String Theory and M-Theory

342 M-theory and string duality
Thus, this wrapping gives a string in nine dimensions, whose tension is
T (11)
(p,q) = L (p,q)TM2. (8.134)
The superscript 11 emphasizes that this is measured in the 11-dimensional
metric. To compare with the type IIB string tensions, we use Eqs (8.130)
and (8.131) to deduce that
T (p,q) = β −2 T (11)
(p,q) . (8.135)
This agrees with the result given earlier, showing that this is a correct interpretation.
Fig. 8.3. In the covering space of the torus, which is the plane tiled by parallelograms,
a closed geodesic curve on the torus is represented by a straight line between
equivalent points.
To match 2-branes in nine dimensions requires wrapping the type IIB
D3-brane on the circle and comparing to the unwrapped M2-brane. The
wrapped D3-brane gives a 2-brane with tension 2πRBTD3. Including the
metric conversion factor, the matching gives
TM2 = 2πRBβ 3 TD3. (8.136)
Combining this with Eqs (8.131) and (8.132) gives the identity
TD3 =
(TF1) 2
2πgs
, (8.137)
in agreement with the tension formulas in Chapter 6. It is remarkable that
the M-theory/type IIB superstring theory duality not only relates M-theory
tensions to type IIB superstring theory tensions, but it even implies a relation
involving only type IIB tensions.
Wrapping the M5-brane on the spatial torus gives a 3-brane in nine dimensions,
which can be identified with the unwrapped type IIB D3-brane
in nine dimensions. This gives
TM5AM = β 4 TD3, (8.138)