LaTeX vs MathPix image — Geometrodynamics of Gauge Fields On the Geometry of Yang-Mills and Gravitational Gauge Theories (Eckehard W. Mielke) (Z-Library)

\ell^{*}:=\sqrt{8 \pi \hbar G_{N} / c^{3}} \simeq 8 \times 10^{-33} \mathrm{~cm},

x_{2} \circ \stackrel{-1}{x}_{1}: \mathbb{R} \rightarrow \mathbb{R}

e^{\mathrm{f}}:=\left.\frac{d f(s(t))}{d t}\right|_{t_{o}=s^{-1}(m)}=\frac{\partial \mathrm{f}}{\partial x^{i}} \cdot \frac{d x^{i}(s(t))}{d t}

e_{\alpha}(m)=e_{, \alpha}^{i}(m) \partial_{i} .

T(M)=\bigcup_{m \in M} T_{m}(M)

\left.e_{\beta}\right\rfloor \vartheta^{\alpha}=\delta^{\alpha}{ }_{\beta} .

\vartheta=\vartheta^{\alpha} P_{\alpha} \in C^{\infty}\left(T^{*}(M)\right),

\vartheta^{\alpha}=\mathrm{E}_{j}^{\alpha}(m) d x^{j} .

E_{i}^{\alpha} e^{i}{ }_{\beta}=\delta^{\alpha}{ }_{\beta} .

T_{p}^{q}(M):=\bigcup_{m \in M} \otimes^{p} T_{m}^{*}(M) \otimes^{q} T_{m}(M) .

T=T_{\alpha_{1} \cdots \alpha_{p}}^{\beta_{1} \cdots \beta_{q}}(m) \vartheta^{\alpha_{1}} \otimes \cdots \otimes \vartheta^{\alpha_{p}} \otimes e_{\beta_{1}} \otimes \cdots \otimes e_{\beta_{q}} \in C^{\infty}\left(T_{p}^{q}(M)\right) .

d s^{2}=g_{\alpha \beta} \vartheta^{\alpha} \otimes_{s} \vartheta^{\beta}=: g_{i j} d x^{i} \otimes_{s} d x^{j} .

d s^{2}=g_{i j} d x^{i} \otimes_{s} d x^{j} \stackrel{(*)}{=}-\sum_{i=1}^{s}\left(\stackrel{\circ}{\vartheta^{i}}\right)^{2}+\sum_{j=s+1}^{n}\left(\stackrel{\circ}{\vartheta}^{j}\right)^{2} .

\alpha^{(p)}=\frac{1}{p!} A_{\alpha_{1} \cdots \alpha_{p}} \vartheta^{\alpha_{1}} \wedge \cdots \wedge \vartheta^{\alpha_{p}} \in C^{\infty}\left(\wedge^{p} T^{*}(M)\right) .

\pi: F \rightarrow M .

\sigma: U \rightarrow F \quad \text { where } \quad \pi \circ \sigma=\mathrm{id}

\iota\left(m, g_{1} g_{2}\right)=\iota\left(m, g_{1}\right) \cdot g_{2}, \quad m \in U, g \in G .

\delta:\left\{\left.\begin{array}{lll} G \times P & \rightarrow & P \\ w & \psi & \psi \\ g_{p} \cdot & p_{\circ}=p \end{array} \right\rvert\, e \cdot p_{\circ}=p_{\circ} \in P\right\} .

\delta_{g}(p, \zeta):=\left(p g, g^{-1} \zeta\right) \subset P \times F, \quad g \in G .

V^{\rho}:=V^{\rho}\left(M, \mathbb{C}^{N}, \rho(G) \subseteq G L(N, \mathbb{C}), P\right),

\phi^{(p)}:=\varphi^{p} \otimes b \in C^{\infty}\left(\wedge^{p} T_{\mathbb{C}^{*}}(M) \otimes V^{\rho}\right), \quad b \in C^{\infty}\left(V^{\rho}\right) .

\varphi^{(p)}=\left\{\varphi^{(p) A} \mid A=1, \ldots, N\right\} .

L_{j}:=\left[\left.\frac{\partial \rho_{A}{ }^{B}(g(\xi))}{\partial \xi^{j}}\right|_{g=e}\right] \in T_{e}(\rho(G))

\left[L_{i}, L_{j}\right]=c_{i j}{ }^{k} L_{k} .

\phi^{(p)}=\frac{1}{p!} \phi_{\alpha_{1} \ldots \alpha_{p}}^{j} L_{j} \otimes \vartheta^{\alpha_{1}} \wedge \cdots \wedge \vartheta^{\alpha_{p}} \in C^{\infty}\left(\wedge^{p} T_{\mathbb{C}}^{*}(M) \otimes V^{A d}\right) .

\left[\phi^{(p)}, \psi^{(q)}\right]:=\phi^{(p)} \wedge \psi^{(q)}-(-1)^{p q} \psi^{(q)} \wedge \phi^{(p)}=(-1)^{p q+1}\left[\psi^{(q)}, \phi^{(p)}\right]

\left[\phi^{(2 k)}, \phi^{(2 k)}\right]=-\left[\phi^{(2 k)}, \phi^{(2 k)}\right]=0 .

V_{p}^{*}=T_{p}^{*}\left(F_{m}(M)\right),

\omega(e g)=g^{-1} \omega(e) g, \quad e \in T(P) .

\omega(p d g)=g^{-1} d g \quad \in \quad T_{e}(G) \approx \mathfrak{g}

D: C^{\infty}\left(V^{\rho}\right) \longrightarrow C^{\infty}\left(T_{\mathbb{C}}^{*}(M) \otimes V^{\rho}\right)

D b=\omega \otimes b, \quad b \in C^{\infty}\left(V^{\rho}\right) .

D\left(\varphi^{(p)} \otimes b\right)=d \varphi^{(p)} \otimes b+(-1)^{p} \varphi^{(p)} \otimes D b .

D\left(\varphi^{(p)} \otimes b\right)=\left\{d \varphi^{p}+(-1)^{p}\left[\varphi^{(p)}, \omega\right]\right\} \otimes b .

D b_{A}=\omega_{A}{ }^{B} b_{B}, A, B=1, \ldots, N .

\Gamma:=\stackrel{*}{\sigma} \omega=\Gamma_{\alpha}{ }^{j} L_{j} \otimes \vartheta^{\alpha}, \quad j=1, \ldots, \operatorname{dim} G,

A_{\alpha}{ }^{j}(m):=\Gamma_{\alpha}{ }^{j}(m)_{\mid G=U(f)}

D D b=: \Omega \otimes b .

\begin{aligned} D D b=D(\omega \otimes b) & =d \omega \otimes b-\omega \wedge D b \\ & =(d \omega-\omega \wedge \omega) \otimes b \end{aligned}

\Omega=d \omega-\omega \wedge \omega=d \omega-\frac{1}{2}[\omega, \omega]

\stackrel{*}{\sigma} \Omega=\frac{1}{2} F_{\alpha} \beta^{j} L_{j} \otimes \vartheta^{\alpha} \wedge \vartheta^{\beta}

F_{\alpha \beta}{ }^{j}=\partial_{\alpha} \Gamma_{\beta}{ }^{j}-\partial_{\beta} \Gamma_{\alpha}{ }^{j}-c_{k l}{ }^{j} \Gamma_{\alpha}{ }^{k} \Gamma_{\beta}{ }^{l},

F_{i j}=\partial_{i} A_{j}-\partial_{j} A_{i}

\Delta b=b-b(\circ)=\oint(d b)_{\mathrm{hor}}

(d b)_{\mathrm{hor}}=D b=\omega \otimes b

\begin{aligned} \Delta b & =\int_{\partial U} \omega \otimes b=\int_{U} d(\omega \otimes b)_{\mathrm{hor}}=\int_{U} D(\omega \otimes b) \\ & =\int_{U} \Omega \otimes b \simeq \Omega \otimes b \end{aligned}

K=\lim _{U \rightarrow 0}\left(\int_{U} \Omega \otimes b\right) /\left(\int_{U} 1 \otimes b\right)

D D D b \equiv 0 .

D \Omega \equiv 0

\begin{aligned} D D D b= & D(\Omega \otimes b)=(d \Omega+[\Omega, \omega]) \otimes b \\ = & (d \Omega+\Omega \wedge \omega-\omega \wedge \Omega) \otimes b \\ = & (d d \omega-d \omega \wedge \omega+\omega \wedge d \omega \\ & \quad+d \omega \wedge \omega-\omega \wedge \omega \wedge \omega \\ & \quad+\omega \wedge \omega \wedge \omega-\omega \wedge d \omega) \otimes b \equiv 0 . \end{aligned}

G(p): P \rightarrow P, \quad G(p) \in \mathscr{G}_{p}

G(g p)=g G(p), \quad g \in G, \quad p \in P ;

\pi \circ G=\pi .

\mathscr{G}_{p} \approx C^{\infty}\left(P \times_{\mathrm{Ad}} G\right) .

G(p)=\exp i \theta^{k}(m) L_{k} \in \mathscr{G}_{p} .

\phi=\varphi \otimes b={G^{-1}} \varphi \otimes^{G} b \quad \in C^{\infty}\left(V^{\rho}\right) .

\varphi \rightarrow^{G^{-1}} \varphi:=G^{-1} \varphi \quad \in C^{\infty}(M, \mathbb{C}), G \in \mathscr{G}_{V},

b \rightarrow^{G} b:=G b \quad \in C^{\infty}\left(V^{\rho}\right) .

\bar{\varphi} \rightarrow{ }^{G^{-1}} \bar{\varphi}:=\bar{\varphi} G \quad \in C^{\infty}\left(V^{\bar{s}}\right) .

\operatorname{det}\left(1+\frac{i}{2 \pi} \Omega\right)=\stackrel{*}{\pi}\left(1+\gamma_{1}+\cdots+\gamma_{m}\right)

{ }^{*}\left(\gamma_{k}\right)=\frac{(-1)^{k}}{(2 \pi i)^{k} k!} \operatorname{Tr}(\Omega \wedge \cdots \wedge \Omega) .

\begin{aligned} & *\left(d \gamma_{k}\right)=d \pi^{*}\left(\gamma_{k}\right)=D \pi^{*}\left(\gamma_{k}\right) \\ & =\frac{(-1)^{k}}{(2 \pi i)^{k}(k-1)!} \operatorname{Tr}(D \Omega \wedge \Omega \wedge \cdots \wedge \Omega)=0 . \end{aligned}

p_{k}\left(V^{\mathbb{R}}\right)=(-1)^{k} c_{2 k}(V) .

\begin{aligned} & \stackrel{*}{\pi}\left(\gamma_{1}\right)=-\frac{1}{2 \pi i} \operatorname{Tr} \Omega \quad(=0) \\ & \stackrel{*}{\pi}\left(\gamma_{2}\right)=-\frac{1}{8 \pi^{2}}\{\operatorname{Tr}(\Omega \wedge \Omega)-\operatorname{Tr}(\Omega) \wedge \operatorname{Tr}(\Omega)\} \\ & \quad\left(=-\frac{1}{32 \pi^{2}} \epsilon^{\alpha \beta \mu \nu} F_{\alpha \beta}^{j} F_{\mu \nu j} \sqrt{|g|} d^{4} x\right) \\ & * \pi^{2}\left(\gamma_{k}\right)=0 \text { for } k \geq 3 . \end{aligned}

c_{2}(M)=-\int_{M}^{*} \pi\left(\gamma_{2}\right)=\frac{1}{8 \pi^{2}} \int_{M} \operatorname{Tr}(\Omega \wedge \Omega) .

\operatorname{Tr}(\Omega \wedge \Omega)=d \operatorname{Tr}\left(\omega \wedge \Omega+\frac{1}{3} \omega \wedge \omega \wedge \omega\right)

\begin{aligned} & \operatorname{Tr}\left\{d\left(\omega \wedge d \omega-\frac{2}{3} \omega \wedge \omega \wedge \omega\right)\right\}=\operatorname{Tr}\{d \omega \wedge d \omega-\omega \wedge d d \omega \\ & \left.-\frac{2}{3} d \omega \wedge \omega \wedge \omega+\frac{2}{3} \omega \wedge d \omega \wedge \omega-\frac{2}{3} \omega \wedge \omega \wedge d \omega\right\} \\ & =\operatorname{Tr}\{d \omega \wedge d \omega-d \omega \wedge \omega \wedge \omega-\omega \wedge \omega \wedge d \omega\} \\ & =\operatorname{Tr}\{(d \omega-\omega \wedge \omega) \wedge(d \omega-\omega \wedge \omega)\}=\operatorname{Tr}(\Omega \wedge \Omega) \end{aligned}

\operatorname{Tr}\{\omega \wedge(\omega \wedge \omega \wedge \omega)\}=-\operatorname{Tr}\{(\omega \wedge \omega \wedge \omega) \wedge \omega\}=0 .

\begin{aligned} c_{2}(M) & =\frac{1}{8 \pi^{2}} \int_{M} d \operatorname{Tr}\left(\omega \wedge \Omega+\frac{1}{3} \omega \wedge \omega \wedge \omega\right) \\ & =\frac{1}{8 \pi^{2}} \int_{\partial M} \operatorname{Tr}\left(\omega \wedge d \omega-\frac{2}{3} \omega \wedge \omega \wedge \omega\right) . \end{aligned}

S_{\mathrm{YM}}=\int_{M} L_{\mathrm{YM}}

\stackrel{\infty}{\omega}:=-G^{-1} d G,

\omega \sim \stackrel{\infty}{\omega}

\begin{aligned} & d \stackrel{\infty}{\omega}-\stackrel{\infty}{\omega} \wedge \stackrel{\infty}{\omega} \\ & =-d\left(G^{-1}\right) \wedge d G-G^{-1} d d G-\left(G^{-1} d G\right) \wedge\left(G^{-1} d G\right) \\ & =-d\left(G^{-1}\right) \wedge d G+G^{-1} G d\left(G^{-1}\right) \wedge d G=0 \end{aligned}

\Omega(\stackrel{\infty}{\omega})=0 .

c_{2}(M)=\frac{1}{24 \pi^{2}} \int_{\partial M} \operatorname{Tr}(\stackrel{\infty}{\omega} \wedge \stackrel{\infty}{\omega} \wedge \stackrel{\infty}{\omega}),

L=L(\varphi, d \varphi) \eta

\delta \int_{M} L=0

\delta \varphi=0 \quad \text { at the boundary } \partial \mathrm{M} \text { of } \mathrm{M}

\delta(d \varphi):=d(\varphi+\delta \varphi)-d \varphi=d \delta \varphi

\pi:=\frac{\partial L}{\partial\left(\partial_{\alpha} \varphi\right)} \vartheta^{\alpha}

\begin{aligned} \delta L & =\delta \varphi \wedge^{*} L_{\varphi}+\delta(d \varphi) \wedge \pi \\ & =\delta \varphi \wedge^{*} L_{\varphi}+d(\delta \varphi) \wedge \pi \\ & =\delta \varphi \wedge\left({ }^{*} L_{\varphi}+d \pi\right)+d(\delta \varphi \wedge \pi), \end{aligned}

\int_{M} d(\delta \varphi \wedge \pi)=\int_{\partial M} \delta \varphi \wedge \pi

\lambda^{\varphi}:=(-1)^{s} L_{\varphi}+{ }^{*} d \pi=0,

\frac{\partial L}{\partial \varphi}-\partial_{\alpha} \frac{\partial L}{\partial\left(\partial_{\alpha} \varphi\right)}=0 .

{ }^{*} H:=L-d \varphi \wedge \pi

d^{*} H=d L+d \varphi \wedge^{*} L_{\varphi}+d \varphi \wedge d \pi-d d \varphi \wedge \pi

d^{*} H \simeq 0

\beta \wedge^{*} D \psi=m \psi \eta .

\square \varphi+m^{2} \varphi=0

\overline{\bar{L}}_{\mathrm{mat}}=\overline{\bar{L}}_{\mathrm{mat}}(\bar{\psi} \psi, \bar{\psi} \beta \cdot d \psi) \eta=\bar{\psi} \beta \wedge^{*} d \psi-m \bar{\psi} \psi \eta .

\mathscr{O} \rightarrow{ }^{G^{-1}} \mathscr{O}=G^{-1} \mathscr{O} G .

L_{\mathrm{mat}}=L\left(\bar{\varphi} \varphi, \bar{\varphi} \beta \cdot\left(d+\omega^{\prime}\right) \varphi\right) \eta=L\left(\bar{\varphi} \varphi, \bar{\varphi} \beta \cdot D^{\prime} \varphi\right) \eta

\omega^{\prime} \rightarrow{ }^{G^{-1}} \omega^{\prime}=G^{-1} \omega^{\prime} G+\left(d G^{-1}\right) G

D\left({ }^{G} b\right)={ }^{G} \omega \otimes{ }^{G} b .

\begin{aligned} D\left({ }^{G} b\right) & =d G \otimes b+G \omega \otimes b \\ & =\left\{d G G^{-1}+G \omega G^{-1}\right\} \otimes^{G} b . \end{aligned}

\omega \rightarrow{ }^{G} \omega=G \omega G^{-1}+d G G^{-1} .

D \rightarrow{ }^{G^{-1}} D=G^{-1} D G .

\Omega \rightarrow{ }^{G^{-1}} \Omega=G^{-1} \Omega G .

\begin{aligned} { }^{\Delta} \omega:={ }^{G^{-1}} \omega-\omega & =G^{-1}[\omega, G]-G^{-1} d G \\ & =-G^{-1} D G=-G^{-1} D . \end{aligned}

F:=d A=\frac{1}{2} F_{i j} d x^{i} \wedge d x^{j} .

A \rightarrow A+d \theta(x) \quad \Rightarrow \quad F \rightarrow d A+d d \theta=d A=F .

\theta=-\frac{e}{\hbar} \int \mathbf{A} \cdot d \mathbf{l}

L=L(A, d A)+L_{\mathrm{mat}}(A, d A, \Psi, d \Psi) .

\frac{\delta L}{\delta A}:=\frac{\partial L}{\partial A}+d \frac{\partial L}{\partial d A}=0,

H:=-\frac{\partial L}{\partial d A}=-\frac{\partial L}{\partial F} \quad \text { and } \quad j:=\frac{\delta L_{\mathrm{mat}}}{\delta A},

d F \equiv 0, \quad d H=j

d j \simeq 0 .

L_{\mathrm{Pontr}}=-\frac{1}{2} F \wedge F=-\frac{1}{2} d C,

L_{\mathrm{Max}}=-\frac{1}{2} F \wedge{ }^{*} F,

\begin{aligned} F:= & E \wedge d t+B \\ = & \left(E_{x} d x+E_{y} d y+E_{z} d z\right) \wedge d t \\ & +B_{x} d y \wedge d z+B_{y} d z \wedge d x+B_{z} d x \wedge d y . \end{aligned}

H:=-\mathfrak{H} \wedge d t+\mathfrak{D}=\frac{1}{2} H_{i j} d x^{i} \wedge d x^{j}

\begin{aligned} j:= & -\mathscr{J} \wedge d t+\hat{\rho} \\ = & -\left(J_{x} d y \wedge d z+J_{y} d z \wedge d x+J_{z} d x \wedge d y\right) \wedge d t \\ & +\rho d x \wedge d y \wedge d z \end{aligned}

H={ }^{*} F .

H_{m n}=\frac{\sqrt{|g|}}{2} \varepsilon_{m n k l} g^{k i} g^{l j} F_{i j} .

d F \equiv 0 \quad\left\{\begin{array}{l} \nabla \times \mathbf{E}+\frac{\partial \mathbf{B}}{\partial t} \equiv 0 \\ \nabla \cdot \mathbf{B} \equiv 0 \end{array}\right.

d H=j\left\{\begin{array}{l} \boldsymbol{\nabla} \times \mathbf{B}-\frac{\partial \mathbf{E}}{c^{2} \partial t}=\mu \mathbf{J} \\ \boldsymbol{\nabla} \cdot \mathbf{E}=\rho / \varepsilon . \end{array}\right.

L_{\mathrm{Max}}:=-\frac{1}{2} F \wedge^{*} F=\frac{1}{2}\left(\mathbf{E}^{2}-\mathbf{B}^{2}\right) \eta,

L_{\mathrm{Pontr}}:=-\frac{1}{2} F \wedge F=\mathbf{E} \cdot \mathbf{B} \eta .

\begin{aligned} \Sigma_{\alpha} & \left.\left.:=e_{\alpha}\right\rfloor L-\left(e_{\alpha}\right\rfloor d A\right) \wedge \frac{\partial L}{\partial(d A)} \\ & \left.\left.=\frac{1}{2}\left[\left(e_{\alpha}\right\rfloor F\right) \wedge H-F \wedge\left(e_{\alpha}\right\rfloor H\right)\right] \end{aligned}

d \Sigma_{\alpha} \simeq\left(e_{\alpha} \downharpoonleft F\right) \wedge j

L_{\mathrm{BF}}=-B \wedge F=-B \wedge d A .

\begin{aligned} \tilde{L}_{\mathrm{BF}} & :=-B \wedge d A+\frac{1}{2} B \wedge B \\ & \cong-B \wedge d A+d C \end{aligned}

B \cong d A, \quad d B \cong d F \equiv 0

C=\frac{1}{2} A \wedge F, \quad d C=\frac{1}{2} F \wedge F,

A \rightarrow A^{\prime}=A+d \theta(x) .

