# Squashing and supersymmetry enhancement in three dimensions

### Submission summary

 As Contributors: Charles Thull Arxiv Link: https://arxiv.org/abs/2107.07151v2 (pdf) Date accepted: 2021-12-01 Date submitted: 2021-11-17 13:31 Submitted by: Thull, Charles Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory 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 ABJ(M) theory on the ellipsoid. We provide further examples of such simplification for various $3d$ supersymmetric theories.

###### Current status:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

We thank the referees for their careful reading of the manuscript and for their valuable suggestions. We have made several changes in the new version following the points made by the referees. These changes are listed in the the appropriate section of our resubmission page.

### List of changes

1. We use more precise language on page 1 as advocated by the first referee.
2. We generalized our discussions so that they are now applicable to ABJ(M) theory.
3. We mention the $N_f$ model at the end of paragraph 5 of the introduction, as well as in the last paragraph of the introduction.
4. We added the frame for the ellipsoid as eq. (2.29).
5. We added a footnote on the gamma matrix convention to equation (2.1).
6. We added a comment on twisted multiplets in the paragraph before (A.6).
7. We fixed typos and changed the notation in appendix A to clarify various expressions. The simplified one-loop determinants for the $N=4$ vector and hyper multiplets are now given in eq. (A.6) through (A.9).
8. In the second to last paragraph of the introduction we have outlined the logic of the paper which addresses the referee’s remark regarding Q-exactness.

Apart from the above changes directly related to the referees’ concerns we have taken the opportunity to fix several typos and make improvements to the draft. The significant changes are listed below.

1. We added a footnote on contracting spinors to eq. (2.6).
2. We added a note on $\mu=-\io\frac{b+b^{-1}}/2$ after (4.20).
3. In (A.3) and the following equations we corrected the charge that couples to $\mu$ as the  flavor charge $F$.
4. We added expressions for the ${\cal N}=4$ vector one-loop partition function, explaining the squashing independence.
5. We added a comment on the flavor charge of the ${\cal N}=4$ multiplet constituents.
6. We revised the $N_f$-model in section (A.2).