String Theory and M-Theory

246 T-duality and D-branes
PROBLEM 6.6
Show that the Born–Infeld action (6.100) gives a finite classical self-energy
for a charged point particle. Hint: show that the solution to the equations
of motion with a point particle of charge e at the origin is given by
Er = Frt =
PROBLEM 6.7
Consider the DBI action in Eq. (6.106).
e
(r 4 + r 4 0 ) , r2 0 = 2πα ′ e.
(i) Derive the equation of motion for the gauge field.
(ii) Expand this equation in powers of k to obtain the leading correction
to the usual Maxwell field equation of electrodynamics in the absence
of sources. You may use the result of Exercise 6.9.
PROBLEM 6.8
Consider a D0–D8 system in the type I ′ theory, where the D0-brane is
coincident with the D8-brane. There are also other D8-branes and O8-planes
parallel to the D8-brane, as described in Section 6.3.
(i) Determine the zero-point energy of a D0–D8 open string in the NS
sector.
(ii) Describe the supersymmetries that are preserved by this configuration.
How many of them are there? Hint: Eq. (6.56) shows which supersymmetries
are preserved by a single D-brane. The problem here
is to determine which ones are preserved by both of the D-branes.
PROBLEM 6.9
Consider a type I ′ configuration in which N1 D8-branes are coincident at
X ′ L = θLR ′ and the remaining N2 = 16 − N1 D8-branes are coincident at
X ′ R = θRR ′ .
(i) What is the gauge symmetry for generic positions X ′ L and X′ R ?
(ii) What is the maximum enhanced gauge symmetry that can be achieved
for N1 = N2 = 8? How are the D8-branes positioned in this case?
PROBLEM 6.10
Show that the right-hand side of Eq. (6.117) is closed.