L_{\mathrm{G}}=-\kappa^{-1}(\Lambda+a \mathcal{R})+L_{\mathcal{R}^{2}}+\kappa^{-1} L_{\mathcal{T}^{2}}L_\mathrm{G} = -\kappa^{-1}(\Lambda + a\mathcal{R}) + L_{\mathcal{R}^2} + \kappa^{-1} L_{\mathcal{T}^2},conf 1.000
\begin{aligned} & L_{\mathcal{R}^{2}}=\alpha_{1} \mathcal{R}^{2}+\alpha_{2} \mathcal{R}_{a b} \mathcal{R}^{a b}+\alpha_{3} \mathcal{R}_{a b} \mathcal{R}^{b a}+\alpha_{4} \mathcal{R}_{a b c d} \mathcal{R}^{a b c d}+\alpha_{5} \mathcal{R}_{a b c d} \mathcal{R}^{a c b d}+\alpha_{6} \mathcal{R}_{a b c d} \mathcal{R}^{c d a b}, \\ & L_{\mathcal{T}^{2}}=\beta_{1} \mathcal{T}_{a b c} \mathcal{T}^{a b c}+\beta_{2} \mathcal{T}_{a b c} \mathcal{T}^{b a c}+\beta_{3} \mathcal{T}_{a} \mathcal{T}^{a}, \end{aligned}\begin{aligned} L_{\mathcal{R}^2} & = & \alpha_1 \mathcal{R}^2 + \alpha_2 \mathcal{R}_{ab}\mathcal{R}^{ab} + \alpha_3 \mathcal{R}_{ab}\mathcal{R}^{ba} + \alpha_4 \mathcal{R}_{abcd}\mathcal{R}^{abcd} + \alpha_5 {\cal R}_{abcd}{\cal R}^{acbd} + \alpha_6 {\cal R}_{abcd}{\cal R}^{cdab}, \\[1mm] L_{{\cal T}^2} & = & \beta_1 {\cal T}_{abc}{\cal T}^{abc} + \beta_2 {\cal T}_{abc}{\cal T}^{bac} + \beta_3 {\cal T}_a{\cal T}^a, \end{aligned}conf 0.774
L_{\mathrm{G}}=\alpha_{1} \mathcal{R}^{2}+\alpha_{2} \mathcal{R}_{a b} \mathcal{R}^{a b}+\alpha_{3} \mathcal{R}_{a b} \mathcal{R}^{b a}+\alpha_{4} \mathcal{R}_{a b c d} \mathcal{R}^{a b c d}+\alpha_{5} \mathcal{R}_{a b c d} \mathcal{R}^{a c b d}+\alpha_{6} \mathcal{R}_{a b c d} \mathcal{R}^{c d a b}+\xi \mathcal{H}_{a b} \mathcal{H}^{a b} \equiv L_{\mathcal{R}^{2}}+L_{\mathcal{H}^{2}}L_\mathrm{G} = \alpha_1 \mathcal{R}^2 + \alpha_2 \mathcal{R}_{ab}\mathcal{R}^{ab} + \alpha_3 \mathcal{R}_{ab}\mathcal{R}^{ba} + \alpha_4 \mathcal{R}_{abcd}\mathcal{R}^{abcd} + \alpha_5 {\cal R}_{abcd}{\cal R}^{acbd} + \alpha_6 {\cal R}_{abcd}{\cal R}^{cdab} + \xi {\cal H}_{ab}{\cal H}^{ab} \equiv L_{{\cal R}^2} + L_{{\cal H}^2},conf 0.855
S_{\mathrm{M}}=\int L_{\mathrm{M}}\left(\varphi, \partial_{\mu} \varphi\right) d^{4} xS_\mathrm{M} = \int L_\mathrm{M}(\varphi,\partial_\mu\varphi)\,d^4x,conf 1.000
x^{\prime \mu}=e^{\rho}\left(\Lambda^{\mu}{ }_{\nu} x^{\nu}+a^{\mu}\right){x'}^\mu=e^\rho({\Lambda^\mu}_\nu x^\nu + a^\mu),conf 0.842
\varphi^{\prime}\left(x^{\prime}\right)=e^{w \rho} \mathrm{~S}(\Lambda) \varphi(x)\varphi'(x')=e^{w\rho}{\text\mathsf{S}}(\Lambda)\varphi(x),conf 0.727
\partial_{\mu}^{\prime} \varphi^{\prime}\left(x^{\prime}\right)=e^{(w-1) \rho} \Lambda_{\mu}{ }^{\nu} \mathrm{S}(\Lambda) \partial_{\nu} \varphi(x)\partial'_\mu\varphi'(x')=e^{(w-1)\rho} {\Lambda_\mu}^\nu {\text\mathsf{S}}(\Lambda)\partial_\nu \varphi(x).conf 0.808
L_{\mathrm{M}}\left(\varphi^{\prime}\left(x^{\prime}\right), \partial_{\mu}^{\prime} \varphi^{\prime}\left(x^{\prime}\right)\right)=e^{-4 \rho} L_{\mathrm{M}}\left(\varphi(x), \partial_{\mu} \varphi(x)\right)L_\mathrm{M}(\varphi'(x'),\partial'_\mu \varphi'(x')) =e^{-4\rho}L_\mathrm{M}(\varphi(x),\partial_\mu \varphi(x)),conf 0.813
L_{\mathrm{KG}}=\frac{1}{2} \eta^{\mu \nu} \partial_{\mu} \phi \partial_{\nu} \phi-\frac{1}{2} m^{2} \phi^{2}L_\mathrm{KG} = {\frac{1}{2}}\eta^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi -{\frac{1}{2}}m^2\phi^2,conf 1.000
\begin{aligned} L_{\mathrm{KG}}^{\prime} & =\frac{1}{2} \gamma^{\prime \mu \nu} \partial_{\mu}^{\prime} \phi^{\prime} \partial_{\nu}^{\prime} \phi^{\prime}-\frac{1}{2} m^{2} \phi^{\prime 2} \\ & =\frac{1}{2} e^{2(w-1) \rho} \eta^{\mu \nu} \partial_{\mu} \phi \partial_{\nu} \phi-\frac{1}{2} m^{2} e^{2 w \rho} \phi^{2} . \end{aligned}\begin{aligned} L'_\mathrm{KG} & = & {\frac{1}{2}}\gamma^{\prime\mu\nu}\partial'_\mu\phi'\,\partial'_\nu\phi' -{\frac{1}{2}}m^2\phi^{\prime 2} \\ & = & {\frac{1}{2}}e^{2(w-1)\rho}\eta^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi -{\frac{1}{2}}m^2e^{2w\rho}\phi^2. \end{aligned}conf 0.783
L_{\mathrm{D}}=\frac{1}{2} i\left[\bar{\psi} \gamma^{\mu} \partial_{\mu} \psi-\left(\partial_{\mu} \bar{\psi}\right) \gamma^{\mu} \psi\right]-m \bar{\psi} \psi \equiv \frac{1}{2} i \bar{\psi} \gamma^{\mu} \overleftrightarrow{\partial_{\mu}} \psi-m \bar{\psi} \psiL_\mathrm{D} = {\frac{1}{2}}i [\bar{\psi}\gamma^\mu\partial_\mu\psi-(\partial_\mu\bar{\psi})\gamma^\mu\psi] -m\bar{\psi}\psi \equiv {\frac{1}{2}}i\bar{\psi}\gamma^\mu{\stackrel{\leftrightarrow}{\partial_\mu}}\psi - m\bar{\psi}\psi.conf 0.956
\begin{aligned} L_{\mathrm{D}}^{\prime} & =\frac{1}{2} i \bar{\psi}^{\prime} \gamma^{\mu} \stackrel{\leftrightarrow}{\partial_{\mu}^{\prime}} \psi^{\prime}-m \bar{\psi}^{\prime} \psi^{\prime} \\ & =\frac{1}{2} i e^{(2 w-1) \rho} \bar{\psi} \gamma^{\mu} \overleftrightarrow{\partial_{\mu}} \psi-m e^{2 w \rho} \bar{\psi} \psi \end{aligned}\begin{aligned} L'_\mathrm{D} & = & {\frac{1}{2}}i \bar{\psi}'\gamma^\mu{\stackrel{\leftrightarrow}{\partial'_\mu}}\psi' - m\bar{\psi}'\psi' \\ &=& {\frac{1}{2}}i e^{(2w-1)\rho} \bar{\psi}\gamma^\mu{\stackrel{\leftrightarrow}{\partial_\mu}}\psi - me^{2w\rho}\bar{\psi}\psi. \end{aligned}conf 0.740
L_{\mathrm{M}}=\frac{1}{2} i \bar{\psi} \gamma^{\mu} \stackrel{\leftrightarrow}{\partial_{\mu}} \psi+\frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi-\mu \phi \bar{\psi} \psiL_\mathrm{M} = {\frac{1}{2}}i\bar{\psi}\gamma^\mu{\stackrel{\leftrightarrow}{\partial_\mu}}\psi + {\frac{1}{2}}\partial_\mu\phi\,\partial^\mu\phi - \mu\phi\bar{\psi}\psi,conf 0.992
L_{\mathrm{EM}}=-\frac{1}{4} F_{\mu \nu} F^{\mu \nu}-J^{\mu} A_{\mu}L_\mathrm{EM} = -{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu} - J^\mu A_\muconf 1.000
\begin{aligned} L_{\mathrm{EM}}^{\prime} & =-\frac{1}{4} F_{\mu \nu}^{\prime} F^{\prime \mu \nu}-J^{\prime \mu} A_{\mu}^{\prime} \\ & =-\frac{1}{4} e^{2(w-1) \rho} F_{\mu \nu} F^{\mu \nu}-e^{\left(w+w_{J}\right) \rho} J^{\mu} A_{\mu}, \end{aligned}\begin{aligned} L'_\mathrm{EM} & = & -{\frac{1}{4}}F^\prime_{\mu\nu}F^{\prime\mu\nu} - J^{\prime\mu} A^\prime_\mu \\ & = & -{\frac{1}{4}}e^{2(w-1)\rho} F_{\mu\nu}F^{\mu\nu} - e^{(w+w_J)\rho} J^\mu A_\mu, \end{aligned}conf 0.897
\varphi^{\prime}\left(x^{\prime}\right)=e^{w \rho(x)} \mathrm{S}(\Lambda(x)) \varphi(x)\varphi'(x')=e^{w\rho(x)}{\text\mathsf{S}}(\Lambda(x))\varphi(x).conf 0.754
\gamma_{\mu \nu}^{\prime}\left(x^{\prime}\right)=\frac{\partial x^{\lambda}}{\partial x^{\prime \mu}} \frac{\partial x^{\sigma}}{\partial x^{\prime \nu}} e^{2 \rho(x)} \gamma_{\lambda \sigma}(x)\gamma'_{\mu\nu}(x') = \frac{\partial x^\lambda}{\partial {x'}^\mu}\frac{\partial x^\sigma}{\partial {x'}^\nu}\,e^{2\rho(x)}\, \gamma_{\lambda\sigma}(x).conf 0.862
\partial_{\mu}^{\prime} \varphi^{\prime}\left(x^{\prime}\right)=\frac{\partial x^{\nu}}{\partial x^{\prime \mu}} e^{w \rho(x)}\left[\mathrm{S}(\Lambda(x)) \partial_{\nu} \varphi(x)+\partial_{\nu} \mathrm{S}(\Lambda(x)) \varphi(x)+w \partial_{\nu} \rho \mathrm{S}(\Lambda(x)) \varphi(x)\right]\partial'_\mu \varphi'(x')=\frac{\partial x^\nu}{\partial {x'}^\mu}e^{w\rho(x)} [{\text\mathsf{S}}(\Lambda(x))\partial_\nu\varphi(x)+\partial_\nu {\text\mathsf{S}}(\Lambda(x)) \varphi(x)+ w\,\partial_\nu\rho\, {\text\mathsf{S}}(\Lambda(x)) \varphi(x)].conf 0.746
\begin{aligned} D_{\mu}^{*} \varphi(x) & \equiv\left[\partial_{\mu}+\frac{1}{2} A^{a b}{ }_{\mu}(x) \Sigma_{a b}+w B_{\mu}(x)\right] \varphi(x) \\ & =\left[D_{\mu}+w B_{\mu}(x)\right] \varphi(x) \end{aligned}\begin{aligned} D^\ast_\mu\varphi(x) &\equiv &[\partial_\mu +{\frac{1}{2}}{A^{ab}}_\mu(x) \Sigma_{ab} + wB_\mu(x)]\varphi(x) \\ &=& [D_\mu + wB_\mu(x)]\varphi(x), \end{aligned}conf 0.949
\left[\Sigma_{a b}, \Sigma_{c d}\right]=\eta_{a d} \Sigma_{b c}-\eta_{b d} \Sigma_{a c}+\eta_{b c} \Sigma_{a d}-\eta_{a c} \Sigma_{b d}[\Sigma_{ab},\Sigma_{cd}] =\eta_{ad}\Sigma_{bc}-\eta_{bd}\Sigma_{ac}+\eta_{bc} \Sigma_{ad}-\eta_{ac}\Sigma_{bd}.conf 1.000
D_{\mu}^{* \prime} \varphi^{\prime}\left(x^{\prime}\right)=\frac{\partial x^{\nu}}{\partial x^{\prime \mu}} e^{w \rho(x)} \mathrm{S}(\Lambda(x)) D_{\nu}^{*} \varphi(x)D^{\ast\prime}_\mu\varphi'(x') = \frac{\partial x^\nu}{\partial {x'}^\mu} e^{w\rho(x)}{\text\mathsf{S}}(\Lambda(x))D^\ast_\nu\varphi(x).conf 0.745
A^{\prime a b}{ }_{\mu}\left(x^{\prime}\right)=\frac{\partial x^{\nu}}{\partial x^{\prime \mu}}\left[\Lambda^{a}{ }_{c}(x) \Lambda^{b}{ }_{d}(x) A^{c d}{ }_{\nu}(x)-\Lambda^{b c}(x) \partial_{\nu} \Lambda^{a}{ }_{c}(x)\right]{{A'}^{ab}}_\mu(x') \!=\! \frac{\partial x^\nu}{\partial {x'}^\mu} [{\Lambda^a}_c(x){\Lambda^b}_d(x) {A^{cd}}_\nu(x)\!-\!\Lambda^{bc}(x) \partial_\nu{\Lambda^a}_c(x)],conf 0.868
B_{\mu}^{\prime}\left(x^{\prime}\right)=\frac{\partial x^{\nu}}{\partial x^{\prime \mu}}\left[B_{\nu}(x)-\partial_{\nu} \rho(x)\right]B'_\mu(x') = \frac{\partial {x}^\nu}{\partial {x'}^\mu} [B_\nu(x) - \partial_\nu \rho(x)].conf 0.834
D_{\mu}^{*} \varphi(x)=\left[\partial_{\mu}^{*}+\frac{1}{2} A^{a b}{ }_{\mu}(x) \Sigma_{a b}\right] \varphi(x)D^\ast_\mu \varphi(x) = [\partial^\ast_\mu + {\frac{1}{2}}{A^{ab}}_\mu(x) \Sigma_{ab}] \varphi(x),conf 0.873
\partial_{\mu}^{*} \equiv \partial_{\mu}+w B_{\mu}(x)\partial^\ast_\mu \equiv \partial_\mu + w B_\mu(x).
conf 0.899
h_{a}^{\prime \mu}\left(x^{\prime}\right)=\frac{\partial x^{\prime \mu}}{\partial x^{\nu}} e^{-\rho(x)} \Lambda_{a}{ }^{b}(x) h_{b}{ }^{\nu}(x){h'_a}^\mu(x') = \frac{\partial {x'}^\mu}{\partial {x}^\nu}\, e^{-\rho(x)}{\Lambda_a}^b(x){h_b}^\nu(x).conf 0.818
h_{a}{ }^{\mu} b^{a}{ }_{\nu}=\delta_{\nu}^{\mu} \quad \text { and } \quad h_{a}{ }^{\mu} b^{c}{ }_{\mu}=\delta_{a}^{c}{h_a}^\mu {b^a}_\nu = \delta_\nu^\mu \quad\text{and}\quad {h_a}^\mu {b^c}_\mu = \delta_a^c.conf 0.945
\begin{aligned} \mathcal{D}_{a}^{*} \varphi(x) & \equiv h_{a}{ }^{\mu}(x) D_{\mu}^{*} \varphi(x) \\ & =h_{a}{ }^{\mu}(x)\left[\partial_{\mu}+\frac{1}{2} A^{c d}{ }_{\mu}(x) \Sigma_{c d}+w B_{\mu}(x)\right] \varphi(x), \end{aligned}\begin{aligned} \hspace*{-5mm}\mathcal{D}^\ast_a\varphi(x) &\equiv &{h_a}^\mu(x)D^\ast_\mu\varphi(x) \\ &=& {h_a}^\mu(x)[\partial_\mu \!+\! {\frac{1}{2}}{A^{cd}}_\mu(x) \Sigma_{cd}\! + \! wB_\mu(x)]\varphi(x), \end{aligned}conf 0.870
\mathcal{D}_{a}^{* \prime} \varphi^{\prime}\left(x^{\prime}\right)=e^{(w-1) \rho(x)} \Lambda_{a}{ }^{b}(x) \mathrm{S}(\Lambda(x)) \mathcal{D}_{b}^{*} \varphi(x)\mathcal{D}^{\ast\prime}_a\varphi'(x') = e^{(w-1)\rho(x)}{\Lambda_a}^b(x) {\text\mathsf{S}}(\Lambda(x)) \mathcal{D}^\ast_b\varphi(x).conf 0.788
L_{\mathrm{M}}\left(\varphi^{\prime}\left(x^{\prime}\right) ; \mathcal{D}_{a}^{* \prime} \varphi^{\prime}\left(x^{\prime}\right)\right)=e^{-4 \rho(x)} L_{\mathrm{M}}\left(\varphi(x) ; \mathcal{D}_{a}^{*} \varphi(x)\right)L_\mathrm{M}(\varphi'(x');\mathcal{D}^{\ast\prime}_a \varphi'(x')) =e^{-4\rho(x)}L_\mathrm{M}(\varphi(x);\mathcal{D}^\ast_a \varphi(x)).conf 0.792
S_{\mathrm{M}}=\int h^{-1} L_{\mathrm{M}}\left(\varphi, \mathcal{D}_{a}^{*} \varphi\right) d^{4} xS_\mathrm{M} = \int h^{-1}L_\mathrm{M}(\varphi,\mathcal{D}^\ast_a\varphi)\,d^4x,conf 0.916
D_{\mu}^{*} h_{a}^{\nu}=\partial_{\mu}^{*} h_{a}^{\nu}+\frac{1}{2} A_{\mu}^{c d}\left(\Sigma_{c d}^{1}\right)_{a}{ }^{b} h_{b}{ }^{\nu}=\partial_{\mu}^{*} h_{a}{ }^{\nu}-A_{a \mu}^{b} h_{b}{ }^{\nu}D^\ast_\mu {h_a}^\nu = \partial^\ast_\mu {h_a}^\nu + {\frac{1}{2}}{A^{cd}}_\mu(\Sigma^1_{cd}{)_a}^b{h_b}^\nu = \partial^\ast_\mu {h_a}^\nu - {A^{b}}_{a\mu}{h_b}^\nu,conf 0.797
D_{\mu}^{*} b^{a}{ }_{\nu}=\partial_{\mu}^{*} b^{a}{ }_{\nu}+A^{a}{ }_{b \mu} b^{b}{ }_{\nu}D^\ast_\mu {b^a}_\nu = \partial^\ast_\mu {b^a}_\nu + {A^a}_{b\mu}{b^b}_\nu.conf 0.836
D_{\mu}^{*} \equiv \partial_{\mu}^{*}+{ }^{0} \Gamma^{\sigma}{ }_{\rho \mu} \mathrm{X}^{\rho}{ }_{\sigma}+\frac{1}{2} A^{a b}{ }_{\mu} \Sigma_{a b}={ }^{0} \nabla_{\mu}^{*}+\frac{1}{2} A^{a b}{ }_{\mu} \Sigma_{a b}D^\ast_\mu \equiv \partial^\ast_\mu + {{}^0}{\Gamma^\sigma}_{\rho\mu} {{\text\mathsf{X}}^\rho}_\sigma + {\frac{1}{2}}{A^{ab}}_\mu\Sigma_{ab} = {{}^0}\nabla^\ast_\mu + {\frac{1}{2}}{A^{ab}}_\mu\Sigma_{ab},conf 0.834
D_{\mu}^{*} h_{a}{ }^{\nu}=\partial_{\mu}^{*} h_{a}{ }^{\nu}+{ }^{0} \Gamma^{\nu}{ }_{\rho \mu} h_{a}{ }^{\rho}-A^{b}{ }_{a \mu} h_{b}{ }^{\nu}D^\ast_\mu {h_a}^\nu = \partial^\ast_\mu {h_a}^\nu + {{}^0}{\Gamma^\nu}_{\rho\mu} {h_a}^\rho - {A^{b}}_{a\mu}{h_b}^\nu.conf 0.842
\left[{ }^{0} \nabla_{\mu},{ }^{0} \nabla_{\nu}\right] \varphi=0[{}^{0}\nabla_\mu,^{0}\nabla_\nu]\varphi=0,conf 0.974
\left[D_{\mu}^{*}, D_{\nu}^{*}\right] \varphi=\frac{1}{2} R^{a b}{ }_{\mu \nu} \Sigma_{a b} \varphi+w H_{\mu \nu} \varphi[D^\ast_\mu,D^\ast_\nu]\varphi = {\frac{1}{2}}{R^{ab}}_{\mu\nu}\Sigma_{ab}\varphi + wH_{\mu\nu}\varphi,conf 0.885
\begin{aligned} R^{a b}{ }_{\mu \nu} & \equiv \partial_{\mu} A^{a b}{ }_{\nu}-\partial_{\nu} A^{a b}{ }_{\mu}+A^{a}{ }_{c \mu} A^{c b}{ }_{\nu}-A^{a}{ }_{c \nu} A^{c b}{ }_{\mu}, \\ H_{\mu \nu} & \equiv \partial_{\mu} B_{\nu}-\partial_{\nu} B_{\mu} . \end{aligned}\begin{aligned} \hspace*{-7mm}{R^{ab}}_{\mu\nu} & \equiv & \partial_\mu {A^{ab}}_\nu \!-\! \partial_\nu {A^{ab}}_\mu \!+\!{A^a}_{c\mu}{A^{cb}}_\nu \!-\! {A^a}_{c\nu}{A^{cb}}_\mu, \\ H_{\mu\nu} & \equiv & \partial_\mu B_\nu - \partial_\nu B_\mu. \end{aligned}conf 0.919
\left[\mathcal{D}_{c}^{*}, \mathcal{D}_{d}^{*}\right] \varphi=\frac{1}{2} \mathcal{R}^{a b}{ }_{c d} \Sigma_{a b} \varphi+w \mathcal{H}_{c d} \varphi-\mathcal{T}^{* a}{ }_{c d} \mathcal{D}_{a}^{*} \varphi[\mathcal{D}^\ast_c,\mathcal{D}^\ast_d]\varphi = {\frac{1}{2}}{\mathcal{R}^{ab}}_{cd}\Sigma_{ab}\varphi + w\mathcal{H}_{cd}\varphi - {\mathcal{T^\ast}^a}_{cd}\mathcal{D}^\ast_a\varphi,conf 0.868
\mathcal{T}^{* a}{ }_{b c} \equiv h_{b}{ }^{\mu} h_{c}{ }^{\nu} T^{* a}{ }_{\mu \nu} \equiv h_{b}{ }^{\mu} h_{c}{ }^{\nu}\left(D_{\mu}^{*} b^{a}{ }_{\nu}-D_{\nu}^{*} b^{a}{ }_{\mu}\right){\mathcal{T^\ast}^a}_{bc} \equiv {h_b}^{\mu}{h_c}^{\nu} {T^{\ast a}}_{\mu\nu} \equiv {h_b}^{\mu}{h_c}^{\nu}(D^\ast_\mu {b^a}_\nu -D^\ast_\nu {b^a}_\mu),conf 0.816
\mathcal{T}^{* a}{ }_{b c}=-b^{a}{ }_{\mu}\left(\mathcal{D}_{b}^{*} h_{c}{ }^{\mu}-\mathcal{D}_{c}^{*} h_{b}{ }^{\mu}\right){\mathcal{T}^{\ast a}}_{bc} = -{b^a}_\mu(\mathcal{D}^\ast_b{h_c}^\mu - \mathcal{D}^\ast_c{h_b}^\mu).conf 0.795
\mathcal{T}^{* a}{ }_{b c}=\mathcal{T}^{a}{ }_{b c}+\delta_{c}^{a} \mathcal{B}_{b}-\delta_{b}^{a} \mathcal{B}_{c}{\mathcal{T}^{\ast a}}_{bc} = {\mathcal{T}^{a}}_{bc} + \delta^a_c\mathcal{B}_b-\delta^a_b\mathcal{B}_c,conf 0.882
