String Theory and M-Theory

216 T-duality and D-branes
µp Ap+1 to the brane. The gauge-invariant field strength is Fp+2 = dAp+1
and its dual is
FD−p−2 = ⋆Fp+2.
Gauss’s law is the statement that if we loop the p-brane once with a sphere
SD−p−2 , then the charge is given by
µp = FD−p−2.
S D−p−2
The magnetic dual of this brane is a (D − p − 4)-brane that can be encircled
by a sphere Sp+2 . Gauss’s law gives its magnetic charge
µD−p−4 = Fp+2.
S p+2
Requiring that both branes have nonnegative dimension gives 0 ≤ p ≤ D−4.
Now let’s consider a probe electric p-brane in the field of a magnetic
(D − p − 4)-brane. For the argument that follows, the topology of the
magnetic brane doesn’t matter, but it is extremely convenient to choose the
electric brane to be topologically a sphere Sp . Let us denote this p-cycle by
β. Then, for the same reason as in the previous exercise, the wave function
of the p-brane has the form
β
ψ(β) = exp iµp Ap+1 ψ0(β)
where ψ0 is gauge invariant. The lower limit is a fixed p-cycle β0 and the
integral is over a region that is a “cylinder” whose topology is a line interval
times S p . As before, it does not matter how this is chosen.
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V
V
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Fig. 6.7. This illustrates, for the case p = 1, how a loop of p-dimensional spheres
can trace out a (p + 1)-dimensional sphere γ.
β0
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