$\begin{split}\delta_{\vartheta}S_{2}=&2\sum_{s_{i}}\int\mho\lambda^{\beta(s_{1})}\hat{\lambda}^{\beta(s_{1})}\lambda^{\tau(s_{2})}\hat{\lambda}^{\tau(s_{2})}\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathcal{A}_{\beta(2s_{1})}{}^{\alpha\,\dot{\gamma}}\\}\\{\mathtt{y}_{\alpha\dot{\sigma}},\epsilon_{\alpha\tau}\vartheta_{\tau(2s_{2}-1)}{}^{\dot{\sigma}}\\}\,,\end{split}$
(A.3)

where we note that the term
$\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathtt{y}^{\alpha\dot{\gamma}}\\}\\{\mathsf{a}_{\alpha\dot{\sigma}},\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\\}$
vanishes in the flat limit using spinorial description of HS-IKKT theory (see
Appendix B) and therefore can be neglected henceforth. The above can be
reduced further to

$\begin{split}\delta_{\vartheta}S_{2}=&\sum_{s_{i}}\int\mho\lambda^{\beta(s_{1})}\hat{\lambda}^{\beta(s_{1})}\lambda^{\tau(s_{2})}\hat{\lambda}^{\tau(s_{2})}\\{\mathtt{y}_{\tau\dot{\gamma}},\mathcal{A}_{\beta(2s_{1})}{}^{\alpha\,\dot{\gamma}}\\}\\{\mathtt{y}_{\alpha\dot{\sigma}},\vartheta_{\tau(2s_{2}-1)}{}^{\dot{\sigma}}\\}\\\
&+\sum_{s_{i}}\int\mho\lambda^{\beta(s_{1})}\hat{\lambda}^{\beta(s_{1})}\lambda^{\tau(s_{2})}\hat{\lambda}^{\tau(s_{2})}\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathcal{A}_{\beta(2s_{1})\tau}{}^{\dot{\gamma}}\\}\\{\mathtt{y}_{\alpha\dot{\sigma}},\vartheta_{\tau(2s_{2}-1)}{}^{\dot{\sigma}}\\}\,.\end{split}$
(A.4)

The first term denoted as I in (A.4) can be written as

$\begin{split}\leavevmode\hbox to13.61pt{\vbox
to13.61pt{\pgfpicture\makeatletter\hbox{\hskip 6.80302pt\lower-6.80302pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.60303pt}{0.0pt}\pgfsys@curveto{6.60303pt}{3.64679pt}{3.64679pt}{6.60303pt}{0.0pt}{6.60303pt}\pgfsys@curveto{-3.64679pt}{6.60303pt}{-6.60303pt}{3.64679pt}{-6.60303pt}{0.0pt}\pgfsys@curveto{-6.60303pt}{-3.64679pt}{-3.64679pt}{-6.60303pt}{0.0pt}{-6.60303pt}\pgfsys@curveto{3.64679pt}{-6.60303pt}{6.60303pt}{-3.64679pt}{6.60303pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.80556pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{I}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=&-\sum_{s_{i}}\int\mho\lambda^{\beta(s_{1})}\hat{\lambda}^{\beta(s_{1})}\lambda^{\tau(s_{2})}\hat{\lambda}^{\tau(s_{2})}\,\vartheta_{\tau(2s_{2}-1)}{}^{\dot{\sigma}}\\{\mathtt{y}_{\alpha\dot{\sigma}},\\{\mathtt{y}_{\tau\dot{\gamma}},\mathcal{A}_{\beta(2s_{1})}{}^{\alpha\,\dot{\gamma}}\\}\\}\,.\end{split}$
(A.5)

after integrating by part. Since

$\displaystyle\\{\mathtt{y}_{\tau\dot{\gamma}},\mathcal{A}_{\beta(2s_{1})}{}^{\alpha\dot{\gamma}}\\}$
$\displaystyle\simeq(\lambda_{\tau}\hat{\lambda}^{\delta}+\lambda^{\delta}\hat{\lambda}_{\tau})\partial_{\delta\dot{\gamma}}\mathcal{A}_{\beta(2s_{1})}{}^{\alpha\dot{\gamma}}\,,$
(A.6)

we get

$\begin{split}\leavevmode\hbox to13.61pt{\vbox
to13.61pt{\pgfpicture\makeatletter\hbox{\hskip 6.80302pt\lower-6.80302pt\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{6.60303pt}{0.0pt}\pgfsys@curveto{6.60303pt}{3.64679pt}{3.64679pt}{6.60303pt}{0.0pt}{6.60303pt}\pgfsys@curveto{-3.64679pt}{6.60303pt}{-6.60303pt}{3.64679pt}{-6.60303pt}{0.0pt}\pgfsys@curveto{-6.60303pt}{-3.64679pt}{-3.64679pt}{-6.60303pt}{0.0pt}{-6.60303pt}\pgfsys@curveto{3.64679pt}{-6.60303pt}{6.60303pt}{-3.64679pt}{6.60303pt}{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}{-1.80556pt}{-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{{I}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&\simeq\int\langle\lambda\,\hat{\lambda}\rangle\lambda^{\beta(s_{1})}\hat{\lambda}^{\beta(s_{1})}\lambda^{\tau(s_{2})}\hat{\lambda}^{\tau(s_{2}-1)}\hat{\lambda}^{\delta}[{\color[rgb]{0.7,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0,0}\lambda_{\alpha}\hat{\lambda}^{\zeta}}+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\hat{\lambda}_{\alpha}\lambda^{\zeta}}]\,\vartheta_{\tau(2s_{2}-1)}{}^{\dot{\sigma}}\partial_{\zeta\dot{\sigma}}\partial_{\delta\dot{\gamma}}\mathcal{A}_{\beta(2s_{1})}{}^{\alpha\,\dot{\gamma}}\\\
&-\int\langle\lambda\,\hat{\lambda}\rangle\lambda^{\beta(s_{1})}\hat{\lambda}^{\beta(s_{1})}\lambda^{\tau(s_{2}-1)}\hat{\lambda}^{\tau(s_{2})}\lambda^{\delta}[{\color[rgb]{0.7,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0,0}\lambda_{\alpha}\hat{\lambda}^{\zeta}}+{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\hat{\lambda}_{\alpha}\lambda^{\zeta}}]\,\vartheta_{\tau(2s_{2}-1)}{}^{\dot{\sigma}}\partial_{\zeta\dot{\sigma}}\partial_{\delta\dot{\gamma}}\mathcal{A}_{\beta(2s_{1})}{}^{\alpha\,\dot{\gamma}}\end{split}$
(A.7)

Observe that by grouping the red and green terms in (A.7) together and using
the identity
$\lambda^{[\alpha}\hat{\lambda}^{\beta]}=-\frac{1}{2}\epsilon^{\alpha\beta}\langle\lambda\,\hat{\lambda}\rangle$,
we can simplify the above to

