DEFORMATION QUANTISATION FOR (−1)-SHIFTED SYMPLECTIC STRUCTURES AND VANISHING CYCLES

J.P.PRIDHAM

Abstract. We construct a map from E_{0} quantisations of (−1)-shifted symplectic structures to power series in de Rham cohomology of derived Artin N-stacks. For a square root of the dualising line bundle, this gives an equivalence between even power series and self-dual quantisations. In particular, there is a canonical quantisation of any such square root, which localises to recover the perverse sheaf of vanishing cycles on derived DM stacks.

Introduction

A (−1)-shifted Poisson structure on a derived scheme X is a Maurer–Cartan element π = ∑_{i≥2} π_{i} with π_{i} ∈ Symm^{i}𝒯_{X}. For a line bundle ℒ, we define an E_{0} quantisation of (π, ℒ) to be a Maurer–Cartan element Δ = ∑_{i≥2} Δ_{i}ℏ^{i−1} ∈ 𝒟_{X}(ℒ)⟦ℏ⟧ such that Δ_{i} is a differential operator of order i lifting π_{i}.

The proof in [Pri4] of the correspondence between n-shifted symplectic and nondegenerate Poisson structures relied on the existence, for all Poisson structures π, of a map μ(−, π) from the de Rham algebra to the algebra T_{π}P̂ol(X, n) of shifted polyvectors with differential twisted by π. Since [π, −] defines a derivation from 𝒪_{X} to T_{π}P̂ol(X, n), it determines a map Ω_{X}^{1} → T_{π}P̂ol(X, n)[1], and μ(−, π) is the resulting morphism of CDGAs.

We adapt this idea to construct (Lemma 1.31), for any E_{0} quantisation Δ of a CDGA A, an A_{∞}-morphism μ(−, Δ) from the de Rham algebra DR(A) to 𝒟_{A}⟦ℏ⟧. Roughly speaking, this is an A-algebra homomorphism, with the restriction to Ω_{A}^{1} corresponding to the derivation [Δ, −]; because 𝒟_{A} is not commutative, we have to define μ explicitly on an associative algebra resolution of the de Rham algebra.

This gives rise to a notion of compatibility between E_{0} quantisations Δ and generalised pre-symplectic structures (power series ω of elements of the de Rham complex): we say that ω and Δ are compatible if

μ(ω, Δ) ≃ ℏ^{2} \frac{∂Δ}{∂ℏ}.

Proposition 1.37 shows that every non-degenerate affine quantisation Δ has a unique compatible generalised pre-symplectic structure, thus giving us a map

QP(A, −1)^{nondeg} → H^{1}(F^{2}DR(A)) × ℏH^{1}(DR(A))⟦ℏ⟧

on the space of non-degenerate E_{0} quantisations.

In fact, much more is true. We have spaces QP(A, −1)/G^{k+1} consisting of E_{0} quantisations of order k, by which we mean Maurer–Cartan elements in ∏_{j≥2}(F_{j}𝒟_{A}/F_{j−k−1})ℏ^{j−1}, for F the order filtration on 𝒟. We then have maps

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

QP(A, −1)^{nondeg}/G^{k+1} → H^{1}(F^{2}DR(A)) × ℏH^{1}(DR(A))[ℏ]/ℏ^{k},

and Proposition 1.40 shows that the resulting map

QP(A, −1)^{nondeg} → (QP(A, −1)^{nondeg}/G^{2}) × ℏ^{2}H^{1}(DR(A))⟦ℏ⟧

underlies an equivalence. Thus quantisation reduces to a first order problem.

Section 2 interposes some abstract nonsense to transfer these results from affine derived schemes to derived DM N-stacks (Propositions 2.23 and 2.20). Section 3 extends the results of §1 to the formalism of bi-CDGAs, and thus to derived Artin N-stacks (Proposition 3.14).

When ℒ is Grothendieck–Verdier self-dual (i.e. a square root of the dualising line bundle ω_{X}), or more generally whenever 𝒟(ℒ) ≃ 𝒟(ℒ)^{opp}, §4 introduces a notion of self-duality for quantisations Δ of ℒ. For self-dual quantisations, the first order obstruction vanishes, and in fact the equivalence class of such quantisations of a nondegenerate (−1)-shifted Poisson structure is canonically isomorphic to

ℏ^{2}H_{dR}^{1}(X)⟦ℏ^{2}⟧.

In particular, there is an ∞-functor from the space of (−1)-shifted symplectic structures to deformation quantisations of ℒ; for the symplectic structure on a derived critical locus, this quantisation is just given by a twisted Hodge complex, so (Proposition 4.8) localising at ℏ recovers the perverse sheaf of vanishing cycles studied in [BBD^{+}].

In §4.3, we discuss how to adapt these results to quantisation of n-shifted symplectic structures for n ≥ −2, including the crucial case n = 0. For n = −2, quantisations should arise as Maurer–Cartan elements of a BV -algebra quantisation of −2-shifted polyvectors. For n ≥ 0, formality of the E_{n+2} operad should allow the construction of a compatibility map μ from de Rham cohomology to quantised Poisson cohomology, leading to a map from non-degenerate E_{n+1}-quantisations of 𝒪_{X} to power series

H^{n+2}(F^{2}DR(A)) × ℏH^{n+2}(DR(A))⟦ℏ⟧.

Again, the only obstruction to deforming a non-degenerate Poisson structure will be first order, and we sketch a notion of self-duality under which the obstruction should vanish.

Contents

QUANTISATION FOR (−1)-SHIFTED SYMPLECTIC STRUCTURES 3

Introduction 1 1. Compatible quantisations on derived affine schemes 3 1.1. Differential operators and quantised polyvectors 3 1.2. Generalised pre-symplectic structures 9 1.3. Comparing quantisations and generalised symplectic structures 14 2. Quantisation for derived DM N-stacks 16 2.1. Quantised polyvectors for diagrams 16 2.2. Towers of obstructions 17 2.3. Descent and line bundles 19 2.4. Comparing quantisations and generalised symplectic structures 20 3. Quantisation for derived Artin N-stacks 22 3.1. Bi-CDGAs 22 3.2. Quantised polyvectors 23 3.3. Generalised symplectic structures and compatible quantisations 24 3.4. Diagrams and derived Artin stacks 25 4. Self-dual quantisations 26 4.1. Duality 27 4.2. Vanishing cycles 29 4.3. Quantisation for n-shifted symplectic structures 30 References 32

1. Compatible quantisations on derived affine schemes

Let R be a graded-commutative differential algebra (CDGA) over ℚ, and fix a CDGA A over R. We will denote the differentials on A and R by δ.

1.1. Differential operators and quantised polyvectors.

1.1.1. Differential operators.

Definition 1.1. Given A-modules M, N in cochain complexes, inductively define the filtered cochain complex Diff(M, N) = Diff_{A/R}(M, N) ⊂ \underline{Hom}_{R}(M, N) of differential operators from M to N by setting

(1) F_{0}Diff(M, N) = \underline{Hom}_{A}(M, N),

(2) F_{k+1}Diff(M, N) = \{u ∈ \underline{Hom}_{R}(M, N) : [a, u] ∈ F_{k}Diff(M, N) ∀a ∈ A\}, where [a, u] = au − (−1)^{deg a deg u}ua.

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

(The reason for the notation F is that F^{p} := F_{−p} gives a Hodge filtration.)

The definitions ensure that the associated gradeds gr_{k}^{F} Diff_{A}(M, N) have the structure of A-modules. Also note that for any u ∈ F_{k+1}Diff(M, N) the commutator [u, −] defines a derivation from A to gr_{k}^{F} Diff(M, N), giving an A-linear map

gr_{k+1}^{F}Diff(M, N) → \underline{Hom}_{A}(Ω_{A}^{1}, gr_{k}^{F} Diff(M, N)).

Proceeding inductively and considering the symmetries involved, this gives maps

gr_{k}^{F} Diff(M, N) → \underline{Hom}_{A}(M ⊗_{A} CoS_{A}^{k}Ω_{A}^{1}, N)

for all k. [Here, CoS_{A}^{p}(M) = CoSymm_{A}^{p}(M) = (M^{⊗_{A}p})^{Σ_{p}} and CoSymm_{A}(M) = ⨁_{p≥0} CoS_{A}^{p}(M).]

These maps will be isomorphisms whenever A is semi-smooth in the sense that the underlying graded algebra A^{#} is isomorphic to (R^{#} ⊗_{R^{0}} S)[P^{#}], for S a smooth R^{0}- algebra and P^{#} a graded projective module over R^{#}⊗_{R^{0}}S (in particular, if A is cofibrant as a CDGA over R), and M^{#} is projective over A^{#}.

Also observe that for A-modules M, N, P, the composition map \underline{Hom}_{R}(N, P) ⊗_{R} \underline{Hom}_{R}(M, N) ⟶^{∘} \underline{Hom}_{R}(M, P) restricts to give F_{l}Diff_{A/R}(N, P)⊗_{R} F_{k}Diff_{A/R}(M, N) → F_{k+l}Diff_{A/R}(M, P).

Definition 1.2. Given an A-module M in cochain complexes, write 𝒟(M) = 𝒟_{A/R}(M) := Diff_{A/R}(M, M), which we regard as a DGAA under the composition above. We simply write 𝒟_{A} = 𝒟_{A/R} for 𝒟_{A/R}(A, A).

For A-modules M, N, inclusion in Hom_{R}(M, N) gives a natural map \underline{Hom}_{A}(M, (N ⊗_{A} 𝒟_{A})^{r}) → Diff(M, N), where (−)^{r} denotes the right A-module structure. This will be an isomorphism whenever A is semi-smooth over R (in particular, if A is cofibrant as a CDGA over R).

1.1.2. Polyvectors. The following is adapted from [Pri4, Definition 1.1], with the introduction of a dummy variable ℏ of cohomological degree 0.

Definition 1.3. Define the complex of (−1)-shifted polyvector fields on A by

P̂ol(A/R, −1) := ∏_{p≥0} \underline{Hom}_{A}(CoS_{A}^{p}(Ω_{A/R}^{1}), A)ℏ^{p−1}.

with graded-commutative multiplication (a, b) ↦ ℏab following the usual conventions for symmetric powers.

The Lie bracket on Hom_{A}(Ω_{A/R}^{1}, A) then extends to give a bracket (the Schouten– Nijenhuis bracket)

[−, −]: P̂ol(A/R, −1) × P̂ol(A/R, −1) → P̂ol(A/R, −1),

determined by the property that it is a bi-derivation with respect to the multiplication operation.

Thus P̂ol(A/R, −1) has the natural structure of a P_{1}-algebra (i.e. a Poisson algebra), and in particular P̂ol(A/R, −1) is a differential graded Lie algebra (DGLA) over R.

Note that the differential δ on P̂ol(A/R, −1) can be written as [δ, −], where δ ∈ P̂ol(A/R, −1)^{1} is the element defined by the derivation δ on A.

Strictly speaking, P̂ol is the complex of multiderivations, as polyvectors are usually defined as symmetric powers of the tangent complex. The two definitions agree (modulo completion) whenever the tangent complex is perfect, and Definition 1.3 is the more natural object when the definitions differ.

Definition 1.4. Define a decreasing filtration F on P̂ol(A/R, −1) by

F^{i}P̂ol(A/R, −1) := ∏_{j≥i} \underline{Hom}_{A}(CoS_{A}^{j}Ω_{A/R}^{1}, A)ℏ^{j−1};

this has the properties that P̂ol(A/R, −1) = \underleftarrow{lim}_{i} P̂ol(A/R, −1)/F^{i}, with [F^{i}, F^{j}] ⊂ F^{i+j−1}, δF^{i} ⊂ F^{i}, and F^{i}F^{j} ⊂ ℏ^{−1}F^{i+j}.

Observe that this filtration makes F^{2}P̂ol(A/R, −1) into a pro-nilpotent DGLA.

Definition 1.5. Define the tangent DGLA of polyvectors by

T P̂ol(A/R, −1) := P̂ol(A/R, −1) ⊕ ∏_{p≥0} \underline{Hom}_{A}(CoS_{A}^{p}(Ω_{A/R}^{1}), A)ℏ^{p}ϵ,

for ϵ of degree 0 with ϵ^{2} = 0. The Lie bracket is given by [u + vϵ, x + yϵ] = [u, x] + [u, y]ϵ + [v, x]ϵ.

Definition 1.6. Given a Maurer–Cartan element π ∈ MC(F^{2}P̂ol(A/R, −1)), define

T_{π}P̂ol(A/R, −1) := ∏_{p≥0} \underline{Hom}_{A}(CoS_{A}^{p}(Ω_{A/R}^{1}), A)ℏ^{p},

with derivation δ + [π, −] (necessarily square-zero by the Maurer–Cartan conditions).

The product on polyvectors makes this a CDGA (with no need to rescale the product by ℏ), and it inherits the filtration F from P̂ol.

Given π ∈ MC(F^{2}P̂ol(A/R, −1)/F^{p}), we define T_{π}P̂ol(A/R, n)/F^{p} similarly. This is a CDGA because F^{i} · F^{j} ⊂ F^{i+j}.

