Purely bosonic vertices were constructed earlier in [4, 5, 6, 7].

Here we use the term “HS gravity” for any consistent full nonlinear theory that includes (some modification of) Einstein gravity with at least one massless field of spin $s>2$ in its spectrum.

See, [20, 21, 22, 23, 24, 25, 26, 27, 28, 29] for some key references on the Fronsdal program in dimensions $d\ge 4$. For completeness, we refer the reader to related parallel programs in the light-cone approach [30, 31, 32, 33, 34] and the Fradkin-Vasiliev (framelike) approach [35, 36, 37, 38]. See also [39, 40, 41, 42, 43] for recent reviews.

See, however, [62] and references therein.

Note that in terms of the generalized Kronecker delta ${\delta }_{{\beta }_1\dots {\beta }_n}^{{\alpha }_1\dots {\alpha }_n}=n!{\delta }_{[{\beta }_1}^{{\alpha }_1}\dots {\delta }_{{\beta }_n]}^{{\alpha }_n}$ this means ${\epsilon }_{\mu \nu \rho }={\delta }_{\mu \nu \rho }^{012}$ and ${\epsilon }^{\mu \nu \rho }=-{\delta }_{012}^{\mu \nu \rho }$.

In general, sending some of the coupling constants to zero might not be possible in the full nonlinear theory, where the independent coupling constants may be related for the consistency of the full theory. However, for the classification of all nontrivial deformations of free action, we do not require the existence of a full nonlinear theory that uses the vertices thus found.

With a small abuse of the terminology, we use the term spin $s$ for the massless Fronsdal fields in three dimensions, described by symmetric tensors of rank $s$, and correspondingly, spin $s+1/2$, for Fang-Fronsdal fields, described by symmetric tensor-spinors of rank $s$.

We use round brackets for symmetrization with weight one [e.g., $A_{(\mu \nu )}=\frac{1}{2}(A_{\mu \nu }+A_{\nu \mu })$] and square brackets for antisymmetrization with weight one [e.g., $A_{[\mu \nu ]}=\frac{1}{2}(A_{\mu \nu }-A_{\nu \mu })$].

This is a complication that arises from the presence of $\gamma$ matrices due to the extra relations involving them, such as $\gamma$ tracelessness of fermions. For example, $\overline{\mathrm{\Psi }}{\gamma }^{\mu }{\mathrm{\Psi }}_{\mu }\approx 0$, but $\overline{\mathrm{\Psi }}{\gamma }^{\mu }{\gamma }^{\rho }{\mathrm{\Psi }}_{\mu }\cancel{\approx }0$. Therefore, not all identities remain valid when one adds $\gamma$ matrices in the vertex operator. An example illustrating this is $W={\epsilon }_{\mu \nu \rho }{A}_3^{\mu }{P}_3^{\nu }{P}_1^{\rho }\approx 0$ [following from (19)], but $\mathrm{\Gamma }W\cancel{\approx }0$.

Note that the holomorphic factorization is specific to massless fields, corresponding to a trivial representation of one of the simple components of the isometry (see, e.g., [94]).