A \rightarrow \widetilde{A}=A+\psi, \quad B \rightarrow \widetilde{B}=B+d \psi,

L_{\mathrm{Max}}=-B \wedge d A+\frac{1}{2} B \wedge{ }^{*} B+L_{\mathrm{matter}},

-d B=d^{*} F \cong j,

\begin{aligned} j:=\frac{\delta L_{\mathrm{matter}}}{\delta A} & =\frac{\partial L_{\mathrm{matter}}}{\partial A}+d \frac{\partial L_{\mathrm{matter}}}{\partial d A} \\ & =\Psi \wedge \frac{\partial L_{\mathrm{matter}}}{\partial D \Psi}+D \frac{\partial L_{\mathrm{matter}}}{\partial B}, \end{aligned}

\mathscr{F}:=\frac{1}{\alpha_{g}} \operatorname{Tr}\left\{{ }^{*}\left(\Omega \wedge{ }^{*} \Omega\right)\right\}=\frac{1}{4 \alpha_{g}} F_{\alpha \beta}{ }^{j} F^{\alpha \beta}{ }_{j}

{ }^{*} \mathscr{F}:=\frac{1}{\alpha_{g}} \operatorname{Tr}\left\{{ }^{*}(\Omega \wedge \Omega)\right\}=\frac{1}{4 \alpha_{g}} F_{\alpha \beta}{ }^{j *} F^{\alpha \beta}{ }_{j} .

\alpha_{g}:=g^{2} / \hbar c

L_{\omega}=L\left(\mathscr{F},{ }^{*} \mathscr{F}\right) \eta .

\delta L_{\omega}=: \delta \Omega \wedge \Pi^{\Omega} .

\Pi^{\Omega}=\frac{2}{\alpha_{g}}\left(\frac{\partial L}{\partial \mathscr{F}}{ }^{*} \Omega+\frac{\partial L}{\partial^{*} \mathscr{F}} \Omega\right)

\begin{aligned} \delta \Omega & =\delta(d \omega-\omega \wedge \omega)=\delta d \omega-\delta \omega \wedge \omega-\omega \wedge \delta \omega \\ & =d \delta \omega-[\omega, \delta \omega]=D \delta \omega . \end{aligned}

\begin{aligned} \delta L_{\omega} & =\delta \Omega \wedge \Pi^{\Omega}=D(\delta \omega) \wedge \Pi^{\Omega} \\ & =-\delta \omega \wedge D \Pi^{\Omega}+d\left(\delta \omega \wedge \Pi^{\Omega}\right) . \end{aligned}

L=L_{\mathrm{mat}}+L_{\omega},

\delta L_{\mathrm{mat}}=: \delta \omega \wedge \tau .

D \Pi^{\Omega}=\tau

D \Omega \equiv 0 .

\begin{aligned} D \tau & =D D \Pi^{\Omega}=\left[\Omega, \Pi^{\Omega}\right] \\ & =\frac{2}{\alpha_{g}}\left\{\left[\Omega,{ }^{*} \Omega\right] \frac{\partial L}{\partial \mathscr{F}}+(-1)^{s}[\Omega, \Omega] \frac{\partial L}{\partial^{*} \mathscr{F}}\right\}=0 . \end{aligned}

\operatorname{div} \iota:=(-1)^{1+s *} d^{*} \iota=0

\iota:=\tau+(-1)^{n+s} *\left[\omega, \Pi^{\Omega}\right]

d \Pi^{\Omega}+\left[\omega, \Pi^{\Omega}\right]=\tau .

Q=Q^{(i)} I_{i}:=\int_{H^{3}} \iota=\int_{\partial H^{3}} \Pi^{\Omega},

Q_{e}=\frac{1}{2} Y+I_{3} .

L_{\mathrm{YM}}=-\frac{1}{\alpha_{g}} \operatorname{Tr}\left(\Omega \wedge^{*} \Omega\right) .

L\left(\mathscr{F},{ }^{*} \mathscr{F}\right)=-\mathscr{F} .

\Pi^{\Omega}=-\frac{2}{\alpha_{g}}{ }^{*} \Omega

L_{\text {Mills }}\left(\mathscr{F},{ }^{*} \mathscr{F}\right)=\mathscr{F}\left[1-\left(1+\frac{1}{b^{2}} \mathscr{F}^{2}\right)^{-1}\right]

L_{\mathrm{BI}}\left(\mathscr{F},{ }^{*} \mathscr{F}\right)=\sqrt{1-2 \mathscr{F}-\left({ }^{*} \mathscr{F}\right)^{2}}-1 .

\Pi^{\Omega}=\zeta \Omega .

\stackrel{(-)}{\sigma} \mu \nu:=\left\{\left.\begin{array}{l} \sigma^{i j}=\frac{1}{2} \varepsilon^{i j k} \sigma_{k} \\ \sigma^{i 0}=(-) \frac{1}{2} \sigma^{i} \end{array} \right\rvert\, i, j, k=1,2,3\right\}

\stackrel{*}{\sigma} \Omega=\frac{1}{4} F_{\alpha \beta \mu \nu} \sigma^{\mu \nu} \otimes \vartheta^{\alpha} \wedge \vartheta^{\beta}

{ }^{*} \sigma^{\mu \nu}=\sigma^{\mu \nu}, \quad{ }^{*} \bar{\sigma}^{\mu \nu}=-\bar{\sigma}^{\mu \nu},

{ }^{*} F_{\alpha \beta \mu \nu}^{*}=(\stackrel{+}{-}) F_{\alpha \beta \mu \nu}

A_{\mu}=i \sigma_{\mu i} \partial^{i} \ln h .

\square h-h^{-1} \partial_{i} h \partial^{i} h=h^{3}

h=\sum_{i=1}^{k+1} \frac{l_{(i)}^{2}}{\left(x-x_{(i)}\right)^{2}},

p_{1}(M)=k .

\langle\infty \mid-\infty\rangle=Z \int_{\mathscr{M}_{\omega}} d \omega \mu[\omega] \exp i \int_{M}\left(L_{\omega}+L_{\infty}\right)

L_{\infty}=(\stackrel{+}{-}) \frac{1}{\alpha_{g}} \operatorname{Tr}(\Omega \wedge \Omega)

L_{\mathrm{YM}}+L_{\infty}=(\stackrel{+}{-}) \frac{1}{2 \alpha_{g}} \operatorname{Tr}\left({ }^{*} \Omega \stackrel{-}{(+)} \Omega\right)^{2} \geq 0

{ }^{*} \Omega=(\stackrel{+}{-}) \Omega ;

\begin{aligned} L_{\mathrm{SW}}= & \frac{1}{2} \overline{D_{ \pm} \psi} \wedge^{*} D_{ \pm} \psi \mp i\left(F^{ \pm}-\frac{1}{2} \bar{\psi} \sigma_{ \pm} \psi\right)^{2} \\ = & \mp i \operatorname{Tr}\left(F^{ \pm} \wedge F^{ \pm}\right)+\frac{1}{2} \overline{D_{ \pm} \psi} \wedge{ }^{*} D_{ \pm} \psi \\ & \pm i \bar{\psi} \sigma_{ \pm} \psi \wedge F^{ \pm} \mp \frac{i}{4} \bar{\psi} \sigma_{ \pm} \psi \wedge \bar{\psi} \sigma_{ \pm} \psi ; \end{aligned}

\mathscr{P}_{ \pm}:=\bar{\psi} \sigma_{ \pm} \psi / 2 m,

D_{ \pm}{ }^{*} D_{ \pm} \psi \mp i\left(F^{ \pm}-\frac{1}{2} \bar{\psi} \sigma_{ \pm} \psi\right) \wedge \sigma_{ \pm} \psi=0

D_{ \pm}\left(F^{ \pm}-\frac{1}{2} \bar{\psi} \sigma_{ \pm} \psi\right)=\mp \frac{i}{2} \bar{\psi}^{*} D_{ \pm} \psi=\mp \frac{1}{2} \bar{\psi}^{*} \gamma \wedge{ }^{*}\left(i^{*} \gamma \wedge D_{ \pm} \psi\right) .

D^{*} D \psi=\left(\nabla^{*} \nabla+\frac{1}{4} R+\frac{1}{2} F^{+}\right) \psi .

i^{*} \gamma \wedge D_{ \pm} \psi=0 \quad \text { and } \quad F^{ \pm}=\frac{1}{2} \bar{\psi} \sigma_{ \pm} \psi=m \mathscr{P}_{ \pm},

\Omega_{ \pm}^{g}=\frac{1}{2 \ell^{2}} \sigma_{ \pm},

L_{\mathrm{H}}=-\bar{\Theta}^{(1)} \wedge{ }^{*} D \phi^{0}+\bar{\Theta}^{(1)} \wedge{ }^{*} \Theta^{(1)}+\frac{\lambda}{4}\left(\frac{\mu^{2}}{\lambda}-\bar{\varphi} \cdot \varphi\right)^{2} \eta .

\begin{aligned} D^{*} \Theta^{(1)} & =\frac{\lambda}{2}\left(\frac{\mu^{2}}{\lambda}-\bar{\varphi} \cdot \varphi\right) \varphi \eta \\ \Theta^{(1)} & =D \phi^{(0)} \end{aligned}

\tau_{\mathrm{H}}=-\left[\phi^{(0)},{ }^{*} \Theta^{(1)}\right]

\begin{aligned} & D^{*} \Theta^{(1)}-\frac{\lambda}{2}\left(\frac{\mu^{2}}{\lambda}-\bar{\varphi} \cdot \varphi\right) \varphi \eta=0, \\ & D \Pi^{\Omega}+\left[\phi^{(0)},{ }^{*} \Theta^{(1)}\right]=0 . \end{aligned}

\bar{\varphi} \cdot \varphi:=\varphi^{* j} \varphi_{j}=\frac{\mu^{2}}{\lambda} .

\left\langle\left. O\right|^{\lambda} \varphi \mid O\right\rangle=\mu / \sqrt{\lambda}

\varphi_{\min }: P \rightarrow C^{\infty}\left(\stackrel{\circ}{V}(M, \rho(G / H), \rho(G), P) \subset V^{\rho}\right) .

H:=\left\{h \in G \mid \rho(h) \mathrm{v}_{0}=\mathrm{v}_{0}\right\},

Q=\left\{p \in P \mid \varphi(p)=\sigma\left(\mathrm{v}_{0}\right)\right\}

D \phi^{(0)}=0 .

\begin{aligned} \Theta^{\perp}:=\Omega_{0} & =d \omega_{0}-(\omega \wedge \omega)_{0}=d \omega_{0}-\omega_{0} \wedge \omega-\omega \wedge \omega_{0} \\ & =d \omega_{0}-\left[\omega_{0}, \omega\right]=D \omega_{0} \end{aligned}

\vec{\Omega}=d \vec{\omega}-(\overrightarrow{\omega \wedge}

d_{0}{ }^{*} \Omega_{0}=0 .

\left(D^{*} \Omega\right)_{0}=\vec{D}^{*} \Omega_{0}=\vec{D}^{*} \Theta^{\perp}=0

\begin{aligned} \left(D^{*} \Omega\right) & =\vec{D}^{*} \vec{\Omega}+\left[\omega_{0}, \Omega_{0}\right] \\ & =\vec{D}^{*} \vec{\Omega}+\left[\omega_{0}, \Theta^{\perp}\right]=0 . \end{aligned}

\omega_{0}=\varphi^{j} L_{j} \otimes \vartheta_{0} .

{ }^{*} \Omega=(\stackrel{+}{-}) \Omega \text {. }

\[ T_{G}: \quad T_{g} \psi(m)=D\left(g_{m}^{-1} g g_{g^{-1} m}, m_{0}\right) \psi\left(g^{-1} m\right) . \] }

x \rightarrow x^{\prime}=g x+\mathrm{a}, \quad g \in G L(n, \mathbb{R}), a \in \mathbb{R}^{n} .

a:=\left[\begin{array}{c|c} g & \mathrm{a} \\ \hline 0 & 1 \end{array}\right]=\left[\begin{array}{c|c} \mathbb{1}_{n} & \mathrm{a} \\ \hline 0 & 1 \end{array}\right]\left[\begin{array}{c|c} g & 0 \\ \hline 0 & 1 \end{array}\right]=a^{\mathrm{T}} \cdot a^{\mathrm{L}} \in A(n, \mathbb{R}) .

A(n, \mathbb{R})=\mathbb{R}^{n} \otimes G L(n, \mathbb{R})

P:=\mathbb{R}^{n} \otimes 0(1, n-1) \subset A(n, \mathbb{R}),

\mathfrak{a}(n, \mathbb{R})=\mathbb{R}^{n} \otimes \mathfrak{g} l(n, \mathbb{R})

\begin{aligned} & {\left[P_{\alpha}, P_{\beta}=0\right] \quad\left(P_{\alpha}:=\partial / \partial x^{\alpha}\right),} \\ & {\left[P_{\alpha}, L_{\gamma \delta}\right]=g_{\alpha \delta} P_{\gamma}-g_{\alpha \gamma} P \delta,} \\ & {\left[L_{\alpha \beta}, L_{\gamma \delta}\right]=c_{\alpha \beta \gamma \delta}{ }^{\varepsilon} L_{\varepsilon \gamma},} \\ & c_{\alpha \beta \delta \gamma}{ }^{\varepsilon \zeta}:=\delta_{(\alpha}^{[\varepsilon} \delta_{\delta}^{\zeta]} \eta_{\beta \gamma)} . \end{aligned}

e_{\alpha}(m)=e_{\alpha}^{i} \partial_{i}

\begin{aligned} e_{\alpha}(m) \rightarrow & e_{\alpha}{ }^{\prime}(m)=g_{\alpha}{ }^{\beta} e_{\beta}(m), \\ & g=\left[g_{\alpha}{ }^{\beta}\right] \in G L(n, \mathbb{R}) . \end{aligned}

L(M)=\bigcup_{m \in M} L_{m}(M)

L(M)=P(M, G L(n, \mathbb{R}), \pi, \delta)

\delta_{g}^{*} \vartheta=g^{-1} \vartheta, \quad g \in G L(n, \mathbb{R}),

\vartheta(e)=0 \Leftrightarrow \pi(e)=0, e \in T_{p}(P),

\stackrel{*}{\sigma}^{*} \vartheta^{\beta}=E_{j}^{\beta} d x^{j}

T(M)=V\left(M, \mathbb{R}^{n}, G L(n, \mathbb{R}), L(M)\right)

A(M):=P(M, A(n, \mathbb{R}), \pi, \delta)=L(M) \times \mathbb{R}^{n} .

\beta: A(n, \mathbb{R}) \rightarrow G L(n, \mathbb{R})=A(n, \mathbb{R}) / \mathbb{R}^{n} .

\delta:\left\{\begin{array}{llll} A(n, \mathbb{R}) & \times A(M) & \longrightarrow & A(M) \\ \downarrow & \downarrow & & \downarrow \\ (g, a) & (e(m), x) & =\left(\text { g.e }(m), g^{-1}(x-a)\right) . \end{array}\right.

T_{\mathrm{A}}(M)=\stackrel{\circ}{V}\left(M, \mathbb{R}^{n}, A(n, \mathbb{R}), A(M)\right) \subset V^{\mathrm{id}}

\stackrel{*}{\chi} \widetilde{\widetilde{\omega}}=\omega^{\mathrm{L}} \otimes \omega^{\mathrm{T}},

\stackrel{* \approx}{\chi}=\left[\begin{array}{c|c} \omega^{\mathrm{L}} & \omega^{\mathrm{T}} \\ \hline 0 & 0 \end{array}\right] .

\begin{aligned} * \widetilde{\widetilde{\Omega}} & =\stackrel{*}{\chi}\left(d \widetilde{\widetilde{\omega}}-\frac{1}{2}[\widetilde{\widetilde{\omega}}, \widetilde{\omega}]\right) \\ & =d \omega^{\mathrm{L}}-\omega^{\mathrm{L}} \wedge \omega^{\mathrm{L}}+d \omega^{\mathrm{T}}-\omega^{\mathrm{T}} \wedge \omega^{\mathrm{L}}-\omega^{\mathrm{L}} \wedge \omega^{\mathrm{T}}+\frac{1}{2}\left[\omega^{\mathrm{T}}, \omega^{\mathrm{T}}\right] \\ & =\Omega^{\mathrm{L}}+\Omega^{\mathrm{T}} \end{aligned}

\Omega^{\mathrm{T}}:=D \omega^{\mathrm{T}}=d \omega^{\mathrm{T}}-\left[\omega^{\mathrm{T}}, \omega^{\mathrm{L}}\right],

\mathscr{A}(n, \mathbb{R}):=C^{\infty}\left(A(M) \times_{\mathrm{Ad}} A(n, \mathbb{R})\right) .

\mathscr{G}(n, \mathbb{R}):=C^{\infty}\left(A(M) \times_{\mathrm{Ad}} G L(n, \mathbb{R})\right)

\mathscr{T}(n, \mathbb{R}):=C^{\infty}\left(A(M) \times_{\mathrm{Ad}} \mathbb{R}^{n}\right)

\widetilde{\widetilde{\omega}} \rightarrow^{A^{-1}} \widetilde{\widetilde{\omega}}=A^{-1} \widetilde{\widetilde{\omega}} A-A^{-1} d A, \quad A \in \mathscr{A}_{p} .

A=\left[\begin{array}{c|c} G & T \\ \hline 0 & 1 \end{array}\right] \in \mathscr{A}_{p}, \quad G \in \mathscr{G}_{p}, T \in \mathscr{T}_{p}

\begin{aligned} & \omega^{\mathrm{L}} \rightarrow{ }^{A^{-1}} \omega^{\mathrm{L}}=G^{-1} \omega^{\mathrm{L}} G-G^{-1} d G, \\ & \omega^{\mathrm{T}} \rightarrow{ }^{A^{-1}} \omega^{\mathrm{T}}=G^{-1} \omega^{\mathrm{T}}-G^{-1} D T . \end{aligned}

\widetilde{\widetilde{\Omega}} \rightarrow^{A^{-1}} \widetilde{\widetilde{\Omega}}=A^{-1} \widetilde{\widetilde{\Omega}} A

\Omega^{\mathrm{L}} \rightarrow^{A^{-1}} \Omega^{\mathrm{L}}=G^{-1} \Omega^{\mathrm{L}} G

\Omega^{\mathrm{T}} \rightarrow{ }^{A^{-1}} \Omega^{\mathrm{T}}=G^{-1}\left(\Omega^{\mathrm{T}}-D D T\right)=G^{-1} \Omega^{\mathrm{T}}-G^{-1} \Omega^{\mathrm{L}} \otimes T .

\widetilde{\widetilde{\varphi}}: A(M) \rightarrow C^{\infty}(\stackrel{\circ}{V})

Q_{A}=\left\{p \in A(M) \mid \widetilde{\widetilde{\varphi}}(p)=\sigma\left(0_{m}\right)\right\}=L(M)

\widetilde{\widetilde{\varphi}} \rightarrow A^{-1} \widetilde{\widetilde{\varphi}}=G^{-1}(\widetilde{\widetilde{\varphi}}-T) .

\omega^{\mathrm{T}}=\widetilde{\widetilde{\vartheta}}-D \widetilde{\widetilde{\varphi}}

\begin{aligned} A^{-1} \omega^{\mathrm{T}} & =A^{-1} \widetilde{\vartheta}-G^{-1} D G^{A^{-1}} \widetilde{\varphi} \\ & =G^{-1} \widetilde{\vartheta}-G^{-1} D \widetilde{\varphi}, \end{aligned}

\widetilde{\widetilde{\vartheta}} \rightarrow^{A^{-1}} \widetilde{\widetilde{\vartheta}}=G^{-1} \widetilde{\vartheta} .

\widetilde{\widetilde{\vartheta}}=f \vartheta .

\Theta:=D \vartheta=d \vartheta-\left[\omega^{\mathrm{L}}, \vartheta\right]

\stackrel{*}{\sigma} \Theta=\frac{1}{2} T_{\alpha \beta}^{. c} P_{c} \otimes \vartheta^{\alpha} \wedge \vartheta^{\beta}=\frac{1}{2} T_{i j}^{. c} P_{c} \otimes d x^{i} \wedge d x^{j},

T_{\alpha \beta}^{. . c}=2 D_{[\alpha} E_{\cdot \mid \beta]}^{c}=2\left(\partial_{[\alpha} E_{\cdot \mid \beta]}^{c}+\Gamma_{[\alpha \mid d}^{. c} E_{\cdot \mid \beta]}^{d}\right) .

