of twisted BV -algebra deformations of square roots ℒ of the dualising line bundle K_{X}, or equivalently deformations of the right 𝒟_{X}-module ℒ ⊗_{𝒪_{X}} 𝒟_{X}.

The purpose of this paper is to unify and generalise these results by looking at quantisations of derived Lagrangians (X, λ) on 0-shifted symplectic derived stacks (Y, ω) (i.e. Lagrangians in the sense of [PTVV]). When X is empty, this recovers the scenario of [Pri4], and when Y is a point it recovers the scenario of [Pri3]. When Y is a smooth variety, this generalises the description [BGKP] of quantisations of pairs (Y, X) for smooth Lagrangians X, but our derived Lagrangians X can also be derived enhancements of singular schemes or stacks. The quantisations we establish are given by curved A_{∞} deformations of the structure sheaf 𝒪_{Y}, equipped with a curved morphism to the ring 𝒟_{X}(ℒ) of differential operators on a line bundle ℒ on X.

Our perspective for studying these quantisations is that the governing DGLA is given by the Hochschild complex CC^{•}(𝒪_{Y}) acting on 𝒟_{X/Y} (ℒ) via the quasi-isomorphism 𝒟_{X/Y} → CC^{•}(𝒪_{Y} , 𝒟_{X}).

A key notion is that of a self-dual (or involutive) quantisation of the pair (Y, ℒ), for ℒ a square root of the dualising line bundle K_{X}. This condition gives us an involution 𝒟_{X}(ℒ) ≃ 𝒟_{X}(ℒ)^{opp}, and a quantisation 𝒪̃_{Y} → 𝒟_{X}(ℒ)⟦ℏ⟧ is said to be self-dual if it is equipped with a compatible involution to its opposite 𝒪̃_{Y}^{opp} → 𝒟_{X}(ℒ)⟦ℏ⟧, semilinear with respect to the transformation ℏ ↦ −ℏ. Our main result is Theorem 4.16, which shows that each suitable formality isomorphism gives a parametrisation of nondegenerate self-dual quantisations by even de Rham power series

This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/I004130/2].

H^{1}(F^{2}cone(DR(Y) → DR(X)))^{nondeg} × ℏ^{2}H^{1}(cone(DR(Y) → DR(X)))⟦ℏ^{2}⟧,

in particular guaranteeing that such quantisations always exist for derived Lagrangians on 0-shifted symplectic structures.

Since each quantisation of (Y, ℒ) leads to a quantisation 𝒪̃_{Y} of 𝒪_{Y} and an 𝒪̃_{Y} - module in right 𝒟_{X}-modules (deforming ℒ ⊗_{𝒪_{X}} 𝒟_{X}), it makes sense to push the module forward to give an 𝒪̃_{Y} − 𝒟_{Y} -bimodule. We can then look at the dg category given by such bimodules coming from self-dual quantisations of proper Lagrangians X → Y. This is an algebraic analogue of the derived category of simple holonomic DQ modules considered by Kashiwara and Schapira in [KS], and enjoys many properties expected for an algebraic analogue of the Fukaya category envisaged in [BF].

Our approach to proving Theorem 4.16 will be familiar from [Pri6, Pri3, Pri4]. For each quantisation Δ, we define a map μ from de Rham power series to a quantised form of Poisson cohomology, giving a filtered quasi-isomorphism when Δ is non-degenerate. To each non-degenerate quantisation, we may then associate a de Rham power series μ^{−1}(ℏ^{2} \frac{∂Δ}{∂ℏ}) whose constant term is a Lagrangian structure. Obstruction calculus shows that this induces an equivalence between self-dual quantisations and even power series.

Our main new technical ingredient is in defining the map μ, where consider the morphism

CC^{•}(𝒪_{Y}) → CC^{•}(𝒟_{X/Y} (ℒ))

of E_{2}-algebras induced by the action of CC^{•}(𝒪_{Y}) on 𝒟_{X/Y} (ℒ). Via formality, we may regard these E_{2}-algebras as P_{2}-algebras, and then each quantisation defines a commutative diagram from the diagram DR(Y) → DR(X) to a deformation of the diagram above. The morphism μ is then given by composing with the map CC^{•}(𝒟_{X/Y} (ℒ)) → 𝒟_{X/Y} (ℒ) and taking cones.

The structure of the paper is as follows.

In Section 1, we establish some technical background results on Hochschild complexes of almost commutative algebras. When equipped with a PBW filtration degenerating to Poisson cohomology, these become almost commutative brace algebras in a suitable sense (§1.2.1). This allows us to construct suitable semidirect products of Hochschild complexes from morphisms of almost commutative algebras in §1.2.2. Section 2 then uses these constructions to define the space QP(A, B; 0) of quantisations associated to a morphism A → B of commutative bidifferential bigraded algebras (i.e. a map Spec B → Spec A of stacky derived affines in the sense of [Pri6]), and more generally the space QP(A, M; 0) for a line bundle M over B.

Section 3 contains the key technical construction of the compatibility map μ in Definition 3.7, with Definition 3.12 then giving the notion of compatibility between a quantisation and a generalised Lagrangian. The main results of this section are Proposition 3.16, giving a map from non-degenerate quantisations to generalised Lagrangians, and Proposition 3.17, which gives an equivalence between Lagrangians and non-degenerate co-isotropic structures. Proposition 3.18 then shows that the obstruction to quantising a co-isotropic structure is first order.

In Section 4, these constructions are globalised via the method introduced in [Pri6]. §4.3 then introduces the notion of self-duality, enabling us to eliminate the first order obstruction and thus lead to Theorem 4.16, the main comparison result. In §4.4, we then explain how the methods and results of the paper should adapt to Lagrangians on positively shifted symplectic stacks. Section 5 outlines an algebraic analogue of the Fukaya category based on self-dual quantisations of line bundles on derived Lagrangians, and sketches a few key properties.

Contents

Introduction 1 1. The centre of an almost commutative algebra 3 1.1. Almost commutative algebras 3 1.2. Hochschild complexes 7 2. Defining quantisations for derived co-isotropic structures 11 2.1. Stacky thickenings of derived affines 11 2.2. Differential operators 14 2.3. Relative quantised polyvectors 14 3. Compatibility of quantisations and isotropic structures 17 3.1. Morphisms from the de Rham algebra 18 3.2. The compatibility map 19 3.3. The equivalences 22 4. Global quantisations 25 4.1. Diagrams of quantised pairs 25 4.2. Descent and line bundles 27 4.3. Self-duality 29 4.4. Quantisations of higher Lagrangians 31 5. A “Fukaya category” for algebraic Lagrangians 35 5.1. DQ modules 35 5.2. Local quantisations of Lagrangians 37 References 39

Notation. Throughout the paper, we will usually denote chain differentials by δ. The graded vector space underlying a chain (resp. cochain) complex V is denoted by V_{#} (resp. V^{#}). Given an associative algebra A in chain complexes, and A-modules M, N in chain complexes, we write \underline{Hom}_{A}(M, N) for the cochain complex given by

\underline{Hom}_{A}(M, N)^{i} = Hom_{A_{#}}(M_{#[i]}, N_{#}),

with differential f ↦ δ_{N} ∘ f ± f ∘ δ_{M}.

1. The centre of an almost commutative algebra

The purpose of this section is to establish a canonical filtration on the Hochschild complex of an almost commutative algebra, and to study the resulting almost commutative brace algebra constructions. The primary motivation is to ensure that these correspond via formality of the E_{2} operad to filtered P_{2}-algebras for which the Lie bracket has weight −1.

1.1. Almost commutative algebras.

1.1.1. Homological algebra of complete filtrations. We now introduce a formalism for working with complete filtered complexes. Although we make little explicit use of these characterisations in the rest of the paper, they implicitly feature in the reasoning for complete filtered functors to have given properties.

Definition 1.1. Given a vector space V with a decreasing filtration F, the Rees module ξ(V, F) is given by ξ(V, F) := ⨁_{p} F^{p}V ℏ^{−p} ⊂ V [ℏ, ℏ^{−1}]. This has the structure of a 𝔾_{m}- equivariant (i.e. graded) ℤ[ℏ]-module, setting ℏ to be of weight 1 for the 𝔾_{m}-action.

The functor ξ gives an equivalence between exhaustively filtered vector spaces and flat 𝔾_{m}-equivariant ℤ[ℏ]-modules — see [Pri5, Lemma 2.1] for instance.

We will be interested in filtrations which are complete, in the sense that V = \underleftarrow{lim}_{i} V/F^{i}. Via the Rees constructions, this amounts to looking at the inverse limit over k of the categories of 𝔾_{m}-equivariant ℤ[ℏ]/ℏ^{k}-modules. However, Koszul duality provides a much more efficient characterisation. The Koszul dual of ℤ[ℏ] is the dg algebra ℤ[d̄] ≃ R\underline{Hom}_{ℤ[ℏ]}(ℤ, ℤ) for d̄ of chain degree −1 and weight 1 with d̄^{2} = 0. Weak equivalences of ℤ[d̄]-modules in graded chain complexes are quasi-isomorphisms of the underlying chain complexes, forgetting d̄, and these correspond to filtered quasiisomorphisms of the associated complete filtered complexes.

Definition 1.2. For a filtered chain complex (V, F), the corresponding 𝔾_{m}-equivariant ℤ[d̄]-module 𝔤𝔯_{F} V is given in weight i by

𝔤𝔯_{F}^{i} V := cone(F^{i+1}V → F^{i}V),

with d̄: 𝔤𝔯_{F}^{i} V → 𝔤𝔯_{F}^{i+1}V_{[−1]} given by the identity on F^{i+1}V and 0 elsewhere.

There is an obvious quasi-isomorphism from 𝔤𝔯_{F} V to the associated graded gr_{F} V, but the latter does not have a natural d̄-action.

The homotopy inverse functor to 𝔤𝔯 can be realised explicitly as follows:

Definition 1.3. Given a ℤ[d̄]-module E in 𝔾_{m}-equivariant chain complexes, define the chain complex 𝔣(E) to be the semi-infinite total complex

𝔣(E) := (⨁_{i>0} E(i) ⊕ ∏_{i≤0} E(i), δ ± d̄),

equipped with the complete exhaustive filtration

F^{p}𝔣(E) := (∏_{i≤p} E(i), δ ± d̄).

This clearly maps weak equivalences to filtered quasi-isomorphisms.

One way of thinking of the category of ℤ[d̄]-modules is that we are allowed to split the filtration on a filtered complex, but only at the expense of having a component d̄ of the differential which does not respect the grading. The associated graded complex is then simply given by forgetting the action of d̄.

Another way of understanding this equivalence is to observe that a cofibrant resolution of ℤ[d̄] as a DGAA is given by the free algebra ℤ⟨d̄_{1}, d̄_{2}, …⟩ with d̄_{m} of chain degree −1 and weight −m, satisfying δd̄_{m} + ∑_{i+j=m} d̄_{i}d̄_{j}. Thus the structure of a ℤ⟨d̄_{1}, d̄_{2}, …⟩- module on a chain complex E is the same as a differential δ + ∑ d̄_{i} on ⨁_{i>0} E(i) ⊕ ∏_{i≤0} E(i) respecting the filtration and agreeing with δ on the associated graded.

Definition 1.4. Given a ring k, a linear algebraic group G over k, and a G-equivariant CDGA R in chain complexes over k, define the category dgMod_{G}(R) to consist of Gequivariant R-modules in chain complexes.

Thus the Rees construction ξ(V, M) of a filtered R-module M lies in dgMod_{𝔾_{m}}(R[ℏ]), while 𝔤𝔯_{F} M ∈ dgMod_{𝔾_{m}}(R[d̄]). When G is linearly reductive, there is a cofibrantly generated model structure on dgMod_{G}(R) in which fibrations are surjections and weak equivalences are quasi-isomorphisms of the underlying chain complexes.

The dg algebra R[d̄] has the natural structure of a dg Hopf R-algebra, by setting d̄ to be primitive.

Definition 1.5. We define a closed symmetric monoidal structure ⊗_{R} on the category dgMod_{𝔾_{m}}(R[d̄]) by giving the chain complex M ⊗_{R} N an R[d̄]-module structure via the comultiplication on the Hopf algebra R[d̄].

With respect to this structure, the functors 𝔤𝔯 and 𝔣 are both lax monoidal. By way of comparison, note that for the usual tensor product of filtered complexes over k, we have gr_{F} (U ⊗_{k} V) = gr_{F} (U) ⊗_{k} gr_{F} (V).

1.1.2. Koszul duality for almost commutative rings. From now on, we fix a chain CDGA R over ℚ. We refer to associative algebras in chain complexes as DGAAs, and commutative algebras in chain complexes as CDGAs. We will also refer to to coassociative coalgebras in chain complexes over R as DGACs over R.

Definition 1.6. We say that a complete filtered DGAA (A, F) is almost commutative if gr_{F} A is a CDGA. Similarly, a filtered DGAC (C, F) is said to be almost cocommutative if the comultiplication on gr_{F} C is cocommutative.

Thus for any almost commutative DGAA (A, F), the Rees construction ξ(A, F) is an algebra over the 𝔹𝔻_{1}-operad over [𝔸^{1}/𝔾_{m}] as described in [CPT^{+}, §3.5.1] (or [CG, §2.4.2] for its completion), corresponding to the filtration on the associative operad Ass given by powers of the augmentation ideal of T (V) → Symm(V). Since we only wish to consider complete filtrations, we are effectively studying algebras 𝔤𝔯(A, F) over the operad 𝔤𝔯(BD_{1}) in dgMod_{𝔾_{m}}(ℚ[d̄]), where we write BD_{1} for the complete filtered operad associated to 𝔹𝔻_{1}.

Definition 1.7. We write B for the bar construction from possibly non-unital DGAAs over R to ind-conilpotent DGACs over R. Explicitly, this is given by taking the tensor coalgebra

BA := T (A_{[−1]}) = ⨁_{i≥0}(A_{[−1]})^{⊗_{R}i},

with chain differential given on cogenerators A_{[−1]} by combining the chain differential and multiplication on A. Write B_{+}A for the subcomplex T_{+}(A_{[−1]}) = ⨁_{i>0} A_{[−1]}^{⊗_{R}i}

Let Ω_{+} be the left adjoint to B_{+}, given by the tensor algebra

Ω_{+}C := ⨁_{j>0}(C_{[1]})^{⊗_{R}i},

with chain differential given on generators C_{[1]} by combining the chain differential and comultiplication on C. We then define ΩC := R ⊕ Ω_{+}C by formally adding a unit.

Definition 1.8. Given an almost commutative DGAA (A, F), we define the filtration βF on BA by convolution with the Poincaré-Birkhoff Witt filtration β. Explicitly, there is a shuffle multiplication ∇ on (BA)_{#} given on cogenerators by the identity maps (A ⊗ R) ⊕ (R ⊗ A) → A, making (BA)_{#} into a Hopf algebra. Writing F as an increasing filtration, we then set β^{j}BA := ∇((B_{+}A)^{⊗j}), and

(βF)_{i}BA := ∑_{j} F_{i+j} ∩ β^{j}BA.

Lemma 1.9. The filtration βF makes BA into an almost cocommutative DGAC.

Proof. The filtration β automatically behaves with respect to the comultiplication, making (BA)_{#} a filtered coalgebra, and so (βF) also gives a filtered coalgebra structure. To see that BA is a filtered DGAC, it only remains to show that the spaces (βF)_{i}BA are closed under the chain differential. Since the latter is a coderivation, it suffices to check that it induces a filtered map on cogenerators.

The filtration induced on cogenerators by β is just A_{[−1]} = gr_{1}^{β}A_{[−1]}, so (βF)_{i}A_{[−1]} = F_{i+1}A_{[−1]}. We also get β^{1}(A_{[−1]}^{⊗2}) = A_{[−1]}^{⊗2}, β^{2}(A_{[−1]}^{⊗2}) = Λ^{2}A_{[−1]}, and β^{3}(A_{[−1]}^{⊗2}) = 0, so

(βF)_{i}(A_{[−1]}^{⊗2}) = F_{i+1}(A_{[−1]}^{⊗2}) + F_{i+2}(Λ^{2}A)_{[−2]}.

Multiplication and the chain differential on A automatically preserve F, so the only remaining condition is that multiplication sends F_{i+2}(Λ^{2}A) to F_{i+1}A — this is precisely the condition that gr_{F} A be commutative.

Finally, observe that on associated gradeds, the multiplication map gr_{i}^{βF} (A ⊗ A) → gr_{i}^{βF} (A) is the map

gr_{i+1}^{F}Symm^{2}(A) ⊕ gr_{i+2}^{F}Λ^{2}A → gr_{i+1}^{F}A

given by multiplication on the first factor and Lie bracket on the second. Thus

gr^{βF} BA_{#}

is the Poisson coalgebra CoSymm_{R}(gr_{∗+1}^{F}CoLie_{R}A)_{#}, the chain differential involving both product and Lie bracket on gr^{F} A. In particular, the comultiplication on BA is cocommutative. □

In fact, observe that we can characterise βF as the smallest almost cocommutative filtration on BA for which the induced filtration on cogenerators is (βF)_{i}A_{[−1]} = F_{i+1}A_{[−1]}.

Definition 1.10. Given an almost cocommutative DGAC (C, F) over R, define the filtration β^{∗}F on ΩC and Ω_{+}C by convolution with the PBW filtration. Explicitly, define a comultiplication Δ on T (C_{[1]}) to be the algebra morphism sending c ∈ C_{[1]} to c ⊗ 1 + 1 ⊗ c, and let β_{r}^{∗} := ker(Δ^{(r+1)} : T (C_{[1]}) → T_{+}(C_{[1]})^{⊗r+1}). We then set

(β^{∗}F)_{i}ΩC := ∑_{j} F_{i−j} ∩ β_{j}^{∗}ΩC,

and similarly for Ω_{+}C. We then define Ω̂_{+}C to be the completion with respect to β^{∗}.

