Supersymmetric Multi-trace Boundary Conditions in AdS

TL;DR

The paper analyzes boundary conditions for massive fermions in AdS$_d$ with $d\ge2$ in the window $0\le |m|<\tfrac{1}{2\ell}$, showing that standard inner products admit multi-trace boundary conditions that map to multi-trace deformations in the dual CFT. By constructing boundary superfields from Fefferman–Graham coefficients, it classifies SUSY-preserving boundary conditions in $d=3,4,5$, distinguishing those that preserve Poincaré SUSY versus full superconformal symmetry. In AdS$_4\times S^7$ there exists a 595-dimensional manifold of double-trace marginal deformations preserving ${\cal N}=1$ on the boundary, while in ${\cal N}=4$ SYM in 3+1 dimensions there are no marginal or relevant multi-trace deformations that preserve even ${\cal N}=1$ SUSY at large $N$ and strong coupling. The results illuminate how bulk boundary conditions translate to exactly marginal deformations in the dual CFT and constrain possible SUSY-preserving deformations in holographic theories.

Abstract

Boundary conditions for massive fermions are investigated in AdS_d for $d \ge 2$. For fermion masses in the range $0 \le |m| < 1/2\ell$ with $\ell$ the AdS length, the standard notion of normalizeability allows a choice of boundary conditions. As in the case of scalars at or slightly above the Breitenlohner-Freedman (BF) bound, such boundary conditions correspond to multi-trace deformations of any CFT dual. By constructing appropriate boundary superfields, for d=3,4,5 we identify joint scalar/fermion boundary conditions which preserve either ${\cal N}=1$ supersymmetry or ${\cal N}=1$ superconformal symmetry on the boundary. In particular, we identify boundary conditions corresponding via AdS/CFT (at large N) to a 595-parameter family of double-trace marginal deformations of the low-energy theory of N M2-branes which preserve ${\cal N} =1$ superconformal symmetry. We also establish that (at large N and large 't Hooft coupling $λ$) there are no marginal or relevant multi-trace deformations of 3+1 ${\cal N} =4$ super Yang-Mills which preserve even ${\cal N}=1$ supersymmetry.

Figures (1)

• Figure 1: The scalar masses $m^2_A, m^2_B$ (solid curves) are plotted against the fermion mass $m$. Dashed lines mark the window between $m^2_{BF}$ and $m^2_{BF}+1$.