FIAN/TD/1-24

Off-Shell Fields and Conserved Currents

E.O. Spirin1,2 and M.A. Vasiliev1,2
1I.E. Tamm Department of Theoretical Physics, Lebedev Physical Institute,
Leninsky prospect 53, 119991, Moscow, Russia
2Moscow Institute of Physics and Technology,
Institutsky lane 9, 141700, Dolgoprudny, Moscow region, Russia

###### Abstract

We study interactions of higher-spin massless fields $\varphi$ with conserved
currents multilinear in the off-shell matter fields $\phi$. Specifically, we
focus on the 3d case where a slight modification of the
$\sigma_{-}$-cohomology technique developed earlier is directly applicable to
control nontriviality of the interaction vertices at the convention that the
vertices removable by a local field redefinition of a higher-spin field or
having schematic form $F(\varphi)G(\phi)$, where $F(\varphi)$ is a gauge
invariant field strength of a free higher-spin field, are called
deformationally trivial. It is demonstrated how the $\sigma_{-}$-cohomology
approach can be applied to the analysis of nonlinear vertices. Generally,
deformationally trivial vertices are not $\sigma_{-}$-closed while the
deformationally non-trivial ones must be in $H(\sigma_{-})$. It is shown that,
at least in the $3d$ case, the relevant cohomology group $H^{1}(\sigma_{-})=0$
and, hence, no deformationally non-trivial off-shell vertices exist. On the
other hand, there exists an infinite class of deformationally trivial
vertices, that includes the vertex recently proposed for spin three. Our
analysis goes beyond the higher-spin vertices allowing to show that, at least
in three dimensions, nonlinear combinations of the off-shell scalar fields and
their derivatives cannot obey non-trivial equations.

###### Contents

  1. 1 Introduction
  2. 2 $\sigma_{-}$-cohomology
  3. 3 $H^{0}(\sigma_{-}$)
  4. 4 $H^{1}(\sigma_{-})$
  5. 5 Spin three example
  6. 6 Conclusion
  7. A Appendix

##  1 Introduction

In the Fronsdal theory of massless fields [1], a spin–$s$ field
$\varphi_{\mu_{1}\dots\mu_{s}}(x)$ obeys the double-tracelessness conditions
$\eta^{\alpha\beta}\eta^{\gamma\delta}\varphi_{\alpha\beta\gamma\delta\mu_{5}\ldots\mu_{s}}=0$.
Spin–$s$ massless field equations

${\cal
R}_{\mu_{1}\dots\mu_{s}}(\varphi):=\Box\varphi_{\mu_{1}\dots\mu_{s}}-s\partial_{(\mu_{1}}\partial^{\alpha}\varphi_{\mu_{2}\dots\mu_{s})\alpha}+\frac{s(s-1)}{2}\partial_{(\mu_{1}}\partial_{\mu_{2}}\varphi_{\mu_{3}\dots\mu_{s})\alpha}^{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\alpha}=0\,,$
(1.1)

where $\partial_{\mu}:=\frac{\partial}{\partial x^{\mu}}$, are invariant under
the gauge transformations

$\delta\varphi_{\mu_{1}\dots\mu_{s}}(x)=\partial_{(\mu_{1}}\xi_{\mu_{2}\ldots\mu_{s})}(x)$
(1.2)

with traceless gauge parameters
$\eta^{\nu\rho}\xi_{\nu\rho\mu_{4}\ldots\mu_{s}}(x)=0$.

It is well known how to construct interactions of higher-spin (HS) massless
fields with conserved currents $J(\phi)$ bilinear in the fluctuating fields
$\phi$ of various spins (see e.g. [2, 3, 4]). In that case, the HS currents
$J(\phi)$ are conserved on shell, i.e. at the conditions that the fields
$\phi$ obey their free field equations.

In the paper [5] a conserved spin-three current $T^{\mu\nu\lambda}[\phi]$
trilinear in the off-shell scalar field $\phi$ was found within the BV-
formalism. An off-shell field is a field that obeys no equations of motion.
Nevertheless, multilinear combinations of derivatives of such fields, which we
call composite fields, still can satisfy some partial differential equations.
For instance, such equations may be associated with the gauge invariance. In
particular, equations for the spin-three current $T^{\mu\nu\lambda}$, built
from an off-shell scalar $\phi$,

$T^{\mu\nu\lambda}=\big{[}\partial^{\mu}\partial^{\nu}\partial^{\lambda}\phi\;\phi^{2}+2\partial^{(\mu}\partial^{\nu}\phi\;\partial^{\lambda)}\phi\;\phi+2\partial^{\mu}\phi\;\partial^{\nu}\phi\;\partial^{\lambda}\phi-\frac{1}{2}\eta^{(\mu\nu}\partial^{\lambda)}\Box\phi\;\phi^{2}-$
$-2\eta^{(\mu\nu}\partial^{\lambda)}\partial_{\sigma}\phi\;\partial^{\sigma}\phi\;\phi-\eta^{(\mu\nu}\partial^{\lambda)}\phi\;\partial_{\sigma}\phi\;\partial^{\sigma}\phi-\Box\phi\;\eta^{(\mu\nu}\partial^{\lambda)}\phi\;\phi\big{]}\,,$
(1.3)

found in [5]111Here (…) denotes the total
symmetrisation:$A_{(\mu_{1}\dots\mu_{k})}=\frac{1}{k!}\sum_{\sigma}A_{\sigma(\mu_{1})\dots\sigma(\mu_{k})}$.,
imply the gauge invariance of the quartic spin-three spin-zero vertex

$S=\int d^{d}x\;\varphi_{\mu\nu\lambda}T^{\mu\nu\lambda}[\phi]$ (1.4)

under the gauge transformations

$\delta\varphi^{\mu\nu\lambda}=\partial^{(\mu}\xi^{\nu\lambda)},\;\;\;\;\;\delta\phi=0,\;\;\;\;\;\eta_{\mu\nu}\xi^{\mu\nu}=0,$
(1.5)

where $\xi^{\mu\nu}$ is a traceless gauge parameter,
$\Box=\partial_{\mu}\partial^{\mu}$, and $\eta_{\mu\nu}$ is the Minkowski
metric in $d$ dimensions. This implies the current conservation equation

$\xi_{\nu\lambda}\partial_{\mu}T^{\mu\nu\lambda}=0\quad\Rightarrow\quad\partial_{\mu}T^{\mu\nu\lambda}=\eta^{\nu\lambda}K[\phi]$
(1.6)

with some $K[\phi]$.

