Namely, the capping operator which relates these two types of vertex functions, satisfies a difference equation, which is a central topic of as a part of a bigger system of difference equations involving quantum Knizhnik–Zamolodchikov equations. This equivariant parameter plays a major role in our approach. These are objects, very close to quantum tautological classes, namely they are equivariant K-theory classes (localized K-theory classes for bare vertices) defined as equivariant pushforwards with nonsingular and relative conditions correspondingly, so that extra equivariant parameter is introduced on a base curve. They can be of two types, bare and capped. 2.5 the most important tools for the computations in our quantum K-theoretic framework are introduced, known as vertex functions. Here we are free of these issue and our quantum classes are generators a priori. Then it is a formidable task in describing the quantum K-ring using generators and relations to verify whether the structure constants are polynomials in Kähler parameters-the property which is given for granted in the quantum cohomology. recent results ), the analogue of our deformed product, known as a small quantum product, is determined by the deformation of the structure constants. We would like to emphasize that in the standard K-theoretic version of Gromov–Witten approach to flag varieties (see e.g. 2.4 we note, that the quantum tautological classes generate the entire quantum K-theory ring. This is different from a standard notion of quantum products defined using stable map theory in the K-theoretic analogue of Gromov–Witten theory. In this paper we will refer to the latter as the quantum product and the resulting unital ring will be referred to as quantum K-theory ring. The second one is the deformed product on equivariant K-theory. The first one is the notion of a quantum tautological class, defined using pushforwards via evaluation map with a relative condition, as a deformation of the corresponding equivariant K-theory tautological class. Unlike stable maps, the quasimap is a combination of a certain vector bundle on a base curve, together with its section, which uses the presentation of Nakajima quiver variety is a GIT quotient. Section 2.2 is devoted to a brief review of theory of nonsingular and relative quasimaps to quiver varieties. 2.1 we remind basic notions of Nakajima quiver varieties as GIT quotients and their equivariant K-theory. 2 we review and generalize main concepts of to a general situation. The construction of can certainly be extended beyond Grassmannians to a large class of Nakajima quiver varieties and this is what the first part of the current work is about. 1.2 Main results and the structure of the paper 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. It was shown that their eigenvalues are the symmetric functions of roots of Bethe Ansatz equations. Such generators of the quantum K-theory ring, which in were called quantum tautological bundles are the deformations (via Kähler parameter) of the exterior powers of these tautological bundles. 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. The Hilbert space of the XXZ spin chain is identified with the space of equivariant localized K-theory of disjoint union of \(T^*Gr(k,n)\) for all k and fixed n, considered in the basis of fixed points. In particular, the relation between quantum equivariant K-theory of cotangent bundles to Grassmannians and the so-called XXZ model (see e.g. Recently the basic example, considered in in the physical context of 3d gauge theories, was described from mathematical point of view. The ideas outlined in these articles gave rise to new developments followed by other important results, see e.g. Early 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. The seminal papers of Nekrasov and Shatashvili paved the road for close interactions between quantum geometry of certain class of algebraic varieties and quantum integrable systems.
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