Knizhnik Zamolodchikov Research Articles

Type $\widehat{\mathrm C}$ Brauer loop schemes and loop model with boundaries

In this paper we study the Brauer loop model on a strip and the associated quantum Knizhnik–Zamolodchikov (qKZ) equation. We show that the minimal degree solution of the Brauer qKZ equation with one of four dierent possible boundary conditions, gives the multidegrees of the irreducible components of generalizations of the Brauer loop scheme of [16, Knutson–Zinn-Justin ’07] with one of four kinds of symplectic-type symmetry. This is accomplished by studying these irreducible components, which are indexed by link patterns, and describing the geometric action of Brauer generators on them. We also provide recurrence relations for the multidegrees and compute the sum rules (multidegrees of the whole schemes).

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.

New differential equations in the six-vertex model

This letter is concerned with the analysis of the six-vertex model with domain-wall boundaries in terms of partial differential equations (PDEs). The model’s partition function is shown to obey a system of PDEs resembling the celebrated Knizhnik–Zamolodchikov equation. The analysis of our PDEs naturally produces a family of novel determinant representations for the model’s partition function.

Theta function solutions of the quantum Knizhnik–Zamolodchikov–Bernard equation for a face model

We consider the quantum Knizhnik–Zamolodchikov–Bernard equation for a face model with elliptic weights, the SOS model. We provide explicit solutions as theta functions. On the so-called combinatorial line, in which the model is equivalent to the three-colour model, these solutions are shown to be eigenvectors of the transfer matrix with periodic boundary conditions.

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.

Matrix product formula for Macdonald polynomials

We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik–Zamolodchikov equations, which arise by considering representations of the Zamolodchikov–Faddeev and Yang–Baxter algebras in terms of t-deformed bosonic operators. These solutions are generalized probabilities for particle configurations of the multi-species asymmetric exclusion process, and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalizations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at q = 1.

ONE-DIMENSIONAL STOCHASTIC MODELS WITH OPEN BOUNDARIES: INTEGRABILITY, APPLICATIONS AND -DEFORMED KNIZHNIK–ZAMOLODCHIKOV EQUATIONS

This thesis contains work on three separate topics, but with common themes running throughout. These themes are drawn together in the asymmetric exclusion process (ASEP) – a stochastic process describing particles hopping on a one-dimensional lattice. The open boundary ASEP is set on a finite length lattice, with particles entering and exiting at both boundaries. The transition matrix of the open boundary ASEP provides a representation of the two boundary Temperley–Lieb algebra, and the integrability of the system allows the diagonalisation of the transition matrix through the Bethe ansatz method. We study the ASEP in the reverse bias regime, where the boundary injection and extraction rates oppose the preferred direction of flow in the bulk. We find the exact asymptotic relaxation rate along the coexistence line by analysing solutions of the Bethe equations. The Bethe equations are first solved numerically, then the form of the resulting root distribution is used as the basis for an asymptotic analysis. The reverse bias induces the appearance of isolated roots, which introduces a modified length scale in the system. We describe the careful treatment of the isolated roots that is required in both the numerical procedure, and in the asymptotic analysis. The second topic of this thesis is the study of a priority queueing system, modelled as an exclusion process with moving boundaries. We call this model the prioritising exclusion process (PEP). In the PEP, the hopping of particles corresponds to high priority customers overtaking low priority customers in order to gain service sooner. Although the PEP is not integrable, techniques from the ASEP allow calculation of exact density profiles in certain phases, and the calculation of approximate average waiting times when the expected queue length is finite. The final topic of this thesis is a study of polynomial solutions of a q-deformed Knizhnik–Zamolodchikov (qKZ) equation with mixed boundaries. The qKZ equation studied here is given in terms of the one boundary Temperley–Lieb algebra, and its solutions have a factorised form in terms of Baxterized elements of the type B Hecke algebra. We find an integral form for certain components of the qKZ solution, along with

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.

Connection Problems for Quantum Affine KZ Equations and Integrable Lattice Models

Cherednik attached to an affine Hecke algebra module a compatible system of difference equations, called quantum affine Knizhnik-Zamolodchikov (KZ) equations. In case of a principal series module we construct a basis of power series solutions of the quantum affine KZ equations. Relating the bases for different asymptotic sectors gives rise to a Weyl group cocycle, which we compute explicitly in terms of theta functions. For the spin representation of the affine Hecke algebra of type C the quantum affine KZ equations become the boundary qKZ equations associated to the Heisenberg spin-1/2 XXZ chain. We show that in this special case the results lead to an explicit 4-parameter family of elliptic solutions of the dynamical reflection equation associated to Baxter's 8-vertex face dynamical R-matrix. We use these solutions to define an explicit 9-parameter elliptic family of boundary quantum Knizhnik-Zamolodchikov-Bernard (KZB) equations.

