The ${\cal N}=4$ Coset Model and the Higher Spin Algebra

Understand

By computing the operator product expansions between the first two ${\cal N}=4$ higher spin multiplets in the unitary coset model, the (anti)commutators of higher spin currents are obtained under the large $(N,k)$ 't Hooft-like limit.

• The free field realization with complex bosons and fermions is presented.
• The (anti)commutators for generic spins $s_1$ and $s_2$ with manifest $SO(4)$ symmetry at vanishing 't Hooft-like coupling constant are completely determined.
• The structure constants can be written in terms of the ones in the ${\cal N}=2$ ${\cal W}_{\infty}$ algebra found by Bergshoeff, Pope, Romans, Sezgin and Shen previously, in addition to the spin-dependent fractional coefficients and two $SO(4)$ invariant tensors.