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

International Mathematics Research Notices

2010

The space of Frobenius manifolds has a natural involutive symmetry on it: there exists a map I which send a Frobenius manifold to another Frobenius manifold. Also, from a Frobenius manifold one may construct a so-called almost dual Frobenius manifold which satisfies almost all of the axioms of a Frobenius manifold. The action of I on the almost dual manifolds is studied, and the action of I on objects such as periods, twisted periods and flows is studied. A distinguished class of Frobenius manifolds sit at the fixed point of this involutive symmetry, and this is made manifest in certain modular properties of the various structures. In particular, up to a simple reciprocal transformation, for this class of modular Frobenius manifolds, the flows are invariant under the action of I .

Section: Commentsmentioning

confidence: 82%

Modular Frobenius Manifolds and their Invariant Flows

Morrison

International Mathematics Research Notices

2010

“…Repeating the argument for the terms that appear in Eqs. (13) and (14) yields no further conditions. 2…”

Section: It Is Easy To Show That ξ a = 0 If And Only If H β = H σ α βmentioning

confidence: 91%

Weyl groups and elliptic solutions of the WDVV equations

Advances in Mathematics

2010

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.

“…where Li3(z) is the tri-logarithm function. A separate theory of trigonometric ⋁-systems has been developed by Feigin 9 (and for the most results, see Ref. 10).…”

Section: B Outlinementioning

confidence: 99%

“…Using the primary differential ω = dz, one can show that the coefficient α k , by evaluating certain residues, is a flat coordinate, which we denote by t ○ (i.e., β k = t ○ ). With this, one may define a change of primary differential 8 dz = {∂ t ○ λ(z)}dz, so z = log(z − z) ○ , and this induces the Legendre transformation between the two Frobenius manifolds, i.e., this change of primary differential induces a change of variable that maps (8) to (9). Thus, the Frobenius manifold structures on C l+1 / W(k) (A l ) and M k,l+1−k are related by a Legendre transformation.…”

Section: A Extended Affine Weyl Orbit Spaces Of Type Amentioning

confidence: 99%

Extended ⋁-systems and trigonometric solutions to the WDVV equations

Stedman

Journal of Mathematical Physics

2021

Rational solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations of associativity are given in terms of configurations of vectors, which satisfy certain algebraic conditions known as ⋁-conditions [A. P. Veselov, Phys. Lett. A 261, 297-302 (1999)]. The simplest examples of such configurations are the root systems of finite Coxeter groups. In this paper, conditions are derived that ensure that an extended configuration-a configuration in a space one-dimension higher-satisfies these ⋁-conditions. Such a construction utilizes the notion of a small orbit, as defined in Serganova [Commun. Algebra, 24, 4281-4299 (1996)]. Symmetries of such resulting solutions to the WDVV equations are studied, in particular, Legendre transformations. It is shown that these Legendre transformations map extended-rational solutions to trigonometric solutions, and for certain values of the free data, one obtains a transformation from extended ⋁-systems to the trigonometric almost-dual solutions corresponding to the classical extended affine Weyl groups.