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Quantum K-theory on flag manifolds, finite-difference Toda lattices and quantum groups

Alexander Givental Department of Mathematics, University of California, Berkeley CA 94709 Yuan-Pin Lee Department of Mathematics, University of Utah, Salt Lake City, Utah, 84112

Algebraic Geometry mathscidoc:2204.45025

2001.8
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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@inproceedings{alexander2001quantum,
  title={Quantum K-theory on flag manifolds, finite-difference Toda lattices and quantum groups},
  author={Alexander Givental, and Yuan-Pin Lee},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220415115120353728043},
  year={2001},
}
Alexander Givental, and Yuan-Pin Lee. Quantum K-theory on flag manifolds, finite-difference Toda lattices and quantum groups. 2001. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220415115120353728043.
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