E-String Theory on Riemann Surfaces

We study compactifications of the 6d E-string theory, the theory of a small E₈ instanton, to four dimensions. In particular we identify N=1 field theories in four dimensions corresponding to compactifications on arbitrary Riemann surfaces with punctures and with arbitrary non-abelian flat connections as well as fluxes for the abelian sub-groups of the E₈ flavor symmetry. This sheds light on emergent symmetries in a number of 4d N=1 SCFTs (including the ‘E7 surprise’ theory) as well as leads to new predictions for a large number of 4-dimensional exceptional dualities and symmetries.