Lie crossed modules and gauge-invariant actions for 2-BF theories
Lie crossed modules and gauge-invariant actions for 2-BF theories
复制标题
DOI：
10.4310/atmp.2011.v15.n4.a4
复制
发表时间：
2010-06
期刊：
影响因子：
--
通讯作者：
João F. Martins;A. Miković
中科院分区：
文献类型：
--
作者：
João F. Martins;A. Miković
我们将BF理论作用推广到一般李交叉模$(H\to G)$的情形，其中$G$和$H$是非阿贝尔李群。我们的构造要求在$G$和$H$的李代数上存在$G$不变的非退化双线性形式，并且利用向量空间的短链复形提供的交叉模的构造，证明了有许多这样的李交叉模的例子。我们还将这种结构推广到有限类型的向量空间的任意链复形。我们构造了2-平坦和伪平坦2-连通及其辅助场的两个规范不变作用。第一个作用量与Girelli，Pfeiffer和Popescu对一类特殊的Lie Cross模引入的BFCG作用量相同，其中$H$是阿贝尔的。第二个动作是包含附加辅助字段的扩展BFCG动作。然而，这两个操作通过字段重新定义而关联。我们还构造了扩展的BFCG作用量的三参数形变，我们认为这与构造嵌入四面体的纽结曲面的非平凡不变量有关。
We generalize the BF theory action to the case of a general Lie crossed module $(H \to G)$, where $G$ and $H$ are non-abelian Lie groups. Our construction requires the existence of $G$-invariant non-degenerate bilinear forms on the Lie algebras of $G$ and $H$ and we show that there are many examples of such Lie crossed modules by using the construction of crossed modules provided by short chain complexes of vector spaces. We also generalize this construction to an arbitrary chain complex of vector spaces, of finite type. We construct two gauge-invariant actions for 2-flat and fake-flat 2-connections with auxiliary fields. The first action is of the same type as the BFCG action introduced by Girelli, Pfeiffer and Popescu for a special class of Lie crossed modules, where $H$ is abelian. The second action is an extended BFCG action which contains an additional auxiliary field. However, these two actions are related by a field redefinition. We also construct a three-parameter deformation of the extended BFCG action, which we believe to be relevant for the construction of non-trivial invariants of knotted surfaces embedded in the four-sphere.