Quantum Knizhnik-Zamolodchikov Equation Research Articles

The open XXZ chain at Δ = −1/2 and the boundary quantum Knizhnik–Zamolodchikov equations

The open XXZ spin chain with the anisotropy parameter and diagonal boundary magnetic fields that depend on a parameter x is studied. For real x > 0, the exact finite-size ground-state eigenvalue of the spin-chain Hamiltonian is explicitly computed. In a suitable normalisation, the ground-state components are characterised as polynomials in x with integer coefficients. Linear sum rules and special components of this eigenvector are explicitly computed in terms of determinant formulas. These results follow from the construction of a contour-integral solution to the boundary quantum Knizhnik–Zamolodchikov equations associated with the R-matrix and diagonal K-matrices of the six-vertex model. A relation between this solution and a weighted enumeration of totally-symmetric alternating sign matrices is conjectured.

Elliptic stable envelopes

We construct stable envelopes in equivariant elliptic cohomology of Nakajima quiver varieties. In particular, this gives an elliptic generalization of the results of Maulik and Okounkov [Astérisque 408 (2019), ix+209]. We apply them to the computation of the monodromy of q q -difference equations arising in the enumerative K-theory of rational curves in Nakajima varieties, including the quantum Knizhnik–Zamolodchikov equations.

QKZ/tRS duality via quantum $K$-theoretic counts

We show that normalized quantum K-theoretic vertex functions for cotangent bundles of partial flag varieties are the eigenfunctions of quantum trigonometric Ruijsenaars-Schneider (tRS) Hamiltonians. Using recently observed relations between quantum Knizhnik-Zamolodchikov (qKZ) equations and tRS integrable system we derive a nontrivial identity for vertex functions with relative insertions.

Reduced qKZ equation: general case

We use the quantum group approach for the investigation of correlation func- tions of integrable vertex models and spin chains. For the inhomogeneous reduced density matrix in case of an arbitrary simple Lie algebra we find functional equations of the form of the reduced quantum Knizhnik–Zamolodchikov equation. This equation is the starting point for the investigation of correlation functions at arbitrary temperature and notably for the ground state.

Grassmann–Grassmann conormal varieties, integrability, and plane partitions

We give a conjectural formula for sheaves supported on (irreducible) conormal varieties inside the cotangent bundle of the Grassmannian, such that their equivariant K-class is given by the partition function of an integrable loop model, and furthermore their K-theoretic pushforward to a point is a solution of the level 1 quantum Knizhnik–Zamolodchikov equation. We prove these results in the case that the Lagrangian is smooth (hence is the conormal bundle to a subGrassmannian). To compute the pushforward to a point, or equivalently to the affinization, we simultaneously degenerate the Lagrangian and sheaf (over the affinization); the sheaf degenerates to a direct sum of cyclic modules over the geometric components, which are in bijection with plane partitions, giving a geometric interpretation to the Razumov–Stroganov correspondence satisfied by the loop model.

Supersymmetric extension of qKZ-Ruijsenaars correspondence

We describe the correspondence of the Matsuo–Cherednik type between the quantum n-body Ruijsenaars–Schneider model and the quantum Knizhnik–Zamolodchikov equations related to supergroup GL(N|M). The spectrum of the Ruijsenaars–Schneider Hamiltonians is shown to be independent of the Z2-grading for a fixed value of N+M, so that N+M+1 different qKZ systems of equations lead to the same n-body quantum problem. The obtained results can be viewed as a quantization of the previously described quantum-classical correspondence between the classical n-body Ruijsenaars–Schneider model and the supersymmetric GL(N|M) quantum spin chains on n sites.

On the calculation of the correlation functions of the -model by means of the reduced qKZ equation

We study the reduced density matrix of the -invariant fundamental exchange model by means of a novel reduced quantum Knizhnik–Zamolodchikov equation. This gives us insight into the algebraic structure and explicit results for correlation functions in the infinite chain ranging over up to three sites.

Integral solutions to boundary quantum Knizhnik–Zamolodchikov equations

We construct integral representations of solutions to the boundary quantum Knizhnik–Zamolodchikov equations. These are difference equations taking values in tensor products of Verma modules of quantum affine sl2, with the K-operators acting diagonally. The integrands in question are products of scalar-valued elliptic weight functions with vector-valued trigonometric weight functions (boundary Bethe vectors). These integrals give rise to a basis of solutions of the boundary qKZ equations over the field of quasi-constant meromorphic functions in weight subspaces of the tensor product.

QKZ–Ruijsenaars correspondence revisited

We discuss the Matsuo–Cherednik type correspondence between the quantum Knizhnik–Zamolodchikov equations associated with GL(N) and the n-particle quantum Ruijsenaars model, with n being not necessarily equal to N. The quasiclassical limit of this construction yields the quantum–classical correspondence between the quantum spin chains and the classical Ruijsenaars models.

Quasimap Counts and Bethe Eigenfunctions

We associate an explicit equivalent descendent insertion to any relative insertion in quantum K-theory of Nakajima varieties. This also serves as an explicit formula for off-shell Bethe eigenfunctions for general quantum loop algebras associated to quivers and gives the general integral solution to the corresponding quantum Knizhnik-Zamolodchikov and dynamical q-difference equations.

Bethe subalgebras in affine Birman–Murakami–Wenzl algebras and flat connections for q-KZ equations**Dedicated to Professor Rodney Baxter on the occasion of his 75th Birthday.

