# Spinorial higher-spin gauge theory from IKKT model
in Euclidean and Minkowski signatures

Harold Steinacker  & Tung Tran

###### Abstract

We explore the semi-classical relation between the fuzzy 4-hyperboloid
$H_{N}^{4}$ and non-compact quantized twistor space $\mathbb{P}_{N}^{1,2}$ at
large $N$. This provides two backgrounds of the IKKT matrix model via two
natural stereographic projections, leading to higher-spin gauge theories with
Euclidean and Minkowski signature denoted by HS-IKKT. The resulting higher-
spin gauge theory can be understood as an uplift of $\mathcal{N}=4$ super
Yang-Mills to twistor space. The action of HS-IKKT is written using a spinor
formalism in both Euclidean and Minkowski signature. We then compute the tree-
level amplitudes of the massless sector within the Yang-Mills part of the HS-
IKKT model in the flat limit in Euclidean signature. All $n$-point tree-level
scattering amplitudes for $n\geq 4$ of this sector are found to vanish in the
flat limit.

##  1 Introduction

The IKKT-matrix model [1] can be considered as an alternative and constructive
description of type IIB superstring theory. It also provides a promising
approach towards a quantum theory of gravity in $3+1$ dimensions, since both
spacetime and physical fields emerge from the same matrix degrees of freedom;
see e.g. [2] for a review. Recently, it was shown that the Einstein-Hilbert
action can be obtained from the one-loop effective action on non-commutative
branes in the presence of fuzzy extra dimensions [3, 4]. From this
perspective, the gravitational theory induced by the IKKT-matrix model is
closely related to the idea of induced gravity of Sakharov [5, 6].

Some natural backgrounds of the IKKT model are given by covariant fuzzy spaces
such as the fuzzy 4-sphere $S^{4}_{N}$ [7] or fuzzy 4-hyperboloid $H^{4}_{N}$
[8]. These are the total spaces of quantized sphere
$\mathbb{P}^{1}_{N}$-bundle over the base manifold $S^{4}$ or $H^{4}$,
respectively. From this point of view, $S^{4}_{N}$ can be viewed as a compact
fuzzy twistor space $\mathbb{P}^{3}_{N}$ while $H^{4}_{N}$ is understood as a
non-compact fuzzy twistor space $\mathbb{P}^{1,2}_{N}$. What is significant
about these fuzzy spaces is that their algebra of functions $\mathscr{C}$ is
truncated. As a result, the higher-spin algebra $\mathfrak{ths}$, which is a
subspace of $\mathscr{C}$, is also truncated for finite $N$ [7, 9]. Since this
truncated symmetry must be accompanied by higher-spin gauge fields,111Note
that the truncated higher-spin algebra $\mathfrak{ths}$ coincides with the
usual higher-spin algebra $\mathfrak{hs}$ of 4-dimensional target space in the
semi-classical (large $N$) limit as discussed in e.g. [2, 10]. it is obvious
that the IKKT-matrix model can induce a higher-spin gauge theory (HS-IKKT)
whose spectrum is finite.

The connection between (HS-)IKKT matrix model and twistor theory naturally
leads to a spinorial formulation of the resulting gauge theory, which was
explored for the case of $S_{N}^{4}$ in [10]. However this only leads to a
background with Euclidean signature. To overcome that limitation, we
generalize in the present paper the analysis in [10] to backgrounds defined
using $H_{N}^{4}$. Indeed, it is known that $H_{N}^{4}$ can be projected to a
fuzzy FLRW cosmological spacetime with Lorentzian signature. Then the IKKT-
matrix model defined on this background becomes a more ‘conventional’ field
theory.

In this work we will consider two projections of $H_{N}^{4}$: (i) a
stereographic projection to a 4-hyperboloid (or equivalently Euclidean AdS) as
the base manifold, and (ii) a $SO(1,3)$-invariant projection which defines a
FLRW type cosmology [11, 12, 13]. In the first case, we essentially recover
the same results of [10] (up to signs) for the spinorial formulation of the
IKKT-matrix model on $H^{4}$. In the second case, we provide a novel
formulation of the IKKT-matrix model in terms of twistor/spinor variables on
the FLRW spacetime. However, in that case the spinorial description turns out
to be rather unconventional, making it hard to study scattering processes in
the flat limit.

As an application of the spinorial formulation, we compute tree-level on-shell
scattering amplitudes of massless fields of the Yang-Mills (YM) sector of HS-
IKKT theory in the complexified Euclidean case, and show that all $n$-point
tree-level amplitudes of this sector vanish for $n\geq 4$ in the flat limit.
To obtain an alternative understanding of this result, we also project all
vertices of this sector to the light-cone gauge, and observe that the MHV
sector can be removed by a local field redefinition. Due to the unconventional
space-like spinor formalism in the Lorentzian case, we compute the tree-level
scattering amplitudes of HS-IKKT theory only in Euclidean signature in the
present paper, and set the stage to elaborate the Lorentzian case elsewhere.

The paper is structured as follows. Section 2 provides a brief review of the
IKKT-matrix model and almost-commutative 4-hyperboloid. Section 3 elaborates
the relation between $H_{N}^{4}$ and $\mathbb{P}_{N}^{1,2}$ using spinor
formalism from the stereographic projection point of view. Section 4 studies
the spinorial description of the $SO(1,3)$-invariant projection. The gauge-
fixing procedure and decompositions of modes are specified in Section 5.
Section 6 derives spacetime action of HS-IKKT. Section 7 computes the
scattering amplitudes of the HS-IKKT. We conclude in Section 8 and collect
some technicalities in the Appendices.

#### Notation.

Throughout the paper, we use $a,b$ as $SO(1,4)$-indices where $a,b=0,1,2,3,4$
while $\hat{a},\hat{b}$ stand for $SO(2,3)$-indices for
$\hat{a},\hat{b}=0,1,2,3,5$. The $Sp(4)$ and twistorial indices will be
denoted as $A,B$ where $A,B=1,2,3,4$. Note that $\alpha,\beta=0,1$ while
$\dot{\alpha},\dot{\beta}=\dot{0},\dot{1}$. We will use the strength-one
symmetrization convention, e.g.
$A_{a}B_{a}=\frac{1}{2}(A_{a_{1}}B_{a_{2}}+A_{a_{2}}B_{a_{1}})$; and write
fully symmetric rank-$s$ tensor as $T_{a(s)}=T_{a_{1}\ldots a_{s}}$ for short.

##  2 Review on the IKKT matrix model and the fuzzy 4-hyperboloid

The field content of the $SO(1,9)$-invariant action functional

$\displaystyle
S=\text{Tr}\Big{(}[Y^{\boldsymbol{I}},Y^{\boldsymbol{J}}][Y_{\boldsymbol{I}},Y_{\boldsymbol{J}}]+\Psi_{\mathcal{A}}(\tilde{\gamma}^{\boldsymbol{I}})^{\mathcal{A}\mathcal{B}}[Y_{\boldsymbol{I}},\Psi_{\mathcal{B}}]\Big{)}\,,\qquad{\boldsymbol{I}}=0,1,\ldots,9\,,$
(2.1)

describing the IKKT model comprises $N\times N$ hermitian matrices
$Y^{\boldsymbol{I}}$, and the matrix-valued Majorana-Weyl spinors
$\Psi_{\mathcal{B}}$ associated with $SO(1,9)$. Since $Y^{\boldsymbol{I}}$ are
matrices, they do not commute. This non-commutativity can be interpreted in
terms of a quantized Poisson structure $\theta^{\boldsymbol{I}\boldsymbol{J}}$
on a brane embedded in target space $\mathbb{R}^{1,9}$ viz.

$\displaystyle[Y^{\boldsymbol{I}},Y^{\boldsymbol{J}}]:=\mathrm{i}\,\theta^{\boldsymbol{I}\boldsymbol{J}}\,.$
(2.2)

To extract classical coordinate functions $\\{y^{\boldsymbol{I}}\\}$
describing $\mathbb{R}^{1,9}$, it is reasonable to assume that there are
localized quasi-coherent states $|y\rangle\in\mathcal{H}$ (with $\mathcal{H}$
being some Hilbert space on which the matrices act) such that the
$y^{\boldsymbol{I}}$ arise as expectation value of $Y^{\boldsymbol{I}}$ viz.

$\displaystyle y^{\boldsymbol{I}}=\langle y|Y^{\boldsymbol{I}}|y\rangle\
\in\mathbb{R}^{1,9}\ .$ (2.3)

The resolution of the coordinate functions $y^{\boldsymbol{I}}$ will increase
with the size of the matrices $Y^{\boldsymbol{I}}$, which suggests to consider
the large $N$ (or semi-classical) limit where matrices are almost-commutative
[14, 15, 16, 17]. In this limit, we can replace (2.2) with Poisson brackets

$\displaystyle\\{y^{\boldsymbol{I}},y^{\boldsymbol{J}}\\}:=\theta^{\boldsymbol{I}\boldsymbol{J}}\,.$
(2.4)

Here $y^{\boldsymbol{I}}$ can be used to define an almost-commutative variety
embedded in target space via
$y^{\boldsymbol{I}}:\mathcal{M}\hookrightarrow\mathbb{R}^{1,9}$. Then
classical functions in terms of $y$ are related to matrices via

$\begin{split}\mathscr{C}(\mathcal{M})\quad&\sim\quad{\rm Mat}(\mathcal{H})\\\
f(y)=\langle y|F(Y)|y\rangle\quad&\sim\quad F(Y)\,.\end{split}$ (2.5)

The matrix algebra $({\rm Mat}(\mathcal{H}),[\,,])$ generated by
$Y^{\boldsymbol{I}}$ is interpreted as quantized version of the Poisson
algebra defined by the pair $(\mathscr{C}(\mathcal{M}),\\{\,,\\})$.
Furthermore, the trace in (2.1) will be replaced by some appropriate integral,
as discussed below.

For our purpose of constructing a higher-spin theory from the IKKT-matrix
model on $H_{N}^{4}$, it is sufficient to consider the semi-classical limit or
-regime, where we can work with ordinary functions as familiar from field
theory. The non-commutativity then reduces to the explicit Poisson brackets

$\displaystyle\\{y^{a},y^{b}\\}=:\theta^{ab}=-\ell_{p}^{2}m^{ab}\,,\qquad
a,b=0,1,2,3,4\,,$ (2.6)

where $m^{ab}$ are functions on twistor space $\mathbb{P}^{1,2}$ (as explained
below) arising from the semi-classical limit of $\mathfrak{so}(1,4)$
generators in certain representations [8], and $\ell_{p}$ is a natural length
scale.

It is worth noting that the Poisson bracket (2.6) involves two derivatives in
$y^{a}$. This means that even though the IKKT-matrix model has the structure
of a Yang-Mills gauge theory, it behaves like a gravitational theory. To study
the IKKT model perturbatively [18], we will consider fluctuations
$y^{a}=\bar{y}^{a}+\mathsf{a}^{a}$ of a background $\bar{y}^{a}$ on
$H^{4}_{N}$ [19]. This defines an almost-commutative Yang-Mills-type gauge
theory on $H_{N}^{4}$ [8] that is invariant under the gauge transformations
$U^{-1}(\bar{y}^{a}+\mathsf{a}^{a})U$ where $U$ is any unitary matrix,
replacing commutators by Poisson brackets.

#### Algebra of functions on semi-classical $H_{N}^{4}$.

Endowing the ambient space $\mathbb{R}^{1,4}$ with the metric
$\eta_{ab}=(-,+,+,+,+)$ allows us to describe a 4-dimensional hyperboloid
$H^{4}_{N}$ of radius $R$ in terms of a space of functions $\mathscr{C}$ with
the following $\mathfrak{so}(1,4)$-covariant semi-classical relations
[8]:222Note the sign difference with the case of $S_{N}^{4}$ [7].

$\displaystyle\\{m_{ab},m_{cd}\\}$
$\displaystyle=+(m_{ad}\eta_{bc}-m_{ac}\eta_{bd}-m_{bd}\eta_{ac}+m_{bc}\eta_{ad})\,,$
(2.7a) $\displaystyle\\{m_{ab},y_{c}\\}$
$\displaystyle=+(y_{a}\eta_{bc}-y_{b}\eta_{ac})\,,$ (2.7b)
$\displaystyle\\{y_{a},y_{b}\\}$ $\displaystyle=-\ell_{p}^{2}\,m_{ab}\,,$
(2.7c) $\displaystyle y_{a}y^{a}$
$\displaystyle=-y_{0}^{2}+y_{\mathtt{a}}y^{\mathtt{a}}=-R^{2}=-\frac{\ell_{p}^{2}N^{2}}{4}\,,\qquad\mathtt{a}=1,2,3,4\,,$
(2.7d) $\displaystyle\epsilon_{abcde}m^{ab}y^{c}$
$\displaystyle=-\frac{4N}{\ell_{p}}m_{de}\,$ (2.7e)

with large $N$. Here, (2.7e) is a self-duality relation, which allows us to
restrict the space of functions on $H^{4}_{N}$ to

$\displaystyle\mathscr{C}(y^{a},m^{ab})=\sum_{k,m}f_{c(k)a(m),b(m)}y^{c(k)}m^{ab}\ldots
m^{ab}=\bigoplus_{k,m}\
\parbox{85.0pt}{{\begin{picture}(70.0,30.0)\put(0.0,3.0){{\begin{picture}(70.0,20.0)\put(0.0,0.0){\begin{picture}(40.0,10.0)\put(0.0,0.0){\line(1,0){40.0}}\put(0.0,0.0){\line(0,1){10.0}}\put(40.0,10.0){\line(-1,0){40.0}}\put(40.0,10.0){\line(0,-1){10.0}}{\put(20.0,2.0){{$m$}}}\end{picture}}\put(0.0,10.0){\begin{picture}(70.0,10.0)\put(0.0,0.0){\line(1,0){70.0}}\put(0.0,0.0){\line(0,1){10.0}}\put(70.0,10.0){\line(-1,0){70.0}}\put(70.0,10.0){\line(0,-1){10.0}}{\put(35.0,2.0){{$m+k$}}}\end{picture}}\end{picture}}}\end{picture}}}\,$
(2.8)

corresponding to two-row Young tableaux, where the convention $y^{a(m)}$ means
$y^{(a_{1}}\ldots y^{a_{m})}$. Within $\mathscr{C}$, we can find a subspace

$\displaystyle\mathfrak{ths}(\mathfrak{so}(1,4))=\sum_{m}g_{a(m),b(m)}m^{ab}\ldots
m^{ab}=\bigoplus_{m}\
\parbox{60.0pt}{{\begin{picture}(70.0,30.0)\put(0.0,3.0){{\begin{picture}(50.0,20.0)\put(0.0,0.0){\begin{picture}(50.0,10.0)\put(0.0,0.0){\line(1,0){50.0}}\put(0.0,0.0){\line(0,1){10.0}}\put(50.0,10.0){\line(-1,0){50.0}}\put(50.0,10.0){\line(0,-1){10.0}}{\put(25.0,2.0){{$m$}}}\end{picture}}\put(0.0,10.0){\begin{picture}(50.0,10.0)\put(0.0,0.0){\line(1,0){50.0}}\put(0.0,0.0){\line(0,1){10.0}}\put(50.0,10.0){\line(-1,0){50.0}}\put(50.0,10.0){\line(0,-1){10.0}}{\put(25.0,2.0){{$m$}}}\end{picture}}\end{picture}}}\end{picture}}}\,,$
(2.9)

which consists of polynomials purely in terms of $m^{ab}$, thus defining a
truncated higher-spin algebra $\mathfrak{ths}$. Note that $\mathfrak{ths}$
coincides with the usual higher-spin algebra $\mathfrak{hs}$ of Euclidean AdS4
in $N\rightarrow\infty$ limit.

The above algebra of function implies that the IKKT-matrix model on a
$H^{4}_{N}$ background induces a higher-spin gauge theory (HS-IKKT) [8].
Remarkably, the higher-spin fields mitigate the violation of Lorentz
invariance through noncommutativity (cf. (2.6)), while parity invariance is
broken due to the presence of $\epsilon_{abcde}$ tensors [7]. This causes the
HS-IKKT to exhibit some sort of ‘chiral’ feature [8], and reflects the fact
that HS-IKKT can be formulated on twistor space [10].

#### Twistor realization of semi-classical $H_{N}^{4}$.

It is manifest that (2.7) is related to the Lie algebra $\mathfrak{so}(2,4)$;
more precisely, they arise in the semiclassical limit of a specific
“doubleton” representation $\mathcal{H}_{N}$ of $\mathfrak{so}(2,4)$.

Instead of working with $\mathfrak{so}(2,4)$, it is sometimes more convenient
to use the $\mathfrak{su}(2,2)$ formulation. As is well-known, the non-compact
twistor space $\mathbb{P}^{1,2}$ can be realized either as
$SO(1,4)$-equivariant bundle over $H^{4}$, or a 6-dimensional co-adjoint orbit
of $SU(2,2)$, i.e. $\mathcal{O}_{\Xi}=\\{g\,\Xi\,g^{-1}\,,\,g\in SO(2,4)\\}$
where $\Xi=(N,0,0)$ has the stabilizer $U(1)\times SU(2)$ [8]. This allows us
to identify $H^{4}_{N}$ with $\mathbb{P}^{1,2}$. More explicitly, consider the
following identifications:

$\displaystyle y^{AB}=-y^{BA}=\ell_{p}^{-1}y^{a}\gamma_{a}^{AB}\,,\qquad
l^{AB}=l^{BA}=\frac{1}{2}m^{ab}\Sigma^{AB}_{ab}\,\,,$ (2.10)

where $\gamma_{a}=-2\Sigma_{a5}$ are the gamma matrices of
$\mathfrak{so}(1,4)$ realized by the following basis adapted to the compact
$SU(2)_{L}\times SU(2)_{R}\subset SU(2,2)$ subgroup:

$\displaystyle(\gamma_{0})^{A}_{\ B}=\begin{pmatrix}\mbox{1
\kern-5.89996pt{\rm l}}_{2}&0\\\ 0&-\mbox{1 \kern-5.89996pt{\rm
l}}_{2}\end{pmatrix}\,,\quad(\gamma_{m})^{A}_{\
B}=\begin{pmatrix}0&-(\sigma_{m})^{\alpha}{}_{\dot{\beta}}\\\
(\sigma_{m})^{\dot{\alpha}}{}_{\beta}&0\end{pmatrix}\,,\quad(\gamma_{4})_{\
B}^{A}=\mathrm{i}\begin{pmatrix}0&\mbox{1 \kern-5.89996pt{\rm l}}_{2}\\\
\mbox{1 \kern-5.89996pt{\rm l}}_{2}&0\end{pmatrix}\,,$ (2.11)

for $m=1,2,3$, and

$\displaystyle\\{\gamma_{a},\gamma_{b}\\}^{A}_{\ B}=-2\eta_{ab}\delta^{A}_{\
B}\,,\qquad\Sigma_{ab}^{AB}=-\Sigma_{ba}^{AB}=\Sigma_{ab}^{BA}=\frac{1}{4\,\mathrm{i}}[\gamma_{a},\gamma_{b}]^{AB}\,.$
(2.12)

This allows us to express the $H_{N}^{4}$ using $\mathfrak{su}(2,2)$
representation as:

$\displaystyle\\{l^{AB},l^{CD}\\}$
$\displaystyle=+(l^{AC}C^{BD}+l^{AD}C^{BC}+l^{BD}C^{AC}+l^{BC}C^{AD})\,,$
(2.13a) $\displaystyle\\{l^{AB},y^{CD}\\}$
$\displaystyle=+(y^{AC}C^{BD}+y^{BC}C^{AD}-y^{AD}C^{BC}-y^{BD}C^{AC})\,,$
(2.13b) $\displaystyle\\{y^{AB},y^{CD}\\}$
$\displaystyle=-(l^{AC}C^{BD}-l^{AD}C^{BC}-l^{BC}C^{AD}+l^{BD}C^{AC})\,,$
(2.13c) $\displaystyle y_{AB}y^{AB}$ $\displaystyle=l_{AB}l^{AB}=-4R^{2}\,,$
(2.13d) $\displaystyle\epsilon_{ABCD}y^{AB}$ $\displaystyle=y_{CD}\,$ (2.13e)

where

$\displaystyle
C^{AB}=-C^{BA}=\,\mathrm{diag}(\epsilon^{\alpha\beta},\epsilon^{\dot{\alpha}\dot{\beta}})$
(2.14)

is the $\mathfrak{so}(1,4)$-invariant matrix and we use
$\epsilon^{01}=-\epsilon^{10}=1\,,\epsilon^{\alpha\beta}=\epsilon_{\alpha\beta}$
as our $\mathfrak{sp}(2)$-invariant matrix. These matrices allow us to raise
and lower $\mathfrak{su}(2,2)$ and/or $\mathfrak{sp}(2)$ indices as

$\displaystyle V_{B}C^{AB}=U^{A}\,,\qquad V^{B}C_{BA}=U_{A}\,,\qquad
u^{\alpha}=u_{\beta}\epsilon^{\alpha\beta}\,,\qquad
u_{\alpha}=u^{\beta}\epsilon_{\beta\alpha}\,.$ (2.15)

The Poisson algebra in $\mathfrak{su}(2,2)$ representation becomes

$\displaystyle\mathscr{C}(y^{AB},l^{AB})=\sum_{k,m}f_{A(k)B(2m)|C(k)}y^{AC}\ldots
y^{AC}l^{BB}\ldots l^{BB}=\bigoplus_{k,m}\
\parbox{75.0pt}{{\begin{picture}(70.0,30.0)\put(0.0,3.0){{\begin{picture}(60.0,20.0)\put(0.0,0.0){\begin{picture}(40.0,10.0)\put(0.0,0.0){\line(1,0){40.0}}\put(0.0,0.0){\line(0,1){10.0}}\put(40.0,10.0){\line(-1,0){40.0}}\put(40.0,10.0){\line(0,-1){10.0}}{\put(20.0,2.0){{$k$}}}\end{picture}}\put(0.0,10.0){\begin{picture}(60.0,10.0)\put(0.0,0.0){\line(1,0){60.0}}\put(0.0,0.0){\line(0,1){10.0}}\put(60.0,10.0){\line(-1,0){60.0}}\put(60.0,10.0){\line(0,-1){10.0}}{\put(30.0,2.0){{$k+2m$}}}\end{picture}}\end{picture}}}\end{picture}}}\,.$
(2.16)

We can identify the following subspace:

$\displaystyle\mathfrak{ths}(\mathfrak{sp}(4))=\sum_{m}g_{B(2m)}l^{BB}\ldots
l^{BB}=\bigoplus_{m}\
\parbox{60.0pt}{{\begin{picture}(60.0,6.0)\put(0.0,-1.5){\begin{picture}(50.0,10.0)\put(0.0,0.0){\line(1,0){50.0}}\put(0.0,0.0){\line(0,1){10.0}}\put(50.0,10.0){\line(-1,0){50.0}}\put(50.0,10.0){\line(0,-1){10.0}}{\put(25.0,2.0){{$2m$}}}\end{picture}}\end{picture}}}\
$ (2.17)

as the truncated higher-spin algbera $\mathfrak{ths}(\mathfrak{sp}(4))$. This
allows us to identify semi-classical $H_{N}^{4}$ with the non-compact semi-
classical twistor space $\mathbb{P}^{1,2}_{N}$ spanned by the
$\mathfrak{su}(2,2)$ vectors $Z^{A}$ and their complex conjugate
$Z_{A}^{\dagger}$.

