String Theory and M-Theory

428 String geometry
which transforms nonlinearly under SL(2, ) transformations in the same
way as the modular parameter of a torus:
τ →
aτ + b
. (9.188)
cτ + 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.
F-theory compactifications involve 7-branes, which end up filling the d
noncompact space-time dimensions and wrapping (8 − d)-cycles in the compact
dimensions. Therefore, before explaining F-theory, it is necessary to
discuss the classification and basic properties of 7-branes. 7-branes in ten
dimensions are codimension two, and so they can be enclosed by a circle,
just as is the case for a point particle in three dimensions and a string in
four dimensions. Just as in those cases, the presence of the brane creates
a deficit angle in the orthogonal plane that is proportional to the tension
of the brane. Thus, a small circle of radius R, centered on the core of the
brane, has a circumference (2π − φ)R, where φ is the deficit angle. In fact,
this property is the key to searching for cosmic strings that might stretch
across the sky.
The fact that fields must be single-valued requires that, when they are
analytically continued around a circle that encloses a 7-brane, they return
to their original values up to an SL(2, ) transformation. The reason for
this is that SL(2, ) is a discrete gauge symmetry, so that the configuration
space is the naive field space modded out by this gauge group. So the
requirement stated above means that fields should be single-valued on this
quotient space. The field τ, in particular, can have a nontrivial monodromy
transformation like that in Eq. (9.188). Other fields, such as B2 and C2,
must transform at the same time, of course.
Since 7-branes are characterized by their monodromy, which is an SL(2, )
transformation, there is an infinite number of different types. In the case of a
D7-brane, the monodromy is τ → τ +1. This implies that 2πC0 is an angular
coordinate in the plane perpendicular to the brane. More precisely, the 7brane
is characterized by the conjugacy class of its monodromy. If there is
another 7-brane present the path used for the monodromy could circle the
other 7-brane then circle the 7-brane of interest, and finally circle the other
7-brane in the opposite direction. This gives a monodromy described by a
different element of SL(2, ) that belongs to the same conjugacy class and
is physically equivalent. The conjugacy classes are characterized by a pair
of coprime integers (p, q). This is interpreted physically as labelling the type