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take positive helicity (15) and (16) can be given by a sum of terms. Each term has the form 

<img src='content_image/320705.jpg'>

where each l$_{i}$ can be any positive helicity gluon, and n 1 $_{1}$, n 1 $_{2}$,..., n 2 $_{1}$, n 2 $_{2}$,..., ..., n N 1 , n N 2 ,... is a permutation of all the gravitons. Following the discussions on amplitudes with one and two gravitons, this term can be given by sum of MHV amplitudes with N + 2 M gluons with appropriate factors. The first line insert two gluons corresponding to n 1 between 1, l$_{1}$ and l$_{1}$ , i respectively, then insert two gluons corresponding to n 1 2 between 1, one gluon corresponding to n 1 and n 1 $_{1}$, the other gluon corresponding to n 1 respectlvely, ... After inserting all the gluons corresponding to gravitons in the first line, we insert two gluons corresponding to n 2 1 between 1, l$_{2}$ and l$_{2}$ , i respectively, then insert two gluons corresponding to n 2 between 1, one gluon corresponding to n 2 1 and n 2 $_{1}$, the other gluon corresponding to n 2 1 respectively, ... In this way, we insert all the 2 M gluons corresponding to the M gravitons into the amplitudes. The phase factor is s$_{l}$$_{1}$$_{,n}$ 1 $_{1}$s$_{n}$ 1 $_{1}$,n 1 $_{2}$...s$_{l}$$_{2}$$_{,n}$ 2 $_{1}$s$_{n}$ 2 $_{1}$,n 2 $_{2}$...s$_{l}$$_{N}$$_{,n}$ N 1 s$_{l}$$_{N}$$_{,n}$ N 2 ... . At last, the amplitudes become MHV tree amplitudes with N + 2 M gluons, where the two gluons corresponding to a same graviton take the same momentum. This is just the D$_{2}$ relations in field theory. 

In string theory, the D$_{2}$ relations are independent of helicity configurations of the legs. Then we expect the D$_{2}$ relations should have helicity-independent form. For example, in the relation for amplitudes with one graviton and N gluons given in Subsection IV A, we only sum over l correspongding to the gluons with positive helicity, and for each l , the two gluons 

