$\displaystyle\mathrm{D}_{\mu}\ =\ \partial_{\mu}+gA_{\mu}\times\ \ ,$

and $S^{\mathrm{b}}_{0}=S^{\mathrm{b}}_{g=0}$. Writing this condition out at
second order gives

$\displaystyle\int\mathrm{d}^{4}x\;\Bigl{\\{}$
$\displaystyle{\textstyle\frac{1}{2}}A^{\prime}_{\mu}|_{\mathcal{O}(g)}\bigl{(}\Box\eta^{\mu\nu}-\partial^{\mu}\partial^{\nu}\bigr{)}A^{\prime}_{\nu}|_{\mathcal{O}(g)}+A_{\mu}\bigl{(}\Box\eta^{\mu\nu}-\partial^{\mu}\partial^{\nu}\bigr{)}A^{\prime}_{\nu}|_{\mathcal{O}(g^{2})}+{\textstyle\frac{1}{2}}\varphi^{\prime}_{i}|_{\mathcal{O}(g)}\Box\varphi^{\prime}_{i}|_{\mathcal{O}(g)}+\varphi_{i}\Box\varphi^{\prime}_{i}|_{\mathcal{O}(g^{2})}\Bigr{\\}}$
(E.3) $\displaystyle\ =\
\int\mathrm{d}^{4}x\;\Bigl{\\{}-{\textstyle\frac{1}{4}}(A_{\mu}\times
A_{\nu})^{2}-{\textstyle\frac{1}{2}}(A_{\mu}\times\varphi_{i})^{2}-{\textstyle\frac{1}{4}}(\varphi_{i}\times\varphi_{j})^{2}\Bigr{\\}}\
,$

after integrating by parts on the left hand side. The free-action condition
(in the Landau gauge) was previously shown for the map in $\mathcal{N}=\,1$
$D=10$ SYM, i.e. before dimensional reduction. It is clear that the condition
remains valid for the reduced map (given by the black terms) and action. We
therefore argue that we only have left to show that the blue terms have no
effect on the l.h.s. of (E.3). Further using
$\partial^{\mu}A^{\prime}_{\mu}=0$ (at all orders), we can drop two terms on
the l.h.s. of (E.3), so we are left with the condition

$\int\mathrm{d}^{4}x\;\Bigl{\\{}A_{\mu}\Box
A^{\prime\mu}|_{\mathcal{O}(g^{2})}+\varphi_{I}\Box\varphi^{\prime}_{I}|_{\mathcal{O}(g^{2})}+\varphi_{I+3}\Box\varphi^{\prime}_{I+3}|_{\mathcal{O}(g^{2})}\Bigr{\\}}_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\text{blue
terms}}}\ =\ 0\ .$ (E.4)

Let us start with the map obtained from $A=4$. We refer to the three
contributions as 1, 2, 3 respectively. Dividing by an overall factor of
${\textstyle\frac{1}{2}}$ and switching to a graphical notation, we find for
the first part

$\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ =\
\epsilon_{\mu\nu\rho\lambda}\Biggl{\\{}\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial^{\nu}A^{\mu}$}; \vertex[below left=of a] (e) {$\varphi_{J}$};
\vertex[above right=of b] (f) {$\varphi_{J+3}$}; \vertex[below right=of b] (g)
{$A^{\lambda}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial^{\nu}A^{\mu}$}; \vertex[below left=of a] (e) {$\varphi_{J+3}$};
\vertex[above right=of b] (f) {$\varphi_{J}$}; \vertex[below right=of b] (g)
{$A^{\lambda}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial^{\nu}A^{\mu}$}; \vertex[below left=of a] (e) {$A^{\rho}$};
\vertex[above right=of b] (f) {$\varphi_{J+3}$}; \vertex[below right=of b] (g)
{$\varphi_{J}$}; \diagram* { (a) -- [edge label=$C^{\lambda}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\Biggr{\\}}\ .$ (E.5)