$\begin{split}\leavevmode\hbox to13.61pt{\vbox
to13.61pt{\pgfpicture\makeatletter\hbox{\hskip 6.80302pt\lower-6.80302pt\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{6.60303pt}{0.0pt}\pgfsys@curveto{6.60303pt}{3.64679pt}{3.64679pt}{6.60303pt}{0.0pt}{6.60303pt}\pgfsys@curveto{-3.64679pt}{6.60303pt}{-6.60303pt}{3.64679pt}{-6.60303pt}{0.0pt}\pgfsys@curveto{-6.60303pt}{-3.64679pt}{-3.64679pt}{-6.60303pt}{0.0pt}{-6.60303pt}\pgfsys@curveto{3.64679pt}{-6.60303pt}{6.60303pt}{-3.64679pt}{6.60303pt}{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}{-1.80556pt}{-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{{I}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\simeq&-\frac{1}{2}\int{\color[rgb]{0.7,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.7,0,0}\lambda_{\alpha}\hat{\lambda}^{\zeta}}\langle\lambda\,\hat{\lambda}\rangle^{2}\lambda^{\beta(s_{1})}\hat{\lambda}^{\beta(s_{1})}\lambda^{\tau(s_{2}-1)}\hat{\lambda}^{\tau(s_{2}-1)}\,\vartheta_{\tau(2s_{2}-2)\delta}{}^{\dot{\sigma}}\partial_{\zeta\dot{\sigma}}\partial^{\delta}{}_{\dot{\gamma}}\mathcal{A}_{\beta(2s_{1})}{}^{\alpha\,\dot{\gamma}}\\\
&+\frac{1}{2}\int{\color[rgb]{0,0.6,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.6,0}\hat{\lambda}_{\alpha}\lambda^{\zeta}}\langle\lambda\,\hat{\lambda}\rangle^{2}\lambda^{\beta(s_{1})}\hat{\lambda}^{\beta(s_{1})}\lambda^{\tau(s_{2}-1)}\hat{\lambda}^{\tau(s_{2}-1)}\,\vartheta_{\tau(2s_{2}-2)\delta}{}^{\dot{\sigma}}\partial_{\zeta\dot{\sigma}}\partial^{\delta}{}_{\dot{\gamma}}\mathcal{A}_{\beta(2s_{1})}{}^{\alpha\,\dot{\gamma}}\,,\end{split}$
(A.8)

which can be reduced one more time to

$\begin{split}\leavevmode\hbox to13.61pt{\vbox
to13.61pt{\pgfpicture\makeatletter\hbox{\hskip 6.80302pt\lower-6.80302pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.60303pt}{0.0pt}\pgfsys@curveto{6.60303pt}{3.64679pt}{3.64679pt}{6.60303pt}{0.0pt}{6.60303pt}\pgfsys@curveto{-3.64679pt}{6.60303pt}{-6.60303pt}{3.64679pt}{-6.60303pt}{0.0pt}\pgfsys@curveto{-6.60303pt}{-3.64679pt}{-3.64679pt}{-6.60303pt}{0.0pt}{-6.60303pt}\pgfsys@curveto{3.64679pt}{-6.60303pt}{6.60303pt}{-3.64679pt}{6.60303pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.80556pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{I}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\simeq\frac{1}{4}\int\langle\lambda\,\hat{\lambda}\rangle^{3}\lambda^{\beta(s_{1})}\hat{\lambda}^{\beta(s_{1})}\lambda^{\tau(s_{2}-1)}\hat{\lambda}^{\tau(s_{2}-1)}\,\vartheta_{\tau(2s_{2}-2)\delta}{}^{\dot{\sigma}}\partial_{\zeta\dot{\sigma}}\partial^{\delta}{}_{\dot{\gamma}}\mathcal{A}_{\beta(2s_{1})}{}^{\zeta\,\dot{\gamma}}\,.\end{split}$
(A.9)

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

$\displaystyle\partial_{\zeta\dot{\sigma}}\partial^{\delta}{}_{\dot{\gamma}}\mathcal{A}_{\beta(2s_{1})}{}^{\zeta\,\dot{\gamma}}=\partial_{\zeta\dot{\sigma}}\partial^{\delta}{}_{\dot{\gamma}}A_{\beta(2s_{1})}{}^{\zeta\,\dot{\gamma}}-\partial_{\beta\dot{\sigma}}\partial^{\delta}{}_{\dot{\gamma}}\tilde{\mathsf{A}}_{\beta(2s_{1}-1)}{}^{\dot{\gamma}}\,.$
(A.10)

The contribution associated with $A_{\beta(2s_{1})}{}^{\zeta\,\dot{\gamma}}$
vanishes since there are a pair of $\lambda^{\beta}\hat{\lambda}^{\beta}$ that
contract with $A_{\beta(2s_{1})}{}^{\zeta\,\dot{\gamma}}$. Therefore, after
integrating out fiber coordinates we obtain a trace in
$A_{\beta(2s_{1})}{}^{\zeta\,\dot{\gamma}}$ which is zero due to
tracelessness. This means that

$\displaystyle\leavevmode\hbox to13.61pt{\vbox
to13.61pt{\pgfpicture\makeatletter\hbox{\hskip 6.80302pt\lower-6.80302pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.60303pt}{0.0pt}\pgfsys@curveto{6.60303pt}{3.64679pt}{3.64679pt}{6.60303pt}{0.0pt}{6.60303pt}\pgfsys@curveto{-3.64679pt}{6.60303pt}{-6.60303pt}{3.64679pt}{-6.60303pt}{0.0pt}\pgfsys@curveto{-6.60303pt}{-3.64679pt}{-3.64679pt}{-6.60303pt}{0.0pt}{-6.60303pt}\pgfsys@curveto{3.64679pt}{-6.60303pt}{6.60303pt}{-3.64679pt}{6.60303pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.80556pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{I}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=-\frac{1}{4}\int\langle\lambda\,\hat{\lambda}\rangle^{3}\lambda^{\beta(s_{1})}\hat{\lambda}^{\beta(s_{1})}\lambda^{\tau(s_{2}-1)}\hat{\lambda}^{\tau(s_{2}-1)}\,\vartheta_{\tau(2s_{2}-2)\delta}{}^{\dot{\sigma}}\partial_{\beta\dot{\sigma}}\partial^{\delta}{}_{\dot{\gamma}}\tilde{\mathsf{A}}_{\beta(2s_{1}-1)}{}^{\dot{\gamma}}\,.$
(A.11)

Notice that $\leavevmode\hbox to13.61pt{\vbox
to13.61pt{\pgfpicture\makeatletter\hbox{\hskip 6.80302pt\lower-6.80302pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.60303pt}{0.0pt}\pgfsys@curveto{6.60303pt}{3.64679pt}{3.64679pt}{6.60303pt}{0.0pt}{6.60303pt}\pgfsys@curveto{-3.64679pt}{6.60303pt}{-6.60303pt}{3.64679pt}{-6.60303pt}{0.0pt}\pgfsys@curveto{-6.60303pt}{-3.64679pt}{-3.64679pt}{-6.60303pt}{0.0pt}{-6.60303pt}\pgfsys@curveto{3.64679pt}{-6.60303pt}{6.60303pt}{-3.64679pt}{6.60303pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.80556pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{I}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\neq 0$. So the only
way $\vartheta$ can be a symmetry of HS-IKKT theory is that I combines with
other contributions in (A.4) to give zero. We show below that it is _not_ the
case.

