FIAN/TD/10-2021

The $\sigma_{-}$ Cohomology Analysis for Symmetric Higher-Spin Fields

A.S. Bychkov1,2, K.A. Ushakov1,2 and M.A. Vasiliev1,2

1 I.E. Tamm Department of Theoretical Physics, Lebedev Physical Institute,
Leninsky prospect 53, 119991, Moscow, Russia

2 Moscow Institute of Physics and Technology,
Institutsky lane 9, 141700, Dolgoprudny, Moscow region, Russia

###### Abstract

In this paper, we present a complete proof of the so-called First On-Shell
Theorem that determines dynamical content of the unfolded equations for free
symmetric massless fields of arbitrary integer spin in any dimension and
arbitrary integer or half-integer spin in four dimensions. This is achieved by
calculation of the respective $\sigma_{-}$ cohomology both in the tensor
language in Minkowski space of any dimension and in terms of spinors in
$AdS_{4}$. In the $d$-dimensional case $H^{p}(\sigma_{-})$ is computed for any
$p$ and interpretation of $H^{p}(\sigma_{-})$ is given both for the original
Fronsdal system and for the associated systems of higher form fields.

###### Contents

  1. 1 Introduction
  2. 2 Fronsdal theory
    1. 2.1 Metric formulation
    2. 2.2 Frame-like formulation
      1. 2.2.1 Tensor formalism
      2. 2.2.2 Spinor language in $AdS_{4}$
  3. 3 The idea of $\sigma_{-}$ cohomology analysis: example of integer spin massless fields
  4. 4 A method for calculating cohomology
  5. 5 $\sigma_{-}$ cohomology in Minkowski space of any dimension
    1. 5.1 Generating functions
    2. 5.2 $\mathrm{GL}(d)$ example
      1. 5.2.1 $H^{0}(\sigma_{-})$
      2. 5.2.2 $H^{p}(\sigma_{-})$, $p>0$
    3. 5.3 $\mathrm{O}(d)$ case
      1. 5.3.1 Irreducibility conditions
      2. 5.3.2 $\sigma_{+}$
      3. 5.3.3 $H^{0}(\sigma_{-})$
      4. 5.3.4 $H^{p}(\sigma_{-})$, $p>0$
      5. 5.3.5 Summary
  6. 6 $\sigma_{-}$ cohomology in $AdS_{4}$ in the spinor language
  7. 7 Bosonic case in $AdS_{4}$
    1. 7.1  $H^{0}(\sigma_{-})$
    2. 7.2  $H^{1}(\sigma_{-})$
      1. 7.2.1 $H^{1}(\sigma_{-})$ in the diagonal sector $N=\overline{N}$
      2. 7.2.2 $H^{1}(\sigma_{-})$ in the far-from-diagonal sector $|N-\overline{N}|>2$
      3. 7.2.3 Subtlety in the near-diagonal sector $|N-\overline{N}|=2$
      4. 7.2.4 $H^{1}(\sigma_{-})$ in the near-diagonal sector $|N-\overline{N}|=2$
    3. 7.3 $H^{2}(\sigma_{-})$
      1. 7.3.1 $H^{2}(\sigma_{-})$ in the far-from-diagonal sector $|N-\overline{N}|>2$
      2. 7.3.2 $H^{2}(\sigma_{-})$ on the diagonal $N=\overline{N}$
      3. 7.3.3 $H^{2}(\sigma_{-})$ in the near-diagonal sector $|N-\overline{N}|=2$
    4. 7.4 Summary for bosonic $H^{0,1,2}(\sigma_{-})$
  8. 8 Fermionic HS fields in $AdS_{4}$
    1. 8.1 Fermionic $H^{0}(\sigma_{-})$
    2. 8.2 Fermionic $H^{1}(\sigma_{-})$
    3. 8.3 Fermionic $H^{2}(\sigma_{-})$
  9. 9 Conclusion

##  1 Introduction

Higher-spin (HS) gauge theory is based on works of Fronsdal [1] and Fang and
Fronsdal [2], where the action and equations of motion for massless gauge
fields of any spin were originally obtained in flat four-dimensional Minkowski
space. Even earlier, important restrictions on low-energy HS vertices were
obtained by Weinberg in [3, 4] and so-called no-go theorems restricting
$S$-matrix possessing too high symmetries in flat space-time were proven in
[5, 6]. (For a review see [7].) The no-go theorems implied the existence of
the $s=2$ barrier suggesting that the construction of an interacting local HS
theory in Minkowski space-time is impossible. The proof of these theorems
essentially uses the specific form of the algebra of isometries of Minkowski
space. The $s=2$ barrier in flat space can be overcome in the space-time with
non-zero sectional curvature, for example, in the anti-de Sitter space [8]. In
these spaces it becomes possible to formulate a consistent nonlinear theory of
fields of all spins [9, 10].

The construction of a nonlinear HS theory is essentially based on the so-
called unfolded approach [11, 12], which is a far-going generalization of the
Cartan formulation of gravity ($s=2$) in terms of differential forms to fields
of any spin $s>2$. Via introducing appropriate auxiliary variables, the
unfolding procedure allows one to replace the system of partial differential
equations of any order on a smooth manifold by a larger system of first-order
equations on vector-valued differential forms. One of the essential features
of this approach, which is very useful for analysing symmetries of a given
system, is that the variables in the equations are valued in one or another
representation of the underlying symmetry algebra.

The dynamical content of the HS theory can be reconstructed from its unfolded
formulation using the $\sigma_{-}$ cohomology technique [13]. As is recalled
below, the dynamical data of the theory are in one-to-one correspondence with
the cohomology of certain linear nilpotent operator $\sigma_{-}$ that can be
read of the unfolded equations in question. The statement that unfolded
equations of free HS fields are equivalent to the Fronsdal equations was made
in the original papers in the spinor [14] and tensor [15] formalisms. In the
tensor formulation of HS theory the idea of the proof was illustrated in [16],
where however the analysis of the trace part of the Fronsdal equations was not
completed, while general arguments for mixed-symmetry HS fields were given in
[17]. In [18] the unfolded equations for massless fields were derived from the
Fronsdal theory by the BRST methods. To the best of our knowledge, no detailed
analysis of the problem in the spinor formalism was available in the
literature.

In this paper we present a complete proof of the so called First On-Shell
Theorem by computation of the cohomology rings of $\sigma_{-}$ for the
physically important cases of the integer-spin symmetric fields both in flat
space-time of any dimension and $AdS_{d}$ as well as for the fields of any
integer and half-integer spin in ${AdS}_{4}$. The computation technique
analogous to the Hodge theory for differential forms is performed in terms of
so-called $\sigma_{-}$ cohomology and provides a complete analysis of the
dynamical content of the free unfolded equations for symmetric massless fields
of any spin. Giving a direct proof of the equivalence between the Fronsdal
formulation of the HS gauge theory and its unfolded formulation this paper
fills in some gaps in the literature also illustrating a general approach
applicable to a broad class of unfolded systems. In addition, in the tensor
case we compute higher $\sigma_{-}$ cohomology groups and interpret them in
terms of higher Bianchi identities and more general dynamical systems. In
particular, we discuss the matching between the Bianchi identities in terms of
one-form gauge fields and zero-form field strengths.

The rest of the paper is organized as follows. In Section 2 we briefly recall
different approaches to the description of HS massless fields. Main idea of
the $\sigma_{-}$ cohomology approach is explained in Section 3. Cohomology
calculation method used in this paper is discussed in Section 4. Section 5
contains derivation of $H^{p}(\sigma_{-})$ in Minkowski space of any
dimension. In particular, the cases of $\mathrm{GL}(d)$ and $\mathrm{O}(d)$
representations are analysed here. In Section 6 calculation of the low-order
cohomology groups in $AdS_{4}$ is performed. Obtained results are discussed in
Section 7. Index conventions and normalisations of the tensor Young diagrams
are presented in the Appendix A.

##  2 Fronsdal theory

###  2.1 Metric formulation

According to Fronsdal [1], a spin-$s$ massless symmetric field can be
described in terms of two symmetric traceless tensors (for index conventions
see Appendix A)

$\phi^{a(s)}\equiv\phi^{a_{1}..a_{s}},\quad\phi^{a(s-2)}\equiv\phi^{a_{1}..a_{s-2}},\quad\eta_{b_{1}b_{2}}\phi^{b_{1}b_{2}a_{3}..a_{s}}\equiv\phi^{a(s-2)b}{}_{b}=0,\quad\phi^{a(s-4)b}{}_{b}=0\,.$
(2.1)

These two fields can be combined into a single rank-$s$ totally symmetric
tensor

$\varphi^{a(s)}=\phi^{a(s)}+\eta^{aa}\phi^{a(s-2)}$ (2.2)

obeying the double-tracelessness condition

$\varphi^{a(s-4)bc}{}_{bc}=0\,.$ (2.3)

The field equations in Fronsdal theory have the form

$R^{a(s)}(\varphi)=\square\varphi^{a(s)}-s\partial^{a}\partial_{k}\varphi^{ka(s-1)}+\frac{s(s-1)}{2}\partial^{a}\partial^{a}\varphi^{a(s-2)k}{}_{k}=0\,,$
(2.4)

where $\partial_{a}=\frac{\partial}{\partial x^{a}}$.

The tensor $R^{a(s)}(\varphi)$ is invariant under the gauge transformations
with a rank-$(s-1)$ traceless gauge parameter $\varepsilon(x)$

$\delta\varphi^{a(s)}=\partial^{a}\varepsilon^{a(s-1)},\qquad\varepsilon^{a(s-3)k}{}_{k}=0\,.$
(2.5)

Fronsdal equation (2.4) is a generalization of the well-known equations of
fields with spins $s=0,1,2$. For the case of $s=1$ the last term in the
Fronsdal tensor disappears and Eq.(2.4) reproduces Maxwell equations for the
field $A^{a}$. Without the last two terms at $s=0$ it gives Klein-Gordon
equation for a massless scalar field. The case of $s=2$ reproduces the
equations of linearized gravity [19]. Gauge transformation (2.5) gives the
known gauge transformations of low-spin fields and its absence for a scalar
field.

###  2.2 Frame-like formulation

####  2.2.1 Tensor formalism

The unified description of massless fields of arbitrary spin can be given in
the so-called frame-like formalism that generalizes Cartan formulation of
gravity, operating in terms of differential forms [14, 15, 20]. Frame-like
formulation of the HS gauge theory in any dimension is given in terms of the
one-form fields $\omega^{a(s-1),b(t)}=dx^{\nu}\omega_{\nu}^{a(s-1),b(t)}$
valued in two-row Young diagrams corresponding to irreducible
$\mathfrak{o}(d-1,1)$ (i.e., traceless) modules [15], obeying conditions

$\displaystyle\omega^{a(s-1),ab(t-1)}=0,$ (2.6)
$\displaystyle\omega^{a(s-3)k}{}_{k}{}^{,b(t)}=0\,.$ (2.7)

(For index conventions see Appendix A.)

By introducing auxiliary fields it is possible to put a system of partial
differential equations into the first-order unfolded form [11, 12]. Generally,
unfolded equations read as

$dW^{A}=\sum_{n=1}^{\infty}G^{A}_{B_{1},..,B_{n}}W^{B_{1}}\wedge...\wedge
W^{B_{n}}\,,\qquad d:=dx^{\nu}\partial_{\nu}\,.$ (2.8)

Here $W^{A}$ is a set of differential forms over some manifold. (Indices are
treated formally and can take an infinite number of values.) The coefficients
$G^{A}_{B_{1},..,B_{n}}$ satisfy the (anti)symmetry condition

$G^{A}_{B_{1},..,B_{i},..,B_{j},..,B_{n}}=(-1)^{|B_{i}||B_{j}|}G^{A}_{B_{1},..,B_{j},..,B_{i},..,B_{n}}\,,$
(2.9)

where $|B_{i}|$ denotes the form-degree of $W^{B_{i}}$. Also
$G^{A}_{B_{1},..,B_{n}}$ are restricted by the integrability conditions
expressing that $d^{2}=0$.

In the tensor language the unfolded HS equations in Minkowski space proposed
in [15] read as

$D_{L}\omega^{a(s-1),b(t)}+h_{m}\wedge\omega^{a(s-1),b(t)m}=0,\quad
t\in\\{0,..,s-2\\},$ (2.10) $D_{L}\omega^{a(s-1),b(s-1)}=h_{n}\wedge
h_{m}\wedge C^{a(s-1)n,b(s-1)m},$ (2.11)

where $h_{n}$ is a soldering form (vielbein, frame field, tetrad) and
$D_{L}=d+\varpi$ is the background Lorentz covariant derivative that satisfies
relations

$D_{L}h^{a}=0,\qquad D_{L}^{2}=0\,.$ (2.12)

In the Cartesian coordinate system with $\varpi=0$ the equations simplify to

$d\omega^{a(s-1),b(t)}+h_{m}\wedge\omega^{a(s-1),b(t)m}=0,\quad
t\in\\{0,..,s-2\\},$ (2.13) $d\omega^{a(s-1),b(s-1)}=h_{n}\wedge h_{m}\wedge
C^{a(s-1)n,b(s-1)m}\,,$ (2.14)

where $C$ satisfies the Lorentz irreducibility conditions

$C^{a(n),ab(m-1)}=0\,,\qquad C^{a(n-2)k}{}_{k}{}^{,b(m)}=0\,.$ (2.15)

The traceless tensor $C$ on the r.h.s. of (2.14) is a generalized Weyl tensor.
There are also unfolded equations on $C$ and on additional auxiliary fields
[10] (for reviews see [16, 21]). This system constitutes an infinite chain of
zero-form equations. Zero-form sector, that contains equations on spin-zero
and spin-one fields, will not be considered in this paper.

Equations (2.13) are invariant under the gauge transformations

$\delta\omega^{a(s-1),b(t)}=d\varepsilon^{a(s-1),b(t)}+h_{m}\varepsilon^{a(s-1),b(t)m},\quad
t\in\\{0,..,s-2\\}$ (2.16)

and eq.(2.14) is invariant under

$\displaystyle\delta\omega^{a(s-1),b(s-1)}=d\varepsilon^{a(s-1),b(s-1)},$
(2.17) $\displaystyle\delta C^{a(s),b(s)}=0,$ (2.18)

where $\varepsilon$ are zero-forms valued in the appropriate two-row
irreducible $\mathfrak{o}(d-1,1)$-modules obeying conditions analogous to
(2.15).

The Fronsdal field is embedded into the frame-like one-form
$e^{a(s-1)}\equiv\omega^{a(s-1)}$ in the following manner. Converting the form
index into the fiber one using vielbein $h$,

$e^{a(s-1)|b}=e^{a(s-1)}_{\mu}h^{\mu b}\,,$ (2.19)

the resulting tensor (2.19) can be decomposed into irreducible
$\mathfrak{o}(d-1,1)$-modules. In terms of traceless Young diagrams this
decomposition is

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\end{picture}\,\,\,\otimes_{so}\,\,\,\begin{picture}(65.0,18.0)(10.0,7.0){\put(25.0,13.0){\scriptsize${s-1}$}
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\end{picture}\,\ \cong\,\,\,\
\begin{picture}(65.0,18.0)(10.0,7.0){\put(30.0,13.0){\scriptsize${s}$}
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\end{picture}\,\ \oplus\,\,\,\
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\end{picture}\,\,\,\ \oplus\,\,\,\ \begin{picture}(10.0,18.0)(0.0,7.0){
\put(5.0,0.0){\line(1,0){10.0}}\put(5.0,10.0){\line(1,0){10.0}}\put(5.0,0.0){\line(0,1){10.0}}\put(15.0,0.0){\line(0,1){10.0}}\par}
\end{picture}\begin{picture}(65.0,18.0)(10.0,7.0){\put(25.0,13.0){\scriptsize${s-1}$}
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\end{picture}\,.$ (2.20)

The first two components give the Fronsdal field, while the third one is an
excess of the components of the frame-like field in comparison with the
Fronsdal field. At the tensor level, this decomposition is represented as:

$e^{a(s-1)|b}=\psi_{1}^{a(s-1)b}+\beta_{1}\eta^{aa}\psi_{2}^{a(s-3)b}+\beta_{2}\eta^{ab}\psi_{2}^{a(s-2)}+\psi_{3}^{a(s-1),b},$
(2.21)

where $\psi_{i}$ are traceless and correspond to the $i$-th diagram. The
relative coefficient $\frac{\beta_{2}}{\beta_{1}}$ is fixed by the
tracelessness condition with respect to indices $a$.

This decomposition shows that the Fronsdal field identifies with the symmetric
part of the frame-like field, since the contribution of the third diagram
disappears upon symmetrization. The resulting field

$\varphi^{a(s)}:=e^{a(s-1)|a}\,$ (2.22)

is symmetric and double-traceless. The extra term $\psi_{3}^{a(s-1),b}$ is
pure gauge. Its contribution can be canceled by the gauge transformation
$\delta e^{a(s-1)|b}=\varepsilon^{a(s-1)|b}$ with suitable gauge parameter.
For detailed discussion of Fronsdal field embedding see [16, 20, 21].

It is not difficult to check [15, 20] (for reviews see [16, 21]) that the
Fronsdal equations and gauge transformations follow from the unfolded system
(2.13), (2.14). A more complicated question is whether the Fronsdal fields and
equations are the only ones that result from (2.13), (2.14). The answer can be
obtained via the $\sigma_{-}$ cohomology technique [13].

####  2.2.2 Spinor language in $AdS_{4}$

The physically important case of the unfolded system for HS connection (2.13),
(2.14) is that of $AdS_{4}$ space-time in which case the language of two-
component spinors is most appropriate. In this language instead of using
Lorentz indices $a,b,...=0,1,2,3$, one uses two pairs of dotted and undotted
spinor indices $\alpha,\beta,...$ and $\dot{\alpha},\dot{\beta},...$ taking
values $\\{1,2\\}$. The two languages are related via Pauli matrices. The
$AdS_{4}$ background geometry is described in terms of the Lorentz connection
$\varpi$ and frame field $h$, that satisfy equations

$\begin{array}[]{l}dh^{\alpha\dot{\beta}}+\varpi^{\alpha}{}_{\gamma}\wedge
h^{\gamma\dot{\beta}}+\overline{\varpi}^{\dot{\beta}}{}_{\dot{\gamma}}\wedge
h^{\alpha\dot{\gamma}}=0\,,\\\
d\varpi^{\alpha\beta}+\varpi^{\alpha}{}_{\gamma}\wedge\varpi^{\gamma\beta}=-\lambda^{2}h^{\alpha}{}_{\dot{\gamma}}\wedge
h^{\beta\dot{\gamma}}\,,\\\
d\overline{\varpi}^{\dot{\alpha}\dot{\beta}}+\overline{\varpi}^{\dot{\alpha}}{}_{\dot{\gamma}}\wedge\overline{\varpi}^{\dot{\gamma}\dot{\beta}}=-\lambda^{2}h_{\gamma}{}^{\dot{\alpha}}\wedge
h^{\gamma\dot{\beta}}\,,\end{array}$ (2.23)

where $\lambda^{2}$ is proportional to the curvature of $AdS_{4}$ and we adopt
the following rules

$A_{\alpha}=A^{\beta}\epsilon_{\beta\alpha}\,,\quad
A^{\alpha}=\epsilon^{\alpha\beta}A_{\beta}\,,\quad\epsilon_{\alpha\beta}\epsilon^{\gamma\beta}=\epsilon_{\alpha}{}^{\gamma}=\delta_{\alpha}^{\gamma}=-\epsilon^{\gamma}{}_{\alpha}\,,$
(2.24)

where

$\epsilon_{\alpha\beta}=-\epsilon_{\beta\alpha}\,,\quad\epsilon_{12}=1\,.$
(2.25)

The spinor version of the unfolded system for one-form $\omega$ reads as
follows. First, the HS curvatures in the spinor language are [14]

$R^{\alpha(n),\dot{\alpha}(m)}=D_{L}\omega^{\alpha(n),\dot{\alpha}(m)}+\lambda^{2}(nh^{\alpha}_{\
\dot{\gamma}}\wedge\omega^{\alpha(n-1),\dot{\gamma}\dot{\alpha}(m)}+mh_{\gamma}^{\
\dot{\alpha}}\wedge\omega^{\gamma\alpha(n),\dot{\alpha}(m-1)})$ (2.26)

and $D_{L}=d+\varpi+\overline{\varpi}$ is a Lorentz-covariant derivative with
Cartan’s spin-connection $(\varpi\oplus\overline{\varpi})$

$D_{L}\omega^{\alpha(n),\dot{\alpha}(m)}=d\omega^{\alpha(n),\dot{\alpha}(m)}+n\varpi_{\alpha}^{\
\beta}\wedge\omega_{\beta\alpha(n-1),\dot{\alpha}(m)}+m\overline{\varpi}_{\dot{\alpha}}^{\
\dot{\beta}}\wedge\omega_{\alpha(n),\dot{\beta}\dot{\alpha}(m-1)}\,.$

The $AdS_{4}$ deformation of the unfolded equations (2.13), (2.14) then takes
the form [12]

$R^{\alpha(n),\dot{\alpha}(m)}=\delta_{0,n}\,h_{\beta\dot{\alpha}}\wedge
h^{\beta}_{\
\dot{\alpha}}\overline{C}^{\dot{\alpha}(m+2)}+\delta_{0,m}\,h_{\alpha\dot{\beta}}\wedge
h_{\alpha}^{\ \dot{\beta}}C^{\alpha(n+2)}.$ (2.27)

The main advantage of the two-component spinor notation is that it makes the
representation theory of the Lorentz group very simple. Namely, every Lorentz
irreducible multispinor representing a traceless tensor is totally symmetric
in its spinor indices. Since the only Lorentz invariant objects are
antisymmetric bispinors $\epsilon_{\alpha\beta}$ and
$\overline{\epsilon}_{\dot{\alpha}\dot{\beta}}$ irreducible multispinors
$X^{\alpha(n),\dot{\alpha}(m)}$ are necessarily symmetric with respect to the
indices in the groups $\alpha(n)$ and $\dot{\alpha}(m)$ separately. Thus,
working with the two-component spinor notation one can happily forget about
painful calculations with the traces of Lorentz-tensors.