For later convenience, we also note that by lowering the indices of the gamma
matrices in (2.11), we obtain

$\displaystyle\gamma^{0}_{AB}=\begin{pmatrix}\epsilon_{\alpha\beta}&0\\\
0&-\epsilon_{\dot{\alpha}\dot{\beta}}\end{pmatrix}\,,\quad\gamma^{m}_{AB}=\begin{pmatrix}0&\tilde{\sigma}^{m}_{\alpha\dot{\beta}}\\\
-\tilde{\sigma}^{m}_{\dot{\beta}\alpha}&0\end{pmatrix}\,,\quad\gamma^{4}_{AB}=\begin{pmatrix}0&\epsilon_{\alpha\dot{\alpha}}\\\
-\epsilon_{\dot{\alpha}\alpha}&0\end{pmatrix}\,.$ (2.18)

The Hilbert space $\mathcal{H}_{N}$ underlying $H_{N}^{4}$ is the lowest-
weight irreducible representation

$\displaystyle(0,0,N)_{\mathfrak{su}(2,2)}$ (2.19)

which can be realized using an oscillator construction in terms of operators
$Z^{A},\bar{Z}_{A}$ subject to the constraint

$\displaystyle\hat{\mathcal{N}}:=\bar{Z}_{A}Z^{A}=Z^{\dagger}_{A}(\gamma^{0})^{A}{}_{B}Z^{B}=N=\frac{2R}{\ell_{p}}\,$
(2.20)

where $\bar{Z}_{A}=Z^{\dagger}_{\bullet}(\gamma^{0})^{\bullet}{}_{A}$ is the
Dirac conjugate of $Z^{A}$; and the “number operator” $\hat{\mathcal{N}}$ is
invariant under $SU(2,2)$ rather than $SU(4)$ (cf. section 3.3 in [8]). Note
that the $A$ indices are chiral wrt. $SU(2,2)$.

The space of operators

$\begin{split}\mathscr{C}(\mathbb{P}^{1,2}_{N})&=\mathrm{End}(\mathcal{H}_{N})=(N,0,0)_{\mathfrak{su}(2,2)}\otimes(0,0,N)_{\mathfrak{su}(2,2)}=\sum_{n=1}^{N}(n,0,n)_{\mathfrak{su}(2,2)}\\\
&=\sum_{n=0}^{N}\,f_{A(n)}{}^{B(n)}Z^{A}\ldots
Z^{A}\bar{Z}_{B}\ldots\bar{Z}_{B}\,\end{split}$ (2.21)

is given by the space of polynomials in $Z^{A},\bar{Z}_{B}$ with equal number
of $Z$ and $\bar{Z}$. Its semi-classical limit is given by the space of
commutative functions generated by $Z^{A},\bar{Z}_{B}$ with equal number of
$Z$ and $\bar{Z}$ subject to the constraint (2.20) modulo $U(1)$ endowed with
the Poisson structure

$\displaystyle\\{Z^{A},\bar{Z}_{B}\\}=\delta^{A}{}_{B}\,$ (2.22)

This is nothing but the space of functions on $\mathscr{C}(\mathbb{P}^{1,2})$.
We can check that

$\displaystyle\\{\hat{\mathcal{N}},Z^{A}\\}=-Z^{A}\,,\qquad\\{\hat{\mathcal{N}},\bar{Z}_{A}\\}=+\bar{Z}_{A}\,.$
(2.23)

Thus $\hat{\mathcal{N}}$ gives the Poisson algebra defined by the pair
$(\mathscr{C}(\mathbb{P}^{1,2}),\\{\,,\\})$ a gradation as shown. Moreover, we
can use $\hat{\mathcal{N}}$ to define an $H^{3,4}\subset\mathbb{C}^{4}$ as

$\displaystyle H^{3,4}\simeq
H^{7}:=\\{Z^{A}\in\mathbb{C}^{4}\,|\,\bar{Z}_{A}Z^{A}=N\\}\,.$ (2.24)

This allows us to understand $\mathbb{P}^{1,2}$ from the point of view of the
following Hopf fibration333note that for $\mathfrak{so}(4,1)$ both indices of
the gamma matrices are equivalent since the spinor representation is self-dual
due to $C_{AB}$, unlike for $\mathfrak{so}(4,2)$.:

$\displaystyle\begin{split}\mathbb{P}^{1}\xhookrightarrow{}\mathbb{P}^{1,2}=H^{3,4}/U(1)&\rightarrow
H^{4}\subset\mathbb{R}^{1,4}\,,\\\ Z^{A}&\mapsto
y^{a}:=\frac{\ell_{p}}{2}\bar{Z}_{A}(\gamma^{a})^{A}{}_{B}Z^{B}\,,\end{split}$
(2.25)

where $\gamma^{a}=2\Sigma^{a5}$. This means that $\mathbb{P}^{1,2}_{N}$ can be
understood as $\mathbb{P}^{1}_{N}$-bundle over $H^{4}$. Here the subscript $N$
indicates not only the radius constraint (2.20) but also the origin of
$\mathbb{P}^{1}_{N}$ as a fuzzy sphere $S^{2}_{N}$ in the non-commutative
regime, as explained further in Section 3. Note that $H^{4}$ has Euclidean
signature even though the ambient space $\mathbb{R}^{1,4}$ is endowed with a
$SO(1,4)$ metric [8]. The $\mathbb{P}^{1}_{N}$ will be described in terms of
‘negative chirality’ spinors which transform under the local $SU(2)_{L}$.

#### Semi-classical $H^{2,2}_{N}$.

For later discussion related to a matrix-model cosmology with Lorentzian
signature, we recall that $\mathbb{P}^{1,2}$ can be also viewed as
$\mathbb{P}^{1}_{N}$-bundle over a split signature 4-hyperboloid $H^{2,2}$.
This is realized by the following Hopf map:

$\displaystyle\begin{split}\mathbb{P}^{1}\xhookrightarrow{}\mathbb{P}^{1,2}=H^{3,4}/U(1)&\rightarrow
H^{2,2}\subset\mathbb{R}^{2,3}\\\ Z^{A}&\mapsto
t_{\hat{a}}=\frac{1}{R}\bar{Z}_{A}(\Sigma_{\hat{a}4})^{A}_{\
B}Z^{B}=\frac{1}{R}m_{\hat{a}4}\,,\qquad\hat{a}=0,1,2,3,5\,,\end{split}$
(2.26)

where $t_{\hat{a}}$ transform as vectors under $SO(2,3)$ whose generators are
$m_{\hat{a}\hat{b}}$. They satisfy the following relations

$\displaystyle\\{t^{\hat{a}},t^{\hat{b}}\\}$
$\displaystyle=\frac{1}{R^{2}}m^{\hat{a}\hat{b}}\,,\qquad\qquad\quad\quad\,\,\hat{a},\hat{b}=0,1,2,3,5\,,$
(2.27a) $\displaystyle\eta_{\hat{a}\hat{b}}t^{\hat{a}}t^{\hat{b}}$
$\displaystyle=-t_{0}^{2}+t_{\hat{i}}t^{\hat{i}}-t_{5}^{2}=\frac{1}{\ell_{p}^{2}}\,,\qquad\hat{i}=1,2,3\,,$
(2.27b) $\displaystyle y_{\hat{a}}t^{\hat{a}}$
$\displaystyle=0=y_{\mu}t^{\mu}\,,\qquad\qquad\quad\quad\ \,\mu=0,1,2,3\,,$
(2.27c)

where the metric of $\mathbb{R}^{2,3}$ is
$\eta_{\hat{a}\hat{b}}=\,\mathrm{diag}(-,+,+,+,-)$. Due to the last relations,
the $t^{\mu}$ can be understood as generators of the internal $S^{2}_{N}$
underlying the higher spin structure. They are associated with the gamma
matrices $\Sigma^{\mu
4}=:\frac{1}{2\,\mathrm{i}}\gamma^{\mu}\gamma^{4}=\frac{1}{2}\underline{\gamma}^{\mu}$
given by

$\displaystyle(\underline{\gamma}^{0})^{A}_{\
B}=\frac{1}{2}\begin{pmatrix}0&\mbox{1 \kern-5.89996pt{\rm l}}_{2}\\\ -\mbox{1
\kern-5.89996pt{\rm
l}}_{2}&0\end{pmatrix}\,,\quad(\underline{\gamma}^{\hat{i}})^{A}_{\
B}\equiv(\underline{\gamma}^{i})^{A}_{\
B}=\frac{1}{2}\begin{pmatrix}\sigma_{i}&0\\\
0&-\sigma_{i}\end{pmatrix}\,,\qquad\hat{i}=1,2,3\,.$ (2.28)

We also note that the above definitions imply $y_{4}=-\ell_{p}R\,t_{5}\,$.

#### Flat limit of semi-classical $H^{4}_{N}$.

In the later sections of this paper, we will focus on the flat limit
$R\to\infty$ of the semi-classical $H^{4}_{N}$. If $y^{\mathtt{a}},\
\mathtt{a}=1,\ldots,4$ are local coordinates near the “south pole”
$y^{a}=(R,0,0,0,0)$, it is natural to consider a second set of vector
generators $t^{\mathtt{a}}$ (different from but analogous to the generators
$t^{\hat{a}}$ in (2.26)) which arises from the underlying representation of
$\mathfrak{so}(2,4)$:

$\displaystyle y^{\mathtt{a}}$ $\displaystyle=\ell_{p}m^{\mathtt{a}5}$
$\displaystyle t^{\mathtt{a}}$ $\displaystyle=\frac{1}{R}m^{\mathtt{a}0}$
(2.29)

for $\mathtt{a}=1,\ldots,4$, which satisfy

$\displaystyle\\{y_{\mathtt{a}},y_{\mathtt{b}}\\}$
$\displaystyle=-\ell_{p}^{2}m_{\mathtt{a}\mathtt{b}}\,,\qquad\qquad\quad\qquad\quad\
\,\mathtt{a},\mathtt{b}=1,2,3,4\,,$ (2.30a) $\displaystyle
y_{\mathtt{a}}y^{\mathtt{a}}$
$\displaystyle=-R^{2}+y_{0}^{2}=-\frac{\ell_{p}^{2}N^{2}}{4}+y_{0}^{2}\,,$
(2.30b) $\displaystyle\\{t^{\mathtt{a}},t^{\mathtt{b}}\\}$
$\displaystyle=\frac{1}{R^{2}}m^{\mathtt{a}\mathtt{b}}\,,$ (2.30c)
$\displaystyle\delta_{\mathtt{a}\mathtt{b}}t^{\mathtt{a}}t^{\mathtt{b}}$
$\displaystyle=\frac{1}{\ell_{p}^{2}}\,,$ (2.30d) $\displaystyle
y_{\mathtt{a}}t^{\mathtt{a}}$ $\displaystyle=0\,.$ (2.30e)

Due to the last relations, $t^{\mathtt{a}}$ can be considered as generators of
the internal $S^{2}_{N}$ underlying the higher spin structure. We will use
this representation of “momentum” on $H^{4}$ to analyze the degrees of freedom
for higher-spin fields in Section 5.

#### Semi-classical $\mathcal{M}^{1,3}_{N}$ spacetime.

To get the desired Lorentzian signature from either $H^{4}$ or $H^{2,2}$, we
can consider the following projections to a $SO(1,3)$-covariant spacetime:

$\displaystyle\pi_{y}\,$ $\displaystyle:m^{ab}\mapsto y^{\mu}=\ell_{p}m^{\mu
5}\,,$ (2.31a) $\displaystyle\pi_{t}\,$
$\displaystyle:m^{\hat{a}\hat{b}}\mapsto t^{\mu}=\frac{1}{R}m^{\mu 4}\,.$
(2.31b)

The projections (2.31) can be realized explicitly as follows:

$\displaystyle\eta_{\mu\nu}y^{\mu}y^{\nu}$
$\displaystyle=-R^{2}-y_{4}^{2}\,\,=-R^{2}\cosh^{2}(\tau)\,,$ (2.32a)
$\displaystyle\eta_{\mu\nu}t^{\mu}t^{\nu}$
$\displaystyle=\frac{1}{\ell_{p}^{2}}+\frac{y_{4}^{2}}{\ell_{p}^{2}R^{2}}=+\ell_{p}^{-2}\cosh^{2}(\tau)\,,$
(2.32b)

where $\eta_{\mu\nu}=\,\mathrm{diag}(-,+,+,+)$ and $\tau$ is a time parameter
that defines a space-like foliation for a cosmological FLRW spacetime with
$k=-1$. In particular, $\tau$ is defined by

$\displaystyle y_{4}=R\sinh(\tau)\,,$ (2.33)

which features a big-bounce at $\tau=0$. Here, $t^{\mu}$ defines internal
space-like sphere $S^{2}$ with radius $\ell_{p}^{-2}\cosh^{2}(\tau)$ (cf.,
(2.27c) and (2.32b)) whose local stabilizer is $SO(3)\simeq SU(2)$. For this
reason, one may use $t^{\mu}$ as auxiliary vectors to describe higher-spin
modes. Further relations include [8, 13]:

$\displaystyle\\{t^{\mu},y^{\nu}\\}$
$\displaystyle=+\frac{\eta^{\mu\nu}}{R}y^{4}=\eta^{\mu\nu}\sinh(\tau)\,,$
(2.34a) $\displaystyle\\{t^{\mu},y^{4}\\}$
$\displaystyle=-\frac{y^{\mu}}{R}\,,$ (2.34b) $\displaystyle m^{\mu\nu}$
$\displaystyle=R^{2}\\{t^{\mu},t^{\nu}\\}=-\frac{1}{\cosh^{2}(\tau)}\Big{(}\sinh(\tau)(y^{\mu}t^{\nu}-y^{\nu}t^{\mu})+\epsilon^{\mu\nu\sigma\rho}y_{\sigma}t_{\rho}\Big{)}\,.$
(2.34c)

The above suggests that we can identify $y^{\mu}$ as coordinates on
$\mathcal{M}^{1,3}_{N}$, and $t^{\mu}$ as momentum generators. In fact, by
virtue of (2.34a), we have

$\displaystyle\\{t_{\mu},\phi(y)\\}=\sinh(\tau)\partial_{\mu}\phi(y)\,.$
(2.35)

There is an important global time-like vector field:

$\displaystyle\mathcal{T}=y^{\mu}\partial_{\mu}$ (2.36)

which describes the time evolution of the FLRW cosmological background with
$k=-1$ and is compatible with $SO(1,3)$ isometry. The coordinates of the FLRW
patch read [11]:

$\displaystyle\begin{pmatrix}x^{0}\\\ x^{1}\\\ x^{2}\\\
x^{3}\end{pmatrix}=R\cosh(\tau)\begin{pmatrix}\cosh(\chi)\\\
\sinh(\chi)\sin(\theta)\cos(\varphi)\\\
\sinh(\chi)\sin(\theta)\sin(\varphi)\\\
\sinh(\chi)\cos(\theta)\end{pmatrix}\,,$ (2.37)

for which the metric can be computed as

$\begin{split}\mathrm{d}s^{2}&=-R^{2}\sinh^{3}(\tau)\mathrm{d}\tau^{2}+R^{2}\sinh(\tau)\cosh^{2}(\tau)\mathrm{d}\Sigma^{2}\\\
&=-\mathrm{d}t^{2}+a^{2}(t)\mathrm{d}\Sigma^{2}\,.\end{split}$ (2.38)

Here,
$\mathrm{d}\Sigma^{2}=\mathrm{d}\chi^{2}+\sinh^{2}(\chi)\mathrm{d}\Omega^{2}$
is the metric on the unit hyperboloid $H^{3}$; and the scale parameter $a(t)$
is determined by

$\displaystyle a(t)^{2}$ $\displaystyle=R^{2}\sinh(\tau)\cosh^{2}(\tau)\,,$
(2.39a) $\displaystyle\mathrm{d}t$
$\displaystyle=R\sinh(\tau)^{\frac{3}{2}}\mathrm{d}\tau\,.$ (2.39b)

One can show that around

early time $\displaystyle:$ $\displaystyle a(t)\sim t^{1/5}\,,$ (2.40a) late
time $\displaystyle:$ $\displaystyle a(t)\sim t\,.$ (2.40b)

The above features a FLRW cosmology that is asymptotically coasting at late
times, and has a Big Bounce at the initial time $t=0$ since the timeline
changes its direction as it ‘jumps’ from one sheet of $\mathcal{M}^{1,3}$ to
the other [20, 21].

##  3 Fuzzy twistor geometry and 4-hyperboloid

We now study the spinorial versions of $H_{N}^{4}$ in a stereographic
projection which naturally admits a smooth flat limit. The spinorial effective
vielbein and metric are also derived. We begin with a brief review of twistor
geometry following [22].

###  3.1 Twistor space

We define the commutative twistor space $\mathbb{PT}$ to be an open subset of
$\mathbb{P}^{1,2}$ (for a review, see e.g. [23] and [24, 25, 26, 27])

$\displaystyle\mathbb{PT}=\\{Z^{A}=(Z^{1},Z^{2},Z^{3},Z^{4})=(\lambda^{\alpha},\mu^{\dot{\alpha}})\in\mathbb{P}^{1,2}\,|\,\hat{\mathcal{N}}\neq
0\\}\,.$ (3.1)

Here $Z^{A}\sim r\,Z^{A}\,,\ \forall r\in\mathbb{C}^{*}$ are homogeneous
coordinates of $\mathbb{P}^{1,2}$, and $\lambda^{\alpha},\mu^{\dot{\alpha}}$
transform in the fundamental of the compact subgroups $SU(2)_{L}\times
SU(2)_{R}$ of $SU(2,2)$. Note that these are not Weyl spinors, as any non-
compact transformation (such as boosts in $SO(1,3)$) will mix these two
spinors. The complex conjugation of the twistor $Z^{A}$ denoted as
$Z_{A}^{\dagger}$:

$\displaystyle Z_{A}^{\dagger}$
$\displaystyle=\begin{pmatrix}{\bar{\lambda}}_{\alpha}\\\
{\bar{\mu}}_{\dot{\alpha}}\end{pmatrix}\,$ (3.2)

transforms in the anti-fundamental representation of $\mathfrak{su}(2,2)$ so
that the Dirac conjugation of $Z^{A}$ reads

$\displaystyle\bar{Z}_{A}=Z^{\dagger}_{B}(\gamma_{0})^{B}{}_{A}=\binom{\bar{\lambda}_{\alpha}}{-\bar{\mu}_{\dot{\alpha}}}\,.$
(3.3)

Restricting ourselves to $\mathfrak{so}(1,4)\subset\mathfrak{su}(2,2)$, we can
also use the anti-symmetric matrix $C^{AB}$ to define the ‘quaternionic’
conjugate twistor $\hat{Z}^{A}$ of $Z^{A}$ as

$\displaystyle\hat{Z}^{A}$
$\displaystyle=(\hat{\lambda}^{\alpha},\hat{\mu}^{\dot{\alpha}})=Z_{B}^{\dagger}C^{AB}\,,$
(3.4)

which is manifestly compatible with $SO(1,4)$. Note that the hat conjugation
acts on spinors with the following rules

$\displaystyle\lambda^{\alpha}$
$\displaystyle=(\lambda^{0},\lambda^{1})\mapsto\hat{\lambda}^{\alpha}=\overline{\lambda}_{\beta}\varepsilon^{\alpha\beta}=(\overline{\lambda_{1}},-\overline{\lambda_{0}})\,,$
$\displaystyle\mu^{\dot{\alpha}}$
$\displaystyle=(\mu^{\dot{0}},\mu^{\dot{1}})\mapsto\hat{\mu}^{\dot{\alpha}}=\overline{\mu}_{\dot{\beta}}\varepsilon^{\dot{\alpha}\dot{\beta}}=(\overline{\mu_{\dot{1}}},-\overline{\mu_{\dot{0}}})\,.$
(3.5)

It is useful to check that

$\displaystyle\langle\lambda\,\bar{\lambda}\rangle$
$\displaystyle=\lambda_{\beta}\epsilon^{\alpha\beta}\bar{\lambda}_{\alpha}=-\lambda_{\beta}\hat{\lambda}^{\beta}=\langle\lambda\,\hat{\lambda}\rangle\,,$
(3.6a) $\displaystyle\hat{\hat{\lambda}}^{\alpha}$
$\displaystyle=-\lambda^{\alpha}\,,$ (3.6b)
$\displaystyle\hat{\hat{\mu}}^{\dot{\alpha}}$
$\displaystyle=-\mu^{\dot{\alpha}}\,.$ (3.6c)

This means at the level of spinors, we can interchangeably use
$(\hat{\lambda},\hat{\mu})$ for $(\bar{\lambda},\bar{\mu}$). In terms of the
hatted spinors, the number operator $\hat{\mathcal{N}}=\bar{Z}_{A}Z^{A}$
becomes

$\displaystyle\hat{\mathcal{N}}=\langle\lambda\,\hat{\lambda}\rangle-[\mu\,\hat{\mu}]=N\,,$
(3.7)

where the angle and square brackets are defined by $\langle
u\,v\rangle=u^{\alpha}v_{\alpha}\,,\ [u\,v]=u^{\dot{\alpha}}v_{\dot{\alpha}}$.

