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

TL;DR

This paper explores the relationship between duality, topological field theories, and quantum groups, providing a unified framework that connects 2d Kramers-Wannier duality with higher-dimensional generalizations and topological invariants.

Contribution

It reformulates 2d Kramers-Wannier duality as a topological claim and introduces a non-abelian generalization using quantum groups, linking duality, TFTs, and quantum invariants.

Findings

Formulation of 2d Kramers-Wannier duality as a 3d topological claim

Introduction of non-abelian generalization with quantum groups

Connection between TFTs, Chern-Simons theory, and quantum invariants

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 Poincare 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. The statistical models live on the boundary of these TFTs, as in the CS/WZW or AdS/CFT correspondence. Classical models (Poisson-Lie T-duality) suggest a non-abelian generalization in the 2dcase, 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.