FIAN/TD/15-2022

Shifted Homotopy Analysis of the Linearized Higher-Spin Equations in Arbitrary
Higher-Spin Background

A.A. Tarusov1,2, 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

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Analysis of the first-order corrections to higher-spin equations is extended
to homotopy operators involving shift parameters with respect to the spinor
$Y$ variables, the argument of the higher-spin connection $\omega(Y)$ and the
argument of the higher-spin zero-form $C(Y)$. It is shown that a relaxed
uniform $(y+p)$-shift and a shift by the argument of $\omega(Y)$ respect the
proper form of the free higher-spin equations and constitute a one-parametric
class of vertices that contains those resulting from the conventional (no
shift) homotopy. A pure shift by the argument of $\omega(Y)$ is shown not to
affect the one-form higher-spin field $W$ in the first order and, hence, the
form of the respective vertices.

###### Contents

  1. 1 Introduction
  2. 2 Higher-spin equations
  3. 3 Perturbative analysis
  4. 4 Shifted homotopy
  5. 5 General shift parameters
  6. 6 Admissible shift parameters
  7. 7 Pure $\omega$-shift
  8. 8 Conclusion

##  1 Introduction

Higher-spin (HS) gauge theory describes an infinite tower of gauge fields of
all spins. Nonlinear field equations for $4d$ massless fields of all spins
were found in [1, 2]. They admit $AdS_{4}$ as the most symmetric vacuum
solution. The presence of $AdS_{4}$ radius as a dimensionful parameter in HS
vertices potentially allows an infinite number of higher-derivative terms.
Because of this HS gauge theory is not a local field theory in the usual
sense. Instead of space-time locality, spin-locality (that is locality for any
finite subset of fields) in the space of auxiliary spinor variables can be
achieved at least in the lowest orders [3, 4, 5, 6, 7, 8, 9]. In the lowest
order, spin-locality in the spinor space is equivalent to space-time spin-
locality. The conditions allowing to extend this property to higher orders
were found recently in [10].

In the approach of [11], HS fields in $AdS_{4}$ are described by the one-form
$\omega(Y;K|x)$ and zero-form $C(Y;K|x)$ that depend on space-time coordinates
$x$, auxiliary variables $Y_{A}=(y_{\mu},\overline{y}_{\dot{\mu}})$,
$\mu,\dot{\mu}=1,2$, and Klein operators $K$. Both $\omega(Y;K|x)$ and
$C(Y;K|x)$ are regular functions of $Y^{A}$ that serve as the generating
functions for the component fields

$F(Y;K|x)=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{1}{n!m!}F^{\mu_{1}...\mu_{n},}{}^{\dot{\mu}_{1}...\dot{\mu}_{m}}(K|x)y_{\mu_{1}}...y_{\mu_{n}}\overline{y}_{\dot{\mu}_{1}}...\overline{y}_{\dot{\mu}_{m}}\,,$
(1.1)

$F=\omega(Y;K|x)$ or $C(Y;K|x)$. The Klein operators $K$ induce the field
doubling that does not matter in the consideration of this section (for more
detail see [11, 12] and Section 2).

Unfolded form of the free HS equations in the gauge sector referred to as
First On-Shell Theorem is [11] (for detailed recent analysis see [13])

$R^{\mu(n),\dot{\mu}(m)}(x)=\delta_{0,n}\,h_{\nu\dot{\mu}}h^{\nu}_{\
\dot{\mu}}\overline{C}^{\dot{\mu}(m+2)}(x)+\delta_{0,m}\,h_{\mu\dot{\nu}}h_{\mu}^{\
\dot{\nu}}C^{\mu(n+2)}(x)\,,$ (1.2)

where only exterior products of differential forms are used (from now on the
wedge symbol is implicit) and111We use a shorthand notation
$\omega_{\mu(n),\dot{\mu}(m)}(x)=\omega_{\mu_{1}...\mu_{n},\dot{\mu}_{1}...\dot{\mu}_{m}}(x)$
for totally symmetric multispinors. Spinor indices are raised and lowered
according to the rules $A^{\mu}=\epsilon^{\mu\nu}A_{\nu}$,
$A_{\mu}=A^{\nu}\epsilon_{\nu\mu}$, $\epsilon_{\nu\mu}=-\epsilon_{\mu\nu}$,
$\epsilon_{12}=\epsilon^{12}=1$ and analogously for dotted indices.

$R^{\mu(n),\dot{\mu}(m)}(x):=D_{L}\omega^{\mu(n),\dot{\mu}(m)}(x)+\lambda(nh^{\mu}_{\
\dot{\rho}}(x)\omega^{\mu(n-1),\dot{\rho}\dot{\mu}(m)}(x)+mh_{\rho}^{\
\dot{\mu}}(x)\omega^{\rho\mu(n),\dot{\mu}(m-1)}(x))\,,$ (1.3)

where $\lambda$ is the inverse $AdS$ radius, and
$D_{L}=\mbox{d}_{x}+\varpi+\overline{\varpi}$ is a Lorentz-covariant
derivative, with space-time de Rham derivative $\mbox{d}_{x}$ and Cartan’s
spin-connection $(\varpi\oplus\overline{\varpi})$ and

$D_{L}\omega_{\mu(n),\dot{\mu}(m)}(x):=\mbox{d}_{x}\omega_{\mu(n),\dot{\mu}(m)}(x)+n\varpi_{\mu}^{\
\nu}(x)\omega_{\nu\mu(n-1),\dot{\mu}(m)}(x)+m\overline{\varpi}_{\dot{\mu}}^{\
\dot{\nu}}(x)\omega_{\mu(n),\dot{\nu}\dot{\mu}(m-1)}(x)\,.$

HS equations are formulated in terms of the zero-forms $C(Y;K|x)$ and one-
forms $\omega(Y;K|x)$. The field variables associated with spin
$s=0,1/2,1,3/2,...$ are

$\omega^{\mu(n),\dot{\mu}(m)}(x):n+m=2(s-1)\,,\qquad
C^{\mu(n),\dot{\mu}(m)}(x):|n-m|=2s\,.$ (1.4)

Fronsdal fields are described in terms of the generalized frame one-form
$\omega^{\mu(n),\dot{\mu}(m)}(x)$ with $n=m$ for bosons and
$\absolutevalue{n-m}=1$ for fermions, the scalar $C(x)$, and the pair of spin
$1/2$ fields: $C^{\mu}(x)$, $C^{\dot{\mu}}(x)$. Fields with other values of
$n,m$ describe derivatives of the Fronsdal field. Specifically, zero-forms
$C(Y;K|x)$ describe gauge invariant combinations of derivatives of the
Fronsdal fields (linearized curvatures) resulting from equation (1.2) and the
equation

$\tilde{D}C(Y;K|x):=\bigg{(}D_{L}-\lambda
h^{\mu\dot{\mu}}\big{(}y_{\mu}\overline{y}_{\dot{\mu}}+\frac{\partial^{2}}{\partial
y^{\mu}\partial\overline{y}^{\dot{\mu}}}\big{)}\bigg{)}C(Y;K|x)=0\,,$ (1.5)

where

$C(Y;K|x)=-C(Y;-K|x)\,,$ (1.6)

which along with (1.2) form a full set of free HS equations for all massless
fields in $AdS_{4}$.

For any fixed spin $s$, the maximal number of derivatives of the Fronsdal
field contained in $\omega^{\mu(n),\dot{\mu}(m)}(x)$ and
$C^{\mu(n),\dot{\mu}(m)}(x)$ is $[s]-1$ and $\frac{n+m}{2}-\\{s\\}$,
respectively. Along with (1.4) this implies that for each spin $s$ there is a
finite number of fields in $\omega(Y;K|x)$ and an infinite number of fields in
$C(Y;K|x)$, which is the source of potential non-locality.

For the analysis of locality it is important to preserve the form (1.2) of the
free HS equations. Indeed, not every scheme of perturbative analysis of the
nonlinear HS equations automatically reproduces free equations in the form
(1.2). As discussed in [11], it may deform the r.h.s. of (1.2) bringing to it
other components of $C(Y;K|x)$. In such a case to reproduce the First On-Shell
Theorem it is necessary to make a field redefinition of the zero-forms
$C(Y;K|x)$, the physical meaning of which is obscure from the locality
perspective. As a consequence, the analysis of (non-)locality of the higher-
order vertices is obstructed either until a field redefinition bringing free
HS equations to the form of First On-Shell Theorem is performed. Thus, it is
vital to keep linearized HS equations in the form (1.2) applying only such
field redefinitions that respect the First On-Shell Theorem.

A useful way to analyse HS equations is to reconstruct interacting vertices in
the unfolded form [11]

$\mbox{d}_{x}\omega=-\omega*\omega+\Upsilon(\omega,\omega,C)+\Upsilon(\omega,\omega,C,C)+...\,,$
(1.7) $\mbox{d}_{x}C=-[\omega,C]_{*}+\Upsilon(\omega,C,C)+...\,,$ (1.8)

where $*$ denotes the Moyal star product underlying the HS algebra [14]

$f(Y)*g(Y)=f(Y)e^{i\epsilon^{AB}\overleftarrow{\partial}_{A}\overrightarrow{\partial}_{B}}g(Y)\,.$
(1.9)

(See [12] for a review and more references.)

In the formulation of HS equations of [2] the derivation of the interaction
vertices amounts to solving first-order differential equations with a
nilpotent differential in the auxiliary spinor space. At each order one faces
the cohomological freedom that effectively encodes the field redefinitions.
The choice of one or another cohomology class is determined by the choice of
the homotopy operator that resolves the differential equation in the spinor
space. Though properly reproducing free HS equations, the seemingly most
natural conventional homotopy of [2] leads to non-local vertices starting from
the second order [15]. In [4] it was suggested that the proper approach is
based on the shifted homotopy, allowing to decrease the level of non-locality
at higher orders, the technique further developed in [5]. The shifted homotopy
operators involve the shifts of arguments of the dynamical HS fields
$\omega(Y;K|x)$ and $C(Y;K|x)$ with some parameters. The shifted homotopy
technique was proven to be efficient by simplifying the analysis of locality
of lower-order vertices [5, 6, 8, 9] (for detail see Section 3). However, in
these papers only the class of shifted homotopy operators dependent on the
derivatives with respect to the spinor arguments $Y$ of $C(Y;K|x)$ was
considered.

The goal of this paper is to fill in this gap by considering a more general
class of shifted homotopy operators involving the derivatives with respect to
the spinor arguments of the HS one-forms $\omega(Y;K|x)$. An important
condition of the linearized analysis in the zero-forms $C(Y;K|x)$ (which
parameterize linearized gauge invariant HS curvatures) considered in this
paper is that it should not affect the form of free HS equations (1.2) since,
otherwise, this would spoil the interpretation of the zero-forms $C(Y;K|x)$ in
terms of derivatives of the HS gauge fields $\omega(Y;K|x)$ ruining the
higher-order locality analysis in terms of $C(Y;K|x)$. The particular form of
the r.h.s. of (1.2) is closely tied to the proper choice of field variables.
In this paper we study the impact of $\omega$-shifts by the argument of
$\omega(Y;K|x)$ on the vertex $\Upsilon(\omega,\omega,C)$ within the shifted
homotopy approach and test a more general linear shift in the homotopy
procedures compared to the ones currently available in the literature, for
instance in [5]. Such shifts do not affect locality of the vertex unless they
change the form of the r.h.s. of (1.2), which is undesirable as explained
above. Our goal is to find the class of shift parameters that preserve the
form of the First On-Shell Theorem. Surprisingly, we find that the free
$\omega$-shift parameters not only respect the form of (1.2) but, in the case
of pure $\omega$-shifts, also do not affect the perturbative corrections to
the HS fields, and thus the nonlinear vertices either, being equivalent to
those resulting from the homotopy with zero $\omega$-shift parameters.We also
find that shifts with respect to $Y$ variables or derivatives $p$ of the $Y$
arguments of $C(Y;K|x)$ can be present in a relaxed uniform way, that is their
shift parameters must be equal to each other at each homotopy procedure step,
but not necessarily shared between the procedures of different homotopy steps
as in [5]. Such shifts preserve the form of the First On-Shell Theorem in the
$AdS_{4}$ background but produce a one-parametric class of pairwise different
vertices for general background HS gauge one-forms.

