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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?

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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.

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