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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adler, M., van Moerbeke, P.: Symmetric random matrices and the Pfaff lattice. arXiv:solv-int/9903009v1
Adler, M., van Moerbeke, P., Shiota, T.: Pfaff τ-functions. Math. Ann. 322, 423–476 (2002). arXiv:nlin/9909010
Chau L.-L., Zaboronsky O.: On the structure of correlation functions in the normal matrix models. Commun. Math. Phys. 196, 203–247 (1998)
Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations. In: Jimbo, M., Miwa, T. (eds.) Nonlinear integrable systems—classical theory and quantum theory, pp. 39–120. World Scientific (1983)
Date E., Jimbo M., Kashiwara M., Miwa T.: Transformation groups for soliton equations, IV. A new hierarchy of soliton equations of KP-type. Physica D 4, 343–365 (1982)
Gerasimov A., Marshakov A., Mironov A., Morozov A., Orlov A.: Matrix models of two-dimensional gravity and Toda theory. Nucl. Phys. B 357(2–3), 565–618 (1991)
Kharchev K., Marshakov A., Mironov A., Orlov A., Zabrodin A.: Matrix models among integrable theories: Forced hierarchies and operator formalism. Nucl. Phys. B 366, 569–601 (1991)
Kharchev S., Marshakov A., Mironov A., Morozov A., Pakuliak S.: Conformal matrix models as an alternative to conventional multi-matrix models. Nucl. Phys. B 404, 717–750 (1993)
Itzykson C., Zuber J.B.: The planar approximation II. J. Math. Phys. 21, 411–421 (1980)
Jimbo M., Miwa T.: Solitons and infinite dimensional lie algebras. Publ. RIMS Kyoto Univ. 19, 943–1001 (1983)
Hall, M.J.W.: Random quantum correlations and density operator distributions. Phys. Lett. A. 242, 123–129 (1998). arXiv:quant-ph/9802052
Harnad, J., Orlov, A.Y.: Fermionic construction of partition functions for two-matrix models and perturbative Schur function expansions. J. Phys. A Math. Gen. 39(28), 8783 (2006)
Hirota R., Ohta Y.: Hierarchies of coupled soliton equations. I. J. Phys. Soc. Japan 60(3), 798–809 (1991)
Kac V., van de Leur J.: The n-component KP hierarchy and representation theory. J. Math. Phys. 44, 3245–3293 (2003)
Kac V., van de Leur J.: The geometry of spinors and the multicomponent BKP and DKP hierarchies. CRM Proc. Lect. Notes 14, 159–202 (1998)
van de Leur J.W.: Matrix integrals and geometry of spinors. J. Nonlinear Math. Phys. 8, 288–311 (2001)
van de Leur, J.W., Yu. Orlov, A.: Random turn walk on a half line with creation of particles at the origin. Phys. Lett. A. pp. 2675–2681 (2009). arXiv:0801.0066 [math-ph]
van de Leur, J.W., Yu. Orlov, A.: Multicomponent BKP tau function and multicomponent KP tau function. Russ. J. Theor. Math. Phys. (2015)
Macdonald, I.G.: Symmetric functions and hall polynomials. Clarendon Press, Oxford (1995)
Mehta, M.L.: Random matrices, 3rd edn. Elsevier, Academic, San Diego (2004)
Mineev-Weinstein M., Wiegmann P., Zabrodin A.: Integrable structure of interface dynamics. Phys. Rev. Lett. 84, 5106–5109 (2000)
Mironov, A., Morozov, A., Semenoff, G.: Unitary matrix integrals in the framework of the generalized kontsevich model. Int. J. Mod. Phys. A. 11, 5031–5080 (1996)
Osipov, V.A., Sommers, H.-J., Zyczkowski, K.: J. Phys. A Math. Theor. 43, 055302 (2010). arXiv:0909.5094
Yu. Orlov, A.: Deformed Ginibre ensembles and integrable systems. Phys. Lett. A. 378, 319–328 (2014)
Yu. Orlov, A., Shiota, T., Takasaki, K.: Pfaffian structures and certain solutions to BKP hierarchies I. Sums over partitions. arXiv:1201.4518
Yu. Orlov, A., Shiota, T., Takasaki, K.: Pfaffian structures and certain solutions to BKP hierarchies II. Multiple integrals. Russ. J. Theor. Math. Phys. (2015)
Sato, M., Sato, Y.: Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold. Nonlinear partial differential equations in applied science (Tokyo, 1982), vol. 81, pp. 259–271. North-Holland Math. Stud., North-Holland (1983)
Takasaki K.: Initial value problem for the Toda lattice hierarchy. Adv. Stud. Pure Math. 4, 139–163 (1984)
Takasaki, K.: Auxilary linear problem, difference Fay identities and dispersionless limit of Pfaff-Toda hierarchy. SIGMA. 5, 109 (2009). arXiv:0908.3569
Takebe T.: Representation theoretical meaning of initial value problem for the Toda lattice hierarchy I. LMP 21, 77–84 (1991)
Takebe T.: Representation theoretical meaning of initial value problem for the Toda lattice hierarchy II. Publ. RIMS Kyoto Univ. 27, 491–503 (1991)
Ueno K., Takasaki K.: Toda lattice hierarchy. Adv. Stud. Pure Math. 4, 1–95 (1984)
Ya. Vilenkin, N., Klimyk, A.U.: Representation of lie groups and special functions. Volume 3: classical and quantum groups and special functions. Kluwer Academic Publishers (1992)
You, Y.: Polynomial solutions of the BKP hierarchy and projective representations of symmetric groups. Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), Adv. Ser. Math. Phys., vol. 7, pp. 449–464. World Sci. Publ., Teaneck (1989)
Zinn-Justin, P.: HCIZ integral and 2D Toda lattice hierarchy. Nucl. Phys. B. 634(3), 417–432 (2002)
Zinn-Justin, P., Zuber, J.B.: On some integrals over the U(N) unitary group and their large limit. arXiv:math-ph/0209019
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
van de Leur, J.W., Orlov, A.Y. Pfaffian and Determinantal Tau Functions. Lett Math Phys 105, 1499–1531 (2015). https://doi.org/10.1007/s11005-015-0786-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-015-0786-6