# Rank $Q$ E-String on Spheres with Flux

### Submission summary

 As Contributors: Chiung Hwang Preprint link: scipost_202104_00009v1 Date submitted: 2021-04-08 10:24 Submitted by: Hwang, Chiung Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Approach: Theoretical

### 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 $\mathrm{E}_8\times \mathrm{SU}(2)$ 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.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission scipost_202104_00009v1 on 8 April 2021

## Reports on this Submission

### Strengths

1 - provides a new construction of a class of theories that opens up potential new vantage points on certain symmetries and features
2 - concrete checks of the proposed construction via detailed computation of anomalies
3 - clarity of the presentation, well written

### Weaknesses

1 - it might be useful to add a more concrete discussion on how these results can be used for future computations and new directions

### Report

In this paper the authors construct compactifications of the rank $Q$ E-string theory on a sphere with different amounts of flux. These theories can be obtained from a 6-dimensional $\mathcal{N}=(1,0)$ SCFT that is realised by $Q$ M5-branes probing an M9-brane. In this paper the authors present a different construction by gluing together different theories on punctured spheres with specific flux: indeed, starting from the so-called tube model (a theory on a two-punctured sphere) considered previously in the literature (by some of the current authors), the authors create the so-called cap model by 'closing' one of the punctures by introducing suitable vevs and adding gauge invariant flipping fields to obtain the correct anomalies. Compactifications of the E-string theories can be obtained by gluing cap- and tube models via adding chiral fields in a suitable representation of various $Usp(2Q)$ symmetries appearing in the tube models. The authors analyse the symmetry content of the resulting theories and calculate the anomalies to check whether they indeed correspond to the rank $Q$ E-string theories.

As far as I can tell, the computations in this paper are genuine and lead to interesting and novel results: they lead to a new construction of a class of interesting theories that gives a new angle on certain of its properties and symmetries, for example a geometric interpretation of the appearance of an $E_6$ symmetry in a sphere model that was previously discussed in the literature. Furthermore, the paper is well structured and the various appendices make it fairly accessible. Based on this, I recommend it for publication.

There are only very few minor points that perhaps the authors could comment on:

-) it would be helpful to discuss in the main text some of the notation that is used throughout. While there is appendix D, it might help, for example to clarify in eq.(3) which chemical potentials $p$ and $q$ encode

-) after eq.(12): it would be very helpful to elaborate to what degree checking that the two models have the same anomalies guarantees that indeed the theory constructed here is a compactification of the E-string theory, or (if so) which possible ambiguities are left

-) eq.(19): comparing with eq.(12), I don't understand why the argument of the $\Gamma_e$ in the first product contains $x_n$ rather than $y_n$

-) I don't understand how the numerical value of $R_b$ in eq.(63) has been obtained (or similar results in subsequent sections)

-) When gluing various cap- and tube theories together by adding chiral fields, are there any constraints on the choice of $\Phi$- or $S$-gluing? While I understand that in order to reproduce the correct E-string models, a particular choice is required, is it clear that other choices lead to inherently inconsistent models or are there a priori also other viable theories? Does this for example produce inconsistencies at the level of the obtained anomalies, thus providing intrinsic selection rules?

-) The authors have not included a Conclusion section: I agree with their decision that they have (notably in the Introduction) outlined their results and put them into context (such that a Conclusion would be a repetition of what has been already said). However, an aspect which might be interesting to discuss is what other possibilities the description of the E-string theories in the current theory entails: since the tube theory is well studied (and runs to a simple WZW model), does this allow other quantities (i.e. other than anomalies) to be computed which are inaccessible by other methods? Can the current methods be applied to other classes of compactifications?

• validity: top
• significance: high
• originality: high
• clarity: top
• formatting: perfect
• grammar: perfect