  Conformal field theory has turned out to be a powerful tool to derive
two-dimensional lattice models displaying fractional quantum Hall physics. So
far most of the work has been for lattices with open boundary conditions in at
least one of the two directions, but it is desirable to also be able to handle
the case of periodic boundary conditions. Here, we take steps in this direction
by deriving analytical expressions for a family of conformal field theory
states on the torus that is closely related to the family of bosonic and
fermionic Laughlin states. We compute how the states transform when a particle
is moved around the torus and when the states are translated or rotated, and we
provide numerical evidence in particular cases that the states become
orthonormal up to a common factor for large lattices. We use these results to
find the S-matrix of the states, which turns out to be the same as for the
continuum Laughlin states. Finally, we show that when the states are defined on
a square lattice with suitable lattice spacing they practically coincide with
the Laughlin states restricted to a lattice.
