  We study the symplectic nature of the moduli stack classifying dormant curves
over a field $K$ of positive characteristic, i.e., proper hyperbolic curves
over $K$ equipped with a dormant indigenous bundle. The central objects of the
present paper are the following two Deligne-Mumford stacks. One is the
cotangent bundle ${^\circledcirc T^{\vee ^\mathrm{Zzz...}}_{g,K}}$ of the
moduli stack ${^\circledcirc \mathfrak{M}^{^\mathrm{Zzz...}}_{g,K}}$
classifying ordinary dormant curves over $K$ of genus $g$. The other is the
moduli stack ${^\circledcirc \mathfrak{S}^{^\mathrm{Zzz...}}_{g,K}}$
classifying ordinary dormant curves over $K$ equipped with an indigenous
bundle. These Deligne-Mumford stacks admit canonical symplectic structures
respectively. The main result of the present paper asserts that a canonical
isomorphism ${^\circledcirc T^{\vee ^\mathrm{Zzz...}}_{g,K}} \rightarrow
{^\circledcirc \mathfrak{S}^{^\mathrm{Zzz...}}_{g,K}}$ preserves the symplectic
structure. This result may be thought of as a positive characteristic analogue
of the works of S. Kawai (in the paper entitled "The symplectic nature of the
space of projective connections on Riemann surfaces"), P. Ar\'{e}s-Gastesi, I.
Biswas, and B. Loustau. Finally, as its application, we construct a
Frobenius-constant quantization on the moduli stack ${^\circledcirc
\mathfrak{S}^{^\mathrm{Zzz...}}_{g,K}}$.
