The quantum Hikita conjecture via quasimaps

Abstract: We propose a refinement of the quantum Hikita conjecture of Kamnitzer, McBreen, and Proudfoot that bridges the representation theory of Coulomb branches with the enumerative geometry of Higgs branches. We also introduce a general framework for proving it, which we carry out for ADE quiver gauge theories with minuscule framings and for the gauge theory corresponding to the Jordan quiver. As an application, we use the resulting quantum Hikita isomorphisms to give a geometric description of graded traces on quantized Coulomb branches.

• The paper proves a refined quantum Hikita conjecture by identifying the PSZ D-module and graded-trace D-module as canonical quotients of a common master D-module, without assuming conicity or goodness.
• It establishes the conjecture for ADE quivers with minuscule framings and the Jordan quiver by matching specialized vertex functions with normalized traces of Verma-type modules using Yangian, Cherednik-algebra, and reverse-plane-partition methods.
• The results geometrically parametrize graded traces, including at singular parameters, and provide explicit rational formulas and natural q-deformations through capped vertices and fixed-point quasimap computations.

Overview and main conjecture

This paper, by Dinkins, Karpov, and Krylov (2608.16746), establishes a refined version of the quantum Hikita conjecture of Kamnitzer–McBreen–Proudfoot (KMBP), connecting the enumerative geometry of Nakajima quiver varieties with the representation theory of quantized Coulomb branches. The setting is a quiver gauge theory $(\Gamma,\mathsf v,\mathsf w)$ with Higgs branch $X=\widetilde{\mathcal M}_H$ (a smooth symplectic Nakajima quiver variety) and Coulomb branch $\mathcal M_C$ in the sense of Braverman–Finkelberg–Nakajima, together with its quantization $\mathcal A$.

The original quantum Hikita conjecture predicts that for good theories the $q=2\hbar$ specialization of Givental's quantum $D$-module $Q^{\mathrm{Giv}}$ is isomorphic to $\operatorname{GrTr}(\mathcal A)$, the $D$-module controlling graded traces on $\mathcal A$. The paper's refinement replaces $X=\widetilde{\mathcal M}_H$0 by the Pushkar–Smirnov–Zeitlin (PSZ) $X=\widetilde{\mathcal M}_H$1-module $X=\widetilde{\mathcal M}_H$2, built from quasimaps to the stacky quotient $X=\widetilde{\mathcal M}_H$3. Two features distinguish the refinement: it requires no conicity ("goodness") assumption, and it identifies both sides as canonical quotients of a single "master" $X=\widetilde{\mathcal M}_H$4-module $X=\widetilde{\mathcal M}_H$5, via the capped vertex on one side and the tautological map on the other. This quotient structure makes solutions correspond naturally: solutions of $X=\widetilde{\mathcal M}_H$6 are given by fixed-point restrictions of vertex functions, while solutions of $X=\widetilde{\mathcal M}_H$7 are normalized graded traces of Verma-type modules.

Reduction theorem and strategy

The central technical device is a reduction theorem: assuming the flavor-fixed locus $X=\widetilde{\mathcal M}_H$8 is finite, if (1) $X=\widetilde{\mathcal M}_H$9 — a condition implied by the classical Hikita conjecture — and (2) there exist $\mathcal M_C$0-modules $\mathcal M_C$1 indexed by fixed points whose normalized graded traces equal the specialized vertex restrictions $\mathcal M_C$2, then the refined quantum Hikita conjecture holds. The proof embeds $\mathcal M_C$3 into a direct sum of formal series via traces, identifies this image with the image of the specialized bare vertex using the trace–vertex identity, and invokes freeness of the truncated modules $\mathcal M_C$4 over $\mathcal M_C$5, established via a dimension comparison between generic and classical specializations.

A key regularity result underpins the whole approach: for any subtorus $\mathcal M_C$6, the pointwise restriction $\mathcal M_C$7 is regular along $\mathcal M_C$8 and lands in nonlocalized equivariant cohomology. The proof analyzes the derived evaluation fiber $\mathcal M_C$9, whose relative obstruction theory has virtual rank zero and is self-dual up to the character $\mathcal A$0; at $\mathcal A$1 the localization factors cancel to signs, yielding regularity. Notably, the componentwise restriction to a positive-dimensional fixed component can have a genuine pole at $\mathcal A$2 — an explicit $\mathcal A$3 Kleinian example exhibits such a pole — so the isolated-fixed-point hypothesis is essential for the classical reformulation.

