1 Introduction↩︎

1.1 Context and overview↩︎

\(3D\) mirror symmetry [1], or symplectic duality [2], [3], dictates that symplectic resolutions should come in pairs and various layers of information should be exchanged for a dual pair of symplectic resolutions. Such duality phenomena can be approached through different lenses, and there is a large body of literature covering different aspects, including quantization, enumerative geometry, and categorification: see [4] for an introduction.

The goal of this paper is to initiate the study of \(3D\) mirror symmetry in characteristic \(p\) where \(p\) is a prime number. Here, mod \(p\) coefficients appear in an asymmetric fashion under symplectic duality. Namely, on one side, we study enumerative geometry of complex symplectic resolutions with mod \(p\) coefficients, in the sense that the counts in the enumeration problems are reduced mod \(p\); on the other side, we study symplectic resolutions defined over a field of characteristic \(p\).

Our formulation of such an “arithmetic” \(3D\) mirror symmetry conjecture is based on the quantum Hikita conjecture [5], which posits that two \(D\)-modules associated to conical symplectic resolutions are isomorphic for dual pairs. On one hand, we have the quantum \(D\)-module coming from enumerative geometry of genus \(0\) curves, and on the other hand, we have the \(D\)-module of graded/twisted traces arising from quantizations of symplectic singularities.

Our conjecture (3) based on the quantum Hikita conjecture proceeds by introducing distinguished central endomorphisms of the \(D\)-modules that only arise in characteristic \(p\), and identifying these under the isomorphism of \(D\)-modules. On the quantum \(D\)-module side, these endomorphisms come from a certain deformation of the classical Steenrod operations in \(\mathbb{F}_p\)-coefficient singular cohomology; on the \(D\)-module of traces side, they arise from a “large center” for quantizations in characteristic \(p\) given by a certain deformation of the Frobenius map. We verify our conjecture (6) for the classical examples of Springer resolutions and hypertoric varieties.

1.2 Motivations and main conjecture↩︎

Our formulation of the conjecture is based on the notable trilogy of the Hikita conjecture [6], the Hikita–Nakajima conjecture [7], and the quantum Hikita conjecture [5]. In characteristic zero, these conjectures predict that given a dual pair of conical symplectic resolutions \((X, X^!)\) over \(\mathbb{C}\) with a Hamiltonian torus \(T\) and \(T^!\) action respectively, the following should be true.

1. Let \(Y\) be the affinization of \(X\), then there exists an isomorphism of graded algebras \[\mathcal{O}(Y^T) \cong H^*(X^!; \mathbb{C}),\] where \(\mathcal{O}(Y^T)\) is the ring of regular functions of the \(T\)-fixed point scheme with the grading induced from the conical \(\mathbb{G}_m\)-action.

2. Let \(\mathcal{A}\) be the universal quantized coordinate ring of \(X\). Then there exists an isomorphism of graded algebras \[B(\mathcal{A}) \cong H^*_{T^! \times \mathbb{G}_m} (X^!; \mathbb{C}),\] where \(B(\mathcal{A})\) is the \(B\)-algebra (42) of \(\mathcal{A}\) with respect to a cocharacter in \(X_{\bullet}(T)\), and the two factors of \(T^! \times \mathbb{G}_m\) act on \(X^{!}\) by Hamiltonian automorphisms of \(X^!\) and conical actions respectively.

3. Finally, the \(D\)-module of graded/twisted traces of \(\mathcal{A}\) (35) should be identified with the (specialized) quantum \(D\)-module of \(X^{!}\) (38): \[\mathscr{B} (\mathcal{A}) \cong QH^*_{T^! \times \mathbb{G}_m}(X^!;\mathbb{C}).\]

We refer the reader to the exposition in Section 2 for a precise discussion of the relevant concepts. Beginning from the original work of [6] verifying the original Hikita conjecture for Hilbert scheme of points in a plane, parabolic type \(A\) Slodowy slices, and hypertoric varieties, extensive efforts (though likely to be an incomplete list) [8]–[17] have been dedicated to prove these conjectural isomorphisms for concrete examples. We refer to [17] for an excellent survey of the current status of the Hikita-type conjectures for various examples.

Our motivation for considering the mod \(p\) quantum Hikita conjecture comes from the manifestation of the “large center" phenomena in \(D\)-modules with mod \(p\) coefficients in both quantization theory and enumerative geometry. Namely, the respective \(D\)-modules in the statement of quantum Hikita conjecture carry extra central endomorphisms in mod \(p\) coefficients with different origins, from deformation quantization in characteristic \(p\) and mod \(p\) curve counts. Our main proposal is to identify these central endomorphisms of \(D\)-modules.

For the statement, let \(k\) be a field of characteristic \(p\). Let \(X, X^!\) be a pair of symplectically dual conical symplectic resolutions equipped with Hamiltonian torus actions of \(T, T^!\) resp., but now assume that \(X\) and the torus \(T\) are defined over the ground field \(k\) (admits a “split form”), while \(X^!\) and \(T^!\) are still both defined over \(\mathbb{C}\). For a more precise formulation of what we mean by symplectically dual pairs in this setting, see the discussion in Section 3.3.

For quantizations of algebraic symplectic varieties over a field of positive characteristic, Bezrukavnikov–Kaledin [18] introduced Frobenius-constant quantizations, which, roughly speaking, are quantizations of the coordinate rings of symplectic varieties carrying a central lift of the Frobenius endomorphism. Such quantizations exist for a large class of examples, including Springer resolutions [19] and Coulomb branches [20] for \(3d\) \(\mathcal{N}=4\) gauge theories [21]. A simple calculation shows that the central elements coming from such a lift of the Frobenius induce central endomorphisms of \(D\)-modules of graded/twisted traces:

On the quantum \(D\)-module side, quantum Steenrod operations [22], [23] and their equivariant generalizations [24], [25] are deformations of the classical Steenrod operations on cohomology with \(\mathbb{F}_p\)-coefficients, in the same way the quantum cohomology is a deformation of the cohomology ring with the cup product. They arise from suitable \(\mathbb{Z}/p\)-equivariant counts of genus \(0\) stable maps. It is proven in [24], [26] that the quantum Steenrod operations are covariantly constant with respect to the quantum connection and commute with each other, thereby giving rise to central endomorphisms of quantum \(D\)-modules (59):

The following is a description of our conjecture. A more precise statement, which we dub the mod p quantum Hikita conjecture, is given in 68.

1.3 Methods↩︎

To provide evidence to our conjecture, we verify its validity in the following examples.

For \(k = \mathbb{C}\), the identification of \(D\)-modules is [5], and establishing such an isomorphism with characteristic \(p\) coefficients is a straightforward extension. Matching the central endomorphisms on two sides is the true content, whose proof should be interesting on its own. We briefly highlight the underlying idea as follows.

Consider the Weyl algebra over \(\mathbb{F}_p\) \[\mathcal{D}_\hbar(\mathbb{A}^1) := \mathbb{F}_p[\hbar]\langle x, \partial\rangle / ([\partial, x] = \hbar).\] Then due to the characteristic \(p\) nature, it is straightforward to check that \(x^p\) and \(\partial^p\) are central elements. A more interesting central element is the so-called Artin–Schreier expression, \[x^p \partial^p = \prod_{i=0}^{p-1} (x \partial - i \hbar) = (x \partial)^p - \hbar^{p-1}(x\partial).\] It arises naturally in the following three contexts.

1. As described above, quantization of Poisson algebras in characteristic \(p\) admit a “large center”: the image of \(xy \in \mathbb{F}_p[x,y] = \mathcal{O}(T^*\mathbb{A}^1)\) under the Frobenius-constant quantization \(\Lambda: \mathcal{O}(T^*\mathbb{A}^1)^{(1)} \to Z(\mathcal{D}_\hbar (\mathbb{A}^1))\) is given by \[\Lambda(xy) = (x\partial)^p - \hbar^{p-1} (x\partial).\]

2. Let \(T\) be the \(1\)-dimensional complex torus; then \(H^*(BT; \mathbb{F}_p) = \mathbb{F}_p [\![u]\!]\) where \(\mathrm{deg}(u) = 2\). Then the total Steenrod operation \(\mathrm{St}_p\) satisfies \[\mathrm{St}_{p}(u) = u^{p} - \hbar^{p-1}u,\] where \(\hbar\) is the degree \(2\) element of the group cohomology of \(\mathbb{Z}/p\) with \(\mathbb{F}_p\)-coefficients.

3. Consider a (\(\hbar\)-)connection over \(\mathbb{F}_p(\!(z)\!)\) with regular singularity at \(z = 0\) \[\nabla_{z\partial_z} = \hbar z\frac{d}{dz} + A(z), \quad \quad \quad A \in \mathrm{Mat}(n, \mathbb{F}_p[\![z]\!]),\] then the \(p\)-curvature of \(\nabla\), which is a \(\mathbb{F}_p(\!(z)\!)\)-linear endomorphism, has the form \[\nabla^p_{z\partial_z} - \hbar^{p-1}\nabla_{z\partial_z}.\]

Our proof of Theorem 6 is based on reducing the covariantly constant endomorphisms of the \(D\)-modules to Artin–Schreier type expressions.

For the \(D\)-module of twisted traces, we show that the image of Frobenius-constant quantization map \(\Lambda: \mathcal{O}(\mathcal{X})^{(1)} \to Z(\mathcal{A})\) from the universal deformation of \(X\) acts by left multiplication (61). The map \(\Lambda\) generalizes Context (1) from quantization theory to the symplectic resolutions we consider. The map \(\Lambda\) can be described explicitly for the Springer resolution \(T^*G/B\) [19] and for hypertoric varieties (Coulomb branches of abelian gauge theories) [21]. These are both eventually described by the ring of crystalline differential operators, in which the Artin–Schreier expressions enter in a direct way.

For quantum \(D\)-modules, Context (3) enters through the conjecture of the second-named author [25]. Here, let \(X^!\) be a \(T^{!}\)-equivariant conical symplectic resolution over \(\mathbb{C}\). In [25], generalizing the work of [22], [23], [26] to the \(T^!\)-equivariant setting, central endomorphisms \(\Sigma^{T^!}\) of the quantum \(D\)-module were constructed and related to the \(p\)-curvature of the quantum \(D\)-module:

Conjecture 8 is established in [25] for Springer resolutions and we verify it for hypertoric varieties in this paper.