Lemma 1.11. The filtration β^{∗}F makes Ω̂A into an almost commutative DGAA.

Proof. The constructions (B, β) and (Ω, β^{∗}) are dual to each other, so the proof of Lemma 1.9 adapts after taking shifts and duals. □

Definition 1.12. Define the functors B_{BD_{1}} and Ω_{BD_{1}} by B_{BD_{1}}(A, F) := (BA, βF) and Ω_{BD_{1}}(C, F) := (Ω̂C, β^{∗}F); define B_{BD_{1},+} and Ω_{BD_{1},+} similarly.

Lemma 1.13. The functor Ω_{BD_{1},+} is left adjoint to the functor B_{BD_{1},+} from complete non-unital almost commutative DGAAs A over R to non-counital almost cocommutative DGACs C over R.

Proof. Given A and C, the sets Hom_{DGAA}(Ω_{+}C, A) and Hom(C, BA) can both be identified with the set

\{f ∈ F_{1}\underline{Hom}_{R}(C, A)^{1} : [δ, f] + f ⌣ f = 0\},

where the product ⌣ combines multiplication on A with comultiplication on C. □

Observe that the product ⌣ makes the complex \underline{Hom}_{R}(C, A) into an almost commutative DGAA, so F_{1}\underline{Hom}_{R}(C, A) is closed under the commutator, hence a differential graded Lie algebra (DGLA).

Lemma 1.14. If A is a complete filtered non-unital almost commutative DGAA with gr_{F} A flat over R, then the co-unit ε_{A}: Ω_{BD_{1},+}B_{BD_{1},+}A → A of the adjunction is a filtered quasi-isomorphism.

Proof. It suffices to show that ε gives quasi-isomorphisms on the graded algebras associated to the filtrations. The functors gr_{β}B_{BD_{1},+} and gr_{β^{∗}}Ω_{BD_{1},+} are then just the bar and cobar functors for the Poisson operad, equipped with a 𝔾_{m}-action setting the commutative multiplication to be of weight 0 and the Lie bracket of weight −1. For ℏ a formal variable of weight 1, the graded Poisson operad can be written as Com ∘ ℏ^{−1}Lie, where (ℏP)(i) := ℏ^{i−1}P(i) for any operad P. The 𝔾_{m}-equivariant Koszul dual of the graded Poisson operad is then (Com ∘ ℏ^{−1}Lie)^{!} = (ℏCom) ∘ Lie = ℏ(Com ∘ ℏ^{−1}Lie), so it is self-dual after a shift in filtrations. This shift is precisely the difference between PBW and lower central series, so grε is a graded quasi-isomorphism by Koszul duality for the Poisson operad. □

1.2. Hochschild complexes. Recall that we are fixing a chain CDGA R over ℚ.

Definition 1.15. For an almost commutative DGAA (A, F) over R and a filtered (A, F)-bimodule (M, F) in chain complexes for which the left and right gr^{F} A-module structures on gr^{F} M agree, we define the filtered chain complex

CC_{R,BD_{1}}(A, M)

to be the completion of the cohomological Hochschild complex CC_{R}(A, M) (rewritten as a chain complex) with respect to the filtration γF defined as follows. We may identify CC_{R}(A, M) with the subcomplex of

\underline{Hom}_{R}(BA, B(A ⊕ M_{[1]}))

consisting of coderivations extending the zero coderivation on BA. The hypotheses on M ensure that A ⊕ M is almost commutative (regarding M as a square-zero ideal), so we have filtrations βF on BA and B(A ⊕ M_{[1]}). We then define (γF)_{i} to consist of coderivations sending (βF)_{j}BA to (βF)_{i+j−1}B(A ⊕ M).

Since a coderivation is determined by its value on cogenerators, and the cogenerators of the bar construction have weight 1 with respect to the PBW filtration β, we may regard (γF)_{i}CC_{R}^{#}(A, M) as the subspace of \underline{Hom}_{R}(BA, M)^{#} consisting of maps sending (βF)_{j}BA to F_{i+j}M.

We also define the subcomplex CC_{R,BD_{1},+}(A, M) to be the kernel of CC_{R,BD_{1}}(A, M) → M, or equivalently \underline{Hom}_{R}(B_{+}A, M)^{#}.

Remark 1.16. When the filtrations F are trivial in the sense that A = gr_{0}^{F} A, M = gr_{0}^{F} M, we simply write γ := γF, and observe that γ_{0}CC_{R}(A, M) = M, while γ_{1}CC_{R}(A, M) is just the Harrison cohomology complex. When A is moreover cofibrant as a CDGA, observe that the HKR isomorphism gives a filtered levelwise quasi-isomorphism (CC_{R}(A, M), τ^{HH}) → (CC_{R,BD_{1}}(A, M), γ), where τ^{HH} denotes good truncation in the Hochschild direction as featured in [Pri4, Definition 1.13].

Lemma 1.17. If ϕ: (A, F) → (D, F) is a morphism of almost commutative DGAAs over R, then CC_{R,BD_{1}}(A, D) is an almost commutative DGAA under the cup product, and CC_{R,BD_{1}}(A, D) → D is a morphism of almost commutative DGAAs.

Proof. This just follows because gr^{γF} CC_{R}(A, D)^{#} = \underline{Hom}(gr^{βF} BA, gr^{F} D)^{#}, with gr^{βF} BA cocommutative and gr^{F} D commutative. □

1.2.1. Brace algebra structures. Recall that a brace algebra B over R is an R-cochain complex equipped with a cup product in the form of a chain map

B ⊗ B ⟶^{⌣} B,

and braces in the form of cochain maps

{−}{−, …, −}_{r}: B ⊗ B^{⊗r} → B[−r]

satisfying the conditions of [Vor, §3.2] with respect to the differential. There is a brace operad Br in cochain complexes, whose algebras are brace algebras.

Definition 1.18. Define an decreasing filtration γ on the brace operad Br by putting the cup product in γ^{0} and the braces {−}{−, …, −}_{r} in γ^{r}.

Thus a (brace, γ)-algebra (A, F) in filtered complexes is a brace algebra for which the cup product respects the filtration, and the r-braces send F_{i} to F_{i−r}. We refer to (brace, γ)-algebras as almost commutative brace algebras.

Beware that the filtration γ is not the same as that featuring in [Saf1, Definition 5.3], since we assign higher weights to higher braces.

In an almost commutative brace algebra A, the brace {−}{−}_{1} is of weight −1; since it gives a homotopy between the cup product and its opposite, it follows that the commutator of the cup product is of weight −1, so A is almost commutative as a DGAA. Moreover, a brace algebra structure on A induces a dg bialgebra structure on BA, as in [Vor, §3.2], and because β^{r}BA ⊂ (A_{[−1]})^{⊗≥r}, the multiplication on BA given by braces preserves the filtration βF on B_{BD_{1}}A, so it is a filtered bialgebra (with almost cocommutative comultiplication).

Lemma 1.19. For any almost commutative DGAA A over R, there is a natural almost commutative brace algebra structure on CC_{R,BD_{1}}(A) over R. In particular, CC_{R,BD_{1}}(A)_{[−1]} is a filtered DGLA over R, and its associated graded DGLA is abelian.

Proof. The formulae of [Vor, §3] define a brace algebra structure on CC_{R}(A). By Lemma 1.17, we know that (CC_{R}(A), γF) is an almost commutative DGAA, so it suffices to show that the brace operations have the required weights.

Given f ∈ (γF)_{p}\underline{Hom}(BA, A) and g_{i} ∈ (γF)_{q_{i}} \underline{Hom}(BA, A), each g_{i} corresponds to a coalgebra coderivation g̃_{i} on BA sending (βF)_{j}BA to (βF)_{j+q_{i}−1}BA.

The element \{f\}\{g_{1}, …, g_{m}\} ∈ \underline{Hom}(BA, A) is the composition

BA ⟶^{Δ^{(m)}} (BA)^{⊗m} ⟶^{g̃_{1}⊗...⊗g̃_{m}} (BA)^{⊗m} ⟶^{∇} BA ⟶^{f} A,

where Δ^{(m)} is the iterated coproduct, and ∇ the shuffle product. The definition of β ensures that ∇ preserves the filtration βF, so we have

\{f\}\{g_{1}, …, g_{m}\} ∈ (γF)_{(p+q_{1}+...+q_{m}−m)}\underline{Hom}(BA, A).
□

Definition 1.20. Given a brace algebra B, define the opposite brace algebra B^{opp} to have the same elements as B, but multiplication b^{opp} ⌣ c^{opp} := (−1)^{deg b deg c}(c ⌣ b)^{opp} and brace operations given by the multiplication (BB^{opp}) ⊗ (BB^{opp}) → BB^{opp} induced by the isomorphism (BB^{opp}) ≅ (BB)^{opp}. Explicitly,

{b^{opp}}{c_{1}^{opp}, …, c_{m}^{opp}} := ±{b}{c_{m}, …, c_{1}}^{opp},

where ± = (−1)^{m(m+1)/2+(deg f−m)(∑_{i} deg c_{i}−m)+∑_{i<j} deg c_{i} deg c_{j}}.

Observe that when a filtered brace algebra B is almost commutative, then so is B^{opp}.

Lemma 1.21. Given DGAAs A, D over R, there is an involution

−i: CC_{R}(A, D)^{opp} → CC_{R}(A^{opp}, D^{opp})

of DGAAs given by

i(f)(a_{1}, …, a_{m}) = −(−1)^{∑_{i<j} deg a_{i} deg a_{j}}(−1)^{m(m+1)/2}f(a_{m}^{opp}, …, a_{1}^{opp})^{opp}.

When A = D, the involution −i is a morphism of brace algebras, and in particular i: CC_{R}(A)_{[−1]} → CC_{R}(A)_{[−1]} is a morphism of DGLAs.

Whenever A is a cofibrant CDGA over R, this involution corresponds under the HKR isomorphism to the involution which acts on \underline{Hom}_{A}(Ω_{A/R}^{p}, A) as scalar multiplication by (−1)^{p−1}.

Proof. This is effectively [Bra, §2.1], adapted along the lines of [Pri4, Lemma 1.15], together with the observation that −i acts on braces in the prescribed manner. □

1.2.2. Semidirect products.

Lemma 1.22. Given a morphism ϕ: A → D of almost commutative filtered DGAAs over R, the almost commutative brace algebra CC_{R,BD_{1}}(A) of Hochschild cochains acts on the almost commutative DGAA CC_{R,BD_{1}}(A, D) in the form of a morphism

B_{BD_{1},+}CC_{R,BD_{1}}(A) → B_{BD_{1},+}CC_{R,BD_{1}}(CC_{R,BD_{1}}(A, D))

of almost cocommutative bialgebras.

Proof. Given g_{1}, …, g_{m} ∈ CC_{R,BD_{1}}(A) and f ∈ CC_{R,BD_{1}}(A, D), the brace operation {f}{g_{1}, …, g_{m}} is well-defined as an element of CC_{R,BD_{1}}(A, D). Reasoning as in [Vor, §3.2], this combines with the morphism ϕ_{∗}: CC_{R,BD_{1}}(A) → CC_{R,BD_{1}}(A, D) to give an action

M_{•,•}: B_{BD_{1}}CC_{R,BD_{1}}(A, D) ⊗_{R} B_{BD_{1}}CC_{R,BD_{1}}(A) → B_{BD_{1}}CC_{R,BD_{1}}(A, D)

of almost cocommutative dg coalgebras, associative with respect to the brace multiplication of [Vor]. This respects the filtrations for the same reason that the multiplication does on the bar construction of an almost commutative brace algebra (Definition 1.18).

Indeed, CC_{R,BD_{1}}(A, D) is a brace CC_{R,BD_{1}}(A)-module in the sense of [Saf1, Definition 3.2]. On restricting to cogenerators, the multiplication above gives a map

B_{BD_{1}}CC_{R,BD_{1}}(A, D) →\underline{Hom}(B_{BD_{1}}CC_{R,BD_{1}}(A), CC_{R,BD_{1}}(A, D))
≅ CC_{R,BD_{1}}(CC_{R,BD_{1}}(A, D)),

and as in [Saf1, Proposition 4.2], this induces a morphism

B_{BD_{1},+}CC_{R,BD_{1}}(A) → B_{BD_{1},+}CC_{R,BD_{1}}(CC_{R,BD_{1}}(A, D))

of almost cocommutative bialgebras, compatibility with the filtrations being automatic from the description above. □

For an E_{2}-algebra C to act on an E_{1}-algebra E is the same as a morphism from C to the Hochschild complex of E. This is what we now construct for Hochschild complexes in the almost commutative setting, so that we will have a BD_{2}-algebra acting on a BD_{1}-algebra. Proposition 1.14 then combines with the adjunction property to give morphisms

CC_{R,BD_{1}}(A) ⟵^{∼} Ω_{BD_{1},+}B_{BD_{1},+}CC_{R,BD_{1}}(A) → CC_{R,BD_{1}}(CC_{R,BD_{1}}(A, D)),

of almost commutative DGAAs, and we need to enhance this to keep track of the brace algebra structures:

Lemma 1.23. If A is a complete filtered non-unital almost commutative brace algebra over R, then there is a natural almost commutative brace algebra structure on the DGAA Ω_{BD_{1},+}B_{BD_{1},+}A. If gr_{F} A is moreover flat over R, then there is a zigzag of filtered quasiisomorphisms of almost commutative brace algebras between A and Ω_{BD_{1},+}B_{BD_{1},+}A.

Proof. As in [Kad], there is a natural brace algebra structure on Ω_{+}C for any bialgebra C; we now show that when C is almost cocommutative, the resulting brace algebra structure on Ω_{BD_{1},+}C is almost commutative. For c ∈ C, the brace operation

{c}{−}: Ω(C) → Ω(C)

is defined by first taking the element ∑_{r} Δ^{(r)}c ∈ T C, then applying the multiplication from C internally within each subspace C^{⊗r}. Since Δ is almost cocommutative and ΩC almost commutative, it follows that when c ∈ F_{p}C, we get {c}{(β^{∗}F)_{i}ΩC} ⊂ (β^{∗}F)_{i+p}ΩC. Equivalently, for y ∈ (β^{∗}F)_{i}ΩC, the map {−}{y} sends (β^{∗}F)_{p}C = F_{p−1}C to (β^{∗}F)_{i+p−1}ΩC.

We automatically have {c}{}_{0} = c, and the higher braces {c}{−}_{n}: Ω(C)^{⊗n} → Ω(C) are then set to be 0 for c ∈ C, and extended to the whole of ΩC via the identities

{xz}{y_{1}, …, y_{n}} = ∑_{i=0}^{n} ±x{y_{1}, …, y_{i}}z{y_{i+1}, …, y_{n}}.

In particular, this means that {−}{y} is a derivation, so must map (β^{∗}F)_{p}ΩC to (β^{∗}F)_{i+p−1}ΩC, since it does so on generators. We can then describe higher braces {−}{y_{1}, …, y_{n}} as the composition

Ω(C) ⟶^{Δ^{(n)}} Ω(C)^{⊗n} ⟶^{{−}{y_{1}}⊗...⊗{−}{y_{n}}} Ω(C)^{⊗n} → Ω(C),

the final map being given by multiplication. By the construction of β^{∗}, the map Δ^{(n)} preserves the filtration (β^{∗}F), so for y_{i} ∈ (β^{∗}F)_{q_{i}}ΩC, we have

{−}{y_{1}, …, y_{n}}: (β^{∗}F)_{p}Ω(C) → (β^{∗}F)_{(p+q_{1}+...+q_{n}−n)}ΩC,

making Ω_{BD_{1},+}C almost commutative

Taking C = B_{BD_{1},+}A gives an almost commutative brace algebra Ω_{BD_{1},+}B_{BD_{1},+}A and an almost commutative DGAA quasi-isomorphism Ω_{BD_{1},+}B_{BD_{1},+}A → A by Lemma 1.14, but this is not a brace algebra morphism in general. If we let Ω_{Br,+} be the left adjoint to B_{BD_{1}} as a functor from almost commutative brace algebras to almost cocommutative bialgebras, then it suffices to establish a filtered brace algebra quasiisomorphism Ω_{BD_{1},+}B_{BD_{1},+}A → Ω_{Br,+}B_{BD_{1},+}A. If we disregard the filtrations, this is the main result of [You], and the filtered case follows by observing that the homotopy of [You, Theorem 3.3] preserves the respective filtrations. □

Combining Lemmas 1.22 and 1.23 gives:

Proposition 1.24. For any morphism ϕ: A → D of almost commutative filtered DGAAs over R, there is a canonical zigzag

CC_{R,BD_{1}}(A) ← C̃ → CC_{R,BD_{1}}(CC_{R,BD_{1}}(A, D))

of almost commutative brace algebras over R.

Definition 1.25. Given an almost commutative brace algebra C over R, and an almost commutative DGAA E over R which is a left brace C-module compatibly with the filtrations, define the semidirect product E_{[1]} ⋊ C to be the almost commutative nonunital brace algebra given by the homotopy fibre product of the diagram

C̃ → CC_{R,BD_{1}}(E) ← CC_{R,BD_{1},+}(E),

for the brace algebra resolution C̃ of C mapping to CC_{R,BD_{1}}(E) via Lemma 1.23 and the proof of Lemma 1.22.

