  We investigate the invariance of the Gibbs measure for the fractional
Schrodinger equation of exponential type (expNLS) $i\partial_t u +
(-\Delta)^{\frac{\alpha}2} u = 2\gamma\beta e^{\beta|u|^2}u$ on $d$-dimensional
compact Riemannian manifolds $\mathcal{M}$, for a dispersion parameter
$\alpha>d$, some coupling constant $\beta>0$, and $\gamma\neq 0$. (i) We first
study the construction of the Gibbs measure for (expNLS). We prove that in the
defocusing case $\gamma>0$, the measure is well-defined in the whole regime
$\alpha>d$ and $\beta>0$ (Theorem 1.1 (i)), while in the focusing case
$\gamma<0$ its partition function is always infinite for any $\alpha>d$ and
$\beta>0$, even with a mass cut-off of arbitrary small size (Theorem 1.1 (ii)).
(ii) We then study the dynamics (expNLS) with random initial data of low
regularity. We first use a compactness argument to prove weak invariance of the
Gibbs measure in the whole regime $\alpha>d$ and $0<\beta < \beta^\star_\alpha$
for some natural parameter $0<\beta^\star_\alpha\sim (\alpha-d)$ (Theorem 1.3
(i)). In the large dispersion regime $\alpha>2d$, we can improve this result by
constructing a local deterministic flow for (expNLS) for any $\beta>0$. Using
the Gibbs measure, we prove that solutions are almost surely global for
$0<\beta \ll\beta^\star_\alpha$, and that the Gibbs measure is invariant
(Theorem 1.3 (ii)). (iii) Finally, in the particular case $d=1$ and
$\mathcal{M}=\mathbb{T}$, we are able to exploit some probabilistic multilinear
smoothing effects to build a probabilistic flow for (expNLS) for
$1+\frac{\sqrt{2}}2<\alpha \leq 2$, locally for arbitrary $\beta>0$ and
globally for $0<\beta \ll \beta^\star_\alpha$ (Theorem 1.5).