c^{* a}{ }_{b c} \equiv h_{b}{ }^{\mu} h_{c}{ }^{\nu}\left(\partial_{\mu}^{*} b^{a}{ }_{\nu}-\partial_{\nu}^{*} b^{a}{ }_{\mu}\right){c^{\ast a}}_{bc} \equiv {h_b}^\mu {h_c}^\nu (\partial^\ast_\mu {b^a}_\nu-\partial^\ast_\nu {b^a}_\mu),conf 0.823
\mathcal{A}_{a b c}=\frac{1}{2}\left(c_{a b c}^{*}+c_{b c a}^{*}-c_{c a b}^{*}\right)-\frac{1}{2}\left(\mathcal{T}_{a b c}^{*}+\mathcal{T}_{b c a}^{*}-\mathcal{T}_{c a b}^{*}\right)\mathcal{A}_{abc} = {\frac{1}{2}}(c^\ast_{abc}+c^\ast_{bca}-c^\ast_{cab}) -{\frac{1}{2}}(\mathcal{T}^\ast_{abc}+\mathcal{T}^\ast_{bca}-\mathcal{T}^\ast_{cab}).conf 0.771
\mathcal{A}_{a b c}={ }^{0} \mathcal{A}_{a b c}^{*}(h, \partial h, B)+\mathcal{K}_{a b c}^{*}(h, \partial h, A, B)\mathcal{A}_{abc} = {^0}\!\mathcal{A}^\ast_{abc}(h,\partial h,B) +\mathcal{K}^\ast_{abc} (h,\partial h,A,B),conf 0.884
{ }^{0} D_{\mu}^{*} \varphi \equiv\left(\partial_{\mu}^{*}+\frac{1}{2}{ }^{0} A^{* a b}{ }_{\mu} \Sigma_{a b}\right) \varphi{^0}\!D^\ast_\mu\varphi \equiv (\partial^\ast_\mu +{\frac{1}{2}} {^0}\!{A^{\ast ab}}_\mu\Sigma_{ab})\varphi,conf 0.798
\mathcal{D}_{a}^{*} \varphi=\left({ }^{0} \mathcal{D}_{a}^{*}+\frac{1}{2} \mathcal{K}^{* b c}{ }_{a} \Sigma_{b c}\right) \varphi\mathcal{D}^\ast_a\varphi = ({^0}\mathcal{D}^\ast_a + {\frac{1}{2}}{\mathcal{K}^{\ast bc}}_a\Sigma_{bc})\varphi, ,conf 0.831
\left[\mathcal{D}_{a}^{*},\left[\mathcal{D}_{b}^{*}, \mathcal{D}_{c}^{*}\right]\right] \varphi+\left[\mathcal{D}_{c}^{*},\left[\mathcal{D}_{a}^{*}, \mathcal{D}_{b}^{*}\right]\right] \varphi+\left[\mathcal{D}_{b}^{*},\left[\mathcal{D}_{c}^{*}, \mathcal{D}_{a}^{*}\right]\right] \varphi=0[\mathcal{D}^\ast_a,[\mathcal{D}^\ast_b,\mathcal{D}^\ast_c]]\varphi + [\mathcal{D}^\ast_c,[\mathcal{D}^\ast_a,\mathcal{D}^\ast_b]]\varphi + [\mathcal{D}^\ast_b,[\mathcal{D}^\ast_c,{\cal D}^\ast_a]]\varphi =0.conf 0.765
\begin{aligned} \mathcal{D}_{[a}^{*} \mathcal{R}^{d e}{ }_{b c]}-\mathcal{T}^{* f}{ }_{[a b} \mathcal{R}^{d e}{ }_{c] f} & =0, \\ \mathcal{D}_{[a}^{*} \mathcal{T}^{* d}{ }_{b c]}-\mathcal{T}^{* e}{ }_{[a b} \mathcal{T}^{* d}{ }_{c] e}-\mathcal{R}^{d}{ }_{[a b c]}+\mathcal{H}_{[a b} \delta_{c]}^{d} & =0, \\ \mathcal{D}_{[a}^{*} \mathcal{H}_{b c]}-\mathcal{T}^{* e}{ }_{[a b} \mathcal{H}_{c] e} & =0 . \end{aligned}
\mathcal{D}_{a}^{*} \mathcal{R}^{a e}{ }_{b c}-2 \mathcal{D}_{[b}^{*} \mathcal{R}^{e}{ }_{c]}-2 \mathcal{T}^{* f}{ }_{a[b} \mathcal{R}^{a e}{ }_{c] f}-\mathcal{T}^{* f}{ }_{b c} \mathcal{R}^{e}{ }_{f}=0\mathcal{D}^\ast_{a}{\mathcal{R}^{ae}}_{bc}-2 \mathcal{D}^\ast_{[b}{\mathcal{R}^{e}}_{c]} -2{\mathcal{T}^{\ast f}}_{a[b} {\mathcal{R}^{ae}}_{c]f} -{\mathcal{T}^{\ast f}}_{bc} {\mathcal{R}^{e}}_{f} = 0.conf 0.869
\mathcal{D}_{a}^{*}\left(\mathcal{R}^{a}{ }_{c}-\frac{1}{2} \delta_{c}^{a} \mathcal{R}\right)+\mathcal{T}^{* f}{ }_{b c} \mathcal{R}^{b}{ }_{f}+\frac{1}{2} \mathcal{T}^{* f}{ }_{a b} \mathcal{R}^{a b}{ }_{c f}=0\mathcal{D}^\ast_{a}({\mathcal{R}^{a}}_{c} -{\frac{1}{2}}\delta^a_c\mathcal{R}) +{\mathcal{T}^{\ast f}}_{bc} {\mathcal{R}^{b}}_{f} +{\frac{1}{2}}{\mathcal{T}^{\ast f}}_{ab} {\mathcal{R}^{ab}}_{cf} = 0.conf 0.869
\mathcal{D}_{a}^{*} \mathcal{T}^{* a}{ }_{b c}+2 \mathcal{D}_{[b}^{*} \mathcal{T}_{c]}^{*}+\mathcal{T}^{* e}{ }_{b c} \mathcal{T}_{e}^{*}+2 \mathcal{R}_{[b c]}+2 \mathcal{H}_{b c}=0\mathcal{D}^\ast_{a}{\mathcal{T}^{\ast a}}_{bc} +2\mathcal{D}^\ast_{[b}\mathcal{T}^\ast_{c]} +{\mathcal{T}^{\ast e}}_{bc}\mathcal{T}^\ast_e +2{\mathcal{R}}_{[bc]} + 2\mathcal{H}_{bc} = 0.conf 0.819
S_{\mathrm{G}}=\int h^{-1} L_{\mathrm{G}}\left(\mathcal{R}_{a b c d}, \mathcal{T}_{a b c}^{*}, \mathcal{H}_{a b}\right) d^{4} xS_\mathrm{G} = \int h^{-1} L_\mathrm{G}(\mathcal{R}_{abcd},\mathcal{T}^\ast_{abc},\mathcal{H}_{ab})\,d^4x,conf 0.934
L_{\mathrm{G}}=L_{\mathcal{R}^{2}}+L_{\mathcal{H}^{2}}L_\mathrm{G} = L_{\mathcal{R}^2} + L_{\mathcal{H}^2},conf 1.000
\mathcal{R}^{2}-4 \mathcal{R}_{a b} \mathcal{R}^{b a}+\mathcal{R}_{a b c d} \mathcal{R}^{c d a b}\mathcal{R}^2-4\mathcal{R}_{ab}\mathcal{R}^{ba}+\mathcal{R}_{abcd}\mathcal{R}^{cdab}conf 1.000
\mathcal{L}_{\mathrm{T}}=\mathcal{L}_{\mathrm{G}}(h, A, \partial A, \partial B)+\mathcal{L}_{\mathrm{M}}(\varphi, \partial \varphi, \phi, \partial \phi, h, \partial h, A, \partial A, B)\mathcal{L}_\mathrm{T} \!=\! \mathcal{L}_\mathrm{G}(h,A,\partial A,\partial B) + \mathcal{L}_\mathrm{M}(\varphi,\partial\varphi,\phi,\partial\phi,h,\partial h, A,\partial A, B),conf 1.000
\begin{aligned} t^{a}{ }_{\mu}+\tau^{a}{ }_{\mu} & =0 \\ s_{a b}{ }^{\mu}+\sigma_{a b}{ }^{\mu} & =0 \\ j^{\mu}+\zeta^{\mu} & =0 \end{aligned}\begin{aligned} {t^a}_\mu + {\tau^a}_\mu & = & 0, \\ {s_{ab}}^\mu + {\sigma_{ab}}^\mu & = & 0,\\ j^\mu+ \zeta^\mu & = & 0. \end{aligned}conf 0.920
\begin{aligned} t^{a}{ }_{b}+\tau^{a}{ }_{b} & =0 \\ s_{a b}{ }^{c}+\sigma_{a b}{ }^{c} & =0 \\ j^{a}+\zeta^{a} & =0 \end{aligned}\begin{aligned} {t^a}_b + {\tau^a}_b & = & 0, \\ {s_{ab}}^c + {\sigma_{ab}}^c & = & 0,\\ j^a+ \zeta^a & = & 0, \end{aligned}conf 0.903
\frac{\delta \mathcal{L}_{\mathrm{M}}}{\delta \varphi} \equiv \frac{\partial \mathcal{L}_{\mathrm{M}}}{\partial \varphi}-\partial_{\mu}\left(\frac{\partial \mathcal{L}_{\mathrm{M}}}{\partial\left(\partial_{\mu} \varphi\right)}\right)=0\frac{\delta \mathcal{L}_\mathrm{M}}{\delta\varphi} \equiv \frac{\partial \mathcal{L}_\mathrm{M}}{\partial \varphi} - \partial_\mu\left(\frac{\partial \mathcal{L}_\mathrm{M}}{\partial (\partial_\mu\varphi)}\right) =0.conf 1.000
\frac{\delta \mathcal{L}_{\mathrm{M}}}{\delta \varphi} \equiv \frac{\bar{\partial} \mathcal{L}_{\mathrm{M}}}{\partial \varphi}-D_{\mu}^{*}\left(\frac{\partial \mathcal{L}_{\mathrm{M}}}{\partial\left(D_{\mu}^{*} \varphi\right)}\right)=0\frac{\delta \mathcal{L}_\mathrm{M}}{\delta\varphi} \equiv \frac{\bar{\partial}{\mathcal{L}_\mathrm{M}}}{\partial\varphi} - D^\ast_\mu\left(\frac{\partial \mathcal{L}_\mathrm{M}}{\partial (D^\ast_\mu\varphi)}\right) =0,conf 0.944
\frac{\bar{\partial} L_{\mathrm{M}}}{\partial \varphi}-\mathcal{D}_{a}^{*}\left(\frac{\partial L_{\mathrm{M}}}{\partial\left(\mathcal{D}_{a}^{*} \varphi\right)}\right)=h D_{\mu}^{*}\left(h^{-1} h_{a}{ }^{\mu}\right) \frac{\partial L_{\mathrm{M}}}{\partial\left(\mathcal{D}_{a}^{*} \varphi\right)}\frac{\bar{\partial} L_\mathrm{M}}{\partial \varphi}-\mathcal{D}^\ast_a\left(\frac{\partial L_\mathrm{M}}{\partial (\mathcal{D}^\ast_a\varphi)}\right)= hD^\ast_\mu(h^{-1}{h_a}^\mu)\frac{\partial L_\mathrm{M}}{\partial (\mathcal{D}^\ast_a\varphi)},conf 0.898
h D_{\mu}^{*}\left(h^{-1} h_{a}{ }^{\mu}\right)=b^{b}{ }_{\mu}\left(\mathcal{D}_{b}^{*} h_{a}{ }^{\mu}-\mathcal{D}_{a}^{*} h_{b}{ }^{\mu}\right)=\mathcal{T}^{* b}{ }_{a b} \equiv \mathcal{T}_{a}^{*}hD^\ast_\mu(h^{-1}{h_a}^\mu)={b^b}_\mu(\mathcal{D}^\ast_b{h_a}^\mu-\mathcal{D}^\ast_a{h_b}^\mu) = {\mathcal{T}^{\ast b}}_{ab} \equiv \mathcal{T}^\ast_a,conf 0.794
\begin{aligned} & \frac{\delta L_{\mathrm{M}}}{\delta \varphi}=\frac{\bar{\partial} L_{\mathrm{M}}}{\partial \varphi}-\left(\mathcal{D}_{a}^{*}+\mathcal{T}_{a}^{*}\right)\left(\frac{\partial L_{\mathrm{M}}}{\partial\left(\mathcal{D}_{a}^{*} \varphi\right)}\right)=0, \\ & \frac{\delta L_{\mathrm{M}}}{\delta \phi}=\frac{\bar{\partial} L_{\mathrm{M}}}{\partial \phi}-\left(\mathcal{D}_{a}^{*}+\mathcal{T}_{a}^{*}\right)\left(\frac{\partial L_{\mathrm{M}}}{\partial\left(\mathcal{D}_{a}^{*} \phi\right)}\right)=0 . \end{aligned}\begin{aligned} \frac{\delta L_\mathrm{M}}{\delta\varphi} = \frac{\bar{\partial} L_\mathrm{M}}{\partial \varphi}-(\mathcal{D}^\ast_a + \mathcal{T}^\ast_a) \left(\frac{\partial L_\mathrm{M}}{\partial (\mathcal{D}^\ast_a\varphi)}\right) & = & 0, \\ \frac{\delta L_\mathrm{M}}{\delta\phi} = \frac{\bar{\partial} L_\mathrm{M}}{\partial \phi}-({\cal D}^\ast_a + {\cal T}^\ast_a) \left(\frac{\partial L_{\rm M}}{\partial ({\cal D}^\ast_a\phi)}\right) & = & 0. \end{aligned}conf 0.750
\begin{aligned} \left(\mathcal{D}_{c}^{*}+\mathcal{T}_{c}^{*}\right)\left(h s_{a b}{ }^{c}\right)+h t_{[a b]} & =0, \\ \left(\mathcal{D}_{c}^{*}+\mathcal{T}_{c}^{*}\right)\left(h t^{c}{ }_{d}\right)-h\left(s_{a b}{ }^{c} \mathcal{R}^{a b}{ }_{c d}-t^{c}{ }_{b} \mathcal{T}^{* b}{ }_{c d}+j^{c} \mathcal{H}_{c d}\right) & =0, \\ \left(\mathcal{D}_{c}^{*}+\mathcal{T}_{c}^{*}\right)\left(h j^{c}\right)-h t^{c}{ }_{c} & =0 . \end{aligned}\begin{aligned} \left(\mathcal{D}_{c}^{*}+\mathcal{T}_{c}^{*}\right)\left(h \sigma_{a b}{ }^{c}\right)+h \tau_{[a b]}+\frac{1}{2} \frac{\delta L_{\mathrm{M}}}{\delta \varphi} \Sigma_{a b} \varphi & =0, \\ \left(\mathcal{D}_{c}^{*}+\mathcal{T}_{c}^{*}\right)\left(h \tau^{c}{ }_{d}\right)-h\left(\sigma_{a b}{ }^{c} \mathcal{R}^{a b}{ }_{c d}-\tau^{c}{ }_{b} \mathcal{T}^{* b}{ }_{c d}+\zeta^{c} \mathcal{H}_{c d}\right)+\frac{\delta L_{\mathrm{M}}}{\delta \phi} \mathcal{D}_{d}^{*} \phi+\frac{\delta L_{\mathrm{M}}}{\delta \varphi} \mathcal{D}_{d}^{*} \varphi & =0, \\ \left(\mathcal{D}_{c}^{*}+\mathcal{T}_{c}^{*}\right)\left(h \zeta^{c}\right)-h \tau^{c}{ }_{c}-\frac{\delta L_{\mathrm{M}}}{\delta \phi} \phi+\frac{\delta L_{\mathrm{M}}}{\delta \varphi} w \varphi & =0 . \end{aligned}
L_{\mathrm{D}}=\frac{1}{2} i \bar{\psi} \gamma^{\mu} \stackrel{\leftrightarrow}{\partial_{\mu}} \psi-\mu \phi \bar{\psi} \psiL_\mathrm{D} = {\frac{1}{2}}i\bar{\psi}\gamma^\mu{\stackrel{\leftrightarrow}{\partial_\mu}}\psi - \mu\phi\bar{\psi}\psi,conf 0.989
\mathcal{L}_{\mathrm{D}}=h^{-1} L_{\mathrm{D}}=h^{-1}\left(\frac{1}{2} i \bar{\psi} \gamma^{a} \dot{\mathcal{D}}_{a}^{*} \psi-\mu \phi \bar{\psi} \psi\right)\mathcal{L}_\mathrm{D} = h^{-1} L_\mathrm{D} = h^{-1}({\frac{1}{2}}i \bar{\psi}\gamma^a{\stackrel{\leftrightarrow}{\mathcal{D}^\dagger_a}}\psi - \mu\phi\bar{\psi}\psi),conf 0.848
\mathcal{L}_{\mathrm{D}}=h^{-1}\left(\frac{1}{2} i \bar{\psi} \gamma^{a} \overleftrightarrow{\mathcal{D}}_{a} \psi-\mu \phi \bar{\psi} \psi\right)\mathcal{L}_\mathrm{D} = h^{-1}({\frac{1}{2}}i \bar{\psi}\gamma^a{\stackrel{\leftrightarrow}{\mathcal{D}_a}}\psi - \mu\phi\bar{\psi}\psi).conf 0.919
i \gamma^{a}\left(\mathcal{D}_{a}^{*}+\frac{1}{2} \mathcal{T}_{a}^{*}\right) \psi-\mu \phi \psi=0i\gamma^a(\mathcal{D}_a+{\frac{1}{2}}\mathcal{T}_a)\psi-\mu\phi\psi = 0.conf 0.937
i \gamma^{a}\left(\mathcal{D}_{a}+\frac{1}{2} \mathcal{T}_{a}\right) \psi-\mu \phi \psi=0i\gamma^a(\mathcal{D}^\ast_a+{\frac{1}{2}}\mathcal{T}^\ast_a)\psi-\mu\phi\psi = 0.conf 0.922
i \gamma^{a}\left({ }^{0} \mathcal{D}_{a}^{*}-\frac{1}{4} \mathcal{T}_{[a b c]}^{*} \Sigma^{b c}\right) \psi-\mu \phi \psi=0i\gamma^a({{}^0}\mathcal{D}_a-{\frac{1}{4}}\mathcal{T}_{[abc]}\Sigma^{bc})\psi-\mu\phi\psi = 0,conf 0.940
i \gamma^{a}\left({ }^{0} \mathcal{D}_{a}-\frac{1}{4} \mathcal{T}_{[a b c]} \Sigma^{b c}\right) \psi-\mu \phi \psi=0i\gamma^a({{}^0}\mathcal{D}^\ast_a-{\frac{1}{4}}\mathcal{T}^\ast_{[abc]}\Sigma^{bc})\psi-\mu\phi\psi = 0.conf 0.929
\begin{aligned} h \tau^{a}{ }_{b} & =\frac{1}{2} i \bar{\psi} \gamma^{a} \overleftrightarrow{\mathcal{D}}_{b} \psi-\delta_{b}^{a} L_{\mathrm{D}} \\ h \sigma_{a b c} & =\frac{1}{4} i \bar{\psi} \gamma_{[a} \gamma_{b} \gamma_{c]} \psi \end{aligned}\begin{aligned} h{\tau^a}_b & = & {\frac{1}{2}}i\bar{\psi}\gamma^a{\stackrel{\leftrightarrow}{\mathcal{D}_b}}\psi - \delta_b^a L_\mathrm{D}, \\ h{\sigma_{abc}} & = & {\frac{1}{4}}i\bar{\psi}\gamma_{[a}\gamma_b\gamma_{c]}\psi, \end{aligned}conf 0.932
\begin{aligned} \mathcal{L}_{\mathrm{M}}=h^{-1}\left[\frac{1}{2} i \bar{\psi} \gamma^{a} \overleftrightarrow{\mathcal{D}}_{a} \psi-\mu \phi \bar{\psi} \psi\right. & +\frac{1}{2} \nu\left(\mathcal{D}_{a}^{*} \phi\right)\left(\mathcal{D}^{* a} \phi\right)-\lambda \phi^{4} \\ & \left.-a \phi^{2} \mathcal{R}+\phi^{2} L_{\mathcal{T}^{* 2}}\right] \end{aligned}\begin{aligned} \mathcal{L}_\mathrm{M} = h^{-1}[{\frac{1}{2}}i\bar{\psi}\gamma^a {\stackrel{\leftrightarrow}{\mathcal{D}_a}}\psi - \mu\phi\bar{\psi}\psi &&+ {\frac{1}{2}}\nu (\mathcal{D}_a\phi) (\mathcal{D}^a \phi) - \lambda\phi^4 \\&&- a\phi^2\mathcal{R} + \phi^2L_{\mathcal{T}^2}], \end{aligned}conf 0.914
L_{\mathrm{M}}=-\frac{1}{4} F_{\mu \nu} F^{\mu \nu}-J^{\mu} A_{\mu}L_\mathrm{M} = -{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu} - J^\mu A_\mu,conf 1.000
\mathcal{L}_{\mathrm{M}}=h^{-1} L_{\mathrm{M}}=-h^{-1}\left(\frac{1}{4} \widehat{\mathcal{F}}_{a b}^{*} \widehat{\mathcal{F}}^{* a b}+\mathcal{J}^{a} \mathcal{A}_{a}\right)\mathcal{L}_\mathrm{M} = h^{-1}L_\mathrm{M} =-h^{-1}({\frac{1}{4}} \widehat{\mathcal{F}}^\ast_{ab}\widehat{\mathcal{F}}^{\ast ab} + \mathcal{J}^a \mathcal{A}_a),conf 0.927
\widehat{\mathcal{F}}_{a b}^{*}=\mathcal{F}_{a b}-\mathcal{T}^{* c}{ }_{a b} \mathcal{A}_{c}\widehat{\mathcal{F}}^\ast_{ab} = \mathcal{F}_{ab}-{\mathcal{T}^{\ast c}}_{ab}\mathcal{A}_c,conf 0.870
\mathcal{L}_{\mathrm{M}}=h^{-1} L_{\mathrm{M}}=-h^{-1}\left(\frac{1}{4} \mathcal{F}_{a b} \mathcal{F}^{a b}+\mathcal{J}^{a} \mathcal{A}_{a}\right)\mathcal{L}_\mathrm{M} = h^{-1}L_\mathrm{M} =-h^{-1}({\frac{1}{4}} \mathcal{F}_{ab}\mathcal{F}^{ab} + \mathcal{J}^a \mathcal{A}_a),conf 1.000
\left(\mathcal{D}_{a}^{*}+\mathcal{T}_{a}^{*}\right) \mathcal{F}^{a c}-\frac{1}{2} \mathcal{T}^{* c}{ }_{a b} \mathcal{F}^{a b}=\mathcal{J}^{c}(\mathcal{D}^\ast_a+\mathcal{T}^\ast_a)\mathcal{F}^{ac}-{\frac{1}{2}}{\mathcal{T}^{\ast c}}_{ab}\mathcal{F}^{ab} = \mathcal{J}^c.conf 0.862
\mathcal{D}_{[a}^{*} \mathcal{F}_{b c]}-\mathcal{T}^{* d}{ }_{[a b} \mathcal{F}_{c] d}=0\mathcal{D}^\ast_{[a}\mathcal{F}_{bc]} -{\mathcal{T}^{\ast d}}_{[ab}\mathcal{F}_{c]d}= 0.conf 0.878
\begin{aligned} \left(\mathcal{D}_{a}+\mathcal{T}_{a}\right) \mathcal{F}^{a c}-\frac{1}{2} \mathcal{T}^{c}{ }_{a b} \mathcal{F}^{a b} & =\mathcal{J}^{c} \\ \mathcal{D}_{[a} \mathcal{F}_{b c]}-\mathcal{T}^{d}{ }_{[a b} \mathcal{F}_{c] d} & =0 \end{aligned}\begin{aligned} (\mathcal{D}_a+\mathcal{T}_a)\mathcal{F}^{ac}-{\frac{1}{2}}{\mathcal{T}^c}_{ab}\mathcal{F}^{ab} & = & \mathcal{J}^c, \\ \mathcal{D}_{[a}\mathcal{F}_{bc]} -{{\cal T}^{d}}_{[ab}{\cal F}_{c]d} & = & 0. \end{aligned}conf 0.953
{ }^{0} \mathcal{D}_{a}^{*} \mathcal{F}^{a c}=\mathcal{J}^{c} \quad \text { and } \quad{ }^{0} \mathcal{D}_{[a}^{*} \mathcal{F}_{b c]}=0{{}^0}\mathcal{D}_a\mathcal{F}^{ac} = \mathcal{J}^c \qquad\text{and}\qquad {{}^0}\mathcal{D}_{[a}\mathcal{F}_{bc]}= 0,conf 0.926
{ }^{0} \mathcal{D}_{a} \mathcal{F}^{a c}=\mathcal{J}^{c} \quad \text { and } \quad{ }^{0} \mathcal{D}_{[a} \mathcal{F}_{b c]}=0{{}^0}\mathcal{D}^\ast_a\mathcal{F}^{ac} = \mathcal{J}^c \qquad\text{and}\qquad {{}^0}\mathcal{D}^\ast_{[a}\mathcal{F}_{bc]}= 0.conf 0.916
