Holography for ${\cal N}=2^*$ on $S^4$

TL;DR

The paper constructs a holographic dual for N=2^* SYM on S^4 by a consistent 5D truncation of N=8 gauged supergravity with three bulk scalars and an S^4 slicing. It derives and solves the BPS equations for a one-parameter family of regular, curved-domain-wall solutions, and performs careful holographic renormalization, including a finite supersymmetry counterterm, to compute the S^4 free energy. The authors show that the third derivative of the free energy with respect to the mass parameter, d^3F/d(ma)^3, matches the field theory result obtained from localization in the large N and large λ limit, independently validating holography in a non-conformal Euclidean setting. This precision test strengthens confidence in the AdS/CFT framework beyond conformal theories and curved-space localization, and points to rich directions for exploring orbifolds, N=1 theories, and ten-dimensional uplifts. The work also clarifies how curvature couplings and additional bulk scalars are essential to correctly capture supersymmetric theories on curved manifolds.

Abstract

We find the gravity dual of $\mathcal{N}=2^*$ super-Yang-Mills theory on $S^4$ and use holography to calculate the universal contribution to the corresponding $S^4$ free energy at large $N$ and large 't Hooft coupling. Our result matches the expression previously computed using supersymmetric localization in the field theory. This match represents a non-trivial precision test of holography in a non-conformal, Euclidean signature setting.

Figures (3)

• Figure 1: Plots of the numerical solutions for $A(r)$, $\eta(r)$, and $\frac{1}{2}(z(r)\pm\tilde{z}(r))$ for $\eta_0=\{1.05,1.10,1.15,1.20\}$ (orange to black). The functions $z$ and $\tilde{z}$ are real in this case. Note that the scalar fields are plotted as a function of $A$ as defined in \ref{['MetricAnsatz']} and not as a function of the radial coordinate $r$.
• Figure 2: Plots of the numerical solutions for $A(r)$, $\eta(r)$, and $\frac{1}{2i}(z(r)\pm\tilde{z}(r))$ for $\eta_0=\{0.95, 0.90, 0.85, 0.80\}$ (orange to black). The functions $z$ and $\tilde{z}$ are pure imaginary in this case. Again the scalar fields are plotted as a function of $A$ as defined in \ref{['MetricAnsatz']}.
• Figure 3: $v(\mu)$ as a function of $\mu$ for both $\eta_0>1$ (left) and $\eta_0<1$ (right). The orange curve is obtained numerically, while the black curve is a plot of the analytical relation \ref{['vofmu']}. Note that for $\eta_0<1$ both $\mu$ and $v$ are pure imaginary.