Abstract
KLT relations almost factorize closed string amplitudes on S 2 by two open string tree amplitudes which correspond to the left- and the right-moving sectors. In this paper, we investigate string amplitudes on D 2 and RP 2 . We find that KLT factorization relations do not hold in these two cases. The relations between closed and open string amplitudes have new forms. On D 2 and RP 2 , the left- and the right-moving sectors are connected into a single sector. Then an amplitude with closed strings on D 2 or RP 2 can be given by one open string tree amplitude except for a phase factor. The relations depends on the topologies of the world-sheets. Under T-duality, the relations on D 2 and RP 2 give the amplitudes between closed strings scattering from D-brane and O-plane respectively by open string partial amplitudes. In the low energy limits of these two cases, the factorization relations for graviton amplitudes do not hold. The amplitudes for gravitons must be given by the new relations instead.
Highlights
ASn2daM nd(NA)Sβ(2N(P)α′β) is the closed are the open string string partial amplitudes on D2 corresponding to the left- and right-moving sectors respectively
A question arises: Do KLT factorization relations hold for any gravity amplitude? In string theory, to calculate the S-matrix, we should sum over all the topologies of worldsheets
We investigated the relations between closed and open strings on D2 and RP2
Summary
We will explore the relations between amplitudes on D2 and open string amplitudes. We should calculate the correlation functions of the vertex operators on D2, integral over the fundamental region for closed strings. The phase factor is used to guarantee the integrals in the right branch cut It only depend on the the orderings of the open strings. Because (2.19) is already an amplitude for open strings on the real axis except for a phase factor, we just increase the number of the open strings on the boundary of D2 and adjust the phase factor to make the integrals in the right branch cuts. We reverse the time in the right sector to get the ordinary form of open string amplitude. In this step, we have used the mass-shell condition. We should use one amplitude for 2N gauge particles instead of the product of two amplitudes for N gauge particles to give an amplitude for N gravitons
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