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We study the directed polymer (DP) of length $t$ in a random potential indimension 1+1 in the continuum limit, with one end fixed and one end free. Thismaps onto the Kardar-Parisi-Zhang growth equation in time $t$, with flatinitial conditions. We use the Bethe Ansatz solution for the replicated problemwhich is an attractive bosonic model. The problem is more difficult than theprevious solution of the fixed endpoint problem as it requires regularizationof the spatial integrals over the Bethe eigenfunctions. We use either a largefixed system length or a small finite slope KPZ initial conditions (wedge). Thelatter allows to take properly into account non-trivial contributions, whichappear as deformed strings in the former. By considering a half-space model ina proper limit we obtain an expression for the generating function of allpositive integer moments $\bar{Z^n}$ of the directed polymer partitionfunction. We obtain the generating function of the moments of the DP partitionsum as a Fredholm Pfaffian. At large time, this Fredholm Pfaffian, valid forall time $t$, exhibits convergence of the free energy (i.e. KPZ height)distribution to the GOE Tracy Widom distribution

The KPZ equation with flat initial condition and the directed polymer with one free end