One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Quantum Inverse Scattering Method and Correlation Functions

A four-dimensional analog of Chern-Simons theory produces integrable lattice models from Wilson lines and surface operators. We show that this theory describes a quasi-topological sector of maximally supersymmetric Yang-Mills theory in six dimensions, topologically twisted and subjected to an \Omega-deformation. By realizing the six-dimensional theory in string theory and applying dualities, we unify various phenomena in which the eight-vertex model and the XYZ spin chain, as well as variants thereof, emerge from supersymmetric gauge theories.

Unification of integrability in supersymmetric gauge theories

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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Quantum K-theory of Quiver Varieties and Many-Body Systems

Let $X$ be a holomorphic symplectic variety with a torus $\mathsf{T}$ action and a finite fixed point set of cardinality $k$. We assume that elliptic stable envelope exists for $X$. Let $A_{I,J}= \operatorname{Stab}(J)|_{I}$ be the $k\times k$ matrix of restrictions of the elliptic stable envelopes of $X$ to the fixed points. The entries of this matrix are theta-functions of two groups of variables: the Kahler parameters and equivariant parameters of $X$. We say that two such varieties $X$ and $X'$ are related by the 3d mirror symmetry if the fixed point sets of $X$ and $X'$ have the same cardinality and can be identified so that the restriction matrix of $X$ becomes equal to the restriction matrix of $X'$ after transposition and interchanging the equivariant and Kahler parameters of $X$, respectively, with the Kahler and equivariant parameters of $X'$. The first examples of pairs of 3d symmetric varieties were constructed in [Rimanyi R., Smirnov A., Varchenko A., Zhou Z., arXiv:1902.03677], where the cotangent bundle $T^*\operatorname{Gr}(k,n)$ to a Grassmannian is proved to be a 3d mirror to a Nakajima quiver variety of $A_{n-1}$-type. In this paper we prove that the cotangent bundle of the full flag variety is 3d mirror self-symmetric. That statement in particular leads to nontrivial theta-function identities.

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Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety

We consider a pair of quiver varieties (X;X') related by 3d mirror symmetry, where X =T*Gr(k,n) is the cotangent bundle of the Grassmannian of k-planes of n-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on X $\times$ X' (the Mother function) whose restrictions to X and X' are the elliptic stable envelopes of those varieties. This implies, that the restriction matrices of the elliptic stable envelopes for X and X' are equal after transposition and identification of the equivariant parameters on one side with the Kahler parameters on the dual side.

3d Mirror Symmetry and Elliptic Stable Envelopes

Baxter Q-operator from quantum K-theory

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.

/pdf/quantum-k-theory-of-quiver-varieties-and-many-body-systems-1481k1j69l.pdf

Petr P. Pushkar

Papers

Baxter Q-operator from quantum K-theory

Quantum K-theory of Quiver Varieties and Many-Body Systems

Quantum K-theory of quiver varieties and many-body systems