The last diagram drops out since we can bring the $\partial^{\nu}$ to the
center through integration by parts and the overall anti-symmetry under
$\mu\leftrightarrow\rho$ (gaining a factor $-1/2$). Then, clearly
$\partial^{\nu}C^{\lambda}=C^{\nu\lambda}$ contracts to zero with the epsilon
symbol as it is symmetric under $\nu\leftrightarrow\lambda$. The second
contribution is

$\displaystyle\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ =$
$\displaystyle{\textstyle\frac{1}{2}}\epsilon_{\mu\nu\rho\lambda}\Biggl{\\{}\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial^{\mu}\varphi_{I}$}; \vertex[below left=of a] (e) {$\varphi_{I+3}$};
\vertex[above right=of b] (f) {$A^{\rho}$}; \vertex[below right=of b] (g)
{$A^{\lambda}$}; \diagram* { (a) -- [edge label=$C^{\nu}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+2\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial^{\mu}\varphi_{I}$}; \vertex[below left=of a] (e) {$A^{\nu}$};
\vertex[above right=of b] (f) {$\varphi_{I+3}$}; \vertex[below right=of b] (g)
{$A^{\lambda}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\Biggr{\\}}$ (E.6)
$\displaystyle+\Biggl{\\{}\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{I}$}; \vertex[below left=of a] (e)
{$\varphi_{I+3}$}; \vertex[above right=of b] (f) {$\varphi_{J+3}$};
\vertex[below right=of b] (g) {$\varphi_{J}$}; \diagram* { (a) -- [edge
label=$C^{\rho}$] (b), (a) -- [photon] (d), (a) -- [photon] (e), (b) --
[photon] (f), (b) -- [photon] (g), }; \; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{I}$}; \vertex[below left=of a] (e) {$\varphi_{J}$};
\vertex[above right=of b] (f) {$\varphi_{I+3}$}; \vertex[below right=of b] (g)
{$\varphi_{J+3}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{I}$}; \vertex[below left=of a] (e)
{$\varphi_{J+3}$}; \vertex[above right=of b] (f) {$\varphi_{I+3}$};
\vertex[below right=of b] (g) {$\varphi_{J}$}; \diagram* { (a) -- [edge
label=$C^{\rho}$] (b), (a) -- [photon] (d), (a) -- [photon] (e), (b) --
[photon] (f), (b) -- [photon] (g), }; \; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\Biggr{\\}}$

and

$\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\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{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{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}{-2.5pt}{-3.22221pt}\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{{3}}
}}\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}}\ =\
-\bigl{[}\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\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{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{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}{-2.5pt}{-3.22221pt}\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{{2}}
}}\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}}\ \text{with}\
(I\leftrightarrow I{+}3)\bigr{]}\ .$ (E.7)

The first respective terms of 2 and 3 cancel each other by means of
integration by parts and symmetry. The second term of 2 cancels the first term
of 1 and the corresponding second term of 3 cancels the second term of 1. We
are left with the second line of (E.6) and the corresponding terms from (E.7):