##  3 The idea of $\sigma_{-}$ cohomology analysis: example of integer spin
massless fields

The l.h.s. ’s of unfolded HS equations and gauge transformations in
$d$-dimensional Minkowski space are [15, 16]

$\displaystyle
R^{a(s-1),b(k)}=D_{L}\omega^{a(s-1),b(k)}+\sigma_{-}(\omega)^{{a(s-1),b(k)}},$
(3.1)
$\displaystyle\delta\omega^{a(s-1),b(k)}=D_{L}\varepsilon^{a(s-1),b(k)}+\sigma_{-}(\varepsilon)^{{a(s-1),b(k)}},$
(3.2)

where

$(\sigma_{-}\omega)^{{a(s-1),b(k)}}:=h_{c}\wedge\omega^{a(s-1),b(k)c}\,,\qquad(\sigma_{-}\varepsilon)^{{a(s-1),b(k)}}:=h_{c}\wedge\varepsilon^{a(s-1),b(k)c}.$
(3.3)

$R^{a(s-1),b(k)}$ is referred to as (linearized) HS curvature. For simplicity
we study the Minkowski case. Since $\sigma_{-}$ in $AdS_{d}$ is defined
analogously, our analysis applies to that case as well.

Due to their definition, HS curvatures obey the Bianchi identities

$D_{L}R^{a(s-1),b(k)}+\sigma_{-}(R)^{a(s-1),b(k)}=0\,.$ (3.4)

The appearance of $\sigma_{-}$ allows one to clarify the role of the fields
$\omega^{a(s-1),b(k)}$ and gauge parameters $\varepsilon^{a(s-1),b(k)}$.
Working with the zero-forms $\varepsilon^{a(s-1),b(k)}$ and one-forms
$\omega^{a(s-1),b(k)}$valued in two-row Young diagrams, we consider the space
$V^{p}$ of $p$-forms valued in two-row Young diagrams with any $p$. Defining
$\sigma_{-}$ to annihilate the forms with an empty second row, we find that
$\sigma_{-}V^{p}\subset V^{p+1}$ and $\sigma_{-}\,\sigma_{-}=0$. As originally
proposed in [13], the $\sigma_{-}$ cohomology
$H(\sigma_{-})=\ker({\sigma_{-}})/{\mathrm{im\,}(\sigma_{-})}$ classifies
fields, their equations and gauge symmetries.

Indeed, those components of the fields $\omega^{a(s-1),b(t)}$, that are not
annihilated by $\sigma_{-}$, can be expressed via derivatives of the fields
with lower $t$ by setting suitable components of the HS curvatures to zero.
Such fields are called auxiliary. Conversely, those components of the fields
$\omega^{a(s-1),b(t)}$, that cannot be expressed in terms of derivatives of
lower fields via zero-curvature conditions, are in $\ker(\sigma_{-})$. By
Stueckelberg fields we mean $\sigma_{-}$-exact fields (i.e. fields of the form
$\sigma_{-}\chi$) as they can be eliminated by an appropriate
$\sigma_{-}$-exact term in the gauge transformation (3.2). Fields that are not
expressed via derivatives of other fields and are not Stueckelberg are called
dynamical. These describe the physical degrees of freedom of the theory. Thus,
the dynamical HS fields are associated with $H^{1}(\sigma_{-})$.

The classification for the gauge parameters is analogous. The parameters, that
are not annihilated by $\sigma_{-}$, describe algebraic Stueckelberg shifts.
The leftover symmetries are described by the parameters in $\ker(\sigma_{-})$.
$\sigma_{-}$-exact parameters correspond to the so called gauge for gauge
transformations. Parameters, which are $\sigma_{-}$-closed and not
$\sigma_{-}$-exact, are referred to as genuine differential gauge parameters.
Note that since in the HS example in question the gauge parameters are zero-
forms there is no room for gauge for gauge symmetries in that case.

Let $V$ be a graded vector space, $\mathcal{C}$ be an element of
$\Lambda^{p}(\mathcal{M}^{d})\otimes V$ over some smooth $d$-dimensional
manifold $\mathcal{M}^{d}$. We demand the grading of $V$ to be bounded from
below, that is $V$ is $\mathbb{N}$-graded. Let $\sigma_{\pm}$ be operators
that act ”vertically”, i.e. do not affect the space-time coordinates, and
shift grading by $\pm 1$, $D_{L}$ be the Grassmann-odd operator that does not
affect the grading and is allowed to act non-trivially on the space-time
coordinates. Consider the covariant constancy condition of a general form
along with the zero-curvature condition

$\mathcal{D}\mathcal{C}=(D_{L}+\sigma_{-}+\sigma_{+})\mathcal{C}=0,\quad\mathcal{D}^{2}=0.$
(3.5)

Notice that eq.(3.5) remains invariant under the gauge transformations

$\delta\mathcal{C}=\mathcal{D}\varepsilon,$ (3.6)

where $\varepsilon\in\Lambda^{p-1}(\mathcal{M}^{d})\otimes V$.

One can prove the following proposition [13] (see also [16, 22, 23, 24]):

######  Theorem 3.1.

The following is true:

1) Differential gauge symmetry parameters $\varepsilon$ span
$H^{p-1}(\sigma_{-})$

2) Nontrivial dynamical fields $\mathcal{C}$ span $H^{p}(\sigma_{-})$

3) Physically distinguishable differential field equations on the nontrivial
dynamical fields, contained in $\mathcal{D}\mathcal{C}=0$, span
$H^{p+1}(\sigma_{-})$

Thus, taking into account that HS gauge fields are described by the one-forms
$\omega$, to prove that the Fronsdal metric formulation is equivalent to the
unfolded one, we have to calculate $H^{0}(\sigma_{-}),H^{1}(\sigma_{-})$ and
$H^{2}(\sigma_{-})$. More generally, higher cohomology $H^{k}(\sigma_{-})$
with $k>p+1$ describes Bianchi identities for dynamical equations at $k=p+2$
and Bianchi for Bianchi identites at $k>p+2$ [24]. Similarly, the lower
cohomology $H^{k}(\sigma_{-})$ with $k<p-1$ describes gauge for gauge
differential symmetries.

##  4 A method for calculating cohomology

Calculation of $\sigma_{-}$ cohomology is of utter importance for the analysis
of unfolded systems of the general form (3.5). The straightforward calculation
of the cohomology can be quite involved. In this paper we find cohomology
using a standard homotopy approach recalled below, that is a generalization of
the Hodge theory for de Rham cohomology extendable to a more general class of
(co)chain complexes. Main details of the construction used in this paper
follow those of [24], where the $\sigma_{-}$ cohomology analysis was applied
to the conformal HS theories of the bosonic fields of any symmetry type.
Unfortunately, some of the methods of [24], based on the fact that
$\sigma_{-}$ in conformal theories has the clear meaning in terms of the
conformal algebra, are not directly applicable to the non-conformal HS
theories discussed in this paper, which makes the analysis of the latter a bit
more involved.

Let $V$ be a graded vector space and $d$ be a linear operator of degree +1 on
$V$ (that is, it raises the grading of a homogeneous element by 1) such that
$d^{2}=0$. Then $H(d)=\ker(d)/\mathrm{im\,}(d)$. Let $\partial$ be another
operator of degree $-1$ on $V$ (i.e., it lowers the grading by 1) such that
$\partial^{2}=0$. The operators $d$ and $\partial$ can be used to compose the
degree 0 operator $\Delta$

$\Delta:=\\{d,\partial\\}=d\,\partial+\partial\,d.$ (4.1)

It is easy to see that $\Delta$ satisfies

$[d,\Delta]=[\partial,\Delta]=0\,.$ (4.2)

######  Lemma 4.1.

If $\Delta$ is diagonalizable on the (graded) vector space $V$, then
$H(d)\hookrightarrow\ker(\Delta)$.

###### Proof.

First of all we should show that $\ker(d)$ is an invariant subspace of
$\Delta$. Suppose $f\in\ker(d)$. Then

$\Delta f=(d\partial+\partial d)f=d\partial f\Rightarrow\Delta
f\in\ker(d)\,,\forall f\in\ker(d)\,.$ (4.3)

Therefore, $\ker(d)$ is an invariant subspace of $\Delta$, because linearity
is obvious.
Since the operator $\Delta$ is diagonalizable by assumption, we can consider
eigenvectors of $\Delta$. Let $g$ be $d$-closed and $\Delta g=\lambda
g,\lambda\neq 0$. Then

$g=\frac{1}{\lambda}\Delta g=\frac{1}{\lambda}\,d\,\partial g\,.$ (4.4)

Hence, $g$ is also $d$-exact for $\lambda\neq 0$, representing a trivial
element of $H(d)$. Thus, every $d$-closed form annihilated by $\Delta$ is not
$d$-exact. In other words, every $d$-closed form $f$ can be written as
$f=h+d\alpha$ with some $h\in\ker(\Delta)$. ∎

If $V$ is a Hilbert space with inner product $\langle\,,\,\rangle$, there
exists such $\partial$ that the converse inclusion
$H(d)\hookleftarrow\ker(\Delta)$ takes place as well, which means that
$H(d)=\ker(\Delta)$.

######  Lemma 4.2.

Let $(V,\langle\,,\,\rangle)$ be a Hilbert space, let $d^{*}$ be the operator
conjugated to $d$ in the usual sense $\langle\alpha,d\beta\rangle=\langle
d^{*}\alpha,\beta\rangle$ and $\Delta=\\{d\,,d^{*}\\}$. Then
$\ker(\Delta)\hookrightarrow H(d)$.

###### Proof.

Take any $f\in\ker(\Delta)$. Then

$0=\langle f,\Delta f\rangle=\langle df,df\rangle+\langle
d^{*}f,d^{*}f\rangle\Leftrightarrow df=0\quad\text{and}\quad d^{*}f=0\,.$
(4.5)

Hence, $f\in\ker(d)$. To show that $f\notin\mathrm{im\,}(d)$ suppose the
opposite. Let $f=dg$. Then due to (4.5)

$d^{*}dg=0\Rightarrow 0=\langle g,d^{*}dg\rangle=\langle
dg,dg\rangle\Rightarrow dg=0\,.$ (4.6)

Thus, $\ker(\Delta)\hookrightarrow H(d)$ ∎

From Lemmas 4.1 and 4.2, it follows that, if all the requirements are met,

$H(d)=\ker(\Delta)\,.$ (4.7)

Thus, in a Hilbert space with a diagonalizable Laplace operator
$\Delta:=\\{d\,,d^{*}\\}$, finding the cohomology is equivalent to finding
$\ker(\Delta)$. Further calculations of $\sigma_{-}$ cohomology will rely on
this fact.

The following important comment [24] is now in order. In the case of interest,
for every unfolded subsystem associated with a fixed spin

$V=\oplus_{n}V_{n}$

with finite-dimensional grade-$n$ subspaces $V_{n}$. In that case $\Delta$
leaves invariant every $V_{n}$ and, being self-adjoint in the finite-
dimensional Hilbert space, is diagonalizable.

It is worth noting the similarity of the above analysis with the Hodge theory
mentioned at the beginning of this section. Indeed, consider the (finite-
dimensional) vector space $V$ endowed with some nilpotent operators $d$ and
$\partial$, $d^{2}=\partial^{2}=0$. The condition of disjointness is also
imposed (see [25] for details), that is,
$\mathrm{im}(d)\cap\ker(\partial)=\mathrm{im}(\partial)\cap\ker(d)=\\{0\\}$.
In other words, it is demanded that

$\displaystyle d\partial x=0\quad\text{implies}\quad\partial x=0\,,$ (4.8a)
$\displaystyle\partial dx=0\quad\text{implies}\quad dx=0.$ (4.8b)

Define the Laplacian $\Delta$ by (4.1). Under these assumptions it can be
shown that

  1. 1.

$\ker(\Delta)=\ker(d)\cap\ker(\partial)$. The harmonic cocycles annihilated by
$\Delta$ are those and only those, that are $d$-closed and $\partial$-closed
simultaneously;

  2. 2.

$V=\mathrm{im}(d)\oplus\mathrm{im}(\partial)\oplus\ker(\Delta)$. In other
words, for any vector $x\in V$ there exists a unique Hodge decomposition
$x=d\alpha+\partial\beta+h$, where $\alpha$ and $\beta$ are some vectors in
$V$, and $h$ is harmonic $\Delta h=0$.

Since by (4.8a) $\partial\beta\neq 0$ implies $d\partial\beta\neq 0$, the
kernel of $d$ consists of vectors of the type $d\alpha+h$, where $h$ is
harmonic,

$\ker(d)=\left\\{x\in V\Big{|}\ x=d\alpha+h,\quad\Delta h=0\right\\}.$ (4.9)

This implies that the harmonic cocycles and cohomology classes of $d$ are
isomorphic as vector spaces, that is (4.7) is true.

In the subsequent sections the operators $\sigma_{-}$ and
$\sigma_{+}:=(\sigma_{-})^{*}$ will play the roles of $d$ and $\partial$.
Moreover, in the following calculations one can spot which Young diagram or
multispinor belongs to $\mathrm{im}(\sigma_{-}),\mathrm{im}(\sigma_{+})$ or
$\ker(\Delta)$ due to the equivariance of the constructed Laplace operators
$\Delta$ with respect to the action of $\mathrm{GL}(d)$ or $\mathrm{O}(d)$ or
$\mathrm{SL}(2;\mathbb{C})$, depending on the problem in question.

##  5 $\sigma_{-}$ cohomology in Minkowski space of any dimension

###  5.1 Generating functions

The problem of finding the $\sigma_{-}$ cohomology in tensor spaces of one or
another type can be conveniently reformulated in terms of differential
operators. To this end two-row Young diagrams in the symmetric basis can be
described as a subset of polynomial ring $\mathbb{R}[Y,Z]$ generated by the
set of $2d$ commuting variables $Y^{a},Z^{b}$ (see [16] for detail). Consider
the ring $\Lambda^{p}(\mathcal{M}^{d})\otimes\mathbb{R}[Y,Z]$. Its homogeneous
elements are differential $p$-forms valued in $\mathbb{R}[Y,Z]$

$\omega_{n,m}(x,dx,Y,Z)=\omega_{a(n),b(m)}(x,dx)Y^{a(n)}Z^{b(m)}\,.$ (5.1)

Consider the generating function

$\omega(x,dx\,|Y,Z)=\sum\limits_{n,m\geq
0}\omega_{n,m}(x,dx\,|Y,Z)=\sum\limits_{n,m\geq
0}\omega_{a(n),b(m)}(x,dx)Y^{a(n)}Z^{b(m)}\,.$ (5.2)

Its expansion in powers of $Y$ and $Z$ yields the tensor-valued forms
$\omega_{a(n),b(m)}$ as the Taylor coefficients. In this language the Young
irreducibility condition reads as

$Y^{a}\partialderivative{\omega}{Z^{a}}=0\quad\Longleftrightarrow\quad\omega_{a(n),ab(m-1)}=0\,.$
(5.3)

The tracelessness condition takes the form

$\eta^{ab}\partial_{Ya}\partial_{Yb}\omega=0\Longleftrightarrow\omega^{k}{}_{ka(n-2),b(m)}=0\,.$
(5.4)

Note that all other traces are also zero as a consequence of (5.3) and (5.4),

$\eta^{ab}\partial_{Ya}\partial_{Zb}\omega=0\Longleftrightarrow\omega^{k}{}_{a(n-1),b(m-1)k}=0\,,$
(5.5)
$\eta^{ab}\partial_{Za}\partial_{Zb}\omega=0\Longleftrightarrow\omega_{a(n),b(m-2)k}{}^{k}=0\,.$
(5.6)

The generators of $\mathfrak{u}(d)$ and $\mathfrak{so}(d)$111 Following [24],
in this section we do not distinguish between different real forms of the same
complex algebra freely going to their compact real (Euclidean) form since, not
affecting the final results, this choice simplifies the analysis by allowing a
positive-definite invariant scalar product on the space of tensors. Results of
the Euclidean case coincide with those of the Lorentz one due to the
equivalence of their representation theory on finite-dimensional modules.
Indeed, suppose that some Lorentz-irreducible tensor $T^{L}$ represented
$\sigma_{-}$ cohomology in the Lorentz case. Then analogous
$\mathfrak{o}(d)$-irreducible tensor $T^{E}$ represents $\sigma_{-}$
cohomology in the compact case and vice versa. The only potential difference
could be related to (anti)self-dual tensors that may exist in one signature
but not in the other. However, these do not play a role in our analysis where
(anti)self-dual tensors always appear in pairs or do not appear at all in
sufficiently high dimensions $d>4$.  are now realized by the first-order
differential operators

$\left(t_{\mathfrak{gl}(d)}{}\right)^{a}_{b}=Y^{a}\partial_{Yb}+Z^{a}\partial_{Zb}+\theta^{a}\partial_{\theta^{b}},\quad\left(t^{\mathfrak{so}(d)}{}\right)_{ab}=\frac{1}{2}\bigg{(}\eta_{ac}t_{\mathfrak{gl}}{}^{c}_{b}-\eta_{bc}t_{\mathfrak{gl}}{}^{c}_{a}\bigg{)}\,,$
(5.7)

where $\theta^{c}$ is a Grassmann-odd element of the exterior algebra
associated with the frame one-form $e^{a}$.

In these terms $\sigma_{-}$ acts as

$\sigma_{-}\omega=\theta^{a}\partialderivative{\omega}{Z^{a}}=m\theta^{c}\omega_{a(n),cb(m-1)}(x,\theta)Y^{a(n)}Z^{b(m-1)}\,.$
(5.8)

It differs from the definition of Section 3 by an additional numerical factor
introduced for future convenience. In the sequel we sometimes do not write
variables $Y,Z,\theta$ explicitly, that are always assumed to be present
implicitly. We adopt the convention that index $a$ is contracted with $Y$, $b$
with $Z$ and $c_{i}$ with $\theta^{c_{i}}$ with $\theta$s ordered as
$c_{1},...,c_{p}$.

The space $\Lambda(\mathcal{M}^{d})\otimes\mathbb{R}[Y,Z]$ can be equipped
with the scalar product

$\langle
f,g\rangle=\frac{1}{\pi^{2n}}\int_{\mathbb{C}^{d}\times\mathbb{C}^{d}}d^{2d}Z\,d^{2d}Y\,d^{d}\theta\,d^{d}\overline{\theta}\,f(Z,Y,\theta)\,\overline{g(Z,Y,\theta)}\,e^{-|Z|^{2}-|Y|^{2}-\overline{\theta}\theta},$
(5.9)

where $f,g\in\Lambda(\mathcal{M}^{d})\otimes\mathbb{R}[Y,Z]$ with complex
$Y,Z,\theta$ and Berezin integral over anticommuting variables. (We work with
the polynomials of complex variables with real coefficients). The space
$\Lambda(\mathcal{M}^{d})\otimes\mathbb{R}[Y,Z]$ with the scalar product (5.9)
is a Hilbert space in the Euclidean metric signature case used in this
section. This scalar product yields the following conjugation rules:

$(Z^{a})^{*}=\partial_{Z}{}_{a}\,,\qquad(Y^{a})^{*}=\partial_{Y}{}_{a}\,,\qquad(\theta^{a})^{*}=\partial_{\theta}{}_{a}\,.$
(5.10)

###  5.2 $\mathrm{GL}(d)$ example

To illustrate the idea of our construction let us first consider a simpler
case where fields and gauge parameters take values in the irreps of
$\mathfrak{gl}(d)$ described by two-row Young diagrams (no tracelessness
conditions are imposed). Define the following operators, that form
$\mathfrak{gl}(2)$

$t_{1}=Y^{a}\frac{\partial}{\partial Z^{a}},\quad
t_{2}=Z^{a}\frac{\partial}{\partial Y^{a}},\quad
t_{0}=Y^{a}\frac{\partial}{\partial Y^{a}}-Z^{a}\frac{\partial}{\partial
Z^{a}},$ (5.11)
$\commutator{t_{1}}{t_{2}}=t_{0},\quad\commutator{t_{0}}{t_{1}}=2t_{1},\quad\commutator{t_{0}}{t_{2}}=-2t_{2},$
(5.12) $h_{1}=Y^{a}\frac{\partial}{\partial Y^{a}},\quad
h_{2}=Z^{a}\frac{\partial}{\partial Z^{a}}\,.$ (5.13)

Namely, $t_{i}$ form $\mathfrak{sl}(2)$ while $h_{1}+h_{2}$ is central.