The correspondence between $\mathbb{PT}$ and $H^{4}$ is given by the incidence
relations

$\displaystyle\mu^{\dot{\alpha}}=\mathtt{x}^{\alpha\dot{\alpha}}\lambda_{\alpha}\qquad\Leftrightarrow\qquad\mathtt{x}^{\alpha\dot{\alpha}}=\frac{\lambda^{\alpha}\hat{\mu}^{\dot{\alpha}}-\hat{\lambda}^{\alpha}\mu^{\dot{\alpha}}}{\langle\lambda\,\hat{\lambda}\rangle}\,,$
(3.8)

which state that a point $\mathtt{x}\in H^{4}$ corresponds to a holomorphic
embedded Riemann sphere. We emphasize that all spinors and the $2\times 2$
matrix $\mathtt{x}^{\alpha\dot{\alpha}}$ are dimensionless [10]. Furthermore,
we have the following reality condition:

$\displaystyle\hat{\mathtt{x}}^{\alpha\dot{\alpha}}=\mathtt{x}^{\alpha\dot{\alpha}}\,.$
(3.9)

The Poisson structure (2.22) in terms of spinors read:

$\displaystyle\\{\lambda^{\alpha},\bar{\lambda}_{\beta}\\}=\delta^{\alpha}{}_{\beta}\,,\qquad\qquad\\{\mu^{\dot{\alpha}},\bar{\mu}_{\dot{\beta}}\\}=\delta^{\dot{\alpha}}{}_{\dot{\beta}}\,,$
(3.10)

which leads us to the following gradations:

$\displaystyle\\{\hat{\mathcal{N}},\lambda^{\alpha}\\}$
$\displaystyle=-\lambda^{\alpha}\,,\qquad\qquad\\{\hat{\mathcal{N}},\mu^{\dot{\alpha}}\\}=-\mu^{\dot{\alpha}}\,,$
(3.11a) $\displaystyle\\{\hat{\mathcal{N}},\bar{\lambda}_{\alpha}\\}$
$\displaystyle=+\bar{\lambda}_{\alpha}\,,\qquad\qquad\\{\hat{\mathcal{N}},\bar{\mu}_{\dot{\alpha}}\\}=+\bar{\mu}_{\dot{\alpha}}\,,$
(3.11b) $\displaystyle\\{\hat{\mathcal{N}},\mathtt{x}^{\alpha\dot{\alpha}}\\}$
$\displaystyle=0\,.$ (3.11c)

The algebra of functions (2.21) on $\mathbb{P}^{1,2}$ in this language
becomes:

$\displaystyle\mathscr{C}(\mathbb{P}^{1,2})=\sum_{n+m=p+q}f_{\alpha(n)\dot{\alpha}(m)}{}^{\beta(p)\dot{\beta}(q)}\lambda^{\alpha(n)}\mu^{\dot{\alpha}(m)}\bar{\lambda}_{\beta(p)}\bar{\mu}_{\dot{\beta}(q)}\,,$
(3.12)

which can be reduced further to polynomials in terms of
$\lambda,\hat{\lambda}$:

$\displaystyle\mathscr{C}(\mathbb{P}^{1,2})=\sum_{n}f_{\alpha(n)}{}^{\beta(n)}(\mathtt{x})\lambda^{\alpha(n)}\bar{\lambda}_{\beta(n)}\,$
(3.13)

by using the incident relation (3.8). Due to the constraint
$\\{\hat{\mathcal{N}},f\\}=0$ where $f\in\mathscr{C}(\mathbb{P}^{1,2})$, the
number of $\lambda$ and $\bar{\lambda}$ generators must be the same at this
point. Therefore, all higher-spin modes can be viewed as functions on the
internal (fuzzy) Riemann sphere $\mathbb{P}^{1}_{N}$.

###  3.2 The stereographic projection to $H^{4}$

Following the lines of [10], we can recover the above twistorial construction
and the incidence relations from the Hopf map (2.25) $\mathbb{P}^{1,2}\to
H^{4}$ (or Euclidean AdS4) followed by a stereographic projection
$H^{4}\to\mathbb{R}^{4}$. Using (2.25), we get

$\displaystyle y^{\mathtt{a}}$
$\displaystyle=\frac{\ell_{p}}{2}\bar{Z}_{A}(\gamma^{\mathtt{a}})^{A}{}_{B}Z^{B}=-\frac{\ell_{p}}{2}\langle\lambda\,\bar{\lambda}\rangle(\hat{\sigma}^{\mathtt{a}})_{\alpha\dot{\alpha}}\mathtt{x}^{\alpha\dot{\alpha}}\,,\qquad\quad\mathtt{a}=1,2,3,4\,,$
(3.14a) $\displaystyle y^{0}$
$\displaystyle=\frac{\ell_{p}}{2}\bar{Z}_{A}(\gamma^{0})^{A}{}_{B}Z^{B}=+\frac{\ell_{p}}{2}([\mu\,\bar{\mu}]+\langle\lambda\,\bar{\lambda}\rangle)=R+\ell_{p}[\mu\,\bar{\mu}]\geq
R$ (3.14b)

where
$\hat{\sigma}^{\mathtt{a}}_{\alpha\dot{\alpha}}=(\tilde{\sigma}^{m}_{\alpha\dot{\alpha}},\epsilon_{\alpha\dot{\alpha}})$
for $m=1,2,3$. Note that when the spinors $\mu^{\dot{\alpha}}=(0,0)$, we will
be at the center $y^{0}=R$ of the projection, reflecting the fact that the
underlying doubleton irrep $\mathcal{H}_{N}$ of $\mathfrak{so}(4,2)$ is a
lowest-weight representation [8]. Furthermore, $y^{0}$ can be also written as

$\displaystyle
y^{0}=-R+\ell_{p}\langle\lambda\,\bar{\lambda}\rangle=-R+\ell_{p}\langle\lambda\,\hat{\lambda}\rangle\,.$
(3.15)

It is then convenient to define

$\displaystyle
x^{\mathtt{a}}=\frac{R}{\ell_{p}\langle\lambda\,\bar{\lambda}\rangle}y^{\mathtt{a}}\,,\qquad
y^{\mathtt{a}}=\Big{(}1+\frac{y^{0}}{R}\Big{)}x^{\mathtt{a}}\,$ (3.16)

for $\mathtt{a}=1,2,3,4$. We obtain

$\displaystyle y^{0}=\frac{R(x^{2}+R^{2})}{(R^{2}-x^{2})}\,,\qquad
y^{\mathtt{a}}=\frac{2R^{2}x^{\mathtt{a}}}{(R^{2}-x^{2})}\,,\qquad
x^{2}=x_{\mathtt{a}}x^{\mathtt{a}}<R^{2}\,.$ (3.17)

It is a simple matter to show that

$\displaystyle-
y_{0}^{2}+y_{\mathtt{a}}y^{\mathtt{a}}=-R^{2}\,,\qquad\mathtt{a}=1,2,3,4\,,$
(3.18)

so that the above $x^{\mathtt{a}}$ define a stereographic projection
$H^{4}\to\mathbb{R}^{4}$. The conformally flat metric corresponding to $H^{4}$
can then be obtained by the pullback:

$\displaystyle\mathrm{d}s^{2}=\Big{(}\frac{\partial y^{a}}{\partial
x^{\mathtt{a}}}\frac{\partial y^{b}}{\partial
x^{\mathtt{b}}}\eta_{ab}\Big{)}\mathrm{d}x^{\mathtt{a}}\mathrm{d}x^{\mathtt{b}}:=g_{\mathtt{a}\mathtt{b}}\mathrm{d}x^{\mathtt{a}}\mathrm{d}x^{\mathtt{b}}=\frac{4R^{4}\eta_{\mathtt{a}\mathtt{b}}\mathrm{d}x^{\mathtt{a}}\mathrm{d}x^{\mathtt{b}}}{(R^{2}-x^{2})^{2}}=\Omega^{2}\eta_{\mathtt{a}\mathtt{b}}\mathrm{d}x^{\mathtt{a}}\mathrm{d}x^{\mathtt{b}}\,$
(3.19)

thus defining the conformal factor $\Omega^{2}$, where
$\eta_{\mathtt{a}\mathtt{b}}=\,\mathrm{diag}(+,+,+,+)$. Although this
coordinate system does not give us the desired Lorentzian signature, the
metric (3.19) can have a smooth flat limit where $R\rightarrow\infty$. Lastly,
it is worth noting that (3.14b) implies

$\displaystyle\langle\lambda\,\hat{\lambda}\rangle$
$\displaystyle=\frac{N}{2}\Big{(}1+\frac{y^{0}}{R}\Big{)}=\frac{NR^{2}}{(R^{2}-x^{2})}=\frac{N}{2}\Omega\,,$
(3.20a) $\displaystyle[\mu\,\hat{\mu}]$
$\displaystyle=\frac{Nx^{2}}{2(R^{2}-x^{2})}\,,$ (3.20b)

which allows us to parametrize $\lambda^{\alpha},\hat{\lambda}^{\alpha}$ fiber
coordinates as

$\displaystyle\lambda^{\alpha}:=\frac{R}{\sqrt{R^{2}-x^{2}}}\binom{z}{-1}\,,\qquad\hat{\lambda}^{\alpha}:=\frac{R}{\sqrt{R^{2}-x^{2}}}\binom{+1}{\bar{z}}\,,$
(3.21)

where $1+|z|^{2}=N$ for $z\in\mathbb{C}^{*}$.

#### Effective vielbein and metric.

Using the explicit form of the gamma matrices we recover the incidence
relation (3.8), which takes a simpler form in terms of

$\displaystyle\mathtt{y}^{\alpha\dot{\alpha}}:=\langle\lambda\,\hat{\lambda}\rangle\mathtt{x}^{\alpha\dot{\alpha}}=\lambda^{\alpha}\hat{\mu}^{\dot{\alpha}}-\hat{\lambda}^{\alpha}\mu^{\dot{\alpha}}\,,\qquad\mathtt{y}^{0}:=\langle\lambda\,\hat{\lambda}\rangle\,.$
(3.22)

The reality condition of $\mathtt{y}^{\alpha\dot{\alpha}}$, which is
$\hat{\mathtt{y}}^{\alpha\dot{\alpha}}=\mathtt{y}^{\alpha\dot{\alpha}}$,
follows directly from (3.9) since $\hat{\hat{\lambda}}=-\lambda$ and
$\hat{\hat{\mu}}=-\mu$.

It is convenient to work with the projective spinor bundle
$\mathbb{P}^{1,2}\simeq\mathbb{PS}=\mathbb{P}^{1}\times\mathbb{R}^{4}$ where
$\mathbb{PS}$ has coordinates
$(\lambda^{\alpha},\mathtt{y}^{\alpha\dot{\alpha}})$ and is a trivial bundle
[10]. Note that we can make a conformal transformation to recover $H^{4}$
whenever it is appropriate. From this consideration, the algebra of functions
$\mathscr{C}(\mathbb{P}^{1,2})$ reduces to:

$\displaystyle\mathscr{C}(\mathbb{PS})=\mathscr{C}(\mathbb{P}^{1})\times
C^{\infty}(\mathbb{R}^{4})\,.$ (3.23)

As in [10], we will consider $H^{4}_{N}$ as a background in the IKKT model via
the matrix configuration given by the (fuzzy version of) $y^{a}$. Such a
background defines an effective frame or vielbein via the Hamiltonian vector
field $\\{y^{a},-\\}$ [8]. Using (3.10), we obtain the spinorial form of the
effective vielbein in analogous to [10]:

$\displaystyle\mathcal{E}^{\alpha\dot{\alpha}|\beta\dot{\beta}}$
$\displaystyle:=\\{\mathtt{y}^{\alpha\dot{\alpha}},\mathtt{y}^{\beta\dot{\beta}}\\}=2(\lambda^{(\alpha}\hat{\lambda}^{\beta)}\epsilon^{\dot{\alpha}\dot{\beta}}+\mu^{(\dot{\alpha}}\hat{\mu}^{\dot{\beta})}\epsilon^{\alpha\beta})\,,$
(3.24a) $\displaystyle\mathcal{E}^{0|\alpha\dot{\alpha}}$
$\displaystyle:=\\{\mathtt{y}^{0},\mathtt{y}^{\alpha\dot{\alpha}}\\}=\hat{\lambda}^{\alpha}\mu^{\dot{\alpha}}+\lambda^{\alpha}\hat{\mu}^{\dot{\alpha}}\,.$
(3.24b)

We also find

$\displaystyle\\{\mathtt{y}^{\alpha\dot{\alpha}},\lambda^{\beta}\\}=-\epsilon^{\alpha\beta}\mu^{\dot{\alpha}}\,,\qquad\qquad\\{\mathtt{y}^{\alpha\dot{\alpha}},\hat{\lambda}^{\beta}\\}=+\epsilon^{\alpha\beta}\hat{\mu}^{\dot{\alpha}}\,.$
(3.25a)

If $\varphi(\mathtt{y})$ is a $\mathfrak{hs}$-valued smooth section of
$\mathscr{C}(\mathbb{PS})$, then

$\displaystyle\begin{split}\\{\mathtt{y}^{\alpha\dot{\alpha}},\varphi(\mathtt{y}|\lambda,\hat{\lambda})\\}:&=\big{(}\\{\mathtt{y}^{\alpha\dot{\alpha}},\mathtt{y}^{\beta\dot{\beta}}\\}\partial_{\beta\dot{\beta}}+\\{\mathtt{y}^{\alpha\dot{\alpha}},\lambda^{\beta}\\}\partial_{\beta}+\\{\mathtt{y}^{\alpha\dot{\alpha}},\hat{\lambda}^{\beta}\\}\hat{\partial}_{\beta}\big{)}\varphi\,\\\
&=\big{(}\mathcal{E}^{\alpha\dot{\alpha}|\beta\dot{\beta}}\partial_{\beta\dot{\beta}}+\mathcal{E}^{\alpha\dot{\alpha}|\beta}\partial_{\beta}+\hat{\mathcal{E}}^{\alpha\dot{\alpha}|\beta}\hat{\partial}_{\beta}\big{)}\varphi\,,\end{split}$
(3.26)

where
$\partial_{\alpha\dot{\alpha}}:=\partial/\partial\mathtt{y}^{\alpha\dot{\alpha}}$,
$\partial_{\alpha}:=\partial/\partial\lambda^{\alpha}$ and
$\hat{\partial}_{\alpha}:=\partial/\partial\hat{\lambda}^{\alpha}$. Similarly,
we define

$\displaystyle\begin{split}\\{\mathtt{y}^{0},\varphi(\mathtt{y}|\lambda,\hat{\lambda})\\}:&=\big{(}\\{\mathtt{y}^{0},\mathtt{y}^{\beta\dot{\beta}}\\}\partial_{\beta\dot{\beta}}+\\{\mathtt{y}^{0},\lambda^{\alpha}\\}\partial_{\alpha}+\\{\mathtt{y}^{0},\hat{\lambda}^{\alpha}\\}\hat{\partial}_{\alpha}\big{)}\varphi\,\\\
&=\big{(}\mathcal{E}^{0|\beta\dot{\beta}}\partial_{\beta\dot{\beta}}+\mathcal{E}^{0|\beta}\partial_{\beta}+\hat{\mathcal{E}}^{0|\beta}\hat{\partial}_{\beta}\big{)}\varphi\,.\end{split}$
(3.27)

Note that all contributions from
$\mathcal{E}^{\alpha\dot{\alpha}|\beta}\,,\,\hat{\mathcal{E}}^{\alpha\dot{\alpha}|\beta}$
and $\mathcal{E}^{0|\beta}\,,\,\hat{\mathcal{E}}^{0|\beta}$ are subleading in
the flat limit as in the case of $S^{4}_{N}$ (cf., [10]). Therefore, whenever
we take a flat limit, they can be neglected. Finally, to compute effective
metric, say $G^{\alpha\dot{\alpha}\beta\dot{\beta}}$, it is sufficient to
consider a scalar field $\vartheta(\mathtt{y})$ whose kinetic Lagrangian reads

$\displaystyle\begin{split}-\mathcal{E}^{0|\alpha\dot{\alpha}}\partial_{\alpha\dot{\alpha}}\vartheta\,\mathcal{E}_{0|\beta\dot{\beta}}\partial^{\beta\dot{\beta}}\vartheta+\mathcal{E}^{\zeta\dot{\zeta}|\alpha\dot{\alpha}}\partial_{\alpha\dot{\alpha}}\vartheta\,\mathcal{E}_{\zeta\dot{\zeta}|\beta\dot{\beta}}\partial^{\beta\dot{\beta}}\vartheta:=G^{\alpha\dot{\alpha}\beta\dot{\beta}}\partial_{\alpha\dot{\alpha}}\vartheta\,\partial_{\beta\dot{\beta}}\vartheta\,.\end{split}$
(3.28)

Proceeding as in [10], we get

$\displaystyle G^{\alpha\dot{\alpha}\beta\dot{\beta}}(\mathtt{y})$
$\displaystyle=N^{2}\epsilon^{\alpha\beta}\epsilon^{\dot{\alpha}\dot{\beta}}-\mathtt{y}^{\alpha\dot{\alpha}}\mathtt{y}^{\beta\dot{\beta}}\,,$
(3.29a) $\displaystyle G^{\alpha\dot{\alpha}\beta\dot{\beta}}(\mathtt{x})$
$\displaystyle=\langle\lambda\,\hat{\lambda}\rangle^{2}\Big{(}\frac{N^{2}}{\langle\lambda\,\hat{\lambda}\rangle^{2}}\epsilon^{\alpha\beta}\epsilon^{\dot{\alpha}\dot{\beta}}+\mathtt{x}^{\alpha\dot{\alpha}}\mathtt{x}^{\beta\dot{\beta}}\Big{)}\,,$
(3.29b)

which matches with the result of [28]. Note that in the flat limit where we
send the dimensionless ratio $\frac{x^{2}}{R^{2}}\to 0$, the effective metric
$G^{\alpha\dot{\alpha}\beta\dot{\beta}}\mapsto
N\,\epsilon^{\alpha\beta}\epsilon^{\dot{\alpha}\dot{\beta}}$, which is simply
the standard metric of flat space in spinor form.

##  4 Spinor description in the $SO(1,3)$-invariant projection

So far we have developed a natural spinor formalism for the fuzzy
4-hyperboloid along with a suitable stereographic projection to $H^{4}$ and
then $\mathbb{R}^{4}$ (when we consider the flat limit), generalizing the
framework in [10]. However, the corresponding fields of the gauge theory
resulting from the IKKT matrix model do not propagate, since the base manifold
$H^{4}$ has all-plus (Euclidean) signature. This can be circumvented by
considering a different $SO(1,3)$-invariant projection (2.31) as in [13],
which leads to a FLRW type spacetime with Lorentzian signature.