Consider the second line of (A.4) for which we denote as II. Using (3.24), we
get

$\begin{split}\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{{II}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&\simeq\int\mho\lambda^{\beta(s_{1})}\hat{\lambda}^{\beta(s_{1})}\lambda^{\tau(s_{2})}\hat{\lambda}^{\tau(s_{2})}\big{[}\lambda^{\alpha}\hat{\lambda}^{\delta}+\lambda^{\delta}\hat{\lambda}^{\alpha}\big{]}\big{[}\lambda_{\alpha}\hat{\lambda}^{\zeta}+\lambda^{\zeta}\hat{\lambda}_{\alpha}\big{]}\times\\\
&\times\partial_{\delta\dot{\gamma}}\mathcal{A}_{\beta(2s_{1})\tau}{}^{\dot{\gamma}}\partial_{\zeta\dot{\sigma}}\vartheta_{\tau(2s_{2}-1)}{}^{\dot{\sigma}}\\\
&\simeq\sum_{s_{i}}\int\mho\lambda^{\beta(s_{1})}\hat{\lambda}^{\beta(s_{1})}\lambda^{\tau(s_{2})}\hat{\lambda}^{\tau(s_{2})}\langle\lambda\,\hat{\lambda}\rangle\big{(}\hat{\lambda}^{\delta}\lambda^{\zeta}-\lambda^{\delta}\hat{\lambda}^{\zeta}\big{)}\partial_{\delta\dot{\gamma}}\mathcal{A}_{\beta(2s_{1})\tau}{}^{\dot{\gamma}}\partial_{\zeta\dot{\sigma}}\vartheta_{\tau(2s_{2}-1)}{}^{\dot{\sigma}}\,,\end{split}$
(A.12)

which can be simplifed further to

$\begin{split}\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{{II}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&\simeq\frac{1}{2}\sum_{s_{i}}\int\mho\lambda^{\beta(s_{1})}\hat{\lambda}^{\beta(s_{1})}\lambda^{\tau(s_{2})}\hat{\lambda}^{\tau(s_{2})}\langle\lambda\,\hat{\lambda}\rangle^{2}\epsilon^{\delta\zeta}\partial_{\delta\dot{\gamma}}\mathcal{A}_{\beta(2s_{1})\tau}{}^{\dot{\gamma}}\partial_{\zeta\dot{\sigma}}\vartheta_{\tau(2s_{2}-1)}{}^{\dot{\sigma}}\\\
&\simeq-\frac{1}{2}\sum_{s_{i}}\int\mho\lambda^{\beta(s_{1})}\hat{\lambda}^{\beta(s_{1})}\lambda^{\tau(s_{2})}\hat{\lambda}^{\tau(s_{2})}\langle\lambda\,\hat{\lambda}\rangle^{2}\mathcal{A}_{\beta(2s_{1})\tau\,\dot{\gamma}}\Box\vartheta_{\tau(2s_{2}-1)}{}^{\dot{\gamma}}\end{split}$
(A.13)

where we have made an integration by part and used the identity
$\partial_{\delta\dot{\gamma}}\partial^{\delta}{}_{\dot{\sigma}}=-\Box\epsilon_{\dot{\gamma}\dot{\sigma}}$.

Since one of the spinors $\lambda^{\tau}$ or $\hat{\lambda}^{\tau}$ is
contracted with $\mathcal{A}_{\alpha(2s_{1})\tau\,\dot{\gamma}}$, it results
in a trace of this coefficient when we integrate out fiber coordinates.
However, based on the decomposition (A.1), it is clear that we will obtain
$A^{\alpha}{}_{\alpha\beta(2s-2)\,\dot{\alpha}}$ which is traceless. The
remaining contribution from the second modes $\tilde{\mathsf{A}}$ gives

$\displaystyle\leavevmode\hbox to16.14pt{\vbox
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\simeq\int\mho\langle\lambda\,\hat{\lambda}\rangle^{2}\lambda^{\beta(s_{1})}\hat{\lambda}^{\beta(s_{1})}\lambda^{\tau(s_{2})}\hat{\lambda}^{\tau(s_{2})}\epsilon_{\beta\tau}\tilde{\mathsf{A}}_{\beta(2s_{1}-1)\,\dot{\gamma}}\Box\vartheta_{\tau(2s_{2}-1)}{}^{\dot{\gamma}}\,,$
(A.14)

which clearly does not vanish by itself. In addition, I and II will not cancel
each others. As such $\vartheta$ is not a symmetry of the HS-IKKT model in the
flat limit.252525Note that $\vartheta$ is nevertheless a symmetry of the self-
dual sector of HS-IKKT in the flat limit [10]. As a result, there are more
propagating degrees of freedom in HS-IKKT compared to massless higher-spin
fields in (A)dS4,5. Intriguingly, in the flat limit, the would-be massive
degrees of freedom modes can be mapped into a helicity basis. We can further
select all the massless modes by imposing the Lorenz gauge condition
$\partial^{\gamma\dot{\gamma}}\mathbb{A}_{\alpha(2s-2)\gamma\,\dot{\gamma}}=0$
as discussed in the main text.

##  Appendix B Averaging over the fibers

The explicit spacetime action of the Yang-Mills sector in the Euclidean case
can be obtained by avaraging over the $\mathbb{P}^{1}$ fiber using (6.31).
Consider the first term in (6.33) where

$\displaystyle
I_{1}=2\int_{\mathbb{P}^{1}}\mathtt{K}\,\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathsf{a}^{\alpha\dot{\gamma}}\\}\\{\mathtt{y}_{\alpha\dot{\sigma}},\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\\}\,,\qquad\mathtt{K}=\frac{\langle\lambda\,\mathrm{d}\lambda\rangle\wedge\langle\hat{\lambda}\,\mathrm{d}\hat{\lambda}\rangle}{\langle\lambda\,\hat{\lambda}\rangle^{2}}\,.$
(B.1)

In the flat limit, the main contribution from the Poisson brackets yield

$\displaystyle
I_{1}=2\sum_{s,s^{\prime}}\int_{\mathbb{P}^{1}}\mathtt{K}\,\mathcal{E}^{\alpha}{}_{\dot{\gamma},\beta\dot{\beta}}\partial^{\beta\dot{\beta}}\mathsf{a}^{\alpha\,\dot{\gamma}}\mathcal{E}_{\alpha\dot{\sigma},\delta\dot{\delta}}\partial^{\delta\dot{\delta}}\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\,.$
(B.2)

where $\mathcal{E}^{\alpha\dot{\alpha}\beta\dot{\beta}}\simeq
2\lambda^{(\alpha}\hat{\lambda}^{\beta)}\epsilon^{\dot{\alpha}\dot{\beta}}$.
The contraction between two effective spinor vielbein gives

