(Non-)Abelian Kramers-Wannier duality and topological field theory

Abstract

We study a connection between duality and topological field theories. First, 2d Kramers-Wannier duality is formulated as a simple 3d topological claim (more or less Poincar\'e duality), and a similar formulation is given for higher-dimensional cases. In this form they lead to simple TFTs with boundary coloured in two colours. Classical models (Poisson-Lie T-duality) suggest a non-abelian generalization in the 2d case, with abelian groups replaced by quantum groups. Amazingly, the TFT formulation solves the problem without computation: quantum groups appear in pictures, independently of the classical motivation. Connection with Chern-Simons theory appears at the symplectic level, and also in the pictures of the Drinfeld double: Reshetikhin-Turaev invariants of links in 3-manifolds, computed from the double, are included in these TFTs. All this suggests nice phenomena in higher dimensions.