Quantum Hikita Conjecture

• Quantum Hikita Conjecture is a refinement of the classical Hikita correspondence, establishing an isomorphism between quantized coordinate rings and specialized quantum D-modules.
• It has been rigorously verified in settings like hypertoric varieties and the Springer resolution, confirming key relationships between quantum and cohomological structures.
• Extensions of the conjecture include K-theoretic and arithmetic frameworks that enhance its role in understanding symplectic duality and mirror symmetry.

The Quantum Hikita Conjecture is a central concept in geometric representation theory, relating the quantized coordinate ring of a conical symplectic singularity to the quantum cohomology ring of a symplectic resolution of its symplectic dual. The conjecture builds on the classical Hikita conjecture, which asserts an isomorphism between the coordinate ring of the torus-fixed point subscheme of a conical symplectic singularity and the cohomology ring of a symplectic resolution of its dual. Quantum versions elevate this correspondence to the field of noncommutative deformations and quantum (co)homological invariants, revealing profound interconnections between algebraic geometry, representation theory, and mathematical physics.

1. Formulation of the Quantum Hikita Conjecture

Given a conical symplectic singularity $X$ and its symplectic dual $X^*$ (equipped with a symplectic resolution), the quantum Hikita conjecture establishes a correspondence between:

• the quantized coordinate ring $A$ of $X$,
• and a specialized quantum $D$-module constructed from the equivariant quantum cohomology of $X^*$.

Formally, two central modules are defined (Kamnitzer et al., 2018):

• The graded trace $D$-module $M = (S \otimes A_0) / J$, where $A_0$ is the degree-zero part of the quantization, $S$ is a semigroup algebra parametrized by positive roots, and $J$ is the trace-type ideal generated by relations $1 \otimes (ab) - q^\lambda \otimes (ba)$ for $a \in A_\lambda$.
• The quantum $D$-module $$Q^* = F^* \otimes H^*_{T^* \times G}(X^*;\mathbb{C})$$, where $F^*$ is analogous to $S$ (built from Kähler roots), and $H^*$ is the equivariant quantum cohomology.

The conjecture posits an isomorphism (after suitable Ore localization/inversion on root parameters): $M_\text{loc} \cong Q^*_\text{loc}$ and their Weyl group invariants: $M^W \cong (Q^*)^W$ with the canonical generator $1 \otimes 1$ mapping to the unit $1 \otimes 1$.

Key structural formulas include:

• $M = (S \otimes A_0)/J$
• $$Q^* = F^* \otimes H^*_{T^* \times G}(X^*;\mathbb{C})$$

This setting interchanges the noncommutative algebraic data from the quantization on one side with quantum cohomological data on the dual side. The module actions are defined over semigroup and symmetric algebras generated on both the equivariant and Kähler parameter rings (Kamnitzer et al., 2018).

2. Verification in Specific Geometric Settings

The quantum Hikita conjecture has been established rigorously for several classes of varieties:

• Hypertoric Varieties: The affine hypertoric variety is realized as a Hamiltonian reduction of $T^*\mathbb{C}^n$ by a torus. Its quantization $A$ is the hypertoric enveloping algebra. The associated roots correspond to minimal cocircuits. On the mirror side, the Gale dual hypertoric variety and its quantum cohomology coincide with the algebraic module structure, affirming the conjecture (Kamnitzer et al., 2018).
• Springer Resolution: Here, $X$ is the nilpotent cone, with the quantization $A$ given by the Rees algebra of the enhanced enveloping algebra and Harish–Chandra homomorphism. The quantum $D$-module is shown to coincide with the module of differential operators after Weyl group invariants are taken, matching the geometric Harish–Chandra images and confirming the conjecture (Kamnitzer et al., 2018).

Ben Webster's appendix provides a geometric description of highest weights for quantizations with isolated torus fixed points, including explicit formulas for highest weight differences in terms of tangent space characters and homological splittings.

3. Relation to Classical Hikita Conjecture and Quantum Extensions

The classical Hikita conjecture, as verified for minimal nilpotent orbits and Kleinian singularities (Shlykov, 2019), states an isomorphism: $H^*\big(\widetilde{\mathbb{C}^2/\Gamma}\big) \cong \mathbb{C}\big[(\overline{\mathcal{O}_{\min})^T\big] \cong \operatorname{Sym}^{\geq 2}(\mathfrak{h})$ where $\mathfrak{h}$ is the Cartan subalgebra and $\Gamma$ is a finite subgroup of $SL(2,\mathbb{C})$.

