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As far as I am aware there are two pictures of branes:

  • for $g_SN\gg1$: branes as solitonic solutions to the field equations of supergravity (i.e. black branes)
  • for $g_SN\ll1$: branes as fundamental objects, on which we can build $\sigma$-models (e.g. the fundamental string).

where $g_S$ is the string coupling and $N$ is the units of flux or number of branes in a stack.

My understanding is that these two are seen as complementary pictures of the same objects, e.g. a condensate of fundamental p-branes turns into a black brane. However, I am not sure whether the black brane picture is valid at strong string coupling too. Some seemingly incompatible excerpts from the literature:

  • In Maldacena's 1997 paper, it is mentioned that the black brane limit $g_SN\gg1$ is understood to be at small $g_S$. Indeed, in the classical review hep-th/9905111, the limit is written as $N>g_SN\gg1$ to emphasise this.
  • From Clifford Johnson's D-brane primer: $g_SN\gg1$ "corresponds to either having $N$ small and $g_S$ large, or vice versa".

So I am not sure what to believe: are black branes (as solutions to supergravity) at small $g_S$, or at large $g_SN$ with $g_S$ free to take any value (as long as $g_SN\gg1$)? I see reasons for both:

  • Since we are talking about (10-d) supergravity solutions, we ought to be at small $g_S$ (in addition to small $\alpha'$), since SUGRA is the low-energy limit of string theory (which is perturbative in $g_S$). However, I guess as we increase $g_S$ we would simply go to 11-d SUGRA, so perhaps $g_S$ need not be small?
  • In the black brane picture, the branes backreact so much that they create the geometry. This makes sense if $g_S$ is large. Also, the large $g_S$ argument seems to me to be at the heart of entropy calculations of black holes.

Or is the whole point that (some) branes are BPS objects? And so even if originally the SUGRA solutions were at small $g_S$ (since it is SUGRA we're talking about), they are protected objects and we may follow them at strong coupling too, and so effectively the black brane picture is valid at any $g_S$?

Many thanks.

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Your question already contains the elements for an answer.

Since we are talking about (10-d) supergravity solutions, we ought to be at small 𝑔𝑆 (in addition to small 𝛼′), since SUGRA is the low-energy limit of string theory (which is perturbative in 𝑔𝑆).

Correct. As $𝑔_𝑆$ increases the correct description becomes the full non perturbative string theory (or M-theory). Do not confuse this with the dimensional reduction 11d --> 10d.

Or is the whole point that (some) branes are BPS objects? And so even if originally the SUGRA solutions were at small 𝑔𝑆 (since it is SUGRA we're talking about), they are protected objects and we may follow them at strong coupling too, and so effectively the black brane picture is valid at any 𝑔𝑆?

It's not that the picture is valid, but that the same solutions of SUGRA are also solutions of the full non perturbative theory. As you say, they are protected from all the quantum corrections that should arise when turning the string coupling on. That is why studying BPS in SUGRA is so useful, it enables us to gather information about the full theory, without dealing with all the complexities.

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  • $\begingroup$ Hi thanks for your answer! When we say that BPS objects are protected, we mean that e.g. their mass receives no quantum corrections, right? But the SUGRA solution itself (i.e. an expression for the graviton $ds^2=\dots$ and whatever gauge field, e.g. $A_3=\dots$) is NOT also a solution of the non-perturbative theory, or is it? i.e. does the solution itself also survive at strong coupling, or just its properties such as mass? $\endgroup$
    – user984949
    May 25, 2022 at 12:14
  • $\begingroup$ Yes, the solution will be expressed in a different way, trough some quantum numbers identifying it. These are usually some quantities that in SUGRA you would describe as mass, charge, angular momentum, etc. The SUGRA description (all the scalar, vector, etc. fields) is not valid anymore. $\endgroup$
    – Rexcirus
    May 25, 2022 at 12:25
  • $\begingroup$ This is often enough to at least count all the solutions, and verify that SUGRA and numbers obtained (indirectly, with some clever trick) from the full theory agree. $\endgroup$
    – Rexcirus
    May 25, 2022 at 12:28
  • $\begingroup$ You may want to check literature around black hole microstates counting for 2-charge black holes, it's the easier case to do calculations. $\endgroup$
    – Rexcirus
    May 25, 2022 at 12:30

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