Symmetry topological field theory and non-abelian Kramers-Wannier dualities of generalised Ising models

TL;DR

This work embeds two-dimensional lattice theories with topological line symmetries into a fully extended 3D TFT framework (SymTFT), enabling explicit boundary realizations and controlled gauging. By mapping between Vec_G and Rep(G) boundary constructions, leveraging module categories and their Morita equivalences, it derives non-abelian generalised Kramers-Wannier dualities and clarifies how gauging subsymmetries and Fourier transforms act on partition functions and topological sectors. The approach yields a precise correspondence between symmetry data, boundary states, and bulk lines, and extends to RG fixed points of gapped symmetric phases via Frobenius algebra objects. The results provide a concrete, tensor-network-friendly pathway to compute dualities in non-abelian Ising-type models, with broad implications for symmetry, duality, and topological order in higher dimensions.

Abstract

For a class of two-dimensional Euclidean lattice field theories admitting topological lines encoded into a spherical fusion category, we explore aspects of their realisations as boundary theories of a three-dimensional topological quantum field theory. After providing a general framework for explicitly constructing such realisations, we specialise to non-abelian generalisations of the Ising model and consider two operations: gauging an arbitrary subsymmetry and performing Fourier transforms of the local weights encoding the dynamics of the theory. These are carried out both in a traditional way and in terms of the three-dimensional topological quantum field theory. Whenever the whole symmetry is gauged, combining both operations recovers the non-abelian Kramers-Wannier duals à la Freed and Teleman of the generalised Ising models. Moreover, we discuss the interplay between renormalisation group fixed points of gapped symmetric phases and these generalised Kramers-Wannier dualities.