(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 $r$-matrix of the 2-Drinfel'd double can be interpreted as a quadratic 2-Casimir, which allows to recover the notion of duality.

40 pages; 0 figures (v2: adjusted \tilde t = t^*, fixed some minor errors and typos)

References in corpus (7)

• Twisted gauge theories in 3D Walker-Wang models
• Three Dimensional Quantum Geometry and Deformed Poincare Symmetry
• Framed Wilson Operators, Fermionic Strings, and Gravitational Anomaly in 4d
• Topological Higher Gauge Theory - from BF to BFCG theory
• Three Dimensional Loop Quantum Gravity: Particles and the Quantum Double
• Canonical transformations generated by the boundary volume: Unimodular and non-Abelian teleparallel gravity
• Discretization of 4d Poincaré BF theory: from groups to 2-groups

Cited by in corpus (5)

• Origin of the quantum group symmetry in 3d quantum gravity
• Higher Gauge Theory and Integrability
• Diffeomorphisms as quadratic charges in 4d BF theory and related TQFTs
• Infinite Dimensional Topological-Holomorphic Symmetry in Three-Dimensions
• 2-Drinfel'd double symmetry of the 4d Kitaev model