nient to describe a torus using two complex parameters „ and ρ, which in

the present case are related to the two radii by

R2

„ =i and ρ = iR1 R2 . (9.87)

R1

The shape, or complex structure, of the torus is described by „ , while the

size is described by ρ. As a result, two transformations can be performed so

that the torus remains a torus. A complex-structure deformation changes

„ , while a K¨hler-structure deformation changes ρ. These deformations are

a

illustrated in Fig. 9.5.

Fig. 9.5. K¨hler structure deformations and complex structure deformations corre-

a

spond to changing the size and shape of a torus respectively.

Recall that the holomorphic one-form on a torus is given by

„¦ = dz. (9.88)

The complex-structure parameter „ can then be written as the quotient of

the two periods

„¦

„= A , (9.89)

„¦

B

where A and B are the cycles shown in Fig. 9.4. This de¬nition is generalized

to Calabi“Yau three-folds in the next section. The rectangular torus is not

the most general torus. There can be an angle θ as shown in Fig. 9.6. When

„ has a real part, mirror symmetry17 only makes sense if ρ has a real part

as well. The imaginary part of ρ then describes the volume, while the real

part descends from the B ¬eld, as explained in Exercise 7.8.

Deformations of Calabi“Yau three-folds

In order to analyze the metric deformations of Calabi“Yau three-folds, let us

use the strategy outlined in the introduction of this section and require that

17 The mirror symmetry transformation that interchanges „ and ρ is discussed in Section 9.9.

9.5 Deformations of Calabi“Yau manifolds 389

gmn and gmn + δgmn both satisfy the Calabi“Yau conditions. In particular,

they describe Ricci-¬‚at backgrounds so that

Rmn (g) = 0 and Rmn (g + δg) = 0. (9.90)

Some metric deformations only describe coordinate changes and are not of

interest. To eliminate them one ¬xes the gauge

1

m m

δgmn = n δgm , (9.91)

2

where δgm = g mp δgmp . Expanding the second equation in (9.90) to linear

m

order in δg and using the Ricci-¬‚atness of g leads to

k

+ 2Rmp nq δgpq = 0.

k δgmn (9.92)

This equation is known as the Lichnerowicz equation, which you are asked

to verify in Problem 9.7. Using the properties of the index structure of the

metric and Riemann tensor of K¨hler manifolds, one ¬nds that the equations

a

for the mixed components δga¯ and the pure components δgab decouple.

b

Consider the in¬nitesimal (1, 1)-form

¯

δga¯dz a § d¯b ,

z (9.93)

b

which is harmonic if (9.92) is satis¬ed, as you are asked to verify in Prob-

lem 9.8. We imagine that after the variation g +δg is a K¨hler metric, which

a

in classical geometry should be positive de¬nite. The K¨hler metric de¬nes

a

¯

the K¨hler form J = iga¯dz a § d¯b , and positivity of the metric is equivalent

a z

b

to

J § · · · § J > 0, r = 1, 2, 3, (9.94)

Mr

r’times

R2

R1

θ

Fig. 9.6. The shape of a torus is described by a complex-structure parameter „ .

The angle θ is the phase of „ .

390 String geometry

where Mr is any complex r-dimensional submanifold of the Calabi“Yau

three-fold. The subset of metric deformations that lead to a K¨hler form

a

satisfying Eq. (9.94) is called the K¨hler cone. This space is a cone since

a

if J satis¬es (9.94), so does rJ for any positive number r, as illustrated in

Fig. 9.7.

Fig. 9.7. The deformations of the K¨hler form that satisfy Eq. (9.94) build the

a

K¨hler cone.

a

The ¬ve ten-dimensional superstring theories each contain an NS“NS two-

form B. After compacti¬cation on a Calabi“Yau three-fold, the internal

(1, 1)-form Ba¯ has h1,1 zero modes, so that this many additional massless

b

scalar ¬elds arise in four dimensions. The real closed two-form B combines

with the K¨hler form J to give the complexi¬ed K¨hler form

a a

J = B + iJ. (9.95)

The variations of this form give rise to h1,1 massless complex scalar ¬elds

in four dimensions. Thus, while the K¨hler form is real from a geometric

a

viewpoint, it is e¬ectively complex in the string theory setting, generalizing

the complexi¬cation of the ρ parameter of the torus. This procedure is called

the complexi¬cation of the K¨hler cone. For M-theory compacti¬cations,

a

discussed later, there is no two-form B, and so the K¨hler form, as well as

a

the corresponding moduli space, is not complexi¬ed.

The purely holomorphic and antiholomorphic metric components gab and

ga¯ are zero. However, one can consider varying to nonzero values, thereby

¯b

changing the complex structure. With each such variation one can associate

the complex (2, 1)-form

¯

„¦abc g cd δgd¯dz a § dz b § d¯e .

z¯ (9.96)

¯e

This is harmonic if (9.90) is satis¬ed. The precise relation to complex-

structure variations is explained in Section 9.6.

9.6 Special geometry 391