q-Opers, QQ-Systems, and Bethe Ansatz (original) (raw)
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q-Opers, QQ-systems, and Bethe Ansatz II: Generalized Minors
2021
In this paper, we describe a certain kind of q-connections on a projective line, namely Z-twisted (G, q)-opers with regular singularities using the language of generalized minors. In part one [FKSZ] we explored the correspondence between these q-connections and QQ-systems/Bethe Ansatz equations. Here we associate to a Z-twisted (G, q)-oper a class of meromorphic sections of a G-bundle, satisfying certain difference equations, which we refer to as generalized q-Wronskians. Among other things, we show that the QQsystems and their extensions emerge as the relations between generalized minors, thereby putting the Bethe Ansatz equations in the framework of cluster mutations known in the theory of double Bruhat cells.
q-opers, QQ-systems, and Bethe Ansatz II: Generalized minors
Crelle's Journal, 2023
In this paper, we describe a certain kind of q-connections on a projective line, namely Z-twisted (G, q)-opers with regular singularities using the language of generalized minors. In part one [FKSZ] we explored the correspondence between these q-connections and QQ-systems/Bethe Ansatz equations. Here we associate to a Z-twisted (G, q)-oper a class of meromorphic sections of a G-bundle, satisfying certain difference equations, which we refer to as (G, q)-Wronskians. Among other things, we show that the QQ-systems and their extensions emerge as the relations between generalized minors, thereby putting the Bethe Ansatz equations in the framework of cluster mutations known in the theory of double Bruhat cells. Contents 1. Introduction 1 2. Group-theoretic data 4 3. Miura (G, q)-opers with regular singularities and QQ-systems 4 4. Z-twisted q-opers and (G, q)-Wronskians 12 References 22
Communications in Mathematical Physics, 2020
A special case of the geometric Langlands correspondence is given by the relationship between solutions of the Bethe ansatz equations for the Gaudin model and opers-connections on the projective line with extra structure. In this paper, we describe a deformation of this correspondence for SL(N). We introduce a difference equation version of opers called q-opers and prove a q-Langlands correspondence between nondegenerate solutions of the Bethe ansatz equations for the XXZ model and nondegenerate twisted qopers with regular singularities on the projective line. We show that the quantum/classical duality between the XXZ spin chain and the trigonometric Ruijsenaars-Schneider model may be viewed as a special case of the q-Langlands correspondence. We also describe an application of q-opers to the equivariant quantum K-theory of partial flag varieties.
KZ equation, G-opers, quantum Drinfeld–Sokolov reduction, and quantum Cayley–Hamilton identity
Journal of Mathematical Sciences, 2009
The Lax operator of the Gaudin type models is a 1-form on the classical level. In virtue of the quantization scheme proposed in [Talalaev04] (hep-th/0404153) it is natural to treat the quantum Lax operator as a connection; this connection is a particular case of the Knizhnik-Zamolodchikov connection [ChervovTalalaev06] (hep-th/0604128). In this paper we find a gauge transformation which produces the "second normal form" or the "Drinfeld-Sokolov" form. Moreover the differential operator naturally corresponding to this form is given precisely by the quantum characteristic polynomial [Talalaev04] of the Lax operator (this operator is called the G-oper or Baxter equation). This observation allows to relate solutions of the KZ and Baxter equations in an obvious way, and to prove that the immanent KZ-equations has only meromorphic solutions. As a corollary we obtain the quantum Cayley-Hamilton identity for the Gaudin-type Lax operators (including the general gl n [t] case). The presented construction sheds a new light on a geometric Langlands correspondence. We also discuss the relation with the Harish-Chandra homomorphism.
Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence
Arxiv preprint hep-th/0604128, 2006
The spectral curve is the key ingredient in the modern theory of classical integrable systems. We develop a construction of the "quantum spectral curve" and argue that it takes the analogous structural and unifying role on the quantum level also. In the simplest but essential case the "quantum spectral curve" is given by the formula "det"(L(z) − ∂ z) [Talalaev04] (hep-th/0404153). As an easy application of our constructions we obtain the following: quite a universal recipe to define quantum commuting hamiltonians from classical ones, in particular an explicit description of a maximal commutative subalgebra in U(gl n [t])/t N and in U(gl n [t −1 ]) ⊗ U(tgl n [t]); the relation (isomorphism) of the constructed commutative subalgebra with the center on the critical level of U(ĝl n) by the AKS-type arguments; an explicit formula for the center generators and conjecture on W-algebra generators; a recipe to obtain the q-deformation of these results; a simple and explicit construction of the Langlands correspondence and new points of view on its higher-dimensional generalization and relation to "D-connections"; a relation between the "quantum spectral curve" and the Knizhnik-Zamolodchikov equation; new generalizations of the KZ-equation; a conjecture on rationality of the solutions of the KZ-equation for special values of level. In simplest cases we observe coincidence of the "quantum spectral curve" and the socalled Baxter equation, our results provide a general construction of the Baxter equation. Connection with the KZ-equation offers a new powerful way to construct the Baxter's Q-operator. Generalizing the known observations on the connection between the Baxter equation and the Bethe ansatz we formulate a conjecture relating the spectrum of the underlying integrable model and the properties of the "quantum spectral curve". Our results are deeply related with the Sklyanin's approach to separation of variables.
Hamiltonian reduction and the construction of q -deformed extensions of the Virasoro algebra
Journal of Physics A: Mathematical and General, 1998
In this paper we employ the construction of Dirac bracket for the remaining current of sl(2) q deformed Kac-Moody algebra when constraints similar to those connecting the sl(2)-WZW model and the Liouville theory are imposed and show that it satisfy the q-Virasoro algebra proposed by Frenkel and Reshetikhin. The crucial assumption considered in our calculation is the existence of a classical Poisson bracket algebra induced, in a consistent manner by the correspondence principle, mapping the quantum generators into commuting objects of classical nature preserving their algebra. 1 Supported by FAPESP 2 Work partially supported by CNPq 3 Supported by CNPq
The Canonical Solutions of the Q -Systems¶ and the Kirillov-Reshetikhin Conjecture
Communications in Mathematical Physics, 2002
We study a class of systems of functional equations closely related to various kinds of integrable statistical and quantum mechanical models. We call them the finite and infinite Q-systems according to the number of functions and equations. The finite Q-systems appear as the thermal equilibrium conditions (the Sutherland-Wu equation) for certain statistical mechanical systems. Some infinite Q-systems appear as the relations of the normalized characters of the KR modules of the Yangians and the quantum affine algebras. We give two types of power series formulae for the unique solution (resp. the unique canonical solution) for a finite (resp. infinite) Q-system. As an application, we reformulate the Kirillov-Reshetikhin conjecture on the multiplicities formula of the KR modules in terms of the canonical solutions of Q-systems.
