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.

  • $\begingroup$ Are you asking about the 2-form B-field? It's always there I think (except in type I when it gets projected out?). $\endgroup$ – Ryan Thorngren Sep 10 '18 at 21:17
  • $\begingroup$ 2-form B-field comes to the theory as a higher gauge field living on spacetime, I think it is permanent ingredient of all WZW theories and do not reflect the existence of any background-brane. If do not agree, please explain me this in little more details. $\endgroup$ – Nary Sep 10 '18 at 21:32
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    $\begingroup$ In string theory it comes from the worldsheet modes $a_i a_j |0\rangle$ with anti-symmetric combinations of indices. Sometimes the GSO projection forbids these. String theory has no fundamental fields. Everything emerges from the string (supposedly). $\endgroup$ – Ryan Thorngren Sep 10 '18 at 21:33
  • $\begingroup$ For the later edit: Are you already in BV fields? Because that will complicate the reduction to a familar SYM or DBI action because you'll need $\mathcal{L}_{g.f.}$ and then probably a messy change of coordinates. $\endgroup$ – AHusain Sep 12 '18 at 1:10
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    $\begingroup$ There are reducible symmetries ($B \to B + d\alpha, \alpha \to \alpha + df$ in the case of 2-form $B$) so when you quantize BRST won't cut it. Then you write the BV extension. nLab. Follow links therein for the procedure. $\endgroup$ – AHusain Sep 12 '18 at 19:31

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