Correction: Bychkov et al. The σ₋ Cohomology Analysis for Symmetric Higher-Spin Fields. Symmetry 2021, 13, 1498

There was an error in the original publication [1]. One set of fermionic cohomology has not been mentioned in the text of Section 8 Fermionic HS Fields in ${AdS}_4$ and the formulas for operators ${\sigma }_{+}$ and ${\Delta }_G^{\mathrm{fermionic}}$ are inaccurate. Therefore, we restore the missing cohomology, correct the inaccuracies in the formulas, and express our gratitude towards Yuri Tatarenko for noticing this error in the Acknowledgement section. There was also a typo in the formula in the first paragraph of Section 7.3.2.

The correct version of the formula in the first paragraph of Section 7.3.2 is ${\Phi }^{\lambda \left(2\right)\left|\nu \left(2\right)\right|\mu (n-2)|\dot{\mu }\left(n\right)}$$(y,\overline{y})=H^{\lambda \lambda }{y}^{\nu }{y}^{\nu }{y}^{\mu (n-2)}{\overline{y}}^{\dot{\mu }\left(n\right)}$

Section 8 and Acknowledgements have to be read as:

8. Fermionic HS Fields in ${\mathit{AdS}}_{\mathbf{4}}$

So far, we considered the bosonic case with even grading $G=|N-\overline{N}|$. By (160) odd G corresponds to fields of half-integer spins, i.e., oddness of G determines the field statistics.

To extend the results for $H^p\left({\sigma }_{-}\right)$ to fermionic fields, we first define the operator ${\sigma }_{-}$ on multispinors of odd ranks. In the fermionic case, the lowest possible odd grading is $G=|N-\overline{N}|=1$. This means that the previously unique lowest grading line on the $(N,\overline{N})$-plane splits into two separate lines $N-\overline{N}=\pm 1$. Therefore, the definition of ${\sigma }_{-}$ and its conjugated ${\sigma }_{+}$ depends on the lowest grading line. We define the action of ${\sigma }_{-}$ to vanish on the both lines. In all other gradings, ${\sigma }_{\pm }$ is defined analogously to the bosonic case. Namely,

$$\begin{array}{cc}\hfill {\sigma }_{-}\omega (y,\overline{y})& :=i{\overline{y}}^{\dot{\alpha }}{h}_{\dot{\alpha }}^{\alpha }{\partial }_{\alpha }\omega (y,\overline{y}),\mathrm{at}N\ge \overline{N}+3,\hfill \end{array}$$

(247a)

$$\begin{array}{cc}\hfill {\sigma }_{-}\omega (y,\overline{y})& :=i{y}^{\alpha }{h}_{\alpha }^{\dot{\alpha }}{\overline{\partial }}_{\dot{\alpha }}\omega (y,\overline{y}),\mathrm{at}N\le \overline{N}-3.\hfill \end{array}$$

(247b)

Analogously, the operator ${\sigma }_{+}$ is defined as

$$\begin{array}{cc}\hfill {\sigma }_{+}\omega (y,\overline{y})& :=-i{y}^{\alpha }{D}_{\alpha }^{\dot{\alpha }}{\overline{\partial }}_{\dot{\alpha }}\omega (y,\overline{y}),\mathrm{at}N\ge \overline{N}+3,\hfill \end{array}$$

(248a)

$$\begin{array}{cc}\hfill {\sigma }_{+}\omega (y,\overline{y})& :=-i{\overline{y}}^{\dot{\alpha }}{D}_{\dot{\alpha }}^{\alpha }{\partial }_{\alpha }\omega (y,\overline{y}),\mathrm{at}N\le \overline{N}-3.\hfill \end{array}$$

(248b)

For the lowest grading lines $N-\overline{N}=1$ and $N-\overline{N}=-1$, ${\sigma }_{+}$ is defined as in the sectors $N\ge \overline{N}+3$ and $N\le \overline{N}-3$, respectively.

