We consider noncompact open SL(2, ℂ) spin chain and construct eigenfunctions of B-element of monodromy matrix for the simplest case of the chain with one site. The reflection operator appearing in this construction can be used to express eigenfunction for n sites in terms of the eigenfunction for n-1 sites, this general result is briefly announced. We prove orthogonality and completeness of constructed eigenfunctions in the case of one site, express them in terms of the hypergeometric function of the complex field and derive the equation for the reflection operator with the general SL(2,ℂ) -invariant ℝ -operator. Bibliography: 23 titles.

In this paper we consider an open SL(2,ℝ) spin chain, mainly the simplest case of one particle. Eigenfunctions of the model can be constructed using the so-called reflection operator. We obtain several representations of this operator and show its relation to the hypergeometric function. Besides, we prove orthogonality and completeness of one-particle eigenfunctions and connect them to the index hypergeometric transform. Finally, we briefly state the formula for the eigenfunctions in many-particle case. Bibliography: 17 titles.

For the noncompact open SL(2, C) spin chain, the eigenfunctions of the special matrix element of monodromy matrix are constructed. The key ingredients of the whole construction are local Yang-Baxter R-operators, Q-operator and raising operators obtained by reduction from the Q-operator. The calculation of various scalar products and the proof of orthogonality are based on the properties of Q-operator and demonstrate its hidden role. The symmetry of eigenfunctions with respect to reflection of the spin variable s -> 1-s is established. The Mellin-Barnes representation for eigenfunctions is derived and equivalence with initial coordinate representation is proved. The transformation from one representation to another is grounded on the application of A-type Gustafson integral generalized to the complex field.

We introduce and study a new family of commuting Baxter operators in the Ruijsenaars hyperbolic system, different from that considered by us earlier. Using a degeneration of Rains integral identity we verify the commutativity between the two families of Baxter operators and explore this fact for the proof of the coupling constant symmetry of the wave function. We also establish a connection between new Baxter operators and Noumi-Sano difference operators.

In the previous paper we showed that the wave functions of the quantum Ruijsenaars hyperbolic system diagonalize Baxter Q-operators. Using this property and duality relation we prove orthogonality and completeness relations for the wave functions or, equivalently, unitarity of the corresponding integral transform.

In these notes we review the technique of Baxter Q-operators in the Ruijsenaars-Sutherland hyperbolic systems in the cases of one and two particles. Using these operators we show in particular that eigenfunctions of these systems admit two dual integral representations and prove their orthogonality and completeness.

We present a proof of hypergeometric identities which play a crucial role in the theory of Baxter operators in the Ruijsenaars model.

We introduce Baxter Q -operators for the quantum Ruijsenaars hyperbolic system. We prove that they represent a commuting family of integral operators and also commute with Macdonald difference operators, which are gauge equivalent to the Ruijsenaars Hamiltonians of the quantum system. The proof of commutativity of the Baxter operators uses a hypergeometric identity on rational functions that generalize Ruijsenaars kernel identities.

In the previous paper, we introduced a commuting family of Baxter Q -operators for the quantum Ruijsenaars hyperbolic system. In the present work, we show that the wave functions of the quantum system found by M. Hallnäs and S. Ruijsenaars also diagonalize Baxter operators. Using this property, we prove the conjectured duality relation for the wave function. As a corollary, we show that the wave function solves bispectral problems for pairs of dual Macdonald and Baxter operators. Besides, we prove the conjectured symmetry of the wave function with respect to spectral variables and obtain new integral representation for it.

The note is devoted to the factorization formula for the matrix with the generators of the group GL(N, ℝ) in the regular representation. The factorization formula helps to evaluate these generators and Casimir operators in the case of arbitrary N and clarifies the connection between GL(N, ℝ) and the quantum Toda chain.

In this paper we study a wide class of planar single-trace four point correlators in the chiral conformal field theory ( χ CFT 4 ) arising as a double scaling limit of the γ -deformed 𝒩 = 4 SYM theory. In the planar (t’Hooft) limit, each of such correlators is described by a single Feynman integral having the bulk topology of a square lattice “fishnet” and/or of an honeycomb lattice of Yukawa vertices. The computation of this class of Feynmann integrals at any loop is achieved by means of an exactly-solvable spin chain magnet with SO(1 , 5) symmetry. In this paper we explain in detail the solution of the magnet model as presented in our recent letter and we obtain a general formula for the representation of the Feynman integrals over the spectrum of the separated variables of the magnet, for any number of scalar and fermionic fields in the corresponding correlator. For the particular choice of scalar fields only, our formula reproduces the conjecture of B. Basso and L. Dixon for the fishnet integrals.

This work develops a new method, based on the use of Gustafson’s integrals and on the evaluation of singular integrals, allowing one to establish the unitarity of the separation of variables transform for infinite-dimensional representations of rank one quantum integrable models. We examine in detail the case of the SL(2,R) spin chains.

