Geometric engineering, mirror symmetry and 6 d 1 0 → 4 d N = 2 $$ 6{\mathrm{d}}_{\left(1,0\right)}\to 4{\mathrm{d}}_{\left(\mathcal{N}=2\right)} $$
Michele Zotto (Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02138, U.S.A.); Cumrun Vafa (Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02138, U.S.A.); Dan Xie (Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02138, U.S.A., CMSA, Harvard University, Cambridge, MA, 02138, U.S.A.)
We study compactification of 6 dimensional (1,0) theories on T 2 . We use geometric engineering of these theories via F-theory and employ mirror symmetry technology to solve for the effective 4d N = 2 $$ \mathcal{N}=2 $$ geometry for a large number of the (1 , 0) theories including those associated with conformal matter. Using this we show that for a given 6d theory we can obtain many inequivalent 4d N = 2 $$ \mathcal{N}=2 $$ SCFTs. Some of these respect the global symmetries of the 6d theory while others exhibit SL(2 , ℤ ) duality symmetry inherited from global diffeomorphisms of the T 2 . This construction also explains the 6d origin of moduli space of 4d affine ADE quiver theories as flat ADE connections on T 2 . Among the resulting 4 d N = 2 $$ \mathcal{N}=2 $$ CFTs we find theories whose vacuum geometry is captured by an LG theory (as opposed to a curve or a local CY geometry). We obtain arbitrary genus curves of class S $$ \mathcal{S} $$ with punctures from toroidal compactification of (1 , 0) SCFTs where the curve of the class S $$ \mathcal{S} $$ theory emerges through mirror symmetry. We also show that toroidal compactification of the little string version of these theories can lead to class S $$ \mathcal{S} $$ theories with no punctures on arbitrary genus Riemann surface.
