Lie crossed modules and gauge-invariant actions for 2BF theories (original) (raw)
We generalize the BF theory action to the case of a general Lie crossed module (HtoG)(H \to G)(HtoG), where GGG and HHH are non-abelian Lie groups. Our construction requires the existence of GGG-invariant non-degenerate bilinear forms on the Lie algebras of GGG and HHH 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 HHH 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.
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