# Rank $Q$ E-string on a torus with flux

## TL;DR

This paper explores compactifications of rank Q E-string theory on a torus with fluxes, introducing a new model E[USp(2Q)] that reveals novel symmetry enhancements and connections to 3d theories like T[SU(Q)].

## Contribution

The paper introduces the E[USp(2Q)] model and demonstrates its role in constructing 4d theories with emergent IR symmetries, linking 6d, 3d, and 4d theories.

## Key findings

- E[USp(2Q)] has global symmetry USp(2Q)×USp(2Q)×U(1)^2.
- IR symmetry enhancement from SU(2)^Q to USp(2Q).
- Connections established between 6d E-string compactifications and 3d T[SU(Q)] theories.

## Abstract

We discuss compactifications of rank $Q$ E-string theory on a torus with fluxes for abelian subgroups of the $E_8$ 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 $USp(2Q)\times USp(2Q)\times U(1)^2$ 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 $4d$ $\mathcal{N}=4$ $SU(Q)$ 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.