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

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)