Weyl groups and Elliptic Solutions of the WDVV equations

Abstract

A functional ansatz is developed which gives certain elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equation. This is based on the elliptic trilogarithm function introduced by Beilinson and Levin. For this to be a solution results in a number of purely algebraic conditions on the set of vectors that appear in the ansatz, this providing an elliptic version of the idea, introduced by Veselov, of a V-system. Rational and trigonometric limits are studied together with examples of elliptic V-systems based on various Weyl groups. Jacobi group orbit spaces are studied: these carry the structure of a Frobenius manifold. The corresponding almost dual structure is shown, in the A_N and B_N and conjecturally for an arbitrary Weyl group, to correspond to the elliptic solutions of the WDVV equations. Transformation properties, under the Jacobi group, of the elliptic trilogarithm are derived together with various functional identities which generalize the classical Frobenius-Stickelburger relations.

Typographical errors corrected. One result weakened (though with changing the main result). Main theorem rewritten

References in corpus (10)

• Frobenius Manifolds: Natural submanifolds and induced bi-Hamiltonian structures
• Locus configurations and $\vee$-systems
• Trigonometric Solutions of WDVV Equations and Generalized Calogero-Moser-Sutherland Systems
• On the geometry of V-systems
• Duality for Jacobi group orbit spaces and elliptic solutions of the WDVV equations
• Universal KZB equations I: the elliptic case
• Trigonometric Solutions of the WDVV Equations from Root Systems
• Frobenius manifolds for elliptic root systems
• WDVV equations for 6d Seiberg-Witten theory and bi-elliptic curves
• Differential and Functional Identities for the Elliptic Trilogarithm

Cited by in corpus (10)

• Many-particle mechanics with D(2,1;alpha) superconformal symmetry
• Trigonometric Solutions of WDVV Equations and Generalized Calogero-Moser-Sutherland Systems
• N=4 Multi-Particle Mechanics, WDVV Equation and Roots
• Superconformal Quantum Mechanics from M2-branes
• Polynomial Modular Frobenius Manifolds
• Trigonometric $\vee$-systems and solutions of WDVV equations
• Extended V-systems and almost-duality for extended affine Weyl orbit spaces
• Differential and Functional Identities for the Elliptic Trilogarithm
• Solutions of $B{C}_n$ Type of WDVV Equations
• Modular Frobenius manifolds and their invariant flows