A combinatorial description of the dormant Miura transformation

A dormant generic Miura ${\mathfrak{s}\mathfrak{l}}_2$-oper is a flat ${\mathrm{P}\mathrm{G}\mathrm{L}}_2$-bundle over an algebraic curve in positive characteristic equipped with some additional data. In the present paper, we give a combinatorial description of dormant generic Miura ${\mathfrak{s}\mathfrak{l}}_2$-opers on a totally degenerate curve. The combinatorial objects that we use are certain branch numberings of $3$-regular graphs. Our description may be thought of as an analogue of the combinatorial description of dormant ${\mathfrak{s}\mathfrak{l}}_2$-opers given by S. Mochizuki, F. Liu, and B. Osserman. It allows us to think of the Miura transformation in terms of combinatorics. As an application, we identify the dormant generic Miura ${\mathfrak{s}\mathfrak{l}}_2$-opers on totally degenerate curves of genus $>0$.