Regarding T_{π}P̂ol(A/R, −1) as an abelian DGLA, observe that MC(T_{π}P̂ol(A/R, −1)) is just the fibre of MC(T P̂ol(A/R, −1)) → MC(P̂ol(A/R, −1)) over π. Evaluation at ℏ = 1 gives an isomorphism from T P̂ol(A/R, −1) to the DGLA P̂ol(A/R, −1) ⊗_{ℚ} ℚ[ϵ] of [Pri4], and the map σ of [Pri4, Definition 1.11] then becomes:

Definition 1.7. Define

σ = −∂_{ℏ^{−1}} : P̂ol(A/R, −1) → T P̂ol(A/R, −1)

by α ↦ α + ϵℏ^{2} \frac{∂α}{∂ℏ}. Note that this is a morphism of filtered DGLAs, so gives a map MC(F^{2}P̂ol(A/R, −1)) → MC(F^{2}T P̂ol(A/R, −1)), with σ(π) ∈ Z^{1}(F^{2}T_{π}P̂ol(A/R, −1)).

1.1.3. Quantised (−1)-shifted polyvectors.

Definition 1.8. Define a strict line bundle over A to be an A-module M in cochain complexes such that M^{#} is a projective module of rank 1 over the graded-commutative algebra A^{#} underlying A. Given b ∈ Z^{1}A, define A_{b} to be the strict line bundle (A, δ+b).

(When A has elements of positive degree, note that M might not be cofibrant in the projective model structure, but this does not affect anything.)

Definition 1.9. Given a strict line bundle M over A, define the complex of quantised (−1)-shifted polyvector fields on M by

QP̂ol(M, −1) := ∏_{p≥0} F_{p}𝒟_{A/R}(M)ℏ^{p−1}.

Multiplication of differential operators gives us a product

QP̂ol(M, −1) × QP̂ol(M, −1) → ℏ^{−1}QP̂ol(M, −1),

but the associated commutator [−, −] takes values in QP̂ol(M, −1), so QP̂ol(M, −1) is a DGLA over R.

Note that the differential δ on QP̂ol(M, −1) can be written as [δ_{M}, −], where δ_{M} ∈ F_{1}𝒟_{A/R}(M)^{1} is the element defined by the differential δ_{M} on M. In particular, QP̂ol(A_{b}, −1) is the graded associative algebra QP̂ol(A, −1) with differential [δ_{A} +b, −].

Definition 1.10. Define a decreasing filtration F̃ on QP̂ol(M, −1) by

F̃^{i}QP̂ol(M, −1) := ∏_{j≥i} F_{j}𝒟_{A/R}(M)ℏ^{j−1};

this has the properties that QP̂ol(M, −1) = \underleftarrow{lim}_{i} P̂ol(M, −1)/F̃^{i}, with [F̃^{i}, F̃^{j}] ⊂ F̃^{i+j−1}, δF̃^{i} ⊂ F̃^{i}, and F̃^{i}F̃^{j} ⊂ ℏ^{−1}F̃^{i+j}.

Observe that this filtration makes F̃^{2}QP̂ol(M, −1) into a pro-nilpotent DGLA.

Definition 1.11. When A is cofibrant, we define an E_{0} quantisation of M over R to be a Maurer–Cartan element

Δ ∈ MC(F̃^{2}QP̂ol(M, −1)).

The associated R⟦ℏ⟧-module M_{Δ} is given by M⟦ℏ⟧ equipped with the differential δ_{M} + Δ, which is necessarily square-zero by the Maurer–Cartan condition. We then have M_{Δ}/(ℏM_{Δ}) = M, because ℏ | Δ.

Remark 1.12. A more conceptual way to interpret such E_{0} quantisations is as deformations of M as a module over the de Rham pro-algebra DR(A/R). Such a deformation of M is the same as a deformation of the right 𝒟_{A}-module M ⊗_{A}𝒟_{A}, and a Maurer–Cartan element Δ gives a deformation x ↦ δ(x) + Δ · x of the differential δ on M ⊗_{A} 𝒟_{A}⟦ℏ⟧.

Equivalently, a (−1)-shifted Poisson structure on 𝒪_{X} is the structure of an 𝒪_{X}⟦Ω_{X}^{1}[−1]⟧ = 𝒪̂_{T X[1]} -module, and an E_{0} quantisation is a lifting of such a structure making 𝒪_{X}⟦ℏ⟧ a module over the Rees pro-algebra ℛ := ∏_{p∈ℤ} ℏ^{−p}F^{p}DR(A/R), via the isomorphism ℛ/ℏℛ ≅ 𝒪̂_{T X[1]} .

In many ways, deformations over DR(A/R) are more natural than R-module deformations, because the de Rham algebra is the natural algebraic analogue of the analytic sheaf of complex constants. For n > 0, E_{n}-algebra deformations over R and over DR(A/R) are the same by the HKR isomorphism, but for n = 0 the DR(A/R)-module structure imposes the condition that deformations be given by differential operators.

Remark 1.13. Observe that when the E_{0} quantisation Δ is linear in ℏ, it is a secondorder differential operator. When M = A and Δ(1) = 0, this gives exactly the structure of a BV -algebra over R⟦ℏ⟧, the associated Lie bracket being given by the image of Δ in ℏ(F_{2}𝒟_{A}/F_{1}𝒟_{A}) ≅ ℏSymm_{A}^{2}𝒯_{A}. In general, for any E_{0} quantisation Δ of A with Δ(1) = 0, the pair (A, Δ) is a commutative BV_{∞}-algebra in the sense of [Kra, Definition 9].

Remark 1.14. For unbounded CDGAs, the hypothesis that A be cofibrant seems unnecessarily strong. Most of our results will hold when A^{#} is free or even when A is semi-smooth. This suggests that the most natural notion of equivalence for CDGAs in this setting might not be quasi-isomorphism, but Morita equivalence of derived categories of the second kind (in the sense of [Pos]). Dealing with semi-smooth CDGAs might provide an alternative approach to the bi-CDGAs featuring in §3 to study Artin stacks.

Definition 1.15. 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.

Definition 1.16. We now define another decreasing filtration G on QP̂ol(M, −1) by setting

G^{i}QP̂ol(M, −1) := ℏ^{i}QP̂ol(M, −1).

We then set G^{i}F̃^{p} := G^{i} ∩ F̃^{p}.

Note that G^{i} ⊂ F̃^{i}, and beware that G^{i}F̃^{p} is not the same as ℏ^{i}F̃^{p} in general, since

G^{i}F̃^{p}QP̂ol(M, −1) = ∏_{j≥p} F_{j−i}𝒟_{A/R}(M)ℏ^{j−1}
ℏ^{i}F̃^{p}QP̂ol(M, −1) = ∏_{j≥p+i} F_{j−i}𝒟_{A/R}(M)ℏ^{j−1}.

We will also consider the convolution G ∗ F̃, given by (G ∗ F̃)^{p} := ∑_{i+j=p} G^{i} ∩ F̃^{j} ; explicitly,

(G ∗ F̃)^{p}QP̂ol(M, −1) = ∏_{j<p} F_{2j−p}𝒟_{A/R}(M)ℏ^{j−1} × ∏_{j≥p} F_{j}𝒟_{A/R}(M)ℏ^{j−1}.

In particular, (G ∗ F̃)^{2}QP̂ol(M, −1) = A ⊕ F̃^{2}QP̂ol(M, −1).

Definition 1.17. Define the space QP(M, −1) of E_{0} quantisations of M over R to be given by the simplicial set

QP(M, −1) := \underleftarrow{lim}_{i} \underline{MC}(F̃^{2}QP̂ol(M, −1)/F̃^{i+2}).

Also write

QP(M, −1)/G^{k} := \underleftarrow{lim}_{i} \underline{MC}(F̃^{2}QP̂ol(M, −1)/(F̃^{i+2} + G^{k})),

so QP(M, −1) = \underleftarrow{lim}_{k} QP(M, −1)/G^{k}.

We will also consider twisted quantisations

Q^{tw}P(M, −1) := \underleftarrow{lim}_{i} \underline{MC}((G ∗ F̃)^{2}QP̂ol(M, −1)/F̃^{i+2});

these are just quantisations of strict line bundles M ⊗_{A} A_{b} for b ∈ Z^{1}(A).

1.1.4. The centre of a quantisation.

Definition 1.18. Define the filtered tangent DGLA of quantised polyvectors by

T QP̂ol(M, −1) := QP̂ol(M, −1) ⊕ ∏_{p≥0} F_{p}𝒟_{A/R}(M)ℏ^{p}ϵ,
F̃^{j}T QP̂ol(M, −1) := F̃^{j}QP̂ol(M, −1) ⊕ ∏_{p≥j} F_{p}𝒟_{A/R}(M)ℏ^{p}ϵ,

for ϵ of degree 0 with ϵ^{2} = 0. The Lie bracket is given by [u + vϵ, x + yϵ] = [u, x] + [u, y]ϵ + [v, x]ϵ.

Definition 1.19. Given a Maurer–Cartan element Δ ∈ MC(F̃^{2}QP̂ol(M, −1)), define the centre of (M, Δ) by

T_{Δ}QP̂ol(M, −1) := ∏_{p≥0} F_{p}𝒟_{A/R}(M)ℏ^{p},

with derivation δ + [Δ, −] (necessarily square-zero by the Maurer–Cartan conditions).

Multiplication of differential operators makes this a DGAA (with no need to rescale the product by ℏ), and it has a filtration

F̃^{i}T_{Δ}QP̂ol(M, −1) := ∏_{p≥i} F_{p}𝒟_{A/R}(M)ℏ^{p},

with F̃^{i} · F̃^{j} ⊂ F̃^{i+j}. Given Δ ∈ MC(F^{2}QP̂ol(M, −1)/F̃^{p}), we define T_{Δ}QP̂ol(M, −1)/F̃^{p} similarly — this is also a DGAA as F̃^{p} is an ideal.

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

Similarly to Definition 1.16, there is a filtration G on T QP̂ol(M, −1), T_{Δ}QP̂ol(M, −1) given by powers of ℏ. Since gr_{G}^{i}F̃^{p−i}QP̂ol = ∏_{j≥p−i} gr_{j−i}^{F}𝒟_{A/R}(M)ℏ^{j−1}, the associated gradeds of the filtration admit maps

gr_{G}^{i}F̃^{p}QP̂ol(M, −1) → ∏_{j≥p} \underline{Hom}_{A}(CoS_{A}^{j−i}(Ω_{A/R}^{1}) ⊗_{A} M, M)ℏ^{j−1}
gr_{G}^{i}F̃^{p}T_{Δ}QP̂ol(M, −1) → ∏_{j≥p} \underline{Hom}_{A}(CoS_{A}^{j−i}(Ω_{A/R}^{1}) ⊗_{A} M, M)ℏ^{j}.

which are isomorphisms when A is semi-smooth (in particular whenever A is cofibrant as a CDGA over R).

For the filtration F of Definition 1.4, we may rewrite these maps as

gr_{G}^{i}F̃^{p}QP̂ol(A, −1) → F^{p−i}P̂ol(A, −1)ℏ^{i},
gr_{G}^{i}F̃^{p}T_{Δ}QP̂ol(M, −1) → F^{p−i}T_{π_{Δ}} P̂ol(A, −1)ℏ^{i},

where π_{Δ} ∈ MC(F^{2}P̂ol(A, −1)) denotes the image of Δ under the map gr_{G}^{0}F̃^{2}QP̂ol(A, −1) → F^{2}P̂ol(A, −1).

Since the cohomology groups of T_{π_{Δ}} P̂ol(A, −1) are Poisson cohomology, we will refer to the cohomology groups of T_{Δ}QP̂ol(M, −1) as quantised Poisson cohomology.

We write T_{Δ}Q^{tw}P(M, −1) := \underline{MC}((G ∗ F̃)^{2}T_{Δ}QP̂ol(M, −1)).

Definition 1.20. Say that an E_{0} quantisation Δ = ∑_{j≥2} Δ_{j}ℏ^{j} is non-degenerate if the map

Δ_{2}^{♯}: M ⊗_{A} Ω_{A}^{1} → \underline{Hom}_{A}(Ω_{A}^{1}, M)[1]

is a quasi-isomorphism and Ω_{A}^{1} is perfect.

Definition 1.21. Define the tangent spaces

T QP(M, −1) := \underleftarrow{lim}_{i} \underline{MC}(F̃^{2}T QP̂ol(M, −1)/F̃^{i+2}),
T Q^{tw}P(M, −1) := \underleftarrow{lim}_{i} \underline{MC}((G ∗ F̃)^{2}T QP̂ol(M, −1)/F̃^{i+2}),

with T QP(M, −1)/G^{k}, T Q^{tw}P(M, −1)/G^{k} defined similarly.

These are simplicial sets over QP(M, −1) (resp. Q^{tw}P(M, −1), QP(M, −1)/G^{k}, Q^{tw}P(M, −1)/G^{k}), fibred in simplicial abelian groups.

Definition 1.22. Define the canonical tangent vector

σ = −∂_{ℏ^{−1}} : QP̂ol(M, −1) → T QP̂ol(M, −1)

by α ↦ α + ϵℏ^{2} \frac{∂α}{∂ℏ}. Note that this is a morphism of filtered DGLAs, so gives a map σ: QP(M, −1) → T QP(M, −1), with σ(Δ) ∈ Z^{1}(F̃^{2}T_{Δ}QP̂ol(M, −1)).

1.2. Generalised pre-symplectic structures.

Definition 1.23. Define the de Rham complex DR(A/R) to be the product total complex of the bicomplex

A ⟶^{d} Ω_{A/R}^{1} ⟶^{d} Ω_{A/R}^{2} ⟶^{d} …,

so the total differential is d ± δ.

We define the Hodge filtration F on DR(A/R) by setting F^{p}DR(A/R) ⊂ DR(A/R) to consist of terms Ω_{A/R}^{i} with i ≥ p. In particular, F^{p}DR(A/R) = DR(A/R) for p ≤ 0.

