Rank Q E-string on a torus with flux

We discuss compactifications of rank Q E-string theory on a torus with fluxes for abelian subgroups of the E₈ global symmetry of the 6d SCFT. We argue that the theories corresponding to such tori are built from a simple model we denote as E[USp(2Q)]. This model has a variety of non trivial properties. In particular the global symmetry is USp(2Q) × USp(2Q) × U(1)² with one of the two USp(2Q) symmetries emerging in the IR as an enhancement of an SU(2)^Q symmetry of the UV Lagrangian. The E[USp(2Q)] model after dimensional reduction to 3d and a subsequent Coulomb branch flow is closely related to the familiar 3d T[SU(Q)] theory, the model residing on an S-duality domain wall of 4d N = 4 SU(Q) SYM. Gluing the E[USp(2Q)] models by gauging the USp(2Q) symmetries with proper admixtures of chiral superfields gives rise to systematic constructions of many examples of 4d 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.