In this penultimate lecture of a series given by Freddy Cachazo, the differential forms corresponding to the massless excitations in the Ramond–Ramond sector signal the existence of p-branes sourcing the associated fields. We find 0- and 2-branes for the type II-A case, and (-1)-, 1- and 3-branes for the type II-B case.

However, for some reason that I didn't follow, we also assume that there also exist "magnetically" charged branes dual to the "electrically" charged ones, yielding the additional 6- and 4-branes for type II-A, and the 7- and 5-branes for type II-B (the dual to the 3-brane is also a 3-brane). We also get an NS5-brane dual to the fundamental string in the NS-NS sector.

Why are these additional objects said to exist? Is it a self-consistency condition, or simply an explicit assumption?

If it's the former, does the same apply for the bosonic theory, i.e. would it need an NS21-brane dual to the fundamental string to be self-consistent?


1 Answer 1


The "charged branes" in superstring theory arise from the democratic formulation of the Ramond-Ramond fields, where to each "naive" Ramond potential $C_p$ (the "electric" potential) there is a dual potential $C_{8-p}$ (the "magnetic" potential) and the two are on-shell related by their field strengths being duals of each other: $$ \mathrm{d}C_{p} = {\star}\mathrm{d}C_{8-p}$$ The names "magnetic" and "electric" arise because this is a generalization of classical electromagnetic duality in vacuum, see e.g. this answer of mine.

All of these RR fields can have source terms, just like electric current in classical EM: $\mathrm{d}{\star}\mathrm{d}C_p = {\star}J_{p}$, where the current is a $p$-form, meaning it gives charge when integrated over a subvolume with $p$ dimensions, which can be interpreted as the worldvolumes of $p-1$-branes, so its "charged objects" are $p-1$-branes. This is exactly analogous to ordinary electric current being a 1-form because it is sourced by the worldlines of particles (=0-branes). Again the "electric"/"magnetic" terminology arises because the $C_p$-charged branes are electric and the $C_{8-p}$-charged branes are magnetic.

Bosonic string theory doesn't have Ramond-Ramond fields and so none of this applies to it.

  • $\begingroup$ I came across a summary of the democratic formulation in section 5.3 of this paper, which seems to say that there is essentially a mathematical equivalence between the "original" and "democratic" field theories after imposing some field strength duality relations. Am I meant to take from this that the "magnetic" branes can be thought of as either fundamental or not? $\endgroup$ Oct 18, 2022 at 14:48
  • $\begingroup$ @turbodiesel4598 I don't think "fundamental" is a useful word here. The dualities between the five string theories and their different brane contents mean that either none of these objects are "fundamental" or all of them are. $\endgroup$
    – ACuriousMind
    Oct 18, 2022 at 14:53
  • $\begingroup$ I see, I suppose I need to learn more about the dualities to better appreciate that. As for the dual NS5-brane in the NS-NS sector, is there a similar "democratic" construction, or does that appear for a different reason? $\endgroup$ Oct 18, 2022 at 15:01
  • 1
    $\begingroup$ @turbodiesel4598 it is exactly the same logic: The string is a 1-brane, so it is charged under some $C_2$, the dual of that is a $C_6$, and the charged objects of that are 5-branes. The field under which it is charged is just the Kalb-Ramond field instead of one of the RR fields. $\endgroup$
    – ACuriousMind
    Oct 18, 2022 at 15:05
  • $\begingroup$ This will apply to the bosonic theory giving an NS21-brane, right? Or does the lack of an appropriate duality or other motivation stop us from following the logic of the superstring case? $\endgroup$ Oct 18, 2022 at 15:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.