Curvature-induced Resolution of Anti-brane Singularities

TL;DR

The paper shows that curvature in a compact AdS$_7 imes S^3$ background with anti-D6-branes and $H_3$ flux induces brane polarization of anti-D6-branes into a D8-brane, resolving the flux singularity that appears in the backreacted solution. This is demonstrated analytically for supersymmetric configurations via the D8-brane potential, which features a local maximum at the brane position and a lower-lying minimum at finite polarization radius; the result persists for nearby non-supersymmetric solutions as suggested by numerical evidence. A key insight is that AdS curvature missing in non-compact Minkowski setups qualitatively changes the polarization potential, enabling a stable polarized endpoint. The findings imply curvature and compactness can cure certain anti-brane singularities and have potential implications for holographic duals and related anti-brane constructions.

Abstract

We study AdS$_7$ vacua of massive type IIA string theory compactified on a 3-sphere with $H_3$ flux and anti-D6-branes. In such backgrounds, the anti-brane backreaction is known to generate a singularity in the $H_3$ energy density, whose interpretation has not been understood so far. We first consider supersymmetric solutions of this setup and give an analytic proof that the flux singularity is resolved there by a polarization of the anti-D6-branes into a D8-brane, which wraps a finite 2-sphere inside of the compact space. To this end, we compute the potential for a spherical probe D8-brane on top of a background with backreacting anti-D6-branes and show that it has a local maximum at zero radius and a local minimum at a finite radius of the 2-sphere. The polarization is triggered by a term in the potential due to the AdS curvature and does therefore not occur in non-compact setups where the 7d external spacetime is Minkowski. We furthermore find numerical evidence for the existence of non-supersymmetric solutions in our setup. This is supported by the observation that the general solution to the equations of motion has a continuous parameter that is suggestive of a modulus and appears to control supersymmetry breaking. Analyzing the polarization potential for the non-supersymmetric solutions, we find that the flux singularities are resolved there by brane polarization as well.

Figures (9)

• Figure 1: The three supersymmetric differential equations \ref{['eee']}--\ref{['uuu']} evaluated for the general solution with $\tilde{a}_0 = 1, \tilde{b}_0 = 1, \tilde{f}_0 = 1, \tilde{\lambda}_0 = -2.5, F_0 = 4$ for positive $x(\theta)$ (light, medium and dark blue) and negative $x(\theta)$ (yellow, orange and red), where the value $0$ means that an equation is solved. One can see that the equations are fulfilled when $x(\theta)$ starts positive at $\theta=0$ and then switches its sign between the two poles. This is consistent with the conventions of Apruzzi:2013yva, where $x=1$ at the north pole and $x=-1$ at the south pole.
• Figure 2: The three supersymmetric differential equations evaluated as in \ref{['f:susydiff']} for the solution with $F_0 = 15$ and $\tilde{\alpha}_0$ from \ref{['t:para_pi']} to get maximal integration range. As the equations have deviations from zero ranging between $10^{-3}$ and $10$, they are not well fulfilled in comparison to the supersymmetric solution in \ref{['f:susydiff']}. In addition, all deviations are large for $\theta \approx 0$ and there is no clear point at which the solution of $x$ switches the sign. This supports the argument that this solution is not supersymmetric although it can be integrated over the whole range.
• Figure 3: The figure shows Matlab results for the integrated $F_2$ Bianchi identity for different values of $F_0$ and $Q_2$. In order that the D6 tadpole is cancelled in a numerical solution, the charge obtained from the integrated $H_3$ flux (blue dots) must equal the charge from the delta function source term, which is read off from a fit of the solution at the south pole (orange dots). Except for the dot at $F_0=4$, all solutions are non-supersymmetric, and all integrate to $\theta\approx\pi$. The main numerical uncertainty comes from the tuning necessary to get the integration range until $\theta=\pi$, which is much easier for supersymmetric solutions. Nevertheless, the non-supersymmetric solutions satisfy the integrated Bianchi identity very well. For the larger values of $F_0$, numerical problems due to small numbers become relevant.
• Figure 4: The function $\mathrm{e}^{2A(\theta)}$ for a set of solutions with fixed $(F_0,Q_2)$.
• Figure 5: The function $\mathrm{e}^{2B(\theta)}\sin^2\theta$ measuring the squared 2-sphere radius for a set of solutions with fixed $(F_0,Q_2)$.