Higher gauge theories based on 3-groups

Abstract

We study the categorical generalizations of a BF theory to 2BF and 3BF theories, corresponding to 2-groups and 3-groups, in the framework of higher gauge theory. In particular, we construct the constrained 3BF actions describing the correct dynamics of Yang-Mills, Klein-Gordon, Dirac, Weyl, and Majorana fields coupled to Einstein-Cartan gravity. The action is naturally split into a topological sector and a sector with simplicity constraints, adapted to the spinfoam quantization programme. In addition, the structure of the 3-group gives rise to a novel gauge group which specifies the spectrum of matter fields present in the theory, just like the ordinary gauge group specifies the spectrum of gauge bosons in the Yang-Mills theory. This allows us to rewrite the whole Standard Model coupled to gravity as a constrained 3BF action, facilitating the nonperturbative quantization of both gravity and matter fields. Moreover, the presence and the properties of this new gauge group open up a possibility of a nontrivial unification of all fields and a possible explanation of fermion families and all other structure in the matter spectrum of the theory.

Global 3-group symmetry and ’t Hooft anomalies in axion electrodynamics

“…Here, higher n-groups are given by a set of n groups with maps between them. For example, a 2-group is given by a set of two groups (G, H), a map H → G, and actions of G on G and H. The higher-groups have been recently applied to various aspects of theoretical physics, such as higher gauge theories where charged objects are extended [75][76][77][78][79][80][81][82][83][84][85][86][87][88], effective theories of gapped or gapless topological matter [89][90][91][92][93][94][95][96][97][98], deformation of current algebra for tensorial currents [99] and quantum chromodynamics [100]. Therefore, higherform symmetries and higher-group structures may provide us with new understandings of the effects in the axion electrodynamics.…”

Hamiltonian Analysis for the Scalar Electrodynamics as 3BF Theory

“…This is not viable in the special case of the 3-group discussed in this work, but instead needs to be done in the case of a generic 3-group, where homomorphisms δ and ∂ and the Peiffer lifting {_ , _} are nontrivial. From the point of view of physics, the obtained results represent the fundamental building blocks for the construction of the quantum theory of scalar electrodynamics coupled to gravity, as well as a convenient model to discuss before proceeding to the Hamiltonian analysis and canonical quantization of the full Standard Model coupled to gravity, formulated as a 3BF action with suitable constraints [8]. Both the Hamiltonian analysis of constrained 3BF models and the corresponding canonical quantization programme need to be further developed in order to achieve these goals.…”

“…As was discussed in detail in [8], one formulates a topological 3BF action by specifying a particular gauge Lie 3-group. It has been proved that any strict 3-group is equivalent to a 2-crossed module [9,10].…”

“…based on their transformation properties as discussed in [8,11], the Lagrangian of the action (9) obtains the form:…”

“…Specifically, the notion of a 3-group in the framework of higher category theory is introduced as a 3-category with only one object where all the morphisms, 2-morphisms and 3-morphisms are invertible. Based on this generalization, recently a constrained 3BF action has been introduced, which describes the full Standard Model coupled to Einstein-Cartan gravity [8].…”