Web of dualities on non-orientable surfaces

Abstract

It is known that a two-dimensional bosonic theory with a non-anomalous ${\mathbb{Z}}_2$ symmetry can be fermionized. Recent work shows that if the bosonic theory also has non-anomalous time-reversal symmetry, fermionization extends to non-orientable surfaces and yields a fermionic theory that depends on a ${\mathrm{Pin}}^{-}$ structure. Besides fermionization, one can define various topological manipulations, such as gauging and stacking invertible phases, which together generate a web of dualities. We prove that their group structure is the dihedral group $D_8$ of order 16. Furthermore, we systematically investigate the web from two perspectives: Symmetry TFT and actions on sectors of the $S^1$ Hilbert space.

35 pages, 5 figures

References in corpus (19)

• Unpaired Majorana fermions in quantum wires
• Generalized Global Symmetries
• Topological phases of fermions in one dimension
• Symmetry-protected topological orders for interacting fermions -- Fermionic topological nonlinear $\sigma$ models and a special group supercohomology theory
• Orbifold groupoids
• Fermionic SPT phases in higher dimensions and bosonization
• State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter
• A Web of 2d Dualities: ${\mathbf{Z}}_2$ Gauge Fields and Arf Invariants
• Topological Transition on the Conformal Manifold
• Majorana chain and Ising model -- (non-invertible) translations, anomalies, and emanant symmetries
• Fermionic minimal models
• Reflection and time reversal symmetry enriched topological phases of matter: path integrals, non-orientable manifolds, and anomalies
• Framed Wilson Operators, Fermionic Strings, and Gravitational Anomaly in 4d
• Two More Fermionic Minimal Models
• Pin TQFT and Grassmann integral
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• Duality and Stacking of Bosonic and Fermionic SPT Phases
• On dimensions of (2+1)D abelian bosonic topological systems on non-orientable manifolds
• Exploring two dimensional ${\mathbb{Z}}_2$ invariant phases with time reversal symmetry and their transitions with topological operations