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|As Contributors:||Usman Naseer|
|Submitted by:||Naseer, Usman|
|Submitted to:||SciPost Physics|
|Subject area:||High-Energy Physics - Theory|
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.
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.
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
See previous version.
See previous version.
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.