The paper is organized as follows: in Section 2, the structure of the HS
equations is briefly recalled, perturbative analysis of which is recalled in
Section 3. Section 4 summarizes the key properties of the shifted homotopy
technique relevant to our analysis. These results are then applied in Section
5 to the derivation of the form of the vertices with the shifts acting on the
argument of $\omega(Y;K|x)$. The possible values of the $y$\- and $p$-shift
parameters that preserve the form of the First On-Shell Theorem are deduced in
Section 6, while the effect of the pure $\omega$-shift is investigated in
Section 7. Section 8 contains a brief conclusion.

##  2 Higher-spin equations

In the frame-like approach to HS equations in $AdS_{4}$, dynamics of the
system is encoded in a one-form $\omega(y,\overline{y};K|x)$ and a zero-form
$C(y,\overline{y};K|x)$ which are regular functions of $sp(4)$ spinors
$Y_{A}=(y_{\mu},\overline{y}_{\dot{\mu}})$. $K=(k,\overline{k})$ is a pair of
Klein operators which will be introduced shortly. HS algebra is defined via
Moyal star product (1.9) with the $sp(4)$-invariant form
$\epsilon^{AB}=(\epsilon^{\mu\nu},\epsilon^{\dot{\mu}\dot{\nu}})$, generated
by the relations

$[y_{\mu},y_{\nu}]_{*}=2i\epsilon_{\mu\nu}\,,\qquad[\overline{y}_{\dot{\mu}},\overline{y}_{\dot{\nu}}]_{*}=2i\epsilon_{\dot{\mu}\dot{\nu}}\,,\qquad[y_{\mu},\overline{y}_{\dot{\nu}}]_{*}=0\,.$
(2.1)

Following [2] we introduce auxiliary variables
$Z_{A}=(z_{\mu},\overline{z}_{\dot{\mu}})$ extending the spinor space. The HS
star product in the extended space is

$(f*g)(Z,Y)=\frac{1}{(2\pi)^{4}}\int
dUdVf(Z+U,Y+U)g(Z-V,Y+V)e^{iU_{A}V^{A}}\,.$ (2.2)

Note that for $Z_{A}$-independent functions it reproduces (1.9). The following
commutation relations hold true

$[Y_{A},Y_{B}]_{*}=-[Z_{A},Z_{B}]_{*}=2i\epsilon_{AB},\qquad[Y_{A},Z_{B}]_{*}=0\,.$
(2.3)

The system of HS equations of [2] is

$\displaystyle\mbox{d}_{x}W+W*W=0,$ (2.4)
$\displaystyle\mbox{d}_{x}S+[W,S]_{*}=0\,,$ (2.5)
$\displaystyle\mbox{d}_{x}B+[W,B]_{*}=0\,,$ (2.6) $\displaystyle
S*S=i(\theta^{A}\theta_{A}+\eta
B*\gamma+\overline{\eta}B*\overline{\gamma})\,,$ (2.7)
$\displaystyle[S,B]_{*}=0\,.$ (2.8)

Here $W(Z,Y;K|x)$ is a one-form that encodes $\omega(Y;K|x)$ while
$B(Z,Y;K|x)$ is a zero-form that encodes $C(Y;K|x)$. The field $S(Z,Y;K|x)$ is
a space-time zero-form but a one-form in additional differentials
$\theta^{A}=(\theta^{\mu},\overline{\theta}^{\dot{\mu}})$ that anticommute
with each other and with the space-time de Rham derivative,

$\\{\theta_{A},\theta_{B}\\}=\\{\theta_{A},\mbox{d}_{x}\\}=0\,.$ (2.9)

The central elements of the HS algebra $\gamma$ and $\overline{\gamma}$ on the
r.h.s. of (2.7) are

$\gamma=e^{iz_{\mu}y^{\mu}}k\theta^{\nu}\theta_{\nu}\,,\qquad\overline{\gamma}=e^{i\overline{z}_{\dot{\mu}}\overline{y}^{\dot{\mu}}}\overline{k}\,\overline{\theta}^{\dot{\nu}}\,\overline{\theta}_{\dot{\nu}}\,$
(2.10)

and $\eta$ is a free complex phase parameter such that
$\eta\,\overline{\eta}=1$. It breaks parity in the interacting HS theory
except for the two cases of $\eta=1$ or $\eta=i$ [16]. The Klein operators
$K=(k,\overline{k})$ satisfy

$\\{k,y_{\mu}\\}=\\{k,z_{\mu}\\}=0\,,\qquad[k,\overline{y}_{\dot{\mu}}]=[k,\overline{z}_{\dot{\mu}}]=0\,,\qquad
k^{2}=1\,,$ (2.11)
$\left\\{\theta_{\mu},k\right\\}=\left[\overline{\theta}_{\dot{\mu}},k\right]=0\,,\quad\left[k,\overline{k}\right]=0\,,$
(2.12)

and analogously for $\overline{k}$. Since $k^{2}=\overline{k}^{2}=1$, the
dependence on $k$ and $\overline{k}$ is at most bilinear. The fields decompose
into physical and topological parts. The former are defined by

$W(Z,Y;K|x)=W(Z,Y;-K|x)\,,\qquad B(Z,Y;K|x)=-B(Z,Y;-K|x)\,.$ (2.13)

##  3 Perturbative analysis

Equations (1.7), (1.8) can be extracted from nonlinear HS system (2.4)-(2.8)
via perturbative expansion. The zero-order vacuum solution of the HS equations
is

$\displaystyle W_{0}$
$\displaystyle=\Omega=\frac{i}{4}\big{(}\omega_{\mu\nu}(x)y^{\mu}y^{\nu}+\overline{\omega}_{\dot{\mu}\dot{\nu}}(x)\overline{y}^{\dot{\mu}}\overline{y}^{\dot{\nu}}+2\lambda
h_{\mu\dot{\mu}}(x)y^{\mu}\overline{y}^{\dot{\mu}}\big{)}\,,$ (3.1)
$\displaystyle B_{0}$ $\displaystyle=0\,,$ (3.2) $\displaystyle S_{0}$
$\displaystyle=\theta^{A}Z_{A}\,.$ (3.3)

The fields $W_{0},B_{0},S_{0}$ satisfy equations (2.4)-(2.8) if
$\omega_{\mu\nu}(x),\overline{\omega}_{\dot{\mu}\dot{\nu}}(x)$ are $AdS_{4}$
spin-connections and $h_{\mu\dot{\mu}}(x)$ is $AdS_{4}$ frame-field. (In the
sequel the inverse $AdS$ radius is set to one, $\lambda=1$.) It is important
to notice that

$[S_{0},f(Z,Y;K)]_{*}=-2i\theta^{A}\frac{\partial}{\partial
Z^{A}}f(Z,Y;K)=-2i\mbox{d}_{Z}f(Z,Y;K)\,.$ (3.4)

In the first order, equation (2.8) yields

$[S_{0},B_{1}]_{*}+[S_{1},B_{0}]_{*}=0\,.$ (3.5)

From (3.2) and (3.4) it follows that $B_{1}$ is $Z$-independent,
$B_{1}=C(Y;K|x)$. Therefore, eq.(2.6) leads to

$\mbox{d}_{x}C(Y;K|x)+[\Omega,C(Y;K|x)]_{*}=0\,,$ (3.6)

that yields (1.5) in the physical sector. In the sector of topological
(auxiliary in terminology of [11]) fields, defined as $C(Y;K|x)=C(Y;-K|x)$,
equation (3.6) yields

$\bigg{(}D_{L}+h^{\mu\dot{\mu}}\big{(}y_{\mu}\overline{\partial}_{\dot{\mu}}+\overline{y}_{\dot{\mu}}\partial_{\mu}\big{)}\bigg{)}C(Y;K|x)=0\,.$
(3.7)

For topological gauge fields, $\omega(Y;K|x)=-\omega(Y;-K|x)$, the First On-
Shell Theorem takes the form [2]

$R^{top}{}_{\alpha_{1}\ldots\alpha_{n},\dot{\beta}_{1}\ldots\dot{\beta}_{m}}(x)=-\bigg{[}\delta_{0,n}m(m-1)h_{\gamma\dot{\beta}_{1}}\wedge
h^{\gamma}{}_{\dot{\beta}_{2}}\overline{C}_{\dot{\beta}_{3}\ldots\dot{\beta}_{m}}(x)+\delta_{0,m}n(n-1)h_{\alpha_{1}\dot{\delta}}\wedge
h_{\alpha_{2}}{}^{\dot{\delta}}C_{\alpha_{3}\ldots\alpha_{n}}(x)\bigg{]}\,,$
(3.8)

where

$R^{top}{}_{\alpha_{1}\ldots\alpha_{n},\dot{\beta}_{1}\ldots\dot{\beta}_{m}}(x)=D_{L}\omega_{\alpha_{1}\ldots\alpha_{n},\dot{\beta}_{1}\ldots\dot{\beta}_{m}}(x)-h_{\alpha_{1}\dot{\beta}_{1}}(x)\omega_{\alpha_{2}\ldots\alpha_{n},\dot{\beta}_{2}\ldots\dot{\beta}_{m}}(x)-\\\
-(n+1)(m+1)h^{\mu\dot{\mu}}(x)\omega_{\mu\alpha_{1}\ldots\alpha_{n},\dot{\mu}\dot{\beta}_{1}\ldots\dot{\beta}_{m}}(x)\,.$
(3.9)

The expression for $S_{1}$ via the field $C$ can be extracted from eq.(2.7)

$-2i\mbox{d}_{Z}S_{1}=i\eta C*\gamma+i\overline{\eta}C*\overline{\gamma}\,.$
(3.10)

Now we have to solve the differential equation with the exterior differential
$\mbox{d}_{Z}$. A solution to such an equation is unique up to the choice of
the cohomology class and its representative. Generally, equation

$\mbox{d}_{Z}f(Z,Y;K;\theta)=g(Z,Y;K;\theta)$ (3.11)

with $\mbox{d}_{Z}g(Z,Y;K;\theta)=0$ can be solved by the homotopy trick. It
can be checked that the r.h.s. of (3.10) is $\mbox{d}_{Z}$-closed. Firstly,
following [5], we choose a nilpotent homotopy operator

$\partial=(Z^{A}+Q^{A})\frac{\partial}{\partial\theta^{A}}\,,$ (3.12)

where

$\frac{\partial Q^{B}}{\partial Z^{A}}=0\,.$ (3.13)

Then we introduce operator

$N=\mbox{d}_{Z}\partial+\partial\mbox{d}_{Z}=\theta^{A}\frac{\partial}{\partial\theta^{A}}+\left(Z^{A}+Q^{A}\right)\frac{\partial}{\partial
Z^{A}}$ (3.14)

and the almost inverse operator

$N^{*}g(Z,Y;\theta):=\int_{0}^{1}\frac{dt}{t}g(tZ-(1-t)Q,Y;t\theta),\quad
g(-Q,Y;0)=0\,.$ (3.15)

The contracting homotopy operator

$\Delta_{Q}:=\partial
N^{*},\quad\Delta_{Q}g(Z,Y;\theta)=\left(Z^{A}+Q^{A}\right)\frac{\partial}{\partial\theta^{A}}\int_{0}^{1}\frac{dt}{t}g(tZ-(1-t)Q,Y;t\theta)$
(3.16)

satisfies the resolution of identity

$\\{\mbox{d}_{Z},\Delta_{Q}\\}=1-h_{Q}\,$ (3.17)

with $h_{Q}$ being a cohomology projector

$h_{Q}f(Z;\theta)=f(-Q;0)\,.$ (3.18)

Hence, resolution of identity yields a particular solution to (3.11)

$f=\Delta_{Q}g\,$ (3.19)

as long as $h_{Q}g=0$, which is true in our case. General solution of (3.11)
is

$f(Z,Y;\theta)=\Delta_{Q}g(Z,Y;\theta)+h(Y)+\mbox{d}_{Z}\epsilon(Z,Y;\theta)\,,$
(3.20)

where $h(Y)$ is a cohomology representative and $\epsilon(Z,Y;\theta)$ is a
parameter of gauge transformation ($\mbox{d}_{Z}$-exact term). Transition from
one $Q$ to another affects the $h$ and $\epsilon$-dependent parts of the
solution. The choice of $Q$ in (3.19) affects the choice of field variables,
that can be essential for the analysis of locality. Originally the choice of
$Q=0$ known as the conventional homotopy was studied [2], which led to the
First On-Shell Theorem. More complex shifts were applied in [4]-[9] for the
analysis of locality problem in the non-linear HS theory.