$\displaystyle\mathcal{E}^{\alpha}{}_{\dot{\gamma},\beta\dot{\beta}}\mathcal{E}_{\alpha\dot{\sigma},\delta\dot{\delta}}=-2\langle\lambda\,\hat{\lambda}\rangle\lambda_{[\beta}\hat{\lambda}_{\delta]}\epsilon_{\dot{\gamma}\dot{\beta}}\epsilon_{\dot{\sigma}\dot{\delta}}=\epsilon_{\beta\delta}\epsilon_{\dot{\gamma}\dot{\beta}}\epsilon_{\dot{\sigma}\dot{\delta}}\,\langle\lambda\,\hat{\lambda}\rangle^{2}\,,$
(B.3)

where
$2\lambda_{[\alpha}\hat{\lambda}_{\beta]}=-\epsilon_{\alpha\beta}\langle\lambda\,\hat{\lambda}\rangle$.
We obtain

$\begin{split}I_{1}=&+2\int_{\mathbb{P}^{1}}\mathtt{K}\,\langle\lambda\,\hat{\lambda}\rangle^{2}\partial_{\circ\dot{\gamma}}\mathsf{a}^{\alpha\,\dot{\gamma}}\,\partial^{\circ}{}_{\dot{\sigma}}\mathsf{a}_{\alpha}{}^{\dot{\sigma}}-8\int_{\mathbb{P}^{1}}\mathtt{K}\,\lambda_{(\alpha_{2}}\hat{\lambda}_{\circ)}\lambda_{(\alpha_{1}}\hat{\lambda}_{\bullet)}\partial^{\circ}{}_{\dot{\gamma}}\mathsf{a}^{\alpha_{1}\dot{\gamma}}\partial^{\bullet}{}_{\dot{\sigma}}\mathsf{a}^{\alpha_{2}\dot{\sigma}}\,.\end{split}$
(B.4)

Observe that the second term above cancels the gauging fixing term

$\begin{split}\int
2\\{\mathtt{y}_{\alpha\dot{\alpha}},\mathsf{a}^{\alpha\dot{\alpha}}\\}^{2}=8\int_{\mathbb{P}^{1}}\lambda_{(\alpha_{2}}\hat{\lambda}_{\circ)}\lambda_{(\alpha_{1}}\hat{\lambda}_{\bullet)}\partial^{\circ}{}_{\dot{\gamma}}\mathsf{a}^{\alpha_{1}\dot{\gamma}}\partial^{\bullet}{}_{\dot{\sigma}}\mathsf{a}^{\alpha_{2}\dot{\sigma}}\,.\end{split}$
(B.5)

Thus,

$\displaystyle I_{1}+\int
2\\{\mathtt{y}_{\alpha\dot{\alpha}},\mathsf{a}^{\alpha\dot{\alpha}}\\}^{2}=2\int_{\mathbb{P}^{1}}\mathtt{K}\,\langle\lambda\,\hat{\lambda}\rangle^{2}\partial_{\circ\dot{\gamma}}\mathsf{a}^{\alpha\,\dot{\gamma}}\,\partial^{\circ}{}_{\dot{\sigma}}\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\,.$
(B.6)

Next, the second contribution in (6.33) reads

$\begin{split}I_{2}&=\int_{\mathbb{P}^{1}}\mathtt{K}\,\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathtt{y}^{\alpha\dot{\gamma}}\\}\\{\mathsf{a}_{\alpha\dot{\sigma}},\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\\}=-2\int_{\mathbb{P}^{1}}\mathtt{K}\\{\mathtt{y}^{\alpha\dot{\gamma}},\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\\}\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathsf{a}_{\alpha\dot{\sigma}}\\}\end{split}$
(B.7)

where we have made an integration by parts and used Jacobi identity. In the
flat limit, $I_{2}$ reduces to

$\displaystyle
I_{2}=\frac{1}{4}\int_{\mathbb{P}^{1}}\mathtt{K}\langle\lambda\,\hat{\lambda}\rangle^{2}\epsilon^{\alpha_{1}\alpha_{2}}\Big{(}\mathsf{a}_{\alpha_{1}}{}^{\dot{\sigma}}\Box\mathsf{a}_{\alpha_{2}\dot{\sigma}}-\mathsf{a}_{\alpha_{2}\dot{\sigma}}\Box\mathsf{a}_{\alpha_{1}}{}^{\dot{\sigma}}\Big{)}=0\,.$
(B.8)

Therefore, the contribution coming from the term
$\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathtt{y}^{\alpha\dot{\gamma}}\\}\\{\mathsf{a}_{\alpha\dot{\sigma}},\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\\}$
vanishes in the flat limit. We conclude that the kinetic action in the flat
limit reduces to

$\displaystyle
S_{2}=\int\mathrm{d}^{4}\mathtt{x}\int\mathtt{K}\,\mathsf{a}^{\alpha\,\dot{\alpha}}\Box\mathsf{a}_{\alpha\,\dot{\alpha}}\,.$
(B.9)

Expanding
$\mathsf{a}_{\alpha\dot{\alpha}}=\lambda^{\beta(s)}\hat{\lambda}^{\beta(s)}\Big{[}A_{\beta(2s)\alpha\,\dot{\alpha}}+\epsilon_{\alpha\beta}\tilde{\mathsf{A}}_{\beta(2s-1)\,\dot{\alpha}}\Big{]}$,
we can elaborate the following explicit cases:

  * $\bullet$

Spin 1. The kinetic action (B.9) consists of the following contributions
coming from spin-1 fields:

$\displaystyle S_{2}^{(1)}=\int A^{\alpha\dot{\alpha}}\Box
A_{\alpha\dot{\alpha}}-\tilde{\mathsf{A}}^{\beta\dot{\alpha}}\Box
A^{\beta}{}_{\dot{\alpha}}\lambda_{\beta}\hat{\lambda}_{\beta}+A_{\beta}{}^{\dot{\alpha}}\Box\tilde{\mathsf{A}}_{\beta\dot{\alpha}}\lambda^{\beta}\hat{\lambda}^{\beta}+\langle\lambda\,\hat{\lambda}\rangle^{2}\tilde{\mathsf{A}}^{\alpha\dot{\alpha}}\Box\tilde{\mathsf{A}}_{\alpha\dot{\alpha}}\,.$
(B.10)

Making an appropriate field redefinition to fit with the form of the integral
(6.31), we get

$\displaystyle S_{2}^{(1)}=-\int A^{\alpha\dot{\alpha}}\Box
A_{\alpha\dot{\alpha}}+\tilde{\mathsf{A}}^{\alpha\dot{\alpha}}\Box\tilde{\mathsf{A}}_{\alpha\dot{\alpha}}\,.$
(B.11)

Thus, the two modes decouple from each others at the quadratic level, and
hence are two independent modes. Intriguingly, we can make the following
change of variables

