The D6-brane

The preceding discussion explained the M-theory origin of the type IIA Dp-

branes for p = 0, 2, 4 in terms of wrapped or unwrapped M-branes. This

raises the question of how one should understand the D6-brane from an

M-theory point of view. Clearly, unlike the other D-branes, it cannot be

related to the M2-brane or the M5-brane in any simple way. The key to

8.3 M-theory 333

answering this question is to recall that the D6-brane is the magnetic dual

of the D0-brane and that the D0-brane is interpreted as a Kaluza“Klein

excitation along the x11 circle. The D0-brane carries electric charge with

respect to the U (1) gauge ¬eld Cµ = gµ11 . Therefore, the D6-brane should

couple magnetically to this same gauge ¬eld.

This problem was solved long ago for the case of pure gravity in ¬ve di-

mensions compacti¬ed on a circle. In this case, the challenge is to construct

the ¬ve-dimensional metric that describes the Kaluza“Klein monopole, that

is, a magnetically charged soliton in four dimensions. By tensoring this so-

lution with 6 , exactly the same construction applies to the 11-dimensional

¡

problem. The extra six ¬‚at dimensions constitute the spatial directions of

the D6-brane world volume.

The relevant ¬ve-dimensional geometry that is Ricci-¬‚at and nonsingular

in ¬ve dimensions is given by

ds2 = ’dt2 + ds2 , (8.109)

5 TN

where the Taub“NUT metric is

1 2

ds2 = V (r) dr2 + r2 d„¦2 + dy + R sin2 (θ/2) dφ . (8.110)

TN 2

V (r)

Here d„¦2 = dθ2 + sin2 θdφ2 is the metric of a round unit two-sphere, and

2

R

V (r) = 1 + . (8.111)

2r

Also, the magnetic ¬eld is given by

Aφ = R sin2 (θ/2),

B=’ V = —A with (8.112)

where we have displayed only the nonvanishing component of the vector

potential. The Taub“NUT metric is nonsingular at r = 0 if the coordinate

y has period 2πR. Thus the actual radius of the circle is

R(r) = V (r)’1/2 R, (8.113)

which approaches R for r ’ ∞ and zero as r ’ 0.

The mass of the soliton described by the Taub“NUT metric can be com-

puted by integrating the energy density T00 . For the purpose of understand-

ing the tension of the D6-brane, we can add six more ¬‚at dimensions and

obtain

2πR

d3 x 2 V.

TD6 = (8.114)

16πG11

334 M-theory and string duality

Since the integral gives 2πR,

(2πR)2 2πR 2π

TD6 = = = , (8.115)

(2π s )7 gs

16πG11 16πG10

where we have used R = gs s . This agrees with the value obtained in

Chapter 6.

There is a simple generalization of the above, the multi-center Taub“NUT

metric, that describes a system of N parallel D6-branes. The metric in this

case is

1 2

ds2 = V (x)dx · dx + dy + A · dx , (8.116)

V (x)

where

N

R 1

B=’ V = —A and V (x) = 1 + . (8.117)

|x ’ x± |

2

±=1

Since this system is BPS, the tension and magnetic charge are just N times

the single D6-brane values.

A similar construction applies to other string theories compacti¬ed on

circles. Indeed, the type IIB superstring theory compacti¬ed on a circle

contains a Kaluza“Klein 5-brane, constructed in the same way as the D6-

brane, which is the magnetic dual of the Kaluza“Klein 0-brane. A T-duality

transformation along the circular dimension transforms the type IIB theory

into the type IIA theory compacti¬ed on the dual circle. The Kaluza“Klein

0-brane is dual to a fundamental type IIA string wound on the dual circle.

Therefore, the Kaluza“Klein 5-brane must map to the magnetic dual of the

fundamental IIA string, which is the type IIA NS5-brane.

E8 — E8 heterotic string theory at strong coupling

Let us brie¬‚y review the Hoˇava“Witten picture of the strongly coupled

r

E8 — E8 heterotic string theory. One starts with the strongly coupled type

IIA superstring theory, or equivalently M-theory on 9,1 — S 1 , and mods

¡

out by a certain 2 symmetry, much like one does in deriving the type I

superstring theory from the type IIB superstring theory. The appropriate

2 symmetry in this case includes the following reversals:

x11 ’ ’x11 A3 ’ ’A3 .

and (8.118)

In particular, modding out by this 2 action implies that the zero mode of

the Fourier expansion of Aµνρ in the x11 direction is eliminated from the

8.3 M-theory 335

spectrum, while the zero mode of

Bµν = Aµν11 (8.119)

survives. This is required, of course, to account for the fact that N = 1

supergravity in ten dimensions contains a massless two-form but no massless

three-form. The heterotic string coupling constant gs is given by

gs = R11 / s , (8.120)

just as in the case of the type IIA theory.

The space S 1 / 2 can be regarded as a line segment from x11 = 0 to

x11 = πR11 . The two end points are the ¬xed points of the orbifold. Their