Finite $N$ holographic conjecture for local operators of six-dimensional superconformal theories

Let $N$ be a positive integer, and let $Obs_N(0)$ denote the local operators of the theory on a stack of $N$ fivebranes wrapping $C^3$ in $R\times C^5$. Let $\mathcal G_N$ be the local $L_\infty$ algebra on $C^3$ obtained from the holographic construction, and let $\mathcal G_{N,c}$ denote its central version. The exceptional super Lie algebra $E(3|6)$ acts on these spaces. Finite $N$ holographic conjecture. There is an equivalence of $E(3|6)$-representations

$$\Obs_N(0)\simeq \clie_\bu(\mathcal G_{N,c})(0).$$

Similarly, for $N\geq 2$, let $\widetilde{\Obs}_N(0)$ denote the local operators with the center-of-mass degrees of freedom removed, and let $\widetilde{\mathcal G}_{N,c}$ be the corresponding central local $L_\infty$ algebra. Then there is an equivalence of $E(3|6)$-representations

$$\widetilde{\Obs}_N(0)\simeq \clie_\bu(\widetilde{\mathcal G}_{N,c})(0).$$

This conjecture identifies the local operators of the six-dimensional theory on $N$ fivebranes with the local operators computed holographically by Lie algebra cohomology. It is motivated by the expected enhancement of the holomorphically twisted superconformal algebra to the exceptional infinite-dimensional super Lie algebra $E(3|6)$ and by the holographic description of the factorization algebra of observables; the claim is presented as an expectation and no resolution is supplied here.