Zooming out of AdS$_4 \times$S$^2 \times$S$^2$: The Branes behind the CFT's

Abstract

We reveal the supersymmetric brane configurations that give rise to AdS$_4\times$S$^2\times$S$^2$ supergravity solutions, which are holographic duals to three-dimensional $N=4$ CFTs or to conformal boundaries and domain walls of four-dimensional $N=4$ SYM. We show that these solutions preserve the same Killing spinors as orthogonal D3, D5 and NS5 branes in flat space, and that the singular sources of these solutions correspond to semi-infinite D3-D5 and D3-NS5 spikes. We track these solutions all the way from the weak-coupling regime of parameters, where the branes do not backreact, to the supergravity regime. We explain how the AdS$_4$ factor arises from certain universal self-similar bending regions of the five-branes, whose steepness is the same as the weak-coupling linking numbers. We also propose a brane configuration that gives rise to the Janus interface solutions. Our construction gives a clear geometric explanation of the Gaiotto-Witten "good-bad-ugly" classification of eight-supercharge theories: only good theories have five-branes that do not cross when back-reacting, and end up sourcing an AdS$_4\times$S$^2\times$S$^2$ solution.

Figures (20)

• Figure 1: Plot of the various functions for the example of the $T^{\sigma = [3,2,2,2]}_{\rho = [2,2,2,2,1]} [SU(9)]$ theory (introduced in Section \ref{['sec:Brane interpretation - field theory']}). In each plot, the horizontal axis is $x$ and the depth axis is $y$ (with $y=0$ in front and $y=\frac{\pi}{2}$ in the back).
• Figure 2: The infinite strip with D5 and NS5 sources. The $3$-cycles around D5 and NS5 sources are defined by contours, $\mathcal{C}_a^{(i)}$, on the Riemann surface.
• Figure 3: The $x$ equation of motion for various values of $\Pi$ for a Janus solution (a), and a solution with no asymptotic AdS$_5 \times S^5$ regions (b). The intersection of a curve with the $x$ gives the position where the D5-D3-brane probe feels no force. (a): Janus solution with parameters $(\alpha_1, \alpha_2,\beta_1,\beta_2)=(1,0.2,-5,3)$. The D3-D5 charge ratio, $\Pi$, takes the values $(150,200,250,300)$ from left to right. There are solutions for all values of $\Pi$. (b): Solution with parameters $(\gamma_1, \gamma_2,\delta_1, \delta_2)=(1,4,1,2)$. Here $\Pi$ takes the values $(\text{Green, Blue, Orange, Red})=(-60,-40,-30,-20)$. There are solutions only for $\Pi \in [-4\pi\,\gamma_2,0]$ as one can easily deduce from \ref{['noAdS5Pi']}.
• Figure 4: The red line denotes a $1+2$-dimensional $\mathcal{N}=4$ theory embedded into $1+3$-dimensional space (the added dimension being the horizontal direction). The three dimensional theory can be coupled to a four-dimensional theory on one side (panel $(b)$) or two sides (panel $(c)$).
• Figure 5: Brane systems and quivers for the 3-dimensional $\mathcal{N}=4$ theory $T^{\sigma = [3,2,2,2]}_{\rho = [2,2,2,2,1]} [SU(9)]$. We depict the D5 branes in blue, the NS5 branes in red, the D3 branes in black. The linking numbers for D5 / NS5 branes are also indicated in blue / red. The black numbers denote the multiplicity of the D3 branes. The top panel shows the partition frame, from which one can reach the Higgs/electric (respectively Coulomb/magnetic frame) by Hanany-Witten moves in such a way that no D3 ends on a D5 brane (respectively an NS5 brane). The bottom left panel shows the stacks of branes ordered by their linking number. This is the frame which reflects the supergravity description. The bottom right panel shows the same system with the asymptotic back-reaction taken into account. Whenever we quote numerical values for the $\gamma$'s we set $\alpha' = 4$, so that the $\gamma$’s take integer values.