Notice that the action of the fermionic Laplace operator is analogous to that of the bosonic one (169) with the grading shifted by one, ${\Delta }_G^{\mathrm{fermionic}}={\Delta }_{G-1}^{\mathrm{bosonic}}$, except for the lowest grading. The final result is

$$\begin{array}{cc}\hfill •{\Delta }_{N>\overline{N}+3}^{\mathrm{fermionic}}& ={\Delta }_{N>\overline{N}+2}^{\mathrm{bosonic}}=N(\overline{N}+2)+y^{\beta }{\partial }_{\alpha }{h}_{\dot{\gamma }}^{\alpha }{D}_{\beta }^{\dot{\gamma }}+{\overline{y}}^{\dot{\alpha }}{\overline{\partial }}_{\dot{\beta }}{h}_{\gamma \dot{\alpha }}{D}^{\gamma \dot{\beta }},\hfill \end{array}$$

(249a)

$$\begin{array}{cc}\hfill •{\Delta }_{N=\overline{N}+3}^{\mathrm{fermionic}}& ={\Delta }_{N=\overline{N}+2}^{\mathrm{bosonic}}={\Delta }_{N>\overline{N}+2}+{\overline{y}}^{\dot{\alpha }}{\overline{y}}^{\dot{\beta }}{\partial }_{\alpha }{\partial }_{\beta }{h}_{\dot{\beta }}^{\beta }{D}_{\dot{\alpha }}^{\alpha },\hfill \end{array}$$

(249b)

$$\begin{array}{cc}\hfill •{\Delta }_{N=\overline{N}+1}^{\mathrm{fermionic}}& ={\overline{y}}^{\dot{\alpha }}{\overline{\partial }}_{\dot{\beta }}{h}_{\gamma \dot{\alpha }}{D}^{\gamma \dot{\beta }}-{\overline{y}}^{\dot{\alpha }}{y}^{\beta }{\partial }_{\alpha }{\overline{\partial }}_{\dot{\beta }}{h}_{\dot{\alpha }}^{\alpha }{D}_{\beta }^{\dot{\beta }},\hfill \end{array}$$

(249c)

$$\begin{array}{cc}\hfill •{\Delta }_{N=\overline{N}-1}^{\mathrm{fermionic}}& =y^{\alpha }{\partial }_{\beta }{h}_{\alpha \dot{\gamma }}{D}^{\beta \dot{\gamma }}-y^{\alpha }{\overline{y}}^{\dot{\beta }}{\partial }_{\beta }{\overline{\partial }}_{\dot{\alpha }}{h}_{\alpha }^{\dot{\alpha }}{D}_{\dot{\beta }}^{\beta }.\hfill \end{array}$$

(249d)

This allows us do deduce the fermionic cohomology from the bosonic one arriving at the following final results.

8.1. Fermionic $H^0\left({\sigma }_{-}\right)$

The space $H^0\left({\sigma }_{-}\right)$ for fermionic HS fields is spanned by two independent zero-forms with $N-\overline{N}=\pm 1:$

$$H^0\left({\sigma }_{-}\right)=\left\{F(y,\overline{y}|x)+\overline{F}(y,\overline{y}|x)=F_{\alpha (n+1),\dot{\alpha }\left(n\right)}\left(x\right)y^{\alpha (n+1)}{\overline{y}}^{\dot{\alpha }\left(n\right)}+{\overline{F}}_{\alpha \left(n\right),\dot{\alpha }(n+1)}\left(x\right)y^{\alpha \left(n\right)}{\overline{y}}^{\dot{\alpha }(n+1)}\right\}.$$

(250)

Recall that, from Theorem 3.1, $H^0\left({\sigma }_{-}\right)$ represents the parameters of differential HS gauge transformations.

8.2. Fermionic $H^1\left({\sigma }_{-}\right)$

In the bosonic case, we had two physically different cocycles in $H^1$ (179a) corresponding to traceless $\varphi (y,\overline{y}|x)$ and trace ${\varphi }^{\mathrm{tr}}(y,\overline{y}|x)$ parts of the Fronsdal field. These belong to the diagonal $N=\overline{N}$.