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{I}$}; \vertex[below left=of a] (e)
{$\varphi_{I+3}$}; \vertex[above right=of b] (f) {$\varphi_{J+3}$};
\vertex[below right=of b] (g) {$\varphi_{J}$}; \diagram* { (a) -- [edge
label=$C^{\rho}$] (b), (a) -- [photon] (d), (a) -- [photon] (e), (b) --
[photon] (f), (b) -- [photon] (g), }; \; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{I}$}; \vertex[below left=of a] (e) {$\varphi_{J}$};
\vertex[above right=of b] (f) {$\varphi_{I+3}$}; \vertex[below right=of b] (g)
{$\varphi_{J+3}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{I}$}; \vertex[below left=of a] (e)
{$\varphi_{J+3}$}; \vertex[above right=of b] (f) {$\varphi_{I+3}$};
\vertex[below right=of b] (g) {$\varphi_{J}$}; \diagram* { (a) -- [edge
label=$C^{\rho}$] (b), (a) -- [photon] (d), (a) -- [photon] (e), (b) --
[photon] (f), (b) -- [photon] (g), }; \; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (E.8)
$\displaystyle-$ $\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{I+3}$}; \vertex[below left=of a] (e)
{$\varphi_{I}$}; \vertex[above right=of b] (f) {$\varphi_{J+3}$};
\vertex[below right=of b] (g) {$\varphi_{J}$}; \diagram* { (a) -- [edge
label=$C^{\rho}$] (b), (a) -- [photon] (d), (a) -- [photon] (e), (b) --
[photon] (f), (b) -- [photon] (g), }; \; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{I+3}$}; \vertex[below left=of a] (e)
{$\varphi_{J}$}; \vertex[above right=of b] (f) {$\varphi_{I}$}; \vertex[below
right=of b] (g) {$\varphi_{J+3}$}; \diagram* { (a) -- [edge label=$C^{\rho}$]
(b), (a) -- [photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) --
[photon] (g), }; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{I+3}$}; \vertex[below left=of a] (e)
{$\varphi_{J+3}$}; \vertex[above right=of b] (f) {$\varphi_{I}$};
\vertex[below right=of b] (g) {$\varphi_{J}$}; \diagram* { (a) -- [edge
label=$C^{\rho}$] (b), (a) -- [photon] (d), (a) -- [photon] (e), (b) --
[photon] (f), (b) -- [photon] (g), }; \; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ =:\
\sum_{I,J}Z_{IJ}\ \stackrel{{\scriptstyle!}}{{=}}\ 0\ .$

We can integrate by parts in the first diagram, which gives two contributions,
one of them canceling the fourth diagram. Similarly, the third and fifth
diagram can be combined into one. In the second and last diagram we can make
use of the anti-symmetry under $I\leftrightarrow J$ to also integrate by parts
in both diagrams (gaining a factor $-1/2$ in each) and then combine them into
one contribution. This way the six diagrams reduce to three:

$-\leavevmode\hbox to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[above left=of a] (d) {$\varphi_{I}$}; \vertex[below left=of a] (e)
{$\varphi_{I+3}$}; \vertex[above right=of a] (f) {$\varphi_{J+3}$};
\vertex[below right=of a] (g) {$\varphi_{J}$}; \diagram* { (a) -- [photon]
(d), (a) -- [photon] (e), (a) -- [photon] (f), (a) -- [photon] (g), }; \;
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[above left=of a] (d) {$\varphi_{I}$}; \vertex[below left=of a] (e)
{$\varphi_{J+3}$}; \vertex[above right=of a] (f) {$\varphi_{J}$};
\vertex[below right=of a] (g) {$\varphi_{I+3}$}; \diagram* { (a) -- [photon]
(d), (a) -- [photon] (e), (a) -- [photon] (f), (a) -- [photon] (g), }; \;
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[above left=of a] (d) {$\varphi_{I}$}; \vertex[below left=of a] (e)
{$\varphi_{J}$}; \vertex[above right=of a] (f) {$\varphi_{I+3}$};
\vertex[below right=of a] (g) {$\varphi_{J+3}$}; \diagram* { (a) -- [photon]
(d), (a) -- [photon] (e), (a) -- [photon] (f), (a) -- [photon] (g), }; \;
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\
\stackrel{{\scriptstyle!}}{{=}}\ 0\ ,$ (E.9)

where we used $\partial_{\rho}C^{\rho}=\Box C=\mathbbm{1}$. The condition
(E.9) follows simply from the Jacobi identity (in color space). In conclusion,
the blue terms of the $A=4$ map indeed have no effect on the free action
condition, at least to the second order.

