Knizhnik-Zamolodchikov Equations Research Articles

Meromorphic tensor equivalence for Yangians and quantum loop algebras

Let ${\mathfrak g}$ be a complex semisimple Lie algebra, and $Y_h({\mathfrak g})$, $U_q(L{\mathfrak g})$ the corresponding Yangian and quantum loop algebra, with deformation parameters related by $q=\exp(\pi i h)$. When $h$ is not a rational number, we constructed in arXiv:1310.7318 a faithful functor $\Gamma$ from the category of finite-dimensional representations of $Y_h ({\mathfrak g})$ to those of $U_q(L{\mathfrak g})$. The functor $\Gamma$ is governed by the additive difference equations defined by the commuting fields of the Yangian, and restricts to an equivalence on a subcategory of $Y_h({\mathfrak g})$ defined by choosing a branch of the logarithm. In this paper, we construct a tensor structure on $\Gamma$ and show that, if $|q|\neq 1$, it yields an equivalence of meromorphic braided tensor categories, when $Y_h({\mathfrak g})$ and $U_q(L{\mathfrak g})$ are endowed with the deformed Drinfeld coproducts and the commutative part of the universal $R$-matrix. This proves in particular the Kohno-Drinfeld theorem for the abelian $q$KZ equations defined by $Y_h({\mathfrak g})$. The tensor structure arises from the abelian $q$KZ equations defined by a appropriate regularisation of the commutative $R$-matrix of $Y_h({\mathfrak g})$.

On the gl(1|1) Wess-Zumino-Witten model

We continue the study of the gl(1|1) Wess-Zumino-Witten model. The Knizhnik-Zamolodchikov equations for the one, two, three and four point functions are analyzed, for vertex operators corresponding to typical and projective representations. We demonstrate their interplay with the logarithmic global conformal Ward identities. We compute the four point function for one projective and three typical representations. Three coupled first order Knizhnik-Zamolodchikov equations are integrated consecutively in terms of generalized hypergeometric functions, and we assemble the solutions into a local correlator. Moreover, we prove crossing symmetry of the four point function of four typical representations at generic momenta. Throughout, the map between the gl(1|1) Wess-Zumino-Witten model and symplectic fermions is exploited and extended.

KZ-Calogero correspondence revisited

We discuss the correspondence between the Knizhnik–Zamolodchikov equations associated with GL(N) and the n-particle quantum Calogero model in the case when n is not necessarily equal to N. This can be viewed as a natural ‘quantization’ of the quantum-classical correspondence between quantum Gaudin and classical Calogero 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.

Cactus Group and Monodromy of Bethe Vectors

Cactus group is the fundamental group of the real locus of the Deligne-Mumford moduli space of stable rational curves. This group appears naturally as an analog of the braid group in coboundary monoidal categories. We define an action of the cactus group on the set of Bethe vectors of the Gaudin magnet chain corresponding to arbitrary semisimple Lie algebra $\mathfrak{g}$. Cactus group appears in our construction as a subgroup in the Galois group of Bethe Ansatz equations. Following the idea of Pavel Etingof, we conjecture that this action is isomorphic to the action of the cactus group on the tensor product of crystals coming from the general coboundary category formalism. We prove this conjecture in the case $\mathfrak{g}=\mathfrak{s}\mathfrak{l}_2$ (in fact, for this case the conjecture almost immediately follows from the results of Varchenko on asymptotic solutions of the KZ equation and crystal bases). We also present some conjectures generalizing this result to Bethe vectors of shift of argument subalgebras and relating the cactus group with the Berenstein-Kirillov group of piecewise-linear symmetries of the Gelfand-Tsetlin polytope.

