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


SIGMA 7 (2011), 023, 21 pages      arXiv:1011.2207      http://dx.doi.org/10.3842/SIGMA.2011.023
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

N=4 Multi-Particle Mechanics, WDVV Equation and Roots

Olaf Lechtenfeld, Konrad Schwerdtfeger and Johannes Thürigen
Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstrasse 2, 30167 Hannover, Germany

Received November 14, 2010, in final form February 24, 2011; Published online March 05, 2011

Abstract
We review the relation of N=4 superconformal multi-particle models on the real line to the WDVV equation and an associated linear equation for two prepotentials, F and U. The superspace treatment gives another variant of the integrability problem, which we also reformulate as a search for closed flat Yang-Mills connections. Three- and four-particle solutions are presented. The covector ansatz turns the WDVV equation into an algebraic condition, for which we give a formulation in terms of partial isometries. Three ideas for classifying WDVV solutions are developed: ortho-polytopes, hypergraphs, and matroids. Various examples and counterexamples are displayed.

Key words: superconformal mechanics; Calogero models; WDVV equation; deformed root systems.

pdf (562 kb)   tex (204 kb)

References

  1. Donets E.E., Pashnev A., Rosales J.J., Tsulaia M.M., N=4 supersymmetric multidimensional quantum mechanics, partial susy breaking and superconformal quantum mechanics, Phys. Rev. D 61 (2000), 043512, 11 pages, hep-th/9907224.
  2. Wyllard N., (Super)conformal many-body quantum mechanics with extended supersymmetry, J. Math. Phys. 41 (2000), 2826-2838, hep-th/9910160.
  3. Bellucci S., Galajinsky A., Krivonos S., New many-body superconformal models as reductions of simple composite systems, Phys. Rev. D 68 (2003), 064010, 7 pages, hep-th/0304087.
  4. Bellucci S., Galajinsky A., Latini E., New insight into the Witten-Dijkgraff-Verlinde-Verlinde equation, Phys. Rev. D 71 (2005), 044023, 8 pages, hep-th/0411232.
  5. Galajinsky A., Lechtenfeld O., Polovnikov K., N=4 superconformal Calogero models, J. High Energy Phys. 2007 (2007), no. 11, 008, 23 pages, arXiv:0708.1075.
  6. Galajinsky A., Lechtenfeld O., Polovnikov K., N=4 mechanics, WDVV equations and roots, J. High Energy Phys. 2009 (2009), no. 3, 113, 28 pages, arXiv:0802.4386.
  7. Bellucci S., Krivonos S., Sutulin A., N=4 supersymmetric 3-particles Calogero model, Nuclear Phys. B 805 (2008), 24-39, arXiv:0805.3480.
  8. Fedoruk S., Ivanov E., Lechtenfeld O., Supersymmetric Calogero models by gauging, Phys. Rev. D 79 (2009), 105015, 6 pages, arXiv:0812.4276.
  9. Krivonos S., Lechtenfeld O., Polovnikov K., N=4 superconformal n-particle mechanics via superspace, Nuclear Phys. B 817 (2009), 265-283, arXiv:0812.5062.
  10. Lechtenfeld O., Polovnikov K., A new class of solutions to the WDVV equation, Phys. Lett. A 374 (2010), 504-506, arXiv:0907.2244.
  11. Witten E., On the structure of the topological phase of two-dimensional gravity, Nuclear Phys. B 340 (1990), 281-332.
  12. Dijkgraaf R., Verlinde H., Verlinde E., Topological strings in d<1, Nuclear Phys. B 352 (1991), 59-86.
  13. Dubrovin B., Geometry of 2D topological field theories, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348, hep-th//9407018.
  14. Martini R., Gragert P.K.H., Solutions of WDVV equations in Seiberg-Witten theory from root systems, J. Nonlinear Math. Phys. 6 (1999), 1-4, hep-th/9901166.