Remark 1.26. Observe that we have a natural morphism E_{[1]} ⋊ C → C of non-unital brace algebras, with homotopy fibre given by the homotopy kernel of CC_{R,BD_{1},+}(E) → CC_{R,BD_{1}}(E). As a complex, this kernel is just E_{[1]}, and the underlying DGLA is just the DGLA underlying the DGAA E. For more discussion of the map CC_{R,+}(E) → CC_{R}(E) of E_{2}-algebras, see [Kon, §2.7].

2. Defining quantisations for derived co-isotropic structures

In this section, we develop a precise notion of quantisation for derived co-isotropic structures in a stacky affine setting. Recall that we are fixing a chain CDGA R over ℚ.

2.1. Stacky thickenings of derived affines. We now recall some definitions and lemmas from [Pri6, §3], as summarised in [Pri3, §3.1]. By default, we will regard the CDGAs in derived algebraic geometry as chain complexes … ⟶^{δ} A_{1} ⟶^{δ} A_{0} ⟶^{δ} … rather than cochain complexes — this will enable us to distinguish easily between derived (chain) and stacky (cochain) structures.

Definition 2.1. 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_{j}^{i} → V_{j−1}^{i}, and the cochain differential by ∂: V_{j}^{i} → V_{j}^{i+1}. 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.

Definition 2.2. 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.

There is a model structure on chain cochain complexes over R in which weak equivalences are levelwise quasi-isomorphisms and fibrations are surjections — this follows by identifying chain cochain complexes with the category dgMod_{𝔾_{m}}(R[∂]) of §1.1.1, for instance, for ∂ of chain degree 0 and weight 1, with ∂^{2} = 0.

The following is [Pri6, Lemma 3.4]:

Lemma 2.3. 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 [Pri6, Lemma 3.5], D^{∗} is a left Quillen functor from the Reedy model structure on cosimplicial chain CDGAs to the model structure of Lemma 2.3.

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}. Stacky CDGAs arise as formal completions of derived Artin Nstacks along affine atlases, as in [Pri6, §3.1]. When 𝔛 is a 1-geometric derived Artin stack (i.e. has affine diagonal), the formal completion of an affine atlas U → 𝔛 is given by the relative de Rham complex

O(U) ⟶^{∂} Ω_{U/𝔛}^{1} ⟶^{∂} Ω_{U/𝔛}^{2} ⟶^{∂} …,

which arises by applying the functor D^{∗} to the Čech nerve of U over 𝔛.

Definition 2.4. Given a chain cochain complex V, define the cochain complex T̂ot V ⊂ Tot^{Π}V by

(T̂ot V)^{m} := (⨁_{i<0} V_{i−m}^{i}) ⊕ (∏_{i≥0} V_{i−m}^{i})

with differential ∂ ± δ.

Definition 2.5. Given a stacky CDGA A and A-modules M, N in chain cochain complexes, we define internal Homs Hom_{A}(M, N) by

Hom_{A}(M, N)_{j}^{i} = Hom_{A_{#}^{#}}(M_{#}^{#}, N_{#[j]}^{#[i]}),

with differentials ∂f := ∂_{N} ∘ f ± f ∘ ∂_{M} and δf := δ_{N} ∘ f ± f ∘ δ_{M}, where V_{#}^{#} denotes the bigraded vector space underlying a chain cochain complex V.

We then define the Hom complex ˆ \underline{Hom}_{A}(M, N) by

ˆ\underline{Hom}_{A}(M, N) := T̂ot Hom_{A}(M, N).

Note that there is a multiplication ˆ \underline{Hom}_{A}(M, N) ⊗ ˆ \underline{Hom}_{A}(N, P) → ˆ \underline{Hom}_{A}(M, P); beware that the same is not true for Tot^{Π}Hom_{A}(M, N) in general.

Definition 2.6. A morphism A → B in DG^{+}dgCAlg(R) is said to be homotopy formally étale when the map

{Tot σ^{≤q}(LΩ_{A}^{1} ⊗_{A}^{L} B^{0})}_{q} → {Tot σ^{≤q}(LΩ_{B}^{1} ⊗_{B}^{L} B^{0})}_{q}

on the systems of brutal cotruncations is a pro-quasi-isomorphism.

Combining [Pri6, Proposition 3.13] with [Pri2, 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).

The constructions of §1 all adapt to chain cochain complexes, by just regarding the cochain structure as a 𝔾_{m}-equivariant ℚ[∂]-module structure; quasi-isomorphisms are only considered in the chain direction. We refer to associative (resp. brace) algebras in chain cochain complexes as stacky DGAAs (resp. stacky brace algebras), and have the obvious notions of almost commutativity for filtered stacky DGAAs and filtered stacky brace algebras. We define bar constructions B generalising Definition 1.7 so that shifts are exclusively in the chain direction.

Definition 2.7. For a stacky DGAA A over R and an A-bimodule M in chain cochain complexes, we define the internal cohomological Hochschild complex CC_{R}(A, M) to be the chain cochain subcomplex of

Hom_{R}(BA, B(A ⊕ M_{[1]}))

consisting of coderivations extending the zero derivation on BA, where the algebra structure on A ⊕ M_{[1]} is defined so that M is a square-zero ideal.

Since a coderivation is determined by its value on cogenerators, the complex CC_{R}(A, M) is given explicitly by

CC_{R}(A, M)_{#} := ∏_{n} Hom_{R}(A^{⊗n}, M)_{[n]},

with chain differential δ ± b, for the Hochschild differential b given by

(bf)(a_{1}, …, a_{n}) =a_{1}f(a_{2}, …, a_{n})

+ ∑_{i=1}^{n−1}(−1)^{i}f(a_{1}, …, a_{i−1}, a_{i}a_{i+1}, a_{i+2}, …, a_{n})

+ (−1)^{n}f(a_{1}, …, a_{n−1})a_{n}.

We simply write CC_{R}(A) for CC_{R}(A, A).

When (A, F) is almost commutative and (M, F) is a filtered A-bimodule for which the left and right gr^{F} A-module structures on gr^{F} M agree, we define the filtered chain cochain complex

CC_{R,BD_{1}}(A, M)

by endowing CC_{R}(A, M) with the filtration γF of Definition 1.15, and completing with respect to it.

2.2. Differential operators. We now fix a stacky CDGA B over a chain CDGA R, and recall the definitions of differential operators from [Pri3, §3.2].

Definition 2.8. Given B-modules M, N in chain cochain complexes, inductively define the filtered chain cochain complex Diff (M, N) = Diff_{B/R}(M, N) ⊂ Hom_{R}(M, N) of differential operators from M to N by setting

(1) F_{0}Diff (M, N) = Hom_{B}(M, N),

(2) F_{k+1}Diff (M, N) = {u ∈ Hom_{R}(M, N) : [b, u] ∈ F_{k}Diff (M, N) ∀b ∈ B}, where [b, u] = bu − (−1)^{deg b deg u}ub.

(3) Diff (M, N) = \underrightarrow{lim}_{k} F_{k}Diff (M, N).

We simply write Diff_{B/R}(M) := Diff_{B/R}(M, M).

We then define the filtered cochain complex D̂iff(M, N) = D̂iff_{B/R}(M, N) ⊂ ˆ\underline{Hom}_{R}(M, N) by D̂iff(M, N) := T̂ot Diff (M, N).

Definition 2.9. Given a B-module M in chain cochain complexes, write 𝒟(M) = 𝒟_{B/R}(M) := D̂iff_{B/R}(M, M), which we regard as a sub-DGAA of ˆ \underline{Hom}_{R}(M, M). We simply write 𝒟_{B} = 𝒟_{B/R} for 𝒟_{B/R}(B, B) and Diff_{B/R} for Diff_{B/R}(B, B).

The definitions ensure that the associated gradeds gr_{k}^{F} Diff_{B}(M, N) have the structure of B-modules. As in [Pri3], there are maps

gr_{k}^{F} Diff (M, N) → Hom_{B}(M ⊗_{B} CoS_{B}^{k}Ω_{B}^{1}, N)

for all k, which are isomorphisms when B is cofibrant. [Here, CoS_{B}^{p}(M) = CoSymm_{B}^{p}(M) = (M^{⊗_{B}p})^{Σ_{p}} and CoSymm_{B}(M) = ⨁_{p≥0} CoS_{B}^{p}(M).]

The following is [Pri3, Definition 3.9]:

Definition 2.10. Define a strict line bundle over B to be a B-module M in chain cochain complexes such that M_{#}^{#} is a projective module of rank 1 over the bigradedcommutative algebra B_{#}^{#} underlying B.

The motivating examples of strict line bundles, and the only ones we will need to consider for our applications in §4.2, are the double complexes B_{c} defined as follows. Given c ∈ Z^{1}Z_{0}B, we just set B_{c}^{#} to be the B-module B^{#} (so the chain differential is still δ), and then we set the cochain differential to be ∂ + c.

2.3. Relative quantised polyvectors.

Definition 2.11. Given a morphism ϕ: A → B of cofibrant stacky CDGAs over R and a strict line bundle M over B, we define the DGLA QP̂ol(A, M; 0)[1] of 0-shifted relative quantised polyvectors as follows. We first note that Definition 1.25 and Proposition 1.24 adapt to double complexes to give a non-unital almost commutative stacky brace algebra

C := CC_{R,BD_{1}}(A, Diff_{B/R}(M))_{[1]} ⋊ CC_{R,BD_{1}}(A),

and then form the DGLA

QP̂ol_{R}(A, M; 0) := ∏_{p≥0} T̂ot (γF)_{p}Cℏ^{p−1}.

We define filtrations F̃ and G on QP̂ol_{R}(A, M; 0) by

F̃^{i}QP̂ol_{R}(A, M; 0) := ∏_{p≥i} T̂ot (γF)_{p}Cℏ^{p−1},
G^{j}QP̂ol_{R}(A, M; 0) := QP̂ol_{R}(A, M; 0)ℏ^{j}.

Note that almost commutativity of C implies that [F̃^{i}QP̂ol, F̃^{j}QP̂ol] ⊂ F̃^{i+j−1}QP̂ol and [G^{i}QP̂ol, G^{j}QP̂ol] ⊂ G^{i+j}QP̂ol.

Remark 2.12. When B = 0, observe that 𝒟_{B/R} = 0, so we just have QP̂ol_{R}(A, 0; 0) ≃ ∏_{p≥0}(T̂ot γ_{p}CC_{R,BD_{1}}(A)ℏ^{p−1}), which admits a filtered quasi-isomorphism from the complex QP̂ol_{R}(A, 0) of 0-shifted quantised polyvectors from [Pri4, Definition 1.16] as in Remark 1.16.

By Remark 1.26, there is always a projection QP̂ol_{R}(A, M; 0)[1] → QP̂ol_{R}(A, 0; 0)[1], and the homotopy fibre over 0 is equivalent to the filtered L_{∞}-algebra underlying the DGAA QP̂ol_{A}(B, −1) := ∏_{p≥0} F_{p}𝒟_{B/A}(M)ℏ^{p−1} when B is cofibrant over A. The latter follows because the HKR isomorphism for A ensures that Diff_{B/A} → CC_{R}(A, Diff_{B/R}) is a filtered quasi-isomorphism.

The following is standard:

Definition 2.13. Given a DGLA L, define the the Maurer–Cartan set by

MC(L) := \{ω ∈ L^{1} | dω + \frac{1}{2}[ω, ω] = 0 ∈ L^{2}\}.

Following [Hin], define the Maurer–Cartan space \underline{MC}(L) (a simplicial set) of a nilpotent DGLA L by

\underline{MC}(L)_{n} := MC(L ⊗_{ℚ} Ω^{•}(Δ^{n})),

where

Ω^{•}(Δ^{n}) = ℚ[t_{0}, t_{1}, …, t_{n}, δt_{0}, δt_{1}, …, δt_{n}]/(∑ t_{i} − 1, ∑ δt_{i})

is the commutative dg algebra of de Rham polynomial forms on the n-simplex, with the t_{i} of degree 0.

Given a pro-nilpotent DGLA L = \underleftarrow{lim}_{i} L_{i}, define \underline{MC}(L) := \underleftarrow{lim}_{i} \underline{MC}(L_{i}).

Definition 2.14. Given a morphism ϕ: A → B of cofibrant stacky CDGAs over R, define the space QP(A, M; 0) of quantisations of the pair (A, M) to be the space

\underline{MC}(F̃^{2}QP̂ol(A, M; 0)[1])

of Maurer–Cartan elements of the pro-nilpotent DGLA F̃^{2}QP̂ol(A, M; 0)[1].

Replacing F̃^{2}QP̂ol(A, M; 0) with its quotient by G^{k} gives a space QP(A, B; 0)/G^{k}; we think of P(A, B; 0) := QP(A, B; 0)/G^{1} as being the space of co-isotropic structures on A → B.

Remark 2.15. Uncoiling the definitions, it follows that each element of QP(A, M; 0) gives rise to a curved almost commutative A_{∞}-deformation Ã of T̂ot A over R⟦ℏ⟧ (coming from elements of MC(CC_{R}(T̂ot A))), together with a curved almost commutative A_{∞}- morphism Ã → 𝒟_{B/R}(M)⟦ℏ⟧ deforming the map T̂ot A → 𝒟_{B/R}(M).

However, there are additional restrictions on the resulting deformations, which remember that they originate from the stacky CDGAs A → B instead of the CDGAs T̂ot A → T̂ot B. When the stacky CDGAs are bounded in the cochain direction, as occurs when they originate from 1-geometric derived Artin stacks, these additional restrictions are vacuous (cf. [Pri4, Example 1.20]).

Definition 2.16. Define the filtered tangent space to quantised polyvectors by

T QP̂ol(A, M; 0) := QP̂ol(A, M; 0) ⊕ ℏQP̂ol_{R}(A, M; 0)ϵ,
F̃^{j}T QP̂ol(A, M; 0) := F̃^{j}QP̂ol(A, M; 0) ⊕ ℏF̃^{j}QP̂ol(A, M; 0)ϵ,

for ϵ of degree 0 with ϵ^{2} = 0. Then T QP̂ol(A, 0)[1] is a DGLA, with Lie bracket given by [u + vϵ, x + yϵ] = [u, x] + [u, y]ϵ + [v, x]ϵ.

Write T QP(A, M; 0) for the space

\underline{MC}(F̃^{2}T QP̂ol(A, M; 0)[1]).

Definition 2.17. Given a Maurer–Cartan element Δ ∈ MC(QP̂ol_{R}(A, M; 0)), define T_{Δ}QP̂ol_{R}(A, M; 0) to be the non-unital brace algebra

(ℏQP̂ol_{R}(A, M; 0)_{#}, δ_{QPol} + [Δ, −]).

We define filtrations F̃ and G on T_{Δ}QP̂ol_{R}(A, M; 0) by

F̃^{i}T_{Δ}QP̂ol_{R}(A, M; 0)_{#} := ℏF̃^{i}QP̂ol_{R}(A, M; 0)_{#},
G^{j}T_{Δ}QP̂ol_{R}(A, M; 0) := ℏ^{j}T_{Δ}QP̂ol_{R}(A, M; 0).

Note that (T_{Δ}QP̂ol_{R}(A, M; 0), F̃) is an almost commutative brace algebra over R.

Observe that T_{Δ}QP(A, M; 0) := \underline{MC}(F̃^{2}T_{Δ}QP̂ol(A, M; 0)[1]) is just the fibre of T QP(A, M; 0) → QP(A, M; 0) over Δ.

Definition 2.18. Given Δ ∈ QP(A, M; 0), define σ(Δ) ∈ Z^{2}(F̃^{2}T_{Δ}QP̂ol(A, M; 0)) to be

−∂_{ℏ^{−1}}Δ = ℏ^{2} \frac{∂Δ}{∂ℏ}.

More generally, define σ: P(A, M; 0) → T QP(A, M; 0) to be the morphism induced by the morphism Δ ↦ Δ − ∂_{ℏ^{−1}}Δϵ of DGLAs from QP̂ol(A, M; 0) to T QP̂ol(A, M; 0).

As in [Pri6, §3.3], we will usually consider stacky CDGAs A ∈ DG^{+}dgCAlg(R) satisfying the following properties:

Assumption 2.19. (1) for any cofibrant replacement Ã → A in the model structure of Lemma 2.3, the morphism Ω_{Ã/R}^{1} → Ω_{A/R}^{1} is a levelwise quasi-isomorphism,

(2) the A^{#}-module (Ω_{A/R}^{1})^{#} 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_{A/R}^{1} ⊗_{A} A^{0})^{i} are acyclic for all i > N.

Lemma 2.20. If A and B are both cofibrant and satisfy Assumption 2.19, then gr_{G}^{i}F̃^{p}QP̂ol(A, M; 0) is quasi-isomorphic to the cocone of

∏_{j≥p} ˆ \underline{Hom}_{A}(Ω_{A/R}^{j−i}, A)ℏ^{j−1}[i − j] → ∏_{j≥p} ˆ \underline{Hom}_{B}(LCoS_{B}^{j−i}𝕃_{B/A}, B)ℏ^{j−1}

coming from the connecting homomorphism S: LΩ_{B/A}^{1} = 𝕃_{B/A} → Ω_{A/R}^{1}[1].

Moreover, gr_{G}^{i}F̃^{p}T_{Δ}QP̂ol(A, M; 0) is quasi-isomorphic to ℏgr_{G}^{i}F̃^{p}T_{Δ}QP̂ol(A, M; 0).

Proof. By construction, gr_{G}^{i}F̃^{p}QP̂ol is the cocone of

∏_{j≥p} T̂ot gr_{j−i}^{γ}CC_{R}^{•}(A)ℏ^{j−1} → ∏_{j≥p} T̂ot gr_{j−i}^{γF}CC_{R}^{•}(A, Diff_{B/R})ℏ^{j−1}.