A bstract We present a basis of eigenvectors for the graph building operators acting along the mirror channel of planar fishnet Feynman integrals in d -dimensions. The eigenvectors of a fishnet lattice of length N depend on a set of N quantum numbers ( u k , l k ), each associated with the rapidity and bound-state index of a lattice excitation. Each excitation is a particle in (1 + 1)-dimensions with O ( d ) internal symmetry, and the wave-functions are formally constructed with a set of creation/annihilation operators that satisfy the corresponding Zamolodchikovs-Faddeev algebra. These properties are proved via the representation, new to our knowledge, of the matrix elements of the fused R-matrix with O ( d ) symmetry as integral operators on the functions of two spacetime points. The spectral decomposition of a fishnet integral we achieved can be applied to the computation of Basso-Dixon integrals in higher dimensions.

Baxter operators are constructed for quantum spin chains with deformed sl2 symmetry. The parallel treatment of Yang-Baxter operators for the cases of undeformed, trigonometrically and elliptically deformed symmetries presented earlier and relying on the factorization regarding parameter permutations is extended to the global chain operators following the scheme worked out recently in the undeformed case. 1 e-mail:chicherin@pdmi.ras.ru 2 e-mail:derkach@pdmi.ras.ru 3 e-mail:karakhan@lx2.yerphi.am 4 e-mail:Roland.Kirschner@itp.uni-leipzig.de

Abstract In this paper we consider systems of quantum particles in the 4d Euclidean space which enjoy conformal symmetry. The algebraic relations for conformal-invariant combinations of positions and momenta are used to construct a solution of the Yang-Baxter equation in the unitary irreducibile representations of the principal series ∆ = 2 + iν for any left/right spins ℓ,$$ \dot{\ell} $$ ℓ ̇ of the particles. Such relations are interpreted in the language of Feynman diagrams as integral star-triangle identites between propagators of a conformal field theory. We prove the quantum integrability of a spin chain whose k-th site hosts a particle in the representation (∆k, ℓk,$$ \dot{\ell} $$ ℓ ̇ k) of the conformal group, realizing a spinning and inhomogeneous version of the quantum magnet used to describe the spectrum of the bi-scalar Fishnet theories [1]. For the special choice of particles in the scalar (1, 0, 0) and fermionic (3/2, 1, 0) representation the transfer matrices of the model are Bethe-Salpeter kernels for the double-scaling limit of specific two-point correlators in the γ-deformed $$ \mathcal{N} $$ N = 4 and $$ \mathcal{N} $$ N = 2 supersymmetric theories.

It was observed recently that relations between matrix elements of certain operators in the SL(2, R) spin chain models take the form of multidimensional integrals derived by R.A. Gustafson. The spin magnets with SL(2, C) symmetry group and L-2(C) as a local Hilbert space give rise to a new type Gamma-function of integrals. In this work we present a direct calculation of two such integrals. We also analyse properties of these integrals and show that they comprise the star-triangle relations recently discussed in the literature. It is also shown that in the quasi-classical limit these integral identities are reduced to the duality relations for Dotsenko-Fateev integrals.

We provide the eigenfunctions for a quantum chain of N conformal spins with nearest-neighbor interaction and open boundary conditions in the irreducible representation of SO(1,5) of scaling dimension Δ=2-iλ and spin numbers ℓ=ℓ[over ˙]=0. The spectrum of the model is separated into N equal contributions, each dependent on a quantum number Y_{a}=[ν_{a},n_{a}] which labels a representation of the principal series. The eigenfunctions are orthogonal and we computed the spectral measure by means of a new star-triangle identity. Any portion of a conformal Feynmann diagram with square lattice topology can be represented in terms of separated variables, and we reproduce the all-loop "fishnet" integrals computed by B. Basso and L. Dixon via bootstrap techniques. We conjecture that the proposed eigenfunctions form a complete set and provide a tool for the direct computation of conformal data in the fishnet limit of the supersymmetric N=4 Yang-Mills theory at finite order in the coupling, by means of a cutting-and-gluing procedure on the square lattice.

We construct the generalised eigenfunctions of the entries of the monodromy matrix of the N-site modular XXZ magnet and show, in each case, that these form a complete orthogonal system in $$L^2(\mathbb {R}^N)$$ . In particular, we develop a new and simple technique, allowing one to prove the completeness of such systems. As a corollary of our analysis, we prove the Bytsko–Teschner conjecture relative to the structure of the spectrum of the $$ \varvec{ \texttt {B} } (\lambda )$$ -operator for the odd length lattice Sinh-Gordon model.

It was shown recently that many of the Gustafson integrals appear in studies of the SL(2, R) spin chain models. One can hope to obtain a generalization of the Gustafson integrals considering spin chain models with a different symmetry group. In this paper we analyse the spin magnet with the SL(2, C) symmetry group in case of open and periodic boundary conditions and derive several new integrals.

We start from known solutions of the Yang-Baxter equation with a spectral parameter defined on the tensor product of two infinite-dimensional principal series representations of the group SL(2, C) or Faddeev's modular double.Then we describe its restriction to an irreducible finite-dimensional representation in one or both spaces.In this way we obtain very simple explicit formulas embracing rational and trigonometric finite-dimensional solutions of the Yang-Baxter equation.Finally, we construct these finite-dimensional solutions by means of the fusion procedure and find a nice agreement between two approaches.