Topological dualities in the Ising model

Freed, Daniel S; Teleman, Constantin

doi:10.2140/gt.2022.26.1907

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Topological dualities in the Ising model

Daniel S Freed and Constantin Teleman

Geometry & Topology 26 (2022) 1907–1984

DOI: 10.2140/gt.2022.26.1907

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/’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.

Keywords

nonabelian Ising model, topological field theory, Turaev–Viro theories, duality

Mathematical Subject Classification 2010

Primary: 57R56, 81T25, 82B20

References

Publication

Received: 1 November 2018

Revised: 14 September 2019

Accepted: 16 January 2021

Published: 12 December 2022

Proposed: Ralph Cohen

Seconded: Jim Bryan, Ian Agol

Authors

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

Volume 26, issue 5 (2022)