### Small Blackholes

I recently wrote about Greg Moore and collaborators’ work computing the entropy of a certain class of $N=2$ blackholes which arise in the compactification of Type IIA on a Calabi-Yau.

Alex Maloney visited us this past week, and gave us a beautiful talk on his work on the geometry of the blackholes in question. The situation is really quite striking. Classically, these blackholes have vanishing horizon area. They have a null singularity, a pathology which, says Alex, is characterized as “naked with the lights off” in the GR literature.

*Umh, whatever* …

One-loop corrections in the string theory induce certain curvature-squared corrections, which can be written as integrals over half of $N=2$ superspace (a holomorphic function of the Kähler moduli times square of the self-dual part of the Weyl curvature, plus Hermitian conjugate).

These corrections have a dramatic effect. They make the singularity time-like and cloak it behind a horizon. In the corrected geometry, one recovers a formula in which the blackhole entropy is proportional to the area. But, instead of the Bekenstein-Hawking formula, $S=\frac{1}{4} A$, you find $S=\frac{1}{2} A$!

Now, at first you might be suspicious: if the one-loop corrections radically change the answer, how about two-loops …? But that’s the wrong way to think about it. For these blackholes, the tree-level answers (for the area and entropy) happen to *vanish*. The one-loop answer is the *leading* nonvanishing contribution. Higher corrections are still controllably small compared to the leading term.

In a little more detail, consider the (tree plus) one-loop Topological String free energy, which I called $F_{\text{pert}}$ in the previous post,

The (generalized) attractor equations are

Let’s solve them in this one-loop approximation, assuming the electric graviphoton charge, $p^0=0$. The computation is fairly easy;

where

Plugging into the formula for the horizon area,

and the (Wald-corrected) formula for the entropy,

Now, in “conventional” four-(or more)-charged blackholes, such that $D=D_{a b c}p^a p^b p^c$ and $\hat{q}_0$ are both large, this just gives

and

the Bekenstein-Hawking result plus a small one-loop correction. However, for these two-charged blackholes, $D=0$, and we get

and $S=\frac{1}{2} A$.

It’s really quite nice that we are seeing *essentially stringy* effects rescuing cosmic censorship, while generating substantial corrections to the Bekenstein-Hawking formula.