A Duality Hopf Algebra for Holomorphic N=1 Special Geometries

We find a self-dual noncommutative and noncocommutative Hopf algebra HF acting as a universal symmetry on the modules over inner Frobenius algebras of modular categories (as used in two dimensional boundary conformal field theory) similar to the GrothendieckTeichmüller groupGT as introduced by Drinfeld as a universal symmetry of quasitriangular quasi-Hopf algebras. We discuss the relationship to a similar self-dual noncommutative and noncocommutative Hopf algebra HGT , previously found as the universal symmetry of trialgebras and three dimensional extended topological quantum field theories. As an application of our result, we get a transitive action of a sub-Hopf algebra HD of HGT on the relative period matrices of holomorphic N = 1 special geometries, i.e. HD appears as a kind of duality Hopf algebra for holomorphic N = 1 special geometries.