Knizhnik-Zamolodchikov Equations Research Articles

Hyperelliptic integrals modulo $p$ and Cartier–Manin matrices

The hypergeometric solutions of the KZ equations were constructed almost 30 years ago. The polynomial solutions of the KZ equations over the finite field $F_p$ with a prime number $p$ of elements were constructed recently. In this paper we consider the example of the KZ equations whose hypergeometric solutions are given by hyperelliptic integrals of genus $g$. It is known that in this case the total $2g$-dimensional space of holomorphic solutions is given by the hyperelliptic integrals. We show that the recent construction of the polynomial solutions over the field $F_p$ in this case gives only a $g$-dimensional space of solutions, that is, a half of what the complex analytic construction gives. We also show that all the constructed polynomial solutions over the field $F_p$ can be obtained by reduction modulo $p$ of a single distinguished hypergeometric solution. The corresponding formulas involve the entries of the Cartier-Manin matrix of the hyperelliptic curve. That situation is analogous to the example of the elliptic integral considered in the classical Y.I. Manin's paper in 1961.

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.

Towers of Solutions of qKZ Equations and Their Applications to Loop Models

Cherednik’s type A quantum affine Knizhnik–Zamolodchikov (qKZ) equations form a consistent system of linear q-difference equations for V_n-valued meromorphic functions on a complex n-torus, with V_n a module over the mathrm{GL}_n-type extended affine Hecke algebra {mathcal {H}}_n. The family ({mathcal {H}}_n)_{nge 0} of extended affine Hecke algebras forms a tower of algebras, with the associated algebra morphisms {mathcal {H}}_nrightarrow {mathcal {H}}_{n+1}, in the Hecke algebra descending of arc insertion at the affine braid group level. In this paper, we consider qKZ towers (f^{(n)})_{nge 0} of solutions, which consist of twisted-symmetric polynomial solutions f^{(n)} (nge 0) of the qKZ equations that are compatible with the tower structure on ({mathcal {H}}_n)_{nge 0}. The compatibility is encoded by the so-called braid recursion relations: f^{(n+1)}(z_1,ldots ,z_{n},0) is required to coincide up to a quasi-constant factor with the push-forward of f^{(n)}(z_1,ldots ,z_{n}) by an intertwiner mu _{n}{:},V_{n}rightarrow V_{n+1} of {mathcal {H}}_{n}-modules, where V_{n+1} is considered as an {mathcal {H}}_{n}-module through the tower structure on ({mathcal {H}}_n)_{nge 0}. We associate with the dense loop model on the half-infinite cylinder with nonzero loop weights, a qKZ tower (f^{(n)})_{nge 0} of solutions. The solutions f^{(n)} are constructed from specialized dual non-symmetric Macdonald polynomials with specialized parameters using the Cherednik–Matsuo correspondence. In the special case that the extended affine Hecke algebra parameter is a third root of unity, f^{(n)} coincides with the (suitably normalized) ground state of the inhomogeneous dense O(1) loop model on the half-infinite cylinder with circumference n.

Stokes Phenomenon and Yang–Baxter Equations

We describe the monodromy of dynamical Knizhnik-Zamolodchikov equations via Stokes phenomenon. It defines a family of braid groups representations by certain Stokes matrices. In particular, these Stokes matrices satisfy the Yang-Baxter equation.

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.

Multiplicative middle convolution for KZ equations

We extend the middle convolution for monodromy of Fuchsian ordinary differential equations due to Katz to an operation for monodromy of KZ equations. This operation, also called the middle convolution, gives a monodromy of KZ equation obtained by the additive middle convolution. Since the fundamental group of the domain of definition of KZ equation is the pure braid group, our middle convolution also gives a recursive way of constructing irreducible representations of the pure braid group. Several applications are considered.

