Gravity dual of dynamically broken supersymmetry

TL;DR

The paper probes dynamical supersymmetry breaking in ABJM theory by embedding it in the warped $A_8$ gravity background and examining RG flows controlled by the parameter $Q = N - \frac{l(l-k)}{2k} - \frac{k}{24}$. For $Q>0$, the gravity solution describes a flow from a YM-like UV fixed point to a superconformal IR with $F \sim Q^{3/2}$; when $Q<0$, the authors study the linearized backreaction of anti-D2 branes around the $Q=0$ background using the Borokhov–Gubser formalism and find no regular solution that satisfies the full set of boundary conditions, leading to a speculative resolution in which curvature-induced charge and anti-D2 matter form a non-supersymmetric clump around a repulson. This picture motivates a phase diagram in $(N/k, l/k)$ with a phase boundary near $igr\epsilon=1/2$ (where $\bigr\epsilon = -kQ/M^2$) and highlights the possible role of the $Q=-k/24$ curvature term in enabling a non-BPS vacuum. The work underscores the richness of gauge/gravity duality in 2+1D, suggesting a non-supersymmetric vacuum stabilized by fluxes and curvature effects, and it points to future directions for understanding anti-brane dynamics in highly curved backgrounds.

Abstract

We study a renormalization group flow of ABJM theory embedded into the warped A_8 geometry and explore the dependence of the vacuum structure on the parameters of the theory. This model has a product group gauge structure U(N)xU(n+l) and comes equipped with discrete parameters N, l and k, a continuous parameter b related to the ratio of the Yang-Mills coupling for the two gauge groups, and one dimensionful parameter gYM^2 setting the overall scale. A supersymmetric supergravity solution exists when Q=N-l(l-k)/2k-k/24 is positive and is interpretable as a RG flow from a Yang-Mills like UV fixed point to a superconformal IR fixed point with free energy of order Q^3/2. The fate of the theory when Q is taken to be negative is less clear. We explore the structure of the possible gravity solution for small negative Q by considering the linearized gravitational back reaction from adding a small number of anti-branes on the Q=0 background. Following the work of Bena, et.al., we find that a sensible solution satisfying appropriate boundary conditions does not appear to exist. This leaves the status of the RG flow for the Q<0 theories a mystery. We offer the following speculative resolution to the puzzle: the -k/24 unit of charge induced by the curvature correction to supergravity should be considered an allowed physical object, and one should be adding an anti brane not to the Q=0 background but rather the Q=-k/24 background. Such a solution has a repulson singularity, and gives rise to a picture of the vacuum configuration where a cluster of anti-branes are floating around the repulson singularity, but are stabilized from being pushed off to infinity by other fluxes. Such a state is non-supersymmetric and appears to describe a vacuum with dynamical breaking of supersymmetry. Based on these considerations, we construct a phase diagram for this theory exhibiting various interesting regions.

Figures (6)

• Figure 1: Hanany-Witten brane diagram for configurations violating the generalized $s$-rule. The configurations (a), (b), and (c) are related by sliding the $(1,k)$ brane around the circle. In this figure, labels such as "$3k-1$" and "$k-1$" refers to the number of D3 brane segments stretched between the 5-branes, as opposed to the counting of integer and fractional branes. The configuration (a) corresponds to $N=k-1$ and $l = 2k$. Configuration (b) corresponds to $N=-1$ and $l = k$. (c) corresponds to $N=-1$ and $l=0$. This figure originally appeared in Hashimoto:2010bq.
• Figure 2: Schematic sketch of the expected minimum energy configuration for the construction illustrated in figure \ref{['figa']}.b including the effect of repulsion between the brane segments. This figure originally appeared in Hashimoto:2010bq.
• Figure 3: The potential experienced by an anti D2-brane in the $Q<0$ background inferred from the DBI action. This plot includes the extreme case $\epsilon = 0$ and a weakly repulsive case $\epsilon = 0.01$.
• Figure 4: The phase diagram of the warped $A_8$ theory as a function of $N/k$ and $l/k$. Here, we have set $b_\infty=1/2$ and $k$ is assumed to be large. The red parabola indicates the region where $Q>0$ and the theory flows to the superconformal fixed point of ABJM. Outside the red parabola, we illustrate the contours of fixed $\epsilon$ in the range $0 < \epsilon < \infty$ in logarithmic intervals. At $\epsilon = 1/2$, the $Q_2^{Maxwell} = Q + M^2/2k$ changes sign, and we expect the theory to transition into a new phase as $\epsilon$ crosses this line. The supergravity approximation should be considered most reliable for large values of $N/k$ and $l/k$ and close to the red parabola corresponding to small values of $\epsilon$.
• Figure 5: The potential of the D0-brane probe for $Q>0$ background inferred from the DBI action. At $Q=0$, the D0 brane is stabilized at a finite radius. For small and positive $Q$, the minimum at finite radius becomes metastable. As $Q$ is increased, the metastable minimum disappears and the D0 is attracted toward the core region at $r=\ell$.