Since B is assumed cofibrant, we have isomorphisms

gr_{k}^{F} Diff_{B/R} → Hom_{B}(CoS_{B}^{k}Ω_{B/R}^{1}, B).

The bar-cobar resolution for A as a commutative algebra then gives quasi-isomorphisms

Hom_{A}(Ω_{A/R}^{j−i}, A)[i − j] → gr_{j−i}^{γ}CC_{R}^{•}(A)
Hom_{A}(CoS_{B}^{j−1}(cocone(Ω_{B/R}^{1} → Ω_{A/R}^{1} ⊗_{A} B)), B) → gr_{j−i}^{γF}CC_{R}^{•}(A, Diff_{B/R}).

Since cocone(Ω_{B/R}^{1} → Ω_{A/R}^{1} ⊗_{A} B) is a model for the cotangent complex 𝕃_{B/A}, the results follow. □

Given an element Δ ∈ QP(A, M; 0), we write Δ_{A} for the image in QP(A, 0) and Δ_{B} for the image in T̂ot CC_{R,BD_{1}}(A, Diff_{B/R}). If we write Δ = ∑_{j≥2} Δ_{j}ℏ^{j−1}, then by working modulo G^{1} + F̃^{3}, Lemma 2.20 allows us to identify Δ_{2} = (Δ_{2,A}, Δ_{2,B}) with a closed element of the cocone of

ˆ\underline{Hom}_{A}(Ω_{A/R}^{2}, A) → R ˆ\underline{Hom}_{B}(LCoS_{B}^{2}𝕃_{B/A}, B)[2].

Now Δ_{2,A} defines a closed element of the first space, and since the composition of this map with

ˆ\underline{Hom}_{B}(LCoS_{B}^{2}𝕃_{B/A}, B) → ˆ \underline{Hom}_{B}(Ω_{B/R}^{1} ⊗_{B}^{L} 𝕃_{B/A}, B)

is homotopic to 0, Δ_{2,B} defines a closed element of the latter.

We then have a diagram

Ω_{A/R}^{1}  ⟶  Ω_{B/R}^{1}
Δ_{2,A}^{♯}↓  ↓^{Δ_{2,B}^{♯}}
ˆ\underline{Hom}_{A}(Ω_{A/R}^{1}, A) ⟶^{S} R ˆ\underline{Hom}_{B}(𝕃_{B/A}, B)[1]

commuting up to a canonical homotopy coming from Δ_{2,B}.

Definition 2.21. Say that a quantisation Δ of the pair (A, M) is non-degenerate if the maps

Δ_{2,A}^{♯}: Tot^{Π}(Ω_{A/R}^{1} ⊗_{A} A^{0}) → ˆ \underline{Hom}_{A}(Ω_{A}^{1}, A^{0})
Δ_{2,B}^{♯}: Tot^{Π}(Ω_{B/R}^{1} ⊗_{B} B^{0}) → R ˆ\underline{Hom}_{B}(𝕃_{B/A}, B^{0})[1]

are quasi-isomorphisms and Tot^{Π}(Ω_{A/R}^{1} ⊗_{A} A^{0}) (resp. Tot^{Π}(Ω_{B/R}^{1} ⊗_{B} B^{0})) is a perfect complex over A^{0} (resp. B^{0}).

3. Compatibility of quantisations and isotropic structures

In this section, we introduce generalised isotropic structures, develop the notion of compatibility between a quantisation and a generalised isotropic structure, and give some preliminary existence results for quantisations of Lagrangians.

3.1. Morphisms from the de Rham algebra.

Definition 3.1. Given a stacky CDGA A over R, define the stacky de Rham algebra of A to be the complete filtered stacky CDGA

DR(A/R)_{i}^{n} := ∏_{j≥0}(Ω_{A}^{j})_{i+j}^{n}

with filtration F^{p}DR(A/R) = ∏_{j≥p}(Ω_{A}^{j})_{[j]}, cochain differential ∂ and chain differential δ ± d, where d is the de Rham differential, and the differentials ∂, δ are induced from those on A.

We then write DR(A/R) := T̂ot DR(A/R).

In particular, beware that the de Rham differential is absorbed in the chain (derived) structure, not the cochain (stacky) structure.

Lemma 3.2. Given a morphism A → gr_{F}^{0} B of stacky CDGAs over R, with A cofibrant and (B, F) a complete filtered stacky CDGA, there is an associated filtered stacky CDGA morphism DR(A/R) → F^{0}B over R, unique up to coherent homotopy.

Proof. We may assume that A is cofibrant, and then DR(A) is cofibrant as a complete filtered stacky CDGA, in the sense that it has the left lifting property with respect to surjections of complete filtered stacky CDGAs over R which are levelwise filtered quasi-isomorphisms. For any filtered A-module (M, F), we may regard M as a DR(A)-module via the projection DR(A) → A. When M = F^{1}M, the double complex Hom_{DR(A),Fil}(Ω_{DR(A)/R}^{1}, M) of filtered derivations from DR(A) to M is then levelwise acyclic, by the construction of DR(A).

Now, the double complex Hom_{DR(A),Fil}(Ω_{DR(A)/R}^{1}, gr_{F}^{r} B) governs the obstruction theory to lifting maps from DR(A) along the square-zero extension F^{0}B/F^{r+1}B → F^{0}B/F^{r}B. Thus the acyclicity above gives the required equivalence of mapping spaces

map_{Fil}(DR(A), B) ≃ map(A, gr_{F}^{0} B)

of filtered stacky CDGAs and of stacky CDGAs, respectively. □

The following is a slight generalisation of [Pri6, Lemma 1.17]:

Lemma 3.3. Take a cofibrant stacky CDGA A over R, a complete filtered CDGA B over R, and a filtered morphism ϕ: DR(A/R) → B. Then for any derivation π ∈ MC(F^{1}\underline{Der}_{R}(B)), there is an associated filtered CDGA morphism

μ(−, π): DR(A/R) → (B, δ + π)

given by μ(a, π) = ϕ(a) and μ(df, π) = ϕ(df) + πϕ(f) for a, f ∈ A.

Proof. The formulae clearly define a filtered morphism μ(−, π): DR(A)^{#} → B^{#} of graded algebras, since ϕ ∘ d + π ∘ ϕ defines a derivation on A with respect to ϕ: A → B. We therefore need only check that μ is a chain map. We have

δμ(a, π) = ϕ(δa) + ϕ(da)
πμ(a, π) = πϕ(a)
(δ + π)μ(a, π) = μ(δa + da, π),

and the calculation above applied to a = f and using that (δ + π)^{2} = 0 gives

(δ + π)μ(df, π) = −(δ + π)μ(δf, π)
= −(δ + π)ϕ(f)
= −ϕ(dδf) − πϕ(δf)
= μ(−dδf, π)
= μ((δ − d)df, π),

as required. □

Combining Lemmas 3.2 and 3.3 gives:

Lemma 3.4. Take a morphism ϕ: A → gr_{F}^{0} B of stacky CDGAs over R, with A cofibrant and B a complete filtered stacky CDGA. Then for any π ∈ \underline{MC}(T̂ot F^{1}Der_{R}(B)), there is an associated morphism

μ(−, π): DR(A/R) → (T̂ot B, δ + π),

of filtered CDGAs, unique up to coherent homotopy.

3.2. The compatibility map. We now develop the notion of compatibility between de Rham data and quantisations of a pair (A → B), generalising the notion of compatibility between generalised 0-shifted pre-symplectic structures and E_{1} quantisations from [Pri4].

A choice of Levi decomposition of the Grothendieck–Teichmüller group over ℚ gives a formality quasi-isomorphism E_{2} ≃ P_{2}. Writing τ for the good truncation filtration τ_{≥p} on a homological operad, a formality quasi-isomorphism automatically gives a filtered quasi-isomorphism (E_{2}, τ) ≃ (P_{2}, τ). The filtration τ on P_{2} gives the commutative multiplication weight 0 and the Lie bracket weight −1, and we refer to (P_{2}, τ)-algebras in complete filtered complexes as almost commutative P_{2}-algebras.

Likewise, the map in [Vor] from the E_{2} operad to the brace operad Br must preserve the good truncation filtrations. Finally, note that the good truncation filtration is contained in the filtration γ on Br from Definition 1.18, since all operations of homological degree r lie in γ^{r}, so in particular the closed operations do so. Thus every almost commutative brace algebra can be regarded as an (E_{2}, τ)-algebra.

Definition 3.5. Given a Levi decomposition w ∈ Levi_{GT}(ℚ) of the Grothendieck– Teichmüller group GT over ℚ, we denote by p_{w} the resulting ∞-functor from almost commutative brace algebras to almost commutative P_{2}-algebras over ℚ.

Note that the ∞-functor p_{w} automatically commutes with the fibre functors A ↦ F_{1}A to the underlying filtered DGLAs.

Definition 3.6. For any of the definitions from §2, we add the subscript w to indicate that we are replacing CC_{R,BD_{1}}(A) with p_{w}CC_{R,BD_{1}}(A) in the construction.

Since the DGLAs underlying CC_{R,BD_{1}}(A) and p_{w}CC_{R,BD_{1}}(A) are filtered quasiisomorphic, in particular we have canonical weak equivalences QP_{w}(A, 0) ≃ QP(A, 0). Properties of the filtration F̃ then ensure that the complexes T_{Δ}QP̂ol_{w}(A, 0) are filtered (P_{2}, τ)-algebras.

Definition 3.7. Given a choice w ∈ Levi_{GT}(ℚ) of Levi decomposition for GT and Δ ∈ QP(A, M; 0)_{w}/G^{j} define

μ_{w}(−, Δ): cocone(DR(A/R) → DR(B/R))⟦ℏ⟧/ℏ^{j} → T_{Δ}QP̂ol_{w}(A; B, 0)/G^{j}

as follows.

Since [B, F_{i}Diff_{B/A}] ⊂ F_{i−1}Diff_{B/A}, we have a map B → gr_{γF}^{0} CC_{R,BD_{1}}(Diff_{B/A}). Combined with the weak equivalence Diff_{B/A} → CC_{R,BD_{1}}(A, Diff_{B/R}), up to coherent homotopy this gives a commutative diagram

A  ⟶  B
↓  ↓
gr0 ; γ̃(p_{w}CC_{R,BD_{1}}(A))⟦ℏ⟧/ℏ^{j}) ⟶ gr0 ; γ̃F (p_{w}CC_{R,BD_{1}}(Diff_{B/A})⟦ℏ⟧/ℏ^{j})

where the filtrations on the bottom row are taken to be (γ̃F)^{p} := ∏_{i≥p}(γF)_{i}ℏ^{i}.

Applying Lemma 3.4 to this diagram and the Maurer–Cartan elements on the bottom line induced by Δ yields a diagram

DR(A) ⟶^{μ_{w}(−,Δ)}  (T̂ot γ^{0}(p_{w}CC_{R,BD_{1}}(A)⟦ℏ⟧/ℏ^{j}), δ + [Δ_{A}, −])
↓  ↓
DR(B) ⟶^{μ_{w}(−,Δ)} (T̂ot γ̃F^{0}(p_{w}CC_{R,BD_{1}}(Diff_{B/A})⟦ℏ⟧/ℏ^{j}), δ + [Δ_{B}, −])
↑  ↑
0  ⟶ (T̂ot γ̃F^{0}(p_{w}CC_{R,BD_{1},+}(Diff_{B/A})⟦ℏ⟧/ℏ^{j}), δ + [Δ_{B}, −]),

and taking homotopy limits of the columns gives the desired map.

Remark 3.8. When B = 0, this recovers the definition of μ_{w} from [Pri4, Definition 2.11]. When R = A, this definition is slightly different from that in [Pri3, Definition 1.31]. The construction there relied on a filtered DGAA resolution DR′(B/R) of DR(B/R), with [Pri3, Lemma 1.30] giving a non-commutative analogue of Lemma 3.4.

Instead, Definition 3.7 effectively constructs the map μ_{w}: DR(B/R) → T_{Δ}𝒟_{B/R} in this setting by first taking

DR(B/R) → p_{w}T̂ot CC_{R,BD_{1}}(Diff_{B/R})

using the commutative structure underlying a P_{2}-algebra, then applying the projection CC_{R,BD_{1}}(Diff_{B/R}) → Diff_{B/R}. The map μ_{w} then converges more quickly than the map μ in [Pri3], but depends on a choice of formality isomorphism.

This raises the question of whether the construction of [Pri3] could be adapted to unshifted symplectic structures, giving equivalences not relying on formality. This would mean establishing an analogue of Lemma 3.2 giving a universal property for DR(B/R) within a suitable category of filtered E_{2}-algebras. The filtered DGAA DR′(B/R) is not almost commutative, but the left and right A-module structures on gr_{F} DR′(B/R) agree. Similarly, DR(B/R) will not have the desired universal property in BD_{2}-algebras, but the analogy raises the possibility that it might do so in some larger category.

3.2.1. Generalised Lagrangians. We now fix a cofibrant stacky CDGA A over R, and a cofibration A → B of stacky CDGAs over R.

Definition 3.9. Recall that a 0-shifted pre-symplectic structure ω on A/R is an element

ω ∈ Z^{2}F^{2}DR(A/R).

It is called symplectic if ω_{2} ∈ Z^{0}Tot^{Π}Ω_{A/R}^{2} induces a quasi-isomorphism

ω_{2}^{♯}: ˆ \underline{Hom}_{A}(Ω_{A/R}^{1}, A^{0}) → Tot^{Π}(Ω_{A/R}^{1} ⊗_{A} A^{0})

and Tot^{Π}(Ω_{A/R}^{1} ⊗_{A} A^{0}) is a perfect complex over A^{0}.

An isotropic structure on B relative to ω is an element (ω, λ) of

Z^{2}cocone(F^{2}DR(A/R) → F^{2}DR(B/R))

lifting ω. This structure is called Lagrangian if ω is symplectic and the image λ̄_{2} of λ in Z^{−1}Tot^{Π}Ω_{B/R}^{1} ⊗_{B} Ω_{B/A}^{1} induces a quasi-isomorphism

λ_{2}^{♯}: ˆ \underline{Hom}_{B}(Ω_{B/A}^{1}, B^{0}) → Tot^{Π}(Ω_{B/R}^{1} ⊗_{B} B^{0})[−1]

and Tot^{Π}(Ω_{B/A}^{1} ⊗_{B} B^{0}) is a perfect complex over B^{0}.

Definition 3.10. Define a decreasing filtration F̃ on DR(A/R)⟦ℏ⟧ by

F̃^{p}DR(A/R) := ∏_{i≥0} F^{p−i}DR(A/R)ℏ^{i}.

Define a further filtration G by G^{k}DR(A/R)⟦ℏ⟧ = ℏ^{k}DR(A/R)⟦ℏ⟧.

Definition 3.11. Define the space of generalised 0-shifted isotropic structures on the pair (A, B) over R to be the simplicial set

GIso(A, B; 0) := \underline{MC}(F̃^{2}cone(DR(A/R)⟦ℏ⟧ → DR(B/R)⟦ℏ⟧)),

where we regard the cochain complexes as a DGLA with trivial bracket.

Also write GIso(A, B; 0)/ℏ^{k} for the obvious truncation in terms of DR[ℏ]/ℏ^{k}, so GIso(A, B; 0) = \underleftarrow{lim}_{k} GIso(A, B; 0)/ℏ^{k}. Write Iso = GIso/ℏ.

Set GLag(A, B; 0) ⊂ GIso(A, B; 0) to consist of the points whose images in Iso(A, B; 0)/ℏ are Lagrangians on symplectic structures — this is a union of pathcomponents.

Thus the components of GIso(A, B; 0) are just elements in H^{1}F̃^{2}cone(DR(A/R) → DR(B/R))⟦ℏ⟧, with equivalence classes of n-morphisms given by elements in H^{1−n} of the same complex.

3.2.2. Compatible structures. In addition to our morphism A → B, we now fix a strict line bundle M over B, in the sense of Definition 2.10.

Definition 3.12. We say that a generalised isotropic structure (ω, λ) and a quantisation Δ of the pair (A, M) are w-compatible (or a w-compatible pair) if

[μ_{w}(ω, Δ)] = [−∂_{ℏ^{−1}}(Δ)] ∈ H^{1}(F̃^{2}T_{Δ}QP̂ol_{w}(A, M; 0)) ≅ H^{1}(F̃^{2}T_{Δ}QP̂ol(A, M; 0)),

where σ = −∂_{ℏ^{−1}} is the canonical tangent vector of Definition 2.18.

Definition 3.13. 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

\xymatrix{X \ar[r]^{s} \ar[r]_{0} & A \ar[r] & Z}.

Definition 3.14. Define the space QComp_{w}(A, M; 0) to be the homotopy vanishing locus of

(μ_{w} − σ): GIso(A, B; 0) × QP_{w}(A, M; 0) → T QP_{w}(A, M; 0)

over QP_{w}(A, M; 0)

We define a cofiltration on this space by setting QComp_{w}(A, M; 0)/G^{j} to be the homotopy vanishing locus of

(μ_{w} − σ): (GIso(A, B; 0)/G^{j}) × (QP_{w}(A, M; 0)/G^{j}) → T QP_{w}(A, M; 0)/G^{j}

over QP_{w}(A, M; 0)/G^{j}.

Thus QComp_{w}(A, M; 0) consists of data (ω, λ, Δ, α), where (ω, λ) is a generalised isotropic structure, Δ a quantisation of (A, M), and α a homotopy between μ_{w}(ω, λ) and σ(Δ).