Note that, though in our case the field $\phi$ is off-shell, the same gauge-
invariant vertex exists for an on-shell scalar field with the standard kinetic
term in the Lagrangian. Analogous interaction vertices both in Minkowski and
AdS space were studied more generally in [6, 7, 8]. According to these works
such vertices must be either on-shell trivial (i.e., removable by a local
field redefinition), or composed from the gauge invariant field strengths in
which case the gauge transformations remain undeformed (Abelian). Both types
of such vertices will be called deformationally trivial to stress that they do
not induce a deformation of the HS gauge symmetry. Note that cubic vertices
for arbitrary spins in $3d$ Minkowski space were classified in [9].

In this paper we develop a technics allowing to check whether there exist
composite fields built from the off-shell fields, that obey some non-trivial
partial differential equations including the conservation conditions like
(1.6). This problem will be analyzed in terms of the $\sigma_{-}$-cohomology
technique [10] allowing to control the dynamical content of the theory in
question. Note that this approach is free from the additional conjectures on
the possibility to use non-local inverse operators as in [6, 7, 8].

Those components $T^{pr}(x)$ of the (composite) fields, that cannot be
expressed via derivatives of the other components, are called dynamical
(primary). Primary components of the fields described by the $p$-forms are
represented by the the $\sigma_{-}$-cohomology $H^{p}(\sigma_{-})$ [10]. The
primary components may or may not obey some differential equations represented
by $H^{p+1}(\sigma_{-})$. The descendants $T^{dec}=L(\partial)T^{pr}$ result
from the action of various differential operators on the primary components.
There are always some descendant fields that obey certain differential
equations (e.g., Bianchi identities) expressing the properties of the
operators $L(\partial)$. Such equations are in a certain sense trivial
expressing the properties of $L(\partial)$ rather than $T^{pr}(x)$. These are
not represented by any $\sigma_{-}$-cohomology. Specifically, in the HS
theory, there are two types of the operators $L(\partial)$ leading to the
gauge invariant current interactions. These are $L_{F}(\partial)$ associated
with the l.h.s. of the Fronsdal equations and $L_{W}(\partial)$ associated
with the gauge invariant combinations of the Fronsdal fields like Faradey
tensor in the spin-one case or Weyl tensor in the spin-two case. The terms
with $L_{F}(\partial)$ are removable by appropriate field redefinitions in the
free action. Those with $L_{W}(\partial)$ describe interactions with the gauge
invariant field strengths. These are not removable by a local field
redefinition but do not affect the gauge transformation law, hence called
deformationally trivial.

Specifically, the $\sigma_{-}$-cohomology analysis of the so-called higher-
rank fields to be used in this paper, has been elaborated in [4] for the $3d$
case. In these terms, a $3d$ off-shell scalar field can be realized as a rank-
two field while its cubic combination is a rank-six field. However, the
approach developed in [4] for fields of any rank operates with the lowest
weight vectors of the Howe-dual algebra (for more detail see Section 4) while,
as discussed in Section 3, the off-shell scalar fields carry zero weights,
that requires some modification of the scheme. Our final result shows that the
$\sigma_{-}$-cohomology representing possible non-trivial partial differential
equations (in particular, conservation conditions) obeyed by cubic
combinations $T^{\mu\nu\lambda}[\phi]$ of the $3d$ off-shell scalar fields
$\phi$ is empty. In turn, in agreement with [6, 7, 8], this implies that
$T^{\mu\nu\lambda}[\phi]$ cannot be a primary conserved current. Hence, it has
to be an improvement while the vertex (1.4) must be removable by a local field
redefinition. The explicit form of the latter is found in Section 5.

The aim of this paper is to illustrate how the $\sigma_{-}$-cohomology
technique can be used for the analysis of (non-)triviality of vertices. As a
consequence it will be shows that, at least in three space-time dimensions,
there are no primary composite fields, that obey non-trivial equations
(including, the conservation conditions) built from any number of the off-
shell scalar fields and their derivatives as well as from HS conserved
currents.

The rest of the paper is organized as follows. The $\sigma_{-}$-cohomology
technique is recalled in Section 2. In Sections 3 and 4, the analysis of the
$\sigma_{-}$-cohomology groups $H^{0}(\sigma_{-})$ and $H^{1}(\sigma_{-})$ is
presented for the composite fields built from the $3d$ off-shell scalar fields
and HS conserved currents. In Section 5, our results are compared against
those of [5]. Section 6 contains conclusions. Some technical details of the
$\sigma_{-}$-cohomology analysis are in Appendix.

##  2 $\sigma_{-}$-cohomology

In this section, we briefly sketch relevant aspects of the HS theory and
$\sigma_{-}$-cohomology approach referring for more detail to [11].

The $Sp(2M)$-invariant equations, that describe $3d$ and $4d$ massless fields
at $M=2$ and $M=4$, respectively, are [12]

$\left(dX^{\alpha\beta}\frac{\partial}{\partial
X^{\alpha\beta}}+\sigma_{-}\right)C(y|X)=0,\;\;\;\;\;\sigma_{-}=dX^{\alpha\beta}\frac{\partial^{2}}{\partial
y^{\alpha}\partial y^{\beta}},\;\;\;\;\;(\sigma_{-})^{2}=0\,,$ (2.1)

where $y^{\alpha}$ are auxilary commuting variables,

$C(y|X)=\sum_{n=0}^{\infty}C_{\alpha_{1}\dots\alpha_{n}}(X)y^{\alpha_{1}}\dots
y^{\alpha_{n}}\,,$ (2.2)

$dX^{\alpha\beta}$ are anticommuting differentials for symmetric matrix space-
time coordinates $X^{\alpha\beta}=X^{\beta\alpha}$ and
$\alpha,\beta=1,\dots,M$. In this paper we confine ourselves to the case of
$M=2$ associated with a $3d$ massless theory.

Let us recall some terminology. A field is called auxillary if it is expressed
in terms of other fields and their derivatives by virtue of equations of
motion or can be gauge fixed to zero by some shift (Stueckelberg) gauge
symmetry. The dynamical sector consists of the fields that are neither
expressed in terms of the other fields nor are pure gauge. Thus, it makes
sense to focus on the dynamical fields and their equations of motion. For
instance, Fronsdal fields are of that type.

It can be shown [10] that dynamical fields and their field equations
associated with the equations (2.1) are contained in zeroth cohomology
$H^{0}(\sigma_{-})$ and first cohomology $H^{1}(\sigma_{-})$, respectively.
Recall that

$H^{p}(\sigma_{-}):=\frac{\ker(\sigma_{-})}{\text{Im}(\sigma_{-})}\bigg{|}_{p},$
(2.3)

where $|_{p}$ implies restriction to the space of $p-$forms. That dynamical
fields are in zeroth cohomology is fairly simple. Indeed, it is obvious from
(2.1) that the fields, that are not in $\ker\sigma_{-}$, are expressed in
terms of space-time derivatives of the fields from $\ker\sigma_{-}$.

