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

### Submission summary

 As Contributors: Sara Pasquetti · Shlomo Razamat · Matteo Sacchi Arxiv Link: https://arxiv.org/abs/1908.03278v2 (pdf) Date accepted: 2020-01-22 Date submitted: 2020-01-15 01:00 Submitted by: Razamat, Shlomo Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Approach: Theoretical

### 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.

### Ontology / Topics

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Published as SciPost Phys. 8, 014 (2020)

We are grateful to the referees for their thoughtful and detailed comments. We have implemented the suggestions of both referees.

### List of changes

Referee I:

1 -- We have rechecked the equation 4.18 and it is OK as written. (One issue that often causes mistakes is that the antisymmetric fields should be remembered to be taken traceless.)

2 -- We have added footnote 1 on page 4 to comment on the needed holonomy.

3 -- We have added the reference to fields $b_n$ in the caption of Figure 3.

4 -- We added footnote 8 to refer to equation 3.19 for the proper relation between the parameters.

We have fixed the typos found by the referee.

Referee II:

1 -- We fixed equation 2.5. We thank the referee for spotting this typo.

2 -- We streamlined the references to different types of models. In particular we now refer only to $FM[SU(Q)]$ and not to $FM[U(Q)]$. The partition function of FM[U(Q)] appears only in intermediate steps of the calculation and we added a comment around 5.24 to explain how it is related to the one of $FM[SU(Q)]$.

4 -- We added a comment below 3.13 explaining how the entry should be determined.

6 -- We have rephrased the discussion around equation 5.2.

We have fixed the typos and parsing issues mentioned by the referee.