Modular extensions of unitary braided fusion categories and 2+1D topological/SPT orders with symmetries

Abstract

A finite bosonic or fermionic symmetry can be described uniquely by a symmetric fusion category $\mathcal{E}$. In this work, we propose that 2+1D topological/SPT orders with a fixed finite symmetry $\mathcal{E}$ are classified, up to $E_8$ quantum Hall states, by the unitary modular tensor categories $\mathcal{C}$ over $\mathcal{E}$ and the modular extensions of each $\mathcal{C}$. In the case $\mathcal{C}=\mathcal{E}$, we prove that the set ${\mathcal{M}}_{ext}(\mathcal{E})$ of all modular extensions of $\mathcal{E}$ has a natural structure of a finite abelian group. We also prove that the set ${\mathcal{M}}_{ext}(\mathcal{C})$ of all modular extensions of $\mathcal{C}$, if not empty, is equipped with a natural ${\mathcal{M}}_{ext}(\mathcal{E})$-action that is free and transitive. Namely, the set ${\mathcal{M}}_{ext}(\mathcal{C})$ is an ${\mathcal{M}}_{ext}(\mathcal{E})$-torsor. As special cases, we explain in details how the group ${\mathcal{M}}_{ext}(\mathcal{E})$ recovers the well-known group-cohomology classification of the 2+1D bosonic SPT orders and Kitaev's 16 fold ways. We also discuss briefly the behavior of the group ${\mathcal{M}}_{ext}(\mathcal{E})$ under the symmetry-breaking processes and its relation to Witt groups.

32 pages; references updated; corrections and improvements; final version to appear in Communications in Mathematical Physics

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Cited by in corpus (47)

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• A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons
• Symmetry as a shadow of topological order and a derivation of topological holographic principle
• Classification of 2+1D topological orders and SPT orders for bosonic and fermionic systems with on-site symmetries
• A classification of 3+1D bosonic topological orders (II): the case when some point-like excitations are fermions
• Classification of topological phases with finite internal symmetries in all dimensions
• Stiefel Liquids: Possible Non-Lagrangian Quantum Criticality from Intertwined Orders
• The center of monoidal 2-categories in 3+1D Dijkgraaf-Witten Theory
• A systematic construction of gapped non-liquid states
• Minimal nondegenerate extensions
• One dimensional gapped quantum phases and enriched fusion categories
• Fermion decoration construction of symmetry protected trivial orders for fermion systems with any symmetries $G_f$ and in any dimensions
• String Condensations in 3+1D and Lagrangian Algebras
• Spontaneous symmetry breaking from anyon condensation
• Coherence for monoidal $G$-categories and braided $G$-crossed categories
• (3+1)D topological orders with only a ${\mathbb{Z}}_2$-charged particle
• Hierarchy construction and non-Abelian families of generic topological orders
• Fractionalizing Global Symmetry on Looplike Topological Excitations
• Categories of quantum liquids II
• Classification of Fermionic Topological Orders from Congruence Representations
• Category of SET orders
• Duality and Stacking of Bosonic and Fermionic SPT Phases
• Gauging or extending bulk and boundary conformal field theories: Application to bulk and domain wall problem in topological matter and their descriptions by (mock) modular covariant
• Drinfeld Centers and Morita Equivalence Classes of Fusion 2-Categories
• Operator-state correspondence in simple current extended conformal field theories: Toward a general understanding of chiral conformal field theories and topological orders
• Fusion rule in conformal field theories and topological orders: A unified view of correspondence and (fractional) supersymmetry and their relation to topological holography
• Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries
• Electric-magnetic duality and ${\mathbb{Z}}_2$ symmetry enriched Abelian lattice gauge theory
• Gapped boundaries of fermionic topological orders and higher central charges
• Matrix formulation for non-Abelian families
• Classes of topological qubits from low-dimensional quantum spin systems
• Modular extension of topological orders from congruence representations
• Stacking Group Structure of Fermionic Symmetry-Protected Topological Phases
• Orbifolds and minimal modular extensions
• The Symmetry Enriched Center Functor is Fully Faithful
• Minimal modular extensions for super-Tannakian categories
• Numerical identification and gapped boundaries of abelian fermionic topological order
• The Reduced Tensor Product of Braided Tensor Categories Containing a Symmetric Fusion Category
• Condensation Completion and Defects in 2+1D Topological Orders
• On the minimal extension and structure of weakly group-theoretical braided fusion categories
• Vertex operator superalgebras and 16-fold way
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Computing the group of minimal non-degenerate extensions of a super-Tannakian category
• Exactly solvable models for fermionic symmetry-enriched topological phases and fermionic 't Hooft anomaly
• Braiding Statistics of Vortices in $2+1$d Topological Superconductors from Stacking