Definition 1.24. When A is a cofibrant CDGA over R, recall that a (−1)-shifted pre-symplectic structure ω on A/R is an element

ω ∈ Z^{1}F^{2}DR(A/R).

In [PTVV], shifted pre-symplectic structures are referred to as closed 2-forms.

A (−1)-shifted pre-symplectic structure ω is called symplectic if ω_{2} ∈ Z^{1}Ω_{A/R}^{2} induces a quasi-isomorphism

ω_{2}^{♯}: Hom_{A}(Ω_{A/R}^{1}, A) → Ω_{A/R}^{1}[−1],

and Ω_{A/R}^{1} is perfect as an A-module.

In order to define compatibility functors for quantisations, we will need to construct A_{∞}-morphisms from the de Rham algebra, which we will do using the following DGAA resolution.

Definition 1.25. Write A^{⊗•+1} for the cosimplicial CDGA n ↦ A^{n+1} given by the Čech nerve, with I the kernel of the diagonal map A^{⊗•+1} → A. This has a filtration F given by powers F^{p} := (I)^{p} of I, and we define the filtered cosimplicial CDGA Â^{⊗•+1} to be the completion

Â^{⊗•+1} := \underleftarrow{lim}_{q} A^{⊗•+1}/F^{q},
F^{p}Â^{⊗•+1} := \underleftarrow{lim}_{q} F^{p}/F^{q}.

We then take the Dold–Kan conormalisation NÂ^{•+1}, which becomes a filtered bi-DGAA via the Alexander–Whitney cup product. Explicitly, N^{n}Â^{•+1} is the intersection of the kernels of all the big diagonals Â^{n+1} → Â^{n}, and the cup product is given by

(a_{0} ⊗ … ⊗ a_{m}) ⌣ (b_{0} ⊗ … ⊗ b_{n}) = a_{0} ⊗ … ⊗ a_{m−1} ⊗ (a_{m}b_{0}) ⊗ b_{1} ⊗ … ⊗ b_{n}.

We then define DR′(A/R) to be the product total complex

DR′(A/R) := Tot^{Π}NÂ^{•+1}

regarded as a filtered DGAA over R, with F^{p}DR′(A/R) := Tot^{Π}NF^{p}Â^{•+1}.

The following is standard:

Lemma 1.26. There is a filtered quasi-isomorphism DR′(A/R) → DR(A/R), given by N^{n}Â^{•+1} → N^{n}Â^{•+1}/F^{n+1} ≅ (Ω_{A/R}^{1})^{⊗_{A}n} → Ω_{A/R}^{n}.

Proof. It suffices to show that the map is an isomorphism on the graded pieces associated to the filtration. Now, gr_{F}^{p} Â^{⊗•+1} = Symm_{A}^{p}D(Ω_{A}^{1}[−1]), where D denotes Dold–Kan denormalisation from cochain complexes to cosimplicial complexes. Thus

Ngr_{F}^{p} Â^{⊗•+1} = NSymm_{A}^{p}D(Ω_{A}^{1}[−1]),

so Tot^{Π}NSymm_{A}^{p}D(Ω_{A}^{1}[−1]) is quasi-isomorphic to Symm_{A}^{p}ND(Ω_{A}^{1}[−1]) = Symm_{A}^{p}(Ω_{A}^{1}[−1]) = Ω_{A}^{p}[−p], combining Dold–Kan with Eilenberg–Zilber. □

Definition 1.27. 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},

where we adopt the convention that F^{j}DR′ = DR′ for all j ≤ 0.

Define further filtrations G, G ∗ F̃ by G^{k}DR′(A/R)⟦ℏ⟧ = ℏ^{k}DR′(A/R)⟦ℏ⟧, and (G ∗ F̃)^{p} := ∑_{i+j=p} G^{i} ∩ F̃^{j}, so

(G ∗ F̃)^{p} = ∏_{i≥0} F^{p−2i}DR′(A/R)ℏ^{i}.

This makes (DR′(A/R)⟦ℏ⟧, G ∗ F̃) into a filtered DGAA, since F̃^{p}F̃^{q} ⊂ F̃^{p+q} and similarly for G.

Definition 1.28. Define a generalised (−1)-shifted pre-symplectic structure on a cofibrant CDGA A/R to be an element

ω ∈ Z^{1}((G ∗ F̃)^{2}DR′(A/R)⟦ℏ⟧) = Z^{1}(F^{2}DR′(A/R)) × ℏZ^{1}DR′(A/R)⟦ℏ⟧.

Call this symplectic if Ω_{A/R}^{1} is perfect as an A-module and the leading term ω_{0} ∈ Z^{1}F^{2}DR′(A/R) induces a quasi-isomorphism

[ω_{0}]^{♯}: Hom_{A}(Ω_{A/R}^{1}, A) → Ω_{A/R}^{1}[−1],

for [ω_{0}] ∈ Z^{−1}Ω_{A/R}^{2} the image of ω_{0} modulo F^{3}.

Definition 1.29. Define the space of generalised (−1)-shifted pre-symplectic structures on A/R to be the simplicial set

GPreSp(A/R, −1) := \underleftarrow{lim}_{i} \underline{MC}((G ∗ F̃)^{2}DR′(A/R)⟦ℏ⟧/F̃^{i+2}),

where we regard the cochain complex DR′(A/R) as a DGLA with trivial bracket. Write PreSp = GPreSp/G^{1}.

Also write GPreSp(A/R, −1)/ℏ^{k} := \underleftarrow{lim}_{i} \underline{MC}(((G ∗ F̃)^{2}DR′(A/R)[ℏ]/(G^{k} + F̃^{i+2)}), so GPreSp(A/R, −1) = \underleftarrow{lim}_{k} GPreSp(A/R, −1)/ℏ^{k}.

Set GSp(A/R, −1) ⊂ GPreSp(A/R, −1) to consist of the symplectic structures — this is a union of path-components.

Note that GPreSp(A/R, −1) is canonically weakly equivalent to the Dold–Kan denormalisation of the good truncation complex τ^{≤0}((G ∗ F̃)^{2}DR(A/R)⟦ℏ⟧[1]) (and similarly for the various quotients we consider), but the description in terms of \underline{MC} will simplify comparisons. In particular, we have

π_{i}GPreSp(A/R, −1) ≅ H^{1−i}(F^{2}DR(A/R)) × ℏH^{1−i}(DR(A/R))⟦ℏ⟧.

1.2.1. Compatible quantisations. We will now develop the notion of compatibility between a (truncated) generalised pre-symplectic structure and a (truncated) E_{0} quantisation. The case k = 1 recovers the notion of compatibility between pre-symplectic and Poisson structures from [Pri4].

Lemma 1.30. Take a complete filtered graded-associative R-algebra (B, Fil^{•}) and a morphism ϕ: A^{#} → Fil^{0}B of graded R-algebras; assume that the left and right Amodule structures on gr_{Fil}B agree. Then for any Δ ∈ Fil^{0}B^{1}, there is an associated morphism

μ(−, Δ): (DR′(A)^{#}, F^{•}) → (B, Fil^{•})

of filtered graded-associative R-algebras induced by the graded algebra map on A^{⊗•+1} determined by μ(1 ⊗ 1, Δ) = Δ and μ(a, Δ) = a for a ∈ A.

Given ρ ∈ Fil^{j}B^{k}, there is then a filtered R-linear derivation

ν(−, Δ, ρ): (DR′(A/R)^{#}, F^{•}) → (B[k], Fil^{•+j})

induced by the μ(−, Δ)-derivation on A^{⊗•+1} determined by ν(1 ⊗ 1, Δ, ρ) = ρ.

Proof. Explicitly, μ(−, Δ) is given on A^{⊗•+1} by

(a_{0} ⊗ a_{1} ⊗ … ⊗ a_{r}) ↦ a_{0}Δa_{1}Δ … Δa_{r},

because a ⊗ b = a ⌣ (1 ⊗ 1) ⌣ b and so on. Similarly, ν(−, Δ, ρ) is given by

(a_{0} ⊗ a_{1} ⊗ … ⊗ a_{r}) ↦ ∑_{i=0}^{r−1} ±a_{0}Δa_{1}Δ … Δa_{i}ρa_{i+1}Δ … Δa_{r}.

We need to show that these respect the filtrations, so giving maps on Â^{⊗•+1} and hence filtered morphisms on DR(A/R)′.

Observe that the filtration on A^{⊗•+1} is generated by that on A^{⊗2}, in the sense that

F^{p}A^{⊗m+1} = ∑_{p_{1}+...+p_{m}=p}(F^{p_{1}}A^{⊗2}) ⌣ … ⌣ (F^{p_{m}}A^{⊗2}).

It therefore suffices to show that μ(−, Δ): A^{⊗2} → B and ν(−, Δ, ρ) are appropriately filtered.

Writing [x, y] := x ⌣ y − (−1)^{deg x deg y}y ⌣ x and · for the internal multiplication on A^{⊗r}, it follows that for a ∈ A and x ∈ A ⊗ A we have [a, x] = (a ⊗ 1 ∓ 1 ⊗ a) · x. Since I = ker(A^{⊗2} → A) is generated by elements of the form (a ⊗ 1 ∓ 1 ⊗ a), this means that [A, J] = I · J for all ideals J in A^{⊗2}. Because F^{p} = I^{·p}, this gives

F^{p}A^{⊗2} = \underbrace{[A, [A, … [A,}^{p} A^{⊗2}] …],

so F is the smallest multiplicative filtration for which left and right A-modules structures on gr_{F} A^{⊗•+1} agree.

Therefore the algebra homomorphism μ(−, Δ) must send F^{p} to Fil^{p}, and the derivation ν(−, Δ, ρ) must send F^{p} to Fil^{p+j}; in particular, the maps descend to the completion DR′(A/R). □

Lemma 1.31. Given Δ ∈ ((G ∗ F̃)^{2}QP̂ol(A, n)/G^{k})^{1}, Lemma 1.30 gives a morphism

μ(−, Δ): DR′(A/R)[ℏ]/G^{k} → T_{Δ}QP̂ol(M, −1)/G^{k}

of graded associative R[ℏ]/ℏ^{k}-algebras, respecting the filtrations (G ∗ F̃).

Given ρ ∈ ((G ∗ F̃)^{p}P̂ol(A, n)/G^{k})^{r}, there is then a R[ℏ]/ℏ^{k}-linear derivation

ν(−, Δ, ρ): (DR′(A/R)[ℏ]/ℏ^{k}, (G ∗ F̃)^{•}) → (T_{Δ}QP̂ol(M, −1)[r]/G^{k}, (G ∗ F̃)^{•+p}).

Proof. It suffices to prove this for the limit over all k, as Δ and ρ always lift to (G ∗ F̃)^{2}QP̂ol(A, n). Set T = T_{0}QP̂ol(M, −1)[ℏ^{−1}], with filtrations F̃ given by powers of ℏ and G^{i}T := ℏ^{i}T_{0}QP̂ol(M, −1). These filtrations are multiplicative, with [G^{i}, G^{j}] ⊂ G^{i+j−1}, so the convolution filtration also satisfies [(G ∗ F̃)^{i}, (G ∗ F̃)^{j}] ⊂ (G ∗ F̃)^{i+j−1}; in particular gr_{G∗F̃}T is commutative, so its left and right A-module structures agree (the same is not true grF̃T, which makes the convolution filtration necessary).

Then QP̂ol(M, −1) = G^{−1}T and T_{0}QP̂ol(M, −1) = G^{0}T, with G^{i}F̃^{j}T = G^{i}F̃^{j}T_{0}QP̂ol(M, −1) and G^{i−1}F̃^{j−1}T = G^{i}F̃^{j}QP̂ol(M, −1) whenever i ≥ 1. Thus (G ∗ F̃)^{p}QP̂ol ⊂ (G ∗ F̃)^{p−2}T, so in particular Δ lies in (G ∗ F̃)^{0}T. Lemma 1.30 now gives filtered morphisms

μ(−, Δ): (DR′(A/R), F^{•}) → (T, (G ∗ F̃)^{•})
ν(−, Δ, ρ): (DR′(A/R), F^{•}) → (T [r], (G ∗ F̃)^{•+p}).

Since Δ ∈ G^{−1}T, we have [Δ, a] ∈ G^{0}T for all a ∈ A, so μ maps to the subalgebra T_{0}QP̂ol(M, −1) of T. Extending linearly gives a map from DR′(A/R)⟦ℏ⟧; since ℏG^{i}F̃^{q}T ⊂ G^{i+1}F̃^{q+1}T, we then see that μ(ℏ^{i}F^{p−2i}, Δ) ⊂ G^{p−r−i}F̃^{r+i}, so

μ((G ∗ F̃)^{p}DR(A/R)′⟦ℏ⟧, Δ) ⊂ (G ∗ F̃)^{p}T_{Δ}QP̂ol(M, −1),

and similarly for ν. □

Lemma 1.32. Take a complete filtered R-DGAA (B, Fil^{•}) and a morphism ϕ: A → Fil^{0}B of R-DGAAs; assume that the left and right A-module structures on gr_{Fil}B agree. Then for any Δ ∈ Fil^{0}B^{1}, and any ω ∈ DR′(A/R), we have

[Δ, μ(ω, Δ)] = μ(dω, Δ) + \frac{1}{2}ν(ω, Δ, [Δ, Δ]),

δ_{Δ}μ(ω, Δ) = μ(Dω, Δ) + ν(ω, Δ, κ(Δ)),

where δ are the structural differentials on A, B, with δ_{Δ} := δ + [Δ, −], D := d ± δ the total differential on DR′(A/R), and κ(Δ) := [δ, Δ] + \frac{1}{2}[Δ, Δ].

