Relations Among Amplitudes On Disk In String Theory And Field Theory Limits

In this thesis, we study the relations between closed and open string amplitudes. We also study the relations between mixed amplitudes with closed strings as well as open strings and pure open string amplitudes. In this thesis, we give an introduction to the KK relations and BCJ relations among pure open string amplitudes on disk. We also give an introduction to the KLT factorization relations between closed string amplitudes on sphere and the disk amplitudes of open strings corresponding to the left-moving sector as well as the right-moving sector. We then study the amplitudes containing closed string external legs on disk and the closed string amplitudes on real projective plane. In these two cases, we find the KLT factorization relations do not hold. However, the amplitudes satisfy new relations. We study the field theory limits of closed string amplitudes. We find that disk relations hold in minimal coupling theory of gauge field and gravity.We first give an introduction to KK relations and BCJ relations among open string amplitudes on disk. Through the discussions on monodromy, KK relations and BCJ rela-tions express any open string partial amplitude in a minimal basis. For any N-point open string amplitude on disk, the minimal basis contain (N - 3)! amplitudes. In field theory limits, KK relations and BCJ relations express any tree partial amplitude in a minimal basis of (N-3)! amplitudes.We then give an introduction to KLT factorization relations for closed string ampli-tudes on sphere. KLT relations factorize any M-point closed string amplitude on sphere into left-and right-moving open string disk amplitudes with M external legs(except for a phase factor). The KLT factorizations hold on sphere because the left-and the right-moving waves are independent of each other. after expressing the disk amplitudes of left-and right-moving open strings as sums of partial amplitudes which depends on the permutations of the external legs, KLT relations express the closed string amplitudes on sphere as sums of products of open string partial amplitudes on disk. The KLT relations can be reduced by contour deformations. This is because the open string partial amplitudes are not inde-pendent of each other. In field theory limits, KLT factorization relations express the tree amplitudes of gravitons by sums of products of two tree partial amplitudes of gauge field. The gauge field has structure of heterotic strings, thus KLT factorization relations can also hold for tree amplitudes in minimal coupling theory of gauge field and gravity.After introducing KK relations, BCJ relations and KLT relations, we introduce our work. First, we study the disk amplitudes containing closed strings as well as closed string amplitudes on real projective plane. Because the reflections at the boundary or crosscap, the left-moving waves interact with the right-moving waves. Thus the factorization relations for amplitudes on sphere do not hold for amplitudes on disk and real projective plane. Since the left-and right-moving sectors are connected into a single one, the amplitudes with N open string legs and M closed string legs on disk can be given by N+2M-point open string amplitudes on disk. In a similar way, the M-point closed string amplitudes on real projective plane can be given by 2M-point open string amplitudes on disk. As in the sphere case, the relations on disk and real projective plane can be reduced because the open string partial amplitudes are not independent of each other. We give some examples of this reduction.Second, we discuss the field theory limits of disk relations. In field theory limits, disk relations give the relations between amplitudes of gauge-gravity minimal coupling and amplitudes of pure gauge field, disk relations are different from KLT factorization relations. However, after some discussions on three-and four-point amplitudes as well as some amplitudes with more external legs, we find the disk relations also hold for tree amplitudes in minimal coupling theory of gauge field and gravity. Disk relations in field theory are based on the disk structure of open strings.