The fields (2.2) describe a so-called rank-one system. The rank-$r$ fields are

$C(y|X)=\sum_{n}f^{i_{1}\dots
i_{n}}_{\alpha_{1}\dots\alpha_{n}}(X)y_{i_{1}}^{\alpha_{1}}\dots
y_{i_{n}}^{\alpha_{n}},$ (2.4)

where $i_{k}=1\dots,r$. The rank-two system with

$(\sigma_{-})_{r=2}=\frac{\partial}{\partial
y_{-}^{\alpha}}\frac{\partial}{\partial y_{+}^{\beta}}dX^{\alpha\beta}$ (2.5)

was studied in [13] where it was shown, in particular, that if the operator

$h=y_{+}^{\alpha}\frac{\partial}{\partial
y_{+}^{\alpha}}-y_{-}^{\alpha}\frac{\partial}{\partial
y_{-}^{\alpha}},\quad[h,(\sigma_{-})_{r=2}]=0$ (2.6)

has zero eigenvalue then the corresponding fields are off-shell scalars. We
will consider them as off-shell fields from which the current (1.3) is built.

In our case the relevant operator $\sigma_{-}$ is

$\sigma_{-}=\sum_{i}\frac{\partial}{\partial
y_{-,i}^{\alpha}}\frac{\partial}{\partial y_{+,i}^{\beta}}dX^{\alpha\beta}.$
(2.7)

We are interested in the trilinear combinations of scalar off-shell fields
because the current (1.3) is cubic in them. Each of the three pairs of
variables $y_{\pm,i}$ is associated with its own scalar field, that in
addition obeys the restriction

$h_{j}C(y_{j}|X)=0\,,\qquad
h_{j}:=\left(y_{-,j}^{\alpha}\frac{\partial}{\partial
y_{-,j}^{\alpha}}-y_{+,j}^{\alpha}\frac{\partial}{\partial
y_{+,j}^{\alpha}}\right)\,,\qquad\forall j\,.$ (2.8)

This is the same restriction as (2.5) in the rank-two case.

Since the operators $h_{j}$ commute with $\sigma_{-}$,

$[h_{j},\sigma_{-}]=0\,,$ (2.9)

the fields corresponding to the different eigenvalues of $h_{j}$ form
different representations. Hence, each eigenvalue corresponds to its own
subsystem. In [13] it was shown that, generally, such systems describe
conserved currents of various spins, that include currents of spin zero
satisfying no conservation conditions thus being off-shell. So, we analyze a
three-linear current built from the spin zero fields, that obey (2.8).

In the end of this section let us recall the Poincaré homotopy trick
conventionally used to calculate the cohomology group. Let $D$ be a linear
operator acting in a Hilbert space $V$, such that $D^{2}=0$. Let there also be
a nilpotent operator $D^{*}$, $(D^{*})^{2}=0$. The homotopy operator is

$\Delta:=\left\\{D,D^{*}\right\\}\,.$ (2.10)

By its definition it obeys

$[D,\Delta]=[D^{*},\Delta]=0\,$ (2.11)

and, hence,

$\Delta\ker D\subset\text{Im}D\,.$

Then it follows that $H(D):=\ker D/\text{Im}D\subset\ker D/\Delta(\ker D)$.
Let the operators $D$ and $D^{*}$ be conjugated in the Hilbert space $V$ and
$V=\sum\oplus V_{A}$ for a set of finite-dimensional $V_{A}$ such that
$\Delta(V_{A})\subset V_{A}$ and $V_{A}$ be orthogonal to $V_{B}$ at $A\neq
B$. Then $\Delta$ can be diagonalized after which it is easily observed that
$\ker D/\Delta(\ker D)=\ker\Delta\cap\ker D$. As a result,

$H(D)\subset\ker\Delta\cap\ker D\,$ (2.12)

and the problem amounts to solution of the homotopy equation

$\Delta f=0\,.$ (2.13)

##  3 $H^{0}(\sigma_{-}$)

Here we analyze $H^{0}(\sigma_{-})$ with $\sigma_{-}$ (2.7), where the index
$i$ ranges from 1 to 3 because we are interested in the cubic combinations of
the off-shell scalar fields. The variables $y_{\pm,i}$ with different $i$ are
associated with different off-shell scalar fields. Let us rewrite $\sigma_{-}$
in the form

$\sigma_{-}=\frac{\partial}{\partial y_{a}^{\alpha}}\frac{\partial}{\partial
y_{b}^{\beta}}\eta_{ab}dX^{\alpha\beta},\;\;\;\;\;\eta_{ab}=\left(\begin{array}[]{cc}0&\mathds{1}_{3\times
3}\\\ \mathds{1}_{3\times 3}&0\end{array}\right)\,,\qquad a,b=1\ldots 6\,,$
(3.1)

where indices are relabelled as follows:
$[(-,1);(-,2);(-,3);(+,1);(+,2);(+,3)]\rightarrow[1,2,3,4,5,6]$.

Recall that the scalar fields are those with $\\#y_{+}=\\#y_{-}$ which
condition is expressed by (2.8). The operators $h_{j}$ (2.8) belong to
$o(3,3)$ with the generators $\tau_{ij}$,

$\tau_{mn}=y_{m}^{\alpha}\frac{\partial}{\partial
y^{\alpha,n}}-y_{n}^{\alpha}\frac{\partial}{\partial
y^{\alpha,m}},\;\;\;\;\;[\tau_{mn},\tau_{kl}]=\eta_{nk}\tau_{ml}+\eta_{ml}\tau_{nk}-\eta_{mk}\tau_{nl}-\eta_{nl}\tau_{mk}\,,$
(3.2)

that commute with $\sigma_{-}$,

$[\sigma_{-},\tau_{mn}]=0\,.$ (3.3)

