AGT/Z2

We relate Liouville/Toda CFT correlators on Riemann surfaces with boundaries and cross-cap states to supersymmetric observables in four-dimensional N = 2 gauge theories. Our construction naturally involves four-dimensional theories with fields de fined on different Z(2) quotients of the sphere (hemisphere and projective space) but nevertheless interacting with each other. The six-dimensional origin is a Z(2) quotient of the setup giving rise to the usual AGT correspondence. To test the correspondence, we work out the RP4 partition function of four-dimensional N = 2 theories by combining a 3d lens space and a 4d hemisphere partition functions. The same technique reproduces known RP2 partition functions in a form that lets us easily check two-dimensional Seiberg-like dualities on this nonorientable space. As a bonus we work out boundary and cross-cap wavefunctions in Toda CFT.