more two-cycles of the K3 manifold degenerate (that is, collapse to a point)

at these loci. In fact, the 2 orbifold described in Section 9.3 is such a limit

in which 16 nonintersecting two-cycles degenerate.

The proof that this is the right moduli space is based on the observation

that the coset space characterizes the alignment of the 19 anti-self-dual and

three self-dual two-forms in the space of two forms. Rather than trying to

explain this carefully, let us con¬rm this structure by physical arguments.

Dual description of M-theory on M7 — K3

The seven-dimensional theory obtained in this way has exactly the same

massless spectrum, the same amount of supersymmetry, and the same mod-

uli space as is obtained by compactifying (either) heterotic string theory on

a three-torus. Recall that in Chapter 7 it was shown that the moduli space

of the heterotic string compacti¬ed on T n is M16+n,n — + , where ¡

M16+n,n = M0

16+n,n /O(16 + n, n; ) (9.175)

and

O(16 + n, n; )

M0

16+n,n = . (9.176)

¡

O(16 + n, ) — O(n, )

¡ ¡

Therefore, it is natural to conjecture, following Witten, that heterotic string

theory on a three-torus is dual to M-theory on K3.

In the heterotic description, the + modulus is associated with the string

¡

coupling constant, which is the vacuum expectation value of exp(¦), where

¦ is the dilaton. Since this corresponds to the K3 volume in the M-theory

description, one reaches the following interesting conclusion: the heterotic-

string coupling constant corresponds to the K3 volume, and thus the strong-

coupling limit of heterotic string theory compacti¬ed on a three-torus cor-

responds to the limit in which the volume of the K3 becomes in¬nite. Thus,

this limit gives 11-dimensional M-theory! This is the same strong-coupling

limit as was obtained in Chapter 8 for ten-dimensional type IIA superstring

theory at strong coupling. The di¬erence is that in one case the size of a

K3 manifold becomes in¬nite and in the other the size of a circle becomes

in¬nite.

An important ¬eld in the heterotic theory is the two-form B, whose

¬eld strength H includes Chern“Simon terms so that dH is proportional

to trR2 ’trF 2 . In the seven-dimensional K3 reduction of M-theory consid-

ered here, the B ¬eld arises as a dual description of A3 . The ¬eld A3 also

9.11 K3 compacti¬cations and more string dualities 421

gives rise to 22 U (1) gauge ¬elds in seven dimensions, as required by the

duality. The Chern“Simons 11-form gives seven-dimensional couplings of

the B ¬eld to these gauge ¬elds of the form required to account for the trF 2

term in the dH equation. To account for the trR2 terms it is necessary to

add higher-dimension terms to the M-theory action of the form A3 § X8 ,

where X8 is quartic in curvature two-forms. Such terms, with exactly the

required structure, have been derived by several di¬erent arguments. These

include anomaly cancellation at boundaries as well as various dualities to

string theories.

Matching BPS branes

As a further test of the proposed duality, one can compare BPS branes in

seven dimensions. One interesting example is obtained by wrapping the

M5-brane on the K3 manifold. This leaves a string in the seven noncompact

dimensions. The only candidate for a counterpart in the heterotic theory

is the heterotic string itself! To decide whether this is reasonable, recall

that in the bosonic description of the heterotic string compacti¬ed on T n

there are 16 + n left-moving bosonic coordinates and n right-moving bosonic

coordinates. To understand this from the point of view of the M5-brane, the

¬rst step is to identify the ¬eld content of its world-volume theory. This is

a tensor supermultiplet in six dimensions, whose bosonic degrees of freedom

consist of ¬ve scalars, representing transverse excitations in 11 dimensions,

and a two-form potential with an anti-self-dual three-form ¬eld strength.30

This anti-self-dual three-form F3 gives zero modes that can be expanded as

a sum of terms

3 19

‚’ X i ω+

i

‚+ X i ω’ ,

i

F3 = + (9.177)

i=1 i=1

where ω± denote the self-dual and anti-self-dual two-forms of K3, and ‚± X i

i

correspond to the left-movers and right-movers on the string world sheet.

Since the latter are self-dual and anti-self-dual, respectively, all terms in

this formula are anti-self-dual. In addition, the heterotic string has ¬ve

more physical scalars, with both left-moving and right-moving components,

describing transverse excitations in the noncompact dimensions. These are

provided by the ¬ve scalars of the tensor multiplet.

Recall that the dimensions of a charged p-brane and its magnetic dual

p -brane are related in D dimensions by

p + p = D ’ 4. (9.178)

30 This ¬eld has three physical degrees of freedom, so the multiplet contains eight bosons and

eight fermions, as is always the case for maximally supersymmetric branes.

422 String geometry

For example, in 11 dimensions, the M5-brane is the magnetic dual of the

M2-brane. It follows that in the compacti¬ed theory, the string obtained by

wrapping the M5-brane on K3 is the magnetic dual of an unwrapped M2-

brane. In the ten-dimensional heterotic string theory, on the other hand, the

magnetic dual of a fundamental string (F1-brane) is the NS5-brane. After

compacti¬cation on T 3 , the magnetic dual of an unwrapped heterotic string

is a fully wrapped NS5-brane. Thus, the heterotic NS5-brane wrapped on

the three-torus corresponds to an unwrapped M2-brane.

The matching of tensions implies that

TF1 = TM5 VK3 and TNS5 VT 3 = TM2 (9.179)

or

1 VK3 VT 3 1

∼ ∼ 3,

and (9.180)

2 6 gs 6

2

s p s p

where the ∼ means that numerical factors are omitted. Combining these

two relations gives the dimensionless relation

3 ’1/2 4 3/4

∼ VK3 /

g s VT 3 / . (9.181)

s p

The left-hand side of this relation is precisely the seven-dimensional heterotic-

string coupling constant. This quanti¬es the earlier claim that gs ’ ∞

corresponds to VK3 ’ ∞.