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Higher-Matter and Landau-Ginzburg Theory of Higher-Group Symmetries

Ruizhi Liu, Ran Luo, Yi-Nan Wang

TL;DR

This work addresses the challenge of extending symmetry concepts to higher-categorical structures, focusing on 2- and 3-group symmetries and the associated higher-matter. It develops a practical Lagrangian formalism by strictifying weak higher-groups, introducing automorphism 2-representations for higher-matter, and formulating higher-gauge theories in the path space. The authors provide explicit constructions: strictification procedures for 2- and select 3-group cases, a dictionary connecting strict and weak gauge fields, and a Landau-Ginzburg model for 2-group symmetries that exhibits spontaneous symmetry breaking, including non-split to split transitions controlled by the Postnikov class. The results offer a concrete, versatile framework for analyzing higher-group symmetries in quantum field theory and condensed matter, enabling both continuum and lattice-inspired realizations, as well as potential connections to center-symmetry physics and brane-like higher-charge objects.

Abstract

Higher-matter is defined by higher-representation of a symmetry algebra, such as the $p$-form symmetries, higher-group symmetries or higher-categorical symmetries. In this paper, we focus on the cases of higher-group symmetries, which are formulated in terms of the strictification of weak higher-groups. We systematically investigate higher-matter charged under 2-group symmetries, defined by automorphism 2-representations. Furthermore, we construct a Lagrangian formulation of such higher-matter fields coupled to 2-group gauge fields in the path space of the spacetime manifold. We interpret such model as the Landau-Ginzburg theory for 2-group symmetries, and discuss the spontaneous symmetry breaking (SSB) of 2-group symmetries under this framework. Examples of discrete and continuous 2-groups are discussed. Interestingly, we find that a non-split 2-group symmetry can admit an SSB to a split 2-group symmetry, where the Postnikov class is trivialized. We also briefly discuss the strictification of weak 3-groups, weak 3-group gauge fields and 3-representations in special cases.

Higher-Matter and Landau-Ginzburg Theory of Higher-Group Symmetries

TL;DR

This work addresses the challenge of extending symmetry concepts to higher-categorical structures, focusing on 2- and 3-group symmetries and the associated higher-matter. It develops a practical Lagrangian formalism by strictifying weak higher-groups, introducing automorphism 2-representations for higher-matter, and formulating higher-gauge theories in the path space. The authors provide explicit constructions: strictification procedures for 2- and select 3-group cases, a dictionary connecting strict and weak gauge fields, and a Landau-Ginzburg model for 2-group symmetries that exhibits spontaneous symmetry breaking, including non-split to split transitions controlled by the Postnikov class. The results offer a concrete, versatile framework for analyzing higher-group symmetries in quantum field theory and condensed matter, enabling both continuum and lattice-inspired realizations, as well as potential connections to center-symmetry physics and brane-like higher-charge objects.

Abstract

Higher-matter is defined by higher-representation of a symmetry algebra, such as the -form symmetries, higher-group symmetries or higher-categorical symmetries. In this paper, we focus on the cases of higher-group symmetries, which are formulated in terms of the strictification of weak higher-groups. We systematically investigate higher-matter charged under 2-group symmetries, defined by automorphism 2-representations. Furthermore, we construct a Lagrangian formulation of such higher-matter fields coupled to 2-group gauge fields in the path space of the spacetime manifold. We interpret such model as the Landau-Ginzburg theory for 2-group symmetries, and discuss the spontaneous symmetry breaking (SSB) of 2-group symmetries under this framework. Examples of discrete and continuous 2-groups are discussed. Interestingly, we find that a non-split 2-group symmetry can admit an SSB to a split 2-group symmetry, where the Postnikov class is trivialized. We also briefly discuss the strictification of weak 3-groups, weak 3-group gauge fields and 3-representations in special cases.
Paper Structure (61 sections, 3 theorems, 307 equations, 3 figures, 1 table)

This paper contains 61 sections, 3 theorems, 307 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Strictification of gauge fields
  3. Higher-matter
  4. Landau-Ginzburg model for strict 2-group symmetries
  5. 3-groups
  6. Strictification of 2-groups
  7. A review of 2-groups
  8. A general procedure of strictification
  9. Example: $\Pi_1=\Pi_2=\mathbb{Z}_2$
  10. Example: $\Pi_1=\mathbb{Z}_N$, $\Pi_2=U(1)$
  11. Example: $\Pi_1=\Pi_2=U(1)$
  12. Strictification of 3-groups
  13. Semi-strict 3-group as a 2-crossed-module
  14. Weak 3-groups with $\Pi_{2}=0$
  15. Example: $\Pi_1=\mathbb{Z}_2$, $\Pi_2=0$, $\Pi_3=\mathbb{Z}_2$
  16. ...and 46 more sections

Key Result

Theorem 1

The cohomology class $H^{4}(B^{2}\Pi_{2};\Pi_{3})$ is in one to one correspondence with $\Pi_{3}$-valued quadratic functions on $\Pi_{2}$.

Figures (3)

  • Figure 1: This figure shows how we construct an automorphism 2-representation. We choose a suitable algebra $A$, and its automorphism 2-group $\mathcal{A}ut(A)$ can be calculated by definition. Then we build an intertwiner $\mathcal{R}$ to embed the 2-group structure into $\mathcal{A}ut(A)$.
  • Figure 2: Here we demonstrate the picture of path space derivative. The above shows the original line in magenta, the below shows both the original line in magenta and the deformed line in cyan. The line in cyan is formed by a deviation from $\sigma$ to $\sigma + \epsilon$ along the $\mu$ direction.
  • Figure 3: From left to right, this figure demonstrates three different cases of loop deformation described above. The orange lines are the loops before deformation and cyan lines are loops after deformation. The lime-coloured region signifies the change of area.

Theorems & Definitions (9)

  • Definition 3.1
  • Theorem 1
  • Theorem 2
  • Example 1
  • Remark 1
  • Definition 5.1
  • Definition 5.2
  • Lemma 1
  • proof