  While gauge symmetry is a well-established requirement for representing
topological orders in projected entangled-pair state (PEPS), its impact on the
properties of low-lying excited states remains relatively unexplored. Here we
perform PEPS simulations of low-energy dynamics in the Kitaev honeycomb model,
which supports fractionalized gauge flux (vison) excitations. We identify gauge
symmetry emerging upon optimizing an unbiased PEPS ground state. Using the PEPS
adapted local mode approximation, we further classify the low-lying excited
states by discerning different vison sectors. Our simulations of spin and
spin-dimer dynamical correlations establish close connections with experimental
observations. Notably, the selection rule imposed by the locally conserved
visons results in nearly flat dispersions in momentum space for excited states
belonging to the 2-vison or 4-vison sectors.
