(2-)Drinfel'd Double and (2-)BF Theory
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 -matrix of the 2-Drinfel'd double can be interpreted as a quadratic 2-Casimir, which allows to recover the notion of duality.
Cite
@article{arxiv.2201.13366,
title = {(2-)Drinfel'd Double and (2-)BF Theory},
author = {Hank Chen and Florian Girelli},
journal= {arXiv preprint arXiv:2201.13366},
year = {2022}
}
Comments
40 pages; 0 figures (v2: adjusted \tilde t = t^*, fixed some minor errors and typos)