It is known that 3D Chern-Simons(C-S) theory has no explicit metric involving in the Lagrangian density:

$$ A \wedge dA + (2/3) A \wedge A \wedge A $$

while the 4D Yang-Mills(Y-M) theory has the metric $g_{\mu\nu}$involves.

$$ F \wedge *F= (dA + A \wedge A) \wedge *(dA + A \wedge A) $$ More explicitly, for Abelian case,

C-S gives the form without the metric(topological), $$d^3 x \epsilon^{\mu\nu\rho} A_\mu \partial_\nu A_\rho$$

Y-M gives the form with the metric:

$$d^4 x (\partial_\mu A_\nu)( \partial^\mu A^\nu)=d^4 xg^{\mu\rho}g^{\nu\lambda}(\partial_\mu A_\nu)( \partial^\nu A^\lambda)$$

How about the statement of coordinate invariance of C-S and Y-M theory? Are C-S or Y-M coordinate invariance? (i.e. does the Lagrangian density look different in other coordinates, such as cylindrical, spherical coordinates? or general curved coordinates? how does that Lagrangian look like in those generic coordinates?)

Is there some relations (if, only if or iff) between:

(1) gauge invariance;

(2) coordinate invariance;

(3) metric independence (=diffeomorphism invariance?);

(4) covariant form in curved background?

for those gauge theories (e.g. Y-M) or for topological field theory (e.g. C-S)?

ps. see also this post, which unfortunately does not directly answer my inquiries yet. Thanks.


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