Toda-lattice model for full quantum cohomology

Prove that, for every smooth Fano Schubert variety \(X_{w,B}\) in a complete flag variety whose Picard number equals the rank of \(\check G\), the full quantum cohomology ring \(QH^*(X_{w,B})\) is isomorphic to the coordinate ring of the scheme-theoretic closure \(\overline{\mathcal L}_{w,B}\) of the associated generalized Peterson variety in the partial compactification of \(\mathcal T^*(T)\).

Background

For factorial full-flag Schubert varieties with full support, the paper maps the generalized Peterson variety into the cotangent bundle T∗(T)\mathcal T^*(T) and defines a scheme-theoretic closure L‾w,B\overline{\mathcal L}_{w,B} after partially compactifying the torus quantum-parameter directions.

The proposed isomorphism is intended as a generalization of the Givental–Kim presentation of the quantum cohomology of complete flag varieties via the degenerate leaf of the Kostant Toda lattice. The conjecture is verified for the smooth Schubert divisor treated in the type AA case study.

References

For smooth Fano Schubert varieties ${w,B}$ in $\check G/\check B$ we furthermore map our generalised Peterson variety to the cotangent bundle $\mathcal T*(T)$ of the maximal torus $T$ of $G$ and construct a partial compactification that we conjecture models the full quantum cohomology ring of ${w,B}$, in analogy with the Givental-Kim presentation of $QH*(\check G/\check B)$ via the degenerate leaf of the Kostant Toda lattice of $\check G$.

— A Peterson program for general Schubert varieties and mirror symmetry  (2609.30242 - Li et al., 24 Sep 2026) in Conjecture 1.2 (labelled conj:Todac), Section 1, “Main results”; also Conjecture 7.1 (labelled c:Toda), Section 7