Abstract
In this paper, we construct the quantum torus symmetry of the KP hierarchy and further derive the quantum torus constraint on the tau function of the KP hierarchy. That means we give a nice representation of the quantum torus Lie algebra in the KP system by acting on its tau function. Comparing to the W ∞ symmetry, this quantum torus symmetry has a nice algebraic structure with double indices. Further by reduction, we also construct the quantum torus symmetries of the KdV and BKP hierarchies and further derive the quantum torus constraints on their tau functions. These quantum torus constraints might have applications in the quantum field theory, supersymmetric gauge theory and so on.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dickey L.A.: Soliton Equations and Hamiltonian Systems, 2nd edn. World Scientific, Singapore (2003)
Orlov A.Yu., Schulman E.I.: Additional symmetries of integrable equations and conformal algebra representation. Lett. Math. Phys. 12, 171–179 (1986)
He J.S., Tian K.L., Forester A., Ma W.X.: Additional symmetries and string equation of the CKP hierarchy. Lett. Math. Phys. 81, 119–134 (2007)
Tian K.L., He J.S., Cheng J.P., Cheng Y.: Additional symmetries of constrained CKP and BKP hierarchies. Sci. China Math. 54, 257–268 (2011)
Li M.H., Li C.Z. et al.: Virasoro type algebraic structure hidden in the constrained discrete Kadomtsev–Petviashvili hierarchy. J. Math. Phys. 54, 043512 (2013)
Block R.: On torsion-free abelian groups and Lie algebras. Proc. Am. Math. Soc. 9, 613–620 (1958)
Dokovic D., Zhao K.: Derivations, isomorphisms and second cohomology of generalized Block algebras. Algebra Colloq. 3, 245–272 (1996)
Su Y.: Quasifinite representations of a Lie algebra of Block type. J. Algebra 276, 117–128 (2004)
Chen, M.R., et al.: W 1+∞ 3-algebra and KP hierarchy. arXiv:1309.4627
Kemmoku R., Saito S.: Discretization of Virasoro algebra. Phys. Lett. B 319, 471–477 (1993)
Mas J., Seco M.: The algebra of q-pseudodifferential symbols and the \({q-W_{KP}^{(n)}}\) algebra. J. Math. Phys. 37, 6510 (1996)
Li C.Z., He J.S., Su Y.C.: Block type symmetry of bigraded Toda hierarchy. J. Math. Phys. 53, 013517 (2012)
Li C.Z., He J.S., Wu K., Cheng Y.: Tau function and Hirota bilinear equations for the extended bigraded Toda hierarchy. J. Math. Phys. 51, 043514 (2010)
Li C.Z.: Solutions of bigraded Toda hierarchy. J. Phys. A 44, 255201 (2011)
Li C.Z., He J.S.: Dispersionless bigraded Toda hierarchy and its additional symmetry. Rev. Math. Phys. 24, 1230003 (2012)
Li, C.Z., He, J.S.: Block algebra in two-component BKP and D type Drinfeld–Sokolov hierarchies. J. Math. Phys. 54, 113501 (2013). arXiv:1210.6498
Fairlie D.B., Fletcher P., Zachos C.K.: Trigonometric structure constants for new infinite-dimensional algebras. Phys. Lett. B 218, 203 (1989)
Maeda T., Nakatsu T., Takasaki K., Tamakoshi T.: Free fermion and Seiberg–Witten differential in random plane partitions. Nucl. Phys. B 715, 275 (2005)
Maeda T., Nakatsu T.: Amoebas and instantons. Int. J. Mod. Phys. A 22, 937 (2007)
Nakatsu T., Takasaki K.: Melting crystal, quantum Torus and Toda hierarchy. Commun. Math. Phys. 285, 445–468 (2009)
Dickey L.A.: Lectures on classical W-algebras. Acta Appl. Math. 47, 243–321 (1997)
Witten E.: Two-dimensional gravity and intersection theory on moduli space. Surv. Differ. Geom. 1, 243–310 (1991)
Kontsevich M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147, 1–23 (1992)
Douglas M.: Strings in less than one dimension and the generalized KdV hierarchies. Phys. Lett. B 238, 176–180 (1990)
Date, E., Kashiwara, M., Jimbo, M., Miwa, T.: Transformation groups for soliton equations. In: Nonlinear Integrable Systems—Classical Theory and Quantum Theory (Kyoto, 1981), 39–119. World Scientific Publishing, Singapore (1983)
Takasaki K.: Quasi-classical limit of BKP hierarchy and W-infinity symmetries. Lett. Math. Phys. 28, 177–185 (1993)
Tu M.H.: On the BKP hierarchy: additional symmetries, Fay identity and Adler-Shiota-van Moerbeke formula. Lett. Math. Phys. 81, 93–105 (2007)
Aoyama S., Kodama Y.: A generalized Sato equation and the W ∞ algebra. Phys. Lett. B 278, 56–62 (1992)
Adler M., Shiota T., Moerbeke P.: From the w ∞-algebra to its central extension: a τ-function approach. Phys. Lett. A 194, 33–43 (1994)
Adler M., Shiota T., Moerbeke P.: A Lax representation for the vertex operator and the central extension. Commun. Math. Phys. 171, 547–588 (1995)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, C., He, J. Quantum Torus Symmetry of the KP, KdV and BKP Hierarchies. Lett Math Phys 104, 1407–1423 (2014). https://doi.org/10.1007/s11005-014-0716-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-014-0716-z