Kramers-Wannier duality from conformal defects

TL;DR

It is demonstrated that the fusion algebra of conformal defects of a two-dimensional conformal field theory contains information about the internal symmetries of the theory and allows one to read off generalizations of Kramers-Wannier duality.

Abstract

We demonstrate that the fusion algebra of conformal defects of a two-dimensional conformal field theory contains information about the internal symmetries of the theory and allows one to read off generalisations of Kramers-Wannier duality. We illustrate the general mechanism in the examples of the Ising model and the three-states Potts model.

Figures (4)

• Figure 1: A conformal defect is transparent to the stress tensor (a), while a bulk field $\phi$ generically becomes a sum of disorder fields (b).
• Figure 2: The TFT-representation of the pulling a defect of type $\sigma$ past a spin field. Collapsing the circular $\sigma$-Wilson line on the rhs generates the TFT-representation of the disorder field $\mu(z)$.
• Figure 3: Taking defects of type $\sigma$ and $\varepsilon$ past field insertions. The TFT-representation of a) is given in figure \ref{['tft-sig-sig']}.
• Figure 4: Order/Disorder duality of a correlator of four spin fields on a sphere, and of two spin fields on a torus, as induced by the $\sigma$-defect.