In terms of these operators the space of $p$-forms valued in two-row Young
diagrams is identified as $\ker(t_{1})$

$V^{p}=\\{F\in\Lambda^{p}(\mathcal{M}^{d})\otimes\mathbb{R}[Y,Z]|t_{1}F=0\\}\,.$
(5.14)

Here $\Lambda^{p}(\mathcal{M}^{d})$ is generated by the Grassmann variables
$\theta^{a}$.

Let us introduce auxiliary operators

$Z_{\theta}=Z^{a}\frac{\partial}{\partial\theta^{a}},\quad
Y_{\theta}=Y^{a}\frac{\partial}{\partial\theta^{a}},\quad
D=\theta^{a}\frac{\partial}{\partial\theta^{a}},\quad\theta_{Y}=\theta^{a}\frac{\partial}{\partial
Y^{a}},\quad\theta_{Z}=\theta^{a}\frac{\partial}{\partial Z^{a}}\,.$ (5.15)

Among the auxiliary operators $D$ plays the most important role as it gives
differential form degree. Now we should construct
$\sigma_{+}:\sigma_{+}^{2}=0$ on $V^{p}$ and $\imaginary(\sigma_{+})\subset
V^{p}$. Consider the following operator:

$\sigma_{+}=f(t_{0})Z_{\theta}+g(t_{0})Y_{\theta}t_{2}\,,$ (5.16)

where $f(t_{0})=\sum_{n=0}^{\infty}f_{n}t_{0}^{n}$ and
$g(t_{0})=\sum_{n=0}^{\infty}g_{n}t_{0}^{n}$. Functions $f$ and $g$ have to be
found from the conditions

$\sigma_{+}^{2}F=0\,,\qquad t_{1}\sigma_{+}F=0\,,\qquad\forall F\in V^{p}\,.$
(5.17)

After some re-ordering of operators this yields two equations

$\displaystyle 0=Y_{\theta}\bigg{(}f(t_{0}-1)F+g(t_{0}-1)t_{0}F\bigg{)},$
(5.18) $\displaystyle
0=Z_{\theta}Y_{\theta}\bigg{(}f(t_{0})g(t_{0}+1)t_{2}F-g(t_{0})f(t_{0}+1)t_{2}F-g(t_{0})g(t_{0}+1)t_{2}F\bigg{)}\,$
(5.19)

verified by

$f(t_{0})=-(t_{0}+1)g(t_{0})$ (5.20)

giving

$\sigma_{+}=-(t_{0}+1)g(t_{0})Z_{\theta}+g(t_{0})Y_{\theta}t_{2}\,.$ (5.21)

The free coefficient $g(t_{0})$ is determined from the conjugacy requirement:

$(f,\sigma_{+}g)=-(\theta_{Z}\bigg{(}g(t_{0})(t_{0}+1)+g(t_{0})\bigg{)}f,g)=(\sigma_{-}f,g)\,$
(5.22)

giving

$g(t_{0})=-\frac{1}{t_{0}+2}$ (5.23)

and hence

$\sigma_{+}:=(\sigma_{-})^{*}=\frac{t_{0}+1}{t_{0}+2}Z_{\theta}-\frac{1}{t_{0}+2}Y_{\theta}t_{2}\,.$
(5.24)

One can notice, that $\sigma_{+}$ in (5.24) differs from what one would expect
from the conjugation rules (5.10). The reason is that in (5.10) we work with
$C:=\Lambda^{p}(\mathcal{M}^{d})\otimes\mathbb{R}[Y,Z]$ complex. In the
$\mathrm{GL}(d)$-case we deal with
$\big{(}\Lambda^{p}(\mathcal{M}^{d})\otimes\mathbb{R}[Y,Z]\big{)}\cap\ker(t_{1})$
complex, therefore one should project on the highest weight vectors of the
underlying $\mathfrak{sl}(2)$ in the complex C. The same procedure applies to
the $\mathrm{O}(d)$-case. Though general formulae for extremal projectors are
known for any simple Lie algebra222We are grateful to the referee for bringing
this fact to our attention. [26, 27, 28, 29](for reviews see [30, 31]), to
keep the paper self-contained we derive the relevant projectors
straightforwardly.

Knowing $\sigma_{+}$, it remains to construct the Laplace operator
$\Delta=\\{\sigma_{-},\sigma_{+}\\}$ and find its zeros. Elementary
computation gives

$\Delta=\frac{t_{0}}{t_{0}+1}(D+h_{2}-1)+\frac{1}{t_{0}+1}Y_{\theta}\theta_{Y}-\frac{1}{(t_{0}+1)(t_{0}+2)}t_{2}\theta_{Z}Y_{\theta}\,.$
(5.25)

Being built from the manifestly $\mathfrak{gl}(d)$-invariant operators,
$\Delta$ commutes with $\mathfrak{gl}(d)$ hence being diagonal on its
irreducible submodules. Thus, it suffices to analyze zeros of $\Delta$ on
$\mathfrak{gl}(d)$ irreducible components of the forms.

####  5.2.1 $H^{0}(\sigma_{-})$

Any element of $V^{0}$ has the form $F=F_{a(n),b(m)}Y^{a(n)}Z^{b(m)}$. It is
easy to see that

$\Delta F=h_{2}F\,.$ (5.26)

Therefore

$H^{0}(\sigma_{-})=\\{F=F_{a(n)}Y^{a(n)}|\forall F_{a(n)}\in\mathbb{R}\\}\,.$
(5.27)

####  5.2.2 $H^{p}(\sigma_{-})$, $p>0$

For $p>0$, a general element of $V^{p}$ is
$F=F_{a(n),b(m)|c_{1},..,c_{p}}Y^{a(n)}Z^{b(m)}\theta^{c_{1}}..\theta^{c_{p}}$.
Generally it forms a reducible $\mathfrak{gl}(d)$-module associated with the
tensor product of two diagrams. In terms of Young diagrams it decomposes into
the following irreducible components:

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\end{picture}\,.\\\ $ (5.28)

At $p=1$ the last diagram is absent. The manifest decomposition of
$F_{a(n),b(m)|c_{1},c_{2},..,c_{p}}$ into irreducible components is

$F_{a(n),b(m)|c_{1},c_{2},..,c_{p}}=F_{1}{}_{a(n)c_{1},b(m),c_{2},..,c_{p}}+\frac{m}{n-m+2}F_{1}{}_{a(n)b,b(m-1)c_{1},c_{2},..,c_{p}}+F_{2}{}_{a(n),b(m)c_{1},c_{2},..,c_{p}}+\\\
+F_{3}{}_{a(n),b(m),c_{1},..,c_{p}}+F_{4}{}_{a(n)c_{1},b(m)c_{2},c_{3},..,c_{p}},$
(5.29)

where $F_{i}$ corresponds to the $i$-th diagram. There are no restrictions on
$n,m,p$ in (5.29) except for $n\geq m$. If for some $n,m$ tensor expression
has a wrong Young shape, it is zero. To simplify calculations we derive
restrictions on $n,m,p$ for each diagram from the condition of being
$\sigma_{-}$-closed. For the second and fourth diagrams we find no
restrictions, but for others we have

$\sigma_{-}\bigg{(}\text{1st
diagram}\bigg{)}=-m\bigg{(}1-\frac{1}{n-m+2}\bigg{)}F_{1}{}_{a(n)c_{0},b(m-1)c_{1},c_{2},..,c_{p}}\Rightarrow
m=0,$ (5.30)
$\sigma_{-}F_{3}{}_{a(n),b(m),c_{1},..,c_{p}}=mF_{3}{}_{a(n),b(m-1)c_{0},c_{1},..,c_{p}}\Rightarrow
m=0\,.$ (5.31)

Using this we obtain the action of $\Delta$ on the rest diagrams:

$\Delta
F_{1}{}_{a(n)c_{1},c_{2},..,c_{p}}=\frac{1}{n+1}\Delta\theta_{Y}F_{1}(Y,Z,\theta)=\frac{n(p-1)}{(n+1)^{2}}\theta_{Y}F_{1}(Y,Z,\theta),$
(5.32) $\Delta
F_{2}{}_{a(n),b(m)c_{1},c_{2},..,c_{p}}=\frac{1}{m+1}\Delta\theta_{Z}F_{2}(Y,Z,\theta)=\frac{m+p}{m+1}\theta_{Z}F_{2}(Y,Z,\theta),$
(5.33) $\Delta
F_{4}{}_{a(n)c_{1},b(m)c_{2},c_{3},..,c_{p}}=\frac{1}{(m+1)(n+1)}\Delta\theta_{Y}\theta_{Z}F_{4}(Y,Z,\theta)\\\
=\frac{(n-m)(p+m-1)}{(n-m+1)(m+1)(n+1)}\theta_{Y}\theta_{Z}F_{4}(Y,Z,\theta)\,.$
(5.34)

As a result,

$H^{1}(\sigma_{-})=\\{\phi=F_{a(n)c}\theta^{c}Y^{a(n)}|h_{2}F=0,F\in
V^{1}\\}\,,$ (5.35)
$H^{p}(\sigma_{-})=\\{W=\theta_{Y}\theta_{Z}C(Y,Z,\theta)|t_{0}C=0,C\in
V^{p-2}\,,\quad p>1\\},$ (5.36)
$C=C{}_{a(n),b(n),c_{1},..,c_{p-2}}Y^{a(n)}Z^{b(n)}\theta^{c_{1}}..\theta^{c_{p-2}}\in
V^{p-2}\,.$

The dynamical interpretation of the obtained results is as follows. The system
has one symmetric gauge field with gauge transformation described by a
symmetric parameter. The second cohomology group $H^{2}(\sigma_{-})$ is
spanned by a single tensor corresponding to the generalized (traceful) Weyl
tensor. If the latter is set to zero, the system becomes topological with the
zero-curvature field equations. Otherwise the unfolded equations encode a set
of constraints expressing all fields and Weyl tensor via derivatives of the
physical fields. Proceeding further with the equations on the Weyl tensor and
its descendants results in an infinite set of constraints with no differential
equations on the physical field. Such off-shell unfolded equations were
considered in [32]. The off-shell systems are for interest in many contexts
such as, e.g., construction of actions and quantization [33, 34]. The lower
cohomology groups (5.27), (5.35) and (5.36) match with those obtained, e.g.,
in [16].

###  5.3 $\mathrm{O}(d)$ case

####  5.3.1 Irreducibility conditions

The $\mathrm{O}(d)$ case is in many respects analogous to that of
$\mathrm{GL}(d)$. The difference is due to the tracelessness condition (5.4).
The algebra of the operators encoding irreducibility conditions is extended
since the metric allows the new types of contractions between $\theta,Y,Z$ and
their derivatives. From the representation theory perspective new terms
associated with traces appear in the diagram decomposition of the form
coefficients, affecting the cohomology analysis.

The following operators form the algebra $\mathfrak{sp}(4)$:

$\displaystyle t_{1}=Y^{a}\frac{\partial}{\partial Z^{a}},\quad
t_{2}=Z^{a}\frac{\partial}{\partial Y^{a}},\quad
h_{1}=Y^{a}\frac{\partial}{\partial Y^{a}},\quad
h_{2}=Z^{a}\frac{\partial}{\partial Z^{a}},$ (5.37) $\displaystyle
t_{0}=Y^{a}\frac{\partial}{\partial Y^{a}}-Z^{a}\frac{\partial}{\partial
Z^{a}},$ (5.38) $\displaystyle f_{1}=\partial_{Y}^{a}\partial_{Ya},\quad
f_{2}=\partial_{Z}^{a}\partial_{Za},\quad
f_{3}=\partial_{Y}^{a}\partial_{Za},$ (5.39) $\displaystyle
e_{1}=Y^{a}Y_{a},\quad e_{2}=Z^{a}Z_{a},\quad e_{3}=Y^{a}Z_{a}\,.$ (5.40)

Evidently, these operators commute with the $\mathfrak{so}(d)$ generators
(5.7). $\mathfrak{sp}(4)$ and $\mathfrak{so}(d)$ form a Howe-dual pair [35].
Young condition (5.3) and tracelessness condition (5.4) impose highest weight
conditions on a $\mathfrak{sp}(4)$-module.

In addition, we introduce the following $\mathrm{O}(d)$ invariant operators:

$\displaystyle Z_{\theta}=Z^{a}\partial_{\theta a},\quad
Y_{\theta}=Y^{a}\partial_{\theta a},$ (5.41) $\displaystyle\partial_{\theta
Z}=\partial_{\theta a}\partial_{Z}^{a},\quad\partial_{\theta
Y}=\partial_{\theta a}\partial_{Y}^{a},$ (5.42)
$\displaystyle\theta_{Z}=\theta^{a}\partial_{Za},\quad\theta_{Y}=\theta^{a}\partial_{Ya}\,,$
(5.43)

which, along with $D$ (5.15) counting differential form degree, extend
$\mathfrak{sp}(4)$ to $\mathfrak{osp}(2|4)$. The simplest way to see this is
to let index $a$ take a single value, treating the operators
$Z,Y,\partial_{Z},\partial_{Y},\theta,\partial_{\theta}$ as creation and
annihilation operators, and apply the oscillator realization of
$\mathfrak{osp}(2|4)$.

In the problem in question, the form space is

$V^{p}=\\{F\in\Lambda^{p}(\mathcal{M}^{d})\otimes\mathbb{R}[Y,Z]|t_{1}F=0,f_{1}F=0\\}\,.$
(5.44)

Note that these restrictions imply the tracelessness over indices $(Y,Z)$ and
$(Z,Z)$ as a consequence of the form of commutators of $t_{1}$ with $f_{1,2}$.

####  5.3.2 $\sigma_{+}$

Let us look for $\sigma_{+}=\sigma_{-}^{*}$ in the form

$\sigma_{+}=g_{1}(h_{1},h_{2})Z_{\theta}+g_{2}(h_{1},h_{2})t_{2}Y_{\theta}+g_{3}(h_{1},h_{2})e_{3}\partial_{\theta
Y}+g_{4}(h_{1},h_{2})e_{1}t_{2}\partial_{\theta
Y}+g_{5}(h_{1},h_{2})e_{2}\partial_{\theta Z}+\\\
+g_{6}(h_{1},h_{2})e_{3}t_{2}\partial_{\theta
Z}+g_{7}(h_{1},h_{2})e_{1}t_{2}^{2}\partial_{\theta Z}\,.$ (5.45)

The condition $\imaginary(\sigma_{+})\subset V^{p}$ gives

$0=f_{1}\sigma_{+}F=\bigg{(}2g_{2}(h_{1}+2,h_{2})+2g_{3}(h_{1}+2,h_{2})+2g_{4}(h_{1}+2,h_{2})(d+2h_{1})\bigg{)}t_{2}\partial_{\theta
Y}F+\\\
+\bigg{(}2g_{6}(h_{1}+2,h_{2})+2g_{7}(h_{1}+2,h_{2})(d+2h_{1})\bigg{)}t_{2}^{2}\partial_{\theta
Z}F,$ (5.46)
$0=t_{1}\sigma_{+}F=\bigg{(}g_{1}(h_{1}-1,h_{2}+1)+g_{2}(h_{1}-1,h_{2}+1)t_{0}\bigg{)}Y_{\theta}F+\bigg{(}-g_{3}(-1,1)+2g_{5}(-1,1)+\\\
+g_{6}(-1,1)t_{0}\bigg{)}e_{3}\partial_{\theta
Z}+\bigg{(}g_{3}(-1,1)+g_{4}(-1,1)(t_{0}-2)\bigg{)}e_{1}\partial_{\theta
Y}+\bigg{(}-g_{4}(-1,1)+g_{6}(-1,1)+\\\
+2g_{7}(-1,1)(t_{0}-1)\bigg{)}e_{1}t_{2}\partial_{\theta Z}\,.$ (5.47)

This imposes the following six equations on seven coefficients

$\displaystyle g_{1}(h_{1},h_{2})+g_{2}(h_{1},h_{2})(t_{0}+2)=0,$ (5.48)
$\displaystyle-
g_{3}(h_{1},h_{2})+2g_{5}(h_{1},h_{2})+g_{6}(h_{1},h_{2})(t_{0}+2)=0,$ (5.49)
$\displaystyle g_{3}(h_{1},h_{2})+g_{4}(h_{1},h_{2})t_{0}=0,$ (5.50)
$\displaystyle-
g_{4}(h_{1},h_{2})+g_{6}(h_{1},h_{2})+2g_{7}(h_{1},h_{2})(t_{0}+1)=0,$ (5.51)
$\displaystyle
g_{2}(h_{1},h_{2})+g_{3}(h_{1},h_{2})+g_{4}(h_{1},h_{2})(d+2h_{1}-4)=0,$
(5.52) $\displaystyle g_{6}(h_{1},h_{2})+g_{7}(h_{1},h_{2})(d+2h_{1}-4)=0\,.$
(5.53)

Choosing $g_{7}(h_{1},h_{2})$ as a free parameter, we obtain

$\displaystyle
g_{1}(h_{1},h_{2})=-(t_{0}+2)(d-4+h_{1}+h_{2})(d-6+2h_{2})g_{7}(h_{1},h_{2}),$
(5.54) $\displaystyle
g_{2}(h_{1},h_{2})=(d-4+h_{1}+h_{2})(d-6+2h_{2})g_{7}(h_{1},h_{2}),$ (5.55)
$\displaystyle g_{3}(h_{1},h_{2})=t_{0}(d-6+2h_{2})g_{7}(h_{1},h_{2}),$ (5.56)
$\displaystyle g_{4}(h_{1},h_{2})=-(d-6+2h_{2})g_{7}(h_{1},h_{2}),$ (5.57)
$\displaystyle
g_{5}(h_{1},h_{2})=(t_{0}+1)(d-4+h_{1}+h_{2})g_{7}(h_{1},h_{2}),$ (5.58)
$\displaystyle g_{6}(h_{1},h_{2})=-(d-4+2h_{1})g_{7}(h_{1},h_{2})\,.$ (5.59)

Now using the conjugation rules (5.10) and highest weight conditions (5.44) we
get

$(F_{1},\sigma_{+}F_{2})=\Big{(}-g_{7}(h_{1},h_{2})(t_{0}+2)(d-4+h_{1}+h_{2})(d-6+2h_{2})\Big{)}(\sigma_{-}F_{1},F_{2})\,.$
(5.60)

The condition $\sigma_{+}=\sigma^{*}_{-}$ demands

$g_{7}(h_{1},h_{2})=-\frac{1}{(t_{0}+2)(d-4+h_{1}+h_{2})(d-6+2h_{2})}$ (5.61)

giving

$\sigma_{+}=Z_{\theta}-\frac{1}{t_{0}+2}t_{2}Y_{\theta}-\frac{t_{0}}{(t_{0}+2)(d-4+h_{1}+h_{2})}e_{3}\partial_{\theta
Y}+\frac{1}{(t_{0}+2)(d-4+h_{1}+h_{2})}e_{1}t_{2}\partial_{\theta Y}-\\\
-\frac{t_{0}+1}{(t_{0}+2)(d-6+2h_{2})}e_{2}\partial_{\theta
Z}+\frac{d-4+2h_{1}}{(t_{0}+2)(d-4+h_{1}+h_{2})(d-6+2h_{2})}e_{3}t_{2}\partial_{\theta
Z}-\\\
-\frac{1}{(t_{0}+2)(d-4+h_{1}+h_{2})(d-6+2h_{2})}e_{1}t_{2}^{2}\partial_{\theta
Z}\,.$ (5.62)

This yields operator $\sigma_{+}$ such that $\sigma_{+}^{2}=0$ on $V$,
$\imaginary(\sigma_{+})\subset V$ and $\sigma_{-}^{*}=\sigma_{+}$.