Recall that the projection (2.31) is realized in the IKKT model by choosing
the background to be (the fuzzy versions of) $t^{\mu}$ defined as

$\displaystyle t^{\mu}=\frac{1}{R}\bar{Z}_{A}(\Sigma^{\mu 4})^{A}_{\
B}Z^{B}=\frac{1}{2R}\bar{Z}_{A}(\underline{\gamma}^{\mu})^{A}{}_{B}Z^{B}\,,\qquad\mu=0,1,2,3\,.$
(4.1)

Furthermore, as discussed in Section 3, the spinors
$\lambda^{\alpha},\mu^{\dot{\alpha}}$ are not Lorentz spinors here, but
spinors of the compact subgroups $SU(2)_{L}$ and $SU(2)_{R}$ of $SU(2,2)$
which are mixed by the $SO(1,3)$ generators $\Sigma^{\mu\nu}$. Using the gamma
matrices (2.28), we obtain explicit spinorial expressions for $t^{\mu}$:

$\displaystyle t^{0}$
$\displaystyle=\frac{1}{4R}\big{(}[\mu\,\bar{\lambda}]+\langle\lambda\,\bar{\mu}\rangle\big{)}\equiv\frac{1}{4R}\big{(}\mu^{\dagger}\lambda+\lambda^{\dagger}\mu\big{)}$
(4.2a) $\displaystyle t^{i}$
$\displaystyle=\frac{1}{4R}\Big{(}\bar{\lambda}_{\alpha}(\sigma^{i})^{\alpha}_{\
\beta}\lambda^{\beta}+\bar{\mu}_{\dot{\alpha}}(\sigma^{i})^{\dot{\alpha}}{}_{\dot{\beta}}\mu^{\dot{\beta}}\Big{)}\,,\qquad
i=1,2,3\,.$ (4.2b)

This leads to the following Poisson brackets

$\displaystyle\\{t^{i},\lambda^{\alpha}\\}$
$\displaystyle=+\frac{\mathrm{i}}{2R}(\sigma^{i})^{\alpha\beta}\lambda_{\beta}\,,\qquad$
$\displaystyle\\{t^{i},\hat{\lambda}^{\alpha}\\}$
$\displaystyle=-\frac{\mathrm{i}}{2R}\hat{\lambda}_{\beta}(\sigma^{i})^{\beta\alpha}\,,$
(4.3a) $\displaystyle\\{t^{i},\mu^{\dot{\alpha}}\\}$
$\displaystyle=-\frac{\mathrm{i}}{2R}(\sigma^{i})^{\dot{\alpha}\dot{\beta}}\mu_{\dot{\beta}}\,,\qquad$
$\displaystyle\\{t^{i},\hat{\mu}^{\dot{\alpha}}\\}$
$\displaystyle=+\frac{\mathrm{i}}{2R}\hat{\mu}_{\dot{\beta}}(\sigma^{i})^{\dot{\beta}\dot{\alpha}}\,.$
(4.3b)

We also provide the explicit spinorial expressions for the $y^{\mu}$ (cf.
(3.14)):

$\displaystyle y^{i}$
$\displaystyle=-\frac{1}{4R}\big{(}\bar{\lambda}_{\alpha}(\sigma^{i})^{\alpha}{}_{\dot{\beta}}\mu^{\dot{\beta}}+\bar{\mu}_{\dot{\alpha}}(\sigma^{i})^{\dot{\alpha}}{}_{\beta}\lambda^{\beta}\big{)}\equiv-\frac{1}{4R}\big{(}\lambda^{\dagger}\sigma^{i}\mu+\mu^{\dagger}\sigma^{i}\lambda\big{)}$
(4.4) $\displaystyle y^{0}$ $\displaystyle=R+\ell_{p}[\mu\,\bar{\mu}]$ (4.5)

It is clear from these formulas that the spinorial representation of
coordinates and derivatives on the FLRW background is quite distinct – and
perhaps less appealing – than the familiar form on Minkowski space, because
our spinors transform under the space-like isometry group $SU(2)\times
SU(2)\subset SO(1,4)$ rather than the local $SL(2,\mathbb{C})$. This reflects
the lack of manifest local Lorentz invariance in the model, which is expected
to be recovered only effectively for the physical fields. We delegate a full
treatment this problem to future work, and avoid this step in the following.
However, we shall illustrate how the local relativistic propagation is
recovered properly in this spinorial setting.

#### Effective vielbein and metric.

To derive the effective frame and metric on the FLRW background444Note that
the present background $t^{\mu}$ play the role of momentum generators, while
the $y^{a}$ in the case of $H^{4}$ are position generators. Nevertheless, both
backgrounds define a higher-spin gauge theory on the respective base manifold.
$\mathcal{M}^{1,3}$ defined by $t^{\mu}$ in the spinorial framework, we need
an explicit realization of general $\mathfrak{hs}$-valued fields on
$\mathcal{M}^{1,3}$ in terms of spinors. We have just seen that in contrast to
the previous case of $H^{4}$, it is no longer possible to define the $t^{\mu}$
globally in terms of a single pair of spinors. However since we are mainly
interested in local scattering amplitudes, we can choose (using the $SO(1,3)$
isometry of the FLRW background) some reference point
$\mathsf{p}=(p^{0},0,0,0)\in\mathcal{M}^{1,3}$. Then $t^{0}=0$ and
$t^{\mu}|_{\mathsf{p}}\rightsquigarrow t^{i}$ spans the local
$\mathbb{P}^{1}$, which can be described effectively in terms of spinors
$\lambda^{\alpha}$. To see this, the following observation is useful:
combining the above expression for $t^{0}$ with (4.4) and recalling that
$t^{0}=0$ at the reference point $y^{\mu}=(y^{0},0,0,0)$, it follows that

$\displaystyle\lambda^{\dagger}\sigma^{\mu}\mu=-\mu^{\dagger}\sigma^{\mu}\lambda$
(4.6)

which implies $\lambda^{\dagger}\otimes\mu=-\mu^{\dagger}\otimes\lambda$ and
therefore $\lambda\propto\mu$. As a consequence, the generator of the local
$\mathbb{P}^{1}$

$\displaystyle
t^{i}=\lambda^{\dagger}\sigma^{i}\lambda+\mu^{\dagger}\sigma^{i}\mu\propto\lambda^{\dagger}\sigma^{i}\lambda$
(4.7)

can be expressed in terms of $\lambda$ only. Hence we can choose an open
subset $U_{\mathsf{p}}\subset\mathcal{M}^{1,3}$ around the reference point
$\mathsf{p}$, with local trivialization
$\mathbb{P}^{1,2}|_{U_{\mathsf{p}}}=\mathbb{P}^{1}\times U_{\mathsf{p}}$, and
write the most general function as (cf. (6.38))

$\begin{split}\varphi&=\varphi(y|t)=\sum_{s=0}^{\infty}t^{i(s)}\varphi_{i(s)}\simeq\sum_{s=0}^{\infty}\lambda^{\beta(s)}\hat{\lambda}^{\beta(s)}\varphi_{\beta(2s)}\,.\end{split}$
(4.8)

This provides a spinorial representation for the $\mathfrak{hs}$ modes on the
FLRW background. Now consider

$\displaystyle\begin{split}\\{t^{\mu},\varphi(y|\lambda,\hat{\lambda})|_{U_{\mathsf{p}}}\\}&=\Big{(}\\{t^{\mu},y^{\nu}\\}\frac{\partial}{\partial
y^{\nu}}+\\{t^{\mu},\lambda^{\alpha}\\}\frac{\partial}{\partial\lambda^{\alpha}}+\\{t^{\mu},\hat{\lambda}^{\alpha}\\}\frac{\partial}{\partial\hat{\lambda}^{\alpha}}\Big{)}\varphi(y|\lambda,\hat{\lambda})|_{U_{\mathsf{p}}}\\\
&=:\Big{(}E^{\mu\nu}\frac{\partial}{\partial
y^{\nu}}+E^{\mu|\alpha}\frac{\partial}{\partial\lambda^{\alpha}}+\hat{E}^{\mu|\alpha}\frac{\partial}{\partial\hat{\lambda}^{\alpha}}\Big{)}\varphi(y|\lambda,\hat{\lambda})|_{U_{\mathsf{p}}}\,,\end{split}$
(4.9)

where
$\varphi(y|\lambda,\hat{\lambda})|_{U_{\mathsf{p}}}\in\mathscr{C}(\mathbb{P}^{1,2})|_{U_{\mathsf{p}}}$
and $E^{\mu|\bullet}$ is the effective vielbein in this coordinate. To compute
effective metric of FLRW patch, it is sufficient to consider a scalar field
$\varphi(y)$ and the kinetic term
$\\{t^{\mu},\varphi(y)\\}\\{t_{\mu},\varphi(y)\\}$. We obtain555Strictly
speaking the effective metric involves an extra conformal factor given by the
dilaton, which we can assume to be constant on the local patch.

$\displaystyle\\{t^{\mu},\varphi(y)\\}\\{t_{\mu},\varphi(y)\\}:=\gamma^{\mu\nu}\partial_{\mu}\varphi(y)\partial_{\nu}\varphi(y)\,,\qquad\gamma^{\mu\nu}=\eta^{\mu\nu}\sinh^{2}(\tau)\,.$
(4.10)

#### Flat limit.

In the limit where $R\rightarrow\infty$, it is obvious that

$\displaystyle E^{\mu\nu}=\eta^{\mu\nu}\sinh(\tau)$ (4.11)

will be the only effective vielbein that survives. Since contributions
resulting from the Poisson bracket (3.10) on fiber coordinates
$\lambda,\bar{\lambda}$ are subleading in the flat limit (except in the
extreme IR regime), they can be neglected to a good approximation; this is
denoted as “asymptotic regime” in [29]. Hence, in the flat limit, we simply
factorize all fiber coordinates $\lambda,\bar{\lambda}$ outside the Poisson
brackets, which implies that the Poisson bracket only acts on functions in the
flat limit. This is also the limit where we can effectively replace $t^{i}$
for a pair of spinors $\lambda^{\alpha},\bar{\lambda}^{\beta}$ as discussed
above.

##  5 Vector description of higher-spin modes in the flat limit

Before analyzing higher-spin modes on semi-classical $\mathbb{P}^{1,2}_{N}$ in
the spinor formalism, it is worthwhile to work out the vector description
using $t^{\mathtt{a}}$ and $t^{\mu}$ as generators of the internal $S^{2}_{N}$
to describe higher-spin modes in the flat limit. The goal of this section is
to simplify some of the results obtained using the group theory approach in
[8].

If we consider $H^{4}$ and $\mathcal{M}^{1,3}$ in the flat limit, then higher-
spin valued functions can be parametrized by the internal generators
$t^{\mathtt{a}}$ and $t^{\mu}$ as:

$\begin{split}\phi&=\phi_{\mathtt{a}(s)}(y)t^{\mathtt{a}(s)}\,,\qquad\,\phi\in\mathscr{C}(\mathbb{R}^{4}\times
S^{2})\,,\\\
\phi&=\phi_{\mu(s)}(y)t^{\mu(s)}\,,\qquad\phi\in\mathscr{C}(\mathbb{R}^{1,3}\times
S^{2})\,,\end{split}$

and we recall that $t^{\mathtt{a}}$ is defined in (2.29) while $t^{\mu}$ is
defined in (2.31). The $\mathfrak{hs}$-valued gauge potential
$\mathcal{A}_{\mathtt{a}}$ and $\mathcal{A}_{\mu}$ can be defined analogously
as

$\displaystyle\mathcal{A}_{\mathtt{a}}=\mathcal{A}_{\mathtt{b}(s)|\mathtt{a}}t^{\mathtt{b}(s)}\,,\qquad\mathcal{A}_{\mu}=\mathcal{A}_{\nu(s)|\mu}t^{\nu(s)}\,.$
(5.1)

Since $t^{\mathtt{a}}$ and $t^{\mu}$ are generators of $SO(3)$, we can use the
Littlewood-Richardson rule to count independent components in
$\mathcal{A}_{\mathtt{a}}$ or $\mathcal{A}_{\mu}$, respectively. It is worth
noting that even though HS-IKKT theory does not possess a mass parameter, its
higher-spin modes should be viewed as “would-be massive”. In the above
representation, the tensors $\mathcal{A}_{\nu(s)|\mu}$ are space-like (due to
(2.30e)) but not divergence-free666It is also possible to represent the modes
in terms of divergence-free but not space-like tensors.. For instance,
$\mathcal{A}_{\mu}$ has a total of $4(2s+1)$ off-shell degrees of freedom as
previously shown in [13].

#### Gauge fixing and propagating dof.

To count the physical degrees of freedom in the higher-spin gauge potential
$\mathcal{A}$, we must impose a gauge fixing condition.777We shall not discuss
the corresponding ghost sector explicitly here but refer the readers to [30].
As usual in Yang-Mills matrix models (see e.g. [30]), a suitable gauge fixing
function on a fluctuating background
$Y^{\boldsymbol{I}}+\mathcal{A}^{\boldsymbol{I}}$ is given by

$\displaystyle\mathcal{G}(\mathcal{A})=\\{y_{\boldsymbol{I}},\mathcal{A}^{\boldsymbol{I}}\\}\
.$ (5.2)

This is a good choice because there is always a gauge such that
$\mathcal{G}(\mathcal{A})=0$, provided
$\Box=\\{y_{\boldsymbol{I}},\\{y^{\boldsymbol{I}},\cdot\\}\\}$ is surjective.
Note that an admissible/integrable fluctuation mode $\mathcal{A}$ satisfies
the gauge-fixing condition
$\\{y_{\boldsymbol{I}},\mathcal{A}^{\boldsymbol{I}}\\}=0$ if and only if it is
orthogonal to all pure gauge modes $\mathcal{A}^{(g)}$, i.e.

$\displaystyle\langle\mathcal{A}^{(g)},\mathcal{A}\rangle=0\
,\qquad\mathcal{A}^{(g)}[\xi]=\\{y^{\boldsymbol{I}},\xi\\}\ .$ (5.3)

for $\xi\in\mathscr{C}(\mathcal{M})$. Since the gauge fixing condition removes
$2s+1$ modes, the gauge-fixed potential
$\mathcal{A}_{\mathtt{b}(s)|\mathtt{a}}$ has $3(2s+1)$ degrees of freedom.
Moreover, we can use (5.3) to remove further $(2s+1)$ on-shell pure gauge
components of $\mathcal{A}_{\nu(s)|\mu}$ on the Lorentzian FLRW background. As
a result, the Lorentzian $\mathcal{A}_{\nu(s)|\mu}$ has $2(2s+1)$ physical
dof. Note that in contrast to more conventional gauge theories, these physical
modes always include the extra dof of the would-be massive higher-spin fields.

#### Mode ansatz and decomposition.

In the flat limit, we can further decompose the above into the following
irreducible modes:

$\begin{split}\mathcal{A}_{\mathtt{a}}&=\Big{[}A_{(\mathtt{b}(s)\mathtt{a})}+\delta_{\mathtt{a}\mathtt{b}}\tilde{\mathsf{A}}_{\mathtt{b}(s-1)}+y_{\mathtt{a}}\xi_{\mathtt{b}(s)}+\partial_{\mathtt{a}}\tilde{\xi}_{\mathtt{b}(s)}\Big{]}t^{\mathtt{b}(s)}\,,\\\
\mathcal{A}_{\mu}&=\Big{[}A_{(\nu(s)\mu)}+\eta_{\mu\nu}\tilde{\mathsf{A}}_{\nu(s-1)}+y_{\mu}\xi_{\nu(s)}+\partial_{\mu}\tilde{\xi}_{\nu(s)}\Big{]}t^{\nu(s)}\,.\end{split}$
(5.4)

Note that the last mode is the pure gauge mode in the Minkowski case, and the
second mode can be written as

$\displaystyle\eta_{\mu\nu}\tilde{\mathsf{A}}_{\nu(s-1)}t^{\nu(s)}\propto
t_{\mu}\tilde{\mathsf{A}}_{\nu(s-1)}t^{\nu(s-1)}$ (5.5)

and similarly in the Euclidean case, up to normalization. This is simpler and
more coherent than the organization of modes in the curved case [8, 31], which
should be useful to study the resulting physics. Since the
$A_{\mathtt{b}(s)\mathtt{a}}$ coefficient has $2s+3$ components while
$\tilde{\mathsf{A}}_{\mathtt{b}(s-1)}$ has $2s-1$ components, we recover
precisely all

$(2s+3)+(2s-1)+2(2s+1)=4(2s+1)$

components of the $\mathfrak{hs}$-valued gauge potential
$\mathcal{A}_{\mathtt{a}}$ (or $\mathcal{A}_{\mu}$, repsectively).888Recall
that a massive spin-$s$ field in $4d$ has $2s+1$ degrees of freedom.
Furthermore, gauge-fixing removes $2\times(2s+1)$ on-shell modes in Minkowski
signature; essentially, the higher-spin modes associated with the coefficients
$\xi$ and $\tilde{\xi}$ will be removed by gauge fixing. To show that the two
modes $A$ and $\tilde{\mathsf{A}}$ are linearly independent, it suffices to
show that their inner product matrix is non-degenerate, which we will verify
for the spin 1 modes below.

#### Kinetic action in the flat limit.

Now, let us focus on the Lorentzian case and consider the kinetic actions of
the $\mathcal{A}_{\mu}$ mode [9]

$\displaystyle
S=\int\\{y^{\mu},\mathcal{A}^{\nu}\\}\\{y_{\mu},\mathcal{A}_{\nu}\\}+2\\{y^{\mu},y^{\nu}\\}\\{\mathcal{A}_{\mu},\mathcal{A}_{\nu}\\}-\\{y_{\mu},\mathcal{A}^{\mu}\\}^{2}\,.$
(5.6)

Using (2.30), we obtain for instance

$\displaystyle\\{y^{\mu},\mathcal{A}^{\nu}\\}\\{y_{\mu},\mathcal{A}_{\nu}\\}$
$\displaystyle=E^{\mu i}E_{\mu}^{\
j}\partial_{i}\mathcal{A}^{\nu}\partial_{j}\mathcal{A}_{\nu}\
=:\partial_{i}\mathcal{A}^{\mu}\partial^{i}\mathcal{A}_{\mu}\,,$ (5.7)

Here, we rise and lower coordinate indices with

$\displaystyle\gamma^{ij}=\eta_{\mu\nu}E^{\mu i}E^{\nu j}\,,\qquad
i,j=0,1,2,3\,,$ (5.8)

which we can consider as (locally constant) effective metric in the flat
limit. On the other hand, the term
$\\{y^{\mu},y^{\nu}\\}\\{\mathcal{A}_{\mu},\mathcal{A}_{\nu}\\}$ is sub-
leading for local scales where $\\{y^{\mu},y^{\nu}\\}\sim\theta^{\mu\nu}$ can
be treated as constant, i.e. for modes with wavelength much shorter than the
curvature scale. This is certainly true in the flat limit, where
$\mathcal{A}_{\mu}\\{\\{y^{\mu},y^{\nu}\\},\mathcal{A}_{\nu}\\}$ (after
partial integration) is a one-derivative term with scale set by the geometric
curvature, and therefore can be neglected. This should be sufficient for the
study of scattering amplitudes. Then the kinetic action without the gauge
fixing term is simply the Box operator, i.e.

$\displaystyle
S_{\text{eff}}=-\int\mathcal{A}_{\mu}\Box\mathcal{A}^{\mu}\,,\qquad\Box:=\partial_{i}\partial^{i}$
(5.9)

as it should, since $\\{y_{\mu},\mathcal{A}^{\mu}\\}^{2}$ is removed upon
gauge fixing. It is then sufficient to analyze the physical modes using the
expansion

$\displaystyle\mathsf{a}_{\mu}=\mathcal{A}_{\nu(s)|\mu}t^{\nu(s)}=\Big{(}A_{(\nu(s)\mu)}+\eta_{\mu\nu}\tilde{\mathsf{A}}_{\nu(s-1)}\Big{)}t^{\nu(s)}$
(5.10)

where $A_{(\nu(s)\mu)}$ and $\tilde{\mathsf{A}}_{\nu(s-1)}$ are traceless,
after gauge fixing and removing pure gauge modes. This encodes the $2(2s+1)$
physical components of the $\mathfrak{hs}$-valued gauge potential
$\mathcal{A}_{\mu}$. Since $t^{\mu}$ is space-like (2.30e), we can demand the
tensors $A_{(\nu(s)\mu)}$ and $\tilde{\mathsf{A}}_{\nu(s-1)}$ to be not only
traceless but also space-like999The time-like component of $\mathcal{A}$ is
contained in the third term in (5.4) which is unphysical.

$\displaystyle
A_{\nu}\mathcal{T}^{\nu}=0=\tilde{\mathsf{A}}_{\nu}\mathcal{T}^{\nu}\,,$
(5.11)

but not divergence-free101010Note that since the fields are not divergence-
free, one cannot introduce a shift-symmetry of type
$\eta_{\mu\nu}\vartheta_{\nu(s-1)}$ to gauge away the second
$\tilde{\mathsf{A}}$ modes as in standard approach in higher-spin literature,
see e.g. Section 3 of [32].. This also shows that the resulting theory is
ghost-free (cf., [31]), even though (local) Lorentz invariance is not
manifest, in accordance with the fact that the matrix model defines a
preferred frame.

