Topological aspects of brane fields: solitons and higher-form symmetries
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2023
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| _version_ | 1866917671488978944 |
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| author | Pace, Salvatore D. Liu, Yu Leon |
| author_facet | Pace, Salvatore D. Liu, Yu Leon |
| contents | 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 $Ω^n X_D$ 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 ${π_{k-n}(\mathcal{E}_n)}$ when ${n>1}$ and by ${π_{k-n}(\mathcal{E}_n)}$ modulo a ${π_{1-n}(\mathcal{E}_n)}$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_09293 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Topological aspects of brane fields: solitons and higher-form symmetries Pace, Salvatore D. Liu, Yu Leon High Energy Physics - Theory Statistical Mechanics Strongly Correlated Electrons Mathematical Physics 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 $Ω^n X_D$ 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 ${π_{k-n}(\mathcal{E}_n)}$ when ${n>1}$ and by ${π_{k-n}(\mathcal{E}_n)}$ modulo a ${π_{1-n}(\mathcal{E}_n)}$ 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. |
| title | Topological aspects of brane fields: solitons and higher-form symmetries |
| topic | High Energy Physics - Theory Statistical Mechanics Strongly Correlated Electrons Mathematical Physics |
| url | https://arxiv.org/abs/2311.09293 |