Conformal Invariance of N=4 SYM Theory and the Yang-Mills Stress Tensor

[BenTheMan]

I am working through a set of notes on conformal field theory by Schellekens and want to show the conformal invariance of N=4 SYM theory in four dimensions. I start with the action
$$S=\frac{1}{4g}\int d^Dx \sqrt{g}Tr\left(F_{\mu\nu}F^{\mu \nu})$$
There's only the metric in the action to worry about, in the Jacobian. (Is this wrong?)

But then the stress tensor I get is this (Abelian case):
$$T^{\alpha\beta} = g^{\alpha\beta}F_{\mu\nu}F^{\mu \nu}$$.

I'm pretty sure that this isn't right because I was assuming I'd use the SYM equation of motion to show the divergence condition on the stress tensor. Instead, I get something like (Abelian case):
$$\partial_{\alpha}T^{\alpha\beta} = \partial^{\beta}F_{\mu\nu}F^{\mu \nu}$$.

Can anyone point me in the right directions? Am I missing something in the actoin (i.e. a hiding metric)?

 

[garrett]

$$F^{\mu \nu} = g^{\mu \rho} g^{\nu \sigma} F_{\rho \sigma}$$

 

[BenTheMan]

Sigh. Then I definitely see how to use the eom to get the divergence condition.

Thanks!