Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 018, 24 pages      arXiv:2008.08182
Contribution to the Special Issue on Representation Theory and Integrable Systems in honor of Vitaly Tarasov on the 60th birthday and Alexander Varchenko on the 70th birthday

Quantum K-Theory of Grassmannians and Non-Abelian Localization

Alexander Givental and Xiaohan Yan
Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720, USA

Received August 25, 2020, in final form February 02, 2021; Published online February 26, 2021

In the example of complex grassmannians, we demonstrate various techniques available for computing genus-0 K-theoretic GW-invariants of flag manifolds and more general quiver varieties. In particular, we address explicit reconstruction of all such invariants using finite-difference operators, the role of the $q$-hypergeometric series arising in the context of quasimap compactifications of spaces of rational curves in such varieties, the theory of twisted GW-invariants including level structures, as well as the Jackson-type integrals playing the role of equivariant K-theoretic mirrors.

Key words: Gromov-Witten invariants; K-theory; grassmannians; non-abelian localization.

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