Positive-characteristic categorification without characteristic-zero categorification

Determine whether an integral fusion ring can admit a categorification over an algebraically closed field of positive characteristic while admitting no categorification in characteristic zero.

Background

The question concerns whether categorifiability can occur exclusively in positive characteristic. The paper notes that no such example is known and explains that any integral fusion ring of rank at most three has a characteristic-zero categorification, so a first example would have to have rank at least four.

The authors construct an infinite family of simple integral vanishing fusion rings of rank four. Because no simple integral fusion ring of rank four admits a characteristic-zero categorification, these rings are identified as candidates for the first examples that might admit a categorification in positive characteristic but not in characteristic zero. The paper does not establish that any member of the family has such a positive-characteristic categorification.