(Non-)Abelian Kramers-Wannier Duality And Topological Field Theory

摘要

We study a connection between duality and topological fieldtheories. First, 2d Kramers-Wannier duality is formulated as a simple3d topological claim (more or less Poincarduality), and a similarformulation is given for higher-dimensional cases. In this form theylead to simple TFTs with boundary coloured in two colours. Classicalmodels (Poisson-Lie T-duality) suggest a non-abelian generalizationin the 2d case, with abelian groups replaced by quantumgroups. Amazingly, the TFT formulation solves the problem withoutcomputation: quantum groups appear in pictures, independently of theclassical motivation. Connection with Chern-Simons theory appears atthe symplectic level, and also in the pictures of the Drinfeld double:Reshetikhin-Turaev invariants of links in 3-manifolds, computed fromthe double, are included in these TFTs. All this suggests nicephenomena in higher dimensions.