The backreaction of anti-D3 branes on the Klebanov-Strassler geometry

TL;DR

The paper advances a controlled, holographically faithful construction of the backreaction of smeared anti-D3 branes on the KS geometry by solving the full space of linearized, $SU(2)\times SU(2)\times \mathbb{Z}_2$-invariant deformations. It employs the Borokhov-Gubser formalism and a two-nested-integral analytic solution to relate IR and UV data, ultimately identifying a unique anti-D3 solution whose fifteen integration constants are fixed in terms of the anti-brane number $\bar{N}$ (up to a subleading IR singularity). The work clarifies how Maxwell and Page charges, as well as B-field integrals and Seiberg-duality cascades, constrain the UV/IR dictionary and the field-theory interpretation, including distinctions between mesonic and baryonic branches and gaugino-mass deformations. It demonstrates that the anti-D3 solution reproduces the expected D3-brane force on probes and discusses the distinct UV boundary conditions that can lead to metastable vacua, illuminating the subtle interplay between IR physics, bulk regularity, and holographic duality in non-supersymmetric contexts.

Abstract

We present the full numerical solution for the 15-dimensional space of linearized deformations of the Klebanov-Strassler background which preserve the SU(2) X SU(2) X Z_2 symmetries. We identify within this space the solution corresponding to anti-D3 branes, (modulo the presence of a certain subleading singularity in the infrared). All the 15 integration constants of this solution are fixed in terms of the number of anti-D3 branes, and the solution differs in the UV from the supersymmetric solution into which it is supposed to decay by a mode corresponding to a rescaling of the field theory coordinates. Deciding whether two solutions that differ in the UV by a rescaling mode are dual to the same theory is involved even for supersymmetric Klebanov-Strassler solutions, and we explain in detail some of the subtleties associated to this.

Figures (5)

• Figure 1: The profile of the field $\tilde{\phi}_8$ corresponding to a shift of the dilaton, for the following choices of integration constants (with e.g. $P=1$). Blue, also labelled $\textcircled{a}$: $X_1 = 1, X_5 = -\frac{15}{2}, X_6=X_7=5, X_8=2, Y_8=-88.05$; Red $\textcircled{b}$: $X_1 = X_6=X_7=1, X_5 = -\frac{7}{6}, X_8=1.8, Y_8=-111.5$; Yellow $\textcircled{c}$: $X_1 = X_7=2, X_5 = -\frac{7}{6}, X_6=8.608, X_8=-0.843, Y_8=-133.9$. In each case, $Y_8$ is fixed so as to ensure that $\tilde{\phi}_8(\infty)=0$.
• Figure 2: The numerical solution for the field $\tilde{\phi}_8$ for $X_1 = 1, X_5 = -\frac{15}{2}, X_6=5, X_7=5, X_8=2, Y_8=-88.05, P=1$ (underlying blue solid line). The red and orange dashed lines correspond respectively to the IR and UV expansions.
• Figure 3: The profile of the first-order Maxwell charge for the anti-D3 solution, setting $\bar{N}=1$.
• Figure 4: Total Maxwell charge for the anti-D3 solution (blue), for the supersymmetric vacuum from the first mesonic branch (red) and for the "would-be supersymmetric vacuum in the baryonic branch" (black dashed line), fixing $\bar{N}=1, M=3 \, (P=\tfrac{3}{4})$ .
• Figure 5: The profile of the first order Maxwell charge for the solution with gaugino masses turned on, satisfying the constraint \ref{['relX2X7']} (blue solid line). The plot is for $\bar{N} =1$ and $X_7 = 1/(24 \, 2^{1/3} P^3)$. The red dashed curve is the profile for $\bar{N}=0$.