\Theta \rightarrow{ }^{A^{-1}} \Theta=G^{-1} \Theta,

\Omega^{\mathrm{T}}=f \Theta+d f \wedge \vartheta-\Omega^{\mathrm{L}} \otimes \widetilde{\widetilde{\varphi}}

\widetilde{\widetilde{D}} \widetilde{\widetilde{\varphi}}=D \widetilde{\widetilde{\varphi}}=0

Q_{\kappa \mu \nu}:=-D_{\kappa} g_{\mu \nu}

\begin{aligned} (\Delta s)^{2} & =-c^{2}(\Delta t)^{2}+(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2} \\ & =g_{i j} \Delta x^{i} \Delta x^{j} . \end{aligned}

\ell(\mathscr{C}):=\int_{t_{0}}^{t_{1}} d s(t)

\ell\left(\mathscr{C}_{1}+\mathscr{C}_{2}\right)=\ell\left(\mathscr{C}_{1}\right)+\ell\left(\mathscr{C}_{2}\right)

d s=F(m, \vartheta) d t,

\begin{aligned} & F(m, f \vartheta)=|f| F(m, \vartheta), \\ & F(m, \vartheta)>0 \quad \text { if } \quad \vartheta \neq 0 . \end{aligned}

g_{\alpha \beta}(m, \vartheta):=\frac{1}{2} \frac{\partial^{2} F^{2}(m, \vartheta)}{\partial \vartheta^{\alpha} \otimes_{s} \partial \vartheta^{\beta}} .

d s^{2}=(F(m, \vartheta) d t)^{2}=g_{\alpha \beta} \vartheta^{\alpha} \otimes \vartheta^{\beta}=g_{i j}(m) d x^{i} \otimes_{s} d x^{j} ;

\cos \phi=\frac{g_{\mu \nu} d V^{\mu} d W^{\nu}}{\sqrt{g_{\alpha \beta} d V^{\alpha} d V^{\beta}} \sqrt{g_{\alpha \beta} d W^{\alpha} d W^{\beta}}}

g_{i j} e_{\alpha}^{\cdot i} e_{\beta}^{j}=g_{\alpha \beta} .

\vartheta \wedge^{*} \vartheta=\eta .

L^{g}(M):=P(M, O(s, n-s), \pi, \delta) .

D e_{\alpha}^{\cdot i}=0

D \phi^{(\circ)}=\vartheta^{\alpha} \nabla_{\alpha} \phi^{(\circ)} .

D_{k} g_{i j}=E_{\cdot k}^{\alpha} \nabla_{\alpha} g_{i j}=0

{ }^{*} \omega^{g}=\frac{1}{2} \omega^{\alpha \beta} L_{\alpha \beta}=\frac{1}{2} \Gamma_{i}^{\cdot \alpha \beta} L_{\alpha \beta} \otimes d x^{i},

\Gamma_{i}^{\cdot \alpha \beta}=-\Gamma_{i}^{\cdot \beta \alpha}

g_{i j} \stackrel{(*)}{=} o_{i j}

\partial_{k} g_{i j} \stackrel{(*)}{=} 0 .

\Omega^{g}=d \omega^{g}-\omega^{g} \wedge \omega^{g}

\stackrel{*}{\sigma} \Omega^{g}=\frac{1}{4} R_{\alpha \beta}^{. . c d} L_{c d} \otimes \vartheta^{\alpha} \wedge \vartheta^{\beta} .

R_{\alpha \beta c}^{. . . d}=2\left\{\partial_{[\alpha} \Gamma_{\beta] c}^{. . d}+\Gamma_{[\alpha \mid h}^{. . d} \Gamma_{\mid \beta] c}^{. . h}\right\}

\omega^{g}=\omega^{\{ \}}-K .

d \vartheta-\left[\omega^{\{ \}}, \vartheta\right]=0

\Theta=[K, \vartheta]

\stackrel{*}{\sigma} K=\frac{1}{2} K_{\alpha}^{. \beta c} L_{\beta c} \otimes \vartheta^{\alpha} .

K_{. \alpha \beta}^{c}=-\frac{1}{2}\left(T_{. \alpha \beta}^{c}-T_{\alpha, \beta}^{. . c}+T_{\beta . \alpha}^{. c}\right)=-K_{. \beta \alpha}^{c} .

\begin{aligned} \Omega^{g} & =\Omega^{\{ \}}-d K+K \wedge \omega^{\{ \}}+\omega^{\{ } \wedge K-K \wedge K \\ & =\Omega^{\{ \}}-D^{\{ } K-K \wedge K=\Omega^{\{ \}}-D K+K \wedge K \end{aligned}

\Gamma_{i j}^{. . k}=e_{b}^{. \kappa} \Gamma_{i \alpha}^{. . b} E_{. j}^{\alpha}+e_{\beta}^{. \kappa} \partial_{i} E_{. j}^{\beta}

\Gamma_{i j}^{\cdot{ }^{\prime}}=\left\{\begin{array}{c} \kappa \\ i j \end{array}\right\}-K_{i j}^{\cdot \kappa},

\left\{\begin{array}{c} \kappa \\ i j \end{array}\right\}=\frac{1}{2} g^{\kappa l}\left(\partial_{i} g_{j l}+\partial_{j} g_{i l}-\partial_{l} g_{i j}\right)

R_{i j c}^{:: d}=R_{i j c}^{\{!\ldots d}-2 D_{[i} K_{j j c}^{\cdot d}+2 K_{[i \mid c}^{\cdot h} K_{i j j h}^{\cdot d} .

\begin{aligned} & R_{(i j) \kappa l}^{\{ \}}=R_{i j(\kappa l)}^{\{ \}}=0 \\ & R_{i j \kappa l}^{\}}=R_{\kappa l i j}^{\}} . \end{aligned}

R_{i[j \kappa l]}^{(l)} \equiv 0

R_{i j}:=R_{\alpha i j}^{\cdots \alpha}

R:=R_{\mu}^{\cdot \mu}=R_{\alpha \mu}^{\cdot \cdot \mu \alpha}={ }^{*}\left(\Omega^{g} \wedge \vartheta \wedge \vartheta\right) .

\left[D_{\alpha}, D_{\beta}\right]=R_{\alpha \beta}^{. . c d} L_{c d}-T_{\alpha \beta}^{. . c} D_{c},

L_{g}=L\left(g_{\alpha \beta} ; \vartheta,\left(\omega^{g}\right), \Theta, \Omega^{g}\right) .

\delta L_{g}=: \delta \vartheta \wedge E+\delta \Theta \wedge \Pi^{\mathrm{T}}+\delta \Omega^{g} \wedge \Pi^{\mathrm{L}},

\delta \Theta=\delta\left(d \vartheta-\left[\omega^{\mathrm{L}}, \vartheta\right]\right)

\begin{aligned} & =d(\delta \vartheta)-\left[\omega^{\mathrm{L}}, \delta \vartheta\right]-\left[\delta \omega^{\mathrm{L}}, \vartheta\right] \\ & =D(\delta \vartheta)-\left[\delta \omega^{\mathrm{L}}, \vartheta\right] . \end{aligned}

\begin{aligned} & \delta L_{g}=\delta \vartheta \wedge E-\delta \vartheta \wedge D \Pi^{\mathrm{T}}+d\left(\delta \vartheta \wedge \Pi^{\mathrm{T}}\right) \\ & -\left[\delta \omega^{g}, \vartheta\right] \wedge\left[\Pi_{[]}^{\mathrm{T}}, \vartheta\right]-\delta \omega^{g} \wedge D \Pi^{\mathrm{L}}+d\left(\delta \omega^{g} \wedge \Pi^{\mathrm{L}}\right), \end{aligned}

S=\int_{M}\left(L_{g}+L_{\mathrm{mat}}\right)

\begin{aligned} & D \Pi^{\mathrm{T}}-E=\Sigma \\ & D \Pi^{\mathrm{L}}+\Pi_{[]}^{\mathrm{T}}=\tau_{\mathrm{s}} \end{aligned}

\delta L_{\text {matter }}=: \delta \vartheta \wedge \Sigma+\delta \omega^{g} \wedge \tau_{s}+\delta_{\varphi} \wedge \lambda^{\varphi}

D \Theta \equiv\left[\vartheta, \Omega^{\mathrm{L}}\right]

D \Omega^{g} \equiv 0,

\begin{aligned} d \Theta & =d d \vartheta-d\left[\omega^{\mathrm{L}}, \vartheta\right] \\ & =-\left[d \omega^{\mathrm{L}}, \vartheta\right]+\left[\omega^{\mathrm{L}}, d \vartheta\right] \\ & =-\left[\Omega^{\mathrm{L}}, \vartheta\right]-\left[\omega^{\mathrm{L}} \wedge \omega^{\mathrm{L}}, \vartheta\right]+\left[\omega^{\mathrm{L}}, \Theta\right]+\left[\omega^{\mathrm{L}},\left[\omega^{\mathrm{L}}, \vartheta\right]\right] \\ & =\left[\omega^{\mathrm{L}}, \Theta\right]+\left[\vartheta, \Omega^{\mathrm{L}}\right] \end{aligned}

D[\Theta, \vartheta] \equiv\left[\vartheta \wedge \vartheta, \Omega^{g}\right]

\begin{aligned} D[\Theta, \vartheta] & =[D \Theta, \vartheta]+[\Theta, D \vartheta] \\ & =\left[\left[\vartheta, \Omega^{g}\right], \vartheta\right]+[\Theta, \Theta] \\ & =\vartheta \wedge \Omega^{g} \wedge \vartheta-\Omega^{g} \wedge \vartheta \wedge \vartheta+\vartheta \wedge \vartheta \wedge \Omega^{g}-\vartheta \wedge \Omega^{g} \wedge \vartheta \\ & =\left[\vartheta \wedge \vartheta, \Omega^{g}\right] . \end{aligned}

D\left(\vartheta \wedge^{*} \Omega^{g}\right)=\Theta \wedge^{*} \Omega^{g}

D \Pi_{[]}^{\mathrm{T}}-\mathrm{E}_{[]}=\Sigma_{[l}

D D \Pi^{\mathrm{L}}+D \Pi_{[]}^{\mathrm{T}}=D \tau_{s} .

\left[\Omega^{g}, \Pi^{\mathrm{L}}\right]+\mathrm{E}_{[]}=D \tau_{s}-\Sigma_{[]} .

\left[\Omega^{g}, \Pi^{L}\right]+\mathrm{E}_{[]}=0

L_{\mathrm{mat}}=L\left(g_{\alpha \beta}, \vartheta^{\alpha}, \Psi, D \Psi\right) .

\Sigma_{\alpha}:=\frac{\partial L}{\partial \vartheta^{\alpha}}, \quad \sigma^{\alpha \beta}:=2 \frac{\partial L}{\partial g_{\alpha \beta}}, \quad \text { and } \quad \tau_{\alpha \beta}:=\frac{\partial L}{\partial \Gamma^{\alpha \beta}}=I_{\alpha \beta} \Psi \wedge \frac{\partial L}{\partial D \Psi}

\begin{aligned} £_{\xi} L & =\left(£_{\xi} g_{\alpha \beta}\right) \wedge \frac{\partial L}{\partial g_{\alpha \beta}}+\left(£_{\xi} \vartheta^{\alpha}\right) \wedge \frac{\partial L}{\partial \vartheta^{\alpha}} \\ & +\left(£_{\xi} \Psi\right) \wedge \frac{\partial L}{\partial \Psi}+\left(£_{\xi} D \Psi\right) \wedge \frac{\partial L}{\partial D \Psi} \\ & \left.\left.\left.=D\left[(\xi\rfloor \vartheta^{\alpha}\right) \wedge \frac{\partial L}{\partial \vartheta^{\alpha}}+(\xi\rfloor \Psi\right) \wedge \frac{\partial L}{\partial \Psi}+(\xi\rfloor D \Psi\right) \wedge \frac{\partial L}{\partial D \Psi}\right] \\ & \left.\left.-\left(\xi \downharpoonleft \vartheta^{\alpha}\right) D \frac{\partial L}{\partial \vartheta^{\alpha}}+(\xi\rfloor T^{\alpha}\right) \wedge \frac{\partial L}{\partial \vartheta^{\alpha}}+(\xi\lrcorner R_{\beta}^{\gamma}\right) \wedge I^{\beta}{ }_{\gamma} \Psi \wedge \frac{\partial L}{\partial D \Psi} \\ & \left.+(\xi\rfloor D \Psi) \wedge \frac{\delta L}{\delta \Psi}+(-1)^{p}(\xi\rfloor \Psi\right) \wedge D \frac{\delta L}{\delta \Psi} \end{aligned}

\xi \downharpoonleft L \equiv\left(\xi \downharpoonleft \vartheta^{\alpha}\right) \wedge \frac{\partial L}{\partial \vartheta^{\alpha}}+(\xi \downharpoonleft \Psi) \wedge \frac{\partial L}{\partial \Psi}+(\xi \downharpoonleft D \Psi) \wedge \frac{\partial L}{\partial D \Psi} .

\left.\Sigma_{\alpha}=e_{\alpha} \downharpoonleft L-\left(e_{\alpha} \downharpoonleft D \Psi\right) \wedge \frac{\partial L}{\partial D \Psi}-\left(e_{\alpha}\right\lrcorner \Psi\right) \wedge \frac{\partial L}{\partial \Psi} .

\begin{aligned} D \Sigma_{\alpha} & \left.\left.=\left(e_{\alpha}\right\rfloor T^{\beta}\right) \wedge \Sigma_{\beta}+\left(e_{\alpha}\right\rfloor R^{\beta \gamma}\right) \wedge \tau_{\beta \gamma}+F_{\alpha} \\ & \left.\left.\simeq\left(e_{\alpha}\right\rfloor T^{\beta}\right) \wedge \Sigma_{\beta}+\left(e_{\alpha}\right\rfloor R^{\beta \gamma}\right) \wedge \tau_{\beta \gamma} . \end{aligned}

F_{\alpha}=\left(e_{\alpha} \downharpoonleft D \Psi\right) \frac{\delta L}{\delta \Psi}+(-1)^{p}\left(e_{\alpha} \downharpoonleft \Psi\right) \wedge D \frac{\delta L}{\delta \Psi}

\delta L=\delta g_{\alpha \beta} \wedge \frac{\partial L}{\partial g_{\alpha \beta}}+\delta \vartheta^{\alpha} \wedge \frac{\partial L}{\partial \vartheta^{\alpha}}+\delta \Psi \wedge \frac{\partial L}{\partial \Psi}+\delta(D \Psi) \wedge \frac{\partial L}{\partial D \Psi} .

\begin{aligned} \delta L & =\delta g_{\alpha \beta} \wedge \frac{\partial L}{\partial g_{\alpha \beta}}+\delta \vartheta^{\alpha} \wedge \frac{\partial L}{\partial \vartheta^{\alpha}} \\ & +\delta \Gamma_{\alpha}{ }^{\beta} \wedge I^{\alpha}{ }_{\beta} \Psi \wedge \frac{\partial L}{\partial D \Psi}+\delta \Psi \wedge \frac{\delta L}{\delta \Psi}+D\left(\delta \Psi \wedge \frac{\partial L}{\partial D \Psi}\right) \end{aligned}

\begin{aligned} & \delta g_{\alpha \beta}=2 \varepsilon_{(\alpha}{ }^{\gamma} g_{\beta) \gamma}, \quad \delta \vartheta^{\alpha}=-\vartheta^{\beta} \varepsilon_{\beta}{ }^{\alpha}, \\ & \delta \Gamma_{\alpha}{ }^{\beta}=D \varepsilon_{\alpha}{ }^{\beta}, \quad \delta \Psi=-\varepsilon_{\alpha}{ }^{\beta} I^{\alpha}{ }_{\beta} \Psi . \end{aligned}

\delta L=-\varepsilon_{\alpha}{ }^{\beta}\left[-\sigma^{\alpha \gamma} g_{\gamma \beta}+\vartheta^{\alpha} \wedge \frac{\partial L}{\partial \vartheta^{\beta}}+D \frac{\partial L}{\partial \Gamma_{\alpha}{ }^{\beta}}+I^{\alpha}{ }_{\beta} \Psi \wedge \frac{\delta L}{\delta \Psi}\right] .

D \tau_{\alpha \beta}+\vartheta_{[\alpha} \wedge \Sigma_{\beta]}=-I_{\alpha \beta} \Psi \wedge \frac{\delta L}{\delta \Psi} \simeq 0 .

\begin{aligned} \widehat{D} \Sigma_{\alpha} & \left.\cong\left(e_{\alpha}\right\rfloor R_{\beta}^{\gamma}\right) \wedge \Delta^{\beta} \gamma \\ & \left.-\frac{1}{2}\left(e_{\alpha}\right\rfloor Q_{\beta \gamma}\right) \wedge \sigma^{\beta \gamma} \end{aligned}

\begin{aligned} & R_{\alpha}{ }^{\beta}=d \Gamma_{\alpha}{ }^{\beta} \\ & -\Gamma_{\alpha}{ }^{\gamma} \wedge \Gamma_{\gamma}{ }^{\beta} \end{aligned}

\begin{gathered} D \Sigma_{\alpha} \simeq 0 \\ D\left(\tau_{\alpha \beta}+x_{[\alpha} \wedge \Sigma_{\beta]}\right) \simeq 0 . \end{gathered}

\varepsilon_{\mathrm{RC}}:=\xi^{\alpha} \Sigma_{\alpha}+\left(e_{\beta} \downharpoonleft \overparen{D} \xi^{\gamma}\right) \tau^{\beta}{ }_{\gamma},

\left.\widetilde{\varepsilon}_{\mathrm{MA}}:=\varepsilon_{\mathrm{RC}}+\xi\right\lrcorner L \cong-\left(\mathscr{L}_{\xi} \Gamma_{\beta}{ }^{\gamma}\right) \wedge H^{\beta}{ }_{\gamma}+d H_{\infty}

\hat{M}:=M^{3} /\left[M^{2}+\frac{\Lambda}{3} J^{2}\right], \quad \hat{J}:=J M^{2} /\left[M^{2}+\frac{\Lambda}{3} J^{2}\right],

C_{\mathrm{AdS}}=\hat{M}^{2}+\frac{\Lambda}{3} \hat{J}^{2}

\widetilde{\widetilde{\Gamma}}:=\Gamma^{(T) \alpha} P_{\alpha}+\Gamma_{\alpha}^{(L) \beta} L^{\alpha}{ }_{\beta}

\begin{aligned} \widetilde{\widetilde{D}} \widetilde{\xi}^{\alpha} & =d \xi^{\alpha}+\Gamma_{\beta}^{(L) \alpha} \wedge \xi^{\beta}+\Gamma^{(T) \alpha} \\ & =D \xi^{\alpha}+\Gamma^{(T) \alpha}=0 \end{aligned}

\begin{aligned} \widetilde{\leftarrow}_{\widetilde{y}} \widetilde{\widetilde{D}} \widetilde{\xi}^{\alpha} & \left.=\widetilde{y}\rfloor\left(\widetilde{\widetilde{D}} \widetilde{\widetilde{D}} \widetilde{\xi}^{\alpha}\right)+\widetilde{\widetilde{D}}(\widetilde{\widetilde{y}}\rfloor \widetilde{\widetilde{D}} \widetilde{\xi}^{\alpha}\right) \\ & =y J\left(D D \xi^{\alpha}+R^{(T) \alpha}\right)=0, \end{aligned}

\begin{aligned} \Delta \xi^{\alpha} & \left.\left.=-\oint_{C} y\right\rfloor\left(D D \xi^{\alpha}\right)=\oint_{C} y\right\rfloor R^{(T) \alpha}=\int_{S} R^{(T) \alpha} \\ & \simeq \frac{1}{2}\left(T_{i j}^{\alpha}-R_{i j \beta}^{\alpha} \xi^{\beta}\right) \int_{S} d y^{i} \wedge d y^{j} \end{aligned}

\Phi(A, \gamma)=P \exp [(i / \hbar) \oint A],

\Phi(\stackrel{\widetilde{\Gamma}}{\Gamma}, \gamma)=P \exp \left[(i / \hbar) \oint\left(\Gamma^{(T) \alpha} P_{\alpha}+\Gamma_{\alpha}^{(L) \beta} L^{\alpha}{ }_{\beta}\right)\right]

\Delta \Phi(\widetilde{\widetilde{\Gamma}}, \gamma) \simeq \frac{i}{\hbar} \int_{S}\left(R^{(T) \alpha} P_{\alpha}+R_{\alpha}^{(L) \beta} L_{\beta}^{\alpha}\right)

