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Topological dualities in the Ising model

Daniel S Freed and Constantin Teleman

Geometry & Topology 26 (2022) 1907–1984

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/’tHooft 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 nonabelian groups; to finite, semisimple Hopf algebras; and, in a different direction, to finite homotopy theories in arbitrary dimension.

nonabelian Ising model, topological field theory, Turaev–Viro theories, duality
Mathematical Subject Classification 2010
Primary: 57R56, 81T25, 82B20
Received: 1 November 2018
Revised: 14 September 2019
Accepted: 16 January 2021
Published: 12 December 2022
Proposed: Ralph Cohen
Seconded: Jim Bryan, Ian Agol
Daniel S Freed
Department of Mathematics
University of Texas
Austin, TX
United States
Constantin Teleman
Department of Mathematics
University of California
Berkeley, CA
United States