  We study dynamical aspects of the $q$-state Potts model on an $n\times n$ box
at its critical $\beta_c(q)$. Heat-bath Glauber dynamics and cluster dynamics
such as Swendsen--Wang (that circumvent low-temperature bottlenecks) are all
expected to undergo "critical slowdowns" in the presence of periodic boundary
conditions: the inverse spectral gap, which in the subcritical regime is
$O(1)$, should at criticality be polynomial in $n$ for $1< q \leq 4$, and
exponential in $n$ for $q>4$ in accordance with the predicted discontinuous
phase transition. This was confirmed for $q=2$ (the Ising model) by the second
author and Sly, and for sufficiently large $q$ by Borgs et al.
  Here we show that the following holds for the critical Potts model on the
torus: for $q=3$, the inverse gap of Glauber dynamics is $n^{O(1)}$; for $q=4$,
it is at most $n^{O(\log n)}$; and for every $q>4$ in the phase-coexistence
regime, the inverse gaps of both Glauber dynamics and Swendsen--Wang dynamics
are exponential in $n$.
  For free or monochromatic boundary conditions and large $q$, we show that the
dynamics at criticality is faster than on the torus (unlike the Ising model
where free/periodic boundary conditions induce similar dynamical behavior at
all temperatures): the inverse gap of Swendsen--Wang dynamics is
$\exp(n^{o(1)})$.
