Simplifying Equation - Quantum String

Question

In this post Quantum String action , the user does the following manipulation :

$$\delta (H_{mnp})H^{mnp} =\partial_m\delta(B_{np})[H^{mnp}+H^{pmn}+H^{npm}] $$

I can't quite understand how he does that, I get where the derivative comes from but how does it appear three combinations of $H^{mnp}$?

Answer 1

If $H_{mnp}=\partial_m B_{np}+\partial_n B_{pm}+\partial_p B_{mn}$, then

$$\delta(H_{mnp})=\partial_m(\delta B_{np})+\partial_n(\delta B_{pm})+\partial_p(\delta B_{mn})$$

and

\begin{align} \delta (H_{mnp})H^{mnp} &=\left[\partial_m(\delta B_{np})+\partial_n(\delta B_{pm})+\partial_p(\delta B_{mn})\right]H^{mnp}\\ &=\partial_m(\delta B_{np})H^{mnp}+\partial_n(\delta B_{pm})H^{mnp}+\partial_p(\delta B_{mn})H^{mnp}\\ &=\partial_m(\delta B_{np})H^{mnp}+\partial_m(\delta B_{np})H^{pmn}+\partial_m(\delta B_{np})H^{npm}\\ &=\partial_m(\delta B_{np})\left[H^{mnp}+H^{pmn}+H^{npm}\right]. \end{align}

So it's just a matter of renaming indices in the last two terms.