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?

This post imported from StackExchange Physics at 2020-11-09 09:44 (UTC), posted by SE-user annie marie heart