We show that General Relativity can be formulated as a constrained topological theory for flat 2-connections associated to the Poincaré 2-group. Matter can be consistently coupled to gravity in this formulation. We also show that the edge lengths of the spacetime manifold triangulation arise as the basic variables in the pathintegral quantization, while the state-sum amplitude is an evaluation of a colored 3-complex, in agreement with the category theory results. A 3-complex amplitude for Euclidean quantum gravity is proposed.

We generalize the BF theory action to the case of a general Lie crossed module (∂ : H → 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.

In this note, we consider the reformulation of the Dirac-Born-Infeld action for a Dirichlet p-brane in Brink-Di Vecchia-Howe-Tucker form, i.e., including an independent non-propagating world-volume metric. When p > 2, the action becomes non-polynomial. A closed expression is derived for p = 3. For selfdual field-strengths, the DBI action is reproduced by an action with a simple F 2 term. We speculate on supersymmetrization of the D 3 -brane action. We also give the governing equations for arbitrary p, and derive an implicit expression for the D 4 -brane lagrangian.