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

TL;DR

The paper explicitly constructs type IIB supergravity solutions describing D3-branes ending on 5-branes in the near-horizon limit, yielding gravity duals to 4d ${\cal N}=4$ SYM on a half-line with half of the supersymmetry preserved. Building on the D’Hoker-Estes-Gutperle framework, the solutions are organized by genus via harmonic functions on a Riemann surface, and degenerations generate NS5- and D5-brane throats as well as multiple 5-brane stacks; flux quantization and Page charges ensure conserved D3-brane charges across complex brane geometries. The work provides a precise holographic realization of boundary conditions for D3-branes ending on 5-branes (per Gaiotto-Witten) and computes strong-coupling one-point functions of chiral operators, finding qualitative differences from weak-coupling expectations and suggesting nontrivial coupling dependence. It also shows that these backgrounds are equivalent to holographic descriptions of 4d ${\cal N}=4$ SYM on AdS4 with corresponding boundary data, and discusses extensions, limitations, and open questions such as spectra, localized gravity, and finite-temperature generalizations.

Abstract

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 4d N=4 SU(N) super-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 N=4 SYM theory on a half-line at strong coupling, and 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 N=4 SYM on AdS_4, with various boundary conditions.

Figures (5)

• Figure 1: In the hyper-elliptic 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 picture 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 two-spheres and $\Sigma$, for the solutions of this subsection corresponding to D3-branes intersecting D5-branes and NS5-branes. This space is non-compact along two $AdS_5\times S^5$ "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 $AdS_4$ space, and is not visible in (a).
• Figure 4: (a) A schematic picture of the six dimensional space made from the two two-spheres and $\Sigma$, for the solutions of this subsection corresponding to D3-branes ending on D5-branes and NS5-branes. This space is non-compact along the $AdS_5\times S^5$ "throat", and has several D5-brane and NS5-brane "throats" coming out of its interior. (b) The dual field theory, which describes the 4d ${\cal N}=4$ 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 figure (a).
• Figure 5: The $u$-plane for solutions of D3-branes ending on NS5-branes and D5-branes, with the $AdS_5\times S^5$ singularity chosen to be at $u=\infty$ and the $AdS_4\times {\mathbb R}^6$ point at $u=0$. We depict the first choice for the surface $\Sigma_1$ on which ${\cal F}_1$ is well-defined, and for the surface $\Sigma_2$ on which ${\cal F}_2$ is well-defined, separated by the curve $\gamma$.