Homological Mirror Symmetry for Hypertoric Varieties I

We consider homological mirror symmetry in the context of hypertoric varieties, showing that appropriate categories of B-branes (that is, coherent sheaves) on an additive hypertoric variety match a category of A-branes on a Dolbeault hypertoric manifold for the same underlying combinatorial data. For technical reasons, the category of A-branes we consider is the modules over a deformation quantization (that is, DQ-modules). We consider objects in this category equipped with an analogue of a Hodge structure, which corresponds to a $\mathbb{G}_m$-action on the dual side of the mirror symmetry. This result is based on hands-on calculations in both categories. We analyze coherent sheaves by constructing a tilting generator, using the characteristic $p$ approach of Kaledin; the result is a sum of line bundles, which can be described using a simple combinatorial rule. The endomorphism algebra $H$ of this tilting generator has a simple quadratic presentation in the grading induced by $\mathbb{G}_m$-equivariance. In fact, we can confirm it is Koszul, and compute its Koszul dual $H^!$. We then show that this same algebra appears as an Ext-algebra of simple A-branes in a Dolbeault hypertoric manifold. The $\mathbb{G}_m$-equivariant grading on coherent sheaves matches a Hodge grading in this category.


Introduction introduction
Toric varieties have proven many times in algebraic geometry to be a valuable testing ground. Their combinatorial flavor and concrete nature has been extremely conducive to calculation. Certainly this is the case in the domain of homological mirror symmetry (see [Abo07,FOOO09]).
Toric varieties have a natural quaternionic generalization (also called a hyperkähler analogue [Pro04]) which we call hypertoric varieties M (some other places in the literature, they are called "toric hyperkähler varieties"). Just as toric varieties can be written as Kähler quotients of complex vector spaces, hypertoric varieties are hyperkähler quotients by tori (see Definition 2.1).
In this paper, and future sequels, we will study the behavior of these, and related varieties, under homological mirror symmetry. Since these are hyperkähler varieties, their mirrors have a very different flavor from those of toric varieties. They are very close to being self-mirror, but this is not quite correct; instead the mirror seems to be a Dolbeault hypertoric manifold, as defined by Hausel and Proudfoot. A Dolbeault hypertoric manifold is a multiplicative analogue of a hypertoric variety. When we need to distinguish, we will call usual hypertoric varieties additive.
Homological mirror symmetry is typically understood to mean an equivalence between the derived category of coherent sheaves (or B-branes) on an algebraic variety and the Fukaya category (the A-branes) of a related symplectic manifold. In this paper we are concerned with a different, but closely related, equivalence.
On the B-side, we consider the derived category of coherent sheaves on the variety M which are equivariant under a conic G m -action. Not all coherent sheaves can be made equivariant, so this is a restriction on our equivalence. On the other hand, we will see that this subcategory has many extra structures which make it interesting in its own right.
On the A-side, we replace the Fukaya category with a certain category of deformation quantization modules on the Dolbeault hypertoric manifold. The latter is conjecturally equivalent to an appropriate version of the Fukaya category, and recent work of authors such as Ganatra-Pardon-Shende [GPSb, GPSa] and Eliashberg-Nadler-Starkston appears to be on the verge of making this a theorem. In fact, we show that the derived category of equivariant coherent sheaves matches the category µm of DQ-modules endowed with a 'microlocal mixed Hodge structure.' The latter carries the information of G m -equivariance: under our ultimate equivalence, shift of G m -weight matches with Tate twist of the Hodge structure.
This equivalence, the central result of our paper, follows from Theorem 4.41 and Corollary 4.42. It may be thought of as homological mirror symmetry for two subcategories of the A and B branes, both of which are enriched with suitable notions of G m -equivariance. The reader may compare with [BMO, MO] and their sequels, where the same G m -action plays a key role.
Moreover, our equivalence identifies interesting t-structures on either side: the 'exotic t-structure' on the derived category of coherent sheaves arising from a quantization in characteristic p, and the natural t-structure on DQ-modules.
As a result of our use of DQ-modules as a substitute for the Fukaya category, this paper contains little about Lagrangian branes, pseudo-holomorphic disks and other staples of symplectic geometry. The reader may wish to compare with the interesting recent preprint [LZ18], which appeared a few days before this paper and treats the problem of non-equivariant mirror symmetry for hypertoric varieties from the perspective of SYZ fibrations.
We will come to understand coherent sheaves on M by using a tilting generator. This is a vector bundle T such that Ext(T, −) defines an equivalence of categories D b (Coh(M)) D b (H -mod op ) where H = End(T).
Our construction of a tilting bundle follows a recipe of Kaledin [Kal08]; thus, the algebra H is an analogue in our context of Bezrukavnikov's noncommutative Springer resolution [Bez06]. While this construction springs from geometry in characteristic p, and the tilting property is checked using this approach, the tilting generators we consider are sums of line bundles and have a simple combinatorial construction, as does the endomorphism ring H. This endomorphism ring inherits a grading from a G m -equivariant structure on T and is Koszul with respect to it. Thus, the category of G m -equivariant coherent sheaves on M is controlled by the derived category of graded H-modules, or equivalently by graded modules over H ! , its Koszul dual. It is this Koszul dual that has a natural counterpart on the mirror side.
These results should generalize in a number of interesting directions. We expect that there is a non-equivariant version of mirror symmetry between complex structures on the Dolbeault hypertoric manifold related by hyperkähler rotation. It is also worth noting that the variety M is the Coulomb branch (in the sense of [BFN18]) with gauge group given by a torus, and that D is expected to be a hyperkähler rotation of the K-theoretic version of this construction. Thus, it is natural to consider how these constructions can be generalized to that case. The analogous calculation of a tilting bundle with explicit endomorphism ring can be generalized in this case, as the second author will shows in [Web], but it is very difficult to even conjecture the correct category to consider on the A-side.
One key motivation for interest in hypertoric varieties is that provide excellent examples of conic symplectic singularities (see [BPW16,BLPW16]), which can be understood in combinatorial terms. Considerations in 3-d mirror symmetry [BLPW16] and calculations in the representation theory of its quantization lead Braden, Licata, Proudfoot and the second author to suggest that these varieties should be viewed as coming in dual pairs, corresponding to Gale dual combinatorial data. In particular, the categories O attached to these two varieties are Koszul dual [BLPW10,BLPW12]. An obvious question in this case is how the categories we have considered, such as coherent sheaves, can be interpreted in terms of the dual variety (they are certainly not equivalent or Koszul dual to the coherent sheaves on the dual variety, as some very simple examples show). Some calculations in quantum field theory suggest that they are the representations of a vertex algebra constructed by a BRST analogue of the hyperkähler reduction, but this is definitely a topic which will need to wait for future research.
Detailed outline of the argument Part 1: coherent sheaves and characteristic p quantizations of the additive hypertoric variety. Section 2.1 defines the additive hypertoric variety M. In Section 2.2 we fix a field K of characteristic p, and review the relation between the quantization of M K , called A λ K , and coherent sheaves on M K . In Section 3.1, we introduce a category of modules A λ K -mod o , along with its graded counterpart A λ K -mod D o . All these objects depend on a quantization parameter λ. In Sections 3.3, 3.2 and 3.5 we classify the projective pro-objects P x of A λ K -mod D o , which also yields a classification of simple objects L x . Both projectives and simples are indexed by the chambers of a periodic hyperplane arrangement A per λ defined in 3.8. We compute the endomorphism algebra End(⊕ x P x ) in 3.13. The latter contains a ring of power series S as a central subalgebra, and we define a variant H λ K (Definition 3.14) in which S is replaced by the corresponding polynomial ring S. We find that A λ K -mod D o is equivalent to the subcategory of H λ Kmodules on which S acts nilpotently. The algebra H λ K has a natural lift to Z, written H λ Z , which we will use to compare with characteristic zero objects on the mirror side.
Corollary 3.19 shows that H λ Z and H λ K are Koszul. We compute the Koszul dual algebra H ! λ,K = Ext(⊕ x L x ) (Definition 3.20 and Theorem 3.21). In Section 3.9 we describe the ungraded category A λ K -mod o in terms of the graded one. Its simples and projectives are indexed by the toroidal hyperplane arrangement A tor λ obtained as the quotient of A per λ by certain translations. We describe the corresponding algebrasH λ K = End( x P x ) andH ! λ,K = Ext( x L x ), where the sums now range over simples (resp projectives) for A λ K -mod o . In Section 3.10 we use the above results to produce a tilting bundle T λ on M with endomorphism ring End(T λ ) =H λ . This gives equivalences (from Corollary 3.37 and Proposition 3.39, respectively): whereH ! λ,Q -perf is the category of perfect dg-modules over this ring.
Remark 1.1. Note that throughout, we will always endow the bounded derived category D b of an abelian category with its usual dg-enhancement using injective resolutions; thus if we write D b (A) C for an abelian category A and a dg-category C, we really mean that this dg-enhancement is quasi-equivalent to C.
Part 2: deformation quantization and microlocal mixed Hodge modules on the Dolbeault hypertoric manifold. The second half of our paper begins with a definition of the Dolbeault hypertoric manifold D (4.3), depending on a moment map parameter ζ. The complex manifold D contains a collection of complex Lagrangian submanifolds X x indexed by the chambers of a toroidal hyperplane arrangement B tor ζ (Definition 4.10 and Proposition 4.11). We also introduce the periodic hyperplane arrangement B per ζ , which bears an analogous relation to the universal cover D of D. In Section 4.5, we define a sheaf O φ of C(( ))-algebras on D quantising the structure sheaf, and for each X x we define a module L x over O φ supported on X x . We consider the subcategory dq of O φ -modules generated by the simple DQ-modules L x , together with the category dg-category DQ of complexes in dq.
When the parameters λ and ζ correspond in a suitable way, the arrangements A tor λ and B tor ζ are identified. We hence have a bijection of chambers, and a corresponding bijection of isomorphism classes of simple objects for the categories dq and A λ K -mod o . Moreover, Theorem 4.27 shows that the Ext algebras of the simples in both categories share a common integral form: Ext(⊕ x L x ) H ! λ,Z ⊗ C(( )). Comparing with the results from the previous section, we see that at this stage there are three obstacles to an equivalence of DQ with D b (Coh(M)).
(1) While D b (Coh(M)) can be defined over any base field, DQ has base ring C(( )).
(2) It is not clear that Ext DQ (⊕ x∈Λ(λ) L x ) is formal as a dg-algebra, which we would need to define a fully-faithful functor DQ → D b (Coh(M)). (2') It is unclear what structure on DQ corresponds to the G m -action on M discussed earlier.
The second and third issue will prove to be one and the same, hence our funny numbering. To resolve these issues, we introduce a new graded abelian category µm (Definition 4.40), and a corresponding triangulated category D b (µM). An object of µm is essentially a O φ -module, such that for each lagrangian X x , the restriction to a Weinstein neighborhood of X x is equipped with the structure of a mixed Hodge module. These structures are required to be compatible in a natural sense whenever two components intersect. In Section 4.11, we show that each object L x has a natural lift to µm, and moreover that any simple object of µm is isomorphic to such a lift. The resulting forgetful functor µm → dq retroactively motivates our definition of dq. Moreover, the grading on µm arising from the mixed Hodge structures allows to conclude formality of the relevant Ext algebra. This allows us to produce a functor D b perf (Coh(M R ) o ) → DQ which becomes an equivalence after tensoring the source with C(( )), and an equivalence of graded categories  [Pro08].
Consider a split algebraic torus T over Z of dimension k (that is, an algebraic group isomorphic to G k m ) and a faithful linear action of T on the affine space A n Z , which we may assume is diagonal in the usual basis. We let D G n m be the group of diagonal matrices in this basis, and write G := D/T.
We have an induced action of T on the cotangent bundle T * A n Z A 2n Z . We'll use z i for the usual coordinates on A n Z , and w i for the dual coordinates. This action has an algebraic moment map µ : T * A n Z → t * Z , defined by a map of polynomial rings Z[t Z ] → Z[z 1 , . . . , z n , w 1 , . . . , w n ] sending a cocharacter χ to the sum n i=1 ǫ i , χ z i w i , where ǫ i is the character on D defined by the action on the ith coordinate line, and −, − is the usual pairing between characters and cocharacters of D.
For us, the main avatar of this action is the (additive) hypertoric variety. This is an algebraic hamiltonian reduction of T * A n Z by T. It comes in affine and smooth flavors, these being the categorical and GIT quotients (respectively) of the scheme-theoretic fiber µ −1 (0) by the group T. More precisely, fix a character α : T → G m whose kernel does not fix a coordinate line.
f:hypertoric Definition 2.1. For a commutative ring K, we let and where t is an additional variable of degree 1 with T-weight −α.
Both varieties carry a residual action of the torus G = D/T, and an additional commuting action of a rank one torus S := G m which scales the coordinates w i linearly while fixing z i .
We say that the sequence T → D → G is unimodular if the image of any tuple of coordinate cocharacters in d Z := Lie(D) Z forming a Q-basis of g Q := Lie(G) Q also forms a Z-basis of g Z .
Let π : M C → N C be the natural map. If we assume unimodularity, then M C is a smooth scheme and π defines a proper T × S-equivariant resolution of singularities of N C . Together with the algebraic symplectic form on M C arising from Hamiltonian reduction, this makes M C a symplectic resolution. Many elements of this paper make sense in the broader context of symplectic resolutions, although we will not press this point here. In the non-unimodular case, M C may have orbifold singularities.
In the description given above, N C appears as the Higgs branch of the N = 4 threedimensional gauge theory attached to the representation of T C on C n . However, it is more natural from the perspective of what is to follow to see N C as the Coulomb branch of the theory attached to the dual action of (D/T) ∨ on C n , in the sense of Braverman-Finkelberg-Nakajima [BFN18,Nak16]. This leads to a different presentation of the hypertoric enveloping algebra, which will be useful for understanding its representation theory. In particular, the multiplicative hypertoric varieties we'll discuss later appear naturally from this perspective as the Coulomb branches of related 4 dimensional theories.
uantizations uantizations 2.2. Quantizations. The ring of functions on the hypertoric variety N Z has a quantization which we call the hypertoric enveloping algebra. We construct it by a quantum analogue of the Hamiltonian reduction which defines M Z . Consider the Weyl algebra W n generated over Z by the elements z 1 , . . . , z n , ∂ 1 , . . . , ∂ n modulo the relations: It is a quantization of the ring of functions on T * A n Z . The torus D acts on W n , scaling z i by the character ǫ i and ∂ i by ǫ −1 i . It thus determines a decomposition into weight spaces . Via the embedding T → D, W n carries an action of the torus T. To this action one can associate a non-commutative moment map, i.e. a map µ q : Z[t Z ] → W n such that [µ q (χ), −] coincides with the action of the Lie algebra t Z . This property uniquely determines µ q up to the addition of a character in t * Z . We make the following choice.
It's worth nothing that in the formula above, we have broken the symmetry between z i and ∂ i ; it would arguably be more natural to use h mid i , but this requires inserting a lot of annoying factors of 1/2 into formulas, not to mention being a bit confusing in positive characteristic.
defHET Definition 2.2. The hypertoric enveloping algebra A Z is the subring W T n ⊂ W n invariant under T. We'll also consider the central quotients of this algebra associated to a character λ ∈ t * Z , given by We will often abbreviate "hypertoric enveloping algebra" to HEA. Let A K := A Z ⊗ Z K be the base change of this algebra to a commutative ring K. The algebra A C was studied extensively in [BLPW12, MV98]. The algebra A K when K has characteristic p was studied in work of Stadnik [Sta13]. Fix a field K of characteristic p for the rest of the paper.
Unlike W n itself, or its base change to a characteristic 0 field, the ring W n ⊗ Z F p has a "big center" generated by the elements z p i , ∂ p i . This central subring can be identified with the function ring H 0 ( 2.3. Coulomb presentation. The algebra A K has a different presentation which is more compatible with the subalgebra K[h ± i ]. The action of D on A K determines a decomposition into weight subspaces. Since A K = W T n , its weights lie in t ⊥ Z = g * Z : For each a ∈ t ⊥ Z , we let Up to scalar multiplication, this is the unique element in A K [a] in of minimal degree. Note that each weight space A K [a] is a module over the D-invariant subalgebra generated by the h + i . Let: and m(a) for a ∈ g * Z , subject to the relations: We call this is the Coulomb presentation, since it matches the presentation of the abelian Coulomb branch in [BFN18, (4.7)], and shows that the algebra A K can also be realized using this dual approach. As mentioned in the introduction, the techniques of this paper generalize to Coulomb branches with non-abelian gauge group as well, whereas it seems very challenging to generalize them to Higgs branches with nonabelian gauge group (that is, hyperkähler reductions by non-commutative groups). char-p-local 2.4. Characteristic p localization. Following [Sta13], in this section we exploit the large center of quantizations in characteristic p to relate modules over A λ K with coherent sheaves on M (1) K . Roughly speaking, upon restriction to fibers of π : M (1) K → N (1) K , the quantization becomes the algebra of endomorphisms of a vector bundle, and thus Morita-equivalent to the structure sheaf of the fiber.
Note that this theorem includes the existence of an injection H 0 (N (1) K . Work of Stadnik shows that the Azumaya algebra A λ splits on fibers of this map after field extension. Fix ξ ∈ d (1) K . Possibly after extending K, we can choose ν such that ν p − ν = ξ, and define the splitting bundle as the quotient A λ / n i=1 A λ (h + i − ν i ); this left module is already supported on the fiber µ −1 (ξ), since We can thicken this to the formal neighborhood of the fiber µ −1 ( ξ) by taking the inverse limit Q ν := lim is an isomorphism.
The sheaf A λ is not globally split; it has no global zero-divisor sections. It still has a close relationship with a tilting vector bundle on M (1) K . We'll fix our attention on the case where ξ = 0, so ν i ∈ F p .
Let T K be a S-equivariant locally free coherent sheaf on M (1) Proposition 2.6. For p sufficiently large and ν i generic, the sheaf T K is a tilting generator on M K and has a lift T Q which is a tilting generator on M Q ; that is, We will later calculate the sheaf T K , once we understand A λ K a bit better. 3. The representation theory of hypertoric enveloping algebras -theory-heas ght-functors 3.1. Module categories and Weight functors. Recall that we have a short exact sequence of tori T → D → G. A λ K is a quotient of W T n , and thus carries a residual action of G, which we will now use to study its modules.
Let o ∈ N (1) be the point defined by z i = w i = 0, i.e. the unique K-valued S-fixed point of N (1) . The following category will play a central role in this paper.
In fact, we will first study the following closely related category.