In the original index notations $\tau_{mn}$ read as

$\tau_{12}=y_{-,1}^{\alpha}\frac{\partial}{\partial
y_{+,2}^{\alpha}}-y_{-,2}^{\alpha}\frac{\partial}{\partial
y_{+,1}^{\alpha}},\;\;\;\;\;\tau_{13}=y_{-,1}^{\alpha}\frac{\partial}{\partial
y_{+,3}^{\alpha}}-y_{-,3}^{\alpha}\frac{\partial}{\partial y_{+,1}^{\alpha}},$
(3.4) $\tau_{14}=y_{-,1}^{\alpha}\frac{\partial}{\partial
y_{-,1}^{\alpha}}-y_{+,1}^{\alpha}\frac{\partial}{\partial
y_{+,1}^{\alpha}},\;\;\;\;\;\tau_{15}=y_{-,1}^{\alpha}\frac{\partial}{\partial
y_{-,2}^{\alpha}}-y_{+,2}^{\alpha}\frac{\partial}{\partial y_{+,1}^{\alpha}},$
(3.5) $\tau_{16}=y_{-,1}^{\alpha}\frac{\partial}{\partial
y_{-,3}^{\alpha}}-y_{+,3}^{\alpha}\frac{\partial}{\partial
y_{+,1}^{\alpha}},\;\;\;\;\;\tau_{23}=y_{-,2}^{\alpha}\frac{\partial}{\partial
y_{+,3}^{\alpha}}-y_{-,3}^{\alpha}\frac{\partial}{\partial y_{+,2}^{\alpha}},$
(3.6) $\tau_{24}=y_{-,2}^{\alpha}\frac{\partial}{\partial
y_{-,1}^{\alpha}}-y_{+,1}^{\alpha}\frac{\partial}{\partial
y_{+,2}^{\alpha}},\;\;\;\;\;\tau_{25}=y_{-,2}^{\alpha}\frac{\partial}{\partial
y_{-,2}^{\alpha}}-y_{+,2}^{\alpha}\frac{\partial}{\partial y_{+,2}^{\alpha}},$
(3.7) $\tau_{26}=y_{-,2}^{\alpha}\frac{\partial}{\partial
y_{-,3}^{\alpha}}-y_{+,3}^{\alpha}\frac{\partial}{\partial
y_{+,2}^{\alpha}},\;\;\;\;\;\tau_{34}=y_{-,3}^{\alpha}\frac{\partial}{\partial
y_{-,1}^{\alpha}}-y_{+,1}^{\alpha}\frac{\partial}{\partial y_{+,3}^{\alpha}},$
(3.8) $\tau_{35}=y_{-,3}^{\alpha}\frac{\partial}{\partial
y_{-,2}^{\alpha}}-y_{+,2}^{\alpha}\frac{\partial}{\partial
y_{+,3}^{\alpha}},\;\;\;\;\;\tau_{36}=y_{-,3}^{\alpha}\frac{\partial}{\partial
y_{-,3}^{\alpha}}-y_{+,3}^{\alpha}\frac{\partial}{\partial y_{+,3}^{\alpha}},$
(3.9) $\tau_{45}=y_{+,1}^{\alpha}\frac{\partial}{\partial
y_{-,2}^{\alpha}}-y_{+,2}^{\alpha}\frac{\partial}{\partial
y_{-,1}^{\alpha}},\;\;\;\;\;\tau_{46}=y_{+,1}^{\alpha}\frac{\partial}{\partial
y_{-,3}^{\alpha}}-y_{+,3}^{\alpha}\frac{\partial}{\partial y_{-,1}^{\alpha}},$
(3.10) $\tau_{56}=y_{+,2}^{\alpha}\frac{\partial}{\partial
y_{-,3}^{\alpha}}-y_{+,3}^{\alpha}\frac{\partial}{\partial y_{-,2}^{\alpha}}$
(3.11)

with mutually commuting Cartan elements

$h_{1}=\tau_{14}\,,\qquad h_{2}=\tau_{25}\,,\qquad h_{3}=\tau_{36}\,.$ (3.12)

Introducing operators $y_{i}^{\alpha}$ and
$W_{\alpha}^{i}=\frac{\partial}{\partial y_{i}^{\alpha}}$, that obey

$[W_{\alpha}^{i},y_{j}^{\beta}]=\delta_{j}^{i}\delta_{\alpha}^{\beta},\;\;\;\;\;[y_{i}^{\alpha},y_{j}^{\beta}]=0,\;\;\;\;\;[W_{\alpha}^{i},W_{\beta}^{i}]=0\,,$
(3.13)

$sp(2M)$ can be represented in the form

$T^{\alpha\beta}=y_{i}^{\alpha}y_{j}^{\beta}\delta^{ij},\;\;\;\;\;T_{\alpha\beta}=W_{\alpha}^{i}W_{\beta}^{j}\delta_{ij},\;\;\;\;\;T_{\beta}^{\alpha}=\frac{1}{2}\left\\{y_{j}^{\alpha},W_{\beta}^{j}\right\\},$
(3.14)
$[T_{\alpha\beta},T^{\gamma\delta}]=\delta_{\alpha}^{\gamma}T_{\beta}^{\delta}+\delta_{\alpha}^{\delta}T_{\beta}^{\gamma}+\delta_{\beta}^{\gamma}T_{\alpha}^{\delta}+\delta_{\beta}^{\delta}T_{\alpha}^{\gamma},$
(3.15)
$[T_{\alpha}^{\beta},T^{\gamma\delta}]=\delta_{\alpha}^{\gamma}T^{\beta\delta}+\delta_{\alpha}^{\delta}T^{\beta\gamma},\;\;\;\;\;[T_{\alpha}^{\beta},T_{\gamma\delta}]=-\delta_{\gamma}^{\beta}T_{\alpha\delta}-\delta_{\delta}^{\beta}T_{\alpha\gamma}.$
(3.16)

Being mutually commuting, algebras $o(3,3)$ and $sp(2M)$ form a Howe dual pair
[14].

The operator $\sigma_{-}^{*}$ conjugated to $\sigma_{-}$ is defined
analogously to [4]:

$\sigma_{-}=T_{\alpha\beta}dX^{\alpha\beta},\quad\sigma_{-}^{*}=T^{\alpha\beta}\frac{\partial}{\partial(dX^{\alpha\beta})},\;\;\;\;\;(\sigma_{-}^{*})^{2}=0\,.$
(3.17)

An elementary computation yields

$\Delta:=\left\\{\sigma_{-},\sigma_{-}^{*}\right\\}=\nu^{\alpha}_{\beta}\nu^{\beta}_{\alpha}+\frac{1}{2}\tau_{mn}\tau^{mn}-(M-2)\nu_{\alpha}^{\alpha},$
(3.18)

where $\frac{1}{2}\tau_{mn}\tau^{mn}$ is the quadratic Casimir of $o(3,3)$,

$\nu_{\beta}^{\alpha}:=\rho^{\alpha}_{\beta}+\Upsilon^{\alpha}_{\beta},\;\;\;\;\;\rho^{\alpha}_{\beta}:=2dX^{\alpha\gamma}\frac{\partial}{\partial(dX^{\beta\gamma})}\;\;\;\;\;\Upsilon^{\alpha}_{\beta}:=T^{\alpha}_{\beta}-\frac{3}{2}\delta^{\alpha}_{\beta}\,,$
(3.19)
$\frac{\partial}{\partial(dX)^{\alpha\beta}}dX^{\gamma\delta}=\frac{1}{2}\left(\delta_{\alpha}^{\gamma}\delta_{\beta}^{\delta}+\delta_{\alpha}^{\delta}\delta_{\beta}^{\gamma}\right)-dX^{\gamma\delta}\frac{\partial}{\partial(dX)^{\alpha\beta}}\,.$
(3.20)

