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Squashing and supersymmetry enhancement in three dimensions
by Joseph Minahan, Usman Naseer, Charles Thull
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Submission summary
| Authors (as registered SciPost users): | Usman Naseer · Charles Thull |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2107.07151v1 (pdf) |
| Date submitted: | 2021-08-11 09:09 |
| Submitted by: | Thull, Charles |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We consider mass-deformed theories with ${\cal N}\geq2$ supersymmetry on round and squashed three-spheres. By embedding the supersymmetric backgrounds in extended supergravity we show that at special values of mass deformations the supersymmetry is enhanced on the squashed spheres. When the $3d$ partition function can be obtained by a limit of a $4d$ index we also show that for these special mass deformations only the states annihilated by extra supercharges contribute to the index. By using an equivalence between partition functions on squashed spheres and ellipsoids, we explain the recently observed squashing independence of the partition function of mass-deformed ABJM theory on the ellipsoid. We provide further examples of such simplification for various $3d$ supersymmetric theories.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2021-11-8 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2107.07151v1, delivered 2021-11-08, doi: 10.21468/SciPost.Report.3810
Report
In this paper, the authors consider mass-deformed 3d ${\cal N} = 2$ theories on a squashed sphere, and they determine certain relations between the mass and the squashing parameter at which there is more supersymmetry. In the cases in which there is supersymmetry enhancement, the authors point out that the $S^3$ partition function simplifies, and in some cases it even becomes independent of the squashing parameter, as had been observed in previous work.
I think the paper is very interesting, and I have a few small comments about potential improvements:
1. When examining the ellipsoid in section 2.2, it would be very useful to write down the frame in which the spinors are given.
2. Similarly, it would be useful to write down the gamma matrices that are used throughout the paper, and also explain the conventions used for multiplying spinors (for instance, in (2.6)).
3. I suspect that in equation (3.3) the right-hand sides of the two lines were accidentally swapped. The authors should double check whether the equations are correct as written.
4. Regarding (A.6): for 3d theories with ${\cal N} = 4$ supersymmetry, one can have vector multiplets and hypermultiplets, but one can also have twisted vector multiplets and twisted hypermultiplets. It would be useful to give the partition functions of the latter two as well, in case they are different from the ones for vector multiplets and hypermultiplets, respectively.
5. Actually, the authors should check whether the formulas in (A.6) are correct. I would've thought that the second line should either have only $\cosh (\pi b \langle \sigma, \alpha \rangle )$ or $\cosh (\pi b^{-1} \langle \sigma, \alpha \rangle )$, with a product over all roots, but I may be wrong. (In other words, I would've thought that this formula should not be symmetric under $b \to 1/b$.)
6. It wasn't clear to me whether the "explaination" of the fact that the partition function of ABJM theory is independent of squasing (provided that the mass is tuned such that there is enhanced supersymmetry) is complete. The authors have shown that there is enhanced supersymmetry, but what I think is still needed is to show that the squashing deformation is Q-exact in this case. It would be great if the authors could add some clarifying comments about whether this Q-exactness is obvious from their analysis, or whether there is still some work to be done.
In any case, I think this paper is of interest to the community of researchers studying supersymmetric field theories on curved manifolds, so I recommend it for publication after the authors have had a chance to consider the improvements/comments mentioned above.
Report #1 by Anonymous (Referee 1) on 2021-8-19 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2107.07151v1, delivered 2021-08-18, doi: 10.21468/SciPost.Report.3412
Strengths
1. Gives a clear geometrical explanation for a recently observed simplification of the squashed sphere free energy for ABJM theory.
Report
This paper explains a recently observed demonstration of the simplification of the mass deformed squashed sphere free energy in ABJM theory when the mass is fixed to a certain function of squashing. The explanation is that supersymmetry is enhanced specifically at this point, which the authors show explicitly. They also show a similar simplification at a different value of the mass, and also similar simplifications for 3d CFTs with less supersymmetry.
Requested changes
1. on page 1, it says "Using holographic duality, these relations imply non-trivial constraints on string theory amplitudes and can be used to rule out certain
higher-derivative terms in string theory effective actions". I would more accurately say that the constraints allow one to fix protected terms in the effective action, which in some case turn out to be zero (but in other cases are nonzero).
2. on page 2, it says "A similar phenomenon was discovered by [17] for ABJM theory". This relation was actually shown for the more general case of ABJ theory (ie nonequal ranks of the gauge group). In general throughout the paper, it would be useful to generalize results to the case of ABJ, which should be straightforward.
3. it would be useful to reference the results of appendix A.2 for the Nf model in the main text, as otherwise readers might not be aware of it.