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Quantum K-theory on flag manifolds, finite-difference Toda lattices and quantum groups
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We conjecture that appropriate K-theoretic Gromov-Witten invariants of complex flag manifolds G/B are governed by finite-difference versions of Toda systems constructed in terms of the Langlands-dual quantized universal enveloping algebras U_q(g'). The conjecture is proved in the case of classical flag manifolds of the series A. The proof is based on a refinement of the famous Atiyah-Hirzebruch argument for rigidity of arithmetical genus applied to hyperquot-scheme compactifications of spaces of rational curves in the flag manifolds.
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Quantum integrable model for the quantum cohomology/K-theory of flag varieties and the double $\beta$-Grothendieck polynomials
Bethe ansatz states of a new GL(n) five vertex model expand into double β-Grothendieck polynomials, and the model's Bethe equations reproduce the quantum Whitney relations of flag varieties.
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