Then, by appealing to the isomorphisms established in the original quantum Hikita conjecture for Springer resolutions and hypertoric varieties [5], we prove Theorem 6 by a direct comparison. Namely, we show that the action from “degree 2” classes can be identified with the \(p\)-curvature of the respective \(D\)-modules, and appeal to the fact that such degree \(2\) classes determine the action in these examples as they generate.

1.4 Related work and speculations↩︎

The fact that the conical symplectic resolutions arising in 3D mirror symmetry have lifts to integer coefficients was observed as early as in Braverman–Finkelberg–Nakajima’s original work in Coulomb branches of gauge theories, [20]. In particular, by taking mod \(p\) reductions, we naturally obtain objects of arithmetic algebraic geometry.

This aspect was leveraged crucially in the inspirational work of Lonergan [21], where Frobenius-constant quantizations of mod \(p\) Coulomb branches are constructed via Steenrod operations (the version for equivariant Borel–Moore homology). Lonergan’s work highlights the important similarity between Frobenius-constant quantizations and Steenrod operations which serves as the starting point of our discussion: for dual pairs of symplectic resolutions given by Higgs/Coulomb branches in supersymmetric gauge theories, our conjecture can be interpreted as the claim that the (quantum) Steenrod operations in the mod \(p\) cohomology of Higgs branches are dual to Lonergan’s Frobenius-constant quantization.

The relationship between quantum Steenrod operations and \(p\)-curvature of quantum connections was noted in [25]. In the context of arithmetic 2D mirror symmetry, the relationship between quantum Steenrod operations and \(p\)-curvature type expression in Cartan calculus in de Rham cohomology is also discussed in [30]. Our observation that Frobenius-constant quantizations act by \(p\)-curvature operators on the \(D\)-module of twisted traces is drawn from the construction of the Frobenius-constant quantizations themselves as \(p\)-curvature in crystalline differential operators in many examples [19], [31], [32].

We collect more speculative connections with other works as remarks below.

1.5 Organization of the paper↩︎

In section 2, we set our notations and conventions for symplectic resolutions (with a conical \(\mathbb{G}_m\)-action and a Hamiltonian torus action), and survey the formulation of symplectic duality via the quantum Hikita conjecture.

In section 3, we introduce the covariantly constant endomorphisms on the \(D\)-modules in the quantum Hikita conjecture in positive characteristic: the quantum Steenrod operators and the multiplication operators from Frobenius-constant quantizations. Then we formulate a precise version of our conjecture.

In section 4, the verification of the conjecture is provided for the case of Springer resolutions and hypertoric varieties (Higgs/Coulomb branches for abelian gauge theories). These go through the identification of the p-curvature of the respective \(D\)-modules and a “generation in degree \(2\)” statement.

1.6 Acknowledgements↩︎

The first author thanks Andrei Okounkov for illuminating discussions on Frobenius structures in enumerative geometry. The second author thanks Dan Pomerleano for introducing the inspiring paper of [21] to him. We thank Zihong Chen, Sanath Devalapurkar, Vasily Krylov, Calder Morton-Ferguson, Nicholas Proudfoot, Mohan Swaminathan, Ben Webster for helpful discussions at various stages of this project. In particular, the suggestion to look into quantum Hikita conjecture [5] was independently given to the second author by Vasily Krylov, Ben Webster, and the first author. We also thank Hunter Dinkins, Ivan Karpov, and Vasily Krylov for explaining their forthcoming work on quantum Hikita conjecture. Finally, we thank Zihong Chen, Vasya Krylov, and the anonymous referee for comments on a draft of this paper which improved the exposition greatly.

2 Quantum Hikita conjecture↩︎

Our formulation of the 3D mirror symmetry conjecture enhances the quantum Hikita conjecture proposed by [5]. In this section, we review the statement of the quantum Hikita conjecture by [5] for symplectic dual pair of symplectic resolutions; for a more detailed description, we refer the reader to the original paper. We begin by describing symplectically dual pairs, the definition of their quantum \(D\)-modules and the \(D\)-modules of twisted traces, and the quantum Hikita conjecture positing the identification of these \(D\)-modules for dual pairs of symplectic resolutions.

2.1 Symplectic resolutions↩︎

The 3D mirror symmetry conjecture or the symplectic duality conjecture describes a duality between pairs of smooth symplectic varieties known as symplectic resolutions.

2.1.1 Definition↩︎

Throughout our discussion, we work over a field \(k\) which may be of positive characteristic. For a discussion of symplectic resolutions in positive characteristic and especially the theory of quantizations in characteristic \(p\), we refer to the excellent [38].

Given a fixed \(Y\), the choice of a resolution \(X \to Y\) may not be unique (they may be related by birational transformations). When we discuss symplectic resolutions, we implicitly fix a choice.

There are two main sources of such varieties: (i) the (generalized) Springer resolution associated to a pair \((G, P)\) of a reductive group \(G\) and a parabolic subgroup \(P\), and (ii) supersymmetric gauge theory associated to a pair \((G,V)\) of a reductive group \(G\) and a complex representation \(V \in \mathrm{Rep}(G)\). For (ii), the data \((G,V)\) give rise to constructions of two large classes of symplectic resolutions, (ii-1) the Higgs branches and (ii-2) the Coulomb branches. In particular, (ii) includes hypertoric varieties, resolutions of Kleinian singularities, Nakajima quiver varieties, etc., as examples. We will review these constructions briefly in the subsequent subsection.

Note that the Poisson bracket is \((-2)\)-shifted with respect to the conical grading, and hence \(\mathcal{O}(X)^2\) forms a Lie algebra with respect to the Poisson bracket.

For a conical Hamiltonian symplectic resolution and its \(X^\bullet(T) \times \mathbb{Z}\)-grading on \(\mathcal{O}(X)\), we denote the \((\lambda, n)\)-graded piece as \(\mathcal{O}(X)^n_\lambda\).

2.1.2 Deformations and quantizations↩︎

Recall the following definition:

Over \(k = \mathbb{C}\), the general theory [39] of conical Hamiltonian symplectic resolutions \((X,\omega)\) guarantees the existence of

• Deformations of \(X\): a deformation family parametrized by \(H^2(X)\), fitting into a diagram \[\begin{tikzcd} \mathcal{X} \dar \rar & H^2(X) \dar \\ \mathcal{Y} \rar & H^2(X)/W \end{tikzcd}\] with \(\mathcal{X}\) as the universal Poisson deformation of \(\mathcal{X}\) and \(W\) is the (Namikawa) Weyl group.

• Universal quantizations of \(Y\): quantizations of \(\mathcal{X}\), whose global sections are denoted by \(\mathcal{A}\). Denoting the quantization parameter by \(\hbar\), this is equipped with a map \(\mathcal{A} / k \hbar \to \mathcal{O}(\mathcal{X})\) to the coordinate ring of the universal deformation.

For a more detailed discussion of the deformations and quantizations of symplectic resolutions, we refer the reader to [4].

If \(k\) is of positive characteristic, we are unaware of such a general theory. However, in many specific examples, both the universal deformation and the universal quantizations can be described explicitly, and may be verified to be split (defined over \(\mathbb{Z}\) or some localization thereof). In this paper, we consider examples for which the existence of universal deformations and quantizations over some ring of integers can be verified explicitly. These provide the examples of conical Hamiltonian symplectic resolutions over fields of positive characteristic.

If \(X\) is Hamiltonian, then there is an induced \(X^\bullet(T)\)-grading on \(\mathcal{O}(\mathcal{X})\) and \(\mathcal{A}\), arising from a fiberwise \(T\)-action for the deformation family \(\mathcal{X} \to H^2(X)\). We again denote the \((\lambda, n) \in X^\bullet(T) \times \mathbb{Z}\)-graded pieces as \(\mathcal{O}(\mathcal{X})_\lambda^n\) and \(\mathcal{A}_\lambda^n\), respectively.

Explicitly, the weight grading takes the form \[\mathcal{A} = \bigoplus_{\lambda \in X^\bullet(T)} \mathcal{A}_\lambda, \quad \mathcal{A}_\lambda := \{ a \in \mathcal{A} : [x, a] = \hbar \langle \lambda, \bar{x} \rangle a, \;x \in \mathcal{A}^2_0\};\] note that the commutator action of \(x \in \mathcal{A}_0^2\) factors through its image \(\bar{x} \in \mathfrak{t}\) under the projection of Lie algebras \(\mathcal{A}_0^2 \to \mathcal{O}(X)_0^2 \cong \mathfrak{t}\).

2.2 Examples of symplectic resolutions↩︎

2.2.1 Springer resolutions↩︎

Let \(G\) be a semisimple, simply-connected split algebraic group defined over \(k\), denote by \(B\) a Borel subgroup and \(T\) a maximal torus; the choice of \(B\) gives an identification of \(T\) with the abstract Cartan group \(H\) of \(G\) which we fix. The flag variety \(G/B\) is smooth regardless of the characteristic.

In positive characteristic, we choose the characteristic of \(k\) to be \(p > h\), where \(h\) is the maximal Coxeter number of \(G\). This in particular ensures that there is a non-degenerate invariant symmetric form on \(\mathfrak{g}\).

The cotangent bundle \(X = T^*(G/B)\) is a smooth variety over \(k\) and admits an affinization map to the nilpotent cone \(X \to \mathcal{N}\). It is called the Springer resolution for the pair \((G,B)\). In positive characteristic, under the assumption \(p > h\) on the characteristic, we may identify \(\mathcal{N}\) as a subvariety of either \(\mathfrak{g}\) or \(\mathfrak{g}^*\), [19].

The Springer resolution \(X = T^*(G/B)\) admits the structure of a conical Hamiltonian symplectic resolution, where the action of \(T\) on \(X\) lifts the left multiplication action of \(T\) on \(G/B\). The affinization map \(X \to \mathcal{N}\) gives an identification of \(\mathcal{O}(X)\) with the coordinate ring on the zero fiber of the adjoint quotient map \(\mathfrak{g}^* \to \mathfrak{t}^*/W\), which carries the Poincaré–Birkhoff–Witt grading so that \(x \in \mathfrak{g} \subseteq k[\mathfrak{g}^*]\) carries degree \(2\).

