Quantum Integrable Models and Discrete Classical Hirota Equations

Krichever, I. M.; Lipan, Ovidiu; Wiegmann, P.; Zabrodin, A.

doi:10.1007/s002200050165

Cited by 246 publications

(372 citation statements)

References 53 publications

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“…In fact, there as been renewed interest in the HBDE (9) for its relationship to quantum field theories and in relating quantum to classical integrable systems [20,38]. In keeping with this trend, we find it useful to describe the HBDE not as a consequence of the analytic theory of the KP hierarchy, but as a natural consequence of the algebraic geometry of the Grassmannian itself.…”

Section: The Geometry Of the Hirota Bilinear Difference Equationmentioning

confidence: 77%

On KP generators and the geometry of the HBDE

Gekhtman

Kasman

2006

Journal of Geometry and Physics

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Abstract. Sato theory provides a correspondence between solutions to the KP hierarchy and points in an infinite dimensional Grassmannian. In this correspondence, flows generated infinitesimally by powers of the "shift" operator give time dependence to the first coordinate of an arbitrarily selected point, making it a tau-function. These tau-functions satisfy a number of integrable equations, including the Hirota Bilinear Difference Equation (HBDE). Here, we rederive the HBDE as a statement about linear maps between Grassmannians. In addition to illustrating the fundamental nature of this equation in the standard theory, we make use of this geometric interpretation of the HBDE to answer the question of what other infinitesimal generators could be used for similarly creating tau-functions. The answer to this question involves a "rank one condition", tying this investigation to the existing results on integrable systems involving such conditions and providing an interpretation for their significance in terms of the relationship between the HBDE and the geometry of Grassmannians.

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Section: The Geometry Of the Hirota Bilinear Difference Equationmentioning

confidence: 77%

On KP generators and the geometry of the HBDE

Gekhtman

Kasman

2006

Journal of Geometry and Physics

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“…Moreover, any problem solvable by the Bethe ansatz method is essentially of the type just described as had been demonstrated rigorously by Gutkin [71]. Therefore, not surprisingly, that there are deep connections between the classical exactly integrable systems of KP type and the quantum mechanical exactly integrable systems solved by the Bethe ansatz method [72]. The Hecke algebra leading to the Yang-Baxter equations providing mathematical justification of the Bethe ansatz method is coming from some particular representation of the symmetric group [73] and, hence, is connected with Schur and related polynomials.…”

Section: Organization Of the Rest Of This Papermentioning

confidence: 84%

Kontsevich–Witten model from 2+1 gravity: new exact combinatorial solution

Kholodenko¹

2002

Journal of Geometry and Physics

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In previous publications ( J. Geom. Phys.38 (2001) 81-139 and references therein ) the partition function for 2+1 gravity was constructed for the fixed genus Riemann surface.With help of this function the dynamical transition from pseudo-Anosov to periodic (Seifert-fibered) regime was studied. In this paper the periodic regime is studied in some detail in order to recover major results of Kontsevich (Comm.Math.Phys. 147 (1992) 1-23 ) inspired by earlier work of Witten on topological two dimensional quantum gravity.To achieve this goal some results from enumerative combinatorics have been used. The logical developments are extensively illustrated using geometrically convincing figures. This feature is helpful for development of some non traditional applications (mentioned through the entire text) of obtained results to fields other than theoretical particle physics.

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“…For example, T(A r ) is a discrete analogue of the Toda field equation and a particular case of the Hirota's bilinear difference equation [Hi1,Hi2,KOS,KLWZ]. See [KLWZ] for more information.…”

Section: Restricted T and Y-systems And Their Periodicitiesmentioning

confidence: 99%

Periodicities of T-systems and Y-systems

Inoue¹,

Iyama²,

Kuniba³

et al. 2010

Nagoya Math. J.

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Abstract. The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of the Yangian or the quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for types A and C, and the direct method for types A, D, and B (level 2).

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Quantum Integrable Models and Discrete Classical Hirota Equations

Cited by 246 publications

References 53 publications

On KP generators and the geometry of the HBDE

On KP generators and the geometry of the HBDE

Kontsevich–Witten model from 2+1 gravity: new exact combinatorial solution

Periodicities of T-systems and Y-systems

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