For the $A=K$ case, most of the calculation can be done in the same way as for
$A=4$ by simply carrying around the sign factors $(-)^{\delta_{KJ}}$ etc.
However, the third and last line in (4.11) require special attention.
Referring to (E.8), the remaining condition for $A=K$ can be written as

$\sum_{I,J}(-)^{\delta_{KI}}Z_{IJ}-2\sum_{I}Z_{IK}\ =\
\sum_{J}\bigl{(}-Z_{KJ}-2Z_{JK}+\sum_{I\neq K}Z_{IJ}\bigr{)}\
\stackrel{{\scriptstyle!}}{{=}}\ 0\ .$ (E.10)

In the following we will show that this condition is satisfied by
demonstrating that

$\sum_{J}Z_{JK}\ =\ -\sum_{J}Z_{KJ}\ ,\quad\text{ for any }\quad K\ =\ 1,2,3\
,$ (E.11)

and using our previous result $\sum_{I,J}Z_{IJ}=0$. In order to make clear
that we are not summing over $K$, we will set $K=1$ in the following, although
the calculation works in the exact same way for $K=2,3$. We start with

$\displaystyle\sum_{J}Z_{J1}\ =$ $\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{J}$}; \vertex[below left=of a] (e)
{$\varphi_{J+3}$}; \vertex[above right=of b] (f) {$\varphi_{4}$};
\vertex[below right=of b] (g) {$\varphi_{1}$}; \diagram* { (a) -- [edge
label=$C^{\rho}$] (b), (a) -- [photon] (d), (a) -- [photon] (e), (b) --
[photon] (f), (b) -- [photon] (g), }; \; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{J}$}; \vertex[below left=of a] (e) {$\varphi_{1}$};
\vertex[above right=of b] (f) {$\varphi_{J+3}$}; \vertex[below right=of b] (g)
{$\varphi_{4}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{J}$}; \vertex[below left=of a] (e) {$\varphi_{4}$};
\vertex[above right=of b] (f) {$\varphi_{J+3}$}; \vertex[below right=of b] (g)
{$\varphi_{1}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (E.12)
$\displaystyle-$ $\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{J+3}$}; \vertex[below left=of a] (e)
{$\varphi_{J}$}; \vertex[above right=of b] (f) {$\varphi_{4}$}; \vertex[below
right=of b] (g) {$\varphi_{1}$}; \diagram* { (a) -- [edge label=$C^{\rho}$]
(b), (a) -- [photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) --
[photon] (g), }; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{J+3}$}; \vertex[below left=of a] (e)
{$\varphi_{1}$}; \vertex[above right=of b] (f) {$\varphi_{J}$}; \vertex[below
right=of b] (g) {$\varphi_{4}$}; \diagram* { (a) -- [edge label=$C^{\rho}$]
(b), (a) -- [photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) --
[photon] (g), }; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{J+3}$}; \vertex[below left=of a] (e)
{$\varphi_{4}$}; \vertex[above right=of b] (f) {$\varphi_{J}$}; \vertex[below
right=of b] (g) {$\varphi_{1}$}; \diagram* { (a) -- [edge label=$C^{\rho}$]
(b), (a) -- [photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) --
[photon] (g), }; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ .$

We can integrate by parts in the first diagram as previously. This gives two
diagrams of which one cancels with the fourth diagram. Further, we integrate
by parts in all of the four other diagrams. This leaves us with the 9 diagrams