h \tau^{a}{ }_{b}=\frac{1}{4} \delta_{b}^{a} \mathcal{F}^{c d} \mathcal{F}_{c d}-\mathcal{F}^{a c} \mathcal{F}_{b c}h{\tau^a}_b = {\frac{1}{4}}\delta^a_b\mathcal{F}^{cd}\mathcal{F}_{cd} -\mathcal{F}^{ac}\mathcal{F}_{bc},conf 0.962
\begin{aligned} \widehat{\varphi} & \equiv\left(\frac{\phi}{\phi_{0}}\right)^{-w} \varphi \\ \widehat{h}_{a}{ }^{\mu} & \equiv\left(\frac{\phi}{\phi_{0}}\right)^{-1} h_{a}{ }^{\mu} \\ \widehat{A}_{\mu}^{a b} & \equiv A_{\mu}^{a b} \\ \widehat{B}_{\mu} & \equiv B_{\mu}-\partial_{\mu} \ln \left(\frac{\phi}{\phi_{0}}\right) \end{aligned}\begin{aligned} \widehat{\varphi}&\equiv& \left(\frac{\phi}{\phi_0}\right)^{-w}\varphi,\\ {\widehat{h}_a}^{\phantom{a}\mu} &\equiv& \left(\frac{\phi}{\phi_0}\right)^{-1}{h_a}^\mu,\\ {\widehat{A}^{ab}}_{\phantom{ab}\mu} &\equiv& {A^{ab}}_\mu,\\ \widehat{B}_\mu &\equiv& B_\mu -\partial_\mu \ln\left(\frac{\phi}{\phi_0}\right), \end{aligned}conf 0.880
\begin{aligned} \mathcal{D}_{a}^{*} \chi & \equiv\left(\mathcal{D}_{a}+w \mathcal{B}_{a}\right) \chi \\ & =\left(\frac{\phi}{\phi_{0}}\right)^{1-w}\left(\widehat{\mathcal{D}}_{a}+w \widehat{\mathcal{B}}_{a}\right) \widehat{\chi} \equiv\left(\frac{\phi}{\phi_{0}}\right)^{1-w} \widehat{\mathcal{D}}_{a}^{*} \widehat{\chi} \end{aligned}\begin{aligned} \hspace*{-5mm}\mathcal{D}^\ast_a\chi &\equiv& (\mathcal{D}_a + w \mathcal{B}_a)\chi \\ &=& \left(\frac{\phi}{\phi_0}\right)^{1-w} (\widehat{\mathcal{D}}_a + w \widehat{\mathcal{B}}_a)\widehat{\chi} \equiv \left(\frac{\phi}{\phi_0}\right)^{1-w} \widehat{\mathcal{D}}^\ast_a \widehat{\chi}, \end{aligned}conf 0.920
\mathcal{L}_{\mathrm{T}}=\mathcal{L}_{\mathrm{G}}(\widehat{h}, \widehat{A}, \partial \widehat{A}, \partial \widehat{B})+\mathcal{L}_{\mathrm{M}}\left(\widehat{\varphi}, \partial \widehat{\varphi}, \phi_{0}, 0, \widehat{h}, \partial \widehat{h}, \widehat{A}, \partial \widehat{A}, \widehat{B}\right)\mathcal{L}_\mathrm{T} \!=\! \mathcal{L}_\mathrm{G}(\widehat{h},\widehat{A},\partial \widehat{A},\partial \widehat{B}) + \mathcal{L}_\mathrm{M}(\widehat{\varphi},\partial\widehat{\varphi},\phi_0,0,\widehat{h},\partial \widehat{h}, \widehat{A}, \partial \widehat{A}, \widehat{B}).conf 1.000
\left.\phi_{0} \frac{\delta \mathcal{L}_{\mathrm{M}}}{\delta \phi}\right|_{\phi=\phi_{0}}=-\left.h_{a}{ }^{\mu} \frac{\delta \mathcal{L}_{\mathrm{M}}}{\delta h_{a}{ }^{\mu}}\right|_{\phi=\phi_{0}}+\partial_{\mu}\left(\frac{\delta \mathcal{L}_{\mathrm{M}}}{\delta B_{\mu}}\right)_{\phi=\phi_{0}}\phi_0\left.\frac{\delta\mathcal{L}_\mathrm{M}}{\delta\phi}\right|_{\phi=\phi_0} = -{h_a}^\mu \left.\frac{\delta\mathcal{L}_\mathrm{M}}{\delta {h_a}^\mu}\right|_{\phi=\phi_0} + \partial_\mu\left(\frac{\delta\mathcal{L}_\mathrm{M}}{\delta B_\mu}\right)_{\phi=\phi_0},conf 0.979
\phi \frac{\delta \mathcal{L}_{\mathrm{M}}}{\delta \phi}=-\widehat{h}_{a}{ }^{\mu} \frac{\delta \mathcal{L}_{\mathrm{M}}}{\delta \widehat{h}_{a}{ }^{\mu}}+\partial_{\mu}\left(\frac{\partial \mathcal{L}_{\mathrm{M}}}{\partial \widehat{B}_{\mu}}\right)\phi \frac{\delta\mathcal{L}_\mathrm{M}}{\delta\phi} = -{\widehat{h}_a}^{\phantom{a}\mu} \frac{\delta\mathcal{L}_\mathrm{M}}{\delta {\widehat{h}_a}^{\phantom{a}\mu}} + \partial_\mu\left(\frac{\partial\mathcal{L}_\mathrm{M}}{\partial \widehat{B}_\mu}\right),conf 0.928
S=-\int d \lambda\left[p_{a} v^{a}-\frac{1}{2} e\left(p_{a} p^{a}-\mu^{2} \phi^{2}\right)\right]S = - \int d\lambda\, [p_a v^{a}-{\frac{1}{2}}e(p_a p^a - \mu^2\phi^2)],conf 1.000
\begin{aligned} v^{a} & =e p^{a} \\ \dot{p}_{a} & =c^{c}{ }_{a b} v^{b} p_{c}+e \mu^{2} \phi \partial_{a} \phi, \\ p^{2} & =\mu^{2} \phi^{2} . \end{aligned}\begin{aligned} v^a & = & ep^a, \\ \dot{p}_a & = & {c^c}_{ab} v^bp_c + e\mu^2 \phi\,\partial_a\phi, \\ p^2 &=& \mu^2\phi^2. \end{aligned}conf 0.979
v^{c}\left(\mathcal{D}_{c}^{*} p_{a}-\mathcal{T}_{c a b}^{*} p^{b}\right)=e \mu^{2} \phi \mathcal{D}_{a}^{*} \phiv^c(\mathcal{D}^\ast_cp_a - \mathcal{T}^\ast_{cab}p^b) =e\mu^2 \phi\,\mathcal{D}^\ast_a\phi.conf 0.821
v^{c}{ }^{0} \mathcal{D}_{c}^{*} p_{a}=e \mu^{2} \phi^{0} \mathcal{D}_{a}^{*} \phiv^c\,{{}^0}\mathcal{D}^\ast_cp_a=e\mu^2 \phi\,{{}^0}\mathcal{D}^\ast_a\phi,conf 0.803
\phi v^{b}{ }^{0} \mathcal{D}_{b}^{*} v_{a}=\left(\delta_{a}^{b}-v_{a} v^{b}\right)^{0} \mathcal{D}_{b}^{*} \phi\phi\,v^b\,{{}^0}\mathcal{D}^\ast_bv_a= (\delta_a^b - v_av^b){{}^0}\mathcal{D}^\ast_b\phi.conf 0.838
\phi v^{b}{ }^{0} \mathcal{D}_{b} v_{a}=\left(\delta_{a}^{b}-v_{a} v^{b}\right) \partial_{b} \phi\phi\,v^b\,{{}^0}\mathcal{D}_bv_a= (\delta_a^b - v_av^b)\partial_b\phi,conf 0.983
v^{b}{ }^{0} \mathcal{D}_{b} v_{a}=0v^b\, {{}^0}\mathcal{D}_bv_a =0.conf 0.958
\begin{aligned} \widehat{\lambda} & \equiv\left(\frac{\phi}{\phi_{0}}\right) \lambda \\ \widehat{p}_{a} & \equiv \widehat{h}_{a}{ }^{\mu} p_{\mu}=\left(\frac{\phi}{\phi_{0}}\right)^{-1} p_{a} \\ \widehat{v}^{a} & \equiv \widehat{b}^{a}{ }_{\mu} \frac{d x^{\mu}}{d \widehat{\lambda}}=v^{a}, \\ \widehat{e} & \equiv\left(\frac{\phi}{\phi_{0}}\right) e, \end{aligned}\begin{aligned} \widehat{\lambda} &\equiv& \left(\frac{\phi}{\phi_0}\right)\lambda,\\ \widehat{p}_a&\equiv& {\widehat{h}_a}^{\phantom{a}\mu} p_\mu = \left(\frac{\phi}{\phi_0}\right)^{-1}p_a,\\ \widehat{v}^a&\equiv& {\widehat{b}^a}_{\phantom{a}\mu} {\frac{d x^\mu}{d \widehat{\lambda}}} =v^a,\\ \widehat{e} &\equiv& \left(\frac{\phi}{\phi_0}\right)e, \end{aligned}conf 0.646
S=-\int d \widehat{\lambda}\left[\widehat{p}_{a} \widehat{v}^{a}-\frac{1}{2} \widehat{\epsilon}\left(\widehat{p}_{a} \widehat{p}^{a}-\mu^{2} \phi_{0}^{2}\right)\right]S = - \int d\widehat{\lambda}\, [\widehat{p}_a \widehat{v}^{a}- {\frac{1}{2}}\widehat{\epsilon}(\widehat{p}_a \widehat{p}^a - \mu^2\phi_0^2)],conf 1.000
\widehat{v}^{b}{ }^{0} \widehat{\mathcal{D}}_{b} \widehat{v}_{a}=0\widehat{v}^b\, {{}^0}\widehat\mathcal{D}_b\widehat{v}_a =0,conf 0.979
\left[{ }^{0} D_{\mu}^{*},{ }^{0} D_{\nu}^{*}\right] \varphi=\frac{1}{2}{ }^{0} R^{* a b}{ }_{\mu \nu} \Sigma_{a b} \varphi+w H_{\mu \nu} \varphi[{^0}\!D^\ast_\mu,{^0}\!D^\ast_\nu]\varphi={\frac{1}{2}} {^0}\!{R^{\ast ab}}_{\mu\nu}\Sigma_{ab}\varphi + wH_{\mu\nu}\varphi,conf 0.820
\left[{ }^{0} \mathcal{D}_{c}^{*},{ }^{0} \mathcal{D}_{d}^{*}\right] \varphi=\frac{1}{2}{ }^{0} \mathcal{R}^{* a b}{ }_{c d} \Sigma_{a b} \varphi+w \mathcal{H}_{c d} \varphi[{{}^0}\mathcal{D}^\ast_c,{{}^0}\mathcal{D}^\ast_d]\varphi={\frac{1}{2}} {^0}{\mathcal{R}^{\ast ab}}_{cd}\Sigma_{ab}\varphi + w\mathcal{H}_{cd}\varphi,conf 0.863
\begin{aligned} \mathcal{R}^{a b}{ }_{c d} & ={ }^{0} \mathcal{R}^{* a b}{ }_{c d}+{ }^{0} \mathcal{D}_{c}^{*} \mathcal{K}^{* a b}{ }_{d}-{ }^{0} \mathcal{D}_{d}^{*} \mathcal{K}^{* a b}{ }_{c}+\mathcal{K}^{* a}{ }_{e c} \mathcal{K}^{* e b}{ }_{d}-\mathcal{K}^{* a}{ }_{e d} \mathcal{K}^{* e b}{ }_{c} \\ \mathcal{R}^{a}{ }_{c} & ={ }^{0} \mathcal{R}^{* a}{ }_{c}+{ }^{0} \mathcal{D}_{c}^{*} \mathcal{K}^{* a b}{ }_{b}-{ }^{0} \mathcal{D}_{b}^{*} \mathcal{K}^{* a b}{ }_{c}+\mathcal{K}^{* a}{ }_{e c} \mathcal{K}^{* e b}{ }_{b}-\mathcal{K}^{* a}{ }_{e b} \mathcal{K}^{* e b}{ }_{c} \\ \mathcal{R} & ={ }^{0} \mathcal{R}^{*}+\frac{1}{4} \mathcal{T}^{* a b c} \mathcal{T}_{a b c}^{*}+\frac{1}{2} \mathcal{T}^{* a b c} \mathcal{T}_{b a c}^{*}-\mathcal{T}^{* a} \mathcal{T}_{a}^{*}-2{ }^{0} \mathcal{D}_{a}^{*} \mathcal{T}^{* a} . \end{aligned}L_{\mathrm{G}}=L_{0_{\mathcal{R}} * 2}+L_{\mathcal{H}^{2}}L_\mathrm{G} = L_{{^0}\mathcal{R}^{\ast 2}} + L_{\mathcal{H}^2},conf 0.875
\begin{array}{r} \mathcal{L}_{\mathrm{M}}=h^{-1}\left[\frac{1}{2} i \bar{\psi} \gamma^{a}{ }^{0} \stackrel{\leftrightarrow}{\mathcal{D}}_{a} \psi-\mu \phi \bar{\psi} \psi+\frac{1}{2} \nu\left({ }^{0} \mathcal{D}_{a}^{*} \phi\right)\left({ }^{0} \mathcal{D}^{* a} \phi\right)-\lambda \phi^{4}\right. \\ \left.-a \phi^{2}{ }^{0} \mathcal{R}^{*}\right],(112) \end{array}\begin{aligned} \mathcal{L}_\mathrm{M} = h^{-1}[{\frac{1}{2}}i\bar{\psi}\gamma^a \,{\stackrel{\leftrightarrow}{{{}^{0}\mathcal{D}}_a}}\psi \!\!-\!\! \mu\phi\bar{\psi}\psi &&+ {\frac{1}{2}}\nu ({{}^{0}\mathcal{D}}^\ast_a\phi) ({{}^{0}\mathcal{D}}^{\ast a} \phi) - \lambda\phi^4 \\&& \hspace{1.5cm} - a\phi^2\,{{}^{0}\mathcal{R}}^\ast ], \end{aligned}conf 0.777
\begin{aligned} \mathcal{L}_{\mathrm{T}}=\mathcal{L}_{\mathrm{G}} & \left(h, \partial h, \partial^{2} h, B, \partial B\right) \\ & +\mathcal{L}_{\mathrm{M}}\left(\varphi, \partial \varphi, \phi, \partial \phi, h, \partial h, \partial^{2} h, B, \partial B\right) . \end{aligned}\begin{aligned} \mathcal{L}_\mathrm{T} & = & \mathcal{L}_\mathrm{G}(h,\partial h, \partial^2 h, B, \partial B) \\ && \hspace{0.5cm} + \mathcal{L}_\mathrm{M}(\varphi,\partial\varphi,\phi,\partial\phi,h,\partial h, \partial^2 h, B, \partial B).\phantom{AAA} \end{aligned}conf 0.924
\hat{\boldsymbol{e}}_{a}=h_{a}{ }^{\mu} \boldsymbol{e}_{\mu}, \quad \boldsymbol{e}_{\mu}=b^{a}{ }_{\mu} \hat{\boldsymbol{e}}_{a}{\hat{\text{$e$}}}_a = {h_a}^\mu\text{$e$}_\mu,\qquad \text{$e$}_\mu = {b^a}_\mu{\hat{\text{$e$}}}_a,conf 0.817
\boldsymbol{e}_{\mu} \cdot \boldsymbol{e}_{\nu}=\eta_{a b} b^{a}{ }_{\mu} b^{b}{ }_{\nu} \equiv g_{\mu \nu}\text{$e$}_\mu \cdot \text{$e$}_\nu = \eta_{ab}{b^a}_\mu {b^b}_\nu \equiv g_{\mu\nu}.conf 0.932
\hat{\boldsymbol{e}}_{a} \cdot \hat{\boldsymbol{e}}_{b}=\eta_{a b}=g_{\mu \nu} h_{a}{ }^{\mu} h_{b}{ }^{\nu}{\hat{\text{$e$}}}_a \cdot {\hat{\text{$e$}}}_b = \eta_{ab} = g_{\mu\nu}{h_a}^\mu {h_b}^\nu.conf 0.867
\delta J^{a}=-\left(A^{a}{ }_{b \mu}+w B_{\mu} \delta_{b}^{a}\right) J^{b} \delta x^{\mu}\delta J^a = - ({A^a}_{b\mu}+wB_\mu\delta^a_b) J^b \,\delta x^\mu,conf 0.946
D_{\mu}^{*} J^{a}=\partial_{\mu} J^{a}+w B_{\mu} J^{a}+A^{a}{ }_{b \mu} J^{b}=\partial_{\mu}^{*} J^{a}+A^{a}{ }_{b \mu} J^{b}D^\ast_\mu J^a = \partial_\mu J^a + wB_\mu J^a + {A^a}_{b\mu}J^b = \partial^\ast_\mu J^a + {A^a}_{b\mu}J^b,conf 0.879
\Delta_{\mu}^{*} \equiv \partial_{\mu}^{*}+\Gamma^{\sigma}{ }_{\rho \mu} \mathrm{X}^{\rho}{ }_{\sigma}+\frac{1}{2} A^{a b}{ }_{\mu} \Sigma_{a b}=\nabla_{\mu}^{*}+D_{\mu}^{*}-\partial_{\mu}^{*}\Delta^\ast_\mu \equiv \partial^\ast_\mu + {\Gamma^\sigma}_{\rho\mu} {{\text\mathsf{X}}^\rho}_\sigma + {\frac{1}{2}}{A^{ab}}_\mu\Sigma_{ab} = \nabla^\ast_\mu + D^\ast_\mu - \partial^\ast_\mu,conf 0.791
J^{a}+\delta J^{a}=\left(J^{\mu}+\delta J^{\mu}\right) e^{a}{ }_{\mu}(x+\delta x)J^a+\delta J^a = (J^\mu+\delta J^\mu)\,{e^a}_\mu(x+\delta x).conf 0.981
\Delta_{\mu}^{*} e^{a}{ }_{\nu} \equiv \partial_{\mu}^{*} e^{a}{ }_{\nu}-\Gamma^{\sigma}{ }_{\nu \mu} e^{a}{ }_{\sigma}+A^{a}{ }_{b \mu} e^{a}{ }_{\nu}=0\Delta^\ast_\mu {e^a}_\nu \equiv \partial^\ast_\mu {e^a}_\nu - {\Gamma^\sigma}_{\nu\mu}{e^a}_\sigma + {A^a}_{b\mu}{e^a}_\nu =0,conf 0.888
\begin{aligned} \Gamma^{\lambda}{ }_{\nu \mu} & =e_{a}{ }^{\lambda}\left(\partial_{\mu}^{*} e^{a}{ }_{\nu}+A^{a}{ }_{b \mu} e^{b}{ }_{\nu}\right), \\ A^{a}{ }_{b \mu} & =e^{a}{ }_{\lambda}\left(\partial_{\mu}^{*} e_{b}{ }^{\lambda}+\Gamma^{\lambda}{ }_{\nu \mu} e_{b}{ }^{\nu}\right) . \end{aligned}\begin{aligned} {\Gamma^\lambda}_{\nu\mu} & = & {e_a}^\lambda (\partial^\ast_\mu {e^a}_\nu + {A^a}_{b\mu} {e^b}_\nu),\\ {A^a}_{b\mu} & = & {e^a}_\lambda (\partial^\ast_\mu {e_b}^\lambda + {\Gamma^\lambda}_{\nu\mu}{e_b}^\nu). \end{aligned}conf 0.900
\nabla_{\sigma} g_{\mu \nu}=-2 B_{\sigma} g_{\mu \nu}\nabla_\sigma g_{\mu\nu} = -2B_\sigma g_{\mu\nu},conf 1.000
\begin{aligned} R_{\sigma \mu \nu}^{\rho} & =2\left(\partial_{[\mu} \Gamma_{|\sigma| \nu]}^{\rho}+\Gamma_{\lambda[\mu}^{\rho} \Gamma_{|\sigma| \nu]}^{\lambda}\right)-H_{\mu \nu} \delta_{\sigma}^{\rho}, \\ T^{* \lambda}{ }_{\mu \nu} & =2 \Gamma_{[\nu \mu]}^{\lambda}, \\ H_{\mu \nu} & =2 \partial_{[\mu} B_{\nu]}, \end{aligned}\widetilde{R}_{\sigma \mu \nu}^{\rho} \equiv R_{\sigma \mu \nu}^{\rho}+H_{\mu \nu} \delta_{\sigma}^{\rho}\left[\nabla_{\mu}^{*}, \nabla_{\nu}^{*}\right] J^{\rho}=\widetilde{R}^{\rho}{ }_{\sigma \mu \nu} J^{\sigma}+w H_{\mu \nu} J^{\rho}-T^{* \sigma}{ }_{\mu \nu} \nabla_{\sigma}^{*} V^{\rho}[\nabla^\ast_\mu,\nabla^\ast_\nu] J^\rho = {\widetilde{R}^{\rho}}_{\phantom{\rho}\sigma\mu\nu}J^\sigma + wH_{\mu\nu} J^\rho - {T^{\ast\sigma}}_{\mu\nu}\nabla^\ast_\sigma V^\rho.conf 0.842
\Gamma^{\lambda}{ }_{\mu \nu}={ }^{0} \Gamma^{* \lambda}{ }_{\mu \nu}+K^{* \lambda}{ }_{\mu \nu}{\Gamma^\lambda}_{\mu\nu} = {^0}{\Gamma^{\ast\lambda}}_{\mu \nu} + {K^{\ast\lambda}}_{\mu\nu},conf 0.894
\begin{aligned} { }^{0} \Gamma^{* \lambda}{ }_{\mu \nu} & =\frac{1}{2} g^{\lambda \rho}\left(\partial_{\mu}^{*} g_{\nu \rho}+\partial_{\nu}^{*} g_{\mu \rho}-\partial_{\rho}^{*} g_{\mu \nu}\right) \\ & ={ }^{0} \Gamma^{\lambda}{ }_{\mu \nu}+\delta_{\nu}^{\lambda} B_{\mu}+\delta_{\mu}^{\lambda} B_{\nu}-g_{\mu \nu} B^{\lambda} \end{aligned}\begin{aligned} {^0}{\Gamma^{\ast\lambda}}_{\mu \nu} &=& {\frac{1}{2}}g^{\lambda\rho}(\partial^\ast_\mu g_{\nu\rho}+\partial^\ast_\nu g_{\mu\rho}-\partial^\ast_\rho g_{\mu\nu}) \\ &=& {^0}{\Gamma^{\lambda}}_{\mu \nu} + \delta^\lambda_\nu B_\mu + \delta^\lambda_\mu B_\nu - g_{\mu\nu}B^\lambda, \end{aligned}conf 0.729
K^{* \lambda}{ }_{\mu \nu}=-\frac{1}{2}\left(T^{* \lambda}{ }_{\mu \nu}-T^{*}{ }_{\nu}{ }^{\lambda}{ }_{\mu}+T^{*}{ }_{\mu \nu}{ }^{\lambda}\right){K^{\ast\lambda}}_{\mu\nu}=-{\frac{1}{2}}({T^{\ast\lambda}}_{\mu\nu}- {{{T^\ast}_\nu}^{\lambda}}_\mu + {{T^\ast}_{\mu\nu}}^{\lambda}),conf 0.836
{ }^{0} \Delta_{\mu}^{*} e^{a}{ }_{\nu} \equiv \partial_{\mu}^{*} e^{a}{ }_{\nu}-{ }^{0} \Gamma^{* \sigma}{ }_{\nu \mu} e^{a}{ }_{\sigma}+{ }^{0} A^{* a}{ }_{b \mu} e^{b}{ }_{\nu}=0{^0}\!\Delta^\ast_\mu {e^a}_\nu \equiv \partial^\ast_\mu {e^a}_\nu - {^0}{\Gamma^{\ast \sigma}}_{\nu\mu}{e^a}_\sigma + {^0}\!{A^{\ast a}}_{b\mu}{e^b}_\nu =0.conf 0.831
\begin{aligned} h_{a}^{\prime \mu}(x) & =e^{-\rho(x)} h_{a}{ }^{\mu}(x), \\ A^{\prime a b}{ }_{\mu}(x) & =A^{a b}{ }_{\mu}(x), \end{aligned}\begin{aligned} {h'_a}^\mu(x)& = & e^{-\rho(x)}{h_a}^\mu(x), \\ {A'^{ab}}_\mu (x) & = & {A^{ab}}_\mu(x) , \end{aligned}conf 0.833
\begin{aligned} h_{a}^{\prime \mu} & =e^{-\rho} h_{a}{ }^{\mu} \\ A^{\prime a b}{ }_{\mu} & =A^{a b}{ }_{\mu}+\theta\left(b^{a}{ }_{\mu} \mathcal{P}^{b}-b^{b}{ }_{\mu} \mathcal{P}^{a}\right), \end{aligned}\begin{aligned} {h'_a}^\mu & = & e^{-\rho}{h_a}^\mu \\ {A'^{ab}}_\mu & = & {A^{ab}}_\mu + \theta ({b^a}_\mu\mathcal{P}^b-{b^b}_\mu\mathcal{P}^a), \end{aligned}conf 0.868
