Lie crossed modules and gauge-invariant actions for 2-BF theories

Understand

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.