Standard Model and 4-groups

TL;DR

This paper explores a categorical generalization of Poincaré symmetry using 4-groups, integrating fermionic, scalar, and Standard Model matter fields into a unified gauge-theoretic framework.

Contribution

It introduces a novel categorification of Poincaré symmetry with 4-groups, enabling natural inclusion of matter fields and Standard Model symmetries in a gauge theory setting.

Findings

Categorical Poincaré symmetry generalizes to 4-groups with natural matter field incorporation.

Fermionic and scalar fields correspond to 3-form and 4-form gauge fields.

Standard Model gauge groups can be embedded into the 4-group structure.

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.