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.