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

SIGMA 6 (2010), 043, 25 pages      arXiv:0909.3109

Supersymmetry of Affine Toda Models as Fermionic Symmetry Flows of the Extended mKdV Hierarchy

David M. Schmidtt
Instituto de Física Teórica, UNESP-Universidade Estadual Paulista, Caixa Postal 70532-2, 01156-970, São Paulo, SP, Brasil

Received December 10, 2009, in final form May 19, 2010; Published online May 27, 2010

We couple two copies of the supersymmetric mKdV hierarchy by means of the algebraic dressing technique. This allows to deduce the whole set of (N,N) supersymmetry transformations of the relativistic sector of the extended mKdV hierarchy and to interpret them as fermionic symmetry flows. The construction is based on an extended Riemann-Hilbert problem for affine Kac-Moody superalgebras with a half-integer gradation. A generalized set of relativistic-like fermionic local current identities is introduced and it is shown that the simplest one, corresponding to the lowest isospectral times t±1 provides the supercharges generating rigid supersymmetry transformations in 2D superspace. The number of supercharges is equal to the dimension of the fermionic kernel of a given semisimple element E ∈ ^g which defines both, the physical degrees of freedom and the symmetries of the model. The general construction is applied to the N=(1,1) and N=(2,2) sinh-Gordon models which are worked out in detail.

Key words: algebraic dressing method; supersymmetry flows; supersymmetric affine Toda models.

pdf (372 kb)   ps (249 kb)   tex (33 kb)


  1. Aratyn H., Gomes J.F., de Castro G.M., Silka M.B., Zimerman A.H., Supersymmetry for integrable hierarchies on loop superalgebras, J. Phys. A: Math. Gen. 38 (2005), 9341-9357, hep-th/0508008.
  2. Aratyn H., Gomes J.F., Zimerman A.H., Integrable hierarchy for multidimensional Toda equations and topological-anti-topological fusion, J. Geom. Phys. 46 (2003), 21-47, Erratum, J. Geom. Phys. 46 (2003), 201, hep-th/0107056.
  3. Aratyn H., Gomes J.F., Zimerman A.H., Supersymmetry and the KdV equations for integrable hierarchies with a half-integer gradation, Nuclear Phys. B 676 (2004), 537-571, hep-th/0309099.
  4. Aratyn H., Gomes J.F., Zimerman A.H., Nisimov E., Pacheva S., Symmetry flows, conservation laws and dressing approach to the integrable models, in Integrable Hierarchies and Modern Physical Theories (Chicago, 2000), NATO Sci. Ser. II Math. Phys. Chem., Vol. 18, Kluwer Acad. Publ., Dordrecht, 2001, 243-275, nlin.SI/0012042.
  5. Au G., Spence B., Hamiltonian reduction and supersymmetric Toda models, Modern Phys. Lett. A 10 (1995), 2157-2168, hep-th/9505026.
  6. Babelon O., Bernard D., Dressing symmetries, Comm. Math. Phys. 149 (1992), 279-306, hep-th/9111036.
  7. Babelon O., Bernard D., Talon M., Introduction to classical integrable systems, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2003.
  8. Dargis P., Mathieu P., Nonlocal conservation laws for supersymmetric KdV equations, Phys. Lett. A 176 (1993), 67-74, hep-th/9301080.
  9. Delduc F., Gallot L., Supersymmetric Drinfeld-Sokolov reduction, J. Math. Phys. 39 (1998), 4729-4745, solv-int/9802013.
  10. Delduc F., Ragoucy E., Sorba P., Super-Toda theories and W-algebras from superspace Wess-Zumino-Witten models, Comm. Math. Phys. 146 (1992), 403-426.
  11. Evans J., Hollowood T., Supersymmetric Toda field theories, Nuclear Phys. B 352 (1991), 723-768.
  12. Evans J.M., Madsen J.O., Integrability versus supersymmetry, Phys. Lett. B 389 (1996), 665-672, hep-th/9608190.
  13. Ferreira L.A., Gervais J.-L., Guillén J.S., Saveliev M.V., Affine Toda systems coupled to matter fields, Nuclear Phys. B 470 (1996), 236-288, hep-th/9512105.
  14. Gervais J.-L., Saveliev M.V., Higher grading generalizations of the Toda systems, Nuclear Phys. B 453 (1995), 449-476, hep-th/9505047.
  15. Gomes J.F., Schmidtt D.M., Zimerman A.H., Super-WZNW with reductions to supersymmetric and fermionic integrable models, Nuclear Phys. B 821 (2009), 553-576, arXiv:0901.4040.
  16. Gomes J.F., Starvaggi Franca G., de Melo G.R., Zimerman A.H., Negative even grade mKdV hierarchy and its soliton solutions, J. Phys. A: Math. Theor. 42 (2009), 445204, 11 pages, arXiv:0906.5579.
  17. Grigoriev M., Tseytlin A., Pohlmeyer reduction of AdS5×S5 superstring sigma model, Nuclear Phys. B 800 (2008), 450-501, arXiv:0711.0155.
  18. Inami T., Kanno H., Lie superalgebraic approach to super Toda lattice and generalized super KdV equations, Comm. Math. Phys. 136 (1991), 519-542.
  19. Madsen J.O., Miramontes J.L., Non-local conservation laws and flow equations for supersymmetric integrable hierarchies, Comm. Math. Phys. 217 (2001), 249-284, hep-th/9905103.
  20. Manin Yu.I., Radul A.O., A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy, Comm. Math. Phys. 98 (1985), 65-77.
  21. Miramontes J.L., Tau-functions generating the conservation laws for generalized integrable hierarchies of KdV and affine Toda type, Nuclear Phys. B 547 (1999), 623-663, hep-th/9809052.
  22. Olshanetsky M.A., Supersymmetric two-dimensional Toda lattice, Comm. Math. Phys. 88 (1983), 63-76.
  23. Schmidtt D.M., Fermionic symmetry flows in non-Abelian Toda models, in preparation.
  24. Sorokin D.P., Toppan F., An n=(1,1) super-Toda model based on OSp(1|4), Lett. Math. Phys. 42 (1997), 139-152, hep-th/9610038.

Previous article   Next article   Contents of Volume 6 (2010)