N=2 gauge theories on unoriented/open four-manifolds and their AGT counterparts

We compute the exact path integral of N=2 supersymmetric gauge theories with general gauge group on ℝℙ4 and a ℤ2-quotient of the hemi-S4. By specializing to SU(2) superconformal quivers, we show that these, together with hemi-S4 partition functions, compute Liouville correlators on unoriented/open Riemann surfaces. We perform explicit checks for Riemann surfaces obtained as ℤ2 quotients of the sphere and the torus. We also discuss the coupled 3d−4d systems associated to Liouville amplitudes with boundary punctures.