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N=4 Multi-Particle Mechanics, WDVV Equation and Roots
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2011, том 7
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N=4 Multi-Particle Mechanics, WDVV Equation and Roots
Інший ID:
2010 Mathematics Subject Classification: 70E55; 81Q60; 17B22; 52B40; 05C65
DOI:10.3842/SIGMA.2011.023
DOI:10.3842/SIGMA.2011.023
Посилання:
N=4 Multi-Particle Mechanics, WDVV Equation and Roots / O. Lechtenfeld, K. Schwerdtfeger, J. Thürigen // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 35 назв. — англ.
Підтримка:
This paper is a contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design” (July 18–30, 2010, Benasque, Spain). The full collection is available at http://www.emis.de/journals/SIGMA/SUSYQM2010.html.
The authors are grateful to Martin Rubey for pointing them to and helping them with hypergraphs and matroids. Of course, all mistakes are ours! O.L. acknowledges fruitful discussions with Misha Feigin, Evgeny Ivanov, Sergey Krivonos, Andrei Smilga and Sasha Veselov. He also thanks the organizers of the Benasque workshop for a wonderful job.
Дата:
2011
Переглядів:
821
Завантажень:
252
Анотація:
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.
