We study compactifications of the 6d E-string theory, the theory of a small E 8 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 E 8 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.

Abstract:We discuss the four dimensional models obtained by compactifying a single M5 brane probing D N singularity (minimal D-type (1, 0) conformal matter in six dimensions) on a torus with flux for abelian subgroups of the SO(4N ) flavor symmetry. We derive the resulting quiver field theories in four dimensions by first compactifying on a circle and relating the flux to duality domain walls in five dimensions. This leads to novel N = 1 dualities in 4 dimensions which arise from distinct five dimensional realizations of the circle compactifications of the D-type conformal matter.

We discuss compactifications of rank QQ E-string theory on a torus with fluxes for abelian subgroups of the E_8E8 global symmetry of the 6d6d SCFT. We argue that the theories corresponding to such tori are built from a simple model we denote as E[USp(2Q)]E[USp(2Q)]. This model has a variety of non trivial properties. In particular the global symmetry is USp(2Q)\times USp(2Q)\times U(1)^2USp(2Q)×USp(2Q)×U(1)2 with one of the two USp(2Q)USp(2Q) symmetries emerging in the IR as an enhancement of an SU(2)^QSU(2)Q symmetry of the UV Lagrangian. The E[USp(2Q)]E[USp(2Q)] model after dimensional reduction to 3d3d and a subsequent Coulomb branch flow is closely related to the familiar 3d3dT[SU(Q)]T[SU(Q)] theory, the model residing on an S-duality domain wall of 4d4d\mathcal{N}=4𝒩=4SU(Q)SU(Q) SYM. Gluing the E[USp(2Q)]E[USp(2Q)] models by gauging the USp(2Q)USp(2Q) symmetries with proper admixtures of chiral superfields gives rise to systematic constructions of many examples of 4d4d theories with emergent IR symmetries. We support our claims by various checks involving computations of anomalies and supersymmetric partition functions. Many of the needed identities satisfied by the supersymmetric indices follow directly from recent mathematical results obtained by E. Rains.

We present a number of investigations of 5d N = 1 supersymmetric gauge theories that make use of 5-brane web constructions and the 5d superconformal index. These include an observation of enhanced global symmetry in the 5d fixed point theory corresponding to SU(N ) gauge theory with Chern-Simons level ±N , enhanced global symmetries in quiver theories, and dualities between quiver theories and non-quiver theories. Instanton contributions play a crucial role throughout.