Gauss law, gauge and global symmetries

Question

I am reading Witten's paper on the confinement/deconfinement phase transition in $\mathcal{N}=4$ $\mathrm{SU}(N)$ SYM theory. I am a bit stuck at section "Confinement" at Finite Volume, page 6, where he states the following

The Gauss law constraint in finite volume says that physical states must be invariant under the global $\mathrm{SU}(N)$.

I understand that on $\mathbb{R}^3$, Gauss's law is related to gauge invariance, and that moreover, at least for $\mathrm{U}(1)$ implies that the electric field at spatial infinity is proportional to the total charge. I'm not quite sure if this directly translates to non-abelian theories.

But at finite volume I am lost. Where does the "global" $\mathrm{SU}(N)$ he talks about comes from. I'm quite sure that the constant part of the gauge $\mathrm{SU}(N)$ should not generate a global $\mathrm{SU}(N)$. Moreover, I don't understand how Gauss's law is related to all of this.

Answer 1

The point being, boundary conditions are such that gauge parameters vanish at infinity, meaning that the gauge group is not a gauge symmetry, but a flavor symmetry, at the boundary. This is the "global SU(N)" Witten is talking about.

On compact manifolds you have gauge transformations at every point, and these kill everything that is charged under the gauge group. On non-compact manifolds, gauge transformations only kill charged stuff in the bulk, while charged objects at the boundary are allowed to exist. (Why we don't observe them, in QCD, even though they are allowed, is precisely the confinement problem).

Answer 2

Gauss's law is sometimes used to signify the condition of zero divergence. I don't who started the usage; historically the culprit was probably the lattice gauge theory community?

And I think this is what Witten intended to say, too. Look at page 4 between eq. (2.1) and (2.2): he said $G'$-invariance is Gauss's law for physical states, and $G'$ appears to be the group of global and ordinary gauge transformations. That means anything physical has no SU(N) charge and transforms trivially.

Answer 3

The Gauss Law is a constraint that appears in Gauge theories of a connection.

Let your gauge theory with compact Lie group G be described by a connection $A$. Then you define your covariant derivative to be:

$D^A\cdot = d\cdot + A\wedge \cdot$

Now one usually go to the Hamiltonian formulation of a theory, where the momentum conjugated to the G-connection $A$ is the G-electric field $\mathcal{E}$. Passing to the phase space formulation, the theory becomes constrained due to the excess of degrees of freedom introduced by the gauge freedom.

The constraint that the connection theories obtain is the Gauss constraint:

$G = D^A \mathcal{E} = d\mathcal{E}+ A\wedge\mathcal{E}=0$

This constraint generates gauge motions (i can expand on this, tell me if i need to edit) and hence at the quantum level, $G$ becomes the operator that enforces gauge invariance of a state:

$\hat{G} \psi = 0$

Solved for $\psi$ yields states that are automatically gauge invariant.

Now as stated by @AccidentalFourierTransform gauge transformations on the boundary are very different from the one in the bulk. The gauss constraint gives gauge invariant states in the bulk, but when applied on the boundary it label different physical sectors, the so called theta sectors, which can be interpreted as a "global" or flavor symmetry of the boundary.

What this means is that in the bulk the Gauss constraint select the gauge invariant states, while on the boundary the Gauss constraint select the states that classify the different flavors (the various "theta sectors"). This is why the transformations on the boundary are called global: flavor symmetries in the standard model are global and not local (ie gauge) symmetries.