$\mathcal{N}=2$ gauge theories on unoriented/open four-manifolds and their AGT counterparts

We compute the exact path integral of $\mathcal{N}=2$ supersymmetric gauge theories with general gauge group on ${\mathbb{R}\mathbb{P}}^4$ and a ${\mathbb{Z}}_2$-quotient of the hemi-$S^4$. By specializing to $SU(2)$ superconformal quivers, we show that these, together with hemi-$S^4$ partition functions, compute Liouville correlators on unoriented/open Riemann surfaces. We perform explicit checks for Riemann surfaces obtained as ${\mathbb{Z}}_2$ quotients of the sphere and the torus. We also discuss the coupled $3d\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}4d$ systems associated to Liouville amplitudes with boundary punctures.