\mathcal{T}^{\prime a}{ }_{b c}=e^{-\rho}\left(\mathcal{T}^{a}{ }_{b c}+\mathcal{P}_{b} \delta_{c}^{a}-\mathcal{P}_{c} \delta_{b}^{a}\right){\mathcal{T}^{\prime a}}_{bc} = e^{-\rho}({\mathcal{T}^a}_{bc}+\mathcal{P}_b\delta_c^a - \mathcal{P}_c\delta_b^a),conf 0.979
\begin{aligned} \mathcal{R}^{\prime a b}{ }_{c d} & =e^{-2 \rho}\left\{\mathcal{R}^{a b}{ }_{c d}+2 \theta \delta_{d}^{[a}\left(\mathcal{D}_{c}-\theta \mathcal{P}_{c}\right) \mathcal{P}^{b]}-2 \theta \delta_{c}^{[a}\left(\mathcal{D}_{d}-\theta \mathcal{P}_{d}\right) \mathcal{P}^{b]}-2 \theta \mathcal{P}^{[a} \mathcal{T}^{b]}{ }_{c d}-2 \theta^{2} \delta_{c}^{[a} \delta_{d}^{b]} \mathcal{P}^{e} \mathcal{P}_{e}\right\}, \\ \mathcal{T}^{\prime a}{ }_{b c} & =e^{-\rho}\left\{\mathcal{T}^{a}{ }_{b c}+2(1-\theta) \mathcal{P}_{[b} \delta_{c]}^{a}\right\} . \end{aligned}\begin{aligned} {\mathcal{R}^{\prime ab}}_{cd} & = & e^{-2\rho}\{{\mathcal{R}^{ab}}_{cd} + 2\theta\delta^{[a}_d(\mathcal{D}_c-\theta\mathcal{P}_c)\mathcal{P}^{b]} - 2\theta\delta^{[a}_c(\mathcal{D}_d-\theta\mathcal{P}_d)\mathcal{P}^{b]} - 2\theta{\cal P}^{[a}{{\cal T}^{b]}}_{cd} - 2\theta^2\delta^{[a}_c\delta^{b]}_d {\cal P}^e {\cal P}_e \}, \\ {{\cal T}^{\prime a}}_{bc} & = & e^{-\rho}\{{{\cal T}^a}_{bc}+2(1-\theta){\cal P}_{[b}\delta_{c]}^a\}. \end{aligned}conf 0.745
A^{\prime a b}{ }_{\mu}=A^{a b}{ }_{\mu}+\left(b^{a}{ }_{\mu} \mathcal{P}^{b}-b^{b}{ }_{\mu} \mathcal{P}^{a}\right){A'^{ab}}_\mu = {A^{ab}}_\mu + ({b^a}_\mu\mathcal{P}^b- {b^b}_\mu\mathcal{P}^a),conf 0.891
A^{\dagger a b}{ }_{\mu} \equiv A^{a b}{ }_{\mu}+\left(\mathcal{V}^{a} b^{b}{ }_{\mu}-\mathcal{V}^{b} b^{a}{ }_{\mu}\right){A^{\dagger ab}}_\mu \equiv {A^{ab}}_\mu + (\mathcal{V}^a{b^b}_\mu - \mathcal{V}^b{b^a}_\mu),conf 0.948
D_{\mu}^{\dagger} \varphi \equiv\left(\partial_{\mu}+\frac{1}{2} A^{\dagger a b}{ }_{\mu} \Sigma_{a b}-w V_{\mu}-\frac{1}{3} w T_{\mu}\right) \varphiD^\dagger_\mu \varphi \equiv (\partial_\mu + {\frac{1}{2}} {A^{\dagger ab}}_\mu\Sigma_{ab} - wV_\mu -{\frac{1}{3}}w T_\mu)\varphi,conf 0.951
V_{\mu}^{\prime}=V_{\mu}+\theta P_{\mu}V'_\mu = V_\mu +\theta P_\mu,
conf 0.852
D_{\mu}^{\dagger} \varphi=\left(\partial_{\mu}^{\dagger}+\frac{1}{2} A^{\dagger a b}{ }_{\mu} \Sigma_{a b}\right) \varphiD^\dagger_\mu \varphi = (\partial^\dagger_\mu + {\frac{1}{2}}{A^{\dagger ab}}_\mu \Sigma_{ab}) \varphi,conf 0.893
\partial_{\mu}^{\dagger} \equiv \partial_{\mu}-w\left(V_{\mu}+\frac{1}{3} T_{\mu}\right)\partial^\dagger_\mu \equiv \partial_\mu - w (V_\mu+{\frac{1}{3}}T_\mu).conf 0.933
\mathcal{D}_{a}^{\dagger} \varphi \equiv h_{a}{ }^{\mu} D_{\mu}^{\dagger} \varphi\mathcal{D}^\dagger_a \varphi \equiv {h_a}^\mu D^\dagger_\mu \varphi,conf 0.883
S_{\mathrm{M}}=\int h^{-1} L_{\mathrm{M}}\left(\varphi, \mathcal{D}_{a}^{\dagger} \varphi\right) d^{4} xS_\mathrm{M} = \int h^{-1} L_\mathrm{M} (\varphi,\mathcal{D}^\dagger_a \varphi)\,d^4x.conf 0.965
D_{\mu}^{\dagger}=\partial_{\mu}^{\dagger}+{ }^{0} \Gamma^{\sigma}{ }_{\rho \mu} \mathrm{X}^{\rho}{ }_{\sigma}+\frac{1}{2} A^{\dagger a b}{ }_{\mu} \Sigma_{a b}={ }^{0} \nabla_{\mu}^{\dagger}+\frac{1}{2} A^{\dagger a b}{ }_{\mu} \Sigma_{a b}D^\dagger_\mu = \partial^\dagger_\mu + {{}^0}{\Gamma^\sigma}_{\rho\mu} {{\text\mathsf{X}}^\rho}_\sigma + {\frac{1}{2}}{A^{\dagger ab}}_\mu\Sigma_{ab} = {{}^0}\nabla^\dagger_\mu + {\frac{1}{2}}{A^{\dagger ab}}_\mu\Sigma_{ab},conf 0.860
\left[D_{\mu}^{\dagger}, D_{\nu}^{\dagger}\right] \varphi=\frac{1}{2} R^{\dagger a b}{ }_{\mu \nu} \Sigma_{a b} \varphi-w H_{\mu \nu}^{\dagger} \varphi[D^\dagger_\mu,D^\dagger_\nu]\varphi = {\frac{1}{2}}{R^{\dagger ab}}_{\mu\nu}\Sigma_{ab}\varphi - w H^\dagger_{\mu\nu}\varphi,conf 0.853
\begin{aligned} R^{\dagger a b}{ }_{\mu \nu} & \equiv \partial_{\mu} A^{\dagger a b}{ }_{\nu}-\partial_{\nu} A^{\dagger a b}{ }_{\mu}+A^{\dagger a}{ }_{c \mu} A^{\dagger c b}{ }_{\nu}-A^{\dagger a}{ }_{c \nu} A^{\dagger c b}{ }_{\mu} \\ & =R^{a b}{ }_{\mu \nu}+4 b^{[b}{ }_{[\nu} D_{\mu]} \mathcal{V}^{a]}+4 \mathcal{V}^{[a} V_{[\mu} b^{b]}{ }_{\nu]}-2 \mathcal{V}^{e} \mathcal{V}_{e} b^{[a}{ }_{\mu} b^{b]}{ }_{\nu}+2 \mathcal{V}^{[a} T^{b]}{ }_{\mu \nu} \end{aligned}\begin{aligned} {R^{\dagger ab}}_{\mu\nu} & \equiv & \partial_\mu {A^{\dagger ab}}_\nu - \partial_\nu {A^{\dagger ab}}_\mu +{A^{\dagger a}}_{c\mu}{A^{\dagger cb}}_\nu - {A^{\dagger a}}_{c\nu}{A^{\dagger cb}}_\mu, \\ & = & {R^{ab}}_{\mu\nu} + 4{b^{[b}}_{[\nu} D_{\mu]} \mathcal{V}^{a]} +4 \mathcal{V}^{[a} V_{[\mu} {b^{b]}}_{\nu]} -2 \mathcal{V}^e \mathcal{V}_e {b^{[a}}_\mu {b^{b]}}_\nu +2 \mathcal{V}^{[a}{T^{b]}}_{\mu\nu} , \end{aligned}conf 0.591
H_{\mu \nu}^{\dagger}=\partial_{\mu}\left(V_{\nu}+\frac{1}{3} T_{\nu}\right)-\partial_{\nu}\left(V_{\mu}+\frac{1}{3} T_{\mu}\right)H^\dagger_{\mu\nu} = \partial_\mu(V_\nu+{\frac{1}{3}}T_\nu) - \partial_\nu(V_\mu+{\frac{1}{3}}T_\mu).conf 0.905
\left[\mathcal{D}_{c}^{\dagger}, \mathcal{D}_{d}^{\dagger}\right] \varphi=\frac{1}{2} \mathcal{R}^{\dagger a b}{ }_{c d} \Sigma_{a b} \varphi-w \mathcal{H}_{c d}^{\dagger} \varphi-\mathcal{T}^{\dagger a}{ }_{c d} \mathcal{D}_{a}^{\dagger} \varphi[\mathcal{D}^\dagger_c,\mathcal{D}^\dagger_d]\varphi = {\frac{1}{2}}{\mathcal{R}^{\dagger ab}}_{cd}\Sigma_{ab}\varphi - w \mathcal{H}^\dagger_{cd}\varphi - {\mathcal{T}^{\dagger a}}_{cd}\mathcal{D}^\dagger_a \varphi,conf 0.937
\mathcal{T}^{\dagger a}{ }_{b c} \equiv h_{b}{ }^{\mu} h_{c}{ }^{\nu}\left(D_{\mu}^{\dagger} b^{a}{ }_{\nu}-D_{\nu}^{\dagger} b^{a}{ }_{\mu}\right) \equiv h_{b}{ }^{\mu} h_{c}{ }^{\nu} T^{\dagger a}{ }_{\mu \nu}{\mathcal{T}^{\dagger a}}_{bc} \equiv {h_b}^\mu {h_c}^\nu (D^\dagger_\mu {b^a}_\nu - D^\dagger_\nu {b^a}_\mu) \equiv {h_b}^\mu {h_c}^\nu {T^{\dagger a}}_{\mu\nu}.conf 0.881
\begin{aligned} \mathcal{R}^{\dagger a b}{ }_{c d} & =\mathcal{R}^{a b}{ }_{c d}+2 \delta_{d}^{[b}\left(\mathcal{D}_{c}+\mathcal{V}_{c}\right) \mathcal{V}^{a]}-2 \delta_{c}^{[b}\left(\mathcal{D}_{d}+\mathcal{V}_{d}\right) \mathcal{V}^{a]}-2 \mathcal{V}^{e} \mathcal{V}_{e} \delta_{c}^{[a} \delta_{d}^{b]}+2 \mathcal{V}^{[a} \mathcal{T}^{b]}{ }_{c d} \\ \mathcal{R}^{\dagger a}{ }_{c} & =\mathcal{R}^{a}{ }_{c}+2\left(\mathcal{D}_{c}+\frac{1}{2} \mathcal{T}_{c}+\mathcal{V}_{c}\right) \mathcal{V}^{a}+\delta_{c}^{a}\left(\mathcal{D}_{b}-2 \mathcal{V}_{b}\right) \mathcal{V}^{b}-\mathcal{T}^{a}{ }_{c b} \mathcal{V}^{b} \\ \mathcal{R}^{\dagger} & =\mathcal{R}+6\left(\mathcal{D}_{a}+\frac{1}{3} \mathcal{T}_{a}-\mathcal{V}_{a}\right) \mathcal{V}^{a}, \end{aligned}\begin{aligned} {\mathcal{R}^{\dagger ab}}_{cd} & = & {\mathcal{R}^{ab}}_{cd} + 2\delta^{[b}_d(\mathcal{D}_c + \mathcal{V}_c)\mathcal{V}^{a]} - 2\delta^{[b}_c(\mathcal{D}_d + \mathcal{V}_d)\mathcal{V}^{a]} -2 {\cal V}^e {\cal V}_e \delta^{[a}_c \delta^{b]}_d +2 {\cal V}^{[a}{{\cal T}^{b]}}_{cd} , \\ {{\cal R}^{\dagger a}}_{c} & = & {{\cal R}^{a}}_{c} + 2({\cal D}_c+{\frac{1}{2}}{\cal T}_c + {\cal V}_c){\cal V}^a + \delta^a_c({\cal D}_b -2{\cal V}_b){\cal V}^b -{{\cal T}^a}_{cb}{\cal V}^b,\\ {\cal R}^\dagger & = & {\cal R} + 6({\cal D}_a+{\frac{1}{3}}{\cal T}_a-{\cal V}_a){\cal V}^a, \end{aligned}conf 0.662
\mathcal{T}^{\dagger a}{ }_{b c}=\mathcal{T}^{a}{ }_{b c}+\frac{1}{3}\left(\delta_{b}^{a} \mathcal{T}_{c}-\delta_{c}^{a} \mathcal{T}_{b}\right){\mathcal{T}^{\dagger a}}_{bc} = {\mathcal{T}^a}_{bc} +{\frac{1}{3}}(\delta^a_b \mathcal{T}_c-\delta^a_c\mathcal{T}_b).conf 0.935
\mathcal{T}_{b}^{\dagger} \equiv \mathcal{T}^{\dagger a}{ }_{b a}=0\mathcal{T}^\dagger_b \equiv {\mathcal{T}^{\dagger a}}_{ba} = 0,conf 0.940
\mathcal{A}_{a b c}^{\dagger}={ }^{0} \mathcal{A}_{a b c}^{\dagger}(h, \partial h, A, V)+\mathcal{K}_{a b c}^{\dagger}(h, \partial h, A)\mathcal{A}^\dagger_{abc} = {^0}\!\mathcal{A}^\dagger_{abc}(h,\partial h,A,V) +\mathcal{K}^\dagger_{abc} (h,\partial h,A),conf 0.871
{ }^{0} D_{\mu}^{\dagger} \varphi \equiv\left(\partial_{\mu}^{\dagger}+\frac{1}{2}{ }^{0} A^{\dagger a b}{ }_{\mu} \Sigma_{a b}\right) \varphi{^0}\!D^\dagger_\mu\varphi \equiv (\partial^\dagger_\mu +{\frac{1}{2}} {^0}\!{A^{\dagger ab}}_\mu\Sigma_{ab})\varphi,conf 0.887
\mathcal{D}_{a}^{\dagger} \varphi=\left({ }^{0} \mathcal{D}_{a}^{\dagger}+\frac{1}{2} \mathcal{K}^{\dagger b c}{ }_{a} \Sigma_{b c}\right) \varphi\mathcal{D}^\dagger_a\varphi = ({^0}\mathcal{D}^\dagger_a + {\frac{1}{2}}{\mathcal{K}^{\dagger bc}}_a\Sigma_{bc})\varphi,conf 0.939
\begin{aligned} \mathcal{D}_{[a}^{\dagger} \mathcal{R}^{\dagger d e}{ }_{b c]}-\mathcal{T}^{\dagger f}{ }_{[a b} \mathcal{R}^{\dagger d e}{ }_{c] f} & =0, \\ \mathcal{D}_{[a}^{\dagger} \mathcal{T}^{\dagger d}{ }_{b c]}-\mathcal{T}^{\dagger e}{ }_{[a b} \mathcal{T}^{\dagger d}{ }_{c] e}-\mathcal{R}^{\dagger d}{ }_{[a b c]}+\mathcal{H}_{[a b}^{\dagger} \delta_{c]}^{d} & =0, \\ \mathcal{D}_{[a}^{\dagger} \mathcal{H}^{\dagger}{ }_{b c]}-\mathcal{T}^{\dagger e}{ }_{[a b} \mathcal{H}^{\dagger}{ }_{c] e} & =0 . \end{aligned}
\mathcal{D}_{a}^{\dagger} \mathcal{R}^{\dagger a e}{ }_{b c}-2 \mathcal{D}_{[b}^{\dagger} \mathcal{R}^{\dagger e}{ }_{c]}-2 \mathcal{T}^{\dagger f}{ }_{a[b} \mathcal{R}^{\dagger a e}{ }_{c] f}-\mathcal{T}^{\dagger f}{ }_{b c} \mathcal{R}^{\dagger e}{ }_{f}=0\mathcal{D}^\dagger_{a}{\mathcal{R}^{\dagger ae}}_{bc}-2 \mathcal{D}^\dagger_{[b}{\mathcal{R}^{\dagger e}}_{c]} -2{\mathcal{T}^{\dagger f}}_{a[b} {\mathcal{R}^{\dagger ae}}_{c]f} -{\mathcal{T}^{\dagger f}}_{bc} {\mathcal{R}^{\dagger e}}_{f} = 0.conf 0.933
\mathcal{D}_{a}^{\dagger}\left(\mathcal{R}^{\dagger a}{ }_{c}-\frac{1}{2} \delta_{c}^{a} \mathcal{R}^{\dagger}\right)+\mathcal{T}^{\dagger f}{ }_{b c} \mathcal{R}^{\dagger b}{ }_{f}+\frac{1}{2} \mathcal{T}^{\dagger f}{ }_{a b} \mathcal{R}^{\dagger a b}{ }_{c f}=0\mathcal{D}^\dagger_{a}({\mathcal{R}^{\dagger a}}_{c} -{\frac{1}{2}}\delta^a_c\mathcal{R}^\dagger) +{\mathcal{T}^{\dagger f}}_{bc} {\mathcal{R}^{\dagger b}}_{f} +{\frac{1}{2}}{\mathcal{T}^{\dagger f}}_{ab} {\mathcal{R}^{\dagger ab}}_{cf} = 0.conf 0.950
\mathcal{D}_{a}^{\dagger} \mathcal{T}^{\dagger a}{ }_{b c}+2 \mathcal{R}^{\dagger}{ }_{[b c]}+2 \mathcal{H}_{b c}^{\dagger}=0\mathcal{D}^\dagger_{a}{\mathcal{T}^{\dagger a}}_{bc}+2{\mathcal{R}^{\dagger}}_{[bc]} + 2\mathcal{H}^\dagger_{bc} = 0.conf 0.926
S_{\mathrm{G}}=\int h^{-1} L_{\mathrm{G}}\left(\mathcal{R}_{a b c d}^{\dagger}, \mathcal{T}_{a b c}^{\dagger}, \mathcal{H}_{a b}^{\dagger}\right) d^{4} xS_\mathrm{G} = \int h^{-1} L_\mathrm{G}(\mathcal{R}^\dagger_{abcd}, \mathcal{T}^\dagger_{abc},\mathcal{H}^\dagger_{ab})\,d^4x,conf 0.864
L_{\mathrm{G}}=L_{\mathcal{R}^{\dagger 2}}+L_{\mathcal{H}^{\dagger 2}}L_\mathrm{G} = L_{\mathcal{R}^{\dagger 2}} + L_{\mathcal{H}^{\dagger 2}},conf 1.000
\begin{aligned} \mathcal{L}_{\mathrm{T}}= & \mathcal{L}_{\mathrm{G}}\left(h, \partial h, \partial^{2} h, A, \partial A, V, \partial V\right) \\ & +\mathcal{L}_{\mathrm{M}}(\varphi, \partial \varphi, \phi, \partial \phi, h, \partial h, A, \partial A, V, \partial V), \end{aligned}\begin{aligned} \mathcal{L}_\mathrm{T} &=& \mathcal{L}_\mathrm{G}(h, \partial h, \partial^2 h, A,\partial A,V,\partial V) \\&&+ \mathcal{L}_\mathrm{M}(\varphi,\partial\varphi,\phi,\partial\phi,h,\partial h,A,\partial A,V,\partial V), \end{aligned}conf 1.000
\begin{aligned} t^{a}{ }_{b}+\tau^{a}{ }_{b} & =0 \\ s_{a b}{ }^{c}+\sigma_{a b}{ }^{c} & =0 \\ j^{a}+\zeta^{a} & =0 \end{aligned}\begin{aligned} {t^a}_b + {\tau^a}_b & = & 0, \\ {s_{ab}}^c + {\sigma_{ab}}^c & = & 0,\\ j^a + \zeta^a & = &0, \end{aligned}conf 0.903
\begin{aligned} & t_{a b}^{\prime}=t_{a b}+2 \theta\left(s_{a c}{ }^{c} \mathcal{P}_{b}-s_{c b a} \mathcal{P}^{c}\right) \\ & \tau_{a b}^{\prime}=\tau_{a b}+2 \theta\left(\sigma_{a c}{ }^{c} \mathcal{P}_{b}-\sigma_{c b a} \mathcal{P}^{c}\right) \end{aligned}\begin{aligned} t'_{ab} & = & t_{ab} + 2\theta({s_{ac}}^c\mathcal{P}_b -s_{cba}\mathcal{P}^c), \\ \tau'_{ab} & = & \tau_{ab} + 2\theta({\sigma_{ac}}^c\mathcal{P}_b -\sigma_{cba}\mathcal{P}^c), \end{aligned}conf 0.917
\tau_{a b}^{\dagger} \equiv \tau_{a b}+2 \sigma_{c b a} \mathcal{V}^{c}-2 \sigma_{a c}{ }^{c} \mathcal{V}_{b}\tau^\dagger_{ab} \equiv \tau_{ab} + 2\sigma_{cba}\mathcal{V}^c - 2{\sigma_{ac}}^c\mathcal{V}_b,conf 0.947
t^{\dagger a}{ }_{b}+\tau^{\dagger a}{ }_{b}=0{t^{\dagger a}}_b + {\tau^{\dagger a}}_b =0.conf 0.941
\begin{aligned} & \frac{\delta L_{\mathrm{M}}}{\delta \varphi}=\frac{\bar{\partial} L_{\mathrm{M}}}{\partial \varphi}-\mathcal{D}_{a}^{\dagger}\left(\frac{\partial L_{\mathrm{M}}}{\partial\left(\mathcal{D}_{a}^{\dagger} \varphi\right)}\right)=0, \\ & \frac{\delta L_{\mathrm{M}}}{\delta \phi}=\frac{\bar{\partial} L_{\mathrm{M}}}{\partial \phi}-\mathcal{D}_{a}^{\dagger}\left(\frac{\partial L_{\mathrm{M}}}{\partial\left(\mathcal{D}_{a}^{*} \phi\right)}\right)=0, \end{aligned}\begin{aligned} \frac{\delta L_\mathrm{M}}{\delta\varphi} = \frac{\bar{\partial} L_\mathrm{M}}{\partial \varphi}-\mathcal{D}^\dagger_a \left(\frac{\partial L_\mathrm{M}}{\partial (\mathcal{D}^\dagger_a\varphi)}\right) & = & 0, \\ \frac{\delta L_\mathrm{M}}{\delta\phi} = \frac{\bar{\partial} L_\mathrm{M}}{\partial \phi}-\mathcal{D}^\dagger_a \left(\frac{\partial L_{\rm M}}{\partial ({\cal D}^\ast_a\phi)}\right) & = & 0, \end{aligned}conf 0.616
\begin{aligned} \mathcal{D}_{c}^{\dagger}\left(h s_{a b}{ }^{c}\right)+h t_{[a b]}^{\dagger} & =0, \\ \mathcal{D}_{c}^{\dagger}\left(h t^{\dagger c}{ }_{d}\right)-h\left(s_{a b}{ }^{c} \mathcal{R}^{\dagger a b}{ }_{c d}-t^{\dagger c}{ }_{b} \mathcal{T}^{\dagger b}{ }_{c d}+j^{\dagger c} \mathcal{H}_{c d}\right) & =0, \end{aligned}\begin{aligned} \mathcal{D}^\dagger_c(h{s_{ab}}^c) + ht^\dagger_{[ab]} & = & 0,\\ \hspace*{-7mm}\mathcal{D}^\dagger_c(h{t^{\dagger c}}_d)\! -\! h({s_{ab}}^c {\mathcal{R}^{\dagger ab}}_{cd} \!-\! {t^{\dagger c}}_b {\mathcal{T}^{\dagger b}}_{cd} \!+\! j^{\dagger c} \mathcal{H}_{cd}) & = & 0, \end{aligned}conf 0.714
j^{\dagger a} \equiv j^{a}-2 s^{a b}{ }_{b}j^{\dagger a} \equiv j^a -2{s^{ab}}_b.conf 0.966
\begin{aligned} D_{c}^{\dagger}\left(h j^{\dagger c}\right) & =0, \\ h t^{\dagger c}{ }_{c} & =0 . \end{aligned}\begin{aligned} D^\dagger_c (hj^{\dagger c}) & = & 0, \\ h{t^{\dagger c}}_c & = & 0. \end{aligned}conf 0.933
