# What does it mean to “wrap” a D-brane around some manifold?

I am getting quite confused with this terminology when I read the papers. Like while constructing the near horizon $AdS_3$ in the $D1-D5$ system one considers $IIB$ on $R^{1,4}\times M^4 \times S^1$ and one "wraps" $N_1$ D1 branes on the $S^1$ and $N_5$ D5-branes on $M^4 \times S^1$. What does it exactly mean?

Coming from reading how D-branes were introduced in Polchinksi's book I would think that in a $9+1$ spacetime Dp branes are some planar streched out stuff with a $p+1$ worldvolume and whose $p$ spatial dimensions are transverse to the $9-p$ spatial dimensions which have been compactified and T-dualized. So are we now saying that its possible that instead of imagining the Dp branes as some set of periodically arranged planes on the T-dual torus we can also think of their spatial world being compactified on some arbitrary p-manifold?

If "wrapping" is really a choice of topology for the p-spatial dimensions of the Dp-brane then what determines this choice? Is this something put in by hand or does this happen naturally?

D-branes are not restricted to planar geometries. They can take on many different forms, and you often encounter branes wrapped around spherical manifolds, like $S^1$ or $S^4$. To determine whether a given configuration is stable, you have to evaluate the action of the D-brane configuration, which is given by the Dirac-Born-Infeld action. For a $Dp$-brane, it is given by

$$S_{DBI}=-T_p\int d^{p+1}x \sqrt{-\text{det}(g_{ab}+2\pi\alpha'F_{ab})},$$

where $T_p$ is the brane-tension, $g_{ab}$ is the metric of the brane, $\alpha'$ is the string length squared and $F_{ab}$ is the field strength of the gauge fields on the brane.

Branes wrapped on nontrivial geometries sometimes have a direct physical interpretation. In holographic QCD, branes wrapped on an $S^4$ correspond to baryons. This can be understood from the fact that this non-trivial topology generates an instanton number which can be identified with the baryon number.

EDIT: For a great introduction to the subject, consult the book "D-Branes" by Johnson, or his shorter lecture notes: http://arxiv.org/abs/hep-th/0007170

• Brunner Thanks for the answer. Can you give some reference where this analysis is done? I mean is anywhere it is explained how Dp branes can take arbitrary shapes? I mean - what is the mechanism for them to attain some shape? The T-duality argument doesn't seem to lead to any shape information? – user6818 Mar 16 '14 at 2:24
• I would recommend Johnson's book "D-Branes", there you should find any answers you want. For a free document, you can also read his shorter lecture notes on D-branes: arxiv.org/abs/hep-th/0007170 – Frederic Brünner Mar 16 '14 at 10:40
• Bruinner I have been reading that book - anything specific you can point out there - where this is explained? - as to how wrapping happens "on its own"? What is the dynamics of it? – user6818 Mar 18 '14 at 18:56
• The books mentions wrapping in chapter 13.3 and 15.4., for example. – Frederic Brünner Apr 11 '14 at 8:27