Near-horizon solutions for D3-branes ending on 5-branes

We construct the type IIB supergravity solutions describing D3-branes ending on 5-branes, in the near-horizon limit of the D3-branes. Our solutions are holographically dual to the four dimensional (4D) $\mathcal{N}=4$ $SU(N)$ supersymmetric-Yang-Mills (SYM) theory on a half line, at large $N$ and large ’t Hooft coupling, with various boundary conditions that preserve half of the supersymmetry. The solutions are limiting cases of the general solutions with the same symmetries constructed in 2007 by D’Hoker, Estes and Gutperle. The classification of our solutions matches exactly with the general classification of boundary conditions for D3-branes ending on 5-branes by Gaiotto and Witten. We use the gravity duals to compute the one-point functions of some chiral operators in the $\mathcal{N}=4$ SYM theory on a half line at strong coupling, and we find that they do not match with the expectation values of the same operators with the same boundary conditions at small ’t Hooft coupling. Our solutions may also be interpreted as the gravity duals of 4D $\mathcal{N}=4$ SYM on ${\mathrm{AdS}}_4$, with various boundary conditions.

Figure 1

In the hyperelliptic ansatz, $\Sigma$ is the lower half of the complex plane. The dots on $\partial \Sigma$ represent the branch points. On each segment connecting two branch points, one (and only one) of the spheres has a vanishing radius. Note that generic points on the real axis $\partial \Sigma$ are not a boundary of the 10D space-time.

Figure 2

The generic genus one solution has four asymptotic regions. The 5-brane solution is achieved by collapsing two such adjacent regions. The figure shows that in the full geometry there is a 3-cycle surrounding the object at $u=k^2$, as expected of a 5-brane.

Figure 3

(a) A schematic picture of the six-dimensional space made from the two 2-spheres and $\Sigma$, for the solutions of this subsection corresponding to D3-branes intersecting D5-branes and NS5-branes. This space is noncompact along two ${\mathrm{AdS}}_5\times S^5$ throats, and has several D5-brane and NS5-brane throats coming out of its interior. (b) The dual field theory, which describes two 4D theories on a half line interacting with a 3D “defect” theory. Note that the $z$ coordinate which parameterizes the half lines in (b) is part of the ${\mathrm{AdS}}_4$ space, and is not visible in (a).

Figure 4

(a) A schematic picture of the six-dimensional space made from the two 2-spheres and $\Sigma$, for the solutions of this subsection corresponding to D3-branes ending on D5-branes and NS5-branes. This space is noncompact along the ${\mathrm{AdS}}_5\times S^5$ throat, and has several D5-brane and NS5-brane throats coming out of its interior. (b) The dual field theory, which describes the 4D $\mathcal{N}=4$ SYM theory on a half line with some boundary condition (that could include interactions with a 3D SCFT at the boundary). As in the previous figure, the 4D boundary component lives infinitely far away along the throat in (a).

Figure 5

The $u$-plane for solutions of D3-branes ending on NS5-branes and D5-branes, with the ${\mathrm{AdS}}_5\times S^5$ singularity chosen to be at $u=\infty$ and the ${\mathrm{AdS}}_4\times {\mathbb{R}}^6$ point at $u=0$. We depict the first choice for the surface ${\Sigma }_1$ on which ${\mathcal{F}}_1$ is well-defined, and for the surface ${\Sigma }_2$ on which ${\mathcal{F}}_2$ is well-defined, separated by the curve $\gamma$.