##  4 Shifted homotopy

For the subsequent analysis we recall, following [5], some properties of the
operators $\Delta_{Q}$ and $h_{Q}$ defined in the previous section. Firstly,
operators $\Delta_{Q}$ and $\Delta_{P}$ anticommute

$\Delta_{Q}\Delta_{P}=-\Delta_{P}\Delta_{Q}\,.$ (4.1)

Analogously,

$h_{P}\Delta_{Q}=-h_{Q}\Delta_{P}\,.$ (4.2)

Confining ourselves to the holomorphic variables
$(Z_{A},Y_{A},K)\rightarrow(z_{\mu},y_{\mu},k)$, let us write down how
$\Delta_{b}\Delta_{a}$ and $h_{c}\Delta_{b}\Delta_{a}$ act

$\Delta_{b}\Delta_{a}f(z,y)\theta^{\mu}\theta_{\mu}=2\int_{[0,1]^{3}}d^{3}\tau\delta(1-\tau_{1}-\tau_{2}-\tau_{3})(z+b)_{\nu}(z+a)^{\nu}f(\tau_{1}z-\tau_{3}b-\tau_{2}a,y)\,,$
(4.3)
$h_{c}\Delta_{b}\Delta_{a}f(z,y)\theta^{\mu}\theta_{\mu}=2\int_{[0,1]^{3}}d^{3}\tau\delta(1-\tau_{1}-\tau_{2}-\tau_{3})(b-c)_{\nu}(a-c)^{\nu}f(-\tau_{1}c-\tau_{3}b-\tau_{2}a,y)\,.$
(4.4)

Note that from (4.4) it follows that for any parameter $\kappa$

$h_{(\kappa+1)q_{2}-\kappa q_{1}}\Delta_{q_{2}}\Delta_{q_{1}}=0\,.$ (4.5)

This identity will have important implications later on.

Application of formulas (4.3), (4.4) to $\gamma$ yields

$\displaystyle\Delta_{b}\Delta_{a}\gamma=2\int_{[0,1]^{3}}d^{3}\tau\delta(1-\tau_{1}-\tau_{2}-\tau_{3})(z+b)_{\nu}(z+a)^{\nu}e^{i(\tau_{1}z-\tau_{2}a-\tau_{3}b)_{\mu}y^{\mu}}k\,,$
(4.6) $\displaystyle
h_{c}\Delta_{b}\Delta_{a}\gamma=2\int_{[0,1]^{3}}d^{3}\tau\delta(1-\tau_{1}-\tau_{2}-\tau_{3})(b-c)_{\nu}(a-c)^{\nu}e^{-i(\tau_{1}c+\tau_{2}a+\tau_{3}b)_{\mu}y^{\mu}}k\,.$
(4.7)

Yet another important property of the operators $\Delta_{Q}$ and $h_{P}$,
implying the $z$-independence of the vertices resulting from equations
(2.4)–(2.8), is

$\left(\Delta_{d}-\Delta_{c}\right)\left(\Delta_{a}-\Delta_{b}\right)\gamma=\left(h_{d}-h_{c}\right)\Delta_{a}\Delta_{b}\gamma\,.$
(4.8)

It has a consequence

$(\Delta_{c}\Delta_{b}-\Delta_{c}\Delta_{a}+\Delta_{b}\Delta_{a})\gamma=h_{c}\Delta_{b}\Delta_{a}\gamma\,.$
(4.9)

Other remarkable properties of the shifted homotopy operators also obtained in
[5] are the so-called star-exchange relations with $z$-independent elements

$\Delta_{q+\alpha
y}(C(y;k)*\phi(z,y;k;\theta))=C(y;k)*\Delta_{q+(1-\alpha)p+\alpha
y}\phi(z,y;k;\theta),$ (4.10) $\Delta_{q+\alpha
y}(\phi(z,y;k;\theta)*k^{m}*C(y;k))=\Delta_{q+(-1)^{m}(1-\alpha)p+\alpha
y}(\phi(z,y;k;\theta))*k^{m}*C(y;k)\,.$ (4.11)

Here

$p_{\mu}C(Y;K)=C(Y;K)p_{\mu}:=-i\frac{\partial}{\partial y^{\mu}}C(Y;K)\,.$
(4.12)

Also, for the central element $\gamma$,

$\Delta_{q}\gamma*C(y;k)=C(y;k)*\Delta_{q+2p}\gamma\,.$ (4.13)

Analogous properties hold true for the cohomology projector $h_{q}$

$h_{q+\alpha y}(C(y;k)*\phi(z,y;k;\theta))=C(y;k)*h_{q+(1-\alpha)p+\alpha
y}\phi(z,y;k;\theta),$ (4.14) $h_{q+\alpha
y}(\phi(z,y;k;\theta)*k^{m}*C(y;k))=h_{q+(-1)^{m}(1-\alpha)p+\alpha
y}(\phi(z,y;k;\theta))*k^{m}*C(y;k)\,.$ (4.15)

##  5 General shift parameters

In this section we calculate $\Upsilon(\omega,\omega,C)$ vertex using the
shifted homotopy operators involving shift parameters that act on the argument
of $\omega$. We adopt notation from [5]

$t_{\mu}\omega(Y;K|x)=-i\frac{\partial}{\partial y^{\mu}}\omega(Y;K|x)\,.$
(5.1)

Let us start with equation (3.10). We choose a solution (3.19) with
$Q^{\mu}=q^{\mu}+\alpha y^{\mu}+\lambda p^{\mu}$ where $q^{\mu}$ and
$\alpha,\lambda$ are free constants. Then, for
$S_{1}=S_{1}^{\eta}+S_{1}^{\overline{\eta}}$, in the
$\overline{\eta}$-independent (holomorphic) sector we obtain

$S_{1}^{\eta}=-\frac{\eta}{2}\Delta_{q+\alpha y+\lambda
p}\big{(}C*\gamma\big{)}\,.$ (5.2)

The next step is to solve eq.(2.5) which yields in the first order

$\mbox{d}_{z}W_{1}^{\eta}=-\frac{i}{2}\big{(}\mbox{d}_{x}S^{\eta}_{1}+\omega*S_{1}^{\eta}+S_{1}^{\eta}*\omega\big{)}\,.$
(5.3)

Equation (5.3) decomposes into two subsystems. This is because, as pointed out
in [11], HS unfolded equations remain consistent with the fields $\omega$ and
$C$ valued in any associative algebra which implies that they are associated
with the so-called $A_{\infty}$-algebra [17, 18]. From this it follows, that
the following equations have to be separately satisfied

$\displaystyle\mbox{d}_{z}W_{1}^{\eta(1)}$
$\displaystyle=-\frac{i}{2}\big{(}\mbox{d}_{x}S^{\eta}_{1}\big{|}_{\omega*C}+\omega*S^{\eta}_{1}\big{)}\,,$
(5.4) $\displaystyle\mbox{d}_{z}W_{1}^{\eta(2)}$
$\displaystyle=-\frac{i}{2}\big{(}\mbox{d}_{x}S^{\eta}_{1}\big{|}_{C*\omega}+S_{1}^{\eta}*\omega\big{)}\,.$
(5.5)

Indeed, using equation (2.6) equations (5.4) and (5.5) can be checked to
separately satisfy consistency conditions. While doing so, it is important to
remember that in the term $\mbox{d}_{x}\mbox{d}_{z}S_{1}^{\eta}$ one must keep
only the term with the chosen order of $\omega$ and $C$ resulting from
equation (3.6).

Hence, we can apply independent shifts for the different components of
$W_{1}^{\eta}$. Let us choose the following solutions to Eqs. (5.4) and (5.5)
with $t$-dependent shifts
$Q_{i}^{\mu}=l_{i}^{\mu}+n_{i}t^{\mu}+\beta_{i}y^{\mu}$. As shown in [5],
uniform shifts $\Delta_{\gamma(p+y)}$ in both $S$ and $W$ do not affect the
form of the vertices. The freedom in the uniform shifts allows us to fix the
$p$ shift for $W_{1}$ to zero, so this is in fact the most general form of a
linear shift for this set of variables. In general, both orderings in $W_{1}$
must result from the same homotopy procedure. However, one can start with
introducing different shifts $n_{i}t$ and $\beta_{i}y$. Any $n_{i}$ respect
the compatibility conditions independently, while, as we show later on,
$\beta_{i}y$ shifts have to vanish. So,

$\displaystyle W^{\eta(1)}_{1}$
$\displaystyle=\frac{1}{2i}\Delta_{l_{1}+n_{1}t+\beta_{1}y}\big{(}\mbox{d}_{x}S^{\eta}_{1}\big{|}_{\omega*C}+\omega*S^{\eta}_{1}\big{)}\,,$
(5.6) $\displaystyle W^{\eta(2)}_{1}$
$\displaystyle=\frac{1}{2i}\Delta_{l_{2}+n_{2}t+\beta_{2}y}\big{(}\mbox{d}_{x}S^{\eta}_{1}\big{|}_{C*\omega}+S_{1}^{\eta}*\omega\big{)}\,$
(5.7)

with $l_{i}^{\mu}$ and $n_{i}$ being some constants. Plugging in (5.2) and
applying star-exchange formulae (4.10), (4.11), (4.13) we obtain

$\displaystyle W^{\eta(1)}_{1}$
$\displaystyle=\frac{\eta}{4i}\omega*C*\Delta_{l_{1}+n_{1}t+\beta_{1}y+(1-\beta_{1})(t+p)}\big{(}\Delta_{\tilde{q}+(1-\alpha+\lambda)p}-\Delta_{\tilde{q}+(1-\alpha+\lambda)(t+p)}\big{)}\gamma\,,$
(5.8) $\displaystyle W^{\eta(2)}_{1}$
$\displaystyle=\frac{\eta}{4i}C*\omega*\Delta_{l_{2}+n_{2}t+\beta_{2}y+(1-\beta_{2})(p+t)}\big{(}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t+p)}-\Delta_{\tilde{q}+(1-\alpha+\lambda)p+2t}\big{)}\gamma\,,$
(5.9)

where $\tilde{q}=q+\alpha y$.

Now consider equation (2.4). In the first order it yields

$\mbox{d}\omega+\omega*\omega+\mbox{d}W_{1}^{\eta}+\omega*W_{1}^{\eta}+W_{1}^{\eta}*\omega=0\,,$
(5.10)

where $W_{1}^{\eta}=W_{1}^{\eta(1)}+W_{1}^{\eta(2)}$. Using (5.8) and (5.9)
and applying formulae (4.9),(4.10),(4.11),(4.13) one can obtain

$\mbox{d}\omega+\omega*\omega+\Upsilon^{\eta}(\omega,\omega,C)+\Upsilon^{\eta}(C,\omega,\omega)+\Upsilon^{\eta}(\omega,C,\omega)=0\,.$
(5.11)

Direct calculation of the vertices yields

$\Upsilon^{\eta}(\omega,\omega,C)=\frac{\eta}{4i}\omega*\omega*C*\big{[}h_{l_{1}+n_{1}(t_{1}+t_{2})+\beta_{1}y+(1-\beta_{1})(p+t_{1}+t_{2})}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\Delta_{\tilde{q}+(1-\alpha+\lambda)p}\gamma+\\\
+h_{l_{1}+n_{1}t_{1}+\beta_{1}y+(1-\beta_{1})(p+t_{1}+t_{2})}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{2}+p)}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\gamma+\\\
+h_{l_{1}+n_{1}t_{2}+\beta_{1}y+(1-\beta_{1})(p+t_{2})}\Delta_{\tilde{q}+(1-\alpha+\lambda)p}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{2}+p)}\gamma+\\\
+h_{\tilde{q}+(1-\alpha+\lambda)p}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{2}+p)}\gamma\big{]}\,,$
(5.12)
$\Upsilon^{\eta}(C,\omega,\omega)=\frac{\eta}{4i}C*\omega*\omega*\big{[}h_{l_{2}+n_{2}t_{2}+\beta_{2}y+(1-\beta_{2})(p+t_{1}+t_{2})}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\gamma+\\\
+h_{l_{2}+n_{2}(t_{1}+t_{2})+\beta_{2}y+(1-\beta_{2})(p+t_{1}+t_{2})}\Delta_{\tilde{q}+(1-\alpha+\lambda)p+2t_{1}+2t_{2}}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\gamma+\\\
+h_{l_{2}+n_{2}t_{1}+\beta_{2}y+(1-\beta_{2})(p+t_{1})+2t_{2}}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\Delta_{\tilde{q}+(1-\alpha+\lambda)p+2t_{1}+2t_{2}}\gamma+\\\
+h_{\tilde{q}+(1-\alpha+\lambda)p+2t_{1}+2t_{2}}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\gamma\big{]}\,,$
(5.13)

