Wolf Space Coset Spectrum in the Large ${\cal N}=4$ Holography

After reviewing the four eigenvalues (the conformal dimension, two $SU(2)$ quantum number, and $U(1)$ charge) in the minimal (and higher) representations in the Wolf space coset where the ${\cal N}=4$ superconformal algebra is realized by $11$ currents in nonlinear way, these four eigenvalues in the higher representations up to three boxes (of Young tableaux) are examined in detail. The eigenvalues associated with the higher spin-$1,2,3$ currents in the (minimal and) higher representations up to two boxes are studied. They are expressed in terms of the two finite parameters $(N, k)$ where the Wolf space coset contains the group $SU(N+2)$ and the affine Kac-Moody spin $1$ current has the level $k$. Under the large $(N,k)$ 't Hooft-like limit, they are simply linear combinations of the eigenvalues in the minimal representations. For the linear case where the ${\cal N}=4$ superconformal algebra is realized by $16$ currents in linear way, the eigenvalues, corresponding to the spin $2$ current and the higher spin $3$ current, which are the only different quantities (compared to the nonlinear case), are also obtained at finite $(N,k)$. They coincide with the results for the nonlinear case above after the large $(N,k)$ 't Hooft-like limit is taken. As a by product, the three-point functions of the higher spin currents with two scalar operators can be determined at finite $(N,k)$.