The irreducibility of the moduli space of pointed stable curves with dormant $\mathrm{PGL}_2^{(N)}$-oper

A $\mathrm{PGL}_n^{(N)}$-oper is a specific type of flat $\mathrm{PGL}_n$-bundle on an algebraic curve in prime characteristic $p$ enhanced by an action of the sheaf of differential operators of level $N-1$. In this paper, we introduce and study a higher-level generalization of the Hitchin-Mochizuki morphism on the moduli space of $\mathrm{PGL}_n^{(N)}$-opers, defined via the characteristic polynomials of their $p^N$-curvatures. As an application, we prove the irreducibility of the moduli space classifying pointed stable curves equipped with dormant $\mathrm{PGL}_2^{(N)}$-opers, i.e., $\mathrm{PGL}_2^{(N)}$-opers with vanishing $p^N$-curvature.