To calculate the Laplace operator
$\Delta=\sigma_{-}\sigma_{+}+\sigma_{+}\sigma_{-}$ on $V^{p}$ we obtain
straightforwardly that

$\sigma_{-}\sigma_{+}=\theta_{Z}Z_{\theta}-\frac{1}{t_{0}+1}t_{2}\theta_{Z}Y_{\theta}-\frac{1}{t_{0}+1}\theta_{Y}Y_{\theta}-\frac{t_{0}-1}{(t_{0}+1)(d-3+h_{1}+h_{2})}\bigg{(}e_{3}\theta_{Z}\partial_{\theta
Y}+\theta^{a}Y_{a}\partial_{\theta Y}\bigg{)}\\\
+\frac{1}{(t_{0}+1)(d-3+h_{1}+h_{2})}\bigg{(}e_{1}t_{2}\theta_{Z}\partial_{\theta
Y}+e_{1}\theta_{Y}\partial_{\theta
Y}\bigg{)}-\frac{t_{0}}{(t_{0}+1)(d-4+2h_{2})}\bigg{(}e_{2}\theta_{Z}\partial_{\theta
Z}+2\theta^{a}Z_{a}\partial_{\theta Z}\bigg{)}\\\
+\frac{d-4+2h_{1}}{(t_{0}+1)(d-3+h_{1}+h_{2})(d-4+2h_{2})}\bigg{(}e_{3}t_{2}\theta_{Z}\partial_{\theta
Z}+e_{3}\theta_{Y}\partial_{\theta Z}+t_{2}\theta^{a}Y_{a}\partial_{\theta
Z}-\theta^{a}Z_{a}\partial_{\theta Z}\bigg{)}\\\
-\frac{1}{(t_{0}+1)(d-3+h_{1}+h_{2})(d-4+2h_{2})}\bigg{(}e_{1}t_{2}^{2}\theta_{Z}\partial_{\theta
Z}+2e_{1}t_{2}\theta_{Y}\partial_{\theta Z}\bigg{)}\,.$ (5.63)

$\sigma_{+}\sigma_{-}=Z_{\theta}\theta_{Z}+\frac{1}{t_{0}+2}t_{2}\theta_{Z}Y_{\theta}+\frac{t_{0}}{(t_{0}+2)(d-4+h_{1}+h_{2})}e_{3}\theta_{Z}\partial_{\theta
Y}-\frac{1}{(t_{0}+2)(d-4+h_{1}+h_{2})}\times\\\ \times
e_{1}t_{2}\theta_{Z}\partial_{\theta
Y}+\frac{t_{0}+1}{(t_{0}+2)(d-6+2h_{2})}e_{2}\theta_{Z}\partial_{\theta
Z}-\frac{d-4+2h_{1}}{(t_{0}+2)(d-4+h_{1}+h_{2})(d-6+2h_{2})}e_{3}t_{2}\theta_{Z}\partial_{\theta
Z}\\\
+\frac{1}{(t_{0}+2)(d-4+h_{1}+h_{2})(d-6+2h_{2})}e_{1}t_{2}^{2}\theta_{Z}\partial_{\theta
Z}\,.$ (5.64)

Since, by construction, both $\sigma_{-}\sigma_{+}$ and $\sigma_{+}\sigma_{-}$
and hence $\Delta$ are $\mathrm{O}(d)$ invariant, $\Delta$ is diagonal on
irreducible $\mathrm{O}(d)$-modules and, to compute $H(\sigma_{-})$, it
suffices to find its zeros on the irreducible components.

####  5.3.3 $H^{0}(\sigma_{-})$

In the sector of zero-forms, all terms that contain
$\frac{\partial}{\partial\theta^{c}}$ trivialize. Hence,

$\Delta F=h_{2}F=mF\,$ (5.65)

and

$H^{0}(\sigma_{-})=\\{F=F_{a(n)}Y^{a(n)}|\forall F_{a(n)}\in\mathbb{R}\\}\,.$
(5.66)

Comparing the resulting differential gauge parameters with (2.5), we find
that, as anticipated, differential gauge symmetries in the unfolded
formulation coincide with those of the Fronsdal theory.

####  5.3.4 $H^{p}(\sigma_{-})$, $p>0$

The main difference between $\mathfrak{o}(d)$\- and $\mathfrak{gl}(d)$\- cases
is due to the traceful terms in the decomposition of the $p$-forms into the
irreducible parts depicted as

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\end{picture}\oplus\begin{picture}(10.0,18.0)(0.0,7.0){
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For $1\leq p\leq 3$, the diagrams carrying negative $p$-dependent labels are
absent. It is important to note that the diagram $(n,m,p-2)$ is present twice:
one copy results from the contraction of one of the form indices with the
first row followed by the symmetrization of another form index with the same
row. Another one results from the application of the same procedure to the
second row. This fact leads to two different tensor implementations. Also note
that some of the diagrams vanish for special dimensions by virtue of the Two-
Column Theorem:

######  Theorem 5.1.

$\mathfrak{so}(d)$ traceless tensors with the symmetry properties of such
Young diagrams that the sum of the heights of the first two columns exceeds
$d$, are identically zero. [36]

The cohomology $H^{p}(\sigma_{-})$ is empty for $p>d$. Analogously, some
potential elements of $H^{p}(\sigma_{-})$ are zero by the Two-Column Theorem
for large $p\leq d$.

Now we are in a position to consider the action of the Laplace operator on
each of the diagrams (5.67) separately. In the following restrictions on
$n,m,p$ will be imposed: if for some $n,m,p$ a tensor has a wrong Young shape,
it is zero. In most cases we will simplify calculation by demanding $p$-forms
be $\sigma_{-}$-closed. Another simplification is due to the fact that $\Delta
F=0$ is equivalent to the two equations $\sigma_{-}F=0$ and $\sigma_{+}F=0$.

Diagram (n,m;p), $n\geq 0,m\geq 0,p\geq 0$ has the tensor form
$T_{a(n),b(m),c_{1},..,c_{p}}$. It is $\sigma_{-}$-closed, if $m=0$. Then

$\Delta T_{a(n),c_{1},..,c_{p}}=pT_{a(n),c_{1},..,c_{p}}\,.$ (5.68)

This diagram is in $\ker(\Delta)$ at $p=0$ that reproduces the already
obtained result for $H^{0}(\sigma_{-})$.

Diagram (n+1,m+1;p-2), $n\geq 0,m\geq 0,p\geq 2$ has the tensor form
$T_{a(n)c_{1},b(m)c_{2},..,c_{p}}$. This diagram is $\sigma_{-}$-closed. The
action of $\Delta$ is

$\Delta
T_{a(n)c_{1},b(m)c_{2},..,c_{p}}=\frac{(n-m)(p+m-1)}{(n-m+1)}T_{a(n)c_{1},b(m)c_{2},..,c_{p}}\,.$
(5.69)

This diagram is in $\ker(\Delta)$, if $n=m$, thus belonging to
$H^{p}(\sigma_{-})$ with $p\geq 2$.

Diagram (n,m+1;p-1), $n\geq 1,m\geq 0,p\geq 1$ has the tensor form
$T_{a(n),b(m)c_{0},c_{1},..,c_{p-1}}$. Then

$\Delta
T_{a(n),b(m)c_{0},c_{1},..,c_{p-1}}=(m+p)T_{a(n),b(m)c_{0},c_{1},..,c_{p-1}}\,.$
(5.70)

This equation admits no solutions since $p\geq 1$ in the case in question.

Diagram (n+1,m;p-1), $n\geq 0,m\geq 0,p\geq 1$ has the tensor form

$T_{a(n)c_{0},b(m),c_{1},..,c_{p-1}}+\frac{m}{n-m+2}T_{a(n)b,b(m-1)c_{0},c_{1},..,c_{p-1}}\,.$

It is $\sigma_{-}$-closed iff $m=0$. Then

$\Delta
T_{a(n)c_{0},c_{1},..,c_{p-1}}=\frac{n(p-1)}{n+1}T_{a(n)c_{0},c_{1},..,c_{p-1}}\,.$
(5.71)

This expression vanishes at $p=1$. Hence,

$T_{a(n)c_{0}}\in H^{1}(\sigma_{-})\,.$ (5.72)

As one can see, $n=0$ in (5.71) also leads to zero Laplace action. However,
this is not a new result, since it has been already accounted in the diagrams
$(n,m;p)$ for $n=m=p=0$ (5.68), $(n+1,m+1;p-2)$ for $n=m=0,p\geq 2$ (5.69) and
the $n=m=0,p=1$ case of $(n+1,m;p-1)$ (5.71). This fact is a simple
consequence of the tensor multiplication of a column by a scalar.

Diagram (n-1,m;p-1), $n\geq 1,m\geq 0,p\geq 1$ has the tensor form

$T=\eta_{ac_{1}}\rho_{a(n-1),b(m),c_{2},..,c_{p}}-\frac{n-1}{d-4+2n}\eta_{aa}\rho_{a(n-2)c_{1},b(m),c_{2},..,c_{p}}+\frac{m(n-1)}{(d-4+m+n)(d-4+2n)}\times\\\
\times\eta_{aa}\rho_{a(n-2)b,b(m-1)c_{1},c_{2},..,c_{p}}-\frac{m}{d-4+m+n}\eta_{ab}\rho_{a(n-1),b(m-1)c_{1},c_{2},..,c_{p}}\,,$
(5.73)

where $\rho$ is an arbitrary $(n-1,m;p-1)$ tensor.

One can check that $T\in\ker(\sigma_{-})$ demands $m=0$ with

$T=\eta_{ac_{1}}\rho_{a(n-1),c_{2},..,c_{p}}-\frac{n-1}{d-4+2n}\eta_{aa}\rho_{a(n-2)c_{1},c_{2},..,c_{p}}.$
(5.74)

After some calculation we obtain

$\Delta T=\frac{(p-1)(d-2+n)}{d-3+n}T\,.$ (5.75)

This implies that $\Delta$ has zero at $p=1$ contributing to
$H^{1}(\sigma_{-})$. The case of $d=2,n=1$ must be considered separately
because of the divergent denominator. The seeming divergence emerges due to
the second term in (5.74), which is absent at $d=2,n=1$,

$T=\eta_{ac_{1}}\rho_{c_{2},..,c_{p}}\Rightarrow\sigma_{+}T=\frac{p-1}{2}\Big{(}\eta_{bc_{1}}\rho_{a,c_{2},..,c_{p-1}}-\eta_{ac_{1}}\rho_{b,c_{2},..,c_{p-1}}\Big{)}\,,$
(5.76)

leading to the same answer with $p=1$.

Diagram (n,m-1;p-1) $n\geq 1,m\geq 1,p\geq 1$ has the tensor form

$T=(n-1)\eta_{aa}\rho_{a(n-2)bc_{1},b(m-1),c_{2},..,c_{p}}-\frac{(n-1)(m-1)}{d-6+2m}\eta_{aa}\rho_{a(n-2)bb,b(m-2)c_{1},c_{2},..,c_{p}}-(d-4+m+n)\times\\\
\times\eta_{ac_{1}}\rho_{a(n-1)b,b(m-1),c_{2},..,c_{p}}-(n-m)\eta_{ab}\rho_{a(n-1)c_{1},b(m-1),c_{2},..,c_{p}}+\frac{(m-1)(d-4+2n)}{d-6+2m}\eta_{ab}\times\\\
\times\rho_{a(n-1)b,b(m-2)c_{1},c_{2},..,c_{p}}+\frac{(n-m+1)(d-4+m+n)}{n}\eta_{bc_{1}}\rho_{a(n),b(m-1),c_{2},..,c_{p}}-\\\
-\frac{(m-1)(n-m+1)(d-4+m+n)}{(d-6+2m)n}\eta_{bb}\rho_{a(n),b(m-2)c_{1},c_{2},..,c_{p}}\,,$
(5.77)

where $\rho$ is an arbitrary $(n,m-1;p-1)$ tensor.

Let us show that this diagram can never be annihilated by $\sigma_{-}$.
Indeed,

$\sigma_{-}T=\bigg{(}d-4+m+n-(n-m)\bigg{)}\eta_{ac_{0}}\rho_{a(n-1)c_{1},b(m-1),c_{2},..,c_{p}}+(m-1)(\emph{lit}),$
(5.78)

where (_lit_) denotes some terms that are linearly independent from the first
one. If $m=1$ the above expression reduces to

$\sigma_{-}T=\big{(}d-2\big{)}\eta_{ac_{0}}\rho_{a(n-1)c_{1},c_{2},..,c_{p}}.$
(5.79)

The expression in brackets vanishes at $d=2$. However, such diagram is zero by
virtue of the Two-column theorem 5.1, since the heights of the first two
columns sum up to $3>d$.

Thus, the nontrivial $T$ is never in $\ker(\sigma_{-})$.

Diagram (n+1,m-1;p-2) $n\geq 1,m\geq 1,p\geq 2$ has the tensor form

$T=\eta_{ac_{1}}\rho_{a(n-1)bc_{2},b(m-1),c_{3},..c_{p}}-\frac{n-m+1}{n}\eta_{bc_{1}}\rho_{a(n)c_{2},b(m-1),c_{3},..c_{p}}+\\\
+\frac{m-1}{n-m+3}\eta_{ac_{1}}\rho_{a(n-1)bb,b(m-2)c_{2},c_{3},..c_{p}}-\frac{(m-1)(n-1)}{(d-6+2m)(n-m+3)}\eta_{aa}\rho_{a(n-2)bbc_{1},b(m-2)c_{2},c_{3},..c_{p}}+\\\
+\frac{2(m-1)(n-m+1)}{(d-6+2m)(n-m+3)}\eta_{ab}\rho_{a(n-1)bc_{1},b(m-2)c_{2},c_{3},..c_{p}}-\frac{(m-1)(n-m+1)}{n(n-m+3)}\times\\\
\times\eta_{bc_{1}}\rho_{a(n)b,b(m-2)c_{2},c_{3},..c_{p}}-\frac{(m-1)(n-m+2)(n-m+1)}{(d-6+2m)(n-m+3)n}\eta_{bb}\rho_{a(n)c_{1},b(m-2)c_{2},c_{3},..c_{p}}\,,$
(5.80)

where $\rho$ is an arbitrary $(n+1,m-1;p-2)$ tensor.

Though the tensor realization (5.80) may look complicated, the problem is
simplified by the observation that all terms except for the first and second
ones carry a factor of $(m-1)$. The action of $\sigma_{-}$ on the first and
second terms produces a factor of $(m-1)$ in front of each $\eta\rho$
combination. It can be checked that $\sigma_{-}T$ has an overall factor of
$(m-1)$ so that the only possible solution for $T\in\ker(\sigma_{-})$ is at
$m=1$ in which case the tensor decomposition acquires the form

$T=\eta_{ac_{1}}\rho_{a(n-1)bc_{2},..,c_{p}}-\eta_{bc_{1}}\rho_{a(n)c_{2},..,c_{p}}.$
(5.81)

At $p=2$, after some calculations one can check that

$\sigma_{+}T=0\,.$ (5.82)

For $p>2$ it is not difficult to see that $\sigma_{+}T\neq 0$. Indeed,

$\sigma_{+}T=(p-2)\bigg{(}1+\frac{1}{n}\bigg{)}\eta_{ac_{1}}\rho_{a(n-1)bc_{2},b,c_{3},..,c_{p-1}}+(\emph{lit})\,,$
(5.83)

where (_lit_) denotes other linearly independent terms. The first term is
never zero.

Thus,

$T=\eta_{ac_{1}}\rho_{a(n-1)bc_{2}}-\eta_{bc_{1}}\rho_{a(n)c_{2}}\in
H^{2}(\sigma_{-})\,.$ (5.84)

Diagram (n-1,m+1;p-2), $n\geq 1,m\geq 0,p\geq 2$ has the tensor form

$T=\eta_{ac_{1}}\rho_{a(n-1),b(m)c_{2},..,c_{p}}-\frac{n-1}{d-4+2n}\eta_{aa}\rho_{a(n-2)c_{1},b(m)c_{2},..,c_{p}}\,,$
(5.85)

where $\rho$ is an arbitrary $(n-1,m+1;p-2)$ tensor.

Though it is obviously in $\ker(\sigma_{-})$ for any $m$, it is not hard to
see that it is never in $\ker(\sigma_{+})$.

$\sigma_{+}T=-\bigg{(}1+\frac{p-2}{m+1}\bigg{)}\eta_{ac_{1}}\rho_{a(n-1),b(m+1),..,c_{p-1}}+(\emph{lit})\,.$
(5.86)

Hence this diagram does not contribute to $H^{p}(\sigma_{-})$.

Diagram (n-1,m-1;p-2), $n\geq 1,m\geq 1,p\geq 2$ has an involved tensor form.
Since the coefficients in the expression below are complicated, we extract the
factor of $(m-1)$ once present denoting the leftover coefficients as
$\alpha_{i}$,

$T=\eta_{ac_{1}}\eta_{bc_{2}}\rho_{a(n-1),b(m-1),..,c_{p}}+(m-1)\alpha_{1}\eta_{aa}\eta_{aa}\rho_{a(n-4)bbc_{1},b(m-2)c_{2},..,c_{p}}+\\\
+\alpha_{2}\eta_{aa}\eta_{ac_{1}}\rho_{a(n-3)bc_{2},b(m-1),..,c_{p}}+(m-1)\alpha_{3}\eta_{aa}\eta_{ac_{1}}\rho_{a(n-3)bb,b(m-2)c_{2},..,c_{p}}+\\\
+(m-1)\alpha_{4}\eta_{aa}\eta_{ab}\rho_{a(n-3)bc_{1},b(m-2)c_{2},..,c_{p}}+\alpha_{5}\eta_{aa}\eta_{bc_{1}}\rho_{a(n-2)c_{2},b(m-1),..,c_{p}}+\\\
+(m-1)\alpha_{6}\eta_{aa}\eta_{bc_{1}}\rho_{a(n-2)b,b(m-2)c_{2},..,c_{p}}+(m-1)\alpha_{7}\eta_{aa}\eta_{bb}\rho_{a(n-2)c_{1},b(m-2)c_{2},..,c_{p}}+\\\
+\alpha_{8}\eta_{ac_{1}}\eta_{ab}\rho_{a(n-2)c_{2},b(m-1),..,c_{p}}+(m-1)\alpha_{9}\eta_{ac_{1}}\eta_{ab}\rho_{a(n-2)b,b(m-2)c_{2},..,c_{p}}+\\\
+(m-1)\alpha_{10}\eta_{ac_{1}}\eta_{bb}\rho_{a(n-1),b(m-2)c_{2},..,c_{p}}+(m-1)\alpha_{11}\eta_{ab}\eta_{ab}\rho_{a(n-2)c_{1},b(m-2)c_{2},..,c_{p}}+\\\
+(m-1)\alpha_{12}\eta_{ab}\eta_{bc_{1}}\rho_{a(n-1),b(m-2)c_{2},..,c_{p}}\,,$
(5.87)

where $\rho$ is an arbitrary $(n-1,m-1;p-2)$ tensor. The explicit form of
$\alpha_{i}$ is given in the Appendix A.

Now we observe that the action of $\sigma_{-}$ on the terms free of the factor
of $(m-1)$ produces such factor. Hence, $\sigma_{-}(T)$ has the form of the
sum of linearly independent terms with the common factor of $(m-1)$.
Consequently,

$\sigma_{-}T=0,\text{ iff\quad}m=1.$ (5.88)

At $m=1$, the only terms that remain are

$T=\eta_{ac_{1}}\eta_{bc_{2}}\rho_{a(n-1),..,c_{p}}+\frac{(n-2)(n-1)}{(d-3+n)(d-4+2n)}\eta_{aa}\eta_{ac_{1}}\rho_{a(n-3)bc_{2},..,c_{p}}+\\\
+\frac{(n-1)(d-2+n)}{(d-3+n)(d-4+2n)}\eta_{aa}\eta_{bc_{1}}\rho_{a(n-2)c_{2},..,c_{p}}-\frac{n-1}{d-3+n}\eta_{ac_{1}}\eta_{ab}\rho_{a(n-2)c_{2},..,c_{p}}.$
(5.89)

It can be checked that for $p=2$

$\sigma_{+}T=0\,.$ (5.90)

For $p>2$ it is not difficult to see that

$\sigma_{+}T=\frac{(p-2)d}{(d-2)(d-3+n)(d-2+n)}e_{3}^{2}\theta_{Y}\rho+(\emph{lit}),$
(5.91)

where
$\rho=\rho_{a(n-1),c_{1},..,c_{p-2}}Y^{a(n-1)}\theta^{c_{1}}..\theta^{c_{p-2}}$.
Therefore,

$H^{2}(\sigma_{-})\ni
T=\eta_{ac_{1}}\eta_{bc_{2}}\rho{}_{a(n-1)}+\frac{(n-2)(n-1)}{(d-3+n)(d-4+2n)}\eta_{aa}\eta_{ac_{1}}\rho{}_{a(n-3)bc_{2}}+\\\
+\frac{(n-1)(d-2+n)}{(d-3+n)(d-4+2n)}\eta_{aa}\eta_{bc_{1}}\rho{}_{a(n-2)c_{2}}-\frac{n-1}{d-3+n}\eta_{ac_{1}}\eta_{ab}\rho{}_{a(n-2)c_{2}}\,.$
(5.92)

Diagram (n+1,m;p-3), $n\geq 1,m\geq 1,p\geq 3$ has the tensor form

$T=\eta_{bc_{1}}\rho_{a(n)c_{2},b(m-1)c_{3},..,c_{p}}-\frac{n}{n-m+1}\eta_{ac_{1}}\rho_{a(n-1)bc_{2},b(m-1)c_{3},..,c_{p}}\,,$
(5.93)

where $\rho$ is an arbitrary $(n+1,m;p-3)$ tensor.

