# No local degrees of freedom when connection is flat

I was studying Chern-Simons theory and variation of action gives us the flatness conditions $\mathrm{d} A + A \wedge A = 0$. I am wondering how to see that this implies there are no local degrees of freedom.

And what precisely does it mean that a degree of freedom is local?

• Possible duplicate: physics.stackexchange.com/q/98484/2451 Sep 4, 2014 at 15:27
• Some comments from an alternative point of view on the problem are given here. Sep 7, 2014 at 6:49

Recall that $\mathrm{d}A + A\wedge A = F = 0$ means that the field strength is vanishing, i.e. the gauge field is always pure gauge locally.
Local degrees of freedom would mean that the equation of motion ($F = 0$) has more than one local solutions that are not related by a symmetry of the theory. But the field being pure gauge locally means that it can always be locally transformed to be $A = 0$, so the local solutions are uniquely zero, thus implying there are no local degrees of freedom.