For a 3-dimensional N=4 gauge theory with Higgs branch M_H, dual Coulomb branch M_C, and quantized Coulomb-branch algebra A_C, the quantum Hikita conjecture predicts an isomorphism between the quantum-geometric data of M_H and the corresponding graded-trace data of A_C. In schematic form, mathsfQGeom(M_H)cong mathsfGrTr(A_C), with the refined conjecture specifying this correspondence via quasimaps.

Quantum Hikita conjecture

Introduced by Kamnitzer, McBreen, and Proudfoot in 2018, the conjecture predicts a correspondence between quantum geometry on a Higgs branch and graded-trace data from the dual Coulomb branch. The latest work refines this correspondence rather than proving the unrestricted original statement.

Known results

Kamnitzer, McBreen, and Proudfoot (2018): proved cases for hypertoric varieties and the Springer resolution.
Février et al. (2023): proved the conjecture for minimal resolutions of ADE surface singularities and corresponding minimal nilpotent-orbit closures, with analogous results in types $B$ , $C$ , $F$ , and $G$ .
A later quiver-variety result established the relevant correspondence for ADE quivers with minuscule framings.

August 2026 refined theorem

Dinkins, Karpov, and Krylov formulate a refined quantum Hikita conjecture via quasimaps and prove it for ADE quiver gauge theories with minuscule framings, including all type $A$ theories, and for the Jordan quiver. The unrestricted refined conjecture, and therefore the general original problem, remains conjectural.

Current status (as of August 2026): Several geometric and gauge-theoretic families are settled, including the newly proved refined cases, but the general quantum Hikita conjecture remains open.

Quantum Hikita conjecture

Progress summary

Primary source

Additional references

Solutions 0