Holography for $\mathcal{N}=1^*$ on $S^4$

Abstract

We construct the five-dimensional supergravity dual of the $\mathcal{N}=1^*$ mass deformation of the $\mathcal{N} =4$ supersymmetric Yang-Mills theory on $S^4$ and use it to calculate the universal contribution to the corresponding $S^4$ free energy at large 't Hooft coupling in the planar limit. The holographic RG flow solutions are smooth and preserve four supercharges. As a novel feature compared to the holographic duals of $\mathcal{N} = 1^*$ on $\mathbb{R}^4$, in our backgrounds the five-dimensional dilaton has a non-trivial profile, and the gaugino condensate is fixed in terms of the mass-deformation parameters. Important aspects of the analysis involve characterizing the ambiguities in the partition function of non-conformal $\mathcal{N}=1$ supersymmetric theories on $S^4$ as well as the action of S-duality on the $\mathcal{N}=1^*$ theory.

Figures (4)

• Figure 1: Plots showing the universal part of the free energy $d^3F/d\mu^3$ as a function of the dimensionless mass $\mu = \pm {\text{i}} ma$ for the one-mass model. The plot on the left (right) shows the results for real (purely imaginary) values of $\mu$. Orange curves are the interpolation function, the black points indicate the data points the interpolation function is based on.
• Figure 2: The holographic variables $w$ encodes the gaugino condensate $\langle \lambda \lambda \rangle \sim w(\mu) N^2$ with $\mu = \pm {\text{i}} ma$. We find numerically that $w(\mu) = 2\mu^3$; the plots show the results of $w$ vs. $\mu$ for $\mu$ real and purely imaginary. Orange curves are the interpolation function, the black points indicate the data points the interpolation function is based on. The orange curves are indistinguishable from $w(\mu) = 2 \mu^3$ within our precision.
• Figure 3: Plots showing the universal part of the free energy $d^3F/d\mu^3$ as a function of the dimensionless mass $\mu = \pm {\text{i}} ma$ for the equal-mass model. For $\mu$ real we find solutions with any value, and $d^3F/d\mu^3 \rightarrow \mp 3N^2$ for large $\mu$. When $\mu$ is purely imaginary, we only find solutions with $-2.318 \lesssim {\text{i}}\mu \lesssim 2.318$; as $\mu$ approaches this values, the interpolation function becomes increasingly noisy and does not appear to be reliably determined beyond $|\mu| \gtrsim 2.1$, hence we restrict the plot to this range. Orange curves are the interpolation function, the black points indicate the data points the interpolation function is based on.
• Figure 4: Scatter plots showing the regions in which we find solutions with smooth IR boundary conditions. On the left, we take real values for $a_0$ and $b_0$. The bounds from (\ref{['IRbnds']}) are the red and blue curves. Note that for real $a_0$ and $b_0$, there is another bound in play too: the region of smooth solutions is bounded by curves that correspond to $s=\pm1$; we do not know an analytic from of these curves in the $b_0,a_0$-plane. The diagonal (purple) are all trivial Euclidean AdS solutions with $\mu=0$. In the plot on the left, the region with $-1< b_0,a_0<-1$ has $-1<s<1$, while the region with $b_0,a_0>1$ has $s<-1$; there is an equivalent region with $b_0,a_0<-1$ in which $s>1$. By the symmetry $(z_i,\tilde{z}_i) \rightarrow (1/z_i,1/\tilde{z}_i)$, the solutions with $-1<a_0,b_0<1$ are equivalent to those with $a_0,b_0 > 1$ or $<-1$.