what is local literature?

I don't know that there is any formal term. Rustic or bucolic come to mind, but, again, i don't know that those are formal terms.

I found a Wikipedia article which speaks of "Pastoral" as a genre, and it also refers to bucolic. However, there is a caveat at the top of the page to the effect that the article reads like a personal essay rather than an encyclopedia article. Pastoral, specifically, refers to shepherds or cowheards, or any activity relating to the grazing of livestock. It is also used as a general term for rural living, so you might be able, reasonably, to use the term pastoral literature.