The time-dependent scaling of the two-time autocorrelation function of spin systems without disorder undergoing phase-ordering kinetics is considered. Its form is shown to be determined by an extension of dynamical scaling to a local scale invariance which turns out to be a new version of conformal invariance. The predicted autocorrelator is in agreement with Monte Carlo data on the autocorrelation function of the 2D kinetic Ising model with Glauber dynamics quenched to a temperature below criticality.
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