Definition 3.2. Let A λ
K -mod D o be the category of modules in A λ K -mod o which are additionally endowed with a compatible D-action, such that T acts via the character λ, and the action of d i ∈ d Z satisfies eq:nilpotent eq:nilpotent (3.5) Note that the difference s i = h + i − d i acts centrally on such a module, since the adjoint action of h + i on A λ K agrees with the action of d i . The operator s i is thus the nilpotent part of the Jordan decomposition of h ± i . The operators s i define an action of the polynomial ring U K (d), which factors through U K (g) since elements of t act by zero. This extends to an action of the completion of U K (g), since s i acts nilpotently by (3.5). Definition 3.3. Let S := U K (g), and let S be its completion at zero.
Let g * ,λ Z ⊂ d * Z be the g * Z -coset of characters of D whose restriction to T coincides with λ. It indexes the D-weights which can occur in an object of A λ K -mod D o . We can construct projectives objects in a slight enlargement of A λ K -mod D o by working with the exact functors picking out weight spaces. That is, for each a ∈ g * ,λ Z , we consider the functor which associates to an object M ∈ A λ K -mod D o the following vector space: W a (M) := {m ∈ M | m has D-weight a}.
Note that even though we are working in characteristic p, the D-weights are valued in g * ,λ Z ⊂ Z n . This functor is exact, and we will show that it is pro-representable. To construct the projective object that represents this functor, we consider the filtration of it by This is endowed with the usual induced topology, and it is a pro-weight module in the sense that its weight spaces are pro-finite dimensional. This is a projective object in the category A λ K -mod D of complete topological A λ K -modules M with compatible D-action in the sense that lim That is, s i acts topologically nilpotently on each D-weight space. This is equivalent to (3.5) if the topology on M is discrete.
In the arguments below, Hom and End will be interpret to mean continuous homomorphisms compatible with D; all objects in A λ K -mod D o will be given the discrete topology, so continuity is a trivial condition for homomorphisms between them.
0 and is thus free over the quotient ring S/ S · s N i . Taking the inverse limit, we see that every weight space of Q a is a free module of rank 1 over S.
Corollary 3.6. We have an isomorphism of rings End(Q a ) W a (Q a ) S. Since S is local, the module Q a is indecomposable (in the category A λ K -mod D ). secisomQ 3.3. Isomorphisms between projectives. In this section, we determine the distinct isomorphism classes of weight functors, i.e. we determine all isomorphisms between the pro-projectives Q a . As we will see, there are typically many distinct weights a ∈ g * ,λ Z that give isomorphic functors. By the results of the previous section, the space W a (Q a+b ) = Hom(Q a , Q a+b ) is free of rank one over S, with generator m(b). Likewise, Hom(Q a+b , Q a ) is generated by m(−b). Thus in order to verify whether Q a and Q a+b are isomorphic, it is enough to check whether the composition m(−b)m(b), viewed as an endomorphism of Q a , is an invertible element of the local ring End(Q a ) S.
By (2.4), we have that where the right-hand side is a product of factors of the form h + i + k with k an integer between 1 2 and b i + 1 2 . To check whether h + i + k defines an invertible element of S, it is enough to compute its action on the weight-space of weight a, on which h i acts by a i + s i . The resulting endomorphism h + i + k = s i + (a i + k) is invertible if and only if k + a i 0 (mod p).
The number of non-invertible factors (each equal to s i ) in [h i ] (−b i ) is therefore the number of integers k divisible by p lying between a i + 1/2 and a i + b i + 1/2. We denote it by δ i (a, a + b).
We can sum up the above computations as follows. Let where y is a formal variable and k ∈ K. Note that q(s i , a i + j)(h + i + j) acts on a D-weight space of weight a by 1 if a i + j is not divisible by p and by s i if it is. Let It is a generator of the S-module W a (Q a+b ). Note that this expression breaks the symmetry between positive and negative; if b i ≤ 0 for all i, then c b a = m(b), since all the products in the definition are over empty sets. c-reverse Lemma 3.7.
Note that for each index i only one of the products is non-unital, depending on the sign, and in either case, we obtain the product of q(s i , a i + j) ranging over integers lying between a i + 1/2 and a i + b i + 1/2. As we noted earlier, [h i ] (−b i ) is the product of h i + j with j ranging over this set. Thus, we obtain the product over this same set of It remains for us to describe which pairs a, a ′ satisfy δ i (a, a ′ ) = 0 for all i and thus index isomorphic projective modules.
fAhyperplane Definition 3.8. Let A per λ be the periodic hyperplane arrangement in g * ,λ Z defined by the hyperplanes d i = kp − 1/2 for k ∈ Z and i = 1, .., n.
By definition, δ i (a, a ′ ) is the minimal number of hyperplanes d i = kp − 1/2 crossed when travelling from a to a ′ . Given x ∈ Z n , let

We have shown
Theorem 3.9. We have an isomorphism Q a Q a ′ if and only if a, a ′ ∈ ∆ x for some x. Let Thus, Λ(λ) canonically parametrizes the set of indecomposable projective modules in the pro-completion of A λ K -mod D o . It follows that Λ(λ) also canonically parametrizes the simple modules in this category.
Let us call the parameter λ smooth if there is a neighborhood U of λ in R ⊗ g * ,λ Z such that for all λ ′ ∈ U, we have Λ(λ) = Λ R (λ ′ ). In particular, if λ is smooth, then the hyperplanes in A per λ must intersect generically.

