AGT/ℤ2

Abstract We relate Liouville/Toda CFT correlators on Riemann surfaces with boundaries and cross-cap states to supersymmetric observables in four-dimensional N=2 $$ \mathcal{N}=2 $$ gauge theories. Our construction naturally involves four-dimensional theories with fields defined on different ℤ2 quotients of the sphere (hemisphere and projective space) but nevertheless interacting with each other. The six-dimensional origin is a ℤ2 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 $$ \mathcal{N}=2 $$ 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