#### Averaging over the fiber.

Following the procedure in [31], we can average over the fiber coordinates
$t^{\nu}$ to obtain the spacetime kinetic action of higher-spin modes in
vectorial description. We illustrate this for the spin 1 fields, whose modes
expansion (5.10) gives

$\displaystyle S_{\text{eff}}$ $\displaystyle=-\int A_{\mu}\Box
A^{\mu}+(A_{\mu}\Box\tilde{\mathsf{A}}_{\sigma}+\tilde{\mathsf{A}}_{\sigma}\Box
A_{\mu})t^{\mu}t^{\sigma}+t_{\nu}t^{\nu}\tilde{\mathsf{A}}_{\rho}\Box\tilde{\mathsf{A}}_{\sigma}t^{\sigma}t^{\rho}\,$
$\displaystyle=-\int A_{\mu}\Box
A^{\mu}+\frac{1}{3\ell_{p}^{2}}(A_{\mu}\Box\tilde{\mathsf{A}}_{\sigma}+\tilde{\mathsf{A}}_{\sigma}\Box
A_{\mu})\kappa^{\mu\sigma}+\frac{\cosh^{2}(\tau)}{3\ell_{p}^{4}}\tilde{\mathsf{A}}_{\rho}\Box\tilde{\mathsf{A}}_{\sigma}\kappa^{\sigma\rho}\
$ (5.12)

using (2.32b), and averaging over fiber coordinates in the flat limit using

$\displaystyle[t^{\mu}t^{\nu}]_{0}$
$\displaystyle=\frac{1}{3\ell_{p}^{2}}\kappa^{\mu\nu}\ .$ (5.13)

Here

$\displaystyle\kappa^{\mu\nu}$
$\displaystyle=\cosh^{2}(\tau)\eta^{\mu\nu}+\frac{{y^{\mu}}{y^{\nu}}}{R^{2}}$
(5.14)

is the space-like projector orthogonal to the time-like FLRW vector field
$\mathcal{T}$ cf. (2.36). By virtue of the space-like constraints (5.11), the
second term in (5.14) will not contribute upon choosing space-like tensors as
discussed above111111The restriction to time-like tensors might be avoided by
absorbing the third mode in (5.4) into the first. This issue will be avoided
in the spinorial approach below., and the kinetic action takes the form

$\displaystyle S_{\text{eff}}=-\int A_{\mu}\Box A^{\mu}+\alpha
A_{\mu}\Box\tilde{\mathsf{A}}^{\mu}+\alpha\tilde{\mathsf{A}}_{\mu}\Box
A^{\mu}+\beta\tilde{\mathsf{A}}^{\mu}\Box\tilde{\mathsf{A}}_{\mu}\,,$ (5.15)

where $\alpha=\cosh^{2}(\tau)/3\ell_{p}^{2}$ and
$\beta=\cosh^{4}(\tau)/3\ell_{p}^{4}$. The inner product of the modes
$(A,\tilde{A})$ is obtained similarly, dropping the $\Box$. This boils down to
the matrix

$\displaystyle M=\begin{pmatrix}1&\alpha\\\ \alpha&\beta\end{pmatrix}$ (5.16)

with non-vanishing determinant $\det
M=\frac{2\cosh^{4}(\tau)}{9\ell_{p}^{4}}\neq 0$. Therefore these two modes are
independent, but not orthogonal121212For a similar analysis of the full
fluctuation spectrum on the curved FLRW background see [31]..

While the analysis can be continued also at non-linear level, we find it
cumbersome to integrate out fiber coordinates using vectorial description.
This problem can be simplified by using spinorial formalism (cf., Sections 3
and 4 and Appendix B). In particular, when the two higher-spin modes $A$ and
$\tilde{\mathsf{A}}$ written in spinoral language will completely disentangle
with each others in the flat limit. As a result, we can make a field
redefinition to bring them into a “helicity” base and perform scattering
amplitude calculation.

##  6 Spinorial reduction to 4 dimensions

In this section we rewrite the IKKT-matrix model in spinorial language and
obtain its spacetime actions for the massless sector from two aforementioned
projections of $H^{4}_{N}$. Note that unlike [10], we will study the IKKT-
matrix model in second-order formalism as in [8]. Moreover, while the
stereographic projection allows us to obtain the spacetime action of HS-IKKT
in a global manner, reducing the twistor action to spacetime one in
$SO(1,3)$-invariant projection can only be achieved locally, i.e. around
certain reference point $\mathsf{p}\in\mathcal{M}$ where we can set the
temporal component $t^{0}$ of the momentum $t^{\mu}$ to zero.

###  6.1 Reduced Euclidean action on $H^{4}$

We start with the $SO(1,4)$-invariant action on the Euclidean hyperboloid
$H^{4}_{N}$, which is embedded along the first $1+4$ coordinates of the target
space. The chiral basis of $\mathfrak{so}(1,4)$ $\gamma$-matrices in (2.11)
allows us to cast $y^{a}$ into spinorial form:

$\displaystyle y^{AB}=q^{AB}+p^{AB}=\begin{pmatrix}q^{\alpha\beta}&0\\\
0&q^{\dot{\alpha}\dot{\beta}}\end{pmatrix}+\begin{pmatrix}0&p^{\alpha\dot{\beta}}\\\
-p^{\dot{\beta}\alpha}&0\end{pmatrix}\,$ (6.1)

where $q^{AB}$ represents the zeroth direction associated to $\gamma_{0}$
matrix. Since the external symmetry group $SO(1,4)$ acts on $q^{0}$,
supersymmetry will be broken by the background, while the underlying matrix
model is of course still supersymmetric. We also note that the reality
condition for $y^{AB}$ is

$\displaystyle(y^{AB})^{*}=-(C^{-1}YC)_{BA}\,=(C^{-1}YC)_{AB}\ .$ (6.2)

In terms of components, we have

$\displaystyle(q^{\alpha\beta})^{*}=-(\epsilon^{-1}q\epsilon)_{\beta\alpha}=(\epsilon^{-1}q\epsilon)_{\alpha\beta}\,,\qquad(p^{\alpha\dot{\alpha}})^{*}=-(\epsilon^{-1}p\epsilon)_{\dot{\alpha}\alpha}\,=(\epsilon^{-1}p\epsilon)_{\alpha\dot{\alpha}}$
(6.3)

due to anti-symmetry. Next, let the remaining 5 coordinates of $SO(1,9)$ in
(2.1), which $SO(1,4)$ does not act on, be the scalar fields
$\widetilde{y}_{i}$ on $H^{4}$ associated to the internal symmetry group
$SO(5)$ [33], where $i=5,6,7,8,9$. The local isomorphism
$\mathfrak{so}(5)\simeq\mathfrak{sp}(4)$ allows us to write:

$\displaystyle\widetilde{y}^{i}\gamma_{i}^{\mathcal{I}\mathcal{J}}\mapsto\phi^{\mathcal{I}\mathcal{J}}\,,\qquad\qquad\mathcal{I},\mathcal{J}=1,2,3,4\,,$
(6.4)

where $\phi^{\mathcal{I}\mathcal{J}}=-\phi^{\mathcal{J}\mathcal{I}}$ can be
written in terms of $2\times 2$ block matrices $\phi^{IJ}$.

Next, consider the fermions $\Psi:=\Psi^{A\mathcal{I}}$ as 4-spinors of
$SO(1,4)$. Let us clarify on the notation of $\Psi$. Here, $A$ is a 4-spinor
index of $SO(1,4)$ and $\mathcal{I}$ is a spinor index of $SO(5)$. Upon
imposing the 9+1-dimensional Majorana-Weyl condition, this amounts to 4 Weyl
spinors on space-time. Using the decomposition (6.1) of bosonic coordinates,
we can write the Yukawa term as

$\displaystyle\leavevmode\hbox to16.14pt{\vbox
to16.14pt{\pgfpicture\makeatletter\hbox{\hskip 8.07158pt\lower-8.07158pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.87158pt}{0.0pt}\pgfsys@curveto{7.87158pt}{4.3474pt}{4.3474pt}{7.87158pt}{0.0pt}{7.87158pt}\pgfsys@curveto{-4.3474pt}{7.87158pt}{-7.87158pt}{4.3474pt}{-7.87158pt}{0.0pt}\pgfsys@curveto{-7.87158pt}{-4.3474pt}{-4.3474pt}{-7.87158pt}{0.0pt}{-7.87158pt}\pgfsys@curveto{4.3474pt}{-7.87158pt}{7.87158pt}{-4.3474pt}{7.87158pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.75pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{Y}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\mathrm{i}\,\bar{\Psi}_{A\mathcal{I}}\\{p^{AB},\Psi_{B}{}^{\mathcal{I}}\\}+\mathrm{i}\,\bar{\Psi}_{A\mathcal{I}}\\{q^{AB},\Psi_{B}{}^{\mathcal{I}}\\}+\mathrm{i}\,\bar{\Psi}_{A\mathcal{I}}\\{\phi^{\mathcal{I}\mathcal{J}},\Psi^{A}{}_{\mathcal{J}}\\}$
(6.5)

In terms of 2-spinors, $\Psi^{A\mathcal{I}}$ and
$\bar{\Psi}_{A\mathcal{I}}:=\Psi_{B\mathcal{I}}^{\dagger}(\gamma_{0})^{B}{}_{A}$
can be decomposed as

$\displaystyle\Psi^{A\mathcal{I}}=(\chi^{\alpha\mathcal{I}},\tilde{\chi}^{\dot{\alpha}\mathcal{I}})\,,\qquad\bar{\Psi}_{A\mathcal{I}}=(\bar{\chi}_{\alpha\mathcal{I}},-\bar{\tilde{\chi}}_{\dot{\alpha}\mathcal{I}})\,$
(6.6)

where the two Weyl spinors for each $\mathcal{I}$ are related by the Majorana
condition $\Psi=C\bar{\Psi}^{t}$ through the charge conjugation matrix $C$ in
$9+1$ -dimensions, which means that

$\displaystyle\tilde{\chi}^{\dot{\alpha}\mathcal{I}}=(C^{3+1})^{\dot{\alpha}}{}_{\beta}(C^{6})^{\mathcal{I}}{}_{\mathcal{J}}\bar{\chi}^{\beta\mathcal{J}}\,.$
(6.7)

Then, the Yukawa term can be written in the following form:

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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=$
$\displaystyle+\mathrm{i}\,\bar{\chi}_{\alpha\mathcal{I}}\\{p^{\alpha\dot{\beta}},\tilde{\chi}_{\dot{\beta}}{}^{\mathcal{I}}\\}-\mathrm{i}\chi_{\alpha\mathcal{I}}\\{p^{\alpha\dot{\beta}},\bar{\tilde{\chi}}_{\dot{\beta}}{}^{\mathcal{I}}\\}+\mathrm{i}\bar{\chi}_{\alpha\mathcal{I}}\\{q^{\alpha\beta},\chi_{\beta}{}^{\mathcal{I}}\\}-\mathrm{i}\bar{\tilde{\chi}}_{\dot{\alpha}\mathcal{I}}\\{q^{\dot{\alpha}\dot{\beta}},\tilde{\chi}_{\dot{\beta}}{}^{\mathcal{I}}\\}$
$\displaystyle+\mathrm{i}\bar{\chi}_{\alpha\mathcal{I}}\\{\phi^{\mathcal{I}\mathcal{J}},\chi^{\alpha}{}_{\mathcal{J}}\\}-\mathrm{i}\bar{\tilde{\chi}}_{\dot{\alpha}\mathcal{I}}\\{\phi^{\mathcal{I}\mathcal{J}},\tilde{\chi}^{\dot{\alpha}}{}_{\mathcal{J}}\\}$
(6.8)

where half of the terms are redundant due to the Majorana condition.

Let the $H^{4}\subset\mathbb{R}^{1,4}$ background be parametrized by
$\mathtt{y}^{\alpha\dot{\alpha}}$ (the tangential direction) and
$\mathtt{y}_{0}$ (the transversal direction). Then $p^{\alpha\dot{\alpha}}$
and $q^{\alpha\beta}$ can be decomposed as:

$\displaystyle\binom{p^{\alpha\dot{\alpha}}}{q^{\alpha\beta}}=\binom{\mathtt{y}^{\alpha\dot{\alpha}}}{\mathtt{y}_{0}\epsilon^{\alpha\beta}}+\binom{\mathsf{a}^{\alpha\dot{\alpha}}}{\hat{\phi}\epsilon^{\alpha\beta}}\,,$
(6.9)

where $(\mathsf{a}^{\alpha\dot{\alpha}},\hat{\phi})$ stand for
$\mathfrak{hs}$-valued fluctuations, which are subject to reality conditions
analogous to (6.3). Together,
$(\mathsf{a}^{\alpha\dot{\alpha}},\hat{\phi},\phi^{IJ})$ constitute the set of
bosonic dynamical fields, while $(\chi,\widetilde{\chi})$ are fermionic
dynamical fields. Note that we do not have a fermionic ‘background’ in our
setup. Using the above organization, we arrive at the following spinorial
action for the IKKT-matrix model in the semi-classical limit on the projective
spinor bundle $\mathbb{PS}$ :

$\begin{split}S=&\int_{\mathbb{PS}}\mho\,\Big{(}\frac{1}{2}\mathsf{f}_{\alpha\alpha}\mathsf{f}^{\alpha\alpha}+\frac{1}{2}\\{p^{\alpha\dot{\alpha}},\hat{\phi}\\}\\{p_{\alpha\dot{\alpha}},\hat{\phi}\\}+\frac{1}{2}\\{p^{\alpha\dot{\alpha}},\phi^{IJ}\\}\\{p_{\alpha\dot{\alpha}},\phi_{IJ}\\}\\\
&-\frac{\mathrm{i}}{2}\bar{\chi}^{\alpha}{}_{\mathcal{I}}\\{p_{\alpha\dot{\beta}},\tilde{\chi}^{\dot{\beta}\mathcal{I}}\\}+\frac{\mathrm{i}}{2}\chi^{\alpha}{}_{\mathcal{I}}\\{p_{\alpha\dot{\beta}},\bar{\tilde{\chi}}_{\dot{\beta}}{}^{\mathcal{I}}\\}+\frac{\mathrm{i}}{2}\bar{\tilde{\chi}}_{\dot{\alpha}\mathcal{I}}\\{\mathtt{y}_{0},\tilde{\chi}^{\dot{\alpha}\mathcal{I}}\\}-\frac{\mathrm{i}}{2}\bar{\chi}_{\alpha\mathcal{I}}\\{\mathtt{y}_{0},\chi^{\alpha\mathcal{I}}\\}\\\
&+\frac{1}{2}\\{\mathtt{y}_{0},p^{\alpha\dot{\alpha}}\\}\\{\mathtt{y}_{0},p_{\alpha\dot{\alpha}}\\}+\frac{1}{2}\\{\mathtt{y}_{0},\hat{\phi}\\}\\{\mathtt{y}_{0},\hat{\phi}\\}+\frac{1}{4}\\{\mathtt{y}_{0},\phi^{IJ}\\}\\{\mathtt{y}_{0},\phi_{IJ}\\}\\\
&+\frac{\mathrm{i}}{2}\bar{\tilde{\chi}}_{\dot{\alpha}\mathcal{I}}\\{\hat{\phi},\tilde{\chi}^{\dot{\alpha}\mathcal{I}}\\}-\frac{\mathrm{i}}{2}\bar{\chi}_{\alpha\mathcal{I}}\\{\hat{\phi},\chi^{\alpha\mathcal{I}}\\}-\frac{\mathrm{i}}{2}\bar{\chi}_{\alpha}{}^{\mathcal{I}}\\{\phi_{\mathcal{I}\mathcal{J}},\chi^{\alpha\mathcal{J}}\\}+\frac{\mathrm{i}}{2}\bar{\tilde{\chi}}_{\dot{\alpha}}{}^{\mathcal{I}}\\{\phi_{\mathcal{I}\mathcal{J}},\tilde{\chi}^{\dot{\alpha}\mathcal{J}}\\}\\\
&+\frac{1}{2}\\{\hat{\phi},\hat{\phi}\\}\\{\hat{\phi},\hat{\phi}\\}+\frac{1}{2}\\{\hat{\phi},\phi^{IJ}\\}\\{\hat{\phi},\phi_{IJ}\\}+\frac{1}{2}\\{\phi^{IJ},\phi^{MN}\\}\\{\phi_{IJ},\phi_{MN}\\}\Big{)}\,,\end{split}$
(6.10)

where

$\displaystyle\mathsf{f}^{\alpha\alpha}=\\{p^{\alpha}{}_{\dot{\gamma}},p^{\alpha\dot{\gamma}}\\}\,,\qquad\hat{f}^{\alpha\alpha}=f^{\alpha\alpha}\,,$
(6.11)

and we have rescaled fields/coordinates appropriately. Here, the commutators
$[\,,]$ have been replaced by the Poisson brackets $\mathrm{i}\\{\,,\\}$, and
the ‘trace’ in (2.1) is substituted by an appropriate integral on
$\mathbb{PS}$ with the measure

$\displaystyle\mho:=\mathrm{D}^{3}Z\wedge\mathrm{D}^{3}\bar{Z}=\frac{R^{8}}{(R^{2}-x^{2})^{4}}\,\mathrm{d}^{4}\mathtt{x}\frac{\langle\lambda\,\mathrm{d}\lambda\rangle\wedge\langle\hat{\lambda}\,\mathrm{d}\hat{\lambda}\rangle}{\langle\lambda\,\hat{\lambda}\rangle^{2}}\,$
(6.12)

which comes from wedging the $SU(4)$-invariant holomorphic measure
$\mathrm{D}^{3}Z$ [34]:

$\displaystyle\mathrm{D}^{3}Z=\epsilon_{ABCD}Z^{A}\mathrm{d}Z^{B}\mathrm{d}Z^{C}\mathrm{d}Z^{D}=\frac{R^{4}\lambda_{\alpha}\lambda_{\beta}\langle\lambda\,\mathrm{d}\lambda\rangle\wedge\mathrm{d}\mathtt{x}^{\alpha\dot{\alpha}}\wedge\mathrm{d}\mathtt{x}^{\beta}{}_{\dot{\alpha}}}{(R^{2}-x^{2})^{2}}\,,$
(6.13)

and the anti-holomorphic measure
$\mathrm{D}^{3}\bar{Z}=\epsilon_{ABCD}\hat{Z}^{A}\mathrm{d}\hat{Z}^{B}\mathrm{d}\hat{Z}^{C}\mathrm{d}\hat{Z}^{D}$.
It can be easily checked that $\\{\hat{\mathcal{N}},\mho\\}=0$ where
$\hat{\mathcal{N}}$ is the number operator defined in (2.20).

It is worth noting that the IKKT-matrix model on $H^{4}$ admits a smooth flat
limit where $R\rightarrow\infty$, and all contributions associated with
$\mathtt{y}_{0}$ can be dropped as shown in [10]. We will assume this limit
from now on to study scattering amplitudes of the IKKT matrix model.

#### A sketch on self-dual sector.

Let us look at the Yang-Mills term in the action (6.10). In terms of
‘background’ coordinates $\mathtt{y}^{\alpha\dot{\alpha}}$ and fluctuations
$\mathsf{a}$, the Yang-Mills term reads:

$\displaystyle\begin{split}L^{\text{YM}}=\frac{1}{2}\mathsf{f}_{\alpha\alpha}\mathsf{f}^{\alpha\alpha}=\bar{L}^{\mathsf{f}}_{\mathtt{BG}}&+2\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathsf{a}^{\alpha\dot{\gamma}}\\}\\{\mathtt{y}_{\alpha\dot{\sigma}},\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\\}+\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathtt{y}^{\alpha\dot{\gamma}}\\}\\{\mathsf{a}_{\alpha\dot{\sigma}},\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\\}\,\\\
&+2\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathsf{a}^{\alpha\dot{\gamma}}\\}\\{\mathsf{a}_{\alpha\dot{\sigma}},\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\\}+\frac{1}{2}\\{\mathsf{a}^{\alpha}{}_{\dot{\gamma}},\mathsf{a}^{\alpha\dot{\gamma}}\\}\\{\mathsf{a}_{\alpha\dot{\sigma}},\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\\}\,,\end{split}$
(6.14)

which is analogous with the field strength term in non-commutative gauge
theory, cf. [9, 30]. Here, the ‘background’ action
$\bar{L}^{\mathsf{f}}_{\mathtt{BG}}$ consists of terms that are 0th order or
1st order in fields. The term
$\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathtt{y}^{\alpha\dot{\gamma}}\\}\\{\mathsf{a}_{\alpha\dot{\sigma}},\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\\}$,
which always occurs in non-commutative field theory, can be absored in the
background by introducing an auxiliary field $\mathsf{b}_{\alpha\alpha}$, and
writing the Yang-Mills term in the first-order form as

$\displaystyle
S^{\text{YM}}=\int_{\mathbb{PS}}\mho\Big{(}\mathsf{b}_{\alpha\alpha}\mathsf{f}^{\alpha\alpha}-\frac{1}{2}\mathsf{b}_{\alpha\alpha}\mathsf{b}^{\alpha\alpha}\Big{)}\,.$
(6.15)

Note that we can also obtain the self-dual sector of (6.15) by dropping the
$\mathsf{b}_{\alpha\alpha}\mathsf{b}^{\alpha\alpha}$ and maintaining gauge-
invariance. Intriguingly, in the flat limit, all higher-spin fields in the
self-dual sector of HS-IKKT theory are effectively massless since there is an
‘emergent’ shift symmetry which can be used to gauge away the higher-spin
modes $\tilde{\mathsf{A}}$ in $\mathcal{A}$ [10]. We discuss below that this
shift-symmetry, however, is _not_ a symmetry of the full HS-IKKT model.

#### Higher-spin modes in spinor formalism.

For concreteness, let us analyze the pure Yang-Mills sector of the IKKT-matrix
model in the flat limit.

If $\mathsf{a}_{\alpha\dot{\alpha}}$ is a section with values in the algebra
$\mathscr{C}(\mathbb{PS})$, it has the following expansion:131313Recall that
the “time-like” fluctuations $\hat{\phi}$ drop out in the flat limit.

