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March 6, 2005

Small Blackholes

I recently wrote about Greg Moore and collaborators’ work computing the entropy of a certain class of N=2N=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=2N=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=14AS=\frac{1}{4} A, you find S=12AS=\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 pertF_{\text{pert}} in the previous post,

(1)F(X A,T 2)=16D abcX aX bX cX 0+12A abX aX b+124c 2aX aX 0T 264 F(X^A,T^2) = \frac{1}{6} D_{a b c} \frac{X^a X^b X^c}{X^0} +\frac{1}{2} A_{a b}X^a X^b +\frac{1}{24} c_{2 a} \frac{X^a}{X^0} \frac{T^2}{64}

The (generalized) attractor equations are

(2)C 2T 2=256,Re(CX A)=p A,Re(CF A)=q A C^2 T^2 = 256,\quad Re(C X^A) = p^A,\quad Re(C F_A)=q_A

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

(3)CX 0 =iD+c 2ap a6q^ 0 CX a =p a+iD+c 2dp d6q^ 0D ab(q bA bcp c) \array{ \arrayopts{\colalign{right left}} C X^0 &= i \sqrt{\frac{D+c_{2 a}p^a}{6\hat{q}_0}} \\ C X^a &= p^a + i \sqrt{\frac{D+c_{2 d}p^d}{6\hat{q}_0}} D^{a b} (q_b - A_{b c} p^c) }


(4)q^ 0 =q 0+12D ab(q aA acp c)(q bA bdp d) D ab =D abcp c,D abD bc=δ c a,D=D abcp ap bp c \array{ \arrayopts{\colalign{right left}} \hat{q}_0 &= q_0 +\frac{1}{2} D^{a b} (q_a - A_{a c} p^c) (q_b - A_{b d} p^d) \\ D_{a b} &= D_{a b c} p^c,\quad D^{a b}D_{b c}= \delta^a_c,\quad D=D_{a b c} p^a p^b p^c }

Plugging into the formula for the horizon area,

(5)A =πi|C| 2(X AF¯ AX¯ AF A) =2πq^ 06(D+c 2ap a)(D+12c 2ap a) \array{ \arrayopts{\colalign{right left}} A &= \pi i |C|^2 (X^A \overline{F}_A - \overline{X}^A F_A) \\ &=2\pi\sqrt{\frac{\hat{q}_0}{6(D+c_{2 a}p^a)}} (D+\frac{1}{2} c_{2 a}p^a) }

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

(6)S =14A+128πi(F¯T¯ 2FT 2) =π2q^ 0(D+c 2ap a)/6 \array{ \arrayopts{\colalign{right left}} S &= \frac{1}{4} A + 128\pi i \left(\frac{\partial \overline{F}}{\partial \overline{T}^2} - \frac{\partial F}{\partial T^2} \right) \\ &= \frac{\pi}{2} \sqrt{\hat{q}_0 (D +c_{2 a}p^a)/6} }

Now, in “conventional” four-(or more)-charged blackholes, such that D=D abcp ap bp cD=D_{a b c}p^a p^b p^c and q^ 0\hat{q}_0 are both large, this just gives

(7)A=2πq^ 0D/6 A = 2\pi \sqrt{\hat{q}_0 D/6}


(8)S=14A(1+12c 2ap a/D) S = \frac{1}{4} A\, \left(1 +\frac{1}{2} c_{2 a} p^a/D\right)

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

(9)A=πq^ 0c 2ap a/6 A = \pi\sqrt{\hat{q}_0 c_{2 a} p^a/6}

and S=12AS=\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.

Posted by distler at March 6, 2005 12:43 AM

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