  The nonequilibrium critical dynamics of the 2D XY model is investigated
numerically through Monte Carlo simulations and analytically in the spin-wave
approximation. We focus in particular on the behaviour of the two-time response
and correlation functions and show that the ageing dynamics depends on the
initial conditions. The presence of critical fluctuations leads to non-trivial
violations of the fluctuation-dissipation theorem apparently reminiscent of the
three dimensional Edwards-Anderson spin glass model. We compute for this reason
the finite-size overlap probability distribution function and find that it is
related to the finite-time fluctuation-dissipation ratio obtained in the out of
equilibrium dynamics, provided that the temperature is not very low.
