Gapped boundary of (4+1)d beyond-cohomology bosonic SPT phase

Abstract

In this work we study gapped boundary states of ${\mathbb{Z}}_N$ bosonic symmetry-protected topological (SPT) phases in 4+1d, which are characterized by mixed ${\mathbb{Z}}_N$-gravity response, and the closely related phases protected by $C_N$ rotation symmetry. We show that if $N\notin \{2,4,8,16\}$, any symmetry-preserving boundary theory is necessarily gapless for the root SPT state. We then propose a (3+1)$d$ ${\mathbb{Z}}_2$ gauge theory coupled to fermionic matter as a candidate boundary theory for $N=2,4,8,16$, where the anomalous symmetry is implemented by invertible topological defects obtained from gauging (2+1)$d$ chiral topological superconductors. For the $C_N$ case, we present an explicit construction for the boundary states for $N=2,4,8,16$, and argue that the construction fails for other values of $N$.

15 pages, 4 figures; v2: added discussions on the group structure of SPT phases, and more on the $N=3$ case. More references added; v3: published. More details on the (4+1)$d$ ${\mathbb{Z}}_N$ SPT classification

References in corpus (11)

• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• Theory of Twist Liquids: Gauging an Anyonic Symmetry
• Construction of bosonic symmetry-protected-trivial states and their topological invariants via $G\times SO(\infty )$ non-linear $\sigma$-models
• Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter
• Codimension-2 defects and higher symmetries in (3+1)D topological phases
• Fermionic symmetry fractionalization in (2+1)D
• Higher-group symmetry in finite gauge theory and stabilizer codes
• Three-dimensional quantum cellular automata from chiral semion surface topological order and beyond
• Exactly Solvable Lattice Hamiltonians and Gravitational Anomalies
• Topological Electromagnetic Responses of Bosonic Quantum Hall, Topological Insulator, and Chiral Semi-Metal phases in All Dimensions
• Anomaly cascade in (2+1)D fermionic topological phases

Cited by in corpus (5)

• Average-exact mixed anomalies and compatible phases
• Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders
• Generalized Statistics on Lattices
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Boundary topological orders of (4+1)d fermionic ${\mathbb{Z}}_{2N}^{\mathrm{F}}$ SPT states