Single- and multi-photon production in $e^+ e^-$ collisions at $\sqrt{s}$ up to 209 GeV

Events containing only energetic photons are analysed in a sample of 628 pb -1 of data recorded from e+e- collisions at centre-of-mass energies between 189 and 209 GeV by the ALEPH detector at LEP. The $\reenunug$ and $\reeggg$ cross sections are measured and found to be in agreement with the Standard Model predictions. The number of light neutrino generations is determined to be N_\nu = 2.86 \pm 0.09. Upper limits are derived on the cross sections for photon production in the context of several supersymmetric models. Limits are also set on the parameters of models with extra spatial dimensions with contact interactions and with excited electrons.