Definition 3.15. Define QComp_{w}(A, M; 0)^{nondeg} ⊂ QComp_{w}(A, M; 0) to consist of w-compatible quantised pairs (ω, Δ) with Δ non-degenerate. This is a union of pathcomponents, and by [Pri6, Lemma 1.22] any pre-symplectic form compatible with a non-degenerate quantisation is symplectic. The same argument shows that any isotropic pair compatible with a non-degenerate quantisation is Lagrangian so there is a natural projection

QComp_{w}(A, M; 0)^{nondeg} → GLag(A, B; 0)

as well as the canonical map

QComp_{w}(A, M; 0)^{nondeg} → QP_{w}(A, M; 0)^{nondeg}.

3.3. The equivalences.

Proposition 3.16. For any Levi decomposition w of GT, the canonical map

QComp_{w}(A, M; 0)^{nondeg} → QP_{w}(A, M; 0)^{nondeg} ≃ QP(A, M; 0)^{nondeg}

is a weak equivalence. In particular, w gives rise to a morphism

QP(A, M; 0)^{nondeg} → GLag(A, B; 0)

(from non-degenerate quantisations to generalised Lagrangians) in the homotopy category of simplicial sets.

Proof. The proof of [Pri6, Proposition 1.26] adapts to this context, along much the same lines as [Pri4, Proposition 2.16]. The essential idea is that non-degeneracy of a quantisation Δ ensures that μ_{w}(−, Δ) is a filtered quasi-isomorphism, so the generalised Lagrangian data (ω, λ) associated to Δ are given by

−μ_{w}(−, Δ)^{−1}(∂_{ℏ^{−1}}Δ).
□

Write P̂ol(A, B; 0) := QP̂ol(A, M; 0)/G^{1}, with a filtration F given by the image of the filtration F̃, then also write Comp := QComp_{w}/G^{1}, P := QP/G^{1}, Lag := GLag/G^{1} and Iso := GIso/G^{1}. In particular, observe that since P̂ol(A, B; 0) is already a P_{2}-algebra, the space Comp is independent of the Levi decomposition w of GT.

The following proposition establishes an equivalence between Lagrangians and nondegenerate co-isotropic Poisson structures in the 0-shifted setting:

Proposition 3.17. The canonical maps

Comp(A, B; 0)^{nondeg} → P(A, B; 0)^{nondeg}
Comp(A, B; 0)^{nondeg} → Lag(A, B; 0)

are weak equivalences.

Proof. The first equivalence is given by observing that the equivalences in Proposition 3.16 respect the cofiltration G. For the second equivalence, we adapt the proofs of [Pri6, Corollary 1.36 and Proposition 1.37], establishing the equivalence by induction on the filtration F.

The space Lag(A, B; 0)/F^{3} is just given by elements (ω, λ) in the cocone of T̂ot Ω_{A/R}^{2} → T̂ot Ω_{A/R}^{2} which are non-degenerate in the sense that the induced map (ω, λ)^{♯} induces a quasi-isomorphism

ˆ\underline{Hom}_{A}(Ω_{A/R}^{1}, A^{0}) ⟶^{S} ˆ \underline{Hom}_{B}(Ω_{B/A}^{1}, B^{0})[1]
ω^{♯}↓  ↓_{λ^{♯}}
T̂ot (Ω_{A/R}^{1} ⊗_{A} A^{0}) ⟶ T̂ot (Ω_{B/R}^{1} ⊗_{B} B^{0})

of diagrams. Since P(A, B; 0)/F^{3} is given by elements (ϖ, π) in the cocone of S: ˆ \underline{Hom}_{A}(Ω_{A/R}^{2}, A) → ˆ \underline{Hom}_{B}(CoS_{B}^{2}Ω_{B/A}^{1}, B)[2], the essentially unique Poisson structure compatible with (ω, λ) is just given by the image of (ω, λ) under the symmetric square of the homotopy inverse of (ω, λ)^{♯}, so

Comp(A, B; 0)^{nondeg}/F^{3} ⟶^{∼} Lag(A, B; 0)/F^{3}.

Adapting the proof of [Pri6, Corollary 1.36], there is a a commutative diagram

(Comp(A, B; 0)^{nondeg}/F^{p+1})_{(ω,λ,ϖ,π)} ⟶  (Lag(A, B; 0)/F^{p+1})_{(ω,λ)}
↓  ↓
(Comp(A, B; 0)^{nondeg}/F^{p})_{(ω,λ,ϖ,π)} ⟶  (Lag(A, B; 0)/F^{p})_{(ω,λ)}
↓  ↓
\underline{MC}(M(ω, λ, ϖ, π, p)[1])  ⟶ \underline{MC}(T̂ot cocone(Ω_{A/R}^{p} → Ω_{B/R}^{p})[2 − p])

of fibre sequences, where M(ω, λ, ϖ, π, p) is defined to be the homotopy limit of the diagram

T̂ot Ω_{A/R}^{p}[1 − p] ⟶  T̂ot Ω_{B/R}^{p}[1 − p]
Λ^{p}(ϖ^{♯})↓  ↓^{Λ^{p}(π^{♯})}
ˆ\underline{Hom}_{A}(Ω_{A/R}^{p}, A)[1 − p] ⟶^{S} ˆ \underline{Hom}_{B}(CoS_{B}^{p}Ω_{B/A}^{1}, B)[1]
ν(ω,ϖ)−(p−1)↑  ↑^{ν(λ,π)−(p−1)}
ˆ\underline{Hom}_{A}(Ω_{A/R}^{p}, A)[1 − p] ⟶^{S} ˆ \underline{Hom}_{B}(CoS_{B}^{p}Ω_{B/A}^{1}, B)[1].

Here ν(ω, ϖ) is the tangent map of μ(ω, −) at ϖ, given by

μ(ω, π + ρϵ) = μ(ω, π) + ν(ω, π)(ρ)ϵ

for ϵ^{2} = 0, with ν(λ, π) defined similarly.

Arguing as in [Pri3, Lemma 1.39], ν(ω, ϖ) ≃ p(ϖ^{♯} ∘ ω^{♯}) and ν(λ, π) ≃ p(π^{♯} ∘ λ^{♯}) in the diagram above. Since we are in the non-degenerate setting, ϖ^{♯} ∘ ω^{♯} and π^{♯} ∘ λ^{♯} are homotopic to the identity maps on their respective spaces, so ν(ω, ϖ) and ν(λ, π) are homotopic to multiplication by p. Because p − (p − 1) is invertible, we then get

M(ω, λ, ϖ, π, p) ≃ T̂ot cocone(Ω_{A/R}^{p} → Ω_{B/R}^{p})[1 − p].

Substituting in the diagram of fibre sequences then gives

(Comp(A, B; 0)^{nondeg}/F^{p+1})
≃ (Comp(A, B; 0)^{nondeg}/F^{p}) ×_{(Lag(A,B;0)/F^{p})}^{h} (Lag(A, B; 0)/F^{p+1}),

from which the desired equivalence (Comp(A, B; 0)^{nondeg}/F^{p+1}) ≃ (Lag(A, B; 0)/F^{p+1}) follows by induction. □

Proposition 3.18. For any Levi decomposition w of GT, the maps

QP_{w}(A, M; 0)^{nondeg}/G^{j}
→ (QP_{w}(A, M; 0)^{nondeg}/G^{2}) ×_{(GLag(A,B;0)/G^{2})}^{h} (GLag(A, B; 0)/G^{j})
≃ (QP_{w}(A, M; 0)^{nondeg}/G^{2}) × ∏_{2≤i<j} \underline{MC}(cone(DR(A/R) → DR(B/R))ℏ^{i})

coming from Proposition 3.16 are weak equivalences for all j ≥ 2.

Proof. The proof of [Pri3, Proposition 1.40] and [Pri4, Proposition 2.17] generalises to this setting. For (ω, λ, ϖ, π) ∈ Comp(A, B; 0), there is a a commutative diagram

(QComp_{w}(A, M; 0)/G^{j+1})_{(ω,λ,ϖ,π)} ⟶  (GIso(A, B; 0)/G^{j+1})_{(ω,λ)}
↓  ↓
(QComp_{w}(A, M; 0)/G^{j})_{(ω,λ,ϖ,π)} ⟶  GIso(A, B; 0)/G^{j})_{(ω,λ)}
↓  ↓
\underline{MC}(N(ω, λ, ϖ, π, j)[1])  ⟶ \underline{MC}(cone(F^{2−j}DR(A/R) → F^{2−j}DR(B/R))ℏ^{j})

of fibre sequences, for a space N(ω, λ, ϖ, π, j) defined as follows.

We set N(ω, λ, ϖ, π, j) to be the homotopy limit of the diagram

cocone(F^{2−j}DR(A/R) → F^{2−j}DR(B/R))ℏ^{j}
↓^{μ(−,−,ϖ,π)}
F^{2−j}T_{(ϖ,π)}P̂ol(A, B; 0)ℏ^{j}
↑^{ν(ω,λ,ϖ,π)+∂_{ℏ^{−1}}}
(F^{2−j}P̂ol(A, B; 0)ℏ^{j}, δ_{ϖ,π}) = F^{2−j}T_{(ϖ,π)}P̂ol(A, B; 0)ℏ^{j−1},

where ν(ω, λ, ϖ, π) is the tangent map of μ(ω, λ, −, −) at (ϖ, π), given by

μ(ω, λ, ϖ + τϵ, π + ρϵ) = μ(ω, π) + ν(ω, π)(τ, ρ)ϵ

with ϵ^{2} = 0.

On the associated graded pieces, the proof of [Pri4, Proposition 2.17] shows that gr_{F}^{p} (ν(ω, λ, ϖ, π) + ∂_{ℏ^{−1}}) is homotopic to (1 − j)ℏ. As this is an isomorphism for all j ≥ 2, the map N(ω, λ, ϖ, π, j) → cocone(F^{2−j}DR(A/R) → F^{2−j}DR(B/R))ℏ^{j} is a quasi-isomorphism, which inductively gives the required weak equivalences from the fibre sequences above.

□

Remark 3.19. Taking the limit over all j, Proposition 3.18 gives an equivalence

QP_{w}(A, M; 0)^{nondeg}
≃ (QP_{w}(A, M; 0)^{nondeg}/G^{2}) × ∏_{i≥2} \underline{MC}(cone(DR(A/R) → DR(B/R))ℏ^{i});

in particular, this means that there is a canonical map

(QP(A, M; 0)^{nondeg}/G^{2}) → QP(A, M; 0)^{nondeg},

dependent on w ∈ Levi_{GT}, corresponding to the distinguished point 0.

Even if π is degenerate, a variant of Proposition 3.18 still holds. Because ϖ^{♯} ∘ ω^{♯} and π^{♯} ∘ λ^{♯} are homotopy idempotent, the map gr_{F}^{p} ν(ω, λ, ϖ, π) has eigenvalues in the interval [0, p], so we just replace (1 − j) with an operator having eigenvalues in the interval [1 − p − j, 1 − j]. Since this is still a quasi-isomorphism for j > 1, we have

QComp_{w}(A, M; 0)
≃ (QComp_{w}(A, M; 0)/G^{2}) × ∏_{i≥2} \underline{MC}(cocone(DR(A/R) → DR(B/R))ℏ^{i}).

giving a sufficient first-order criterion for degenerate quantisations to exist.

4. Global quantisations

As in [Pri3, §3] and [Pri4, §3], in order to pass from stacky CDGAs to derived Artin stacks, we will exploit a form of étale functoriality. We then introduce the notion of self-duality and thus establish the existence of quantisations for derived Lagrangians.

4.1. Diagrams of quantised pairs.

Definition 4.1. Given a small category I, an I-diagram (A, F) in almost commutative stacky DGAAs over R, and a filtered A-bimodule M in I-diagrams of chain cochain complexes for which the left and right gr^{F} A-module structures on gr^{F} M agree, we define the filtered chain cochain complex

CC_{R,BD_{1}}(A, M)

to be the equaliser of the obvious diagram

∏_{i∈I} CC_{R,BD_{1}}^{•}(A(i), M(i)) ⟹ ∏_{f: i→j in I} CC_{R,BD_{1}}^{•}(A(i), M(j)),

for the BD_{1} Hochschild complexes of Definition 2.7.

We then write CC_{R,BD_{1}}^{•}(A) := CC_{R,BD_{1}}^{•}(A, A), which inherits the structure of a stacky brace algebra from each CC_{R,BD_{1}}^{•}(A(i), A(i)).

Note that if u: I → J is a morphism of small categories and A is a J-diagram of almost commutative stacky DGAAs over R, with B = A ∘ u, then we have a natural map CC_{R}^{•}(A) → CC_{R}^{•}(B).

In order to ensure that CC_{R}^{•}(A, M) has the correct homological properties, we now consider categories of the form [m] = (0 → 1 → … → m). Similarly to [Pri4, Lemma 3.2], the construction CC_{R}^{•}(A, M) preserves weak equivalences proved we restrict to pairs (A, M) for which each A(i) is cofibrant as an R-module and M is fibrant for the injective model structure (i.e. the maps M(i) → M(i + 1) are all surjective).

As in [Pri3, §3.4.1], we can do much the same for differential operators:

Definition 4.2. Given a small category I, an I-diagram B of stacky CDGAs over R, and B-modules M, N in chain cochain complexes, define the filtered chain cochain complex Diff_{B/R}(M, N) to be the equaliser of the obvious diagram

∏_{i∈I} Diff_{B(i)/R}(M(i), N(i)) ⟹ ∏_{f: i→j in I} Diff_{B(i)/R}(M(i), f_{∗}N(j)),

and write Diff_{B/R} for Diff_{B/R}(B, B)

If B is an [m]-diagram in DG^{+}dgCAlg(R) which is cofibrant and fibrant for the injective model structure (i.e. each B(i) is cofibrant in the model structure of Lemma 2.3 and the maps B(i) → B(i + 1) are surjective), then observe that gr_{k}^{F} Diff_{B/R} is a model for the derived Hom-complex RHom_{B}(CoS_{B}^{k}Ω_{B/R}^{k}, B).

The constructions in §2 now all carry over verbatim, generalising from morphisms of cofibrant stacky CDGAs to morphisms A → B of [m]-diagrams of stacky CDGAs which are cofibrant and fibrant for the injective model structure. In particular, for any such morphism and a strict line bundle M over B, we have a DGLA

QP̂ol(A, M; 0)[1]

of 0-shifted relative quantised polyvectors as in Definition 2.11, and a space

QP(A, M; 0)

of quantisations of the pair (A, M) as in Definition 2.14.

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]-diagrams A, B of stacky CDGAs, each A(j), B(j) satisfies Assumption 2.19, so there exists N for which the chain complexes (Ω_{A(j)/R}^{1} ⊗_{A(j)} A(j)^{0})^{i} are acyclic for all i > N, and similarly for B.

Definition 4.3. Given a morphism A → B of fibrant cofibrant [m]-diagrams in stacky CDGAs (for the injective model structure) define

GIso(A, B; 0) := GIso(A(0), B(0); 0) = \underleftarrow{lim}_{i∈[m]} GIso(A(i), B(i); 0),

for the space GIso of generalised isotropic structures of Definition 3.11, and define the space GLag(A, B; 0) of generalised Lagrangians similarly.

Given a choice w ∈ Levi_{GT}(ℚ) of Levi decomposition for GT, define

μ_{w}: GIso(A, B; 0) × QP_{w}(A, M; 0) → T QP_{w}(A, M; 0)

by setting μ_{w}(ω, λ, Δ)(i) := μ_{w}(ω(i), λ(i), Δ(i)) ∈ T QP_{w}(A(i), B(i); 0) for i ∈ [m], and let QComp_{w}(A, M; 0) be the homotopy vanishing locus of

(μ_{w} − σ): GIso(A, B; 0) × QP_{w}(A, M; 0) → T QP_{w}(A, M; 0).

over QP_{w}(A, M; 0).

As in [Pri6, §3.4.2], if we let (DG^{+}dgCAlg(R)^{[1]})^{ét} ⊂ DG^{+}dgCAlg(R)^{[1]} be the wide subcategory of the arrow category with only homotopy formally étale morphisms (see Definition 2.6) between arrows, then for any of the constructions F based on QP, [Pri6, Definition 2.7] adapts to give an ∞-functor

RF: L(DG^{+}dgCAlg(R)^{[1]})^{ét} → LsSet

from the ∞-category of stacky CDGAs and homotopy formally étale morphisms to the ∞-category of simplicial sets. This construction has the property that (RF)(ϕ: A → B) ≃ F (ϕ: A → B) for all morphisms ϕ of cofibrant stacky CDGAs A over R.

Immediate consequences of Propositions 3.16 and 3.18 are that for any w ∈ Levi_{GT}(ℚ), the canonical maps

QComp_{w}(A, M; 0)^{nondeg} → QP_{w}(A, M; 0)^{nondeg} ≃ QP(A, M; 0)^{nondeg};
QP_{w}(A, M; 0)^{nondeg}/G^{j}
→ (QP_{w}(A, M; 0)^{nondeg}/G^{2}) × ∏_{2≤i<j} \underline{MC}(cocone(DR(A/R) → DR(B/R))ℏ^{i}[1])

are weak equivalences of ∞-functors on the full subcategory of (LDG^{+}dgCAlg(R)^{[1]})^{ét} consisting of objects satisfying the conditions of Assumption 2.19, for all j ≥ 2.

4.2. Descent and line bundles. We now extend the constructions above to line bundles, via 𝔾_{m}-equivariance exactly as in [Pri3, §3.4.2].