Explicit computation gives

$\Delta
T=\frac{t_{0}}{t_{0}+1}\bigg{(}p+h_{2}-1\bigg{)}T-\frac{2t_{0}}{(t_{0}+1)(d-4+2h_{2})}\theta^{a}Z_{a}\partial_{\theta
Z}T+\\\
+\frac{d-4+2h_{1}}{(t_{0}+1)(d-3+h_{1}+h_{2})(d-4+2h_{2})}\bigg{(}t_{2}\theta^{a}Y_{a}\partial_{\theta
Z}-\theta^{a}Z_{a}\partial_{\theta Z}\bigg{)}T-\\\
-\frac{t_{0}-1}{(t_{0}+1)(d-3+h_{1}+h_{2})}\theta^{a}Y_{a}\partial_{\theta
Y}T\,.$ (5.94)

This expression vanishes at $n=m$. Indeed, in this case
$t_{0}T=0,h_{1}T=h_{2}T=nT,t_{2}T=0$ so that

$\Delta T=\frac{1}{d-3+2n}\bigg{(}t_{2}\theta^{a}Y_{a}\partial_{\theta
Z}-\theta^{a}Z_{a}\partial_{\theta
Z}\bigg{)}T+\frac{1}{d-3+2n}\theta^{a}Y_{a}\partial_{\theta Y}T\,.$ (5.95)

Since

$t_{2}\theta^{a}Y_{a}\partial_{\theta Z}T=\theta^{a}Y_{a}\partial_{\theta
Z}t_{2}T-\theta^{a}Y_{a}\partial_{\theta Y}T+\theta^{a}Z_{a}\partial_{\theta
Z}T=-\theta^{a}Y_{a}\partial_{\theta Y}T+\theta^{a}Z_{a}\partial_{\theta
Z}T\,,$ (5.96)

it follows that $\Delta T=0$ at $n=m$. To check that $\Delta T\neq 0$ at
$n\neq m$ one should substitute the expression for $T$ noticing that different
linearly independent terms have no common factor to vanish, that implies
nontriviality of $\Delta T$.

Diagram (n,m+1;p-3), $n\geq 1,m\geq 0,p\geq 3$ has the tensor form
$T=\eta_{ac_{1}}\rho_{a(n-1)c_{2},b(m)c_{3},..,c_{p}}$. It belongs to
$\ker(\sigma_{-})$, but not to $\ker(\sigma_{+})$.

$\sigma_{+}T=\bigg{(}1+\frac{p-3}{m+1}\bigg{)}\eta_{ac_{1}}\rho_{a(n-1)c_{2},b(m+1),..,c_{p}}+(\emph{lit})\,.$
(5.97)

Diagram (n,m;p-2) $n\geq 1,m\geq 0,p\geq 2$ admits two tensor realizations
$T_{i}$ due to the double presence of this diagram in the result of tensor
product. The tensors

$T_{1}=\eta_{ac_{1}}\rho_{1}{}_{a(n-1)c_{2},b(m),c_{3},..,c_{p}}+\frac{m}{n}\eta_{bc_{1}}\rho_{1}{}_{a(n),b(m-1)c_{2},c_{3},..,c_{p}},$
(5.98)
$T_{2}=\eta_{aa}\rho_{2}{}_{a(n-2)bc_{1},b(m-1)c_{2},c_{3},..,c_{p}}-\frac{n-m}{n-1}\eta_{ab}\rho_{2}{}_{a(n-1)c_{1},b(m-1)c_{2},c_{3},..,c_{p}}-\frac{d-4+n+m}{n-1}\eta_{ac_{1}}\times\\\
\times\rho_{2}{}_{a(n-1)b,b(m-1)c_{2},c_{3},..,c_{p}}+\frac{(n-m+1)(d-4+m+n)}{n(n-1)}\eta_{bc_{1}}\rho_{2}{}_{a(n),b(m-1)c_{2},c_{3},..,c_{p}}$
(5.99)

are linearly independent. That Laplace operator acts diagonally on $T_{i}$,
$\Delta T_{i}=\lambda_{i}(n,m,p)T_{i}$, allows us to separately consider each
of these diagrams. Firstly, we check if these are in $\ker(\sigma_{-})$
computing

$\displaystyle\sigma_{-}T_{1}=m\eta_{ac_{1}}\rho_{1}{}_{a(n-1)c_{2},b(m-1)c_{3},..,c_{p+1}},$
(5.100)
$\displaystyle\sigma_{-}T_{2}=\frac{d-4+2m}{n-1}\eta_{ac_{1}}\rho_{2}{}_{a(n-1)c_{2},b(m-1)c_{3},..,c_{p+1}}.$
(5.101)

$T_{1}\in\ker(\sigma_{-})$ at $m=0$. Formally, $T_{2}$ is annihilated by
$\sigma_{-}$ at $d=2,m=1$, but this is not allowed by the Two-column theorem.
So, the only candidate for cohomology is $T_{1}$. However,

$\sigma_{+}T_{1}=\frac{1}{n+1}\Big{(}1+\frac{p-2}{n}\Big{)}\eta_{bc_{1}}\rho_{1}{}_{a(n),c_{2},..,c_{p-1}}+(\emph{lit})\,,$
(5.102)

which is never zero.

Diagram (n-1,m;p-3), $n\geq 2,m\geq 1,p\geq 3$ has the tensor form

$T=\eta_{ac_{1}}\eta_{bc_{2}}\rho_{a(n-1),b(m-1)c_{3},..,c_{p}}-\frac{(n-1)(d-3+m+n)}{(d-4+n+m)(d-4+2n)}\eta_{aa}\eta_{bc_{1}}\rho_{a(n-2)c_{2},b(m-1)c_{3},..,c_{p}}-\\\
-\frac{(n-1)(n-2)}{(d-4+m+n)(d-4+2n)}\eta_{aa}\eta_{ac_{1}}\rho_{a(n-3)bc_{2},b(m-1)c_{3},..,c_{p}}+\frac{n-1}{d-4+m+n}\times\\\
\times\eta_{ab}\eta_{ac_{1}}\rho_{a(n-2)c_{2},b(m-1)c_{3},..,c_{p}}\,,$
(5.103)

where $\rho$ is an arbitrary $(n-1,m;p-3)$ tensor.

Obviously, $T\in\ker(\sigma_{-})$. However, $T\notin\ker(\sigma_{+})$.

$\sigma_{+}T=\bigg{(}1+\frac{p-3}{m}\bigg{)}\eta_{ac_{1}}\eta_{bc_{2}}\rho_{a(n-1),b(m),c_{3},..,c_{p-1}}+(\emph{lit})\,,$
(5.104)

hence not contributing to cohomology.

Diagram (n,m-1;p-3), $n\geq 1,m\geq 1,p\geq 3$ has the tensor form

$T=\eta_{ac_{1}}\eta_{bc_{2}}\rho_{a(n-1)c_{3},b(m-1),..,c_{p}}+(m-1)(\emph{lit})\,$
(5.105)

with all terms except for the first one carrying a factor of $(m-1)$. The
action of $\sigma_{-}$ on the first term brings a factor of $(m-1)$ in front
of $\eta\rho$. Since all $\eta\rho$ terms in the decomposition are linearly
independent we conclude that

$\sigma_{-}T=0,\text{ if }m=1.$ (5.106)

At $p=3$ one can check that $\sigma_{+}T=0$. However, for $p>3$ $T$ does not
belong to $\ker(\sigma_{+})$,

$\sigma_{+}T=-(p-3)\eta_{ac_{1}}\eta_{bc_{2}}\rho_{a(n-1)c_{3},b,..,c_{p-1}}+(\emph{lit})\,.$
(5.107)

Consequently, the only contribution to $H^{3}(\sigma_{-})$ is

$T=\eta_{ac_{1}}\eta_{bc_{2}}\rho_{a(n-1)c_{3}}\in H^{3}(\sigma_{-})\,.$
(5.108)

Diagram (n,m;p-4), $n\geq 1,m\geq 1,p\geq 4$ has the tensor form

$T=\eta_{ac_{1}}\eta_{bc_{2}}\rho_{a(n-1)c_{3},b(m-1)c_{4},..,c_{p}}\,.$
(5.109)

This is obviously annihilated by $\sigma_{-}$, but not by $\sigma_{+}$.

$\sigma_{+}T=-\bigg{(}1+\frac{p-4}{m}\bigg{)}\eta_{ac_{1}}\eta_{bc_{2}}\rho_{a(n-1)c_{3},b(m),..,c_{p-1}}+(\emph{lit})\,.$
(5.110)

Hence it does not contribute to $H^{p}(\sigma_{-})$.

####  5.3.5 Summary

Summarizing the results of Sections 5.3.3 and 5.3.4 we found the following
cohomology groups:

$H^{0}(\sigma_{-})=\\{F=F_{a(n)}Y^{a(n)}|F\in V^{0}\\}\,,$ (5.111)

$H^{1}(\sigma_{-})=\\{\phi=F_{1}{}_{a(n)c}Y^{a(n)}\theta^{c},F_{1}\in
V^{1};\\\
\phi^{tr}=\Big{[}(n-1)\eta_{aa}F_{2}{}_{a(n-2)c}-(d-4+2n)\eta_{ac}F_{2}{}_{a(n-1)}\Big{]}Y^{a(n)}\theta^{c}\in
V^{1}\\}\,,$ (5.112)

$H^{2}(\sigma_{-})=\\{W=\theta_{Y}\theta_{Z}C(Y,Z):t_{0}C=0,C\in V^{0};\\\
\mathcal{E}_{A}=\Big{[}\eta_{ac_{1}}\rho_{1}{}_{a(n-1)bc_{2}}-\eta_{bc_{1}}\rho_{1}{}_{a(n)c_{2}}\Big{]}Y^{a(n)}Z^{b}\theta^{c_{1}}\theta^{c_{2}}\in
V^{2};\\\
\mathcal{E}_{B}=\Big{[}\eta_{ac_{1}}\eta_{bc_{2}}\rho_{2}{}_{a(n-1)}+\frac{(n-2)(n-1)}{(d-3+n)(d-4+2n)}\eta_{aa}\eta_{ac_{1}}\rho_{2}{}_{a(n-3)bc_{2}}+\\\
+\frac{(n-1)(d-2+n)}{(d-3+n)(d-4+2n)}\eta_{aa}\eta_{bc_{1}}\rho_{2}{}_{a(n-2)c_{2}}-\frac{n-1}{d-3+n}\eta_{ac_{1}}\eta_{ab}\rho_{2}{}_{a(n-2)c_{2}}\Big{]}Y^{a(n)}Z^{b}\theta^{c_{1}}\theta^{c_{2}}\in
V^{2}\\}\,,$ (5.113)
$H^{3}(\sigma_{-})=\\{B^{fr}=\eta_{ac_{1}}\eta_{bc_{2}}\rho_{a(n-1)c_{3}}Y^{a(n)}Z^{b}\theta^{c_{1}}\theta^{c_{2}}\theta^{c_{3}}\in
V^{3};\\\
B_{1}=\theta_{Y}\theta_{Z}C(Y,Z,\theta):t_{0}C=0,C=C_{a(n),b(n),c}Y^{a(n)}Z^{b(n)}\theta^{c}\in
V^{1};\\\
B_{2}=\Big{[}\eta_{bc_{1}}\rho_{a(n)c_{2},b(n-1)c_{3}}-n\eta_{ac_{1}}\rho_{a(n-1)bc_{2},b(n-1)c_{3}}\Big{]}Y^{a(n)}Z^{b(n)}\theta^{c_{1}}\theta^{c_{2}}\theta^{c_{3}}\in
V^{3}\\}\,.$ (5.114)

At $p>3$

$H^{p}(\sigma_{-})=\Big{\\{}B_{1}=\theta_{Y}\theta_{Z}C(Y,Z,\theta):t_{0}C=0,C=C_{a(n),b(n),c_{1},..,c_{p-2}}Y^{a(n)}Z^{b(n)}\theta^{c_{1}}..\theta^{c_{p-2}}\in
V^{p-2};\\\
B_{2}=\Big{[}\eta_{bc_{1}}\rho_{a(n)c_{2},b(n-1)c_{3},..,c_{p}}-n\eta_{ac_{1}}\rho_{a(n-1)bc_{2},b(n-1)c_{3},..,c_{p}}\Big{]}Y^{a(n)}Z^{b(n)}\theta^{c_{1}}..\theta^{c_{p}}\in
V^{p}\Big{\\}}.$ (5.115)

According to Theorem 3.1, the differential gauge transformation parameters are
described by $H^{0}(\sigma_{-})$ (5.111). The gauge parameter in the Fronsdal
theory is known to be a symmetric traceless tensor. Since tensors $(n,0,0)$
constitute the cohomology group $H^{0}(\sigma_{-})$, the unfolded differential
gauge transformation is shown to coincide with the Fronsdal one.

As recalled in Section 2.1, the Fronsdal field consists of two symmetric
traceless fields (2.2). These fields are represented by the cohomology groups
$H^{1}(\sigma_{-})$ (5.112). Cohomology group $H^{1}(\sigma_{-})$ consists of
two elements $(n+1,0,0)$ and $(n-1,0,0)$ matching the components of the
Fronsdal field. Thus, the physical fields in the unfolded approach indeed
coincide with the Fronsdal field.

The cohomology group $H^{2}(\sigma_{-})$ (5.113) describes gauge invariant
combinations of derivatives of the physical fields that can be used to impose
differential equations on the latter. The Fronsdal cohomology classes
$\mathcal{E}_{A}$ and $\mathcal{E}_{B}$ match with the Fronsdal equations:
$\mathcal{E}_{A}$ is associated with the traceless part of the Fronsdal
equations, while $\mathcal{E}_{B}$ with the trace part, that is the equations
$\mathcal{E}_{A}=0,\mathcal{E}_{B}=0$ just reproduce the Fronsdal equations.
Note that the number of resulting equations is the same as the number of
fields, as it should be in a Lagrangian system.

$W$ in (5.113) represents ”Weyl” cohomology. Imposing $W=0$ in the case of
gravity one gets conformally flat metrics and in the case of higher spins
”conformally flat” fields. In Einstein gravity and HS theory, the equation
$W=0$ is not imposed. Instead, elements of $W$ are interpreted as new fields
$C$ that describe generalized Weyl tensors by virtue of the unfolded equations
(2.14). Thus, calculation of the cohomology group $H^{2}(\sigma_{-})$ shows
that unfolded equations (2.13), (2.14) contain Fronsdal equations along with
constraints on auxiliary fields.

In accordance with the general discussion of Section 3 elements of
$H^{3}(\sigma_{-})$ (5.114) correspond to Bianchi identities. Class $B^{fr}$
describes the Bianchi identities for the Fronsdal equations. Note that their
number coincides with the number of differential gauge parameters. The
remaining classes $B_{1}$ and $B_{2}$ correspond to the Bianchi identities on
the Weyl tensor. It is noteworthy that the latter can be checked to coincide
with the first $\widetilde{\sigma}_{-}$ cohomology in the Weyl sector of zero-
forms of [10] for $s>1$. This fact exhibits the connection between the gauge
and Weyl sectors.

For $p>3$ cohomology groups $H^{p}(\sigma_{-})$ describe the higher Bianchi
identities for Bianchi identities on the Weyl tensor also interpreted as
syzygies [24].

Obtained lower cohomology groups match with the results of [16, 17, 18].

In HS theory the fields are realized by one-forms. Formally, one can consider
field equations (2.13), (2.14) for $p$-forms $\omega_{a(n),b(m)}$ valued in a
two-row irreducible $\mathfrak{o}(d)$-module. From our results and physical
interpretation of the $\sigma_{-}$ cohomology groups it follows that for $p>1$
the unfolded system in the gauge sector is off-shell. To answer the question
whether the full unfolded system including both the gauge $p$-form sector and
the Weyl $(p-1)$-form sector is off-shell, the analysis of
$H(\tilde{\sigma}_{-})$ has to be performed in the Weyl sector. The case of
$p>1$ may be somewhat similar to the $s=1$ case, where the equation on
$A_{\mu}$ lies in the Weyl sector.

Finally, let us stress that the results of this section for HS fields in
Minkowski space admit a straightforward deformation to $AdS_{d}$ with the same
operator $\sigma_{-}$. This is because in that case dynamical fields are
described by rectangular diagrams of the $AdS_{d}$ algebra
$\mathfrak{o}(d-1,2)$ [37]. In general, in the flat limit, irreducible
massless (gauge) fields in $AdS_{d}$ decompose into nontrivial sets of
irreducible flat space massless fields [38, 39, 40] and there is no one-to-one
correspondence between massless fields in Minkowski space and $AdS_{d}$.
Namely, a generic irreducible field in Minkowski space may admit no
deformation to $AdS_{d}$ (see also [41]).

##  6 $\sigma_{-}$ cohomology in $AdS_{4}$ in the spinor language

$4d$ HS theories admit a description in terms of two-component spinors instead
of tensors. That is, instead of using the generating functions in the tensor
form $\omega(x,dx\,|Y,Z)$, where $Y^{a}$ and $Z^{a}$ carried vector Lorentz
indices $a=\\{0,1,2,3\\}$, we will use

$\omega(y,\overline{y}\,|x,dx)=\sum\limits_{k,m}\omega^{\alpha_{1}\dots\alpha_{k},\dot{\alpha}_{1}\dots\dot{\alpha}_{m}}(x,dx)\,y_{\alpha_{1}}\dots
y_{\alpha_{k}}\,\overline{y}_{\dot{\alpha}_{1}}\dots\overline{y}_{\dot{\alpha}_{m}}\,,$
(6.1)

where the indices $\alpha$ and $\dot{\alpha}$ of the two-component commuting
spinors $y^{\alpha}$ and $\overline{y}^{\dot{\alpha}}$ take two values
$\\{1,2\\}$.

Analogously to Sections 3 and 4, we have to introduce the grading on the space
of $\Lambda^{\bullet}(M)\otimes\mathbb{C}[[y,\overline{y}]]$. Consider the
homogeneous element of
$\Lambda^{\bullet}(M)\otimes\mathbb{C}[[y,\overline{y}]]$ of degree $N$ and
$\overline{N}$ in $y$ and $\overline{y}$, respectively

$\omega(\mu
y,\overline{\mu}\overline{y}\,|x,dx)=\mu^{N}\,\overline{\mu}{}^{\overline{N}}\,\omega(y,\overline{y}\,|x,dx).$
(6.2)

Define the grading operator $G$ on
$\Lambda^{\bullet}(M)\otimes\mathbb{C}[[y,\overline{y}]]$ as follows:

$G\omega(y,\overline{y}\,|x,dx)=|N-\overline{N}|\,\omega(y,\overline{y}\,|x,dx)\equiv|\deg_{y}(\omega(y,\overline{y}\,|x,dx))-\deg_{\overline{y}}(\omega(y,\overline{y}\,|x,dx))|.$
(6.3)

Note that in the bosonic sector the frame-like fields $e^{m_{1}\dots m_{s-1}}\
\leftrightarrow\
e^{\alpha_{1}\dots\alpha_{s-1},\dot{\alpha}_{1}\dots\dot{\alpha}_{s-1}}$ have
the lowest possible grading $G=0$. For our later computations to match with
the Fronsdal theory, we define the action of $\sigma_{-}$ on
$\omega(y,\overline{y})$ to decrease the $G$-grading:

$\displaystyle\sigma_{-}\omega(y,\overline{y})$
$\displaystyle:=i\,\overline{y}^{\dot{\alpha}}h^{\alpha}_{\
\dot{\alpha}}\partial_{\alpha}\ \omega(y,\overline{y}),\quad\quad\text{at
}\deg_{y}(\omega)>\deg_{\overline{y}}(\omega),$ (6.4a)
$\displaystyle\sigma_{-}\omega(y,\overline{y})$
$\displaystyle:=i\,y^{\alpha}h_{\alpha}^{\
\dot{\alpha}}\overline{\partial}_{\dot{\alpha}}\
\omega(y,\overline{y}),\quad\quad\text{at
}\deg_{y}(\omega)<\deg_{\overline{y}}(\omega),$ (6.4b)
$\displaystyle\sigma_{-}\omega(y,\overline{y})$
$\displaystyle:=0\qquad\qquad\qquad\qquad\quad\,\,\text{at
}\deg_{y}(\omega)=\deg_{\overline{y}}(\omega)\,,$ (6.4c)

where

$\displaystyle\partial_{\alpha}:=\frac{\partial}{\partial
y^{\alpha}}\,,\qquad\overline{\partial}_{\dot{\alpha}}:=\frac{\partial}{\partial\overline{y}^{\dot{\alpha}}}\,,$
(6.5)

and the dependence on $x$ and $dx$ in
$\omega(y,\overline{y})=\omega(y,\overline{y}\,|x,dx)$ is always implicit.