Proof. Both [Δ, μ(−, Δ)] and δ_{Δ}μ(−, Δ) are derivations with respect to μ(−, Δ), so it suffices to verify these identities on the generators a, df of the denormalisation of DR′(A/R), for a, f ∈ A.

In these cases, we have

[Δ, μ(a, Δ)] = [Δ, a] = μ(1 ⊗ a ∓ a ⊗ 1, Δ) = μ(da, Δ),
[Δ, μ(1 ⊗ f − f ⊗ 1, Δ)] = Δ^{2}f ∓ fΔ^{2} = \frac{1}{2}ν(df, Δ, [Δ, Δ]).

Because ν(a, Δ, [Δ, Δ]) = 0 (ν being A-linear) and ddf = 0, this gives the required results, the second set of equalities following by adding δ. □

In particular, Lemma 1.32 implies that when Δ ∈ MC((G ∗ F̃)^{2}QP̂ol(A, n)/G^{k}) is an E_{0} quantisation, μ(−, Δ) is a chain map (since T_{Δ}QP̂ol(M, −1) = (T_{0}QP̂ol(M, −1), δ_{Δ})), so μ(−, Δ) defines a map from de Rham cohomology to quantised Poisson cohomology.

Definition 1.33. We say that a generalised (−1)-shifted pre-symplectic structure ω and an E_{0} quantisation Δ of a strict line bundle M are compatible (or a compatible pair) if

[μ(ω, Δ)] = [−∂_{ℏ^{−1}}(Δ)] ∈ H^{1}((G ∗ F̃)^{2}T_{Δ}QP̂ol(M, −1)),

where σ = −∂_{ℏ^{−1}} is the canonical tangent vector of Definition 1.22.

Definition 1.34. 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 1.35. Define the space QComp(M, −1) of compatible quantised (−1)- shifted pairs to be the homotopy vanishing locus of

(μ − σ): GPreSp(A/R, −1) × QP(M, −1) → T Q^{tw}P(M, −1)

over Q^{tw}P(M, −1)

We define a cofiltration on this space by setting QComp(M, −1)/G^{k} to be the homotopy vanishing locus of

(μ − σ): (GPreSp(A/R, −1)/G^{k}) × (QP(M, −1)/G^{k}) → T Q^{tw}P(M, −1)/G^{k}

over Q^{tw}P(M, −1)/G^{k}.

When k = 1, note that this recovers the notion of compatible (−1)-shifted pairs from [Pri4].

Definition 1.36. Define QComp(M, −1)^{nondeg} ⊂ QComp(M, −1) to consist of compatible quantised pairs (ω, Δ) with Δ non-degenerate. This is a union of path-components, and by [Pri4, Lemma 1.22] has a natural map

QComp(M, −1)^{nondeg} → GSp(A/R, −1)

as well as the canonical map

QComp(M, −1)^{nondeg} → QP(M, −1)^{nondeg}.

Proposition 1.37. For any strict line bundle M, the canonical map

QComp(M, −1)^{nondeg} → QP(M, −1)^{nondeg}

is a weak equivalence. In particular, there is a morphism

QP(M, −1)^{nondeg} → GSp(A/R, −1)

in the homotopy category of simplicial sets.

Proof. We adapt the proof of [Pri4, Proposition 1.26]. For any Δ ∈ QP(M, −1), the homotopy fibre of QComp(A/R, −1)^{nondeg} over Δ is just the homotopy fibre of

μ(−, Δ): GPreSp(A/R, −1) → T_{Δ}Q^{tw}P(M, −1)

over −∂_{ℏ^{−1}}(Δ).

The map μ(−, Δ): DR′(A/R)⟦ℏ⟧ → T_{Δ}QP̂ol(M, −1) is a morphism of complete G∗F̃- filtered R⟦ℏ⟧-CDGAs by Lemma 1.32. Since the morphism is R⟦ℏ⟧-linear, it maps G^{k}(G ∗ F̃)^{p}DR′(A/R)⟦ℏ⟧ to G^{k}(G ∗ F̃)^{p}T_{Δ}QP̂ol(M, −1). Non-degeneracy of Δ_{0} modulo F_{1} implies that μ(−, Δ) induces quasi-isomorphisms

Ω^{p−2k}ℏ^{k}[2k − p] → \underline{Hom}_{A}(CoS^{p−2k}Ω_{A/R}^{1} ⊗_{A} M, M)ℏ^{p−k}

on the associated gradeds gr_{G}^{k}gr_{(G∗F̃)}^{p}. We therefore have a quasi-isomorphism of bifiltered complexes, so we have isomorphisms on homotopy groups:

π_{j}GPreSp(A/R, −1) → π_{j}T_{Δ}Q^{tw}P(M, −1)
H^{1−j}((G ∗ F̃)^{2}DR(A/R)⟦ℏ⟧) → H^{1−j}((G ∗ F̃)^{2}T_{Δ}QP̂ol(M, −1)).
□

1.3. Comparing quantisations and generalised symplectic structures.

Definition 1.38. Given a compatible pair (ω, π) ∈ Comp(A, −1) = QComp(M, −1)/G^{1}, and k ≥ 0, define the complex N(ω, π, k) to be the cocone of the map

gr_{G}^{k}(G ∗ F̃)^{2}(DR′(A/R)⟦ℏ⟧ ⊕ gr_{G}^{k}F̃^{2}QP̂ol(M, −1) → gr_{G}^{k}(G ∗ F̃)^{2}T_{π}QP̂ol(M, −1)

given by combining

gr_{G}^{k}μ(−, π): gr_{G}^{k}(G ∗ F̃)^{2}DR′(A/R)⟦ℏ⟧ → gr_{G}^{k}(G ∗ F̃)^{2}T_{π}QP̂ol(M, −1)
F^{2−2k}DR(A/R)ℏ^{k} → ∏_{i≥(2−2k),0} \underline{Hom}_{A}(CoS_{A}^{i}(Ω_{A/R}^{1}), A)ℏ^{i+k}

with the maps

gr_{G}^{k}ν(ω, π) + ∂_{ℏ^{−1}} : gr_{G}^{k}F̃^{2}QP̂ol(M, −1) → gr_{G}^{k}(G ∗ F̃)^{2}T_{π}QP̂ol(M, −1)
∏_{i≥(2−k),0} \underline{Hom}_{A}(CoS_{A}^{i}(Ω_{A/R}^{1}), A)ℏ^{i+k−1} → ∏_{i≥(2−2k),0} \underline{Hom}_{A}(CoS_{A}^{i}(Ω_{A/R}^{1}), A)ℏ^{i+k},

where

ν(ω, π)(b) := ν(ω, π, b).

It follows from the proof of Proposition 1.37 that the maps gr_{G}^{k}μ(−, π) are all F -filtered quasi-isomorphisms when π is non-degenerate, so the projection maps N(ω, π, k) → gr_{G}^{k}F̃^{2}QP̂ol(M, −1) are also quasi-isomorphisms. The behaviour of the other projection is more subtle for low k, but it behaves well thereafter:

Lemma 1.39. The projection maps

N(ω, π, k) → ℏ^{k}DR(A/R)

are F -filtered quasi-isomorphisms for all k ≥ 2.

Proof. This amounts to showing that the map

gr_{G}^{k}ν(ω, π) + ∂_{ℏ^{−1}}

is a filtered quasi-isomorphism. It suffices to show that the associated maps

gr_{F}^{p} gr_{G}^{k}ν(ω, π) + ∂_{ℏ^{−1}} : gr_{F}^{p} gr_{G}^{k}(G ∗ F̃)^{2}QP̂ol(M, −1) → gr_{F}^{p} gr_{G}^{k}(G ∗ F̃)^{2}T_{π}QP̂ol(M, −1)
\underline{Hom}_{A}(CoS_{A}^{p}(Ω_{A/R}^{1}), A)ℏ^{p+k−1} → \underline{Hom}_{A}(CoS_{A}^{p}(Ω_{A/R}^{1}), A)ℏ^{p+k},

are quasi-isomorphisms for all p ≥ 0.

Reasoning as in [Pri4, Remark 1.34], gr_{F} gr_{G}ν(ω, π) is an R⟦ℏ⟧-linear derivation on gr_{F} gr_{G}QP̂ol(M, −1) ≅ P̂ol(A, −1)⟦ℏ⟧ with respect to the commutative multiplication. It is given on generators \underline{Hom}_{A}(Ω_{A/R}^{1}, A) by ℏπ^{♯} ∘ ω^{♯}. Compatibility of ω and π implies that π^{♯} ∘ ω^{♯} is homotopy idempotent by [Pri4, Example 1.19]. Thus ℏ^{−1}gr_{F}^{p} gr_{G}^{k}ν(ω, π) is homotopy diagonalisable, with integral eigenvalues in the interval [0, p].

On the other hand, ∂_{ℏ^{−1}} coincides on gr_{F}^{p} gr_{G}^{k} with multiplication by (1 − p − k)ℏ, so the eigenvalues of ℏ^{−1}gr_{F}^{p} gr_{G}^{k}ν(ω, π) + ℏ^{−1}∂_{ℏ^{−1}} lie in [1 − p − k, 1 − k], giving a quasi-isomorphism when k > 1. □

Proposition 1.40. The maps

QP(M, −1)^{nondeg}/G^{k} → (QP(M, −1)^{nondeg}/G^{2}) ×_{(GSp(A,−1)/G^{2})}^{h} (GSp(A, −1)/G^{k})

≃ (QP(M, −1)^{nondeg}/G^{2}) × ∏_{i=2}^{k−1} \underline{MC}(DR(A/R)ℏ^{i})

coming from Proposition 1.37 are weak equivalences for all k ≥ 2.

Proof. Proposition 1.37 gives equivalences between QP^{nondeg} and QComp^{nondeg}. Fix (ω, π) ∈ Comp(A, −1) and denote homotopy fibres by subscripts. Arguing as in the proof of [Pri4, Proposition 1.35], but with abelian (rather than central) extensions of DGLAs as in [Pri2, Lemma 3.3] gives a commutative diagram

(QComp(M, −1)/G^{k+1})_{(ω,π)} ⟶ (QComp(M, −1)/G^{k})_{(ω,π)} ⟶  \underline{MC}(N(ω, π, k)[1])
↓  ↓  ↓
(GPreSp(A, −1)/G^{k+1})_{ω} ⟶ (GPreSp(A, −1)/G^{k})_{ω} ⟶ \underline{MC}(F^{2−2k}ℏ^{k}DR(A/R)[1])

of fibre sequences.

The right-hand map is a weak equivalence for k ≥ 2, by Lemma 1.39, so QComp(M, −1)/G^{k+1} is equivalent to the homotopy fibre product

(QComp(M, −1)/G^{k}) ×_{GPreSp(A,−1)/G^{k}}^{h} GPreSp(A, −1)/G^{k+1},

and the result follows by induction. □

Remark 1.41. Taking the limit over all k, Proposition 1.40 gives an equivalence

QP(M, −1)^{nondeg} ≃ (QP(M, −1)^{nondeg}/G^{2}) × ∏_{i≥2} \underline{MC}(DR(A/R)ℏ^{i});

in particular, this means that there is a canonical map

(QP(M, −1)^{nondeg}/G^{2}) → QP(M, −1)^{nondeg},

corresponding to the distinguished point 0 ∈ \underline{MC}(ℏ^{2}DR(A/R)⟦ℏ⟧).

Thus to quantise a non-degenerate (−1)-shifted Poisson structure π = ∑_{j≥2} π_{j} (or equivalently, by [Pri4, Corollary 1.38], a (−1)-shifted symplectic structure), it suffices to lift the power series ∑_{j≥2} π_{j}(−ℏ)^{j−1} to a Maurer–Cartan element of ∏_{j≥2}(F_{j}𝒟(M, −1)/F_{j+2})ℏ^{j−1}.

Even in the degenerate case, the proof of Proposition 1.40 gives a sufficient first-order criterion for quantisations to exist:

QComp(M, −1) ≃ (QComp(M, −1)/G^{2}) × ∏_{i≥2} \underline{MC}(DR(A/R)ℏ^{i}).

2. Quantisation for derived DM N-stacks

In order to pass from derived affine schemes to derived DM stacks, we will exploit étale functoriality using Segal spaces.

The basic idea is that given a small category I, an I-diagram A of CDGAs, and an A-module M in I-diagrams, we can construct a DGAA 𝒟_{A}(M) of differential operators of M. When M is a strict line bundle, 𝒟_{A}(M) then gives rise to a filtered DGLA QP̂ol(M, −1) governing E_{0} quantisations of the diagram M.

If we could choose appropriate restrictions on (A, M) to ensure that 𝒟_{A}(M) had the correct homotopical properties, passage to Maurer–Cartan spaces would then naturally give a presentation of the ∞-category of E_{0}-quantisations as a derivator. However, this is not straightforward, since in order to compute both de Rham and Poisson cohomology correctly, we need the A-modules Ω_{A}^{m} to be both fibrant and cofibrant.

When I is a category of the form [m] = (0 → 1 → … → m), and A is fibrant and cofibrant in the injective model structure, this condition is satisfied, so we can construct Maurer–Cartan spaces of [m]-diagrams, providing all the data necessary to form Segal spaces.

2.1. Quantised polyvectors for diagrams.

Definition 2.1. Given a small category I, an I-diagram A of R-CDGAs, and Amodules M, N in I-diagrams of cochain complexes, define the filtered cochain complex Diff(M, N) = Diff_{A/R}(M, N) ⊂ \underline{Hom}_{R}(M, N) of differential operators from M to N as the equaliser of the obvious diagram

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

with the filtration F_{k}Diff(M, N) defined similarly.