$\displaystyle\sum_{J}Z_{J1}\ =$ $\displaystyle-\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[above left=of a] (d) {$\varphi_{J}$}; \vertex[below left=of a] (e)
{$\varphi_{J+3}$}; \vertex[above right=of a] (f) {$\varphi_{4}$};
\vertex[below right=of a] (g) {$\varphi_{1}$}; \diagram* { (a) -- [photon]
(d), (a) -- [photon] (e), (a) -- [photon] (f), (a) -- [photon] (g), }; \;
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[above left=of a] (d) {$\varphi_{J}$}; \vertex[below left=of a] (e)
{$\varphi_{1}$}; \vertex[above right=of a] (f) {$\varphi_{J+3}$};
\vertex[below right=of a] (g) {$\varphi_{4}$}; \diagram* { (a) -- [photon]
(d), (a) -- [photon] (e), (a) -- [photon] (f), (a) -- [photon] (g), }; \;
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[above left=of a] (d) {$\varphi_{J}$}; \vertex[below left=of a] (e)
{$\varphi_{4}$}; \vertex[above right=of a] (f) {$\varphi_{1}$}; \vertex[below
right=of a] (g) {$\varphi_{J+3}$}; \diagram* { (a) -- [photon] (d), (a) --
[photon] (e), (a) -- [photon] (f), (a) -- [photon] (g), }; \;
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (E.13)
$\displaystyle+\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[above left=of a] (d) {$\varphi_{J+3}$}; \vertex[below left=of a] (e)
{$\varphi_{1}$}; \vertex[above right=of a] (f) {$\varphi_{J}$}; \vertex[below
right=of a] (g) {$\varphi_{4}$}; \diagram* { (a) -- [photon] (d), (a) --
[photon] (e), (a) -- [photon] (f), (a) -- [photon] (g), }; \;
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[above left=of a] (d) {$\varphi_{J+3}$}; \vertex[below left=of a] (e)
{$\varphi_{4}$}; \vertex[above right=of a] (f) {$\varphi_{J}$}; \vertex[below
right=of a] (g) {$\varphi_{1}$}; \diagram* { (a) -- [photon] (d), (a) --
[photon] (e), (a) -- [photon] (f), (a) -- [photon] (g), }; \;
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d) {$\varphi_{J}$};
\vertex[below left=of a] (e) {$\partial_{\rho}\varphi_{1}$}; \vertex[above
right=of b] (f) {$\varphi_{J+3}$}; \vertex[below right=of b] (g)
{$\varphi_{4}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle+\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d) {$\varphi_{J}$};
\vertex[below left=of a] (e) {$\partial_{\rho}\varphi_{4}$}; \vertex[above
right=of b] (f) {$\varphi_{J+3}$}; \vertex[below right=of b] (g)
{$\varphi_{1}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d) {$\varphi_{J+3}$};
\vertex[below left=of a] (e) {$\partial_{\rho}\varphi_{1}$}; \vertex[above
right=of b] (f) {$\varphi_{J}$}; \vertex[below right=of b] (g)
{$\varphi_{4}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d) {$\varphi_{J+3}$};
\vertex[below left=of a] (e) {$\partial_{\rho}\varphi_{4}$}; \vertex[above
right=of b] (f) {$\varphi_{J}$}; \vertex[below right=of b] (g)
{$\varphi_{1}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ ,$

of which the first three cancel through the Jacobi identity and the fourth and
fifth can be combined into one using the Jacobi identity once more. We find

$\displaystyle\sum_{J}Z_{J1}\ =$ $\displaystyle+\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[above left=of a] (d) {$\varphi_{J+3}$}; \vertex[below left=of a] (e)
{$\varphi_{J}$}; \vertex[above right=of a] (f) {$\varphi_{1}$}; \vertex[below
right=of a] (g) {$\varphi_{4}$}; \diagram* { (a) -- [photon] (d), (a) --
[photon] (e), (a) -- [photon] (f), (a) -- [photon] (g), }; \;
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d) {$\varphi_{J}$};
\vertex[below left=of a] (e) {$\partial_{\rho}\varphi_{1}$}; \vertex[above
right=of b] (f) {$\varphi_{J+3}$}; \vertex[below right=of b] (g)
{$\varphi_{4}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (E.14)
$\displaystyle+\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d) {$\varphi_{J}$};
\vertex[below left=of a] (e) {$\partial_{\rho}\varphi_{4}$}; \vertex[above
right=of b] (f) {$\varphi_{J+3}$}; \vertex[below right=of b] (g)
{$\varphi_{1}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d) {$\varphi_{J+3}$};
\vertex[below left=of a] (e) {$\partial_{\rho}\varphi_{1}$}; \vertex[above
right=of b] (f) {$\varphi_{J}$}; \vertex[below right=of b] (g)
{$\varphi_{4}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d) {$\varphi_{J+3}$};
\vertex[below left=of a] (e) {$\partial_{\rho}\varphi_{4}$}; \vertex[above
right=of b] (f) {$\varphi_{J}$}; \vertex[below right=of b] (g)
{$\varphi_{1}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ .$