A taxicab metric.
We can endow Λ(λ) with a metric given by the taxicab distance |x − y| 1 = i |x i − y i | for all x, y ∈ Λ(λ). We can add a graph structure to Λ(λ) by adding in a pair of edges between any two chambers satisfying |x − y| 1 = 1; generically, this is the same as requiring that ∆ R x and ∆ R y are adjacent across a hyperplane. We say that this adjacency is across i if x, y differ in the ith coordinate. For every x, let α(x) be the set of neighbors of x in Λ(λ). Generically, this is the same as the number of facets of ∆ R x ; we let α i (x) ⊂ α(x) be those facets adjacent across i. Note that in some degenerate cases, we may have that is typically 0 or 1, but could be 2.
Lemma 3.11. The module P x has a unique simple quotient L x , and L x for x ∈ Z n such that Proof. We show that Q a has a unique simple quotient by showing the sum of two proper submodules is proper; this then shows that there is a unique maximal proper submodule, and L x is the quotient by it.
This shows that the sum of two proper submodules is proper, and so L x is well-defined.
Using the isomorphism Q a Q b if a, b ∈ ∆ x , we can extend this to the observation that a submodule M ⊂ P x is proper if and only if W a (M) ⊂ mW a (P x ) for all a ∈ ∆ x . By Lemma 3.7, we can check that there is a unique submodule M in P x such that By the observation above, this must be the maximal proper submodule, so L x = P x /M. This shows that L x has the claimed dimensions of weight spaces. Furthermore, this shows that we can recover the set ∆ x for L x , so we must have L x L y if x y.
For any simple L, we must have W a (L) 0 for some a. This induces a map P x → L where a ∈ ∆ x . Since L x is the unique simple quotient of P x , this shows that L x L. This shows that they give a complete list and completes the proof. assicexample Example 3.12. An interesting example to keep in mind is the following. Let T be the scalar matrices acting on A 3 . In this case, n = 3, k = 1. The space g * ,λ Z is an affine space on which d 1 , d 2 give a set of coordinates, with d 3 related by the relation d 3 = −d 1 − d 2 + λ for some λ ∈ Z. Thus, the hyperplane arrangement that interests us is given by In particular, we have that ∆ x ∅ if and only if, there exist integers a 1 , a 2 such that The values of −a 1 − a 2 + λ for a 1 , a 2 satisfying the first two inequalities range from −( . If λ ≡ −1, −2 mod p, then there are 2, and the parameter λ is not smooth. Of course, the numbers −1 and −2 have another significance in terms of P 2 : the line bundles O(−1) and O(−2) on P 2 are the unique ones that have trivial pushforward. This is not coincidence. Let λ + be the unique integer in the range 0 ≤ λ + < p congruent to λ (mod p) and ) and H 1 (P 2 ; O(λ − )). If λ −1, −2 mod p, then the latter group is trivial, so one of the simple representations is "missing." Note that the final simple can be identified with the first cohomology of the kernel of the
3.6. The endomorphism algebra of a projective generator. Having developed this structure theory, we can easily give a presentation of our category. For each x, y with ∆ x ∅ and ∆ y ∅, we can define c x,y to be c a ′ −a a for a ∈ ∆ y , a ′ ∈ ∆ x . For each i, let resentation1 Theorem 3.13. The algebra x,y Hom(P x , P y ) is generated by the idempotents 1 x and the elements c x,y over S modulo the relation: Note that this relation is homogeneous if deg c x,y = |x − y| 1 and deg s i = 2.
Proof. The relation holds by an easy extension of Lemma 3.7. To see that these elements and relations are sufficient, note that in the algebra H with this presentation, the Hom-space 1 x H1 y is cyclically generated over S by c x,y . The image of c x,y under induced map 1 x H1 y → Hom(P y , P x ) generates the target space over S. Since the target is free of rank 1 as a S-module, the map must be an isomorphism.
definingH Definition 3.14. Let S Z := U Z (g). Let H λ Z be the graded algebra over S Z with presentation given in Theorem 3.13. Let H λ Since x∈Λ(λ) P x is a faithfully projective module, we have the following result.
ndomorphisms Theorem 3.15. The functor defines an equivalence of categories between A λ K -mod D o and the category of finitely generated representations of H λ K , on which each s i acts nilpotently.
In fact, we will see that when λ is smooth, H λ K admits a presentation as a quadratic algebra. We begin by producing some generators. Let These elements correspond to the adjacencies in the graph structure of Λ(λ). Thus, we have a homomorphism from the path algebra of Λ(λ) sending each length 0 path to the corresponding 1 x and each edge to the corresponding c ±i x . We'll be interested in the particular cases of (3.9) which relate these length 1 paths.
Note, we can view this as saying that the length 2 paths that cross a hyperplane and return satisfy the same linear relations as the normal vectors to the corresponding hyperplanes.
, then the corresponding chambers fit together as in the picture below: In this situation, we find that either way of going around the codimension 2 subspace gives the same result, and that more generally any two paths between chambers that never cross the same hyperplane twice give equal elements of the algebra.
If λ is a smooth parameter, then as the following theorem shows, these are the only relations needed.
resentation2 Theorem 3.16. If λ is a smooth parameter, then the algebra x,y Hom(P x , P y ) is generated by the idempotents 1 x and the elements c ±i x for all x ∈ Λ(λ) over S modulo the relations (3.10a-3.10c).
Proof. Since these relations are a consequence of Theorem 3.13, it suffices to show that the elements c ±i x generate, and that the relations (3.9) are a consequence of (3.10a-3.10c).
We show that c ±i x generate c x,y by induction on the L 1 -norm |x − y|. If |x − y| 1 = 1, then c x,y = c ±i y . On the other hand, if |x − y| 1 > 1, then there is some we can consider the line segment joining generic points in∆ x and∆ y , and let x ′ be any chamber this line segment passes through. The smoothness hypothesis is needed to conclude that there is such a chamber that lies in Λ(λ). Since c x,y = c x,x ′ c x ′ ,y , this proves generation by induction.
We must now check that the relations (3.9) are satisfied. First, consider the situation where , and y (0) = x, . . . , y (m) = y a path with the same conditions. These two paths differ by a finite number of applications of the relations (3.10b-3.10c).
What it remains to show is that if x (0) = x, . . . , x (m) = y is a path of minimal length between these points with |x (i) − x (i+1) | 1 = 1, and we have similar paths y (0) = y, . . . , y (n) = u and u (0) = x, . . . , u (p) = u, then product product (3.11) We'll prove this by induction on min(m, n). If m = 0 or n = 0, then this is tautological. Assume m = 1, and x = y + σǫ j for σ ∈ {1, −1}. If σ(y j − u j ) ≥ 0, then η j (x, y, u) = 0, so this follows from the statement about minimal length paths. If σ(y j − u j ) < 0, then η j (x, y, u) = 1, and we can assume that y (1) = x, . . . , y (n) is a minimal length path from x to u. Thus c x,y c y,x · · · c y (n−1) ,y (n) = c x,y (2) · · · c y (n−1) ,y (n) s j as desired. The argument if n = 1 is analogous. Now consider the general case. Assume for simplicity that n ≥ m. Consider the path x (m−1) , y (0) , . . . , y (n) = u. Either this is a minimal path, or by induction, we have that In the former case, by induction, the relation (3.11) for the paths x (0) = x, . . . , x (m−1) and x (m−1) , y (0) , . . . , y (n) = u holds. This is just a rebracketing of the desired case of (3.11). In the latter, after rebracketing, we have applying (3.11) to the shorter paths. atic-duality 3.7. Quadratic duality and the Ext-algebra of the sum of all simple modules. The algebra H λ Z for smooth parameters has already appeared in the literature in [BLPW10]; it is the "A-algebra" of the hyperplane arrangement defined by d i = pk − 1/2 for all k ∈ Z. This is slightly outside the scope of that paper, since only finite hyperplane arrangements were considered there, but the results of that paper are easily extended to the locally finite case. In particular, we have that the algebra H is quadratic, and its quadratic dual also has a geometric description, given by the "B-algebra." We will use this to produce a description of the Ext-algebra of the sum of all simple representations of H λ Z . If we fix an integer m, we may consider the hyperplane arrangement given by be the A-algebra associated to this arrangement as in [BLPW12, §8.3] (in that paper, it is denoted by A(η, −)). We leave the dependence on λ and the ground ring implicit.
By definition, H [m] is obtained by considering the chambers of the arrangement we have fixed above, putting a quiver structure on this set by connecting chambers adjacent across a hyperplane, and then imposing the same local relations (3.10a-3.10c). One result which will be extremely important for us is:

between homogeneous elements of degree q and an isomorphism Ext
Proof. An element of (1 x H [m] 1 y ) q can be written as a sum of length n paths from x to y.
is clearly surjective in this case, and injective as well, since any relation used in H is also a relation in H [m] .
Thus, if we take a projective resolution of L x over H [m] and tensor it with H λ , we can choose m sufficiently large that the result is still exact in degrees below 2q. Since H [m] is Koszul, with global dimension ≤ 2n, every simple over H [m] has a linear resolution of length less than ≤ 2n. This establishes that the tensor product complex is a projective resolution for m ≫ 0.
This establishes that we have an isomorphism Ext In fact, since the Koszul dual of a quadratic algebra is its quadratic dual, we can use this result to identify the Koszul dual of H λ . Continue to assume that Λ(λ) is smooth. If we dualize the short exact sequence x for x, x ± ǫ i ∈ Λ(λ) with trivial differential and subject to the quadratic relations: ( For each x and each i, we have: ll-crossbang ll-crossbang (3.12a)