We then write 𝒟(M) = 𝒟_{A/R}(M) := Diff_{A/R}(M, M), which we regard as a DGAA under composition. We simply write 𝒟_{A} = 𝒟_{A/R} for 𝒟_{A/R}(A, A).

For f: i → j a morphism in I, the maps

gr_{k}^{F} Diff_{A(i)/R}(M(i), f_{∗}N(j)) → \underline{Hom}_{A(i)}(M(i) ⊗_{A(i)} CoS_{A(i)/R}^{k}Ω_{A(i)}^{1}, f_{∗}N(j))

are isomorphisms whenever A(i) is semi-smooth and M(i)^{♯} projective over A(i)^{♯}. When these conditions hold for all i, the maps

gr_{k}^{F} Diff_{A/R}(M, N) → \underline{Hom}_{A}(M ⊗_{A} CoS_{A}^{k}Ω_{A/R}^{1}, N)

are thus also isomorphisms.

We now have analogues of all the constructions in §§1.1.3, 1.1.4.

Definition 2.2. Given an I-diagram A of R-CDGAs, and an I-diagram M of strict line bundles over A, define the filtered DGLA of quantised (−1)-shifted polyvector fields on M by

QP̂ol(M, −1) := ∏_{j≥0} F_{j}𝒟_{A/R}(M)ℏ^{j−1}
F̃^{i}QP̂ol(M, −1) := ∏_{j≥i} F_{j}𝒟_{A/R}(M)ℏ^{j−1}
G^{k}QP̂ol(M, −1) := ℏ^{k}QP̂ol(M, −1).

Definition 2.3. We then define QP(M, −1), T QP̂ol(M, −1), T_{Δ}QP̂ol(M, −1), T QP(M, −1), T_{Δ}QP(M, −1), σ = −∂_{ℏ^{−1}} : QP̂ol(M, −1) → T QP̂ol(M, −1) and σ: QP(M, −1) → T QP(M, −1) as before, replacing Definition 1.9 with Definition 2.2.

Note that if u: I → J is a morphism of small categories and A is a functor from J to R-CDGAs with B = A ∘ u, then we have natural maps F (A) → F (B) for all the constructions F of Definition 2.3.

The following is [Pri4, Lemma 2.2]:

Lemma 2.4. If A is [n]-diagram in R-CDGAs which is cofibrant and fibrant for the injective model structure (i.e. each A(i) is cofibrant and the maps A(i) → A(i + 1) are surjective), then \underline{Hom}_{A}(CoS_{A}^{k}Ω_{A}^{1}, A) is a model for the derived Hom-complex R\underline{Hom}_{A}(LCoS_{A}^{k}LΩ_{A}^{1}, A), and \underline{Hom}_{A}(A, Ω_{A}^{m}) ≃ ho\underleftarrow{lim}_{i} LΩ_{A(i)}^{m}.

When A satisfies the conditions of Lemma 2.4, the lemma combines with the observation above to show that every strict line bundle M over A satisfies

gr_{k}^{F} 𝒟_{A/R}(M) ≃ R\underline{Hom}_{A}(LCoS_{A}^{k}LΩ_{A/R}^{1}, A).

Definition 2.5. Given an [m]-diagram A satisfying the conditions of Lemma 2.4, define

GPreSp(A/R, −1) := GPreSp(A(0)/R, −1) = \underleftarrow{lim}_{i∈[m]} GPreSp(A(i)/R, −1),

for the space GPreSp of generalised pre-symplectic structures of Definition 1.29. For a strict line bundle M over A, define

μ: GPreSp(A/R, −1) × QP(M, −1) → T Q^{tw}P(M, −1)

by setting μ(ω, Δ)(i) := μ(ω(i), Δ(i)) ∈ T QP(M(i), −1) for i ∈ [m], and let QComp(M, −1) be the homotopy vanishing locus of

(μ − σ): GPreSp(A/R, −1) × QP(M, −1) → T Q^{tw}P(M, −1).

over Q^{tw}P(M, −1).

The following is [Pri4, Lemma 2.6]:

Lemma 2.6. If D = (A → B) is a fibrant cofibrant [1]-diagram of R-CDGAs which is formally étale in the sense that the map Ω_{A}^{1} ⊗_{A} B → Ω_{B}^{1} is a quasi-isomorphism, then the map

\underline{Hom}_{D}(CoS_{D}^{k}Ω_{D}^{1}, D) → \underline{Hom}_{A}(CoS_{A}^{k}Ω_{A}^{1}, A),

is a quasi-isomorphism.

2.2. Towers of obstructions.

Definition 2.7. For an [m]-diagram A and k ≥ 1, define

Ob(QP, A, k) := \underline{MC}(F^{2}P̂ol(A, −1) ⊕ ℏ^{k}F^{2−k}P̂ol(A, −1)[1]),
Ob(Q^{tw}P, A, k) := \underline{MC}(F^{2}P̂ol(A, −1) ⊕ ℏ^{k}F^{2−2k}P̂ol(A, −1)[1]),

where the DGLA structure is defined by regarding the second term as a module over the first. Note that these expressions only differ for k = 1, as F^{<0} = F^{0}.

Projection gives a fibration Ob(QP, A, k) → P(A, −1) = QP(A, −1)/G^{1}, with the fibre over π being \underline{MC}(ℏ^{k}F^{2−k}T_{π}P̂ol(A, −1)[1]), and similarly for Ob(Q^{tw}P, A, k).

For strict line bundles M over I-diagrams A, the extension F̃^{2}QP̂ol(M, −1)/G^{k+1} → F̃^{2}QP̂ol(M, −1)/G^{k} of DGLAs, with abelian kernel ℏ^{k}F^{2−k}P̂ol(A, −1), and its analogue for (G ∗ F̃)^{2} give rise to canonical fibration sequences

QP(M, −1)/G^{k+1} → QP(M, −1)/G^{k} ⟶^{ob} Ob(QP, A, k),
Q^{tw}P(M, −1)/G^{k+1} → Q^{tw}P(M, −1)/G^{k} ⟶^{ob} Ob(Q^{tw}P, A, k)

over P(A, −1).

Similarly, we have a fibration sequence

GPreSp(A/R, −1)/G^{k+1} → GPreSp(A/R, −1)/G^{k} ⟶^{ob} \underline{MC}(ℏ^{k}DR(A(0)/R)[1]).

We also have a map σ = −∂_{ℏ^{−1}} : Ob(QP, A, k) → ℏOb(QP, A, k), and maps

PreSp(A/R, −1) × Ob(QP, A, k) ⟶^{ν} ℏOb(Q^{tw}P, A, k)

\underline{MC}(ℏ^{k}DR(A(0)/R)[1]) × P(A, −1) ⟶^{μ} ℏOb(Q^{tw}P, A, k)

given by ν(ω, π, u)(i) = (π(i), ν(ω(i), π(i), u(i))) and μ(v, π)(i) = (π(i), μ(v(i), π(i))) for i ∈ [m]. Definition 2.8. For k ≥ 1, define Ob(QComp/QP, A, k) to be the homotopy vanishing locus of

μ: \underline{MC}(ℏ^{k}DR(A(0)/R)[1]) × P(A, −1) → ℏOb(Q^{tw}P, A, k).

over P(A, −1)

Combining the earlier fibration sequences with the definition of QComp, we have Lemma 2.9. There is a natural obstruction map ob: (QComp(M, −1)/G^{k})×_{(QP(M,−1)/G^{k})}^{h}(QP(M, −1)/G^{k+1}) → Ob(QComp/QP, A, k) over P(A, −1), whose homotopy vanishing locus is

QComp(M, −1)/G^{k+1}.

Definition 2.10. Define Ob(QComp/QS, A, k) to be the homotopy vanishing locus of

(ν − σ ∘ pr_{2}): Comp(A, −1) ×_{P(A,−1)} Ob(QP, A, k) → ℏOb(Q^{tw}P, A, k) over P(A, −1).

Combining the earlier fibration sequences with the definition of QComp, we also have: Lemma 2.11. There is a natural obstruction map ob: (QComp(M, −1)/G^{k})×_{(GPreSp(A,−1)/G^{k})}^{h}(GPreSp(A, −1)/G^{k+1}) → Ob(QComp/QS, A, k) over Comp(A, −1), whose homotopy vanishing locus is

QComp(M, −1)/G^{k+1}.

2.3. Descent and line bundles.

Definition 2.12. Write dgCAlg(R) for the category of CDGAs over R, and let dgCAlg(R)_{c,↠} ⊂ dgCAlg(R) be the subcategory with all cofibrant R-CDGAs as objects, and only surjective morphisms.

We already have simplicial set-valued functors GPreSp(−, −1) and GSp(−, −1) from dgCAlg(R) to sSet, mapping quasi-isomorphisms in dgCAlg(R)_{c} to weak equivalences. Poisson structures and their quantisations are only functorial with respect to formally étale morphisms, in an ∞-functorial sense which we now make precise.

Observe that, when F is any of the constructions QP(−, −1), QComp(−, −1), GPreSp(−, −1), Ob(QP, −, k), Ob(QComp/QP, −, k) or Ob(QComp/QS, −, k) applied to [m]-diagrams in dgCAlg(R)_{c,↠}, Lemmas 2.4 and 2.6 combine with the obstruction calculus of §2.2 to show that [Pri4, Properties 2.5] are satisfied:

Properties 2.13. (1) the maps from F (A(0) → … → A(m)) to

F (A(0) → A(1)) ×_{F (A(1)}^{h} F (A(1) → A(2)) ×_{F (A(2)}^{h} … ×_{F (A(n−1)}^{h} F (A(n − 1) → A(n))

are weak equivalences;

(2) if the [1]-diagram A → B is a quasi-isomorphism, then the natural maps from F (A → B) to F (A) and to F (B) are weak equivalences.

(3) if the [1]-diagram A → B is formally étale, then the natural map from F (A → B) to F (A) is a weak equivalence.

The first two properties ensure that the simplicial classes ∐_{A∈B_{m}dgCAlg(R)_{c,↠}} F (A) fit together to give a complete Segal space ∫ F over the nerve BdgCAlg(R)_{c,↠}. Taking Segal spaces as our preferred model of ∞-categories, we define LdgCAlg(R)_{c,↠} and LdgCAlg(R) to be the ∞-categories obtained by localising the respective (∞-)categories at quasi-isomorphisms or weak equivalences, and let LdgCAlg(R)^{ét} ⊂ LdgCAlg(R) the be the ∞-subcategory of homotopy formally étale morphisms.

For any construction F satisfying the conditions above, [Pri4, Definition 2.7] then gives an ∞-functor

RF: LdgCAlg(R)^{ét} → LsSet

to the ∞-category of simplicial sets, with the property that

(RF)(A) ≃ F (A)

for all cofibrant R-CDGAs A.

Definition 2.14. Given a derived Deligne–Mumford N-stack 𝔛 and any of the constructions F above, define F (𝔛) to be the homotopy limit of RF (A) over the ∞-category (DGAff_{ét} ↓ 𝔛) consisting of derived affines Spec A equipped with homotopy étale (i.e. étale in the sense of [TV]) maps to 𝔛, and all homotopy étale morphisms between them.

When 𝔛 ≃ Spec B is a derived affine, note that it is final in the category of derived affines over 𝔛, so RF (𝔛) = RF (B) = F (B). In general, it suffices to take the homotopy limit over any subcategory of (DGAff_{ét} ↓ 𝔛) with colimit 𝔛, so this definition also coincides with [Pri4, Definition 2.18], by applying it to a suitable hypergroupoid.

Definition 2.14 is insufficient for our purposes, as we wish to consider line bundles. Since DM stacks only involve CDGAs with non-positive cohomology, the line bundles we encounter will be locally trivial, so for now we only need to set up 𝔾_{m}-equivariance.

Definition 2.15. Define the functor 𝔾_{m} from CDGAs to groups by

𝔾_{m}(A) := Z^{0}(A)^{×}.

Given a string A = (A(0) → … → A(n)) of CDGAs, we write 𝔾_{m}(A) := 𝔾_{m}(A(0)), regarded as \underleftarrow{lim}_{i} 𝔾_{m}(A(i)).

Now, the group 𝔾_{m}(A)^{×} acts by conjugation on QP̂ol(A, −1), corresponding to automorphisms of A as a line bundle over A. This action preserves all the filtrations, so it acts on the simplicial set QP(A, −1) = \underline{MC}(F̃^{2}QP̂ol(A, −1)) respecting the cofiltrations. Note that the action is trivial on the quotient QP(A, −1)/G^{1} = P(A, −1).

Definition 2.16. For any of the constructions F above, let R(F/^{h}𝔾_{m}) be the ∞- functor on LdgCAlg(R)^{ét} given by applying the construction of [Pri4, Definition 2.18] to the homotopy quotient F/^{h}𝔾_{m}, then taking étale hypersheafification.

Up to now, hypersheafification has not been necessary because all our functors have been hypersheaves — this follows because the associated gradeds gr_{F} of obstruction functors can be written in terms of tangent sheaves and sheaves of differential forms. However, B𝔾_{m} requires hypersheafification because the simplicial presheaf B𝔾_{m} does not preserve weak equivalences or satisfy étale descent.

For any derived line bundle ℒ on a derived stack 𝔛, there is an associated 𝔾_{m}-torsor given locally by the disjoint union of spaces of quasi-isomorphisms from 𝒪_{𝔛}[m] to ℒ for all m ∈ ℤ.

