String Theory and M-Theory

430 String geometry
The tension of the 7-brane is given by
T7 = 1
d
2
2 x ∂τ ·
∂¯τ 1
=
(Im τ) 2 2
d 2 x ∂τ ¯ ∂¯τ + ¯ ∂τ∂¯τ
(Im τ) 2 . (9.195)
Now let us evaluate this for the solution proposed in Eq. (9.193). Since τ is
holomorphic
T7 = 1
d
2
2 x ∂τ ¯
∂¯τ 1 d
=
(Im τ) 2 2 F
2τ π
= . (9.196)
(Im τ) 2 6
This has used the fact that the inverse image of the complex plane is the
fundamental region F. The volume of the moduli space was evaluated in
Exercise 3.9.
The integrand in Eq. (9.196) is the energy density that acts as a source
for the gravitational field in the Einstein equation
R00 − 1
2 g00R = − 1
2 g00e −A ∂τ ¯ ∂¯τ
. (9.197)
(Im τ) 2
Evaluating the curvature for the metric in Eq. (9.190), one obtains the equation
∂ ¯ ∂A = − 1 ∂τ
2
¯ ∂¯τ
(τ − ¯τ) 2 = ∂ ¯ ∂ log Im τ. (9.198)
The energy density is concentrated within a string-scale distance of the
origin, where the supergravity equations aren’t reliable. The total energy is
reliable because of supersymmetry (saturation of the BPS bound), however.
So, to good approximation, we can take A = α log r and use ∇ 2 log r =
2πδ 2 (x) to approximate the energy density by a delta function at the core.
Doing this, one then matches the integrals of the two sides to determine
α = −1/6. This gives a result that is correct for large r, namely
A ∼ − 1
log r. (9.199)
6
By the change of variables ρ = r11/12 this brings the two-dimensional metric
to the asymptotic form
ds 2 ∼ dρ 2 + ρ 2
11
12 dθ
2 , (9.200)
which shows that there is a deficit angle of π/6 in the Einstein frame.
A more accurate solution, applicable for multiple 7-branes at positions
zi, i = 1, . . . , N, can be constructed as follows. The general solution of
Eq. (9.198) is
e A = |f(z)| 2 Im τ (9.201)