For the fermionic case, the situation is almost analogous. The lowest grading is now $G=|N-\overline{N}|=1$. So, in this sector, there are four (not two) different 1-cocycles inherited from the bosonic case: $\psi$, ${\psi }^{\mathrm{tr}}$, $\overline{\psi }$, and ${\overline{\psi }}^{\mathrm{tr}}$ and two additional cocycles, ${\psi }^{ext}$, ${\overline{\psi }}^{ext}$, given by

$$\begin{array}{cc}\hfill \psi (y,\overline{y}|x)& ={\psi }_{\mu (n+2),\dot{\mu }(n+1)}\left(x\right)h^{\mu \dot{\mu }}{y}^{\mu (n+1)}{\overline{y}}^{\dot{\mu }\left(n\right)},\hfill \end{array}$$

(251a)

$$\begin{array}{cc}\hfill \overline{\psi }(y,\overline{y}|x)& ={\overline{\psi }}_{\mu (n+1),\dot{\mu }(n+2)}\left(x\right)h^{\mu \dot{\mu }}{y}^{\mu \left(n\right)}{\overline{y}}^{\dot{\mu }(n+1)},\hfill \end{array}$$

(251b)

$$\begin{array}{cc}\hfill {\psi }^{\mathrm{tr}}(y,\overline{y}|x)& ={\psi }_{\mu \left(n\right),\dot{\mu }(n-1)}^{\mathrm{tr}}\left(x\right)h_{\nu \dot{\nu }}{y}^{\nu }{y}^{\mu \left(n\right)}{\overline{y}}^{\dot{\nu }}{\overline{y}}^{\dot{\mu }(n-1)},\hfill \end{array}$$

(251c)

$$\begin{array}{cc}\hfill {\overline{\psi }}^{\mathrm{tr}}(y,\overline{y}|x)& ={\overline{\psi }}_{\mu (n-1),\dot{\mu }\left(n\right)}^{\mathrm{tr}}\left(x\right)h_{\nu \dot{\nu }}{y}^{\nu }{y}^{\mu (n-1)}{\overline{y}}^{\dot{\nu }}{\overline{y}}^{\dot{\mu }\left(n\right)},\hfill \end{array}$$

(251d)

$$\begin{array}{cc}\hfill {\psi }^{ext}(y,\overline{y}|x)& ={\psi }_{\mu \left(n\right),\dot{\mu }(n+1)}^{ext}\left(x\right){h_{\nu }}^{\dot{\mu }}{y}^{\nu }{y}^{\mu \left(n\right)}{\overline{y}}^{\dot{\mu }\left(n\right)},\hfill \end{array}$$

(251e)

$$\begin{array}{cc}\hfill {\overline{\psi }}^{ext}(y,\overline{y}|x)& ={\overline{\psi }}_{\mu (n+1),\dot{\mu }\left(n\right)}^{ext}\left(x\right){h^{\mu }}_{\dot{\nu }}{y}^{\mu \left(n\right)}{\overline{y}}^{\dot{\nu }}{\overline{y}}^{\dot{\mu }\left(n\right)}\hfill \end{array}$$

(251f)

with a non-negative integer n (positive for ${\psi }^{\mathrm{tr}}$ and ${\overline{\psi }}^{\mathrm{tr}}$). Cocycles $\psi$, ${\psi }^{\mathrm{tr}}$ and ${\psi }^{ext}$ belong to the upper near-diagonal line $N=\overline{N}+1$, whereas $\overline{\psi }$, ${\overline{\psi }}^{\mathrm{tr}}$ and ${\overline{\psi }}^{ext}$ belong to the lower near-diagonal line $N=\overline{N}-1$. All of them have a grading of $G=1$. $\psi$ and $\overline{\psi }$ are mutually conjugated.