Commutative sets of Jucys–Murphy elements for affine braid groups of types were defined. Construction of R-matrix representations of the affine braid group of type and its distinguished commutative subgroup generated by the -type Jucys–Murphy elements are given. We describe a general method to produce flat connections for the two-boundary quantum Knizhnik–Zamolodchikov equations as necessary conditions for Sklyanin's type transfer matrix associated with the two-boundary multicomponent Zamolodchikov algebra to be invariant under the action of the -type Jucys–Murphy elements. We specify our general construction to the case of the Birman–Murakami–Wenzl algebras (BMW algebras for short). As an application we suggest a baxterization of the Dunkl–Cherednik elements in the double affine Hecke algebra of type A.

Quasimap Counts and Bethe Eigenfunctions

We associate an explicit equivalent descendent insertion to any relative insertion in quantum K-theory of Nakajima varieties. This also serves as an explicit formula for off-shell Bethe eigenfunctions for general quantum loop algebras associated to quivers and gives the general integral solution to the corresponding quantum Knizhnik-Zamolodchikov and dynamical q-difference equations.

Quiver varieties and the quantum Knizhnik–Zamolodchikov equation

We show how equivariant volumes of tensor product quiver varieties of type A are given by matrix elements of vertex operators of centrally extended doubles of Yangians, and how they satisfy in some cases the rational, level 1, quantum Knizhnik--Zamolodchikov equation.

Quiver varieties and the quantum Knizhnik-Zamolodchikov equation

Показано, как эквивариантные объемы тензорных произведений колчанных переменных типа A задаются матричными элементами вершинных операторов центрально расширенных янгианных дублей и как эти элементы удовлетворяют в некоторых случаях рациональному квантовому уравнению Книжника-Замолодчикова уровня один.

Boundary transfer matrices and boundary quantum KZ equations

A simple relation between inhomogeneous transfer matrices and boundary quantum Knizhnik-Zamolodchikov (KZ) equations is exhibited for quantum integrable systems with reflecting boundary conditions, analogous to an observation by Gaudin for periodic systems. Thus, the boundary quantum KZ equations receive a new motivation. We also derive the commutativity of Sklyanin’s boundary transfer matrices by merely imposing appropriate reflection equations, in particular without using the conditions of crossing symmetry and unitarity of the R-matrix.

Boundary Quantum Knizhnik–Zamolodchikov Equations and Fusion

In this paper we extend our previous results concerning Jackson integral solutions of the boundary quantum Knizhnik-Zamolodchikov equations with diagonal K-operators to higher-spin representations of quantum affine $\mathfrak{sl}_2$. First we give a systematic exposition of known results on $R$-operators acting in the tensor product of evaluation representations in Verma modules over quantum $\mathfrak{sl}_2$. We develop the corresponding fusion of $K$-operators, which we use to construct diagonal $K$-operators in these representations. We construct Jackson integral solutions of the associated boundary quantum Knizhnik-Zamolodchikov equations and explain how in the finite-dimensional case they can be obtained from our previous results by the fusion procedure.

Boundary Quantum Knizhnik–Zamolodchikov Equations and Bethe Vectors

Solutions to boundary quantum Knizhnik–Zamolodchikov equations are constructed as bilateral sums involving “off-shell” Bethe vectors in case the reflection matrix is diagonal and only the 2-dimensional representation of $${{\mathcal{U}}_q(\widehat{{\mathfrak{sl}}}_2)}$$ is involved. We also consider their rational and classical degenerations.

Formula omitted] scattering amplitudes and the deformed Graßmannian

Some time ago the general tree-level scattering amplitudes of N=4 Super Yang–Mills theory were expressed as certain Graßmannian contour integrals. These remarkable formulas allow to clearly expose the super-conformal, dual super-conformal, and Yangian symmetries of the amplitudes. Using ideas from integrability it was recently shown that the building blocks of the amplitudes permit a natural multi-parameter deformation. However, this approach had been criticized by the observation that it seemed impossible to reassemble the building blocks into Yangian-invariant deformed non-MHV amplitudes. In this note we demonstrate that the deformations may be succinctly summarized by a simple modification of the measure of the Graßmannian integrals, leading to a Yangian-invariant deformation of the general tree-level amplitudes. Interestingly, the deformed building blocks appear as residues of poles in the spectral parameter planes. Given that the contour integrals also contain information on the amplitudes at loop-level, we expect the deformations to be useful there as well. In particular, applying meromorphicity arguments, they may be expected to regulate all notorious infrared divergences. We also point out relations to Gelfand hypergeometric functions and the quantum Knizhnik–Zamolodchikov equations.

Connectivity Patterns in Loop Percolation I: the Rationality Phenomenon and Constant Term Identities

Loop percolation, also known as the dense O(1) loop model, is a variant of critical bond percolation in the square lattice Z^2 whose graph structure consists of a disjoint union of cycles. We study its connectivity pattern, which is a random noncrossing matching associated with a loop percolation configuration. These connectivity patterns exhibit a striking rationality property whereby probabilities of naturally-occurring events are dyadic rational numbers or rational functions of a size parameter n, but the reasons for this are not completely understood. We prove the rationality phenomenon in a few cases and prove an explicit formula expressing the probabilities in the "cylindrical geometry" as coefficients in certain multivariate polynomials. This reduces the rationality problem in the general case to that of proving a family of conjectural constant term identities generalizing an identity due to Di Francesco and Zinn-Justin. Our results make use of, and extend, algebraic techniques related to the quantum Knizhnik-Zamolodchikov equation.

Higher Spin Polynomial Solutions of Quantum Knizhnik–Zamolodchikov Equation

We provide explicit formulae for highest-weight to highest-weight correlation functions of perfect vertex operators of $U_q(\hat{\mathfrak{sl}(2)})$ at arbitrary integer level $\ell$. They are given in terms of certain Macdonald polynomials. We apply this construction to the computation of the ground state of higher spin vertex models, spin chains (spin $\ell/2$ XXZ) or loop models in the root of unity case $q=-e^{-i\pi/(\ell+2)}$.