On DG^{+}dgAlg(ℚ), we consider the functor (B𝔾_{m})^{Δ} ∘ D, which sends B to the nerve of the groupoid

TLB(B) := [Z^{1}(Z_{0}B)/(Z_{0}B^{0})^{×}]

of trivial line bundles, where f ∈ (B^{0})^{×} acts on Z^{1}B by addition of ∂ log f = f^{−1}∂f.

For any morphism A → B of cofibrant stacky CDGAs over R, we can extend QP(A, B; 0) to a simplicial representation of the groupoid TLB(B) above by sending an object c ∈ Z^{1}(Z_{0}B) to QP(A, B_{c}; 0), with (Z_{0}B^{0})^{×} acting via functoriality for line bundles. Note that the quotient representation QP(−, −; 0)/G^{1} = P(−, 0) is trivial; we also set GIso to be a trivial representation c ↦ GIso(A, B; 0).

Definition 4.4. For any of the constructions F of §4.1, let R(F/^{h}𝔾_{m}) be the ∞-functor on LdgCAlg(R)^{ét} given by applying the construction of [Pri6, §3.4.2] to the right-derived functor of the Grothendieck construction

B ↦ holim_{\overrightarrow{c∈TLB(B)}} F (A, B_{c}),

then taking hypersheafification with respect to homotopy formally étale coverings.

Given a derived Artin N-stack X, and A ∈ DG^{+}dgCAlg(R), we say that an element f ∈ ho\underleftarrow{lim}_{i} X(D^{i}A) is homotopy formally étale if the induced morphism

N_{c}f_{0}^{∗}𝕃_{X/R} → {Tot σ^{≤q}LΩ_{A/R}^{1} ⊗_{A}^{L} A^{0}}_{q}

from [Pri6, §3.2.2] is a pro-quasi-isomorphism.

Given a morphism X → Y of derived Artin N-stacks, we then write (dg_{+}DGAff_{ét}^{[1]} ↓ X/Y) for the ∞-category consisting of morphisms Spec B → Spec A in dg_{+}DGAff_{R}, equipped with homotopy formally étale elements of ho\underleftarrow{lim}_{i} X(D^{i}B) ×_{Y (D^{i}B)}^{h} Y (D^{i}A); morphisms in this ∞-category are given by compatible homotopy formally étale maps A → A′, B → B′ .

Definition 4.5. Given a map X → Y of strongly quasi-compact derived Artin N-stacks over R, a line bundle ℒ on X and any of the functors F above, define F (Y, ℒ) to be the homotopy limit of

R(F/^{h}𝔾_{m})(A, B) ×_{R(∗/^{h}𝔾_{m})(B)}^{h} {ℒ |_{B}}

over objects Spec B → Spec A in the ∞-category (dg_{+}DGAff_{ét}^{[1]} ↓X/Y).

Remark 4.6. In many cases, we can take smaller categories than (dg_{+}DGAff_{ét}^{[1]} ↓ X/Y) on which to calculate the homotopy limit. When the 𝔾_{m}-action on F is trivial, we can restrict to compatible hypergroupoid resolutions as in [Pri6, §3.4.2]. When X and Y are derived Deligne–Mumford N-stacks, we do not need stacky CDGAs, and can just work over (DGAff_{ét}^{[1]} ↓X/Y).

When X and Y are 1-geometric derived Artin stacks, we may just consider the ∞- category of commutative diagrams

\begin{matrix}U & ⟶^{f} & X \\ ↓ & & ↓ \\ V & ⟶^{g} & Y \end{matrix}

with U, V derived affines and the maps f, g being smooth; to this we associate the morphism Ω_{U/X}^{•} → Ω_{V/Y}^{•} of stacky CDGAs as in §2.1, giving an object of (dg_{+}DGAff_{ét}^{[1]} ↓ X/Y). Following Remark 2.15, this means that an object of QP(Y, ℒ; 0) is a form of curved A_{∞} deformation of the presheaf V ↦ Tot Ω_{V/Y}^{•}, acting on a deformation of the presheaf U ↦ Tot Ω_{U/X}^{•} ⊗_{f^{−1}𝒪_{X}} f^{−1}ℒ given by R-linear differential operators.

Adapting [Pri3, Definition 2.21] along the lines of Definition 2.21 gives:

Definition 4.7. Say that a quantisation Δ ∈ QP(Y, ℒ; 0)/G^{k} is non-degenerate if the induced maps

Δ_{2,Y}^{♯} : 𝕃_{Y/R} → Rℋ om_{𝒪_{Y}} (𝕃_{Y/R}, 𝒪_{Y/R})
Δ_{2,X}^{♯}: 𝕃_{X/R} → Rℋ om_{𝒪_{X}} (𝕃_{X/Y} , 𝒪_{X})[1]

are quasi-isomorphisms and and 𝕃_{X}, 𝕃_{Y} are perfect.

Propositions 3.17 and 3.18 now readily generalise (substituting the relevant results from [Pri6, §3] to pass from local to global), giving:

Proposition 4.8. For any X → Y, any line bundle ℒ on X and any w ∈ Levi_{GT}(ℚ), the canonical maps

Comp(Y, X; 0)^{nondeg} → P(Y, X; 0)^{nondeg}
Comp(Y, X; 0)^{nondeg} → Lag(Y, X; 0)
QComp_{w}(Y, ℒ; 0)^{nondeg} → QP(Y, ℒ; 0)^{nondeg}
QComp_{w}(Y, ℒ; 0) → (QComp_{w}(Y, ℒ; 0)/G^{2}) ×_{(GIso(Y,X;0)/G^{2})}^{h} GIso(Y, X; 0) ≃

(QComp_{w}(Y, ℒ; 0)/G^{2}) × ∏_{i≥2} \underline{MC}(cone(DR(Y/R) → DR(X/R)ℏ^{i}) are filtered weak equivalences. In particular, w gives rise to a morphism

QP(Y, ℒ; 0)^{nondeg} → GLag(Y, X; 0)

in the homotopy category of simplicial sets.

Remark 4.9. The results of Proposition 4.8 are compatible with those of [BGKP, Theorem 1.1.4], which fixes a quantisation 𝒪̃_{Y} of a smooth variety Y and describes quantisations of line bundles ℒ on smooth Lagrangians X, compatible with 𝒪̃_{Y} . They show that the obstructions to quantising ℒ in this way are a class c_{1}(ℒ)− \frac{1}{2}c_{1}(K_{X})−At(𝒪̃_{Y} , X) ∈ H^{2}F^{1}DR(X) and a power series in ℏ^{2}H^{2}DR(X)⟦ℏ⟧ determined by 𝒪̃_{Y} . Their first condition corresponds to our first-order obstruction, i.e. the obstruction to lifting the co-isotropic structure from P(Y, X; 0)^{nondeg} to QP(Y, ℒ; 0)^{nondeg}/G^{2}. There are no further obstructions to quantising the pair (𝒪_{Y} , ℒ), but their second condition is to ensure that the resulting quantisation of 𝒪_{Y} is 𝒪̃_{Y} , with the obstruction then coming from the higher-order coefficients of the exact sequence

H^{1}(cone(DR(Y) → DR(X))⟦ℏ⟧ → H^{2}DR(Y)⟦ℏ⟧ → H^{2}DR(X)⟦ℏ⟧.

When ℒ^{⊗2} has a right 𝒟-module structure, the Chern class c_{1}(ℒ) − \frac{1}{2}c_{1}(K_{X}) vanishes. Moreover, whenever there is an isomorphism 𝒪̃_{Y} ≃ 𝒪̃_{Y}^{opp} of quantisations which is semilinear with respect to the transformation ℏ ↦ −ℏ, the calculations of [BGKP, Remark 5.3.4] show that At(𝒪̃_{Y} , X) = 0. Thus their obstruction does indeed vanish in the scenario of Theorem 4.16 below.

4.3. Self-duality. In order to eliminate the potential first order obstruction to quantising a generalised Lagrangian in Proposition 4.8, we now introduce the notion of self-duality, combining the ideas of [Pri3, §4] and [Pri4, §1.6].

We wish to consider line bundles ℒ on X equipped with an involutive equivalence (−)^{t}: 𝒟(ℒ) ≃ 𝒟(ℒ)^{opp}. Such an equivalence is the same as a right 𝒟-module structure on ℒ^{⊗2}. Since a dualising line bundle K_{X} on X naturally has the structure of a right 𝒟-module (see for instance [GR, §2.4] for a proof in the derived setting), we will typically take ℒ to be a square root of K_{X}. In this case, the equivalence 𝒟(ℒ) ≃ 𝒟(ℒ)^{opp} comes from the equivalences ℒ ≃ ℒ^{∨} and 𝒟(ℰ)^{opp} ≃ 𝒟(ℰ^{∨}), where ℰ^{∨} := Rℋ om_{𝒪_{X}} (ℰ , K_{X}).

Definition 4.10. Given a morphism ϕ: A → B of cofibrant stacky CDGAs over R and a strict line bundle M over B, equipped with a contravariant involution (−)^{t} of Diff_{B/R}(M), we define an involution (−)^{∗} on the DGLA QP̂ol(A, M; 0)[1] by

Δ^{∗}(ℏ) := i(Δ)(−ℏ)^{t},

for the brace algebra involution

−i: (CC_{R,BD_{1}}(A, Diff_{B/R}(M))_{[1]} ⋊ CC_{R,BD_{1}}(A))^{opp}
→ CC_{R,BD_{1}}(A, Diff_{B/R}(M)^{opp})_{[1]} ⋊ CC_{R,BD_{1}}(A)

adapted from Lemma 1.21.

Since (−)^{∗} is a quasi-isomorphism of filtered DGLAs, it gives rise to an involutive weak equivalence

(−)^{∗}: QP(A, M; 0) → QP(A, M; 0)

Lemma 4.11. For the filtration G induced on F̃^{p}QP̂ol(A, M; 0)^{sd} by the corresponding filtration on F̃^{p}QP̂ol(A, M; 0), we have

gr_{G}^{k}F̃^{p}QP̂ol(A, M; 0)^{sd} ≃ \begin{cases}gr_{G}^{k}F̃^{p}QP̂ol(A, M; 0) & k even \\ 0 & k odd. \end{cases}

Proof. This combines [Pri3, Lemma 4.4] and [Pri4, Lemma 1.35]. It follows because Lemma 1.21 ensures that the involution acts trivially on gr_{G}^{0}QP̂ol(A, M; 0). It therefore acts as multiplication by (−1)^{k} on gr_{G}^{k}QP̂ol(A, M; 0) = ℏ^{k}gr_{G}^{0}QP̂ol(A, M; 0). □

Definition 4.12. For a line bundle ℒ on X with a right 𝒟-module structure on ℒ^{⊗2}, we define the space

QP(Y, ℒ; 0)^{sd}

of self-dual quantisations to be the space of homotopy fixed points of the ℤ/2-action on QP(Y, ℒ; 0) generated by (−)^{∗}.

Remark 4.13. Following Remark 2.15, a self-dual quantisation of (X ⟶^{ϕ} Y, ℒ) gives rise to a curved A_{∞}-deformation 𝒪̃_{Y} of T̂ot 𝒪_{Y} over R⟦ℏ⟧ ,equipped with a contravariant involution ∗ which is semilinear under the transformation ℏ ↦ −ℏ, together with a curved involutive A_{∞}-morphism ϕ^{−1}𝒪̃_{Y} → 𝒟_{𝒪_{X}/R} (ℒ)⟦ℏ⟧.

More is true: by [Pri4, Proposition 1.25], a quantisation gives a curved A_{∞} deformation of the dg category per_{dg}(𝒪_{Y}) of perfect complexes on Y, with self-dual quantisations incorporating a semilinear lift of the involution Rℋ om_{𝒪_{Y}} (−, 𝒪_{Y}). A self-dual quantisation of the pair (Y, ℒ) thus gives a curved semilinearly involutive A_{∞}-deformation of the involutive category per_{dg}(𝒪_{Y}) fibred over per_{dg}(𝒪_{X}) via the functor

(per_{dg}(𝒪_{Y}), Rℋ om_{𝒪_{Y}} (−, 𝒪_{Y})) → (per_{dg}(𝒪_{X}), Rℋ om_{𝒪_{X}} (−, ℒ^{⊗2}))
ℱ ↦ ϕ^{∗}ℱ ⊗ ℒ ,

with an additional restriction of the curvature of the deformation in terms of differential operators.

Adapting [Pri4, Remark 1.34], we can extend the input data from the space RΓ(X, B𝔾_{m}) of line bundles to the space RΓ(Y, B^{2}𝔾_{m})×_{RΓ(X,B^{2}𝔾_{m})}^{h}{1} of pairs (𝒢 , ℒ) with 𝒢 a 𝔾_{m}-gerbe on Y, and ℒ a trivialisation of ϕ^{∗}𝒢. There is then a notion of selfdual quantisation for pairs (𝒢 , ℒ) with 𝒢 a μ_{2}-gerbe and ℒ a trivialisation of the 𝔾_{m}- gerbe associated to ϕ^{∗}𝒢, with a right 𝒟-module structure on the line bundle ℒ^{⊗2}. In particular, we may consider involutive quantisations of (per_{dg}(𝒪_{Y}), Rℋ om_{𝒪_{Y}} (−, ℳ)) for any line bundle ℳ, the criterion for self-duality now being that ℒ^{⊗2} ⊗ ϕ^{∗}ℳ be a right 𝒟-module, so that we consider the involution Rℋ om_{𝒪_{X}} (−, ℒ^{⊗2} ⊗ ϕ^{∗}ℳ) on per_{dg}(𝒪_{X}).

The natural example to take for ℳ is the dualising line bundle K_{Y} = det 𝕃_{Y}, but when X is Lagrangian, ϕ^{∗}K_{Y} will be trivial, so the resulting quantisations are quite similar. In any case, the 𝔾_{m}-actions on our filtered DGLAs are all unipotent, so extend to 𝔾_{m} ⊗_{ℤ} ℚ-actions. Since μ_{2} ⊗ ℚ = 0, this means there are canonical equivalences between the spaces of self-dual quantisations for varying (𝒢 , ℒ).

Definition 4.14. As in [Pri4, Remark 2.21], write t ∈ GT(ℚ) for the (−1)-Drinfel’d associator which induces the involution of Lemma 1.21. We then denote by Levi_{GT}^{t} the space of Levi decompositions w of GT with w(−1) = t; these form a torsor for the subgroup (GT^{1})^{t} of t-invariants in the pro-unipotent radical GT^{1}.

Definition 4.15. Define GLag(Y, X; 0)^{sd} to be the homotopy fixed points of the involution of GLag(Y, X; 0) given by ℏ ↦ −ℏ. Explicitly, we set GIso(A, B; 0)^{sd} to be

\underline{MC}(cone(F^{2}DR(A/R) → F^{2}DR(B/R))) × ∏_{i>0} \underline{MC}(cone(DR(A/R) → DR(B/R))ℏ^{2i}),

with GLag(A, B; 0)^{sd} the subspace of non-degenerate elements.

Theorem 4.16. Take a morphism X → Y of strongly quasi-compact Artin N-stacks over R, and a line bundle ℒ on X with a right 𝒟-module structure on ℒ^{⊗2} (such as when ℒ is any square root of K_{X}). For any w ∈ Levi_{GT}^{t}(ℚ), the induced map

QP(Y, ℒ; 0)^{nondeg,sd} → GLag(Y, X; 0)^{sd}

(from non-degenerate self-dual quantisations to generalised self-dual Lagrangians) coming from Proposition 4.8 is a weak equivalence.

In particular, w associates a canonical choice of self-dual quantisation of (Y, ℒ) to every Lagrangian structure of X over Y.

Proof. This is much the same as [Pri3, Proposition 4.5]. Lemma 4.11 implies that w gives rise to weak equivalences

QP(Y, ℒ; 0)^{sd}/G^{2i} → QP(Y, ℒ; 0)^{sd}/G^{2i−1}
QP(Y, ℒ; 0)^{sd}/G^{2i+1} → (QP(Y, ℒ; 0)^{sd}/G^{2i}) ×_{(QP(Y,ℒ;0)/G^{2i})}^{h} (QP(Y, ℒ; 0)/G^{2i+1}).

Combined with Proposition 4.8, these give weak equivalences from QP(Y, ℒ; 0)^{nondeg,sd}/G^{2i+1} to

(QP(Y, ℒ; 0)^{nondeg,sd}/G^{2i}) × \underline{MC}(ℏ^{2i}cone(DR(Y/R) → DR(X/R))

for all i > 0. Moreover, [Pri4, Remark 2.21] ensures that for our choice of Levi decomposition w, the map μ_{w} is equivariant under the involutions ∗, so these equivalences are just given by taking homotopy ℤ/2-invariants. The result then follows by induction, the base case holding because ∗ acts trivially on QP(Y, ℒ; 0)/G^{1} = P(Y, X; 0), so QP(Y, ℒ; 0)^{sd}/G^{1} ≃ P(Y, X; 0). □

4.4. Quantisations of higher Lagrangians. Given a Lagrangian (X, λ) with respect to an n-shifted symplectic structure (Y, ω) for n > 0, we now discuss how the techniques of this paper should adapt to give a notion of quantisations and to establish their existence. The broad picture is that we should have an E_{n+1}-algebra deformation of 𝒪_{Y} acting on an E_{n}-algebra deformation of 𝒪_{X}.

