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

Abstract

By computing the operator product expansions between the first two $\mathcal{N}=4$ 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 $\mathcal{N}=2$ ${\mathcal{W}}_{\infty }$ algebra found by Bergshoeff, Pope, Romans, Sezgin and Shen previously, in addition to the spin-dependent fractional coefficients and two $SO(4)$ invariant tensors. We also describe the $\mathcal{N}=4$ higher spin generators, by using the above coset construction results, for general super spin $s$ in terms of oscillators in the matrix generalization of $Ad{S}_3$ Vasiliev higher spin theory at nonzero 't Hooft-like coupling constant. We obtain the $\mathcal{N}=4$ higher spin algebra for low spins and present how to determine the structure constants, which depend on the higher spin algebra parameter, in general, for fixed spins $s_1$ and $s_2$.

arXiv comment99 pages; Minor corrections in 3,4,5,7,8,9,10,19,22,42,and 43 pages added

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Cited by in corpus (5)

• Toward A Supersymmetric $w_{1+\infty }$ Symmetry in the Celestial Conformal Field Theory
• The $\mathcal{N}=4$ Higher Spin Algebra for Generic $\mu$ Parameter
• The Structure of the $\mathcal{N}=4$ Supersymmetric Linear $W_{\infty }[\lambda ]$ Algebra
• Adding Complex Fermions to the Grassmannian-like Coset Model
• The Grassmannian-like Coset Model and the Higher Spin Currents