Trigonometric Gaudin subalgebras conjecturally describe specialized quantum cohomology for slices in the affine Grassmannian

Establish that for slices in the affine Grassmannian Gr_G with G of ADE type, the specialized equivariant quantum cohomology algebra QH^•_{θ,1}(X) is given by the action of the family of trigonometric Gaudin subalgebras, i.e., the maximal commutative subalgebras of (U(𝔤)^{⊗ n})^{𝔥} containing the trigonometric Gaudin Hamiltonians.

Background

In the affine Grassmannian setting, quantum multiplication on specialized equivariant cohomology has operators identified with trigonometric Gaudin Hamiltonians.

The full algebraic description is conjectural: the maximal commutative trigonometric Gaudin subalgebras are expected to act to generate the quantum cohomology, and while parameter-space compactifications are understood, the algebraic identification remains unproven in general.