  We study the model of a discrete directed polymer (DP) on the square lattice
with homogeneous inverse gamma distribution of site random Boltzmann weights,
introduced by Seppalainen. The integer moments of the partition sum,
$\overline{Z^n}$, are studied using a transfer matrix formulation, which
appears as a generalization of the Lieb-Liniger quantum mechanics of bosons to
discrete time and space. In the present case of the inverse gamma distribution
the model is integrable in terms of a coordinate Bethe Ansatz, as discovered by
Brunet. Using the Brunet-Bethe eigenstates we obtain an exact expression for
the integer moments of $\overline{Z^n}$ for polymers of arbitrary lengths and
fixed endpoint positions. Although these moments do not exist for all integer
n, we are nevertheless able to construct a generating function which reproduces
all existing integer moments, and which takes the form of a Fredholm
determinant (FD). This suggests an analytic continuation via a Mellin-Barnes
transform and we thereby propose a FD ansatz representation for the probability
distribution function (PDF) of $Z$ and its Laplace transform. In the limit of
very long DP, this ansatz yields that the distribution of the free energy
converges to the GUE Tracy-Widom distribution up to a non-trivial average and
variance that we calculate. Our asymptotic predictions coincide with a result
by Borodin et al. based on a formula obtained by Seppalainen using the gRSK
correspondence. In addition we obtain the dependence on the endpoint position
and the exact elastic coefficient at large time. We argue the equivalence
between our formula and the one of Borodin et al. As we discuss, this open the
way to explore the connections between quantum integrability and tropical
geometry.
