  We study the statistical properties of one-dimensional directed polymers in a
short-range random potential by mapping the replicated problem to a many body
quantum boson system with attractive interactions. We find the full set of
eigenvalues and eigenfunctions of the many-body system and perform the
summation over the entire spectrum of excited states. The analytic continuation
of the obtained exact expression for the replica partition function from
integer to non-integer replica parameter N turns out to be ambiguous.
Performing the analytic continuation simply by assuming that the parameter N
can take arbitrary complex values, and going to the thermodynamic limit of the
original directed polymer problem, we obtain the explicit universal expression
for the probability distribution function of free energy fluctuations.