Definition 2.17. Given a derived Deligne–Mumford N-stack 𝔛, a derived line bundle ℒ on 𝔛 and any of the constructions F above, define F (ℒ) to be the homotopy limit of R(F/^{h}𝔾_{m})(A) ×_{R(∗/^{h}𝔾_{m})(A)}^{h} {ℒ |_{A}} over the ∞-category (DGAff_{ét} ↓𝔛).

Remarks 2.18. If we fix π ∈ P(𝔛, −1), then observe that the homotopy fibres of QP(ℒ , −1) → QP(ℒ , −1)/G^{1} = P(M, −1) over π can be combined and enhanced to a dg category whose objects are E_{0} quantisations (ℒ , Δ) over π, with dg morphisms (ℒ , Δ) → (ℒ′, Δ′) given by the complex (∏_{j≥0} F_{j}𝒟iff_{𝔛/R}(ℒ , ℒ′)ℏ^{j}, (δ + Δ′)_{∗} ∓ f(δ + Δ)^{∗}) and the obvious composition law.

Also note that the action of 𝔾_{m}(A)^{×} is unipotent, as it is trivial on P(A, −1), so it extends naturally to an action of 𝔾_{m}(A)^{×}⊗_{ℤ}ℚ, and we can therefore define quantisations for étale B(𝔾_{m}/μ_{∞})-torsors. In fact, more is true: line bundles ℳ with left 𝒟-module structure give equivalences 𝒟(ℒ) ≃ 𝒟(ℒ ⊗ ℳ) and hence QP(ℒ , −1) ≃ QP(ℒ ⊗ ℳ , −1). By sheafifying these equivalences, we get a notion of quantisation for all elements of \underline{MC}(F^{1}DR(𝔛/R)[1]) via Chern classes.

2.4. Comparing quantisations and generalised symplectic structures. We now fix a strongly quasi-compact derived DM N-stack 𝔛 over R.

Lemma 2.19. For (ω, π) ∈ Comp(𝔛, −1), the homotopy fibre

Ob(QComp/QS, 𝔛, k)_{(ω,π)} of Ob(QComp/QS, 𝔛, k) → Comp(𝔛, −1)

over (ω, π) is contractible for all k ≥ 2.

Proof. As in the proof of Lemma 1.39, the map ℏ^{−1}ν(ω, π, −) − (p − 1) on

ℏ^{p−1}Ext_{𝒪_{𝔛}}^{2−i}(LCoS_{𝒪_{𝔛}}^{p−k}LΩ_{𝔛/R}^{1}, 𝒪_{𝔛})

is invertible for k ≥ 2, so Lemma 2.11 gives contractibility of the homotopy fibre. □

Proposition 2.20. For any line bundle ℒ on 𝔛, the map

QComp(ℒ , −1) → (QComp(ℒ , −1)/G^{2}) ×_{(GPreSp(𝔛,−1)/G^{2})}^{h} GPreSp(𝔛, −1)

≃ (QComp(ℒ , −1)/G^{2}) × ∏_{i≥2} \underline{MC}(DR(𝔛/R)ℏ^{i}).

is a weak equivalence. Proof. This is essentially the same as the proof of Proposition 1.40. Lemma 2.19 combines with the obstruction maps (QComp(ℒ , −1)/G^{k})×_{(GPreSp(𝔛,−1)/G^{k})}^{h}(GPreSp(𝔛, −1)/G^{k+1}) → Ob(QComp/QS, 𝔛, k) to give the weak equivalences (QComp(ℒ , −1)/G^{k+1}) ≃ (QComp(ℒ , −1)/G^{k})×_{(GPreSp(𝔛,−1)/G^{k})}^{h}(GPreSp(𝔛, −1)/G^{k+1}).

□

Definition 2.21. Given a (−1)-shifted Poisson structure π ∈ P(𝔛, −1), we say that π is non-degenerate if the induced map

π^{♯}: LΩ_{𝔛}^{1} → RHom_{𝒪_{X}} (LΩ_{𝔛}^{1}, 𝒪_{𝔛})[1]

is a quasi-isomorphism of sheaves on 𝔛, and LΩ_{𝔛}^{1} is perfect. Lemma 2.22. If π is a non-degenerate (−1)-shifted Poisson structure on 𝔛, then the homotopy fibre Ob(QComp/QP, 𝔛, k)_{π} of

Ob(QComp/QP, 𝔛, k) → P(𝔛, −1)

over π is contractible. Proof. The map

μ(−, π): H^{2−i}(𝔛, LΩ_{𝔛/R}^{p−2k}) → ℏ^{p}Ext_{𝒪_{𝔛}}^{2−i}(LCoS_{𝒪_{𝔛}}^{p−2k}LΩ_{𝔛/R}^{1}, 𝒪_{𝔛})

is given by Λ^{p−2k}π^{♯}, so is an isomorphism by the non-degeneracy of π. Lemma 2.9 then gives contractibility of Ob(QComp/QP, 𝔛, k)_{π}. □ Proposition 2.23. For any line bundle ℒ on 𝔛, the canonical map

QComp(ℒ , −1)^{nondeg} → QP(ℒ , −1)^{nondeg}

is a weak equivalence. In particular, there is a morphism

QP(ℒ , −1)^{nondeg} → GSp(𝔛, −1)

in the homotopy category of simplicial sets. Proof. This is much the same as Proposition 1.37. Lemma 2.22 combines with the obstruction maps (QComp(ℒ , −1)/G^{k}) ×_{(QP(ℒ ,−1)/G^{k})}^{h} (QP(ℒ , −1)/G^{k+1}) → Ob(QComp/QP, 𝔛, k) to give the weak equivalences

(QComp(ℒ , −1)^{nondeg}/G^{k+1}) ≃

(QComp(ℒ , −1)/G^{k}) ×_{(QP(ℒ ,−1)/G^{k})}^{h} (QP(ℒ , −1)^{nondeg}/G^{k+1}).
□

3. Quantisation for derived Artin N-stacks

In order to proceed further, we will make use of the étale resolutions of derived Artin stacks by bi-CDGAs given in [Pri4, §3]. We just extend the results of §1 from CDGAs to bi-CDGAs, and then the étale descent approach of §2 adapts immediately.

3.1. Bi-CDGAs. We now recall some definitions and lemmas from [Pri4, §3]. From now on, we will regard the CDGAs encountered so far 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 3.1. A bi-CDGA is a chain cochain complex A_{•}^{•} equipped with a commutative product A ⊗ A → A and unit k → A. Given a chain CDGA R, a bi-CDGA over R is then a morphism R → A of bi-CDGAs. We write DGdgCAlg(R) for the category of bi-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}.

Definition 3.2. 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 bi-CDGAs is a levelwise quasi-isomorphism if the underlying morphism of chain cochain complexes is so.

The following is [Pri4, Lemma 3.4]:

Lemma 3.3. There is a cofibrantly generated model structure on bi-CDGAs over R in which fibrations are surjections and weak equivalences are levelwise quasi-isomorphisms.

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 bi-CDGA in non-negative cochain degrees. By [Pri4, Lemma 3.5], D^{∗} is a left Quillen functor from the Reedy model structure on cosimplicial chain CDGAs to the model structure of Lemma 3.3.

Since DA is a pro-nilpotent extension of A^{0}, when H_{<0}(A) = 0 we think of the hypersheaf RSpec DA as a stacky derived thickening of the derived affine scheme RSpec A^{0}.

Definition 3.4. 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 3.5. Given a bi-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).

Definition 3.6. 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 [Pri4, Proposition 3.12] with [Pri3, Theorem 4.15 and Corollary 6.35], every strongly quasi-compact derived Artin N-stack over R can be resolved by a homotopy formally étale cosimplicial diagram in DG^{+}dgCAlg(R).

3.2. Quantised polyvectors. We now fix a bi-CDGA A over a chain CDGA R.

Definition 3.7. Given A-modules M, N in chain cochain complexes, inductively define the filtered chain cochain complex Diff (M, N) = Diff_{A/R}(M, N) ⊂ Hom_{R}(M, N) of differential operators from M to N by setting

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

(2) F_{k+1}Diff (M, N) = {u ∈ Hom_{R}(M, N) : [a, u] ∈ F_{k}Diff (M, N) ∀a ∈ A}, where [a, u] = au − (−1)^{deg a deg u}ua.

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

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

Definition 3.8. Given an A-module M in chain cochain complexes, write 𝒟(M) = 𝒟_{A/R}(M) := D̂iff_{A/R}(M, M), which we regard as a sub-DGAA of ˆ \underline{Hom}_{R}(M, M). We simply write 𝒟_{A} = 𝒟_{A/R} for 𝒟_{A/R}(A, A).

Definition 3.9. Define a strict line bundle over A to be an A-module M in chain cochain complexes such that M_{#}^{#} is a projective module of rank 1 over the bigradedcommutative algebra A_{#}^{#} underlying A.

Definitions 1.9 and 1.10 then carry over verbatim to define quantised polyvectors over a bi-CDGA, and the filtrations F̃, G, and G ∗ F̃.

We now follow [Pri4, §3.3] in making the following assumptions on A ∈ DG^{+}dgCAlg(R):

(1) for any cofibrant replacement Ã → A in the model structure of Lemma 3.3, 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. These conditions are satisfied by D^{∗}O(X) for DG Artin hypergroupoids X.

For bi-CDGAs of this form, Definition 1.17 adapts verbatim to define the space QP(M, −1) of E_{0} quantisations of a strict line bundle M, and its twisted counterpart Q^{tw}P(M, −1). The second assumption gives us isomorphisms

F^{p−2i}P̂ol(A, −1)ℏ^{i} → gr_{G}^{i}(G ∗ F̃)^{p}QP̂ol(M, −1),
T_{π_{Δ}}F^{p−2i}P̂ol(A, −1)ℏ^{i} → gr_{G}^{i}(G ∗ F̃)^{p}T_{Δ}QP̂ol(M, −1).

The following is a slight generalisation of [Pri4, Definition 3.20]:

Definition 3.10. Say that an E_{0} quantisation Δ = ∑_{j≥2} Δ_{j}ℏ^{j} of a strict line bundle M over A is non-degenerate if the map

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

of is a quasi-isomorphism, and Tot^{Π}(Ω_{A}^{1} ⊗_{A} A^{0}) is a perfect complex over A^{0}.

Definitions 1.21 and 1.22 adapt verbatim, giving tangent spaces T QP(M, −1), T QP(M, −1)/G^{k}, T Q^{tw}P(M, −1), T Q^{tw}P(M, −1)/G^{k}and a canonical tangent vector

σ = −∂_{ℏ^{−1}} : QP̂ol(M, −1) → T QP̂ol(M, −1).

3.3. Generalised symplectic structures and compatible quantisations. The following is [Pri4, Definition 3.22]

Definition 3.11. Define the de Rham complex DR(A/R) to be the product total complex of the bicomplex

Tot^{Π}A ⟶^{d} Tot^{Π}Ω_{A/R}^{1} ⟶^{d} Tot^{Π}Ω_{A/R}^{2} ⟶^{d} …,

so the total differential is d ± δ ± ∂.

We define the Hodge filtration F on DR(A/R) by setting F^{p}DR(A/R) ⊂ DR(A/R) to consist of terms Tot^{Π}Ω_{A/R}^{i} with i ≥ p.

We then similarly define DR′(A/R) to be the (triple) product total complex

DR′(A/R) := Tot^{Π}NÂ^{•+1}

regarded as a filtered DGAA over R, with F^{p}DR′(A/R) := Tot^{Π}NF^{p}Â^{•+1}. Definition 1.27 then carries over to give a filtration F̃ on DR′(A/R)⟦ℏ⟧.

Definition 1.29 carries over to give a space GPreSp(A/R, −1) of generalised (−1)- shifted pre-symplectic structures on A/R. We say that a generalised pre-symplectic structure ω is symplectic if its leading term ω_{0} ∈ PreSp(A/R, −1) is symplectic in the sense of [Pri4, Definition 3.23]; explicitly, this says that Tot^{Π}(Ω_{A}^{1} ⊗_{A} A^{0}) is a perfect complex over A^{0} and the map

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

is a quasi-isomorphism. We then let Sp(A/R, n) ⊂ PreSp(A/R, n) consist of the symplectic structures — this is a union of path-components.

Lemmas 1.31 and 1.32 then adapt to give compatible maps

μ(−, Δ): DR′(A/R)⟦ℏ⟧/G^{k} → T_{Δ}QP̂ol(M, −1)/G^{k}

respecting the filtrations (G ∗ F̃).

We may now define the space QComp(A/R, n) of compatible quantisations as in Definition 1.35, with Proposition 1.37 adapting to show that

QComp(M, −1)^{nondeg} → QP(M, −1)^{nondeg}

is a weak equivalence and Proposition 1.40 adapting to show that the resulting maps

QP(M, −1)^{nondeg}/G^{k} → (QP(M, −1)^{nondeg}/G^{2}) ×_{(GSp(A,−1)/G^{2})}^{h} (GSp(A, −1)/G^{k})

≃ (QP(M, −1)^{nondeg}/G^{2}) × ∏_{i≥2} \underline{MC}(DR(A/R)ℏ^{i})

are weak equivalences for all k ≥ 2.