$\Upsilon^{\eta}(\omega,C,\omega)=\frac{\eta}{4i}\omega*C*\omega*\big{[}h_{\tilde{q}+(1-\alpha)(t_{1}+t_{2}+p)}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{2}+p)}\gamma+\\\
+h_{\tilde{q}+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\Delta_{\tilde{q}+(1-\alpha+\lambda)p+2t_{2}}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{2}+p)}\gamma+\\\
+h_{l_{1}+n_{1}t_{1}+\beta_{1}y+(1-\beta_{1})(t_{1}+t_{2}+p)}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{2}+p)}\gamma+\\\
+h_{l_{2}+n_{2}t_{2}+\beta_{2}y+(1-\beta_{2})(p+t_{1}+t_{2})}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\gamma+\\\
+h_{l_{1}+n_{1}t_{1}+\beta_{1}y+(1-\beta_{1})(p+t_{1})+2t_{2}}\Delta_{\tilde{q}+(1-\alpha+\lambda)p+2t_{2}}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\gamma+\\\
+h_{l_{2}+n_{2}t_{2}+\beta_{2}y+(1-\beta_{2})(p+t_{2})}\Delta_{\tilde{q}+(1-\alpha+\lambda)(t_{2}+p)}\Delta_{\tilde{q}+(1-\alpha+\lambda)p+2t_{2}}\gamma\big{]}\,.$
(5.14)

To simplify further analysis we set $l_{i}^{\mu}=q^{\mu}=0$, which is anyway
necessary since non-zero constant spinors like $l_{i}^{\mu}$ and $q^{\mu}$
violate Lorentz covariance. In practice, the presence of such constant
parameters would result in terms containing $\omega h$ in the vertices so that
the Lorentz connection would not enter solely via the Lorentz covariant
derivative.

Now we use (4.7) and evaluate star products in the vertices (5.12)-(5.13)
using the following notations for the argument of the exponent

$\displaystyle\varkappa_{\omega\omega C}(y,t_{i},p)$
$\displaystyle=y^{\mu}(t_{1}+t_{2}+p)_{\mu}+t_{1}^{\mu}t_{2}{}_{\mu}+(t_{1}+t_{2})^{\mu}p_{\mu}\,,$
(5.15) $\displaystyle\varkappa_{\omega C\omega}(y,t_{i},p)$
$\displaystyle=y^{\mu}(t_{1}+t_{2}+p)_{\mu}+t_{1}^{\mu}p_{\mu}+(t_{1}+p)^{\mu}t_{2}{}_{\mu}\,,$
(5.16) $\displaystyle\varkappa_{C\omega\omega}(y,t_{i},p)$
$\displaystyle=y^{\mu}(t_{1}+t_{2}+p)_{\mu}+p^{\mu}t_{1}{}_{\mu}+(t_{1}+p)^{\mu}t_{2}{}_{\mu}\,.$
(5.17)

This yields

1) $\omega*\omega*C$-terms

$\omega*\omega*C*h_{n_{1}(t_{1}+t_{2})+\beta_{1}y+(1-\beta_{1})(p+t_{1}+t_{2})}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\Delta_{\alpha
y+(1-\alpha+\lambda)p}\gamma=\\\
=2\omega\overline{*}\omega\overline{*}C\int_{[0,1]^{3}}d^{3}\tau_{i}\delta(1-\sum_{i=1}^{3}\tau_{i})(1-\alpha+\lambda)\Bigg{[}(\alpha-\beta_{1})y^{\mu}(t_{1}+t_{2})_{\mu}-\lambda
p_{\mu}(t_{1}+t_{2})^{\mu}\Bigg{]}\exp{i\varkappa_{\omega\omega
C}(y,t_{i},p)}\\\
\text{exp}\bigg{[}-i(y+t_{1}+t_{2}+p)^{\nu}\big{(}\tau_{1}[n_{1}(t_{1}+t_{2})+(1-\beta_{1})(p+t_{1}+t_{2})]+\tau_{2}[(1-\alpha+\lambda)p]+\tau_{3}[(1-\alpha+\lambda)(t_{1}+t_{2}+p)]\big{)}_{\nu}\bigg{]}k\,,$
(5.18)

where $\overline{*}$ is the star product in the antiholomorphic variables
$\overline{y}_{\dot{\mu}}$

$f(\overline{y})\overline{*}g(\overline{y})=f(\overline{y})e^{i{\epsilon}^{\dot{\mu}\dot{\nu}}\overleftarrow{\overline{\partial}}_{\dot{\mu}}\overrightarrow{\overline{\partial}}_{\dot{\nu}}}g(\overline{y})\,,$
(5.19)
$\omega*\omega*C*h_{n_{1}t_{1}+\beta_{1}y+(1-\beta_{1})(p+t_{1}+t_{2})}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{2}+p)}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\gamma=\\\
=2\omega\overline{*}\omega\overline{*}C\int_{[0,1]^{3}}d^{3}\tau_{i}\delta(1-\sum_{i=1}^{3}\tau_{i})(\alpha-1-\lambda)\Bigg{[}(\alpha-\beta_{1})y^{\mu}t_{1}{}_{\mu}-\lambda(p+t_{2})_{\mu}t_{1}^{\mu}\Bigg{]}\exp{i\varkappa_{\omega\omega
C}(y,t_{i},p)}\\\
\text{exp}\bigg{[}-i(y+t_{1}+t_{2}+p)^{\nu}\big{(}\tau_{1}[n_{1}t_{1}+(1-\beta_{1})(p+t_{1}+t_{2})]+\tau_{2}[(1-\alpha+\lambda)(t_{1}+t_{2}+p)]+\tau_{3}[(1-\alpha+\lambda)(t_{2}+p)]\big{)}_{\nu}\bigg{]}k\,,$
(5.20)
$\omega*\omega*C*h_{n_{1}t_{2}+\beta_{1}y+(1-\beta_{1})(p+t_{2})}\Delta_{\alpha
y+(1-\alpha+\lambda)p}\Delta_{\alpha y+(1-\alpha+\lambda)(t_{2}+p)}\gamma=\\\
=2\omega\overline{*}\omega\overline{*}C\int_{[0,1]^{3}}d^{3}\tau_{i}\delta(1-\sum_{i=1}^{3}\tau_{i})(\alpha-1-\lambda)\Bigg{[}(\alpha-\beta_{1})(y+t_{1})^{\mu}t_{2}{}_{\mu}-\lambda
p_{\mu}t_{2}^{\mu}\Bigg{]}\exp{i\varkappa_{\omega\omega C}(y,t_{i},p)}\\\
\text{exp}\bigg{[}-i(y+t_{1}+t_{2}+p)^{\nu}\big{(}\tau_{1}[n_{1}t_{2}+(1-\beta_{1})(p+t_{2})]+\tau_{2}[(1-\alpha+\lambda)(t_{2}+p)]+\tau_{3}[(1-\alpha+\lambda)p]\big{)}_{\nu}\bigg{]}k\,,$
(5.21) $\omega*\omega*C*h_{\alpha y+(1-\alpha+\lambda)p}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{2}+p)}\gamma=\\\
=2\omega\overline{*}\omega\overline{*}C\int_{[0,1]^{3}}d^{3}\tau_{i}\delta(1-\sum_{i=1}^{3}\tau_{i})(\alpha-1-\lambda)^{2}t_{2}{}^{\mu}t_{1}{}_{\mu}\exp{i\varkappa_{\omega\omega
C}(y,t_{i},p)}\\\
\text{exp}\bigg{[}-i(y+t_{1}+t_{2}+p)^{\nu}\big{(}\tau_{1}[(1-\alpha+\lambda)p]+\tau_{2}[(1-\alpha+\lambda)(t_{2}+p)]+\tau_{3}[(1-\alpha+\lambda)(t_{1}+t_{2}+p)]\big{)}_{\nu}\bigg{]}k\,.$
(5.22)

2) $\omega*C*\omega$-terms

$\omega*C*\omega*h_{\alpha y+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{2}+p)}\gamma=\\\
=2\omega\overline{*}C\overline{*}\omega\int_{[0,1]^{3}}d^{3}\tau_{i}\delta(1-\sum_{i=1}^{3}\tau_{i})(1+\alpha-\lambda)(\alpha-1-\lambda)t_{1}{}^{\mu}t_{2}{}_{\mu}\exp{i\varkappa_{\omega
C\omega}(y,t_{i},p)}\\\
\text{exp}\bigg{[}-i(y+t_{1}+t_{2}+p)^{\nu}\big{(}\tau_{1}[(1-\alpha+\lambda)(t_{1}+t_{2}+p)]+\tau_{2}[(1-\alpha+\lambda)(t_{2}+p)]+\tau_{3}[(1-\alpha+\lambda)(t_{1}+p)+2t_{2}]\big{)}_{\nu}\bigg{]}k\,,$
(5.23) $\omega*C*\omega*h_{\alpha
y+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)p+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{2}+p)}\gamma=\\\
=2\omega\overline{*}C\overline{*}\omega\int_{[0,1]^{3}}d^{3}\tau_{i}\delta(1-\sum_{i=1}^{3}\tau_{i})(\alpha+1-\lambda)(\alpha-1-\lambda)t_{1}{}^{\mu}t_{2}{}_{\mu}\exp{i\varkappa_{\omega
C\omega}(y,t_{i},p)}\\\
\text{exp}\bigg{[}-i(y+t_{1}+t_{2}+p)^{\nu}\big{(}\tau_{1}[(1-\alpha+\lambda)(t_{1}+p)+2t_{2}]+\tau_{2}[(1-\alpha+\lambda)(t_{2}+p)]+\tau_{3}[(1-\alpha+\lambda)p+2t_{2}]\big{)}_{\nu}\bigg{]}k\,,$
(5.24)
$\omega*C*\omega*h_{n_{1}t_{1}+\beta_{1}y+(1-\beta_{1})(p+t_{1}+t_{2})}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{2}+p)}\gamma=\\\
=-2\omega\overline{*}C\overline{*}\omega\int_{[0,1]^{3}}d^{3}\tau_{i}\delta(1-\sum_{i=1}^{3}\tau_{i})(\alpha-1-\lambda)\Bigg{[}(\alpha-\beta_{1})y{}^{\mu}t_{1}{}_{\mu}+\lambda(p+t_{2})^{\mu}t_{1}{}_{\mu}\Bigg{]}\exp{i\varkappa_{\omega
C\omega}(y,t_{i},p)}\\\
\text{exp}\bigg{[}-i(y+t_{1}+t_{2}+p)^{\nu}\big{(}\tau_{1}[n_{1}t_{1}+(1-\beta_{1})(p+t_{1}+t_{2})]+\tau_{2}[(1-\alpha+\lambda)(t_{2}+p)]+\tau_{3}[(1-\alpha+\lambda)(t_{1}+t_{2}+p)]\big{)}_{\nu}\bigg{]}k\,,$
(5.25)
$\omega*C*\omega*h_{n_{2}t_{2}+\beta_{2}y+(1-\beta_{2})(p+t_{1}+t_{2})}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\gamma=\\\
=2\omega\overline{*}C\overline{*}\omega\int_{[0,1]^{3}}d^{3}\tau_{i}\delta(1-\sum_{i=1}^{3}\tau_{i})(\alpha+1-\lambda)\bigg{[}(\alpha-\beta_{2})y{}^{\mu}t_{2}{}_{\mu}+\lambda(p+t_{1})^{\mu}t_{2}{}_{\mu}\bigg{]}\exp{i\varkappa_{\omega
C\omega}(y,t_{i},p)}\\\
\text{exp}\bigg{[}-i(y+t_{1}+t_{2}+p)^{\nu}\big{(}\tau_{1}[n_{2}t_{2}+(1-\beta_{2})(t_{1}+t_{2}+p)]+\tau_{2}[(1-\alpha+\lambda)(t_{1}+t_{2}+p)]+\\\
+\tau_{3}[(1-\alpha+\lambda)(t_{1}+p)+2t_{2}]\big{)}_{\nu}\bigg{]}k\,,$ (5.26)
$\omega*C*\omega*h_{n_{1}t_{1}+\beta_{1}y+(1-\beta_{1})(p+t_{1})+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)p+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\gamma=\\\
=2\omega\overline{*}C\overline{*}\omega\int_{[0,1]^{3}}d^{3}\tau_{i}\delta(1-\sum_{i=1}^{3}\tau_{i})(\alpha-1-\lambda)\bigg{[}(\alpha-\beta_{1})(y{}+t_{2}{})^{\mu}t_{1}{}_{\mu}+\lambda
p^{\mu}t_{1}{}_{\mu}\bigg{]}\exp{i\varkappa_{\omega C\omega}(y,t_{i},p)}\\\
\text{exp}\bigg{[}-i(y+t_{1}+t_{2}+p)^{\nu}\big{(}\tau_{1}[n_{1}t_{1}+(1-\beta_{1})(p+t_{1})+2t_{2}]+\tau_{2}[(1-\alpha+\lambda)p+2t_{2}]+\\\
+\tau_{3}[(1-\alpha+\lambda)(t_{1}+p)+2t_{2}]\big{)}_{\nu}\bigg{]}k\,,$ (5.27)
$\omega*C*\omega*h_{n_{2}t_{2}+\beta_{2}y+(1-\beta_{2})(p+t_{2})}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{2}+p)}\Delta_{\alpha
y+(1-\alpha+\lambda)p+2t_{2}}\gamma=\\\
=-2\omega\overline{*}C\overline{*}\omega\int_{[0,1]^{3}}d^{3}\tau_{i}\delta(1-\sum_{i=1}^{3}\tau_{i})(\alpha+1-\lambda)\bigg{[}(\alpha-\beta_{2})(y{}+t_{1}{})^{\mu}t_{2}{}_{\mu}+\lambda
p^{\mu}t_{2}{}_{\mu}\bigg{]}\exp{i\varkappa_{\omega C\omega}(y,t_{i},p)}\\\
\text{exp}\bigg{[}-i(y+t_{1}+t_{2}+p)^{\nu}\big{(}\tau_{1}[n_{2}t_{2}+(1-\beta_{2})(p+t_{2})]+\tau_{2}[(1-\alpha+\lambda)p+2t_{2}]+\tau_{3}[(1-\alpha+\lambda)(t_{2}+p)]\big{)}_{\nu}\bigg{]}k\,.$
(5.28)