It is easy to check that so defined $\sigma_{-}$ is nilpotent,
$(\sigma_{-})^{2}=0$.

Note that $\sigma_{\pm}$ change the grading $G$ by 2. This agrees in
particular with the fact that the bosonic and fermionic sectors, where the
grading is even and odd respectively, are independent. We consider in detail
the more complicated bosonic case, observing in the end that the computation
for fermionic fields is quite similar.

Note that the analysis of $\sigma_{-}$ cohomology in the $4d$ conformal HS
theory was also performed in terms of two-component spinors in [42, 43]. It is
more complicated since the generating functions of conformal HS theory depend
on twice as many independent spinors, but simpler since it is free of the
module factors like $|N-N|$ in the grading definition (6.3).

Next, we define a scalar product333Being $\mathrm{SL}(2,\mathbb{C})$-invariant
this scalar product is not positive-definite. Analogously to the tensorial
case, without affecting the $\sigma_{-}$ cohomology analysis it can be made
positive-definite by going to the $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$
algebra, which is the compact real form of
$\mathfrak{sl}(2,\mathbb{C})\oplus\mathfrak{sl}(2,\mathbb{C})$ with altered
conjugation rules $\overline{y}^{\alpha}=y_{\alpha}$,
$\overline{y}^{\dot{\alpha}}=y_{\dot{\alpha}}$. on generating elements of
$\Lambda^{\bullet}(M)\otimes\mathbb{C}[[y,\overline{y}]]$ by

$\displaystyle\langle q^{\alpha}|q^{\beta}\rangle=\langle
y^{\alpha}|y^{\beta}\rangle=i\epsilon^{\alpha\beta}\,,$
$\displaystyle\langle\overline{p}_{\dot{\alpha}}|\overline{p}_{\dot{\beta}}\rangle=-\langle\overline{\partial}_{\dot{\alpha}}|\overline{\partial}_{\dot{\beta}}\rangle=i\overline{\epsilon}_{\dot{\alpha}\dot{\beta}}\,,$
$\displaystyle\langle
h_{\alpha\dot{\alpha}}|h_{\beta\dot{\beta}}\rangle=\epsilon_{\alpha\beta}\epsilon_{\dot{\alpha}\dot{\beta}}\,,$
(6.6)

where $q^{\alpha}=y^{\alpha}$, $p_{\alpha}=i\partial_{\alpha}$ and
$h^{\alpha\dot{\beta}}$ is a vierbein one-form.

In some local coordinates $x^{\mu}$ on the base manifold (which in our case is
$AdS_{4}$) the vierbein one-forms $h^{\alpha}_{\ \dot{\alpha}}$ can be
expressed as

$h^{\alpha}_{\ \dot{\alpha}}=\left(h_{\mu}\right)^{\alpha}_{\
\dot{\alpha}}\,dx^{\mu}.$ (6.7)

The $AdS_{4}$ vierbein $\left(h_{\mu}\right)^{\alpha}_{\ \dot{\alpha}}$ is
demanded to be non-degenerate at any point of $AdS_{4}$.

The next step is to obtain $\sigma_{+}:=\sigma_{-}^{\dagger}$ with respect to
the scalar product $\langle\,,\,\rangle$, i.e.,
$\langle\phi|\sigma_{-}\psi\rangle=\langle\sigma_{+}\phi|\psi\rangle$. It is
not hard to check that

$\displaystyle\sigma_{+}\omega(y,\overline{y})$
$\displaystyle:=-iy^{\alpha}D_{\alpha}^{\
\dot{\alpha}}\overline{\partial}_{\dot{\alpha}}\
\omega(y,\overline{y})\quad\quad\text{at
}\deg_{y}(\omega)>\deg_{\overline{y}}(\omega)\,,$ (6.8a)
$\displaystyle\sigma_{+}\omega(y,\overline{y})$
$\displaystyle:=-i\overline{y}^{\dot{\alpha}}D^{\alpha}_{\
\dot{\alpha}}\partial_{\alpha}\ \omega(y,\overline{y})\quad\quad\text{at
}\deg_{y}(\omega)<\deg_{\overline{y}}(\omega)\,,$ (6.8b)
$\displaystyle\sigma_{+}\omega(y,\overline{y})$
$\displaystyle:=-i\left(y^{\alpha}D_{\alpha}^{\
\dot{\alpha}}\overline{\partial}_{\dot{\alpha}}+\overline{y}^{\dot{\alpha}}D^{\alpha}_{\
\dot{\alpha}}\partial_{\alpha}\right)\omega(y,\overline{y})\quad\quad\text{at
}\deg_{y}(\omega)=\deg_{\overline{y}}(\omega)\,,$ (6.8c)

where

$\displaystyle D_{\alpha}^{\ \dot{\alpha}}:=\frac{\partial}{\partial
h^{\alpha}_{\ \dot{\alpha}}}\,,\qquad D^{\alpha}_{\
\dot{\alpha}}:=\frac{\partial}{\partial h_{\alpha}^{\ \dot{\alpha}}}\,.$ (6.9)

By $\omega=\omega(y,\overline{y}\,|x,h)$ we mean a general $p$-form polynomial
in $y$ and $\overline{y}$ with the coordinate one-forms $dx$ replaced by $h$
via (6.7), that is

$\omega(y,\overline{y}\,|x,h)=\sum_{n,m}\omega_{\alpha_{1}\dots\alpha_{p}|\mu(n)|\dot{\alpha}_{1}\dots\dot{\alpha}_{p}|\dot{\mu}(m)}(x)\,h^{\alpha_{1}\dot{\alpha}_{1}}\wedge\dots\wedge
h^{\alpha_{p}\dot{\alpha}_{p}}\,y^{\mu(n)}\,\overline{y}^{\dot{\mu}(m)}\,.$
(6.10)

So defined $\sigma_{+}$ increases the grading. The Laplace operator

$\Delta:=\sigma_{-}\sigma_{+}+\sigma_{+}\sigma_{-}$ (6.11)

is by construction self-adjoint with respect to $\langle\,|\,\rangle$ and non-
negative definite for the compact version of the space-time symmetry algebra.

##  7 Bosonic case in $AdS_{4}$

To calculate cohomology of $\sigma_{-}$ we have to compute the action of
$\Delta$. Since $\sigma_{-}$ and $\sigma_{+}$ are defined differently in the
different regions of the $(N,\overline{N})$ plane, we compute the Laplacian
action in the these regions separately. Direct computation yields:

$\displaystyle\Delta_{N>\overline{N}+2}$
$\displaystyle=N(\overline{N}+2)+y^{\beta}\partial_{\alpha}h^{\alpha}_{\
\dot{\gamma}}D_{\beta}^{\
\dot{\gamma}}+\overline{y}^{\dot{\alpha}}\overline{\partial}_{\dot{\beta}}h_{\gamma\dot{\alpha}}D^{\gamma\dot{\beta}}\,,$
(7.1a) $\displaystyle\Delta_{N<\overline{N}-2}$
$\displaystyle=\overline{N}(N+2)+y^{\alpha}\partial_{\beta}h_{\alpha\dot{\gamma}}D^{\beta\dot{\gamma}}+\overline{y}^{\dot{\alpha}}\overline{\partial}_{\dot{\beta}}h_{\gamma}^{\
\dot{\beta}}D^{\gamma}_{\ \dot{\alpha}}\,,$ (7.1b)
$\displaystyle\Delta_{N=\overline{N}+2}$
$\displaystyle=\Delta_{N>\overline{N}+2}+\overline{y}^{\dot{\alpha}}\overline{y}^{\dot{\beta}}\partial_{\alpha}\partial_{\beta}h^{\beta}_{\
\dot{\beta}}D^{\alpha}_{\ \dot{\alpha}}\,,$ (7.1c)
$\displaystyle\Delta_{N=\overline{N}-2}$
$\displaystyle=\Delta_{N<\overline{N}-2}+y^{\alpha}y^{\beta}\overline{\partial}_{\dot{\alpha}}\overline{\partial}_{\dot{\beta}}h_{\beta}^{\
\dot{\beta}}D_{\alpha}^{\ \dot{\alpha}}\,,$ (7.1d)
$\displaystyle\Delta_{N=\overline{N}}$
$\displaystyle=\overline{y}^{\dot{\alpha}}\overline{\partial}_{\dot{\beta}}h_{\gamma\dot{\alpha}}D^{\gamma\dot{\beta}}+y^{\alpha}\partial_{\beta}h_{\alpha\dot{\gamma}}D^{\beta\dot{\gamma}}-\overline{y}^{\dot{\alpha}}y^{\beta}\partial_{\alpha}\overline{\partial}_{\dot{\beta}}h^{\alpha}_{\
\dot{\alpha}}D_{\beta}^{\
\dot{\beta}}-y^{\alpha}\overline{y}^{\dot{\beta}}\partial_{\beta}\overline{\partial}_{\dot{\alpha}}h_{\alpha}^{\
\dot{\alpha}}D^{\beta}_{\ \dot{\beta}}\,.$ (7.1e)

The computation of the cohomology $H^{p}(\sigma_{-})$ will be performed as
follows. Taking a general $p$-form $\omega(y,\overline{y}\,|x)$, we decompose
it into Lorentz irreducible components. As we will observe, the projectors
onto irreducible parts will commute with the action of the Laplacian. Thus,
instead of involved calculation of the action of $\Delta$ on all of the
irreducible components of $\omega(y,\overline{y}\,|x)$ we can first calculate
its action on the general $\omega(y,\overline{y}\,|x)$ and then project.

###  7.1  $H^{0}(\sigma_{-})$

Evidently, $\Delta_{N=\overline{N}}\Big{|}_{\text{0-forms}}=0$ since all the
terms in (7.1e) contain derivatives in $h$’s. At the same time,
$\Delta_{N\neq\overline{N}}\Big{|}_{\text{0-forms}}>0$. Thus, we conclude

$H^{0}(\sigma_{-})=\ker\left(\Delta\Big{|}_{\text{0-forms}}\right)=\left\\{F(y,\overline{y})=F_{\alpha(n),\dot{\alpha}(n)}y^{\alpha(n)}\overline{y}^{\dot{\alpha}(n)},\
n\in\mathbb{N}_{0}\right\\}.$ (7.2)

By Theorem 3.1, elements of this cohomology space correspond to the parameters
of differential (non-Stueckelberg) linearized HS gauge transformations. This
result fits the pattern of the spin-$s$ Fronsdal gauge symmetry parameters
with $n=s-1$.

###  7.2  $H^{1}(\sigma_{-})$

The decomposition of a one-form $\Theta(y,\overline{y}\,|x)$ into Lorentz
irreps reads as

$\Theta(y,\overline{y}\,|x)=\underbrace{\Theta_{\mu(n+1)|\dot{\mu}(m+1)}\
h^{\mu\dot{\mu}}y^{\mu(n)}y^{\dot{\mu}(m)}}_{\Theta_{\mathrm{A}}(y,\overline{y})}-\frac{1}{2}\underbrace{\Theta_{\mu(n-1)|\dot{\mu}(m+1)}\
h_{\nu}^{\
\dot{\mu}}y^{\nu}y^{\mu(n-1)}\overline{y}^{\dot{\mu}(m)}}_{\Theta_{\mathrm{B}}(y,\overline{y})}-\\\
-\frac{1}{2}\underbrace{\Theta_{\mu(n+1)|\dot{\mu}(m-1)}\ h^{\mu}_{\
\dot{\nu}}y^{\mu(n)}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-1)}}_{\Theta_{\mathrm{C}}(y,\overline{y})}+\frac{1}{4}\underbrace{\Theta_{\mu(n-1)|\dot{\mu}(m-1)}\
h_{\nu\dot{\nu}}y^{\nu}y^{\mu(n-1)}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-1)}}_{\Theta_{\mathrm{D}}(y,\overline{y})}.$
(7.3)

Thus, for fixed $n$ and $m$, there are four Lorentz-irreducible one-forms:
$\Theta_{\text{A}}$, $\Theta_{\mathrm{B}}$, $\Theta_{\mathrm{C}}$,
$\Theta_{\mathrm{D}}$.
For direct computations it will be convenient to separate two of the indices
of the $y$ group:

$\Theta(y,\overline{y})=\Theta_{\lambda,\nu,\mu(n-1)|\dot{\lambda},\dot{\nu},\dot{\mu}(m-1)}\
h^{\lambda\dot{\lambda}}y^{\nu}y^{\mu(n-1)}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-1)}.$
(7.4)

In terms of the basis one-forms
$h^{\lambda\dot{\lambda}}y^{\nu}y^{\mu(n-1)}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-1)}$
the projectors onto irreducible components are

$\displaystyle\mathcal{P}_{\mathrm{A}}$
$\displaystyle=\mathscr{S}_{(\lambda,\mu,\nu)}\mathscr{S}_{(\dot{\lambda},\dot{\mu},\dot{\nu})},$
$\displaystyle\mathcal{P}_{\mathrm{C}}=\mathscr{S}_{(\lambda,\mu,\nu)}\epsilon_{\dot{\lambda}\dot{\nu}},$
(7.5) $\displaystyle\mathcal{P}_{\mathrm{B}}$
$\displaystyle=\epsilon_{\lambda\nu}\mathscr{S}_{(\dot{\lambda},\dot{\mu},\dot{\nu})},$
$\displaystyle\mathcal{P}_{\mathrm{D}}=\epsilon_{\lambda\nu}\epsilon_{\dot{\lambda}\dot{\nu}},$
(7.6)

where $\mathscr{S}_{(\lambda,\mu,\nu)}$ implies symmetrization over indices
$\lambda,\mu,\nu$ and similarly for the dotted indices.

####  7.2.1 $H^{1}(\sigma_{-})$ in the diagonal sector $N=\overline{N}$

In the diagonal sector with $n=m$ the Laplacian is a sum of the following four
terms:

$\Delta_{N=\overline{N}}\Theta(y,\overline{y})=\underbrace{\overline{y}^{\dot{\alpha}}\overline{\partial}_{\dot{\beta}}h_{\gamma\dot{\alpha}}D^{\gamma\dot{\beta}}\Theta(y,\overline{y})}_{T_{1}(y,\overline{y})}+\underbrace{y^{\alpha}\partial_{\beta}h_{\alpha\dot{\gamma}}D^{\beta\dot{\gamma}}\Theta(y,\overline{y})}_{T_{2}(y,\overline{y})}-\\\
\underbrace{-\overline{y}^{\dot{\alpha}}y^{\beta}\partial_{\alpha}\overline{\partial}_{\dot{\beta}}h^{\alpha}_{\
\dot{\alpha}}D_{\beta}^{\
\dot{\beta}}\Theta(y,\overline{y})}_{T_{3}(y,\overline{y})}\underbrace{-y^{\alpha}\overline{y}^{\dot{\beta}}\partial_{\beta}\overline{\partial}_{\dot{\alpha}}h_{\alpha}^{\
\dot{\alpha}}D^{\beta}_{\
\dot{\beta}}\Theta(y,\overline{y})}_{T_{4}(y,\overline{y})}.$ (7.7)

Consider the first term in (7.7).

$\overline{y}^{\dot{\alpha}}\overline{\partial}_{\dot{\beta}}h_{\gamma\dot{\alpha}}D^{\gamma\dot{\beta}}\Theta(y,\overline{y})=\Theta_{\lambda,\nu,\mu(n-1)|\dot{\lambda},\dot{\nu},\dot{\mu}(m-1)}\
\underbrace{\Big{[}\overline{y}^{\dot{\alpha}}\overline{\partial}_{\dot{\beta}}h_{\gamma\dot{\alpha}}D^{\gamma\dot{\beta}}\big{(}h^{\lambda\dot{\lambda}}y^{\nu}y^{\mu(n-1)}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-1)}\big{)}\Big{]}}_{T_{1}^{\lambda,\nu,\mu(n-1)|\dot{\lambda},\dot{\nu},\dot{\mu}(n-1)}}\,.$
(7.8)

The expression in square brackets is denoted by
$T_{1}^{\lambda,\nu,\mu(n-1)|\dot{\lambda},\dot{\nu},\dot{\mu}(n-1)}$. The
notation for other irreducible components $T_{2}$, $T_{3}$ and $T_{4}$ is
analogous. Straightforward computation yields

$\displaystyle
T_{1}^{\lambda,\nu,\mu(n-1)|\dot{\lambda},\dot{\nu},\dot{\mu}(n-1)}$
$\displaystyle=$ $\displaystyle-h^{\lambda}_{\
\dot{\alpha}}\epsilon^{\dot{\nu}\dot{\lambda}}y^{\nu}y^{\mu(n-1)}\overline{y}^{\dot{\alpha}}\overline{y}^{\dot{\mu}(n-1)}-{}$
(7.9a) $\displaystyle-(n-1)h^{\lambda}_{\
\dot{\alpha}}\epsilon^{\dot{\mu}\dot{\lambda}}y^{\nu}y^{\mu(n-1)}\overline{y}^{\dot{\alpha}}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(n-2)}\,,$
$\displaystyle
T_{2}^{\lambda,\nu,\mu(n-1)|\dot{\lambda},\dot{\nu},\dot{\mu}(n-1)}$
$\displaystyle=$ $\displaystyle-h_{\alpha}^{\
\dot{\lambda}}\epsilon^{\nu\lambda}y^{\alpha}y^{\mu(n-1)}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(n-1)}-{}$
(7.9b) $\displaystyle-(n-1)h_{\alpha}^{\
\dot{\lambda}}\epsilon^{\mu\lambda}y^{\alpha}y^{\nu}y^{\mu(n-2)}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(n-1)}\,,$
$\displaystyle
T_{3}^{\lambda,\nu,\mu(n-1)|\dot{\lambda},\dot{\nu},\dot{\mu}(n-1)}$
$\displaystyle=$ $\displaystyle-\left(h^{\nu}_{\
\dot{\alpha}}y^{\lambda}y^{\mu(n-1)}+(n-1)h^{\mu}_{\
\dot{\alpha}}y^{\lambda}y^{\nu}y^{\mu(n-2)}\right){}\times$ (7.9c)
$\displaystyle\times\left(\epsilon^{\dot{\nu}\dot{\lambda}}\overline{y}^{\dot{\alpha}}\overline{y}^{\dot{\mu}(n-1)}+(n-1)\epsilon^{\dot{\mu}\dot{\lambda}}\overline{y}^{\dot{\alpha}}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(n-2)}\right)\,,$
$\displaystyle
T_{4}^{\lambda,\nu,\mu(n-1)|\dot{\lambda},\dot{\nu},\dot{\mu}(n-1)}$
$\displaystyle=$ $\displaystyle-\left(h_{\alpha}^{\
\dot{\nu}}\overline{y}^{\dot{\lambda}}\overline{y}^{\dot{\mu}(n-1)}+(n-1)h_{\alpha}^{\
\dot{\nu}}\overline{y}^{\dot{\lambda}}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(n-2)}\right){}\times$
(7.9d)
$\displaystyle\times\left(\epsilon^{\nu\lambda}y^{\alpha}y^{\mu(n-1)}+(n-1)\epsilon^{\mu\lambda}y^{\alpha}y^{\nu}y^{\mu(n-2)}\right).$

Projecting onto the irreducible parts of $\Theta(y,\overline{y})$ we find

$\displaystyle\Delta_{N=\overline{N}}\left(\Theta_{\mathrm{A}}\right)$
$\displaystyle=0\,,$ (7.10a)
$\displaystyle\Delta_{N=\overline{N}}\left(\Theta_{\mathrm{B}}\right)$
$\displaystyle=(n+1)^{2}\Theta_{\mathrm{B}}\neq 0\,,$ (7.10b)
$\displaystyle\Delta_{N=\overline{N}}\left(\Theta_{\text{C}}\right)$
$\displaystyle=(n+1)^{2}\Theta_{\text{C}}\neq 0\,,$ (7.10c)
$\displaystyle\Delta_{N=\overline{N}}\left(\Theta_{\mathrm{D}}\right)$
$\displaystyle=0.$ (7.10d)

Thus, the only elements of the kernel of
$\Delta^{\text{1-forms}}\Big{|}_{N=\overline{N}}$ are
$\Theta_{\mathrm{A}}(y,\overline{y})$ and
$\Theta_{\mathrm{D}}(y,\overline{y})$. By the Hodge theorem of Section 4 this
yields that
$H^{1}(\sigma_{-})=\ker\left(\Delta^{\text{1-forms}}\Big{|}_{N=\overline{N}}\right)$
is

$\displaystyle H^{1}(\sigma_{-})$ $\displaystyle=\bigoplus_{n\geq
0}H^{1}_{(n)}(\sigma_{-}),$ (7.11a) $\displaystyle H^{1}_{(n)}(\sigma_{-})$
$\displaystyle=\Big{\\{}\phi_{(n)}(y,\overline{y}\,|x)+\phi_{(n)}^{\text{tr}}(y,\overline{y}\,|x),$
(7.11b) $\displaystyle\phi_{(n)}(y,\overline{y}\,|x)$
$\displaystyle:=\phi_{\mu(n+1),\dot{\mu}(n+1)}(x)\,h^{\mu\dot{\mu}}\,y^{\mu(n)}\overline{y}^{\dot{\mu}(n)},$
(7.11c) $\displaystyle\phi_{(n)}^{\text{tr}}(y,\overline{y}\,|x)$
$\displaystyle:=\phi^{\mathrm{tr}}_{\mu(n-1),\dot{\mu}(n-1)}(x)\,h_{\nu\dot{\nu}}\,y^{\nu}y^{\mu(n-1)}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(n-1)}\,,\text{
if }n>0\Big{\\}},$ (7.11d)

where $n$ is the number of indices of the corresponding cocycles.

