Standard Model and 4-groups

Abstract

We show that a categorical generalization of the the Poincar\'e symmetry which is based on the n-crossed modules becomes natural and simple when n=3 and that the corresponding 3-form and 4-form gauge fields have to be a Dirac spinor and a Lorentz scalar, respectively. Hence by using a Poincar\'e 4-group we naturally incorporate fermionic and scalar matter into the corresponding 4-connection. The internal symmetries can be included into the 4-group structure by using a 3-crossed module based on the $SL(2,\mathbb{C}) \times K$ group, so that for $K=U(1)\times SU(2) \times SU(3)$ we can include the Standard Model into this categorification scheme.