3) $C*\omega*\omega$-terms

$C*\omega*\omega*h_{n_{2}t_{2}+\beta_{2}y+(1-\beta_{2})(t_{1}+t_{2}+p)}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\gamma=\\\
=-2C\overline{*}\omega\overline{*}\omega\int_{[0,1]^{3}}d^{3}\tau_{i}\delta(1-\sum_{i=1}^{3}\tau_{i})(\alpha+1-\lambda)\Bigg{[}(\alpha-\beta_{2})y^{\mu}t_{2}{}_{\mu}-\lambda(p+t_{1})_{\mu}t_{2}{}^{\mu}\Bigg{]}\exp{i\varkappa_{C\omega\omega}(y,t_{i},p)}\\\
\text{exp}\bigg{[}-i(y+t_{1}+t_{2}+p)^{\nu}\big{(}\tau_{1}[n_{2}t_{2}+(1-\beta_{2})(t_{1}+t_{2}+p)]+\tau_{2}[(1-\alpha+\lambda)(t_{1}+p)+2t_{2}]+\\\
+\tau_{3}[(1-\alpha+\lambda)(t_{1}+t_{2}+p)]\big{)}_{\nu}\bigg{]}k\,,$ (5.29)
$C*\omega*\omega*h_{n_{2}(t_{1}+t_{2})+\beta_{2}y+(1-\beta_{2})(p+t_{1}+t_{2})}\Delta_{\alpha
y+(1-\alpha+\lambda)p+2t_{1}+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\gamma=\\\
=2C\overline{*}\omega\overline{*}\omega\int_{[0,1]^{3}}d^{3}\tau_{i}\delta(1-\sum_{i=1}^{3}\tau_{i})(\alpha+1-\lambda)\Bigg{[}(\alpha-\beta_{2})y^{\mu}(t_{1}+t_{2})_{\mu}-\lambda
p_{\mu}(t_{1}+t_{2})^{\mu}\Bigg{]}\exp{i\varkappa_{C\omega\omega}(y,t_{i},p)}\\\
\text{exp}\bigg{[}-i(y+t_{1}+t_{2}+p)^{\nu}\big{(}\tau_{1}[n_{2}(t_{1}+t_{2})+(1-\beta_{2})(p+t_{1}+t_{2})]+\tau_{2}[(1-\alpha+\lambda)(t_{1}+t_{2}+p)]+\\\
+\tau_{3}[(1-\alpha+\lambda)p+2t_{1}+2t_{2}]\big{)}_{\nu}\bigg{]}k\,,$ (5.30)
$C*\omega*\omega*h_{n_{2}t_{1}+\beta_{2}y+(1-\beta_{2})(p+t_{1})+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)p+2t_{1}+2t_{2}}\gamma=\\\
=-2C\overline{*}\omega\overline{*}\omega\int_{[0,1]^{3}}d^{3}\tau_{i}\delta(1-\sum_{i=1}^{3}\tau_{i})(\alpha+1-\lambda)\Bigg{[}(\alpha-\beta_{2})(y+t_{2})^{\mu}t_{1}{}_{\mu}-\lambda
p_{\mu}t_{1}{}^{\mu}\Bigg{]}\exp{i\varkappa_{C\omega\omega}(y,t_{i},p)}\\\
\text{exp}\bigg{[}-i(y+t_{1}+t_{2}+p)^{\nu}\big{(}\tau_{1}[n_{2}t_{1}+(1-\beta_{2})(p+t_{1})+2t_{2}]+\tau_{2}[(1-\alpha+\lambda)p+2t_{1}+2t_{2}]+\\\
+\tau_{3}[(1-\alpha+\lambda)(t_{1}+p)+2t_{2}]\big{)}_{\nu}\bigg{]}k\,,$ (5.31)
$C*\omega*\omega*h_{\alpha y+(1-\alpha+\lambda)p+2t_{1}+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\gamma=\\\
=2C\overline{*}\omega\overline{*}\omega\int_{[0,1]^{3}}d^{3}\tau_{i}\delta(1-\sum_{i=1}^{3}\tau_{i})(1+\alpha-\lambda)^{2}t_{2}^{\mu}t_{1}{}_{\mu}\exp{i\varkappa_{C\omega\omega}(y,t_{i},p)}\\\
\text{exp}\bigg{[}-i(y+t_{1}+t_{2}+p)^{\nu}\big{(}\tau_{1}[(1-\alpha+\lambda)p+2t_{1}+2t_{2}]+\tau_{2}[(1-\alpha+\lambda)(t_{1}+t_{2}+p)]+\tau_{3}[(1-\alpha+\lambda)(t_{1}+p)+2t_{2}]\big{)}_{\nu}\bigg{]}k\,.$
(5.32)

One can notice similarities in different vertices resulting from the
antiautomorphism $\rho$ of the HS star-product algebra,

$\rho\bigg{(}f(Z,Y;K;\theta)\bigg{)}=f(-iZ,iY;K;-i\theta)\,,$ (5.33)

that leaves invariant non-linear HS equations (2.4)-(2.8) [12]. Indeed it is
easy to see that application of such antiautomorphism along with the
substitution $\alpha\leftrightarrow-\alpha$, $t_{1}\leftrightarrow t_{2}$
$n_{1}\leftrightarrow-n_{2}$, $\lambda\leftrightarrow-\lambda$ and
$\beta_{1}\leftrightarrow-\beta_{2}$ maps some pairs of terms to each other.
Namely, the terms of the vertex $\Upsilon^{\eta}(\omega,\omega,C)$ are mapped
to those of $\Upsilon^{\eta}(C,\omega,\omega)$, while a half of the terms in
$\Upsilon^{\eta}(\omega,C,\omega)$ is mapped to the other half.

##  6 Admissible shift parameters

As explained above, non-zero constant spinors $q^{\mu}$ or $l_{i}^{\mu}$
manifestly violate Lorentz invariance and hence are not allowed. The analysis
of the role of the parameters $\alpha$, $\lambda$ and $\beta_{i}$ requires a
bit more work. To respect the form of First On-Shell Theorem for the AdS
background $\omega=\Omega$ vertices should have the $y$-independent form
$h_{\mu}^{\
\dot{\mu}}h^{\mu\dot{\nu}}\,\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}C(0,\overline{y}\,|x)$
or $h_{\mu}^{\
\dot{\mu}}h^{\mu\dot{\nu}}\,\overline{y}_{\dot{\mu}}\overline{y}_{\dot{\nu}}C(0,\overline{y}\,|x)$
in the $\eta$-sector. Therefore, it is instrumental to analyse the
$y$-dependence of the $C$-field in the vertex. To this end let us inspect all
results of multiplication of two fields $\Omega$ and a single field $C$ paying
attention to the terms of the form
$hh\overline{\partial}\overline{\partial}C$. Recall that in the previous
analysis, arguments of both $\Omega$ and $C$ were uplifted into a single
exponent by virtue of the Taylor formula

$f(a)=\exp(a\frac{\mbox{d}}{\mbox{d}b})f(b)\bigg{|}_{b=0}\,$ (6.1)

with an auxiliary variable $b$. Proceeding this way, let us assign the
auxiliary variables $y_{1}$ and $y_{2}$ to the first and second factors of
$\Omega$ in the ordered product, respectively. We will use the fact that a
product of two frame fields can be decomposed into irreducible parts as

$h^{\nu\dot{\nu}}h^{\lambda\dot{\lambda}}=\frac{1}{2}H^{\nu\lambda}\epsilon^{\dot{\nu}\dot{\lambda}}+\frac{1}{2}\overline{H}^{\dot{\nu}\dot{\lambda}}\epsilon^{\nu\lambda},$
(6.2)

where

$\displaystyle H^{\nu\lambda}=H^{(\nu\lambda)}:=h^{\nu}_{\
\dot{\gamma}}h^{\lambda\dot{\gamma}}\,,\qquad\overline{H}^{\dot{\nu}\dot{\lambda}}=H^{(\dot{\nu}\dot{\lambda})}:=h^{\
\dot{\nu}}_{\gamma}h^{\gamma\dot{\lambda}}\,.$ (6.3)

In the $\eta$-sector only the second term of (6.2) is nontrivial. One then
gets

$\Omega\overline{*}\Omega\overline{*}C\bigg{|}_{\overline{H}\overline{\partial}^{2}}=C\overline{*}\Omega\overline{*}\Omega\bigg{|}_{\overline{H}\overline{\partial}^{2}}=\frac{1}{8}\overline{H}^{\dot{\mu}\dot{\nu}}y_{1\nu}y_{2}^{\nu}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}C(0,\overline{y};k,\overline{k})\,,$
(6.4)
$\Omega\overline{*}C\overline{*}\Omega\bigg{|}_{\overline{H}\overline{\partial}^{2}}=-\frac{1}{8}\overline{H}^{\dot{\mu}\dot{\nu}}y_{1\nu}y_{2}^{\nu}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}C(0,\overline{y};-k,-\overline{k})\,.$
(6.5)

The role of the auxiliary variables $y_{1,2}$ is that the action of bilinears
$y^{\mu}t_{i}{}_{\mu}$, $p^{\mu}t_{i}{}_{\mu}$ and $t_{1}^{\mu}t_{2}{}_{\mu}$
replaces $y_{1,2}$ with actual variables $y^{\mu}$, derivatives $p^{\mu}$ or
organizes the index contraction via $t_{1}^{\mu}t_{2}{}_{\mu}$. In all
vertices the pre-exponent contains one of the bilinear factors
$y^{\mu}t_{i}{}_{\mu}$, $p^{\mu}t_{i}{}_{\mu}$ or $t_{1}^{\mu}t_{2}{}_{\mu}$.
The action of $y^{\mu}t_{i}{}_{\mu}$ and $p^{\mu}t_{i}{}_{\mu}$ on two
$\Omega$ and single $C$ yields

