Quantum Lax Operator Research Articles

Intersecting defects in gauge theory, quantum spin chains, and Knizhnik-Zamolodchikov equations

We propose an interesting BPS/CFT correspondence playground: the correlation function of two intersecting half-BPS surface defects in four-dimensional mathcal{N} = 2 supersymmetric SU(N) gauge theory with 2N fundamental hypermultiplets. We show it satisfies a difference equation, the fractional quantum T-Q relation. Its Fourier transform is the 5-point conformal block of the {hat{mathfrak{sl}}}_N current algebra with one of the vertex operators corresponding to the N-dimensional {mathfrak{sl}}_N representation, which we demonstrate with the help of the Knizhnik-Zamolodchikov equation. We also identify the correlator with a state of the {XXX}_{{mathfrak{sl}}_2} spin chain of N Heisenberg-Weyl modules over Y ( {mathfrak{sl}}_2 ). We discuss the associated quantum Lax operators, and connections to isomonodromic deformations.

On quantum separation of variables beyond fundamental representations

We describe the extension, beyond fundamental representations of the Yang-Baxter algebra, of our new construction of separation of variables bases for quantum integrable lattice models. The key idea underlying our approach is to use the commuting conserved charges of the quantum integrable models to generate bases in which their spectral problem is separated, i.e. in which the wave functions are factorized in terms of specific solutions of a functional equation. For the so-called “non-fundamental” models we construct two different types of SoV bases. The first is given from the fundamental quantum Lax operator having isomorphic auxiliary and quantum spaces and that can be obtained by fusion of the original quantum Lax operator. The construction essentially follows the one we used previously for fundamental models and allows us to derive the simplicity and diagonalizability of the transfer matrix spectrum. Then, starting from the original quantum Lax operator and using the full tower of the fused transfer matrices, we introduce a second type of SoV bases for which the proof of the separation of the transfer matrix spectrum is naturally derived. We show that, under some special choice, this second type of SoV bases coincides with the one associated to Sklyanin’s approach. Moreover, we derive the finite difference type (quantum spectral curve) functional equation and the set of its solutions defining the complete transfer matrix spectrum. This is explicitly implemented for the integrable quantum models associated to the higher spin representations of the general quasi-periodic Y(gl_{2})Y(gl2) Yang-Baxter algebra. Our SoV approach also leads to the construction of a QQ-operator in terms of the fused transfer matrices. Finally, we show that the QQ-operator family can be equivalently used as the family of commuting conserved charges enabling to construct our SoV bases.

Quantum integrable models of interacting bosons and classical r-matrices with spectral parameters

Using the technique of classical r -matrices with spectral parameters we construct a general form of quantum Lax operators of interacting boson systems corresponding to an arbitrary simple (or reductive) Lie algebra. We prove quantum integrability of these models in the physically important case of g = g l ( n ) and “diagonal” in the root basis classical r -matrices. We consider in detail two classes of non-skew-symmetric classical r -matrices with spectral parameters and obtain the corresponding quantum Lax operators and quantum integrable many-boson hamiltonians that generalize Bose–Hubbard dimer hamiltonians.

General integrable n-level, many-mode Janes-Cummings-Dicke models and classical r-matrices with spectral parameters

Using the technique of classical r-matrices and quantum Lax operators, we construct the most general form of the quantum integrable “n-level, many-mode” spin-boson Jaynes-Cummings-Dicke-type hamiltonians describing an interaction of a molecule of N n-level atoms with many modes of electromagnetic field and containing, in general, additional non-linear interaction terms. We explicitly obtain the corresponding quantum Lax operators and spin-boson analogs of the generalized Gaudin hamiltonians and prove their quantum commutativity. We investigate symmetries of the obtained models that are associated with the geometric symmetries of the classical r-matrices and construct the corresponding algebra of quantum integrals. We consider in detail three classes of non-skew-symmetric classical r-matrices with spectral parameters and explicitly obtain the corresponding quantum Lax operators and Jaynes-Cummings-Dicke-type hamiltonians depending on the considered r-matrix.

Elliptic three-boson system, “two-level three-mode” JCD-type models and non-skew-symmetric classical r-matrices

Using the technique of the classical r-matrices and quantum Lax operators we construct the most general form of quantum integrable multi-boson and spin-multi-boson models associated with linear Lax algebras and sl ( 2 ) ⊗ sl ( 2 ) -valued classical non-dynamical r-matrices with spectral parameters. We consider example of non-skew-symmetric elliptic r-matrix and explicitly obtain one-, two- and three-boson integrable models and the corresponding one-, two- and three-mode two-level Jaynes–Cummings–Dicke-type models. We show that integrable “elliptic” two-level one-mode Jaynes–Cummings–Dicke Hamiltonian is hermitian and contains both rotating and counter-rotating terms.

