  We consider the point-to-point log-gamma polymer of length $2N$ in a
half-space with i.i.d. $\operatorname{Gamma}^{-1}(2\theta)$ distributed bulk
weights and i.i.d. $\operatorname{Gamma}^{-1}(\alpha+\theta)$ distributed
boundary weights for $\theta>0$ and $\alpha>-\theta$. We establish the KPZ
exponents ($1/3$ fluctuation and $2/3$ transversal) for this model when
$\alpha=N^{-1/3}\mu$ for $\mu\in \mathbb{R}$ fixed (critical regime) and when
$\alpha>0$ is fixed (supercritical regime). In particular, in these two
regimes, we show that after appropriate centering, the free energy process with
spatial coordinate scaled by $N^{2/3}$ and fluctuations scaled by $N^{1/3}$ is
tight. These regimes correspond to a polymer measure which is not pinned at the
boundary.
  This is the first instance of establishing the $2/3$ transversal exponent for
a positive temperature half-space model, and the first instance of the $1/3$
fluctuation exponent besides precisely at the boundary where recent work of
arXiv:2204.08420 applies and also gives the exact one-point fluctuation
distribution (our methods do not access exact fluctuation distributions).
  Our proof relies on two inputs -- the relationship between the half-space
log-gamma polymer and half-space Whittaker process (facilitated by the
geometric RSK correspondence as initiated in arXiv:1110.3489, arXiv:1210.5126),
and an identity in arXiv:2108.08737 which relates the point-to-line half-space
partition function to the full-space partition function for the log-gamma
polymer.
  The primary technical contribution of our work is to construct the half-space
log-gamma Gibbsian line ensemble and develop, in the spirit of work initiated
in arXiv:1108.2291, a toolbox for extracting tightness and absolute continuity
results from minimal information about the top curve of such half-space line
ensembles. This is the first study of half-space line ensembles.