These results can be put into the following concise form

$$\begin{array}{cc}\hfill \psi (y,\overline{y}|x)& =h^{\mu \dot{\mu }}{\partial }_{\mu }{\overline{\partial }}_{\dot{\mu }}{F}_1(y,\overline{y}|x),\hfill \end{array}$$

(252a)

$$\begin{array}{cc}\hfill {\psi }^{\mathrm{tr}}(y,\overline{y}|x)& =h_{\mu \dot{\mu }}{y}^{\mu }{\overline{y}}^{\dot{\mu }}{F}_2(y,\overline{y}|x),\hfill \end{array}$$

(252b)

$$\begin{array}{cc}\hfill {\psi }^{ext}(y,\overline{y}|x)& ={h_{\nu }}^{\dot{\mu }}{y}^{\nu }{\overline{\partial }}_{\dot{\mu }}{F}_3(y,\overline{y}|x),\hfill \end{array}$$

(252c)

where $F_{1,2}(y,\overline{y}|x)$ are of the homogeneity degree $N-\overline{N}=1$, i.e.,

$$\left(y^{\alpha }{\displaystyle \frac{\partial }{\partial y^{\alpha }}}-{\overline{y}}^{\dot{\alpha }}{\displaystyle \frac{\partial }{\partial {\overline{y}}^{\dot{\alpha }}}}\right)F_{1,2}(y,\overline{y}|x)=F_{1,2}(y,\overline{y}|x),$$

(253)

and $F_3(y,\overline{y}|x)$ is of the homogeneity degree $N-\overline{N}=-1$, i.e.,

$$\left(y^{\alpha }{\displaystyle \frac{\partial }{\partial y^{\alpha }}}-{\overline{y}}^{\dot{\alpha }}{\displaystyle \frac{\partial }{\partial {\overline{y}}^{\dot{\alpha }}}}\right)F_3(y,\overline{y}|x)=-F_3(y,\overline{y}|x).$$

(254)

For $\overline{\psi }$, ${\overline{\psi }}^{\mathrm{tr}}$ and ${\overline{\psi }}^{ext}$, the results are analogous except that ${\overline{F}}_{1,2}(y,\overline{y}|x)$ as degree $N-\overline{N}=-1$ and ${\overline{F}}_3(y,\overline{y}|x)$ has degree $N-\overline{N}=1$.

8.3. Fermionic $H^2\left({\sigma }_{-}\right)$

According to the same arguments, the fermionic $H^2$ is almost analogous to the bosonic one. Recall that the bosonic 2-cocycles are represented by three different two-forms: Weyl tensor, traceless and traceful parts of the generalized Einstein tensors (near diagonal, $G=3$).

The fermionic Weyl cohomology is given by the same formula as the bosonic one:

$$W^{\mathrm{ferm}}(y,\overline{y}|x)=H^{\mu \nu }{\partial }_{\mu }{\partial }_{\nu }C(y,0|x)+{\overline{H}}^{\dot{\mu }\dot{\nu }}{\overline{\partial }}_{\dot{\mu }}{\overline{\partial }}_{\dot{\nu }}C(0,\overline{y}|x),$$

(255)

where $C(y,0|x)$ and $C(0,\overline{y}|x)$ are polynomials of y and $\overline{y}$, respectively.

The two bosonic Fronsdal cocycles (246) were represented by the two zero-forms $C^{\mathrm{diag}}(y,\overline{y})$ with the support on the diagonal $N=\overline{N}$. In the fermionic case the two Fronsdal cocycles split into four. The bosonic diagonal polynomial $C^{\mathrm{diag}}(y,\overline{y})$ is replaced by a pair of near-diagonal $C^{\mathrm{near}\text{-}\mathrm{diag}}(y,\overline{y})$ and ${\overline{C}}^{\mathrm{near}\text{-}\mathrm{diag}}(y,\overline{y})$ satisfying the relations

$$\begin{array}{cc}\hfill \left(y^{\alpha }{\displaystyle \frac{\partial }{\partial y^{\alpha }}}-{\overline{y}}^{\dot{\alpha }}{\displaystyle \frac{\partial }{\partial {\overline{y}}^{\dot{\alpha }}}}\right)C^{\mathrm{near}\text{-}\mathrm{diag}}(y,\overline{y}|x)& =C^{\mathrm{near}\text{-}\mathrm{diag}}(y,\overline{y}|x),\hfill \end{array}$$