$y^{\mu}t_{1}{}_{\mu}\Omega\overline{*}\Omega\overline{*}C\bigg{|}_{\overline{H}\overline{\partial}^{2}}=y^{\mu}t_{1}{}_{\mu}C\overline{*}\Omega\overline{*}\Omega\bigg{|}_{\overline{H}\overline{\partial}^{2}}=\frac{i}{8}\overline{H}^{\dot{\mu}\dot{\nu}}y_{\nu}y_{2}^{\nu}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}C(0,\overline{y};k,\overline{k})\,,$
(6.6)
$y^{\mu}t_{1}{}_{\mu}\Omega\overline{*}C\overline{*}\Omega\bigg{|}_{\overline{H}\overline{\partial}^{2}}=-\frac{i}{8}\overline{H}^{\dot{\mu}\dot{\nu}}y_{\nu}y_{2}^{\nu}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}C(0,\overline{y};-k,-\overline{k})\,,$
(6.7)
$p^{\mu}t_{1}{}_{\mu}\Omega\overline{*}\Omega\overline{*}C\bigg{|}_{\overline{H}\overline{\partial}^{2}}=p^{\mu}t_{1}{}_{\mu}C\overline{*}\Omega\overline{*}\Omega\bigg{|}_{\overline{H}\overline{\partial}^{2}}=\frac{i}{8}\overline{H}^{\dot{\mu}\dot{\nu}}p_{\nu}y_{2}^{\nu}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}C(0,\overline{y};k,\overline{k})\,,$
(6.8)
$p^{\mu}t_{1}{}_{\mu}\Omega\overline{*}C\overline{*}\Omega\bigg{|}_{\overline{H}\overline{\partial}^{2}}=-\frac{i}{8}\overline{H}^{\dot{\mu}\dot{\nu}}p_{\nu}y_{2}^{\nu}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}C(0,\overline{y};-k,-\overline{k})\,.$
(6.9)

For $y^{\mu}t_{2}{}_{\mu}$ and $p^{\mu}t_{2}{}_{\mu}$ the situation is
analogous up to the exchange of $y_{2}$ with $y_{1}$ and an additional minus
sign. Examining the exponents in all vertices, constructed in Section 5, one
observes that it is impossible to obtain the desired form of the First On-
Shell Theorem from the terms with $y^{\mu}t_{i}{}_{\mu}$ and
$p^{\mu}t_{i}{}_{\mu}$ in the pre-exponent since the First On-Shell Theorem
does not contain terms with $y_{\mu}$ or $p_{\mu}$ contracted with the frame
field $h$. The combination $t_{1}^{\mu}t_{2}{}_{\mu}$ leads to the correct
contraction of two frame fields

$t_{1}^{\mu}t_{2}{}_{\mu}\Omega\overline{*}\Omega\overline{*}C\bigg{|}_{\overline{H}\overline{\partial}^{2}}=t_{1}^{\mu}t_{2}{}_{\mu}C\overline{*}\Omega\overline{*}\Omega\bigg{|}_{\overline{H}\overline{\partial}^{2}}=-\frac{1}{4}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}C(0,\overline{y};k,\overline{k})\,,$
(6.10)
$t_{1}^{\mu}t_{2}{}_{\mu}\Omega\overline{*}C\overline{*}\Omega\bigg{|}_{\overline{H}\overline{\partial}^{2}}=\frac{1}{4}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}C(0,\overline{y};-k,-\overline{k})\,.$
(6.11)

Plugging these expressions into the vertex components (5.18)-(5.32), that
contain the pre-exponential factor $t_{1}^{\mu}t_{2}{}_{\mu}$, and integrating
out $\tau_{i}$ we obtain

$\Omega*\Omega*C*h_{n_{1}t_{1}+\beta_{1}+(1-\beta_{1})(p+t_{1}+t_{2})}\Delta_{\alpha
y+(1-\alpha+\lambda)(p+t_{2})}\Delta_{\alpha
y+(1-\alpha+\lambda)(p+t_{1}+t_{2})}\gamma\bigg{|}_{\overline{H}\overline{\partial}^{2}}=\frac{\lambda(\alpha-1-\lambda)}{4}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}\\\
\bigg{[}C(0,\overline{y};k,\overline{k})+2\frac{\beta_{1}^{n+2}+(\alpha-\lambda)^{n+1}((\alpha-\lambda)(n+1)-\beta_{1}(n+2))}{(n+1)(n+2)(\alpha-\lambda-\beta_{1})^{2}}y^{\mu_{1}}...y^{\mu_{n}}\overline{y}^{\dot{\mu}_{1}}...\overline{y}^{\dot{\mu}_{m}}C_{\mu_{1}...\mu_{n},\dot{\mu}_{1}...\dot{\mu}_{m}}(k,\overline{k})\bigg{]}k\,,$
(6.12) $\Omega*\Omega*C*h_{n_{1}t_{2}+(1-\beta_{1})(p+t_{2})}\Delta_{\alpha
y+(1-\alpha+\lambda)p}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{2}+p)}\gamma\bigg{|}_{\overline{H}\overline{\partial}^{2}}=-\frac{(\alpha-\beta_{1})(\alpha-1-\lambda)}{4}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}\\\
\bigg{[}C(0,\overline{y};k,\overline{k})+2\frac{\beta_{1}^{n+2}+(\alpha-\lambda)^{n+1}((\alpha-\lambda)(n+1)-\beta_{1}(n+2))}{(n+1)(n+2)(\alpha-\lambda-\beta_{1})^{2}}y^{\mu_{1}}...y^{\mu_{n}}\overline{y}^{\dot{\mu}_{1}}...\overline{y}^{\dot{\mu}_{m}}C_{\mu_{1}...\mu_{n},\dot{\mu}_{1}...\dot{\mu}_{m}}(k,\overline{k})\bigg{]}k\,,$
(6.13) $\Omega*\Omega*C*h_{\alpha y+(1-\alpha+\lambda)p}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{2}+p)}\gamma\bigg{|}_{\overline{H}\overline{\partial}^{2}}=\\\
=\frac{(1-\alpha+\lambda)^{2}}{4}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}C((\alpha-\lambda)y,\overline{y};k,\overline{k})k\,,$
(6.14) $\Omega*C*\Omega*h_{\alpha
y+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{2}+p)}\gamma\bigg{|}_{\overline{H}\overline{\partial}^{2}}=\\\
=\frac{((\alpha-\lambda)^{2}-1)}{4}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}C((\alpha-\lambda)y,\overline{y};-k,-\overline{k})k\,,$
(6.15) $\Omega*C*\Omega*h_{\alpha
y+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)p+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{2}+p)}\gamma\bigg{|}_{\overline{H}\overline{\partial}^{2}}=\\\
=\frac{((\alpha-\lambda)^{2}-1)}{4}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}C((\alpha-\lambda)y,\overline{y};-k,-\overline{k})k\,,$
(6.16)
$\Omega*C*\Omega*h_{n_{1}t_{1}+\beta_{1}y+(1-\beta_{1})(p+t_{1})+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)p+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\gamma\bigg{|}_{\overline{H}\overline{\partial}^{2}}=\\\
=-\frac{(\alpha-\beta_{1})(\alpha-1-\lambda)}{4}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}\bigg{[}C(0,\overline{y};-k,-\overline{k})+\\\
+2\frac{\beta_{1}^{n+2}+(\alpha-\lambda)^{n+1}((\alpha-\lambda)(n+1)-\beta_{1}(n+2))}{(n+1)(n+2)(\alpha-\beta_{1}-\lambda)^{2}}y^{\mu_{1}}...y^{\mu_{n}}\overline{y}^{\dot{\mu}_{1}}...\overline{y}^{\dot{\mu}_{m}}C_{\mu_{1}...\mu_{n},\dot{\mu}_{1}...\dot{\mu}_{m}}(-k,-\overline{k})\bigg{]}k\,,$
(6.17)
$\Omega*C*\Omega*h_{n_{1}t_{1}+\beta_{1}y+(1-\beta_{1})(p+t_{1}+t_{2})}\Delta_{\alpha
y+(1-\alpha+\lambda)(p+t_{1}+t_{2})}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{2}+p)}\gamma\bigg{|}_{\overline{H}\overline{\partial}^{2}}=\\\
=\frac{\lambda(\alpha-1-\lambda)}{4}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}\bigg{[}C(0,\overline{y};-k,-\overline{k})+\\\
+2\frac{\beta_{1}^{n+2}+(\alpha-\lambda)^{n+1}((\alpha-\lambda)(n+1)-\beta_{1}(n+2))}{(n+1)(n+2)(\alpha-\beta_{1}-\lambda)^{2}}y^{\mu_{1}}...y^{\mu_{n}}\overline{y}^{\dot{\mu}_{1}}...\overline{y}^{\dot{\mu}_{m}}C_{\mu_{1}...\mu_{n},\dot{\mu}_{1}...\dot{\mu}_{m}}(-k,-\overline{k})\bigg{]}k\,,$
(6.18)
$\Omega*C*\Omega*h_{n_{2}t_{2}+\beta_{2}y+(1-\beta_{2})(p+t_{2})}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{2}+p)}\Delta_{\alpha
y+(1-\alpha+\lambda)p+2t_{2}}\gamma\bigg{|}_{\overline{H}\overline{\partial}^{2}}=\\\
=-\frac{(\alpha-\beta_{2})(\alpha+1-\lambda)}{4}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}\bigg{[}C(0,\overline{y};-k,-\overline{k})+\\\
+2\frac{\beta_{2}^{n+2}+(\alpha-\lambda)^{n+1}((\alpha-\lambda)(n+1)-\beta_{2}(n+2))}{(n+1)(n+2)(\alpha-\beta_{2}-\lambda)^{2}}y^{\mu_{1}}...y^{\mu_{n}}\overline{y}^{\dot{\mu}_{1}}...\overline{y}^{\dot{\mu}_{m}}C_{\mu_{1}...\mu_{n},\dot{\mu}_{1}...\dot{\mu}_{m}}(-k,-\overline{k})\bigg{]}k\,,$
(6.19)
$\Omega*C*\Omega*h_{n_{2}t_{2}+\beta_{2}y+(1-\beta_{2})(p+t_{1}+t_{2})}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)(p+t_{1}+t_{2})}\gamma\bigg{|}_{\overline{H}\overline{\partial}^{2}}=\\\
=\frac{\lambda(\alpha+1-\lambda)}{4}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}\bigg{[}C(0,\overline{y};-k,-\overline{k})+\\\
+2\frac{\beta_{2}^{n+2}+(\alpha-\lambda)^{n+1}((\alpha-\lambda)(n+1)-\beta_{2}(n+2))}{(n+1)(n+2)(\alpha-\beta_{2}-\lambda)^{2}}y^{\mu_{1}}...y^{\mu_{n}}\overline{y}^{\dot{\mu}_{1}}...\overline{y}^{\dot{\mu}_{m}}C_{\mu_{1}...\mu_{n},\dot{\mu}_{1}...\dot{\mu}_{m}}(-k,-\overline{k})\bigg{]}k\,,$
(6.20)
$C*\Omega*\Omega*h_{n_{2}t_{2}+\beta_{2}y+(1-\beta_{2})(p+t_{1}+t_{2})}\Delta_{\alpha
y+(1-\alpha+\lambda)(p+t_{1}+t_{2})}\Delta_{\alpha
y+(1-\alpha+\lambda)(p+t_{1})+2t_{2}}\gamma\bigg{|}_{\overline{H}\overline{\partial}^{2}}=\\\
=\frac{\lambda(\alpha+1-\lambda)}{4}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}\bigg{[}C(0,\overline{y};k,\overline{k})+\\\
+2\frac{\beta_{2}^{n+2}+(\alpha-\lambda)^{n+1}((\alpha-\lambda)(n+1)-\beta_{2}(n+2))}{(n+1)(n+2)(\alpha-\beta_{2}-\lambda)^{2}}y^{\mu_{1}}...y^{\mu_{n}}\overline{y}^{\dot{\mu}_{1}}...\overline{y}^{\dot{\mu}_{m}}C_{\mu_{1}...\mu_{n},\dot{\mu}_{1}...\dot{\mu}_{m}}(k,\overline{k})\bigg{]}k\,,$
(6.21)
$C*\Omega*\Omega*h_{n_{2}t_{1}+\beta_{2}y+(1-\beta_{2})(p+t_{1})+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)p+2t_{1}+2t_{2}}\gamma\bigg{|}_{\overline{H}\overline{\partial}^{2}}=\\\
=-\frac{(\alpha+1-\lambda)(\alpha-\beta_{2})}{4}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}\bigg{[}C(0,\overline{y};k,\overline{k})+\\\
+2\frac{\beta_{2}^{n+2}+(\alpha-\lambda)^{n+1}((\alpha-\lambda)(n+1)-\beta_{2}(n+2))}{(n+1)(n+2)(\alpha-\beta_{2}-\lambda)^{2}}y^{\mu_{1}}...y^{\mu_{n}}\overline{y}^{\dot{\mu}_{1}}...\overline{y}^{\dot{\mu}_{m}}C_{\mu_{1}...\mu_{n},\dot{\mu}_{1}...\dot{\mu}_{m}}(k,\overline{k})\bigg{]}k\,,$
(6.22) $C*\Omega*\Omega*h_{\alpha
y+(1-\alpha+\lambda)p+2t_{1}+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+p)+2t_{2}}\Delta_{\alpha
y+(1-\alpha+\lambda)(t_{1}+t_{2}+p)}\gamma\bigg{|}_{\overline{H}\overline{\partial}^{2}}=\\\
=\frac{(1+\alpha-\lambda)^{2}}{4}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}C((\alpha-\lambda)y,\overline{y};k,\overline{k})k\,.$
(6.23)