The universal deformation \(\mathcal{X}\) of \(X\) is given by the Grothendieck–Springer resolution \(\widetilde{\mathfrak{g}}^*\), whose affinization agrees with \(\mathfrak{g}^* \times_{\mathfrak{t}^*/W} \mathfrak{t}^*\). The quantization \(\mathcal{A}= U_\hbar \mathfrak{g} \otimes_{Z(U_\hbar\mathfrak{g})} \mathrm{S} ( \mathfrak{t}) [\hbar]\) is the Rees algebra with respect to the PBW filtration, where the Rees parameter is denoted by \(\hbar\).

For non-maximal parabolic \(P \le G\), one still has a conical Hamiltonian symplectic resolution \(T^*(G/P)\) resolving its affinization, the generalized Springer resolution. In this paper, we exclusively consider the case \(P = B\).

2.2.2 Gauge theory: Higgs branches↩︎

Let \(G\) be a complex reductive connected Lie group and \(V\) be a finite-dimensional complex representation. From these data, one can construct two symplectic singularities called the Higgs branch and the Coulomb branch. We first briefly discuss the construction of the Higgs branch.

Take the cotangent bundle \(T^* V \cong V \times V^*\) equipped with the standard symplectic form. Then the \(G\)-action on \(V\) can be lifted to a Hamiltonian action on \(T^* V\) with moment map \(\Phi: V \times V^* \to \mathfrak{g}^*\).

Deformations and resolutions of Higgs branches are both controlled by characters of \(G\), as follows. After choosing a character \(\chi: G \to \mathbb{G}_m\), we can take the GIT quotient \[T^* V /\!\!/\!\!/\!\!/_{0,\chi} G := \Phi^{-1}(0) /\!\!/_{\chi} G,\] with affinization map \[\Phi^{-1}(0) /\!\!/_{\chi} G \to \Phi^{-1}(0) /\!\!/_{0} G,\] which in nice cases yield a (partial) resolution of the Higgs branch.

Deformations of Higgs branches arise from allowing the moment map parameter to vary, in particular for \((\mathfrak{g}^{ab})^* := \mathrm{Hom}(\mathfrak{g}, k) \subseteq \mathfrak{g}^*\), \[\Phi^{-1}((\mathfrak{g}^{ab})^*)/\!\!/_\chi G \to \Phi^{-1}((\mathfrak{g}^{ab})^*)/\!\!/_0 G\] is a model for the universal deformation and its affinization.

2.2.3 Gauge theory: Coulomb branches↩︎

Continuing our notations introduced above, let \(G\) be a complex reductive connected Lie group and suppose \(V\) is a finite-dimensional complex representation. Braverman–Finkelberg–Nakajima introduced the construction of Coulomb branches associated to the data \((G,V)\). The construction may be regarded as an analog of the (generalized) Springer resolutions in the affine, parabolic setting.

There is a modular description of both \(\mathrm{Gr}_G\) and \(\mathcal{T}_{G,V}\). The affine Grassmannian \(\mathrm{Gr}_G\) can be thought of as the space of pairs \(\{ (\mathcal{P}, \phi) \}\) where \(\mathcal{P}\) is a principal \(G\)-bundle over the formal disc and \(\phi\) is a trivialization over the punctured disc, considered up to isomorphism of such pairs. The identification is given by describing the action of \(G(\!(z)\!)\) on such space by multiplying the trivialization by an element of \(G(\!(z)\!)\); this is a transitive action with stabilizer \(G[\![z]\!]\).

Similarly, the BFN Springer resolution can be described as isomorphism classes of triples \(\{ (\mathcal{P}, \phi, s) \}\) where \(s\) is a section of the associated \(V\)-bundle over the formal disc. The action map \(\mathcal{T}_{G,V} \to V(\!(z)\!)\) is given by using the trivialization over the punctured disc to consider \(s\) as an element of \(V(\!(z)\!)\).

By definition, \(\mathcal{R}_{G, V}\) consists of elements in \(\mathcal{T}_{G, V}\) such that the section \(s\) is mapped to a formal power series (as opposed to Laurent series) under the trivialization \(\phi\). The advantage of considering \(\mathcal{R}_{G, V}\) is that the BFN Steinberg variety equipped with the \(G(\!(z)\!)\)-action can be identified as stacks with \(\mathcal{R}_{G,V}\) equipped with the \(G[\![z]\!]\)-action. The fact that \(G[\![z]\!]\) is pro-algebraic (in particular, much smaller than \(G(\!(z)\!)\)) is leveraged to define the commutative convolution product structure on equivariant (Borel–Moore) homology of \(\mathcal{R}_{G,V}\) through the convolution diagram, see [20].

The group \(\mathbb{G}_m = \mathbb{C}^*\) acts on \(\mathcal{R}_{G, V}\) as automorphisms of the (punctured) formal disc, and the equivariant (Borel–Moore) homology \[\mathcal{A}_{\hbar}(G,V) := H_*^{G[\![z]\!]\rtimes \mathbb{G}_m}(\mathcal{R}_{G, V}; k)\] with the convolution product structure defines a noncommutative deformation of the Coulomb branch algebra \(H_*^{G[\![z]\!]}(\mathcal{R}_{G, V}; k)\), inducing a Poisson structure on \(\mathcal{M}_C(G, V)\).

Deformations and resolutions of Coulomb branches may be obtained by introducing an additional torus action \(T_F \to \mathrm{GL}(V)\) commuting with the \(G\)-action, or more generally an extension of \(T_F\) by \(G\) which fits into a short exact sequence \[1 \longrightarrow G \longrightarrow \tilde{G} \longrightarrow T_F \longrightarrow 1,\] see [20]. The point is that if \(V\) is a \(\tilde{G}\) representation, \(\mathcal{M}_C(G, V)\) is a Hamiltonian reduction of \(\mathcal{M}_C(\tilde{G}, V)\) under the dual group \(T_F^{\vee}\)-action, and \(\mathcal{A}_{\hbar}(G,V)\) is obtained from \(\mathcal{A}_{\hbar}(\tilde{G},V)\) by quantum Hamiltonian reduction. The deformation space is given by \(\mathrm{Lie}(T_F)\). On the other hand, upon choosing a character of \(T_F^{\vee}\), one can define a (partial) resolution of \(\mathcal{M}_C(G, V)\), which has been conjectured to give rise to a symplectic resolution in [20] and verified in [40].

2.3 Symplectic duality↩︎

Symplectic duality posits that conical Hamiltonian symplectic resolutions arise in pairs, where certain structures are interchanged for dual pairs. Depending on what the structure of concern is, there are distinct formulations of the symplectic duality phenomena, as surveyed in [4]. In this subsection, we explain the formulation of the quantum Hikita conjecture which identifies the quantum \(D\)-module and the \(D\)-module of twisted traces for dual pairs.

2.3.1 Quantization exact sequence and cohomology exact sequence↩︎

Consider a conical Hamiltonian symplectic resolution \(X\) with a \(T \times \mathbb{G}_m\)-action. We may construct two short exact sequences of vector spaces involving the canonical quantization of \(X\) and the \(T \times \mathbb{G}_m\)-equivariant cohomology of \(X\) respectively, which are interchanged for symplectically dual pairs.

For symplectically dual pairs \((X, X^!)\), we expect the quantization exact sequence and the cohomology exact sequence to be interchanged, with isomorphisms even over integers \(H^2(X^!;\mathbb{Z}) \cong \mathfrak{t}_{\mathbb{Z}}\) and \(H^2_{T^!}(\mathrm{pt};\mathbb{Z}) \cong (\mathfrak{t}^!)^*_{\mathbb{Z}} \cong H_2(X;\mathbb{Z})\). Both of the exact sequences are (non-canonically) split: for the former, see [5]; for the latter, it follows from equivariant formality.

2.3.2 Equivariant and Kähler roots↩︎

Consider a conical Hamiltonian symplectic resolution \(X\) with a \(T \times \mathbb{G}_m\)-action. We may construct finite collections of weights in \(X^\bullet(T)\) and curve classes in \(H_2(X; \mathbb{Z})\) dubbed the equivariant roots and the Kähler roots respectively, which are interchanged for symplectically dual pairs.

We first define the equivariant roots. Denote the canonical quantization as \(\mathcal{A}\). We denote \(\mathcal{A}^+ \subseteq \mathcal{A}\) to be the two-sided ideal of elements with positive conical grading.

A choice of a cocharacter \(\sigma : \mathbb{G}_m \to T\) determines a distinguished subset of \(\Sigma_{eq}\):

The positive equivariant roots generate an additive semigroup denoted by \(\mathbb{Z}_{\ge 0} \Sigma_{eq,+}\).

We now define the Kähler roots. The definition of Kähler roots depends on a (hypothetical) structure theorem for the equivariant quantum cohomology of symplectic resolutions conjectured by Okounkov.

Denote by \(\Lambda' := k [z^\alpha : \alpha \in H_2^{\mathrm{eff}}(X;\mathbb{Z})]\) to be the polynomial Novikov ring, generated over \(k\) by symbols \(z^\alpha\) where \(\alpha\) indexes the additive semigroup of effective curve classes in \(H_2(X;\mathbb{Z})\). Let \(\hat{\Lambda}\) be the completion of \(\Lambda'\) at the maximal ideal generated by \(z^\alpha\).

Recall that the (equivariant) quantum cohomology of \(X\) is the \(k\)-vector space \[QH^*(X):=H^*_{T \times \mathbb{G}_m}(X) \otimes \hat{\Lambda}\] endowed with a graded commutative product \(\star\) which is constructed using genus \(0\), \(3\)-pointed equivariant Gromov–Witten invariants. Denote by \(\hbar\) the generator of \(H^2_{\mathbb{G}_m}(pt)\), which represents the conical weight of the holomorphic symplectic form.

