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SIGMA 5 (2009), 088, 10 pages arXiv:0802.0532
https://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
- 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.
- 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.
- Hoevenaars L.K., Martini R.,
On the WDVV equations in five-dimensional gauge theories,
Phys. Lett. B 557 (2003), 94-104,
math-ph/0212016.
- Martini R., Hoevenaars L.K.,
Trigonometric solutions of the WDVV equations from root systems,
Lett. Math. Phys. 65 (2003), 15-18,
math-ph/0302059.
- Pavlov M.,
Explicit solutions of the WDVV equation determined by the "flat" hydrodynamic reductions of the Egorov hydrodynamic chains,
nlin.SI/0606008.
- 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.
- 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.
- Riley A.,
Frobenius manifolds: caustic submanifolds and discriminant almost duality, Ph.D. Thesis, Hull University, 2007.
- 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.
- 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.
- 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.
- Feigin M.V., Veselov A.P.,
Logarithmic Frobenius structures and Coxeter discriminants,
Adv. Math. 212 (2007), 143-162,
math-ph/0512095.
- 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.
- Feigin M.V.,
Bispectrality for deformed Calogero-Moser-Sutherland systems,
J. Nonlinear Math. Phys. 12 (2005), suppl. 2, 95-136,
math-ph/0503020.
- 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.
- Strachan I.A.B.,
Weyl groups and elliptic solutions of the WDVV equations,
arXiv:0802.0388.
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