String Theory and M-Theory

232 T-duality and D-branes
By T-duality there should be a dual interpretation in terms of a D0-brane
on a dual circle. In this case it was shown in Chapter 5 that
A1 = − 1
2πα ′ X 1 , (6.102)
where X 1 is the coordinate on the dual circle. This gives a field strength
F01 = − 1
2πα ′ v where v = ˙ 1
X . (6.103)
Here v is the velocity of the D0-brane on the dual circle. The spatial integration
gives a constant factor, and one is left with the action for a relativistic
particle (compare with Chapter 2)
1
−m − v2 dt, (6.104)
for the choice
k = 2πα ′ . (6.105)
Thus the Born–Infeld structure is required for Lorentz invariance of the
T-dual description.
Generalizing to p + 1 dimensions, the Born–Infeld structure combines
nicely with the usual Nambu–Goto structure for a Dp-brane (discussed in
Chapter 2) to give the action
S1 = −TDp
d p+1 σ
− det(Gαβ + kFαβ), (6.106)
where TDp is the tension (or energy density), and k = 2πα ′ . For type
II superstrings in Minkowski space-time supersymmetry is incorporated by
defining
where
Gαβ = ηµνΠ µ αΠ ν β , (6.107)
Π µ α = ∂αX µ − ¯ Θ A Γ µ ∂αΘ A . (6.108)
This is the same supersymmetric combination introduced in Chapter 5.
Also,
Fαβ = Fαβ + bαβ, (6.109)
where F = dA is the usual Maxwell field strength and the two-form b is a
Θ-dependent term that is required in order that F is supersymmetric. The
concrete expression, whose verification is a homework problem, is
b = ( ¯ Θ 1 ΓµdΘ 1 − ¯ Θ 2 ΓµdΘ 2 )(dX µ − 1
2 ¯ Θ A Γ µ dΘ A ). (6.110)