5D super Yang-Mills theory and the correspondence to AdS$_7$/CFT$_6$

TL;DR

This work analyzes the relation between 5D $\mathcal{N}=1$ Yang-Mills with matter on $S^5$ and the holographic description of the 6D $(2,0)$ theory by applying localization to obtain a matrix model for the $\mathcal{N}=1^*$ theory, and then studying its large-$N$ behavior. The authors derive the strong-coupling free energy $F=-\frac{g_{\mathrm{YM}}^2 N^3}{96\pi r}\left(\frac{9}{4}+m^2\right)^2$ and the strong-coupling Wilson loop $\langle W\rangle\sim\exp\left(\frac{9}{4}+m^2\right)\frac{\lambda}{8\pi}\right)$, and connect these to AdS$_7$/CFT$_6$ via a specific identification $R_6=\frac{g_{\mathrm{YM}}^2}{16\pi^2}\left(\frac{9}{4}+m^2\right)$ (with extensions to quivers). A Euclidean rotation of the adjoint mass is required for localization, and numerical results corroborate the analytic large-$N$ predictions, though a full Euclidean $(2,0)$ formulation remains an open issue. Overall, the paper provides a controlled 5D–6D holographic link, clarifies localization requirements, and highlights $N^3$ scaling and mass-dependent matching with supergravity predictions. The findings offer a concrete arena to test the $(2,0)$ holographic dictionary and motivate further work on Euclidean 6D theories and UV completions.

Abstract

We study the relation between 5D super Yang-Mills theory and the holographic description of 6D (2,0) superconformal theory. We start by clarifying some issues related to the localization of N=1 SYM with matter on $S^5$. We concentrate on the case of a single adjoint hypermultiplet with a mass term and argue that the theory has a symmetry enlargement at mass M=1/(2r), where r is the $S^5$ radius. However, in order to have a well-defined localization locus it is necessary to rotate M onto the imaginary axis, breaking the enlarged symmetry. Based on our prescription, the imaginary mass values are physical and we show how the localized path integral is consistent with earlier results for 5D SYM in flat space. We then compute the free energy and the expectation value for a circular Wilson loop in the large N limit. The Wilson loop calculation shows a mass dependent constant rescaling between weak and strong coupling. The Wilson loop continued back to to the enlarged symmetry point is consistent with a supergravity computation for an M2 brane using the standard identification of the compactification radius and the 5D coupling. If we continue back to the physical regime and use this value of the mass to determine the compactification radius, then we find agreement between the SYM free energy and the corresponding supergravity calculation. We also verify numerically some of our analytic approximations.

Figures (4)

• Figure 1: Density of eigenvalues $\rho(\phi)$ for $m=0$, $N=200$, $\beta\equiv\frac{g_{YM}^2}{r}=2$(left) and $m=\frac{1}{2}$, $N=160$, $\beta\equiv\frac{g_{YM}^2}{r}=2$(right). The dashed blue lines are the analytical strong coupling solutions.
• Figure 2: Density of states for pure $\mathcal{N}=1$ SYM with $\beta=2$.
• Figure 3: (left)$N$-dependence of free energies for different values of hypermultiplet mass $m$: $m=0$orange, $m=\frac{1}{2}$purple, $m=1$blue; (right) Free energy $N$-dependence for pure $\mathcal{N}=1$ SYM. ($\beta=2$ for all plots.)
• Figure 4: $m$-dependence of $N^3$ coefficient in the matrix model free energy.