# AGT/ℤ2

Journal of High Energy Physics, Dec 2017

We relate Liouville/Toda CFT correlators on Riemann surfaces with boundaries and cross-cap states to supersymmetric observables in four-dimensional $$\mathcal{N}=2$$ 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 $$\mathcal{N}=2$$ 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.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP12%282017%29099.pdf

Bruno Le Floch, Gustavo J. Turiaci. AGT/ℤ2, Journal of High Energy Physics, 2017, 99, DOI: 10.1007/JHEP12(2017)099