Black Branes from String Theory

LECTURE 4

The Kaluza-Klein picture we have presented above is classical and, just as we need quantum mechanics to understand the electron (which is a point particle, i.e. a singularity), we need quantum mechanics to understand a black hole. Unfortunately (because it is very complicated) the only quantum theory of gravity we possess is string theory. To avoid inﬁnities in the construction of quantum gravity, string theory complicates things:

• Strings live in ten dimensions

• Strings vibrate. The greater the frequency, the higher the mass. Also an inﬁnite number of vibrational modes should correspond to an inﬁnite number of particles. The massless modes must be the particles we see. The electron’s mass is 0.5 MeV , the proton’s is 1 GeV , and the W particle has a mass of 80 GeV , but the vibrational modes have mass/energy on the order of the Planck mass/energy 1019 GeV . Presumably these particles appeared only at the beginning of the universe.

• We need supersymmetry: every particle must have a super partner.

Example 1. Type IIA. Low energy (massless strings). In ten dimensions specify a metric gµν , a scalar ϕ, an anti-symmetric tensor Bµν , a vector potential Aµ, and an anti-symmetric tensor Aµνρ. As usual Fµν = ∂µAν − ∂ν Aµ; we can do a similar thing for Bµν and Aµνρ to get anti-symmetric tensors

Hµνρ = ∂µBνρ + ∂ρBµν + ∂ν Bρµ

Fµνρσ = ∂µAνρσ. 20 LECTURE 4: Black Branes from String Theory

Also we can anti-symmetrize Fµνρσ + HµνρAσ to get a Gµνρσ. The action is Z µ ¶ 1 √ R − 1 (∂ ϕ)2 − 1 eϕ/2H Hµνρ− 10 2 ¡ µ 12 µνρ ¢ S10 = d x −g 1 5ϕ/2 µν 1 µνρσ 16πG10 e Fµν F − GµνρσG Z 4 12 1 µ1...µ10 − ² Bµ1µ2 Fµ3...µ6 Fµ7...µ10 32πG10 where ² is the Levi-Civita tensor (zero if any indices are repeated, ±1 if the permutation of the indices is even (odd)). In eleven dimensions the situation is simpler. We only require an anti-symmetric tensor AABC with anti-symmetric FABCD as above. The action is Z p µ ¶ 1 11 (11) (11) 1 ABCD S11 = d x −g R + FABCDF . 16πG11 18 (11) If the tenth dimension is tightly wrapped, we can partition the matrix gAB into a 9 by (10) 9 matrix gµν , a 9 by 1 column Aµ (and the same 1 by 9 row) and a scalar ϕ:

µ (10) ¶ (11) gµν Aµ gAB = . Aµ ϕ

If we restrict the indices of AABC to be from 0 to 9, we get a 10-dimensional tensor Aµνρ and setting Bµν = Aµν(10), we obtain all the ﬁelds for the 10-dimensional case and S(11) gives S(10). Example 2. Type IIB. p-brane solutions. We are given an anti-symmetric tensor

Aµ1...µp+1 and a metric µ ¶ ¡ ¢ 1 ds2 = −H(p−7)/8 −f(r)dt2 + dy2 + H(p+1)/8 dr2 + r2dΩ2 f(r) 8−p where ρ7−p r7−p H = 1 + , e2ϕ = H(3−p)/2, f(r) = 1 − + , and y ∈ Rp. r7−p r7−p This would be a black hole except for the presence of dy. It appears as if a black hole is stretched through other dimensions and is called a black brane around hyi. In the extremal limit r+ → 0, ¡ ¢ ds2 = −H(p−7)/8 −dt2 + dy2 + H(p+1)/8dx2 where x ∈ R9−p. As r → 0, H has a singularity and ³ρ´−(7−p)2/8 ¡ ¢ ³ρ´(7−p)(p+1)/8 ds2 ≈ −dt2 + dy2 + dx2. r r When p = 3, we get something special, the dilaton ϕ is constant and r2 ¡ ¢ ρ2 ds2 ≈ −dt2 + dy2 + dx2 ρ2 r2 r2 ¡ ¢ ρ2 = −dt2 + dy2 + dr2 + ρ2dΩ2. ρ2 r2 5 Black Branes from String Theory 21

The last term is just the ﬁve sphere S5 and r would not fall out if p 6= 3. The other terms are the anti-deSitter space AdS5 (3 spatial and 1 time dimension). The situation is similar to two dimensional conformal theory on the boundary giving quantum mechanics. 22 LECTURE 4: Black Branes from String Theory