We can use $\mathbbm{1}=C\indices{{}^{\rho}_{\rho}}$, symmetry and integration
by parts to modify the first diagram

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[above left=of a] (d) {$\varphi_{J+3}$}; \vertex[below left=of a] (e)
{$\varphi_{J}$}; \vertex[above right=of a] (f) {$\varphi_{1}$}; \vertex[below
right=of a] (g) {$\varphi_{4}$}; \diagram* { (a) -- [photon] (d), (a) --
[photon] (e), (a) -- [photon] (f), (a) -- [photon] (g), }; \;
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ =\ \leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[above left=of a] (d) {$\varphi_{1}$}; \vertex[below left=of a] (e)
{$\varphi_{4}$}; \vertex[above right=of a] (f) {$\varphi_{J+3}$};
\vertex[below right=of a] (g) {$\varphi_{J}$}; \diagram* { (a) -- [photon]
(d), (a) -- [photon] (e), (a) -- [photon] (f), (a) -- [photon] (g), }; \;
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ =$ (E.15)
$\displaystyle\ =$ $\displaystyle-\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{1}$}; \vertex[below left=of a] (e) {$\varphi_{4}$};
\vertex[above right=of b] (f) {$\varphi_{J+3}$}; \vertex[below right=of b] (g)
{$\varphi_{J}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d) {$\varphi_{1}$};
\vertex[below left=of a] (e) {$\partial_{\rho}\varphi_{4}$}; \vertex[above
right=of b] (f) {$\varphi_{J+3}$}; \vertex[below right=of b] (g)
{$\varphi_{J}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

Using antisymmetry to flip the external lines appropriately, it is now easy to
see that

$\displaystyle\sum_{J}Z_{J1}\ =$ $\displaystyle-\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{1}$}; \vertex[below left=of a] (e) {$\varphi_{4}$};
\vertex[above right=of b] (f) {$\varphi_{J+3}$}; \vertex[below right=of b] (g)
{$\varphi_{J}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{4}$}; \vertex[below left=of a] (e) {$\varphi_{1}$};
\vertex[above right=of b] (f) {$\varphi_{J+3}$}; \vertex[below right=of b] (g)
{$\varphi_{J}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{1}$}; \vertex[below left=of a] (e) {$\varphi_{J}$};
\vertex[above right=of b] (f) {$\varphi_{4}$}; \vertex[below right=of b] (g)
{$\varphi_{J+3}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (E.16)
$\displaystyle+\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{4}$}; \vertex[below left=of a] (e) {$\varphi_{J}$};
\vertex[above right=of b] (f) {$\varphi_{1}$}; \vertex[below right=of b] (g)
{$\varphi_{J+3}$}; \diagram* { (a) -- [edge label=$C^{\rho}$] (b), (a) --
[photon] (d), (a) -- [photon] (e), (b) -- [photon] (f), (b) -- [photon] (g),
}; \; \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{1}$}; \vertex[below left=of a] (e)
{$\varphi_{J+3}$}; \vertex[above right=of b] (f) {$\varphi_{4}$};
\vertex[below right=of b] (g) {$\varphi_{J}$}; \diagram* { (a) -- [edge
label=$C^{\rho}$] (b), (a) -- [photon] (d), (a) -- [photon] (e), (b) --
[photon] (f), (b) -- [photon] (g), }; \; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} \feynman \vertex(a);
\vertex[right=of a] (b); \vertex[above left=of a] (d)
{$\partial_{\rho}\varphi_{4}$}; \vertex[below left=of a] (e)
{$\varphi_{J+3}$}; \vertex[above right=of b] (f) {$\varphi_{1}$};
\vertex[below right=of b] (g) {$\varphi_{J}$}; \diagram* { (a) -- [edge
label=$C^{\rho}$] (b), (a) -- [photon] (d), (a) -- [photon] (e), (b) --
[photon] (f), (b) -- [photon] (g), }; \; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\ =\ -\sum_{J}Z_{1J}\
,$

