Higher structure of non-invertible symmetries from Lagrangian descriptions

TL;DR

The paper develops a Lagrangian framework to access the higher-structure of non-invertible symmetries in QFTs, focusing on defect networks, local fusion junctions, and associators (F-symbols) in both 2d and 4d settings. By constructing explicit defect actions for the 2d Tambara-Yamagami TY(ℤ_N,+1) category and 4d Maxwell theory’s non-invertible duality/triality defects, the authors recover all TY F-symbols and obtain 2d TFT realizations for the 4d associators, with some dependence on boundary data. The results are cross-validated against a group-theoretical approach, showing agreement between Lagrangian and algebraic constructions and yielding new explicit associator data in Maxwell theory. Overall, the work demonstrates how Lagrangian descriptions can encode and compute higher-category data for non-invertible symmetries, with potential implications for understanding symmetry, dualities, and gapped phases in QFTs.

Abstract

The symmetry structure of a quantum field theory is determined not only by the topological defects that implement the symmetry and their fusion rules, but also by the topological networks they can form, which is referred to as the higher structure of the symmetry. In this paper, we consider theories with non-invertible symmetries that have an explicit Lagrangian description, and use it to study their higher structure. Starting with the 2d free compact boson theory and its non-invertible duality defects, we will find Lagrangian descriptions of networks of defects and use them to recover all the $F$-symbols of the familiar Tambara-Yamagami fusion category $\operatorname{TY}(\mathbb{Z}_N,+1)$. We will then use the same approach in 4d Maxwell theory to compute $F$-symbols associated with its non-invertible duality and triality defects, which are 2d topological field theories. In addition, we will also compute some of the $F$-symbols using a different (group theoretical) approach that is not based on the Lagrangian description, and find that they take the expected form.

Figures (22)

• Figure 1: (a) Gauging some symmetry $\mathbb{A}$ in $\operatorname{End}_{\mathfrak{C}^{[3]}}(\mathcal{D})$ on half-space with Dirichlet boundary condition creates a non-trivial topological interface between $\mathcal{D}$ and $\mathcal{D}/\mathbb{A}$. (b) Stacking a decoupled 3d TFT $\mathcal{T}$ with the gapped boundary $\mathcal{B}$ creates a non-trivial topological interface between $\mathcal{D}$ and $\mathcal{D}\mathcal{T}$. $\mathcal{D}$, $\mathcal{D}/\mathbb{A}$, and $\mathcal{D}\mathcal{T}$ are in the same equivalence class, i.e., $[\![ \mathcal{D} ]\!] = [\![ \mathcal{D}/\mathbb{A} ]\!] = [\![ \mathcal{D} \mathcal{T} ]\!]$.
• Figure 2: Given with two codim-$1$ defects $\mathcal{D}_a$ and $\mathcal{D}_b$ where $\mathcal{D}_a = [\![\mathcal{D}_a]\!]_0, \mathcal{D}_b = [\![ \mathcal{D}_b]\!]_0$, one can always construct a topological local fusion junction into $[\![\mathcal{D}_a \otimes \mathcal{D}_b]\!]_0$ as follows. Starting with two parallel $\mathcal{D}_a$ and $\mathcal{D}_b$, fusing the upper part leads to a topological local fusion junction from $\mathcal{D}_a$ and $\mathcal{D}_b$ to $\mathcal{D}_a \otimes \mathcal{D}_b$. Then, gauging the corresponding symmetry on the world-volume of $\mathcal{D}_a \otimes \mathcal{D}_b$ with Dirichlet boundary condition on the fusion junction leads to a topological local fusion junction with the outgoing line being $[\![ \mathcal{D}_a \otimes \mathcal{D}_b ]\!]_0$.
• Figure 3: Description of an $\eta^k$ line defect using two fields, $\phi_L$ and $\phi_R$, from the two sides of the defect and a field $\varphi$ which is supported on the defect.
• Figure 4: Topological construction of the local fusion junction for Hom($a\times b,c$). $L_{1,2}$ in the second diagram are the semi-infinite lines from the bottom that end on the local fusion junction at $t=0$ (blue dot).
• Figure 5: Topological construction of the local fusion junction for Hom($T\times \eta^k,T$).