In particular, the conjecture implies that the quantum cohomology can be defined over \(H^*_{T \times \mathbb{G}_m}(X) \otimes \Lambda\), where \(\Lambda = \Lambda' [(1-z^\alpha)^{-1}: \alpha \in \Sigma_{Kah, +}] \subseteq \hat{\Lambda}\) is the localization which is much more restricted than a general power series in \(z^\alpha\).

For symplectically dual pairs \((X, X^!)\), symplectic duality predicts that the equivariant and the Kähler roots are interchanged.

2.3.3 \(D\)-module of twisted traces and the quantum \(D\)-module↩︎

The quantum Hikita conjecture identifies two \(D\)-modules associated to a conical Hamiltonian symplectic resolution for a dual pair. We first describe the relevant rings of differential operators for both constructions, over which we will define the respective \(D\)-modules.

Fix a conical Hamiltonian symplectic resolution \((X, \omega)\) with its \(T \times \mathbb{G}_m\)-action, and choose a cocharacter \(\sigma : \mathbb{G}_m \to T\). Denote the corresponding positive equivariant roots by \(\Sigma_{eq,+}\). Also, assuming Conjecture 31, write the set of positive Kähler roots as \(\Sigma_{Kah, +}\).

Here, the \(k[\hbar]\)-module structure on \(R_{eq}\) comes from the quantization exact sequence. Although \(R_{eq}\) is not commutative, one can show that there exists a well-defined localization \(R_{eq, reg} := R_{eq} [(1-z^\mu)^{-1}]\) obtained by inverting \((1-z^\mu)\) for \(\mu \in \Sigma_{eq,+}\), see [5].

Now we define the \(D\)-module of twisted traces from the universal quantization \(\mathcal{A}\) of \(X\).

The fact that the \(D\)-module of twisted traces is well-defined is verified in [5]. By inverting \((1-z^\mu)\) for \(\mu \in \Sigma_{eq,+}\), we obtain a \(R_{eq, reg}\)-module \(M_{eq, reg} := R_{eq, reg} \otimes_{R_{eq}} M_{eq}\).

The module \(\mathscr{B}(\mathcal{A})\) is considered as a quantum (i.e. \(z\)-parameter) deformation of the \(B\)-algebra \(B(\mathcal{A})\). It does not carry a natural algebra structure, unless \(\mathcal{A}\) itself is commutative.

The counterpart of the ring of differential operators on the quantum cohomology side is the following.

Similar to the definition of \(R_{eq, reg}\), by inverting \((1-z^\alpha)\) for \(\alpha \in \Sigma_{Kah, +}\), we obtain \(R_{Kah, reg}\).

The quantum \(D\)-module is defined using the equivariant quantum cohomology of \(X\), whose definition in the present form depends on assuming Conjecture 31.

Note that the quantum \(D\)-module only exists as a \(R_{Kah, reg}\)-module, as the quantum product \(\star\) involves denominators of the form \((1-z^{\alpha})^{-1}\) for \(\alpha \in \Sigma_{Kah, +}\), see 31. The quantum \(D\)-module is closely related to the object known as the quantum connection, where the action of \(x\) is interpreted as the “covariant derivative” \(\hbar \partial_{\bar{x}} + x \star\) where \(\partial_{\bar{x}}\) acts on the variable \(z^\beta\) by \(\partial_{\bar{x}} z^{\beta} = \langle \beta, \bar{x} \rangle z^\beta\).

For symplectically dual pairs \((X, X^!)\), we expect that the equivariant and Kähler ring of differential operators are interchanged, identifying the \(D\)-module of twisted traces and the (specialized) quantum \(D\)-module. We spell this conjectural correspondence more precisely in the next subsection.

2.4 Quantum Hikita conjecture↩︎

Here, we survey the formulation of symplectic duality via the quantum Hikita conjecture, following [5]. In this subsection, we assume that \(k = \mathbb{C}\) and that the symplectic resolutions \(X \to Y\) and the torus acting on \(X\) are defined over \(\mathbb{C}\). Assume that \(X\) and \(X^!\) are symplectically dual pairs of conical Hamiltonian symplectic resolutions, equipped with actions of algebraic tori \(T\) and \(T^!\). We denote the equivariant/Kahler roots, ring of differential operators, and the \(D\)-modules constructed for \(X^!\) with the superscript \(!\).

The Hikita conjecture, originally formulated as [6], states that the cohomology ring of \(X^!\) is isomorphic to the coordinate ring of the (scheme-theoretic) \(T\)-fixed locus of \(Y\).

To better relate the Hikita conjecture with its equivariant and quantum generalizations, we take a different perspective. Namely, the coordinate ring \(\mathcal{O}(Y^T)\) can be described in terms of the \(B\)-algebra construction.

In the commutative case, where \(A\) is the coordinate ring of an affine variety with a \(T\)-action, the \(0\)-weight part corresponds to the coordinate ring of the orbit space (affine GIT quotient). Quotienting out by \(A_\lambda A_{-\lambda}\) removes the non-trivial orbits in the orbit space, so that \(\mathrm{Spec}B(A)\) is the (scheme-theoretic) fixed locus. As \(X \to Y\) is the affinization map, we know that \(\mathcal{O}(X) = \mathcal{O}(Y)\) as commutative algebras. Therefore, Conjecture 41 can be restated as the isomorphism \[B(\mathcal{O}(X)) \cong H^*(X^!; k).\]

The equivariant Hikita conjecture (the Hikita–Nakajima conjecture, cf. [7]) extends the Hikita conjecture to a conjectural isomorphism between the \(B\)-algebra of the universal quantization \(\mathcal{A}\) of \(\mathcal{O}(X)\) and the equivariant cohomology of \(X^!\):

In particular, Conjecture 43 predicts isomorphisms \(H^2(X) \cong \mathfrak{t}^!\) and \(\mathfrak{t} \cong H^2(X^!)\) (the identification of equivariant and Kähler parameters). Also note that \(B(\mathcal{A})\) is not a priori commutative, but the existence of the isomorphism with the commutative \(H^*_{T^! \times \mathbb{G}_m} (X^! ;k)\) implies that the ring structure is in fact commutative.

The quantum Hikita conjecture introduced by [5] extends the Hikita–Nakajima conjecture further to include the data of roots and \(D\)-modules.

The compatibility of the isomorphisms refers to the following. First, there is an identification of the equivariant roots and the Kähler roots of \(X\), \(X^!\) respectively, inducing an isomorphism of monoids \(\mathbb{Z}_{\ge 0} \Sigma_{eq, +} \cong \mathbb{Z}_{\ge 0} \Sigma_{Kah, +}^!\). Together with the isomorphism \(\mathcal{A}_0^2 \cong H^2_{T^! \times \mathbb{G}_m}(X^!)\) depicted by the equivariant Hikita conjecture, we have an isomorphism of graded \(k[\hbar]\)-algebras \[R_{eq} := k[z^\mu : \mu \in \mathbb{Z}_{\ge 0} \Sigma_{eq, +}] \otimes \mathrm{Sym} \mathcal{A}_0^2 \cong k [z^\alpha : \alpha \in \mathbb{Z}_{\ge 0 } \Sigma_{Kah, +}^!] \otimes \mathrm{Sym}H^2_{T^! \times \mathbb{G}_m}(X^!) =: R_{Kah}^!.\] After localization, this isomorphism implies the isomorphism of graded algebras \[R_{eq, reg} \cong R_{Kah, reg}^!.\] Lastly, we should have an isomorphism \[M_{eq, reg} := \mathscr{B}(\mathcal{A}) [(1-z^\mu)^{-1}] \cong \Lambda \otimes H^*_{T^! \times \mathbb{G}_m}(X^{!}) =: M_{Kah, reg}^!\] of graded \(R_{eq, reg} \cong R_{Kah, reg}^!\)-modules.

3 Quantum Hikita conjecture mod \(p\): Statement↩︎

In this section, we provide the statement of our mod \(p\) 3D mirror symmetry conjecture, inspired by the quantum Hikita proposal of [5], as an isomorphism of the mod \(p\) quantum \(D\)-module and the mod \(p\) \(D\)-module of twisted traces for symplectically dual pairs, together with the action of Frobenius-linear covariantly constant operators.

For the mod \(p\) quantum \(D\)-module, these Frobenius-linear operators are given by the quantum Steenrod operators [26] (or equivariant generalizations thereof, [25]). We emphasize that for the quantum \(D\)-module the symplectic resolution itself is still defined over \(\mathbb{C}\), and the positive characteristic only enters through the coefficients of quantum cohomology algebra.

For the mod \(p\) \(D\)-module of twisted traces, these Frobenius-linear operators are given by the multiplication operators arising from Frobenius-constant quantizations of the underlying symplectic singularity. Such Frobenius-constant quantizations exist for a large class of symplectic singularities, including Springer resolutions [19] and all BFN Coulomb branches [21].

We fix \(k\) an algebraically closed field of characteristic \(p\), which is assumed to be \(\gg 0\) if necessary.

Denote by \(\mu_p := \mu_p(\mathbb{C}) \subseteq \mathbb{G}_m(\mathbb{C})\) the multiplicative group of complex \(p^\mathrm{th}\) roots of unity. By choosing a generator \(\zeta = e^{2\pi i/p} \in \mu_p\), we obtain (and fix) an isomorphism \(\mu_p \cong \mathbb{Z}/p\mathbb{Z}\) of abelian groups.

We now describe the candidate descriptions for the Frobenius-linear actions mentioned above.

3.1 Quantum Steenrod operations↩︎

In characteristic \(p\), the quantum \(D\)-module admits an action of the (Frobenius-twisted) mod \(p\) quantum cohomology algebra, known as the quantum Steenrod operations. These are (Frobenius-)linear operators that deform the usual action of Steenrod powers in mod \(p\) singular cohomology by taking counts of certain holomorphic spheres with a \(\mathbb{Z}/p\)-symmetry into account.

3.1.1 A heuristic description↩︎

We discuss the quantum Steenrod operators of [26], which reformulates the quantum Steenrod operations of [22] as linear endomorphisms of the mod \(p\) quantum cohomology. We also need their torus-equivariant generalizations introduced in [25] for applications to conical Hamiltonian symplectic resolutions.