\Phi(\widetilde{\widetilde{\Gamma}}, \gamma)=\left(2 \pi \hbar \mathscr{M}_{n} G / \hbar \ell c^{2}\right)=2 \pi n

L_{\mathrm{D}}=L(\gamma, \psi, D \psi)=\frac{i}{2}\left\{\bar{\psi}^{*} \gamma \wedge D \psi+\overline{D \psi} \wedge^{*} \gamma \psi\right\}-m \bar{\psi} \psi \eta,

j:=\bar{\psi}^{*} \gamma \psi=j^{\mu} \eta_{\mu} \quad \text { and } \quad j_{5}:=\bar{\psi}^{*} \gamma \gamma_{5} \psi=j_{5}^{\mu} \eta_{\mu},

\begin{aligned} L_{\mathrm{D}} & =L\left(\gamma, \psi, D^{\{ \}} \psi\right)-\frac{i}{2} \bar{\psi}\left({ }^{*} \gamma \wedge K-K \wedge^{*} \gamma\right) \psi \\ & =L\left(\gamma, \psi, D^{\{ } \psi\right)+\frac{1}{4} \mathscr{A} \wedge j_{5} . \end{aligned}

\mathscr{A}:=\frac{1}{4}{ }^{*} \operatorname{Tr}(\gamma \wedge D \gamma)={ }^{*}\left(\vartheta^{\alpha} \wedge T_{\alpha}\right)=\frac{1}{2} T^{[\alpha \beta \gamma]} \eta_{\alpha \beta \gamma}=\mathscr{A}_{i} d x^{i},

\tau_{\alpha \beta}:=\frac{\partial L_{\mathrm{D}}}{\partial \Gamma^{\alpha \beta}}=\frac{1}{8} \bar{\Psi}\left({ }^{*} \gamma \sigma_{\alpha \beta}+\sigma_{\alpha \beta}{ }^{*} \gamma\right) \Psi=\frac{1}{4} \eta_{\alpha \beta \gamma \delta} \bar{\Psi} \gamma^{\delta} \gamma_{5} \Psi \eta^{\gamma}=\tau_{\alpha \beta \gamma} \eta^{\gamma}

\left.d U=T^{\alpha} \wedge \mu_{\alpha}-\vartheta^{\alpha} \wedge D \mu_{\alpha}=\vartheta^{\alpha} \wedge\left[e_{\beta}\right\rfloor\left(T^{\beta} \wedge \mu_{\alpha}\right)-D \mu_{\alpha}\right]

\begin{aligned} \Sigma_{\alpha} \rightarrow \sigma_{\alpha} & \left.:=\Sigma_{\alpha}-D \mu_{\alpha}+e_{\beta}\right\rfloor\left(T^{\beta} \wedge \mu_{\alpha}\right), \\ \tau_{\alpha \beta} \rightarrow \hat{\tau}_{\alpha \beta} & :=\tau_{\alpha \beta}-\vartheta_{[\alpha} \wedge \mu_{\beta]}=0 \end{aligned}

L_{\mathrm{EC}}:=-\frac{1}{2 \kappa} R^{\alpha \beta} \wedge \eta_{\alpha \beta},

G_{\alpha}:=\frac{1}{2} R^{\beta \gamma} \wedge \eta_{\alpha \beta \gamma}=\kappa \Sigma_{\alpha},

\left.\widehat{D} G_{\alpha} \equiv \frac{1}{2}\left(e_{\alpha}\right\rfloor R^{\beta \gamma}\right) \wedge \eta_{\beta \gamma \mu} \wedge T^{\mu}

D^{\{ \}} G_{\alpha}^{\{ \}} \equiv 0

\Theta:=D \gamma \quad \text { and } \quad \Omega^{g}:=d \Gamma+\Gamma \wedge \Gamma .

D \Theta \equiv\left[\Omega^{g}, \gamma\right], \quad D \Omega^{g} \equiv 0,

G:=G_{\alpha} \gamma^{\alpha}:=\frac{1}{2} R^{\mu \nu} \wedge \eta_{\mu \nu \lambda} \gamma^{\lambda}=-i \gamma_{5}\left(\Omega^{g} \wedge \gamma+\gamma \wedge \Omega^{g}\right)=i\left[\gamma, \gamma_{5} \Omega^{g}\right] .

D[\gamma, \Theta] \equiv 2 i\left[\sigma, \Omega^{g}\right], \quad D\left[\gamma, \Omega^{g}\right] \equiv\left[\Theta, \Omega^{g}\right],

D G \equiv i\left[\Theta, \gamma_{5} \Omega^{g}\right]

\eta_{\alpha \beta \gamma} \wedge T^{\gamma}=2 \kappa \tau_{\alpha \beta}

\mathrm{T}_{\mu \nu}:=\frac{2}{\sqrt{g}} \frac{\delta L}{\delta g_{\mu \nu}}

\begin{aligned} \widetilde{T}_{\mu \nu} & =T_{\mu \nu}+\ell^{* 2}\left\{-4 \tau_{\mu \cdot[\lambda]}^{\cdot \kappa} \tau_{\nu \cdot \mid \kappa]}^{\cdot \lambda}-2 \tau_{\mu}^{\cdot \kappa \lambda} \tau_{\nu \kappa \lambda}+\tau_{\cdot, \mu}^{\kappa \lambda} \tau_{\kappa \lambda \nu}\right. \\ & \left.+\frac{1}{2} g_{\mu \nu}\left(4 \tau_{\delta \cdot[\lambda}^{\cdot \kappa} \tau_{\cdot, \kappa]}^{\delta \lambda}+\tau^{\delta \kappa \lambda} \tau_{\delta \kappa \lambda}\right)\right\} . \end{aligned}

R^{\alpha \beta}=R^{\{j \alpha \beta}-D^{\{ \}} K^{\alpha \beta}-K^{\alpha}{ }_{\mu} \wedge K^{\mu \beta}

\begin{aligned} R^{\{\alpha \beta \beta} \wedge \eta_{\alpha \beta} \equiv & R^{\alpha \beta} \wedge \eta_{\alpha \beta}-K^{\alpha \mu} \wedge K_{\mu}{ }^{\beta} \wedge \eta_{\alpha \beta}+K^{\alpha \beta} \wedge T^{\gamma} \wedge \eta_{\alpha \beta \gamma} \\ & +d\left(K^{\alpha \beta} \wedge \eta_{\alpha \beta}\right) \\ = & R^{\alpha \beta} \wedge \eta_{\alpha \beta}+T^{\alpha} \wedge{ }^{*}\left(-{ }^{(1)} T_{\alpha}+2^{(2)} T_{\alpha}+\frac{1}{2}{ }^{(3)} T_{\alpha}\right)+2 d C_{\mathrm{TT}^{*}} \end{aligned}

{ }^{(3)} T_{\alpha}=\frac{(-1)^{s}}{3} *\left(\vartheta_{\alpha} \wedge \mathscr{A}\right)

L_{\mathrm{EC}}:=-\frac{1}{2 \kappa} R^{\alpha \beta} \wedge \eta_{\alpha \beta}=L_{\mathrm{HE}}+\frac{1}{12 \kappa} \mathscr{A}^{*} \mathscr{A}

\begin{aligned} G_{\alpha} & :=\frac{1}{2} R^{\beta \gamma} \wedge \eta_{\alpha \beta \gamma} \\ & \left.\left.=G_{\alpha}^{\{ \}}+\frac{(-1)^{s}}{12}\left(e_{\alpha}\right\rfloor \mathscr{A} \wedge{ }^{*} \mathscr{A}-\frac{1}{3} \mathscr{A} \wedge e_{\alpha}\right\rfloor^{*} \mathscr{A}\right)+\frac{(-1)^{s}}{6} \vartheta_{\alpha} \wedge d \mathscr{A} \end{aligned}

{ }^{*} \mathscr{A}=\frac{\kappa}{2} j_{5},

\Delta \widetilde{L}:=\kappa\left(L_{\mathrm{EC}}-L_{\mathrm{HE}}\right) \simeq \frac{\kappa^{2}}{48} j_{5} \wedge^{*} j_{5}=4 f^{2} j_{5} \wedge^{*} j_{5},

k \partial_{k} \Gamma_{k}=\frac{1}{2} \mathrm{~S} \operatorname{Tr}\left\{\left[\Gamma_{k}^{(2)}+R_{k}\right]^{-1}\left(k \partial_{k} R_{k}\right)\right\},

L_{\mathrm{HE}}:=L_{\mathrm{HE}}+\frac{\Lambda}{\kappa} \eta

g_{\mathrm{N}}:=\kappa k^{2}, \quad \lambda:=\Lambda / k^{2},

k \frac{\partial}{\partial k} g_{\mathrm{N}}=\beta_{1}\left(g_{\mathrm{N}}, \lambda\right)=\left(2+d_{\mathrm{N}}\right) g_{\mathrm{N}}, \quad k \frac{\partial}{\partial k} \lambda=\beta_{2}\left(g_{\mathrm{N}}, \lambda\right),

\mu=\frac{4}{3} \kappa \Lambda=\frac{1}{3}(2 \kappa)^{2} \rho_{\Lambda} \leq \frac{4}{3} g_{\mathrm{N} *} \lambda_{*} \simeq 0.2

f^{2}=\frac{1}{192} \kappa^{2}=3 \times 2^{-10}\left(\frac{\mu}{\Lambda}\right)^{2}=2^{-8} \frac{\mu}{\rho_{\Lambda}}

\left\langle d j_{5}\right\rangle=2 i m\left\langle\bar{\psi} \gamma_{5} \psi\right\rangle \eta-\frac{1}{96 \pi^{2}}\left[2 R_{\alpha \beta}^{\{ \}} \wedge R^{\{\beta \alpha \beta}+d \mathscr{A} \wedge d \mathscr{A}\right]

C_{\mathrm{TT}}:=\frac{1}{2 \ell^{2}}\left(\vartheta^{\alpha} \wedge T_{\alpha}\right)=-\frac{(-1)^{\mathrm{sig}}}{2 \ell^{2}} * \mathscr{A},

\begin{aligned} C_{\mathrm{RR}} & :=-\operatorname{Tr}\left(\Gamma \wedge \Omega-\frac{1}{3} \Gamma \wedge \Gamma \wedge \Gamma\right) \\ & =-\frac{1}{2}\left(\Gamma_{\alpha}{ }^{\beta} \wedge R_{\beta}{ }^{\alpha}+\frac{1}{3} \Gamma_{\alpha}{ }^{\beta} \wedge \Gamma_{\beta}{ }^{\gamma} \wedge \Gamma_{\gamma}{ }^{\alpha}\right) . \end{aligned}

\hat{C}=C_{\mathrm{RR}}-2 C_{\mathrm{TT}}

d C_{\mathrm{TT}} \equiv \frac{1}{2 \ell^{2}}\left(T^{\alpha} \wedge T_{\alpha}+R_{\alpha \beta} \wedge \vartheta^{\alpha} \wedge \vartheta^{\beta}\right),

d C_{\mathrm{RR}}=-\operatorname{Tr}(\Omega \wedge \Omega)=\frac{1}{2} R^{\alpha \beta} \wedge R_{\alpha \beta}

D R_{\alpha \beta} \equiv 0,

D T^{\alpha} \equiv R_{\beta}{ }^{\alpha} \wedge \vartheta^{\beta}

L_{\mathrm{HE}}=-\frac{1}{2 \kappa} R_{\alpha \beta}^{\{ \}} \wedge^{*}\left(\vartheta^{\alpha} \wedge \vartheta^{\beta}\right)

L_{\|}:=-\frac{1}{2 \kappa} T^{\alpha} \wedge{ }^{*}\left({ }^{(1)} T_{\alpha}-2^{(2)} T_{\alpha}-\frac{1}{2}{ }^{(3)} T_{\alpha}\right),

H_{\alpha}^{\|}:=-\partial L_{\|} / \partial T^{\alpha}=(1 / \kappa) \eta_{\alpha \beta \gamma} \wedge K^{\beta \gamma}

L_{\|} \equiv L_{\mathrm{HE}}+\frac{1}{2 \kappa} R_{\alpha \beta} \wedge^{*}\left(\vartheta^{\alpha} \wedge \vartheta^{\beta}\right)+\frac{2 \ell^{2}}{\kappa} d C_{\mathrm{TT}^{*}},

\tilde{L}_{\|}=L_{\|}-R^{\alpha \beta} \wedge \lambda_{\alpha \beta} .

\begin{gathered} D H_{\alpha}^{\|}-E_{\alpha}^{\|}=\Sigma_{\alpha}, \\ D \lambda_{\alpha \beta}+\vartheta_{[\alpha} \wedge H_{\beta]}^{\|}=\tau_{\alpha \beta}, \end{gathered}

R^{\alpha \beta}=0 .

D D \lambda_{\alpha \beta}=-2 R_{[\alpha \mid}{ }^{\gamma} \wedge \lambda_{\gamma \mid \beta]}=0

D\left(\tau_{\alpha \beta}-\vartheta_{[\alpha} \wedge H_{\beta]}^{\|}\right)=0

\stackrel{( \pm)}{L}_{\mathrm{EC}}:=L_{\mathrm{EC}} \pm i d C_{\mathrm{TT}}=-\frac{1}{2 \ell^{2}} \stackrel{( \pm)}{R}_{\alpha \beta} \wedge *^{*}\left(\vartheta^{\alpha} \wedge \vartheta^{\beta}\right)+\frac{\Lambda}{\ell^{2}} \eta .

R_{\alpha \beta}=\xi^{*} R_{\alpha \beta}^{(\star)}+\left(\gamma / \ell^{2}\right)\left[\vartheta_{\alpha} \wedge \vartheta_{\beta} \pm i^{*}\left(\vartheta_{\alpha} \wedge \vartheta_{\beta}\right)\right],

\stackrel{( \pm)}{L}_{\|}:=L_{\|} \pm i \frac{2 \ell^{2}}{\kappa} d C_{\mathrm{TT}}=L_{\mathrm{HE}}-L_{\mathrm{EC}} \pm i \frac{2 \ell^{2}}{\kappa} d \stackrel{(\mp)}{C}_{\mathrm{TT}} .

\stackrel{( \pm)}{C}_{\mathrm{TT}}:=\frac{1}{2 \ell^{2}}\left(\vartheta^{\alpha} \wedge \stackrel{( \pm)}{T}_{\alpha}\right),

\stackrel{( \pm)}{\Pi}_{\alpha}=-\partial \stackrel{( \pm)}{L} / \partial T^{\alpha}=H_{\alpha}^{\|} \mp(i / \kappa) T_{\alpha}

\stackrel{(+)}{\Pi}^{\beta} \wedge \stackrel{(-)}{\Pi}_{\beta}=H^{\beta} \wedge H_{\beta}+\ell^{-4} T^{\beta} \wedge T_{\beta}=0 .

\left.\stackrel{( \pm)( \pm)}{D} \stackrel{(-)}{\Pi}_{\alpha} \mp \frac{i}{4} \ell^{2} e_{\alpha}\right\rfloor\left(\stackrel{(+)}{\Pi}^{\beta} \wedge \stackrel{(-)}{\Pi}_{\beta}\right)=\Sigma_{\alpha},

\stackrel{( \pm)( \pm)}{D} \Pi_{\alpha}=\Sigma_{\alpha} .

\begin{aligned} \stackrel{( \pm)}{L}= & -\frac{i}{4} \ell^{2}\left[\stackrel{(+)}{\Pi}^{\alpha}-\stackrel{(-)}{\Pi^{\alpha}}\right] \wedge \stackrel{( \pm)}{\Pi}_{\alpha} \\ & -\frac{i}{4}\left[\stackrel{(+)}{\Pi}^{\alpha \beta}-\stackrel{(-)}{\Pi}^{\alpha \beta}\right] \wedge\left(\stackrel{( \pm)}{\Pi}_{\alpha \beta} \mp \frac{i}{\ell^{2}} \vartheta_{\alpha} \wedge \vartheta_{\beta}-\frac{a_{0}}{2 \ell^{2}} \eta_{\alpha \beta}\right) \end{aligned}

\stackrel{( \pm)}{L}_{\|}=\mp \frac{i}{4} \ell^{2} \stackrel{( \pm)}{\Pi}^{\alpha} \wedge \stackrel{( \pm)}{\Pi}_{\alpha}, \quad \stackrel{\overline{+)}}{L}_{\|}=\stackrel{(-)}{L}_{\|}

\stackrel{( \pm)}{D} \vartheta^{\alpha}= \pm \frac{i}{2} \ell^{2} g^{\alpha \beta} \stackrel{( \pm)}{\Pi}_{\beta},

\stackrel{( \pm)}{L}_{\|}=-\frac{1}{2}\left[\vartheta^{\alpha} \wedge \stackrel{( \pm)( \pm)}{D}_{\alpha}+d\left(\vartheta^{\alpha} \wedge \stackrel{( \pm)}{\Pi}_{\alpha}\right)\right]=-\frac{1}{2}\left(\stackrel{( \pm)}{D} \vartheta^{\alpha} \wedge \stackrel{( \pm)}{\Pi}_{\alpha},\right.

D D \vartheta^{\alpha}=R_{\beta}{ }^{\alpha} \wedge \vartheta^{\beta}=0, \stackrel{( \pm)}{D} \stackrel{( \pm)}{D} \stackrel{( \pm)}{\Pi}_{\alpha}=-\stackrel{( \pm)}{R}_{\alpha}{ }^{\beta} \wedge \stackrel{( \pm)}{\Pi}_{\beta}= \pm \frac{i}{2} \ell^{2}\left(e_{\alpha} \downharpoonleft \stackrel{( \pm)}{\Pi^{\beta}}\right) \wedge \Sigma_{\beta} \cong 0 .

\stackrel{( \pm)}{L}_{\mathrm{YM}}=\mp(1 / 4) \operatorname{Tr}(\stackrel{( \pm)}{F} \wedge \stackrel{( \pm)}{F})

\stackrel{( \pm)( \pm)}{D} \Pi_{\alpha}=0,

\stackrel{( \pm)( \pm)}{D} \Pi_{\alpha}:=D \stackrel{( \pm)}{\Pi}_{\alpha} \pm \frac{i}{2} \ell^{2}\left(e_{\alpha} \downharpoonleft \stackrel{(\mp)}{\Pi}\right) \wedge \stackrel{( \pm)}{\Pi}_{\beta},

\stackrel{( \pm)}{\Gamma}_{\alpha}{ }^{\beta}:=\Gamma_{\alpha}{ }^{\beta}+i \stackrel{(\mp)}{A^{\beta}}, \quad \stackrel{( \pm)}{A^{\beta}}{ }^{\beta}:= \pm \frac{1}{2} \ell^{2} e_{\alpha} \downharpoonleft \stackrel{( \pm)}{\Pi^{\beta}},

\check{E}_{\alpha}:=E_{\alpha}+\Gamma_{\alpha}{ }^{\beta} \wedge H_{\beta}+\Sigma_{\alpha} \cong d H_{\alpha},

H_{\alpha}^{\|}=\frac{1}{\ell^{2}}\left[\vartheta^{\beta} \wedge{ }^{*}\left(d \vartheta_{\beta} \wedge \vartheta_{\alpha}\right)-\frac{1}{2} \vartheta_{\alpha} \wedge{ }^{*}\left(d \vartheta^{\beta} \wedge \vartheta_{\beta}\right)\right]=\frac{1}{2 \ell^{2}} \Gamma^{\{ \} \beta \gamma} \wedge \eta_{\alpha \beta \gamma}

\stackrel{( \pm)}{S}_{\alpha}:=\stackrel{( \pm)}{\Gamma}_{\alpha} \beta \wedge \stackrel{( \pm)}{\Pi}_{\beta}+\Sigma_{\alpha} \cong d \stackrel{( \pm)}{\Pi}_{\alpha}, \quad \stackrel{(\mathrm{A}}{S}_{\alpha}=\stackrel{(\mp)}{S}_{\alpha}

n:=\partial_{t}-N^{A} \partial / \partial x^{A}, \quad A=1,2,3,

\left.{ }^{\perp} \Psi:=d t \wedge \Psi_{\perp} ; \quad \Psi_{\perp}:=n\right\rfloor \Psi,

\left.\underline{\Psi}:=n\rfloor(d t \wedge \Psi)=\left(1-{ }^{\perp}\right) \Psi, \quad n\right\rfloor \underline{\Psi} \equiv 0 .