Ribbon Braided Module Categories, Quantum Symmetric Pairs and Knizhnik–Zamolodchikov Equations

Let $\mathfrak u$ be a compact semisimple Lie algebra, and $\sigma$ be a Lie algebra involution of $\mathfrak u$. Let Rep$_q(\mathfrak u)$ be the ribbon braided tensor C*-category of $U_q(\mathfrak u)$-representations for $0<q<1$. We introduce three module C*-categories over Rep$_q(\mathfrak u)$ starting from the input data $(\mathfrak u,\sigma)$. The first construction is based on the theory of cyclotomic KZ-equations. The second construction uses the notion of quantum symmetric pair as developed by G. Letzter. The third construction uses a variation of Drinfeld twisting. In all three cases the module C*-category is ribbon twist-braided in the sense of A. Brochier---this is essentially due to B. Enriquez in the first case, is proved by S. Kolb in the second case, and is closely related to work of J. Donin, P. Kulish, and A. Mudrov in the third case. We formulate a conjecture concerning equivalence of these ribbon twist-braided module C*-categories, and confirm it in the rank one case.

KZ equations and Bethe subalgebras in generalized Yangians related to compatible $R$-matrices

Abstract The notion of compatible braidings was introduced in Isaev et al. (1999, J. Phys. A, 32, L115–L121). On the base of this notion, the authors of Isaev et al. (1999, J. Phys. A, 32, L115–L121) defined certain quantum matrix algebras generalizing the RTT algebras and Reflection Equation ones. They also defined analogues of some symmetric polynomials in these algebras and showed that these polynomials generate commutative subalgebras, called Bethe. By using a similar approach, we introduce certain new algebras called generalized Yangians and define analogues of some symmetric polynomials in these algebras. We claim that they commute with each other and thus generate a commutative Bethe subalgebra in each generalized Yangian. Besides, we define some analogues (also arising from couples of compatible braidings) of the Knizhnik–Zamolodchikov equation—classical and quantum. Communicated by: Alexander Veselov

Generalized sℓ(2) Gaudin algebra and corresponding Knizhnik–Zamolodchikov equation

The Gaudin model has been revisited many times, yet some important issues remained open so far. With this paper we aim to properly address its certain aspects, while clarifying, or at least giving a solid ground to some other. Our main contribution is establishing the relation between the off-shell Bethe vectors with the solutions of the corresponding Knizhnik–Zamolodchikov equations for the non-periodic sℓ(2) Gaudin model, as well as deriving the norm of the eigenvectors of the Gaudin Hamiltonians. Additionally, we provide a closed form expression also for the scalar products of the off-shell Bethe vectors. Finally, we provide explicit closed form of the off-shell Bethe vectors, together with a proof of implementation of the algebraic Bethe ansatz in full generality.

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.

Matrix KP: tropical limit and Yang\u2013Baxter maps

We study soliton solutions of matrix Kadomtsev–Petviashvili (KP) equations in a tropical limit, in which their support at fixed time is a planar graph and polarizations are attached to its constituting lines. There is a subclass of “pure line soliton solutions” for which we find that, in this limit, the distribution of polarizations is fully determined by a Yang–Baxter map. For a vector KP equation, this map is given by an R-matrix, whereas it is a nonlinear map in the case of a more general matrix KP equation. We also consider the corresponding Korteweg–deVries reduction. Furthermore, exploiting the fine structure of soliton interactions in the tropical limit, we obtain an apparently new solution of the tetrahedron (or Zamolodchikov) equation. Moreover, a solution of the functional tetrahedron equation arises from the parameter dependence of the vector KP R-matrix.

Integrable Stochastic Dualities and the Deformed Knizhnik–Zamolodchikov Equation

AbstractWe present a new method for obtaining duality functions in multi-species asymmetric exclusion processes (mASEP), from solutions of the deformed Knizhnik–Zamolodchikov (KZ) equations. Our method reproduces, as a special case, duality functions for the self-dual single species ASEP on the integer lattice.

BPS/CFT correspondence IV: sigma models and defects in gauge theory

Quantum field theory $L_1$ on spacetime $X_{1}$ can be coupled to another quantum field theory $L_2$ on a spacetime $X_{2}$ via the third quantum field theory $L_{12}$ living on $X_{12} = X_{1} \cap X_{2}$. We explore several such constructions with two and four dimensional $X_{1}, X_{2}$'s and zero and two dimensional $X_{12}$'s, in the context of $\mathcal{N}=2$ supersymmetry, non-perturbative Dyson-Schwinger equations, and BPS/CFT correspondence. The companion paper will show that the BPZ and KZ equations of two dimensional conformal field theory are obeyed by the half-BPS surface defects in quiver $\mathcal{N}=2$ gauge theories.

