  We study the second quantized -or guiding center- description of the torus
Laughlin state. Our main focus is the change of the guiding center degrees of
freedom with the torus geometry, which we show to be generated by a two-body
operator. We demonstrate that this operator can be used to evolve the full
torus Laughlin state at given modular parameter \tau\ from its simple
(Slater-determinant) thin torus limit, thus giving rise to a new presentation
of the torus Laughlin state in terms of its "root partition" and an exponential
of a two-body operator. This operator therefore generates in particular the
adiabatic evolution between Laughlin states on regular tori and the
quasi-one-dimensional thin torus limit. We make contact with the recently
introduced notion of a "Hall viscosity" for fractional quantum Hall states, to
which our two-body operator is naturally related, and which serves as a
demonstration of our method to generate the Laughlin state on the torus.
