Jacobi Groups, Jacobi Forms and their Applications

Marco Bertola

Abstract: Jacobi groups are discrete infinite groups. They arise in the study of the period mapping of simple elliptic singularities in a similar way as the Weyl groups of A-D-E type arise from simple hypersurface singularities. They act faithfully on a cone; (quasi)-invariant functions under this action are called Jacobi forms. Also, Jacobi groups can be realized as "Weyl"-groups of suitable infinite dimensional Lie algebras. We describe these objects and in particulare the construction of Jacobi forms pointing out the parallel with the construction of the invariant polynomials for certain Coxeter groups. Two relevant applications are provided; the first to the theory of Frobenius manidolds (which appear in the study of two-dimensional topological field theory); the second to the Chern-Simons states in the context of the Wess-Zumino-Novikov-Witten theory over elliptic curves.