\mathscr{H}=\mathscr{K}-\dot{\Psi} \wedge\left(\frac{\partial \mathscr{K}}{\partial \dot{\Psi}}\right),

\mathscr{H}=n\rfloor L-\ell_{n} \underline{\Psi} \wedge\left(\frac{\partial L}{\partial d \Psi}\right) .

\begin{aligned} \Sigma_{\alpha}:=\frac{\delta L}{\delta \vartheta^{\alpha}}= & \left.\left.e_{\alpha}\right\rfloor L-\left(e_{\alpha} \downharpoonleft D \Psi\right) \wedge \frac{\partial L}{\partial D \Psi}-\left(e_{\alpha}\right\rfloor \Psi\right) \wedge \frac{\partial L}{\partial \Psi} \\ & -\left(e_{\alpha} \downharpoonleft T^{\beta}\right) \wedge \frac{\partial L}{\partial T^{\beta}}+D \frac{\partial L}{\partial T^{\alpha}}-\left(e_{\alpha} \downharpoonleft R_{\beta}^{\gamma}\right) \wedge \frac{\partial L}{\partial R_{\beta}^{\gamma}} . \end{aligned}

\left.\left.n^{\alpha} \Sigma_{\alpha}=n \downharpoonleft L-(n\rfloor D \Psi\right) \wedge \frac{\partial L}{\partial D \Psi}-(n\rfloor \Psi\right) \wedge \frac{\partial L}{\partial \Psi} .

\begin{aligned} n^{\alpha} \Sigma_{\alpha} & \left.\left.=n \downharpoonleft L-\left(\mathrm{t}_{n} \Psi\right) \wedge \frac{\partial L}{\partial D \Psi}-(n\rfloor \Psi\right) \wedge \frac{\delta L}{\delta \Psi}+D[(n\rfloor \Psi) \wedge \frac{\partial L}{\partial D \Psi}\right] \\ & \left.\cong n\rfloor L-\left(\ell_{n} \Psi+\Gamma_{\perp}^{\alpha \beta} \rho\left(I_{\alpha \beta}\right) \Psi\right) \wedge \frac{\partial L}{\partial D \Psi}+D[(n\rfloor \Psi) \wedge \frac{\partial L}{\partial D \Psi}\right] . \end{aligned}

\left.\mathscr{H} \cong n^{\alpha} \underline{\Sigma}_{\alpha}+\Gamma_{\perp}^{\alpha \beta} \rho\left(I_{\alpha \beta}\right) \underline{\Psi} \wedge \frac{\partial L}{\partial D \Psi}-\underline{d}[(n\rfloor \Psi) \wedge \frac{\partial L}{\partial D \Psi}\right] .

\Sigma_{\alpha}:=\frac{\partial L}{\partial \vartheta^{\alpha}}, \quad \tau_{\alpha \beta}:=\frac{\partial L}{\partial \Gamma^{\alpha \beta}}=\rho\left(I_{\alpha \beta}\right) \Psi \wedge \frac{\partial L}{\partial D \Psi}

\begin{aligned} \mathscr{H} \cong & n^{\alpha} \frac{\delta L}{\frac{\delta \vartheta^{\alpha}}{}}+\Gamma_{\perp}^{\alpha \beta} \frac{\delta L}{\frac{\delta \Gamma^{\alpha \beta}}{\partial L}} \\ & \left.-\underline{d}[(n\rfloor \Psi) \wedge \frac{\partial L}{\partial D \Psi}+n^{\alpha} \frac{\partial L}{\partial T^{\alpha}}+\Gamma_{\perp}^{\alpha \beta} \frac{\partial L}{\partial R^{\alpha \beta}}\right] . \end{aligned}

\mathscr{H} \cong n^{\alpha} \mathscr{G}_{\alpha}+\Gamma_{\perp}{ }^{\alpha \beta} \mathscr{G}_{\alpha \beta},

\begin{aligned} \left\{\mathscr{G}_{\alpha}(t, \mathbf{x}), \mathscr{G}_{\beta}(t, \mathbf{y})\right\} & =\left(-T_{\alpha \beta}{ }^{\gamma} \mathscr{G}_{\gamma}+R_{\alpha \beta \gamma}{ }^{\delta} \mathscr{G}^{\gamma}{ }_{\delta}\right) \cdot \delta(\mathbf{x}-\mathbf{y}), \\ \left\{\mathscr{G}^{\alpha}{ }_{\beta}(t, \mathbf{x}), \mathscr{G}_{\gamma}(t, \mathbf{y})\right\} & =\delta_{\beta}^{\gamma} \mathscr{G}_{\beta} \delta(\mathbf{x}-\mathbf{y}), \\ \left\{\mathscr{G}^{\alpha}{ }_{\beta}(t, \mathbf{x}), \mathscr{G}^{\gamma}{ }_{\delta}(t, \mathbf{y})\right\} & =\left(\delta_{\delta}^{\alpha} \mathscr{G}^{\gamma}{ }_{\beta}-\delta_{\beta}^{\gamma} \mathscr{G}^{\alpha}{ }_{\delta}\right) \cdot \delta(\mathbf{x}-\mathbf{y}) . \end{aligned}

\begin{aligned} {\left[D_{\alpha}, D_{\beta}\right] } & =-T_{\alpha \beta}{ }^{\gamma}(x) D_{\gamma}+R_{\alpha \beta \gamma}{ }^{\delta}(x) L^{\gamma}{ }_{\delta}, \\ {\left[L^{\alpha}{ }_{\beta}, D_{\gamma}\right] } & =\delta_{\gamma}^{\alpha} D_{\beta}, \\ {\left[L^{\alpha}{ }_{\beta}, L^{\gamma}{ }_{\delta}\right] } & =\delta_{\delta}^{\alpha} L^{\gamma}{ }_{\beta}-\delta_{\beta}^{\gamma} L^{\alpha}{ }_{\delta}, \end{aligned}

\stackrel{( \pm)}{\mathscr{G}}_{\alpha}:=\stackrel{( \pm)}{\underline{D}}^{( \pm)} \underline{\Pi}_{\alpha}=\stackrel{( \pm)}{\underline{D}} \stackrel{*}{ }_{\underline{A}_{\alpha}}^{( \pm)}=0,

\mathscr{H}_{\|}=\stackrel{( \pm)}{\underline{D}} \stackrel{*}{-\mathscr{A}}_{\hat{0}}^{( \pm)}=0,

\begin{array}{ll} q: & \underline{*}^{\vartheta^{\beta}} \Psi_{\|}(\vartheta)=\underline{*}^{\prime} \underline{\vartheta}^{\beta} \Psi_{\|}(\vartheta), \\ \stackrel{ \pm}{p}: & \stackrel{*}{+}_{\underline{\Pi}} \Psi_{\|}(\vartheta)=-i \ell^{2} \frac{\delta}{\delta \underline{*}_{\alpha}} \Psi_{\|}(\vartheta), \end{array}

\Psi_{\|}(\vartheta)=\exp \left(-\int_{M_{3}} \stackrel{( \pm)}{C}_{\mathrm{TT}}\right)=\exp \left(\frac{-1}{2 \ell^{2}} \int_{M_{3}}\left[\stackrel{*}{+}_{\underline{\vartheta}_{B}} \wedge \stackrel{*}{T}_{\bar{T}}^{B}\right]\right)

-\ell^{2} \frac{\delta}{\delta \underline{*}_{\underline{0}}} \Psi_{\|}(\vartheta)=\stackrel{*}{\underline{T}}^{( \pm)}{ }^{B} \Psi_{\|}(\vartheta) .

-\ell^{2} \underline{D}^{( \pm)}{ }^{*}\left(\frac{\delta}{\delta \underline{*}_{B}}\right)=\underline{D}^{( \pm)} \underline{T}^{B} \equiv \underline{R}_{C}^{( \pm)}{ }^{B} \stackrel{( \pm)}{\vartheta}^{C}=0,

\stackrel{( \pm)( \pm)}{\underline{D}}^{B} \underline{T}^{B}=0

\Psi_{\Lambda}(A)=\exp \left(\frac{3}{\Lambda} \int_{M_{3}} \stackrel{( \pm)}{C}_{\mathrm{RR}}( \}\right),

d s^{2}=h^{2} d r^{2}+f^{2}\left[d \psi^{2}+\sin ^{2} \psi\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)\right],

T^{A}=\frac{1}{f}\left(d f \wedge \vartheta^{A}-2 \eta^{0 A \beta \gamma} \vartheta_{\beta} \wedge \vartheta_{\gamma}\right)=-\frac{2}{f} \eta^{A}{ }_{\mu \nu} \vartheta^{\mu} \wedge \vartheta^{\nu}= \pm{ }^{*} T^{A}, \quad T^{\hat{0}}=0,

\ell^{2} \underline{C}_{\mathrm{TT}}=\frac{1}{2} \underline{\vartheta}^{\alpha} \wedge d \underline{\vartheta}_{\alpha}=\frac{1}{2} \underline{\vartheta}^{A} \wedge d \underline{\vartheta}_{A}=3 \underline{\vartheta}^{\hat{1}} \wedge \underline{\vartheta}^{\hat{2}} \wedge \underline{\vartheta}^{\hat{3}}

n_{\mathrm{NY}}:=\int_{R^{4}} d C_{\mathrm{TT}}=\int_{S_{\infty}^{3}} \underline{C}_{\mathrm{TT}}=3 \operatorname{Vol}\left(S^{3}\right) k=6 \pi^{2} k .

f=\frac{a r^{2}}{r^{2}+c^{2}}

W(A, \gamma)=\operatorname{Tr} \mathbf{P} \exp \oint A=\operatorname{Tr} \mathbf{P} \exp \oint d s \dot{\gamma}^{i}(s) A_{i}^{J}(\gamma(s)) \lambda_{J},

\Psi(\gamma)=\int_{\mathscr{C} / \mathscr{G}} \mathscr{D} A W(A, \gamma) \Psi(A) .

\begin{aligned} \langle 0| W(\gamma)|0\rangle^{(1)} & =\int_{\mathscr{C} \mid \mathscr{G}} \mathscr{D} A e^{i C} W(A, \gamma) \\ & =\frac{1}{4 \pi} \oint d s \oint d t \dot{\gamma}^{\mu}(s) \dot{\gamma}^{\nu}(t) \eta_{0 \mu \nu \kappa} \frac{\gamma^{\kappa}(s)-\gamma^{\kappa}(t)}{|\gamma(s)-\gamma(t)|^{3}} \end{aligned}

\langle 0| W(\gamma)|0\rangle e^{\Lambda G(\gamma)} V(\gamma),

\int T=\int d \sigma=2 \pi N

W^{N}:=S^{3} \#^{N}\left(S^{1} \times S^{2}\right) .

L:=L_{\mathrm{EC}}+L_{\theta}+L_{\mathrm{D}},

L_{\theta}=\theta_{\mathrm{T}} d C_{\mathrm{TT}}+\theta_{\mathrm{L}} d C_{\mathrm{RR}}

H_{\alpha}:=-\frac{\partial L}{\partial T^{\alpha}}=-\frac{\theta_{\mathrm{T}}}{\ell^{2}} T_{\alpha},

H_{\alpha \beta}:=-\frac{\partial L}{\partial R^{\alpha \beta}}=\frac{1}{2 \kappa} \eta_{\alpha \beta}-\frac{\theta_{\mathrm{T}}}{2 \ell^{2}} \vartheta_{\alpha} \wedge \vartheta_{\beta}-\theta_{\mathrm{L}} R_{\alpha \beta}

D \theta_{\mathrm{T}} \wedge T_{\alpha}+\theta_{\mathrm{T}} D T_{\alpha}+\ell^{2} E_{\alpha}=-\ell^{2} \Sigma_{\alpha} .

\begin{aligned} \frac{1}{2 \kappa} T^{\gamma} \wedge \eta_{\alpha \beta \gamma} & -\frac{D \theta_{\mathrm{T}}}{2 \ell^{2}} \wedge \vartheta_{\alpha} \wedge \vartheta_{\beta}+D \theta_{\mathrm{L}} \wedge R_{\alpha \beta}=\vartheta_{[\alpha} \wedge \mu_{\beta]} \\ & =\frac{1}{4} \vartheta_{\alpha} \wedge \vartheta_{\beta} \wedge^{*} j_{5} . \end{aligned}

C_{\mathrm{TT}} \cong \frac{\kappa}{4 \ell^{2}} j_{5}

\mu_{\alpha}=\frac{1}{4} \vartheta_{\alpha} \wedge{ }^{*} j_{5},

T^{\gamma} \wedge \eta_{\gamma \beta}+\kappa D \theta_{\mathrm{L}} \wedge D T_{\beta}=0 .

\kappa D \theta_{\mathrm{L}} \wedge D T_{\beta}=-T \wedge \eta_{\beta}=D \eta_{\beta},

\kappa d \theta_{\mathrm{L}} \wedge T_{\beta}=-\eta_{\beta}

\kappa \ell^{2} d \theta_{\mathrm{L}} \wedge C_{\mathrm{TT}}=2 \eta .

D T^{\gamma} \wedge \eta_{\alpha \beta \gamma}+\kappa D \theta_{\mathrm{T}} \wedge T_{[\alpha} \wedge \vartheta_{\beta]} / \ell^{2}=2 \kappa\left(T_{[\alpha} \wedge \mu_{\beta]}-\vartheta_{[\alpha} \wedge D \mu_{\beta]}\right) .

W=\int\left[L\left(\vartheta^{\alpha}, T^{\alpha}, R^{\alpha \beta}, \Psi, D \Psi\right)+L_{g}\left(\vartheta^{\alpha}, T^{\alpha}, R^{\alpha \beta}\right)\right] .

\begin{gathered} \frac{\delta L}{\delta \Psi}=\frac{\partial L}{\partial \Psi}-(-1)^{p} D \frac{\partial L}{\partial D \Psi}=0, \quad \text { (MATTER) } \\ D H_{\alpha}-E_{\alpha}=\Sigma_{\alpha}, \quad \text { (FIRST) } \\ D H_{\alpha \beta}-E_{\alpha \beta}=\tau_{\alpha \beta} . \quad \text { (SECOND) } \end{gathered}

H_{\alpha}:=-\frac{\partial L_{g}}{\partial d \vartheta^{\alpha}}=-\frac{\partial L_{g}}{\partial T^{\alpha}}, \quad \text { and } \quad H_{\alpha \beta}:=-\frac{\partial L_{g}}{\partial d \Gamma^{\alpha \beta}}=-\frac{\partial L_{g}}{\partial R^{\alpha \beta}} .

\Sigma_{\alpha}:=\frac{\delta L}{\delta \vartheta^{\alpha}}=\frac{\partial L}{\partial \vartheta^{\alpha}}+D \frac{\partial L}{\partial T^{\alpha}}

\tau_{\alpha \beta}:=\frac{\delta L}{\delta \Gamma^{\alpha \beta}}=\rho\left(I_{\alpha \beta}\right) \Psi \wedge \frac{\partial L}{\partial(D \Psi)}+\vartheta_{[\alpha} \wedge \frac{\partial L}{\partial T^{\beta]}}+D \frac{\partial L}{\partial R^{\alpha \beta}},

E_{\alpha}:=\frac{\partial L_{g}}{\partial \vartheta^{\alpha}}=e_{\alpha} \downharpoonleft L_{g}+\left(e_{\alpha} \downharpoonleft T^{\beta}\right) \wedge H_{\beta}+\left(e_{\alpha} \downharpoonleft R^{\beta \gamma}\right) \wedge H_{\beta \gamma},

E_{\alpha \beta}:=-\vartheta_{[\alpha} \wedge H_{\beta]}

\vartheta^{\alpha} \wedge E_{\alpha}=4 L_{g}+2 T^{\alpha} \wedge H_{\alpha}+2 R^{\alpha \beta} \wedge H_{\alpha \beta} .

\left.D \Sigma_{\alpha} \cong\left(e_{\alpha} \downharpoonleft T^{\gamma}\right) \wedge \Sigma_{\gamma}+\left(e_{\alpha}\right\rfloor R^{\gamma \delta}\right) \wedge \tau_{\gamma \delta}

D \tau_{\alpha \beta}+\vartheta_{[\alpha} \wedge \Sigma_{\beta]} \cong 0,

L_{\mathrm{qPG}}=-\frac{1}{2} T^{\alpha} \wedge H_{\alpha}-\frac{1}{2} R^{\alpha \beta} \wedge\left(H_{\alpha \beta}-\frac{a_{0}}{2 \ell^{2}} \eta_{\alpha \beta}\right),

H_{\alpha}=-\frac{1}{\ell^{2}} *\left(\sum_{i=1}^{3} a_{i}{ }^{(i)} T_{\alpha}\right),

H_{\alpha \beta}=-\frac{a_{0}}{2 \ell^{2}} \eta_{\alpha \beta}-\frac{1}{\kappa} *\left(\sum_{j=1}^{6} b_{j}{ }^{(j)} R_{\alpha \beta}\right) .

d s^{2}=-\left(1-\frac{2 M}{r}+\frac{Q^{2}}{r^{2}}\right) d t^{2}+\left(1-\frac{2 M}{r}+\frac{Q^{2}}{r^{2}}\right)^{-1} d r^{2}+r^{2} d \Omega^{2},

\begin{aligned} \stackrel{\theta}{L} & =\frac{1}{2 \ell}\left[T^{\alpha} \wedge{ }^{*} T_{\alpha}+\theta_{\mathrm{T}}\left(T^{\alpha} \wedge T_{\alpha}+R_{\alpha \beta} \wedge \vartheta^{\alpha} \wedge \vartheta^{\beta}\right)\right] \\ & +\frac{1}{2}\left[R^{\alpha \beta} \wedge{ }^{*} R_{\alpha \beta}+\theta_{\mathrm{L}} R^{\alpha \beta} \wedge R_{\alpha \beta}\right] . \end{aligned}

\eta={ }^{*} 1=\frac{1}{4!} \eta_{\alpha \beta \gamma \delta} \vartheta^{\alpha} \wedge \vartheta^{\beta} \wedge \vartheta^{\gamma} \wedge \vartheta^{\delta},

p=\frac{1}{8 \pi^{2}} \int R_{\alpha}{ }^{\beta} \wedge R_{\beta}{ }^{\alpha} .

{ }^{*} \Omega^{g(\star)}:=\frac{1}{4}{ }^{*} R_{\alpha \beta}^{(\star) c d} L_{c d} \otimes \vartheta^{\alpha} \wedge \vartheta^{\beta},

\Omega^{g}=\stackrel{+}{\Omega}^{g}+\bar{\Omega}^{g}

\stackrel{+}{\Omega}^{g}:=\frac{1}{2}\left(\Omega^{g} \pm{ }^{*} \Omega^{g(\star)}\right),

C_{. . c d}^{a b}=R_{. . c d}^{a b}-2 R_{.[c}^{[a \cdot} \delta_{d]}^{b]}+\frac{R}{6} \delta_{[c}^{a} \delta_{d]}^{b} .

\[ L_{a b}^{(\star)}:=\frac{1}{2} \varepsilon_{a b c d} L^{c d}=L_{a b} . \] }

\Omega^{g}=\Omega_{\mathrm{C}}+\bar{\Omega}^{g}-\frac{1}{6} \vartheta \wedge \vartheta R .

\bar{\Omega}^{g}=\vartheta \wedge^{*}\left(\vartheta \wedge \Omega^{g}\right)-\frac{1}{4} \vartheta \wedge \vartheta R,

\bar{R}_{. . c d}^{a b}=2 R_{.[c}^{[a} \delta_{d]}^{b]} ; \quad R_{. b}^{a}:=R_{. b}^{a}-\frac{1}{4} \delta_{b}^{a} R,

\operatorname{Tr}\left(\vartheta \wedge \vartheta \wedge \Omega_{\mathrm{C}}\right)=\operatorname{Tr}\left(\vartheta \wedge \vartheta \wedge \bar{\Omega}^{g}\right)=0

\mathfrak{s o}(4) \approx \mathfrak{s o}(3) \times \mathfrak{s o}(3) \approx \mathfrak{s u}(2) \times \mathfrak{s u}(2) .

