Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 036, 28 pages      arXiv:1102.0087
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”

CKP Hierarchy, Bosonic Tau Function and Bosonization Formulae

Johan W. van de Leur a, Alexander Yu. Orlov b and Takahiro Shiota c
a) Mathematical Institute, University of Utrecht, P.O. Box 80010, 3508 TA Utrecht, The Netherlands
b) Nonlinear Wave Processes Laboratory, Oceanology Institute, 36 Nakhimovskii Prospect, Moscow 117851, Russia
c) Mathematics Department, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan

Received January 17, 2012, in final form June 07, 2012; Published online June 22, 2012

We develop the theory of CKP hierarchy introduced in the papers of Kyoto school [Date E., Jimbo M., Kashiwara M., Miwa T., J. Phys. Soc. Japan 50 (1981), 3806-3812] (see also [Kac V.G., van de Leur J.W., Adv. Ser. Math. Phys., Vol. 7, World Sci. Publ., Teaneck, NJ, 1989, 369-406]). We present appropriate bosonization formulae. We show that in the context of the CKP theory certain orthogonal polynomials appear. These polynomials are polynomial both in even and odd (in Grassmannian sense) variables.

Key words: integrable system; Pfaffian; Hafnian; symmetric functions; Schur type functions.

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  1. Aratyn H., van de Leur J.W., The CKP hierarchy and the WDVV prepotential, in Bilinear Integrable Systems: from Classical to Quantum, Continuous to Discrete, NATO Sci. Ser. II Math. Phys. Chem., Vol. 201, Springer, Dordrecht, 2006, 1-11, nlin.SI/0302004.
  2. Date E., Jimbo M., Kashiwara M., Miwa T., Transformation groups for soliton equations. III. Operator approach to the Kadomtsev-Petviashvili equation, J. Phys. Soc. Japan 50 (1981), 3806-3812.
  3. Date E., Jimbo M., Kashiwara M., Miwa T., Transformation groups for soliton equations. VI. KP hierarchies of orthogonal and symplectic type, J. Phys. Soc. Japan 50 (1981), 3813-3818.
  4. Date E., Kashiwara M., Jimbo M., Miwa T., Transformation groups for soliton equations, in Nonlinear Integrable Systems - Classical Theory and Quantum Theory (Kyoto, 1981), World Sci. Publishing, Singapore, 1983, 39-119.
  5. Harnad J., Orlov A.Yu., Fermionic construction of tau functions and random processes, Phys. D 235 (2007), 168-206, arXiv:0704.1157.
  6. Harnad J., van de Leur J.W., Orlov A.Yu., Multiple sums and integrals as neutral BKP tau functions, Theoret. and Math. Phys. 168 (2011), 951-962, arXiv:1101.4216.
  7. Ishikawa M., Kawamuko H., Okada S., A Pfaffian-Hafnian analogue of Borchardt's identity, math.CO/0408364.
  8. Kac V., Vertex algebras for beginners, University Lecture Series, Vol. 10, 2nd ed., American Mathematical Society, Providence, RI, 1998.
  9. Kac V.G., van de Leur J.W., Super boson-fermion correspondence of type B, in Infinite-Dimensional Lie Algebras and Groups (Luminy-Marseille, 1988), Adv. Ser. Math. Phys., Vol. 7, World Sci. Publ., Teaneck, NJ, 1989, 369-406.
  10. Kac V.G., van de Leur J.W., The n-component KP hierarchy and representation theory, J. Math. Phys. 44 (2003), 3245-3293, hep-th/9308137.
  11. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995.
  12. Nimmo J.J.C., Hall-Littlewood symmetric functions and the BKP equation, J. Phys. A: Math. Gen. 23 (1990), 751-760.
  13. Orlov A.Yu., Hypergeometric functions related to Schur Q-polynomials and the BKP equation, Theoret. and Math. Phys. 137 (2003), 1574-1589, math-ph/0302011.
  14. Orlov A.Yu., Shiota T., Takasaki K., Pfaffian structures and certain solutions to BKP hierarchies. I. Sums over partitions, arXiv:1201.4518.
  15. Orlov A.Yu., Scherbin D.M., Multivariate hypergeometric functions as τ-functions of Toda lattice and Kadomtsev-Petviashvili equation, Phys. D 152/153 (2001), 51-65, math-ph/0003011.
  16. Sato M., Soliton equations as dynamical systems on an infinite dimensional Grassmann manifolds, in Random Systems and Dynamical Systems (Kyoto, 1981), RIMS Kokyuroku, Vol. 439, Kyoto, 1981, 30-46.
  17. Takasaki K., Initial value problem for the Toda lattice hierarchy, in Group Representations and Systems of Differential Equations (Tokyo, 1982), Adv. Stud. Pure Math., Vol. 4, North-Holland, Amsterdam, 1984, 139-163.
  18. van de Leur J.W., Orlov A.Yu., Random turn walk on a half line with creation of particles at the origin, Phys. Lett. A 373 (2009), 2675-2681, arXiv:0801.0066.
  19. You Y., Polynomial solutions of the BKP hierarchy and projective representations of symmetric groups, in Infinite-Dimensional Lie Algebras and Groups (Luminy-Marseille, 1988), Adv. Ser. Math. Phys., Vol. 7, World Sci. Publ., Teaneck, NJ, 1989, 449-464.

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