$\displaystyle\mathbb{A}_{\pm}^{\alpha\dot{\alpha}}=A^{\alpha\dot{\alpha}}\pm\mathrm{i}\,\tilde{\mathsf{A}}^{\alpha\dot{\alpha}}\,,$
(B.12)

to bring the kinetic term at level-1 to a more standard form:

$\displaystyle
S_{2}^{(1)}=-\int\mathbb{A}_{-}^{\alpha\dot{\alpha}}\Box\mathbb{A}^{+}_{\alpha\dot{\alpha}}\,.$
(B.13)

  * $\bullet$

Spin 2. There are the following contributions at level 2:

$\begin{split}S_{2}^{(2)}=&\int A^{\beta(2)\alpha\,\dot{\alpha}}\Box
A_{\zeta(2)\alpha\,\dot{\alpha}}\lambda_{\beta}\hat{\lambda}_{\beta}\lambda^{\zeta}\hat{\lambda}^{\zeta}-\tilde{\mathsf{A}}^{\beta(3)\dot{\alpha}}\Box
A^{\beta}{}_{\zeta(2)\,\dot{\alpha}}\lambda_{\beta(2)}\hat{\lambda}_{\beta(2)}\lambda^{\zeta}\hat{\lambda}^{\zeta}\\\
+&\int
A_{\beta}{}^{\zeta(2)\,\dot{\alpha}}\Box\tilde{\mathsf{A}}_{\beta(3)\,\dot{\alpha}}\lambda^{\beta(2)}\hat{\lambda}^{\beta(2)}\lambda_{\zeta}\hat{\lambda}_{\zeta}+\langle\lambda\,\hat{\lambda}\rangle^{2}\lambda_{\beta}\hat{\lambda}_{\beta}\tilde{\mathsf{A}}^{\beta(2)\alpha\,\dot{\alpha}}\Box\tilde{\mathsf{A}}_{\zeta(2)\alpha\,\dot{\alpha}}\lambda^{\zeta}\hat{\lambda}^{\zeta}\,,\end{split}$
(B.14)

which gives

$\displaystyle S_{2}^{(2)}=-\int A^{\alpha(3)\dot{\alpha}}\Box
A_{\alpha(3)\,\dot{\alpha}}+\tilde{\mathsf{A}}^{\alpha(3)\,\dot{\alpha}}\Box\tilde{\mathsf{A}}_{\alpha(3)\,\dot{\alpha}}\,.$
(B.15)

Inductively, we get the full kinetic term as

$\begin{split}S_{2}&=-\sum_{s\geq
1}\int\mathrm{d}^{4}\mathtt{x}\Big{(}A^{\alpha(2s-1)\,\dot{\alpha}}\Box
A_{\alpha(2s-1)\,\dot{\alpha}}+\tilde{\mathsf{A}}^{\alpha(2s-1)\,\dot{\alpha}}\Box\tilde{\mathsf{A}}_{\alpha(2s-1)\,\dot{\alpha}}\Big{)}\\\
&=-\sum_{s\geq
1}\int\mathrm{d}^{4}\mathtt{x}\,\mathbb{A}_{-}^{\alpha(2s-1)\,\dot{\alpha}}\Box\mathbb{A}^{+}_{\alpha(2s-1)\,\dot{\alpha}}\,.\end{split}$
(B.16)

where

$\displaystyle\mathbb{A}^{\alpha(2s-1)\,\dot{\alpha}}_{\pm}:=A^{\alpha(2s-1)\,\dot{\alpha}}\pm\mathrm{i}\,\tilde{\mathsf{A}}^{\alpha(2s-1)\,\dot{\alpha}}\,.$
(B.17)

The same change of variables can also be made at cubic and quartic
interactions. Therefore, we may interpret $\mathbb{A}_{\pm}$ as higher-spin
fields with positive/negative “helicity”.

We can now understand better the role of the “would-be massive” higher-spin
fields. These extra dof arise because the above fields
$A^{\alpha(2s-1)\,\dot{\alpha}}$ etc. are not divergence-free. Hence they
contain extra dof arising as pure divergence mode, which should behave like
ordinary pure gauge modes in the flat limit, and are thus expected to
decouple. This is consistent with the lack of a scale parameter in the flat
limit.

Next, the cubic vertices read:

$\begin{split}I_{3}&=2\int_{\mathbb{P}^{1}}\mathtt{K}\,\\{\mathtt{y}^{\alpha}{}_{\dot{\gamma}},\mathsf{a}^{\alpha\dot{\gamma}}\\}\\{\mathsf{a}_{\alpha\dot{\sigma}},\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\\}=2\int_{\mathbb{P}^{1}}\mathtt{K}\,\mathcal{E}^{\alpha}{}_{\dot{\gamma},\beta\dot{\beta}}\partial^{\beta\dot{\beta}}\mathsf{a}^{\alpha\dot{\gamma}}\,\mathcal{E}^{\circ\dot{\circ},\bullet\dot{\bullet}}\partial_{\circ\dot{\circ}}\mathsf{a}_{\alpha\dot{\sigma}}\,\partial_{\bullet\dot{\bullet}}\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\,\\\
&=-2\int_{\mathbb{P}^{1}}\mathtt{K}\big{(}\lambda^{\alpha}\hat{\lambda}_{\beta}+\lambda_{\beta}\hat{\lambda}^{\alpha}\big{)}\partial^{\beta}{}_{\dot{\gamma}}\mathsf{a}^{\alpha\dot{\gamma}}\big{(}\lambda^{\circ}\hat{\lambda}^{\bullet}+\lambda^{\bullet}\hat{\lambda}^{\circ}\big{)}\partial_{\circ\dot{\circ}}\mathsf{a}_{\alpha\dot{\sigma}}\partial_{\bullet}{}^{\dot{\circ}}\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\,.\end{split}$
(B.18)

This means that

$\displaystyle
I_{3}=-2\int(\lambda^{\alpha}\hat{\lambda}_{\beta}\lambda^{\circ}\hat{\lambda}^{\bullet}+\lambda^{\alpha}\hat{\lambda}_{\beta}\lambda^{\bullet}\hat{\lambda}^{\circ}+\lambda_{\beta}\hat{\lambda}^{\alpha}\lambda^{\circ}\hat{\lambda}^{\bullet}+\lambda_{\beta}\hat{\lambda}^{\alpha}\lambda^{\bullet}\hat{\lambda}^{\circ})\partial^{\beta}{}_{\dot{\gamma}}\mathsf{a}^{\alpha\dot{\gamma}}\partial_{\circ\dot{\circ}}\mathsf{a}_{\alpha\dot{\sigma}}\partial_{\bullet}{}^{\dot{\circ}}\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\,.$
(B.19)

From here, we can reposition indices to obtain $\epsilon$ symbols using
(6.31). While there is no shortcut to contract indices, we observe that there
are many contributions cancel each others, as well as contributions vanish on-
shell (cf., (7.4)). For instance, contributions such as

$\displaystyle
A\partial^{\alpha\dot{\alpha}}B\partial_{\alpha\dot{\alpha}}C\,,$ (B.20)

can be discarded since it vanishes on-shell on support of momentum
conservation in the flat limit. Furthermore, contributions that produce both
types of angled and square brackets in terms of physical (but complex) spinors
at 3-points will also vanish (see discussion in Section 7). The vertices that
produce these contributions will be ‘irrelevant’ when computing scattering
amplitudes, and thus can be neglected.