\Omega_{\mathrm{C}}=\stackrel{(+)}{\Omega}_{\mathrm{C}}+\stackrel{(-)}{\Omega}_{\mathrm{C}}, \quad \stackrel{( \pm)}{\Omega}_{\mathrm{C}}:=\frac{1}{2}\left(\Omega_{\mathrm{C}} \pm \Omega_{\mathrm{C}}^{(*)}\right)

g_{i j} \rightarrow \bar{g}_{i j}=\phi^{L} g_{i j},

\stackrel{(\mp)}{\Omega}_{\mathrm{C}}=0

L_{\mathrm{SKY}}=\operatorname{Tr}\left(\Omega^{g} \wedge^{*} \Omega^{g}\right)=-\frac{1}{2} R^{\alpha \beta} \wedge{ }^{*} R_{\alpha \beta},

D^{*} \Omega^{g}=0,

D^{*} R_{\alpha \beta}=0,

\left.\left.E_{\alpha}:=\partial L_{\mathrm{SKY}} / \partial \vartheta^{\alpha}=\frac{1}{2}\left(e_{\alpha}\right\rfloor R^{\mu \nu} \wedge{ }^{*} R_{\mu \nu}-R^{\mu \nu} \wedge e_{\alpha}\right\rfloor{ }^{*} R_{\mu \nu}\right)=0

G_{\alpha}^{\{ \}}:=\frac{1}{2} R^{\{ \} \beta \gamma} \wedge \eta_{\alpha \beta \gamma}, \quad G_{i j}:=\operatorname{Ric}_{i j}^{\{ \}}-\frac{1}{2} g_{i j},

\begin{aligned} L_{\mathrm{SKY}}^{(\star)} & =-\frac{1}{2} R^{\alpha \beta} \wedge{ }^{*} R_{\alpha \beta}-\frac{(-1)^{s}}{2} R^{\alpha \beta} \wedge R_{\alpha \beta}^{(\star)} \\ & =-\frac{1}{4}\left(R_{\alpha \beta}+{ }^{*} R_{\alpha \beta}^{(\star)}\right) \wedge{ }^{*}\left(R^{\alpha \beta}+{ }^{*} R^{\alpha \beta(\star)}\right) \\ & =\frac{1}{2} \operatorname{Tr}\left(\bar{\Omega}^{g} \wedge{ }^{\star} \bar{\Omega}^{g}\right) . \end{aligned}

R_{\alpha \beta}=-{ }^{*} R_{\alpha \beta}^{(\star)},

R_{\alpha \beta}={ }^{*} R_{\alpha \beta}^{(\star)},

{ }^{*} \Omega^{g(*)}=\zeta \Omega^{g},

\bar{\Omega}^{\{ \}}=0 \Longleftrightarrow \Omega_{\mathrm{C}}^{\{ \}}=0 \text { and } R^{\{ \}}=0 .

\stackrel{十}{\Omega}^{(t)}=0 \Longleftrightarrow R_{i j}^{\{ \}}=0 \Longleftrightarrow R_{i j}^{\{ \}}=\Lambda^{\prime} g_{i j} .

\begin{aligned} & \int \mathscr{D} \Gamma \exp \left[\int L_{\mathrm{SKY}}^{(\star)} / \hbar\right] \\ = & \int \mathscr{D} \Gamma \exp \left[-\int\left(R_{\alpha \beta}+{ }^{*} R_{\alpha \beta}^{(\star)}\right) \wedge{ }^{*}\left(R^{\alpha \beta}+{ }^{*} R^{\alpha \beta(\star)}\right) / 4 \hbar\right] \end{aligned}

R_{\alpha \beta} \equiv C_{\alpha \beta}+\frac{1}{2} R i^{\tau}{ }_{\alpha \beta}-\frac{R}{12} \vartheta_{\alpha} \wedge \vartheta_{\beta},

{ }^{*} R_{\alpha \beta}^{(\star)}=-C_{\alpha \beta}+\frac{1}{2} R i^{\tau}{ }_{\alpha \beta}+\frac{R}{12} \vartheta_{\alpha} \wedge \vartheta_{\beta} .

C_{\alpha \beta}=0 \quad \text { and } \quad R=0,

g_{i j}=\Omega^{2} o_{i j}^{\mathrm{Mink}}

\operatorname{Ric}_{\alpha \beta}:=\operatorname{Ric}_{\alpha \beta}-\frac{1}{4} \operatorname{Rg}_{\alpha \beta}=0 .

G_{\alpha}-\Lambda \eta_{\alpha}=0

\square^{c}:=\square-\frac{1}{6} R, \quad \square:=(1 / \sqrt{g}) \partial_{i}\left(\sqrt{g} g^{i j} \partial_{j}\right) .

\square^{c} g_{i j}=\square^{c}\left(\Omega^{2} o_{i j}^{\text {Mink }}\right)=\Omega^{6} \square^{c} o_{i j}^{\text {Mink }}=0

\left(\square-\frac{2}{3} \Lambda\right) h_{i j}=0

\partial^{i} h_{i j}-\partial_{j} h=0

\Delta m=\sqrt{2 \Lambda / 3}>0

g_{i j} \rightarrow \widetilde{g}_{i j}=g_{i j}+a \operatorname{Ric}_{i j}+b g_{i j} \operatorname{Ric}_{k}^{k}

\begin{aligned} \vartheta^{\alpha} & \left.\rightarrow \widetilde{\vartheta}^{\alpha}:=\vartheta^{\alpha}+\ell^{2} e_{\beta}\right\rfloor^{*} H^{\alpha \beta}, \\ \Gamma_{\alpha}{ }^{\beta} & \left.\rightarrow \widetilde{\Gamma}_{\alpha}{ }^{\beta}:=\Gamma_{\alpha}{ }^{\beta}+\ell^{2} e_{\alpha}\right\rfloor^{*} H^{\beta} . \end{aligned}

\left.\left.\left.\eta_{\alpha \beta} \quad \rightarrow \quad \tilde{\eta}_{\alpha \beta}=\eta_{\alpha \beta}+\ell^{2} \eta_{\alpha \beta \gamma} \wedge e_{\mu}\right\rfloor^{*} H^{\mu \gamma}+\frac{\ell^{4}}{2} \eta_{\alpha \beta \gamma \delta}\left(e_{\mu}\right\rfloor^{*} H^{\gamma \mu}\right) \wedge\left(e_{\nu}\right\rfloor^{*} H^{\delta \nu}\right) .

L_{\mathrm{EC}}=-\frac{1}{2 \kappa^{2}} R^{\alpha \beta} \wedge \eta_{\alpha \beta}

\widetilde{L}=-\sum_{k=0}^{K}(1 / 2 k) R^{\alpha \beta} \wedge \stackrel{(2 k)}{H}_{\alpha \beta}+L_{\theta},

\begin{gathered} \left.\left.-\widetilde{E}_{\alpha}:=-e_{\alpha}\right\rfloor \widetilde{L}-\left(e_{\alpha}\right\rfloor R^{\beta \gamma}\right) \wedge \widetilde{H}_{\beta \gamma}=\Sigma_{\alpha}, \\ D \widetilde{H}_{\alpha \beta}=\tau_{\alpha \beta} . \end{gathered}

\widetilde{L} \rightarrow L=-\frac{1}{2}\left(R^{\alpha \beta} \wedge \widetilde{H}_{\alpha \beta}-\widetilde{L}\right)

H_{\alpha \beta}:=-\frac{\partial L}{\partial R^{\alpha \beta}}=\widetilde{H}_{\alpha \beta}+\frac{1}{2} R^{\mu \nu} \wedge\left(\partial \widetilde{H}_{\mu \nu} / \partial R^{\alpha \beta}\right)

\widetilde{H}_{\alpha \beta \mu \nu}:=\frac{\partial^{2} \widetilde{L}}{\partial R^{\mu \nu} \partial R^{\alpha \beta}}=-\frac{\partial \widetilde{H}_{\mu \nu}}{\partial R^{\alpha \beta}}

\widetilde{\eta}_{\alpha \beta}:=\frac{\theta_{\mathrm{T}}^{*}}{2} \eta_{\alpha \beta}-\frac{\theta_{\mathrm{T}}}{2} \vartheta_{\alpha} \wedge \vartheta_{\beta}-\ell^{2} \widetilde{H}_{\alpha \beta}=0

{ }^{*}\left(\vartheta^{\alpha} \wedge \vartheta^{\beta} \wedge \vartheta^{\gamma} \wedge \vartheta^{\delta}\right)=\eta^{\alpha \beta \gamma \delta}, \text { where } \eta_{\alpha \beta \gamma \delta}:=+\delta_{\alpha \beta \gamma \delta}^{0123},

\eta:=\eta_{\alpha \beta \gamma \delta} \vartheta^{\alpha} \wedge \vartheta^{\beta} \wedge \vartheta^{\gamma} \wedge \vartheta^{\delta} / 4!

d s^{2}=o_{\alpha \beta} \vartheta^{\alpha} \otimes \vartheta^{\beta}=g_{i j} d x^{i} \otimes d x^{j}, \quad o_{\alpha \beta}:=\operatorname{diag}\{1, \underbrace{-1, \cdots,-1}_{\text {sig }}\},

{ }^{*} \Phi:=\frac{1}{(4-p!) p!} \sqrt{\operatorname{det} g_{i j}} \varepsilon^{i_{1} \cdots i_{p}}{ }_{j_{1} \cdots j_{4-p}} \Phi_{i_{1} \cdots i_{p}} d x^{j_{1}} \wedge \cdots \wedge d x^{j_{4-p}}

\gamma_{\alpha} \gamma_{\beta}+\gamma_{\beta} \gamma_{\alpha}=2 o_{\alpha \beta} \mathbf{1}_{4} .

{ }^{*} \sigma=\frac{1}{2} \sigma_{\alpha \beta}{ }^{*}\left(\vartheta^{\alpha} \wedge \vartheta^{\beta}\right)=\frac{1}{2} \sigma_{\alpha \beta} \eta^{\alpha \beta}=: \sigma^{(\star)}=i \gamma_{5} \sigma .

\gamma:=\gamma^{\alpha} \vartheta_{\alpha}

{ }^{*} \gamma=\gamma^{\alpha} \eta_{\alpha}=\frac{i}{6} \gamma_{5} \gamma \wedge \gamma \wedge \gamma,

\sigma:=\frac{1}{2} \sigma_{\alpha \beta} \vartheta^{\alpha} \wedge \vartheta^{\beta}=\frac{i}{2} \gamma \wedge \gamma, \quad{ }^{*} \sigma=\frac{1}{2} \sigma_{\alpha \beta} \eta^{\alpha \beta}=: \sigma^{(\star)}=i \gamma_{5} \sigma .

\sigma^{(\star)}:=i \gamma_{5} \sigma, \quad \sigma^{(\star \star)}=i^{2} \gamma_{5}^{2} \sigma=-\sigma .

\left[\gamma, \sigma^{(\star)}\right]=\gamma \wedge i \gamma_{5} \sigma-i \gamma_{5} \sigma \wedge \gamma=-i \gamma_{5}(\gamma \wedge \sigma+\sigma \wedge \gamma)=2 \gamma_{5} \gamma \wedge \gamma \wedge \gamma

\gamma_{5}:=(i / 4!)^{*}(\gamma \wedge \gamma \wedge \gamma \wedge \gamma)

\sigma_{ \pm}:=\left(\sigma \pm i^{*} \sigma\right) / 2=\frac{1}{2}\left(1 \mp \gamma_{5}\right) \sigma, \quad \text { with } \quad i^{*} \sigma_{ \pm}= \pm \sigma_{ \pm}

\gamma \rightarrow \gamma^{\theta}=e^{i \gamma^{5} \theta} \gamma e^{-i \gamma^{5} \theta}

D=d+[\Gamma, \quad]

\gamma:=\vartheta^{\alpha} \gamma_{\alpha}, \quad \Gamma:=\frac{i}{4} \Gamma^{\alpha \beta} \sigma_{\alpha \beta}=\Gamma^{\{ \}}-K,

\begin{aligned} \Theta:=D \gamma=d \gamma+[\Gamma, \gamma] & =\left(d \vartheta^{\alpha}+\Gamma_{\beta}^{\alpha} \wedge \vartheta^{\beta}\right) \gamma_{\alpha} \\ & =T^{\alpha} \gamma_{\alpha}=\frac{1}{2} T_{i j}^{\alpha} \gamma_{\alpha} d x^{i} \wedge d x^{j} \end{aligned}

\begin{aligned} \Omega^{g}:=d \Gamma+\Gamma \wedge \Gamma & =\frac{i}{4}\left(d \Gamma_{\alpha}{ }^{\beta}-\Gamma_{\alpha}{ }^{\gamma} \wedge \Gamma_{\gamma}{ }^{\beta}\right) \sigma^{\alpha}{ }_{\beta} \\ & =\frac{i}{4} R^{\alpha \beta} \sigma_{\alpha \beta}=\frac{i}{8} R_{i j}{ }^{\alpha \beta} \sigma_{\alpha \beta} d x^{i} \wedge d x^{j}, \end{aligned}

\Omega^{(\star)}=\frac{1}{4} R_{\alpha \beta}^{(\star)} \sigma^{\alpha \beta}, \quad R_{\alpha \beta}^{(\star)}:=\frac{1}{2} \eta_{\alpha \beta \gamma \delta} R^{\gamma \delta} .

D \sigma=\frac{i}{2}[\Theta, \gamma]

\left.\left.T:=\frac{1}{4} \operatorname{Tr}(\check{\gamma}\rfloor \Theta\right)=e_{\alpha}\right\rfloor T^{\alpha}, \quad \mathscr{A}:=\frac{1}{4}{ }^{*} \operatorname{Tr}(\gamma \wedge \Theta)={ }^{*}\left(\vartheta_{\alpha} \wedge T^{\alpha}\right) .

\widetilde{T}=T-3 d \varphi, \quad \mathscr{A}^{\theta}=\mathscr{A}-i d \theta .

d^{*} \gamma=0, \quad d^{*} \Gamma=0

D D \Phi=\left[\Omega^{g}, \Phi\right],

D \Theta \equiv\left[\Omega^{g}, \gamma\right], \quad D \Omega^{g} \equiv 0,

G:=G_{\alpha} \gamma^{\alpha}:=\frac{1}{2} R^{\mu \nu} \wedge \eta_{\mu \nu \lambda} \gamma^{\lambda}=-i \gamma_{5}\left(\Omega^{g} \wedge \gamma+\gamma \wedge \Omega^{g}\right)=i\left[\gamma, \gamma_{5} \Omega^{g}\right] .

D[\gamma, \Theta] \equiv 2 i\left[\sigma, \Omega^{g}\right], \quad D\left[\gamma, \Omega^{g}\right] \equiv\left[\Theta, \Omega^{g}\right], \quad D G \equiv i\left[\Theta, \gamma_{5} \Omega^{g}\right],

\begin{gathered} K^{(\star)}:=\eta_{\alpha \beta \gamma} \wedge K^{\beta \gamma} \gamma^{\alpha}, \\ \Omega^{g(\star)}:=\frac{i}{8} R_{\alpha \beta} \eta^{\alpha \beta \gamma \delta} \sigma_{\gamma \delta}=-\frac{1}{4} R^{\alpha \beta} \gamma_{5} \sigma_{\alpha \beta}=i \gamma_{5} \Omega^{g} . \end{gathered}

D R_{\alpha \beta}^{(\star)} \equiv 0,

\Theta^{ \pm}:=\frac{1}{2}\left(\Theta \pm{ }^{*} \Theta\right), \quad \Omega^{g \pm}:=\frac{1}{2}\left(\Omega^{g} \pm{ }^{*} \Omega^{g}\right), \quad \Omega^{( \pm)}:=\frac{1}{2}\left(\Omega^{g} \pm \Omega^{g(\star)}\right),

\begin{aligned} C_{\mathrm{TT}} & :=\frac{1}{8 \ell^{2}} \operatorname{Tr}(\gamma \wedge \Theta)=\frac{1}{2 \ell^{2}} \vartheta^{\alpha} \wedge T_{\alpha}=-\frac{(-1)^{\mathrm{sig}}}{2 \ell^{2}} * \mathscr{A}, \\ C_{\mathrm{RR}} & :=\operatorname{Tr}\left(\Gamma \wedge \Omega^{g}-\frac{1}{3} \Gamma \wedge \Gamma \wedge \Gamma\right), \end{aligned}

L_{\mathrm{NY}}:=d C_{\mathrm{TT}}=\frac{1}{2 \ell^{2}}\left(T^{\alpha} \wedge T_{\alpha}+R_{\alpha \beta} \wedge \vartheta^{\alpha} \wedge \vartheta^{\beta}\right),

\begin{aligned} L_{\mathrm{Pontr}}:= & d C_{\mathrm{RR}}=\frac{1}{2} R_{\alpha}{ }^{\beta} \wedge R_{\beta}{ }^{\alpha}=-\frac{1}{2} R_{\alpha \beta}^{\{ \}} \wedge R^{\{j \alpha \beta} \\ & -\frac{1}{12} d\left[{ }^{*} \mathscr{A} \wedge R^{\{ \}}-\frac{1}{3} \mathscr{A} \wedge d \mathscr{A}+\frac{1}{9}{ }^{*} \mathscr{A} \wedge^{*}\left(\mathscr{A} \wedge{ }^{*} \mathscr{A}\right)\right] . \end{aligned}

\begin{aligned} 2 \ell^{2} d C_{\mathrm{T} \mathrm{~T}^{*}}: & =2 d\left(\vartheta^{\alpha} \wedge{ }^{*} T_{\alpha}\right) \\ & =R^{\{f \alpha \beta} \wedge \eta_{\alpha \beta}-R^{\alpha \beta} \wedge \eta_{\alpha \beta} \\ & \left.\left.-T^{\alpha} *\left[T_{\alpha}-\vartheta_{\alpha} \wedge\left(e_{\beta}\right\rfloor T^{\beta}\right)-\frac{1}{2} e_{\alpha}\right\rfloor\left(T^{\beta} \wedge \vartheta_{\beta}\right)\right] \\ & =R^{\{f \alpha \beta} \wedge \eta_{\alpha \beta}-R^{\alpha \beta} \wedge \eta_{\alpha \beta}-T^{\alpha} \wedge *\left(-{ }^{(1)} T_{\alpha}+2^{(2)} T_{\alpha}+\frac{1}{2}{ }^{(3)} T_{\alpha}\right), \end{aligned}

\begin{aligned} L_{\text {Euler }}: & =(-1)^{\text {sig+1 }} \operatorname{Tr}\left\{\Omega \wedge \Omega^{(\star)}\right\}=\frac{(-1)^{\text {sig }}}{2} R^{\alpha \beta} \wedge R_{\alpha \beta}^{(\star)} \\ & =d C_{\mathrm{RR}(\star)}=\frac{(-1)^{\text {sig }}}{2} d\left(\Gamma_{\alpha \beta} \wedge R^{\alpha \beta(\star)}-\frac{1}{3} \Gamma_{\alpha}{ }^{\beta(\star)} \wedge \Gamma_{\beta}{ }^{\gamma} \wedge \Gamma_{\gamma}{ }^{\alpha}\right) \\ & \equiv \frac{1}{2} R_{\alpha \beta} \wedge{ }^{*} R^{\alpha \beta}-2 \operatorname{Ric}_{\alpha \beta} \wedge{ }^{*} \operatorname{Ric}^{\alpha \beta}+\frac{1}{2} \operatorname{Ric}_{\alpha}{ }^{\alpha} \wedge{ }^{*} \operatorname{Ric}_{\beta}{ }^{\beta} \\ & \equiv-L_{\mathrm{SKY}} \\ & -2\left(\operatorname{Ric}_{\alpha \beta} \wedge{ }^{*} \operatorname{Ric}^{\alpha \beta}-\frac{1}{4} \operatorname{Ric}_{\alpha}{ }^{\alpha} \wedge{ }^{*} \operatorname{Ric}_{\beta}{ }^{\beta}\right) \end{aligned}

\left.{ }^{*} R_{\alpha \beta}^{(\star)} \equiv(-1)^{\mathrm{sig}} R_{\alpha \beta}+e_{[\alpha}\right\rfloor G_{\beta]}+\frac{1}{4} R \eta_{\alpha \beta}+(-1)^{\mathrm{sig}+1} D_{[\alpha} T_{\beta]}

\left.\left.R:=e_{\beta}\right\rfloor e_{\alpha}\right\rfloor R^{\alpha \beta}, \quad{ }^{*} R^{\alpha \beta} \wedge \vartheta_{\alpha} \wedge \vartheta_{\beta} \equiv R^{\alpha \beta} \wedge \eta_{\alpha \beta} \equiv-R \eta,

{ }^{(6)} R^{\alpha \beta}=-\frac{1}{12} R \vartheta^{\alpha} \wedge \vartheta^{\beta}

\begin{aligned} G & :=\frac{1}{2} R^{\beta \gamma} \wedge \eta_{\alpha \beta \gamma} \gamma^{\alpha} \\ & \left.\left.=G^{(\beta)}+\frac{(-1)^{\mathrm{sig}}}{12}\left(e_{\alpha}\right\rfloor \mathscr{A} \wedge{ }^{*} \mathscr{A}-\frac{1}{3} \mathscr{A} \wedge e_{\alpha}\right\rfloor^{*} \mathscr{A}\right) \gamma^{\alpha}+\frac{(-1)^{\mathrm{sig}}}{6} \gamma \wedge d \mathscr{A} \end{aligned}

\begin{aligned} s \Gamma & =\Psi-D C \\ s C & =\Phi-\frac{1}{2}[C, C] \\ s \Omega^{g} & =-D \Psi-\left[C, \Omega^{g}\right] \\ s \Psi & =-D \Phi-[C, \Psi] \\ s \Phi & =-[C, \Phi] \end{aligned}

\widetilde{\Gamma}:=\Gamma \oplus C, \quad \widetilde{\Omega}^{g}:=\Omega^{g} \oplus \Psi \oplus \Phi .

\begin{aligned} (d \oplus s) \tilde{\Gamma}+\frac{1}{2}[\widetilde{\Gamma}, \widetilde{\Gamma}] & =\widetilde{\Omega}^{g} \\ (d \oplus s) \widetilde{\Omega}^{g}+\left[\widetilde{\Gamma}, \widetilde{\Omega}^{g}\right] & \equiv 0 \end{aligned}

\begin{array}{ll} s \bar{C}=\mathbb{B}, & s \mathbb{B}=0, \\ s \bar{\chi}=\beta, & s \beta=0, \\ s \bar{\Phi}=\bar{\eta}, & s \bar{\eta}=0 . \end{array}

s_{\mathrm{loc}}(\alpha+\lambda)=-d \lambda, \quad s_{\mathrm{loc}} \lambda=0

\left(d \oplus s_{\mathrm{loc}}\right)(\alpha+\lambda)=d \alpha .

\begin{aligned} (d \oplus s) \gamma+[\widetilde{\Gamma}, \gamma] & =\widetilde{\Theta} \\ (d \oplus s) \widetilde{\Theta}+[\widetilde{\Gamma}, \widetilde{\Theta}] & \equiv\left[\widetilde{\Omega}^{g}, \gamma\right] \end{aligned}

s \gamma=\psi-[C, \gamma]

s \sigma:=\frac{i}{2} s(\gamma \wedge \gamma)=\frac{i}{2}([\psi, \gamma]-[[C, \gamma], \gamma])=\frac{i}{2}[\psi, \gamma]-[C, \sigma] .

s \bar{c}=b, \quad s b=0 .

