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

TL;DR

This paper explores the deep connection between duality in physics and topological field theories, revealing how 2d Kramers-Wannier duality can be understood through 3d TFTs and extending these ideas to higher dimensions with quantum groups and Chern-Simons theory.

Contribution

It provides a novel topological field theory formulation of dualities, including non-abelian cases, using quantum groups and links to Chern-Simons theory, without requiring explicit computations.

Findings

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

Extension of duality concepts to higher dimensions with quantum groups.

Connection established between dualities, topological field theories, and link 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 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.