Separation of variables for quantum integrable systems on elliptic curves

Felder, Giovanni; Schorr, Anke

doi:10.1088/0305-4470/32/46/302

Cited by 19 publications

(40 citation statements)

References 12 publications

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“…It characterises the set of eigenvalues for which multi-valued analytic solutions can exist in terms of the opers associated to the Lie algebra L g. In all the cases where the Separation of Variables (SOV) approach has been developed it gives a concrete realisation of a correspondence between opers and eigenfunctions of the quantised Hitchin Hamiltonians. This has been fully realised when the surface C has genus g = 0 [11] or g = 1 [22,23,24] with any number of punctures. The SOV approach therefore offers an alternative approach to the geometric Langlands correspondence which is similar to the first construction of such a correspondence due to Drinfeld [25], as has been pointed out in [11].…”

Section: Quantum Separation Of Variablesmentioning

confidence: 99%

Quantization of the Quantum Hitchin System and the Real Geometric Langlands Correspondence

Teschner¹

2018

Geometry and Physics: Volume I

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Main resultsWe are going to propose a natural quantisation condition for the Hitchin system, and explain how it can be reformulated in terms of a function Y(a, t). The function Y(a, t) relevant for this task is found to be the generating function for the variety of opers within the space of all local systems as predicted in [6,9]. However, the condition on Y expressing the quantisation condition turns out to be different from the types of conditions considered in [1]. Our derivation is essentially complete for Hitchin systems associated to the Lie algebra sl 2 in genus 0 and 1, which may be called the Gaudin and elliptic Calogero-Moser models assciated to the group SL(2, C). It reduces to a conjecture of E. Frenkel [11] for g > 1, as will be discussed below.Reformulating the quantisation conditions in terms of Y can be done using the Separation of Variables (SOV) method pioneered by Sklyanin [12]. This method may be seen as a more concrete procedure to construct the geometric Langlands correspondence relating opers to Dmodules (eigenvalue equations), as was pointed out in [11]. In our case it will be found that the SOV method relates single-valued solutions of the eigenvalue equations to opers having real holonomy. This problem is closely related to the classification of projective structures on C with real holonomy which has been studied in [13]. Using complex Fenchel-Nielsen coordinates we will reformulate this description in terms of the generating function for the variety of opers.From the point of view of the geometric Langlands correspondence we obtain a correspondence between opers with real holonomy and D-modules admitting single-valued solutions. We expect that a generalisation to more general local systems with real holonomy will exist. We propose to call such correspondences the real geometric Langlands correspondence. Separation of variables for the classical Hitchin integrable system 2.1 Integrability and special geometryA complex symplectic manifold M with holomorphic symplectic form Ω is called an algebraic integrable system if it can be described as a Lagrangian torus fibration π : M → B with fibres being principally polarised abelian varieties. Algebraic integrability is equivalent to the fact that the base B is a special Kähler manifold satisfying certain integrality conditions [14].These connections may be reformulated conveniently in terms of a covering of M with local charts carrying action-angle coordinates consisting of a tuple a = (a 1 , . . . , a d ) of coordinates for the base B, and complex coordinates z = (z 1 , . . . , z d ) for the torus fibres( 2.3)The transformation z D := τ −1 b · z gives an equivalent representation of the torus fibres Θ b . It

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Section: Quantum Separation Of Variablesmentioning

confidence: 99%

Quantization of the Quantum Hitchin System and the Real Geometric Langlands Correspondence

Teschner¹

2018

Geometry and Physics: Volume I

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“…We use the following property about the elliptic polynomials [13,81] presented below. A character is a group homomorphism χ from multiplicative groups Γ = Z + τ Z to C × .…”

Section: Foda-wheeler-zuparic (Elliptic Felderhof ) Modelmentioning

confidence: 99%

“…The elements of the space Θ N (χ) are called elliptic polynomials. The space Θ N (χ) is Ndimensional [13,81] and the following fact holds for the elliptic polynomials.…”