\begin{aligned} \mathcal{D}_{c}^{\dagger}\left(h \sigma_{a b}{ }^{c}\right)+h \tau_{[a b]}^{\dagger}+\frac{1}{2} \frac{\delta L_{\mathrm{M}}}{\delta \varphi} \Sigma_{a b} \varphi & =0, \\ \mathcal{D}_{c}^{\dagger}\left(h \tau^{\dagger c}{ }_{d}\right)-h\left(\sigma_{a b}{ }^{c} \mathcal{R}^{\dagger a b}{ }_{c d}-\tau^{\dagger c}{ }_{b} \mathcal{T}^{\dagger b}{ }_{c d}+\zeta^{\dagger c} \mathcal{H}_{c d}\right)+\frac{\delta L_{\mathrm{M}}}{\delta \phi} \mathcal{D}_{d}^{\dagger} \phi+\frac{\delta L_{\mathrm{M}}}{\delta \varphi} \mathcal{D}_{d}^{\dagger} \varphi & =0, \\ \mathcal{D}_{c}^{\dagger}\left(h \zeta^{\dagger c}\right) & =0, \\ h \tau^{\dagger c}{ }_{c}+\frac{\delta L_{\mathrm{M}}}{\delta \phi} \phi-\frac{\delta L_{\mathrm{M}}}{\delta \varphi} w \varphi & =0, \end{aligned}\begin{aligned} \mathcal{D}^\dagger_c(h{\sigma_{ab}}^c) + h\tau^\dagger_{[ab]} +{\frac{1}{2}}\frac{\delta L_\mathrm{M}}{\delta\varphi} \Sigma_{ab}\varphi & = & 0, \\ \mathcal{D}^\dagger_c(h{\tau^{\dagger c}}_d) - h({\sigma_{ab}}^c {\mathcal{R}^{\dagger ab}}_{cd} - {\tau^{\dagger c}}_b {\mathcal{T}^{\dagger b}}_{cd} + \zeta^{\dagger c} \mathcal{H}_{cd}) + \frac{\delta L_\mathrm{M}}{\delta\phi} \mathcal{D}^\dagger_d \phi + \frac{\delta L_{\rm M}}{\delta\varphi} {\cal D}^\dagger_d\varphi & = & 0,\\ {\cal D}^\dagger_c(h\zeta^{\dagger c}) & = & 0,\\ h{\tau^{\dagger c}}_c +\frac{\delta L_{\rm M}}{\delta\phi} \phi - \frac{\delta L_{\rm M}}{\delta\varphi} w\varphi & = & 0, \end{aligned}conf 0.767
\zeta^{\dagger a} \equiv \zeta^{a}-2 \sigma^{a b}{ }_{b}\zeta^{\dagger a} \equiv \zeta^a -2{\sigma^{ab}}_b,conf 0.976
\mathcal{D}_{a}^{\natural} \varphi \equiv h_{a}{ }^{\mu} D_{\mu}^{\natural} \varphi \equiv h_{a}{ }^{\mu}\left(\partial_{\mu}+\frac{1}{2} A^{\dagger b c}{ }_{\mu} \Sigma_{b c}\right) \varphi\mathcal{D}^\natural_a\varphi \equiv {h_a}^\mu D^\natural_\mu \varphi \equiv {h_a}^\mu (\partial_\mu + {\frac{1}{2}}{A^{\dagger bc}}_\mu\Sigma_{bc})\varphi.conf 0.931
\mathcal{T}^{\natural a}{ }_{b c} \equiv h_{b}{ }^{\mu} h_{c}{ }^{\nu}\left(D_{\mu}^{\natural} b^{a}{ }_{\nu}-D_{\nu}^{\natural} b^{a}{ }_{\mu}\right){\mathcal{T}^{\natural a}}_{bc} \equiv {h_b}^\mu {h_c}^\nu (D^\natural_\mu {b^a}_\nu - D^\natural_\nu {b^a}_\mu),conf 0.863
\mathcal{D}_{a}^{\dagger} \varphi=\left(\mathcal{D}_{a}^{\natural}-\frac{1}{3} w \mathcal{T}_{a}^{\natural}\right) \varphi\mathcal{D}^\dagger_a\varphi = (\mathcal{D}^\natural_a-{\frac{1}{3}}w\mathcal{T}^\natural_a)\varphi,conf 0.929
\begin{aligned} \mathcal{L}_{\mathrm{T}}= & \mathcal{L}_{\mathrm{G}}\left(h, \partial h, \partial^{2} h, A^{\dagger}, \partial A^{\dagger}\right) \\ & +\mathcal{L}_{\mathrm{M}}\left(\varphi, \partial \varphi, \phi, \partial \phi, h, \partial h, A^{\dagger}, \partial A^{\dagger}\right), \end{aligned}\begin{aligned} \mathcal{L}_\mathrm{T} &=& \mathcal{L}_\mathrm{G}(h, \partial h, \partial^2 h, A^\dagger,\partial A^\dagger) \\&&+ \mathcal{L}_\mathrm{M}(\varphi,\partial\varphi,\phi,\partial\phi,h,\partial h,A^\dagger,\partial A^\dagger), \end{aligned}conf 1.000
\begin{aligned} \left(\frac{\delta \mathcal{L}_{\mathrm{M}}}{\delta h_{c}{ }^{\mu}}\right)_{\dagger} & =\tau^{c}{ }_{\mu}+2 \sigma_{a b}{ }^{c} \mathcal{V}^{a} b_{\mu}^{b}-2 \sigma^{c b}{ }_{b} V_{\mu}=\tau^{\dagger c}{ }_{\mu},(184 \mathrm{a}) \\ \left(\frac{\delta \mathcal{L}_{\mathrm{M}}}{\delta A^{\dagger a b}{ }_{\mu}}\right)_{\dagger} & =\sigma_{a b}{ }^{\mu}, \\ \left(\frac{\delta \mathcal{L}_{\mathrm{M}}}{\delta V_{\mu}}\right)_{\dagger} & =\zeta^{\mu}-2 h_{a}{ }^{\mu} \sigma^{a b}{ }_{b}=\zeta^{\dagger \mu}=0 \\ \left(\frac{\delta \mathcal{L}_{\mathrm{M}}}{\delta \varphi}\right)_{\dagger} & =\frac{\delta \mathcal{L}_{\mathrm{M}}}{\delta \varphi}, \\ \left(\frac{\delta \mathcal{L}_{\mathrm{M}}}{\delta \phi}\right)_{\dagger} & =\frac{\delta \mathcal{L}_{\mathrm{M}}}{\delta \phi}, \end{aligned}\zeta^{a}=2 \sigma^{a b}{ }_{b}\zeta^a = 2{\sigma^{ab}}_b,conf 0.955
j^{a}=2 s^{a b}{ }_{b}j^a = 2{s^{ab}}_b.conf 0.923
\mathcal{L}_{\mathrm{D}}=h^{-1} L_{\mathrm{D}}=h^{-1}\left(\frac{1}{2} i \bar{\psi} \gamma^{a} \stackrel{\leftrightarrow}{\mathcal{D}}_{a}^{\dagger} \psi-\mu \phi \bar{\psi} \psi\right)\mathcal{L}_\mathrm{D} = h^{-1} L_\mathrm{D} = h^{-1}({\frac{1}{2}}i \bar{\psi}\gamma^a{\stackrel{\leftrightarrow}{\mathcal{D}^\ast_a}}\psi - \mu\phi\bar{\psi}\psi),conf 0.936
i \gamma^{a} \mathcal{D}_{a}^{\dagger} \psi-\mu \phi \psi=0i\gamma^a\mathcal{D}^\dagger_a\psi-\mu\phi\psi= 0.conf 0.957
\tau^{a}{ }_{b}=\frac{1}{2} i h^{-1} \bar{\psi} \gamma^{a} \stackrel{\leftrightarrow}{\mathcal{D}}_{b}^{\dagger} \psi-\delta_{b}^{a} \mathcal{L}_{\mathrm{D}}-2 \sigma_{c b}{ }^{a} \mathcal{V}^{c}{\tau^a}_b = {\frac{1}{2}}ih^{-1}\bar{\psi}\gamma^a{\stackrel{\leftrightarrow}{\mathcal{D}^\dagger_b}}\psi -\delta_b^a \mathcal{L}_\mathrm{D} -2{\sigma_{cb}}^a\mathcal{V}^c,conf 0.955
\tau^{\prime a}{ }_{b}=\tau^{a}{ }_{b}-2 \theta \sigma_{c d}{ }^{a} \mathcal{P}^{c} \delta_{b}^{d}{\tau^{\prime a}}_b = {\tau^a}_b - 2\theta {\sigma_{cd}}^a \mathcal{P}^c\delta^d_b.conf 0.926
\tau^{\dagger a}{ }_{b}=\tau^{a}{ }_{b}+2 \sigma_{c b}{ }^{a} \mathcal{V}^{c}{\tau^{\dagger a}}_{b} = {\tau^a}_{b} + 2{\sigma_{cb}}^a\mathcal{V}^c.conf 0.943
\begin{aligned} \mathcal{L}_{\mathrm{M}}=h^{-1}\left[\frac{1}{2} i \bar{\psi} \gamma^{a} \overleftrightarrow{\mathcal{D}}_{a} \psi-\mu \phi \bar{\psi} \psi\right. & +\frac{1}{2} \nu\left(\mathcal{D}_{a}^{\dagger} \phi\right)\left(\mathcal{D}^{\dagger a} \phi\right)-\lambda \phi^{4} \\ & \left.-a \phi^{2} \mathcal{R}^{\dagger}+\phi^{2} L_{\mathcal{T}^{\dagger 2}}\right],(192) \end{aligned}\begin{aligned} \mathcal{L}_\mathrm{M} = h^{-1}[{\frac{1}{2}}i\bar{\psi}\gamma^a {\stackrel{\leftrightarrow}{\mathcal{D}_a}}\psi - \mu\phi\bar{\psi}\psi &&+ {\frac{1}{2}}\nu (\mathcal{D}^\dagger_a\phi) (\mathcal{D}^{\dagger a} \phi) - \lambda\phi^4 \\&&- a\phi^2\mathcal{R}^\dagger + \phi^2L_{\mathcal{T}^{\dagger 2}}], \end{aligned}conf 0.898
\mathcal{L}_{\mathrm{M}}=h^{-1} L_{\mathrm{M}}=-h^{-1}\left(\frac{1}{4} \mathcal{F}_{a b} \mathcal{F}^{a b}+\mathcal{J}^{a} \mathcal{A}_{a}\right)\mathcal{L}_\mathrm{M} = h^{-1}L_\mathrm{M} =-h^{-1}({\frac{1}{4}} \mathcal{F}_{ab}\mathcal{F}^{ab} + \mathcal{J}^a \mathcal{A}_a),conf 1.000
\mathcal{D}_{a}^{\dagger} \mathcal{F}^{a c}-\frac{1}{2} \mathcal{T}^{\dagger c}{ }_{a b} \mathcal{F}^{a b}=\mathcal{J}^{c}\mathcal{D}^\dagger_a\mathcal{F}^{ac}-{\frac{1}{2}}{\mathcal{T}^{\dagger c}}_{ab}\mathcal{F}^{ab} = \mathcal{J}^c.conf 0.965
\mathcal{D}_{[a}^{\dagger} \mathcal{F}_{b c]}-\mathcal{T}^{\dagger d}{ }_{[a b} \mathcal{F}_{c] d}=0\mathcal{D}^\dagger_{[a}\mathcal{F}_{bc]} -{\mathcal{T}^{\dagger d}}_{[ab}\mathcal{F}_{c]d}= 0.conf 0.926
\widehat{\varphi} \equiv\left(\frac{\phi}{\phi_{0}}\right)^{-w} \varphi, \quad \widehat{h}_{a}{ }^{\mu} \equiv\left(\frac{\phi}{\phi_{0}}\right)^{-1} h_{a}{ }^{\mu}, \quad \widehat{A}_{\mu}^{\dagger a b} \equiv A^{\dagger a b}{ }_{\mu}\widehat{\varphi}\equiv \left(\frac{\phi}{\phi_0}\right)^{-w}\varphi, \quad {\widehat{h}_a}^{\phantom{a}\mu} \equiv \left(\frac{\phi}{\phi_0}\right)^{-1}{h_a}^\mu, \quad {\widehat{A}^{\dagger ab}}_{\phantom{ab}\mu} \equiv {A^{\dagger ab}}_\mu,conf 0.794
\mathcal{D}_{a}^{\dagger} \chi=\left(\frac{\phi}{\phi_{0}}\right)^{1-w} \widehat{\mathcal{D}}_{a}^{\dagger} \widehat{\chi}\mathcal{D}^\dagger_a\chi = \left(\frac{\phi}{\phi_0}\right)^{1-w} \widehat{\mathcal{D}}^\dagger_a \widehat{\chi},conf 0.953
\begin{aligned} \mathcal{L}_{\mathrm{T}}= & \mathcal{L}_{\mathrm{G}}\left(\widehat{h}, \partial \widehat{h}, \partial^{2} \widehat{h}, \widehat{A}^{\dagger}, \partial \widehat{A}^{\dagger}\right) \\ & +\mathcal{L}_{\mathrm{M}}\left(\widehat{\varphi}, \partial \widehat{\varphi}, \phi_{0}, 0,, \widehat{h}, \partial \widehat{h}, \widehat{A}^{\dagger}, \partial \widehat{A}^{\dagger}\right) . \end{aligned}\begin{aligned} \mathcal{L}_\mathrm{T} &=& \mathcal{L}_\mathrm{G}(\widehat{h}, \partial\widehat{h},\partial^2\widehat{h},\widehat{A}^\dagger,\partial \widehat{A}^\dagger) \\&&+ \mathcal{L}_\mathrm{M}(\widehat{\varphi},\partial\widehat{\varphi},\phi_0,0,,\widehat{h},\partial \widehat{h}, \widehat{A}^\dagger, \partial \widehat{A}^\dagger). \end{aligned}conf 1.000
\begin{aligned} \widehat{\varphi} & \equiv\left(\frac{\phi}{\phi_{0}}\right)^{-w} \varphi \\ \widehat{h}_{a}{ }^{\mu} & \equiv\left(\frac{\phi}{\phi_{0}}\right)^{-1} h_{a}{ }^{\mu} \\ \widehat{A}_{\mu}^{\dagger a b} & \equiv A_{\mu}^{a b}+\left(\mathcal{V}^{a} b_{\mu}^{b}-\mathcal{V}^{b} b_{\mu}^{a}\right) \\ \widehat{V}_{\mu} & \equiv V_{\mu}+\frac{1}{3} T_{\mu}+\partial_{\mu} \ln \left(\frac{\phi}{\phi_{0}}\right) \end{aligned}\begin{aligned} \widehat{\varphi}&\equiv& \left(\frac{\phi}{\phi_0}\right)^{-w}\varphi,\\ {\widehat{h}_a}^{\phantom{a}\mu} &\equiv &\left(\frac{\phi}{\phi_0}\right)^{-1}{h_a}^\mu,\\ {\widehat{A}^{\dagger ab}}_{\phantom{ab}\mu} &\equiv& {A^{ab}}_\mu + (\mathcal{V}^a{b^b}_\mu - \mathcal{V}^b{b^a}_\mu),\\ \widehat{V}_\mu &\equiv& V_\mu +{\frac{1}{3}}T_\mu + \partial_\mu\ln\left(\frac{\phi}{\phi_0}\right), \end{aligned}conf 0.735
\phi v^{b}{ }^{0} \mathcal{D}_{b}^{\dagger} v_{a}=\left(\delta_{a}^{b}-v_{a} v^{b}\right)^{0} \mathcal{D}_{b}^{\dagger} \phi\phi\,v^b\,{{}^0}\mathcal{D}^\dagger_bv_a= (\delta_a^b - v_av^b){{}^0}\mathcal{D}^\dagger_b\phi.conf 0.914
\left[{ }^{0} D_{\mu}^{\dagger},{ }^{0} D_{\nu}^{\dagger}\right] \varphi=\frac{1}{2}{ }^{0} R^{\dagger a b}{ }_{\mu \nu} \Sigma_{a b} \varphi+w H_{\mu \nu}^{\dagger} \varphi[{^0}\!D^\dagger_\mu,{^0}\!D^\dagger_\nu]\varphi={\frac{1}{2}} {^0}\!{R^{\dagger ab}}_{\mu\nu}\Sigma_{ab}\varphi + wH^\dagger_{\mu\nu}\varphi,conf 0.818
\left[{ }^{0} \mathcal{D}_{c}^{\dagger},{ }^{0} \mathcal{D}_{d}^{\dagger}\right] \varphi=\frac{1}{2}{ }^{0} \mathcal{R}^{\dagger a b}{ }_{c d} \Sigma_{a b} \varphi+w \mathcal{H}_{c d}^{\dagger} \varphi[{{}^0}\mathcal{D}^\dagger_c,{{}^0}\mathcal{D}^\dagger_d]\varphi={\frac{1}{2}} {^0}{\mathcal{R}^{\dagger ab}}_{cd}\Sigma_{ab}\varphi + w\mathcal{H}^\dagger_{cd}\varphi,conf 0.921
L_{\mathrm{G}}=L_{0_{\mathcal{R}^{\dagger 2}}}+L_{\mathcal{H}^{\dagger 2}}L_\mathrm{G} = L_{{^0}\mathcal{R}^{\dagger 2}} + L_{\mathcal{H}^{\dagger 2}},conf 0.943
\begin{array}{r} \mathcal{L}_{\mathrm{M}}=h^{-1}\left[\frac{1}{2} i \bar{\psi} \gamma^{a}{ }^{0} \stackrel{\leftrightarrow}{\mathcal{D}}_{a} \psi-\mu \phi \bar{\psi} \psi+\frac{1}{2} \nu\left({ }^{0} \mathcal{D}_{a}^{\dagger} \phi\right)\left({ }^{0} \mathcal{D}^{\dagger a} \phi\right)-\lambda \phi^{4}\right. \\ \left.-a \phi^{2}{ }^{0} \mathcal{R}^{\dagger}\right] .(204) \end{array}\begin{aligned} \mathcal{L}_\mathrm{M} = h^{-1}[{\frac{1}{2}}i\bar{\psi}\gamma^a \,{\stackrel{\leftrightarrow}{{{}^{0}\mathcal{D}}_a}}\psi \!\!-\!\! \mu\phi\bar{\psi}\psi &&+ {\frac{1}{2}}\nu ({{}^{0}\mathcal{D}}^\dagger_a\phi) ({{}^{0}\mathcal{D}}^{\dagger a} \phi) - \lambda\phi^4 \\&& \hspace{1.5cm} - a\phi^2\,{{}^{0}\mathcal{R}}^\dagger ]. \end{aligned}conf 0.813
\begin{aligned} \mathcal{L}_{\mathrm{T}}= & \mathcal{L}_{\mathrm{G}}\left(h, \partial h, \partial^{2} h, A, \partial A, V, \partial V\right) \\ & +\mathcal{L}_{\mathrm{M}}\left(\varphi, \partial \varphi, \phi, \partial \phi, h, \partial h, \partial^{2} h, A, \partial A, V, \partial V\right) . \end{aligned}\begin{aligned} \mathcal{L}_\mathrm{T} & = & \mathcal{L}_\mathrm{G}(h,\partial h, \partial^2 h, A, \partial A, V, \partial V) \\ && \hspace{0.2cm} + \mathcal{L}_\mathrm{M}(\varphi,\partial\varphi,\phi,\partial\phi,h,\partial h, \partial^2 h, A, \partial A, V, \partial V).\phantom{AAA} \end{aligned}conf 0.934
\begin{aligned} \mathcal{L}_{\mathrm{T}}= & \mathcal{L}_{\mathrm{G}}\left(h, \partial h, \partial^{2} h, A^{\dagger}, \partial A^{\dagger}\right) \\ & +\mathcal{L}_{\mathrm{M}}\left(\varphi, \partial \varphi, \phi, \partial \phi, h, \partial h, \partial^{2} h, A^{\dagger}, \partial A^{\dagger}\right) \end{aligned}\begin{aligned} \mathcal{L}_\mathrm{T} & = & \mathcal{L}_\mathrm{G}(h,\partial h, \partial^2 h, A^\dagger, \partial A^\dagger) \\ && \hspace{0.2cm} + \mathcal{L}_\mathrm{M}(\varphi,\partial\varphi,\phi,\partial\phi,h,\partial h, \partial^2 h, A^\dagger, \partial A^\dagger),\phantom{AAA} \end{aligned}conf 0.934
\delta J^{a}=-\left[A^{\dagger a}{ }_{b \mu}-w\left(V_{\mu}+\frac{1}{3} T_{\mu}\right) \delta_{b}^{a}\right] J^{b} \delta x^{\mu}\delta J^a = - [{A^{\dagger a}}_{b\mu}-w(V_\mu+{\frac{1}{3}}T_\mu)\delta^a_b] J^b \,\delta x^\mu.conf 0.962
\begin{aligned} D_{\mu}^{\dagger} J^{a} & =\partial_{\mu} J^{a}-w\left(V_{\mu}+\frac{1}{3} T_{\mu}\right) J^{a}+A_{b \mu}^{\dagger a} J^{b} \\ & =\partial_{\mu}^{\dagger} J^{a}+A_{b \mu}^{\dagger a} J^{b} \end{aligned}\begin{aligned} D^\dagger_\mu J^a &=& \partial_\mu J^a - w(V_\mu+{\frac{1}{3}}T_\mu) J^a + {A^{\dagger a}}_{b\mu}J^b \\ &=& \partial^\dagger_\mu J^a + {A^{\dagger a}}_{b\mu}J^b, \end{aligned}conf 0.805
\Delta_{\mu}^{\dagger} \equiv \partial_{\mu}^{\dagger}+\Gamma^{\sigma}{ }_{\rho \mu} \mathrm{X}^{\rho}{ }_{\sigma}+\frac{1}{2} A^{a b}{ }_{\mu} \Sigma_{a b}=\nabla_{\mu}^{\dagger}+D_{\mu}^{\dagger}-\partial_{\mu}^{\dagger}\Delta^\dagger_\mu \equiv \partial^\dagger_\mu + {\Gamma^\sigma}_{\rho\mu} {{\text\mathsf{X}}^\rho}_\sigma + {\frac{1}{2}}{A^{ab}}_\mu\Sigma_{ab} = \nabla^\dagger_\mu + D^\dagger_\mu - \partial^\dagger_\mu,conf 0.838
\Delta_{\mu}^{\dagger} e^{a}{ }_{\nu} \equiv \partial_{\mu}^{\dagger} e^{a}{ }_{\nu}-\Gamma^{\sigma}{ }_{\nu \mu} e^{a}{ }_{\sigma}+A^{\dagger a}{ }_{b \mu} e^{b}{ }_{\nu}=0\Delta^\dagger_\mu {e^a}_\nu \equiv \partial^\dagger_\mu {e^a}_\nu - {\Gamma^\sigma}_{\nu\mu}{e^a}_\sigma + {A^{\dagger a}}_{b\mu}{e^b}_\nu =0,conf 0.892
\nabla_{\sigma} g_{\mu \nu}=2\left(V_{\sigma}+\frac{1}{3} T_{\sigma}\right) g_{\mu \nu}\nabla_\sigma g_{\mu\nu} = 2(V_\sigma + {\frac{1}{3}}T_\sigma)g_{\mu\nu}.conf 1.000
\begin{aligned} R^{\dagger \rho}{ }_{\sigma \mu \nu} & =2\left(\partial_{[\mu} \Gamma^{\rho}{ }_{|\sigma| \nu]}+\Gamma^{\rho}{ }_{\lambda[\mu} \Gamma^{\lambda}{ }_{|\sigma| \nu]}\right)+H_{\mu \nu}^{\dagger} \delta_{\sigma}^{\rho}, \\ T^{\dagger \lambda}{ }_{\mu \nu} & =2 \Gamma^{\lambda}{ }_{[\nu \mu]}, \\ H_{\mu \nu}^{\dagger} & =\partial_{\mu}\left(V_{\nu}+\frac{1}{3} T_{\nu}\right)-\partial_{\nu}\left(V_{\mu}+\frac{1}{3} T_{\mu}\right) . \end{aligned}\begin{aligned} {R^\rho}_{\sigma\mu\nu} & = & 2(\partial_{[\mu}{\Gamma^\rho}_{|\sigma|\nu]} +{\Gamma^\rho}_{\lambda[\mu}{\Gamma^\lambda}_{|\sigma|\nu]})- H_{\mu\nu}\delta_\sigma^\rho,\phantom{AAA} \\ {T^{\ast\lambda}}_{\mu\nu} & = & 2{\Gamma^\lambda}_{[\nu\mu]}, \\ H_{\mu\nu} & = & 2\partial_{[\mu} B_{\nu]}, \end{aligned}conf 0.773