(256a)

$$\begin{array}{cc}\hfill \left(y^{\alpha }{\displaystyle \frac{\partial }{\partial y^{\alpha }}}-{\overline{y}}^{\dot{\alpha }}{\displaystyle \frac{\partial }{\partial {\overline{y}}^{\dot{\alpha }}}}\right){\overline{C}}^{\mathrm{near}\text{-}\mathrm{diag}}(y,\overline{y}|x)& =-{\overline{C}}^{\mathrm{near}\text{-}\mathrm{diag}}(y,\overline{y}|x)\hfill \end{array}$$

(256b)

These support the fermionic 2-cocycles associated with the $\mathit{l}.\mathit{h}.\mathit{s}.$’s of the fermionic field equations for spin $s\ge 3/2$ massless fields as follows

$$\begin{array}{cc}\hfill {\mathcal{E}}_{\mathrm{A}}^{\mathrm{ferm}}(y,\overline{y}|x)& =\left(H^{\mu \nu }{\partial }_{\mu }{\partial }_{\nu }+{\overline{H}}^{\dot{\mu }\dot{\nu }}{\overline{\partial }}_{\dot{\mu }}{\overline{\partial }}_{\dot{\nu }}\right)C^{\mathrm{near}\text{-}\mathrm{diag}}(y,\overline{y}|x),\hfill \end{array}$$

(257a)

$$\begin{array}{cc}\hfill {\overline{\mathcal{E}}}_{\mathrm{A}}^{\mathrm{ferm}}(y,\overline{y}|x)& =\left(H^{\mu \nu }{\partial }_{\mu }{\partial }_{\nu }+{\overline{H}}^{\dot{\mu }\dot{\nu }}{\overline{\partial }}_{\dot{\mu }}{\overline{\partial }}_{\dot{\nu }}\right){\overline{C}}^{\mathrm{near}\text{-}\mathrm{diag}}(y,\overline{y}|x),\hfill \end{array}$$

(257b)

$$\begin{array}{cc}\hfill {\mathcal{E}}_{\mathrm{B}}^{\mathrm{ferm}}(y,\overline{y}|x)& =\left(H^{\mu \nu }{y}_{\mu }{y}_{\nu }+{\overline{H}}^{\dot{\mu }\dot{\nu }}{\overline{y}}_{\dot{\mu }}{\overline{y}}_{\dot{\nu }}\right)C^{\mathrm{near}\text{-}\mathrm{diag}}(y,\overline{y}|x),\hfill \end{array}$$

(257c)

$$\begin{array}{cc}\hfill {\overline{\mathcal{E}}}_{\mathrm{B}}^{\mathrm{ferm}}(y,\overline{y}|x)& =\left(H^{\mu \nu }{y}_{\mu }{y}_{\nu }+{\overline{H}}^{\dot{\mu }\dot{\nu }}{\overline{y}}_{\dot{\mu }}{\overline{y}}_{\dot{\nu }}\right){\overline{C}}^{\mathrm{near}\text{-}\mathrm{diag}}(y,\overline{y}|x).\hfill \end{array}$$

(257d)

As in the case of 1-forms, additional cocycles appear in the cohomology

$$\begin{array}{cc}\hfill {\mathcal{E}}_{\mathrm{C}}^{\mathrm{ferm}}(y,\overline{y}|x)& =H^{\mu \nu }{y}_{\mu }{\partial }_{\nu }{C}^{\mathrm{near}\text{-}\mathrm{diag}}(y,\overline{y}|x),\hfill \end{array}$$

(258a)

$$\begin{array}{cc}\hfill {\overline{\mathcal{E}}}_{\mathrm{C}}^{\mathrm{ferm}}(y,\overline{y}|x)& ={\overline{H}}^{\dot{\mu }\dot{\nu }}{\overline{y}}_{\dot{\mu }}{\overline{\partial }}_{\dot{\nu }}{\overline{C}}^{\mathrm{near}\text{-}\mathrm{diag}}(y,\overline{y}|x).\hfill \end{array}$$

(258b)