As a result,

$\Upsilon^{\eta}(\Omega,\Omega,C)+\Upsilon^{\eta}(\Omega,C,\Omega)+\Upsilon^{\eta}(C,\Omega,\Omega)\bigg{|}_{\overline{H}\overline{\partial}^{2}}=\frac{\eta}{8i}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}\bigg{[}((\alpha-\lambda)^{2}+1)C((\alpha-\lambda)y,\overline{y};k,\overline{k})+\\\
+((\alpha-\lambda)^{2}-1)C((\alpha-\lambda)y,\overline{y};-k,-\overline{k})\bigg{]}k-\frac{\eta}{16i}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}\Bigg{[}(\alpha-\beta_{1}-\lambda)(\alpha-1-\lambda)+\\\
+(\alpha-\beta_{2}-\lambda)(\alpha+1-\lambda)\Bigg{]}C(0,\overline{y};k,\overline{k})k-\frac{\eta}{16i}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}\Bigg{[}(\alpha-\beta_{1}-\lambda)(\alpha-1-\lambda)+\\\
+(\alpha-\beta_{2}-\lambda)(\alpha+1-\lambda)\Bigg{]}C(0,\overline{y};-k,-\overline{k})k-\\\
-\frac{\eta(\alpha-1-\lambda)}{8i}\frac{\beta_{1}^{n+2}+(\alpha-\lambda)^{n+1}((\alpha-\lambda)(n+1)-\beta_{1}(n+2))}{(n+1)(n+2)(\alpha-\beta_{1}-\lambda)}\\\
\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}y^{\mu_{1}}...y^{\mu_{n}}\overline{y}^{\dot{\mu}_{1}}...\overline{y}^{\dot{\mu}_{m}}\Bigg{[}C_{\mu_{1}...\mu_{n},\dot{\mu}_{1}...\dot{\mu}_{m}}(k,\overline{k})+C_{\mu_{1}...\mu_{n},\dot{\mu}_{1}...\dot{\mu}_{m}}(-k,-\overline{k})\Bigg{]}k-\\\
-\frac{\eta(\alpha+1-\lambda)}{8i}\frac{\beta_{2}^{n+2}+(\alpha-\lambda)^{n+1}((\alpha-\lambda)(n+1)-\beta_{2}(n+2))}{(n+1)(n+2)(\alpha-\beta_{2}-\lambda)}\\\
\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}y^{\mu_{1}}...y^{\mu_{n}}\overline{y}^{\dot{\mu}_{1}}...\overline{y}^{\dot{\mu}_{m}}\Bigg{[}C_{\mu_{1}...\mu_{n},\dot{\mu}_{1}...\dot{\mu}_{m}}(k,\overline{k})+C_{\mu_{1}...\mu_{n},\dot{\mu}_{1}...\dot{\mu}_{m}}(-k,-\overline{k})\Bigg{]}k\,.$
(6.24)

Now, since the subsystems for the components of $C(Y;K|x)$ that are even and
odd in $K$ are independent, we have to respect the First On-Shell Theorem for
both physical (1.2) and topological fields (3.8). This results in the doubling
of shift parameters $\alpha^{e,o},\beta_{i}^{e,o},\lambda^{e,o}$.

For odd components $C(Y;k,\overline{k}|x)=-C(Y;-k,-\overline{k}|x)$:

$\Upsilon^{\eta}(\Omega,\Omega,C)+\Upsilon^{\eta}(\Omega,C,\Omega)+\Upsilon^{\eta}(C,\Omega,\Omega)\bigg{|}_{\overline{H}\overline{\partial}^{2}}=-\frac{i\eta}{4}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}C((\alpha^{o}-\lambda^{o})y,\overline{y};K)k\,.$
(6.25)

The form of the First On-Shell Theorem in the physical sector is respected if
$\alpha^{o}=\lambda^{o}$.

For even components $C(Y;k,\overline{k})=C(Y;-k,-\overline{k})$:

$\Upsilon^{\eta}(\Omega,\Omega,C)+\Upsilon^{\eta}(\Omega,C,\Omega)+\Upsilon^{\eta}(C,\Omega,\Omega)\bigg{|}_{\overline{H}\overline{\partial}^{2}}=\frac{\eta}{4i}(\alpha^{e}-\lambda^{e})^{2}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}C((\alpha^{e}-\lambda^{e})y,\overline{y};K)k-\\\
-\frac{\eta}{8i}\bigg{(}(\alpha^{e}-\lambda^{e})(2\alpha^{e}+\beta_{1}^{e}-\beta_{2}^{e}-2\lambda^{e})+(\beta_{1}^{e}-\beta_{2}^{e})\bigg{)}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}\\\
C(0,\overline{y};K)k+\frac{\eta}{4i}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}y^{\mu_{1}}...y^{\mu_{n}}\overline{y}^{\dot{\mu}_{1}}...\overline{y}^{\dot{\mu}_{m}}C_{\mu_{1}...\mu_{n},\dot{\mu}_{1}...\dot{\mu}_{m}}(k,\overline{k})\\\
\bigg{[}(\alpha^{e}-1-\lambda^{e})\frac{(\beta_{1}^{e})^{n+2}+(\alpha^{e}-\lambda^{e})^{n+1}((\alpha^{e}-\lambda^{e})(n+1)-\beta_{1}^{e}(n+2))}{(n+1)(n+2)(\alpha^{e}-\beta_{1}^{e}-\lambda^{e})}+\\\
+(\alpha^{e}+1-\lambda^{e})\frac{(\beta_{2}^{e})^{n+2}+(\alpha^{e}-\lambda^{e})^{n+1}((\alpha^{e}-\lambda^{e})(n+1)-\beta^{e}_{2}(n+2))}{(n+1)(n+2)(\alpha^{e}-\beta^{e}_{2}-\lambda^{e})}\bigg{]}k\,.$
(6.26)

Since the First On-Shell Theorem for topological fields features no such
terms, they must vanish. The decomposition of the field $C(Y;K)$ into power
series in $Y$ yields an infinite chain of equations on the parameters
$\alpha^{e},\beta_{i}^{e},\lambda^{e}$.

$(\alpha^{e}-1-\lambda^{e})\frac{(\beta^{e}_{1})^{n+2}+(\alpha^{e}-\lambda^{e})^{n+1}((\alpha^{e}-\lambda^{e})(n+1)-\beta^{e}_{1}(n+2))}{(n+1)(n+2)(\alpha^{e}-\beta^{e}_{1}-\lambda^{e})}+\\\
+(\alpha^{e}+1-\lambda^{e})\frac{(\beta^{e}_{2})^{n+2}+(\alpha^{e}-\lambda^{e})^{n+1}((\alpha^{e}-\lambda^{e})(n+1)-\beta^{e}_{2}(n+2))}{(n+1)(n+2)(\alpha^{e}-\beta^{e}_{2}-\lambda^{e})}+(\alpha^{e}-\lambda^{e})^{n+2}=0\,,\forall
n\in\mathds{N}\,,$ (6.27)

that demand $\alpha^{e}=\lambda^{e}$ and $\beta_{1}^{e}=\beta_{2}^{e}$. The
origin of these conditions is that in the AdS background the dependence on
$t_{i}$ can be at most bilinear, so that the terms with $t_{1}t_{2}$ in the
pre-exponent must have matching exponents at $t_{i}=0$ to respect the First
On-Shell Theorem. Note that these constraints do not reduce the vertices to
those given by the conventional homotopy for general $\omega$ which allow
higher orders in $t_{i}$.

To find the possible solutions for $\beta_{1}^{e,o}$ and $\beta_{2}^{e,o}$,
the terms in the $(y^{\mu}t_{i\mu}+p^{\mu}t_{i\mu})$
$hh\overline{\partial}\overline{\partial}C$ sector have to be inspected

$\Upsilon^{\eta}(\Omega,\Omega,C)+\Upsilon^{\eta}(\Omega,C,\Omega)+\Upsilon^{\eta}(C,\Omega,\Omega)\bigg{|}_{\overline{H}\overline{\partial}^{2}}=\\\
-\frac{\eta}{16}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}\bigg{[}C_{\rho}(\overline{y};k,\overline{k})y^{\rho}\bigg{(}-\frac{\beta_{1}}{2}-\frac{\beta_{2}}{2}+\frac{\beta_{1}^{2}}{6}-\frac{\beta_{2}^{2}}{6}\bigg{)}-C_{\rho}(\overline{y};-k,-\overline{k})y^{\rho}\bigg{(}-\frac{\beta_{1}}{2}-\frac{\beta_{2}}{2}-\frac{\beta_{1}^{2}}{6}+\frac{\beta_{2}^{2}}{6}\bigg{)}-\\\
-\frac{1}{(n+1)(n+2)}C_{\rho_{1}...\rho_{n}}(\overline{y};k,\overline{k})y^{\rho_{1}}...y^{\rho_{n}}\bigg{(}-\beta_{1}^{n}(n+2-\beta_{1}n)-\beta_{2}^{n}(n+2+\beta_{2}n)\bigg{)}+\\\
+\frac{1}{(n+1)(n+2)}C_{\rho_{1}...\rho_{n}}(\overline{y};-k,-\overline{k})y^{\rho_{1}}...y^{\rho_{n}}\bigg{(}-\beta_{1}^{n}(n+2+\beta_{1}n)-\beta_{2}^{n}(n+2-\beta_{2}n)\bigg{)}\bigg{]}\,.$
(6.28)

Here no terms with parameters $\alpha=\lambda$ are present since they are
accompanied by the factors of the form $a_{\mu}a^{\mu}=0$ with some spinors
$a_{\mu}$. Since the parameters contribute to the argument of the field
$C(Y;K)$, to respect the First On-Shell Theorem one has to expand $C(Y;K)$ in
power series that yields an infinite chain of algebraic equations on
$\beta_{1}$ and $\beta_{2}$. The projection onto the odd sector yields

$\Upsilon^{\eta}(\Omega,\Omega,C)+\Upsilon^{\eta}(\Omega,C,\Omega)+\Upsilon^{\eta}(C,\Omega,\Omega)\bigg{|}_{\overline{H}\overline{\partial}^{2}}=\\\
=-\frac{\eta}{16}\overline{H}^{\dot{\mu}\dot{\nu}}\overline{\partial}_{\dot{\mu}}\overline{\partial}_{\dot{\nu}}\bigg{[}C_{\mu_{1}}y^{\mu_{1}}(\overline{y};k,\overline{k})\biggl{(}\beta^{o}_{1}+\beta^{o}_{2}\biggr{)}+\frac{2}{(n+1)}C_{\mu_{1}...\mu_{n}}y^{\mu_{1}}...y^{\mu_{n}}(\overline{y};k,\overline{k})\biggl{(}(\beta^{o}_{1})^{n}+(\beta^{o}_{2})^{n}\biggr{)}\bigg{]}\,,$
(6.29)

that only obeys the First On-Shell Theorem at $\beta_{1}^{o}=\beta_{2}^{o}=0$.

The same reasoning in the even sector gives

$\frac{1}{3}C_{\mu_{1}}(\overline{y};k,\overline{k})\biggl{(}(\beta^{e}_{1})^{2}-(\beta^{e}_{2})^{2}\biggr{)}+\frac{2n}{(n+1)(n+2)}C_{\mu_{1}...\mu_{n}}(\overline{y};k,\overline{k})\biggl{(}(\beta^{e}_{1})^{n+1}-(\beta^{e}_{2})^{n+1}\biggr{)}=0\,,$
(6.30)

implying $\beta^{e}_{1}=\beta^{e}_{2}$.

