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


SIGMA 5 (2009), 088, 10 pages      arXiv:0802.0532      http://dx.doi.org/10.3842/SIGMA.2009.088

Trigonometric Solutions of WDVV Equations and Generalized Calogero-Moser-Sutherland Systems

Misha V. Feigin
Department of Mathematics, University of Glasgow, G12 8QW, UK

Received May 18, 2009, in final form September 07, 2009; Published online September 17, 2009

Abstract
We consider trigonometric solutions of WDVV equations and derive geometric conditions when a collection of vectors with multiplicities determines such a solution. We incorporate these conditions into the notion of trigonometric Veselov system (∨-system) and we determine all trigonometric ∨-systems with up to five vectors. We show that generalized Calogero-Moser-Sutherland operator admits a factorized eigenfunction if and only if it corresponds to the trigonometric ∨-system; this inverts a one-way implication observed by Veselov for the rational solutions.

Key words: Witten-Dijkgraaf-Verlinde-Verlinde equations, ∨-systems, Calogero-Moser-Sutherland systems.

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References

  1. Marshakov A., Mironov A., Morozov A., WDVV-like equations in N=2 SUSY Yang-Mills theory, Phys. Lett. B 389 (1996), 43-52, hep-th/9607109.
  2. Marshakov A., Mironov A., Morozov A., More evidence for the WDVV equations in N=2 SUSY Yang-Mills theories, Internat. J. Modern Phys. A 15 (2000), 1157-1206, hep-th/9701123.
  3. Hoevenaars L.K., Martini R., On the WDVV equations in five-dimensional gauge theories, Phys. Lett. B 557 (2003), 94-104, math-ph/0212016.
  4. Martini R., Hoevenaars L.K., Trigonometric solutions of the WDVV equations from root systems, Lett. Math. Phys. 65 (2003), 15-18, math-ph/0302059.
  5. Pavlov M., Explicit solutions of the WDVV equation determined by the "flat" hydrodynamic reductions of the Egorov hydrodynamic chains, nlin.SI/0606008.
  6. 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.
  7. Dubrovin B., On almost duality for Frobenius manifolds, in Geometry, Topology, and Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 212, Amer. Math. Soc., Providence, RI, 2004, 75-132, math.DG/0307374.
  8. Riley A., Frobenius manifolds: caustic submanifolds and discriminant almost duality, Ph.D. Thesis, Hull University, 2007.
  9. Riley A., Strachan I.A.B., A note on the relationship between rational and trigonometric solutions of the WDVV equations, J. Nonlinear Math. Phys. 14 (2007), 82-94, nlin.SI/0605005.
  10. 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.
  11. Veselov A.P., On generalizations of the Calogero-Moser-Sutherland quantum problem and WDVV equations, J. Math. Phys. 43 (2002), 5675-5682, math-ph/0204050.
  12. Feigin M.V., Veselov A.P., Logarithmic Frobenius structures and Coxeter discriminants, Adv. Math. 212 (2007), 143-162, math-ph/0512095.
  13. Feigin M.V., Veselov A.P., On the geometry of ∨-systems, in Geometry, Topology, and Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 224, Amer. Math. Soc., Providence, RI, 2008, 111-123, arXiv:0710.5729.
  14. Feigin M.V., Bispectrality for deformed Calogero-Moser-Sutherland systems, J. Nonlinear Math. Phys. 12 (2005), suppl. 2, 95-136, math-ph/0503020.
  15. Braden H., Marshakov A., Mironov A., Morozov A., Seiberg-Witten theory for a non-trivial compactification from five to four dimensions, Phys. Lett. B 448 (1999), 195-202, hep-th/9812078.
  16. Strachan I.A.B., Weyl groups and elliptic solutions of the WDVV equations, arXiv:0802.0388.

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