A Combinatorial Description of the Dormant Miura Transformation

The aim of the present paper is to describe Miura sl2-opers and Miura transformations in terms of graph-theoretic objects. We construct a bijective correspondence between dormant generic Miura sl2-opers on a totally degenerate curve in positive characteristic and certain branch numberings on a 3-regular graph. This correspondence allows us to completely identify dormant generic Miura sl2-opers on totally degenerate curves. Also, we investigate how this result can be related to the combinatorial description of dormant sl2-opers given by S. Mochizuki, F. Liu, and B. Osserman.