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Topological aspects of brane fields: solitons and higherform symmetries
by Salvatore D. Pace, Yu Leon Liu
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Submission summary
Authors (as registered SciPost users):  Salvatore Pace 
Submission information  

Preprint Link:  https://arxiv.org/abs/2311.09293v2 (pdf) 
Date accepted:  20240506 
Date submitted:  20240109 16:16 
Submitted by:  Pace, Salvatore 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
In this note, we classify topological solitons of $n$brane fields, which are nonlocal fields that describe $n$dimensional extended objects. We consider a class of $n$brane fields that formally define a homomorphism from the $n$fold loop space $\Omega^n X_D$ of spacetime $X_D$ to a space $\mathcal{E}_n$. Examples of such $n$brane fields are Wilson operators in $n$form gauge theories. The solitons are singularities of the $n$brane field, and we classify them using the homotopy theory of ${\mathbb{E}_n}$algebras. We find that the classification of codimension ${k+1}$ topological solitons with ${k\geq n}$ can be understood using homotopy groups of $\mathcal{E}_n$. In particular, they are classified by ${\pi_{kn}(\mathcal{E}_n)}$ when ${n>1}$ and by ${\pi_{kn}(\mathcal{E}_n)}$ modulo a ${\pi_{1n}(\mathcal{E}_n)}$ action when ${n=0}$ or ${1}$. However, for ${n>2}$, their classification goes beyond the homotopy groups of $\mathcal{E}_n$ when ${k< n}$, which we explore through examples. We compare this classification to $n$form $\mathcal{E}_n$ gauge theory. We then apply this classification and consider an ${n}$form symmetry described by the abelian group ${G^{(n)}}$ that is spontaneously broken to ${H^{(n)}\subset G^{(n)}}$, for which the order parameter characterizing this symmetry breaking pattern is an ${n}$brane field with target space ${\mathcal{E}_n = G^{(n)}/H^{(n)}}$. We discuss this classification in the context of many examples, both with and without 't Hooft anomalies.
Published as SciPost Phys. 16, 128 (2024)
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2024425 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2311.09293v2, delivered 20240425, doi: 10.21468/SciPost.Report.8934
Report
The paper discusses the classification of topological solitons in theories with extended degrees of freedom or brane fields. This requires a more sophisticated mathematical analysis that goes beyond homotopy groups and utilises homotopy algebraic structures. The classification is illustrated and compared to numerous concrete examples, which complement the more formal mathematical parts of the paper. An interesting result of the analysis is the prediction of topological solitons in higher form gauge theories that go beyond the usual gauge theory classification.
To summarise, the paper initiates a systematic study of the classification of topological solitons in theories with brane fields. The paper is exceptionally clearly written with concrete examples complementing formal calculations. I recommend it for publication.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Report #1 by YuAn Chen (Referee 1) on 2024416 (Invited Report)
 Cite as: YuAn Chen, Report on arXiv:2311.09293v2, delivered 20240416, doi: 10.21468/SciPost.Report.8894
Report
I have completed a review of the manuscript "Topological Aspects of Brane Fields: Solitons and HigherForm Symmetries" by Salvatore D. Pace and Yu Leon Liu. This manuscript contributes significantly to the theoretical physics discourse by articulating the classification of topological solitons in nbrane fields via the homotopy theory of Enalgebras. The authors proficiently elucidate the classification of singularities within these fields, effectively demonstrating the connection between topological solitons and their codimensions, leveraging the framework of homotopy groups of En.
The manuscript's principal strengths are as follows:
1. Applying a comprehensive and rigorous theoretical framework for classifying topological solitons, augmenting the existing academic literature on the topic.
2. An exhaustive examination of nbrane fields and nform gauge theories, enhancing their foundational understanding.
3. A thorough explanation for the practical implications of studying spontaneously broken higherform symmetries.
Moreover, the manuscript underscores the role of topological solitons in comprehending generalized symmetries and analyzing phase transitions, both of which are crucial areas of current and future research.
Given the manuscript's clear presentation, relevance to current research trends, and potential impact on future theoretical studies, I recommend its acceptance for publication in SciPost Physics. The authors have maintained a rigorous academic standard, presenting robust and wellsubstantiated results. This work advances the theoretical constructs it addresses and is a valuable resource for researchers exploring the broader implications of brane field dynamics and topological phenomena. It is poised to capture significant interest from the SciPost Physics readership and to stimulate further research and discussion within the academic community. Therefore, I strongly endorse its publication.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)