# (1,0) gauge theories on the six-sphere

### Submission summary

 As Contributors: Usman Naseer Arxiv Link: https://arxiv.org/abs/1809.06272v3 Date submitted: 2018-12-04 Submitted by: Naseer, Usman Submitted to: SciPost Physics Domain(s): Theoretical Subject area: High-Energy Physics - Theory

### Abstract

We construct gauge theories with a vector-multiplet and hypermultiplets of $(1,0)$ supersymmetry on the six-sphere. The gauge coupling on the sphere depends on the polar angle. This has a natural explanation in terms of the tensor branch of $(1,0)$ theories on the six-sphere. For the vector-multiplet we give an off-shell formulation for all supersymmetries. For hypermultiplets we give an off-shell formulation for one supersymmetry. We show that the path integral for the vector-multiplet localizes to solutions of the Hermitian-Yang-Mills equation, which is a generalization of the (anti-)self duality condition to higher dimensions. For the hypermultiplet, the path integral localizes to configurations where the field strengths of two complex scalars are related by an almost complex structure.

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

### Author comments upon resubmission

Dear Editor,

In this revised version I have fixed several typos pointed out by all three referees. An important change is the discussion about the regularity of the vector-multiplet action at the end of section 3. This point was raised by two referees.

Thank you,
-Usman

### List of changes

1. "vector-multiplet" changed to "vector multiplet" in a lot of places.
2. end of intro: "furhter"->"further"
3. in sec.2, new reference is added [arXiv:1209.5408], which also discussed 4d N=1 susy on S^4.
4.before eq.(4.6): "cupling"-->"coupling"
5.after eq.(4.12): "in above arguments"-->"in the above argument"
6.intro to sec.5: "except the south pole"-->"except at the south pole"
7.before eq.(5.3): "contirbute"-->"contribute"
8.middle of sec.6: "be achieve by"-->"be achieved by"
9.end of sec.6: "à la localization" with "à".
10. Typo in eq. 5.9 is fixed.
11. At the end of section 3, a field redefinition is discussed which makes the Lagrangian manifestly regular everywhere.
12. chirality condition is explicitly mentioned in the beginning of section 2 after 2.2

### Submission & Refereeing History

Resubmission 1809.06272v3 (4 December 2018)
Submission 1809.06272v2 (29 October 2018)

## Invited Reports on this Submission

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### Strengths

See previous version.

### Weaknesses

See previous version.

### Report

This is an interesting and topical paper. See my review of the previous version. The new version has only minor changes. This is a good paper that I would recommend for publication.

### Requested changes

none.

• validity: good
• significance: ok
• originality: ok
• clarity: good
• formatting: good
• grammar: good

### Report

The modifications address the minor issues raised in the previous report.

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