(2-)Drinfel'd Double and (2-)BF Theory

TL;DR

This paper explores the extension of Drinfel'd doubles from 3D BF theories to 4D using 2-BF actions and strict Lie 2-algebras, revealing a higher gauge symmetry structure called 2-Drinfel'd double.

Contribution

It introduces the concept of 2-Drinfel'd doubles in 4D BF theories, generalizing the symmetry structure from Lie algebras to Lie 2-algebras and demonstrating their role as gauge symmetries.

Findings

Identification of 2-Drinfel'd double as gauge symmetry in 4D BF theory

Extension of duality concepts via quadratic 2-Casimir

Connection between 2-gauge transformations and dual crossed-modules

Abstract

The gauge symmetry and shift/translational symmetry of a 3D BF action, which are associated to a pair of dual Lie algebras, can be combined to form the Drinfel'd double. This combined symmetry is the gauge symmetry of the Chern-Simons action which is equivalent to the BF action, up to some boundary term. We show that something similar happens in 4D when considering a 2-BF action (aka BFCG action), whose symmetries are specified in terms of a pair of dual strict Lie 2-algebras (ie. crossed-modules). Combining these symmetries gives rise to a 2-Drinfel'd double which becomes the gauge symmetry structure of a 4D BF theory, up to a boundary term. Concretely, we show how using 2-gauge transformations based on dual crossed-modules, the notion of 2-Drinfel'd double defined in Ref. arXiv:1109.1344 appears. We also discuss how, similarly to the Lie algebra case, the symmetric contribution of the $r$-matrix of the 2-Drinfel'd double can be interpreted as a quadratic 2-Casimir, which allows to recover the notion of duality.