# Comments on Foliated Gauge Theories and Dualities in 3+1d

### Submission summary

 As Contributors: Po-Shen Hsin · Kevin Slagle Arxiv Link: https://arxiv.org/abs/2105.09363v3 (pdf) Date accepted: 2021-08-11 Date submitted: 2021-07-15 02:35 Submitted by: Slagle, Kevin Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory High-Energy Physics - Theory Approach: Theoretical

### Abstract

We investigate the properties of foliated gauge fields and construct several foliated field theories in 3+1d that describe foliated fracton orders both with and without matter, including the recent hybrid fracton models. These field theories describe Abelian or non-Abelian gauge theories coupled to foliated gauge fields, and they fall into two classes of models that we call the electric models and the magnetic models. We show that these two classes of foliated field theories enjoy a duality. We also construct a model (using foliated gauge fields and an exactly solvable lattice Hamiltonian model) for a subsystem-symmetry protected topological (SSPT) phase, which is analogous to a one-form symmetry protected topological phase, with the subsystem symmetry acting on codimension-two subregions. We construct the corresponding gauged SSPT phase as a foliated two-form gauge theory. Some instances of the gauged SSPT phase are a variant of the X-cube model with the same ground state degeneracy and the same fusion, but different particle statistics.

Published as SciPost Phys. 11, 032 (2021)

We thank the referees and editors for their time and for thoroughly reviewing and improving our work.

### List of changes

We have implemented many referee suggestions:
1. Clarify equation (A.3).
2. Elaborate on 3-loop braiding.
3. Conjecture D_8 model equivalence.
4. Clarify Fig. 7 commutivity.
6. Clarify E and M subscripts in Sec 2.2.1.
7. Clarify Table 1 caption.
9. Fixed typos

A more detailed overview of changes can be found in this diff:

### Submission & Refereeing History

Resubmission 2105.09363v3 on 15 July 2021
Submission 2105.09363v2 on 9 June 2021

## Reports on this Submission

### Anonymous Report 1 on 2021-7-21 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2105.09363v3, delivered 2021-07-20, doi: 10.21468/SciPost.Report.3272

### Report

I am happy with the changes and recommend the manuscript for publication. I only have some further minor comments for the authors to take into consideration below.

### Requested changes

In Fig. 1, I agree that all the terms commute and that the GSD is equal to 1. However, could the authors clarify how the subsystem symmetries are defined on the lattice? If the symmetry is defined as an application of $X$ on rigid planes as usual, then the terms are not symmetric. Perhaps the authors are defining it as an alternating product of $X$ and $X^\dagger$ along the plane?

Regarding the description for three-loop braiding, the new paragraph added is still quite hard to follow and I would still encourage an accompanying figure. However, I will leave this to the discretion of the authors.

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

### Author:  Kevin Slagle  on 2021-07-21  [id 1600]

(in reply to Report 1 on 2021-07-21)

Thank you for clarifying. We have added daggers to Fig 1 so that the subsystem symmetry is simply a product of X operators along a plane. We also added a couple sentences to the figure caption to explicitly define the symmetry. Attached is the new figure.

### Author:  Kevin Slagle  on 2021-07-27  [id 1619]

(in reply to Report 1 on 2021-07-21)

We have also created a new three-loop braiding figure (attached) for the next version.