Rank $Q$ E-String on Spheres with Flux

TL;DR

This paper explores compactifications of rank Q E-string theory on spheres with flux, constructing models with symmetry enhancements and dualities, and relating 4D symmetries to 6D isometries.

Contribution

It introduces a new class of 4D models from E-string compactifications, demonstrating symmetry enhancements, dualities, and geometric interpretations of symmetries.

Findings

Construction of the cap model for sphere with one puncture.

Identification of IR symmetry enhancements and dualities.

Evidence linking 4D SU(2) symmetry to 6D sphere isometry.

Abstract

We consider compactifications of rank $Q$ E-string theory on a genus zero surface with no punctures but with flux for various subgroups of the $\text{E}_8\times \text{SU}(2)$ global symmetry group of the six dimensional theory. We first construct a simple Wess-Zumino model in four dimensions corresponding to the compactification on a sphere with one puncture and a particular value of flux, the cap model. Using this theory and theories corresponding to two punctured spheres with flux, one can obtain a large number of models corresponding to spheres with a variety of fluxes. These models exhibit interesting IR enhancements of global symmetry as well as duality properties. As an example we will show that constructing sphere models associated to specific fluxes related by an action of the Weyl group of $\text{E}_8$ leads to the S-confinement duality of the $\text{USp}(2Q)$ gauge theory with six fundamentals and a traceless antisymmetric field. Finally, we show that the theories we discuss possess an $\text{SU}(2)_{\text{ISO}}$ symmetry in four dimensions that can be naturally identified with the isometry of the two-sphere. We give evidence in favor of this identification by computing the `t Hooft anomalies of the $\text{SU}(2)_{\text{ISO}}$ in 4d and comparing them with the predicted anomalies from 6d.