To calculate the part of $H^{0}(\sigma_{-})$ associated with the off-shell
scalar fields we (i) choose a Chevalley basis in $o(3,3)$, consisting of
raising $e_{i}$, lowering $f_{i}$ and Cartan operators $h_{i}$, (ii) find the
lowest vectors annihilated by the lowering operators, (iii) act by the raising
operators on the lowest vectors to obtain the vectors annihilated by the
Cartan operators

$f_{1}|vac\rangle=f_{2}|vac\rangle=f_{3}|vac\rangle=0\,,$ (3.21)
$h_{1}F(e_{1},e_{2},e_{3})|vac\rangle=h_{2}F(e_{1},e_{2},e_{3})|vac\rangle=h_{3}F(e_{1},e_{2},e_{3})|vac\rangle=0.$
(3.22)

If $|vac\rangle\in\ker\Delta$, from the commutativity of $o(3,3)$ and $sp(2M)$
it follows that any vector $F(e_{1},e_{2},e_{3})|vac\rangle$ also belongs to
$\ker\Delta$. Hence all solutions of (3.22) also belong to
$H^{0}(\sigma_{-})$.

The Chevalley decomposition is ($\alpha_{i}$ are simple roots)

$\tilde{h}_{i}=\frac{2}{\langle\alpha_{i}\alpha_{i}\rangle}h_{\alpha_{i}},\;\;\;\;\;e_{i}=e_{\alpha_{i}},\;\;\;\;\;f_{i}=\frac{2}{\langle\alpha_{i}\alpha_{i}\rangle(e_{\alpha_{i}},e_{-\alpha_{i}})}e_{-\alpha_{i}},\;\;\;\;\;(e_{i},f_{i})=\frac{2}{\langle\alpha_{i}\alpha_{i}\rangle}.$
(3.23)

In this basis the commutation relations are

$[\tilde{h}_{i},\tilde{h}_{j}]=0,\;\;\;\;\;[\tilde{h_{i}},e_{i}]=a_{ij}e_{j},\;\;\;\;\;[\tilde{h}_{i},f_{j}]=-a_{ij}f_{j},\;\;\;\;\;[e_{i},f_{j}]=\delta_{ij}\tilde{h}_{i},$
(3.24)

where $a_{ij}$ is the Cartan matrix,

$a_{ij}=\frac{2\langle\alpha_{j}\alpha_{i}\rangle}{\langle\alpha_{i}\alpha_{i}\rangle}\,.$
(3.25)

In the case of interest it reads as

$a_{ij}=\left(\begin{array}[]{ccc}2&-1&-1\\\ -1&2&0\\\
-1&0&2\end{array}\right)$ (3.26)

and the Chevalley basis is

$\tilde{h}_{1}=h_{1}+h_{2},\;\;\;\;\;\tilde{h}_{2}=-h_{1}+h_{3},\;\;\;\;\;\tilde{h}_{3}=-h_{1}-h_{3},$
(3.27)
$e_{1}=\tau_{12},\;\;\;\;\;e_{2}=\tau_{34},\;\;\;\;\;e_{3}=\tau_{46},\;\;\;\;\;f_{1}=-\tau_{45},\;\;\;\;\;f_{2}=\tau_{16},\;\;\;\;\;f_{3}=-\tau_{13}.$
(3.28)

It is worth mentioning that the zero-weight constraints (3.22) are equivalent
for $\tilde{h}_{i}$ and $h_{i}$.

Let us start with the lowest weight vacuum conditions (3.21)

$\begin{cases}\left(y_{-,1}^{\alpha}\frac{\partial}{\partial
y_{-,3}^{\alpha}}-y_{+,3}^{\alpha}\frac{\partial}{\partial
y_{+,1}^{\alpha}}\right)C=0,\\\ \left(y_{-,1}^{\alpha}\frac{\partial}{\partial
y_{+,3}^{\alpha}}-y_{-,3}^{\alpha}\frac{\partial}{\partial
y_{+,1}^{\alpha}}\right)C=0,\\\ \left(y_{+,1}^{\alpha}\frac{\partial}{\partial
y_{-,2}^{\alpha}}-y_{+,2}^{\alpha}\frac{\partial}{\partial
y_{-,1}^{\alpha}}\right)C=0,\\\
\tilde{h_{1}}C=\mu_{1}C,\;\;\tilde{h_{2}}C=\mu_{2}C,\;\;\tilde{h_{3}}C=\mu_{3}C\,,\end{cases}$
(3.29)

where $\mu_{i}$ are the lowest weights.

According to the method of characteristics, the general solution of a system
of first-order equations is a function of elementary solutions. The elementary
solutions to the first three equations (3.29) are

$G_{1}^{\alpha}=y_{+,2}^{\alpha},\quad
G_{2}^{\alpha\alpha}=y_{+,1}^{\alpha}y_{-,1}^{\alpha}+y_{+,2}^{\alpha}y_{-,2}^{\alpha}+y_{-,3}^{\alpha}y_{+,3}^{\alpha}.$
(3.30)

The proof that these are indeed elementary solutions will be given in Appendix
A.

The solution $G_{1}$ is obvious because there are no $y_{+,2}^{\alpha}$
derivatives in (3.29). The solution $G_{2}$ is irrelevant because it does not
belong to the kernel of $\sigma_{-}$ while $G_{1}$ does hence belonging to
$H^{0}(\sigma_{-})$. Now we are in a position to analyze how the raising
operators act on this solution. As mentioned above, functions of elemetrary
solutions also satisfy the set of equations. So we will take for example
$\psi_{1}^{\alpha(2)}=y_{+,2}^{\alpha}y_{+,2}^{\alpha}$ and
$\psi_{2}^{\alpha(4)}=y_{+,2}^{\alpha}y_{+,2}^{\alpha}y_{+,2}^{\alpha}y_{+,2}^{\alpha}$.
Searching through all the possible combinations of raising operators, we find
the solutions, annihilated by the Cartan operators:

$e_{3}e_{1}e_{2}e_{1}\psi_{1}^{\alpha(2)}=2y_{+,1}^{\alpha}y_{-,1}^{\alpha}-2y_{-,3}^{\alpha}y_{+,3}^{\alpha},\;\;\;\;\;e_{1}e_{3}e_{2}e_{1}\psi_{1}^{\alpha(2)}=2y_{+,1}^{\alpha}y_{-,1}^{\alpha}-2y_{-,2}^{\alpha}y_{+,2}^{\alpha},$
(3.31)
$e_{2}e_{1}e_{3}e_{1}\psi_{1}^{\alpha(2)}=2y_{+,1}^{\alpha}y_{-,1}^{\alpha}-2y_{+,3}^{\alpha}y_{-,3}^{\alpha}.$
(3.32)

A linear combination of these solutions can be written as

$\Psi_{1}^{\alpha(2)}=Ay_{+,1}^{\alpha}y_{-,1}^{\alpha}+By_{-,2}^{\alpha}y_{+,2}^{\alpha}+Cy_{+,3}^{\alpha}y_{-,3}^{\alpha},\;\;\;\;\;A+B+C=0.$
(3.33)

It is easy to check that $\Psi_{1}^{\alpha(2)}$ belongs to the kernel of
operator $\sigma_{-}$. Now we examine $\psi_{2}^{\alpha(4)}$:

$\Psi_{2}^{\alpha(4)}=e_{3}e_{2}e_{1}e_{1}e_{3}e_{2}e_{1}e_{1}\psi_{2}^{\alpha(4)}=192(y_{-,1}^{\alpha}y_{+,1}^{\alpha})^{2}-192y_{-,1}^{\alpha}y_{+,1}^{\alpha}y_{-,2}^{\alpha}y_{+,2}^{\alpha}-$
$-576y_{-,1}^{\alpha}y_{+,1}^{\alpha}y_{-,3}^{\alpha}y_{+,3}^{\alpha}+192y_{-,2}^{\alpha}y_{+,2}^{\alpha}y_{-,3}^{\alpha}y_{+,3}^{\alpha}+96(y_{-,3}^{\alpha}y_{+,3}^{\alpha})^{2}.$
(3.34)

It can be checked straightforwardly that $\Psi_{2}^{\alpha(4)}$ also belongs
to $H^{0}(\sigma_{-})$.

Thus, checking all possible combinations of the raising operators acting on
the vacuum solutions $(y_{+,2}^{\alpha})^{n}$, new functions, annihilated by
Cartan operators, can be found, associated with the off-shell scalar field ,

$h_{1}F(e_{1},e_{2},e_{3})(y_{+,2}^{\alpha})^{n}=h_{2}F(e_{1},e_{2},e_{3})(y_{+,2}^{\alpha})^{n}=h_{3}F(e_{1},e_{2},e_{3})(y_{+,2}^{\alpha})^{n}=0\,.$
(3.35)

##  4 $H^{1}(\sigma_{-})$

In the case of $H^{1}(\sigma_{-})$ we consider the lowest weight sector
following [4], where this problem has been solved. Note that in [4]
$\tau_{mn}$ were representing algebra $o(6)$ rather than $o(3,3)$ as in this
paper. However, for the finite-dimensional modules, the representation theory
of orthogonal Lie algebras is equivalent (up to possible (anti)selfduality
conditions irrelevant to the analysis of this paper) to that of the pseudo-
orthogonal ones. This allows us to use the results of [4] directly. In [4],
the general case of $sp(2M)$ and $o(r)$ was considered while in our problem we
confine ourselves to $M=2$, implying $d=3$, and $r=6$, implying that the
current is trilinear in the off-shell scalars.

The problem is to solve the homotopy equation (2.13) with the homotopy
operator (3.18) expressed as a sum of quadratic Casimirs of $o(r)$ and
$gl(M)$. The representation is described in terms of the $gl(M)$ Young
diagrams (YD) which determine the symmetry of the l.h.s.’s of equations of
motion and dynamical fields. These $gl(M)$ YD are associated with traceless
tensors with respect to the color indices $a,b$ in (3.1) that have the same
symmetry properties as the $gl(M)$ YD. We say that the respective traceless
symmetric $o(r)$ tensors are described by the traceless symmetric YD.

In [4] it was shown that $H^{1}(\sigma_{-})$ consists of the traceless tensors
described by the YD ${\bf
Y}_{1}(n_{1},\dots,n_{m},\underbrace{2,\dots,2}_{r+1-2m-q},\underbrace{1,\dots,1}_{q})$,
where $2m+q\leqslant r$. In our case of $r=6$ these are

1) m=0, q=0…6, boxsize=1em 2,2,2,2,2,2,2 2,2,2,2,2,2,1 2,2,2,2,2,1,1 …
2,2,1,1,1,1,1 2,1,1,1,1,1,1 ,

2) m=1, q=0…4, 6,2,2,2,2,2 6,2,2,2,2,1 … 6,2,1,1,1,1 ,

3) m=2, q=0,1,2, 6,4,2,2,2 6,4,2,2,1 6,4,2,1,1 ,

4) m=3, q=0, 9,7,5,2 .

According to the two-column theorem (see, e.g., [15]) traceless tensors
corresponding to Young diagrams, in which the sum of the heigths of the first
two columns exceeds $r$, vanish. This impliess that all these diagrams are
trivial, which manifests the comment from [4] that for $M<r$ many diagrams
trivialize. In turn, this implies that, in the case of interest,
$H^{1}(\sigma_{-})=0$. Therefore, there are no non-trivial equations obeyed by
the trilinear combinations of the off-shell scalar fields.

Naively, this contradicts the results of [5]. Before explaining the origin of
this seeming disagreement in Section 5 let us make the following comment. It
is easily observed that such YD are trivial also for higher ranks ($r>6$),
which stands for arbitrary nonlinear combinations of currents. Indeed, the sum
of lengths of the first two columns $l$ of diagrams ${\bf
Y}_{1}(n_{1},\dots,n_{m},\underbrace{2,\dots,2}_{r+1-2m-q},\underbrace{1,\dots,1}_{q})$
is $l=2r+2-2m-q$. Taking into account that $2m+q\leqslant r$ it is obvious
that $l>r$. Thus, all these diagrams and corresponding equations on the
multilinear composite fields must be trivial. It also should be mentioned that
$H^{1}(\sigma_{-})$ describes the equations of motion for fields of arbitrary
spin. According to [13], the subsystems associated with different eigenvalues
of operators $h_{i}$ (2.8) describe conserved currents of different spins.
That $H^{1}(\sigma_{-})$ is empty means that actually there are no dynamical
equations not only for multilineal combinations of scalar currents, but also
for multilinear combinations of spin-$s$ conserved currents in three
dimensions.

