论文标题
（2-）Drinfel'd Double和（2-）BF理论
(2-)Drinfel'd Double and (2-)BF Theory
论文作者
论文摘要
3D BF动作的规格对称性和移位/翻译对称性与一对双重代数相关，可以组合形成drinfel的双重。这种组合的对称性是Chern-Simons动作的量规对称性，其等效于BF动作，直到某个边界项。我们表明，当考虑2-BF动作（又称BFCG动作）时，在4D中发生了类似的事情，该动作的对称性是根据一对双重严格的谎言2-代数（即交叉模块）指定的。结合这些对称性会产生一个2个drinfel'd double，这成为4D BF理论的量规对称结构，直到边界项。具体而言，我们展示了如何基于双重交叉模块的2个规范变换，这是参考文献中2 drinfel'd的概念。 ARXIV：1109.1344出现。我们还讨论了如何与Lie代数案例相似，2 drinfel'd double的$ r $ -Matrix的对称贡献可以解释为二次2-Casimir，这允许恢复二元性概念。
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.
