Towards All-Order Locality in the Higher-Spin Holomorphic Zero-Form Sector

Abstract: This work addresses the construction of local interaction vertices in the holomorphic zero-form sector obtained from Vasiliev equations. We focus on a requirement on zero-form vertices, previously proposed in a less restrictive form in arXiv:2208.02004, which, together with spinor spin-locality, guarantees space-time spin-locality. Necessary and sufficient conditions on the holomorphic one-form sector vertices are derived that ensure the existence of holomorphic zero-form vertices satisfying this requirement. As a concrete demonstration, the proposed scheme is applied to compute the spin-local holomorphic $η<sup>2$ zero-form sector vertex, a result that goes beyond previous work where only spinor spin-locality was established. These findings lay the groundwork for a systematic all-order construction of local vertices in higher-spin theory.