which concludes our check of the free action condition for $A=K$.

Check of the determinant matching: The map has to satisfy

$\operatorname{log}\ \operatorname{det}\
\frac{\delta\mathscr{A}^{\prime}}{\delta\mathscr{A}}\ =\ \operatorname{log}\
\Delta_{\mathrm{MSS}}[A]\Delta_{\mathrm{FP}}[A]\ ,$ (E.17)

which has been checked for the $\mathcal{N}=\,1$ $D=10$ result (to fourth
order). It is easy to convince oneself that the condition is preserved under
dimensional reduction to $\mathcal{N}=\,4$ $D=4$. Hence, we will again only
show that the blue terms have no effect on the l.h.s. of the condition. Using
$\operatorname{log}\ \operatorname{det}=\mathrm{tr}\ \operatorname{log}$, we
see that they first enter at the second order through the first term of

$\operatorname{log}\ \operatorname{det}\
\frac{\delta\mathscr{A}^{\prime}}{\delta\mathscr{A}}\Bigr{|}_{\mathcal{O}(g^{2})}\
=\ \mathrm{tr}\
\frac{\delta\mathscr{A}^{\prime}}{\delta\mathscr{A}}\Bigr{|}_{\mathcal{O}(g^{2})}-{\textstyle\frac{1}{2}}\mathrm{tr}\
\frac{\delta\mathscr{A}^{\prime}}{\delta\mathscr{A}}\Bigr{|}_{\mathcal{O}(g)}\frac{\delta\mathscr{A}^{\prime}}{\delta\mathscr{A}}\Bigr{|}_{\mathcal{O}(g)}\
.$ (E.18)

More specifically, we have

$\displaystyle\mathrm{tr}\
\frac{\delta\mathscr{A}^{\prime}}{\delta\mathscr{A}}$ $\displaystyle\ =\
\int\;\mathrm{d}^{4}x\;\mathrm{d}^{4}y\
\delta^{(4)}(x-y)\delta^{ab}\delta\indices{{}^{\Sigma}_{\Delta}}\frac{\delta\mathscr{A}^{\prime
a}_{\Sigma}(x)}{\delta\mathscr{A}^{b}_{\Delta}(y)}$ (E.19) $\displaystyle\ =\
\int\;\mathrm{d}^{4}x\;\mathrm{d}^{4}y\
\delta^{(4)}(x-y)\delta^{ab}\Bigl{\\{}\frac{\delta A^{\prime
a}_{\mu}(x)}{\delta
A^{b}_{\nu}(y)}\delta\indices{{}^{\mu}_{\nu}}+\frac{\delta\varphi^{\prime
a}_{I}(x)}{\delta\varphi_{J}^{b}(y)}\delta_{IJ}+\frac{\delta\varphi^{\prime
a}_{I+3}(x)}{\delta\varphi_{J+3}^{b}(y)}\delta_{IJ}\Bigr{\\}}\ .$

