String Theory and M-Theory

8.3 M-theory 329
since an SL(2, ) transformation that turns an F-string into a (p, q) string
leaves the D3-brane invariant.
• There exist stable supersymmetric (p, q) 5-branes, which are the magnetic
duals of (p, q) strings. Their SL(2, ) properties are quite similar to those
of the (p, q) strings.
• The D7-brane couples magnetically to C0. This field transforms in a
rather complicated way under SL(2, ), so it is not immediately obvious
how to classify 7-branes. Although this issue won’t be pursued here, the
classification is important, because certain nonperturbative vacua of type
IIB superstring theory (described by F-theory) contain various 7-branes.
This is addressed later.
The definition of a D-brane as a p-brane on which an F-string can end
has to be interpreted carefully for p = 1. A naive interpretation of “a
fundamental string ending on a D-string” would suggest a junction of three
string segments, one of which is (1, 0) and two of which are (0, 1). This
is not correct, however, because the charge on the end of the fundamental
string results in flux that must go into one or the other of the attached
string segments, changing the string charge in the process. In short, the
three-string junction must satisfy charge conservation. This means that an
allowed junction of three strings with charges (p (i) , q (i) ) with i = 1, 2, 3 has
to satisfy
p (i) =
q (i) = 0. (8.102)
i
Mathematically, this is just like momentum conservation at a vertex in a
Feynman diagram (in two dimensions). The junction configuration is stable
if the angles are chosen so that the three tensions, treated as vectors, add
to zero. It is possible to build complex string webs using such junctions.
8.3 M-theory
The term M-theory was introduced by Witten to refer to the “mysterious”
or “magical” quantum theory in 11 dimensions whose leading low-energy
effective action is 11-dimensional supergravity. M-theory is not yet fully
formulated, but the evidence for its existence is very compelling. It is as
fundamental (but not more) as type IIB superstring theory, for example.
In fact, the latter is somewhat better understood precisely because it is a
string theory and therefore admits a well-defined perturbation expansion.
This section describes a duality that relates M-theory compactified on a
torus to type IIB superstring theory compactified on a circle. Since this
i