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.

Globally, the solutions are given by the finite-dimensional space of flat connections modulo the gauge transformations.

Note that we are talking about 3D Chern-Simons here, the higher dimensional CS theories *do* exhibit local degrees of freedom, see arXiv/hep-th/9506187.

This post imported from StackExchange Physics at 2014-09-05 08:27 (UCT), posted by SE-user ACuriousMind