After the duality revolution, we learned about the important role of both, black p-branes and p-branes. However, sometimes their fundamental definition is confused in the literature. I mean the following:

  1. Black p-brane: solitonic solution to (super)gravity field equations (and / or superstring/supergravity theories), describing a black hole with p-spatial dimensions.

  2. Fundamental p-branes: p-dimensional extended object, described by a Nambu-Goto action (or any other action equivalent at the level of the equation of motion, e.g., the Howe-Tucker action).


  • Is any black brane fundamental somehow, and so, can it be described by some kind of Nambu-Goto action (or Polyakov-like, etc), and viceversa, ...Can we consider any fundamental p-brane like some kind of "black-hole"-like solution? I have never read a clarification of this stuff in the literature, at least, not in a transparent way. In particular, I can not see any reason why the Nambu-Goto action is relevant to black hole p-branes, BUT, seemingly, the counting of Dp-branes providing the BH entropy in some BH solutions supports the idea of having fundamental p-branes as degrees of freedom entangled with this problem. I am not sure, however, of the equivalence or not of these objects.

1 Answer 1


That's an excellent question. Indeed, despite plenty of interesting relations, there is a crucial distinction between fundamental and black branes, and the standard informal string theory literature does tend to blur this distinction. Of course a key reason is that an actual non-perturbative theory that would allow to precisely phrase the relation has been missing.

But a precise formulation of fundamental/black branes at singularities that captures both their difference as well as their close relation is offered in our recent article

There M-branes are are argued to be modeled by cocycles in real ADE-equivariant cohomotopy (much as like plain D-branes are modeled by (co)cycles in real K-theory) and the rich component structure of such cocycles involves three kinds of data:

  1. there is data associated to real ADE-singularities in spacetime, corresponding to bound states/intersections of black branes at singularities;
  2. there are WZW terms for Green-Schwarz sigma models of fundamental branes attached to spacetime and to the strata of these singularities;
  3. there is compatibility data between the two exhibited by appearance of the full Green-Schwarz action functional wherever a fundamental super $p$-brane WZW term is restricted to the corresponding black $p$-brane singular locus.

Here is a snippet from Table 3 in the article, showing two such cocycles exhibiting "bound states" of black and fundamental branes:

enter image description here

See also the discussion in Example 2.10.


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