to 1/gs .

Black-hole mass formula

When the type IIB theory is compacti¬ed on a Calabi“Yau three-fold, four-

dimensional supersymmetric black holes can be realized by wrapping D3-

branes on special Lagrangian three-cycles. In the present case the bound

for the mass of the black holes takes the form

2,1 /2 2,1 /2

M ≥ eK „¦ = eK „¦§“ , (9.157)

C M

where “ is the three-form that is Poincar´ dual to the cycle C. Here we are

e

assuming that the mass distribution on the D3-brane is uniform. Letting

“ = q I ±I ’ pI β I , (9.158)

we can introduce special coordinates and use the expansion (9.110) to obtain

the BPS bound

2,1 /2

M ≥ eK | p I X I ’ q I FI | . (9.159)

For BPS states the inequality is saturated, and the mass is equal to the

absolute value of the central charge Z in the supersymmetry algebra. Thus

Eq. (9.157) is also a formula for |Z|. As a result, BPS states become massless

when a cycle shrinks to zero size. The above expression relating the central

charge to the special coordinates plays a crucial role in the discussion of the

attractor mechanism for black holes which will be presented in chapter 11.

Holomorphic cycles

In the case of type II theories other supersymmetric cycles also can con-

tribute. For example, some supersymmetry can be preserved if a Euclidean

type IIA string world sheet wraps a holomorphic cycle. This means that the

embedding satis¬es

¯

‚X a = 0 ‚X a = 0,

¯

and (9.160)

µ

in addition to X µ = X0 . Thus, the complex structure of the Euclideanized

string world sheet is aligned with that of the Calabi“Yau manifold. In this

case, one says that it is holomorphically embedded. Recall that the type IIA

theory corresponds to M-theory compacti¬ed on a circle. Therefore, from the

M-theory viewpoint this example corresponds to a solution on M4 — S 1 — M

9.8 Nonperturbative e¬ects in Calabi“Yau compacti¬cations 409

in which a Euclidean M2-brane wraps the circle and a holomorphic two-cycle

of the Calabi“Yau.

EXERCISES

EXERCISE 9.14

Show that the submanifold X = X is a supersymmetric three-cycle inside

the Calabi“Yau three-fold given by a quintic hypersurface in P 4 .

£

SOLUTION

To prove the above statement, we should ¬rst check that the pullback of

the K¨hler form is zero. This is trivial in this case, because X ’ X under

a

the transformation J ’ ’J. On the other hand, the pullback of J onto the

¬xed surface X = X should give J ’ J, so the pullback of J is zero.

Now let us consider the second condition, and compute the pullback of

the holomorphic three-form. The equation for a quintic hypersurface in P 4

£

discussed in Section 9.3 is

5

(X m )5 = 0.

m=1

De¬ning inhomogeneous coordinates Y k = X k /X 5 , with k = 1, 2, 3, 4, on

the open set X 5 = 0, the holomorphic three-form can be written as

dY 1 § dY 2 § dY 3

„¦= .

(Y 4 )4

The norm of „¦ is

1 1

abc

2

„¦ = „¦abc „¦ = ,

g |Y 4 |8

6 ˆ

where g = det ga¯. Using Eqs (9.104) and (9.129), as well as Exercise 9.8,

ˆ b

one has

1

2,1 1,1

e’K = i „¦ § „¦ = V „¦ 2 = e’K „¦ 2

8

which implies that

2

= 8e2K ,

„¦

410 String geometry

where K = 1 (K1,1 ’ K2,1 ). It follows that

2

e’2K

g=

ˆ .

8|Y 4 |8

The pullback of the metric gives

¯

h±β = 2‚± Y a ga¯‚β Y b

b

so

√ e’K

8ˆ | det(‚Y )| = | det(‚Y )| 4 4 .

h= g

|Y |

Now we can calculate the pullback of the holomorphic (3, 0)-form

√

µabc ‚± Y a ‚β Y b ‚γ Y c ’iφ K

a b c

‚± Y ‚β Y ‚γ Y „¦abc = = e e h µ±βγ ,

(Y 4 )4

which is what we wanted to show. 2

EXERCISE 9.15

Derive the equivalence between Eq. (9.148) and Eqs (9.149) and (9.150).

For M-theory on M5 —M , where M is a Calabi“Yau three-fold, the M-theory

spinor µ has the decomposition

µ = » — · + + »— — ·’ ,

where » is a spinor on M5 , and ·± are Weyl spinors on the Calabi“Yau

manifold. So the condition (9.148) takes the form

i

e’iθ ·+ + c.c. = 0,

1 ’ µ±βγ ‚± X m ‚β X n ‚γ X p γmnp

6

where m, n, p label real coordinates of the internal Calabi“Yau manifold.

Let us focus on the ·+ terms and take account of the complex-conjugate

terms at the end of the calculation.

¯¯

The formula can be simpli¬ed by using complex coordinates X a and X a ,

as in the text, and the conditions γa ·+ = 0. This implies that γabc ·+ = 0

and γab¯·+ = 0. The nonzero terms are

c

γa¯c ·+ = ’2iJa[¯γc] ·+

b¯

b¯