##  5 Spin three example

Thus it is shown that composite fields in question obey no differential
equations. On the other hand, the current (1.3) found in [5] does obey the
conservation condition. This seeming contradiction can be resolved however if
the current (1.3) is actually deformationally trivial in which case the action
term (1.4) can be compensated by a local field redefinition of the free
Fronsdal action.

The spin-three Fronsdal tensor is [1]:

${\cal
R}_{\mu\nu\lambda}[\varphi]=\Box\varphi_{\mu\nu\lambda}-3\partial_{(\mu}\partial^{\alpha}\varphi_{\nu\lambda)\alpha}+3\partial_{(\mu}\partial_{\nu}\varphi_{\;\;\lambda)\alpha}^{\alpha}.$
(5.1)

By plugging

$T^{pr}_{\mu\nu\lambda}:=\eta_{(\mu\nu}\partial_{\lambda)}\phi\;\phi^{2}$
(5.2)

into this expression we get the tensor $T_{\mu\nu\lambda}$ (1.3):

${\cal R}_{\mu\nu\lambda}[T^{pr}]=(-2)T_{\mu\nu\lambda}.$ (5.3)

This implies that there exists such a local transformation of the spin-three
field $\varphi_{\mu\nu\lambda}$ that eliminates the vertex (1.4). Indeed, by
substituting

$\varphi_{\mu\nu\lambda}\to\varphi_{\mu\nu\lambda}+\delta\varphi_{\mu\nu\lambda}$
(5.4)

into the Fronsdal action

$S[\varphi]:=\frac{(-1)^{s+1}}{2}\int
d^{d}x(\varphi^{\mu_{1}\dots\mu_{s}}{\cal
R}_{\mu_{1}\dots\mu_{s}}(\varphi)-\frac{s(s-1)}{4}\varphi_{\nu}^{\;\;\;\nu\mu_{3}\dots\mu_{s}}{\cal
R}^{\beta}_{\;\;\;\beta\mu_{3}\dots\mu_{s}}(\varphi))\,$ (5.5)

and using that, as one can easily check, it is a symmetric functional,

$\int d^{d}x\left(\delta\varphi^{\mu_{1}\dots\mu_{s}}{\cal
R}_{\mu_{1}\dots\mu_{s}}(\varphi)\right)=\int
d^{d}x\left(\varphi^{\mu_{1}\dots\mu_{s}}{\cal
R}_{\mu_{1}\dots\mu_{s}}(\delta\varphi)\right)\,,$ (5.6)

one observes that the substitution (5.4) induces the vertex (1.4).

Clearly, there exists an infinite number of such transformations and
corresponding nonlinear conserved currents, because an arbitrary combination
of derivatives and fields
$P_{\alpha_{1}\alpha_{2}\dots}[\partial_{\alpha};\phi]$ would produce a
current in the same way as for $T^{pr}_{\mu\nu\lambda}$. Being removable by
local field redefinitions all such vertices are dynamically trivial, however,
which conclusion is in agreement with the results of [6, 7, 8].

Also let us stress that the resulting currents, including that of (5.3), are
of the form $L_{F}(\partial)T^{pr}$ discussed in Introduction, where
$L_{F}(\partial)$ is the Fronsdal operator on the RHS of (5.1).

##  6 Conclusion

In this paper, interaction vertices of spin-$s$ massless fields and composite
fields identified with nonlinear combinations of other fields $\phi$, that may
obey no field equations thus being off-shell, are considered. It is
illustrated how the $\sigma_{-}$-cohomology technique can be applied to the
analysis of the interaction vertices. To this end, composite fields are
interpreted as primaries of the tensor products of the unfolded modules
associated with $\phi$. It is explained that, in general, there may be two
types of vertices. The deformationally trivial ones, that do not demand the
deformation of the transformation law, are associated with the descendants of
the unfolded vertex modules. The non-trivial vertices, associated in
particular with usual conserved currents, are in the $\sigma_{-}$-cohomology.

For simplicity we focus on the case of three dimensions, though some general
conclusions are made for any dimension. The $\sigma_{-}$-cohomology technique,
which allows one to control the dynamical sector of the theory in question,
was applied to find out whether composite fields obey any non-trivial
equations of motion as well as the nontriviality of their interaction
vertices. As explained in Section 2 the problem amounts to the solution of the
homotopy equation with the restriction (2.8) that the eigenvalues of the
operators $h_{i}$ are zero. The key observation, that reduces the problem to
the Lie algebra representation theory, is that $h_{i}$ form the Cartan
subalgebra of the $o(3,3)$ Howe dual [14] to the space-time algebra. It is
explained in detail how to calculate $H^{0}(\sigma_{-})$ in order to find the
independent composite fields in the theory. Howe duality plays instrumental
role in this analysis.

Using the results of [4] we have shown that the first cohomology is empty,
$H^{1}(\sigma_{-})=0$, which means that there are no non-trivial equations of
motion obeyed by the composite fields. In particular, this implies that the
conserved current of [5] must be deformationally trivial, that was checked
directly in any dimension. Moreover, that $H^{1}(\sigma_{-})=0$ implies that
no non-trivial equations of motion exist for all (not only trilinear)
multilinear combinations of spin-$s$ currents with spin-zero current
identified with the off-shell scalar field. Thus, all possible interaction
vertices between spin-$s$ fields and multilinear currents must be
deformationally trivial at least in three dimensions. In agreement with the
earlier papers [6, 7, 8], we anticipate this result to be true in any
dimension. Note however that our analysis is free from the conjectures of [6,
7, 8] on the possibility to use certain non-local operators in the proof.
Another comment is that our results go far beyond the specific case of the
conservation conditions related to HS dynamics, stating that primary
components of the composite combinations of the off-shell fields and their
derivatives cannot obey any non-trivial partial differential equations at
least in three dimensions.

## Acknowledgments

The authors are grateful to Olga Gelfond for useful discussions and comments
and to Dmitry Ponomarev for the correspondence. The comments by Anatoly Shabad
are also appreciated. MV is grateful to Petr Lavrov and Yuri Zinoviev for the
illuminating conversation and wishes to thank Ofer Aharony, Theoretical High
Energy Physics Group of Weizmann Institute of Science for hospitality during
accomplishment of this work.