Note that this implies that if
(3) If x and u are chambers such that |x − u| = 2 and there is only one length 2 path (x, y, u) in Λ(λ) from x to u, then . We can map this to d K by sending the unit vector corresponding to u to s i where x = u ± ǫ i . The relations are the preimage of t K .
By standard linear algebra, the annihilator of a preimage is the image of the annihilator under the dual map. Thus, we must consider the dual map t ⊥ K ⊂ d * K → K α(x) , and identify its image with the relations in H ! . These are exactly the relations imposed by taking linear combinations of the relations in (3.12a) such that the RHS is 0.
ingoutKoszul Corollary 3.22. We have a quasi-isomorphism of dg-algebras Ext(⊕ x L x , ⊕ y L y ) H ! λ,K , with individual summands given by Ext(L y , L x ) e x H ! λ e y . Proof. Here, we apply Theorem 3.15; this equivalence of abelian categories implies that we can replace the computation of Ext A λ (L x , L y ) with that of the corresponding 1-dimensional simple modules over H λ in the subcategory of modules on which s i acts nilpotently.
If we instead did the same computation in the bounded derived category of all finitely generated modules, then we would know the result is e x H ! λ e y by Koszul duality. The formality of the Ext-algebra follows from the consistency of A ∞ -operations with the internal grading, so this is a quasi-isomorphism of dg-algebras. Thus, we need to know that the inclusion of the category on which s i acts nilpotently induces a fully-faithful functor on derived categories.
For this, it's enough to show that every pair of objects A, B has an object C (all in the subcategory) and a surjective morphism ψ : C → A such that the induced map Ext n (A, B) → Ext n (C, B) is trivial for all n. We can accomplish this with C a sum of quotients of H λ 1 z 's by the ideal generated by s N i for N ≫ 0; this is clear for degree reasons if A and B are gradeable, and since gradeable objects dg-generate, this is enough.
This gives us a combinatorial realization of the Ext-algebra of the simple modules in this category. We can restate it in terms of Stanley-Reisner rings as follows.
For every x, y, we have a polytope∆ R x ∩∆ R y , which has an associated Stanley-Reisner ring SR(x, y) K . The latter is the quotient of K[t 1 , . . . , t n ] by the relation that t i 1 · · · t i k = 0 if the intersection of∆ R x ∩∆ R y with the hyperplanes defined by a i j = pn for n ∈ Z is empty. Let SR(x, y) K be its quotient modulo the system of parameters defined by the image of t ⊥ K . We can define SR(x, y) Z and SR(x, y) Z by the same prescription, replacing K by Z everywhere. In [BLPW10,4.1], the authors define a product on the sum SR Z ⊕ x,y∈Λ SR(x, y) Z , which they call the "B-algebra." The same definition works over K.
The result [BLPW10,4.14] shows that this algebra is isomorphic to the "A-algebra" (that defined by the relations (3.10a-3.10c)) for a Gale dual hyperplane arrangement. Unfortunately, for a periodic arrangement, the Gale dual is an arrangement on an infinite dimensional space, which we will not consider. We can easily restate this theorem in a way which will generalize for us. Assume that λ is a smooth parameter.
terpretation 3.8. Interpretation as the cohomology of a toric variety. For our purposes, the key feature of the quadratic dual of H λ Z is its topological interpretation, which is exactly as in [BLPW10,§4.3]. This interpretation will allow us to match the Ext-algebras which appear on the mirror side, in the second half of this paper.
Indeed, the periodic hyperplane arrangement A per λ defines a tiling of g * ,λ R by the polytopes∆ R x . To each such polytope we can associate a G-toric variety X x [CdS01, Chapter XI]. Each facet of the polytope defines a toric subvariety of X x . In particular, the facet ∆ R x ∩ ∆ R y defines a toric subvariety X x,y of both X x and X y . Moreover, the Stanley-Reisner ring SR(x, y) K is identified with H * G (X x,y ; K), and the quotient SR(x, y) K is identified with H * (X x,y ; K). Composing this identification with Proposition 3.23, we have an identification In this presentation, multiplication in the Ext-algebra is given by a natural convolution on cohomology groups [BLPW10, §4.3]. ec:degrading 3.9. Degrading. So far, we have only considered A λ K -modules which are endowed with a D-action. Now, we use the results of the preceding sections to describe the category A λ K -mod o of modules without this extra structure.

Proposition 3.24. Assume that L is a simple module in the category
In L, there thus must exist a simultaneous eigenvector v for all h + i 's, and a such that This shows that L x gives a complete list of simples. The module P x represents the a generalized eigenspace of h + i , and thus still projective. In fact, there are redundancies in this list, but they are easy to understand.
We writex for the image of x inΛ(λ). Recall that g * ,λ Z is a torsor for the lattice g * Z . The action of the sublattice p · g * Z preserves the periodic arrangement A per λ . The quotient A tor λ = A per λ /p · g * Z is an arrangement on the quotient g * ,λ Z /p · g * Z , andΛ(λ) is the set of chambers of A tor λ . Example 3.26. In the setting of Example 3.12, A tor λ has three chambers. A set of representatives is given by those chambers of the periodic arrangement lying within the pictured square.
Theorem 3.27. As A λ K -modules L x L y if and only if x ∼ y. That is, the simple modules in A λ K -mod o are in bijection withΛ(λ). Proof. If x ∼ y, then P x and P y are canonically isomorphic as A λ K -modules, since (3.6) is only sensitive to the coset of a under the action of p · t ⊥ Z . It follows that L x L y as A λ K -modules. On the other hand, if L x L y as A λ K -modules, their weights modulo p must agree. This is only possible if x| t Z = y| t Z .
When convenient, we will write Lx for the simple attached tox ∈Λ(λ). We can understand the Ext-algebra of simples using the degrading functor D : K -mod o which forgets the action of D. Theorem 3.28. We have a canonical isomorphism of algebras Proof. This is immediate from the fact that P x remains projective in A λ K -mod o , so the degrading of a projective resolution of L x remains projective.
One can easily see that this implies that, just like A λ K -mod D o , the category A λ K -mod o has a Koszul graded lift, since the coincidence of the homological and internal gradings is unchanged.
We can deduce a presentation of Lx Indeed, we think of H ! λ,K as the path algebra of the quiver Λ(λ) (over the base ring U K (t * )) satisfying the relations in definition 3.20, and then apply the quotient map toΛ(λ), keeping the arrows and relations in place. This is well-defined since the relations (3.12a-3.12c) are unchanged by adding a character of G to x.
Likewise, we have the following definition, describing the endomorphism algebra of the projectives.  (3.10a-3.10c). LetH λ Z be the natural lift to Z. Example 3.30. We continue Example 3.12. The setΛ(λ) has 3 elements corresponding to the chambers A where

. We have adjacencies between A and B across 3 hyperplanes, and between B and C across 3 hyperplanes, with none between A and C.
Thus, our quiver is We use x i to the path from A to B across the d i hyperplane, and y i the path from C to B across the d i hyperplane. Our relations thus become: Note that there are only finitely many elements ofΛ(λ). In fact, the number of such elements has an explicit upper bound. A basis of the inclusion T ⊂ D is a set of coordinates such that the corresponding coweights form a basis of d Q /t Q . For generic parameters, taking the intersection of the corresponding coordinate subtori defines a bijection of the bases with the vertices of A tor λ . Lemma 3.31. The number of elements ofΛ R (λ) is less than or equal to the number of bases for the inclusion T ⊂ D.
Proof. Choose a generic cocharacter ξ ∈ t ⊥ Q ⊂ d * Q . Note that a real number c satisfies the equations x i p ≤ c < x i p + p if and only if it satisfies x i p − ǫ < c < x i p + p − ǫ for ǫ sufficiently small. Thus, we will have no fewer nonempty regions if we consider the chambers x , there is a maximal point for this cocharacter, that is, a point a such that for all b a ∈∆ R x , we have ξ(b−a) < 0. By standard convex geometry, this is only possible if there are hyperplanes in our arrangement passing through a defined by coordinates that are a basis. In fact, by the genericity of the elements ǫ i , we can assume that the point a is hit by exactly a basis of hyperplanes. This gives a map fromΛ(λ) R to the set of bases and this map is injective, since all but one of the chambers that contain a in its closure will contain points higher than a.
Since the number of elements ofΛ R (λ) is lower-semicontinuous in λ, we see immediately that λ is smooth if the size ofΛ(λ) is the number of bases. g-generators 3.10. Tilting generators for coherent sheaves. We can also interpret these results in terms of coherent sheaves. In particular, we can consider the coherent sheaf on the formal completion of the fiber µ −1 (0). Here, as before, we assume that a i ∈ F p , so a p i − a i = 0. On this formal subscheme, this is an equivariant splitting bundle for the Azumaya algebra A λ K by [Sta13,4.3.4]. If we think of A λ K | µ −1 (0) as a left module over itself, it decomposes according the eigenvalues of h + i acting on the right. By construction, each generalized eigenspace defines a copy of Q a for some weight a. If we let g * ,λ F p be the set of characters of d F p which agree with λ (mod p) on t F p , then these are precisely the simultaneous eigenvalues of the Euler operators h + i that occur. Thus, we have In particular, given an A λ K -module M over the formal neighborhood of µ −1 (0), we have an isomorphism of coherent sheaves The elements of A λ K act on Q a on the left as endomorphisms of the underlying coherent sheaf; in particular, Q a naturally decomposes as the sum of the generalized eigenspaces for the Euler operators h + i . In fact, each eigenspace for the action of h + i defines a line bundle, so that the sheaf Q a is the sum of these line bundles. The next few results will provide a description of these line bundles. We begin with some preliminaries. Recall that M K is defined as a free quotient of a D-stable subset of T * A n K by T. Given any character of x ∈ D, the associated bundle construction defines a D-line bundle on M K . If we forget the D-equivariance, then the underlying line bundle depends only on the imagex of x in d * Z /t ⊥ Z . Definition 3.32. Given x ∈ d * Z , let ℓ D (x) be the associated D-equivariant line bundle line bundle on M K . We write ℓ(x) or ℓ(x) for the resulting non-equivariant line-bundle.
Recall that the Weyl algebra W K defines a coherent sheaf over the spectrum of its center, namely (T * A (1) K ) n . As a coherent sheaf, it is simply a direct sum of copies of the structure sheaf. Consider a monomial m(k, l) := n i=1 ∂ k i i z l i i , viewed as a section of the structure sheaf. We have the following description of its D-weight x ∈ d * Z . Write ǫ i for the generators of d * Z , so that x = n i=1 δ i ǫ i . Let δ + i be the maximal power of z p i dividing m(l, k) and δ − i be the maximal power of ∂ p i dividing m(l, k). Then In the notation of Section 3.3, we can write this as x = n i=1 δ i (0, l − k)ǫ i . We conclude the following. The following proposition holds over the formal neighborhood π −1 (0). characQ Proposition 3.34.
Z , so that the sum is well defined. The different isomorphism classes of line bundles that appear are in bijection with the chambers ofΛ(λ), but not canonically so, since we must choose a.
Proof of Proposition 3.34. The second isomorphism follows from the first by (3.13). To construct the first isomorphism, we recall that A λ It has a section given by the element m(b − a) ∈ A λ K . By Lemma 3.33, it is the line bundle defined via the associated bundle construction by the character For another commutative ring R, let T λ R be the corresponding bundle on M R , the base change to Spec(R). Every line bundle which appears has a canonical S-equivariant structure (induced from the trivial structure S-equivariant structure on O T * A n Z ), and we endow T λ Z with the induced S-equivariant structure. Note that any lift ofΛ(λ) to Λ(λ) determines a D × S-equivariant structure, although we do not need it here. The S-weights make End(T λ Z ) into a Z ≥0 graded algebra. Let x ∈ d * Z . Consider the monomial Note the similarity with (2.2), with the key difference that we do not require x ∈ t ⊥ Z . After Hamiltonian reduction, this defines a section of ℓ(x) with S-weight equal to |x| 1 . By the same token, it defines an element of Hom(ℓ(y), ℓ(y ′ )) whenever y ′ = y + x.
op:equiv-coh Proposition 3.35. For all λ, we have an isomorphism of graded algebrasH λ Proof. We first check that the map is well-defined. The map s i → z i w i is well-defined since the linear relations satisfied by s i exactly match the relations on z i w i coming from restriction to the zero fiber of the T-moment map. The map c x,y → m p (y − x) is welldefined if the elements m p (y − x) satisfy relations (3.10a) and (3.10b-3.10c). Relation The relations (3.10b-3.10c) are clear from the commutativity of multiplication. Thus, we have defined an algebra mapH λ Z → End(T λ Z ). This is a map of graded algebras, since both c x,y and m p (y − x) have degree |y − x| 1 .
This map is a surjection, since homomorphisms from one line bundle to another are spanned over C[z 1 w 1 , . . . , z n w n ] by m p (x). SinceH λ Z is torsion free over Z, it's enough to check that it is injective modulo sufficiently large primes, which follows from Theorems 3.13 and 3.16.
This allows us to understand more fully the structure of the bundle T λ Z . Note that the bundle T λ Z depends on λ, but only through the structure of the setΛ(λ). ooth-tilting Proposition 3.36. The bundle T λ Q is a tilting generator on M Q if and only if λ is smooth. Proof. T λ Q is tilting by Theorem 2.5, so we need only check if it is a generator. In order to check this over Q, it is enough to check it modulo a large prime p. By [Kal08, 4.2], for a fixed affine line Z in g * ,λ Z , there is an integer N, independent of p, such that T λ F p is a tilting generator when λ lies outside a set of size ≤ N in the reduction Z F p of Z modulo p. If λ is smooth, then the set of λ ′ ∈ Z F p that give the sameΛ(λ) grows asymptotically to Ap where A is the volume in Z R/Z of the real points such that Λ(λ) R for the induced torus arrangement is the same asΛ(λ) for λ. Thus, whenever p ≥ N/A, there must be a point whereΛ(λ) (and thus T λ Q ) is the same as for our original λ, and T λ Q is a tilting generator. If λ is not smooth, thenH λ Q has fewer simple modules than at a smooth parameter, so T λ Q cannot be a generator. Combining the above results yields the following equivalence of categories. In the following, we view T λ Q as a coherent sheaf ofH λ Q -modules. -equivalence Corollary 3.37. For smooth λ, the adjoint functors define equivalences between the derived categories of coherent sheaves over M Q and finitely generated rightH λ Q -modules.