  15. Veselov A.P., Deformations of the root systems and new solutions to generalised WDVV equations, Phys. Lett. A 261 (1999), 297-302, hep-th/9902142.
  16. Strachan I.A.B., Weyl groups and elliptic solutions of the WDVV equations, Adv. Math. 224 (2010), 1801-1838, arXiv:0802.0388.
  17. Chalykh O.A., Veselov A.P., Locus configurations and -systems, Phys. Lett. A 285 (2001), 339-349, math-ph/0105003.
  18. Feigin M.V., Veselov A.P., Logarithmis Frobenius structures and Coxeter discriminants, Adv. Math. 212 (2007), 143-162, math-ph/0512095.
  19. Feigin M.V., Veselov A.P., On the geometry of -systems, Amer. Math. Soc. Transl. (2), Vol. 224, Amer. Math. Soc., Providence, RI, 2008, 111-123, arXiv:0710.5729.
  20. Bonelli G., Matone M., Nonperturbative relations in N=2 susy Yang-Mills and WDVV equation, Phys. Rev. Lett. 77 (1996), 4712-4715, hep-th/9605090.
  21. Marshakov A., Mironov A., Morozov A., WDVV-like equations in N=2 SUSY Yang-Mills theory, Phys. Lett. B 189 (1996), 43-52, hep-th/9607109.
    Marshakov A., Mironov A., Morozov A., WDVV equations from algebra of forms, Modern Phys. Lett. A 12 (1997), 773-788, hep-th/9701014.
  22. Mironov A., WDVV equations in Seiberg-Witten theory and associative algebras, Nuclear Phys. B Proc. Suppl. 61A (1998), 177-185, hep-th/9704205.
  23. Lechtenfeld O., WDVV solutions from orthocentric polytopes and Veselov systems, in Problems of Modern Theoretical Physics, Editor V. Epp, Tomsk State Pedagogical University Press, 2008, 265-265, arXiv:0804.3245.
  24. Iohara K., Koga Y., Central extensions of Lie superalgebras, Comment. Math. Helv. 76 (2001), 110-154.
  25. Ivanov E., Krivonos S., Leviant V., Geometric superfield approach to superconformal mechanics, J. Phys. A: Math. Gen. 22 (1989), 4201-4222.
  26. Ivanov E., Krivonos S., Lechtenfeld O., N=4, d=1 supermultiplets from nonlinear realizations of D(2,1;α), Classical Quantum Gravity 21 (2004), 1031-1050, hep-th/0310299.
  27. Delduc F., Ivanov E., Gauging N=4 supersymmetric mechanics. II. (1,4,3) models from the (4,4,0) ones, Nuclear Phys. B 770 (2007), 179-205, hep-th/0611247.
  28. Ivanov E., Lechtenfeld O., N=4 supersymmetric mechanics in harmonic superspace, J. High Energy Phys. 2003 (2003), no. 8, 073, 33 pages, hep-th/0307111.
  29. Bellucci S., Krivonos S., Supersymmetric mechanics in superspace, in Supersymmetric Mechanics, Lecture Notes in Phys., Vol. 698, Springer, Berlin, 2006, 49-96, hep-th/0602199.
  30. Voloshin V.I., Introduction to graph and hypergraph theory, Nova Science Publishers, Inc., New York, 2009.
  31. Wikipedia entry: Matroid, available at http://en.wikipedia.org/wiki/Matroid.
  32. Oxley J., What is a matroid?, Cubo Mat. Educ. 5 (2003), 179-218, Revised version is available at https://www.math.lsu.edu/~oxley/survey4.pdf.
  33. Dukes W.M.B., On the number of matroids on a finite set, Sém. Lothar. Combin. 51 (2004), Art. B51g, 12 pages, math.CO/0411557 (see also Dukes' lists of matroids at http://www.stp.dias.ie/~dukes/matroid.html).
  34. Schwerdtfeger K.W., Über Lösungen zu den WDVV-Gleichungen, Diploma Thesis, unpublished, http://www.itp.uni-hannover.de/~lechtenf/Theses/schwerdtfeger.pdf.
  35. Schwerdtfeger K.W., A Mathematica notebook with the tools and the computation, available at http://www.itp.uni-hannover.de/~lechtenf/vsystems.nb.

Previous article   Next article   Contents of Volume 7 (2011)