  We derive universal constraints on $(1+1)d$ rational conformal field theories
(CFTs) that can arise as transitions between topological theories protected by
a global symmetry. The deformation away from criticality to the trivially
gapped phase is driven by a symmetry preserving relevant deformation and under
renormalization group flow defines a conformal boundary condition of the CFT.
When a CFT can make a transition between distinct trivially gapped phases the
spectrum of the CFT quantized on an interval with the associated boundary
conditions has degeneracies at each energy level. Using techniques from
boundary CFT and modular invariance, we derive universal inequalities on all
such degeneracies, including those of the ground state. This establishes a
symmetry enriched $c$-theorem, effectively a lower bound on the central charge
which is strictly positive, for this class of CFTs and symmetry protected
flows. We illustrate our results for the case of flows protected by
$SU(M)/\mathbb{Z}_{M}$ symmetry. In this case, all SPT transitions can arise
from the WZW model $SU(M)_{1},$ and we develop a dictionary between conformal
boundary conditions and relevant operators.
