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

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 ($\vee$-system) and we determine all trigonometric $\vee$-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 $\vee$-system; this inverts a one-way implication observed by Veselov for the rational solutions.

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Cited by in corpus (11)

• Many-particle mechanics with D(2,1;alpha) superconformal symmetry
• Weyl groups and Elliptic Solutions of the WDVV equations
• Generalized Legendre transformations and symmetries of the WDVV equations
• Superconformal Quantum Mechanics from M2-branes
• Trigonometric $\vee$-systems and solutions of WDVV equations
• Intertwining operator for $A{G}_2$ Calogero-Moser-Sutherland system
• Extended V-systems and almost-duality for extended affine Weyl orbit spaces
• Legendre transforms for type $A_n$ and $B_n$ $\vee$-systems
• Differential and Functional Identities for the Elliptic Trilogarithm
• Solutions of $B{C}_n$ Type of WDVV Equations
• Frobenius algebras and root systems: the trigonometric case