Weyl groups and elliptic solutions of the WDVV equations

Abstract

A functional ansatz is developed which gives certain elliptic solutions of the Witten-Dijkgraaf-VerlindeVerlinde (or WDVV) equations. This ansatz 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 ∨-system.Rational and trigonometric limits are studied together with examples of elliptic ∨-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 cases 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-Stickelberger relations.

Löwner equations, Hirota equations and reductions of the universal Whitham hierarchy

“…We expect to find a prototype in differential geometry of Hurwitz spaces [31,32] and associated Whitham-type hierarchies [13,27], because they are generalizations of the rational reductions that we considered as a prototype of general multi-variable reductions for the genus zero case. Presumably, we should start with the genus one case, for which an explicit description of Hurwitz spaces are available in the literature [31,33,34,35] along with a candidate of Löwner-type equations [36].…”

“…As in the case of one-variable reduction, the dual equations (53) imply that the z-functions are functionally related with each other by functions f α (z) of one variable z as (34) shows. Though Guil et al [9] further assumed the special form (35), the following consideration is not limited to that case.…”

N=4 Multi-Particle Mechanics, WDVV Equation and Roots

“…Over the past decade, there has been substantial progress in the construction of N =4 superconformal multi-particle mechanics (in one space dimension) [1,2,3,4,5,6,7,8,9,10]. In 2004 a deep connection between these physical systems and the so-called WDVV equation [11,12] was discovered [4], relating Calogero-type models with D(2, 1; α) superconformal symmetry to a branch of mathematics concerned with solving this equation [13,14,15,16,17,18,19]. Here, we describe physicists' attempts to take advantage of the mathematical literature on this subject and to develop it further towards constructing and classifying such multi-particle models.…”

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

“…rational case [7,12]). Comparison with a recent work on the elliptic solutions [16] might also be interesting. We also hope that the series conditions would allow understanding and eventually classification of the trigonometric ∨-systems.…”