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

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.

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• Les Houches Lecture Notes on Tensor Networks
• A non-semisimple non-invertible symmetry