moduli and complex-structure moduli are interchanged. While this is anal-

ogous to what we found for Calabi“Yau three-fold compacti¬cation, there

are also some signi¬cant di¬erences. For one thing, the two sets of mod-

uli are incorporated in a single moduli space rather than a product of two

separate spaces. Also, type IIA is related to type IIA, whereas in the Calabi“

Yau three-fold case type IIA was related to type IIB. In that case, we used

the SYZ argument to show that, when the Calabi“Yau has a T 3 ¬bration,

this could be understood in terms of T-duality along the ¬bers. The cor-

responding statement now is that, when K3 has a T 2 ¬bration, the mirror

description can be deduced by a T-duality along the ¬bers. The reason type

IIA is related to type IIA is that this is an even number (two) of T-duality

transformations.

Let us now investigate the relationship between the two dilatons, or equiv-

alently the two string coupling constants, by matching branes. The analysis

is very similar to that considered for the previous duality. For the purpose

of this argument, let us denote the string coupling and string scale of the

type IIA theory by gA and A and those of the heterotic theory by gH and

H . Equating tensions of the type IIA NS5-brane wrapped on K3 and the

heterotic string as well as the heterotic NS5-brane wrapped on T 4 and the

426 String geometry

type IIA string gives the relations

1 VK3 VT 4 1

∼ ∼ 2.

and (9.183)

2 gA 6

2 gH 6

2

H A H A

Let us now de¬ne six-dimensional string coupling constants by

4 ’1 4 ’1

2 2 2 2

g6H = gH VT 4 / and g6A = gA VK3 / . (9.184)

H A

Then these relations can be combined to give

’2

2

g6H ∼ g6A . (9.185)

This means that the relation between the two six-dimensional theories is

an S-duality that relates weak coupling and strong coupling, just like the

duality relating the two SO(32) superstring theories in ten dimensions.

Type IIB superstring theory on K3

Compacti¬cation of type IIB superstring theory on K3 gives a chiral the-

ory with 16 unbroken supersymmetries in six dimensions. The two six-

dimensional supercharges have the same chirality. The massless sector in

six dimensions consists of a chiral N = 2 supergravity multiplet coupled

to 21 tensor multiplets. This is the unique number of tensor multiplets for

which anomaly cancellation is achieved. The chiral N = 2 supergravity has

a U Sp(4) ≈ SO(5) R symmetry, and there is an SO(21) symmetry that

rotates the tensor multiplets. In fact, in the supergravity approximation,

these combine into a noncompact SO(21, 5) symmetry. However, as always

happens in string theory, this gets broken by string and quantum corrections

to the discrete duality subgroup SO(21, 5; ).

The gravity multiplet contains ¬ve self-dual three-form ¬eld strengths,

while each of the tensor multiplets contains one anti-self-dual three-form

¬eld strength and ¬ve scalars. This is the same multiplet that appears on

the world volume of an M5-brane, discussed a moment ago. It is the only

massless matter multiplet that exists for chiral N = 2 supersymmetry in six

dimensions. Most of the three-form ¬eld strengths come from the self-dual

¬ve-form in ten dimensions as a consequence of the fact that K3 has three

self-dual two-forms (b+ = 3) and 19 anti-self-dual two-forms (b’ = 19).

2 2

The additional two self-dual and anti-self-dual three-forms are provided by

F3 = dC2 and H3 = dB2 . The 5 — 21 = 105 scalar ¬elds arise as follows: 58

from the metric, 1 from the dilaton ¦, 1 from C0 , 22 from B2 , 22 from C2 ,

and 1 from C4 .

The symmetries and the moduli counting described above suggest that

9.11 K3 compacti¬cations and more string dualities 427

the moduli space for K3 compacti¬cation of the type IIB theory should be

M21,5 . The natural question is whether this has a dual heterotic string

interpretation. The closest heterotic counterpart is given by toroidal com-

pacti¬cation to ¬ve dimensions, for which the moduli space is

+

— M21,5 . (9.186)

¡

The extra modulus, corresponding to the + factor, is provided by the

¡

heterotic dilaton. Therefore, it is tempting to identify the heterotic string

theory compacti¬ed to ¬ve dimensions on T 5 with the type IIB superstring

compacti¬ed to ¬ve dimensions on K3 — S 1 . In this duality the heterotic-

string coupling constant corresponds to the radius of the type IIB circle.

Thus, the strong coupling limit of the toroidally compacti¬ed heterotic string

theory in ¬ve dimensions gives the K3 compacti¬ed type IIB string in six

dimensions. The relationship is analogous to that between the type IIA

theory in ten dimensions and M-theory in 11 dimensions.

This picture can be tested by matching branes, as in the previous exam-

ples. However, the analysis is more complicated this time. The essential

fact is that in ¬ve dimensions both constructions give 26 U (1) gauge ¬elds,

with ¬ve of them belonging to the supergravity multiplet and 21 belong-

ing to vector multiplets. Thus, point particles can carry 26 distinct electric

charges. Their magnetic duals, which are strings, can also carry 26 distinct

string charges. By matching the BPS formulas for their tensions one can

deduce how to map parameters between the two dual descriptions and verify

that, when the heterotic string coupling becomes large, the type IIB circle

decompacti¬es.

Compacti¬cation of F-theory on K3

Type IIB superstring theory admits a class of nonperturbative compacti¬ca-

tions, ¬rst described by Vafa, that go by the name of F-theory. The dilaton

is not constant in these compacti¬cations, and there are regions in which

it is large. Therefore, since the value of the dilaton ¬eld determines the

string coupling constant, these solutions cannot be studied using perturba-

tion theory (except in special limits that correspond to orientifolds). This

is the sense in which F-theory solutions are nonperturbative.

The crucial fact that F-theory exploits is the nonperturbative SL(2, )

symmetry of type IIB superstring theory in ten-dimensional Minkowski

space-time. Recall that the R“R zero-form potential C0 and the dilaton

¦ can be combined into a complex ¬eld

„ = C0 + ie’¦ , (9.187)

428 String geometry

which transforms nonlinearly under SL(2, ) transformations in the same

way as the modular parameter of a torus:

a„ + b

„’ . (9.188)

c„ + d

The two two-forms B2 and C2 transform as a doublet at the same time,

while C4 and the Einstein-frame metric are invariant.