  We find a breakdown of the critical dynamic scaling in the coarsening
dynamics of an antiferromagnetic {\em XY} model on the kagome lattice when the
system is quenched from disordered states into the Kosterlitz-Thouless ({\em
KT}) phases at low temperatures. There exist multiple growing length scales:
the length scales of the average separation between fractional vortices are
found to be {\em not} proportional to the length scales of the quasi-ordered
domains. They are instead related through a nontrivial power-law relation. The
length scale of the quasi-ordered domains (as determined from optimal collapse
of the correlation functions for the order parameter $\exp[3 i \theta (r)]$)
does not follow a simple power law growth but exhibits an anomalous growth with
time-dependent effective growth exponent. The breakdown of the critical dynamic
scaling is accompanied by unusual relaxation dynamics in the decay of
fractional ($3\theta$) vortices, where the decay of the vortex numbers is
characterized by an exponential function of logarithmic powers in time.