Quantum extensions, as elucidated in (Chen et al., 2023), assert the isomorphism at the level of quantized algebras: $QH_{C^\times}(\widetilde{\mathbb{C}^2/\Gamma}) \cong Q\left(\mathcal{A}[\overline{\mathcal{O}_{\min})]\right)$ where $QH_{C^\times}$ is the equivariant quantum cohomology and $Q(\mathcal{A}[\overline{\mathcal{O}_{\min})])$ denotes the $D$-module of graded traces for the quantization via Joseph ideals.

This has been extended to singularities of BCFG types through the consideration of "minimal special nilpotent orbits" and symmetry-breaking automorphisms.

4. Combinatorial and Recursive Structures

The paper of Hikita polynomials in combinatorial frameworks (notably for $(m,3)$ parameters) reveals explicit formulas (Kaliszewski et al., 2016): $$\mathcal{H}_{m,3}(X;q,t) = C_{m,3}(q,t) \cdot s_{(1,1,1)}(X) + [K_{m-1,3}(q,t) + K_{m-2,3}(q,t)] \cdot s_{(2,1)}(X) + C_{m-3,3}(q,t) \cdot s_{(3)}(X)$$ where $C_{m,3}(q,t)$ are rational $q,t$-Catalan polynomials (enumerating Dyck paths), $K_{m,3}$ are modified Catalan polynomials, and $s_{(1,1,1)}(X), s_{(2,1)}(X), s_{(3)}(X)$ are Schur functions.

Recursive relations among $K_{m,3}(q,t)$ encode intricate combinatorial symmetry, corresponding to geometric decomposition principles mirrored in the quantum setting. The polynomials exhibit $q,t$-symmetry, paralleling Poincaré duality and mirror symmetry phenomena encountered in geometric representation theory (Kaliszewski et al., 2016).

5. Extensions: K-Theoretic, Arithmetic, and Mirror Symmetry Aspects

K-Theoretic Extension

Recent work generalizes the conjecture to K-theoretic Coulomb branches (Dumanski et al., 7 Sep 2025). Appropriate completions of K-theoretic and homological Coulomb branches are shown to be isomorphic, connecting results to foundational Riemann–Roch-type isomorphisms after Todd class correction. This affords K-theoretic analogs of classical and quantum Hikita maps.

Positive Characteristic and Power Operations

The arithmetic aspect in characteristic $p$ fields is developed in (Bai et al., 30 Mar 2025). Frobenius-constant quantizations map via central endomorphisms to quantum Steenrod operations on the enumerative geometry side: $$\Lambda(a) = a^p - \hbar^{p-1} a^{[p]} \quad \longleftrightarrow \quad \Sigma_x^T = \nabla_x^p - t^{p-1} \nabla_x$$ where $\nabla_x = t \partial_{(x + \star)}$ is the quantum connection.

This formulation intertwines arithmetic invariants with quantum enumerative operators, fundamentally enhancing the quantum Hikita isomorphism in 3D mirror symmetry contexts.

6. Symplectic Duality, Localization, and Future Directions

Symplectic duality, both in the classical and quantum setting, underpins the conceptual framework of the Quantum Hikita Conjecture. Techniques such as abelian localization (Hoang et al., 27 Oct 2024), Kirwan surjectivity for quiver varieties (Setiabrata, 21 Oct 2024), and explicit combinatorial presentations (e.g., generalized coinvariant algebras in classical Lie algebras (Hoang, 20 Sep 2024)) all contribute to establishing isomorphisms predicted by the conjecture.

Broader implications include:

• Deep links between quantizations, highest weight representations, and geometric categorifications.
• Robustness of the conjecture across ADE and BCFG types, Nakajima quiver varieties, Springer and parabolic Slodowy settings.
• Extensions to K-theory, arithmetic structures, and the interplay with mirror symmetry.

Future research focuses on K-theoretic refinements, global module structures, extension to new duality classes (affine Grassmannians, Coulomb branches), and the detailed analysis of categorified invariants and their mod $p$ properties.

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The Quantum Hikita Conjecture represents a landmark unification of quantum algebraic, combinatorial, and geometric structures, catalyzing ongoing investigations into duality phenomena, enumerative invariants, and representation theory in algebraic geometry and mathematical physics.