If we exploit Koszul duality for P_{n+1}-algebras, we may replace the filtered Hochschild complexes of §1 with Poisson coalgebra coderivations on bar complexes to give P_{n+2}- algebras of derived multiderivations acting on P_{n+1}-algebras (instead of E_{2}-algebras acting on E_{1}-algebras). Proposition 3.17 then generalises to give an alternative proof of the equivalence, announced by Costello and Rozenblyum and now proved by Melani and Safronov [MS], between n-shifted Lagrangians and non-degenerate n-shifted co-isotropic structures. By adapting the methods of this paper, [MS] also established quantisations for n-shifted co-isotropic structures for n > 1. We now sketch a parametrisation of quantisations for higher Lagrangians, including the case n = 1 not addressed in [MS]. Following [Pri4, Remark 2.24], these constructions might lead to parametrisations of degenerate n-shifted co-isotropic structures.

4.4.1. Almost commutative E_{k}-algebras. We begin with the notion of a BD_{k}-algebra as a higher analogue of an almost commutative algebra. There is a filtration on the Lie operad given by arity, inducing a filtration on the free Lie algebra generated by any filtered complex. Taking the universal enveloping E_{k}-algebra of this Lie algebra then gives a filtered E_{k}-algebra, and this construction corresponds to a filtration on the E_{k} operad. We can then define the BD_{k} operad to be the E_{k} operad equipped with this completed filtration, for k ≥ 1.

Explicitly, BD_{1} is just the operad defined in [CPT^{+}, §3.5.1], whose algebras are almost commutative DGAAs. For k ≥ 2, the operad BD_{k} is just given by the reindexed good truncation filtration F^{p}BD_{k} = τ_{≥p(k−1)}E_{k} — this agrees with [CPT^{+}, §3.5.1] for k = 2, but differs by the reindexation for higher k. In particular, almost commutative brace algebras are equivalent to BD_{2}-algebras.

Informally, an n-shifted quantisation of a morphism A → B of CDGAs consists of a BD_{n+1}-algebra deformation Ã of A acting on a BD_{n}-algebra deformation B̃ of B in a sense we now attempt to make precise. An n-shifted quantisation of a morphism A → B of stacky CDGAs will be an n-shifted quantisation of T̂ot A → T̂ot B subject to additional boundedness constraints.

4.4.2. Centres. From now on, we refer to BD_{k}-algebras in complete filtered cochain chain complexes as stacky BD_{k}-algebras. Adapting [Lur, Theorem 5.3.1.14] from ∞- operads to the operads BD_{k} in filtered chain complexes will give a stacky BD_{k}-algebra

RCC_{BD_{k},R}(A, D)

associated to any morphism A → D of stacky BD_{k}-algebras over R, universal with the property that there is a BD_{k}-algebra morphism RCC_{BD_{k},R}(A, D) ⊗_{R}^{L} A → D in the associated ∞-category. Explicitly, these centres should be given by E_{k} Hochschild complexes equipped with a PBW filtration. The associated graded grRCC_{BD_{k},R}(A, D) is necessarily the centre of the morphism grA → grB of graded P_{k}-algebras, so is given by derived P_{k} multiderivations from grA to grB.

The universal property implies that RCC_{BD_{k},R}(A) := RCC_{BD_{k},R}(A, A) is naturally an E_{1}-algebra in stacky BD_{k}-algebras, i.e. a stacky E_{1} ⊗_{BV}^{L} BD_{k}-algebra for the Boardman–Vogt tensor product ⊗_{BV}. Moreover, for any morphism A → D, the centre RCC_{BD_{k},R}(A, D) will then become a RCC_{BD_{k}}(A)-module in stacky BD_{k}-algebras.

For any morphism A_{1}×A_{2} → D, the idempotents in the domain give a decomposition D = D_{1} × D_{2}, and by universality for each morphism A → D we thus have

RCC_{BD_{k},R}(R×A, R×D) ≃ RCC_{BD_{k},R}(R, R)×RCC_{BD_{k},R}(A, D) = R×RCC_{BD_{k},R}(A, D).

The centre of R × A → R × D as in the category of augmented stacky BD_{k}-algebras over R is just

RCC_{BD_{k},R}(R × A, R × D) ×_{(R×D)}^{h} R,

so the reasoning above shows that

RCC_{BD_{k},R,+}(A, D) := RCC_{BD_{k},R}(A, D) ×_{D}^{h} 0

is naturally a non-unital stacky BD_{k}-algebra, with RCC_{BD_{k},R,+}(D) a non-unital stacky E_{1} ⊗_{BV}^{L} BD_{k}-algebra.

Adapting Lemma 1.25, we then have:

Definition 4.17. Given a stacky E_{1} ⊗_{BV}^{L} BD_{k}-algebra C over R and a C-module E in stacky BD_{k}-algebras over R, we define E_{[1]}⋊C to be the non-unital stacky E_{1}⊗_{BV}^{L} BD_{k}- algebra

C ×_{RCC_{BD_{k},R}(E)}^{h} RCC_{BD_{k},R,+}(E),

the morphism C → RCC_{BD_{k},R}(E) existing by universality.

4.4.3. Quantised n-shifted relative polyvectors for n > 0. Given a morphism ϕ: A → B of stacky CDGAs over R, now consider the non-unital E_{1} ⊗_{BV}^{L} BD_{n+1}-algebra

(C, F) := RCC_{BD_{n+1},R}(A, RCC_{BD_{n},R}(B))_{[1]} ⋊ RCC_{BD_{n+1},R}(A)

in complete filtered cochain chain complexes. Definition 2.11 then adapts verbatim to give a complex QP̂ol(A, B; n) equipped with filtrations F̃ and G.

Since we wish QP̂ol(A, B; n)[n + 1] to have the structure of a DGLA with [F̃^{i}, F̃^{j}] ⊂ F̃^{i+j−1}Q and [G^{i}, G^{j}] ⊂ G^{i+j}, acting as derivations on the bifiltered E_{1} ⊗_{BV}^{L} BD_{n+1}- algebra ℏQP̂ol(A, B; n), we need to know that RCC_{BD_{k}}(A) has the structure of a BD_{k+1}- algebra. The analogous statement for k = 1 is the content of Lemma 1.19. In general, the property would follow from the following conjecture:

Conjecture 4.18. For k ≥ 1, the additivity isomorphism E_{k+1} ≃ E_{1} ⊗_{BV}^{L} E_{k} of [Lur, Theorem 5.1.2.2] induces a map BD_{k+1} ≃ E_{1} ⊗_{BV}^{L} BD_{k} of operads in complete filtered chain complexes.

Here, ⊗_{BV}^{L} denotes the derived Boardman–Vogt tensor product, so the conjecture amounts to saying that an A_{∞}-algebra in BD_{k}-algebras is naturally a BD_{k+1}-algebra. On passing to associated graded complex, the equivalence would give P_{k+1} → E_{1}⊗_{BV}^{L} P_{k}, which has been proved to be an equivalence by Rozenblyum (unpublished, cf. [CPT^{+}, §3.4]) and independently by Safronov [Saf2]; thus the map in the conjecture is necessarily an equivalence if it exists. A proof of Conjecture 4.18 has also been announced by Rozenblyum (cf. [CPT^{+}, comment after Conjecture 3.5.7]). For k ≥ 2, the conjecture would follow if additivity is compatible with the action of the Grothendieck–Teichmüller group.

The conjecture would also ensure that the centres RCC_{BD_{k},R}(A, D) above all exist by appealing directly to [Lur, Theorem 5.3.1.14] for k ≥ 1, regarding BD_{k}-algebras as E_{k−1}-algebras in BD_{1}-algebras.

The definitions of §§2, 3 all then adapt, replacing QP̂ol(A, M; 0) with QP̂ol(A, B; n) and taking appropriate shifts. The space QP(A, B; n) of n-shifted quantisations of the pair (A, B) is just

\underline{MC}(F̃^{2}QP̂ol(A, B; n)[n + 1]),

elements of which give rise to curved E_{n+1}-algebra deformations of T̂ot A acting on curved E_{n}-algebra deformations of B.

The space GIso(A, B; n) of n-shifted isotropic structures is

\underline{MC}(F̃^{2}cone(DR(A/R)⟦ℏ⟧ → DR(B/R)⟦ℏ⟧)[n]),

and Definition 3.7 then adapts to give a compatibility map

μ_{w}(−, Δ): cocone(DR(A/R) → DR(B/R)⟦ℏ⟧/ℏ^{j} → T_{Δ}QP̂ol_{w}(A; B, 0)/G^{j}

for each quantisation Δ; Definition 3.14 adapts to give a space QComp_{w}(A, B; n) for each w ∈ Levi_{GT}(ℚ).

Propositions 3.16, 3.17 and 3.18 will all carry over directly, in particular giving a map QP(A, B; n)^{nondeg} → GLag(A, B; n), the non-degenerate locus in GIso(A, B; n). The techniques of §4 then extend these to global constructions for Artin N-stacks.

4.4.4. Self-duality. The functor D ↦ D^{opp} sending an almost commutative algebra to its opposite gives an involutive endofunctor of the category of BD_{1}-algebras, and hence of the categories of E_{1} ⊗_{BV}^{L} BD_{k}-algebras. The universal property of centres then gives an involution

−i: RCC_{BD_{k},R}(A, D)^{opp} → RCC_{BD_{k},R}(A^{opp}, D^{opp}),

which in the k = 1 case is the involution −i of Lemma 1.21. Defining an involutive E_{1} ⊗_{BV}^{L} BD_{k}-algebra to be a homotopy fixed point of the involutive endofunctor (−)^{opp}, the involution above makes RCC_{BD_{k},R}(A, D) a stacky involutive BD_{k}-algebra whenever A and D are stacky involutive BD_{k}-algebras. In fact, this is necessarily the centre of A → D in the category of stacky involutive BD_{k}-algebras — the operad governing involutive BD_{k}-algebras is BD_{k} ∘ (0, ℚ.(ℤ/2), 0, …), with distributivity transformation given by the involution.

As in §4.3, we then have an involution (−)^{∗} on the DGLA QP̂ol(A, D; 0)[n + 1] given by Δ^{∗}(ℏ) := i(Δ)(−ℏ)^{t}, and we can define QP(A, B; n)^{sd} to be the fixed points of the resulting ℤ/2-action, so its points give rise to involutive quantisations.

The proof of Theorem 4.16 then adapts to give:

Theorem 4.19. Take a morphism X → Y of strongly quasi-compact Artin N-stacks over R. If Conjecture 4.18 holds, then for any w ∈ Levi_{GT}^{t}(ℚ), the induced map

QP(Y, X; n)^{nondeg,sd} → GLag(Y, X; n)^{sd}

(from non-degenerate self-dual quantisations to generalised self-dual Lagrangians) is a weak equivalence for all n > 0.

In particular, w associates a canonical choice of self-dual quantisation of (Y, X) to every n-shifted Lagrangian structure of X over Y.

Remark 4.20 (Twisted quantisations). One significant difference between Theorems 4.16 and 4.19 is that the former incorporates the data of a line bundle. Similar input data are not essential for positively shifted quantisations because a commutative algebra is canonically isomorphic to its opposite E_{1}-algebra, whereas 𝒪_{X} is not in general a right 𝒟-module.

However, by generalising Remark 4.13 we still expect a sensible notion of twisted quantisations for n-shifted Lagrangians, fibred over the space RΓ(Y, B^{n+2}𝔾_{m}) ×_{RΓ(X,B^{n+2}𝔾_{m})}^{h} {1} of pairs (𝒢 , ℒ) with 𝒢 a B^{n+1}𝔾_{m}-torsor on Y, and ℒ a trivialisation of ϕ^{∗}𝒢 on X. Self-dual (i.e. involutive) quantisations would then be parametrised by RΓ(Y, B^{n+2}μ_{2})×_{RΓ(X,B^{n+2}μ_{2})}^{h} {1}. Adapting [Lur, Theorem 5.3.2.5] from filtered E_{n+2}-algebras to BD_{n+2}-algebras would establish the required actions of (n + 2)-groupoids ho\underleftarrow{lim}_{i∈Δ} B^{n+2}D^{i}(A)^{×} generalising TLB from §4.2.

However, since these spaces will come from unipotent group actions on quantised polyvectors, the actions of the torsion groups B^{n+1}μ_{2}(A), B^{n+1}μ_{2}(B) must be trivial, so the spaces of twisted self-dual quantisations will be canonically equivalent as (𝒢 , ℒ) varies.

5. A “Fukaya category” for algebraic Lagrangians

In [BF, §5.3], Behrend and Fantechi discussed the construction of a dg category whose objects are local systems on Lagrangian submanifolds of a complex symplectic variety. An extensive survey of related results is given in [BBD^{+}, Remark 6.15], where Joyce et al. discuss possible approaches to constructing such a “Fukaya category” with complexes of vanishing cycles as morphisms. On a complex symplectic manifold, Kashiwara and Schapira give a likely candidate for this category for smooth Lagrangians in [KS], as the derived category of simple holonomic DQ modules for a DQ algebroid quantisation of the sheaf of analytic functions, and it is this approach which generalises naturally in our setting.

5.1. DQ modules. Since we are working algebraically rather than analytically, the analogue of a DQ module is an 𝒪̃_{Y} − 𝒟_{Y} -bimodule, where 𝒪̃_{Y} is a quantisation of 𝒪_{Y}. For a line bundle ℒ on a derived Lagrangian ϕ: X → Y, each quantisation (𝒪̃_{Y} , ℒ̃) of (𝒪_{Y} , ℒ) gives rise to such a bimodule as follows.

When the quantisation ℒ̃ is given by a differential operator Δ ∈ 𝒟_{X}(ℒ)⟦ℏ⟧, we write ℒ̃ˆ⊗_{𝒪_{X}} 𝒟_{X} for the associated right 𝒟-module

(ℒ ⊗_{𝒪_{X}} 𝒟_{X}⟦ℏ⟧, δ + Δ · −),

and similarly for related constructions. This is an abuse of notation because Δ is not 𝒪_{X}-linear, and in fact ℒ̃ˆ⊗_{𝒪_{X}} 𝒟_{X} is a more fundamental object than the R-linear deformation ℒ̃ of ℒ, because

ℒ̃ = (ℒ̃ˆ⊗_{𝒪_{X}} 𝒟_{X}) ⊗_{𝒟_{X}} 𝒪_{X}.

The associated 𝒪̃_{Y} − 𝒟_{Y} -bimodule is then given by taking

ϕ_{†}(ℒ̃ˆ⊗_{𝒪_{X}} ϕ^{−1}𝒟_{Y}) = 𝒟_{𝒪_{Y}} (𝒪_{Y} , Rϕ_{∗}ℒ̃),

which is naturally equipped with a left 𝒪̃_{Y} -module structure. Here ˆ⊗ denotes the ℏcompleted tensor product, since we regard 𝒪̃_{Y} , ℒ̃ as inverse systems over {R[ℏ]/ℏ^{i}}_{i}.

Definition 5.1. Fix a non-degenerate involutive quantisation 𝒪̃_{Y} ∈ QP(Y, 0)^{nondeg,sd} quantising a symplectic structure ω ∈ H^{2}F^{2}DR(Y/R), and assume that 𝒪̃_{Y} is wcompatible with ω · a for some w ∈ Levi_{GT}^{t}(ℚ) and a ∈ H^{0}DR(Y/R)⟦ℏ^{2}⟧. Now define a dg category F(𝒪̃_{Y} ) as follows.

Objects are given by morphisms ϕ: X → Y equipped with a square root ℒ of the dualising complex K_{X}, together with an element Δ ∈ QP(Y, ℒ; 0)^{nondeg,sd} lifting 𝒪̃_{Y} . We then define the complex

\underline{Hom}_{F(𝒪̃_{Y} )}((ℒ_{1}, Δ_{1}), (ℒ_{2}, Δ_{2}))

of morphisms to be the complex

RΓ(Y, (T̂ot CC_{R}(𝒪_{Y} , Diff_{𝒪_{Y}} (Rϕ_{1∗}ℒ_{1}, Rϕ_{2∗}ℒ_{2}))⟦ℏ⟧, δ + Δ_{Y} + Δ_{1} ∓ Δ_{2})),

with composition of morphisms defined in the obvious way.

Here, Δ_{Y} is the Hochschild element defining the quantisation 𝒪̃_{Y} , which thus acts on T̂ot CC_{R}(𝒪_{Y} , M) for all 𝒪_{Y} -bimodules M. The elements Δ_{i} giving the quantisations of ℒ_{i} lie in the complexes T̂ot CC_{R}(ϕ_{i}^{−1}𝒪_{Y} , Diff_{ϕ_{i}^{−1}𝒪_{Y} /R} (ℒ_{i})) twisted by Δ_{Y}, so act on the complex above by left and right multiplication respectively.

Remarks 5.2. The condition that 𝒪̃_{Y} is w-compatible with ω·a for some w ∈ Levi_{GT}^{t}(ℚ) and a ∈ H^{0}DR(Y/R)⟦ℏ^{2}⟧ is probably independent of w, and ensures that every self-dual line bundle on a Lagrangian (X, λ) over (Y, ω) admits a self-dual quantisation extending 𝒪̃_{Y} , since the generalised symplectic structure ω · a extends to a generalised Lagrangian (ω · a, λ · a).