The actual definition of quantum Steenrod operators involves constructions in symplectic enumerative geometry, due to technical issues in curve-counting with positive characteristic coefficients; in particular, an algebro-geometric definition is not currently available. Here we give a heuristic description of how the definition of these operators would look like in a hypothetical situation where these foundational issues do not exist, and point out the actual difficulties. For the actual construction in symplectic geometry which allows one to overcome such difficulties, we refer to [26] (for the original definition) and [24], [25] (for the torus-equivariant versions).

Fix \(C =\mathbb{P}^1 = \mathbb{C} \cup \{ \infty \}\) to be a curve with the following \((p+2)\) distinguished marked points \[z_0 = 0, \;z_1 = \zeta = e^{2\pi i / p}, \dots, z_p = \zeta^p = 1,\; z_\infty = \infty.\] We consider parametrized maps from \(C\) to a smooth (quasi)projective complex variety \(X\).

Let \(\tau\) be the rotation of the curve \(C\) by the angle of \(2\pi/p\) which fixes \(z_0\) and \(z_\infty\), then, it cyclically permutes the marked points as \[\tau : (C, z_0, z_1 , z_2 \dots, z_p, z_\infty) \mapsto (C, z_0, z_2, z_3, \dots, z_1, z_\infty).\] Using \(\tau\), one defines an action of the cyclic group \({\mathbb{Z}/p}\) on \(C\), whose generator is identified with \(\zeta = e^{2\pi i/p}\). For \(\eta = \zeta^j \in \mathbb{Z}/p\), we denote by \(\tau(\eta) = \tau^j\) the corresponding automorphism of \(C\).

There is also an action of \(\mathbb{Z}/p\) which cyclically permutes the product \(X^p\): \[\begin{align} \label{eqn:X94p-cyclic-permutation} \sigma_{X} :& \;\mathbb{Z}/p \times X \times X^p \times X \to X \times X^p \times X, \\ & [\eta; (x_0; x_1, x_2, \dots, x_{p}; x_\infty)] \mapsto (x_0; x_{1+\eta}, x_{2+\eta}, \dots, x_{p+\eta}; x_\infty), \end{align}\tag{1}\] where the numbers \(k+\eta\) are read mod \(p\).

The moduli space of parametrized maps \(u: C \cong \mathbb{P}^1 \to X\), \[\mathcal{M}_\alpha := \{ u : C \to X : u_*[C] = \alpha \in H_2(X;\mathbb{Z}) \},\] carries an evaluation map \[\begin{align} \mathrm{ev}: \mathcal{M}_\alpha &\to X \times X^p \times X \\ (u: C \to X) &\mapsto \left( u(z_0); u(z_1), \dots, u(z_p); u(z_\infty) \right) \end{align}\] that is \(\mathbb{Z}/p\)-equivariant with respect to the action on \(\mathcal{M}_\alpha\) induced by source rotation \(\tau\) and the action \(\sigma_X\) on the target. The moduli space \(\mathcal{M}_\alpha\) is noncompact in general, and must be suitably compactified by stable maps as is well known (see, e.g. [43]). When the target \(X\) is non-compact, the properness of the stable maps and the evaluation map \(\mathrm{ev}\) require additional hypotheses on \(X\) such as the existence of the conical \(\mathbb{G}_m\)-action, see e.g. [44]. Let us assume that \(\mathcal{M}_\alpha\) admits a \(\mathbb{Z} / p\)-equivariant compactification \(\overline{\mathcal{M}}_\alpha\) (by stable maps) over which \(\mathrm{ev}\) extends to a \(\mathbb{Z} / p\)-equivariant proper map, and continue to denote the extension by \(\mathrm{ev}\) (such an extension is indeed provided in [24], [25]).

Fix a cohomology class \(x \in H^*(X;k)\), and consider its image \(x_{eq} := x \otimes x \otimes \cdots \otimes x \in H^*_{\mu_p}(X^p ; k)\) under the (non-linear) map that sends \(x \in H^*(X;k)\) to its \(p\)-fold tensor power. Assuming the existence of a suitable equivariant virtual fundamental cycle of \(\overline{\mathcal{M}}_\alpha\) over \(k\)-coefficients \[[\overline{\mathcal{M}}_\alpha]^{vir}_{\mu_p} \in H_{-*}^{\mu_p} (\overline{\mathcal{M}}_\alpha ; k),\] one would define the corresponding “correlator” by the push-pull diagram

as \[\langle x_0 ; x_{eq}; x_\infty \rangle_{0, \alpha} := (\mathrm{proj})_* \left( [\overline{\mathcal{M}}_\alpha]^{vir}_{\mu_p} \cap \mathrm{ev}^* (x_0 \otimes x_{eq} \otimes x_\infty) \right) \in H^*_{\mu_p}(\mathrm{pt}) \cong k [\![t, \theta]\!].\] Here, \(H^*_{\mu_p}(\mathrm{pt}) \cong k [\![t, \theta]\!]\) is the group cohomology of \(\mu_p(\mathbb{C})\) over \(k\), the graded polynomial algebra on two commuting variables \(t, \theta\) of degree \(|t| = 2\) and \(|\theta| = 1\). (The notation \(k[\![t,\theta]\!]\) refers to the \(\mathbb{Z}\)-graded completion, and since \(t, \theta\) carry nontrivial degrees, \(k[\![t,\theta]\!] \cong k [t] \otimes\Lambda(\theta)\).)

Now recall that the quantum cohomology ring is an algebra over the (completed) Novikov ring \(\hat{\Lambda} = k[\![z^\alpha : \alpha \in H_2^{\mathrm{eff}}(X;\mathbb{Z})]\!]\) (cf. 2.3.2). The quantum Steenrod operators are defined as linear operators on quantum cohomology over \(k\), whose structure constants are given by the correlators above. For the definition, we fix a \(k\)-linear basis \(B\) of \(H^*(X;k)\) (and the definition is independent of the basis by linearity):

By extending \(t, \theta\)-linearly, this can be made into an endomorphism of \(H^*(X;\hat{\Lambda}) [\![t, \theta]\!]\).

The actual construction, outlined in [26] or [24], has to go through symplectic enumerative geometry and moduli space of pseudo-holomorphic maps due to the lack of theory providing the required equivariant VFC \([\overline{\mathcal{M}}_\alpha]^{vir}_{\mu_p}\) to define the correlators \(\langle x_0; x_{eq}; x_\infty \rangle_{0, \alpha}\).

In the presence of a \(T\)-action on \(X\), the equivariant generalizations of these operators are given in [25]. The operators are indexed by \(b \in H^*_T(X;k)\), and are defined as \(\hat{\Lambda}[\![t,\theta]\!]\)-linear endomorphisms \(\Sigma_b^T \in \mathrm{End} \left(H^*_{T, loc}(X;\hat{\Lambda})[\![t,\theta]\!]\right)\) of localized equivariant cohomology. The main technical result of [25] can be summarized as follows:

3.1.2 Properties of quantum Steenrod operations↩︎

In this subsection, we record the important properties of the (\(T\times \mathbb{G}_m\)-equivariant) quantum Steenrod operators, in particular its equivalence with the \(p\)-curvature of the quantum \(D\)-module. For our proofs, we only need these algebraic properties rather than their actual constructions.

Consider the localized equivariant cohomology with Novikov coefficients, \(H^*_{T \times \mathbb{G}_m, loc} (X; \hat{\Lambda}) [\![{t, \theta}]\!] := H^*_{T \times \mathbb{G}_m} (X; \hat{\Lambda}) [\![{t, \theta}]\!] \otimes \mathrm{Frac}(H^*_{T \times \mathbb{G}_m}(\mathrm{pt}) )\). The quantum Steenrod operator \(\Sigma_x^T\) is a \(\hat{\Lambda} \otimes k[\![t, \theta]\!] \otimes H^*_{T \times \mathbb{G}_m}(\mathrm{pt})\)-linear endomorphism of this vector space. Note that this group also carries a (\(T \times \mathbb{G}_m\)-equivariant) quantum product, which we denote by \(\star_{T}\).

The quantum Steenrod operators are related to the usual Steenrod operators as follows. Recall that given a cohomology class \(x \in H^*_T(X;k)\), one has the (non-linear) map \[\begin{align} H^*_T(X;k) &\to H^*_{T^p \rtimes \mu_p} (X^p;k) \\ x &\mapsto x_{eq} := x \otimes x \otimes \cdots \otimes x. \end{align}\] Note that there is an action of \(T \times \mu_p\) on \(X\), where \(\mu_p\) acts trivially. The diagonal inclusion \(\Delta: X \to X^p\), which is \((T \times \mu_p, T^p \rtimes \mu_p)\)-equivariant, induces a pullback map in equivariant cohomology. Through this pullback we define

The coefficients in \(H^*_T(X;k)\) of \(\mathrm{St}^T(x)\) as a polynomial in \(t, \theta\) are (up to a sign, see [26] or [21]) the usual Steenrod operations on \(x\) in singular cohomology with positive characteristic coefficients, hence the name.

Property (ii) is just Frobenius-linearity. Property (iii) claims that quantum Steenrod operators are equivariant deformations of the \(p\)-th power operation in quantum cohomology, and (iv) claims that the quantum Steenrod operators are quantum deformations of the Steenrod operations in mod \(p\) cohomology. Property (v) is called the quantum Cartan relation, which shows that quantum Steenrod operators \(\Sigma_x^T\) collectively define an algebra action from the (Frobenius-twisted) mod \(p\) (\(T\)-equivariant) quantum cohomology algebra. In particular, the actions of \(\Sigma_x^T\) and \(\Sigma_{x'}^T\) graded commute for any \(x, x'\) by the graded commutativity of \(\star_T\).