Equivalently,

$H^{1}(\sigma_{-})=\Big{\\{}h^{\mu\dot{\mu}}\,\partial_{\mu}\overline{\partial}_{\dot{\mu}}\,F_{1}(y,\overline{y}\,|x)+h_{\mu\dot{\mu}}\,y^{\mu}\overline{y}^{\dot{\mu}}F_{2}(y,\overline{y}\,|x)\Big{\\}},$
(7.12)

where $F_{1,2}(y,\overline{y}\,|x)$ belongs to the diagonal $N=\overline{N}$,
that is,

$\left(y^{\alpha}\frac{\partial}{\partial
y^{\alpha}}-\overline{y}^{\dot{\alpha}}\frac{\partial}{\partial\overline{y}^{\dot{\alpha}}}\right)F_{1,2}(y,\overline{y}\,|x)=0.$
(7.13)

The fields $\phi(y,\overline{y})$ and $\phi^{\text{tr}}(y,\overline{y})$
exactly correspond to the irreducible components of the double-traceless
Fronsdal field.

It remains to prove that there are no other nontrivial cocycles in
$H^{1}(\sigma_{-})$ at $N\neq\overline{N}$.

####  7.2.2 $H^{1}(\sigma_{-})$ in the far-from-diagonal sector
$|N-\overline{N}|>2$

Consider the action of the Laplace operator $\Delta_{N>\overline{N}+2}$ on
general one-forms at $N>\overline{N}+2$

$\Delta_{N>\overline{N}+2}\Theta(y,\overline{y})=\left(n(m+2)+y^{\beta}\partial_{\alpha}h^{\alpha}_{\
\dot{\gamma}}D_{\beta}^{\
\dot{\gamma}}+\overline{y}^{\dot{\alpha}}\overline{\partial}_{\dot{\beta}}h_{\gamma\dot{\alpha}}D^{\gamma\dot{\beta}}\right)\Theta(y,\overline{y})=\\\
=\underbrace{n(m+2)\Theta(y,\overline{y})}_{T_{1}(y,\overline{y})}+\underbrace{y^{\beta}\partial_{\alpha}h^{\alpha}_{\
\dot{\gamma}}D_{\beta}^{\
\dot{\gamma}}\Theta(y,\overline{y})}_{T_{2}(y,\overline{y})}+\underbrace{\overline{y}^{\dot{\alpha}}\overline{\partial}_{\dot{\beta}}h_{\gamma\dot{\alpha}}D^{\gamma\dot{\beta}}\Theta(y,\overline{y})}_{T_{3}(y,\overline{y})}.$
(7.14)

Analogously to (7.8), we denote

$T_{2}^{\lambda,\nu,\mu(n-1)|\dot{\lambda},\dot{\nu},\dot{\mu}(m-1)}=y^{\beta}\partial_{\alpha}h^{\alpha}_{\
\dot{\gamma}}D_{\beta}^{\
\dot{\gamma}}\big{(}h^{\lambda\dot{\lambda}}y^{\nu}y^{\mu(n-1)}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-1)}\big{)}$
(7.15)

and similarly for $T_{1}$ and $T_{3}$. In this sector we obtain

$\displaystyle
T_{1}^{\lambda,\nu,\mu(n-1)|\dot{\lambda},\dot{\nu},\dot{\mu}(m-1)}$
$\displaystyle=n(m+2)h^{\lambda\dot{\lambda}}y^{\nu}y^{\mu(n-1)}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-1)}\,,$
(7.16a) $\displaystyle
T_{2}^{\lambda,\nu,\mu(n-1)|\dot{\lambda},\dot{\nu},\dot{\mu}(m-1)}$
$\displaystyle=-h^{\nu\dot{\lambda}}y^{\lambda}y^{\mu(n-1)}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-1)}-(n-1)h^{\mu\dot{\lambda}}y^{\lambda}y^{\nu}y^{\mu(n-2)}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-1)}\,,$
(7.16b) $\displaystyle
T_{3}^{\lambda,\nu,\mu(n-1)|\dot{\lambda},\dot{\nu},\dot{\mu}(m-1)}$
$\displaystyle=-h^{\lambda}_{\
\dot{\alpha}}\epsilon^{\dot{\nu}\dot{\lambda}}y^{\nu}y^{\mu(n-1)}\overline{y}^{\dot{\alpha}}\overline{y}^{\dot{\mu}(m-1)}-(m-1)h^{\lambda}_{\
\dot{\alpha}}\epsilon^{\dot{\lambda}\dot{\mu}}y^{\nu}y^{\mu(n-1)}\overline{y}^{\dot{\alpha}}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-2)}\,.$
(7.16c)

Projection onto the irreducible components according to (7.5) yields

$\displaystyle\Delta_{N>\overline{N}+2}\left(\Theta_{\mathrm{A}}\right)$
$\displaystyle=n(m+1)\Theta_{\mathrm{A}}\neq 0\,,$ (7.17a)
$\displaystyle\Delta_{N>\overline{N}+2}\left(\Theta_{\mathrm{B}}\right)$
$\displaystyle=(nm+2n+1)\Theta_{\mathrm{B}}\neq 0\,,$ (7.17b)
$\displaystyle\Delta_{N>\overline{N}+2}\left(\Theta_{\mathrm{C}}\right)$
$\displaystyle=(nm-n+2m)\Theta_{\mathrm{C}}\neq 0\,,$ (7.17c)
$\displaystyle\Delta_{N>\overline{N}+2}\left(\Theta_{\mathrm{D}}\right)$
$\displaystyle=(nm+2n-m+4)\Theta_{\mathrm{D}}\neq 0\,.$ (7.17d)

Thus, there are no nontrivial cocycles in this sector. For $N<\overline{N}-2$
the computation is analogous. Thus, $H^{1}(\sigma_{-})=0$ in the far-from-
diagonal sector.

####  7.2.3 Subtlety in the near-diagonal sector $|N-\overline{N}|=2$

In this case we face certain peculiarity. Denote the space of $p$-forms ($p=1$
for $H^{1}$ ) with $N$ chiral and $\overline{N}$ anti-chiral indices by
$\mathcal{V}_{(N,\overline{N})}$. Recall that the grading operator is
$G=|N-\overline{N}|$. Consider the case with $N-\overline{N}=2$. Namely, let
$N=n+1$ and $\overline{N}=n-1$. At $G=2$ the operator $\sigma_{-}$ maps a
state $X\in\mathcal{V}_{(n+1,n-1)}$ onto the diagonal,
$\sigma_{-}(X)\in\mathcal{V}_{(n,n)}$, where in accordance with (6.8c),
$\sigma_{+}$ acts ’both up and down’:

$\mathcal{V}_{(n+1,n-1)}\xrightarrow{\sigma_{-}}\mathcal{V}_{(n,n)}\xrightarrow{\sigma_{+}}\mathcal{V}_{(n-1,n+1)}\oplus\mathcal{V}_{(n+1,n-1)}\,.$
(7.18)

Thus,

$\mathcal{V}_{(n+1,n-1)}\xrightarrow{\sigma_{+}\sigma_{-}}\mathcal{V}_{(n-1,n+1)}\oplus\mathcal{V}_{(n+1,n-1)}\,.$
(7.19)

As a result,

$\displaystyle\Delta_{(n+1,n-1)}:\mathcal{V}_{(n+1,n-1)}\longrightarrow\mathcal{V}_{(n-1,n+1)}\oplus\mathcal{V}_{(n+1,n-1)}\,,$
(7.20)
$\displaystyle\Delta_{(n-1,n+1)}:\mathcal{V}_{(n-1,n+1)}\longrightarrow\mathcal{V}_{(n-1,n+1)}\oplus\mathcal{V}_{(n+1,n-1)}\,.$
(7.21)

Consequently, $\ker(\Delta)$ should be searched in the form of a linear
combination of the vectors both from $\mathcal{V}_{(n+1,n-1)}$ and from
$\mathcal{V}_{(n-1,n+1)}$.

Indeed, let $X$ be a vector in $\mathcal{V}_{(n+1,n-1)}$. Consider the complex
conjugated vector $\overline{X}\in\mathcal{V}_{(n-1,n+1)}$ and compute the
action of the Laplacian on them. Let

$\displaystyle\Delta X=\Delta_{(n+1,n-1)}X=\alpha(n)X+\beta(n)\overline{X}\,,$
(7.22)
$\displaystyle\Delta\overline{X}=\Delta_{(n-1,n+1)}\overline{X}=\gamma(n)X+\delta(n)\overline{X}$
(7.23)

with some coefficients $\alpha$, $\beta$, $\gamma$, and $\delta$. That $X$ and
$\overline{X}$ are conjugated and operator $\Delta$ is self-adjoint implies
the relations $\alpha=\overline{\delta}$ and $\beta=\overline{\gamma}$.
Looking for $\ker(\Delta)$ in the form

$Y=F(n)X+G(n)\overline{X}\in\ker(\Delta)$ (7.24)

and acting on $Y$ by the Laplace operator we find that the condition $\Delta
Y=0$ yields

$\Delta Y=F(n)\Delta_{(n+1,n-1)}X+G(n)\Delta_{(n-1,n+1)}\overline{X}=\\\
=\big{(}\alpha(n)F(n)+\overline{\beta(n)}G(n)\big{)}X+\big{(}\beta(n)F(n)+\overline{\alpha(n)}G(n)\big{)}\overline{X}=0\,.$
(7.25)

Since $X$ and $\overline{X}$ are linearly independent, the problem of finding
such $Y=\alpha X+\beta\overline{X}$ that $\Delta Y=0$, amounts to the linear
system

$\begin{bmatrix}\alpha(n)&\overline{\beta(n)}\\\
\beta(n)&\overline{\alpha(n)}\end{bmatrix}\begin{bmatrix}F(n)\\\
G(n)\end{bmatrix}=\begin{bmatrix}0\\\ 0\end{bmatrix}\,,$ (7.26)

which admits non-trivial solutions iff

$\det\begin{bmatrix}\alpha(n)&\overline{\beta(n)}\\\
\beta(n)&\overline{\alpha(n)}\end{bmatrix}=|\alpha(n)|^{2}-|\beta(n)|^{2}=0.$
(7.27)

Hence, we conclude that

$\alpha(n)=\beta(n)\cdot e^{i\chi},\quad\quad\chi\in[0,2\pi)\,.$ (7.28)

In the next section coefficients $\alpha(n)$ and $\beta(n)$ will be shown to
be real, i.e., $e^{i\chi}=\pm 1$.

Summarizing, if we find that the coefficients $\alpha(n)$ and $\beta(n)$
coincide up to a sign, $\alpha(n)=\pm\beta(n)$, this would imply the existence
of a non-trivial $\sigma_{-}$-cocycle

$Y=X\mp\overline{X}\,.$ (7.29)

Otherwise the cohomology is trivial.

####  7.2.4 $H^{1}(\sigma_{-})$ in the near-diagonal sector
$|N-\overline{N}|=2$

To compute $H^{1}(\sigma_{-})$ in the leftover sector of $N=\overline{N}+2$
(analysis at $N=\overline{N}-2$ is analogous) consider a general one-form
$\Theta(y,\overline{y})$ (7.4) with $N=\overline{N}+2$.

In this sector, the Laplacian differs form that at $N>\overline{N}+2$ by the
$T_{4}(y,\overline{y})$ term in

$\Delta_{N=\overline{N}+2}\Theta(y,\overline{y})=\Big{(}\underbrace{N(\overline{N}+2)+y^{\beta}\partial_{\alpha}h^{\alpha}_{\
\dot{\gamma}}D_{\beta}^{\
\dot{\gamma}}+\overline{y}^{\dot{\alpha}}\overline{\partial}_{\dot{\beta}}h_{\gamma\dot{\alpha}}D^{\gamma\dot{\beta}}}_{\Delta_{N>\overline{N}+2}}\Big{)}\Theta(y,\overline{y})+\underbrace{\overline{y}^{\dot{\alpha}}\overline{y}^{\dot{\beta}}\partial_{\alpha}\partial_{\beta}h^{\beta}_{\
\dot{\beta}}D^{\alpha}_{\
\dot{\alpha}}\Theta(y,\overline{y})}_{T_{4}(y,\overline{y})}.$ (7.30)

Consequently, it is essential to compute the action of this additional term.
As before (cf. (7.8)), denote

$T_{4}^{\lambda,\nu,\mu(n-1)|\dot{\lambda},\dot{\nu},\dot{\mu}(m-1)}=\overline{y}^{\dot{\alpha}}\overline{y}^{\dot{\beta}}\partial_{\alpha}\partial_{\beta}h^{\beta}_{\
\dot{\beta}}D^{\alpha}_{\
\dot{\alpha}}\big{(}h^{\lambda\dot{\lambda}}y^{\nu}y^{\mu(n-1)}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-1)}\big{)}\,.$

This yields

$T_{4}^{\lambda,\nu,\mu(n-1)|\dot{\lambda},\dot{\nu},\dot{\mu}(m-1)}=(n-1)\epsilon^{\mu\lambda}h^{\nu}_{\
\dot{\beta}}y^{\mu(n-2)}\overline{y}^{\dot{\lambda}}\overline{y}^{\dot{\beta}}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-1)}+(n-1)\epsilon^{\nu\lambda}h^{\mu}_{\
\dot{\beta}}y^{\mu(n-2)}\overline{y}^{\dot{\lambda}}\overline{y}^{\dot{\beta}}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-1)}+\\\
+(n-1)(n-2)\epsilon^{\mu\lambda}h^{\mu}_{\
\dot{\beta}}y^{\nu}y^{\mu(n-3)}\overline{y}^{\dot{\lambda}}\overline{y}^{\dot{\beta}}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-1)}.$
(7.31)

Projecting $T_{4}$ onto the irreducible parts, we find:

$\displaystyle\text{(A)}:$
$\displaystyle\mathscr{S}_{(\lambda,\nu,\mu)}\mathscr{S}_{(\dot{\lambda},\dot{\nu},\dot{\mu})}T_{4}^{\lambda,\nu,\mu(n-1)|\dot{\lambda},\dot{\nu},\dot{\mu}(m-1)}$
$\displaystyle=0\,,$ (7.32) $\displaystyle\text{(B)}:$ $\displaystyle\ \
\mathscr{S}_{(\dot{\lambda},\dot{\nu},\dot{\mu})}\epsilon_{\lambda\nu}T_{4}^{\lambda,\nu,\mu(n-1)|\dot{\lambda},\dot{\nu},\dot{\mu}(m-1)}$
$\displaystyle=-(n-1)(2n-1)h^{\mu}_{\
\dot{\beta}}y^{\mu(n-2)}\overline{y}^{\dot{\beta}}\overline{y}^{\dot{\mu}(m+1)}\,,$
(7.33) $\displaystyle\text{(C)}:$ $\displaystyle\ \
\epsilon_{\dot{\lambda}\dot{\nu}}\mathscr{S}_{(\lambda,\nu,\mu)}T_{4}^{\lambda,\nu,\mu(n-1)|\dot{\lambda},\dot{\nu},\dot{\mu}(m-1)}$
$\displaystyle=0\,,$ (7.34) $\displaystyle\text{(D)}:$ $\displaystyle\ \
\epsilon_{\dot{\lambda}\dot{\nu}}\epsilon_{\lambda\nu}T_{4}^{\lambda,\nu,\mu(n-1)|\dot{\lambda},\dot{\nu},\dot{\mu}(m-1)}$
$\displaystyle=0\,.$ (7.35)

We observe that the action of the Laplacian in this sector differs from the
previously computed one only in the type-(B) sector, namely,

$\Delta_{N=\overline{N}+2}\left(\Theta_{\mathrm{B}}\right)=\left(n(m+3)\Theta_{\mathrm{B}}+(2n^{2}-3n+1)\Theta_{\mathrm{C}}\right)\Big{|}_{m=n-2}=\underbrace{(n^{2}+n)}_{\alpha(n)}\Theta_{\mathrm{B}}+\underbrace{(2n^{2}-3n+1)}_{\beta(n)}\Theta_{\mathrm{C}}\,.$
(7.36)

That $|\alpha(n)|\neq|\beta(n)|$ at integer $n$ implies triviality of
$H^{1}(\sigma_{-})$ in the near-diagonal sector.

