# 3d N=4 Bootstrap and Mirror Symmetry

### Submission summary

 As Contributors: Chi-Ming Chang · Ying-Hsuan Lin · Yifan Wang Arxiv Link: https://arxiv.org/abs/1910.03600v2 (pdf) Date submitted: 2020-10-21 03:08 Submitted by: Chang, Chi-Ming Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Approaches: Theoretical, Computational

### Abstract

We investigate the non-BPS realm of 3d ${\cal N} = 4$ superconformal field theory by uniting the non-perturbative methods of the conformal bootstrap and supersymmetric localization, and utilizing special features of 3d ${\cal N} = 4$ theories such as mirror symmetry and a protected sector described by topological quantum mechanics (TQM). Supersymmetric localization allows for the exact determination of the conformal and flavor central charges, and the latter can be fed into the mini-bootstrap of the TQM to solve for a subset of the OPE data. We examine the implications of the $\mathbb{Z}_2$ mirror action for the SCFT single- and mixed-branch crossing equations for the moment map operators, and apply numerical bootstrap to obtain universal constraints on OPE data for given flavor symmetry groups. A key ingredient in applying the bootstrap analysis is the determination of the mixed-branch superconformal blocks. Among other results, we show that the simplest known self-mirror theory with $SU(2) \times SU(2)$ flavor symmetry saturates our bootstrap bounds, which allows us to extract the non-BPS data and examine the self-mirror $\mathbb{Z}_2$ symmetry thereof.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission scipost_202102_00025v2 on 3 April 2021
Resubmission scipost_202102_00025v1 on 17 February 2021

Submission 1910.03600v2 on 21 October 2020

## Reports on this Submission

### Anonymous Report 1 on 2021-1-25 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1910.03600v2, delivered 2021-01-25, doi: 10.21468/SciPost.Report.2465

### Strengths

1. This paper considers 3d SCFTs with N=4 supersymmetries taking inputs from both localization techniques and the numerical CFT bootstrap.
2. New results are found for the non-BPS aspects of mirror symmetry.
3. The protected 1d subsector of topological quantum mechanics is studied using the bootstrap.

### Weaknesses

1. Subject in general is very technical in nature and for a non-expert, the dense nature of the writing can be very difficult to follow.

### Report

The paper is a useful addition to the 3d SCFT literature and provides several benchmarks for future research in this direction. As such I feel the paper can be published in SciPost.

### Requested changes

1. In table 1 (and appendix) the C_T entry for E_6 is 160.2. Is this approximately 160.2 (and if so to what accuracy?).
2. In equation 5.21 an error value of (2) is quoted. How was this obtained? Same question for 5.22. Why is there no error value in 5.23?
3. Owing to the extremely technical nature of the paper, I feel that it would make it somewhat more readable if the main text were to focus on the results and technical details and background material were relegated to the appendices. However, I leave to the authors' better judgment how to improve the readability.

• validity: high
• significance: high
• originality: high
• clarity: good
• formatting: good
• grammar: excellent

### Author:  Chi-Ming Chang  on 2021-02-17  [id 1248]

(in reply to Report 1 on 2021-01-25)

We thank the referee for the comments, and we have made corrections and adjustments accordingly, please see our new version. Our responses are below.

1- Please see footnote 30 on page 56 of the new version. 2- Please see footnote 25 on page 43 and the footnote 28 on page 48 of the new version. 3- We added a brief summary of the main results of this paper in the introduction section of the new version.