3d mirror of the intersection U\T*G × T*G // P_I

Establish that the intersection U \!\backslash\!\backslash T*G ×_{[\mathfrak{g}^*/G]} T*G // P_I is 3d mirror to {T} \!\backslash\!\backslash T*{G} ×_{[{\mathfrak{g}^*/{G}]} T*{G} //_{e_I} {U}_{e_I}, where {G} is the Langlands dual group, {T} is its maximal torus, e_I is the regular nilpotent element associated to the Levi subgroup L_I on the dual side, and {U}_{e_I} is the corresponding unipotent subgroup.

Background

Intersections of 4d branes are central to the paper’s approach to 3d mirror symmetry. Earlier sections relate known pairs such as parabolic Slodowy varieties and bow varieties through intersections of twisted cotangent bundles.

In the concluding section, the authors extend this philosophy by conjecturing a specific 3d mirror pairing between an intersection involving the unipotent reduction on G and a parabolic reduction, and a dual intersection involving the maximal torus and a unipotent-twisted reduction on the Langlands dual group. This proposal aims to reconcile fixed-point computations and mirror symmetry expectations for such intersections.