3.4. Diagrams and derived Artin stacks.

3.4.1. Diagrams. We may now proceed as in [Pri4, §3.4.2]. For any small category I, any I-diagram A in DG^{+}dgCAlg(R), and A-modules M, N in I-diagrams of chain cochain complexes, we define Diff_{A/R}(M, N) to be the equaliser of

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

The constructions QP(−, −1), GPreSp(−, −1) and QComp(−, −1) all adapt to such diagrams, and behave well for [m]-diagrams A which are fibrant and cofibrant for the injective [m]-diagram model structure on bi-CDGAs, so A(i) is cofibrant for the model structure of Lemma 3.3 and the maps A(i) → A(i + 1) are all surjective. In particular, these constructions satisfy the conditions of [Pri4, §3.4.2], so for each construction F we have an ∞-functor

RF: LDG^{+}dgCAlg(R)^{ét} → LsSet

on the ∞-categories given by localising weak equivalences, with (RF)(A) ≃ F (A) for all cofibrant bi-CDGAs A over R. Here, DG^{+}dgCAlg(R)^{ét} ⊂ DG^{+}dgCAlg(R) is the subcategory of morphisms A → B which are homotopy formally étale in the sense of Definition 3.6.

By naturality of these constructions and the equivalences above, we then have weak equivalences of ∞-functors

RQComp(−, −1)^{nondeg} → RQP(−, −1)^{nondeg}

and

RQP(−, −1)^{nondeg}/G^{k} → (RQP(−, −1)^{nondeg}/G^{2}) ×_{(RGSp(−,−1)/G^{2})}^{h} (RGSp(−, −1)/G^{k})

for all k ≥ 2.

The approach of [Pri4, §3.4.2] now applies immediately to associate to any of the constructions above an ∞-functor on derived Artin N-stacks, with natural transformations and equivalences carrying over. However, this is not quite sufficient for our purposes, since we wish to consider quantisations of non-trivial line bundles.

3.4.2. Descent and line bundles. We say that a morphism A → B in DG^{+}dgCAlg(R) is a covering if A^{0} → B^{0} is faithfully flat. In particular, this implies that

ho\underrightarrow{lim}_{i} RSpec D^{i}B → ho\underrightarrow{lim}_{i} RSpec D^{i}A

is a surjection of étale hypersheaves. Note that when X → Y is a relative trivial derived Artin hypergroupoid, X_{0} → Y_{0} is faithfully flat, so the morphism D^{∗}O(Y) → D^{∗}O(X) is a covering in the sense above.

In §2.3, we were able to extend the functor QP to line bundles solely by making use of the 𝔾_{m}-action on it. For Artin stacks, the situation is more subtle, because for any A ∈ DG^{+}Alg(ℚ), we have Hom(D^{∗}O(B𝔾_{m}), A) ≅ Z^{1}A.

The most naïve simplicial set-valued functor we can consider on DG^{+}dgAlg(ℚ) is (B𝔾_{m})^{Δ} ∘D, which is represented by the cosimplicial CDGA D^{∗}O((B𝔾_{m})^{Δ}), and sends A to the nerve B[Z^{1}(Z_{0}A)/(Z_{0}A^{0})^{×}], of the groupoid

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

where f ∈ (A^{0})^{×} acts on Z^{1}A by addition of ∂ log f = f^{−1}∂f. We think of TLB(A) as the groupoid of trivial line bundles.

For any cofibrant A ∈ DG^{+}dgAlg(R), we can extend QP to a simplicial representation of the groupoid TLB(A) above by sending an object b ∈ Z^{1}(Z_{0}A) to QP(A_{b}, −1), with (Z_{0}A^{0})^{×} acting via functoriality for line bundles. Note that the quotient representation QP(−, −1)/G^{1} = P(−, −1) is trivial; we also set GPreSp to be a trivial representation b ↦ GPreSp(A).

Definition 3.12. For any of the constructions F of §3.4.1, let R(F/^{h}𝔾_{m}) be the ∞- functor on LdgCAlg(R)^{ét} given by applying the construction of [Pri4, §3.4.2] to the right-derived functor of the Grothendieck construction

A ↦ holim_{\overrightarrow{b∈TLB(A)}} F (A_{b}),

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 [Pri4, §3.2.2] is a pro-quasi-isomorphism. We then write (dg_{+}DGAff_{ét} ↓ 𝔛) for the ∞-category of homotopy formally étale elements f ∈ ho\underleftarrow{lim}_{i} 𝔛(D^{i}A) with homotopy formally étale maps A → B between them.

Definition 3.13. Given a derived Artin N-stack 𝔛, a line bundle ℒ on 𝔛 and any of the functors F above, define F (ℒ) to be the homotopy limit of

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

over the ∞-category (dg_{+}DGAff_{ét} ↓𝔛).

If we now fix a derived Artin N-stack 𝔛, Definition 2.21 carries over verbatim to give a notion of non-degeneracy for a (−1)-shifted Poisson structure π ∈ P(𝔛, −1), and Propositions 2.20 and 2.23 readily adapt (substituting the relevant results from [Pri4, §3]), giving

Proposition 3.14. For any line bundle ℒ on 𝔛, the canonical maps

QComp(ℒ , −1)^{nondeg} → QP(ℒ , −1)^{nondeg}
QComp(ℒ , −1) → (QComp(ℒ , −1)/G^{2}) ×_{(GPreSp(𝔛,−1)/G^{2})}^{h} GPreSp(𝔛, −1)

≃ (QComp(ℒ , −1)/G^{2}) × ∏_{i≥2} \underline{MC}(DR(𝔛/R)ℏ^{i}).

are filtered weak equivalences. In particular, there is a morphism

QP(ℒ , −1)^{nondeg} → GSp(𝔛, −1)

in the homotopy category of simplicial sets.

4. Self-dual quantisations

We now introduce the notion of duality for quantisations, and indicate how it leads to canonical quantisations for line bundles which are Grothendieck–Verdier self-dual, giving rise to the perverse sheaf PV of vanishing cycles from [BBD^{+}]. From our point of view, the key property of this sheaf is that it is Verdier self-dual [BBD^{+}, Equation (2.6)], while the object it quantises is Grothendieck–Verdier self-dual.

4.1. Duality. We wish to consider line bundles ℒ equipped with an involutive equivalence 𝒟(ℒ) ≃ 𝒟(ℒ)^{opp}. A dualising line bundle ω_{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), and if we write ℰ^{∨} := Rℋ om_{𝒪_{X}} (ℰ , ω_{X}) for the Grothendieck–Verdier dual of a perfect complex ℰ, then the right 𝒟-module structure of ω_{X} gives a quasi-isomorphism of DGAAs between 𝒟(ℰ)^{opp} and 𝒟(ℰ^{∨}).

Thus an equivalence between 𝒟(ℒ) and 𝒟(ℒ)^{opp} is the same as 𝒟(ℒ) ≃ 𝒟(ℒ^{∨}), which is satisfied when ℒ ≃ ℒ^{∨} (so ℒ is a square root of ω_{X}). Indeed, an equivalence will exist whenever ℒ has the structure of a right 𝒟(ℒ)-module, or equivalently whenever ℒ^{⊗2} has the structure of a right 𝒟-module; the equivalence will automatically be involutive as ℒ has rank 1.

We begin by noting that for any ℰ, quantisations Δ(ℏ) of ℰ give rise to quantisations of ℰ^{∨}:

Definition 4.1. For a line bundle ℰ, writing (−)^{t}: 𝒟(ℰ)^{opp} → 𝒟(ℰ^{∨}) for the natural quasi-isomorphism, define

(−)^{∗}: QP̂ol(ℰ , −1) → QP̂ol(ℰ^{∨}, −1)

by

Δ^{∗}(ℏ) := −Δ^{t}(−ℏ).

Since this is a quasi-isomorphism of filtered DGLAs, it gives rise to a weak equivalence

(−)^{∗}: QP̂ol(ℰ , −1) → QP̂ol(ℰ^{∨}, −1),

and hence

QP(ℰ , −1) → QP(ℰ^{∨}, −1)

The reason for the choice of sign −ℏ in the definition of Δ^{∗} is that on the associated graded gr_{p}^{F} 𝒟_{X}(ℰ) ≅ Symm^{p}𝒯_{X}, the operation (−)^{t} is given by (−1)^{p}. Thus the underlying Poisson structures satisfy π_{Δ^{∗}} = π_{Δ}.

Definition 4.2. For a line bundle ℒ with ℒ^{⊗2} a right 𝒟-module, the map (−)^{∗} becomes a (homotopy) involution of QP̂ol(ℒ , −1), and we define QP̂ol(ℒ , −1)^{sd} to be the space of homotopy fixed points of the resulting ℤ/2-action.

Similarly, we define the space

QP(ℒ , −1)^{sd}

of self-dual quantisations to be the space of homotopy fixed points of the ℤ/2-action on QP(ℒ , −1) generated by (−)^{∗}.

Remark 4.3. Following Remark 1.12, to each E_{0} quantisation Δ ∈ QP(ℒ , −1) there corresponds a right 𝒟_{X}⟦ℏ⟧-module ℳ_{ℏ} := (ℒ ⊗_{𝒪_{X}}^{L} 𝒟_{X}⟦ℏ⟧, δ + Δ · {−}). Definition 4.2 says that the quantisation is self-dual with respect to the right 𝒟-module structure on ℒ^{⊗2} when Rℋ om_{𝒟_{X}^{opp}⟦ℏ⟧} (ℳ_{ℏ}, 𝒟_{X}⟦ℏ⟧) ⊗_{𝒪_{X}}^{L} ℒ^{⊗2} is equivalent to ℳ_{−ℏ} as a right 𝒟-module. This can be phrased as a symmetric perfect pairing

(ℳ_{−ℏ} ⊗_{𝒪_{X}}^{L} ℒ^{−1}) ⊗_{𝒟_{X}(ℒ)⟦ℏ⟧}^{L} (ℳ_{ℏ} ⊗_{𝒪_{X}}^{L} ℒ^{−1})^{opp} → 𝒟(ℒ)⟦ℏ⟧.

Lemma 4.4. For the filtration G induced on F̃^{p}QP̂ol(ℒ , −1)^{sd} by the corresponding filtration on F̃^{p}QP̂ol(ℒ , −1)^{sd}, we have

gr_{G}^{k}F̃^{p}QP̂ol(ℒ , −1)^{sd} ≃ \begin{cases}gr_{G}^{k}F̃^{p}QP̂ol(ℒ , −1) & k even \\ 0 & k odd. \end{cases}

Proof. As already observed, the involution acts trivially on gr_{G}^{0}P̂ol(ℒ , −1). It therefore acts as multiplication by (−1)^{k} on gr_{G}^{k}P̂ol(ℒ , −1) = ℏ^{k}gr_{G}^{0}P̂ol(ℒ , −1). □

In particular, this means that Ob(QP, ℒ , 1)^{sd} ≃ 0, so the map

QP(ℒ , −1)^{sd}/G^{2} → QP(ℒ , −1)/G^{1} ≃ P(X, −1)

is a weak equivalence. In other words, Poisson structures correspond to first order self-dual quantisations. We can say much more in non-degenerate cases:

Proposition 4.5. For a line bundle ℒ with ℒ^{⊗2} a right 𝒟-module (such as any square root of ω_{X}), there is a canonical weak equivalence

QP(ℒ , −1)^{nondeg,sd} ≃ P(X, −1)^{nondeg} × \underline{MC}(ℏ^{2}DR(X/R)⟦ℏ^{2}⟧).

In particular, every non-degenerate (−1)-shifted Poisson structure gives a canonical choice of self-dual quantisation of ℒ.

Proof. Lemma 4.4 implies that we have weak equivalences

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

Combined with Propositions 1.40 and 2.20, these give weak equivalences

QP(ℒ , −1)^{nondeg,sd}/G^{2i+1} → (QP(ℒ , −1)^{nondeg,sd}/G^{2i}) × \underline{MC}(ℏ^{2i}DR(X/R))

for all i > 0, so

QP(ℒ , −1)^{nondeg,sd}/G^{2i+1} ≃ (QP(ℒ , −1)^{nondeg,sd}/G^{2i}) × \underline{MC}(ℏ^{2i}DR(X/R))
≃ QP(ℒ , −1)^{nondeg}/G^{2i−1} × \underline{MC}(ℏ^{2i}DR(X/R)),

and we have seen that ∗ acts trivially on QP(ℒ , −1)/G^{1} = P(ℒ , −1), so QP(ℒ , −1)^{sd}/G^{1} ≃ P(ℒ , −1). □

Remark 4.6. The proof of Proposition 4.5 only shows that for a self-dual quantisation of a non-degenerate (−1)-shifted Poisson structure, the corresponding generalised symplectic structure is determined by its even coefficients. In fact, the odd coefficients must be homotopic to 0, with the following reasoning.

As μ is multiplicative and the de Rham algebra is commutative, we have a homotopy μ(ω, Δ)^{t} ≃ μ(ω, −Δ^{t}) for any ω and Δ, so μ(ω, Δ)^{t}(−ℏ) ≃ μ(ω, Δ^{∗})(ℏ). We also have σ(Δ)^{t}(−ℏ) ≃ σ(Δ^{∗})(ℏ), so ω(ℏ) is compatible with Δ if and only if ω(−ℏ) is compatible with Δ^{∗}. When Δ is self-dual and non-degenerate, this implies that ω(ℏ) ≃ ω(−ℏ).

For a more explicit description of the generalised symplectic structure ω corresponding to a non-degenerate self-dual quantisation Δ, observe that we then have an isomorphism

μ(−, Δ): H^{∗}(F^{2}DR(X/R) × ℏ^{2}DR(X/R)⟦ℏ^{2}⟧)
→ {v ∈ H^{∗}(T_{Δ}(G ∗ F̃)^{2}QP̂ol(ℒ , −1)) : v(−ℏ) = v^{t}(ℏ)},

and that [ω] must be the inverse image of [ℏ^{2} \frac{∂Δ}{∂ℏ}].

