  We define the Jacobian of a Riemann surface with analytically parametrized
boundary components. These Jacobians belong to a moduli space of ``open abelian
varieties'' which satisfies gluing axioms similar to those of Riemann surfaces,
and therefore allows a notion of ``conformal field theory'' to be defined on
this space. We further prove that chiral conformal field theories corresponding
to even lattices factor through this moduli space of open abelian varieties.
