# Mistake or Rewriting of Yang-Mills in Nakahara

I am familiar with Yang-Mills equation of motion E.O.M. (without matter or source fields) in differential form.

$$D * F =0$$ and Bianchi identity $$D F=0$$ where $$F= dA + A \wedge A$$ and $$D=d + [A, ]$$ as the covariant derivative version of exterior derivative $$d$$.

However, in Nakahar book Geometry, Topology and Physics, Second Edition ,

we can compare E.O.M. to his (1.269) below,

and Bianchi identity to his (1.266) below.

My question is that: Did Nakahara make any mistake? Or are his equations the rewriting of my Yang-Mills Equations above? If so, how do we convert to make the rewriting precise?

• Note that the indices in Nakahara's formulae are contracted and have only one free index, while yours are equations for 3-forms and would have 3 free indices. Now think about what transforms a 3-form to a 1-form... – ACuriousMind May 19 at 9:38
• ...in 4 spacetime dimensions – Kosm May 19 at 9:45
• You gotta remember that to get a "normal" contraction you always have to throw in a Hodge star. For example, the action contains $F \wedge \star F$, not $F \wedge F$ even though the $F$'s are just contracted in index notation. You can show this by just expanding everything in components. – knzhou May 19 at 12:43

Apply Hodge star to equations of motion $$d*F=0$$: $$*d*F=*(\frac{1}{2}\partial_\mu\tilde{F}_{\nu\rho}~dx^\mu\wedge dx^\nu\wedge dx^\rho)=\frac{1}{2}\partial_\mu \tilde{F}_{\nu\rho}~dx^\sigma \eta_{\sigma\omega}\epsilon^{\omega\mu\nu\rho}=\\ =-\partial_\mu F^{\omega\mu}\eta_{\omega\sigma}dx^\sigma=0~\rightarrow~ \partial_\mu F^{\omega\mu}=0,$$ where I used $$\tilde{F}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}$$ and Levi-Civita contractions. Bianchi identities can be "derived" in the same way (by expressing $$F_{\mu\nu}$$ in terms of $$\tilde{F}_{\mu\nu}$$).
Or you can simply convince yourself that the following expressions of Bianchi identities are equivalent $$\partial_\mu F_{\nu\rho}+\partial_\rho F_{\mu\nu}+\partial_\nu F_{\rho\mu}=0~\Leftrightarrow~\epsilon^{\sigma\mu\nu\rho} \partial_\mu F_{\nu\rho}=0.$$ Substitute $$\tilde{F}$$ above for equations of motion.