The same functors define an equivalence between the derived categories of graded modules and equivariant sheaves:
is defined as a path algebra modulo relations, its graded simple modules are just the 1-dimensional modules L op x := Hom(⊕ y∈Λ(λ) L y , L x ); we denote the corresponding complexes of coherent sheaves by The induced t-structure on D b (Coh G m (M)) is what's often called an "exotic tstructure. " We also have a Koszul dual description of coherent sheaves as dg-modules over the quadratic dualH ! λ,Q . SinceH λ Q is an infinite dimensional algebra, we have to be a bit careful about finiteness properties here. We let Coh(M Q ) o be the category of coherent sheaves set theoretically supported on the fiber π −1 (o), andH This ideal contains all elements of sufficiently large degree (since the quotient by it is finite dimensional and graded), so each cohomology module of the image is a finite extensions of the graded simples. Thus the complex itself is an iterated extension of shifts of these modules.

LetH !
λ,Q -perf be the category of perfect dg-modules overH ! λ,Q . As usual, we will abuse notation and let D b (Coh(M Q )) to denote the usual dg-enhancement of this category, and similarly with D b (Coh G m (M Q )). Combining the equivalence of Corollary 3.37 with Koszul duality: This shows that smooth parameters also have an interpretation in terms of A λ K ; this is effectively a restatement of Proposition 3.36, so we will not include a proof.
Proposition 3.40. The functor RΓ : is an equivalence of categories if and only if the parameter λ is smooth.

Mirror symmetry via microlocal sheaves
In the previous sections, the conical G m -action on hypertoric varieties played a key role in our study of coherent sheaves. This is what allowed us to construct a tilting bundle based on a quantization in characteristic p. This conic action also plays a crucial role in the study of enumerative invariants of these varieties [BMO, MO,MS12]. The quantum connection and quantum cohomology which appear in those papers lose almost all of their interesting features if one does not work equivariantly with respect to the conic action. We are thus interested in a version of mirror symmetry which remembers this conic action.
We expect the relevant A-model category to be a subcategory of a Fukaya category of the Dolbeault hypertoric manifold D, built from Lagrangian branes endowed with an extra structure corresponding to the conical G m -action on M. However, rather than working directly with the Fukaya category, we will replace it below by a category of DQ-modules on D . The calculations presented there should also be valid in the Fukaya category, but this requires some machinery for comparing Fukaya categories and categories of constructible sheaves. Recent and forthcoming work of Ganatra, Pardon and Shende [GPSb, GPSa] makes us optimistic about the prospects of having such machinery soon, so let us have some fuel ready for it.
After defining the relevant spaces and categories of DQ modules, we state our main equivalence in Theorem 4.41 and Corollary 4.42.
There are a few obvious related questions. What corresponds to the category of all (not necessarily equivariant) coherent sheaves on M? What corresponds to the full Fukaya category of D? We plan to address these questions in a future publication.

Dolbeault hypertoric manifolds.
In this section, we introduce Dolbeault hypertoric manifolds, whose definition we learned from unpublished work of Hausel and Proudfoot.
Dolbeault hypertoric manifolds are complex manifolds attached to the data of a toric hyperplane arrangement (i.e. a collection of codimension one affine subtori), in much the same way that an additive hypertoric variety is attached to an affine hyperplane arrangement, and a toric variety is attached to a polytope. They carry a complex symplectic form, and a proper fibration whose generic fibers are complex lagrangian abelian varieties.

Our construction of Dolbeault manifolds parallels the construction of toric varieties as Hamiltonian reductions of powers of a basic building block.
For toric varieties, this building block is C with the usual Hamiltonian action of U 1 . Its polytope is a ray in R. Other toric varieties are constructed by taking the Hamiltonian reduction of C n by a subtorus of U n 1 . Additive hypertoric varieties are similarly constructed from the basic building block T * C with its hyperhamiltonian action of U 1 . The affine hyperplane arrangement associated to this building block is a single point in R. For Dolbeault manifolds, our basic building block will be the Tate curve Z with a (quasi)-hyperhamiltonian action of U 1 . Its toric hyperplane arrangement is a single point in U 1 .
We give a construction of Z suited to our purposes below, culminating in Definition 4.2.
Let C * = Spec C[q, q −1 ], and let D * be the punctured disk defined by 0 < q < 1. Let Z * be the family of elliptic curves over D * defined by (C * × D * )/Z, where 1 ∈ Z acts by multiplication by q × 1.
We will define an extension of Z * to a family Z over D with central fiber equal to a nodal elliptic curve.
Let W n := Spec C[x, y] for n ∈ Z. Consider the birational map f : W n → W n+1 defined by f * (x) = 1 y , f * (y) = xy 2 . This defines an automorphism of the subspace W 0 \ {xy = 0}, and identifies the y-axis in W n with the x-axis in W n+1 birationally, so they glue to a P 1 . If we let q := xy, then we can rewrite this automorphism as (x, y) → (q −1 x, qy). Note that this map preserves the product xy and commutes with the C * -action on W n defined by τ · x = τx, τ · y = τ −1 y; we let T denote this copy of C * .
Definition 4.1. Let W be the quotient of the union n∈Z W n by the equivalence relations that identify the points x ∈ W n and f (x) ∈ W n+1 .
The variety W is smooth of infinite type, with a map q := xy : W → C and an action of C * preserving the fibers of q. The map W 0 \ {xy = 0} → W \ q −1 (0) is easily checked to be an isomorphism.
W carries a Z-action defined by sending W n to W n+1 via the identity map. The action of n ∈ Z is the unique extension of the automorphism of W 0 \ {xy = 0} given by (x, y) → (q −n x, q n y). Thus, n fixes a point (x, y) if and only if q is an nth root of unity. In particular, the action of Z on is free. Combining this with the paragraph above, we see that q −1 (D * ) = {(x, y) ∈ W 0 | xy ∈ D * }; since we can choose x ∈ C * and q ∈ D * , with y = q/x uniquely determined, we have an isomorphism q −1 (D * ) C * × D * . Note that transported by this isomorphism, the C * -action we have defined acts by scalar multiplication on the first factor, and trivially on the second.
Thus, we obtain the following commutative diagram of spaces: (4.17) The fiber Z 0 := q −1 (0) is an infinite chain of CP 1 's with each link connected to the next by a single node.
The action of C * on Z 0 scales each component, matching the usual action of scalars on CP 1 , thought of as the Riemann sphere. The action of the generator of Z translates the chain by one link. The manifold Z will be our basic building block. We now study various group actions and moment maps for Z, in order to eventually define a symplectic reduction of Z n .
The action of C * on Z descends to an action on Z; note that on any nonzero fiber of the map to D, it factors through a free action of the quotient group C * /q Z , which is transitive unless q = 0. Thus the generic fiber of q is an elliptic curve. The fiber Z 0 := q −1 (0) is a nodal elliptic curve. We write n for the node.
The action of U 1 ⊂ C * on Z is Hamiltonian with respect to a hyperkahler symplectic form and metric described in [GW00,Prop. 3.2], where one also finds a description of the Z-invariant moment map. This moment map descends to Hence µ is the quasi-hamiltonian moment map for the action of U 1 on Z. We may arrange that µ(n) = 1 ∈ U 1 . The nodal fiber Z 0 is the image of a U 1 -equivariant immersion ι : CP 1 → Z, which is an embedding except that 0 and ∞ are both sent to n. We have a commutative diagram: moment-map-1 moment-map-1 (4.18) The action of U 1 and map µ × q form a kind of 'multiplicative hyperkahler hamiltonian action' of U 1 . In particular, (µ × q) −1 (a, b) is a single U 1 orbit, which is free unless a = 1, b = 0, which case it is just the node n. It's worth comparing this with the hyperkähler moment map on T * C for the action of U 1 : this is given by the map The fibers over non-zero elements of R × C are circles, and the fiber over zero is the origin. In a neighborhood of n, µ × q is analytically isomorphic to this map. Without seeking to formalize the notion, we will simply mimic the notion of hyperkhaler reduction in this setting. Recall that a hypertoric variety M is defined using an embedding of tori (C * ) k = T → D = (C * ) n . Let T R , D R be the corresponding compact tori in these groups, and T ∨ R t * R /t * Z the Langlands dual torus; the usual inner product induces an isomorphism D R D ∨ R which we will leave implicit. Thus, we have an action of T R on Z n and a T R -invariant map For the rest of this paper, we make the additional assumption that the torus embedding T → D is unimodular, meaning that if e k are the coordinate basis of d Z , then any collection of e k whose image spans d Q /t Q also spans d Z /t Z . As with toric varieties, this guarantees that for generic ζ the action of T on Φ −1 (ζ ′ ) is actually free. We expect that this assumption can be lifted without significant difficulties, but it will help alleviate notation in what follows.
The following definition is due to Hausel and Proudfoot. We will also need to consider the universal cover D; this can also be constructed as a reduction. We have a hyperkähler moment mapΦ : Z n → t * R ⊕ t * . Letζ ′ be a preimage of ζ ′ . D carries a natural action of g * Z , the subgroup of Z n which preserves the level Φ −1 (ζ ′ ). The quotient by this map is D, and the quotient map ν : D → D is a universal cover. Note that D is a (non-multiplicative) hyperkähler reduction, and the action of g * Z preserves the resulting complex symplectic form. This gives one way of defining the complex symplectic form on D.
The T R action on Z n and the hyperkähler moment mapΦ both extend to the infinite type algebraic variety W n . Definition 4.7. We define the core of D to be C := q −1 D (0), and denote by C its preimage in D.
We can organise the various spaces above into the following diagram: eq:thefam eq:thefam (4.20) The left-hand horizontal arrows are closed embeddings, and the right-hand horizontal arrow is an open embedding. Whereas D is merely a complex manifold, we will see that C is naturally an algebraic variety. It is a free quotient of D, whose components, as we shall see, are smooth complex Lagrangians. We can give an explicit description of D as follows, in the spirit of the combinatorial description of toric varieties in terms of their moment polytopes. In our setting, polytopes are replaced by toroidal arrangements.
We have the map D * R be the periodic hyperplane arrangement defined by the preimage of B tor ζ in ζ ′ + g * R . Let Λ R (ζ) be the set of chambers of B per ζ . We write ∆ R x ⊂ g * ,ζ R for the (closed) chamber indexed by x ∈ Λ R (ζ). As in section 3.8, let X x be the toric variety obtained from the polytope ∆ R x by the Delzant construction.

strone
(1) The irreducible components of C are smooth toric varieties X x indexed by x ∈ Λ R (ζ).

