The geometry of spinors and the
                    multicomponent BKP and DKP hierarchies

Kač, Victor G.; Leur, Johan van de

doi:10.1090/crmp/014/13

Cited by 96 publications

(225 citation statements)

References 14 publications

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“…Using results of [9] and doing similar calculations in the framework of BKP hierarchy constructed in [6] (see Appendix) one can show that the operator e H(s) applied to the left vacuum generates all basis left Fock vectors, quite similar eH (s) applied to the right vacuum vector generates all basis right Fock vectors. Namely, we have the following left and right coherent states (see also (6) and (7))…”

Section: Relation To the Bkp Tau Functionmentioning

confidence: 98%

“…The hierarchy of of Kadomtsev-Petviashvili equations of type B (the BKP hierarchy) was introduced in [5]. As we have already mentioned we use its modification suggested in [6]. These is a semi-infinite set of compatible nonlinear differential equations which may be viewed as a set of commutative time flows.…”

Section: Relation To the Bkp Tau Functionmentioning

confidence: 99%

“…Expression (57) where U may be chosen as an arbitrary set of numbers, provides an example of the BKP tau function constructed in [6], where s = (s 1 , s 3 , s 5 . .…”

Section: Relation To the Bkp Tau Functionmentioning

confidence: 99%

“…We follow [6], where the action of φ 0 (which is not a number) is different. See Appendix for some more details.…”

Section: Introductionmentioning

confidence: 99%

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Random turn walk on a half line with creation of particles at the origin

Leur¹,

Orlov²

2009

Physics Letters A

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We consider a version of random motion of hard core particles on the semi-lattice 1, 2, 3, . . ., where in each time instant one of three possible events occurs, viz., (a) a randomly chosen particle hops to a free neighboring site, (b) a particle is created at the origin (namely, at site 1) provided that site 1 is free and (c) a particle is eliminated at the origin (provided that the site 1 is occupied). Relations to the BKP equation are explained. Namely, the tau functions of two different BKP hierarchies provide generating functions respectively (I) for transition weights between different particle configurations and (II) for an important object: a normalization function which plays the role of the statistical sum for our non-equilibrium system. As an example we study a model where the hopping rate depends on two parameters (r and β). For time T → ∞ we obtain the asymptotic configuration of particles obtained from the initial empty state (the state without particles) and find an analog of the first order transition at β = 1.

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Section: Relation To the Bkp Tau Functionmentioning

confidence: 98%

Section: Relation To the Bkp Tau Functionmentioning

confidence: 99%

“…Expression (57) where U may be chosen as an arbitrary set of numbers, provides an example of the BKP tau function constructed in [6], where s = (s 1 , s 3 , s 5 . .…”

Section: Relation To the Bkp Tau Functionmentioning

confidence: 99%

“…We follow [6], where the action of φ 0 (which is not a number) is different. See Appendix for some more details.…”

Section: Introductionmentioning

confidence: 99%

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Random turn walk on a half line with creation of particles at the origin

Leur¹,

Orlov²

2009

Physics Letters A

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“…The Pfaff lattice of Adler and van Moerbeke [1,2] was discovered by Jimbo and Miwa ( [10], section 7) it was rediscovered by Hirota and Ohta [13] as the coupled KP hierarchy and studied in a paper by Kac and one of the authors [15] as the charged DKP hierarchy. The Pfaff lattice studied in [16] is slightly bigger than the one studied by Adler, Shiota and van Moerbeke in [2] it is the charged BKP hierarchy of [15], called the large BKP in [25] to make difference to the small BKP of [5].…”

Section: Introductionmentioning

confidence: 99%

Pfaffian and Determinantal Tau Functions

Leur

Orlov

2015

Lett Math Phys

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Abstract. Adler, Shiota and van Moerbeke observed that a tau function of the Pfaff lattice is a square root of a tau function of the Toda lattice hierarchy of Ueno and Takasaki. In this paper, we give a representation theoretical explanation for this phenomenon. We consider 2-BKP and two-component 2-KP tau functions. We shall show that a square of a BKP tau function is equal to a certain two-component KP tau function and a square of a 2-BKP tau function is equal to a certain two-component 2-KP tau function.Mathematics Subject Classification. 17B65, 17B80, 35Q51, 35Q53, 35Q55.

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Links Between Quantum Chaos and Counting Problems

Yurievich

2019

Trends in Mathematics

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We show that Hurwitz numbers may be generated by certain correlation functions which appear in quantum chaos.First, in short we present two different topics: Hurwitz numbers which appear in counting of branched covers of Riemann and Klein surfaces, and the study of spectral correlation functions of products of random matrices which belong to independent (complex) Ginibre ensembles.There are a lot of studies on extracting information about Hurwitz numbers, on the one hand side, from integrable systems, as it was done in [51], [52], [21] and further developed in [43], [44], [6], [7], [23], [48], [27], [66], [15], [18], [49] (see also reviews [29] and [33]) and from matrix integrals [39], [22], [34] on the other hand. (Actually the point that there is a special family of tau functions which were introduced in [35] and in [55] and studied in [56], [25], [53], [24], [59], [58], [26] where links with matrix models were written down which describes perturbation series in coupling constants of a number of matrix models, and these very tau functions, called hypergeometric ones, count also special types of Hurwitz numbers. This article is based on [48], [59] and [54] and it was the content of my talk in Bialowzie meeting "XXXVI Workshop in Geometric Method in Physics, 3-8 June 2017". In the last paper we put known results in quantum chaos [1], [2], [3], The results of our the work should be compared to ones obtained in [31], [5] and [12].

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The geometry of spinors and the multicomponent BKP and DKP hierarchies

Abstract: We develop a formalism of multicomponent BKP hierarchies using elementary geometry of spinors. The multicomponent KP and the modified KP hierarchy (hence all their reductions like KdV, NLS, AKNS or DS) are reductions of the multicomponent BKP.

Cited by 96 publications

References 14 publications

Random turn walk on a half line with creation of particles at the origin

Random turn walk on a half line with creation of particles at the origin

Pfaffian and Determinantal Tau Functions

Links Between Quantum Chaos and Counting Problems

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