Unfortunately, the brane configuration in figure12 considered above is not correct. The reason for this is that, when one fivebrane ends on another fivebrane, the branes are deformed; they do not meet at convenient right angles [56].
The D5-brane ending on the NS5-brane
The D5-brane extends in the (x1, x2, x7, x8, x9)-directions, and the NS50-brane that it ends on extends in the (x1, x2, x3, x8, x9)-directions [55]. The D5 and NS50-branes can be imagined as deforming one another at the point of intersection. Before including this deformation the picture looks like:
Figure 13: The NS50-brane and D5-brane intersection before deformations are ac- counted for.
The NS50-brane has four transverse directions (x4, x5, x6, x7). The x7-direction is the only transverse direction of the NS50-brane that the D5-brane extends along.
Similarly the D5-brane has the transverse directions (x3, x4, x5, x6), out of which the NS5-brane extends along x3. As a result, it is expected that the x3 position of the NS50-brane depends on thex7 position of the D5-brane. x7 can be written as a function of x3, where the function is required to minimize the worldvolume of the NS50-brane. For largex3 the two positions of the branes are related by the one-dimensional Laplace equation [55]:
∇2x7 =δ(x3) (6.18)
The solution to this equation is [55]:
x7= 1
2|x3|+cx3+d (6.19)
wherec and dare constants. At large and negativex3 the solution is expected to correspond to a the NS50-brane located at x7 = 0. With this in mind the constants are chosen to take the valuesc= 1/2 and d= 0. This gives:
x7= 1 2|x3|+
1
2x3 (6.20)
Therefore, for a D5-brane ending on an NS50-brane at x3 =x7 = 0, the configuration is drawn [55]:
Figure 14: The NS50-brane and D5-brane intersection results in a (1,1)-brane bound- state.
The diagonal brane is interpreted as a (1,1)-brane. A (1,1)-brane is a (p, q)-brane withp=q = 1, and a (p, q)-brane is a bound state of p NS5-branes andq D5-branes [55]. The (1,1)-brane is required to extend at 90◦ in the (x3, x7) plane in order for supersymmetry not to be broken. It also extends along (x1, x2, x8, x9). In general a (p, q)-brane must be oriented at an angleθ where [55]:
in order to preserve supersymmetry.
It is clear that the brane configuration in figure12 is not drawn accurately. The D5-branes should actually split along the NS5-brane as in the diagram below [37]:
Figure 15: The correct splitting of D5-branes on the NS50-brane.
The (p, q)-brane extends at an angle θ= tan−1(1/Nf) in the (x3, x7) plane, and also extends along (x1, x2, x8, x9).
Note that not all of the D5-branes need be split as above. An aribitrary number of theNf D5-branes can be formed in to the NS50-D5-(p, q) ‘web’, whilst the remaining
can be left in their original position intersecting the D3-branes.
The Bare FI-term from NS5-brane Separations in (x7,x8,x9)
The Fayet-Iliopoulos D-term coefficient ζ is given by the separation of the NS5-branes in the (x7, x8, x9)-directions [48]:
~
ζ =w~1−w~2 (6.22)
Here w1 and w2 are the (x7, x8, x9) positions of the two NS5-branes. The FI-term is associated with the center of the U(1)∈U(Nc) of the gauge group U(Nc) that arises
from D3-D3 strings. In this case the (p, q)-web introduces a displacement of the two halves of the NS50-branes in thex7-direction. The top NS50-brane moves to positve x7 whilst the bottom one moves to negativex7. The result is that the bare value ofζ associated with theU(Nc) group is given by the difference in x7 positions of the two NS50-branes. This is the sameζ that appears in equation4.4, and in subsequent equations in section4. The value of this coupling in the effective theory is then adjusted by integrating out massive matter, according to equation4.15.
This FI-termζ is actually mirror dual to the mass term that corresponds to the D5-brane position inx3. The mirror dual of an NS5-brane extending along
(x1, x2, x3, x4, x5) and at position (x7, x8, x9) = (a, b, c) is a D5-brane extending along (x1, x2, x7, x8, x9) and at position (x3, x4, x5) = (a, b, c). As well as exchanging
NS5-branes with D5-branes, the duality exchanges the (x3, x4, x5) positions with (x7, x8, x9). The D5-brane position in (x3, x4, x5) corresponds to mass terms for the fundamental hypermultiplet associated with NS5-D3 strings. Therefore mirror symmetry also corresponds tom~ ↔~ζ [34,57]. The (p, q)-web gives rise to D5s displaced inx3. This displacement corresponds to the real massm found in the background vector multiplet (see equation4.2). The mirror dual is an NS5-brane displaced inx7, corresponding to a real FI parameterζ [35].
D3-branes ending on Fivebranes
When considering brane deformations, the situation is different for a D3-brane ending on a fivebrane [56]. Consider the D3-brane ending on the NS50-brane: The D3-brane extends along (x1, x2, x6) whilst the NS50-brane extends along (x1, x2, x3, x8, x9). The NS50-brane has one transverse direction x6 that the D3-brane extends in, whilst the D3-brane has three transverse directions (x3, x8, x9) which the NS50-brane extends in. As a result thex6 position of the NS50-brane is dependendent on (x3, x8, x9), and the resulting Laplacian is 3d. Such a solution to the Laplacian gives a constant as
x3, x8, x9→ ∞. This constant just corresponds to the NS50-brane position and the brane is not drawn any differently. The same reasoning applies to the NS5-brane. The NS5-brane extends along the (x1, x2, x3, x4, x5). This has a transverse directionx6 that the D3-brane extends along, whilst the D3-brane has the transverse directions (x3, x4, x5) that the NS5-bane extends along. Again, this results in a 3d Laplacian and the position of the NS5-brane is just a constant at large (x3, x4, x5). Finally, the D5-brane extends along (x1, x2, x7, x8, x9). This has a transverse directionx6 that the D3-brane extends along, whilst the D3-brane has the transverse directions (x7, x8, x9) that the D5-bane extends along. So, again, the D5-brane position is just a constant at large (x7, x8, x9) and the brane is not drawn any differently.