###  7.3 $H^{2}(\sigma_{-})$

The calculation of $H^{2}(\sigma_{-})$ is in main features analogous to that
of $H^{1}(\sigma_{-})$. To decompose a general two-form
$\Omega(y,\overline{y}\,|x)$ into irreducible parts we use the following
useful identity

$h^{\nu\dot{\nu}}\wedge
h^{\lambda\dot{\lambda}}=\frac{1}{2}H^{\nu\lambda}\epsilon^{\dot{\nu}\dot{\lambda}}+\frac{1}{2}\overline{H}^{\dot{\nu}\dot{\lambda}}\epsilon^{\nu\lambda},$
(7.37)

where

$\displaystyle H^{\nu\lambda}=H^{(\nu\lambda)}:=h^{\nu}_{\ \dot{\gamma}}\wedge
h^{\lambda\dot{\gamma}}\,,\qquad\overline{H}^{\dot{\nu}\dot{\lambda}}=H^{(\dot{\nu}\dot{\lambda})}:=h^{\
\dot{\nu}}_{\gamma}\wedge h^{\gamma\dot{\lambda}}\,.$ (7.38)

The decomposition reads

$\Omega(y,\overline{y}\,|x)=\underbrace{\Omega^{\mathrm{A}}_{\mu(n+2)|\dot{\mu}(m)}\
H^{\mu\mu}y^{\mu(n)}\overline{y}^{\dot{\mu}(m)}}_{\Phi_{\mathrm{A}(n,m)}}+\underbrace{\overline{\Omega}^{\mathrm{A}}_{\mu(n)|\dot{\mu}(m+2)}\
\overline{H}^{\dot{\mu}\dot{\mu}}y^{\mu(n)}\overline{y}^{\dot{\mu}(m)}}_{\overline{\Phi}_{\mathrm{A}(n,m)}}+\\\
+\underbrace{\Omega^{\mathrm{B}}_{\mu(n-2)|\dot{\mu}(m)}\
H_{\nu\nu}y^{\nu}y^{\nu}y^{\mu(n-2)}\overline{y}^{\dot{\mu}(m)}}_{\Phi_{\mathrm{B}(n,m)}}+\underbrace{\overline{\Omega}^{\mathrm{B}}_{\mu(n)|\dot{\mu}(m-2)}\overline{H}_{\dot{\nu}\dot{\nu}}y^{\mu(n)}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-2)}}_{\overline{\Phi}_{\mathrm{B}(n,m)}}+\\\
+\underbrace{\Omega^{\mathrm{C}}_{\mu(n)|\dot{\mu}(m)}\ H_{\nu}^{\
\mu}y^{\nu}y^{\mu(n-1)}\overline{y}^{\dot{\mu}(m)}}_{\Phi_{\mathrm{C}(n,m)}}+\underbrace{\overline{\Omega}^{\mathrm{C}}_{\mu(n)|\dot{\mu}(m)}\
\overline{H}_{\dot{\nu}}^{\
\dot{\mu}}y^{\mu(n)}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-1)}}_{{\overline{\Phi}_{\mathrm{C}(n,m)}}}.$
(7.39)

Consider now the reducible two-forms

$\displaystyle\Phi^{\lambda(2),\nu(2),\mu(n-2)|\dot{\mu}(m)}=H^{\lambda\lambda}y^{\nu}y^{\nu}y^{\mu(n-2)}\overline{y}^{\dot{\mu}(m)},$
(7.40a)
$\displaystyle\overline{\Phi}^{\mu(n)|\dot{\lambda}(2),\dot{\nu}(2),\dot{\mu}(m-2)}=\overline{H}^{\dot{\lambda}\dot{\lambda}}y^{\mu(n)}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-2)}.$
(7.40b)

In these terms, the projectors onto irreducible components are

$\displaystyle\mathcal{P}_{\mathrm{A}}$
$\displaystyle=\mathscr{S}_{(\lambda,\nu,\mu)}\,,$
$\displaystyle\mathcal{P}_{\mathrm{B}}=\epsilon_{\lambda\nu}\epsilon_{\lambda\nu}\,,$
$\displaystyle\mathcal{P}_{\mathrm{C}}=\mathscr{S}_{(\lambda,\nu,\mu)}\circ\epsilon_{\lambda\nu}\,,$
(7.41) $\displaystyle\overline{\mathcal{P}}_{\mathrm{A}}$
$\displaystyle=\mathscr{S}_{(\dot{\lambda},\dot{\nu},\dot{\mu})}\,,$
$\displaystyle\overline{\mathcal{P}}_{\mathrm{B}}=\epsilon_{\dot{\lambda}\dot{\nu}}\epsilon_{\dot{\lambda}\dot{\nu}}\,,$
$\displaystyle\overline{\mathcal{P}}_{\mathrm{C}}=\mathscr{S}_{(\dot{\lambda},\dot{\nu},\dot{\mu})}\circ\epsilon_{\dot{\lambda}\dot{\nu}}\,$
(7.42)

and the decomposition (7.39) reads as

$\displaystyle\Phi_{\mathrm{A}}$
$\displaystyle=\mathcal{P}_{\mathrm{A}}\Phi\,,$
$\displaystyle\Phi_{\mathrm{B}}=\mathcal{P}_{\mathrm{B}}\Phi\,,$
$\displaystyle\Phi_{\mathrm{C}}=\mathcal{P}_{\mathrm{C}}\Phi\,,$ (7.43)
$\displaystyle\overline{\Phi}_{\mathrm{A}}$
$\displaystyle=\overline{\mathcal{P}}_{\mathrm{A}}\overline{\Phi}\,,$
$\displaystyle\overline{\Phi}_{\mathrm{B}}=\overline{\mathcal{P}}_{\mathrm{B}}\overline{\Phi}\,,$
$\displaystyle\overline{\Phi}_{\mathrm{C}}=\overline{\mathcal{P}}_{\mathrm{C}}\overline{\Phi}\,.$
(7.44)

For practical calculations we have to find the result of the action of the
operator $D=\frac{\partial}{\partial h}$ on the two-form $H$. The result is

$D_{\alpha\dot{\beta}}H^{\nu\lambda}=D_{\alpha\dot{\beta}}\left(h^{\nu}_{\
\dot{\gamma}}\wedge h^{\lambda\dot{\gamma}}\right)=\epsilon_{\alpha}^{\
\nu}\epsilon_{\dot{\beta}\dot{\gamma}}h^{\lambda\dot{\gamma}}-h^{\nu}_{\
\dot{\gamma}}\,\epsilon_{\alpha}^{\ \lambda}\epsilon_{\dot{\beta}}^{\
\dot{\gamma}}=-\epsilon_{\alpha}^{\ \nu}h^{\lambda}_{\
\dot{\beta}}-\epsilon_{\alpha}^{\ \lambda}h^{\nu}_{\
\dot{\beta}}=-2\epsilon_{\alpha}^{\ (\nu}h^{\lambda)}_{\ \dot{\beta}}\,,$
(7.45)

or, in the condensed notation for symmetrized indices,

$D_{\alpha\dot{\beta}}H^{\nu\nu}=-2\epsilon_{\alpha}^{\ \nu}h^{\nu}_{\
\dot{\beta}}.$ (7.46)

####  7.3.1 $H^{2}(\sigma_{-})$ in the far-from-diagonal sector
$|N-\overline{N}|>2$

Compute $\Delta_{N>\overline{N}+2}$ on the general two-forms $\Phi$ and
$\overline{\Phi}$,

$\Delta_{N>\overline{N}+2}(\Phi)=\underbrace{n(m+2)\Phi(y,\overline{y})}_{T_{1}(y,\overline{y})}+\underbrace{y^{\beta}\partial_{\alpha}h^{\alpha}_{\
\dot{\gamma}}D_{\beta}^{\
\dot{\gamma}}\Phi(y,\overline{y})}_{T_{2}(y,\overline{y})}+\underbrace{\overline{y}^{\dot{\alpha}}\overline{\partial}_{\dot{\beta}}h_{\gamma\dot{\alpha}}D^{\gamma\dot{\beta}}\Phi(y,\overline{y})}_{T_{3}(y,\overline{y})}\,.$
(7.47)

As in (7.8), denote

$T_{2}^{\lambda(2),\nu(2),\mu(n-2)|\dot{\mu}(m)}=y^{\beta}\partial_{\alpha}h^{\alpha}_{\
\dot{\gamma}}D_{\beta}^{\
\dot{\gamma}}H^{\lambda\lambda}y^{\nu}y^{\nu}y^{\mu(n-2)}\overline{y}^{\dot{\mu}(m)}$
(7.48)

and similarly for $T_{1}$ and $T_{3}$. Straightforward computation yields

$\displaystyle T_{1}^{\lambda(2),\nu(2),\mu(n-2)|\dot{\mu}(m)}$
$\displaystyle=n(m+2)\,H^{\lambda\lambda}y^{\nu}y^{\nu}y^{\mu(n-2)}\overline{y}^{\dot{\mu}(m)}\,,$
(7.49a) $\displaystyle T_{2}^{\lambda(2),\nu(2),\mu(n-2)|\dot{\mu}(m)}$
$\displaystyle=-4H^{\nu\lambda}y^{\lambda}y^{\nu}y^{\mu(n-2)}\overline{y}^{\dot{\mu}(m)}-2(n-2)\,H^{\mu\lambda}y^{\lambda}y^{\nu(2)}y^{\mu(n-3)}\overline{y}^{\dot{\mu}(m)}\,,$
(7.49b) $\displaystyle T_{3}^{\lambda(2),\nu(2),\mu(n-2)|\dot{\mu}(m)}$
$\displaystyle=m\,H^{\lambda\lambda}y^{\nu(2)}y^{\mu(n-2)}\overline{y}^{\dot{\mu}(m)}.$
(7.49c)

Projecting onto the irreducible part $\Phi_{\mathrm{A}}$ we obtain

$\Delta_{N>\overline{N}+2}\Phi_{\mathrm{A}}=\mathcal{P}_{\mathrm{A}}\left(T_{1}+T_{2}+T_{3}\right)=\left[n(m+2)-2n+m\right]\Phi_{\mathrm{A}}=m(n+1)\Phi_{\mathrm{A}}.$
(7.50)

We see that $\Phi_{\mathrm{A}}\in\ker(\Delta)$ whenever $m=0$. This gives a
2-cocycle of the form $H^{\mu\mu}y^{\mu(n)}$. It can be represented in terms
of the generating function as follows. Contract all the indices in
$H^{\mu\mu}y^{\mu(n)}$ with some symmetric coefficients
$\Omega^{\mathrm{A}}_{\mu\mu\mu(n)}$ to obtain

$\Omega^{\mathrm{A}}_{\mu(n+2)}H^{\mu\mu}y^{\mu(n)}\equiv h^{(\lambda}_{\
\dot{\gamma}}\wedge
h^{\nu\dot{\gamma})}\Omega_{(\lambda\nu\mu(n))}y^{\mu(n)}\Rightarrow
h^{\lambda}_{\ \dot{\gamma}}\wedge
h^{\nu\dot{\gamma}}\,\partial_{\lambda}\partial_{\nu}C(y,0\,|x)\in\ker\left(\Delta_{N>\overline{N}+2}\Big{|}_{\text{2-forms}}\right),$
(7.51)

where $C(y,0\,|x)=\Omega_{\mu\mu\mu(n)}y^{\mu}y^{\mu}y^{\mu(n)}$. Summarizing,
we found a part of the kernel of $\Delta$ represented by the two-forms

$W(y,0\,|x)=H^{\mu\nu}\partial_{\mu}\partial_{\nu}C(y,0\,|x)$ (7.52)

with $C(y,0|x)$ being a general polynomial of $y$’s of degree $\geq 4$.

Let us now project (7.47) onto the second irreducible part
$\Phi_{\mathrm{B}}$,

$\Delta_{N>\overline{N}+2}\Phi_{\mathrm{B}}=\mathcal{P}_{\mathrm{B}}\left(T_{1}+T_{2}+T_{3}\right)=\left[n(m+2)+0+m\right]\Phi_{\mathrm{B}}\neq
0\quad\forall n,m\in\mathbb{N}_{0}.$ (7.53)

Since $\Phi_{B(n,m)}$ is proportional to
$H_{\nu\nu}y^{\nu}y^{\nu}y^{\mu(n-2)}\overline{y}^{\dot{\mu}(m)}$, the case
$n=m=0$ is beyond the allowed region. Thus, $\Phi_{\mathrm{B}}$ does not
contribute to $H^{2}(\sigma_{-})$.

Projecting (7.47) onto $\Phi_{\mathrm{C}}$, we find

$\Delta_{N>\overline{N}+2}\Phi_{\mathrm{C}}=\mathcal{P}_{\mathrm{C}}\left(T_{1}+T_{2}+T_{3}\right)=[n(m+2)-2-(n-2)+m]\Phi_{\mathrm{C}}=(nm+n+m)\Phi_{\mathrm{C}}\,.$
(7.54)

Again, $\Phi_{\mathrm{C}}$ does not contribute to $H^{2}(\sigma_{-})$ since
$m>0$, $n\geq 0$.

Next, we consider the anti-holomorphic two-form $\overline{\Phi}$. The action
of the Laplacian yields

$\Delta_{N>\overline{N}+2}(\overline{\Phi})=\underbrace{n(m+2)\overline{\Phi}(y,\overline{y})}_{T_{1}(y,\overline{y})}+\underbrace{y^{\beta}\partial_{\alpha}h^{\alpha}_{\
\dot{\gamma}}D_{\beta}^{\
\dot{\gamma}}\overline{\Phi}(y,\overline{y})}_{T_{2}(y,\overline{y})}+\underbrace{\overline{y}^{\dot{\alpha}}\overline{\partial}_{\dot{\beta}}h_{\gamma\dot{\alpha}}D^{\gamma\dot{\beta}}\overline{\Phi}(y,\overline{y})}_{T_{3}(y,\overline{y})}\,.$
(7.55)

As in (7.8) we set

$T_{2}^{\mu(n)|\dot{\lambda}(2),\dot{\nu}(2),\dot{\mu}(m-2)}=y^{\beta}\partial_{\alpha}h^{\alpha}_{\
\dot{\gamma}}D_{\beta}^{\
\dot{\gamma}}\,\overline{H}^{\dot{\lambda}\dot{\lambda}}y^{\mu(n)}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-2)}$
(7.56)

and analogously for $T_{1}$ and $T_{3}$. The computation in components yields

$\displaystyle T_{1}^{\mu(n)|\dot{\lambda}(2),\dot{\nu}(2),\dot{\mu}(m-2)}$
$\displaystyle=n(m+2)\overline{H}^{\dot{\lambda}\dot{\lambda}}y^{\mu(n)}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-2)},$
(7.57a) $\displaystyle
T_{2}^{\mu(n)|\dot{\lambda}(2),\dot{\nu}(2),\dot{\mu}(m-2)}$
$\displaystyle=-n\,\overline{H}^{\dot{\lambda}\dot{\lambda}}y^{\mu(n)}\overline{y}^{\dot{\nu}(2)}\overline{y}^{\dot{\mu}(m-2)},$
(7.57b) $\displaystyle
T_{3}^{\mu(n)|\dot{\lambda}(2),\dot{\nu}(2),\dot{\mu}(m-2)}$
$\displaystyle=-4\,\epsilon^{\dot{\nu}\dot{\lambda}}\overline{H}_{\dot{\alpha}}^{\
\dot{\lambda}}y^{\mu(n)}\overline{y}^{\dot{\alpha}}\overline{y}^{\dot{\nu}}\overline{y}^{\dot{\mu}(m-2)}-2(m-2)\,\epsilon^{\dot{\mu}\dot{\lambda}}\overline{H}_{\dot{\alpha}}^{\
\dot{\lambda}}y^{\mu(n)}\overline{y}^{\dot{\alpha}}\overline{y}^{\dot{\nu}(2)}\overline{y}^{\dot{\mu}(m-3)}.$
(7.57c)

Projecting onto the irreducible components we obtain

$\displaystyle\Delta_{N>\overline{N}+2}\overline{\Phi}_{\mathrm{A}}$
$\displaystyle=\overline{\mathcal{P}}_{\mathrm{A}}\left(T_{1}+T_{2}+T_{3}\right)=[n(m+1)]\overline{\Phi}_{\mathrm{A}}\,,$
(7.58) $\displaystyle\Delta_{N>\overline{N}+2}\overline{\Phi}_{\mathrm{B}}$
$\displaystyle=\overline{\mathcal{P}}_{\mathrm{B}}\left(T_{1}+T_{2}+T_{3}\right)=(nm+n+4m)\overline{\Phi}_{\mathrm{B}}\,,$
(7.59) $\displaystyle\Delta_{N>\overline{N}+2}\overline{\Phi}_{\mathrm{C}}$
$\displaystyle=\overline{\mathcal{P}}_{\mathrm{C}}\left(T_{1}+T_{2}+T_{3}\right)=(nm+n-m)\overline{\Phi}_{\mathrm{C}}\,.$
(7.60)

The condition $n>m+2$ valid in the far-from-diagonal sector does not allow
$\overline{\Phi}_{\mathrm{A,B,C}}$ to be in the kernel of $\Delta$.

The analysis of the opposite sector $N<\overline{N}-2$ is analogous via
swapping dotted and undotted indices. As a result, the final answer for the
under-diagonal sector is

$W(0,\overline{y}\,|x)=\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}C(0,\overline{y}\,|x).$
(7.61)

This completes the analysis of $H^{2}(\sigma_{-})$ in the sector
$|N-\overline{N}|>2$. The cohomology is represented by the two-forms

$W(y,\overline{y}\,|x)=h^{\mu}_{\ \dot{\gamma}}\wedge
h^{\nu\dot{\gamma}}\,\partial_{\mu}\partial_{\nu}C(y,0\,|x)+h_{\gamma}^{\
\dot{\mu}}\wedge
h^{\gamma\dot{\nu}}\,\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}C(0,\overline{y}\,|x).$
(7.62)

These two-forms are known to represent the so-called Weyl cocycle in the HS
theory. It is thus shown that there are no other non-trivial 2-cocycles in
this sector.

####  7.3.2 $H^{2}(\sigma_{-})$ on the diagonal $N=\overline{N}$

Now we prove that there are no non-trivial cocycles at $N=\overline{N}$ except
for the Weyl cohomology (7.62). As before, act by the operator
$\Delta_{N=\overline{N}}$ on the two-form
$\Phi^{\lambda(2)|\nu(2)|\mu(n-2)|\dot{\mu}(n)}(y,\overline{y})=H^{\lambda\lambda}y^{\nu}y^{\nu}y^{\mu(n-2)}\overline{y}^{\dot{\mu}(n)}$

$\Delta_{N=\overline{N}}\Phi(y,\overline{y})=\underbrace{\overline{y}^{\dot{\alpha}}\overline{\partial}_{\dot{\beta}}h_{\gamma\dot{\alpha}}D^{\gamma\dot{\beta}}\Phi(y,\overline{y})}_{T_{1}(y,\overline{y})}+\underbrace{y^{\alpha}\partial_{\beta}h_{\alpha\dot{\gamma}}D^{\beta\dot{\gamma}}\Phi(y,\overline{y})}_{T_{2}(y,\overline{y})}-\\\
\underbrace{-\overline{y}^{\dot{\alpha}}y^{\beta}\partial_{\alpha}\overline{\partial}_{\dot{\beta}}h^{\alpha}_{\
\dot{\alpha}}D_{\beta}^{\
\dot{\beta}}\Phi(y,\overline{y})}_{T_{3}(y,\overline{y})}\underbrace{-y^{\alpha}\overline{y}^{\dot{\beta}}\partial_{\beta}\overline{\partial}_{\dot{\alpha}}h_{\alpha}^{\
\dot{\alpha}}D^{\beta}_{\
\dot{\beta}}\Phi(y,\overline{y})}_{T_{4}(y,\overline{y})}\,.$ (7.63)

Denoting

$T_{1}^{\lambda(2),\nu(2),\mu(n-2)|\dot{\mu}(n)}=\overline{y}^{\dot{\alpha}}\overline{\partial}_{\dot{\beta}}h_{\gamma\dot{\alpha}}D^{\gamma\dot{\beta}}\,H^{\lambda\lambda}y^{\nu(2)}y^{\mu(n-2)}\overline{y}^{\dot{\mu}(n)}$
(7.64)

and analogously for $T_{2}$, $T_{3}$ and $T_{4}$, straightforward computation
yields

$\displaystyle T_{1}^{\lambda(2),\nu(2),\mu(n-2)|\dot{\mu}(n)}$
$\displaystyle=$ $\displaystyle
n\,H^{\lambda\lambda}y^{\nu(2)}y^{\mu(n-2)}\overline{y}^{\dot{\mu}(n)}\,,$
(7.65a) $\displaystyle T_{2}^{\lambda(2),\nu(2),\mu(n-2)|\dot{\mu}(n)}$
$\displaystyle=$ $\displaystyle-4\epsilon^{\nu\lambda}H_{\alpha}^{\
\lambda}y^{\alpha}y^{\nu}y^{\mu(n-2)}\overline{y}^{\dot{\mu}(n)}-{}$ (7.65b)
$\displaystyle-2(n-2)\epsilon^{\mu\lambda}H_{\alpha}^{\
\lambda}y^{\alpha}y^{\nu(2)}y^{\mu(n-3)}\overline{y}^{\dot{\mu}(n)}\,,$
$\displaystyle T_{3}^{\lambda(2),\nu(2),\mu(n-2)|\dot{\mu}(n)}$
$\displaystyle=$ $\displaystyle 4n\,h^{\nu}_{\ \dot{\alpha}}\wedge
h^{\lambda\dot{\mu}}\,y^{\lambda}y^{\nu}y^{\mu(n-2)}\overline{y}^{\dot{\alpha}}\overline{y}^{\dot{\mu}(n-2)}+{}$
(7.65c) $\displaystyle+2n(n-2)\,h^{\mu}_{\ \dot{\alpha}}\wedge
h^{\lambda\dot{\mu}}y^{\lambda}y^{\nu(2)}y^{\mu(n-3)}\overline{y}^{\dot{\alpha}}\overline{y}^{\dot{\mu}(n-1)}\,,$
$\displaystyle T_{4}^{\lambda(2),\nu(2),\mu(n-2)|\dot{\mu}(n)}$
$\displaystyle=$ $\displaystyle 4n\,\epsilon^{\nu\lambda}\,h_{\alpha}^{\
\dot{\mu}}\wedge h^{\lambda}_{\
\dot{\beta}}\,y^{\alpha}y^{\nu}y^{\mu(n-2)}\overline{y}^{\dot{\beta}}\overline{y}^{\dot{\mu}(n-1)}+{}$
(7.65d) $\displaystyle+2n(n-2)\,\epsilon^{\mu\lambda}\,h_{\alpha}^{\
\dot{\mu}}\wedge h^{\lambda}_{\
\dot{\beta}}\,y^{\alpha}y^{\nu(2)}y^{\mu(n-3)}\overline{y}^{\dot{\beta}}\overline{y}^{\dot{\mu}(n-1)}.$

and