end we will identify the leading part of the action comparing the result with the expansions -. We can identify the operators $A_0$ and $A_1$ appearing in the as follow: $$A_0 =\left(\begin{array}{cc}
L_\star (V_{A,\phi})\textbf{I} + L_\star(F_A^{\mu\nu})\Gamma^\Omega_{\mu\nu}
&i(\Gamma^\mu + \Omega\Gamma^{\mu+4})\Gamma_9 L_\star (D_\mu \phi)\\
i(\Gamma^\mu + \Omega \Gamma^{\mu+4} )\Gamma_9 L_\star (\overline{D_\mu \phi})&
L_\star (V_{B,\phi})\textbf{I} + L_\star(F_B^{\mu\nu})\Gamma^\Omega_{\mu\nu}
\end{array}\right) \label{A0}$$ $$A_1 =\left(\begin{array}{cc}
2i(1+\Omega^2)L_\star(A^\mu)\nabla_\mu^{(\Omega)} &0\\
0& 2i(1+\Omega^2)L_\star(B^\mu)\nabla_\mu^{(\Omega)}
\end{array}\right)\label{A1}$$ with $$V_{A,\phi}= \phi\star\bar{\phi}+(1+\Omega^2)(i \partial_\mu A^\mu +A_\mu\star A^\mu) , \  V_{B,\phi}= \bar{\phi}\star\phi+(1+\Omega^2)(i\partial_\mu B^\mu+B_\mu\star A^\mu)$$ $\nabla_\mu^{(\Omega)}$ are define as $\nabla_\mu^{(\Omega)}=\frac{1}{1+\Omega^2}\left(\partial_\mu-i\Omega^2 M_\bullet(\tilde{x}_\mu)\right) $. We are allowed to split the traces in two parts a matrix trace and the continuous one. After the matrices trace computations we obtain [@non-com-tri] for the $A$ field: $$\begin{aligned}
&&\textrm{Tr}(e^{-t\mathcal{D}_A^2})\nonumber \\
&&=\Bigg\{16\cosh^4(t\Omega)\textrm{tr}(e^{-tH_4^2})-tT(16V_{A,\phi})-t\mathcal{T}_\mu(32i(1+\Omega^2)A^\mu)\nonumber \\
&&+\frac{t^2}{2}\mathcal{T}\Big(16V_{A,\phi}\star V_{A,\phi}+16(1+\Omega^2)D_\mu\phi\star\overline{D^\mu\phi}+8(1+\Omega^2)F_{\mu\nu}^A F^{\mu\nu}_A\nonumber \\
&&+ 32i(1+\Omega^2)A^\mu\star\partial_\mu V_{A,\phi}\Big)\nonumber \\
&&+ \frac{t^2}{2}\mathcal{T}_\mu\Big(32i(1+\Omega^2)A^\mu\star V_{A,\phi}+32i(1+\Omega^2)V_{A,\phi}\star A^\mu\nonumber \\
&&-64(1+\Omega^2)^2A^\nu\star\partial_\nu A^\mu\Big)+\frac{t^2}{2}\mathcal{T}_{\mu\nu}\left(-64(1+\Omega^2)^2 A^\mu\star A^\nu \right)\nonumber \\
&&- \frac{t^3}{6}\mathcal{T}_{\mu\nu}\Big(-64(1+\Omega^2)^2V_{A,\phi}\star \partial^\mu A^\nu -64i(1+\Omega^2)^2A^\nu\star\partial^\nu V_{A,\phi}\nonumber \\
&&+64(1+\Omega^2)^3A_\rho\star(\delta^{\rho\mu}\triangle A^\nu +2\partial^\rho\partial^\mu A^\nu)\nonumber \\
&&- 64(1+\Omega^2)^2(V_{A,\phi}\star A^\mu\star A^\nu + A^\mu\star V_{A,\phi}\star A^\nu +A^\mu\star A^\nu\star V_{A,\phi})\nonumber \\
&&-128i(1+\Omega^2)^3\left(A^\rho\star(\partial_\rho A^\mu)\star A^\nu +A^\rho\star A^\mu(\partial_\rho A^\nu\right)\Big)\nonumber \\
&&- \frac{t^3}{6}\mathcal{T}_{\mu\nu\rho}\left(128(1+\Omega^2)^3A^\mu\star\partial^\nu A^\rho -128i(1+\Omega^2)^3A^\nu\star A^\mu\star A^\rho\right)\nonumber \\
&&+ \frac{t^3}{6}\tilde{\mathcal{T}}_{\mu\nu}\left(512i\Omega^2(1+\Omega^2)(\Theta^{-1})^{\rho\nu}A^\mu\star A_\rho\right)\nonumber \\
&&+ \frac{t^4}{24}\mathcal{T}_{\mu\nu\rho\sigma}\Big(-256(1+\Omega^2)^4A^\mu\star\partial^\nu\partial^\rho A^\sigma +512i(1+\Omega^2)^3 A^\mu\star A^\nu\star \partial^\rho A^\sigma\nonumber \\
&&+256i(1+\Omega^2)^4 A^\mu\star(\partial^\nu A^\rho)\star A^\sigma +256(1+\Omega^2)^4 A^\mu\star A^\nu\star A^\rho\star A^\sigma\Big)\Bigg\}\nonumber \\
&& + \mbox{$B$ 