  We calculate exactly the first cumulants of the free energy of a directed
polymer in a random medium for the geometry of a cylinder. By using the fact
that the n-th moment <Z^n> of the partition function is given by the ground
state energy of a quantum problem of n interacting particles on a ring of
length L, we write an integral equation allowing to expand these moments in
powers of the strength of the disorder gamma or in powers of n. For n small and
n of order (L gamma)^(-1/2), the moments <Z^n> take a scaling form which allows
to describe all the fluctuations of order 1/L of the free energy per unit
length of the directed polymer. The distribution of these fluctuations is the
same as the one found recently in the asymmetric exclusion process, indicating
that it is characteristic of all the systems described by the
Kardar-Parisi-Zhang equation in 1+1 dimensions.