strtwo
(2) The intersection X x ∩ X y is the toric subvariety of either component indexed by The image under the G R -moment map of X x is precisely the polytope ∆ R x . strfour (4) All components meet with normal crossings.
Proof. We begin by noting that C is the image in D of Φ −1 (ζ ′ ) ∩ Z n 0 . The irreducible components of Z n 0 are copies of (CP 1 ) n indexed by x ∈ Z n . The moment map µ n : Z n 0 → R n , restricted to the component (CP 1 ) n x , has image the translation [0, 1] n x of the unit cube by x. We writeΦ x : (CP 1 ) n → t ∨ R for the restriction the T R moment map. It is given by be the composition of µ n CP 1 : (CP 1 ) n → [0, 1] n x with the projection p : x is a polytope, given by g * ,ζ R ∩ [0, 1] n x . It is non-empty precisely when x ∈ Λ R (ζ), in which case it is the chamber ∆ R x . The irreducible components of C are thus the quotientsΦ −1 x (ζ)/T R for x ∈ Λ R (ζ). The claims (1), (2) and (3) now follow from standard toric geometry. Claim (4) follows from the corresponding property for Z n 0 . In fact, the singular points of C are analytically locally a product of m nodes, and a d − m-dimensional affine space.
:torBarrange Definition 4.10. Let B tor ζ ⊂ G * ,ζ R be the toric hyperplane arrangement defined by the coordinate subtori of D * R . LetΛ R (ζ) be the set of chambers of B tor ζ . Given x ∈Λ R (ζ), we write ∆ R x ⊂ G * ,ζ R for the corresponding chamber. The toric arrangement B tor ζ is simply the quotient of the periodic arrangement B per ζ by the action of the lattice g * Z . The restriction of the quotient map to a fixed chamber ∆ x ⊂ g * ,ζ R is one-to-one on the interior, but may identify certain smaller strata. Correspondingly, the composition X x → D → D is in general only an immersion. The following is easily deduced from 4.9.

strone
(1) The irreducible components of C are immersed toric varietiesX x indexed by x ∈Λ R (ζ).  Consider the union W 0 ∪ W 1 ⊂ W. This is a Zariski open subset of W isomorphic to T * CP 1 . Let Z 0 be its intersection with Z. This is an open submanifold, isomorphic to a tubular neighborhood of CP 1 in its cotangent bundle.
These identifications map the function q to the function induced by the vector field z d dz for z the usual coordinate on CP 1 . The induced mapŨ → Z is an immersion. Applying the action of Z gives neighborhoods W k of each component of q −1 (0) ⊂ W. Repeating the same construction for the product W n , we obtain for each x ∈ Z n an open neighborhoodW x of (CP 1 ) n x in W n , isomorphic to T * (CP 1 ) n . This neighborhood is preserved by the hyperhamiltonian action of T R . Consider its hyperhamiltonian reduction D alg It is an open neighborhood of X x in D alg , naturally symplectomorphic to T * X x . Intersecting with D ⊂ D alg , we obtain an open neighborhood D x of the zero section in T * X x mapping by a symplectic immersion The set of such lifts is a torsor over g * Z . 4.3. Scaling actions. The scaling C * -action on T * X x extends to an action of C * on D alg , which does not preserve D. We first describe this action in the basic case of W. Fix p ∈ Z and let S p be the copy of C * which acts on W k giving x degree 1 − k + p and y degree k − p. One can easily check on that this action descends to an action on W and gives the Poisson bracket degree one. On W p ∪ W p+1 T * CP 1 , it acts by the scaling action on the fibers. Note that S p does preserve the open subset Z ⊂ W.
The action of S p × T does not commute with the translation action of Z. Instead, the Z-action intertwines the actions of S p × T for different p. In particular, all such actions are given by precomposing an isomorphism S p × T → S 0 × T with the action of the latter torus on W.
We can upgrade all these structures to the general case: for each x, we have a copy S x of C * which acts on D alg such that on D alg x ⊂ T * X x it matches the scaling action. As before, these actions do not commute with the g * Z -action. Instead, they are intertwined by this action. In particular, all such actions factor through an isomorphism S x × G → S 0 × G with the action of the latter torus on D alg . We make the following (purely notational) definition, to emphasise this independence of choices.

4.4.
Other flavors of multiplicative hypertoric manifold. In this paper, starting from the data of an embedding of tori T → G n m , we have constructed both an additive hypertoric variety M and a Dolbeault hypertoric manifold D. We view the latter as a multiplicative analogue of M. One can attach to the same data another, better known multiplicative analogue B, which however plays only a motivational role in this paper. For a definition, see [Gan18]. B is often simply known as a multiplicative hypertoric variety. For generic parameters, it is a smooth affine variety, of the same dimension as M and D. In fact, work of Zsuzsanna Dancso, Vivek Shende and the first author [DMS19] constructs a smooth open embedding D → B, such that B retracts smoothly onto the image. The embedding does not, however, respect complex structures; for instance, the complex Lagrangians considered here map to real submanifolds of the multiplicative hypertoric variety. Instead, B and D play roles analogous to the Betti and Dolbeault moduli of a curve.
In the sequel [GMW] to this paper, joint with Ben Gammage, we show that the core C ⊂ D becomes the Liouville skeleton of B, thought of as a Liouville manifold with respect to the affine Liouville structure. Microlocal sheaves on this skeleton compute the wrapped Fukaya category of B. In the next section, we will introduce a category of deformation quantization modules on D, which roughly corresponds to microlocal sheaves on B with an extra G m equivariant structure. This helps place our main results in the usual context of homological mirror symmetry. The relationship between the two papers is explained in more detail in [GMW].
sec:defquant 4.5. Deformation quantization of D. In the next few sections, we define a deformation quantization of D over C(( )), and compare modules over this quantization with the category A λ K -mod o from the first half of the paper. We'll also discuss how the structure of G m -equivariance of coherent sheaves can be recaptured by considering a category µm of deformation quantization modules equipped with the additional structure of a 'microlocal mixed Hodge module.' Consider the sheaf of analytic functions O W n on W n . We'll endow the sheaf O W n := O W n (( 1 /2 )) with the Moyal product multiplication f ⋆ g := f g + Note that if f or g is a polynomial this formula only has finitely many terms, but for a more general meromorphic function, we will have infinitely many. Following the conventions of [BPW16], we let O W n (0) = O W n [[ 1 /2 ]], which is clearly a subalgebra. We'll clarify later why we have adjoined a square root of . Sending x → 1/y, y → xy 2 induces an algebra automorphism of this sheaf on the subset W n \ {xy = 0}, since This shows that we have an induced star product on the sheaf O W , and thus on O Z . We now use non-commutative Hamiltonian reduction to define a star product on O D . This depends on a choice of non-commutative moment map κ : t → O Z n . We fix φ ∈ d * . Given (a 1 , . . . , a n ) ∈ d, define κ (a 1 , . . . , a n ) := a i x i y i + φ(a).
Our quantum moment map is the restriction of κ to t ⊂ d. Note that this agrees mod with the pullback of functions from t * under Φ.
be the quotient of O Z n by the left ideal generated by these functions. Note that this is supported on the subset Φ −1 (T ∨ R × {0}). We have an endomorphism sheaf End(C φ ) of this sheaf of modules over O Z n .
Definition 4.14. Let O φ be the sheaf of algebras on D defined by restricting End(C φ ) to Φ −1 (ζ ′ ) and pushing the result forward to D.
One can easily check, as in [KR08], that O φ defines a deformation quantization of D, that is, this sheaf is free and complete over C

the sections on any G-invariant open set is locally finite, i.e. it is spanned by its generalized weight spaces for this torus.
A pre-weak equivariant structure can be upgraded to a weak equivariant structure as follows: we can assume that M is indecomposable, so all weights appearing are in a single coset of the character lattice of G. We can take the semi-simple part of the action of each element of g, and globally shift by a character of the Lie algebra to make all weights appearing integral. The resulting action integrates to a weak G-equivariant structure (but we do not want to fix a specific one); we call such an action compatible with the O φ -module structure. Note that pre-weakly G-equivariant modules are a Serre subcategory.  Unfortunately, the action of SG on D alg does not preserve D. We can nevertheless speak of SG-equivariance on D and D, as follows.
Let M be a pre-weakly G-equivariant O φ -module. Let ν * M be the pullback of this module to D. We write (ν * M) alg for the pushforward of ν * M along the inclusion D → D alg . Note that by Lemma 4.16, ν * M is supported on q −1 D (0) ⊂ D, and this subset remains closed in D alg . Thus the support is not enlarged.
Definition 4.17. A pre-weakly SG-equivariant structure on a pre-weakly G-equivariant O φ -module M is an action of the Lie algebra × x 0 commuting with g which integrates to an equivariant structure for S x 0 on (ν * M) alg . We write O φ -mod SG for the category of such modules.
As with pre-weakly G-equivariant modules, after making some auxiliary choices, we can endow a pre-weakly SG-equivariant-module with a 'compatible' action of the torus SG, which integrates the semisimple part of (a shift of) the infinitesimal action. Proof. Again, we can reduce to the case where M is indecomposable. By construction, any two compatible G-equivariant structures on M differ by tensor product with a character of the group G, so the induced S y structures differ by tensor product with a character of S y , which we can think of as the integer weight w. Since has weight 1 under S y , multiplication by w intertwines these two actions, and gives an isomorphism between the two S y -equivariant structures.
4.7. The deformation quantization near a component of C. Given φ ∈ t * Q , we can define a fractional line bundle ℓ φ on any quotient by a free T-action. The component X x was defined by a free T R -action; by standard toric geometry, it also carries a canonical presentation as a free T-quotient. Applying this construction to X x thus yields a bundle ℓ φ,x . If φ ∈ t * Z , the set of honest characters, then this is an honest line bundle; otherwise, it gives a line bundle over a gerbe, but we can still define an associated Picard groupoid, and thus a sheaf of twisted differential operators (TDO) on X x . Let Ω x be the canonical line bundle on X x , and Ω 1/2 x the half-density fractional line bundle. It is a classical fact that Ω x = l −φ 0 ,x where φ 0 is the sum of all T-characters of C n induced by the map T → D.
We let D φ,x denote the TDO associated to the fractional line bundle ℓ φ,x ⊗ Ω 1/2 x , and let W φ,x be its microlocalization on T * X x . That is, W φ,x is a sheaf in the classical topology on T * X x whose sections on T * U for U ⊂ X x is the Rees algebra for the order filtration on D φ,x (U); for an open subset V ⊂ T * U (where we can assume WLOG that U is affine), we further invert any element of the Rees algebra whose image under the map W φ,x (U)/ W φ,x (U) O T * U (T * U) is invertible on V. The construction of this algebra is discussed in more detail in [BPW16, §4.1]. We'll be more interested in its localisation: If we equip a module M over the TDO D φ,x with a good filtration, which for technical reasons we'll index with 1 2 Z, its Rees module M(0) generated by −k M ≤k for k ∈ 1 2 Z is a coherent module over the Rees algebra (we can use this as a definition of good filtration). That is, it is a coherent sheaf of W φ,x -modules, equipped with a C * -equivariant structure for the squared scaling C * -action (or equivalently, a grading of its sections on T * U). Inverting , we obtain a W φ,x -module M = M(0)[ − 1 /2 ] which is independent of the choice of good filtration, which is good in the sense of [BPW16,§4], that is, it admits a coherent, C * -equivariant W φ,x -lattice. By [BPW16,Prop. 4.5], this is an equivalence between coherent D φ,x -modules and good W φ,x -modules.
hm:micro-iso Theorem 4.20. We have an isomorphism of algebra sheaves ι * Proof. First, we check that this holds in the base case, i.e. when D = Z. It is convenient to check this on the universal cover Z. By the Z-symmetry of the latter, it is enough to check for a single component of the core. Hence, consider the copy of CP 1 in the union of W 0 ∪ W 1 . Using superscripts to indicate which W * we work on, we have birational coordinates y (0) = 1/x (1) and x (0) = y (1) (x (1) ) 2 . We thus have an isomorphism of W 0 ∪ W 1 to T * CP 1 with coordinate z and dual coordinate ξ sending We can quantize this to a map from O W to W φ by the corresponding formulas: This induces an isomorphism of sheaves, which in turn restricts to an isomorphism ι * x O Z → W φ,x | Z 0 . Note that under this isomorphism, q = x (1) y (1) → z d dz − /2. To proceed to the general case, we consider Z n and its quantized T-moment map κ . Fix as above an open subset of isomorphic to (T * P 1 ) n . Applying the above morphism to the image of κ , we obtain the following.
The result then follows from the compatibility of twisted microlocal differential operators with symplectic reduction as in [BPW16,Prop. 3.16]. We can identify the twist of a TDO from its period by [BPW16,Prop. 4.4].
Thus, given aÕ φ -module M, we can pull it back to an W φ, If we additionally choose a S x -equivariant structure which makes M| D x into a good module, then the equivalence of [BPW16,Prop. 4.5] will give a corresponding module over the TDO D φ,x , with a choice of good filtration. The modules σ x (M) for different x are compatible in the following sense: As discussed previously, the intersection X x ∩ D alg y is precisely the conormal bundle N x,y = N * X y (X x ∩ X y ) to X x ∩ X y in T * X y . Thus the intersection D alg x,y = D alg x ∩ D alg y can be identified with T * (N x,y ) or swapping the roles of x, y with T * (N y,x ).
Since the vector bundles N * X y (X x ∩X y ) and N * X x (X x ∩X y ) are dual, so Fourier transform F y,x gives an equivalence between the categories of pre-weakly G-equivariant Dmodules on these spaces, and between constructible sheaves with R-coefficients, which are compatible with respect to the solution functor. By construction, we thus have eq:Fourier1 eq:Fourier1 (4.22) F y,x σ x (M)| N x,y σ y (M)| N y,x .

