Topological non-linear $σ$-model, higher gauge theory, and a realization of all 3+1D topological orders for boson systems

TL;DR

<3-5 sentence high-level summary>Topological non-linear σ-models provide a unifying framework to realize and classify 3+1D bosonic topological orders by discretizing both spacetime and target spaces. The paper shows how 1-gauge theories (Dijkgraaf–Witten) arise from classifying spaces and how higher gauge theories emerge from 2-group targets, yielding a complete realization of EF1 orders via 2-gauge theories and EF2 orders via topological non-linear σ-models built from fusion-2-category data. It further connects the boundary data (unitary fusion 2-categories) to bulk topological orders, and proposes a holographic viewpoint that the boundary fully determines the bulk. Additionally, it develops explicit cocycle-based constructions, dualities, and canonical boundaries, arguing that exactly soluble higher gauge theories realize all bosonic topological orders with gapped boundaries in 3+1D.

Abstract

A discrete non-linear $σ$-model is obtained by triangulate both the space-time $M^{d+1}$ and the target space $K$. If the path integral is given by the sum of all the complex homomorphisms $φ: M^{d+1} \to K$, with an partition function that is independent of space-time triangulation, then the corresponding non-linear $σ$-model will be called topological non-linear $σ$-model which is exactly soluble. Those exactly soluble models suggest that phase transitions induced by fluctuations with no topological defects (i.e. fluctuations described by homomorphisms $φ$) usually produce a topologically ordered state and are topological phase transitions, while phase transitions induced by fluctuations with all the topological defects give rise to trivial product states and are not topological phase transitions. If $K$ is a space with only non-trivial first homotopy group $G$ which is finite, those topological non-linear $σ$-models can realize all 3+1D bosonic topological orders without emergent fermions, which are described by Dijkgraaf-Witten theory with gauge group $π_1(K)=G$. Here, we show that the 3+1D bosonic topological orders with emergent fermions can be realized by topological non-linear $σ$-models with $π_1(K)=$ finite groups, $π_2(K)=Z_2$, and $π_{n>2}(K)=0$. A subset of those topological non-linear $σ$-models corresponds to 2-gauge theories, which realize and classify bosonic topological orders with emergent fermions that have no emergent Majorana zero modes at triple string intersections. The classification of 3+1D bosonic topological orders may correspond to a classification of unitary fully dualizable fully extended topological quantum field theories in 4-dimensions.

Figures (14)

• Figure 1: "Topology" in topological insulator/superconductor (2005) corresponds to the twist in the band structure of orbitals, which is similar to the topological structure that distinguishes a sphere from a torus. This kind of topology is classical topology.
• Figure 2: "Topology" in topological order (1989) corresponds to pattern of many-body entanglement in many-body wave function $\Psi(m_1,m_2,\cdots,m_N)$, that is robust against any local perturbations that can break any symmetry. Such robustness is the meaning of "topological" in topological order. This kind of topology is quantum topology.
• Figure 3: X-ray scattering is a universal probe for all crystal orders.
• Figure 4: A string configuration in the bulk described by a triple $(\chi_{g_1^f},\chi_{g_2^f},\chi_{g_3^f})$, where $\chi_{g^f}$ is a conjugacy class in $G_f$ containing $g^f\in G_f$ and the triple satisfy $g_1^f g_2^f=g_3^f$.
• Figure 5: The tensor ${C}_{v_{0}v_{1}v_{2}v_{3}; t_{023} t_{013} t_{123} }^{ l_{01} l_{02} l_{03} l_{12} l_{13} l_{23}; t_{012} }$ is associated with a tetrahedron, which has a branching structure. If the vertex-0 is above the triangle-123, the tetrahedron has an orientation $s_{0123}=*$. If the vertex-0 is below the triangle-123, the tetrahedron has an orientation $s_{0123}=1$. The branching structure gives the vertices a local order: the $i^{th}$ vertex has $i$ incoming links.

Theorems & Definitions (1)

• Lemma 4.1