4.2. Vanishing cycles. Given a smooth scheme Y of dimension m over ℂ, and a function f: Y → 𝔸^{1}, we can consider the derived critical locus X of f, which is equipped with a canonical (−1)-shifted symplectic structure ω. Explicitly, 𝒪_{X} is the CDGA given by the alternating algebra 𝒪_{Y} [𝒯_{Y} [1]], with differential δ given by contraction with df.

Now, the line bundle Ω_{Y}^{m} pulls back to give a square root i^{∗}Ω_{Y}^{m} of the dualising sheaf ω_{X} on X. This complex can be written explicitly as i^{∗}Ω_{Y}^{m} ≅ (Ω_{Y}^{∗} [m], df∧). There is a canonical (−1)-shifted symplectic structure ω on X, and we write Δ_{ω} for the unique compatible self-dual E_{0} quantisation of Proposition 4.5.

Lemma 4.7. On the derived critical locus X, the quantisation Δ_{ω} is given by ℏd: Ω_{Y}^{∗} → Ω_{Y}^{∗} ⟦ℏ⟧, for the de Rham differential d on Y.

Proof. We first need to check that Δ := Δ_{ω} is self-dual, but this follows because we have a pairing

(i^{∗}Ω_{Y}^{m}⟦ℏ⟧, δ + ℏd) ⊗_{R} (i^{∗}Ω_{Y}^{m}⟦ℏ⟧, δ − ℏd) → (i^{∗}Ω_{Y}^{m})^{⊗2}⟦ℏ⟧ = ω_{X}⟦ℏ⟧

given by combining the cup product Ω_{Y}^{∗} [m] ⊗ Ω_{Y}^{∗} [m] → Ω_{Y}^{∗} [2m] with projection to Ω_{Y}^{m}[m] followed by inclusion in ω_{X} ≅ Ω_{Y}^{m} ⊗_{𝒪_{Y}} Ω_{Y}^{∗} [m].

Now, for y ∈ 𝒪_{Y} ⊂ 𝒪_{X}, the differential operator [Δ, y] acts on Ω_{Y}^{∗} [m] as multiplication by ℏdy. For η ∈ 𝒯_{Y} ⊂ 𝒪_{X}, the differential operator [Δ, η] acts on Ω_{Y}^{∗} [m] as the Lie derivative ℏLie_{η}. Thus the ring homomorphism μ(−, Δ) is given on generators of Ω_{X}^{1} by

μ(dy, Δ) = ℏdy∧ , μ(dη, Δ) = ℏLie_{η},

and we need to show that it maps ω to ℏ^{2} \frac{∂Δ}{∂ℏ} = ℏ^{2}d, up to homotopy.

There is a canonical representative ω ∈ Z^{−1}Γ(X, Ω_{X}^{2}) with dω = 0, and on any affine open we may lift it to an element of ω̃ ∈ Z^{1}F^{2}DR′(X/R) lying in N^{2}𝒪̂_{X}^{⊗•+1} . Any such choices differ by dα for α ∈ F^{2}N^{1}𝒪̂_{X}^{⊗•+1} ; in other words, α ∈ I^{2}, for I = ker(𝒪̂_{X}^{⊗2} → 𝒪_{X}). We now just observe that μ(α, Δ) is anti-self-dual, since μ(da.db, Δ) = ℏ⟨a, b⟩_{π} ∈ ℏ𝒪_{X}. Therefore μ(ω̃, Δ) + μ(ω̃, Δ)^{∗} is strictly independent of the choice of lift.

It thus suffices to show that for some such local choice we have μ(ω̃, Δ) = σ(Δ). To make things explicit, we now take local co-ordinates y_{1}, …, y_{m}, and write η_{i} ∈ 𝒪_{X} for the element given by ∂_{y_{i}} ∈ 𝒯_{Y}, so X has co-ordinates y_{1}, …, y_{m}, η_{1}, …, η_{m}. The generator dy_{1} ∧ … ∧ dy_{m} then gives an isomorphism i^{∗}Ω_{Y}^{m} ≅ 𝒪_{X}, and Δ corresponds to the quantisation of 𝒪_{X} given by Δ := ℏ ∑_{i} ∂_{y_{i}}∂_{η_{i}}.

The calculations above reduce to

μ(y_{i} ⊗ 1 − 1 ⊗ y_{i}, Δ) = μ(dy_{i}, Δ) = ℏ∂_{η_{i}}
μ(η_{i} ⊗ 1 − 1 ⊗ η_{i}, Δ) = μ(dη_{i}, Δ) = ℏ∂_{y_{i}}.

We now choose the lift ω̃ := ∑_{i} dy_{i} ⌣ dη_{i} ∈ Z^{1}F^{2}DR′(X/R) of the canonical (−1)- shifted symplectic structure ω = ∑_{i} dy_{i} ∧ dη_{i} ∈ Z^{1}F^{2}DR(X/R). Since μ(−, Δ) is multiplicative, it follows that

μ(dy_{i} ⌣ dη_{i}, Δ) = ℏ^{2}∂_{η_{i}}∂_{y_{i}},

so

μ(ω̃, Δ) = μ(∑_{i} dy_{i} ⌣ dη_{i}, Δ) = ∑_{i} ℏ^{2}∂_{η_{i}}∂_{y_{i}} = ℏ^{2} \frac{∂Δ}{∂ℏ} = σ(Δ).

□

Proposition 4.8. On the derived critical locus X, the localisation

RΓ(X, (M⟦ℏ⟧, δ + Δ) ⊗_{ℂ[ℏ]} ℂ[ℏ, ℏ^{−1}]

is quasi-isomorphic to

RΓ(X, PV_{Y,f}^{•})((ℏ)),

for the perverse sheaf PV of vanishing cycles from [BBD^{+}, §2.4].

Proof. By [Sab, Theorem 1.1],

RΓ(X, PV_{Y,f}^{•})((u)) ≃ (Ω_{Y}^{∗} ((u)), d − u^{−1}df)[m].

Multiplying by ℏ^{i} in degree i, and setting ℏ = −u, we get

RΓ(X, PV_{Y,f}^{•})((ℏ)) ≃ (Ω_{Y}^{∗} ((ℏ)), df + ℏd)[m];

Lemma 4.7 completes the proof. □

4.3. Quantisation for n-shifted symplectic structures. We now discuss how these results should generalise to more general n, including the non-trivial cases n = 0, −2.

4.3.1. Unshifted Poisson structures. To address the case n = 0, replace the filtered DGAA 𝒟_{A} of differential operators with the filtered DGLA 𝒟_{A}^{poly}[1] of polydifferential operators, setting QP̂ol(A, 0) := ∏_{p≥0} F_{p}𝒟_{A}^{poly}ℏ^{p−1}. As in [VdB], the HKR isomorphism leads to a quasi-isomorphism between 𝒟_{A}^{poly} and the Hochschild complex of A over R.

For a quantisation

Δ ∈ QP(A, 0) := \underline{MC}(QP̂ol(A, 0)[1]),

the centre T_{Δ}QP̂ol(A, 0) := (ℏQP̂ol(A, 0), δ +[Δ, −]) then has the canonical structure of an E_{2}-algebra. A choice of formality isomorphism for E_{2} will therefore give a P_{2}-algebra structure on T_{Δ}QP̂ol(A, 0), and we may define a CDGA map

μ(−, Δ): DR(A) → T_{Δ}QP̂ol(A, 0)

determined on generators by μ(a, Δ) = a, μ(df, Δ) = [Δ, f] for a, f ∈ A. This will be a quasi-isomorphism when Δ is non-degenerate.

Such a construction yields analogues of all the results in §§1, 2, with a cohomological shift. In particular, there is a map from quantisations to ∏_{k≥0} H^{2}(F^{2−2k}DR(A))ℏ^{k}, and the analogue of Propositions 1.40, 2.20 says that a deformation quantisation of a non-degenerate Poisson structure exists whenever it can be quantised to first order. The analogue of self-dual E_{0}-quantisations are DQ algebroid quantisations ℬ_{ℏ} equipped with involutions ℬ_{−ℏ} ≃ ℬ_{ℏ}^{opp} — for details, see [Pri5].

4.3.2. Positively shifted Poisson structures. For n ≥ 1, we can likewise define QP̂ol(A, n) in terms of shifted differential operators or E_{n+1}-Hochschild complexes over Rees(DR(A)). However, formality of E_{n+1} should yield equivalences QP̂ol(A, n) ≃ P̂ol(A, n)⟦ℏ⟧ and QP(A/R, n) ≃ QP(A⟦ℏ⟧/R⟦ℏ⟧, n), making the analogues of Propositions 1.37 and 1.40 less interesting.

[We might also wish to quantise higher analogues of line bundles: for derived stacks, these should be classes in H^{n+2}(𝔛, 𝔾_{m}). For unbounded CDGAs A, the analogue of the strict line bundle A_{b} of Definition 1.8 is the curved E_{n+1}-algebra (A, c) for c ∈ Z^{n+2}(A)

— because A is an E_{∞} algebra, its Lie bracket is trivial, so we still have δ^{2} = [c, −].] 4.3.3. (−2)-shifted Poisson structures. For n ≥ −1, there is a canonical E_{n+2}-algebra quantisation ℏQPol(A, n) of ℏPol(A, n) given by (n + 1)-shifted differential operators, which is equipped with an E_{n+2}-algebra morphism A → ℏQPol(A, n). In order to adapt the techniques of this paper to (−2)-shifted symplectic structures, we would need to start with the data of an E_{0} quantisation

ℏQP̂ol(A, −2) := (ℏP̂ol(A, −2)⟦ℏ⟧, δ + Δ)

of the P_{0}-algebra ℏP̂ol(A, −2), with Δ ∈ QP(ℏPol(A, −2), −1) satisfying Δ(A) = 0 (but not necessarily A-linear) and lifting the canonical Poisson bracket on ℏP̂ol(A, −2).

In particular, the condition Δ(A) = 0 implies Δ(1) = 0, so ℏQP̂ol(A, −2) would be a BV_{∞}-algebra. It would thus suffice to have a right 𝒟-module structure on A, or equivalently a left 𝒟-module structure on the dualising bundle, and to define ℏQP̂ol(A, −2) in terms of the right de Rham complex of A, as in [Sch].

Given a (−2)-shifted Poisson structure π ∈ P(A, −2), we then define an E_{−1} quantisation of π to be a lift of ℏπ to an element ℏS of the BV_{∞}-algebra ℏF̃^{2}QP̂ol(A, −2) satisfying the L_{∞} Maurer–Cartan equation, or equivalently the quantum master equation (δ + Δ)e^{S} = 0.

Writing Δ_{S} := ad_{(e^{−S})}Δ+δS, define the centre T_{S}QP̂ol(A, −2) of S to be the complex (ℏP̂ol(A, −2)⟦ℏ⟧, δ + Δ_{S}). We could then define a compatibility map μ by composing the map

DR′(A) → DR′(Pol(A, −2)) ⟶^{μ(−,Δ_{S})} T_{Δ_{S}}QP̂ol(Pol(A, −2), −1)

from this paper with the evaluation of differential operators at 1. Since Δ_{S}(1) = 0 by the quantum master equation, this would then give a filtered map

μ(−, S): DR′(A) → T_{S}QP̂ol(A, −2),

and we could then say that a (−2)-shifted pre-symplectic structure ω is compatible with S when μ(ω, S) ≃ ℏ^{2} \frac{∂S}{∂ℏ}.

To first order, we have Δ_{S} ≅ Δ + {π, −}; since Δ(A) = 0, the map μ(−, S): DR′(A) → T_{S}QP̂ol(A, −2)/G^{1} = T_{π}P̂ol(A, −2) is then just the compatibility map μ(−, π) from [Pri4].

4.3.4. Self-duality. In §4, the key to self-duality for E_{0} quantisations of \sqrt{ω}_{X} is the filtered involution (−)^{t}: 𝒟_{X}(\sqrt{ω}_{X}) ≃ 𝒟_{X}(\sqrt{ω}_{X})^{opp}. For n ≥ 0, HKR isomorphisms mean that the analogue of 𝒟_{A} is the higher Hochschild cohomology complex HH^{E_{n+1}}(A) with its E_{n+2}-algebra structure. In order to define self-dual quantisations, we would thus need a filtered involution (−)^{t}: HH^{E_{n+1}}(A) ≃ HH^{E_{n+1}}(A)^{opp}, lifting the P_{n+2}-algebra involution of P̂ol(X, n) given by (−1)^{m} on m-vectors.

Of course, when n > 0, the equivalence QP̂ol(A, n) ≃ P̂ol(A, n)⟦ℏ⟧ coming from formality of E_{n+1} allows us to transfer the involution on P̂ol(A, n) to QP̂ol(A, n). We then have QP̂ol(A, n)^{sd} ≃ P̂ol(A, n)⟦ℏ^{2}⟧, giving a sense in which the canonical quantisations coming from formality of E_{n+1} are self-dual.

Whereas Verdier self-duality for a line bundle ℒ is an involutive equivalence ℒ ≃ RHom_{𝒪_{X}} (−, ω_{X}), the obvious notion of self duality for an algebroid A is an involution A ≃ A^{opp}. When n = 0, an involutive filtered E_{2}-equivalence on the Hochschild complex of X gives an analogue of Proposition 4.5, generating self-dual quantisations from symplectic structures. This amounts to looking for DQ algebroid quantisations A equipped with involutions A(−ℏ) ≃ A(ℏ)^{opp} deforming a chosen contravariant involution on the Picard algebroid (or even any 2-line bundle). Such involutions correspond to line bundles, the obvious choices being RHom_{𝒪_{X}} (−, 𝒪_{X}) and RHom_{𝒪_{X}} (−, ω_{X}), and involutive 2-line bundles are μ_{2}-gerbes. For more details, see [Pri5].

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