It is well known that if one constructs ordinary WZW type sigma model for string action, it is possible whenever they find a cocycle in appropriate Chevalley-Eilenberg algebra. If I understand it correctly, this approach gives the right actions in cases, when we do not consider so-called "background branes" (this terminology is taken from the article https://arxiv.org/abs/1308.5264 ). When we do consider also background branes, our theory effectively obtains new (higher) gauge field on its world-volume.
If I work with theory of open strings and impose D-branes as a background branes to this theory, what is the relevant gauge field on its world-sheet and why?
According to my former understanding, it should be Chan-Paton gauge field, but Chan-Paton gauge field probably lives on the world-volume of D-branes and has more likely something to do with terms in DBI action. But I am pretty confused about it, I would be glad for proper explanation.
Later edit: More careful reading of mentioned article has shown that on this very background brane there appears the higher gauge field in question, not on the world-sheet of a string. It sounds plausible, because then commonly known (super) Yang-Mills theories on the world-volume of D-brane are probably just low-energy limits of such higher WZW theories with higher gauge fields on their world-volume.
Where my current understanding wanes is the description of suitable theory for strings with such background-brane environment. One has to consider some physics of string's boundaries, which are charged under these higher gauge fileds. But I am not aware of good arguments for that.