Download this article
Download this article For screen
For printing
Recent Issues

Volume 26
Issue 7, 2855–3306
Issue 6, 2405–2853
Issue 5, 1907–2404
Issue 4, 1435–1905
Issue 3, 937–1434
Issue 2, 477–936
Issue 1, 1–476

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
 
Other MSP Journals
Topological dualities in the Ising model

Daniel S Freed and Constantin Teleman

Geometry & Topology 26 (2022) 1907–1984
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