Gravity and Strings

19.6 Intersections 567
outside the string in the presence of other fields, the equation of motion has additional terms
and the homotopy-invariant definition of charge is
−2 ˆϕ⋆
e ˆH − ⋆ ˆG (3) Ĉ (0) − ˆG (5) Ĉ (2) , (19.140)
qF1 ∼
S 7
where S7 surrounds the string. Let us consider a semi-infinite string. At a large enough
distance L from the endpoint, boundary effects are not important and the charge is still
approximately given by the first of Eqs. (19.102). The larger L is for a fixed value of the S7 radius R7, the better the approximation. Closer to the endpoint, the additional terms must
contribute (otherwise, we are back in the previous case), but we can obtain the same value
for the integral by making R7 → 0keeping R7/L constant until the only contribution to the
integral comes from the endpoint. The degenerate S7 can be decomposed, for convenience,
into the product S5 × S2 ,ifweassume that the contribution to qF1B comes from the last
term in the above integral. The integral decomposes into a product of integrals,
ˆG (5)
Ĉ (2) . (19.141)
S 5
The first integral gives the D3-brane charge (assuming, as we are doing here, that ˆH does
not contribute), ˆG (5) = ⋆ ˆG (5) ,and thus the string endpoint must be at a D3-brane. If there is
no D1-brane present, then, ˆG (3) ∼ dĈ (2) = 0 inside the D3-brane and, locally, Ĉ (2) = dV,
where V is a vector that lives in the D3-brane worldvolume. Then
qF1B ∼ dV. (19.142)
The interpretation is clear: an F1B can end on a D3-brane and at the intersection point
there is an excited worldvolume vector field (the dual BI vector field) whose magnetic
charge is proportional to the F1B charge. This is the same result as we obtained before.
Asimilar reasoning indicates that, if it is the second term that contributes to the charge
integral, the F1B can also end on a D-string and at the intersection the BI vector field is
excited so its dual field strength is a constant. 27
These arguments that determine the opening of branes seem to depend on field redefinitions
(the charge integrand is defined up to total derivatives). However, the different
expressions for the charge are just choices that are more or less adequate to describe a
given physical situation. The most symmetric expression for qF1B can be obtained by using
Eq. (17.9). Each of the four possible terms corresponds to the F1B ending on one of the
four Dp-branes p = 1, 3, 5, and 7 and exciting the dual BI field magnetically.
19.6.2 Marginally bound supersymmetric states and intersections
We are considering only supersymmetric brane intersections, in which the branes that intersect
do not interact and are in supersymmetric equilibrium. These intersections can be
27 This is similar to viewing the mass parameter of Romans’ N = 2A, d = 10 SUEGRA as the dual of the RR
10-form field strength.
S 2
S 2