Abstract We define quantum equivariant K-theory of Nakajima quiver varieties. We discuss type A in detail as well as its connections with quantum XXZ spin chains and trigonometric Ruijsenaars-Schneider models. Finally we study a limit which produces a K-theoretic version of results of Givental and Kim, connecting quantum geometry of flag varieties and Toda lattice.
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Moduli Spaces of Generalized Hyperpolygons
Abstract We introduce the notion of generalized hyperpolygon, which arises as a representation, in the sense of Nakajima, of a comet-shaped quiver. We identify these representations with rigid geometric figures, namely pairs of polygons: one in the Lie algebra of a compact group and the other in its complexification. To such data, we associate an explicit meromorphic Higgs bundle on a genus-g Riemann surface, where g is the number of loops in the comet, thereby embedding the Nakajima quiver variety into a Hitchin system on a punctured genus-g Riemann surface (generally with positive codimension). We show that, under certain assumptions on flag types, the space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system of Gelfand–Tsetlin type, inherited from the reduction of partial flag varieties. In the case where all flags are complete, we present the Hamiltonians explicitly. We also remark upon the discretization of the Hitchin equations given by hyperpolygons, the construction of triple branes (in the sense of Kapustin–Witten mirror symmetry), and dualities between tame and wild Hitchin systems (in the sense of Painlevé transcendents).
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- Award ID(s):
- 1749013
- PAR ID:
- 10232976
- Date Published:
- Journal Name:
- The Quarterly Journal of Mathematics
- ISSN:
- 0033-5606
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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