Analogous analysis can be applied to the $t^{\mu}_{1}t_{2}{}_{\mu}$ and
$(y^{\mu}t_{i\mu}+p^{\mu}t_{i\mu})$-dependent terms in the
$\overline{H}\overline{y}\overline{y}C$ sector. Due to the sign change in the
products $\Omega\overline{*}\Omega\overline{*}C|_{\overline{Hyy}}$ and
$C\overline{*}\Omega\overline{*}\Omega|_{\overline{Hyy}}$ compared to
$\Omega\overline{*}\Omega\overline{*}C|_{\overline{H\partial^{2}}}$ and
$C\overline{*}\Omega\overline{*}\Omega|_{\overline{H\partial^{2}}}$ we find a
permutation of the even and odd projections of a slightly changed versions of
(6.24) and (6.28). This yields $\alpha^{e,o}=\lambda^{e,o}$,
$\beta_{1}^{e}=\beta_{2}^{e}=0$ and $\beta_{1}^{o}=\beta_{2}^{o}$.

One can also check that the terms with
$H_{\alpha\beta}y_{1}^{\alpha}y_{2}^{\beta}$ in all vertices (recall that
$y_{1}$ and $y_{2}$ are the auxiliary variables assigned, respectively, to the
first and second factors of $\Omega$ in the ordered product) impose no
restrictions on the parameters. The resulting restrictions on the parameters
are $\alpha^{e,o}=\lambda^{e,o}$ in the both sectors, which means that they
are otherwise free. At the same time the $y$-shift parameters in $W_{1}$ are
necessarily vanishing $\beta^{e,o}_{1}=\beta^{e,o}_{2}=0$.

The obtained results imply that one can use two independent homotopy operators
when resolving $S_{1}$ and $W_{1}$, provided the $y$ and $p$ shifts are equal
within each homotopy procedure:

$\displaystyle
S_{1}=-\frac{\eta}{2}\Delta_{a(y+p)}\big{(}C*\gamma\big{)}+c.c.\,,$ (6.31)
$\displaystyle
W_{1}=-\frac{i}{2}\Delta_{b(y+p)}(d_{x}S_{1}+\omega*S_{1}+S_{1}*\omega)+c.c.\,$
(6.32)

with independent $a$ and $b$. Such a homotopy procedure generalizes uniform
shifts considered in [5], where only the shifts with $a=b$ were considered,
that preserve the form of the conventional homotopy vertices. The case of
different $a$ and $b$ is referred to as the relaxed uniform shift. We have
shown that the relaxed uniform shifts produce vertices that differ from those
resulting from the conventional homotopy in general HS background but still
respect the First On-Shell Theorem in $AdS_{4}$ background.

It is worth noticing that $n_{i}$-parameters are not present in the above
considerations. This suggests that there is no interplay between the
$y,p$-shifts and $\omega$-shifts, which indicates that the latter do not
affect the First On-Shell Theorem at all. In the particular case of a pure
$\omega$-shift ($\alpha=\lambda=\beta_{i}=0$) these parameters do not
contribute even beyond the level of free HS equations in $AdS_{4}$, as they do
not affect the HS fields, being equivalent to those resulting from the
conventional (i.e. zero shift) homotopy.

##  7 Pure $\omega$-shift

Now we consider the effect of the pure shift by the arguments of $\omega$ on
the full $\omega^{2}C$ vertices beyond the $AdS_{4}$ background. To this end
we set $q^{\mu}=l_{i}^{\mu}=\beta_{i}=\alpha=\lambda=0$ leaving the
$\omega$–shifts with parameters $n_{i}$ free. From pre-exponential factors
(5.18) - (5.32) one can see that the only non-zero terms at
$\alpha=\lambda=\beta_{i}=0$ are

$\Upsilon^{\eta}(\omega,\omega,C)=\frac{\eta}{4i}\omega*\omega*C*h_{p}\Delta_{t_{1}+t_{2}+p}\Delta_{t_{2}+p}\gamma\,,$
(7.1)
$\Upsilon^{\eta}(\omega,C,\omega)=\frac{\eta}{4i}\omega*C*\omega*\big{[}h_{t_{1}+t_{2}+p}\Delta_{t_{1}+p+2t_{2}}\Delta_{t_{2}+p}\gamma+h_{t_{1}+p+2t_{2}}\Delta_{p+2t_{2}}\Delta_{t_{2}+p}\gamma\big{]}\,,$
(7.2)
$\Upsilon^{\eta}(C,\omega,\omega)=\frac{\eta}{4i}C*\omega*\omega*h_{p+2t_{1}+2t_{2}}\Delta_{t_{1}+p+2t_{2}}\Delta_{t_{1}+t_{2}+p}\gamma\,.$
(7.3)

This yields the equation

$\mbox{d}\omega+\omega*\omega+\omega*\omega*C*h_{p}\Delta_{t_{1}+t_{2}+p}\Delta_{t_{2}+p}\gamma+\omega*C*\omega*h_{t_{1}+t_{2}+p}\Delta_{t_{1}+p+2t_{2}}\Delta_{t_{2}+p}\gamma+\\\
+\omega*C*\omega*h_{t_{1}+p+2t_{2}}\Delta_{p+2t_{2}}\Delta_{t_{2}+p}\gamma+C*\omega*\omega*h_{p+2t_{1}+2t_{2}}\Delta_{t_{1}+p+2t_{2}}\Delta_{t_{1}+t_{2}+p}\gamma=0\,.$
(7.4)

Using (4.7) and partial star-product (5.19) we obtain

$\Upsilon^{\eta}(\omega,\omega,C)=\frac{\eta}{2i}\int_{[0,1]^{3}}d^{3}\tau\delta\left(1-\tau_{1}-\tau_{2}-\tau_{3}\right)e^{i\left(1-\tau_{3}\right)\partial_{1}^{\mu}\partial_{2\mu}}\partial^{\nu}\omega\left(\left(1-\tau_{1}\right)y,\overline{y};K\right)\overline{*}\\\
\overline{*}\partial_{\nu}\omega\left(\tau_{2}y,\overline{y};K\right)\overline{*}C\left(-i\tau_{1}\partial_{1}-i\left(1-\tau_{2}\right)\partial_{2},\overline{y};K\right)k\,,$
(7.5)
$\Upsilon^{\eta}(C,\omega,\omega)=\frac{\eta}{2i}\int_{[0,1]^{3}}d^{3}\tau\delta\left(1-\tau_{1}-\tau_{2}-\tau_{3}\right)e^{i\left(1-\tau_{3}\right)\partial_{1}^{\mu}\partial_{2\mu}}\\\
C\left(i\tau_{1}\partial_{2}+i\left(1-\tau_{2}\right)\partial_{1},\overline{y};K\right)\overline{*}\partial^{\nu}\omega\left(\tau_{2}y,\overline{y};K\right)\overline{*}\partial_{\nu}\omega\left(-\left(1-\tau_{1}\right)y,\overline{y};K\right)k\,,$
(7.6)
$\Upsilon^{\eta}(\omega,C,\omega)=\frac{\eta}{2i}\int_{[0,1]^{3}}d^{3}\tau\delta\left(1-\tau_{1}-\tau_{2}-\tau_{3}\right)e^{i\left(1-\tau_{3}\right)\partial_{1}^{\mu}\partial_{2\mu}}\partial^{\nu}\omega\left(\tau_{1}y,\overline{y};K\right)\overline{*}\\\
\overline{*}C\left(i\left(1-\tau_{2}\right)\partial_{2}-i\left(1-\tau_{1}\right)\partial_{1},\overline{y};K\right)\overline{*}\partial_{\nu}\omega\left(-\left(1-\tau_{2}\right)y,\overline{y};K\right)k+\\\
+\frac{\eta}{2i}\int_{[0,1]^{3}}d^{3}\tau\delta\left(1-\tau_{1}-\tau_{2}-\tau_{3}\right)e^{-i\tau_{2}\partial_{1}^{\mu}\partial_{2\mu}}\partial^{\nu}\omega\left(\left(1-\tau_{1}\right)y,\overline{y};K\right)\overline{*}\\\
\overline{*}C\left(-i\tau_{1}\partial_{1}+i\tau_{3}\partial_{2},\overline{y};K\right)\overline{*}\partial_{\nu}\omega\left(-\left(1-\tau_{3}\right)y,\overline{y};K\right)k\,.$
(7.7)

Remarkably, $n_{1,2}$ do not contribute to the vertices (7.5)-(7.7), which
coincide with those resulting from the conventional homotopy procedure with
zero shift parameters [5].

The same result can be obtained in a simpler way using the property of the
$\Delta_{Q}$ and $h_{Q}$ (4.9) presented in Section 4. Moreover, the absence
of restrictions on the parameters $n_{i}$ can already be established at the
level of the field $W_{1}^{\eta}$. For instance, consider the field
$W_{1}^{(1)}$ with the parameter $n_{1}$

$W^{\eta(1)}_{1}=\frac{\eta}{4i}\omega*C*\Delta_{(n_{1}+1)t+p}\big{(}\Delta_{p}-\Delta_{(t+p)}\big{)}\gamma\,.$
(7.8)

Using (4.9), one gets

$W^{\eta(1)}_{1}=\frac{\eta}{4i}\omega*C*\big{(}h_{(n_{1}+1)t+p}\Delta_{p}\Delta_{(t+p)}-\Delta_{p}\Delta_{(t+p)}\big{)}\gamma\,.$
(7.9)

Inspecting the seemingly $n_{1}$-dependent first term, we find that it
vanishes by virtue of (4.5) which proves independence of $W^{\eta(1)}_{1}$ of
$n_{1}$. Analogously, $W_{1}^{\eta(2)}$ is $n_{2}$-independent. Thus, for any
$n_{i}$, the field $W_{1}^{\eta}$ is the same as in the case of the
conventional homotopy, i.e. at $n_{1,2}=0$, and the form of the First On-Shell
Theorem is intact. The output of this analysis is that, being equivalent to
the conventional homotopy, pure $\omega$-shifts do not affect higher-order
corrections to the fields and non-linear HS equations.

##  8 Conclusion

In this paper we have analysed an extension of the homotopy procedure
elaborated in [5] to the homotopy operators with the shift parameters acting
on the arguments of the one-form HS gauge fields $\omega$, the arguments of
the zero-form HS fields $C$ and proportional to spinor variables $Y^{A}$. We
have found general restrictions on the shift parameters that respect the
canonical form of the free unfolded HS equations known as First On-Shell
theorem [11], which is necessary to preserve the interpretation of zero-forms
$C$ as derivatives of the HS gauge fields. To put it differently, this
condition is demanded to provide locality of the unfolded HS equations at the
free field level. The conditions that respect canonical form of the First On-
Shell theorem are shown to leave six free parameters ($n_{i}^{e,o}$,
$\alpha^{e,o}$), four of which ($n_{i}^{e,o}$) are associated with the shifts
of the arguments of the one-form $\omega$, and the other two ($\alpha^{e,o}$)
of the $(p+y)$-shift.

Thus, in the perturbative analysis, one can use different homotopy operators
$\Delta_{a(y+p)}$ and $\Delta_{n_{i}t+b(y+p)}$ to resolve for $S_{1}$ and
$W_{1}$, respectively, still preserving the form of the First On-Shell
Theorem. In the particular case of $y$ and $p$ shifts, this results generalize
the uniform shifts of [5] with $a=b$. Relaxing this condition to relaxed
uniform shifts with independent $a$ and $b$ we have shown that the relaxed
uniform shifts produce (ultralocal) vertices that differ from those obtained
by the conventional homotopy in general HS background but still respect the
First On-Shell Theorem in $AdS_{4}$ background.

In the particular case of a pure $\omega$-shift, surprisingly enough, not only
the form of free HS field equations in $AdS_{4}$ is not affected by the
$\omega$-shift parameters, but also all vertices
$\Upsilon^{\eta}(\omega,\omega,C)$, $\Upsilon^{\eta}(\omega,C,\omega)$ and
$\Upsilon^{\eta}(C,\omega,\omega)$ remain intact. Moreover, by virtue of
identities (4.5) originally obtained in [5] this is shown to be a consequence
of the fact that the first-order corrections to the one-form fields
$W_{1}^{\eta}(Z;Y|x)$ turn out to be independent of the $\omega$-shift
parameters.

## Acknowledgement

We are grateful to Olga Gelfond, Anatoly Korybut for useful comments and the
referee for a stimulating question. This research was supported by the Russian
Science Foundation grant 18-12-00507.

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