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

TL;DR

This paper computes the operator product expansions and (anti)commutators of ${ m N}=4$ higher spin currents in a coset model, revealing their algebraic structure and relation to known ${ m W}_{ m infinity}$ algebras, with implications for higher spin theories.

Contribution

It provides a detailed derivation of the ${ m N}=4$ higher spin algebra and structure constants, extending previous ${ m W}_{ m infinity}$ results to the ${ m N}=4$ case with explicit oscillator realizations.

Findings

Derived (anti)commutators for generic spins with $SO(4)$ symmetry.

Expressed structure constants in terms of ${ m N}=2$ ${ m W}_{ m infinity}$ algebra constants.

Connected coset model results to ${ m AdS}_3$ Vasiliev higher spin theory.

Abstract

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. We also describe the ${\cal N}=4$ higher spin generators, by using the above coset construction results, for general super spin $s$ in terms of oscillators in the matrix generalization of $AdS_3$ Vasiliev higher spin theory at nonzero 't Hooft-like coupling constant. We obtain the ${\cal N}=4$ 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 $s_2$.