\left.\left.\left.\exp (\zeta\rfloor):=\mathbf{1}+\zeta\rfloor+\frac{1}{2!} \zeta\right\rfloor \zeta\right\rfloor+\frac{1}{3!} \zeta ل \zeta ل \zeta\right\rfloor+\cdots .

d \rightarrow \exp (\zeta\rfloor) d \exp (-\zeta\rfloor) .

d s^{2}=\frac{1}{4} \operatorname{Tr}\{\gamma \otimes \gamma\} \stackrel{*}{=} o_{\alpha \beta} D \xi^{\alpha} \otimes D \xi^{\beta}

\begin{aligned} L_{\mathrm{Pontr}} & :=d C_{\mathrm{RR}}=\operatorname{Tr}\left\{\Omega^{g} \wedge \Omega^{g}\right\}=\frac{1}{2} R_{\alpha}{ }^{\beta} \wedge R_{\beta}{ }^{\alpha} \\ & =\frac{1}{2} d\left(\Gamma_{\alpha}{ }^{\beta} \wedge R_{\beta}{ }^{\alpha}-\frac{1}{3} \Gamma_{\alpha}{ }^{\beta} \wedge \Gamma_{\beta}{ }^{\gamma} \wedge \Gamma_{\gamma}{ }^{\alpha}\right) \end{aligned}

s L_{\text {Pontr }}=-2 d \operatorname{Tr}\left\{\Psi \wedge \Omega^{g}\right\}=-d\left(\Psi_{\alpha}{ }^{\beta} \wedge R_{\beta}{ }^{\alpha}\right) .

\begin{aligned} L_{\mathrm{FP}} & :=-s \operatorname{Tr}\left\{\bar{\chi} \wedge^{*} \stackrel{ }{\Omega}^{g}+\bar{\Phi} D^{*} \Psi+\frac{1}{2} \rho \bar{\chi} \wedge^{*} \beta+\bar{C} d^{*} \Gamma+\frac{1}{2} \bar{C} \wedge{ }^{*} \mathbb{B}\right\} \\ & =\frac{1}{2} s\left[\bar{\chi}^{\alpha \beta} \wedge{ }^{*} R_{\alpha \beta}^{( \pm)}+\bar{\Phi}_{\alpha \beta} D^{*} \Psi^{\alpha \beta}+\frac{1}{2} \rho \bar{\chi}^{\alpha \beta} \wedge{ }^{*} b_{\alpha \beta}+\widetilde{\Psi}_{\mathrm{F}}\right] \end{aligned}

L_{\mathrm{TG}}=d C_{\mathrm{RR}}+L_{\mathrm{FP}}=d C_{\mathrm{RR}}+s\{\cdots\}

\begin{aligned} L_{\mathrm{FP}}= & -\operatorname{Tr}\left\{\beta \wedge{ }^{*} \bar{\Omega}^{g}-\bar{\chi} \wedge^{*} D \Psi^{( \pm)}-\bar{\chi} \wedge^{*}\left[C, \stackrel{ \pm}{\Omega}^{g}\right]\right. \\ & +\bar{\eta} D^{*} \Psi+\bar{\Phi} D^{*} D \Phi+\bar{\Phi}\left[\Psi,{ }^{*} \Psi\right]+\mathbb{B} d^{*} \Gamma \\ & \left.-\bar{C} d^{*} \Psi+\bar{C} d^{*}(D C)+\frac{1}{2} \rho \beta \wedge^{*} \beta+\frac{1}{2} \mathbb{B} \wedge{ }^{*} \mathbb{B}\right\} . \end{aligned}

\stackrel{+}{\Omega}^{g}:=\frac{1}{2}\left(\Omega^{g} \pm{ }^{*} \Omega^{g(\star)}\right)=0

\begin{aligned} L_{\mathrm{SKY}}^{(\star)} & =\frac{1}{2} \operatorname{Tr}\left(\stackrel{ \pm}{\Omega}^{g} \wedge{ }^{*} \stackrel{ }{\circ}^{g}\right) \\ & =-\frac{1}{2} R^{\alpha \beta} \wedge{ }^{*} R_{\alpha \beta} \mp \frac{(-1)^{\mathrm{sig}}}{2} R^{\alpha \beta} \wedge R_{\alpha \beta}^{(\star)} \\ & =-\frac{1}{4}\left(R_{\alpha \beta} \pm{ }^{*} R_{\alpha \beta}^{(\star)}\right) \wedge{ }^{*}\left(R^{\alpha \beta} \pm{ }^{*} R^{\alpha \beta(\star)}\right) . \end{aligned}

D^{*} \stackrel{ \pm}{\Omega}^{g}=0

\left.\left.E_{\alpha}=\frac{1}{2} \operatorname{Tr}\left(\stackrel{ \pm}{\Omega}^{g} \wedge e_{\alpha}\right\rfloor^{*} \stackrel{ \pm}{\Omega}^{g}-e_{\alpha}\right\rfloor^{ \pm} \stackrel{ }{\Omega}^{g} \wedge{ }^{*} \stackrel{ \pm}{\Omega}^{g}\right)=0,

R_{\alpha \beta}=\mp{ }^{*} R_{\alpha \beta}^{(\star)}

\left\langle d j_{5}\right\rangle=2 i^{*} m\left\langle\bar{\psi} \gamma_{5} \psi\right\rangle-\frac{1}{48 \pi^{2}}\left(R_{\alpha \beta}^{\{ \}} \wedge R^{\{\beta \alpha \beta}+\frac{1}{2} d \mathscr{A} \wedge d \mathscr{A}\right),

\Delta L_{\mathrm{FP}}=s \operatorname{Tr}\left\{\frac{\theta_{\mathrm{T}}^{*} i}{4 \ell^{2}} \bar{\chi} \wedge^{*} \sigma-\bar{c} d^{*} \gamma\right\}

\sigma:=\frac{i}{2} \gamma \wedge \gamma .

\stackrel{ \pm}{\Omega^{g}}=i \frac{\theta_{\mathrm{T}}^{*}}{4 \ell^{2}} \sigma

{ }^{*} \sigma^{(\star)}=(-1)^{\mathrm{sig}} \sigma,

\begin{aligned} \widetilde{L}_{\mathrm{SKY}}^{(\star)} & =\frac{1}{2} \operatorname{Tr}\left[\left(\mathcal{\Omega}^{g}-i \frac{\theta_{\mathrm{T}}^{*}}{4 \ell^{2}} \sigma\right) \wedge^{*}\left(\stackrel{t}{\Omega}^{g}-i \frac{\theta_{\mathrm{T}}^{*}}{4 \ell^{2}} \sigma\right)\right] \\ & =-\frac{\theta_{\mathrm{T}}^{*}}{2 \ell^{2}} R^{\alpha \beta} \wedge \eta_{\alpha \beta}+\theta_{\mathrm{T}}^{*} \Lambda_{\text {eff }} \eta-\frac{1}{2} R^{\alpha \beta} \wedge{ }^{*} R_{\alpha \beta} \mp \frac{(-1)^{\mathrm{sig}}}{2} R^{\alpha \beta} \wedge R_{\alpha \beta}^{(\star)} \end{aligned}

\Omega^{g(\theta)}:=\sqrt{2}\left(\Omega^{g} \sin \theta+{ }^{*} \Omega^{g(\star)} \cos \theta\right)

L_{\mathrm{qPG}}=\frac{\Lambda}{\ell^{2}} \eta-\frac{a_{0}}{4 \ell^{2}} R^{\alpha \beta} \wedge \eta_{\alpha \beta}-\frac{1}{2} T^{\alpha} \wedge H_{\alpha}-\frac{1}{2} R^{\alpha \beta} \wedge H_{\alpha \beta} .

H_{\alpha}:=-\partial L_{\mathrm{qPG}} / \partial T^{\alpha}=\frac{1}{\ell^{2}} *\left(\sum_{M=1}^{3} a_{(M)}{ }^{(M)} T_{\alpha}\right),

H_{\alpha \beta}:=-\partial L_{\mathrm{qPG}} / \partial R^{\alpha \beta}=\frac{a_{0}}{2 \ell^{2}} \eta_{\alpha \beta}+{ }^{*}\left(\sum_{N=1}^{6} b_{(N)}{ }^{(N)} R_{\alpha \beta}\right)

\begin{aligned} H_{\alpha \beta}(* *) & =\theta_{\mathrm{L}} R_{\alpha \beta}+\theta_{\mathrm{L}}^{\star} R_{\alpha \beta}^{(\star)}+\frac{\theta_{\mathrm{T}}^{*}}{2 \ell^{2}} \eta_{\alpha \beta}-\frac{\theta_{\mathrm{T}}}{2 \ell^{2}} \vartheta_{\alpha} \wedge \vartheta_{\beta} \\ & \simeq-\partial L_{\theta} / \partial R^{\alpha \beta} \end{aligned}

L_{\theta}:=\theta_{\mathrm{T}} d C_{\mathrm{TT}}+\theta_{\mathrm{T}}^{*} d C_{\mathrm{TT}^{*}}+\theta_{\mathrm{L}} d C_{\mathrm{RR}}+\theta_{\mathrm{L}}^{\star}(-1)^{\mathrm{sig}+1} d C_{\mathrm{RR}^{*}}=d C_{\theta} .

\begin{aligned} D H_{\alpha}-E_{\alpha} & =\Sigma_{\alpha}, \\ D H_{\alpha \beta}+\vartheta_{[\alpha} \wedge H_{\beta]} & =\tau_{\alpha \beta}, \end{aligned}

\left.E_{\alpha}:=\partial L / \partial \vartheta^{\alpha}=e_{\alpha}\right\rfloor L+\left(e_{\alpha} \downharpoonleft T^{\beta}\right) \wedge H_{\beta}+\left(e_{\alpha} \downharpoonleft R^{\beta \gamma}\right) \wedge H_{\beta \gamma}

\frac{\theta_{\mathrm{T}}^{*}}{2 \ell^{2}} \eta_{\alpha \beta \gamma} T^{\gamma}+\frac{\theta_{\mathrm{T}}}{\ell^{2}} \vartheta_{[\alpha} \wedge T_{\beta]}+\vartheta_{[\alpha} \wedge H_{\beta]}(* *)=\tau_{\alpha \beta} .

\begin{aligned} H_{\alpha}(* *) & =\frac{\theta_{\mathrm{T}}^{*}}{2 \ell^{2}} K^{\beta \gamma} \wedge \eta_{\alpha \beta \gamma}-\frac{\theta_{\mathrm{T}}}{\ell^{2}} T_{\alpha}=\frac{1}{\ell^{2}}\left[\theta_{\mathrm{T}}^{*} K_{\alpha}^{(\star)}-\theta_{\mathrm{T}} T_{\alpha}\right] \\ & \left.\left.=\frac{\theta_{\mathrm{T}}^{*}}{\ell^{2}} *\left[T_{\alpha}-\vartheta_{\alpha} \wedge\left(e_{\beta}\right\rfloor T^{\beta}\right)-\frac{1}{2} e_{\alpha}\right\rfloor\left(T^{\beta} \wedge \vartheta_{\beta}\right)\right]-\frac{\theta_{\mathrm{T}}}{\ell^{2}} T_{\alpha} \\ & \simeq-\partial L_{\theta} / \partial T^{\alpha} . \end{aligned}

\begin{aligned} L_{\mathrm{qPG}}(* *)= & \frac{\Lambda}{\ell^{2}} \eta-\frac{a_{0}}{4 \ell^{2}} R^{\alpha \beta} \wedge \eta_{\alpha \beta}-\frac{1}{2} T^{\alpha} \wedge H_{\alpha}(* *)-\frac{1}{2} R^{\alpha \beta} \wedge H_{\alpha \beta}(* *) \\ = & \frac{\Lambda}{\ell^{2}} \eta-\frac{\theta_{\mathrm{T}}^{*}+a_{0}}{4 \ell^{2}} R^{\alpha \beta} \wedge \eta_{\alpha \beta}-\frac{1}{2 \ell^{2}} T^{\alpha} \wedge\left[\theta_{\mathrm{T}}^{*} K_{\alpha}^{(\star)}-\theta_{\mathrm{T}} T_{\alpha}\right] \\ & -\frac{1}{2} R^{\alpha \beta} \wedge\left[\theta_{\mathrm{L}} R_{\alpha \beta}+\theta_{\mathrm{L}}^{\star} R_{\alpha \beta}^{(\star)}-\frac{\theta_{\mathrm{T}}}{2 \ell^{2}} \vartheta_{\alpha} \wedge \vartheta_{\beta}\right] . \end{aligned}

L_{\mathrm{eff}}:=L_{\mathrm{qPG}}(* *)-d C_{\theta}=-\frac{\theta_{\mathrm{T}}^{*}}{2 \ell^{2}} R^{\{ \} \alpha \beta} \wedge \eta_{\alpha \beta}+\frac{\theta_{\mathrm{T}}^{*} \Lambda_{\mathrm{eff}}}{\ell^{2}} \eta .

\left.\left.R:=e_{\beta}\right\rfloor e_{\alpha}\right\rfloor R^{\alpha \beta}=\frac{6\left(\theta_{\mathrm{T}}^{*}-a_{0}\right)}{\ell^{2}\left(\theta_{\mathrm{L}}^{\star}-b_{6}\right)}

\Lambda_{\mathrm{eff}}=\frac{1}{\theta_{\mathrm{T}}^{*}}\left[\Lambda-\frac{1}{4}\left(\theta_{\mathrm{T}}^{*}-a_{0}\right) R\right]=\frac{\Lambda}{\theta_{\mathrm{T}}^{*}}-\Lambda_{\theta}

-E_{\alpha}=\frac{\theta_{\mathrm{T}}^{*}}{2} R^{\{ \} \beta \gamma} \wedge \eta_{\alpha \beta \gamma}-\theta_{\mathrm{T}}^{*} \Lambda_{\mathrm{eff}} \eta_{\alpha}=\ell^{2} \overparen{\Sigma}_{\alpha} .

\widetilde{L}_{\mathrm{qPG}}=L_{\mathrm{qPG}}+\lambda_{\alpha} \wedge T^{\alpha} .

\left.\left.\left.2 D\left(e^{\beta}\right\rfloor D H_{\alpha \beta}-\frac{1}{4} \vartheta_{\alpha} \wedge e^{\gamma}\right\rfloor e^{\delta}\right\lrcorner D H_{\gamma \delta}\right)-E_{\alpha}=\Sigma_{\alpha}-D \mu_{\alpha},

G_{\alpha}^{\{ \}}-\Lambda_{\text {eff }} \eta_{\alpha}=\kappa_{\text {eff }} \overparen{\Sigma}_{\alpha} .

\widehat{\Sigma}_{\alpha}:=\Sigma_{\alpha}-D^{\{ \}} \mu_{\alpha},

\Lambda_{\theta}=\frac{3\left(\theta_{\mathrm{T}}^{*}-a_{0}\right)^{2}}{2 \ell^{2}\left(\theta_{\mathrm{L}}^{\star}-b_{6}\right) \theta_{\mathrm{T}}^{*}}

d s^{2}=-e^{v} d t^{2}+e^{\lambda} d r^{2}+r^{2} d \Omega^{n-2}

\begin{aligned} e^{\nu}=e^{-\lambda} & =1-\left(\frac{2 \mu}{r}\right)^{n-3}-\frac{2 \Lambda_{\text {eff. }}}{(n-1)(n-2)} r^{2} \\ \mu & =\frac{M}{m^{* 2}} ; \quad d \Omega^{2}:=d \vartheta^{2}+\sin ^{2} \vartheta d \phi^{2} \end{aligned}

d s^{2}=\frac{n-2}{2 \Lambda_{\text {eff }} t^{2}}\left(-d t^{2}+d r^{2}\right)+\frac{(n-2)(n-3)}{2 \Lambda_{\text {eff. }}} d \Omega^{n-2}

d s^{2}=\left(\frac{N(r \pm t)}{r}\right)^{2}\left(-d t^{2}+d r^{2}+r^{2} d \Omega^{2}\right),

d r^{*}:=e^{(\lambda-v) / 2)} d r .

u^{2}-v^{2}=e^{r * / 4 \mu}

(u+v) /(u-v)=e^{t / 2 \mu},

d s^{2}=16 \mu^{2} e^{\nu-r^{* / 2 \mu}}\left(-d \nu^{2}+d u^{2}\right)+r^{2} d \Omega^{2},

\Omega^{\{ \}}=\Omega_{\mathrm{C}}^{\{ \}}+\bar{\Omega}^{\{ \}}-\frac{1}{6} R^{\{ \}} \vartheta \wedge \vartheta,

\stackrel{+}{\Omega}^{g}=\frac{\gamma \kappa}{\ell^{* 2}} \vartheta \wedge \vartheta .

R={ }^{*}\left(\Omega^{g} \wedge \vartheta \wedge \vartheta\right)=-\frac{6 \kappa \gamma}{\ell^{* 2}}=4 \Lambda_{\text {eff }} .

\bar{\Omega}^{\{ \}}=0 .

R^{\{ \}}=-\frac{6 \kappa \gamma}{\ell^{* 2}} .

(\stackrel{+}{D} K)-K \wedge K=\stackrel{+}{\Omega}_{\mathrm{C}}\{ \} .

