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relations for three- and four-point amplitudes are given as 

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In field theory limits, KLT relations factorize the pure-graviton tree amplitudes into products of tree amplitudes for gluons corresponding to left- and right-moving sectors. The tree amplitudes where gravitons are minimally coupled to gluons can also be factorized by KLT relations. In this situation, the two sectors of a graviton with helicity ± 2 correspond to two gluons with helicity ± 1, while the two sectors of a gluon with helicity ± 1 correspond to one gluon with helicity ± 1 and one scalar particle. The gauge degrees of freedom are taken by the scalar field in one sector. With these correspondences, KLT relations express the amplitudes for N gluons and M gravitons by products of amplitudes in left- and right-moving sectors. The amplitudes in the left-moving sector are pure-gluon partial tree amplitudes with N + M legs while the amplitudes in the right-moving sector are partial tree amplitudes for M gluons and N scalar particles. Using the Feynman rules given in [17], one can calculate the partial amplitudes in the left- and right-moving sectors, then the amplitudes where gluons are minimally coupled with gravitons can be given by KLT relations. 

The KLT factorization relations in string theory hold on S$_{2}$ . But they do not hold on D$_{2}$ . In D$_{2}$ case, the left- and right-moving sectors of closed strings are connected into a single one[5]. The relations between amplitudes for N open strings in addition to M closed strings on D$_{2}$ and open string tree amplitudes are given as 

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In the relation (3), we do not introduce the Chan-Paton degrees of freedom. On D$_{2}$ , one can add Chan-Paton factor to the ends of open strings to incorporate gauge degree of freedom. 

