I am not sure if this question makes sense "intuitively" but here it is. In Klebanov-Witten theory one puts branes on the tip of a cone over a Calabi-Yau manifold. My question is what does it mean to put a "D-brane on a tip" which is a singularity? I cannot understand where this tip is spatially located. Nor can I understand if the cone breaks or not Lorentz symmetry in the 10d space.


It might help to think of obtaining Klebanov-Witten theory in a limiting proceedure. Imagine a dynamical setting where the branes don't start out at the singularity, but instead at some smooth point near the singularity. Then move them to the conifold point. And the conifold point can be located because it's a singular point. If we choose coordinates so that the conifold is at $r=0$, then the metric is

$ds^2 = dr^2 + r^2 ds^2(T^{1,1}),$

which is a cone. Because the second factor isn't a round sphere with unit radius, this has a curvature singularity at $r=0$. The conifold is also often described as the quartic in $\mathbb{C}^4$: $\sum_{i=1}^4 (z^i)^2 = 0$.

The cone doesn't affect the Lorentz symmetry of the Minkowski space. The metric for the full 10D solution looks like

$ds^2 = H^{-1/2} dx_{\mu}dx^{\mu} + H^{1/2} ds_6^2$,

where in the usual case $ds_6^2 = ds^2(\mathbb{R}^6)$. In Klebanov-Witten (or Klebanov-Tseytlin) it is the space written above with the $\mathbb{T}^{1,1}$ factor.

By the way, the singularity of the conifold in the Klebanov-Tseytlin solution get's ``fixed'' by replacing the conifold with the deformed conifold in Klebanov-Strassler.

  • $\begingroup$ Thanks. This is interesting but I still fail to see the essence of my question. The branes are not supposed to be finite, right? I will try to interpret your answer but I am still confused. $\endgroup$ – Marion Feb 19 '15 at 19:27
  • $\begingroup$ What do you mean, finite? They extend infinitely in the $x^{\mu}$ directions, and sit at a point in the transverse ones. Ordinarily this point is just a smooth point, and this for example leads to $AdS_5 \times S^5$ as a near horizon limit, but in the case of the conifold they sit at a singular point. $\endgroup$ – Surgical Commander Feb 19 '15 at 19:44
  • $\begingroup$ So I should be thinking of this singularity as a 0 dimensional one over the cone but as a 4-dimensional one (say for D3-branes) on the $x^{\mu}$. This makes some sense now. Do you have some good reference? It would be great, thanks! $\endgroup$ – Marion Feb 19 '15 at 20:14
  • $\begingroup$ For the purpose of understanding this, I would actually just forget about the brane directions $x^{\mu}$ and think about the D3 brane as a point in the 6-dimensional transverse space. I would think about the fact that D3-branes can be added in superposition to help understand this--if you're not familiar with this fact, it can be found in the review article arxiv.org/abs/hep-th/9905111. $\endgroup$ – Surgical Commander Feb 19 '15 at 20:18
  • $\begingroup$ Indeed, the approach you propose indeed looks as a better way to look at it. I know this review not sure if I remember anything about cones though, I ll look at it. $\endgroup$ – Marion Feb 19 '15 at 21:33

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