The main result of [25] is the identification of quantum Steenrod operators on degree \(2\) classes for conical Hamiltonian symplectic resolutions with the \(p\)-curvature of the quantum connection. Recall from 2.3.3 that the quantum connection is a \(k[\![t, \theta]\!] \otimes H^*_{T \times \mathbb{G}_m}(\mathrm{pt})\)-linear operator \[\nabla_x := t \partial_{\bar{x}} + x \;\star_T : H^*_{T \times \mathbb{G}_m, loc} (X; \hat{\Lambda}) [\![{t, \theta}]\!] \to H^*_{T \times \mathbb{G}_m, loc} (X; \hat{\Lambda}) [\![{t, \theta}]\!]\] where \(\partial_{\bar{x}}\) acts on \(z^\alpha \in \hat{\Lambda}\) by \(\partial_{\bar{x}} z^\alpha = \langle \alpha, \bar{x} \rangle z^\alpha\). Here, \(t\) is the equivariant parameter for the \(\mu_p(\mathbb{C})\)-action on the source curve for holomorphic maps \(u : \mathbb{P}^1 \to X\) counted for defining the quantum connection, which we later identify with the equivariant parameter for the \(\mathbb{G}_m(\mathbb{C})\)-action on the target \(X\) (see 39).

The right hand side expression, \(\nabla_x^p - t^{p-1}\nabla_x\), is the operator known as the \(p\)-curvature of the quantum connection. Although the connection \(\nabla_x\) is not \(\hat{\Lambda}\)-linear, the \(p\)-curvature \(\nabla_x^p - t^{p-1} \nabla_x\) remarkably is \(\hat{\Lambda}\)-linear.

Conjecturally the nilpotent error always vanishes, that is, the quantum Steenrod operators (on degree \(2\) classes) are expected to be always equal to the \(p\)-curvature of the quantum connection. Current proof for this conjecture assumes generic semisimplicity, as follows. Recall that a collection of commuting linear operators \(\{F_i\}\) (with \([F_i, F_j] = 0\)) acting on a vector space \(V\) over a field \(K\) have jointly simple spectrum if the associated simultaneous eigendecomposition of \(V \otimes_K \overline{K}\) is into \(1\)-dimensional subspaces.

The following theorem establishes the covariant constancy of quantum Steenrod operators, which is also used in the proof of 56.

3.2 Frobenius-constant quantizations↩︎

In characteristic \(p\), the notion of a quantization as a flat \(k[\![\hbar]\!]\)-deformation still makes sense, but a striking new feature is that the resulting quantization may admit a large center. This observation motivates the definition of a Frobenius-constant quantization, the definition of which is due to [18]. Here, we observe that Frobenius-constant quantizations give rise to an action of the (Frobenius-twisted) coordinate ring of the symplectic resolution on the \(D\)-module of twisted traces. Throughout our discussion, for any vector space, algebra, etc. \(M\) over a field \(k\) of characteristic \(p\), we denote by \[M^{(1)} := M \otimes_k k\] the Frobenius pullback with respect to the Frobenius map \(\mathrm{Fr}: k \to k\) which maps \(x\) to \(x^p\).

3.2.1 Frobenius-constant quantization operators↩︎

First we recall the following definition:

Let us refer to the map \(\Lambda\) in the data of a Frobenius-constant quantization as a quantized Frobenius, as it lifts the usual, commutative Frobenius map \(\mathcal{O}(X)^{(1)} \to \mathcal{O}(X)\) in characteristic \(p\) to the quantization \(A\).

Now we fix a conical Hamiltonian symplectic resolution \(X \to Y\), its universal deformation \(\mathcal{X} \to \mathcal{Y}\) and the canonical quantization \(\mathcal{A}\) of \(\mathcal{O}(\mathcal{X})\). We assume that this quantization is equipped with the structure of a Frobenius-constant quantization, that is there exists a quantized Frobenius \[\Lambda : \mathcal{O}(\mathcal{X})^{(1)}_{\lambda} \to Z (\mathcal{A}_{p\lambda})\] where \(\mathcal{O}(\mathcal{X})_\lambda\) denotes the \(T\)-weight \(\lambda\) functions on the universal deformation and \(\mathcal{A}_{p\lambda}\) is the \(T\)-weight \(p \cdot \lambda\) part of the quantization. Such maps indeed exist for:

• Springer resolutions: classically the Frobenius center of asymptotic universal enveloping algebra (crystalline differential operators) gives the desired quantized Frobenius [48] [19],

• Coulomb branches for a gauge theory associated with \((G,V)\) (for abelian \(G\), these recover hypertoric varieties), by the construction of [21].

We use the map \(\Lambda\) to define multiplication operators acting on the \(D\)-module of twisted traces as follows. Fix \(\bar{x} \in \mathcal{O}(\mathcal{X})_0\), an element of the universal coordinate ring of torus weight \(0\).

Since the quantized Frobenius \(\Lambda\) is by definition an algebra map, the collection of multiplication operators \(\Lambda({\bar{x}})\) define an algebra action of \(\mathcal{O}(\mathcal{X})_0\), i.e. \[\Lambda(\bar{x} \bar{y}) = \Lambda(\bar{x}) \circ \Lambda(\bar{y}).\] This is mirror to the quantum Cartan relation from 54 (v).

We can extend the action to \(k[z^\mu : \mu \in \mathbb{Z}_{\ge 0} \Sigma_{eq, +} ] \otimes \mathcal{O}(\mathcal{X})_0\) by letting \(z^\mu\) act by left multiplication by \(z^{p\mu} \otimes 1\). For this extended action, the following is an immediate consequence of the definition of \(D\)-module of twisted traces:

3.2.2 Lonergan’s construction of quantized Frobenius for Coulomb branches↩︎

There is a general construction of the Frobenius-constant quantization of Coulomb branches for a gauge theory \((G,V)\) due to Lonergan, cf. [21]. We briefly sketch this construction, as (i) we later use the description to compute the quantized Frobenius for hypertoric varieties, and (ii) we would like to highlight the similarity of this construction and the construction of quantum Steenrod operations. For a complete description, the reader is referred to the original paper [21].

The starting point is to consider the following generalization of the affine Grassmannian. Let \(X\) be a curve, \(S' \subseteq S\) a finite set and a subset. Denote by \(\Delta_S(x)\) the union of formal disc neighborhoods of \(x_s \in X\) for all \(s \in S\). Denote by \(\Delta_{S}^{S'}(x)\) the complement of the graphs of \(\{ x_{s} \}_{s \in S'}\) in \(\Delta_S(x)\), i.e. introducing punctures at points indexed by \(s \in S'\).

The construction of such a space as a fpqc-quotient of an inductive limit of “homogeneous” spaces (generalizing the description of the affine Grassmannian as a \(G(\!(z)\!)\)-homogeneous space) is given in [21], where it is denoted \(G_S^{S'}/G_S\).

Similarly, we may introduce the moduli spaces \(\mathcal{T}_{G, V | S, S'}\) with its projection \(\pi: \mathcal{T}_{G,V|S,S'} \to \mathrm{Gr}_{G|S,S'}\) and the action map \(\mu_{S,S'}: \mathcal{T}_{G,V|S, S'} \to V_{S, S'}\). Here, \(\mathcal{T}_{G, V | S, S'}\) parametrizes isomorphism classes of quadruples \((x, \mathcal{P}, \phi, v)\) where \(x \in X^S\), \(\mathcal{P}\) is a principal \(G\)-bundle over \(\Delta_S(x)\), \(\phi\) is a trivialization of \(\mathcal{P}\) over \(\Delta_S^{S'}(x)\), and \(v\) is a section of the associated bundle \(\mathcal{P} \times_{G} V\). Then, \(\pi\) is the projection map defined by forgetting the section. The space \(V_{S, S'}\) can be thought of as the product bundle over \(\Delta_S^{S'}(x)\), which admits an action by the group \(G_S^{S'}\). The action map \(\mu_{S, S'}\) is then defined analogously to \(\mu: \mathcal{T}_{G, V} \to V(\!(z)\!)\), using the action by \(G_{S}^{S'}\) on sections.

Correspondingly, \(\mathcal{R}_{G, V|S, S'}\) can be described by taking the fiber product of \(\mu_{S,S'}\) with \(V_S \to V_{S, S'}\), where \(V_S\) can be interpreted as the sections of the product bundle which have no poles. We refer the reader to [21] for the accurate constructions.

As a special case, the Coulomb branches are constructed as convolution algebras of \(\mathcal{R}_{\{1\},\{1\}}\) with \(G_{\{1\}} = G[\![z]\!]\)-action. The idea due to Beilinson–Drinfeld, which is employed in Lonergan’s construction of the Frobenius-constant quantization, is that this convolution algebra structure can be expressed as a composition of a (manifestly commutative) multiplication map and a specialization map (cf. [21]).

Lonergan’s construction of the quantized Frobenius uses the generalized Beilinson–Drinfeld Grassmannian corresponding to \(X = \mathbb{G}_a = \mathrm{Spec} \;\mathbb{C}[t]\). Note that there is a projection map \(\pi: X \to Y = X /\!\!/_0 \mu_p = \mathrm{Spec} \;\mathbb{C} [t^p]\) which is a quotient by \(\mu_p\) action on \(X\) given by a character \(\chi\). By considering the generalized Beilinson–Drinfeld Grassmannian with \(p\) points supported on the \(\mu_p\)-equivariant embedding \(X \to X^p\) given by \[t \mapsto (\zeta \cdot t, \dots, \zeta^{p-1} \cdot t)\] where \(\zeta\) denotes the generator of the action corresponding to \(\chi\) and \(X^p\) is equipped with the cyclic permutation action, and taking \(\mu_p\)-quotients, we obtain the following moduli space fibering over \(Y = X /\!\!/_0 \mu_p\). For a \(\mathbb{C}\)-point \(y\) of \(Y\), denote by \[\pi^* \Delta_{S}^{S'}(y) := \mathrm{Spec} \left( \mathbb{C}[t] \left[\!\left[ \prod_{s \in S} (t^p - y_s) \right]\!\right]\left[ \prod_{s \in S'} (t^p - y_s)^{-1} \right] \right).\]

There is a natural action of \(G_{(p)} = \{ (y, g) : y \in Y, g : \pi^* \Delta_1(y) \to G \}\) acting on \(\mathcal{R}_{(p)}^{(p)}\) where we multiply the trivialization. (This is the analog of usual \(G[\![z]\!]\)-action on \(\mathcal{R}\)). This can be extended to an action of \(G_{(p)} \rtimes \mathbb{C}^*\) where \(\mathbb{C}^*\) is the usual loop rotation action.

