String Theory and M-Theory

298 M-theory and string duality
belong to a long supermultiplet. The zeroes that appear in the supersymmetry
algebra when M = |Z1| are responsible for the multiplet shortening.
A further refinement in the description of BPS states keeps track of the
number of central charges that equal the mass. Thus, for example, in the
N = 4 case, states with M = |Z1| = |Z2| are called half-BPS and ones with
M = |Z1| > |Z2| are called quarter-BPS. These fractions refer to the number
of supersymmetries that are unbroken when these particles are present.
The preceding discussion is specific to point particles in four dimensions,
but it generalizes to p-branes in D dimensions. The important point to
remember from Chapter 6 is that a charged p-brane has a (p + 1)-form
conserved current, and hence a p-form charge. To analyze such cases the
supersymmetry algebra needs to be generalized to cases appropriate to D
dimensions and p-form central charges. Calling them central is a bit of a
misnomer in this case, because for p > 0 they carry Lorentz indices and
therefore do not commute with Lorentz transformations.
One very important conclusion from the BPS bound given above is that
BPS states, which have M = |Z1| and belong to a short multiplet, are stable.
The mass is tied to a central charge, and this relation does not change as
parameters are varied if the supersymmetry is unbroken. The only way in
which this could fail is if another representation becomes degenerate with the
BPS multiplet, so that they can pair up to give a long representation. The
idea is actually more general than supersymmetry. This is what happens
in the Higgs mechanism, where a massless vector (a short representation of
the Lorentz group) joins up with a scalar to give a massive vector (a long
representation) as a parameter in the Higgs potential is varied. The thing
that is different about supersymmetric examples is that short multiplets can
be massive. In any case, the conclusion is that so long as such a joining of
multiplets does not happen, it is possible to follow BPS states from weak
coupling to strong coupling with precise control. This is very important for
testing conjectures about the behavior of string theories at strong coupling,
as we shall see in this chapter.
EXERCISES
EXERCISE 8.1
The N = 1 supersymmetry algebra in four dimensions does not have a
central extension. The explicit form of this algebra, with the supercharges