Topological dualities in the Ising model

TL;DR

<3-5 sentence high-level summary> We establish a unified framework connecting the 2D Kramers-Wannier duality of the Ising model with electromagnetic duality in 3D finite gauge theories by treating the Ising model as a boundary theory for a 3D bulk TQFT. The approach recasts lattice models as fully extended topological field theories with boundaries and defects, enabling a generalization to nonabelian groups, finite Hopf algebras, and finite homotopy theories; it also provides an explicit duality theorem exchanging Wilson and 't Hooft operators and order/disorder operators on the boundary. The work further develops a nonabelian Ising dual via Turaev-Viro theory, and formulates a bi-colored boundary/ polarization framework that unifies boundary data, domain walls, and boundary lattice models. Finally, it outlines a program for classifying gapped Ising-like phases through unbroken subgroups and central extensions, and discusses higher-dimensional generalizations via finite path integrals and spectrum-valued dualities.

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.

Figures (22)

• Figure 1: Wilson loop/order operator (left); 't Hooft loop/disorder operator (right)
• Figure 2: Flow on moduli space of one-dimensional Ising models
• Figure 3: Three bordisms with boundary theory labeled by $x\in C$. The arrows indicate incoming vs. outgoing boundary components.
• Figure 4: $D^1$ with boundary colored red by the regular boundary theory
• Figure 5: Seven bordisms to evaluate under $(\mathrsfs{F}^{\newline} _{\!\mathscr{T}},\mathrsfs{B}^{\newline} _{\mathscr{T}})$

Theorems & Definitions (83)

• Remark 1.7
• Theorem 1.13
• Theorem 1.21
• Remark 1.23: Polarization of $\mathrsfs{F}$
• Remark 1.24
• Remark 1.25
• Remark 1.29
• Example 2.3: $\mathscr{C}=\mathscr{T}ens\mathscr{C}at$
• Definition 2.5
• Remark 2.9