Preliminaries on the Ext-algebra of the simples.
Assume that φ is chosen so that ℓ φ ⊗ Ω 1/2 is an honest line bundle for all x. From now on, we use the abbreviations E x := E φ,x , D x := D φ,x and ℓ x := ℓ φ,x , since the dependence on φ will not play any further role in this paper.

Remark 4.22.
Recall that Ω 1/2 x equals ℓ −φ 0 /2,x where φ 0 is the sum of all T-characters induced by the embedding T → D. Thus our assumption will be satisfied whenever φ(a) ∈ Z + 1 2 a i for all a ∈ t Z . For example, we can let φ be the restriction of the element ( 1 2 , · · · , 1 2 ) ∈ d * . Nothing we do will depend on this choice; in fact, the categories of O φ -modules for φ in a fixed coset of t * Z are all equivalent via tensor product with quantizations of line bundles on D (as in [BPW16, §5.1]), so our calculations will be independent of this choice.
In this case, the sheaf W x naturally acts on L ′ x := ℓ x ⊗ Ω 1/2 x (( )) as a sheaf on X x pushed forward into D x ; under the equivalence of [BPW16,Prop. 4.5] mentioned above, this corresponds to the twisted D-module ℓ x ⊗ Ω 1/2 x . Of course, this sheaf is equivariant for the action of S x , and pre-weakly G-equivariant.
Via the maps we can define modules over O φ andÕ φ : Using S x -equivariance, and the pre-weak G-equivariant of this module, we obtain a twisted D-module σ yLx . Recall that we have a universal cover map ν : D → D.
The first category we will consider on the A-side of our correspondence is DQ, the dg-subcategory of O φ -mod SG generated by L x for all x. As observed before, since weakly G-equivariant modules form a Serre subcategory, any finite length object in this category is pre-weakly G-equivariant, and so we can define the D-modules σ y (M) for any module M in this category.
This has a natural t-structure, whose heart is an abelian category dq. We similarly let DQ be the dg-subcategory generated byL x , and dq the heart of the natural tstructure. This definition might seem slightly ad hoc, but we will later see that it is motivated by our notion of microlocal mixed Hodge modules.
Since the Ext sheaf betweenL x andL y is supported on the intersection between X x and X y , we have In fact, if we replaceL y by an injective resolution, we see that this induces a homotopy equivalence between the corresponding Ext complexes. Since L ′ x is supported on the zero section, Ext to it is unchanged by passing to an open subset containing this support and eq:micro-ext eq:micro-ext (4.23) Ext DQ (L y ,L x ) Ext D(X x ) (σ xLy , ℓ x ⊗ Ω 1/2 x ) where the latter Ext is computed in the category of D-modules on the toric variety X x . In the toric variety X x , the preimage of the intersection with the image of X y is a toric subvariety corresponding to the intersection of the corresponding chambers in B per ζ .
nctionmodule Lemma 4.25. The microlocalization σ xLy is the line bundle ℓ y ⊗ Ω 1/2 y pulled back to X x ∩ X y and pushed forward to X x as a D x -module.
Proof. Consider the intersection of X y with D alg x . This is a closed S x -invariant Lagrangian closed subset, so it is the conormal to its intersection with the zero section X x ∩ X y . The D-module σ xLy has singular support on this subvariety, and thus must be a local system on X x ∩ X y , which is necessarily ℓ y ⊗ Ω 1/2 y .
Since X x ∩ X y is a smooth toric subvariety, the sheaf Ext between σ xLy and ℓ x ⊗ Ω 1/2 where k is the codimension of X x ∩ X y in X x . This shows that we have an isomorphism eq:ext-coh eq:ext-coh (4.24) Ext m We will be interested in the class d y,x in the left-hand space corresponding to the identity in H * X y ∩ X x ; C . Unfortunately, this is only well-defined up to scalar. We will only need the case where |x − y| 1 = 1. In this case, we can define d y,x (without scalar ambiguity) as follows.
Consider the inclusions X x \ (X y ∩ X x ) j ֒→ X x i ←֓ X y ∩ X x and the corresponding sequence of D-modules eq:ext1 eq:ext1 (4.25) Any identification of the right-hand D-module with i ! O X y ∩X x defines a class d y, . Such an identification is obtained by picking the germ of a function g on X x in the formal neighborhood of X y ∩X x that vanishes on this divisor with order 1. Given such a function, the map f →f /g wheref is an extension of a meromorphic function on X y ∩ X x to the formal neighborhood defines an isomorphism of D- We can arrange our choice of chart in Z n so that X y ∩ X x is defined by the vanishing of one of the coordinate functions; note that in this case, X y ∩ X x is defined inside X y by the vanishing of the symplectically dual coordinate function (eg, if the first is defined by the vanishing of x i , then the latter will be defined by y i ). We choose this as the function to define d x,y .

