The periodic Fock bundle

The Fock bundle is an Hermitean vector bundle over Siegel's generalized upper halfplane, the fibers of which can be realized as Hilbert spaces of entire functions. In this paper a “periodic” version of the Fock bundle is constructed, that is, we factor the fibers of the (usual) Fock bundle by a maximal isotropic discrete subgroup of the underlying symplectic vector space. Applications to theta functions are obtained. In fact, it is our intention to work out, in a subsequent publication, major parts of the classical theory of theta functions on the basis ofthe corresponding “doubly periodic” object, obtained by instead factoring by a symplectic lattice.