Fundamental Solutions of the Knizhnik-Zamolodchikov Equation of One Variable and the Riemann-Hilbert Problem

In this article, we derive multiple polylogarithms from multiple zeta values by using a recursive Riemann-Hilbert problem of additive type. Furthermore we show that this Riemann-Hilbert problem is regarded as an inverse problem for the connection problem of the KZ equation of one variable, so that the fundamental solutions to the equation are derived from the Drinfel'd associator by using a Riemann-Hilbert problem of multiplicative type. These results say that the duality relation for the Drinfel'd associator can be interpreted as the solvability condition for this inverse problem.

(q, t)-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces

We describe the general strategy for lifting the Wess-Zumino-Witten model from the level of one-loop Kac-Moody {U}_q{left(widehat{mathfrak{g}}right)}_k to generic quantum toroidal algebras. A nearly exhaustive presentation is given for both {U}_{q,t}left({widehat{widehat{mathfrak{gl}}}}_1right) and {U}_{q,t}left({widehat{widehat{mathfrak{gl}}}}_nright) when the screenings do not exist and thus all the correlators are purely algebraic, i.e. do not include additional hypergeometric type integrations/summations.Generalizing the construction of the intertwiner (refined topological vertex) of the Ding-Iohara-Miki (DIM) algebra, we obtain the intertwining operators of the Fock representations of the quantum toroidal algebra of type An. The correlation functions of these operators satisfy the (q, t)-Knizhnik-Zamolodchikov (KZ) equation, which features the ℛ-matrix. The matching with the Nekrasov function for the instanton counting on the ALE space is worked out explicitly.We also present an important application of the DIM formalism to the study of 6d gauge theories described by the double elliptic integrable systems. We show that the modular and periodicity properties of the gauge theories are neatly explained by the network matrix models providing solutions to the elliptic (q, t)-KZ equations.

Integrable time-dependent Hamiltonians, solvable Landau–Zener models and Gaudin magnets

We solve the non-stationary Schrödinger equation for several time-dependent Hamiltonians, such as the BCS Hamiltonian with an interaction strength inversely proportional to time, periodically driven BCS and linearly driven inhomogeneous Dicke models as well as various multi-level Landau–Zener tunneling models. The latter are Demkov–Osherov, bow-tie, and generalized bow-tie models. We show that these Landau–Zener problems and their certain interacting many-body generalizations map to Gaudin magnets in a magnetic field. Moreover, we demonstrate that the time-dependent Schrödinger equation for the above models has a similar structure and is integrable with a similar technique as Knizhnik–Zamolodchikov equations. We also discuss applications of our results to the problem of molecular production in an atomic Fermi gas swept through a Feshbach resonance and to the evaluation of the Landau–Zener transition probabilities.

Homological and Monodromy Representations of Framed Braid Groups

In this paper, we introduce two new classes of representations of the framed braid groups. One is the homological representation constructed as the action of a mapping class group on a certain homology group. The other is the monodromy representation of the confluent KZ equation, which is a generalization of the KZ equation to have irregular singularities. We also give a conjectural equivalence between these two classes of representations.

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.

Multi-soft gluon limits and extended current algebras at null-infinity

In this note we consider aspects of the current algebra interpretation of multisoft limits of tree-level gluon scattering amplitudes in four dimensions. Building on the relation between a positive helicity gluon soft-limit and the Ward identity for a level-zero Kac-Moody current, we use the double-soft limit to define the Sugawara energy-momentum tensor and, by using the triple- and quadruple-soft limits, show that it satisfies the correct OPEs for a CFT. We study the resulting Knizhnik-Zamolodchikov equations and show that they hold for positive helicity gluons in MHV amplitudes. Turning to the sub-leading soft-terms we define a one-parameter family of currents whose Ward identities corresponding to the universal tree-level sub-leading soft-behaviour. We compute the algebra of these currents formed with the leading currents and amongst themselves. Finally, by parameterising the ambiguity in the double-soft limit for mixed helicities, we introduce a non-trivial OPE between the holomorphic and anti-holomorphic currents and study some of its implications.