I. INTRODUCTION Superstring theories are theories containing both gravitational and gauge interactions[1, 2]. They offer a possible way to unify the gauge field and gravity. In string theory, gravitons and gauge particles can correspond to massless states of closed and open strings. Then the relations between closed and open strings imply the relations between gravity and gauge field. There are many relationships between gauge field and gravity in string theory such as AdS/CFT[3] correspondence and the perturbative relations for amplitudes on sphere( S$_{2}$ )[4], disk( D$_{2}$ )[5–10] and real projective plane( RP$_{2}$ )[5]. The perturbative relations in S$_{2}$ case named KLT relation[4] factorize the amplitudes on S$_{2}$ into products of two amplitudes for open strings corresponding to the left- and the right-moving sectors of closed strings. However, the amplitudes on D$_{2}$ cannot be factorized into two sectors, because the boundary of D$_{2}$ connect the two sectors into a single one. We should use the new relations instead of KLT factorization relations on D$_{2}$ . KLT factorization relations and D$_{2}$ relations hold on worldsheets with different topologies. In field theory limit, KLT factorization relations allow one to obtain gravity amplitudes from gauge theory ones[11–16]. They can also be used in many theories of gravity-matter couplings[17, 18]. An important application is that the low energy limits of KLT relations can be used to calculate the tree amplitudes for gluons minimally coupled to gravitons. In this case, using the relations, the amplitudes with gluons and gravitons can be factorized into products of amplitudes in left- and right-moving sectors. Amplitudes in one sector are pure gauge partial amplitudes while those in the other sector are partial amplitudes with gluons and scalars. These relations for gauge-gravity minimal coupling are based on the 2