Infinitesimal 2-braidings and differential crossed modules

We categorify the notion of an infinitesimal braiding in a linear strict symmetric monoidal category, leading to the notion of a (strict) infinitesimal 2-braiding in a linear symmetric strict monoidal 2-category. We describe the associated categorification of the 4-term relations, leading to six categorified relations. We prove that any infinitesimal 2-braiding gives rise to a flat and fake flat 2-connection in the configuration space of n particles in the complex plane, hence to a categorification of the Knizhnik–Zamolodchikov connection. We discuss infinitesimal 2-braidings in a certain monoidal 2-category naturally assigned to every differential crossed module, leading to the notion of a symmetric quasi-invariant tensor in a differential crossed module. Finally, we prove that symmetric quasi-invariant tensors exist in the differential crossed module associated to Wagemann's version of the String Lie-2-algebra. As a corollary, we obtain a more conceptual proof of the flatness of a previously constructed categorified Knizhnik–Zamolodchikov connection with values in the String Lie-2-algebra.

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.

Local Systems on Configuration Spaces, KZ Connections and Conformal Blocks

We describe a relationship between homological representations of the braid groups due to R. Lawrence, D. Krammer, and S. Bigelow and the monodromy representations of the Knizhnik-Zamolodchikov (KZ) connection. First, we describe the comparison theorem in the case of generic parameters. Then, we discuss the nongeneric case from the viewpoint of conformal field theory.

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.

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 $q=-e^{-i\pi/(\ell+2)}$.

The Brauer loop scheme and orbital varieties

A. Joseph invented multidegrees in Joseph (1984) to study orbital varieties, which are the components of an orbital scheme, itself constructed by intersecting a nilpotent orbit with a Borel subalgebra. Their multidegrees are known as Joseph polynomials, and these polynomials give a basis of a (Springer) representation of the Weyl group. In the case of the nilpotent orbit {M2=0}, the orbital varieties can be indexed by noncrossing chord diagrams in the disk.In this paper we study the normal cone to the orbital scheme inside this nilpotent orbit {M2=0}. This gives a better-motivated construction of the Brauer loop scheme we introduced in Knutson and Zinn-Justin (2007), whose components are indexed by all chord diagrams (now possibly with crossings) in the disk.The multidegrees of its components, the Brauer loop varieties, were shown to reproduce the ground state of the Brauer loop model in statistical mechanics (Di Francesco and Zinn-Justin, 2006). Here, we reformulate and slightly generalize these multidegrees in order to express them as solutions of the rational quantum Knizhnik–Zamolodchikov equation associated to the Brauer algebra. In particular, the vector of the multidegrees satisfies two sets of equations, corresponding to the ei and fi generators of the Brauer algebra. The proof of the analogous statement in Knutson and Zinn-Justin (2007) was slightly roundabout; we verified the fi equation using the geometry of multidegrees, and used algebraic results of Di Francesco and Zinn-Justin (2006) to show that it implied the ei equation. We describe here the geometric meaning of both ei and fi equations in our slightly extended setting.We also describe the corresponding actions at the level of orbital varieties: while only the ei equations make sense directly on the Joseph polynomials, the fi equations also appear if one introduces a broader class of varieties. We explain the connection of the latter with matrix Schubert varieties.

Connection coefficients for basic Harish-Chandra series

Basic Harish-Chandra series are asymptotically free meromorphic solutions of the system of basic hypergeometric difference equations associated to root systems. The associated connection coefficients are explicitly computed in terms of Jacobi theta functions. We interpret the connection coefficients as the transition functions for asymptotically free meromorphic solutions of Cherednikʼs root system analogs of the quantum Knizhnik–Zamolodchikov equations. They thus give rise to explicit elliptic solutions of root system analogs of dynamical Yang–Baxter and reflection equations. Applications to quantum c-functions, basic hypergeometric functions, reflectionless difference operators and multivariable Baker–Akhiezer functions are discussed.

Spherical and Whittaker functions via DAHA I

This paper is focused on the spinor (nonsymmetric) Whittaker functions in the rank one, related q-Toda–Dunkl operators, and other aspects of the spinor construction, including one-dimensional Bessel functions and the isomorphism between the affine Knizhnik–Zamolodchikov equation and the Quantum many-body problem (the Heckman–Opdam system).

Multiple Schramm–Loewner evolutions for conformal field theories with Lie algebra symmetries

We provide multiple Schramm–Loewner evolutions (SLEs) to describe the scaling limit of multiple interfaces in critical lattice models possessing Lie algebra symmetries. The critical behavior of the models is described by Wess–Zumino–Witten (WZW) models. Introducing a multiple Brownian motion on a Lie group as well as that on the real line, we construct the multiple SLE with additional Lie algebra symmetries. The connection between the resultant SLE and the WZW model can be understood via SLE martingales satisfied by the correlation functions in the WZW model. Due to interactions among SLE traces, these Brownian motions have drift terms which are determined by partition functions for the corresponding WZW model. As a concrete example, we apply the formula to the suˆ(2)k-WZW model. Utilizing the fusion rules in the model, we conjecture that there exists a one-to-one correspondence between the partition functions and the topologically inequivalent configurations of the SLE traces. Furthermore, solving the Knizhnik–Zamolodchikov equation, we exactly compute the probabilities of occurrence for certain configurations (i.e. crossing probabilities) of traces for the triple SLE.

Virasoro Action on Imaginary Verma Modules and the Operator Form of the KZ-Equation

We define the Virasoro algebra action on imaginary Verma modules for affine $${\mathfrak{sl}(2)}$$ and construct an analogue of the Knizhnik–Zamolodchikov equation in the operator form. Both these results are based on a realization of imaginary Verma modules in terms of sums of partial differential operators.