  Using the QCD dipole picture of BFKL dynamics and the conformal invariance
properties of the BFKL kernel in transverse coordinate space, we show that the
1--to--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.