A generalizedQ-operator forUqvertex models
Journal of Physics A: Mathematical and General, 2002
In this paper, we construct a Q-operator as a trace of a representation of the universal R-matrix of U q (sl 2) over an infinite-dimensional auxiliary space. This auxiliary space is a four-parameter generalization of the q-oscillator representations used previously. We derive generalized T-Q relations in which 3 of these parameters shift. After a suitable restriction of parameters, we give an explicit expression for the Q-operator of the 6-vertex model and show the connection with Baxter's expression for the central block of his corresponding operator. His approach was to start with the 6-vertex model Bethe ansatz, and to derive certain functional relations between the transfer matrix T (v) and a matrix Q(v)-the elements of both matrices being entire functions. He went on to show that the reverse argument could be used in order to start from the functional relations (and some other properties of T (v) and Q(v)) and derive the Bethe equations. He then considered the 8-vertex model, constructed a Q(v) operator that obeyed the correct requirements, and used the reverse argument to derive Bethe equations. The approach is described clearly in Baxter's book [9]. Later on in the 70s, the quantum inverse scattering method (QISM) was developed and used to produce a rather simpler derivation of the same Bethe equations for the 8-vertex model (the algebraic Bethe ansatz approach) [10]. Baxter also invented his corner transfer matrix technique for the 8-vertex model [9]. So, remarkably successful though it was, the Q-operator approach perhaps came to be considered by many as a historical curiosity. However, in the last few years there has been something of a revival of interest in Q. The reasons for this include the following: • Some understanding has been obtained into how Q fits into the QISM/quantum-groups picture of solvable lattice models [11, 12, 13, 14, 15, 16, 17, 18]. • The discovery of the mysterious ODE/IM models correspondence-relating functional relations obeyed by the solutions and spectral determinants of certain ODEs to Bethe ansatz functional relations [19, 20, 21]. • The role of Q in classical integrable systems as a generator of Backlünd transformation has been understood in certain cases (see [22] and references therein). In this paper, we are concerned with the first point. The key to the QISM approach to solvable lattice models is to understand them in terms of an underlying algebra A. The generators of A are matrix elements L ij (z), where i, j ∈ {0, 1} (in the simplest case) and z is a spectral parameter. The set of relations amongst the generators are given by the matrix relation R(z/z ′)L 1 (z)L 2 (z ′) = L 2 (z ′)L 1 (z)R(z/z ′), (1.1) where L 1 (z) = L(z) ⊗ 1, L 2 (z) = 1 ⊗ L(z), and R(z) is a 4 × 4 matrix. This QISM description was later refined in terms of quantum groups. In this picture A is recognised as a quasi-triangular Hopf algebra (aka a quantum group). For the vertex models of the title, the algebra A is U q (sl 2). Families of R-matrices and L-operators are then all given in terms of representations of a universal R-matrix R ∈ U q (b +) ⊗ U q (b −), where U q (b ±) are two Borel subalgebras of U q (sl 2). The relevant U q (sl 2) representations are the spin-n/2 evaluation representations (π (n) z , V (n) z) defined in Section 3 (in this paper, a representation of an algebra A is specified by a pair (π, V), consisting of an A module V and the associated map π : A → End(V)). Then we have R(z/z ′) ≡ (π (1) z ⊗ π (1) z ′)R, L(z) ≡ (π (1) z ⊗ 1)R,
The Canonical Solutions of the Q-Systems¶ and the Kirillov–Reshetikhin Conjecture
Communications in Mathematical Physics, 2002
We study a class of systems of functional equations closely related to various kinds of integrable statistical and quantum mechanical models. We call them the finite and infinite QQQ-systems according to the number of functions and equations. The finite Q-systems appear as the thermal equilibrium conditions (the Sutherland–Wu equation) for certain statistical mechanical systems. Some infinite Q-systems appear as the relations of the normalized characters of the KR modules of the Yangians and the quantum affine algebras. We give two types of power series formulae for the unique solution (resp. the unique canonical solution) for a finite (resp. infinite) Q-system. As an application, we reformulate the Kirillov–Reshetikhin conjecture on the multiplicities formula of the KR modules in terms of the canonical solutions of Q-systems.
Quantization of Lie bialgebras, Part IV: The coinvariant construction and the quantum KZ equations
Selecta Mathematica, 2000
This paper is a continuation of . In [EK3], we introduced the Hopf algebra F (R) z associated to a quantum R-matrix R(z) with a spectral parameter defined on a 1-dimensional connected algebraic group Σ, and a set of points z = (z 1 , . . . , z n ) ∈ Σ n . This algebra is generated by entries of a matrix power series T i (u), i = 1, . . . , n, subject to Faddeev-Reshetikhin-Takhtajan type commutation relations, and is a quantization of the group GL N [[t]] n .