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.

Highlights

  • 1.1 Some prehistory and earlier resultsThe seminal papers of Nekrasov and Shatashvili [40,41] paved the road for close interactions between quantum geometry of certain class of algebraic varieties and quantum integrable systems

  • Signs of such a fruitful collaboration between quantum cohomology/quantum K-theory and integrability were noted in mathematics literature in the works of Givental et al [20,23]

  • The generating function for such quantum tautological bundles is known in the theory of integrable systems as Baxter Q-operator which contains information about the spectrum of genuine physical Hamiltonians

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Summary

Some prehistory and earlier results

The seminal papers of Nekrasov and Shatashvili [40,41] paved the road for close interactions between quantum geometry of certain class of algebraic varieties and quantum integrable systems. Signs of such a fruitful collaboration between quantum cohomology/quantum K-theory and integrability were noted in mathematics literature in the works of Givental et al [20,23]. Using a different method than in standard Gromov–Witten-inspired approach to quantum products, the quantum K-theory ring was defined, as well as the generators using the theory of quasimaps to GIT quotients [13,42].

Main results and the structure of the paper
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Connections to physics literature and beyond
Classical equivariant K-theory
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Quasimaps
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Picture notation
Virtual structure
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Gluing operator
Quantum K-theory ring
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Vertex functions
Capping operator and difference equation
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Computations for partial flags
Bare vertex for partial flags
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Bethe equations and baxter operators
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Construction of the tRS Lax matrix
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Dual tRS model from XXZ chain
Compact limit of XXZ Bethe Ansatz and of tRS model
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Five-vertex model and quantum toda chain
Compact limit of tRS model
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Full Text
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