$\displaystyle\mathsf{a}_{\alpha\dot{\alpha}}=\sum_{s=0}^{\infty}\lambda^{\beta(s)}\hat{\lambda}^{\beta(s)}\mathcal{A}_{\beta(2s)|\alpha\dot{\alpha}}\,,\qquad\lambda^{\beta(s)}:=\lambda^{(\beta_{1}}\ldots\lambda^{\beta_{s})}\quad\text{etc.}$
(6.16)

Observe that

$\displaystyle\hat{\mathsf{a}}_{\alpha\dot{\alpha}}$
$\displaystyle=\sum_{s=0}^{\infty}(-)^{s}\lambda^{\beta(s)}\hat{\lambda}^{\beta(s)}\hat{\mathcal{A}}_{\beta(2s)|\alpha\dot{\alpha}}\,,$
(6.17) $\displaystyle\mathsf{a}_{\alpha\dot{\alpha}}^{\dagger}$
$\displaystyle=\sum_{s=0}^{\infty}(-)^{s}\lambda^{\beta(s)}\hat{\lambda}^{\beta(s)}\
(\mathcal{A}_{\beta(2s)|\alpha\dot{\alpha}})^{\dagger}\,.$ (6.18)

Therefore, if we want $\mathsf{a}_{\alpha\dot{\alpha}}$ to be real
$\mathfrak{hs}$-valued field, $\mathcal{A}_{\beta(2s)|\alpha\dot{\alpha}}$
need to be complex, subject to the reality conditions

$\displaystyle\hat{\mathcal{A}}_{\beta(2s)|\alpha\dot{\alpha}}=\mathrm{i}^{2s}\mathcal{A}_{\beta(2s)|\alpha\dot{\alpha}}\,,\qquad(\mathcal{A}_{\beta(2s)|\alpha\dot{\alpha}})^{\dagger}=\mathrm{i}^{2s}\mathcal{A}_{\beta(2s)|\alpha\dot{\alpha}}$
(6.19)

As a consequence, if we expand the $\mathfrak{hs}$-valued field strength
$\mathsf{f}^{\alpha\alpha}$ in terms of $\mathcal{A}$ will become complex.
This fact will be exploited in Appendix B when we obtained the interacting
vertices from the self-dual sector.

For now, we will decompose the coefficients
$\mathcal{A}_{\beta(2s)|\alpha\dot{\alpha}}$ into the following irreps:

$\displaystyle\mathcal{A}_{\beta(2s)|\alpha\dot{\alpha}}=A_{(\beta(2s)\alpha)\,\dot{\alpha}}+\epsilon_{\alpha\beta}\tilde{\mathsf{A}}_{\beta(2s-1)\,\dot{\alpha}}\,.$
(6.20)

We therefore have $\sum_{s}2(2s+2)+2(2s)=\sum_{s}4(2s+1)$ off-shell degrees of
freedom, in complete agreement with the organization using the vector
formalism in Section 5. Notice that when we integrate out all fiber
coordinates, the coefficients $A_{\alpha(2s-1)\,\dot{\alpha}}$ and
$\tilde{\mathsf{A}}_{\alpha(2s-3)\,\dot{\alpha}}$ will become tensorial fields
in the maximally unbalanced/chiral representation [35, 36].141414 A
representation features fields with more un-dotted and dotted indices and
allows us to work with lowest possible number of derivatives in the
interactions. This is in perfect agreement with the results in vector
description [8] where the $\epsilon_{abcde}$ tensor breaks parity-invariance
(see the discussion in Section 2).

#### Gauge fixing and dof.

In spinorial language and in the flat limit, the gauge-fixing condition takes
the form

$\begin{split}0=\\{\mathtt{y}_{\alpha\dot{\alpha}},\mathcal{A}^{\beta(2s)|\alpha\dot{\alpha}}\\}&=\\{\mathtt{y}_{\alpha\dot{\alpha}},A^{(\beta(2s)\alpha)\dot{\alpha}}\\}+\varepsilon^{\alpha\beta}\\{\mathtt{y}_{\alpha\dot{\alpha}},\tilde{\mathsf{A}}^{\beta(2s-1)\dot{\alpha}}\\}\\\
&=\\{\mathtt{y}_{\alpha\dot{\alpha}},A^{(\beta(2s)\alpha)\dot{\alpha}}\\}-\\{\mathtt{y}^{\beta}{}_{\dot{\alpha}},\tilde{\mathsf{A}}^{\beta(2s-1)\dot{\alpha}}\\}\,.\end{split}$
(6.21)

The linearized gauge transformation for higher-spin gauge potentials
$\mathcal{A}_{\beta(2s)|\alpha\dot{\alpha}}$ reads

$\displaystyle\delta_{\xi}\mathcal{A}_{\beta(2s)|\alpha\dot{\alpha}}=\\{\mathtt{y}_{\alpha\dot{\alpha}},\xi_{\beta(2s)}\\}\,\equiv\mathcal{A}^{(g)}[\xi_{\beta(2s)}]\
.$ (6.22)

We can then use (6.21) and (6.22) to remove $(2s+1)$ components of
$\mathcal{A}^{\beta(2s)|\alpha\dot{\alpha}}$ which leaves us with $3(2s+1)$
off-shell degrees of freedom of $\mathcal{A}^{\beta(2s)|\alpha\dot{\alpha}}$
on $H^{4}$ (without the ghosts). To understand this, we recall from Section 5
and [8] that even though the IKKT-matrix model does not possess a mass
parameter, it leads to three off-shell towers of “would-be massive” higher-
spin degrees of freedom on $H^{4}$. These extra dof. arise from divergence-
like components, which are physical on the curved background [31], but
expected to decouple in the flat limit151515It would be interesting to
understand whether they can be described by a more complicated net of gauge
symmetry as in Zinoviev’s system [37]. .

#### On shift symmetry.

Note that in other constructions of $3d$ or $4d$ higher-spin gauge theories
using spinor formalism (see e.g. [35, 36, 38]), one often imposes by hand an
extra “shift” gauge symmetry of the form

$\displaystyle\delta_{\vartheta}\mathcal{A}_{\beta(2s)|\alpha\dot{\alpha}}=\epsilon_{\alpha\beta}\,\vartheta_{\beta(2s-1)\,\dot{\alpha}}\,$
(6.23)

to remove the second mode $\tilde{\mathsf{A}}_{\beta(2s-1)\,\dot{\alpha}}$
from $\mathcal{A}_{\beta(2s)|\alpha\dot{\alpha}}$ given that
$\mathsf{a}_{\alpha\dot{\alpha}}$ is divergence-free. This symmetry was shown
to be a symmetry of the self-dual Yang-Mills sector in HS-IKKT theory in the
flat limit [10]. However, in the full matrix model, there is no such gauge
symmetry (see Appendix A). As a result, we cannot use $\vartheta$ to remove
the second higher-spin modes
$\tilde{\mathsf{A}}_{\alpha(2s-3)\,\dot{\alpha}}$. This is consistent with the
fact that all higher-spin modes are not divergence-free a priori.

#### On kinetic action.

Let us first look at the kinetic part of the action (6.14) in the flat limit
for the $\mathfrak{hs}$-valued gauge potential
$\mathsf{a}_{\alpha\dot{\alpha}}$:

$\displaystyle
S_{2}=\int\mho\,\Big{(}2\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathsf{a}^{\alpha\dot{\gamma}}\\}\\{\mathtt{y}_{\alpha\dot{\sigma}},\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\\}+\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathtt{y}^{\alpha\dot{\gamma}}\\}\\{\mathsf{a}_{\alpha\dot{\sigma}},\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\\}\Big{)}\,.$
(6.24)

Let us first check gauge invariance explicitly: Under a local gauge
transformation, we get

$\displaystyle\delta_{\xi}S_{2}=\int\mho\,\Big{(}2\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathsf{a}^{\alpha\dot{\gamma}}\\}\\{\mathtt{y}_{\alpha\dot{\sigma}},\\{\mathtt{y}_{\alpha}{}^{\dot{\sigma}},\xi\\}\\}+2\mathsf{a}_{\alpha\dot{\sigma}}\\{\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathtt{y}^{\alpha\dot{\gamma}}\\},\\{\mathtt{y}_{\alpha}{}^{\dot{\sigma}},\xi\\}\\}\Big{)}\,.$
(6.25)

where we have made an integration by part. The second term can be rewritten as

$\displaystyle\int\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathtt{y}^{\alpha\dot{\gamma}}\\}\\{\mathsf{a}_{\alpha\dot{\sigma}},\\{\mathtt{y}_{\alpha}{}^{\dot{\sigma}},\xi\\}\\}=-\int\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathtt{y}^{\alpha\dot{\gamma}}\\}\big{(}\\{\mathtt{y}_{\alpha}{}^{\dot{\sigma}},\\{\xi,\mathsf{a}_{\alpha\dot{\sigma}}\\}\\}+\\{\xi,\\{\mathsf{a}_{\alpha\dot{\sigma}},\mathtt{y}_{\alpha}{}^{\dot{\sigma}}\\}\\}\big{)}$
$\displaystyle=\int\\{\mathtt{y}_{\alpha}{}^{\dot{\sigma}},\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathtt{y}^{\alpha\dot{\gamma}}\\}\\}\\{\xi,\mathsf{a}_{\alpha\dot{\sigma}}\\}+\\{\xi,\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathtt{y}^{\alpha\dot{\gamma}}\\}\\}\\{\mathsf{a}_{\alpha\dot{\sigma}},\mathtt{y}_{\alpha}{}^{\dot{\sigma}}\\}$
(6.26)

using Jacobi identity. The first term in (6.1) vanishes due to background eom

$\displaystyle\\{\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathtt{y}^{\alpha\dot{\gamma}}\\},\mathtt{y}_{\alpha}{}^{\dot{\sigma}}\\}=0\,.$
(6.27)

Furthermore, the second term in (6.1) cancels with the first tem in (6.25) if
we use Jacobi identity to write

$\displaystyle\int\\{\xi,\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathtt{y}^{\alpha\dot{\gamma}}\\}\\}\\{\mathsf{a}_{\alpha\dot{\sigma}},\mathtt{y}_{\alpha}{}^{\dot{\sigma}}\\}=-\int\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\\{\mathtt{y}^{\alpha\dot{\gamma}},\xi,\\}\\}\\{\mathsf{a}_{\alpha\dot{\sigma}},\mathtt{y}_{\alpha}{}^{\dot{\sigma}}\\}\,.$
(6.28)

noting that $\int\\{\xi,-\\}=0$. This establishes the gauge invariance of
(6.24).

#### Gauge-fixed kinetic action.

Note that the quadratic action (6.24) can be simplified further in the flat
limit by dropping the term term
$\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathtt{y}^{\alpha\dot{\gamma}}\\}\\{\mathsf{a}_{\alpha\dot{\sigma}},\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\\}$
in the flat limit (see Appendix B). Therefore, we can consider the following
gauge-fixed quadratic action

$\displaystyle
S_{2}^{\mathtt{gf}}=S_{2}+2\int\\{\mathtt{y}_{\alpha\dot{\alpha}},\mathsf{a}^{\alpha\dot{\alpha}}\\}^{2}\,.$
(6.29)

Here, the contributions from the gauge-fixing term removes certain
contributions from $S_{2}$ in the flat limit leaving us with the standard
kinetic $\Box$ terms in the quadratic action as shown in Appendix B. In
particular, we obtain

$\displaystyle
S_{2}^{\mathtt{gf}}=\int\mathrm{d}^{4}x\,\mathbb{A}_{-}^{\alpha(2s-1)\,\dot{\alpha}}\Box\mathbb{A}^{+}_{\alpha(2s-1)\,\dot{\alpha}}\,,\qquad\mathbb{A}^{\pm}_{\alpha(2s-1)\,\dot{\alpha}}=A_{\alpha(2s-1)\,\dot{\alpha}}\pm\mathrm{i}\tilde{\mathsf{A}}_{\alpha(2s-1)\,\dot{\alpha}}\,.$
(6.30)

This fact is convenient for us to compute the propagator of HS-IKKT theory
using spinor formalism. We detail this fact further below.

#### Euclidean action of massless sector.

A complete action of the (HS-) IKKT matrix model on $H_{N}^{4}$ can be
obtained using the following integral over $\mathbb{P}^{1}$ [39, 40, 41]:

$\displaystyle\int_{\mathbb{P}^{1}}\mathtt{K}\,\frac{\hat{\lambda}_{\alpha(m)}\,\lambda^{\beta(m)}}{\langle\lambda\,\hat{\lambda}\rangle^{m}}:=\int_{\mathbb{P}^{1}}\frac{\langle\lambda\,\mathrm{d}\lambda\rangle\wedge\langle\hat{\lambda}\,\mathrm{d}\hat{\lambda}\rangle}{\langle\lambda\,\hat{\lambda}\rangle^{2}}\,\frac{\hat{\lambda}_{\alpha(m)}\,\lambda^{\beta(m)}}{\langle\lambda\,\hat{\lambda}\rangle^{m}}=-\frac{2\pi
i}{(m+1)}\epsilon^{\ \beta}_{\alpha}\ldots\epsilon^{\ \beta}_{\alpha}\,.$
(6.31)

As discussed in [10, 22], in the present almost-commutative twistor
construction, there are more contributions (after integrating out fiber
coordinates) than the usual twistor construction using only holomorphic data
on twistor space, see e.g. [42, 43]. The reason is that the Poisson bracket
(cf., (3.10)) acts both on spacetime fields and fiber coordinates. However in
flat limit, the effective vielbeins
$\mathcal{E}^{\alpha\dot{\alpha}|\beta},\,\hat{\mathcal{E}}^{\alpha\dot{\alpha}|\beta}$
and $\mathcal{E}^{0|\beta},\hat{\mathcal{E}}^{0|\beta}$ are sub-leading, and
all contributions associated to $\mu,\hat{\mu}$ spinors will be suppressed
since they scale as $1/\sqrt{R}$ [10]. For this reason, we will only need to
consider terms where all fiber coordinates factorized outside the Poisson
brackets. For example,

$\displaystyle\lambda^{\alpha(s-1)}\hat{\lambda}^{\alpha(s-1)}\\{\mathtt{y}_{\alpha\dot{\alpha}},\mathbb{A}_{\alpha(2s-1)}{}^{\dot{\alpha}}\\}=\lambda^{\alpha(s-1)}\hat{\lambda}^{\alpha(s-1)}\mathcal{E}_{\alpha\dot{\alpha},\beta\dot{\beta}}\partial^{\beta\dot{\beta}}\mathbb{A}_{\alpha(2s-1)}{}^{\dot{\alpha}}$
(6.32)

will be the leading contribution in the higher-spin extension for
$\\{\mathtt{y}_{\alpha}{}^{\dot{\alpha}},\mathsf{a}_{\alpha\dot{\alpha}}\\}$
in the flat limit. Below we only present the final result of the spacetime
action of HS-IKKT on $H^{4}$ for massless sector, delegating the detail of the
computation to Appendix B.

The Euclidean gauge-fixed kinetic action for the $\mathfrak{hs}$-valued gauge
potential $\mathsf{a}_{\alpha\dot{\alpha}}$ reads

$\begin{split}S_{2}&=\int_{\mathbb{PS}}\mho\,\Big{(}2\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathsf{a}^{\alpha\dot{\gamma}}\\}\\{\mathtt{y}_{\alpha\dot{\sigma}},\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\\}+\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathtt{y}^{\alpha\dot{\gamma}}\\}\\{\mathsf{a}_{\alpha\dot{\sigma}},\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\\}+2\int\\{\mathtt{y}_{\alpha\dot{\alpha}},\mathsf{a}^{\alpha\dot{\alpha}}\\}^{2}\Big{)}\,\\\
&\simeq
2\sum_{s}\int\mathrm{d}^{4}\mathtt{x}\,\mathbb{A}^{\alpha(2s-1)\,\dot{\alpha}}_{-}\Box\mathbb{A}^{+}_{\alpha(2s-1)\,\dot{\alpha}}\,.\end{split}$
(6.33)

where the composite field $\mathbb{A}$, see (6.30). Here the
$\Box=\partial^{\alpha\dot{\alpha}}\partial_{\alpha\dot{\alpha}}$ in the flat
limit, cf. (3.28), and we have integrated out all fiber coordinates in the
second line of (6.33). We also note that the
$\tilde{\mathsf{A}}^{\alpha(2s-1)\,\dot{\alpha}}$ component originate from the
second higher-spin mode of the spin-$(s+1)$ gauge potential
$\mathcal{A}_{\alpha(2s+1)|\beta\dot{\beta}}$. This combination of higher-spin
fields allows us to diagonalize the $\Box$ operator in the flat limit in a
trivial way using spinorial formalism. Intriguingly, one may think of
$\mathbb{A}_{+}$ as a positive helicity field while $\mathbb{A}_{-}$ as a
negative helicity field in the flat limit. It is worth emphasizing once again
that even though $\mathbb{A}_{\pm}$ may look massless, they have more degrees
of freedom than usual massless higher-spin fields due to the fact that their
original twistor fields are not divergence-free a priori.

Next, we find at cubic order that

$S_{3}\simeq
4\sum_{s_{2}+s_{3}=s_{1}+2}\int\mathrm{d}^{4}\mathtt{x}\,\partial_{\alpha\dot{\alpha}}\mathbb{A}^{\pm}_{\alpha(2s_{1}-1)}{}^{\dot{\alpha}}\partial_{\alpha\dot{\gamma}}\mathbb{A}_{\pm}^{\alpha(2s_{2}-1)\,\dot{\sigma}}\partial_{\alpha}{}^{\dot{\gamma}}\mathbb{A}_{\pm}^{\alpha(2s_{3}-1)}{}_{\dot{\sigma}}+\widetilde{S}_{3}\,,$
(6.34)

where $\tilde{S}_{3}$ are terms that are irrelevant for scattering process of
massless modes as they vanish upon plugging in the plane-wave solutions with
higher-spin polarization tensors in (7.4). Unlike the usual story of higher-
spin Yang-Mills [36], here all helicity configurations are allowed. However,
most of the 3-pt amplitudes resulting from gluing the cubic vertices (6.34)
with external states vanish on-shell (see Section 7). One can check that the
above cubic vertex has two transverse derivatives
$\partial^{0\dot{1}}=\bar{\partial}$ in the light-cone gauge by following the
procedure in [35, 44]. For this reason, the HS-IKKT matrix model can be
referred to as a two-derivative higher-spin theory. Lastly, the quartic term
reads

$\displaystyle S_{4}\simeq
2\int\mathrm{d}^{4}x\Big{(}\partial_{\alpha\dot{\gamma}_{1}}\mathbb{A}_{\pm}^{\alpha(2s_{1}-1)\,\dot{\sigma}}\partial_{\alpha}{}^{\dot{\gamma}_{1}}\mathbb{A}_{\pm}^{\alpha(2s_{2}-1)}{}_{\dot{\sigma}}\Big{)}\Big{(}\partial^{\alpha}{}_{\dot{\gamma}_{2}}\mathbb{A}^{\pm}_{\alpha(2s_{3}-1)}{}^{\dot{\tau}}\partial^{\alpha\dot{\gamma}_{2}}\mathbb{A}^{\pm}_{\alpha(2s_{4}-1)\,\dot{\tau}}\Big{)}+\widetilde{S}_{4}\,.$
(6.35)

Similar to $\tilde{S}_{3}$, $\tilde{S}_{4}$ contains irrelevant terms that do
not contribute in scattering amplitudes upon plugging plane-wave solutions
(7.4); and thus can be ignored in the flat limit. We use the above action to
compute scattering amplitudes of the HS-IKKT in Section 7. Note that while the
above action is defined on a background with Euclidean signature, it is
nevertheless possible to use some of the standard techniques in quantum field
theory to compute the scattering amplitudes if we work with complexified
kinematics. In fact, it is natural to do so since our ‘spacetime’ fields are
generically chiral, i.e. they have more un-dotted indices than dotted ones.

It is worth emphasizing that although there are more structures after
integrating out fiber coordinates in almost-commutative twistor approach
compared to the conventional twistor construction (see e.g. [43]), the non-
vanishing contributions of the massless sector in spacetime action on flat
space $S$-matrix in both approaches turn out to be the same.161616We leave the
study of the scattering of extra divergence modes for future study. The main
advantage of the present non-commutative twistor approach in [7, 9, 8, 13, 10]
is that it allows us to study quantization of twistor space, and to consider
models which appear to define a well-defined quantum theory.

###  6.2 Reduced action on the FLRW spacetime

Now we consider similarly the HS-IKKT theory on the FLRW-like spacetime after
doing an $SO(1,3)$-invariant projection as described in Section 4. Using the
spinorial description of $t^{\mu}$ in (4.2) where we set $t^{0}=0$ at the
referent point $\mathsf{p}\in\mathcal{M}^{1,3}$, we can effectively replace
the “momentum” generators $t^{i}$ by a pair of fiber coordinates
$(\lambda^{\alpha},\hat{\lambda}^{\alpha})$ in the flat limit. Recall that by
flat limit, we simply mean the limit where we can factorize all other fiber
coordinates $(\lambda,\hat{\lambda})$ outside the Poisson brackets, and drop
the explicit $\mu,\hat{\mu}$ spinors for the $x$-dependent fields.

