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Rank $Q$ E-String on Spheres with Flux

by Chiung Hwang, Shlomo S. Razamat, Evyatar Sabag, Matteo Sacchi

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Submission summary

Authors (as Contributors): Chiung Hwang · Shlomo Razamat
Submission information
Preprint link: scipost_202104_00009v2
Date accepted: 2021-08-20
Date submitted: 2021-07-27 18:25
Submitted by: Hwang, Chiung
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


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 $\mathrm{E}_8\times \mathrm{SU}(2)$ 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 $\mathrm{E}_8$ leads to the S-confinement duality of the $\mathrm{USp}(2Q)$ gauge theory with six fundamentals and a traceless antisymmetric field. Finally, we show that the theories we discuss possess an $\mathrm{SU}(2)_{\text{ISO}}$ 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 $\mathrm{SU}(2)_{\text{ISO}}$ in 4d and comparing them with the predicted anomalies from 6d.

Published as SciPost Phys. 11, 044 (2021)

Author comments upon resubmission

We would like to thank the referees for thoughtful and detailed comments that help us to improve the manuscript. We have implemented the referees' suggestions in the resubmitted version.

List of changes

Report I:

1. We have added the definition of the superconformal index in eq. (3) with a brief explanation of the notation after that.

2. We have commented in the paragraph after eq. (14) on the evidence we provide to support our cap model, which is supposed to flow to the same IR fixed point as the E-string theory compactified on a sphere with a puncture.

3. We have corrected the typos the referee pointed out.

4. $R_b$ in eq. (65), which was (63) in the previous version, is obtained by the a-maximization as we explained in the paragraph before eq. (64).

5. We have added footnote 14 explaining the gluing should be taken in such a way that the resulting theory is anomaly-free.

6. We have added discussions for the possible extension of our result in the last paragraph of introduction.

Report II:

1. We have corrected the typos the referee pointed out.

2. We have briefly commented after eq. (66) why the rational approximation of the mixing coefficients is used in our analysis.

3. We have added an explanation for the quantization of the flux in footnote 5, which was footnote 4 in the previous version.

We have also fixed the typos in eqs. (6-7) because we missed $i \sigma_2$ in (6) and the vev $\mathsf{J}_Q$ in (7) is for $\mathsf{H}$ instead of $\mathsf{O_H}$, which is the flip field of $\mathsf{H}$. We have added eq. (5) to elaborate the explanation accordingly.

Reports on this Submission

Anonymous Report 2 on 2021-8-12 (Invited Report)


In this revised version, the authors have fully addressed all the (minor) points that had been raised in the previous reports. I therefore recommend to publish the article in its current form.

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Anonymous Report 1 on 2021-8-9 (Invited Report)


In this resubmitted version, the authors have carefully implemented the referees' suggestions. The paper is therefore recommended for publication without further modifications.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
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