Geodesic
The periodic Fock bundle
Algebra i analiz, Tome 3 (1991) no. 5, pp. 135-154.

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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.
Keywords: Fock space, Heisenberg group, Siegel's generalized upper halfplane, reproducing kernel, theta function, Hermitean vector bundle, connection. 
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     author = {Jaak Peetre},
     title = {The periodic {Fock} bundle},
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Jaak Peetre. The periodic Fock bundle. Algebra i analiz, Tome 3 (1991) no. 5, pp. 135-154. http://geodesic.mathdoc.fr/item/AA_1991_3_5_a5/