Notice that due to gravitational interactions coming from the Poisson
structure (3.10), the lowest possible spins entering the cubic vertices should
be $(1,1,2)$. In expanding higher-spin modes, we obtain

$\displaystyle
I_{3}=4\sum_{s_{i}}\int_{\mathbb{P}^{1}}\digamma^{\alpha,\beta|\circ,\bullet|\zeta(2s_{2})|\rho(2s_{3})}_{\tau(2s_{1})}\partial_{\beta}{}^{\dot{\beta}}\mathcal{A}^{\tau(2s_{1})\alpha}{}_{\dot{\beta}}\partial_{\circ\dot{\diamond}}\mathcal{A}_{\zeta(2s_{2})\alpha\,\dot{\sigma}}\partial_{\bullet}{}^{\dot{\diamond}}\mathcal{A}_{\rho(2s_{3})\alpha}{}^{\dot{\sigma}}\,,$
(B.21)

where we have introduced the notation

$\displaystyle\digamma^{\alpha,\beta|\circ,\bullet|\zeta(2s_{2})|\rho(2s_{3})}_{\tau(2s_{1})}=\lambda^{(\alpha}\hat{\lambda}^{\beta)}\lambda^{(\circ}\hat{\lambda}^{\bullet)}\lambda^{\zeta(s_{2})}\hat{\lambda}^{\zeta(s_{2})}\lambda^{\rho(s_{3})}\hat{\lambda}^{\rho(s_{3})}\lambda_{\tau(s_{1})}\hat{\lambda}_{\tau(s_{1})}$
(B.22)

etc., for convenience. The results can be summarized into the form

$\displaystyle\partial A\\{A,A\\}+\partial
A\\{A,\tilde{\mathsf{A}}\\}+\partial
A\\{\tilde{\mathsf{A}},A\\}+\partial\tilde{\mathsf{A}}\\{A,A\\}$
$\displaystyle+\partial\tilde{\mathsf{A}}\\{\tilde{\mathsf{A}},A\\}+\partial\tilde{\mathsf{A}}\\{A,\tilde{\mathsf{A}}\\}+\partial
A\\{\tilde{\mathsf{A}},\tilde{\mathsf{A}}\\}+\partial\tilde{\mathsf{A}}\\{\tilde{\mathsf{A}},\tilde{\mathsf{A}}\\}$

where $\\{\,,\\}$ indicate the contributions coming from the Poisson brackets.
Here, the positions of fields are important since we are working with almost-
commutative field theory. In terms of $\mathbb{A}_{\pm}$ we have all possible
configuration of “helicities” at cubic order. Very roughly,

$\displaystyle I_{3}\sim
V_{+++}+V_{-++}+V_{+-+}+V_{++-}+V_{--+}+V_{-+-}+V_{+--}+V_{---}\,.$ (B.23)

Without specifying the $\pm$ subscript below $\mathbb{A}$, we can write the
cubic action as

$S_{3}\simeq 4\sum_{s_{2}+s_{3}=s_{1}+2}\int
d^{4}\mathtt{x}\,\partial_{\alpha\dot{\alpha}}\mathbb{A}_{\alpha(2s_{1}-1)}{}^{\dot{\alpha}}\partial_{\alpha\dot{\gamma}}\mathbb{A}^{\alpha(2s_{2}-1)\,\dot{\sigma}}\partial_{\alpha}{}^{\dot{\gamma}}\mathbb{A}^{\alpha(2s_{3}-1)}{}_{\dot{\sigma}}+\widetilde{S}_{3}\,,$
(B.24)

where $\widetilde{S}_{3}$ denotes the irrelevant part of the action for
massless sector, i.e. contributions that vanish on-shell. Here, the un-dotted
indices in the partial derivatives are contracted to those of the gauge
potentials in all possible way. This way of writing is possible due to our
symmetrization convention, i.e. the sum of all coefficients coming from
contracting of $\epsilon$ tensors in (6.31) with fields and derivatives is
one. While the non-commutative twistor approach produces more contributions
than the standard approach in twistor literature, we observe that the non-
vanishing contributions in non-commutative twistor approach coincide with the
ones using the standard approach of twistor theory in the flat limit.

Finally, let us look at the quartic term:

$\begin{split}I_{4}&=\frac{1}{2}\int_{\mathbb{P}^{1}}\mathtt{K}\,\\{\mathsf{a}^{\alpha}{}_{\dot{\gamma}},\mathsf{a}^{\alpha\dot{\gamma}}\\}\\{\mathsf{a}_{\alpha\dot{\sigma}},\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\\}=\frac{1}{2}\int_{\mathbb{P}^{1}}\mathtt{K}\,\mathcal{E}^{\circ\dot{\circ},\bullet\dot{\bullet}}\partial_{\circ\dot{\circ}}\mathsf{a}^{\alpha}{}_{\dot{\gamma}}\partial_{\bullet\dot{\bullet}}\mathsf{a}^{\alpha\dot{\gamma}}\,\mathcal{E}^{\diamond\dot{\diamond},\blacklozenge\dot{\blacklozenge}}\partial_{\diamond\dot{\diamond}}\mathsf{a}_{\alpha\dot{\sigma}}\partial_{\blacklozenge\dot{\blacklozenge}}\mathsf{a}_{\alpha}{}^{\dot{\sigma}}\\\
&=2\sum_{s_{i}}\int_{\mathbb{P}^{1}}\mathtt{K}\,\digamma^{\circ,\bullet|\diamond,\blacklozenge|\zeta(2s_{3})|\tau(2s_{4})}_{\rho(2s_{1})|\delta(2s_{2})}\partial_{\circ\dot{\circ}}\mathcal{A}^{\rho(2s_{1})\alpha}{}_{\dot{\gamma}}\partial_{\bullet}{}^{\dot{\circ}}\mathcal{A}^{\delta(2s_{2})\alpha\dot{\gamma}}\partial_{\diamond\dot{\diamond}}\mathcal{A}_{\zeta(2s_{3})\,\dot{\sigma}}\partial_{\blacklozenge}{}^{\dot{\diamond}}\mathcal{A}_{\tau(2s_{4})\alpha}{}^{\dot{\sigma}}\,.\end{split}$
(B.25)

Thanks to the plane-wave solutions (7.4) and the specific gauge choice we made
in Section 7, all contributions resulting from the integral over
$\mathbb{P}^{1}$ vanish on-shell. For this reason, we can either ignore the
quartic, or write it as

$\displaystyle S_{4}\simeq 2\int
d^{4}\mathtt{x}\,\Big{(}\partial_{\alpha\dot{\gamma}_{1}}\mathbb{A}^{\alpha(2s_{1}-1)\,\dot{\sigma}}\partial_{\alpha}{}^{\dot{\gamma}_{1}}\mathbb{A}^{\alpha(2s_{2}-1)}{}_{\dot{\sigma}}\Big{)}\Big{(}\partial^{\alpha}{}_{\dot{\gamma}}\mathbb{A}_{\alpha(2s_{3}-1)}{}^{\dot{\tau}}\partial^{\alpha\dot{\gamma}_{2}}\mathbb{A}_{\alpha(2s_{4}-1)\,\dot{\tau}}\Big{)}+\widetilde{S}_{4}\,.$
(B.26)

Unlike the construction of higher-spin Yang-Mills [36], all possible
configuration of helicities are allowed due to the helicity-like solutions
$\mathbb{A}_{\pm}$.

