  Strontium titanate SrTiO$_3$ is one of the most studied and paradigmatic
transition metal oxides. Recently, a breakthrough has been achieved with the
fabrication of freestanding SrTiO$_3$ ultrathin films down to the monolayer
limit. However, the many-body effects on the quasiparticle and optical
properties of monolayer SrTiO$_3$ remain unexplored. Using state-of-the-art
many-body perturbation theory in the GW approximation combined with the
Bethe-Salpeter equation, we study the quasiparticle band structure, optical and
excitonic properties of monolayer SrTiO$_3$. We show that quasiparticle
corrections significantly alter the band structure topology; however, the
widely used diagonal $G_0W_0$ approach yields unphysical band dispersions. The
correct band dispersions are restored only by taking into account the
off-diagonal elements of the self-energy. The optical properties are studied
both in the optical limit and for finite momenta by computing the electron
energy loss spectra. We find that the imaginary part of dielectric function at
the long wavelength limit is dominated by three strongly bound excitonic peaks
and the direct optical gap is associated to a bright exciton state with a large
binding energy of 0.93 eV. We discuss the character of the excitonic peaks via
the contributing interband transitions, and reveal that the lowest bound
excitonic state becomes optical inactive for finite momenta along $\Gamma$-M,
while the other two excitonic peaks disperse to higher energies and eventually
merge for momenta close to M.
