and

γa¯c ·+ = e’K „¦a¯c ·’ .

¯b¯ ¯b¯

The ¬rst of these relations follows from the {γa , γ¯} = 2ga¯ and Ja¯ = iga¯.

b b b b

The second one is an immediate consequence of the complex conjugate of

9.9 Mirror symmetry 411

„¦abc = e’K ·’ γabc ·’ and ·+ ·’ = 1. The dependence on K re¬‚ects a choice

T T

of normalization of „¦. The arbitrary phase θ could have been absorbed

into ·+ earlier, but then it would reappear in this equation re¬‚ecting an

arbitrariness in the phase of „¦.

Now we can write the above condition as

i

e’iθ ·+ + eiθ µ±βγ ‚± X a ‚β X b ‚γ X c e’K „¦abc ·+

6

¯

’e’iθ µ±βγ ‚± X a ‚β X b ‚γ X c Ja¯γc ·+ + c.c. = 0.

¯

b¯

Because ·’ , γa ·’ , ·+ , γa ·+ are linearly independent, this is equivalent to

¯

the following two conditions:

¯

µ±βγ ‚± X a ‚β X b ‚γ X c Ja¯ = 0

¯

b

and

i

e’iθ + eiθ µ±βγ ‚± X a ‚β X b ‚γ X c e’K „¦abc = 0.

6

Because the ¬rst equation is satis¬ed for all c, we have

¯

¯

‚[± X a ‚β] X b Ja¯ = 0,

b

which is exactly Eq. (9.149). The second equation can be written as

‚± X a ‚β X b ‚γ X c „¦abc = ’ie’2iθ eK µ±βγ .

Setting e’i• = ’ie’2iθ gives Eq. (9.150). 2

9.9 Mirror symmetry

As T-duality illustrated, the geometry probed by point particles is di¬erent

from the geometry probed by strings. In string geometry a circle of radius R

can be equivalent to a circle of radius ± /R, providing a simple example of the

surprising properties of string geometry. A similar phenomenon for Calabi“

Yau three-folds, called mirror symmetry, is the subject of this section.

The mirror map associates with almost28 any Calabi“Yau three-fold M

another Calabi“Yau three-fold W such that

H p,q (M ) = H 3’p,q (W ). (9.161)

This conjecture implies, in particular, that h1,1 (M ) = h2,1 (W ) and vice

28 In the few cases where this fails, there still is a mirror, but it is not a Calabi“Yau manifold.

However, it is just as good for string theory compacti¬cation purposes. This happens, for

example, when M has h2,1 = 0, since any Calabi“Yau manifold W has h11 ≥ 1.

412 String geometry

versa. An early indication of mirror symmetry was that the space of thou-

sands of string theory vacua appears to be self-dual in the sense that if a

Calabi“Yau manifold with Hodge numbers (h1,1 , h2,1 ) exists, then another

Calabi“Yau manifold with ¬‚ipped Hodge numbers (h2,1 , h1,1 ) also exists.

The set of vacua considered were known to be only a sample, so perfect

matching was not expected. In fact, a few examples in this set had no

candidate mirror partners. This was shown in Fig. 9.1.

These observations lead to the conjecture that the type IIA superstring

theory compacti¬ed on M is exactly equivalent to the type IIB superstring

theory compacti¬ed on W . This implies, in particular, an identi¬cation of

the moduli spaces:

M1,1 (M ) = M2,1 (W ) and M1,1 (W ) = M2,1 (M ). (9.162)

This is a highly nontrivial statement about how strings see the geometry of

Calabi“Yau manifolds, since M and W are in general completely di¬erent

from the classical geometry point of view. Indeed, even the most basic

topology of the two manifolds is di¬erent, since the Euler characteristics are

related by

χ(M ) = ’χ(W ). (9.163)

Nonetheless, the mirror symmetry conjecture implies that the type IIA the-

ory compacti¬ed on M and the type IIB theory compacti¬ed on W are dual

descriptions of the same physics, as they give rise to isomorphic string the-

ories. A second, and genuinely di¬erent, possibility is given by the type IIA

theory compacti¬ed on W , which (by mirror symmetry) is equivalent to the

type IIB theory compacti¬ed on M .

Mirror symmetry is a very powerful tool for understanding string geome-

try. To see this note that the prepotential of the type IIB vector multiplets

is independent of the K¨hler moduli and the dilaton. As a result, its depen-

a

dence on ± and gs is exact. Mirror symmetry maps the complex-structure

moduli space of type IIB compacti¬ed on W to the K¨hler-structure moduli

a

space of type IIA on the mirror M . The type IIA side does receive cor-

rections in ± . As a result, a purely classical result is mapped to an (in

general) in¬nite series of quantum corrections. In other words, a classical

computation of the periods of „¦ in W is mapped to a problem of counting

holomorphic curves in M . Both sides should be exact to all orders in gs ,

since the IIA dilaton is not part of M1,1 (M ) and the IIB dilaton is not part

of M2,1 (W ).

Let us start by discussing mirror symmetry for a circle and a torus. These

simple examples illustrate the basic ideas.

9.9 Mirror symmetry 413

R

R/1

Fig. 9.8. T-duality transforms a circle of radius R into a circle of radius 1/R. This

duality is probably the origin of mirror symmetry.

The circle

The simplest example of mirror symmetry has already been discussed ex-

tensively in this book. It is T-duality. Chapter 6 showed that, when the

bosonic string is compacti¬ed on a circle of radius R, the perturbative string

spectrum is given by

2 2

K WR

2

+ 2NL + 2NR ’ 4,

±M =± + (9.164)

R ±

with

NR ’ NL = W K. (9.165)

These equations are invariant under interchange of W and K, provided

that one simultaneously sends R ’ ± /R as illustrated in Fig. 9.8. This

turns out to be exactly true for the full interacting string theory, at least

perturbatively.

The torus

One can also illustrate mirror symmetry for the two-torus T 2 = S 1 — S 1 ,

where the ¬rst circle has radius R1 and the second circle has radius R2 .

This torus may be regarded as an S 1 ¬bration over S 1 . It is characterized

414 String geometry

by complex-structure and K¨hler-structure parameters