\Gamma^{\lambda}{ }_{\mu \lambda}=\Gamma^{\lambda}{ }_{\lambda \mu}{\Gamma^\lambda}_{\mu\lambda} = {\Gamma^\lambda}_{\lambda\mu}.conf 0.966
\widehat{R}^{\rho}{ }_{\sigma \mu \nu} \equiv R^{\dagger \rho}{ }_{\sigma \mu \nu}-H_{\mu \nu}^{\dagger} \delta_{\sigma}^{\rho}{\widetilde{R}^{\rho}}_{\phantom{\rho}\sigma\mu\nu} \equiv {R^\rho}_{\sigma\mu\nu} + H_{\mu\nu}\delta_\sigma^\rho.conf 0.804
\left[\nabla_{\mu}^{\dagger}, \nabla_{\nu}^{\dagger}\right] J^{\rho}=\widehat{R}^{\rho}{ }_{\sigma \mu \nu} J^{\sigma}-w H_{\mu \nu}^{\dagger} J^{\rho}-T^{\dagger \sigma}{ }_{\mu \nu} \nabla_{\sigma}^{\dagger} V^{\rho}[\nabla^\dagger_\mu,\nabla^\dagger_\nu] J^\rho = {\widehat{R}^{\rho}}_{\phantom{\rho}\sigma\mu\nu}J^\sigma - wH^\dagger_{\mu\nu} J^\rho - {T^{\dagger\sigma}}_{\mu\nu}\nabla^\dagger_\sigma V^\rho.conf 0.821
\Gamma^{\lambda}{ }_{\mu \nu}={ }^{0} \Gamma^{\dagger \lambda}{ }_{\mu \nu}+K^{\dagger \lambda}{ }_{\mu \nu}{\Gamma^\lambda}_{\mu\nu} = {^0}{\Gamma^{\dagger\lambda}}_{\mu \nu} + {K^{\dagger\lambda}}_{\mu\nu},conf 0.957
\begin{aligned} { }^{0} \Gamma^{\dagger \lambda}{ }_{\mu \nu}= & \frac{1}{2} g^{\lambda \rho}\left(\partial_{\mu}^{\dagger} g_{\nu \rho}+\partial_{\nu}^{\dagger} g_{\mu \rho}-\partial_{\rho}^{\dagger} g_{\mu \nu}\right) \\ = & { }^{0} \Gamma^{\lambda}{ }_{\mu \nu}-\delta_{\nu}^{\lambda}\left(V_{\mu}+\frac{1}{3} T_{\mu}\right)-\delta_{\mu}^{\lambda}\left(V_{\nu}+\frac{1}{3} T_{\nu}\right) \\ & +g_{\mu \nu}\left(V^{\lambda}+\frac{1}{3} T^{\lambda}\right) \end{aligned}\begin{aligned} {^0}{\Gamma^{\dagger\lambda}}_{\mu \nu} &=& {\frac{1}{2}}g^{\lambda\rho}(\partial^\dagger_\mu g_{\nu\rho}+\partial^\dagger_\nu g_{\mu\rho}-\partial^\dagger_\rho g_{\mu\nu}) \\ &=& {^0}{\Gamma^{\lambda}}_{\mu \nu} - \delta^\lambda_\nu (V_\mu+{\frac{1}{3}}T_\mu) - \delta^\lambda_\mu (V_\nu+{\frac{1}{3}}T_\nu) \\ &&\hspace{35mm}+ g_{\mu\nu}(V^\lambda +{\frac{1}{3}}T^\lambda),\phantom{ABC} \end{aligned}conf 0.774
K^{\dagger \lambda}{ }_{\mu \nu}=-\frac{1}{2}\left(T^{\dagger \lambda}{ }_{\mu \nu}-T^{\dagger}{ }_{\nu}{ }^{\lambda}{ }_{\mu}+T^{\dagger}{ }_{\mu \nu}{ }^{\lambda}\right){K^{\dagger\lambda}}_{\mu\nu}=-{\frac{1}{2}}({T^{\dagger\lambda}}_{\mu\nu}- {{{T^\dagger}_\nu}^{\lambda}}_\mu + {{T^\dagger}_{\mu\nu}}^{\lambda}).conf 0.947
\begin{aligned} & L_{\mathcal{R}^{\dagger 2}}=\alpha_{1} \mathcal{R}^{\dagger 2}+\alpha_{2} \mathcal{R}_{a b}^{\dagger} \mathcal{R}^{\dagger a b}+\alpha_{3} \mathcal{R}_{a b}^{\dagger} \mathcal{R}^{\dagger b a}+\alpha_{4} \mathcal{R}_{a b c d}^{\dagger} \mathcal{R}^{\dagger a b c d}+\alpha_{5} \mathcal{R}_{a b c d}^{\dagger} \mathcal{R}^{\dagger a c b d}+\alpha_{6} \mathcal{R}_{a b c d}^{\dagger} \mathcal{R}^{\dagger c d a b}, \\ & L_{\mathcal{H}^{\dagger 2}}=\frac{1}{2} \xi \mathcal{H}_{a b}^{\dagger} \mathcal{H}^{\dagger a b}, \end{aligned}\begin{aligned} L_{\mathcal{R}^{\dagger 2}} & = & \alpha_1 \mathcal{R}^{\dagger 2} + \alpha_2 \mathcal{R}^\dagger_{ab}\mathcal{R}^{\dagger ab} + \alpha_3 \mathcal{R}^\dagger_{ab}\mathcal{R}^{\dagger ba} + \alpha_4 \mathcal{R}^\dagger_{abcd}\mathcal{R}^{\dagger abcd} + \alpha_5 {\cal R}^\dagger_{abcd}{\cal R}^{\dagger acbd} + \alpha_6 {\cal R}^\dagger_{abcd}{\cal R}^{\dagger cdab}, \\ L_{{\cal H}^{\dagger 2}} & = & {\frac{1}{2}}\xi{\cal H}^\dagger_{ab} {\cal H}^{\dagger ab}, \end{aligned}conf 0.731
L_{\varphi}+L_{\phi}+\phi^{2} L_{\mathcal{R}^{\dagger}}+\phi^{2} L_{\mathcal{T}^{\dagger 2}}
\begin{aligned} L_{\varphi} & =L_{\varphi}(\varphi, \partial \varphi, h, \partial h, A, V, \phi) \\ L_{\phi} & =\frac{1}{2} \nu \mathcal{D}_{a}^{\dagger} \phi \mathcal{D}^{\dagger a} \phi-\lambda \phi^{4} \\ L_{\mathcal{R}^{\dagger}} & =-\frac{1}{2} a \mathcal{R}^{\dagger} \\ L_{\mathcal{T}^{\ddagger 2}} & =\beta_{1} \mathcal{T}_{a b c}^{\dagger} \mathcal{T}^{\dagger a b c}+\beta_{2} \mathcal{T}_{a b c}^{\dagger} \mathcal{T}^{\dagger b a c}, \end{aligned}\begin{aligned} L_\varphi & = & L_\varphi(\varphi,\partial\varphi,h,\partial h,A,V,\phi), \\ L_\phi & = & {\frac{1}{2}}\nu\mathcal{D}^\dagger_a\phi\,\mathcal{D}^{\dagger a}\phi -\lambda\phi^4, \\ L_{\mathcal{R}^\dagger} & = & -{\frac{1}{2}}a\mathcal{R}^\dagger, \\ L_{\mathcal{T}^{\dagger 2}} & = & \beta_1\mathcal{T}^\dagger_{abc}\mathcal{T}^{\dagger abc} + \beta_2 \mathcal{T}^\dagger_{abc}{\cal T}^{\dagger bac}, \end{aligned}conf 0.823
\left(t_{\mathcal{R}^{2}}^{\dagger}\right)^{a}{ }_{b}+\left(t_{\mathcal{H}^{2}}^{\dagger}\right)^{a}{ }_{b}+\left(\tau_{\varphi}^{\dagger}\right)^{a}{ }_{b}+\left(\tau_{\phi}^{\dagger}\right)^{a}{ }_{b}+\left(\tau_{\mathcal{R}}^{\dagger}\right)^{a}{ }_{b}+\left(\tau_{\mathcal{T}^{2}}^{\dagger}\right)^{a}{ }_{b}=0(j_{\mathcal{R}^2})_a + (j_{\mathcal{H}^2})_a + (\zeta_\varphi)_a + (\zeta_\phi)_a + (\zeta_\mathcal{R})_a + (\zeta_{\mathcal{T}^2})_a = 0,conf 0.671
\begin{aligned} h\left(t_{\mathcal{R}^{2}}^{\dagger}\right)^{a}{ }_{b}= & \alpha_{1} \mathcal{R}^{\dagger}\left(4 \mathcal{R}^{\dagger a}{ }_{b}-\delta_{b}^{a} \mathcal{R}^{\dagger}\right) \\ & +\alpha_{2}\left[2\left(\mathcal{R}^{\dagger c a} \mathcal{R}_{c b}^{\dagger}-\mathcal{R}^{\dagger c d} \mathcal{R}^{\dagger a}{ }_{c d b}\right)-\delta_{b}^{a} \mathcal{R}_{c d}^{\dagger} \mathcal{R}^{\dagger c d}\right] \\ & +\alpha_{3}\left[2\left(\mathcal{R}^{\dagger a c} \mathcal{R}_{c b}^{\dagger}-\mathcal{R}^{\dagger d c} \mathcal{R}^{\dagger a}{ }_{c d b}\right)-\delta_{b}^{a} \mathcal{R}_{c d}^{\dagger} \mathcal{R}^{\dagger d c}\right] \\ & +\alpha_{4}\left[4 \mathcal{R}^{\dagger c d e a} \mathcal{R}_{c d e b}^{\dagger}-\delta_{b}^{a} \mathcal{R}^{\dagger c d e f} \mathcal{R}_{c d e f}^{\dagger}\right] \\ & +\alpha_{5}\left[2\left(\mathcal{R}^{\dagger a c d e}-\mathcal{R}^{\dagger e c d a}\right) \mathcal{R}_{c d e b}^{\dagger}-\delta_{b}^{a} \mathcal{R}^{\dagger c e d f} \mathcal{R}_{c d e f}^{\dagger}\right] \\ & +\alpha_{6}\left[4 \mathcal{R}^{\dagger e a c d} \mathcal{R}_{c d e b}^{\dagger}-\delta_{b}^{a} \mathcal{R}^{\dagger e f c d} \mathcal{R}_{c d e f}^{\dagger}\right] \\ h\left(t_{\mathcal{H}^{2}}^{\dagger}\right)^{a}{ }_{b}= & \frac{1}{2} \xi\left[4 \mathcal{H}_{b c}^{\dagger} \mathcal{H}^{\dagger a c}-\delta_{b}^{a} \mathcal{H}_{c d}^{\dagger} \mathcal{H}^{\dagger c d}+\frac{4}{3} \mathcal{D}_{b}^{\dagger}\left(\mathcal{D}_{c}^{\dagger} \mathcal{H}^{\dagger c a}-\frac{1}{2} \mathcal{T}^{\dagger a}{ }_{c d} \mathcal{H}^{\dagger c d}\right)\right], \end{aligned}\begin{aligned} h{(t^\dagger_{\mathcal{R}^2})^a}_b & = & \alpha_1 \mathcal{R}^\dagger(4{\mathcal{R}^{\dagger a}}_b-\delta^a_b \mathcal{R}^\dagger) \\ & & + \alpha_2[2(\mathcal{R}^{\dagger ca}\mathcal{R}^\dagger_{cb} \!\!-\!\! \mathcal{R}^{\dagger cd}{\mathcal{R}^{\dagger a}}_{cdb}) - \delta_b^a {\cal R}^{\dagger}_{cd} {\cal R}^{\dagger cd}] \\ & & + \alpha_3[2({\cal R}^{\dagger ac}{\cal R}^\dagger_{cb} \!\!-\!\! {\cal R}^{\dagger dc}{{\cal R}^{\dagger a}}_{cdb}) - \delta_b^a {\cal R}^{\dagger}_{cd} {\cal R}^{\dagger dc}] \\ & & + \alpha_4[4{\cal R}^{\dagger cdea} {\cal R}^\dagger_{cdeb} \!-\delta^a_b{\cal R}^{\dagger cdef}{\cal R}^\dagger_{cdef}] \\ & & + \alpha_5[2({\cal R}^{\dagger acde}\!\!-\!\!{\cal R}^{\dagger ecda}) {\cal R}^\dagger_{cdeb} \!- \delta_b^a {\cal R}^{\dagger cedf}{\cal R}^\dagger_{cdef}] \\ & & + \alpha_6[4{\cal R}^{\dagger eacd} {\cal R}^\dagger_{cdeb} -\delta^a_b{\cal R}^{\dagger efcd}{\cal R}^\dagger_{cdef}],\\ [1mm] h {(t^\dagger_{{\cal H}^2})^a}_b & = & {\frac{1}{2}}\xi [4{\cal H}^\dagger_{bc}{\cal H}^{\dagger ac}-\delta_b^a {\cal H}^\dagger_{cd}{\cal H}^{\dagger cd}+ {\frac{4}{3}}{\cal D}^\dagger_b({\cal D}^\dagger_c{\cal H}^{\dagger ca}-{\frac{1}{2}}{{\cal T}^{\dagger a}}_{cd}{\cal H}^{\dagger cd})], \end{aligned}conf 0.559
\begin{aligned} h\left(\tau_{\phi}^{\dagger}\right)^{a}{ }_{b}= & \frac{1}{2} \nu\left[\frac{4}{3} \mathcal{D}^{\dagger a} \phi \mathcal{D}_{b}^{\dagger} \phi-\frac{1}{3} \delta_{b}^{a} \mathcal{D}^{\dagger c} \phi \mathcal{D}_{c}^{\dagger} \phi+\frac{2}{3} \phi\left(\delta_{b}^{a} \mathcal{D}_{c}^{\dagger} \mathcal{D}^{\dagger c} \phi-\mathcal{D}_{b}^{\dagger} \mathcal{D}^{\dagger a} \phi\right)\right]+\lambda \delta_{b}^{a} \phi^{4} \\ h\left(\mathcal{T}_{\mathcal{R}}^{\dagger}\right)^{a}{ }_{b}= & -a \phi^{2}\left(\mathcal{R}^{\dagger a}{ }_{b}-\frac{1}{2} \delta_{b}^{a} \mathcal{R}^{\dagger}\right) \\ h\left(\tau_{\mathcal{T}^{2}}^{\dagger}\right)^{a}{ }_{b}= & \beta_{1}\left[\phi^{2}\left(4 \mathcal{T}^{\dagger c d a} \mathcal{T}_{c d b}^{\dagger}-2 \mathcal{T}^{\dagger a d e} \mathcal{T}_{b d e}^{\dagger}-\delta_{b}^{a} \mathcal{T}^{\dagger c d e} \mathcal{T}_{c d e}^{\dagger}\right)+4 \mathcal{D}_{c}^{\dagger}\left(\phi^{2} \mathcal{T}_{b}^{\dagger c a}\right)\right] \\ & +\beta_{2}\left[\phi^{2}\left(2 \mathcal{T}^{\dagger d c a} \mathcal{T}_{c d b}^{\dagger}+2 \mathcal{T}^{\dagger a d c} \mathcal{T}_{c d b}^{\dagger}-2 \mathcal{T}^{\dagger a c d} \mathcal{T}_{c b d}^{\dagger}-\delta_{b}^{a} \mathcal{T}^{\dagger d c e} \mathcal{T}_{c d e}^{\dagger}\right)+4 \mathcal{D}_{c}^{\dagger}\left(\phi^{2} \mathcal{T}^{\dagger[c}{ }_{b}{ }^{a]}\right)\right] \end{aligned}\begin{aligned} \left(s_{\mathcal{R}^{2}}\right)_{a b}{ }^{c} & +\left(s_{\mathcal{H}^{2}}\right)_{a b}{ }^{c} \\ & +\left(\sigma_{\varphi}\right)_{a b}{ }^{c}+\left(\sigma_{\phi}\right)_{a b}{ }^{c}+\left(\sigma_{\mathcal{R}}\right)_{a b}{ }^{c}+\left(\sigma_{\mathcal{T}^{2}}\right)_{a b}{ }^{c}=0, \end{aligned}\begin{aligned} \hspace*{-8mm}{(s_{\mathcal{R}^2})_{ab}}^c\! &+& \!{(s_{\mathcal{H}^2})_{ab}}^c \\ &+& \!{(\sigma_\varphi)_{ab}}^c \!+\! {(\sigma_\phi)_{ab}}^c \!+\! {(\sigma_\mathcal{R})_{ab}}^c \!+\! {(\sigma_{\mathcal{T}^2})_{ab}}^c =0, \end{aligned}conf 0.919
\begin{aligned} h\left(s_{\mathcal{R}^{2}}\right)_{a b}{ }^{c}= & 4 \alpha_{1}\left(\delta_{[a}^{c} \mathcal{D}_{b]}^{\dagger}+\frac{1}{2} \mathcal{T}^{\dagger c}{ }_{a b}\right) \mathcal{R}^{\dagger} \\ & +4 \alpha_{2}\left(\delta_{[d}^{c} \mathcal{D}_{e]}^{\dagger}+\frac{1}{2} \mathcal{T}^{\dagger c}{ }_{d e}\right) \mathcal{R}_{[a}^{\dagger}{ }_{\delta_{b]}^{e}} \\ & +4 \alpha_{3}\left(\delta_{[d}^{c} \mathcal{D}_{e]}^{\dagger}+\frac{1}{2} \mathcal{T}^{\dagger c}{ }_{d e}\right) \mathcal{R}^{\dagger d}{ }_{[a} \delta_{b]}^{e} \\ & +4 \alpha_{4}\left(\delta_{[d}^{c} \mathcal{D}_{e]}^{\dagger}+\frac{1}{2} \mathcal{T}^{\dagger c}{ }_{d e}\right) \mathcal{R}_{a b}^{\dagger}{ }_{d e} \\ & +4 \alpha_{5}\left(\delta_{[d}^{c} \mathcal{D}_{e]}^{\dagger}+\frac{1}{2} \mathcal{T}^{\dagger c}{ }_{d e}\right) \mathcal{R}^{\dagger[d}{ }_{[a b]} \\ & +4 \alpha_{6}\left(\delta_{[d}^{c} \mathcal{D}_{e]}^{\dagger}+\frac{1}{2} \mathcal{T}^{\dagger c}{ }_{d e}\right) \mathcal{R}^{\dagger d e}{ }_{a b} \\ h\left(s_{\mathcal{H}^{2}}\right)_{a b}{ }^{c}= & \frac{2}{3} \xi \eta_{f[b} \delta_{a]}^{c}\left(\mathcal{D}_{e}^{\dagger} \mathcal{H}^{\dagger e f}-\frac{1}{2} \mathcal{T}^{\dagger f}{ }_{d e} \mathcal{H}^{\dagger d e}\right), \end{aligned}\begin{aligned} h{(s_{\mathcal{R}^2})_{ab}}^c & = & 4\alpha_1 (\delta_{[a}^c \mathcal{D}^\dagger_{b]} + {\frac{1}{2}}{\mathcal{T}^{\dagger c}}_{ab})\mathcal{R}^\dagger \\ & & + 4\alpha_2(\delta_{[d}^c \mathcal{D}^\dagger_{e]} + {\frac{1}{2}}{\mathcal{T}^{\dagger c}}_{de}){\mathcal{R}^{\dagger\;\, d}_{[a}}\delta_{b]}^e \\ & & + 4\alpha_3(\delta_{[d}^c \mathcal{D}^\dagger_{e]} + {\frac{1}{2}}{{\cal T}^{\dagger c}}_{de}){{\cal R}^{\dagger d}}_{[a}\delta_{b]}^e \\ & & +4\alpha_4 (\delta^c_{[d}{\cal D}^\dagger_{e]} + {\frac{1}{2}} {{\cal T}^{\dagger c}}_{de}) {{\cal R}^\dagger_{ab}}^{de} \\ & & +4\alpha_5(\delta^c_{[d}{\cal D}^\dagger_{e]} + {\frac{1}{2}} {{\cal T}^{\dagger c}}_{de}) {\cal R}^{\dagger [d\phantom{[ab]}e]}_{\phantom{\dagger r}\,\,[ab]} \\ & & +4\alpha_6(\delta^c_{[d}{\cal D}^\dagger_{e]} + {\frac{1}{2}} {{\cal T}^{\dagger c}}_{de}) {{\cal R}^{\dagger de}}_{ab}, \\[1mm] h{(s_{{\cal H}^2})_{ab}}^c & = & {\frac{2}{3}}\xi\eta_{f[b}\delta_{a]}^c({\cal D}^\dagger_e {\cal H}^{\dagger ef}-{\frac{1}{2}}{{\cal T}^{\dagger f}}_{de}{\cal H}^{\dagger de}), \end{aligned}conf 0.700
\begin{aligned} h\left(\sigma_{\phi}\right)_{a b}{ }^{c} & =-\frac{1}{3} \nu \phi \delta_{[a}^{c} \mathcal{D}_{b]}^{\dagger} \phi \\ h\left(\sigma_{\mathcal{R}}\right)_{a b}{ }^{c} & =-a\left(\delta_{[a}^{c} \mathcal{D}_{b]}^{\dagger}+\frac{1}{2} \mathcal{T}^{\dagger c}{ }_{a b}\right) \phi^{2} \\ h\left(\sigma_{\mathcal{T}^{2}}\right)_{a b}{ }^{c} & =-4 \beta_{1} \phi^{2} \mathcal{T}^{\dagger}{ }_{[a b]}{ }^{c}+2 \beta_{2} \phi^{2}\left(\mathcal{T}^{\dagger c}{ }_{a b}+\mathcal{T}^{\dagger}{ }_{[a b]}{ }^{c}\right) \end{aligned}\left(j_{\mathcal{R}^{2}}\right)_{a}+\left(j_{\mathcal{H}^{2}}\right)_{a}+\left(\zeta_{\varphi}\right)_{a}+\left(\zeta_{\phi}\right)_{a}+\left(\zeta_{\mathcal{R}}\right)_{a}+\left(\zeta_{\mathcal{T}^{2}}\right)_{a}=0{(t^\dagger_{\mathcal{R}^2})^a}_b + {(t^\dagger_{\mathcal{H}^2})^a}_b + {(\tau^\dagger_\varphi)^a}_b + {(\tau^\dagger_\phi)^a}_b + {(\tau^\dagger_\mathcal{R})^a}_b + {(\tau^\dagger_{\mathcal{T}^2})^a}_b=0,conf 0.671
\begin{aligned} h\left(j_{\mathcal{R}^{2}}\right)_{a}= & -12 \alpha_{1} \mathcal{D}_{a}^{\dagger} \mathcal{R}^{\dagger} \\ & -4 \alpha_{2}\left[\mathcal{D}_{b}^{\dagger}\left(\mathcal{R}_{a}^{\dagger b}+\frac{1}{2} \delta_{a}^{b} \mathcal{R}^{\dagger}\right)-\frac{1}{2} \mathcal{T}^{\dagger b}{ }_{a c} \mathcal{R}_{b}^{\dagger c}\right] \\ & -4 \alpha_{3}\left[\mathcal{D}_{b}^{\dagger}\left(\mathcal{R}^{\dagger b}{ }_{a}+\frac{1}{2} \delta_{a}^{b} \mathcal{R}^{\dagger}\right)-\frac{1}{2} \mathcal{T}^{\dagger b}{ }_{a c} \mathcal{R}^{\dagger c}{ }_{b}\right] \\ & -8 \alpha_{4}\left[\mathcal{D}_{b}^{\dagger} \mathcal{R}_{a}^{\dagger b}-\frac{1}{2} \mathcal{T}^{\dagger b c d} \mathcal{R}_{a b c d}^{\dagger}\right] \\ & -2 \alpha_{5}\left[\mathcal{D}_{b}^{\dagger}\left(\mathcal{R}_{a}^{\dagger b}+\mathcal{R}^{\dagger b}{ }_{a}\right)-\mathcal{T}^{\dagger b c d}\left(\mathcal{R}_{c b a d}^{\dagger}+\mathcal{R}_{c a b d}^{\dagger}\right)\right] \\ & -8 \alpha_{6}\left[\mathcal{D}_{b}^{\dagger} \mathcal{R}^{\dagger b}{ }_{a}-\frac{1}{2} \mathcal{T}^{\dagger b c d} \mathcal{R}_{c d a b}^{\dagger}\right] \\ h\left(j_{\mathcal{H}^{2}}\right)_{a}= & -2 \xi\left[\mathcal{D}_{b}^{\dagger} \mathcal{H}^{\dagger b}{ }_{a}-\frac{1}{2} \mathcal{T}^{\dagger}{ }_{a c d} \mathcal{H}^{\dagger c d}\right] \end{aligned}\begin{aligned} h(j_{\mathcal{R}^2})_a & = & -12\alpha_1\mathcal{D}^{\dagger}_a\mathcal{R}^\dagger \\ & & - 4\alpha_2[\mathcal{D}^\dagger_b({\mathcal{R}^{\dagger \,b}_a} +{\frac{1}{2}}\delta_a^b\mathcal{R}^\dagger)-{\frac{1}{2}}{\mathcal{T}^{\dagger b}}_{ac} {\mathcal{R}^{\dagger c}_b}] \\ & & - 4\alpha_3[{\cal D}^\dagger_b({{\cal R}^{\dagger b}}_a+{\frac{1}{2}}\delta_a^b{\cal R}^\dagger)-{\frac{1}{2}}{{\cal T}^{\dagger b}}_{ac} {{\cal R}^{\dagger c}}_b] \\ & & - 8\alpha_4[{\cal D}^\dagger_b{\cal R}^{\dagger\, b}_a-{\frac{1}{2}}{\cal T}^{\dagger bcd}{\cal R}^{\dagger}_{abcd}] \\ & & -2\alpha_5[{\cal D}^\dagger_b({\cal R}^{\dagger \,b}_a \!+\! {{\cal R}^{\dagger b}}_a)\!-\!{\cal T}^{\dagger bcd}({\cal R}^{\dagger}_{cbad} \!+\!{\cal R}^{\dagger}_{cabd})] \\ & & - 8\alpha_6[{\cal D}^\dagger_b{{\cal R}^{\dagger b}}_a-{\frac{1}{2}}{\cal T}^{\dagger bcd}{\cal R}^{\dagger} _{cdab}],\\ h(j_{{\cal H}^2})_a & = & -2\xi[{\cal D}^\dagger_b{{\cal H}^{\dagger b}}_a-{\frac{1}{2}} {{\cal T}^{\dagger}}_{acd}{\cal H}^{\dagger cd}], \end{aligned}conf 0.558
