I am quoting the following from the Wikipedia article on the projective unitary group:
In the pure Yang–Mills $SU(n)$ gauge theory, which is a gauge theory with only gluons and no fundamental matter, all fields transform in the adjoint of the gauge group $SU(n)$. The $Z/n$ center of $SU(n)$ commutes, being in the center, with $SU(n)$-valued fields and so the adjoint action of the center is trivial. Therefore the gauge symmetry is the quotient of $SU(n)$ by $Z/n$, which is $PU(n)$ and it acts on fields using the adjoint action described above.
In this context, the distinction between $SU(n)$ and $PU(n)$ has an important physical consequence. $SU(n)$ is simply connected, but the fundamental group of $PU(n)$ is $Z/n$, the cyclic group of order $n$. Therefore a $PU(n)$ gauge theory with adjoint scalars will have nontrivial codimension 2 vortices in which the expectation values of the scalars wind around $PU(n)$'s nontrivial cycle as one encircles the vortex. These vortices, therefore, also have charges in $Z/n$, which implies that they attract each other and when $n$ come into contact they annihilate. An example of such a vortex is the Douglas–Shenker string in $SU(n)$ Seiberg–Witten gauge theories.
- What is the center of $SU(n)$?
- What does it mean the adjoint action to be trivial? Which action are they talking about?
- I struggle to understand why the resulting gauge symmetry is $PU(n)$.
- What are the "codimension 2 vertices" of the scalars?
- Does this apply for pure $SU(3)$ QCD?