The moduli space \(\mathcal{R}_{(p)}^{(p)}\) naturally fibers over \(Y\). Correspondingly, there is a diagram \[\begin{tikzcd} \mathcal{R} \rar \dar & \mathcal{R}_{(p)}^{(p)} \dar & \lar \mathcal{R}^p \dar \\ 0 \rar & Y = \mathbb{G}_a & \lar 1 \end{tikzcd}\] where the \(G_{(p)} \rtimes \mathbb{C}^*\)-action restricts to the usual \(G[\![z]\!]\rtimes \mathbb{C}^*\)-action on the zero fiber \(\mathcal{R}\) and \(G [\![z]\!]^p \rtimes \mu_p\)-action on the generic fiber \(\mathcal{R}^p\) (where \(\mu_p\) acts by cyclic permutation of factors).

Lonergan uses this geometry to construct the Frobenius-constant quantization. Namely, one gets \[\Lambda_{Coulomb} : \mathcal{O}(X) := H_*^{G [\![z]\!]} (\mathcal{R}) \to H_*^{G [\![z]\!]^p \rtimes \mu_p} (\mathcal{R}^p) \to H_*^{G_{(p)} \rtimes \mathbb{C}^*}((\mathcal{R}_{(p)}^{(p)})^* ) \to H_*^{G [\![z]\!]\rtimes \mu_p} (\mathcal{R}) = A\] by the composition of the following maps.

The first map is the Steenrod power map, defined for Borel–Moore homology using Lonergan’s reinterpretation of the Steenrod operations in equivariant derived constructible categories. The second map is the descent isomorphism, induced by identifying the nonzero fiber over \(1 \in Y\) as the slice of the \(\mathbb{C}^*\)-action on \(\mathcal{R}_{(p)}^{(p)}\) over \(Y^* = Y - \{0\}\)). The last map is the specialization map in Borel–Moore homology.

3.3 The mod \(p\) Quantum Hikita conjecture↩︎

We are now ready to state the mod \(p\) version of the quantum Hikita conjecture. As in the beginning of the section, fix \(k\) an algebraically closed field of characteristic \(p \gg 0\) (say \(k = \bar{\mathbb{F}}_p\)).

Assume that there is a pair \(X, X^!\) of symplectically dual conical Hamiltonian symplectic resolutions, equipped with actions of algebraic tori \(T\) and \(T^!\).

Our interpretation of a symplectically dual pair \((X/k, X^!/\mathbb{C})\) is the following. We first assume that the pair \((X/\mathbb{C}, X^!/\mathbb{C})\) is symplectically dual in the usual sense (cf. 40). Then we assume there exists \(R\), a finite localization of \(\mathbb{Z}\), over which \(X\) is actually defined such that \(X/\mathbb{C}\) is pulled back from \(R \to \mathbb{C}\). When our \(X/k\) is obtained by pulling back \(X/R\) under \(R \to R/\mathfrak{m} \cong k\) where \(\mathfrak{m}\) is a maximal ideal of \(R\), we say that \(X/k\) is dual to \(X^!/\mathbb{C}\). (In particular, the dual of \(X^!\) is not unique, corresponding to the choice of coefficients for invariants of \(X^!\)).

This way, we assume that \(X\) and the torus \(T\) are defined over the ground field \(k\), but assume that \(X^!\) and \(T^!\) are still both defined over \(\mathbb{C}\).

We assume the existence of a preferred Frobenius-constant quantization structure for the canonical quantization \(\mathcal{A}\) of \(X\); see 3.2.1.

Consider the \(D\)-module of twisted traces \(M_{eq, reg}\) for \(X\), and the quantum \(D\)-module \(M_{Kah, reg}^!\) for \(X^!\) with \(k\)-linear coefficients. In particular, under our assumptions, both \(D\)-modules are defined over \(k\).

4 Quantum Hikita conjecture mod \(p\): Proofs↩︎

In this section, we verify the mod \(p\) 3D mirror symmetry conjecture in the case of Springer resolutions and hypertoric varieties. We describe a restricted quantized algebra structure on the ring of differential operators, giving rise to the action of \(p\)-curvature on both the quantum \(D\)-module and \(D\)-module of twisted traces. Then we identify the action of Frobenius-linear operators for degree \(2\) elements with the action of \(p\)-curvature. The general result follows when the algebras acting on the \(D\)-modules are generated in degree \(2\).

Our description of the Frobenius-constant quantization operators, and the proof that they are equal to the \(p\)-curvature, uses the theory of restricted Lie algebras (can be found in standard textbooks such as [49]) and the closely related concept of restricted quantized algebras (due to [18]). For more detailed discussions of these structures we refer to the references above.

4.1 Restricted quantized algebra and \(p\)-curvature↩︎

The action of the Frobenius-linear operators on the \(D\)-modules can be understood as the action from the restricted powers in the ring of differential operators.

We introduce an additional flatness assumption to the definition in [18] so that the following characterization holds. Note that by [18], we have \((\hbar x)^{[p]} = \hbar x^p\) given that \(s\) defined below is an algebra map using the flatness assumption. By passing to the classical limit \(\hbar = 0\), one obtains a notion of a restricted Poisson algebra [18].

In particular, we assume that the canonical quantization \(\mathcal{A} \cong \mathcal{A}_{\mathbb{F}_p} \otimes_{\mathbb{F}_p} k\) so that the projection \((\mathcal{A}_0^2)_{\mathbb{F}_p} \to \mathfrak{t}_{\mathbb{F}_p}\) in the quantization exact sequence is compatible with the \(\mathbb{F}_p\)-forms.

The image of the Artin–Schreier map \(s\) gives rise to central endomorphisms of the \(D\)-module:

The fact that the action of \(s(R)\) is an endomorphism of \(D\)-modules (that is, covariant constancy of the \(p\)-curvature) is an immediate consequence of the centrality of the image of \(s\).

In the context of 68, the isomorphism of \(D\)-modules \(M_{eq, reg} \cong M^!_{Kah, reg}\) for symplectically dual pairs \(X/k\) and \(X^!/\mathbb{C}\) compatible with \(R_{eq} \cong R^!_{Kah}\) implies the identification of \(p\)-curvature operators for both \(D\)-modules. Hence, in the case where \(\mathcal{A}_0\) and \(H^*_{T^! \times \mathbb{G}_m}(X^!)\) are generated by (conical and cohomological, resp.) degree \(2\) elements, 68 can be reduced to the following claims.

The strategy for verifying the mod \(p\) quantum Hikita conjecture (68), modulo the identification of \(D\)-modules, is to verify 76 and 77 for 3D mirror pairs. Namely, the isomorphism of \(D\)-modules in characteristic \(p\) in particular identifies the \(p\)-curvature of the two \(D\)-modules, which we interpret as the central endomorphisms we construct.

4.2 Springer resolutions↩︎

As in 2.2.1, fix the Springer resolution \(X = T^*G/B \to \mathcal{N}\), where \(\mathcal{N} \subseteq \mathfrak{g}^*\) is the nilpotent cone. This is a conical Hamiltonian symplectic resolution, with the universal deformation \(\mathcal{X}\) given by the Grothendieck–Springer simultaneous resolution; the deformation fits into the picture

where the vertical maps are the affinizations. In particular, the affinization of \(\mathcal{X}\) is given by \(\mathfrak{g}^* \times_{\mathfrak{t}^*/W} \mathfrak{t}^*\) where we take the adjoint quotient (i.e. characteristic polynomial) map \(\mathfrak{g}^* \to \mathfrak{t}^*/W\) and the Weyl quotient \(\mathfrak{t}^* \to \mathfrak{t}^*/W\).

To describe the canonical quantization, note that the universal enveloping algebra \(U\mathfrak{g}\) carries the PBW filtration with \(\mathfrak{g}\) in degree \(2\). With respect to this filtration, one can form the corresponding Rees algebra (with Rees parameter \(\hbar\)), denoted \(U_\hbar \mathfrak{g}\), with the commutator given by \([x, y] = \hbar \{ x, y\}\) for \(x, y \in \mathfrak{g}\). The canonical quantization is then given by \(\mathcal{A} = U_{\hbar} \mathfrak{g} \otimes_{Z_{\mathrm{HC}}(U_\hbar \mathfrak{g})} \mathrm{S} (\mathfrak{t})[\hbar]\) where \(\mathrm{S}(\mathfrak{t}) := \mathcal{O}(\mathfrak{t}^*)\) is the symmetric algebra on \(\mathfrak{t}\); here, the structure map \[Z_{\mathrm{HC}}(U_\hbar \mathfrak{g}) \cong\mathrm{S} (\mathfrak{t}) ^W [\hbar] \subseteq \mathrm{S} (\mathfrak{t})[\hbar]\] is the Harish–Chandra isomorphism.

We denote by \(T\) the maximal torus in \(G\), which acts in a Hamiltonian fashion on \(T^*(G/B)\), and \(T \times \mathbb{G}_m\) for the product of the Hamiltonian torus and the copy of \(\mathbb{G}_m\) which scales the holomorphic symplectic form.

The cohomology \(H^*_{T \times \mathbb{G}_m}(X)\) and the weight degree \(0\) part of the quantization \(\mathcal{A}_0\) are both generated by elements of (cohomological and conical, resp.) degree \(2\), see [5]. Hence it suffices to verify 77, 76.

4.2.1 The \(D\)-module of twisted traces↩︎

The affinization of the universal deformation of \(T^*G/B\) is given by \[\mathcal{O}(\mathcal{X}) \cong \mathcal{O}(\mathfrak{g}^*) \otimes_{\mathrm{S}(\mathfrak{t})^W } \mathrm{S}(\mathfrak{t}),\] where \(\mathrm{S}(\mathfrak{t})^W \cong k [\mathfrak{t}^*]^W \cong k [\mathfrak{g}^*]^G \hookrightarrow \mathcal{O}(\mathfrak{g}^*)\) is the inclusion of \(G\)-invariant polynomials.