Consider a local trivialization at $\mathsf{p}\in\mathcal{M}^{1,3}$ such that
$\mathbb{P}^{1,2}|_{U_{\mathsf{p}}}=\mathbb{P}^{1}\times U_{\mathsf{p}}$,
where $U_{\mathsf{p}}$ is an open subset around $\mathsf{p}$. Then the measure
on our twistor space is given by [13]

$\displaystyle\mho=\mathtt{K}\,\rho_{\mathcal{M}}\,d^{4}y=\mathtt{K}\,\frac{1}{R|\sinh(\tau)|}\,d^{4}y$
(6.36)

where $\mathtt{K}$ is again the top form on $\mathbb{P}^{1}$ fiber (cf.,
(6.31)). This measure is invariant under symplectomorphism on $\mathbb{PT}$,
and it is in fact globally well defined. The local split allows us to average
over fiber coordinates of $\mathbb{P}^{1}$ on an open subset $U_{\mathsf{p}}$
around $\mathsf{p}\in\mathcal{M}$, and to obtain a spacetime action for the
HS-IKKT matrix model on our background defining a FLRW cosmology.

Recalling that the background $\mathcal{M}^{1,3}$ is defined in terms of
‘momentum’ generators $t^{\mu}$, the action for the fluctuations
$\mathsf{a}^{\mu}$ of the (HS-)IKKT model in the semi-classical limit reads

$\begin{split}S=\int_{\mathbb{P}^{1,2}}&\mho\,\Big{(}\frac{1}{2}\\{t^{\mu},\mathsf{a}^{\nu}\\}\\{t_{\mu},\mathsf{a}_{\nu}\\}+\frac{1}{2}\\{t_{\mu},t_{\nu}\\}\\{\mathsf{a}^{\mu},\mathsf{a}^{\nu}\\}+\\{t^{\mu},\mathsf{a}^{\nu}\\}\\{\mathsf{a}_{\mu},\mathsf{a}_{\nu}\\}+\frac{1}{4}\\{\mathsf{a}^{\mu},\mathsf{a}^{\nu}\\}\\{\mathsf{a}_{\mu},\mathsf{a}_{\nu}\\}\\\
&-\frac{\mathrm{i}}{4}\bar{\Psi}\gamma^{\mu}\\{t_{\mu},\Psi\\}-\frac{\mathrm{i}}{4}\bar{\Psi}\gamma^{\mu}\\{\mathsf{a}_{\mu},\Psi\\}+\frac{1}{4}\\{\phi^{\hat{\mu}},\phi^{\hat{\nu}}\\}\\{\phi_{\hat{\mu}},\phi_{\hat{\nu}}\\}-\frac{\mathrm{i}}{4}\bar{\Psi}\gamma^{\hat{\mu}}\\{\phi_{\hat{\mu}},\Psi\\}\Big{)}+S_{\mathtt{BG}}\end{split}$
(6.37)

where $\hat{\mu}=4,5,\ldots,9$ indicates the 6 extra dimensions and
$S_{\mathtt{BG}}$ are the ‘background’ action which consists of zeroth or
first order in fluctuations. Note that we have identified
$t^{\hat{\mu}}\equiv\phi^{\hat{\mu}}$ as scalar fields in non-commutative
$\mathcal{N}=4$ SYM.

#### Higher-spin modes and dof.

As pointed out above, we can realize $t^{i}$ in terms of the pair of
$(\lambda^{\alpha}\,,\hat{\lambda}^{\alpha})$ fiber coordinates. This allows
to parametrize the higher-spin modes of the Yang-Mills gauge potential
$\mathsf{a}_{\mu}$ around a reference point $\mathsf{p}\in\mathcal{M}$ (cf.,
(4.2)) as follows

$\begin{split}\mathsf{a}_{\mu}&=\sum_{s=0}^{\infty}t^{i(s)}\mathcal{A}_{i(s)|\mu}\simeq\sum_{s=0}^{\infty}\lambda^{\beta(s)}\hat{\lambda}^{\beta(s)}\mathcal{A}_{\beta(2s)|\mu}\,.\end{split}$
(6.38)

cf. (5.10), which encodes $\sum_{s}4(2s+1)$ off-shell degrees of freedom of
the higher-spin valued gauge potential $\mathsf{a}_{\mu}$. The degeneracy of
the kinetic term is removed as usual by imposing the gauge-fixing condition

$\begin{split}0=\\{t^{\mu},\mathsf{a}_{\mu}\\}\end{split}$ (6.39)

and factoring out the pure gauge modes, defining the physical Hilbert space as

$\displaystyle\mathcal{H}_{\rm phys}=\\{\mbox{gauge-fixed on-shell
modes}\\}/_{\\{\mbox{pure gauge modes}\\}}$ (6.40)

(at ghost number zero). This removes two towers of higher-spin modes, leaving
us with $\sum_{s}2(2s+1)$ physical degrees of freedom171717recall that these
spinorial modes are not divergence-free here.. A more detailed analysis in
vectorial form is given in [31], where $\mathcal{H}_{\rm phys}$ was shown to
be free of negative modes, i.e. ghosts. The basic reason is that $t^{0}=0$
around the reference point $\mathsf{p}$, so that there are no time-like
higher-spin components in (6.38).

In the following, we will focus on certain specific degrees of freedom among
these modes.

#### Spacetime action in the flat limit.

Let us focus on the Yang-Mills sector of the action (6.37) in quadratic, cubic
and quatic orders as:

$\begin{split}S=\int_{\mathbb{P}^{1,2}}\mho\,\Big{(}\frac{1}{2}\\{t^{\mu},\mathsf{a}^{\nu}\\}\\{t_{\mu},\mathsf{a}_{\nu}\\}&+\frac{1}{2}\\{t_{\mu},t_{\nu}\\}\\{\mathsf{a}^{\mu},\mathsf{a}^{\nu}\\}+\\{t^{\mu},a_{\mu}\\}\\{t^{\nu},a_{\nu}\\}\\\
&+\\{t^{\mu},\mathsf{a}^{\nu}\\}\\{\mathsf{a}_{\mu},\mathsf{a}_{\nu}\\}+\frac{1}{4}\\{\mathsf{a}^{\mu},\mathsf{a}^{\nu}\\}\\{\mathsf{a}_{\mu},\mathsf{a}_{\nu}\\}+\ldots\Big{)}\,.\end{split}$
(6.41)

where un-hatted indices are contracted with $\eta^{\mu\nu}$. The last term in
the first line drops out upon gauge fixing. Using the effective metric (4.10)
and (2.35), we can write the gauge-fixed kinetic term explicitly as

$\displaystyle
S_{2}=\int_{\mathbb{P}^{1,2}}d^{4}y\,\mathtt{K}\,\Big{[}\frac{1}{2}\frac{\sinh(\tau)}{R}\Big{(}\gamma^{\mu\nu}\partial_{\mu}\mathsf{a}^{\rho}\partial_{\nu}\mathsf{a}_{\rho}+(\partial_{\mu}\mathsf{a}^{\mu})^{2}\Big{)}+\frac{m^{\mu\nu}}{2R^{3}|\sinh(\tau)|}\\{\mathsf{a}_{\mu},\mathsf{a}_{\nu}\\}\Big{]}\,.$
(6.42)

where $m^{\mu\nu}=R^{2}\\{t^{\mu},t^{\nu}\\}$ defined in (2.34c), and the
metric $\gamma^{\mu\nu}$ is given in (4.10), which locally reduces to
$\eta^{\mu\nu}$. As explained above, the term
$\frac{1}{2}\\{t_{\mu},t_{\nu}\\}\\{\mathsf{a}^{\mu},\mathsf{a}^{\nu}\\}$ is
suppressed in the flat limit of the FLRW matrix model-like spacetime, where
$R\rightarrow\infty$ and $\sinh(\tau)$ can be treated as (large) constant at
late time $\tau$. Therefore, in the flat limit

$\displaystyle
S_{2}\simeq-\frac{1}{2}\int_{\mathbb{PS}}\mathrm{d}^{4}y\,\mathtt{K}\Big{(}\,\mathsf{a}^{\mu}\Box\mathsf{a}_{\mu}+(\partial_{\mu}\mathsf{a}^{\mu})^{2}\Big{)}\,,\qquad\mu=0,1,2,3\,$
(6.43)

where the last term drops out upon gauge fixing. The fluctuations
$\mathsf{a}^{\mu}$ are real functions on $\mathbb{PT}$, since the matrices of
the IKKT model are hermitian. In particular, the d’Alembertian is given by

$\displaystyle\Box=\gamma^{\mu\nu}\partial_{\mu}\partial_{\nu}$ (6.44)

where $\mu,\nu=0,1,2,3\,$ dropping a conformal factor from the FLRW
background, which can be considered as locally constant in the present
context.

The appropriate spinorial formulation of this action is not evident, since the
spinors on the present background transforms under the $SU(2)_{L}\times
SU(2)_{R}$ space-like isometry group rather than $SL(2,\mathbb{C})$. Then the
time-like components take a non-standard form, and local Lorentz-invariance is
not manifest. Nevertheless, a close relation with the Euclidean case on
$H^{4}$ is expected, since the underlying space of functions on $\mathbb{PT}$
is the same, given by (principal series) unitary irreps of
$\mathfrak{so}(2,4)$. Therefore the interactions arising from the matrix model
are the same in both signatures, while the kinetic terms should be related by
some sort of Wick rotation. We expect that this relation should work most
naturally for the physical fields, because then the time-like components of
$\mathsf{a}_{\mu}$ are unphysical, while for the space-like components we do
recover the ususal spinorial representation due to (4.4). We could thus
declare that the spinors transforms as $SL(2,\mathbb{C})$ spinors under the
local Lorentz group, thereby extending the local $SO(3)$ to $SO(1,3)$. Since
the averaging over the local $S^{2}$ fiber is uniquely defined by $SO(3)$
invariance for irreducible fields, it should respect the local Lorentz
invariance automatically.

Another possible strategy would be to perform an analytic continuation in the
$y^{0}y^{4}$ plane, so that the actions on two different coordinates can be
analytically continued into each others in the flat limit. Moreover for
irreducible (divergence-free) tensor fields, there is a natural map from the
FLRW background to the $H^{4}$ background. This could also provide some sort
of Wick rotation, relating these two backgrounds with different signature via
the embedding in $\mathbb{R}^{1,4}$. However, a thorough treatment of this
issue is left for future work.

##  7 Amplitudes of the Yang-Mills massless sector in Euclidean signature

In this section, we start with the reduced action of the HS-IKKT matrix model
on $H^{4}$ in the flat limit elaborated in Section 6 (see also Appendix B),
and study tree-level scattering amplitudes of the higher-spin modes

$\displaystyle\mathbb{A}^{\pm}_{\alpha(2s-1)\,\dot{\alpha}}=A_{\alpha(2s-1)\,\dot{\alpha}}\pm\mathrm{i}\,\tilde{\mathsf{A}}_{\alpha(2s-1)\,\dot{\alpha}}\,.$
(7.1)

Of course, fields do not propagate on a manifold with Euclidean signature,
instead they “decay” with the distance. Nevertheless, it is possible to study
scattering amplitudes by analytic continuation of the real kinematics into the
complex domain. This will allow us to compute scattering amplitudes using
well-known recursion techniques as in [45]. Note that all higher-spin fields
with $s>1$ thereby become complex-valued.

We recall from section 5 that the bosonic higher-spin gauge fields carry
$3(2s+1)$ degrees of freedom in Euclidean signature, and $2(2s+1)$ propagating
dof on the physical FLRW space-time. Since these are more degrees of freedom
compared to the usual cases of massless with 2 and massive with $2s+1$ degrees
of freedom in 4 dimension, there will be more higher-spin modes to consider
when computing scattering amplitudes. For simplicity, we will only consider
the massless sector in this section which contains the fields that satisfy the
Lorenz gauge condition
$\partial^{\alpha\dot{\alpha}}\mathbb{A}_{\beta(2s-2)\alpha\,\dot{\alpha}}=0$.
Our notation for an $n$-point scattering amplitude is then
$\mathcal{M}_{n}(1_{s_{1}}^{h_{1}},\ldots,n_{s_{n}}^{h_{n}})$, where
$h_{i}=\pm$ indicates whether the $i$th particle of spin-$s_{i}$ has positive
or negative helicity.

To determine helicity of the external states, we note that the kinetic action
can be written in the form [36]:

$S_{2}=\,\int_{\mathbb{M}}\mathrm{d}^{4}\mathtt{x}\,\mathbb{A}^{\alpha(2s-1)\,\dot{\alpha}}_{-}\Box\mathbb{A}^{+}_{\alpha(2s-1)\,\dot{\alpha}}\,,\qquad
s\geq 1\,,$ (7.2)

for the appropriately chosen integration domain $\mathbb{M}$, in terms of the
complex fields (7.1), where $\pm$ denotes positive/negative helicity fields.
Then by imposing the Lorenz gauge

$\partial^{\gamma\dot{\alpha}}\mathbb{A}_{\alpha(2s-2)\gamma\,\dot{\alpha}}=0\,.$
(7.3)

on the base manifold $\mathbb{M}$, we can select _only_ massless modes out of
$3(2s+1)$ components of the $\mathbb{A}_{\pm}^{\alpha(2s-1)\,\dot{\alpha}}$
fields. Suppose
$k^{\alpha\dot{\alpha}}=\kappa^{\alpha}\tilde{\kappa}^{\dot{\alpha}}$ is an
on-shell complex 4-momentum. We define positive and negative helicity
polarization tensors associated to external higher-spin states as [36]:

$\displaystyle\epsilon^{+}_{\alpha(2s-1)\,\dot{\alpha}}=\frac{\zeta_{\alpha(2s-1)}\tilde{\kappa}_{\dot{\alpha}}}{\langle\kappa\,\zeta\rangle^{2s-1}}\,,\qquad\epsilon^{-}_{\alpha(2s-1)\,\dot{\alpha}}$
$\displaystyle=\frac{\kappa_{\alpha(2s-1)}\,\tilde{\zeta}_{\dot{\alpha}}}{[\tilde{\kappa}\,\tilde{\zeta}]}\,,$
(7.4)

where $\zeta_{\alpha},\tilde{\zeta}_{\dot{\alpha}}$ are constant/reference
spinors. The above representatives for polarization tensors are chosen such
that they obey the normalization [46]

$\epsilon^{+}_{\alpha(2s-1)\,\dot{\alpha}}\epsilon_{-}^{\alpha(2s-1)\,\dot{\alpha}}=-1\,.$
(7.5)

With the choice of polarization tensors in (7.4), it can be checked that

$\displaystyle\partial_{\alpha}{}^{\dot{\gamma}}\mathbb{A}^{+}_{\alpha(2s-1)\,\dot{\gamma}}$
$\displaystyle=0\,,$ (7.6a)
$\displaystyle\partial^{\beta\dot{\alpha}}\partial_{(\beta}{}^{\dot{\gamma}}\mathbb{A}^{-}_{\alpha(2s-1))\,\dot{\gamma}}$
$\displaystyle=0\,.$ (7.6b)

The propagator between positive and negative helicity fields in the Lorenz
gauge (7.3) is

$\displaystyle\langle\mathbb{A}^{+}_{\alpha(2s-1)\,\dot{\alpha}}(p)\mathbb{A}_{-}^{\beta(2s^{\prime}-1)\,\dot{\beta}}(p^{\prime})\rangle=\delta^{4}(p+p^{\prime})\tilde{\delta}(s-s^{\prime})\frac{\delta_{(\alpha_{1}}{}^{(\beta_{1}}\ldots\delta_{\alpha_{2s-1})}{}^{\beta_{2s^{\prime}-1})}\delta_{\dot{\alpha}}{}^{\dot{\beta}}}{p^{2}}\,,$
(7.7)

where $\tilde{\delta}$ is a Kronecker delta:

$\tilde{\delta}(x)=\begin{cases}0\,,\qquad x\neq 0\,,\\\ 1\,,\qquad
x=0\,.\end{cases}$ (7.8)

Note that since the standard linearized gauge transformation
$\delta\mathbb{A}_{\alpha(2s-1)\,\dot{\alpha}}=\partial_{\alpha\dot{\alpha}}\xi_{\alpha(2s-2)}$
is not a symmetry of HS-IKKT theory a priori,181818Recall that all higher-spin
fields in HS-IKKT are “would-be massive” fields with more degrees of freedom
than the massless ones. there is no restriction on the positive helicity of a
massless field compared to the cases studied in [36, 47]. To this end, we
recall that since $\mathbb{A}_{\alpha(2s-1)\,\dot{\alpha}}$ has more un-dotted
than dotted spinorial indices, they belong to what is so-called chiral
representation (see footnote 14).

###  7.1 Tree-level amplitudes

Since we are working with complex-valued fields and complex kinematics, the
on-shell tree-level 3-point amplitudes are _not_ vanishing a priori. They will
act as seeds to construct higher-point tree-level $S$-matrices.

#### 3-point amplitudes.

There are eight possible helicity configurations:

$\displaystyle(+,+,+)\,,\ (-,+,+)\,,\ (+,-,+)\,,\ (+,+,-)\,,\ (-,-,+)\,,\
(-,+,-)\,,\ (+,-,-)\,,\ (-,-,-)\,,$

at 3-points, where we recall that the positions of fields in the cubic
vertices are important. Since we work with complex kinematics where
$\tilde{\kappa}^{\dot{\alpha}}$ is not the complex conjugate of
$\kappa^{\alpha}$, the momentum conservation implies:191919It is important to
recall that $k_{i}$ is dimensionless since we work with dimensionless spinors.

$\displaystyle\sum_{i=1}^{3}k_{i}=\sum_{i=1}^{3}\kappa_{i}^{\alpha}\tilde{\kappa}_{i}^{\dot{\alpha}}=0\qquad\Leftrightarrow\qquad\langle
i\,j\rangle=0\quad\text{or}\quad[i\,j]=0\,,\quad\forall\ i,j=1,2,3\,.$ (7.9)

As a result, there can be non-vanishing 3-point amplitudes with complex
kinematics whose forms must be written explicitly only in terms of angled or
square brackets (cf., [48, 49, 50]). This analytic continuation between real
momentum and complex kinematics is significant for constructing higher-
multiplicity scattering amplitudes from 3-point building blocks [45].

The tree-level 3-point amplitudes of the Yang-Mills sector are given by
substituting the polarization tensors (7.4) to the cubic interaction:

$\tilde{\delta}(2-(s_{2}+s_{3}-s_{1}))\int_{\mathbb{M}}\mathrm{d}^{4}\mathtt{x}\,\left(\partial_{\alpha}{}^{\dot{\gamma}}\mathbb{A}_{\alpha(2s_{1}-1)\,\dot{\gamma}}\,\partial_{\alpha\dot{\sigma}}\mathbb{A}^{\alpha(2s_{2}-1)\,\dot{\beta}}\partial_{\alpha}{}^{\dot{\sigma}}\mathbb{A}^{\alpha(2s_{3}-1)}{}_{\dot{\beta}}\right)\,.$
(7.10)

It is a simple computation to show that both
$\mathcal{M}_{3}(1^{+},2^{+},3^{+})$ and $\mathcal{M}_{3}(1^{-},2^{-},3^{-})$
vanish. Therefore, we can concentrate on the six others helicity
configurations.

Notice that $\mathcal{M}_{3}(1^{+},2^{h_{2}},3^{h_{3}})$ vanishes on-shell,
which leaves us with only three possible non-vanishing contributions at cubic
order. Namely,

$\displaystyle(-,+,+)\,,\quad(-,-,+)\,,\quad(-,+,-)\,.$ (7.11)

These are the vertices that feature minimal couplings – the couplings with
lowest number of derivatives given a triplet of external spins
$(s_{1},s_{2},s_{3})$.

Upon substituting (7.4) to the cubic vertex (7.10), we obtain the $(-,+,+)$ or
$\overline{\mbox{MHV}}_{3}$ scattering amplitude as:

$\mathcal{M}_{3}(1_{s_{1}}^{-},2_{s_{2}}^{+},3_{s_{3}}^{+})=\tilde{\delta}(s_{1}-s_{2}-s_{3}+2)\,\frac{[2\,3]^{2}\,\langle\zeta_{2}\,1\rangle^{2s_{2}-2}\,\langle\zeta_{3}\,1\rangle^{2s_{3}-2}}{\langle\zeta_{2}\,2\rangle^{2s_{2}-2}\,\langle\zeta_{3}\,3\rangle^{2s_{3}-2}}\,,$
(7.12)

where the overall momentum conserving delta function has been suppressed, and
we ignored the overall factor of $\mathrm{i}$. The following useful relation:

$\langle\zeta_{2}\,1\rangle\,[1\,3]+\langle\zeta_{2}\,2\rangle\,[2\,3]=0\,,\qquad\langle\zeta_{3}\,1\rangle\,[1\,2]+\langle\zeta_{3}\,3\rangle\,[3\,2]=0\,,$
(7.13)

can be obtained on the support of momentum conservation. Then taking advantage
of (7.13), we arrive at the following result for the
$\overline{\mbox{MHV}}_{3}$ amplitude:

$\mathcal{M}_{3}(1_{s_{1}}^{-},2_{s_{2}}^{+},3_{s_{3}}^{+})=\tilde{\delta}(s_{1}-s_{2}-s_{3}+2)\,\frac{[2\,3]^{2s_{2}+2s_{3}-2}}{[1\,2]^{2s_{3}-2}\,[3\,1]^{2s_{2}-2}}\,.$
(7.14)

This is in agreement with the 3-pt amplitudes of the self-dual higher-spin
gravity [35, 10]. Next, we find the following MHV3 amplitudes:

$\mathcal{M}_{3}(1_{s_{1}}^{-},2_{s_{2}}^{-},3_{s_{3}}^{+})=\tilde{\delta}(s_{1}-s_{2}-s_{3}+2)\,\frac{1}{2}\,\frac{\langle
1\,2\rangle^{2(s_{2}+s_{3})-2}[3\,1]}{\langle 2\,3\rangle^{2s_{3}-2}\langle
3\,1\rangle}+(1\leftrightarrow 2)\,,$ (7.15)

where we have symmetrized the positions of two negative helicity external
fields. The result of
$\mathcal{M}_{3}(1^{-}_{s_{1}},2^{+}_{s_{2}},3^{-}_{s_{3}})$ is similar where
we simply swap $(2\leftrightarrow 3)$.

