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Open & Closed vs. Pure Open String Disk Amplitudes

We establish a relation between disk amplitudes involving N_o open and N_c closed strings and disk amplitudes with only N_o+2N_c open strings. This map, which represents a sort of generalized KLT relation on the disk, reveals important structures between open & closed and pure open string disk amplitudes: it relates couplings of brane and bulk string states to pure brane couplings. On the string world-sheet this becomes a non-trivial monodromy problem, which reduces the disk amplitude of N_o open and N_c closed strings to a sum of many color ordered partial subamplitudes of N_o+2N_c open strings. This sum can be further reduced to a sum over (N_o+2N_c-3)! subamplitudes of N=N_o+2N_c open strings only. Hence, the computation of disk amplitudes involving open and closed strings is reduced to computing these subamplitudes in the open string sector. In this sector we find a string theory generalization and proof of the Kleiss-Kuijf and Bern-Carrasco-Johanson relations: All order alpha' identities between open string subamplitudes are derived, which reproduce these field-theory relations in the limit alpha'->0. These identities allow to reduce the number of independent subamplitudes of an open string N-point amplitude to (N-3)!. This number is identical to the dimension of a minimal basis of generalized Gaussian hypergeometric functions describing the full N-point open string amplitude.