General integrable two-level, one-mode Jaynes–Cummings–Dicke models and classicalr-matrices with spectral parameters

Using the technique of classicalr-matrices and quantum Lax operators we construct the most general form of the quantum integrable‘two-level, one-mode’ spin-boson Jaynes–Cummings–Dicke-type model. We consider in detailtwo subclasses of this model and the corresponding examples associated with concrete classicalr-matrices with spectral parameters. We show that the so-called ‘diagonal’ in the root basisr-matrices produces integrable Jaynes–Cummings–Dicke-type models withthe ‘rotating wave approximation’, while all other types of classicalr-matrices produce integrable JCD models without the rotating wave approximation. Weconstruct an explicit example of an integrable Hermitian Jaynes–Cummings–Dicke-typeHamiltonian containing both ‘rotating’ and ‘counter-rotating’ terms. In the case of the diagonalr-matrices we find the spectrum of the corresponding quantum Hamiltonians using thealgebraic Bethe ansatz.

Integrability and superintegrability of the generalized n-level many-mode Jaynes–Cummings and Dicke models

We analyze symmetries of the integrable generalizations of Jaynes–Cummings and Dicke models associated with simple Lie algebras g and their reductive subalgebras gK [T. Skrypnyk, “Generalized n-level Jaynes-Cummings and Dicke models, classical rational r-matrices and nested Bethe ansatz,” J. Phys. A: Math. Theor. 41, 475202 (2008)]. We show that their symmetry algebras contain commutative subalgebras isomorphic to the Cartan subalgebras of g, which can be added to the commutative algebras of quantum integrals generated with the help of the quantum Lax operators. We diagonalize additional commuting integrals and constructed with their help the most general integrable quantum Hamiltonian of the generalized n-level many-mode Jaynes–Cummings and Dicke-type models using nested algebraic Bethe ansatz.

KZ equation, G-opers, quantum Drinfeld–Sokolov reduction, and quantum Cayley–Hamilton identity

The Lax operator of Gaudin-type models is a 1-form at the classical level. In virtue of the quantization scheme proposed by D. Talalaev, it is natural to treat the quantum Lax operator as a connection; this connection is a partcular case of the Knizhnik–Zamolodchikov connection. In this paper, we find a gauge trasformation that produces the “second normal form,” or the “Drinfeld–Sokolov” form. Moreover, the differential operator nurally corresponding to this form is given precisely by the quantum characteristic polynomial of the Lax operator (this operator is called the G-oper or Baxter operator). This observation allows us to relate solutions of the KZ and Baxter equations in an obvious way, and to prove that the immanent KZ equation has only meromorphic solutions. As a corollary, we obtain the quantum Cayley–Hamilton identity for Gaudin-type Lax operators (including the general $ \mathfrak{gl}_{n}[t] $ case). The presented construction sheds a new light on the geometric Langlands correspondence. We also discuss the relation with the Harish-Chandra homomorphism. Bibliography: 19 titles.

Quantum Lax Operators and Discrete 2+1-Dimensional Integrable Models

A specific quantum Lax operator, forming a quantum U q (sl n) chain, is suggested. The intertwining R L L relation is proved. The invariant local 2+1-dimensional reformulation of the auxiliary linear problem for the L-operator is discussed.

Quantum relativistic Toda chain

Investigated is the quantum relativistic periodic Toda chain, to each site of which the ultra‐local Weyl algebra is associated. Weyl’s q we are considering here is restricted to be inside the unit circle. Quantum Lax operators of the model are intertwined by six‐vertex R‐matrix. Both independent Baxter’s Q‐operators are constructed explicitly as seria over local Weyl generators. The operator‐valued Wronskian of R‐matrix. Both independent Baxter’s Qs is also calculated.

On trigonometric intertwining vectors and non-dynamical R-matrix for the Ruijsenaars model

We elaborate on the trigonometric version of intertwining vectors and factorized L-operators. The starting point is the corresponding elliptic construction with Belavin's R-matrix. The naive trigonometric limit is singular and a careful analysis is needed. It is shown that the construction admits several different trigonometric degenerations. As a by-product, a quantum Lax operator for the trigonometric Ruijsenaars model intertwined by a non-dynamical R-matrix is obtained. The latter differs from the standard trigonometric R-matrix of A n type. A connection with the dynamical R-matrix approach is discussed.

Toda Theories [formula omitted]-algebras, and minimal models

We discuss the classical W -algebra symmetries of Toda field theories in terms of the pseudo-differential Lax operator associated with the Toda Lax pair. We then show how the W -algebra transformations can be understood as the non-abelian gauge transformations which preserve the form of the Lax pair. This provides a new understanding of the W -algebras, and we discuss their closure and co-cycle structure using this approach. The quantum Lax operator is investigated, and we show that this operator, which generates the quantum W -algebra currents, is concerned in the conformally extended Toda theories. The W -algebra minimal model primary fields are shown to arise naturally in these theories, leading to the conjecture that the conformally extended Toda theories provide a lagrangian formulation of the W -algebra minimal models.