Geometric Engineering, Mirror Symmetry and 6d (1,0) -> 4d, N=2

TL;DR

This paper explores the compactification of 6d (1,0) theories on T^2 using geometric engineering and mirror symmetry, revealing new 4d N=2 SCFTs with diverse symmetries and geometric structures, including connections to class S theories and LG models.

Contribution

It introduces a novel geometric and mirror symmetry approach to derive a wide variety of 4d N=2 theories from 6d (1,0) theories, including those with complex moduli spaces and duality properties.

Findings

Many inequivalent 4d N=2 SCFTs can be obtained from a single 6d theory.

Some 4d theories exhibit SL(2,Z) duality symmetry inherited from T^2 diffeomorphisms.

The construction explains the origin of moduli spaces of 4d affine ADE quiver theories.

Abstract

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 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 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 4d 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 with punctures from toroidal compactification of (1,0) SCFTs where the curve of the class 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 theories with no punctures on arbitrary genus Riemann surface.