The complex \underline{Hom}_{F(𝒪̃_{Y} )}((ℒ_{1}, Δ_{1}), (ℒ_{2}, Δ_{2})) is effectively the Hochschild complex of 𝒪̃_{Y} with coefficients in the bimodule 𝒟_{𝒪_{Y}} (Rϕ_{1∗}ℒ_{1}, Rϕ_{2∗}ℒ_{2}), so can be thought of as a model for

R\underline{Hom}_{𝒪̃_{Y} ˆ⊗_{R}𝒟_{Y}^{opp}} (Rϕ_{1†}(ℒ̃_{1} ˆ⊗𝒟_{X_{1}}), Rϕ_{2†}(ℒ̃_{2} ˆ⊗𝒟_{X_{2}})),

Since we are permitting all derived Lagrangians to give rise to elements of F(𝒪̃_{Y} ), we cannot expect all morphisms in this dg category to be related to vanishing cycles. However, when Grothendieck–Verdier duality applies (such as for proper morphisms) we have the following:

Lemma 5.3. When the functor Rϕ_{1∗} has a derived right adjoint ϕ_{1}^{!} on quasi-coherent complexes given by ϕ_{1}^{!}ℱ = ϕ_{1}^{∗}ℱ ⊗ K_{X_{1}}[−d], then the complex \underline{Hom}_{F(𝒪̃_{Y} )}((ℒ_{1}, Δ_{1}), (ℒ_{2}, Δ_{2}))[d] is given by derived global sections of a deformation over R⟦ℏ⟧ of the self-dual line bundle

ϕ_{2}^{∗}ℒ_{1} ⊗ ϕ_{1}^{∗}ℒ_{2}

on X_{1} ×_{Y}^{h} X_{2}.

Proof. By definition, \underline{Hom}_{F(𝒪̃_{Y} )}((ℒ_{1}, Δ_{1}), (ℒ_{2}, Δ_{2})) reduces modulo ℏ to the complex

RΓ(Y, T̂ot CC_{R}(𝒪_{Y} , Diff_{𝒪_{Y}} (Rϕ_{1∗}ℒ_{1}, Rϕ_{2∗}ℒ_{2}))).

We now observe that the inclusion

RHom_{𝒪_{Y}} (Rϕ_{1∗}ℒ_{1}, Rϕ_{2∗}ℒ_{2}) → Diff_{𝒪_{Y}} (Rϕ_{1∗}ℒ_{1}, Rϕ_{2∗}ℒ_{2})

naturally extends to a morphism

RHom_{𝒪_{Y}} (Rϕ_{1∗}ℒ_{1}, Rϕ_{2∗}ℒ_{2}) → \underline{Hom}_{F(𝒪̃_{Y} )}((ℒ_{1}, Δ_{1}), (ℒ_{2}, Δ_{2})).

This is a quasi-isomorphism, because

𝒪_{Y} → CC_{R}(𝒪_{Y} , Diff_{𝒪_{Y}} (𝒪_{Y}))

is a quasi-isomorphism as in Remark 2.12.

Finally, we have

RHom_{𝒪_{Y}} (Rϕ_{1∗}ℒ_{1}, Rϕ_{2∗}ℒ_{2}) ≃ R\underline{Hom}_{𝒪_{X_{1}×^{h}_{Y} X_{2}}} (ϕ_{2}^{∗}ℒ_{1}, ϕ_{1}^{!}ℒ_{2})
≃ R\underline{Hom}_{𝒪_{X_{1}×^{h}_{Y} X_{2}}} (ϕ_{2}^{∗}ℒ_{1}, ϕ_{1}^{∗}ℒ_{2} ⊗ K_{X_{1}}[−d])
≃ ϕ_{2}^{∗}ℒ_{1} ⊗ ϕ_{1}^{∗}ℒ_{2}[−d],

because self-duality of ℒ_{1} says ℋ om_{𝒪_{X_{1}}} (ℒ_{1}, K_{X_{1}}) ≃ ℒ_{1}. Since X_{1} is Lagrangian, ϕ_{1}^{∗}K_{X} is trivial, so ϕ_{2}^{∗}ℒ_{1}⊗ϕ_{1}^{∗}ℒ_{2} is indeed self-dual with respect to K_{X_{1}×_{Y}^{h} X_{2}} = ϕ_{2}^{∗}K_{X_{1}} ⊗ ϕ_{1}^{∗}K_{X_{2}}. □

Remarks 5.4. The argument sketched in [Saf1, Proposition 5.6 and Theorem 5.7] gives a map

QP(Y, ℒ_{1}; 0) ×_{(−)^{opp},QP(Y,0)}^{h} QP(Y, ℒ_{1}; 0) → QP(ϕ_{2}^{∗}ℒ_{1} ⊗ ϕ_{1}^{∗}ℒ_{2}, −1)
(𝒪̃_{Y}^{opp} , ℒ̃_{1} ; 𝒪̃_{Y} , ℒ̃_{2} ) ↦ ℒ̃_{1} ⊗L ; 𝒪̃_{Y} ℒ̃_{2} ,

by ensuring that the differential on the right hand side is a power series of differential operators of the correct orders on X_{1} ×_{Y}^{h} X_{2}, via properties of the shuffle product. Existence of this map would also be an immediate consequence of the k = 0 analogue of Conjecture 4.18, since it would allow us to regard 𝒪̃_{Y} as an E_{1}-algebra in 𝔹𝔻_{0}-algebras, with the bar construction realising the derived tensor product as a twisted 𝔹𝔻_{0}-algebra.

Thus the deformation in Lemma 5.3 is an E_{0}-quantisation in the sense of [Pri3], i.e. an element of QP(ϕ_{2}^{∗}ℒ_{1} ⊗ ϕ_{1}^{∗}ℒ_{2}, −1); it will also be non-degenerate and self-dual.

Furthermore, there is a generalisation of Lemma 5.3 to Lagrangian correspondences. Say we had non-degenerate quantisations 𝒪̃_{Y} , 𝒪̃_{Z} of derived Artin N-stacks Y, Z, a morphism ψ: T → Y × Z, and a self-dual line bundle ℳ on T with a quantisation Δ ∈ QP(Y × Z, ℳ; 0)^{nondeg,sd} lifting the quantisation 𝒪̃_{Y} ⊗ 𝒪̃_{Z} of Y × Z.

The definition of \underline{Hom}_{F(𝒪̃_{Y} )} then adapts to give a a dg functor from F(𝒪̃_{Y} )^{opp} to 𝒪̃_{Z} ˆ⊗_{R}𝒟_{Z}^{opp}-modules, roughly given by

R\underline{Hom}_{𝒪̃_{Y} ˆ⊗𝒟_{Y}^{opp}} (−, (pr_{Y} ψ)_{†}(ℳ̃⊗ 𝒟_{Y ×Z})).

After a shift, this should also give a dg functor F(𝒪̃_{Y} )^{opp} → F(𝒪̃_{Z} ), at least after restricting to proper objects and provided T is proper over Y.

Similarly, for a quantisation of ψ lifting 𝒪̃_{Y}^{opp} ⊗ 𝒪̃_{Z} , we should have a dg functor F(𝒪̃_{Y} ) → F(𝒪̃_{Z} ) given by taking the tensor product with 𝒪̃_{T} over 𝒪̃_{Y} ˆ⊗𝒟_{Y}^{opp}.

5.2. Local quantisations of Lagrangians. The Fukaya category envisaged in [BF, §5.3] had an object for each local system on a Lagrangian submanifold L. By contrast, the category outlined in [BBD^{+}, Remark 6.15] only had one object for each square root of K_{L}. Our approach in Definition 5.1 is closest to [KS], which considers all simple DQ modules supported on smooth Lagrangians. Once we have fixed our quantisation 𝒪̃_{Y} in QP(Y, 0)^{nondeg,sd} and a compatible Lagrangian in (ω, λ) ∈ Lag(Y, X; 0), the homotopy fibre of

QP(Y, ℒ; 0)^{nondeg,sd} → Lag(Y, X; 0) ×_{Lag(Y,∅;0)}^{h} QP(Y, 0)^{nondeg,sd}

over (𝒪̃_{Y} , λ) parametrises self-dual 𝒪̃_{Y} -module quantisations of the line bundle ℒ on the Lagrangian (X, λ). We now explain how this homotopy fibre can be regarded as a torsor for the group of self-dual rank 1 local systems, so comes close to the intention of [BF].

By Theorem 4.16, components of the homotopy fibre are a torsor for the even de Rham power series

H^{1}(F^{2}DR(X/R))^{nondeg} × ∏_{i>0} H^{1}DR(X/R)ℏ^{2i},

although the parametrisation depends on w ∈ Levi_{GT}^{t}.

As in [Pri3, Remark 4.3], quantisations (ℒ ⟦ℏ⟧, δ + Δ) of ℒ correspond to deformations ℰ_{ℏ} := (ℒ ⊗_{𝒪_{X}}^{L} 𝒟_{X}⟦ℏ⟧, δ + Δ · {−}) of ℒ ⊗_{𝒪_{X}}^{L} 𝒟_{X} as a right 𝒟_{X}-module. Other deformations of this form can be obtained by tensoring with deformations 𝒪′_{ℏ} of 𝒪_{X}⟦ℏ⟧ as a left 𝒟_{X}-module. When ℒ^{⊗2} = K_{X}, the self-duality condition for ℰ_{ℏ} is

ℰ_{−ℏ} ≃ Rℋ om_{𝒟_{X}^{opp}⟦ℏ⟧} (ℰ_{ℏ}, 𝒟_{X}⟦ℏ⟧) ⊗_{𝒪_{X}} K_{X}

as right 𝒟_{X}⟦ℏ⟧-modules. The condition for 𝒪′_{ℏ} ⊗ ℰ_{−ℏ} to also be self-dual is then

𝒪′_{−ℏ} ≃ Rℋ om_{𝒪_{X}⟦ℏ⟧} (𝒪′_{ℏ}, 𝒪_{X}⟦ℏ⟧)

as left 𝒟_{X}-modules.

The parametrisation in terms of de Rham cohomology strongly suggests that the homotopy fibre above is a torsor for this group of self-dual rank 1 local systems, and this is the original motivation behind the construction of μ in [Pri3]. Making this precise would be quite cumbersome, so we give a statement which implies it will be true on formal neighbourhoods:

Lemma 5.5. If R = H_{0}R and H^{1}DR(X/R) = 0, then the non-empty homotopy fibres of

QP(Y, X; 0)^{nondeg,sd} → Lag(Y, X; 0) ×_{Lag(Y,∅;0)}^{h} QP(Y, 0)^{nondeg,sd}

are connected, the space of automorphisms of each quantisation being the group

{g ∈ 1 + ℏR⟦ℏ⟧ : g(ℏ)^{−1} = g(−ℏ)},

acting by scalar multiplication.

Proof. Theorem 4.16 implies that the homotopy fibre is connected, because H^{1}DR(X/R) = 0. Morphisms are ∏_{i>0} H^{0}DR(X/R)ℏ^{2i}, and we need to understand what these map to via the equivalences between generalised Lagrangians and quantisations.

If we take ℏ^{2}a(ℏ^{2}) ∈ ℏ^{2}R⟦ℏ^{2}⟧, then linearity gives

μ_{w}(ℏ^{2}a(ℏ^{2})) = ℏ^{2}a(ℏ^{2}) ∈ τ_{≥0}G^{2}T_{Δ}QP̂ol(ℒ , −1)^{sd}.

The corresponding gauge automorphism in τ_{≥0}QP(ℒ , −1)_{π}^{nondeg,sd} is then an element g with −∂_{ℏ^{−1}}(g)g^{−1} = ℏ^{2}a(ℏ^{2}), so

g = exp(∫ adℏ).

Gauge elements are thus precisely exponentials of odd power series, giving the group described above. □

Remark 5.6 (Vanishing cycles). Following Remarks 5.4, when we restrict to proper Lagrangians, morphisms in the dg category F(𝒪̃_{Y} ) are given by elements in QP(ϕ_{2}^{∗}ℒ_{1} ⊗ ϕ_{1}^{∗}ℒ_{2}, −1)^{nondeg,sd} over the canonical (−1)-shifted symplectic structure on a fibre product of Lagrangians.

On the derived critical locus i: Crit(f) → Z of f: Z → 𝔸^{1}, we know from [Pri3, Lemma 4.7] that the vanishing cycles quantisation Δ_{f} is an element of QP(i^{∗}Ω_{Z}^{d}, −1)_{λ_{f}}^{nondeg,sd} for the canonical (−1)-shifted symplectic structure λ_{f}; it gives the perverse sheaf of vanishing cycles on localising at ℏ.

Since the behaviour of μ_{w} on cohomology is independent of w, the proof of [Pri3, Lemma 4.7] even adapts to show that λ_{f} is w-compatible with Δ_{f} étale locally. In settings where the shifted Darboux theorems of [BBBBJ, BG] apply, Lemma 5.5 then allows us to regard QP(i^{∗}Ω_{Z}^{d}, −1)_{λ_{f}}^{nondeg,sd} as the space of those self-dual quantisations which correspond to vanishing cycles on formal neighbourhoods.

Note that in our approach to defining F(𝒪̃_{Y} ), it does not seem sensible to restrict to the locus of QP(Y, ℒ; 0)^{nondeg,sd} corresponding to Lagrangians rather than generalised Lagrangians under the equivalence of Theorem 4.16, because we cannot control the interaction of μ_{w} with the comparison of Lemma 5.3.

References

[BBBBJ] 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., 19:1287–1359, 2015. arXiv:1312.0090 [math.AG].

[BBD^{+}] C. Brav, V. Bussi, D. Dupont, D. Joyce, and B. Szendröi. Symmetries and stabilization for sheaves of vanishing cycles. J. Singul., 11:85–151, 2015. arXiv:1211.3259 [math.AG]. With an appendix by Jörg Schürmann.

[BF] Kai Behrend and Barbara Fantechi. Gerstenhaber and Batalin-Vilkovisky structures on Lagrangian intersections. In Yuri Tschinkel and Yuri Zarhin, editors, Algebra, Arithmetic, and Geometry, volume 269 of Progress in Mathematics, pages 1–47. Birkhuser Boston, 2009.

[BG] E. Bouaziz and I. Grojnowski. A d-shifted Darboux theorem. arXiv:1309.2197v1 [math.AG], 2013.

[BGKP] Vladimir Baranovsky, Victor Ginzburg, Dmitry Kaledin, and Jeremy Pecharich. Quantization of line bundles on lagrangian subvarieties. Selecta Math. (N.S.), 22(1):1–25, 2016. arXiv:1403.3493v2 [math.AG].

[Bra] Christopher Braun. Involutive A_{∞}-algebras and dihedral cohomology. J. Homotopy Relat. Struct., 9(2):317–337, 2014. arXiv:1209.1261v2 [math.QA].

[CG] Kevin Costello and Owen Gwilliam. Factorization algebras in quantum field theory. Vol. 2. http://people.mpim-bonn.mpg.de/gwilliam/vol2may8.pdf, to be published by C.U.P., 2016.

[CPT^{+}] D. Calaque, T. Pantev, B. Toën, M. Vaquié, and G. Vezzosi. Shifted Poisson structures and

deformation quantization. J. Topol., to appear. arXiv:1506.03699v4 [math.AG].

[GR] Dennis Gaitsgory and Nick Rozenblyum. Notes on geometric Langlands: crystals and Dmodules. arXiv:1111.2087, 2011.

[Hin] Vladimir Hinich. DG coalgebras as formal stacks. J. Pure Appl. Algebra, 162(2-3):209–250, 2001.

[Kad] T. Kadeishvili. On the cobar construction of a bialgebra. Homology Homotopy Appl., 7(2):109– 122, 2005.

[Kon] Maxim Kontsevich. Operads and motives in deformation quantization. Lett. Math. Phys., 48(1):35–72, 1999. Moshé Flato (1937–1998).

[KS] Masaki Kashiwara and Pierre Schapira. Deformation quantization modules. Astérisque, (345):xii+147, 2012.

[Lur] Jacob Lurie. Higher algebra. Available at: http://www-math.mit.edu/∼lurie/papers/HA.pdf, September 2014.

[MS] V. Melani and P. Safronov. Derived coisotropic structures II: stacks and quantization. arXiv:1704.03201 [math.AG], April 2017.

[Pri1] J. P. Pridham. Unifying derived deformation theories. Adv. Math., 224(3):772–826, 2010. arXiv:0705.0344v6 [math.AG].

[Pri2] J. P. Pridham. Presenting higher stacks as simplicial schemes. Adv. Math., 238:184–245, 2013. arXiv:0905.4044v4 [math.AG].

[Pri3] J. P. Pridham. Deformation quantisation for (−1)-shifted symplectic structures and vanishing cycles. arXiv:1508.07936v2 [math.AG], 2015.

[Pri4] J. P. Pridham. Deformation quantisation for unshifted symplectic structures on derived Artin stacks. arXiv: 1604.04458v2 [math.AG], 2016.

[Pri5] J. P. Pridham. Real non-abelian mixed Hodge structures for quasi-projective varieties: formality and splitting. Mem. Amer. Math. Soc., 243(1150):vi+178 pp., 2016. arXiv: 1104.1409v3 [math.AG].

[Pri6] J. P. Pridham. Shifted Poisson and symplectic structures on derived N-stacks. J. Topol., 10(1):178–210, 2017. arXiv:1504.01940v5 [math.AG].

[PTVV] T. Pantev, B. Toën, M. Vaquié, and G. Vezzosi. Shifted symplectic structures. Publ. Math. Inst. Hautes Études Sci., 117:271–328, 2013. arXiv: 1111.3209v4 [math.AG].

[Saf1] Pavel Safronov. Poisson reductions as a coisotropic intersection. arXiv:1509.08081v1 [math.AG], 2015.

[Saf2] Pavel Safronov. Braces and Poisson additivity. arXiv:1611.09668v1[math.AG], 2016.

[Vor] Alexander A. Voronov. Homotopy Gerstenhaber algebras. In Conférence Moshé Flato 1999, Vol. II (Dijon), volume 22 of Math. Phys. Stud., pages 307–331. Kluwer Acad. Publ., Dordrecht, 2000. arXiv:9908040 [math.QA].

[You] Justin Young. Brace bar-cobar duality. arXiv:1309.2820v1 [math.AT], 2013.