##  Appendix A Appendix

In this appendix we explain in detail how the system (3.29) can be solved. Let
us examine the first two equations in this system:

$\begin{cases}\left(y_{-,1}^{\alpha}\frac{\partial}{\partial
y_{-,3}^{\alpha}}-y_{+,3}^{\alpha}\frac{\partial}{\partial
y_{+,1}^{\alpha}}\right)C=0,\\\ \left(y_{-,1}^{\alpha}\frac{\partial}{\partial
y_{+,3}^{\alpha}}-y_{-,3}^{\alpha}\frac{\partial}{\partial
y_{+,1}^{\alpha}}\right)C=0.\\\ \end{cases}$ (A.1)

We will look for a solution in the form
$C=\sum_{n=0}(y_{-,1}^{\alpha})^{n}C_{n}$. In the zeroth order in
$y_{-,1}^{\alpha}$ we get $\frac{\partial C_{0}}{\partial
y_{+,1}^{\alpha}}=0$, that is

$C_{0}=C_{0}(y_{\pm,3})\,.$ (A.2)

In the first order in $y_{-,1}^{\alpha}$ we obtain

$\begin{cases}\frac{\partial}{\partial
y_{-,3}^{\alpha}}C_{0}=y_{+,3}^{\alpha}\frac{\partial}{\partial
y_{+,1}^{\alpha}}C_{1},\\\ \frac{\partial}{\partial
y_{+,3}^{\alpha}}C_{0}=y_{-,3}^{\alpha}\frac{\partial}{\partial
y_{+,1}^{\alpha}}C_{1}.\end{cases}$ (A.3)

Suppose that $C_{0}$ can be expressed as
$C_{0}=y_{-,3}^{\alpha}y_{+,3}^{\alpha}C_{0}^{\prime}(y_{\pm,3})$. Then

$\begin{cases}C_{0}^{\prime}+y_{-,3}^{\alpha}\frac{\partial}{\partial
y_{-,3}^{\alpha}}C_{0}^{\prime}=\frac{\partial}{\partial
y_{+,1}^{\alpha}}C_{1},\\\
C_{0}^{\prime}+y_{+,3}^{\alpha}\frac{\partial}{\partial
y_{+,3}^{\alpha}}C_{0}^{\prime}=\frac{\partial}{\partial
y_{+,1}^{\alpha}}C_{1}\,.\end{cases}$ (A.4)

Since the left-hand side is $y_{+,1}^{\alpha}$ independent, the equations can
be easily integrated

$\begin{cases}C_{1}=y_{+,1}^{\alpha}(C_{0}^{\prime}+y_{-,3}^{\alpha}\frac{\partial}{\partial
y_{-,3}^{\alpha}}C_{0}^{\prime})+c_{1}(y_{\pm,3}),\\\
C_{1}=y_{+,1}^{\alpha}(C_{0}^{\prime}+y_{+,3}^{\alpha}\frac{\partial}{\partial
y_{+,3}^{\alpha}}C_{0}^{\prime})+c_{2}(y_{\pm,3})\,.\end{cases}$ (A.5)

By subtracting the second equation from the first, we get

$y_{-,3}^{\alpha}\frac{\partial}{\partial
y_{-,3}^{\alpha}}C_{0}^{\prime}=y_{+,3}^{\alpha}\frac{\partial}{\partial
y_{+,3}^{\alpha}}C_{0}^{\prime},\quad c_{1}=c_{2}=C_{0}^{1}(y_{\pm,3})\,.$
(A.6)

This implies

$C_{0}=\sum_{n}\alpha_{n}(y_{-,3}^{\alpha}y_{+,3}^{\alpha})^{n}.$ (A.7)

Plugging (A.7) into (A.5) we obtain

$C_{1}=y_{+,1}^{\alpha}C^{1}_{1}+C_{0}^{1},\;\;\;\;\;C_{1}^{1}=\sum_{n}n\alpha_{n}(y_{-,3}^{\alpha}y_{+,3}^{\alpha})^{n-1},\;\;\;\;\;C_{0}^{1}=C_{0}^{1}(y_{\pm,3}).$
(A.8)

The equations for other $C_{k}^{n}$ are analogous. Finally, the solution can
be written in the form of $y$ polynomials:

$C_{b}^{m}=\sum_{k=0}^{n-m-b}(k+b)_{b}\alpha_{k,m-b}(y_{-,3}^{\alpha}y_{+,3}^{\alpha})^{k},\;\;\;\;\;(k+b)_{b}:=\prod_{\gamma=0}^{b-1}(k+b-\gamma)=\frac{(k+b)!}{k!},\;\;\;\;\;C_{n}=\sum_{b=0}^{n}\frac{(y_{+,1}^{\alpha})^{b}}{b!}C_{b}^{n},$
(A.9)
$C=\sum_{b=0}^{m}\sum_{k=0}^{n-m}\sum_{m=0}^{n}(y_{-,1}^{\alpha})^{m}\frac{(y_{+,1}^{\alpha})^{b}}{b!}(k+b)_{b}\alpha_{k+b,m-b}(y_{-,3}^{\alpha}y_{+,3}^{\alpha})^{k}.$
(A.10)

Since the equations (A.1) are first-order partial differential equations they
can be solved by the method of characteristics implying that any function of
some solutions is a solution. As is not hard to check straightforwardly, there
are the following two simple solutions

$g_{1}^{\alpha}=y_{-,1}^{\alpha},\;\;\;g_{2}^{\alpha_{1}\alpha_{2}}=y_{-,3}^{\alpha_{1}}y_{+,3}^{\alpha_{2}}+y_{-,1}^{\alpha_{1}}y_{+,1}^{\alpha_{2}}\,.$
(A.11)

At $n=1$ a solution (A.10) can be written in the form

$C=(\alpha_{00}+(\alpha_{10})_{\alpha\alpha}y_{-,3}^{\alpha}y_{+,3}^{\alpha})+y_{-,1}^{\alpha}((\alpha_{01})_{\alpha}+y_{+,1}^{\alpha}(\alpha_{10})_{\alpha\alpha})=\alpha_{00}+(\alpha_{10})_{\alpha\alpha}g_{2}^{\alpha\alpha}+(\alpha_{01})_{\alpha}g_{1}^{\alpha}\,.$
(A.12)

It is not hard to make sure by induction that the obtained solution can be
expressed as a polynomial of $g_{1}$ and $g_{2}$ at any order. Assuming that
this is true at the $n$-th step one can easily check that it is also true at
the ($n+1$)-th step. Thus it is shown that the solution (A.10) is a polynomial
of $g_{1}^{\alpha}$ and $g_{2}^{\alpha_{1}\alpha_{2}}$ (A.11).

Solutions of the full system of equations (3.21) can be searched in the form
$g_{1,2}+P(y_{\pm 2})$. After some straightforward calculations the elementary
solutions can be written down as:

$G_{1}^{\alpha}=y_{+,2}^{\alpha},\quad
G_{2}^{\alpha\alpha}=y_{+,1}^{\alpha}y_{-,1}^{\alpha}+y_{+,2}^{\alpha}y_{-,2}^{\alpha}+y_{-,3}^{\alpha}y_{+,3}^{\alpha}\,.$
(A.13)

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