\begin{aligned} h\left(\zeta_{\phi}\right)_{a} & =\nu \phi \mathcal{D}_{a}^{\dagger} \phi \\ h\left(\zeta_{\mathcal{R}}\right)_{a} & =3 a \mathcal{D}_{a}^{\dagger} \phi^{2} \\ h\left(\zeta_{\mathcal{T}^{2}}\right)_{a} & =0 \end{aligned}\begin{aligned} h(\zeta_\phi)_a & = & \nu \phi \mathcal{D}^{\dagger}_a\phi, \\ h(\zeta_\mathcal{R})_a & = & 3a\mathcal{D}^{\dagger}_a\phi^2,\\ h(\zeta_{\mathcal{T}^2})_a & = & 0. \end{aligned}conf 0.955
\partial_{\phi} L_{\mathcal{R}^{2}}+\partial_{\phi} L_{\mathcal{H}^{2}}+\partial_{\phi} L_{\varphi}+\delta_{\phi} L_{\phi}+\partial_{\phi} L_{\mathcal{R}}+\partial_{\phi} L_{\mathcal{T}^{2}}=0\partial_\phi L_{\mathcal{R}^2} + \partial_\phi L_{\mathcal{H}^2} + \partial_\phi L_\varphi + \delta_\phi L_\phi + \partial_\phi L_\mathcal{R} + \partial_\phi L_{\mathcal{T}^2} = 0,conf 1.000
\begin{aligned} \partial_{\phi} L_{\mathcal{R}^{2}} & =0, \\ \partial_{\phi} L_{\mathcal{H}^{2}} & =0, \end{aligned}\begin{aligned} \partial_\phi L_{\mathcal{R}^2} & = & 0,\\ \partial_\phi L_{\mathcal{H}^2} & = & 0, \end{aligned}conf 1.000
\begin{aligned} \delta_{\phi} L_{\phi} & =-\nu \mathcal{D}_{a}^{\dagger}\left(\mathcal{D}^{\dagger a} \phi\right)-4 \lambda \phi^{3} . \\ \partial_{\phi} L_{\mathcal{R}} & =-a \phi^{2} \mathcal{R}^{\dagger}, \\ \partial_{\phi} L_{\mathcal{T}^{2}} & =2 \beta_{1} \phi \mathcal{T}_{a b c}^{\dagger} \mathcal{T}^{\dagger a b c}+2 \beta_{2} \phi \mathcal{T}_{a b c}^{\dagger} \mathcal{T}^{\dagger b a c} . \end{aligned}\begin{aligned} \delta_\phi L_\phi & = & -\nu \mathcal{D}^\dagger_a(\mathcal{D}^{\dagger a}\phi) - 4\lambda\phi^3. \\ \partial_\phi L_\mathcal{R} & = & -a\phi^2\mathcal{R}^\dagger, \\ \partial_\phi L_{\mathcal{T}^2} & = & 2\beta_1\phi \mathcal{T}^\dagger_{abc}\mathcal{T}^{\dagger abc} + 2\beta_2 \phi \mathcal{T}^\dagger_{abc}{\cal T}^{\dagger bac}. \end{aligned}conf 0.861
\begin{aligned} & \nu \mathcal{D}_{a}^{\dagger}\left(\mathcal{D}^{\dagger a} \phi\right)+4 \lambda \phi^{3}-\partial_{\phi} L_{\varphi} \\ & \quad+\phi\left(a \mathcal{R}^{\dagger}-2 \beta_{1} \mathcal{T}_{a b c}^{\dagger} \mathcal{T}^{\dagger a b c}-2 \beta_{2} \mathcal{T}_{a b c}^{\dagger} \mathcal{T}^{\dagger b a c}\right)=0 \end{aligned}\begin{aligned} \hspace*{-10mm}\nu \mathcal{D}^\dagger_a(\mathcal{D}^{\dagger a}\phi) &+& 4\lambda\phi^3 - \partial_\phi L_\varphi \\ &&\hspace*{-17mm}+ \phi(a\mathcal{R}^\dagger- 2\beta_1\mathcal{T}^\dagger_{abc}\mathcal{T}^{\dagger abc} - 2\beta_2 \mathcal{T}^\dagger_{abc}\mathcal{T}^{\dagger bac}) = 0. \end{aligned}conf 0.777
\mathcal{D}_{c}^{\prime} \varphi^{\prime}=e^{(w-1) \rho}\left[\mathcal{D}_{c} \varphi+w \mathcal{P}_{c} \varphi+\theta \mathcal{P}^{[b} \delta_{c}^{a]} \Sigma_{a b} \varphi\right]\mathcal{D}_c^\prime \varphi^\prime = e^{(w-1)\rho}[\mathcal{D}_c \varphi + w \mathcal{P}_c\varphi +\theta\mathcal{P}^{[b}\delta_c^{a]}\Sigma_{ab}\varphi],conf 1.000
\begin{aligned} \mathcal{R}^{\prime a b}{ }_{c d} & =e^{-2 \rho}\left\{\mathcal{R}^{a b}{ }_{c d}+2 \theta \delta_{d}^{[a}\left(\mathcal{D}_{c}-\theta \mathcal{P}_{c}\right) \mathcal{P}^{b]}-2 \theta \delta_{c}^{[a}\left(\mathcal{D}_{d}-\theta \mathcal{P}_{d}\right) \mathcal{P}^{b]}-2 \theta \mathcal{P}^{[a} \mathcal{T}^{b]}{ }_{c d}-2 \theta^{2} \delta_{c}^{[a} \delta_{d}^{b]} \mathcal{P}^{e} \mathcal{P}_{e}\right\}, \\ \mathcal{R}^{\prime a}{ }_{c} & =e^{-2 \rho}\left\{\mathcal{R}^{a}{ }_{c}-2 \theta \mathcal{D}_{c} \mathcal{P}^{a}-\theta \delta_{c}^{a} \mathcal{D}_{b} \mathcal{P}^{b}+2 \theta^{2} \mathcal{P}^{a} \mathcal{P}_{c}-2 \theta^{2} \delta_{c}^{a} \mathcal{P}^{2}+\theta \mathcal{P}^{b} \mathcal{T}^{a}{ }_{c b}-\theta \mathcal{P}^{a} \mathcal{T}_{c}\right\}, \\ \mathcal{R}^{\prime} & =e^{-2 \rho}\left\{\mathcal{R}-6 \theta\left(\mathcal{D}_{a}+\frac{1}{3} \mathcal{T}_{a}\right) \mathcal{P}^{a}-6 \theta^{2} \mathcal{P}^{2}\right\}, \end{aligned}\begin{aligned} {\mathcal{R}^{\prime ab}}_{cd} & = & e^{-2\rho}\{{\mathcal{R}^{ab}}_{cd} + 2\theta\delta^{[a}_d(\mathcal{D}_c-\theta\mathcal{P}_c)\mathcal{P}^{b]} - 2\theta\delta^{[a}_c(\mathcal{D}_d-\theta\mathcal{P}_d)\mathcal{P}^{b]} - 2\theta{\cal P}^{[a}{{\cal T}^{b]}}_{cd} - 2\theta^2\delta^{[a}_c\delta^{b]}_d {\cal P}^e {\cal P}_e \}, \\ {{\cal R}^{\prime a}}_{c} & = & e^{-2\rho}\{{{\cal R}^{a}}_{c} -2\theta{\cal D}_c{\cal P}^a - \theta\delta_c^a{\cal D}_b{\cal P}^b + 2\theta^2 {\cal P}^a{\cal P}_c -2\theta^2\delta_c^a{\cal P}^2 + \theta{\cal P}^b{{\cal T}^a}_{cb} - \theta {\cal P}^a{\cal T}_c\},\\ {\cal R}^\prime & = & e^{-2\rho}\{{\cal R} -6\theta({\cal D}_a +{\frac{1}{3}}{\cal T}_a){\cal P}^a-6\theta^2{\cal P}^2\}, \end{aligned}conf 0.679
\begin{aligned} \mathcal{T}^{\prime a}{ }_{b c} & =e^{-\rho}\left\{\mathcal{T}^{a}{ }_{b c}+2(1-\theta) \mathcal{P}_{[b} \delta_{c]}^{a}\right\}, \\ \mathcal{T}^{\prime}{ }_{b} & =e^{-\rho}\left\{\mathcal{T}_{b}+3(1-\theta) \mathcal{P}_{b}\right\} . \end{aligned}\begin{aligned} {\mathcal{T}^{\prime a}}_{bc} & = & e^{-\rho}\{{\mathcal{T}^a}_{bc}+2(1-\theta)\mathcal{P}_{[b}\delta_{c]}^a\},\\ {\mathcal{T}^{\prime}}_{b} & = & e^{-\rho}\{{\mathcal{T}}_{b}+3(1-\theta)\mathcal{P}_b\}. \end{aligned}conf 0.981
\begin{aligned} \mathcal{L}_{\mathrm{M}}=h^{-1}\left[\frac{1}{2} i \bar{\psi} \gamma^{a} \stackrel{\leftrightarrow}{\mathcal{D}}_{a} \psi-\mu \phi \bar{\psi} \psi\right. & +\frac{1}{2} \nu\left(\mathcal{D}_{a} \phi\right)\left(\mathcal{D}^{a} \phi\right)-\lambda \phi^{4} \\ & \left.-a \phi^{2} \mathcal{R}+\phi^{2} L_{\mathcal{T}^{2}}\right], \end{aligned}\begin{aligned} \mathcal{L}_\mathrm{M} = h^{-1}[{\frac{1}{2}}i\bar{\psi}\gamma^a {\stackrel{\leftrightarrow}{\mathcal{D}_a}}\psi - \mu\phi\bar{\psi}\psi &&+ {\frac{1}{2}}\nu (\mathcal{D}^\ast_a\phi) (\mathcal{D}^{\ast a} \phi) - \lambda\phi^4 \\&&- a\phi^2\mathcal{R} + \phi^2L_{\mathcal{T}^{\ast 2}}], \end{aligned}conf 0.959
\begin{aligned} \mathcal{L}_{\phi}^{\prime} & =\mathcal{L}_{\phi}+\frac{1}{2} \nu h^{-1}\left(\phi^{2} \mathcal{P}^{2}-2 \phi \mathcal{P}^{a} \mathcal{D}_{a} \phi\right) \\ \left(\phi^{2} \mathcal{L}_{\mathcal{R}}\right)^{\prime} & =\phi^{2} \mathcal{L}_{\mathcal{R}}+a \phi^{2} h^{-1} \theta\left[6\left(\mathcal{D}_{a}+\frac{1}{3} \mathcal{T}_{a}\right) \mathcal{P}^{a}+6 \theta \mathcal{P}^{2}\right] \\ \left(\phi^{2} \mathcal{L}_{\mathcal{T}^{2}}\right)^{\prime} & =\phi^{2} \mathcal{L}_{\mathcal{T}^{2}}+\left(2 \beta_{1}+\beta_{2}+3 \beta_{3}\right) \phi^{2} h^{-1}(1-\theta)\left[2 \mathcal{P}_{a} \mathcal{T}^{a}+3(1-\theta) \mathcal{P}^{2}\right] \end{aligned}\begin{aligned} \mathcal{L}'_\phi & = & \mathcal{L}_\phi +{\frac{1}{2}}\nu h^{-1}(\phi^2\mathcal{P}^2-2\phi\mathcal{P}^a\mathcal{D}_a\phi), \\ (\phi^2\mathcal{L}_\mathcal{R})' & = & \phi^2\mathcal{L}_{\cal R} +a\phi^2h^{-1}\theta[6({\cal D}_a+{\frac{1}{3}}{\cal T}_a){\cal P}^a + 6\theta{\cal P}^2], \\ (\phi^2{\cal L}_{{\cal T}^2})' & = & \phi^2{\cal L}_{{\cal T}^2} +(2\beta_1+\beta_2+3\beta_3)\phi^2 h^{-1}(1-\theta) [2{\cal P}_a{\cal T}^a+3(1-\theta){\cal P}^2]. \end{aligned}conf 0.850
\frac{1}{2} \nu=-6 a \theta=3(\theta-1)\left(2 \beta_{1}+\beta_{2}+3 \beta_{3}\right){\frac{1}{2}}\nu = -6a\theta = 3(\theta-1)(2\beta_1+\beta_2+3\beta_3),conf 1.000
\mathcal{L}_{\mathrm{M}}^{\prime}=\mathcal{L}_{\mathrm{M}}+6 a \theta h^{-1}\left(\mathcal{D}_{a}+\mathcal{T}_{a}\right)\left(\phi^{2} \mathcal{P}^{a}\right)\mathcal{L}'_\mathrm{M} = \mathcal{L}_\mathrm{M} + 6a\theta h^{-1} (\mathcal{D}_a+\mathcal{T}_a)(\phi^2 \mathcal{P}^a).conf 0.953
\mathcal{L}_{\mathrm{M}}=h^{-1}\left[\frac{1}{2} i \bar{\psi} \gamma^{a} \stackrel{\leftrightarrow}{\mathcal{D}}_{a} \psi-\mu \phi \bar{\psi} \psi-\lambda \phi^{4}-a \phi^{2} \mathcal{R}+\phi^{2} L_{\mathcal{T} * 2}\right]\mathcal{L}_\mathrm{M} = h^{-1}[{\frac{1}{2}}i\bar{\psi}\gamma^a {\stackrel{\leftrightarrow}{\mathcal{D}_a}}\psi - \mu\phi\bar{\psi}\psi - \lambda\phi^4 - a\phi^2\mathcal{R} + \phi^2L_{\mathcal{T}^{\ast 2}}],conf 0.974
\mathcal{L}_{\mathrm{M}}=h^{-1}\left[\frac{1}{2} i \bar{\psi} \gamma^{a} \stackrel{\leftrightarrow}{\mathcal{D}}_{a} \psi-\mu \phi \bar{\psi} \psi+\frac{1}{2} \nu\left(\mathcal{D}_{a}^{\dagger} \phi\right)\left(\mathcal{D}^{\dagger a} \phi\right)-\lambda \phi^{4}+\frac{1}{12} \nu \phi^{2} \mathcal{R}^{\dagger}+\phi^{2} L_{\mathcal{T}{ }^{\dagger} 2}\right]-\frac{1}{2} \nu h^{-1}\left(\mathcal{D}_{a}+\mathcal{T}_{a}\right)\left(\phi^{2} \mathcal{V}^{a}\right)\mathcal{L}_\mathrm{M} = h^{-1}[{\frac{1}{2}}i\bar{\psi}\gamma^a {\stackrel{\leftrightarrow}{\mathcal{D}_a}}\psi - \mu\phi\bar{\psi}\psi + {\frac{1}{2}}\nu (\mathcal{D}^\dagger_a\phi) (\mathcal{D}^{\dagger a} \phi) - \lambda\phi^4 + {\frac{1}{12}}\nu\phi^2\mathcal{R}^\dagger + \phi^2L_{\mathcal{T}^{\dagger 2}}] - {\frac{1}{2}}\nu h^{-1} (\mathcal{D}_a+{\cal T}_a)(\phi^2 {\cal V}^a),conf 0.946
{ }^{0} \mathcal{D}_{c}^{\prime} \varphi^{\prime}=e^{(w-1) \rho}\left[{ }^{0} \mathcal{D}_{c} \varphi+w \mathcal{P}_{c} \varphi+\mathcal{P}^{[b} \delta_{c}^{a]} \Sigma_{a b} \varphi\right]{^0}\mathcal{D}_c^\prime \varphi^\prime = e^{(w-1)\rho}[{^0}\mathcal{D}_c \varphi + w \mathcal{P}_c\varphi +\mathcal{P}^{[b}\delta_c^{a]}\Sigma_{ab}\varphi],conf 0.986
\begin{aligned} { }^{0} \mathcal{R}^{\prime a b}{ }_{c d} & =e^{-2 \rho}\left\{{ }^{0} \mathcal{R}^{a b}{ }_{c d}+2 \delta_{d}^{[a}\left({ }^{0} \mathcal{D}_{c}-\mathcal{P}_{c}\right) \mathcal{P}^{b]}-2 \delta_{c}^{[a}\left({ }^{0} \mathcal{D}_{d}-\mathcal{P}_{d}\right) \mathcal{P}^{b]}-2 \delta_{c}^{[a} \delta_{d}^{b]} \mathcal{P}^{e} \mathcal{P}_{e}\right\}, \\ { }^{0} \mathcal{R}^{\prime a}{ }_{c} & =e^{-2 \rho}\left\{^{0} \mathcal{R}^{a}{ }_{c}-2^{0} \mathcal{D}_{c} \mathcal{P}^{a}-\delta_{c}^{a}{ }^{0} \mathcal{D}_{b} \mathcal{P}^{b}+2 \mathcal{P}^{a} \mathcal{P}_{c}-2 \delta_{c}^{a} \mathcal{P}^{2}\right\}, \\ { }^{0} \mathcal{R}^{\prime} & =e^{-2 \rho}\left\{{ }^{0} \mathcal{R}-6^{0} \mathcal{D}_{a} \mathcal{P}^{a}-6 \mathcal{P}^{2}\right\}, \end{aligned}\begin{aligned} {{}^0}{\mathcal{R}^{\prime ab}}_{cd} & = & e^{-2\rho}\{{{}^0}{\mathcal{R}^{ab}}_{cd} + 2\delta^{[a}_d({{}^0}\mathcal{D}_c-\mathcal{P}_c)\mathcal{P}^{b]} - 2\delta^{[a}_c({{}^0}\mathcal{D}_d-\mathcal{P}_d)\mathcal{P}^{b]} - 2\delta^{[a}_c\delta^{b]}_d {\cal P}^e {\cal P}_e \}, \\ {{{}^0}{\cal R}^{\prime a}}_{c} & = & e^{-2\rho}\{{{}^0}{{\cal R}^{a}}_{c} -2\,{{}^0}{\cal D}_c{\cal P}^a - \delta_c^a\,{{}^0}{\cal D}_b{\cal P}^b + 2{\cal P}^a{\cal P}_c -2\delta_c^a{\cal P}^2\},\\ {{}^0}{\cal R}^\prime & = & e^{-2\rho}\{{{}^0}{\cal R} -6\,{{}^0}{\cal D}_a{\cal P}^a-6{\cal P}^2\}, \end{aligned}conf 0.624
L_{\mathrm{G}}=\alpha_{1}{ }^{0} \mathcal{R}^{2}+\alpha_{2}{ }^{0} \mathcal{R}_{a b}{ }^{0} \mathcal{R}^{a b}+\alpha_{3}{ }^{0} \mathcal{R}_{a b c d}{ }^{0} \mathcal{R}^{a b c d}L_\mathrm{G} = \alpha_1\, {^0}\mathcal{R}^2 + \alpha_2\, {^0}\mathcal{R}_{ab}{^0}\mathcal{R}^{ab} + \alpha_3\, {^0}\mathcal{R}_{abcd}{^0}\mathcal{R}^{abcd},conf 0.956
L_{\mathrm{G}}=\alpha\left({ }^{0} \mathcal{R}_{a b c d}{ }^{0} \mathcal{R}^{a b c d}-2{ }^{0} \mathcal{R}_{a b}{ }^{0} \mathcal{R}^{a b}+\frac{1}{3}{ }^{0} \mathcal{R}^{2}\right)L_\mathrm{G} = \alpha( {^0}\mathcal{R}_{abcd}{^0}\mathcal{R}^{abcd} - 2\, {^0}\mathcal{R}_{ab}{^0}\mathcal{R}^{ab} + {\frac{1}{3}}\, {^0}\mathcal{R}^2),conf 0.952
L_{\mathrm{G}}=2 \alpha\left({ }^{0} \mathcal{R}_{a b}{ }^{0} \mathcal{R}^{a b}-\frac{1}{3}{ }^{0} \mathcal{R}^{2}\right)L_\mathrm{G} = 2\alpha({^0}\mathcal{R}_{ab}{^0}\mathcal{R}^{ab} - {\frac{1}{3}}\, {^0}\mathcal{R}^2),conf 0.956
\begin{aligned} { }^{0} \mathcal{C}_{a b c d}={ }^{0} \mathcal{R}_{a b c d} & -\frac{1}{2}\left(\eta_{a c}{ }^{0} \mathcal{R}_{b d}-\eta_{a d}{ }^{0} \mathcal{R}_{b c}-\eta_{b c}{ }^{0} \mathcal{R}_{a d}+\eta_{b d}{ }^{0} \mathcal{R}_{a c}\right) \\ & +\frac{1}{6}\left(\eta_{a c} \eta_{b d}-\eta_{a d} \eta_{b c}\right)^{0} \mathcal{R} . \end{aligned}\begin{aligned} {^0}\mathcal{C}_{abcd} \!=\! {^0}\mathcal{R}_{abcd} &&-{\frac{1}{2}}( \eta_{ac}{^0}\mathcal{R}_{bd} \!\!-\!\eta_{ad}{^0}\mathcal{R}_{bc} \!\!-\!\eta_{bc}{^0}\mathcal{R}_{ad} \!+\!\eta_{bd}{^0}\mathcal{R}_{ac}) \\ &&+{\frac{1}{6}}(\eta_{ac}\eta_{bd}-\eta_{ad}\eta_{bc}){^0}\mathcal{R}. \end{aligned}conf 0.964
\begin{array}{r} \mathcal{L}_{\mathrm{M}}=h^{-1}\left[\frac{1}{2} i \bar{\psi} \gamma^{a}{ }^{0} \stackrel{\leftrightarrow}{\mathcal{D}}_{a} \psi-\mu \phi \bar{\psi} \psi+\frac{1}{2} \nu\left({ }^{0} \mathcal{D}_{a} \phi\right)\left({ }^{0} \mathcal{D}^{a} \phi\right)-\lambda \phi^{4}\right. \\ \left.-a \phi^{2}{ }^{0} \mathcal{R}\right] . \end{array}\begin{aligned} \mathcal{L}_\mathrm{M} = h^{-1}[{\frac{1}{2}}i\bar{\psi}\gamma^a \,{\stackrel{\leftrightarrow}{{{{}^0}\mathcal{D}_a}}}\psi\!-\! \mu\phi\bar{\psi}\psi &&+ {\frac{1}{2}}\nu\, ({{}^0}\mathcal{D}_a\phi) \,({{}^0}\mathcal{D}^a \phi) \!-\! \lambda\phi^4 \\ &&\hspace{12mm}- a\phi^2\,{{}^0}\mathcal{R}]. \end{aligned}conf 0.795
\begin{aligned} \mathcal{L}_{\mathrm{M}}^{\prime}=\mathcal{L}_{\mathrm{M}}-h^{-1}\left[\frac{1}{2} \nu \mathcal{P}^{a}{ }^{0} \mathcal{D}_{a} \phi^{2}\right. & -6 a \phi^{2}{ }^{0} \mathcal{D}_{a} \mathcal{P}^{a} \\ & \left.+\left(\frac{1}{2} \nu+6 a\right) \phi^{2} \mathcal{P}_{a} \mathcal{P}^{a}\right] \end{aligned}\begin{aligned} \hspace*{-5mm}\mathcal{L}_\mathrm{M}^\prime \!=\! \mathcal{L}_\mathrm{M} \!-\! h^{-1}[{\frac{1}{2}}\nu \mathcal{P}^a \,{^0}\mathcal{D}_a\phi^2 &&- 6a \phi^2 \,{^0}\mathcal{D}_a \mathcal{P}^a \\&& + ({\frac{1}{2}}\nu+6a)\phi^2 {\cal P}_a {\cal P}^a]. \end{aligned}conf 0.912
