This section explains how to calculate nonperturbative e¬ects due to Eu-

clideanized branes wrapping supersymmetric cycles. The world volume of a

Euclideanized p-brane has p + 1 spatial dimensions, and it only exists for an

instant of time. Note that only the world-volume time, and not the time in

the physical Minkowski space, is Euclideanized. If a Euclideanized p-brane

can wrap a (p + 1)-cycle in such a way that some supersymmetry is pre-

served, then the corresponding cycle is called supersymmetric. This gives a

9.8 Nonperturbative e¬ects in Calabi“Yau compacti¬cations 405

contribution to the path integral that represents a nonperturbative instan-

ton correction to the theory. More precisely, fundamental-string instantons

give contributions that are nonperturbative in ± , whereas D-branes and

NS5-branes give contributions that are also nonperturbative in gs .23 If the

internal manifold is a Calabi“Yau three-fold, the values of p for which there

are nontrivial (p + 1)-cycles are p = ’1, 1, 2, 3, 5.24

As was discussed in Chapter 6, type IIA superstring theory contains

even-dimensional BPS D-branes, whereas the type IIB theory contains odd-

dimensional BPS D-branes. Each of these D-branes carries a conserved

R“R charge. So, in addition to fundamental strings wrapping a two-cycle

and NS5-branes wrapping the entire manifold, one can consider wrapping

D2-branes on a three-cycle in the IIA case. Similarly, one can wrap D1,

D3 and D5-branes, as well as D-instantons, in the IIB case. These con¬g-

urations give nonperturbative instanton contributions to the moduli-space

geometry, that need to be included in order for string theory to be consis-

tent. As explained in Section 9.9, these e¬ects are crucial for understanding

fundamental properties of string theory, such as mirror symmetry. There

are di¬erent types of supersymmetric cycles in the context of Calabi“Yau

compacti¬cations, which we now discuss.25

Special Lagrangian submanifolds

For Calabi“Yau compacti¬cation of M-theory, which gives a ¬ve-dimensional

low-energy theory, the possible instanton con¬gurations arise from M2-

branes wrapping three-cycles and M5-branes wrapping the entire Calabi“

Yau manifold. Let us ¬rst consider a Euclidean M2-brane, which has a

three-dimensional world volume. The goal is to examine the conditions un-

der which a Euclidean membrane wrapping a three-cycle of the Calabi“Yau

manifold corresponds to a stationary point of the path-integral-preserving

supersymmetry. Once this is achieved, the next step is to determine the cor-

responding nonperturbative contribution to the low-energy ¬ve-dimensional

e¬ective action.

The M2-brane in 11 dimensions has a world-volume action, with global

supersymmetry and local κ symmetry, whose form is similar to the actions

for fundamental superstrings and D-branes described in Chapters 5 and 6.

As in the other examples, in ¬‚at space-time this action is invariant under

23 The gs dependence is contained in the tension factor that multiplies the world-volume actions.

24 A p-brane with p = ’1 is the D-instanton of the type IIB theory.

25 A vanishing potential for the tensor ¬elds is assumed here. The generalization to a nonvanishing

potential is known.

406 String geometry

global supersymmetry

δµ X M = i¯“M ˜,

δµ ˜ = µ and µ (9.144)

where X M (σ ± ), with M = 0, . . . , 10, describes the membrane con¬guration

in space-time. ˜ is a 32-component Majorana spinor, and µ is a constant

in¬nitesimal Majorana spinor. However, the question arises how much of this

supersymmetry survives if a Euclideanized M2-brane wraps a three-cycle of

the compacti¬cation. The M2-brane is also invariant under fermionic local

κ symmetry, which acts on the ¬elds according to

¯

δκ X M = 2i˜“M P+ κ(σ),

δκ ˜ = 2P+ κ(σ) and (9.145)

where κ is an in¬nitesimal 32-component Majorana spinor, and P± are or-

thogonal projection operators de¬ned by

1 i

1 ± µ±βγ ‚± X M ‚β X N ‚γ X P “M N P

P± = . (9.146)

2 6

The key to the analysis is the observation that a speci¬c con¬guration

X M (σ ± ) (and ˜ = 0) preserves the supersymmetry corresponding to a par-

ticular µ transformation, if this transformation can be compensated by a κ

transformation. In other words, there should exist a κ(σ) such that

δµ ˜ + δκ ˜ = µ + 2P+ κ(σ) = 0. (9.147)

By acting with P’ this implies

P’ µ = 0. (9.148)

This equation is equivalent to the following two conditions:26

• The 11 coordinates X M consist of X a and X a , which refer to Calabi“Yau

¯

coordinates, and X µ , which is the coordinate in ¬ve-dimensional space-

µ

time. In the supersymmetric instanton solution, X µ = X0 is a constant,

and the nontrivial embedding involves the other coordinates. The ¬rst

condition is

¯

‚[± X a ‚β] X b Ja¯ = 0. (9.149)

b

The meaning of this equation is that the pullback of the K¨hler form of

a

the Calabi“Yau three-fold to the membrane world volume vanishes.

• The second condition is27

‚± X a ‚β X b ‚γ X c „¦abc = e’i• eK µ±βγ . (9.150)

26 The equivalence of Eq. (9.148) and the conditions (9.149) and (9.150) is proved in Exercise 9.15.

√

27 µ±βγ is understood to be a tensor here. Otherwise a factor of G, where G±β is the induced

metric, would appear.

9.8 Nonperturbative e¬ects in Calabi“Yau compacti¬cations 407

The meaning of this equation is that the pullback of the holomorphic

(3, 0)-form „¦ of the Calabi“Yau manifold to the membrane world volume

is proportional to the membrane volume element. The complex-conjugate

equation implies the same thing for the (0, 3)-antiholomorphic form „¦.

The phase • is a constant that simply re¬‚ects an arbitrariness in the

de¬nition of „¦. The factor eK , where K is given by

1 1,1

(K ’ K2,1 ),

K= (9.151)

2

is a convenient normalization factor. The term K 2,1 is a function of the

complex moduli belonging to h2,1 hypermultiplets. K1,1 is a function of

the real moduli belonging to h1,1 vector supermultiplets.

The supersymmetric three-cycle conditions (9.149) and (9.150) de¬ne a

special Lagrangian submanifold. When these conditions are satis¬ed, there

exists a nonzero covariantly constant spinor of the form µ = P+ ·. Thus,

the conclusion is that a Euclidean M2-brane wrapping a special Lagrangian

submanifold of the Calabi“Yau three-fold gives a supersymmetric instanton

contribution to the ¬ve-dimensional low-energy e¬ective theory.

The conditions (9.149) and (9.150) imply that the membrane has mini-

mized its volume. In order to derive a bound for the volume of the membrane

consider

†

µ† P’ P’ µ d3 σ ≥ 0, (9.152)

Σ

where Σ is the membrane world volume. Since

†

P’ P’ = P ’ P’ = P ’ , (9.153)

the inequality becomes

2V ≥ e’K ei• „¦ + e’i• „¦, (9.154)

Σ Σ

where • is a phase which can be adjusted so that we obtain

V ≥ e’K „¦. (9.155)

Σ

The bound is saturated whenever the membrane wraps a supersymmetric

cycle C, in which case

V = e’K „¦. (9.156)

C

Type IIA or type IIB superstring theory, compacti¬ed on a Calabi“Yau

three-fold, also has supersymmetric cycles, which can be determined in a

408 String geometry

similar fashion. As in the case of M-theory, the type IIA theory receives

instanton contributions associated with a D2-brane wrapping a special La-

grangian manifold. These contributions have a coupling constant depen-