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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?

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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.

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  • $\begingroup$ 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? $\endgroup$
    – tomdodd4598
    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? $\endgroup$
    – tomdodd4598
    Oct 18, 2022 at 15:01
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    $\begingroup$ @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? $\endgroup$
    – tomdodd4598
    Oct 18, 2022 at 15:10

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