Correspondingly, the universal quantization of \(T^*G/B\) is given by \[\mathcal{A} \cong U_\hbar \mathfrak{g} \otimes_{Z(U_\hbar \mathfrak{g})} \mathrm{S} ( \mathfrak{t})[\hbar];\] the structure map \(\varphi: Z(U_\hbar \mathfrak{g}) \to \mathrm{S}( \mathfrak{t})^W [\hbar]\) is the Harish–Chandra isomorphism.

Then the \(0\)-weight part of the quantization is given by \[\mathcal{A}_0 = (U_\hbar \mathfrak{g})_0 \otimes_{Z (U_\hbar \mathfrak{g})} \mathrm{S}(\mathfrak{t}) [\hbar]\] where the \(Z(U_\hbar \mathfrak{g}) \to (U_\hbar \mathfrak{g})_0\) is the obvious inclusion.

The following is the key observation that can be extracted from [19]:

We assume the following simple lemma.

In particular, the computation of the map immediately verifies 76 for the Springer resolution:

4.2.2 The quantum \(D\)-module↩︎

The quantum \(D\)-module of the Springer resolution is computed in [47], where the quantum connection is identified with the affine KZ connection (or the Dunkl connection). In characteristic \(p\), the mod \(p\) quantum \(D\)-module of the Springer resolution was studied in [25], and the following result was obtained as a special case of 56. Denote by \(\nabla_x = t \partial_{\bar{x}} + x \star_{T}\) the (\(T \times \mathbb{G}_m\)-equivariant) quantum connection for \(X = T^*(G/B)\).

This result verifies 77 for the Springer resolution, hence concludes the proof of the mod \(p\) quantum Hikita conjecture for the Springer resolution.

4.3 Hypertoric varieties↩︎

We begin by introducing two descriptions of hypertoric varieties as Higgs branches and Coulomb branches of gauge theories with abelian group \(G\). In both descriptions, one can explicitly understand the structure of quantizations and the Frobenius-constant quantization operators: in the Higgs description, this is given by (crystalline) differential operators which is well-studied in [32], and in the Coulomb description, this is due to [21] as reviewed in [50]. We review the computation of Frobenius-constant quantizations for both descriptions. The Coulomb branch perspective is an instance of Remark 70.

4.3.1 Definition: the Higgs description↩︎

Hypertoric varieties are analogs of toric varieties in the quaternionic setting, and can be obtained as a hyperKähler quotient of an abelian group action on an affine space in the Higgs description.

Consider \(\mathbb{A}^n\) together with the coordinate scaling action of the split algebraic torus \((\mathbb{G}_m)^n\). For a fixed map \((\mathbb{G}_m)^n \to T\) for \(T\) some split algebraic torus, there is an associated exact sequence of algebraic groups \[\begin{tikzcd} 1 \rar & K \rar & (\mathbb{G}_m)^n \rar & T \rar & 1. \end{tikzcd}\] By restriction, \(\mathbb{A}^n\) admits a \(K\)-action, which extends to a Hamiltonian \(K\)-action on \(T^*\mathbb{A}^n\) (with its standard algebraic symplectic form). Denote the corresponding moment map by \(\mu : T^*\mathbb{A}^n \to \mathfrak{k}^*\), where \(\mathfrak{k}^* := \mathrm{Hom}(K, \mathbb{G}_m)\) is the Lie coalgebra.

A resolution of \(Y_H\) can be obtained by choosing a generic stability condition \(\chi \in \mathfrak{k}^*\) and performing the associated projective GIT quotient \(X_H : = \mu^{-1}(0)/\!\!/_\chi K\); note there is a natural map \(X_H \to Y_H\).

The universal deformation \(\mathcal{X}_H\) of \(X_H\) is obtained by varying the moment map parameter, and its affinization is given by \(\mathcal{Y}_H = T^*\mathbb{A}^n /\!\!/_0 K = \mathrm{Spec}\;\mathcal{O}(T^*\mathbb{A}^n)^K = \mathrm{Spec}\; k[x_i, y_i]^K\).

These fit into the following picture:

The canonical quantization in the Higgs description is obtained by taking the Weyl quantization of \(\mathcal{O}(T^*\mathbb{A}^n)^K\). Namely, one considers the ring of (asymptotic) differential operators \(\mathcal{D}(\mathbb{A}^n)\) on \(\mathbb{A}^n\), generated by \(x_i\) and \(\partial_i\) for \(i = 1, \dots, n\) with relation \([\partial_i, x_i] = \hbar\). The \((\mathbb{G}_m)^n\) action on \(\mathbb{A}^n\) induces a grading by \(\mathbb{Z}^n \cong X^\bullet(\mathbb{G}_m^n)\) on \(\mathcal{D}(\mathbb{A}^n)\). Then the canonical quantization is obtained by taking the \(K\)-invariant part, namely \(\mathcal{A} := \bigoplus_{\lambda \in \mathfrak{t}^*} \mathcal{D}(\mathbb{A}_n)_\lambda\).

In general, we have generators \(m^\lambda := x^{\lambda_+} \partial^{\lambda_-}\) so that \(\mathcal{D}(\mathbb{A}_n)_{\lambda} = \mathcal{D}(\mathbb{A}_n)_0 m^\lambda\) for \(\lambda = \lambda_+ - \lambda_-\).

The cohomology \(H^*_{T \times \mathbb{G}_m}(X_H)\) and the weight degree \(0\) part of the quantization \(\mathcal{A}_0\) are both generated by elements of (cohomological and conical, resp.) degree \(2\), see [5]. Hence it suffices to verify 77, 76.

4.3.2 Definition: the Coulomb description↩︎

Alternatively, hypertoric varieties can also be constructed using BFN Coulomb branches. Again, let \(V = \mathbb{A}^n\) be equipped with its coordinate-wise scaling action \((\mathbb{G}_m)^n\), and consider it as a representation of the abelian group \(G=K\) by the exact sequence \[\begin{tikzcd} 1 \rar & K \rar & (\mathbb{G}_m)^n \rar & T \rar & 1. \end{tikzcd}\]

First, note that \(\mathrm{Gr}_K := K(\!(z)\!)/K[\![z]\!]\) is just a copy of the cocharacter lattice \(X_\bullet(K)\). We denote the generators by \(z^\nu\) for \(\nu \in X_\bullet(K)\). It is then easy to identify the BFN Steinberg variety and the moduli of triples for \((G, V) = (K, \mathbb{A}^n)\) as \[\mathcal{T}_{K, \mathbb{A}^n} := \bigsqcup_{\nu \in X_\bullet(K)} \{z ^\nu \} \times z^\nu V[\![z]\!], \quad \mathcal{R}_{K, \mathbb{A}^n} := \bigsqcup_{\nu \in X_\bullet(K)} \{z ^\nu \} \times (V[\![z]\!]\cap z^\nu V[\![z]\!]).\] For the description of \(\mathcal{R}_{K, \mathbb{A}^n}\), we used the fact that the action map \(\mu: \mathcal{T}_{G, V} \to V(\!(z)\!)\) simply takes \(\{z^\nu\} \times z^\nu V[\![z]\!]\) to \(z^\nu V[\![z]\!]\). We denote the fundamental classes (in the Borel–Moore homology) of components in \(\mathcal{R}_{G,V}\) as \(r_\nu\); these are also called the monopole operators. The following statement is evident.

The following is a computation of Braverman–Finkelberg–Nakajima of the ring structure of the Coulomb branch in the abelian case.

The Coulomb branch is therefore a hypertoric variety obtained by the hyperKähler reduction construction on \(T^*\mathbb{A}^n\) associated to the dual sequence \[\begin{tikzcd} 1 \rar & T^\vee \rar & (\mathbb{G}_m)^n \rar & K^\vee \rar & 1. \end{tikzcd}\]

Note that for \((G,V) = (K, \mathbb{A}^n)\), the generators of the Coulomb branch algebra are given by \(\nu \in \mathfrak{k}\) (cocharacters of \(K\)). In the Higgs case with the same data \((G,V) = (K, \mathbb{A}^n)\), the generators were indexed by \(\mathfrak{t}^*\) (characters of \(T\)). This is a manifestation of Gale duality as a special case of 3D mirror symmetry.

Again by choosing a generic stability condition \(\xi \in \mathfrak{t}\) one can take a resolution of the BFN Coulomb branch in the abelian case. The universal deformation \(\mathcal{X}_C\) of \(X_C\) is obtained by varying the \((T^\vee)\)-moment map parameter, and its affinization is given by \(\mathcal{Y}_C = T^*\mathbb{A}^n /\!\!/_0 T^\vee = \mathrm{Spec}\;\mathcal{O}(T^*\mathbb{A}^n)^{T^\vee} = \mathrm{Spec}\; k[x_i, y_i]^{T^\vee}\).

These fit into the following diagram (cf. [20]):

This diagram is compatible with the description of the deformation of Coulomb branches (see the discussion at the end of 2.2.3) in terms of flavor symmetries: here we consider the extension of \(K^{\vee}\) by \(T^{\vee}\) into \((\mathbb{G}_m)^n\) acting on \(\mathbb{A}^n\).

4.3.3 The \(D\)-module of twisted traces↩︎

The Frobenius-constant quantization structure on hypertoric varieties can be constructed in either the Higgs or the Coulomb description. In the Higgs case, the construction is simple, as the quantization is again given by (crystalline) differential operators, not unlike the case of Springer resolutions.

Recall that the universal deformation \(\mathcal{X}_H\) has coordinate ring given by \(\mathcal{O}(T^*\mathbb{A}^n)^K = k[x_i, y_i]^K\), with the zero \(T\)-weight part given by \(\mathcal{O}(\mathcal{X}_H)_0 = k[x_iy_i]\).

The construction of the Frobenius-constant quantization structure for the hypertoric varieties in their description as Coulomb branches is given by a special case of Lonergan’s theorem.

These computations verify 76 for hypertoric varieties.

4.3.4 The quantum \(D\)-module↩︎

In light of 57, to show that the quantum Steenrod operations agree with the \(p\)-curvature, it suffices to verify that the quantum multiplication operators by divisor classes have jointly simple spectrum. We use the explicit description of \(H^2(X)\) for smooth hypertoric varieties \(X\) obtained by [52].

This result verifies 77 for smooth hypertoric varieties.

References↩︎