# What's the difference between a gauge theory with group $G$ and one with its universal cover?

Consider a gauge theory with gauge group $G$, which is not simply connected. What is the difference between this theory, and one with gauge group $\tilde G$, the universal cover of $G$?

Sharing the algebra, it seems to me that these two theories are identical at the perturbative level. Is this correct? In any case, it is obvious that the two theories are generically different at the non-perturbative level. But how exactly does that happen? What are the observables that are affected by the topology of the gauge group? How does one determine, from an experimental point of view, which of the groups $G,\tilde G$ is the correct gauge group of the theory?

• Related paper here. – knzhou May 3 '18 at 20:20
• @knzhou What a nice, concise and well-written paper. Right into my "papers" folder :-) – AccidentalFourierTransform May 3 '18 at 20:48

(This answer refers to Yang-Mills theory in 4 (Euclidean) dimensions and not to gauge theories in general, but the principles are general and can be repeated to any other case)

Perturbatively:

A pure gauge theory is necessarily and $\mathrm{Ad}G$ , because the gauge fields transform according to the adjoint representation. But the representations of the adjoint group include only a part of the (integrable) representations of the corresponding Lie algebra. For example, the representation$\mathbf{3}$ is not a representation of $\mathrm{Ad}SU(3)$. Thus the adjoint theory can be coupled to adjoint fermions, but no fundamental fermions can be included in this theory without enlarging its gauge group to the universal cover. This fact is crucial for example to the symmetry breakdown patterns of the theory, and the possibility of the existence of monopoles, please see my answer on the following PSE question.

Non-Perturbatively:

The possible gauge field configurations (for example those included in the path integrals depend on the principal bundle classification with the corresponding gauge group. This classification is known for compact oriented $4$ dimensional manifolds, please see lecture no. 13 of the gauge theory lectures by Ben Mares.

This classification is known as the Dold-Whitney theorem, which states that principal compact simple group bundles over compact oriented manifolds are classified by:

1. The instanton number.
2. The 't Hooft discrete magnetic flux which is an element of $H^2(M, \pi_1(G))$.

(subject to a consistency condition).

Both parameters depend on the group's center; thus they are different between the group and its universal cover. The instanton number is fractional for non-simply connected groups, and the 't Hooft flux depends explicitly on the center.