#### Vertices from the self-dual sector.

To obtain the above result, one can also start from the self-dual sector. This
provides a simpler way to contract indices in the interactions. But we first
need to know how the higher-spin modes of the $\mathsf{b}_{\alpha\alpha}$
field talk to each others. Recall that the action of the YM sector in first
order formalism has the form cf. Section 6

$\displaystyle
S^{\text{YM}}=\int_{\mathbb{PS}}\mho\Big{(}\mathsf{b}_{\alpha\alpha}\mathsf{f}^{\alpha\alpha}-\frac{1}{2}\mathsf{b}_{\alpha\alpha}\mathsf{b}^{\alpha\alpha}\Big{)}$
(B.27)

Now, we let $\mathsf{b}_{\alpha\alpha}$ to have the following higher-spin
expansion (see the discussion around (6.19) for the reality condition of
$\mathfrak{hs}$-valued $\mathsf{f}^{\alpha\alpha}$ and
$\mathsf{b}_{\alpha\alpha}$):

$\displaystyle\mathsf{b}_{\alpha\alpha}=\Big{(}B_{\beta(2s)\alpha\alpha}-\mathrm{i}\epsilon_{\beta\alpha}\tilde{\mathsf{B}}_{\beta(2s-1)\alpha}\Big{)}\lambda^{\beta(s)}\hat{\lambda}^{\beta(s)}\,.$
(B.28)

It can be checked that there is no mixed term between $B$ and
$\tilde{\mathsf{B}}$. For instance, consider spin-1 case

$\displaystyle\int\mathsf{b}_{\alpha\alpha}\mathsf{b}^{\alpha\alpha}\big{|}_{s=1}$
$\displaystyle=\int
B_{\alpha\alpha}B^{\alpha\alpha}+\mathrm{i}B_{\alpha\alpha}\epsilon^{\beta\alpha}\tilde{\mathsf{B}}^{\beta\alpha}\lambda_{\beta}\hat{\lambda}_{\beta}-\mathrm{i}B_{\alpha\alpha}\epsilon^{\beta\alpha}\tilde{\mathsf{B}}^{\beta\alpha}\lambda_{\beta}\hat{\lambda}_{\beta}+\lambda^{\beta}\hat{\lambda}^{\beta}\epsilon_{\beta\alpha}\epsilon^{\gamma\alpha}\tilde{\mathsf{B}}_{\beta\alpha}\tilde{\mathsf{B}}^{\gamma\alpha}\lambda_{\gamma}\hat{\lambda}_{\gamma}$
$\displaystyle=\int
B_{\alpha\alpha}B^{\alpha\alpha}+\lambda^{\beta}\hat{\lambda}^{\beta}\epsilon_{\beta\alpha}\epsilon^{\gamma\alpha}\tilde{\mathsf{B}}_{\beta\alpha}\tilde{\mathsf{B}}^{\gamma\alpha}\lambda_{\gamma}\hat{\lambda}_{\gamma}\,.$
(B.29)

We can shorten the last term in (B) to

$\displaystyle\int-\frac{1}{4}\langle\lambda\,\hat{\lambda}\rangle^{2}\tilde{\mathsf{B}}_{\alpha\alpha}\tilde{\mathsf{B}}^{\alpha\alpha}+\frac{1}{2}\lambda^{\beta}\hat{\lambda}^{\beta}\tilde{\mathsf{B}}_{\beta}{}^{\gamma}\tilde{\mathsf{B}}_{\beta}{}^{\gamma}\lambda_{\gamma}\hat{\lambda}_{\gamma}\sim\int\mathrm{d}^{4}x\,\tilde{\mathsf{B}}_{\alpha\alpha}\tilde{\mathsf{B}}^{\alpha\alpha}\,.$
(B.30)

Therefore, the $B$ and $\tilde{\mathsf{B}}$ modes do not couple to each other
(other higher-spin cases are analogous). Observe that we can use
$\tilde{\mathsf{B}}$ as Lagrangian multipliers for the fuzzy Lorenz gauge
condition (6.21)

$\displaystyle\int\mho\,\tilde{\mathsf{B}}_{\beta(2s)}\\{\mathtt{y}_{\alpha\dot{\alpha}},\mathcal{A}^{\beta(2s)|\alpha\dot{\alpha}}\\}$
(B.31)

so that only the first modes $B_{\alpha(2s)}$ propagate and
$\tilde{\mathsf{B}}$ becomes non-dynamical. Therefore, it is suggestive to
treat $B_{\alpha(2s)}$ as negative helicity modes. To proceed, we will look at
the kinetic terms

$\displaystyle\sum_{s_{1},s_{2}}\int\mho\lambda^{\beta(s_{1})}\hat{\lambda}^{\beta(s_{1})}B_{\beta(2s_{1})\alpha\alpha}\\{\mathtt{y}^{\alpha}{}_{\dot{\alpha}},\mathcal{A}^{\gamma(2s_{2})|\alpha\dot{\alpha}}\\}\lambda_{\gamma(s_{2})}\hat{\lambda}_{\gamma(s_{2})}\,.$
(B.32)

By decomposing
$\mathcal{A}^{\beta(2s)|\alpha\dot{\alpha}}=A^{\beta(2s)\alpha\,\dot{\alpha}}+\mathrm{i}\epsilon^{\alpha\beta}\tilde{\mathsf{A}}^{\beta(2s-1)\,\dot{\alpha}}$,
we can check that

$\displaystyle L_{2}$
$\displaystyle=\int\mho\lambda^{\beta(s)}\hat{\lambda}^{\beta(s)}\lambda_{\gamma(s)}\hat{\lambda}_{\gamma(s)}B_{\beta(2s)\alpha\alpha}\Big{(}\\{\mathtt{y}^{\alpha}{}_{\dot{\alpha}},A^{\gamma(2s)\alpha\,\dot{\alpha}}\\}+\mathrm{i}\epsilon^{\alpha\gamma}\\{\mathtt{y}^{\alpha}{}_{\dot{\alpha}},\tilde{\mathsf{A}}^{\gamma(2s+1)\,\dot{\alpha}}\\}\Big{)}\,$
$\displaystyle=\int\mathrm{d}^{4}xB_{\alpha(2s)}\partial^{\alpha}{}_{\dot{\alpha}}\mathbb{A}_{+}^{\alpha(2s-1)\,\dot{\alpha}}\,.$
(B.33)

