Relation to the standard three-dimensional higher-spin construction

Determine how the ambient-oscillator factorisation at d=3 is related to the standard three-dimensional higher-spin construction, including the corresponding value of the parameter lambda where applicable.

Background

At d=3, the ambient oscillator construction yields a single member of the one-parameter family of three-dimensional higher-spin algebras rather than the full family. The paper explains that recovering the full family would require enlarging the generating-variable system, for example through deformed oscillators.

The precise relationship between the factorisation used in the paper and the standard three-dimensional construction is not identified, and the corresponding parameter value is likewise not determined.

References

How the family is reproduced in the ambient-oscillator language, and how the factorisation used here is related to the standard three-dimensional construction at the undeformed point, remains an open question.

— An explicit product for the on-shell higher-spin algebra in $AdS_d$-space  (2610.26704 - Povarnin, 19 Oct 2026) in Section 4.5, 'The case d=3'; Conclusion

The additional requirements for lifting the factorisation to $m_3$ remain to be established.

— An explicit product for the on-shell higher-spin algebra in $AdS_d$-space  (2610.26704 - Povarnin, 19 Oct 2026) in Section 5.5, 'Towards the ideal I'; Conclusion