We need to show that the contributions from the blue terms in (E.19) vanish.
We again refer to the three contributions as 1, 2 and 3 respectively and start
with the case $A=4$. It is easy to see that $\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=0$, since every term
contains $\epsilon_{\mu\lambda\rho\sigma}\eta^{\mu\sigma}=0$. For 2, we note
that the first line of the blue terms in (4.10) contains no fields
$\varphi_{I}$, i.e. drops out when varying w.r.t. $\varphi_{I}$. The second
term in the last line gives no contribution since $f^{aac}=0$, whereas the
remaining two contributions cancel each other. In an analogous fashion, one
finds $\leavevmode\hbox to14.18pt{\vbox
to14.18pt{\pgfpicture\makeatletter\hbox{\hskip 7.09111pt\lower-7.09111pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.89111pt}{0.0pt}\pgfsys@curveto{6.89111pt}{3.8059pt}{3.8059pt}{6.89111pt}{0.0pt}{6.89111pt}\pgfsys@curveto{-3.8059pt}{6.89111pt}{-6.89111pt}{3.8059pt}{-6.89111pt}{0.0pt}\pgfsys@curveto{-6.89111pt}{-3.8059pt}{-3.8059pt}{-6.89111pt}{0.0pt}{-6.89111pt}\pgfsys@curveto{3.8059pt}{-6.89111pt}{6.89111pt}{-3.8059pt}{6.89111pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=0$, so that in total,
the blue terms leave the determinant matching condition (E.17) invariant. The
$A=K$ case works in almost the same way by carrying around the sign factors
and additionally taking care of the last line in (4.11).

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# Sensor-based Temporal Logic Planning
in Uncertain Semantic Maps

Yiannis Kantaros, Qi Jin, and George J. Pappas The authors are with the School
of Engineering and Applied Sciences, GRASP Lab, University of Pennsylvania,
Philadelphia, PA, 19104, USA.
<EMAIL_ADDRESS>This
material is based upon work supported by the Air Force Research Laboratory
(AFRL) and the Defense Advanced Research Projects Agency (DARPA) under
Contract No. FA8750-18-C-0090.

###### Abstract

This paper addresses a multi-robot mission planning problem in partially
unknown semantic environments. The environment is assumed to have known
geometric structure (e.g., walls) and to be occupied by static labeled
landmarks with uncertain positions and classes. This modeling approach gives
rise to an uncertain semantic map generated by semantic SLAM algorithms. Our
goal is to design control policies for robots equipped with perception systems
so that they can accomplish complex collaborative high level tasks captured by
global temporal logic specifications. To account for environmental and
perceptual uncertainty, we extend a fragment of Linear Temporal Logic (LTL),
called co-safe LTL, by including perception-based predicates allowing us to
incorporate uncertainty and probabilistic satisfaction requirements directly
into the task specification. The perception-based LTL planning problem gives
rise to an optimal control problem, solved by a novel sampling-based
algorithm, that generates open-loop control policies that can be updated
online to adapt to the map that is continuously learned by existing semantic
SLAM methods. We provide extensive experiments that corroborate the
theoretical analysis and show that the proposed algorithm can address large-
scale planning tasks in the presence of uncertainty.

##  I Introduction

##  II Problem Definition

##  III Planning In Probabilistic Semantic Maps

##  IV Completeness & Optimality

##  V Numerical Experiments

##  VI Conclusion

In this paper we addressed a multi-robot mission planning problem in partially
unknown semantic environments. The environment is assumed to have known
geometric structure (e.g., walls) and to be occupied by static labeled
landmarks with uncertain positions and classes. We proposed a sampling-based
approach to design control policies for robots equipped with perception
systems so that they can accomplish complex collaborative high level tasks
captured by global temporal logic specifications that take into account
perceptual and environmental uncertainty. We showed that the proposed
sampling-based algorithm is probabilistically complete, asymptotically
optimal. Finally, we provided extensive experimental case studies verifying
the efficiency of the proposed algorithm. Our future work will focus on
designing control policies that can efficiently react to both metric/semantic
and geometric environmental uncertainty and adapt to potentially dynamic
changes in the environmental structure.

##  Appendix A Designing Biased Sampling Functions

##  Appendix B Proof of Completeness & Optimality