Non-Negative Integral Level Affine Lie Algebra Tensor Categories and Their Associativity Isomorphisms

For a finite-dimensional simple Lie algebra $\mathfrak{g}$, we use the vertex tensor category theory of Huang and Lepowsky to identify the category of standard modules for the affine Lie algebra $\hat{\mathfrak{g}}$ at a fixed level $\ell\in\mathbb{N}$ with a certain tensor category of finite-dimensional $\mathfrak{g}$-modules. More precisely, the category of level $\ell$ standard $\hat{\mathfrak{g}}$-modules is the module category for the simple vertex operator algebra $L_{\hat{\mathfrak{g}}}(\ell,0)$, and as is well known, this category is equivalent as an abelian category to $\mathbf{D}(\mathfrak{g},\ell)$, the category of finite-dimensional modules for the Zhu's algebra $A(L_{\hat{\mathfrak{g}}}(\ell,0))$, which is a quotient of $U(\mathfrak{g})$. Our main result is a direct construction using Knizhnik-Zamolodchikov equations of the associativity isomorphisms in $\mathbf{D}(\mathfrak{g},\ell)$ induced from the associativity isomorphisms constructed by Huang and Lepowsky in $L_{hat{\mathfrak{g}}}(\ell,0)-\mathbf{mod}$. This construction shows that $\mathbf{D}(\mathfrak{g},\ell)$ is closely related to the Drinfeld category of $U(\mathfrak{g})[[\hbar]]$-modules used by Kazhdan and Lusztig to identify categories of $\hat{\mathfrak{g}}$-modules at irrational and most negative rational levels with categories of quantum group modules.

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.

“Quantization” of an isomonodromic Hamiltonian Garnier system with two degrees of freedom

We construct solutions of analogues of the nonstationary Schr\"odinger equation corresponding to the polynomial isomonodromic Hamiltonian Garnier system with two degrees of freedom. This solutions are obtained from solutions of systems of linear ordinary differential equations whose compatibility condition is the Garnier system. This solutions upto explicit transform also satisfy the Belavin --- Polyakov --- Zamolodchikov equations with four time variables and two space variables.

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.

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.

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

GL(3) -Based Quantum Integrable Composite Models. I. Bethe Vectors

We consider a composite generalized quantum integrable model solvable by the nested algebraic Bethe ansatz. Using explicit formulas of the action of the monodromy matrix elements onto Bethe vectors in the GL(3)-based quantum integrable models we prove a formula for the Bethe vectors of composite model. We show that this representation is a particular case of general coproduct property of the weight functions (Bethe vectors) found in the theory of the deformed Knizhnik-Zamolodchikov equation.

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.

Wall-crossing invariants: from quantum mechanics to knots

We offer a pedestrian level review of the wall-crossing invariants. The story begins from the scattering theory in quantum mechanics where the spectrum reshuffling can be related to permutations of S-matrices. In non-trivial situations, starting from spin chains and matrix models, the S-matrices are operator-valued and their algebra is described in terms of R- and mixing (Racah) U-matrices. Then, the Kontsevich-Soibelman invariants are nothing but the standard knot invariants made out of these data within the Reshetikhin-Turaev-Witten approach. The R- and Racah matrices acquire a relatively universal form in the quasiclassical limit, where the basic reshufflings with the change of moduli are those of the Stokes line. Natural from this point of view are matrices provided by the modular transformations of conformal blocks (with the usual identification R=T and U=S), and in the simplest case of the first degenerate field (2,1), when the conformal blocks satisfy a second order Shroedinger-like equation, the invariants coincide with the Jones (N=2) invariants of the associated knots. Another possibility to construct knot invariants is to realize the cluster coordinates associated with reshufflings of the Stokes lines immediately in terms of check-operators acting on the solutions to the Knizhnik-Zamolodchikov equations. Then, the R-matrices are realized as products of successive mutations in the cluster algebra and are manifestly described in terms of quantum dilogarithms ultimately leading to the Hikami construction of knot invariants.

W-algebras and higher analogues of the Knizhnik-Zamolodchikov equations

The key role in the derivation of the Knizhnik-Zamolodchikov equations in the Wess-Zumino-Witten model is played by the energy-momentum tensor, which is constructed from a second-order Casimir element in the universal enveloping algebra of the corresponding Lie algebra. We investigate the possibility of constructing analogues of Knizhnik-Zamolodchikov equations using higher-order central elements. We consider the Casimir element of the third order for the Lie algebra \(\mathfrak{s}\mathfrak{l}_N\) and of the fourth order for \(\mathfrak{o}_N\). The construction is impossible in the first case, but we succeed in obtaining the sought equation in the second case.