Minimal nondegenerate extensions

Abstract

We prove that every slightly degenerate braided fusion category admits a minimal nondegenerate extension, and hence that every pseudo-unitary super modular tensor category admits a minimal modular extension. This completes the program of characterizing minimal nondegenerate extensions of braided fusion categories. Our proof relies on the new subject of fusion 2-categories. We study in detail the Drinfel'd centre Z(Mod-B) of the fusion 2-category Mod-B of module categories of a braided fusion 1-category B. We show that minimal nondegenerate extensions of B correspond to certain trivializations of Z(Mod-B). In the slightly degenerate case, such trivializations are obstructed by a class in ${\mathrm{H}}^5(K({\mathbb{Z}}_2,2);k^{\times })$ and we use a numerical invariant -- defined by evaluating a certain two-dimensional topological field theory on a Klein bottle -- to prove that this obstruction always vanishes. Along the way, we develop techniques to explicitly compute in braided fusion 2-categories which we expect will be of independent interest. In addition to the model of Z(Mod-B) in terms of braided B-module categories, we develop a computationally useful model in terms of certain algebra objects in B. We construct an S-matrix pairing for any braided fusion 2-category, and show that it is nondegenerate for Z(Mod-B). As a corollary, we identify components of Z(Mod-B) with blocks in the annular category of B and with the homomorphisms from the Grothendieck ring of the Müger centre of B to the ground field.

66 pages. final version

References in corpus (7)

• Fusion 2-categories and a state-sum invariant for 4-manifolds
• Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions
• Constructing symmetric monoidal bicategories
• Framed Wilson Operators, Fermionic Strings, and Gravitational Anomaly in 4d
• Constructing symmetric monoidal bicategories functorially
• On dualizable objects in monoidal bicategories, framed surfaces and the Cobordism Hypothesis
• On the minimal extension and structure of weakly group-theoretical braided fusion categories

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• String Condensations in 3+1D and Lagrangian Algebras
• Enriched string-net models and their excitations
• Finite Semisimple Module 2-Categories
• Categories of quantum liquids II
• Fusion 3-Categories for Duality Defects
• Classification of Fermionic Topological Orders from Congruence Representations
• Higher Gauge Theory and Integrability
• Characterization and Classification of Fermionic Symmetry Enriched Topological Phases
• Local Modules in Braided Monoidal 2-Categories
• Gapped boundaries of fermionic topological orders and higher central charges
• Compact Semisimple 2-Categories
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• Modular extension of topological orders from congruence representations
• On the Dualizability of Fusion 2-Categories
• Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders
• Ordering the topological order in the fractional quantum Hall effect
• 2-Drinfel'd double symmetry of the 4d Kitaev model
• Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd)
• Compact Semisimple Tensor 2-Categories are Morita Connected
• Classification of integral modular data up to rank 13
• Exactly solvable models for fermionic symmetry-enriched topological phases and fermionic 't Hooft anomaly