Galois Orbits of TQFTs: Symmetries and Unitarity

Abstract

We study Galois actions on $2+1$D topological quantum field theories (TQFTs), characterizing their interplay with theory factorization, gauging, the structure of gapped boundaries and dualities, 0-form symmetries, 1-form symmetries, and 2-groups. In order to gain a better physical understanding of Galois actions, we prove sufficient conditions for the preservation of unitarity. We then map out the Galois orbits of various classes of unitary TQFTs. The simplest such orbits are trivial (e.g., as in various theories of physical interest like the Toric Code, Double Semion, and 3-Fermion Model), and we refer to such theories as unitary "Galois fixed point TQFTs." Starting from these fixed point theories, we study conditions for preservation of Galois invariance under gauging 0-form and 1-form symmetries (as well as under more general anyon condensation). Assuming a conjecture in the literature, we prove that all unitary Galois fixed point TQFTs can be engineered by gauging 0-form symmetries of theories built from Deligne products of certain abelian TQFTs.

arXiv comment88 pages; 10 figures; 2 appendices; v2: typos fixed; reference added;

References in corpus (53)

• Anyons in an exactly solved model and beyond
• Structure Constants and Conformal Bootstrap in Liouville Field Theory
• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• On gauging finite subgroups
• Topological boundary conditions in abelian Chern-Simons theory
• On 2-Group Global Symmetries and Their Anomalies
• Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d
• On the classification of topological orders
• Finite Group Modular Data
• A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons
• Congruence Property In Conformal Field Theory
• The kernel of the modular representation and the Galois action in RCFT
• Generalized string-net models: A thorough exposition
• Higher central charges and topological boundaries in 2+1-dimensional TQFTs
• On Gauging Symmetry of Modular Categories
• In and around Abelian anyon models
• Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter
• Microscopic models of interacting Yang-Lee anyons
• Large Fourier transforms never exactly realized by braiding conformal blocks
• The center of monoidal 2-categories in 3+1D Dijkgraaf-Witten Theory
• Rank-finiteness for modular categories
• Lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups
• Time-reversal and spatial reflection symmetry localization anomalies in (2+1)D topological phases of matter
• Galois Modular Invariants of WZW Models
• M-theoretic Genesis of Topological Phases
• Twisted Quantum Double Model of Topological Orders with Boundaries
• Classification and construction of unitary topological field theories in two dimensions
• Galois Conjugates of Topological Phases
• Non-unitary TQFTs from 3D $\mathcal{N}=4$ rank 0 SCFTs
• Galois conjugated tensor fusion categories and non-unitary CFT
• Rationalizing CFTs and Anyonic Imprints on Higgs Branches
• Topological Quantum Computation with Gapped Boundaries
• A (Dummy's) Guide to Working with Gapped Boundaries via (Fermion) Condensation
• A study of time reversal symmetry of abelian anyons
• A Defect Verlinde Formula
• Galois Symmetry Induced by Hecke Relations in Rational Conformal Field Theory and Associated Modular Tensor Categories
• Spontaneous symmetry breaking from anyon condensation
• On Arithmetic Modular Categories
• Knots, links, and long-range magic
• Braided zesting and its applications
• Uniqueness of unitary structure for unitarizable fusion categories
• Circuit Complexity of Knot States in Chern-Simons theory
• Non-cyclotomic fusion categories
• Vanishing of categorical obstructions for permutation orbifolds
• Modular invariants and fusion rule automorphisms from Galois theory
• $a\times b=c$ in $2+1$D TQFT
• Galois Conjugation and Multiboundary Entanglement Entropy
• Galois conjugates of String-net Model
• Complexity for link complement States in Chern Simons Theory
• Morita equivalence of pointed fusion categories of small rank
• Classification of Metaplectic Fusion Categories
• Integral Metaplectic Modular Categories
• Zesting produces modular isotopes and explains their topological invariants

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