Section: Foda-wheeler-zuparic (Elliptic Felderhof ) Modelmentioning

confidence: 99%

Scalar products of the elliptic Felderhof model and elliptic Cauchy formula

Motegi

2018

Journal of Geometry and Physics

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We analyze the scalar products of the elliptic Felderhof model introduced by Foda-Wheeler-Zuparic as an elliptic extension of the trigonometric face-type Felderhof model by Deguchi-Akutsu. We derive the determinant formula for the scalar products by applying the Izergin-Korepin technique developed by Wheeler to investigate the scalar products of integrable lattice models. By combining the determinant formula for the scalar products with the recently-developed Izergin-Korepin technique to analyze the wavefunctions, we derive a Cauchy formula for elliptic Schur functions. *

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“…Proposition 2.1. [30,65] Suppose there are two elliptic polynomials P (y) and Q(y) in Θ n (χ), where χ(1) = (−1) n and χ(τ ) = (−1) n e α . If these two polynomials are equal at n points y j , j = 1, .…”

Section: Preliminariesmentioning

confidence: 99%

Elliptic free-fermion model with OS boundary and elliptic Pfaffians

Motegi

2018

Lett Math Phys

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We introduce and study a class of partition functions of an elliptic free-fermionic face model. We study the partition functions with a triangular boundary using the off-diagonal K-matrix at the boundary (OS boundary), which was introduced by Kuperberg as a class of variants of the domain wall boundary partition functions. We find explicit forms of the partition functions with OS boundary using elliptic Pfaffians. We find two expressions based on two versions of Korepin's method, and we obtain an identity between two elliptic Pfaffians as a corollary. * E-mail: kmoteg0@kaiyodai.ac.jp [18,19,20,21,22,23,24,25,26], in which more simplified factorized representations of the partition functions were found, even for elliptic models.Studying elliptic generalizations of the DWBPF is interesting, and it is particularly interesting to find determinant and Pfaffian representations. In particular, finding representations using Pfaffians of a matrix whose matrix entries are elliptic functions is interesting, since there are only a few studies on elliptic Pfaffians. For example, Rosengren [27] introduced a family of elliptic Pfaffians and showed that the partition functions of the Andrews-Baxter-Forrester (ABF) model [28] at the supersymmetric point are expressed as a sum of two elliptic Pfaffians. We mention that expressions of the DWBPF of the ABF model which hold in generic parameters are derived in [29,30,31], a factorized expression at the free-fermion point is derived in [23,24,25]. and a single determinant representation was recently derived in [32].As for the properties of elliptic Pfaffians, Okada [33], Rosengren [34,35] and Rains [36] discovered several elliptic generalizations of the Pfaffian counterpart [37] of the Cauchy determinant formulas. The properties of elliptic determinants have been extensively studied. For example, several generalizations of the Cauchy determinant formula [38] have been discovered [39,40,41,42]. On the other hand, there are only a few results on elliptic Pfaffians by Okada, Rosengren and Rains.In this paper, we study partition functions of an elliptic free-fermionic face model with a triangular boundary, and show that they can be explicitly expressed using elliptic Pfaffians. The face model we treat can be regarded as degenerations of the ABF model [28], Okado-Deguchi-Martin (elliptic Perk-Schultz) model [43,44] and Foda-Wheeler-Zuparic (elliptic Felderhof) model [23], which are face-type counterparts of the elliptic vertex models [45,46], and are elliptic analogues of the trigonometric models of the U q (sl 2 ) six-vertex model, Perk-Schultz model and the Felderhof free-fermion model [47,48,49,50,51,52]. In this paper, we treat a fundamental example of the variations of the DWBPF introduced by Kuperberg [12]. Kuperberg introduced a class of partition functions of the U q (sl 2 ) six-vertex model with a triangular boundary using an off-diagonal boundary K-matrix at the boundary, and showed that they have explicit expressions using Pfaffians. He called this boundary condition the OS boun...

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Separation of variables for quantum integrable systems on elliptic curves

Cited by 19 publications

References 12 publications

Quantization of the Quantum Hitchin System and the Real Geometric Langlands Correspondence

Quantization of the Quantum Hitchin System and the Real Geometric Langlands Correspondence

Scalar products of the elliptic Felderhof model and elliptic Cauchy formula

Elliptic free-fermion model with OS boundary and elliptic Pfaffians

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