Main results

The reduction is carried out in two families:

ADE quivers with minuscule framings. For $\mathcal A$4 of type ADE with framing supported at minuscule nodes, Verma-type modules are constructed via comultiplication from chamber ("extremal") modules over shifted Yangians. Their normalized traces factor as products over decompositions $\mathcal A$5, matching the factorization of the specialized vertex across fixed-point components (Proposition on signed vertices: $\mathcal A$6). For point quiver varieties ($\mathcal A$7 minuscule), both sides reduce to sums over reverse plane partitions on the heap $\mathcal A$8, known from the Peterson–Proctor hook-product formula and prior work on chamber modules. The dimension assumption holds with equality via the Hikita isomorphism for generalized affine Grassmannian slices.

Jordan quiver. Here $\mathcal A$9 is the Gieseker variety; the quantized Coulomb branch is the spherical cyclotomic rational Cherednik algebra of type $q=2\hbar$0. The trace–vertex identity follows by comparing eigenvalues of Cartan generators on Jack bases of Verma modules with the reverse-plane-partition formula for vertices at fixed points indexed by $q=2\hbar$1-tuples of partitions.

Consequently, the refined quantum Hikita conjecture is proved in these cases without any goodness assumption.

Applications: geometric description of graded traces

The isomorphism yields a description of all graded traces on central quotients $q=2\hbar$2, including singular parameters where Verma traces no longer span. For integral $q=2\hbar$3 coming from a cocharacter $q=2\hbar$4, every graded trace is given by integration against the capped vertex:

$q=2\hbar$5

where $q=2\hbar$6 is the Lagrangian core. In particular, every graded trace admits a natural $q=2\hbar$7-deformation. The sphere trace $q=2\hbar$8 of Gaiotto–Okazaki fits this framework via a class $q=2\hbar$9, though describing that class explicitly remains open; note also that $D$0 exists only for good or ugly theories whereas the capped vertex always exists.

Explicit vertex computations

For skew subheaps of dominant-minuscule heaps, $D$1-fixed based quasimaps are in bijection with reverse plane partitions, giving

$D$2

with rationality following from the Naruse–Okada skew hook formula. Examples beyond dominant-minuscule elements show that none of dominance, minuscule, or full commutativity characterizes the "simple spectrum" condition that suffices; finding the correct combinatorial criterion is posed as open. For types $D$3 and $D$4 with single minuscule framing, differential operators (conjugated Sekiguchi-type operators) insert descendants, proving that descendant vertex functions are rational with poles only at $D$5 for roots $D$6 — strengthening Etingof–Stryker's rationality theorem for Verma characters, with a deformation to Jack/Macdonald settings also available.

Limitations and open questions

Several caveats are explicit. The specialization of the bare vertex to $D$7 depends on the choice of torus $D$8 and does not commute with restriction to subtori, reflecting the distinctness of Verma-type modules at singular parameters. The equivalence of the PSZ and Givental $D$9-modules away from $Q^{\mathrm{Giv}}$0 is only conjectured, for ADE theories with dominant $Q^{\mathrm{Giv}}$1, on the regular Kähler torus; an ugly-theory example shows it must fail in general. Whether all graded traces can be described purely via the (simpler) bare vertex, whether the class $Q^{\mathrm{Giv}}$2 can be identified, and whether the Botta–Tamagni modules categorify the trace–vertex correspondence are left open. Extensions to $Q^{\mathrm{Giv}}$3-theory and general oriented cohomology theories, positive characteristic compatibility with Steenrod operations, and a Hochschild-homological interpretation via a quantized Drinfeld–Gaitsgory interpolation bimodule are proposed but not carried out.

Conclusion

The paper proves a torus-equivariant, stack-level refinement of the quantum Hikita conjecture for ADE quivers with minuscule framings and the Jordan quiver, reducing the statement to concrete trace–vertex identities verified by reverse-plane-partition combinatorics. It thereby gives a geometric parametrization of graded traces on quantized Coulomb branches, including at singular quantization parameters, and supplies explicit rational formulas for Calabi–Yau-specialized descendant vertex functions in types $Q^{\mathrm{Giv}}$4 and $Q^{\mathrm{Giv}}$5.