  Recently introduced reparametrization mode operators in CFTs have been shown
to govern stress tensor interactions $via$ the shadow operator formalism and
seem to govern the effective dynamics of chaotic systems. We initiate a study
of Ward identities of reparametrization mode operators $i.e.$ how two
dimensional CFT Ward identities govern the behaviour of insertions of
reparametrization modes $\epsilon$ in correlation functions:
$\langle\epsilon\epsilon\phi\phi\rangle$. We find that in the semi-classical
limit of large $c$ they dictate the leading $\mathcal{O}(c^{-1})$ behaviour.
While for the $4$pt function this reproduces the same computation as done by
Heahl, Reeves \& Rozali in \cite{Haehl:2019eae}, in the case of 6pt function of
pair-wise equal operators this provides an alternative way of computing the
Virasoro block in stress-tensor comb channel. We compute a maximally out of
time ordered correlation function in a thermal background and find the expected
behaviour of an exponential growth governed by Lyapunov index
$\lambda_L=2\pi/\beta$ lasting for twice the scrambling time of the system
$t^*=\frac{\beta}{2\pi}\log\,c$ for the maximally braided type of
$out$-$of$-$time$-$ordering$. However when only the internal operators of the
comb channel are \emph{out-of-time-ordered}, the correlator sees no exponential
behaviour despite the inclusion of the Virasoro contribution. From a bulk
perspective for the \emph{out-of-time-ordered} $4$pt function we find that the
Casimir equation for the stress tensor block reproduces the linearised back
reaction in the bulk.