Definition 4.26.
For any x, y such that |x − y| 1 = 1, let d y,x ∈ Ext 1 DQ (L y ,L x ) be the class defined by the above prescription.
4.9. Mirror symmetry. We are almost ready to compare the first and second halves of this paper. First, we need to match the parameters entering into our constructions. Recall that D depends on a choice of generic stability parameter ζ ∈ T * R , Likewise, the hypertoric enveloping algebra in characteristic p depends on a central character λ ∈ t * F p . The algebra H ! λ which describes the Ext groups of its simple modules thereby also depends on λ.
In order to match ζ and λ, we identify t * F p with t * Z /pt * Z and thereby embed it in T * R = t * R /t * Z via λ → 1 p λ. From now on we suppose that λ is smooth, and that ζ is its image in T * R . It follows that B Proof. We need to check that the rule d x,y → d x,y defines a homomorphism, i.e. that the relations (3.12a-3.12d) hold in Ext DQ x∈Λ(λ)L x . (1) The relation (3.12a) follows from the fact that when |x − y| 1 = 1, the element d x,y D y,x is the class in H 2 X x ; Q dual to the divisor X x ∩ X y , while the class t i is defined by the Chern class of the corresponding line bundle, for which a natural section vanishes with order one on X x ∩ X y for y ∈ α(i) and nowhere else.
(2) Note that the relations (3.12b) and (3.12c) equate two elements of the 1dimensional space Ext 2 (L x , L w ) H 0 (X x ∩ X w ; C). Thus, we only need check that we have the scalars right, and this can be done after restricting to any small neighborhood where all the classes under consideration have non-zero image.
Thus ultimately we can reduce to assuming X x = C 2 , and X y and X w are the conormals to the coordinate lines, and X z the cotangent fiber over 0. Let r 1 , r 2 be the usual coordinates on C 2 , and ∂ 1 , ∂ 2 be the directional derivatives for these coordinates. Thus, we are interested in comparing the Ext 2 's given by the sequences in the first and third row of the diagram below. Both sequences are quotients of the free Koszul resolution in the second row: The opposite signs in the leftmost column confirm that we have d z,w d w,x = −d z,y d y,x . Hence the elements d x,y satisfy the relations (3.12b-3.12c). (3) The relations (3.12d) follows from the fact that in this case X x ∩ X z = ∅.
Recall that the complex dimension of e x H ! λ,C e y coincides with that of H * (X x ∩ X y ; C), as we discussed in Section 3.8. Thus the spaces e x H ! λ,C e y and H * (X x ∩ X y ; C) are vector spaces of the same rank. Thus, in order to show that our map is an isomorphism, it is enough to show that it is surjective.
By Kirwan surjectivity, the fundamental class generates H * (X x ∩ X y ; C) as a module over the Chern classes of line bundles associated to representations of T. Since the fundamental classes are images of ±d x,y and the Chern classes are images of C[t 1 , . . . , t n ], we have a surjective map. As noted before, comparing dimensions shows that it is also injective, which concludes the proof.
Comparing this result with Proposition 3.39, we see that the categories DQ and D b (Coh(M)) are rather similar. We would immediately obtain a fully-faithful functor DQ → D b (Coh(M)) if we knew that Ext DQ (⊕ x∈Λ(λ) L x ) were formal as a dg-algebra, but it is not clear that this is the case. This issue has another, closely related manifestation: finding a structure on DQ that corresponds to the G m -action on M discussed earlier. We propose to resolve both of these issues by introducing a new structure on DQmodules, closely related to Saito's theory of mixed Hodge modules. This will result in a graded category, whose relationship to D b (Coh G m (M)) is analogous to that between DQ and D b (Coh(M)). 4.10. Microlocal mixed Hodge modules. We will need the notion of a unipotent mixed R-Hodge structure on σ x (M); see [Sai] for a reference. "Unipotent" simply means that the monodromy on every piece of a stratification on which the D-module is smooth is unipotent. Mixed Hodge modules are a very deep subject, but one which we can use in a mostly black-box manner. The important thing for us is that given a holonomic regular D-module M, a mixed Hodge structure can be encoded as real form and a pair of filtrations, a good filtration (often called the Hodge filtration) and the weight filtration (by submodules) on M. As discussed previously, we are allowing good filtrations indexed by 1 2 Z. Note that while most references on mixed Hodge modules only consider untwisted D-modules, since a Hodge structure is given by local data, the definition extends to twisted D-modules in an obvious way. We will only be using twists by honest line bundles (as opposed fractional powers), so we have an even easier definition available to us: a mixed/pure Hodge structure on a module M over differential operators twisted by a line bundle L is the same structure on the untwisted D-module L * ⊗ M.
Since we will be working with fixed twists in what follows, we will conceal this choice and simply speak of mixed Hodge modules on X x rather than twisted mixed Hodge modules.
An R-form of σ x (M) is a perverse sheaf L on X x with coefficients in R with a fixed isomorphism L ⊗ R C Sol(σ x M). We wish to define a R-form of M analogously, but we need to think carefully about compatibility between different x.  Generalizing this definition to other cases is, of course, a quite interesting question but not one on which we can provide much insight at the moment. ssMHMsimples 4.11. Classification of simple objects in µm. One natural operation on mixed Hodge DQ-modules is that of Tate twist, which shifts the filtrations by F i M(k) = F i+k M and W i M(k) = W i+2k M for k ∈ 1 2 Z. Note that defining Tate twists for half-integers requires using good filtrations which are indexed by k ∈ 1 2 Z, this explains our cryptic introduction of half-integers in earlier sections. We're only interesting in understanding simple modules up to this operation. We can easily check that: Proof. The trivial local system on X x has the structure of a variation of Hodge structure which is pure of weight 0. This is unique by [Del87, Prop. 1.13]. Of course, any mixed Hodge structure of weight 0 on L x must be induced by this VMHS, which shows uniqueness. Thus, we only need to show that the induced lattice L x (0), real form, and (trivial) weight filtration induce mixed Hodge structures on the microlocalizations σ y (L x ) for each y. Recall that σ y (L x ) is the pushforward of the trivial line bundle on X x ∩ X y , so the result follows from the compatibility of mixed Hodge structure with pushforward.
4.12. Projectives. Unfortunately, while the Hodge structure on a simple module is unique up to Tate twist, there are "too many" different Hodge structures on other objects in dq. For example, L x ⊕ L x (k) has a non-trivial moduli of Hodge structures, induced by the same phenomenon on R ⊕ R(k).
Thus, we need to find a way of avoiding these sort of deformations of Hodge structure. We do this by constructing a natural Hodge structure on certain projective type modules.
As usual, let use first construct these onZ. We define a DQ-module on this space as follows: Consider A = C[x, y, ] with the usual Moyal star product defined above. There are unique dq-modules P (k) * , P (k) ! over C 2 whose sections are the quotients Identifying A with the Rees algebra of differential operators D x on C[x] (sending y → ∂ ∂x ), these modules become the Rees modules of D-modules Π (k) * , Π (k) ! on A 1 with coordinate x. We can identify these with the *and !-pushforwards of the Dmodule L (k) on C * = Spec(C[x, x −1 ]) defined by the connection ∇ = d − N x on the trivial bundle with fiber C k , where N is the regular nilpotent matrix Both Π (k) * and Π (k) ! are projective in the category of D-modules on A 1 , smooth away from the origin, whose monodromy around the origin has nilpotent part of length ≤ n. The D-module Π (k) ! is the projective cover of the D-module of polynomials on A 1 , and P (k) * is the projective cover of the delta functions at the origin. As mentioned above, our presentation of these D-modules induces a good filtration on them; in DQ-module terms, this is an equivariant structure for the cotangent scaling S which has weight 0 on x and weight 1 on y. In fact, we will want to use shifts of this filtration, corresponding to P (k) * and 1 /2 P (k) ! (note that the latter is only equivariant under the squared scaling). In D-module terms, this means that we endow Π (k) * with the good filtration such that the image of These might seem like slightly strange choices: they are deliberately chosen so that in both cases, the unique simple quotient carries a pure Hodge structure of weight 0. Now, we consider Hodge structures on these DQ-modules extending the good filtrations defined above on Π (k) * and Π (k) ! . Their real form is the obvious one where x and y are conjugation invariant; this corresponds to the obvious real form of L (k) . We define the weight filtration on Π (k) * by Proof. First, let's consider Π (k) * . By the definition above, W p Π (k) * /W p−1 Π (k) * D x /D x x if p is even and 0 ≥ p ≥ 2k+1; this is equipped the good filtration where the image of ∂ r x for r < s span F s+ p /2 . On the other hand, the V-filtration of this D-module for the function x has V ℓ spanned by y r for r ≥ −ℓ. Thus, the vanishing cycles Φ = φ(W p Π (k) * /W p−1 Π (k) * ) are spanned by the image of 1, i.e. they are 1-dimensional. Accounting for the shift of good filtration (as in [Sai, (2.1.7)]) they are equipped with the good filtration This means that W p Π (k) * /W p−1 Π (k) * is isomorphic to the usual Tate pure Hodge structure of weight p on R, pushed forward at the origin x = 0. If p is odd, then we have exactly as above, the generic fiber of this local system has the Tate Hodge structure of weight p − 1, and so gives a pure Hodge module of weight p.
For Π (k) ! , the calculations are the same, but odd and even cases swap roles. In particular, we see that half-integral filtrations are needed so that we can endow R with a Tate Hodge structure of odd weight (i.e. a half integral Tate twist).
We will need certain morphisms between these DQ-modules: (1) The linear map N on C k induces an endomorphism on L (k) and hence of P (k) * and P (k) ! . This is the same as right multiplication by y ⋆ x or x ⋆ y, respectively.
(2) We have a c − : P (k) * → P (k) ! , induced by multiplication on the right by y. Note that this map becomes an isomorphism if we invert y, and consider these as D-modules on Spec C[y, y −1 ].
(3) In the opposite direction we have a map c + : P (k) ! → P (k) * , induced by multiplication on the right by x; this is also induced by the identity on the local system L (k) . Similarly, this map becomes an isomorphism if we invert x.
Note that the morphisms c − and c + shift the good filtration by 1 2 . By [Ara10, Th. 2.12], we can identify these maps with the logarithm of the monodromy around the origin, the canonical map from nearby to vanishing cycles and the modified variation map discussed in [Ara10, §2.7].
The morphisms we discussed above preserve the mixed Hodge structure up to Tate twist. These become morphisms of mixed Hodge modules N : P (k) * → P Proof. By construction, Hom(P (k) ! , M) is the kernel of the kth power of the logarithm of the monodromy on the stalk of M at a generic point, given by the image of (0, . . . , 0, 1) in this stalk. In particular, for Hom(P (k) ! , P (k) ! ), this is C k itself, and the map sending (0, . . . , 0, 1) to (a 1 , . . . , a k ) is a k + a k−1 N + . . . a 1 N k−1 . Similarly for Hom(P (k) ! , P (k) * ), this stalk is the same, but now the map sending (0, . . . , 0, 1) to (a 1 , . . . , a k ) is (a k + a k−1 N + · · · + a 1 N k−1 )c + . A symmetric argument holds with * and ! reversed.
As noted before, the map τ induces an isomorphism C 2 \ {y = 0} C 2 \ {x = 0}. We can construct a DQ-module on W i ∪ W i+1 glued using τ, and placing P (k) * or P (k) ! on each W i .
• If the two modules are different, i.e. P (k) * on W i and P (k) ! on W i+1 or vice versa then we use the natural isomorphism induced by swapping the roles of x and y.
• If they are the same, i.e. P (k) * or P (k) ! on both W i and W i+1 , then we use the isomorphisms of multiplication by y ±1 on W i or equivalently x ±1 on W i+1 .
Iterating this process, we can construct a dq-module onZ associated to a choice of integer k and a map ℘ : Z → { * , !}, isomorphic to P (k) ℘(i) on W i . To endow this dq-module with a global S-action, we will need to shift the natural S-action on the local components P (k) ℘(i) by a certain amount, determined as follows. We can associate to P (k) * a variation of mixed Hodge structure on each of the two components of its singular support, {x = 0} and {y = 0}, both described in terms of the mixed Hodge structure R (k) R ⊕ R(1) ⊕ · · · ⊕ R(k − 1). At a generic point of {x = 0}, the fiber is R (k) 1 2 (so we obtain local systems of weights 0, 2, . . . , 2(k − 1)), and at a generic point of {y = 0}, the fiber is R (k) (1); for P (k) ! , these swap roles. Thus, in order to have matching S-actions (or equivalently, good filtrations), we need to choose a function ς : Z → 1 2 Z with the property that: Proof. We can reduce to the case where i = 0 using the Z action. First, we must prove that L 0 is the unique simple quotient of P (k) i . On W 0 ∪ W 1 T * P 1 , this module is the pushforward j ! L (k) where j : C * ֒→ P 1 is the inclusion of the complement of the north and south poles. This has unique simple quotient given the intermediate extension of the 1-d local system with the standard connection. This matches the simple L 0 . Any other simple quotient must be L m with m 0. If m < 0, this would induce a map on W m of P (k) ! to the delta function D-module; similarly, if m > 0, it would induce a map on W m+1 of P (k) * to the function D-module. No such map exists, so indeed L 0 is the unique quotient.
on T gives us the quotient S/(s k i )(|x − y| 1 /2). This is generated by the image of c x,y , so our homomorphism is surjective, and the fact that H is free as an S-module shows it is also injective.
4.13. The category of mixed Hodge modules. As discussed above, looking at all mixed Hodge structures on DQ-modules results in "too many" objects. We will restrict the structures we consider to those which arise as a quotient of the objects P (k) x ; it's worth noting that while these objects have a projective property in dq (subject to a restriction on monodromy), they are not projective amongst mixed Hodge DQmodules with this monodromy. The important effect this has is that it forces the local systems on the open part of X x to be Tate as mixed Hodge structures; typically, the structures we wish to avoid will not have this property.
def:mhm Definition 4.40. We let µm and µm be the categories of mixed Hodge DQ-modules in dq and dq which are quotients of a sum of the form q p=1 P (k) x p (ℓ p ) for some k ≥ 0, ℓ p ∈ 1 2 Z and {x 1 , . . . , x q } ⊂ Λ.
We let µM and µM be the standard dg-enhancements of the derived categories D b (µm) and D b ( µm) (the quotient of the dg-category of all complexes modulo that of acyclic complexes). Now, assume that M is a finite dimensional graded right H λ R -module. Recall that M(ℓ) denotes M with the grading shifted down by ℓ. Assume k is chosen large enough that s k i kills M. We can thus write M as a quotient of Proof. First, we must show this functor is well-defined. Of course, for morphisms between projective modules, this is just the isomorphism of Lemma 4.39.
If f : M → M ′ is a homogeneous map of modules, then we can choose k so that s k i = 0 on both M and M ′ . Thus, we can write R modules. By the projective property, we have a chain map Thus, we define m( f ) is the map on cokernels induced by m( f 0 ). As usual, applying this for f = id M and two different presentations of M as a cokernel also shows that the m is independent of the choice of the presentation of M.
We wish to show that this map is fully faithful. Since the map of Lemma 4.39 is an isomorphism, this is true for projective H (k) R modules. Thus shows fullness immediately, because any map m(M) → mM ′ is induced by a map mP 0 → mP ′ 0 . Now, we turn to faithfulness. If m( f ) = 0, then we must have that the map m( f 0 ) maps m(P 0 ) to m(P ′ 1 ), i.e. m( f 0 ) is in the image of the map Hom(mP 0 , mP ′ 1 ) → Hom(mP 0 , mP ′ 0 ). Since we already know that the functor is fully faithful on projectives, this implies that f is in the image of the map Hom(P 0 , P ′ 1 ) → Hom(P 0 , P ′ 0 ), and so f = 0.
Finally we need to show that this functor is essentially surjective. By definition, any module M in µm is a quotient of m(P 0 ) for some P 0 . Thus, we need to show that the kernel K is also an object in µm. The object K has a largest semi-simple quotient, i.e. its cosocle. This is a finite sum of objects of the form L y r (ν r ). This shows that K is generated by the images of maps (of DQ-modules, ignoring Hodge structure) from P (k) y r for r = 1, . . . , s. Note that Hom(P (k) y r , K) carries a mixed Hodge structure which is a subobject of Hom(P (k) y r , m(P 0 )), the former has Tate type since the latter does as well. Thus, there is a module M 1 such that m(P 1 ) = s r=1 Hom(P (k) y r , K) ⊗ C P (k) y r as mixed Hodge DQ-modules; of course, the image of the induced map m(P 1 ) → m(P 0 ) is exactly K, and so M = m(M) where M is the cokernel of the map P 1 → P 0 .
Thus, combining with Corollary 3.37, we see our version of homological mirror symmetry in this context, as promised in the introduction: We conclude with a few questions raised by this result. Under the second equivalence, the G m -action on M C corresponds to the weight grading on µM. This action, which dilates the symplectic form, is key to the enumerative geometry of hypertoric varieties. Indeed, the symplectic structure on M C implies that the non-equivariant quantum connection of M C is essentially trivial. Its G m -equivariant version, on the other hand, is the hypergeometric system studied in [MS12]. The same is true for more general symplectic resolutions : for instance, the G m -equivariant quantum connection of the Springer resolution is the decidedly non-trivial affine KZ connection [BMO]. Our result thus suggests that the mirror description of these connections can be approached via microlocal Hodge structures.
We also note that whereas the left-hand side of both of our equivalences is a geometrically defined category, the right-hand sides are defined by picking certain generators inside the ambient category of deformation-quantization modules. This is in contrast to the equivalence proven in the sequel to this paper [GMW], which equates coherent sheaves on M C with the wrapped Fukaya category of its mirror. A more direct geometric definition of µM and its grading, in particular, would be of great interest.