Observe that while all constant spinors have dropped out of the final MHV3
amplitudes, due to the appearance of both angled and square brackets in
(7.15), it is obvious that the above MHV3 amplitude vanishes even with complex
kinematic. In fact, we can directly verify this statement by writing

$\displaystyle\frac{[3\,1]}{\langle
3\,1\rangle}\sim\frac{(k_{1}+k_{3})^{2}}{\langle 3\,1\rangle^{2}}=0$ (7.16)

by virtue of (7.9) and momentum conservation. In the next subsection, by
projecting the MHV3 amplitudes to light-cone gauge, we argue that they are
spurious and can be removed by a local field redefinition. This is the general
feature of any (higher-spin) gauge theories described by the chiral
representation whose interactions have two or higher number of derivatives
[47]. It would be interesting to investigate the gauge-matter and matter-
matter sectors in the HS-IKKT matrix model to see if this pattern persists. We
leave this investigation for future work.

#### 4-point amplitudes.

Besides contributions from the exchanges, we also have potential contributions
from the contact interaction:

$\tilde{\delta}(s_{1}+s_{2}-s_{3}-s_{4})\,\int_{\mathbb{M}}\mathrm{d}^{4}\mathtt{x}\,\Big{(}\partial_{\alpha\dot{\gamma}_{1}}\mathbb{A}^{\alpha(2s_{1}-1)\,\dot{\sigma}}\partial_{\alpha}{}^{\dot{\gamma}_{1}}\mathbb{A}^{\alpha(2s_{2}-1)}{}_{\dot{\sigma}}\Big{)}\Big{(}\partial^{\alpha}{}_{\dot{\gamma}_{2}}\mathbb{A}_{\alpha(2s_{3}-1)}{}^{\dot{\tau}}\partial^{\alpha\dot{\gamma}_{2}}\mathbb{A}_{\alpha(2s_{4}-1)\,\dot{\tau}}\Big{)}\,$
(7.17)

when considering 4-point scattering amplitudes. Below we consider different
cases of the 4-point scattering amplitudes between massless modes in the Yang-
Mills sector of HS-IKKT theory.

Case 1: Consider the 4-point amplitudes with helicity configuration
$(+,+,+,+)$. The exchange channels of this amplitude are obtained by gluing
$(+,+,+)$ and $(-,+,+)$ vertices together. Since
$\mathcal{M}_{3}(1_{s_{1}}^{+},2_{s_{2}}^{+},2_{s_{1}}^{+})=0$, one can show
that all
$\mathcal{M}_{4}(1_{s_{1}}^{+},2_{s_{2}}^{+},3_{s_{3}}^{+},4_{s_{4}}^{+})$-related
exchange channels in the Yang-Mills sector of HS-IKKT theory are zero. By
making suitable choices for the reference spinors, the contributions from the
contact terms also vanish. As a result, the 4-point amplitude
$\mathcal{M}_{4}(1^{+},2^{+},3^{+},4^{+})=0$.

Case 2: Next, consider the 4-point $\mathcal{M}_{4}(1^{-},2^{+},3^{+},4^{+})$
amplitude:

$\mathcal{M}_{4}(1_{s_{1}}^{-},2_{s_{2}}^{+},3_{s_{3}}^{+},4_{s_{4}}^{+})=\mathcal{A}_{4}^{s}+\mathcal{A}_{4}^{t}+\mathcal{A}_{4}^{u}+\mathcal{A}_{4}^{\mathrm{cont}}\,,$

where
$\mathcal{A}_{4}^{\bullet}=\\{\mathcal{A}_{4}^{s}\,,\mathcal{A}^{t}_{4}\,,\mathcal{A}^{u}_{4}\,,\mathcal{A}^{\mathrm{cont}}\\}$
is the set of contributions coming from the $s,t,u$-channel exchanges and the
contact interaction. Let us keep the spins arbitrary for now, and denote the
spin of the exchange as $\omega$. It is a simple computation to show that:

$\displaystyle\mathcal{A}_{4}^{s}=(-)^{\Lambda_{4}}\tilde{\delta}(4-s_{2}-s_{3}-s_{4}+s_{1})\frac{[12]^{-s_{1}+s_{2}+\omega}[34]^{s_{3}+s_{4}-\omega}}{(k_{1}+k_{2})^{2}}f(\zeta_{2},\zeta_{3},\zeta_{4})\,,$
(7.18)

where $\Lambda_{4}=-s_{1}+s_{2}+s_{3}+s_{4}$, and

$\displaystyle
f(\zeta_{2},\zeta_{3},\zeta_{4})=\Big{(}\frac{\langle\zeta_{2}\,1\rangle}{\langle\zeta_{2}\,2\rangle}\Big{)}^{s_{2}}\Big{(}\frac{\langle\zeta_{4}\,1\rangle}{\langle\zeta_{4}\,2\rangle}\Big{)}^{s_{1}}\Big{(}\frac{\langle\zeta_{3}\,4\rangle}{\langle\zeta_{3}\,3\rangle}\Big{)}^{s_{3}}\Big{(}\frac{\langle\zeta_{4}\,3\rangle}{\langle\zeta_{4}\,4\rangle}\Big{)}^{s_{4}}\Big{(}\frac{\langle\zeta_{3}\,1\rangle\langle\zeta_{4}\,2\rangle}{\langle\zeta_{3}\,3\rangle\langle\zeta_{4}\,4\rangle}\Big{)}^{\omega}\,$
(7.19)

is a rational function $f$ whose homogeneity in reference spinors of the
positive helicity external particles is zero.

Using residue gauge freedom to set
$\zeta_{2}^{\alpha}=\zeta_{3}^{\alpha}=\zeta_{4}^{\alpha}=\kappa_{1}^{\alpha}$,
it follows that $f(\zeta_{2},\zeta_{3},\zeta_{4})=0$. Thus,
$\mathcal{A}_{4}^{s}=0$. Similarly, we also obtain $\mathcal{A}^{t}_{4}=0$ and
$\mathcal{A}^{u}_{4}=0$. Lastly, the contribution of contact interaction reads

$\tilde{\delta}(s_{1}+s_{2}-s_{3}-s_{4})\,\frac{[1\,2][\tilde{\zeta}\,2][3\,4]^{2}\langle
1\,\zeta_{3}\rangle^{2s_{3}-2s_{2}}\langle
1\,\zeta_{4}\rangle^{2s_{4}-2}\langle\zeta_{2}\,\zeta_{3}\rangle^{2s_{2}-2}}{[1\,\tilde{\zeta}_{1}]\langle
2\,\zeta_{2}\rangle^{2s_{2}-2}\langle 3\,\zeta_{3}\rangle^{2s_{3}-2}\langle
4\,\zeta_{4}\rangle^{2s_{4}-2}}-\,(2\leftrightarrow 3)\,.$ (7.20)

Since we have chosen $\zeta_{i}=\kappa_{1}$ (for $i\neq 1$), the contributions
from the contact terms also vanish in this case. Therefore, the final result
is
$\mathcal{M}_{4}(1_{s_{1}}^{-},2_{s_{2}}^{+},3_{s_{3}}^{+},4_{s_{4}}^{+})=0$.
The vanishing of this amplitude reflects the deep connection between chiral
field representations and self-dual theories such as self-dual Yang-Mills,
self-dual gravity, or self-dual/chiral higher-spin theories [51, 52, 53, 54,
55, 56, 57]. Namely, tree-level amplitudes compose of $(-,+,+)$ vertices with
all but one positive (or negative) helicity vanish for any number of external
legs greater than or equal to three.

Case 3: Next, we consider the 4-point MHV amplitude
$\mathcal{M}_{4}(1^{-}_{s_{1}},2^{-}_{s_{2}},3^{+}_{s_{3}},4^{+}_{s_{4}})$. We
shall fix

$\begin{split}\zeta_{3}^{\alpha}=\zeta_{4}^{\alpha}=\kappa_{1}^{\alpha}\,,\\\
\tilde{\zeta}_{1}^{\dot{\alpha}}=\tilde{\zeta}_{2}^{\dot{\alpha}}=\tilde{\kappa}_{4}^{\dot{\alpha}}\,,\end{split}$
(7.21)

to simplify the computation. The $s$-channel contribution is given
by:202020Once again, we ignore overall factor.

$\mathcal{A}_{4}^{s}=\frac{\tilde{\delta}(s_{1}-s_{2})\tilde{\delta}(4-s_{3}-s_{4})}{2}\,\frac{\langle
1\,2\rangle^{2s_{1}-2}\,\langle\zeta_{3}\,1\rangle^{2}\,\langle\zeta_{3}\,2\rangle^{2}\,[3\,4]^{2}}{\langle
3\,\zeta_{3}\rangle^{2s_{3}-2}\,\langle
4\,\zeta_{3}\rangle^{2s_{4}-2}}\,+\>(1\leftrightarrow 2)\,.$ (7.22)

Here the spin constraints fix the two negative helicity particles to have
identical spin, while the spins of the two positive helicity particles have to
sum up to four. Since $\zeta_{3}^{\alpha}=\kappa_{1}^{\alpha}$, the
$s$-channel vanishes. Similar computations also lead to the vanishing of the
$t$\- and $u$\- channels, as well as the contributions coming from the contact
terms. As a result, the 4-point MHV amplitude is

$\mathcal{M}_{4}(1^{-}_{s_{1}},2^{-}_{s_{2}},3^{+}_{s_{3}},4^{+}_{s_{4}})=0\,.$
(7.23)

We find similar results in the case where the negative helicity particles are
not consecutive, e.g.
$\mathcal{M}_{4}(1^{-}_{s_{1}},2^{+}_{s_{2}},3^{-}_{s_{3}},4^{+}_{s_{4}})=0$.

#### n-point amplitudes.

As a consequence of the above considerations, we conclude that all $n$-point
tree-level amplitudes of the Yang-Mills sector in HS-IKKT model vanish, since
they cannot be constructed from lower point amplitudes.

###  7.2 HS-IKKT vertices in the light-cone gauge

The fact that (7.23) is vanishing for generic higher-spin fields with $s>1$
is, in fact, not a surprise. As observed in [47], what determines the
existence of non-trivial higher-spin tree-level scattering is not spin but
rather the number of transverse derivatives in the cubic interactions of type
$\mathcal{V}_{3}^{-\pm+}$. In particular, if the number of transverse
derivatives is greater than or equal to two, non-trivial tree-level scattering
amplitudes would be very unlikely to exist. To support this statement, let us
project the cubic interactions (7.10) to the light-cone gauge in momentum
space using the dictionary in [58, 59]. In particular, the map between spinors
and momenta in the light-cone gauge are:

$\displaystyle
i]=2^{1/4}\binom{\bar{\boldsymbol{k}}_{i}\,\beta_{i}^{-1/2}}{-\beta_{i}^{1/2}}\,,\qquad
i\rangle=2^{1/4}\binom{\boldsymbol{k}_{i}\,\beta_{i}^{-1/2}}{-\beta_{i}^{1/2}}\,,$
(7.24)

where $\boldsymbol{k}_{i}^{+}\equiv\beta_{i}$ and

$\displaystyle
k_{i}^{\alpha\dot{\alpha}}=\begin{pmatrix}\beta_{i}&\bar{\boldsymbol{k}}_{i}\\\
\boldsymbol{k}_{i}&\boldsymbol{k}_{i}^{-}\end{pmatrix}\,.$ (7.25)

Note that $\bar{\boldsymbol{k}}_{i}$ and $\boldsymbol{k}_{i}$ are referred to
as _transverse_ derivatives. Using the above, we can express the square and
angle brackets as

$\displaystyle[i\,j]=\sqrt{\frac{2}{\beta_{i}\beta_{j}}}\overline{\mathbb{P}}_{ij}\,,\qquad\langle
i\,j\rangle=\sqrt{\frac{2}{\beta_{i}\beta_{j}}}\mathbb{P}_{ij}\,$ (7.26)

for
$\overline{\mathbb{P}}_{ij}=\bar{\boldsymbol{k}}_{i}\beta_{j}-\bar{\boldsymbol{k}}_{j}\beta_{i}$
and $\mathbb{P}_{ij}=\boldsymbol{k}_{i}\beta_{j}-\boldsymbol{k}_{j}\beta_{i}$.
By virtue of momentum conservation, one can show that

$\displaystyle\overline{\mathbb{P}}_{12}=\overline{\mathbb{P}}_{23}=\overline{\mathbb{P}}_{31}=\overline{\mathbb{P}}=\frac{1}{3}\left[(\beta_{1}-\beta_{2})\bar{\boldsymbol{k}}_{3}+(\beta_{2}-\beta_{3})\bar{\boldsymbol{k}}_{1}+(\beta_{3}-\beta_{1})\bar{\boldsymbol{k}}_{2}\right]\,$
(7.27)

at the level of 3-point amplitudes.212121Note that this effect can also be
achieved in the spinor formalism if we use the parametrization (3.21). Thus,
in terms of these new variables, the cubic vertices (7.10) reduce to

$\displaystyle V_{3}^{\text{HS-
IKKT}}=(x\,\overline{\mathbb{P}}+y\,\mathbb{P}^{2s_{2}-1})\overline{\mathbb{P}}\,,$
(7.28)

schematically, and $x,y$ are coefficients in terms of $\beta_{i}$ variables.
According to light-cone recipe (see e.g. [51, 52, 60, 53]), one can always
make a local field redefinition at cubic order if there is a combination of
type $\mathbb{P}\overline{\mathbb{P}}$ to remove unnecessary data from the
interactions. This goes hand in hand with the fact that there must be only one
type of bracket in the final form of cubic amplitudes (cf., (7.9)): either
angled or square bracket, but not both of them at the same time. As a result,
the cubic interaction of HS-IKKT theory reduces further to

$\displaystyle V_{3}^{\text{HS-IKKT}}\rightsquigarrow
x\,\overline{\mathbb{P}}^{2}\simeq V_{3}^{\text{self-dual HSGR}}$ (7.29)

Thus, HS-IKKT theory in the flat limit is ‘secretly’ a supersymmetric chiral
higher-spin theory with two-derivative interactions (see e.g. [55, 56,
57]).222222 We choose the word _chiral_ instead of self-dual here because the
HS-IKKT theory has scalar fields in the spectrum.

##  8 Discussion

In this work, we established the connections between the almost-commutative
4-hyperboloid and non-compact semi-classical twistor space
$\mathbb{P}^{1,2}_{N}$ in the spinor formalism. Using this as a background in
the IKKT model, we studied the resulting higher-spin gauge theory for two
spaces with distinct signatures: $(i)$ a Euclidean 4-hyperboloid; $(ii)$ and a
FLRW-like cosmological spacetime. We also furnished a simple vectorial
description of HS-IKKT in the flat limit of these two cases, which simplifies
some of the technical steps in counting and organizing the degrees of freedom
of higher-spin modes in HS-IKKT in [8] and [31].

Armed with this setup, we then shown that all tree-level $n$-point on-shell
amplitudes (for $n\geq 4$) with an appropriate analytic continuation of the
massless sector within the Yang-Mills part of the Euclidean HS-IKKT theory
vanish in the flat limit. This result is expected, since HS-IKKT is a ‘parity-
violating’ higher-spin theory featuring two-derivative interactions when we
express it in terms of chiral representation.232323This representation gives
the lowest number of derivatives in the interactions, and unveils an
intriguing relation between standard No-go theorems [61, 62] in flat space and
the number of derivatives in the interaction. Indeed, as observed in [47],
massless higher-spin theories constructed from the chiral representation in
the flat space with the number of derivatives in the cubic vertices higher
than one must have trivial tree-level scattering amplitudes. Thus, the
massless sector of HS-IKKT is also in argreement with the result of [47]. This
can be explained by projecting the cubic vertices of the Yang-Mills sector in
the flat limit to the light-cone gauge, we observe that the MHV3 amplitudes
vanish so no other higher-point amplitudes can be formed using just the non-
vanishing $\overline{\text{MHV}}_{3}$ amplitudes.

This allows to identify the massless YM sector with self-dual higher-spin
gravity with 2-derivative interactions considered in [35]. In this sense, the
massless sector of HS-IKKT theory in the flat limit falls into the class of
(quasi-)chiral theories. We conclude that $S=1$ at tree-level, at least for
the modes under consideration. However, the triviality of the $S$-matrix at
tree level does not imply that the theory is trivial or not interesting;
rather, it should be viewed as a consistency check. It would be then
interesting to study the scattering amplitudes of HS-IKKT or any other higher-
spin theories at loop-level, see the discussions in e.g. [63, 36, 64].

The structure of the higher-spin gauge theory under consideration is quite
interesting. In contrast to more conventional attempts to formulate higher-
spin gauge theory using free differential algebra (see e.g. [65, 66, 67, 68])
and the chiral formulation (see e.g. [69, 70, 44, 71, 72, 38, 73]), the
framework of (HS-)IKKT matrix model leads naturally to a local action, albeit
at the expense of manifest local Lorentz invariance and a larger number of
propagating degrees of freedom compared to massless or massive higher-spin
gravities in 4-dimensional spacetime. This is due to the fact that the higher-
spin fields are not divergence-free as discussed242424Recall that in
Lorentzian signature we have $2(2s+1)$ dof. and in Euclidean signature we have
$3(2s+1)$ dof. for the would-be massive higher-spin fields of HS-IKKT theory.
in Section 5. The lack of local Lorentz invariance is also reflected in the
preferred frame, which encodes not only a metric but also a dilaton and an
axion. The Levi-Civita connection is accordingly replaced by the Weitzenböck
connection, whose torsion encodes the Riemannian curvature; see e.g. [29, 74,
75]. Nevertheless, the propagation of all modes is governed by a universal
effective metric, and the preferred frame naturally leads to a Cartan–type
framework to describe gravitational couplings in (HS-)IKKT model. While the
bare action is of Yang-Mills type, the Einstein-Hilbert action does arise at
one loop, under certain assumptions for the background [4, 3]. Non-Abelian
gauge theory would then arise on a stack of such background branes, coupled to
the effective metric. It is hence clear that the theory does contain
interesting physics, even if the local scattering of its abelian sector
vanishes at tree level.

From the physics point of view, it would be important to elaborate in more
detail the spinorial formulation on the FLRW background with Minkowski
signature. This leads to an unusual type of spinors adapted to the space-like
isometries rather than the local Lorentz invariance, which we have only
briefly touched upon. Due to the common origin from $\mathbb{PT}$, a close
relation with the present Euclidean computation is expected. Moreover, the
rather complicated organization of modes in [31, 13] is expected to simplify
in the spinorial formalism, which needs to be studied in more detail. This may
allow to compute amplitudes directly in Minkowski signature, while avoiding
complexification of the Euclidean case. This problem will be addressed in
future work.

## Acknowledgments

We appreciates useful discussions with Zhenya Skvortsov. TT is grateful to Tim
Adamo for useful discussion during the collaboration of [36]. This research
was partially completed at Corfu Summer Institute 2022 and the Humboldt Kolleg
on “Noncommutative and generalized geometry in string theory, gauge theory and
related physical models”. The work of TT is partially supported by the Fonds
de la Recherche Scientifique under Grants No. F.4503.20 (HighSpinSymm), No.
F.4544.21 (Higher- SpinGraWave), and the funding from the European Research
Council (ERC) under Grant No. 101002551. The work of HS is supported by the
Austrian Science Fund (FWF) grant P32086.

##  Appendix A On the shift symmetry in the flat limit

We verify that the $\vartheta$ transformation (6.23) is _not_ a symmetry of
HS-IKKT model in the flat limit. Recall that

$\displaystyle\mathsf{a}_{\alpha\dot{\alpha}}=\sum_{s=0}^{\infty}=\lambda^{\beta(s)}\hat{\lambda}^{\beta(s)}\mathcal{A}_{\beta(2s)|\alpha\dot{\alpha}}=\sum_{s=0}^{\infty}\lambda^{\beta(s)}\hat{\lambda}^{\beta(s)}\Big{[}A_{(\beta(2s)\alpha)\,\dot{\alpha}}+\epsilon_{\alpha\beta}\tilde{\mathsf{A}}_{\beta(2s-1)\,\dot{\alpha}}\Big{]}\,,$
(A.1)

and

$\displaystyle\delta_{\vartheta}\mathcal{A}_{\beta(2s)|\alpha\dot{\alpha}}=\epsilon_{\alpha\beta}\,\vartheta_{\beta(2s-1)\,\dot{\alpha}}\,.$
(A.2)

Using the above, the linearized action in the flat limit transforms as