  Using the dipole picture describing the $1/N_C$ limit of QCD at small $x$ and
the conformal invariance properties of the BFKL kernel in transverse coordinate
space, we show that the 1->p dipole densities can be expressed in terms of dual
Shapiro-Virasoro amplitudes B_{2p+2} and their generalization including
non-zero conformal spins. We discuss the possibility of an effective closed
string theory of interacting QCD dipoles.
