K-theoretic Hikita conjecture for quiver gauge theories
Ilya Dumanski, Vasily Krylov
TL;DR
The paper advances the mathematical understanding of the Hikita conjecture in the setting of quiver gauge theories by establishing an equivariant bridge from the non-equivariant theory, proposing and deriving a K-theoretic variant, and proving the conjecture in finite ADE types through a robust isomorphism of completed Coulomb-branch algebras. Central to the approach is an equivariant Riemann–Roch framework that yields a precise correspondence between completed K-theoretic and homological Coulomb branches, with explicit formulas on monopole generators. The results yield concrete consequences such as fixed-point parametrizations on deformed K-theoretic Coulomb branches and connect to affine Grassmannian geometry and shifted Yangians in ADE types. By linking all three layers—homological, K-theoretic, and their completions—the work provides a unified, calculable picture of dual pairs in 3d mirror symmetry and opens avenues for elliptic and quantum refinements. The methods offer new tools for studying the structure of Coulomb branches and their representations, with potential impact on the study of category ${\mathcal O}$ and cluster structures in the quantum setting.
Abstract
We study variants of Hikita conjecture for Nakajima quiver varieties and corresponding Coulomb branches. First, we derive the equivariant version of the conjecture from the non-equivariant one for a set of gauge theories. Second, we suggest a variant of the conjecture, with K-theoretic Coulomb branches involved. We show that this version follows from the usual (homological) one for a set of theories. We apply this result to prove the conjecture in finite ADE types. In the course of the proof, we show that appropriate completions of K-theoretic and homological (quantized) Coulomb branches are isomorphic.