Topological defects

TL;DR

Topological defects provide a unifying framework across lattice models, 2d QFT, and higher-dimensional QFT by organizing into higher categories. The survey develops the 2d defect QFT framework (pivotal 2-categories, fusion categories) and demonstrates gauging via symmetric Frobenius algebras, plus the 3d/4d TFT perspectives (defect TFTs, condensations, orbifold completions) and the symmetry-TFT approach to dualities. It then extends these ideas to $d>2$ by examining invertible $p$-form defects, correlator identities, and non-invertible duality defects, with Maxwell/QED as concrete examples. Collectively, the work provides a cohesive, category-theoretic picture of defects, gauging, dualities, and symmetry in QFTs of varied dimensions, highlighting the role of non-invertible defects in constraining vacua, correlators, and phase structure. The results illuminate how higher-categorical structures underpin known dualities and predict new non-invertible symmetry phenomena in four-dimensional theories.

Abstract

This is a survey article for the Encyclopedia of Mathematical Physics, 2nd Edition. Topological defects are described in the context of the 2-dimensional Ising model on the lattice, in 2-dimensional quantum field theory, in topological quantum field theory in arbitrary dimension, and in higher-dimensional quantum field theory with a focus on 4-dimensional quantum electrodynamics.

Figures (5)

• Figure 3.1: a) Example of a surface $S$ with labels $a,b,c,d \in D_2$ for 2d patches and $v,w,x,y,z \in D_1$ for line components such that $s(v)=a$, $t(v)=b$, etc. b) Circle with marked points labelled by $\underline x^*$ and $\underline y$ to which $Q$ assigns the state space $\mathcal{H}_r(\underline x, \underline y)$. c) Example of a surface $S$ entering the condition for scale and translation invariant states.
• Figure 3.2: Surfaces defining (a) the vertical composition $\underline x \xrightarrow{u} \underline y \xrightarrow{v} \underline z$ and (b) the horizontal composition $(\underline y \xrightarrow{g} \underline y') \otimes (\underline x \xrightarrow{f} \underline x')$, as well as (c) the adjunction 2-morphism $1_b \to \underline x \otimes \underline x^*$.
• Figure 3.3: a) Part of a triangulation with oriented edges and the corresponding dual $A$-network. b) Invariance under orientation flip of the internal edge requires $\mathcal{A}$ to be symmetric. c) The Nakayama automorphism $N_\mathcal{A}$ of $\mathcal{A}$ as a string diagram.
• Figure 3.4: a) A surface defect $\mathcal{M}$ in the Reshetikhin--Turaev TFT for the modular fusion category $\mathcal{C}$, and line defects $U,V$ starting at the holomorphic and antiholomorphic boundary and meeting at the topological junction $\varphi$. b) Description of the surface defect via the gaugeable topological symmetry $\mathcal{A} = (A,\mu,\Delta)$. c) The folded picture of a) gives a symmetry TFT description of 2d CFT correlators.
• Figure 5.1: Example of an identity among correlators induced by a topological defect. Here, $\mathcal{D} = \mathcal{D}^{(0)}$ is a codimension-1 topological defect of $\mathcal{T}$ for which we assume that $\mathcal{D}(S^{d-1}) = N \cdot \mathbf{1}_{\mathcal{T}}$ for some $N\in\mathds C^\times$. (1) Insert $\mathcal{D}(S^{d-1})$ and multiply the correlators by $N^{-1}$; (2) expanding the bubble across $S^d$, the topological defect $\mathcal{D}$ wraps the supports of the operators $\mathcal{O}_i$ and moves to the opposite side of the sphere; (3) the topological defect shrinks on the opposite side of $S^d$, and the local operators $\mathcal{O}_i$ become twist operators $\mathcal{O}_i'$ sitting at the end of topological line defects $\mathcal{L}^{(d-2)}$; (4) the remaining $\mathcal{D}$-sphere shrinks into a topological point defect $\mathcal{J}(y)$, into which we also absorb the prefactor $N^{-1}$.

Theorems & Definitions (12)

• Proposition 3.1
• Example 3.3
• Proposition 3.4
• proof
• Example 3.5
• Theorem 3.7
• Example 3.8
• Remark 5.1
• Example 5.2
• Example 5.3