\mathcal{L}_{\mathrm{M}}^{\prime}=\mathcal{L}_{\mathrm{M}}-h^{-1}\left[\frac{1}{2} \nu^{0} \mathcal{D}_{a}\left(\phi^{2} \mathcal{P}^{a}\right)\right]\mathcal{L}_\mathrm{M}^\prime = \mathcal{L}_\mathrm{M} - h^{-1}[{\frac{1}{2}}\nu \,{^0}\mathcal{D}_a(\phi^2 \mathcal{P}^a)],conf 0.988
L_{\mathrm{D}}=\frac{1}{2} i\left[\bar{\psi} \gamma^{\mu} \partial_{\mu} \psi-\left(\partial_{\mu} \bar{\psi}\right) \gamma^{\mu} \psi\right]-m \bar{\psi} \psi \equiv \Re(i \bar{\psi} \not \partial \psi)-m \bar{\psi} \psiL_\mathrm{D} = {\frac{1}{2}}i [\bar{\psi}\gamma^\mu\partial_\mu\psi-(\partial_\mu\bar{\psi})\gamma^\mu\psi] -m\bar{\psi}\psi \equiv \Re(i\bar{\psi}\slashed{\partial}\psi)- m\bar{\psi}\psi,conf 0.967
\bar{\psi} \mathscr{J}=J^{\mu} \bar{\psi} \gamma_{\mu}=\bar{\psi} \gamma^{\mu} \psi \bar{\psi} \gamma_{\mu}\bar{\psi}\slashed{J} = J^\mu\bar{\psi}\gamma_\mu = \bar{\psi}\gamma^\mu\psi\bar{\psi}\gamma_\mu.conf 0.927
\begin{aligned} \psi \bar{\psi}=\frac{1}{4}\left(\bar{\psi} \psi+\bar{\psi} \gamma^{\nu} \psi \gamma_{\nu}-\bar{\psi} \gamma^{\nu} \gamma^{5} \psi \gamma_{\nu} \gamma^{5}\right. & +\frac{1}{2} \bar{\psi} \sigma^{\nu \rho} \psi \sigma_{\nu \rho} \\ & \left.+\bar{\psi} \gamma^{5} \psi \gamma^{5}\right) \end{aligned}\begin{aligned} \hspace*{-5mm}\psi\bar{\psi} = {\frac{1}{4}}(\bar{\psi}\psi+\bar{\psi}\gamma^\nu\psi\gamma_\nu -\bar{\psi}\gamma^\nu\gamma^5\psi\gamma_\nu\gamma^5 &&+{\frac{1}{2}}\bar{\psi}\sigma^{\nu\rho}\psi\sigma_{\nu\rho} \\&& +\bar{\psi}\gamma^5\psi\gamma^5), \end{aligned}conf 0.959
\bar{\psi} \mathscr{J}=\bar{\psi}\left(\bar{\psi} \psi+\bar{\psi} i \gamma^{5} \psi i \gamma^{5}\right)\bar{\psi}\slashed{J} = \bar{\psi}(\bar{\psi}\psi+\bar{\psi}i\gamma^5\psi i\gamma^5).conf 0.915
\bar{\psi}=\frac{\bar{\psi}\left(\rho+\beta i \gamma^{5}\right)}{}=\frac{}{\rho^{2}+\beta^{2}}\bar{\psi} = \frac{\bar{\psi}(\rho+\beta i\gamma^5)\slashed{J}}{\rho^2+\beta^2}.conf 0.894
L_{\mathrm{D}}=\frac{\Re\left[i \bar{\psi}\left(\bar{\psi} \psi+\bar{\psi} i \gamma^{5} \psi i \gamma^{5}\right), \mathbb{J} \not \partial \psi\right]}{(\bar{\psi} \psi)^{2}+\left(\bar{\psi} i \gamma^{5} \psi\right)^{2}}-m \bar{\psi} \psiL_\mathrm{D} = \frac{\Re[i\bar{\psi} (\bar{\psi}\psi+\bar{\psi}i\gamma^5\psi i\gamma^5) \slashed{J}\slashed{\partial}\psi]}{(\bar{\psi}\psi)^2+ (\bar{\psi}i\gamma^5\psi)^2}- m\bar{\psi}\psi.conf 0.935
\nexists \not \partial \psi=J^{\mu} \partial_{\mu} \psi-i \sigma^{\mu \nu} J_{\mu} \partial_{\nu} \psi\slashed{J}\slashed{\partial}\psi = J^\mu\partial_\mu\psi-i\sigma^{\mu\nu}J_\mu\partial_\nu\psi.conf 0.879
S=\int d \lambda\left[\Re(i \bar{R} \dot{R})-p_{\mu}\left(\dot{x}^{\mu}-m e \bar{R} \gamma^{\mu} R\right)-m^{2} e\right]S = \int d\lambda\, [\Re(i\bar{R}\dot{R})-p_\mu(\dot{x}^{\mu}-me\bar{R}\gamma^\mu R) -m^2e].conf 1.000
S=-\int d \lambda\left[p_{\mu} \dot{x}^{\mu}-\frac{1}{2} e\left(p_{\mu} p^{\mu}-m^{2}\right)\right]S = - \int d\lambda\, [p_\mu \dot{x}^{\mu}-{\frac{1}{2}}e(p_\mu p^\mu - m^2)],conf 1.000
L_{\mathrm{G}}=-\kappa^{-1}(\Lambda+a \mathcal{R})+L_{\mathcal{R}^{2}}+\kappa^{-1} L_{\mathcal{T}^{2}}L_\mathrm{G} = -\kappa^{-1}(\Lambda + a\mathcal{R}) + L_{\mathcal{R}^2} + \kappa^{-1}L_{\mathcal{T}^2},conf 1.000
\mathcal{L}_{\mathrm{T}}=\mathcal{L}_{\mathrm{G}}(h, \partial h, A, \partial A)+\mathcal{L}_{\mathrm{M}}(\varphi, \partial \varphi, h, A)\mathcal{L}_\mathrm{T} = \mathcal{L}_\mathrm{G}(h,\partial h, A,\partial A) + \mathcal{L}_\mathrm{M}(\varphi,\partial\varphi,h,A),conf 1.000
L_{\mathrm{G}}=-\kappa^{-1}\left(\Lambda+a^{0} \mathcal{R}\right)+L_{0^{\prime}} \mathcal{R}^{2}L_\mathrm{G} = -\kappa^{-1}(\Lambda + a\,{^0}\mathcal{R}) + L_{{^0}\mathcal{R}^2},conf 0.902
L_{0} \mathcal{R}^{2}=\alpha_{1}{ }^{0} \mathcal{R}^{2}+\alpha_{2}{ }^{0} \mathcal{R}_{a b}{ }^{0} \mathcal{R}^{a b}+\alpha_{3}{ }^{0} \mathcal{R}_{a b c d}{ }^{0} \mathcal{R}^{a b c d} \text {, }L_{{^0}\mathcal{R}^2} = \alpha_1\, {^0}\mathcal{R}^2 + \alpha_2\, {^0}\mathcal{R}_{ab}{^0}\mathcal{R}^{ab} + \alpha_3\, {^0}\mathcal{R}_{abcd}{^0}\mathcal{R}^{abcd},conf 0.937
L_{\mathrm{G}}=-\kappa^{-1}\left(\Lambda+a^{0} \mathcal{R}\right)+\alpha_{1}{ }^{0} \mathcal{R}^{2}+\alpha_{2}{ }^{0} \mathcal{R}_{a b}{ }^{0} \mathcal{R}^{a b}L_\mathrm{G} = -\kappa^{-1}(\Lambda + a\,{^0}\mathcal{R}) + \alpha_1\, {^0}\mathcal{R}^2 + \alpha_2\, {^0}\mathcal{R}_{ab}{^0}\mathcal{R}^{ab}.conf 0.962
\begin{aligned} R={ }^{0} R+\frac{1}{4} T^{\mu \nu \lambda} T_{\mu \nu \lambda} & +\frac{1}{2} T^{\mu \nu \lambda} T_{\nu \mu \lambda}-T^{\mu} T_{\mu} \\ & -\frac{2}{\sqrt{-g}} \partial_{\mu}\left(\sqrt{-g} T^{\mu}\right), \end{aligned}\begin{aligned} R = {{}^0}R+{\frac{1}{4}}T^{\mu\nu\lambda}T_{\mu\nu\lambda} &&+ {\frac{1}{2}}T^{\mu\nu\lambda}T_{\nu\mu\lambda}-T^\mu T_\mu \\&& - \frac{2}{\sqrt{-g}}\partial_\mu (\sqrt{-g}T^\mu), \end{aligned}conf 0.993
\mathcal{L}_{\mathrm{G}}=-\frac{1}{2 \kappa} h^{-1} \mathcal{R}\mathcal{L}_\mathrm{G} = -\frac{1}{2\kappa}h^{-1}\mathcal{R},conf 1.000
\begin{aligned} \mathcal{R}^{a c}{ }_{b c}-\frac{1}{2} \delta_{b}^{a} \mathcal{R} & =\kappa h \tau^{a}{ }_{b}, \\ h b^{c}{ }_{\mu} D_{\nu}\left[h^{-1}\left(h_{a}{ }^{\mu} h_{b}{ }^{\nu}-h_{a}{ }^{\nu} h_{b}{ }^{\mu}\right)\right] & =2 \kappa h \sigma_{a b}{ }^{c} . \end{aligned}\begin{aligned} {\mathcal{R}^{ac}}_{bc}-{\frac{1}{2}}\delta_b^a \mathcal{R} & = & \kappa h{\tau^a}_b, \\ h{b^c}_\mu D_\nu[h^{-1}({h_a}^\mu {h_b}^\nu - {h_a}^\nu {h_b}^\mu)] & = & 2\kappa h {\sigma_{ab}}^c. \end{aligned}conf 0.944
\mathcal{T}^{c}{ }_{a b}+\delta_{a}^{c} \mathcal{T}_{b}-\delta_{b}^{c} \mathcal{T}_{a}=2 \kappa h \sigma_{a b}{ }^{c}{\mathcal{T}^c}_{ab} + \delta_a^c \mathcal{T}_b-\delta_b^c\mathcal{T}_a =2\kappa h {\sigma_{ab}}^c.conf 0.975
\mathcal{T}_{a}=-\kappa h \sigma_{a b}{ }^{b}\mathcal{T}_a = -\kappa h {\sigma_{ab}}^b.conf 0.969
\mathcal{T}^{c}{ }_{a b}=\kappa h\left(2 \sigma_{a b}{ }^{c}+\delta_{a}^{c} \sigma_{b d}{ }^{d}-\delta_{b}^{c} \sigma_{a d}{ }^{d}\right){\mathcal{T}^c}_{ab} = \kappa h (2{\sigma_{ab}}^c + \delta_a^c {\sigma_{bd}}^d - \delta_b^c {\sigma_{ad}}^d).conf 0.953
\begin{aligned} \mathcal{A}_{a b c}= & \frac{1}{2}\left(c_{a b c}+c_{b c a}-c_{c a b}\right) \\ & +\kappa h\left(\sigma_{a b c}-\sigma_{b c a}-\sigma_{c a b}+\eta_{a c} \sigma_{b d}{ }^{d}-\eta_{b c} \sigma_{a d}{ }^{d}\right), \end{aligned}\begin{aligned} \hspace*{-7mm}\mathcal{A}_{abc} = &&{\frac{1}{2}}(c_{abc}+c_{bca}-c_{cab}) \\&&+ \kappa h(\sigma_{abc}\!-\!\sigma_{bca}\!-\!\sigma_{cab} +\eta_{ac}{\sigma_{bd}}^d -\eta_{bc}{\sigma_{ad}}^d), \end{aligned}conf 0.931
\mathcal{D}_{b} h_{a}{ }^{\mu}-\mathcal{D}_{a} h_{b}{ }^{\mu}=0\mathcal{D}_b{h_a}^\mu-\mathcal{D}_a{h_b}^\mu=0.conf 0.951
\mathcal{T}_{a}=-\kappa h \sigma_{a b}{ }^{b}=0\mathcal{T}_a = -\kappa h {\sigma_{ab}}^b = 0,conf 0.971
i \gamma^{a} \mathcal{D}_{a} \psi-m \psi=0i\gamma^a\mathcal{D}_a\psi-m\psi = 0,conf 1.000
\begin{aligned} R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} R & =\kappa \frac{\tau_{\mu \nu}}{\sqrt{-g}} \\ T^{\lambda}{ }_{\mu \nu} & =\frac{\kappa}{2 \sqrt{-g}}\left(2 \sigma_{\mu \nu}{ }^{\lambda}+\delta_{\mu}^{\lambda} \sigma_{\nu \rho}{ }^{\rho}-\delta_{\nu}^{\lambda} \sigma_{\mu \rho}{ }^{\rho}\right), \end{aligned}\begin{aligned} \hspace*{-8mm}R_{\mu\nu}-{\frac{1}{2}}g_{\mu\nu}R & = & \kappa \frac{\tau_{\mu\nu}}{\sqrt{-g}},\\ {T^\lambda}_{\mu\nu} & = & \frac{\kappa}{2\sqrt{-g}} (2{\sigma_{\mu\nu}}^\lambda + \delta_\mu^\lambda {\sigma_{\nu\rho}}^\rho -\delta^\lambda_\nu{\sigma_{\mu\rho}}^\rho), \end{aligned}conf 0.898
L_{\mathrm{G}}=-\frac{1}{4} \xi \mathcal{H}_{a b} \mathcal{H}^{a b}L_\mathrm{G} = -{\frac{1}{4}}\xi\mathcal{H}_{ab}\mathcal{H}^{ab},conf 1.000
\begin{aligned} L_{\mathrm{M}} & \equiv L_{\varphi}+L_{\phi}+\phi^{2} L_{\mathcal{R}} \\ & =L_{\varphi}+\frac{1}{2} \nu \mathcal{D}_{a}^{*} \phi \mathcal{D}^{* a} \phi-\lambda \phi^{4}-\frac{1}{2} a \phi^{2} \mathcal{R}, \end{aligned}\begin{aligned} L_\mathrm{M} &\equiv& L_\varphi + L_\phi + \phi^2 L_\mathcal{R} \\ &=& L_\varphi + {\frac{1}{2}}\nu \mathcal{D}^\ast_a\phi\,\mathcal{D}^{\ast a}\phi -\lambda\phi^4 - {\frac{1}{2}}a\phi^2\mathcal{R}, \end{aligned}conf 0.943
\begin{aligned} L_{T} & =L_{\mathrm{M}}+L_{\mathrm{G}} \\ & =L_{\varphi}+\frac{1}{2} \nu \mathcal{D}_{a}^{*} \phi \mathcal{D}^{* a} \phi-\lambda \phi^{4}-\frac{1}{2} a \phi^{2} \mathcal{R}-\frac{1}{4} \xi \mathcal{H}_{a b} \mathcal{H}^{a b} . \end{aligned}\begin{aligned} \hspace*{-12mm}L_T &=& L_\mathrm{M} + L_\mathrm{G} \\ &=& L_\varphi \!+\! {\frac{1}{2}}\nu\mathcal{D}^\ast_a\phi\,\mathcal{D}^{\ast a}\phi \!-\!\lambda\phi^4 \!\!-\! {\frac{1}{2}}a\phi^2\mathcal{R} \!-\! {\frac{1}{4}}\xi \mathcal{H}_{ab}\mathcal{H}^{ab}.\phantom{A} \end{aligned}conf 0.872
\begin{aligned} a \phi^{2}\left(\mathcal{R}^{a}{ }_{b}-\frac{1}{2} \delta_{b}^{a} \mathcal{R}\right)= & h\left(\tau_{\varphi}\right)^{a}{ }_{b}+\left[\nu\left(\mathcal{D}^{* a} \phi\right)\left(\mathcal{D}_{b}^{*} \phi\right)-\delta_{b}^{a} L_{\phi}\right] \\ & -\xi\left(\mathcal{H}^{a c} \mathcal{H}_{b c}-\frac{1}{4} \delta_{b}^{a} \mathcal{H}_{c d} \mathcal{H}^{c d}\right), \end{aligned}\begin{aligned} a\phi^2 ({\mathcal{R}^{a}}_{b} -{\frac{1}{2}}\delta_b^a\mathcal{R}) = && h {(\tau_\varphi)^a}_b + [\nu(\mathcal{D}^{\ast a}\phi) (\mathcal{D}^\ast_b\phi)-\delta^a_b L_\phi] \\ &&-\xi (\mathcal{H}^{ac}\mathcal{H}_{bc}-{\frac{1}{4}}\delta^a_b \mathcal{H}_{cd}\mathcal{H}^{cd}), \end{aligned}conf 0.900
a\left[\phi^{2} \mathcal{T}^{* c}{ }_{a b}+\delta_{a}^{c}\left(\mathcal{D}_{b}^{*}+\mathcal{T}_{b}^{*}\right) \phi^{2}-\delta_{b}^{c}\left(\mathcal{D}_{a}^{*}+\mathcal{T}_{a}^{*}\right) \phi^{2}\right]=2 h\left(\sigma_{\varphi}\right)_{a b}{ }^{c}a[\phi^2{\mathcal{T}^{\ast c}}_{ab} +\delta_a^c (\mathcal{D}^\ast_b + \mathcal{T}^\ast_b)\phi^2 -\delta_b^c (\mathcal{D}^\ast_a + \mathcal{T}^\ast_a)\phi^2] = 2h {(\sigma_\varphi)_{ab}}^c,conf 0.850
\xi\left[\frac{1}{2} \mathcal{T}^{* c}{ }_{a b} \mathcal{H}^{a b}-\left(\mathcal{D}_{a}^{*}+\mathcal{T}_{a}^{*}\right) \mathcal{H}^{a c}\right]-\nu \phi \mathcal{D}^{* c} \phi=h\left(\zeta_{\varphi}\right)^{c}\xi[{\frac{1}{2}} {\mathcal{T}^{\ast c}}_{ab} \mathcal{H}^{ab} - (\mathcal{D}^\ast_a+\mathcal{T}^\ast_a)\mathcal{H}^{ac}] - \nu \phi \mathcal{D}^{\ast c} \phi = h (\zeta_\varphi)^cconf 0.875
\frac{\partial L_{\varphi}}{\partial \phi}-\nu\left(\mathcal{D}_{a}^{*}+\mathcal{T}_{a}^{*}\right) \mathcal{D}^{* a} \phi-4 \lambda \phi^{3}-a \phi \mathcal{R}=0\frac{\partial L_\varphi}{\partial \phi}-\nu (\mathcal{D}^\ast_a+\mathcal{T}^\ast_a)\mathcal{D}^{\ast a}\phi -4\lambda\phi^3 - a\phi\mathcal{R} = 0.conf 0.901
\mathcal{R}^{a}{ }_{b}-\frac{1}{2} \delta_{b}^{a} \mathcal{R}-\delta_{b}^{a} \Lambda=\kappa\left[\left.h\left(\tau_{\varphi}\right)^{a}{ }_{b}\right|_{\phi=\phi_{0}}+\nu\left(\mathcal{B}^{a} \mathcal{B}_{b}-\frac{1}{2} \delta_{b}^{a} \mathcal{B}^{c} \mathcal{B}_{c}\right)-\xi\left(\mathcal{H}^{a c} \mathcal{H}_{b c}-\frac{1}{4} \delta_{b}^{a} \mathcal{H}_{c d} \mathcal{H}^{c d}\right)\right]{\mathcal{R}^{a}}_{b} -{\frac{1}{2}}\delta_b^a\mathcal{R} - \delta^a_b \Lambda = \kappa \left[\left.h{(\tau_\varphi)^a}_b\right|_{\phi=\phi_0} +\nu (\mathcal{B}^a\mathcal{B}_b-{\frac{1}{2}}\delta^a_b \mathcal{B}^c\mathcal{B}_c) - \xi (\mathcal{H}^{ac}\mathcal{H}_{bc}-{\frac{1}{4}}\delta^a_b {\cal H}_{cd}{\cal H}^{cd})\right].conf 0.785
\mathcal{T}^{c}{ }_{a b}+\delta_{a}^{c} \mathcal{T}_{b}-\delta_{b}^{c} \mathcal{T}_{a}=\left.2 \kappa h\left(\sigma_{\varphi}\right)_{a b}{ }^{c}\right|_{\phi=\phi_{0}}{\mathcal{T}^c}_{ab} + \delta_a^c \mathcal{T}_b-\delta_b^c\mathcal{T}_a =\left.2\kappa h {(\sigma_\varphi)_{ab}}^c\right|_{\phi=\phi_0},conf 0.981
\xi\left[\frac{1}{2} \mathcal{T}^{c}{ }_{a b} \mathcal{H}^{a b}-\left(\mathcal{D}_{a}+\mathcal{T}_{a}\right) \mathcal{H}^{a c}\right]+\nu \phi_{0}^{2} \mathcal{B}^{c}=\left.h\left(\zeta_{\varphi}\right)^{c}\right|_{\phi=\phi_{0}}\xi\left[{\frac{1}{2}} {\mathcal{T}^{c}}_{ab} \mathcal{H}^{ab} - (\mathcal{D}_a+\mathcal{T}_a)\mathcal{H}^{ac}\right] + \nu \phi_0^2 \mathcal{B}^c = \left.h (\zeta_\varphi)^c\right|_{\phi=\phi_0},conf 0.993
\mathcal{R}+4 \Lambda=\kappa\left[\left.\phi_{0} \partial_{\phi} L_{\varphi}\right|_{\phi=\phi_{0}}+\nu \phi_{0}^{2}\left(\mathcal{D}_{a}+\mathcal{T}_{a}+\mathcal{B}_{a}\right) \mathcal{B}^{a}\right]\mathcal{R} + 4 \Lambda = \kappa \left[\phi_0\left.\partial_\phi L_\varphi\right|_{\phi=\phi_0} + \nu\phi_0^2 (\mathcal{D}_a+\mathcal{T}_a + \mathcal{B}_a)\mathcal{B}^a\right]conf 0.992
L_{\psi}=\frac{1}{2} i \bar{\psi} \gamma^{a} \stackrel{\leftrightarrow}{\mathcal{D}}_{a} \psi-\mu \phi \bar{\psi} \psiL_\psi = {\frac{1}{2}}i\bar{\psi}\gamma^a {\stackrel{\leftrightarrow}{\mathcal{D}_a}}\psi - \mu\phi\bar{\psi}\psi,conf 0.988
i \gamma^{a} \mathcal{D}_{a} \psi-\mu \phi \psi=0i\gamma^a\mathcal{D}_a\psi-\mu\phi\psi = 0,conf 1.000
\begin{aligned} \mathcal{D}_{a} \mathcal{D}^{a} \mathcal{B}^{c} & -\mathcal{R}^{a c} \mathcal{B}_{a}+m^{2} \mathcal{B}^{c} \\ & +2 \kappa\left[\left(\delta_{b}^{c} \mathcal{D}_{a}-\kappa h \sigma_{a b}{ }^{c}\right)\left(h \sigma^{a b d} \mathcal{B}_{d}\right)\right]=0 \end{aligned}\begin{aligned} \mathcal{D}_a\mathcal{D}^{a} \mathcal{B}^{c} &&-\mathcal{R}^{ac}\mathcal{B}_a + m^2\mathcal{B}^c \\&&+ 2\kappa[(\delta_b^c \mathcal{D}_a-\kappa h {\sigma_{ab}}^c)(h \sigma^{abd} \mathcal{B}_d)]=0,\phantom{AB} \end{aligned}conf 0.958
A^{\dagger a b}{ }_{\mu}=A^{a b}{ }_{\mu}+\left(\mathcal{V}^{a} b^{b}{ }_{\mu}-\mathcal{V}^{b} b^{a}{ }_{\mu}\right){A^{\dagger ab}}_\mu = {A^{ab}}_\mu + (\mathcal{V}^a{b^b}_\mu - \mathcal{V}^b{b^a}_\mu),conf 0.944
\mathcal{D}_{a}^{\natural} \varphi \equiv h_{a}{ }^{\mu} D_{\mu}^{\natural} \varphi \equiv h_{a}{ }^{\mu}\left(\partial_{\mu}+\frac{1}{2} A^{\dagger b c}{ }_{\mu} \Sigma_{b c}\right) \varphi\mathcal{D}^\natural_a\varphi \equiv {h_a}^\mu D^\natural_\mu \varphi \equiv {h_a}^\mu (\partial_\mu + {\frac{1}{2}}{A^{\dagger bc}}_\mu\Sigma_{bc})\varphi.conf 0.931
\left[D_{\mu}^{\natural}, D_{\nu}^{\natural}\right] \varphi=\frac{1}{2} R^{\dagger a b}{ }_{\mu \nu} \Sigma_{a b} \varphi[D^\natural_\mu, D^\natural_\nu]\varphi = {\frac{1}{2}}{R^{\dagger ab}}_{\mu\nu}\Sigma_{ab}\varphi,conf 0.905
V_{\mu}^{\prime}=V_{\mu}+\theta P_{\mu}V_\mu^\prime = V_\mu + \theta P_\mu,
conf 1.000