Thus, as always, the kinetic terms decrible the coupling between negative and
positive helicity fields. From here, we can express $A$ and
$\tilde{\mathsf{A}}$ in terms of $\mathbb{A}_{\pm}$ as

$\displaystyle
A^{\alpha(2s-1)\,\dot{\alpha}}=\frac{\mathbb{A}_{+}^{\alpha(2s-1)\,\dot{\alpha}}+\mathbb{A}_{-}^{\alpha(2s-1)\,\dot{\alpha}}}{2}\,,\quad\tilde{\mathsf{A}}^{\alpha(2s-1)\,\dot{\alpha}}=\frac{\mathbb{A}_{+}^{\alpha(2s-1)\,\dot{\alpha}}-\mathbb{A}_{-}^{\alpha(2s-1)\,\dot{\alpha}}}{2\mathrm{i}}$
(B.34)

to study the cubic interactions $\int\mathsf{b}\\{\mathcal{A},\mathcal{A}\\}$.

In this first-order formalism, the interaction terms have much less structure
compared to the second-order case. In particular, after integrating out fiber
coordinates, we end up with three main structures

$\displaystyle
B^{-}_{\alpha(2s_{1})}\partial^{\alpha}{}_{\dot{\gamma}}\mathbb{A}_{\pm}^{\alpha(2s_{2}-1)\,\dot{\alpha}}\partial^{\alpha\dot{\gamma}}\mathbb{A}_{\pm}^{\alpha(2s_{3}-1)}{}_{\dot{\alpha}}\,,$
(B.35a) $\displaystyle
B^{-}_{\alpha(2s_{1})}\partial_{\alpha\dot{\gamma}}\mathbb{A}_{\pm}^{\alpha(2s_{2}-1)\,\dot{\alpha}}\partial^{\alpha\dot{\gamma}}\mathbb{A}_{\pm}^{\alpha(2s_{3}-1)}{}_{\dot{\alpha}}\,,$
(B.35b) $\displaystyle
B^{-}_{\alpha(2s_{1})}\partial_{\alpha\dot{\gamma}}\mathbb{A}_{\pm}^{\alpha(2s_{2}-1)\,\dot{\alpha}}\partial_{\alpha}{}^{\dot{\gamma}}\mathbb{A}_{\pm}^{\alpha(2s_{3}-1)}{}_{\dot{\alpha}}\,,$
(B.35c)

where, as always, all un-dotted indices of the lower level are understood to
be contracted in all possible way with the indices of the upper level. Notice
that the difference between the above structures are merely the positions of
the undotted indices in the partial derivatives.

Besides the polarization tensors (7.4) for the $\mathbb{A}_{\pm}$ fields, we
set

$\displaystyle\epsilon_{\alpha(2s)}^{-}=\lambda_{\alpha(2s)}$ (B.36)

to the the polarization tensor associated with the $B_{\alpha(2s)}$ negative
helicity fields [35]. Note that there is no auxiliary/reference spinors
$\epsilon^{-}_{\alpha(2s)}$. Then, upon plugging in the plane-wave solutions,
it can be checked that (B.35c) is the only structure that survives. It is
worth mentioning that there are no combination such as

$\displaystyle
B_{\alpha(2s_{1})}\\{\mathbb{A}^{\alpha(2s_{2}-1-m)\beta(m)\,\dot{\alpha}},\mathbb{A}^{\alpha(2s_{3}-1-m)}{}_{\beta(m)\,\dot{\alpha}}\\}$
(B.37)

where un-dotted indices of the $\mathbb{A}$ fields contracted with each others
since it results in trivial amplitudes. In fact, this can also be understood
from the light-cone point of view where we set
$\mathbb{A}^{\alpha(2s-2)0\,\dot{0}}=0$ and choose
$\mathbb{A}^{1(2s-1)\,\dot{0}}$ to be the components carry physical degrees of
the $\mathbb{A}$ fields. Then, for any combination such as (B.37), it will
vanish on the nose in the light-cone gauge.

Further scrutiny shows that non-trivial cubic amplitudes will come from

$\displaystyle
B^{-}_{\alpha(2s_{1})}\partial_{\alpha\dot{\gamma}}\mathbb{A}_{+}^{\alpha(2s_{2}-1)\,\dot{\alpha}}\partial_{\alpha}{}^{\dot{\gamma}}\mathbb{A}_{+}^{\alpha(2s_{3}-1)}{}_{\dot{\alpha}}\,.$
(B.38)

These are the vertices that give us the $\overline{\text{MHV}}_{3}$ amplitudes
(7.14) in the main text. As a result, the
$\mathsf{b}_{\alpha\alpha}\mathsf{f}^{\alpha\alpha}$ can be written as

$\displaystyle\int\mho\,\mathsf{b}_{\alpha\alpha}\mathsf{f}^{\alpha\alpha}\sim\int\mathrm{d}^{4}xB^{-}_{\alpha(2s)}F^{\alpha(2s)}$
(B.39)

where for convenience we have defined

$\displaystyle
F_{+}^{\alpha(2s)}:=\partial^{\alpha\dot{\alpha}}\mathbb{A}_{+}^{\alpha(2s-1)}{}_{\dot{\alpha}}+\frac{1}{2}\sum_{m+n=s+2}\partial_{\alpha\dot{\gamma}}\mathbb{A}_{+}^{\alpha(2s_{2}-1)\,\dot{\sigma}}\partial_{\alpha}{}^{\dot{\gamma}}\mathbb{A}_{+}^{\alpha(2s_{3}-1)}{}_{\dot{\sigma}}\,.$
(B.40)

Of course, the above BF action (B.39) in flat space is only the self-dual
sector which features the $\overline{\text{MHV}}_{3}$ amplitudes. To go back
to the second-order formalism, we can integrate out the $B^{-}_{\alpha(2s)}$
which results in

$\displaystyle S_{\text{YM}}=\int F_{\alpha(2s)}F^{\alpha(2s)}$ (B.41)

As this stage, one can give the higher-spin fields
$\mathbb{A}^{\alpha(2s-1)\,\dot{\alpha}}$ all possible helicity when we
consider cubic and quartic interactions. The reason is that the negative
helicity modes $B^{-}_{\alpha(2s)}$ need to be replaced by
$\mathbb{A}_{-}^{\alpha(2s-1)\,\dot{\alpha}}$. Lastly, so as not to forget
about other structures, we can write the final action as

$\displaystyle
S^{\text{YM}}=\sum_{s}\int\mathrm{d}^{4}\mathtt{x}\,F_{\alpha(2s)}F^{\alpha(2s)}+\widetilde{S}_{3}+\widetilde{S}_{4}\,.$
(B.42)

where $\widetilde{S}_{i}$ for $i=3,4$ denote terms that are irrelevant when
studying scattering amplitudes.

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