Topological dualities in the Ising model

Abstract

We relate two classical dualities in low-dimensional quantum field theory: Kramers-Wannier duality of the Ising and related lattice models in $2$ dimensions, with electromagnetic duality for finite gauge theories in $3$ dimensions. The relation is mediated by the notion of boundary field theory: Ising models are boundary theories for pure gauge theory in one dimension higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects the multiplicity of topological boundary states. In the process we describe lattice theories as (extended) topological field theories with boundaries and domain walls. This allows us to generalize the duality to non-abelian groups; finite, semi-simple Hopf algebras; and, in a different direction, to finite homotopy theories in arbitrary dimension.

62 pages, 22 figures; v2 adds important reference [S]; v2 has reworked introduction, additional reference [KS], and minor changes; v4 for publication in Geometry and Topology has all new figures and a few minor changes and additional references

References in corpus (5)

• Generalized Global Symmetries
• Turaev-Viro invariants as an extended TQFT
• Gapped boundary theories in three dimensions
• Symmetries of categorical representations and the quantum Ngô action
• The low-energy TQFT of the generalized double semion model

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• Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness
• Fermionization of fusion category symmetries in 1+1 dimensions
• Towards Non-Invertible Anomalies from Generalized Ising Models
• Non-invertible topological defects in 4-dimensional ${\mathbb{Z}}_2$ pure lattice gauge theory
• Boundary states of Three Dimensional Topological Order and the Deconfined Quantum Critical Point
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• Orientation twisted homotopy field theories and twisted unoriented Dijkgraaf-Witten theory
• The low-energy TQFT of the generalized double semion model
• Generalized Symmetries of Non-SUSY and Discrete Torsion String Backgrounds
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
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