Quantum Hikita Conjecture for Conical Theories

Establish the quantum Hikita conjecture by proving that for conical three-dimensional N=4 gauge theories specified by a complex reductive group G and representation N, the specialized quantum D-modules on Higgs branches are isomorphic, as D-modules, to the D-modules of graded traces on the corresponding quantum Coulomb branches.

Background

The paper constructs explicit operator representations and integral formulas for twisted traces on quantized Coulomb branches of conical quiver gauge theories, providing a concrete realization aligned with expectations from conformal field theory. These results are positioned as complementary to prior work on twisted traces for quantum Higgs branches.

Within this context, the authors highlight the quantum Hikita conjecture, which predicts a deep equivalence between structures arising on Higgs and Coulomb sides—specifically, an isomorphism of D-modules. Their construction of twisted traces is presented as a potential tool to attack a weaker form of the conjecture, underscoring the conjecture’s status as an open problem and its relevance to the paper’s main theme.

References

The twisted trace is closely related to the quantum Hikita conjecture. The quantum Hikita conjecture Conjecture 1.1 states that for conical theories (see \Cref{1.1}), the specialized quantum D-modules on Higgs branches and the D-modules of graded traces for Coulomb branches are isomorphic as D-modules.

— Analytical Traces on Coulomb Branches of Quiver Gauge Theories  (2510.20045 - Zhang, 22 Oct 2025) in Subsection “Future Directions”

We propose the following conjecture. There is an isomorphism of $D$-modules $Q_{q=2\hbar}{\mathrm{PSZ} \simeq \operatorname{GrTr}(\mathcal{A})$ as quotients of $\mathscr H_\hbar[[{\boldsymbol{z}]]$.

— The quantum Hikita conjecture via quasimaps  (2608.16746 - Dinkins et al., 17 Aug 2026) in Conjecture 1.5, Section 1.1; also Section 2.7

For every $\tau \in \mathscr{H}\hbar[[{\boldsymbol{z}]]$ $\widetilde{\operatorname{tr}{\Theta(p\vee)}(\tau) = V{(\tau)}_{q=2\hbar,p}.$

— The quantum Hikita conjecture via quasimaps  (2608.16746 - Dinkins et al., 17 Aug 2026) in Conjecture 1.7, Section 1.3; also Conjecture 4.13