More than just strings
To understand the presence of objects in string
theory that are not strings, but higher dimensional objects, or even
points, it helps to know the formulation of Maxwell's equations in the
language of differential forms, because this is what tells us that the
sources of charge in the Maxwell equations are zerodimensional objects.
Gauge field strengths that are p+2forms turn out to have sources that
are pdimensional objects. We call these pbranes.
In the regular maxwell equations in d=4 spacetime
dimension, the electric and magentic fields are packed together into
the field strength F, which satisfies
the equation F=dA,
d is the exterior derivative, and A
is the vector potential, a oneform. The twoform *F
is the dual of F relative to the spacetime
volume fourform v.
(The subscripts on F, etc, below
are just to indicate the degree of the differential form.)
The charge sources enter through the equation
d*F=*J, where *J is the threeform dual
to the current fourvector J=(r,j).
In the rest frame of the charge density r,
J=(r,0),
so *J is r
times the the volume element for threedimensional space. In a threedimensional
space, a surface that can be localized in three dimensions (has codimension
three) must be a zerodimensional surface, also known as a point.
This is the math that tells us that the Maxwell
equations couple electrically to sources that are points, or zerobranes,
as zerodimensional objects are now called in string theory. (For magnetic
couplings, the roles of F and *F
are interchanged, but that won't be covered here.) This same math works
for twoforms in any spacetime dimension, so we know that Maxwell's
equations couple to point charges in any spacetime dimension.
Superstring theories contain electromagnetism,
but they also contain field strengths that are threeforms, fourforms
and on up. These field strengths obey equations just like the Maxwell
equations, and their sources can be analyzed in the same manner as above.
Suppose we start in d spacetime dimensions
with a vector potential A that is a
p+1form. Then F is a p+2form, v
is a dform (because it's the volume element of ddimensional spacetime),
*F is a (dp2)form, and d*F
is a (dp1)form. (Once again, the subscripts are just to indicate
the degree of the differential form.)
The equations of motion tell us that the source
term *J is also a (dp1)form. In the
rest frame of an isolated source, *J
is proportional to a volume element of a (d1p)dimensional subspace
of (d1)dimensional space. The codimension of the source is therefore
(dp1), and since space has dimension d1, the charges that serve as
sources must be objects with p dimensions, known as pbranes.
So a (p+2)form field strength couples to sources
that are pbranes. This little fact has turned out to be extremely
important in string theory.
Superstring theories are theories with gravity,
so these pdimensional localizations of charge must lead to spacetime
curvature. A pbrane spacetime whose metric solves the equations of
motion for a (p+2)form field strength in d spacetime dimensions can
be described using p space coordinates {y^{i}} along the pbrane
and (d1p) space coordinates {x^{a}} orthogonal to the pbrane.
The isometries of this spacetime consist of translations (shifting
the coordinate by a constant) and Lorentz transformations in the (p+1)dimensional
worldvolume, plus spatial rotations in the (d1p)dimensional space
orthogonal to the pbrane.
There's a problem with adding gravity, however.
Most pbrane spacetimes turn out to be unstable. Supersymmetry stabilizes
pbranes, but only for the certain values of p and d. Two of the most
important pbranes in string theory are the twobrane
in d=11 and the fivebrane in d=10.
Since we're talking about a spacetime metric,
we're obviously in the low energy limit of string theory. But pbranes
can be protected from quantum corrections by supersymmetry, if they
satisfy an equality between mass and charge known as the BPS
condition. These branes are then known as BPS
branes.
Next:
From pbranes to Dbranes>>
