We give a family of solutions of Witten–Dijkgraaf–Verlinde– Verlinde equations in n n -dimensional space. It is defined in terms of B C n BC_{n} root system and n + 2 n+2 independent multiplicity parameters. We also apply these solutions to define some N = 4 {\mathcal N}=4 supersymmetric mechanical systems.

We give a family of solutions of Witten-Dijkgraaf-Verlinde-Verlinde equations in n-dimensional space. It is defined in terms of BC n root system and n + 2 independent multiplicity parameters. We also apply these solutions to define some N = 4 supersymmetric mechanical systems.

We consider a class of trigonometric solutions of Witten–Dijkgraaf–Verlinde–Verlinde equations determined by collections of vectors with multiplicities. We show that such solutions can be restricted to special subspaces to produce new solutions of the same type. We find new solutions given by restrictions of root systems, as well as examples which are not of this form. Further, we consider a closely related notion of a trigonometric ∨-system, and we show that its subsystems are also trigonometric ∨-systems. Finally, while reviewing the root system case we determine a version of (generalised) Coxeter number for the exterior square of the reflection representation of a Weyl group.

We consider commutativity equations F i F j = F j F i for a function F (x 1 , . . . , x N ), where F i is a matrix of the third order derivatives F ikl . We show that under certain nondegeneracy conditions a solution F satisfies the WDVV equations. Equivalently, the corresponding family of Frobenius algebras has the identity field e.We study trigonometric solutions F determined by a finite collection of vectors with multiplicities, and we give an explicit formula for e for all the known such solutions. The corresponding collections of vectors are given by non-simply laced root systems or are related to their projections to the intersection of mirrors.