SIGMA 4 (2008), 041, 16 pages      arXiv:0805.0656      https://doi.org/10.3842/SIGMA.2008.041
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP(n) Groups

Victor D. Gershun
ITP, NSC Kharkiv Institute of Physics and Technology, Kharkiv, Ukraine

Received October 30, 2007, in final form April 22, 2008; Published online May 06, 2008

Abstract
We considered two types of string models: on the Riemmann space of string coordinates with null torsion and on the Riemman-Cartan space of string coordinates with constant torsion. We used the hydrodynamic approach of Dubrovin, Novikov to integrable systems and Dubrovin solutions of WDVV associativity equation to construct new integrable string equations of hydrodynamic type on the torsionless Riemmann space of chiral currents in first case. We used the invariant local chiral currents of principal chiral models for SU(n), SO(n), SP(n) groups to construct new integrable string equations of hydrodynamic type on the Riemmann space of the chiral primitive invariant currents and on the chiral non-primitive Casimir operators as Hamiltonians in second case. We also used Pohlmeyer tensor nonlocal currents to construct new nonlocal string equation.

Key words: string; integrable models; Poisson brackets; Casimir operators; chiral currents.

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References

1. Magri F., A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), 1156-1162.
2. Okubo S., Das A., The integrability condition for dynamical systems, Phys. Lett. B 209 (1988), 311-314.
3. Dubrovin B.A., Novikov S.P., Hamiltonian formalism of one-dimensional systems of hydrodynamic type and the Bogolyubov-Whitham averaging method, Dokl. Akad. Nauk SSSR 270 (1983), 781-785 (in Russian).
4. Dubrovin B.A., Novikov S.P., Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory, Russian Math. Surveys 44 (1989), no. 6, 35-124.
5. Ferapontov E.V., Differential geometry of nonlocal Hamiltonian of hydrodynamic type, Funct. Anal. Appl. 25 (1991), 195-204.
6. Mokhov O.I., Ferapontov E.V., Nonlocal Hamiltonian operators of hydrodynamic type that are connected with metrics of constant curvature, Russian Math. Surveys 45 (1990), no. 3, 218-219.
7. Mokhov O.I., Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems, Russian Math. Surveys 53 (1998), no. 3, 515-622.
8. Tsarev S.P., Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type, Dokl. Akad. Nauk SSSR 282 (1985), 534-537.
9. Maltsev A.Ya., On the compatible weakly nonlocal Poisson brackets of hydrodynamic type, Int. J. Math. Math. Sci. 32 (2002), 587-614, nlin.SI/0111015.
10. Ferapontov E.V., Nonlocal Hamiltonian operators of hydrodynamic type: differential geometry and applications, Amer. Math. Soc. Transl. (2) 170 (1995), 33-58.
11. Mokhov O.I., The Liouville canonical form of compatible nonlocal Poisson brackets of hydrodynamic type, and integrable hierarchies, Funct. Anal. Appl. 37 (2003), no 2, 103-113, math.DG/0201223.
12. Mokhov O.I., Compatible Dubrovin-Novikov Hamiltonian operators, Lie derivative and integrable systems of hydrodynamic type, Theoret. and Math. Phys. 133 (2002), 1557-1564, math.DG/0201281.
13. Pavlov M.V., Multi-Hamiltonian structures of the Whitham equations, Russian Acad. Sci. Dokl. Math. 50 (1995), no 2, 220-223.
14. Pavlov M.V., Integrability of Egorov systems of hydrodynamic type, Theoret. and Math. Phys. 150 (2007), no. 2, 225-243, nlin.SI/0606017.
15. Maltsev A.Ya., Novikov S.P., On the local systems Hamiltonian in the weakly non-local Poisson brackets, Phys. D 156 (2001), 53-80, nlin.SI/0006030.
16. Goldschmidt Y.Y., Witten E., Conservation laws in some two-dimensional models, Phys. Lett. B 91 (1980), 392-396.
17. Aref'eva I.Ya., Kulish P.P., Nissimov E R., Pacheva S.J., Infinite set of conservation laws of the quantum chiral field in 2D spacetime, preprint LOMI E-I-1978, unpublished.
18. Gershun V.D., Integrable string models of hydrodynamic type, J. Kharkov University, Phys. Ser. Nucl., Part., Fields 657 (2005), no. 1 (26), 109-113.
19. Gershun V.D., Integrable string models and sigma-models of hydrodynamic type in terms of invariant chiral currents, Problems of Atomic Science and Technology 3(1) (2007), Part 1, 16-21.
20. Dijkgraaf R., Witten E., Mean field theory, topological field theory, and multi-matrix models, Nuclear Phys. B 342 (1990), 486-522.
21. Dijkgraaf R., Verlinde H., Verlinde E., Topological strings in d < 1, Nuclear Phys. B 352 (1991), 59-86.
22. Dubrovin B.A., Geometry of 2D topological field theories, Lecture Notes in Math. 1620 (1996), 120-348, hep-th/9407018.
23. Dubrovin B., Zhang Y., Bi-Hamiltonian hierarchies in 2D topological field theory at one-loop approximation, Comm. Math. Phys. 198 (1998), 311-361, hep-th/9712232.
24. Evans J.M., Hassan M., MacKay N.J., Mountain A.J., Local conserved charges in principal chiral models, Nuclear Phys. B 561 (1999), 385-412, hep-th/9902008.
25. de Azcarraga J.A., Macfarlane A.J., Mountain A.J., Perez Bueno J.C., Invariant tensors for simple groups, Nuclear Phys. B 510 (1998), 657-687, physics/9706006.
26. Pohlmeyer K., A group-theoretical approach to the quantization of the free relativistic closed string, Phys. Lett. B 119 (1982), 100-104.
27. Meusburger C., Rehren K.-H., Algebraic quantization of the closed bosonic string, Comm. Math. Phys. 237 (2003), 69-85, math-ph/0202041.
28. Thiemann T., The LQS string-loop quantum gravity quantization of string theory. I. Flat target space, Classical Quantum Gravity 23 (2006), 1923-1970, math-ph/0401172.
29. Klein A., Invariant operators of the unitary unimodular group in n dimensions, J. Math. Phys. 4 (1963), 1283-1284.
30. Sudbery A., Some aspects of chiral [d]u(3) × [d]u(3) symmetry in hadron dynamics, PhD Thesis, Cambridge University, 1970.
    Sudbery A., Computer-friendly d-tensor identities for SU(n), J. Phys. A: Math. Gen. 23 (1990), L705-L710.