# $G_{\mu\nu}$'s field equations in electromagnetic duality

Given $$G^{\mu\nu} = -2 \frac{\partial L}{\partial F_{\mu\nu}}$$ it is written in http://arxiv.org/abs/hep-th/9506035 that the field equations are (eq 1.8) $$\partial_{[\alpha} \star G_{\beta \gamma]}= 0$$

How is that? Why the dual and not the $G_{\mu\nu}$ itself? Here L is general form of Lagrangian which is a function of $F_{\mu\nu}$ and $g_{\mu\nu}$ and $\partial_{[\alpha} \star G_{\beta \gamma]}= 0$ is the anti-symmetrized way of writing the equation of motion. Please note that here we are considering duality.

• Please incorporate the relevant content of the link into the post - in this case: What does $\partial_{[\alpha\star G_\beta\gamma]}$ mean? What kind of $L$ are we looking at? – ACuriousMind Dec 4 '14 at 16:53
• I edited it, hope it is clearer now! – Fluctuations Dec 4 '14 at 17:26
• I don't know what "the antisymmetrization way of writing the e.o.m." is, and which duality are we considering? Is the $\star$ supposed to be the Hodge dual? – ACuriousMind Dec 4 '14 at 17:32
• @ACuriousMind: I have corrected his equation, using the arXiv reference. He just forgot to move the $G$ and $\star$ outside the subscript. – JamalS Dec 4 '14 at 17:36
• @ACuriousMind: Yes, the exterior derivative is an anti-symmetrized differentiation operator, so it's exactly that. In other words, the codifferential $\delta G = 0$. The OP's choice of notation is unfortunate since he combines tensor calculus and exterior calculus :) – JamalS Dec 4 '14 at 17:39