3D mirror symmetry in positive characteristic
Shaoyun Bai, Jae Hee Lee
TL;DR
This work develops an arithmetic enhancement of 3D mirror symmetry by studying symplectic dual pairs in characteristic $p$. Building on the quantum Hikita conjecture, it introduces Frobenius-constant quantizations and quantum Steenrod operations as central endomorphisms that intertwine under the duality between the trace D-module and the quantum D-module. The authors prove the mod $p$ conjecture for Springer resolutions and hypertoric varieties by identifying these endomorphisms with $p$-curvature operators on the respective D-modules, reducing the comparison to degree-two generation. The results reveal a precise, Frobenius-rooted dictionary between trace- and quantum-theoretic structures across dual pairs, and point toward deeper arithmetic aspects of HMS with potential K-theoretic extensions. Overall, the paper provides concrete mod $p$ verifications and a conceptual framework linking Steenrod operations, Frobenius-constant quantizations, and 3D mirror symmetry in positive characteristic.
Abstract
Via the formulation of (quantum) Hikita conjecture with coefficients in a characteristic $p$ field, we explain an arithmetic aspect of the theory of 3D mirror symmetry. Namely, we propose that the action of Steenrod-type operations and Frobenius-constant quantizations intertwine under the (quantum) Hikita isomorphism for 3D mirror pairs, and verify this for the Springer resolutions and hypertoric varieties.