\left[\mathrm{T}_{\alpha \beta}{ }^{\gamma}\right]=\left[\begin{array}{cccc} f & -h & \cdot & \cdot \\ \cdot & \cdot & -k & \cdot \\ \cdot & \cdot & \cdot & -k \\ \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & -g \\ \cdot & & g & \cdot \\ 0 & 1 & 2 & 3 \end{array} \begin{array}{c} 01 \\ 02 \\ 03 \\ 23 \\ 31 \\ 12 \end{array}\right.

f=-h=g=k=-\frac{\mu}{r^{2}} e^{-\nu / 2} .

\begin{aligned} & f=-\frac{1}{r}\left(1+\frac{\kappa}{2 \ell^{* 2}} r^{2}\right) e^{-\nu / 2}, g=\frac{1}{r} e^{\nu / 2} \\ & h=-\frac{1}{(+) \ell *} \frac{\sqrt{-\kappa}}{3}, \quad k=+\frac{1}{(-) 2 \ell *} \sqrt{-3 \kappa} \end{aligned}

\begin{aligned} d s_{\mathrm{Sds}}^{2} & =\left(1-\frac{2 \mu}{r}+\frac{\kappa}{4 \ell^{* 2}}\right) d \tau^{2} \\ & +\left(1-\frac{2 \mu}{r}+\frac{\kappa}{4 \ell^{* 2}}\right)^{-1} d r^{2}+r^{2} d \Omega^{2} \end{aligned}

\begin{aligned} & f=\frac{2 \mu}{r^{2}}\left(1-\frac{2 \mu}{r}+\frac{\kappa}{4 \ell^{* 2}}\right)^{-1 / 2} \\ & g=h=k=0 \end{aligned}

\begin{aligned} p_{1}\left(M^{4}\right) & =\frac{1}{2(2 \pi)^{2}} \int_{M^{4}} \operatorname{Tr}\left(\Omega^{\mathrm{L}} \wedge \Omega^{\mathrm{L}}\right) \\ & =\frac{(-1)^{s / 2}}{(4 \pi)^{2}} \int_{M^{4}} R_{\alpha \beta \gamma \delta}{ }^{*} R^{\alpha \beta \gamma \delta} \sqrt{|g|} d^{4} x \end{aligned}

p_{1}\left(M^{4}\right)=-3 \tau

\begin{aligned} L_{\text {Euler }}, & ={ }^{*}\left(\gamma_{2}^{g}\right)=\frac{1}{(4 \pi)^{2}} \operatorname{Tr}\left(\Omega^{g} \wedge \Omega^{g(*)}\right) \\ & =\frac{(-1)^{s / 4}}{32 \pi^{2}} R^{\alpha \beta \gamma \delta *} R_{\alpha \beta \gamma \delta}^{*} \sqrt{|g|} d^{4} x . \end{aligned}

\left.\chi(M):=\sum_{p=0}^{n}(-1)^{p} b_{p}=\int_{M}{ }^{*} \pi^{( } \gamma_{2}^{g}\right)

\chi(M)=\int_{M}{ }^{*}\left(\gamma_{2}^{g}\right)-\frac{1}{(4 \pi)^{2}} \int_{\partial M}\left\{\alpha \wedge \Omega^{g(\star)}+\frac{2}{3} \alpha \wedge(\alpha \wedge \alpha)^{(\star)}\right\} .

\begin{aligned} \chi(M \# N) & =\chi(M)+\chi(N)-\chi\left(S^{n}\right), \\ \chi(M \times N) & =\chi(M) \cdot \chi(N), \\ \chi\left(M^{2}\right) & =2(g-1), \\ \chi\left(\mathbb{C} P^{2}\right) & =3 \end{aligned}

\chi\left(M^{2 k+1}\right)=0

\begin{aligned} & \left(1+\zeta^{2}\right) \int_{M^{4}} L_{\mathrm{SKY}}-(-1)^{s} \zeta(4 \pi)^{2} \cdot \chi\left(M^{4}\right) \\ & =\frac{1}{4} \int_{M^{4}}\left(\Omega^{g}-\zeta^{*} \Omega^{g(\star)}\right) \wedge^{*}\left(\Omega^{g}-\zeta^{*} \Omega^{g(\star)}\right) \geq 0, \end{aligned}

\begin{aligned} d C_{\mathrm{RR}^{(\star)}} & :=\frac{1}{2} d\left(\Gamma_{\alpha \beta} \wedge R^{\alpha \beta(\star)}-\frac{1}{3} \Gamma_{\alpha}{ }^{\beta(\star)} \wedge \Gamma_{\beta}{ }^{\gamma} \wedge \Gamma_{\gamma}{ }^{\alpha}\right) \\ & \equiv-L_{\mathrm{SKY}}-2 \operatorname{Ric}_{\alpha \beta} \wedge{ }^{*} \operatorname{Ric}^{\alpha \beta}+\frac{1}{2} \operatorname{Ric}_{\alpha}{ }^{\alpha} \wedge{ }^{*} \operatorname{Ric}_{\beta}{ }^{\beta} \end{aligned}

L_{\mathrm{SKY}}:=-\frac{1}{2} R_{\alpha \beta} \wedge{ }^{*} R^{\alpha \beta},

L_{\mathrm{EC}}=-\frac{\chi}{\ell} \vartheta^{\alpha} \wedge R_{\alpha}^{\star}

\vartheta^{\alpha}=E_{i}^{\alpha} d x^{i} \quad \text { and } \quad \Gamma_{\alpha}^{\star}:=\frac{1}{2} \eta_{\alpha \beta \gamma} \Gamma^{\beta \gamma} .

T^{\alpha}:=d \vartheta^{\alpha}-(-1)^{s} \eta^{\alpha \beta} \wedge \Gamma_{\beta}^{\star}

R_{\alpha}^{\star}=\frac{1}{2} \eta_{\alpha \beta \gamma} R^{\beta \gamma}:=d \Gamma_{\alpha}^{\star}+\frac{(-1)^{s}}{2} \eta_{\alpha}^{\beta \gamma} \Gamma_{\beta}^{\star} \wedge \Gamma_{\gamma}^{\star},

D T^{\alpha} \equiv(-1)^{s} \eta^{\alpha \beta} \wedge R_{\beta}^{\star}

D R_{\alpha}^{\star} \equiv 0 .

C_{\mathrm{T}}:=\frac{1}{2 \ell^{2}} \vartheta^{\alpha} \wedge T_{\alpha}=-\frac{(-1)^{s}}{\ell^{2}} \eta^{\alpha} \wedge K_{\alpha}^{\star},

C_{\mathrm{L}}:=(-1)^{s} \Gamma^{\star \alpha} \wedge R_{\alpha}^{\star}-\frac{1}{3!} \eta_{\alpha \beta \gamma} \Gamma^{\star \alpha} \wedge \Gamma^{\star \beta} \wedge \Gamma^{\star \gamma} .

\frac{\delta C_{\mathrm{T}}}{\delta \vartheta^{\alpha}}=\frac{1}{\ell^{2}} T_{\alpha} \quad \text { and } \quad \frac{\delta C_{\mathrm{T}}}{\delta \Gamma^{\star \alpha}}=\frac{(-1)^{s}}{\ell^{2}} \eta_{\alpha},

\frac{\delta C_{\mathrm{L}}}{\delta \vartheta^{\alpha}}=0 \quad \text { and } \quad \frac{\delta C_{\mathrm{L}}}{\delta \Gamma^{\star \alpha}}=(-1)^{s} 2 R_{\alpha}^{\star},

C_{\mathrm{TL}}:=\frac{1}{\ell}\left(\Gamma^{\star \alpha} \wedge T_{\alpha}-\frac{(-1)^{s}}{2} \eta_{\alpha \beta \gamma} \Gamma^{\star \alpha} \wedge \Gamma^{\star \beta} \wedge \vartheta^{\gamma}\right) .

\frac{\delta C_{\mathrm{TL}}}{\delta \vartheta^{\alpha}}=\frac{1}{\ell} R_{\alpha}^{\star} \quad \text { and } \quad \frac{\delta C_{\mathrm{TL}}}{\delta \Gamma^{\star \alpha}}=\frac{1}{\ell} T_{\alpha},

D H_{\alpha}-E_{\alpha}=\Sigma_{\alpha},

D H_{\alpha}^{\star}-S_{\alpha}=\tau_{\alpha}^{\star},

H_{\alpha}:=-\frac{\partial L}{\partial T^{\alpha}}, \quad H_{\alpha}^{\star}:=-\frac{(-1)^{s}}{2} \frac{\partial L}{\partial R^{\star \alpha}},

E_{\alpha}:=\frac{\partial L}{\partial \vartheta^{\alpha}}=e_{\alpha} \downharpoonleft L+\left(e_{\alpha} \downharpoonleft T^{\beta}\right) \wedge H_{\beta}+2(-1)^{s}\left(e_{\alpha} \downharpoonleft R^{\star \beta}\right) \wedge H_{\beta}^{\star},

S_{\alpha}:=\frac{(-1)^{s}}{2} \frac{\partial L}{\partial \Gamma^{\star \alpha}}=\frac{1}{2} \eta_{\alpha \beta} \wedge H^{\beta} .

\Sigma_{\alpha}:=\frac{\delta L_{\mathrm{MAT}}}{\delta \vartheta^{\alpha}}, \quad \tau_{\alpha}^{\star}:=\frac{(-1)^{s}}{2} \frac{\delta L_{\mathrm{MAT}}}{\delta \Gamma^{\star \alpha}},

L_{\mathrm{EC}}:=-\frac{\chi}{\ell} \vartheta^{\alpha} \wedge R_{\alpha}^{\star} \equiv-\chi C_{\mathrm{TL}}-\frac{\chi}{\ell} d\left(\Gamma_{\alpha}^{\star} \wedge \vartheta^{\alpha}\right)

L_{\mathrm{MB}}\left(\vartheta^{\alpha}, \Gamma_{\alpha}^{\star}\right)=\theta_{\mathrm{T}} C_{\mathrm{T}}+\theta_{\mathrm{L}} C_{\mathrm{L}}+\theta_{\mathrm{TL}} C_{\mathrm{TL}},

-\theta_{\mathrm{TL}} R_{\alpha}^{\star}-\frac{1}{\ell} \theta_{\mathrm{T}} T_{\alpha}=\ell \Sigma_{\alpha}

-(-1)^{s} \theta_{\mathrm{TL}} T_{\alpha}-\frac{1}{2 \ell} \theta_{\mathrm{T}} \eta_{\alpha}-\theta_{\mathrm{L}} \ell R_{\alpha}^{\star}=\ell \tau_{\alpha}^{\star} ;

T_{\alpha}-\frac{\varepsilon}{\ell} \eta_{\alpha}=\frac{2}{A} \ell\left(\theta_{\mathrm{TL}} \tau_{\alpha}^{\star}-\theta_{\mathrm{L}} \ell \Sigma_{\alpha}\right)

R_{\alpha}^{\star}-\frac{\rho}{\ell^{2}} \eta_{\alpha}=\frac{2}{A}\left(\theta_{\mathrm{TL}} \ell \Sigma_{\alpha}-\theta_{\mathrm{T}} \tau_{\alpha}^{\star}\right)

T_{\alpha}=\frac{\varepsilon}{\ell} \eta_{\alpha}, \quad R_{\alpha}^{\star}=\frac{\rho}{\ell^{2}} \eta_{\alpha},

\Gamma_{\alpha}^{\star}=\Gamma_{\alpha}^{\{ \} \star}-K_{\alpha}^{\star}

R_{\alpha}^{\star} \equiv R_{\alpha}^{\{ \} \star}-D K_{\alpha}^{\star}+\frac{1}{2} \eta_{\alpha \beta \gamma} K^{\star \beta} \wedge K^{\star \gamma}

R_{\alpha}^{\{i \star}=-\Lambda_{\text {eff }} \eta_{\alpha},

\Lambda_{\mathrm{eff}}=-\left[4 \rho-\left(1+2(-1)^{s}\right) \varepsilon^{2}\right] / 4 \ell^{2}=\theta_{\mathrm{T}}^{2}\left[\left(9+2(-1)^{s}\right) \theta_{\mathrm{TL}}^{2}+8 \theta_{\mathrm{T}} \theta_{\mathrm{L}}\right] /(2 A \ell)^{2}

d s^{2}=-\left(1-\Lambda_{\text {eff }} r^{2}\right) d t^{2}+\left(1-\Lambda_{\text {eff }} r^{2}\right)^{-1} d r^{2}+r^{2} d \phi^{2}

d s^{2}=-N^{2}(r) d t^{2}+N^{-2}(r) d r^{2}+r^{2}\left[d \phi+N^{\phi}(r) d t\right]^{2}

N^{2}(r)=-M-\Lambda_{\mathrm{eff}} r^{2}+\frac{J^{2}}{4 r^{2}}, \quad N^{\phi}(r)=-\frac{J}{2 r^{2}},

\widetilde{\Gamma}_{\alpha}^{\star}=\Gamma_{\alpha}^{\star}-(-1)^{s} \frac{\varepsilon}{2 \ell} \vartheta_{\alpha},

\widetilde{T}_{\alpha}=T_{\alpha}-\frac{\varepsilon}{\ell} \eta_{\alpha}, \quad \widetilde{R}_{\alpha}^{\star}=R_{\alpha}^{\star}-(-1)^{s} \frac{\varepsilon}{2 \ell} T_{\alpha}+(-1)^{s} \frac{\varepsilon^{2}}{4 \ell^{2}} \eta_{\alpha}

\Gamma_{\alpha}^{\star}=(-1)^{s} \frac{\varepsilon}{2 \ell} \vartheta_{\alpha},

S_{\alpha}=\frac{\ell}{\varepsilon} E_{\alpha}

H_{\alpha}^{\star}=\frac{\ell}{\varepsilon} H_{\alpha}

T_{\alpha}=\frac{\varepsilon}{\ell} \eta_{\alpha}, \quad R_{\alpha}^{\star}=\frac{\rho}{\ell^{2}} \eta_{\alpha}

H_{\alpha}^{\star}=\delta \ell H_{\alpha}+\frac{\gamma}{\ell} \vartheta_{\alpha}

\delta \ell D H_{\alpha}+\frac{\gamma}{\ell} T_{\alpha}-\frac{1}{2} \eta_{\alpha \beta} \wedge H^{\beta}=\tau_{\alpha}^{\star}

H_{\alpha}=-\frac{\gamma \varepsilon}{\ell^{2}(1+\delta \varepsilon)} \vartheta_{\alpha} \quad \Leftrightarrow \quad T_{\alpha}=\frac{\varepsilon}{\ell} \eta_{\alpha},

H_{\alpha}^{\star}=\frac{\gamma}{\ell(1+\delta \varepsilon)} \vartheta_{\alpha}=-\frac{\ell}{\varepsilon} H_{\alpha},

\left.\left.E_{\alpha}=e_{\alpha} \downharpoonleft L+\left[\left(e_{\alpha} \downharpoonleft T^{\beta}\right)+(-1)^{s} 2 \delta \ell\left(e_{\alpha}\right\rfloor R^{\star \beta}\right)\right] \wedge H_{\beta}+(-1)^{s} \frac{2 \gamma}{\ell}\left(e_{\alpha}\right\rfloor R^{\star \beta}\right) \wedge \vartheta_{\beta},

\left.\left.E_{\alpha}=e_{\alpha}\right\rfloor L(\star)-\frac{2 \gamma \varepsilon^{2}}{\ell^{3}(1+\delta \varepsilon)} \eta_{\alpha}+(-1)^{s} \frac{2 \gamma}{\ell(1+\delta \varepsilon)}\left[\left(e_{\alpha}\right\rfloor R^{\star \beta}\right) \wedge \vartheta_{\beta}\right] \text {, }

\left.\left.E_{\alpha}=e_{\alpha}\right\rfloor L(\star)+(-1)^{s} \frac{2 \gamma}{\ell}\left[\left(e_{\alpha}\right\rfloor R^{\star \beta}\right) \wedge \vartheta_{\beta}\right]

\left.E_{\alpha}=e_{\alpha}\right\rfloor L(\star)-\frac{2 \gamma}{\delta \ell^{2}} T_{\alpha},

{ }^{*} D\left(\vartheta^{\alpha} \wedge H_{\alpha}\right)-{ }^{*}\left(T^{\alpha} \wedge H_{\alpha}\right)=\frac{\gamma}{\delta \ell^{2}}{ }^{*}\left(\vartheta^{\alpha} \wedge T_{\alpha}\right)+\frac{1}{\delta \ell}{ }^{*}\left(\eta^{\alpha} \wedge H_{\alpha}\right) .

H={ }^{*}\left(\vartheta^{\alpha} \wedge H_{\alpha}\right),

t={ }^{*}\left(T^{\alpha} \wedge H_{\alpha}\right), \quad \mathscr{A}:={ }^{*}\left(\vartheta^{\alpha} \wedge T_{\alpha}\right), \quad h={ }^{*}\left(\eta^{\alpha} \wedge H_{\alpha}\right),

\square H-(-1)^{s}{ }^{*} D^{*} D H=d t+\frac{\gamma}{\delta \ell^{2}} d \mathscr{A}+\frac{1}{\delta \ell} d h,

\square:=(-1)^{p n+s}\left[{ }^{*} D{ }^{*} D+(-1)^{n} D^{*} D^{*}\right]

d s^{2}=(-1)^{s} d t^{2}+d \mathscr{A}^{2}+d h^{2}

\square \vartheta_{\alpha}=\frac{\ell}{\gamma}{ }^{*} D^{*}\left[\tau_{\alpha}^{\star}-S_{\alpha}-\delta \ell\left(\Sigma_{\alpha}+E_{\alpha}\right)\right]+\frac{\ell}{\gamma} D^{*}\left(D^{*} H_{\alpha}^{\star}-\delta \ell D^{*} H_{\alpha}\right) .

D^{*} \vartheta_{\alpha}:=\frac{\ell}{\gamma}\left[D^{*} H_{\alpha}^{\star}-\delta \ell D^{*} H_{\alpha}\right]

\square \vartheta_{\alpha}+\frac{\ell}{\gamma}{ }^{*} D^{*}\left(\delta \ell E_{\alpha}+S_{\alpha}\right)=0,

\left[\square+(-1)^{s} m^{2}\right] \vartheta^{\alpha} \cong 0,

D^{*} \vartheta^{\alpha} \cong 0,

m=\frac{\theta_{\mathrm{T}} \theta_{\mathrm{TL}}}{2\left(\theta_{\mathrm{TL}}^{2}+\theta_{\mathrm{T}} \theta_{\mathrm{L}}\right) \ell}, \quad \Lambda_{\mathrm{ind}}=\frac{\theta_{\mathrm{T}}^{2}\left[-(-1)^{s} 3 \theta_{\mathrm{TL}}^{2}+8 \theta_{\mathrm{T}} \theta_{\mathrm{L}}\right]}{4\left[(-1)^{s} \theta_{\mathrm{TL}}^{2}-2 \theta_{\mathrm{T}} \theta_{\mathrm{L}}\right]^{2} \ell^{2}},

\check{\mathscr{E}}_{\alpha}:=d H_{\alpha} \cong \Sigma_{\alpha}+E_{\alpha}-(-1)^{s} \eta_{\alpha \beta \gamma} \Gamma^{\star \beta} \wedge H^{\gamma}

\check{\mathscr{J}}_{\alpha}^{\star}:=d H_{\alpha}^{\star} \cong \tau_{\alpha}^{\star}+E_{\alpha}^{\star}-(-1)^{s} \eta_{\alpha \beta \gamma} \Gamma^{\star \beta} \wedge H^{\star \gamma},

P_{\alpha}=\int \check{\mathscr{E}}_{\alpha}, \quad J_{\alpha}=\int \check{\mathscr{J}}_{\alpha}^{\star} .

H_{\alpha}=-\frac{\theta_{\mathrm{T}}}{2 \ell^{2}} \vartheta_{\alpha}-\frac{\theta_{\mathrm{TL}}}{\ell} \Gamma_{\alpha}^{\star}, \quad H_{\alpha}^{\star}=-\frac{\theta_{\mathrm{L}}}{2} \Gamma_{\alpha}^{\star},

\check{\mathscr{E}}_{\alpha}=-\frac{\theta_{\mathrm{T}} \theta_{\mathrm{TL}}}{2 \ell^{2}}\left(\frac{1}{\theta_{\mathrm{TL}}} d \vartheta_{\alpha}-\frac{4 \ell}{\theta_{\mathrm{T}} \theta_{\mathrm{L}}} \check{\mathscr{J}}_{\alpha}^{\star}\right),

\check{\mathscr{J}}_{\alpha}^{\star}=-\frac{\theta_{\mathrm{L}}}{2} d \Gamma_{\alpha}^{\star}

\check{\mathscr{E}}_{\alpha}^{\mathrm{MB}}=-(-1)^{s} \frac{\theta_{\mathrm{T}} \theta_{\mathrm{L}}}{2 \chi \ell}\left(d \Gamma_{\alpha}^{\star}+\frac{2}{\theta_{\mathrm{L}}} \check{\mathscr{J}}_{\alpha}^{\mathrm{MB} \star}\right),

\check{\mathscr{J}}_{\alpha}^{\mathrm{MB} \star}=(-1)^{s} \frac{\chi}{2 \ell} d \vartheta_{\alpha}-\frac{\theta_{\mathrm{L}}}{2} d \Gamma_{\alpha}^{\star}

\check{\mathscr{E}}_{\alpha}^{\mathrm{MB}}-\check{\mathscr{E}}_{\alpha}=\frac{\theta_{\mathrm{TL}}}{\ell} d \Gamma_{\alpha}^{\star},

\widetilde{C}_{\mathrm{L}}=(-1)^{s} \widetilde{\Gamma}^{\star \alpha} \wedge \widetilde{R}_{\alpha}^{\star}-\frac{1}{3!} \eta_{\alpha \beta \gamma} \widetilde{\Gamma}^{\star \alpha} \wedge \widetilde{\Gamma}^{\star \beta} \wedge \widetilde{\Gamma}^{\star \gamma},

\theta_{\mathrm{L}} C_{\mathrm{L}}-\tilde{\theta}_{\mathrm{L}} \widetilde{C}_{\mathrm{L}}=-\frac{\chi}{\ell} \vartheta^{\alpha} \wedge R_{\alpha}^{\star}-\frac{\Lambda}{\ell} \eta+\frac{\theta_{\mathrm{T}}}{2 \ell} \vartheta^{\alpha} \wedge T_{\alpha}+\Delta \theta_{\mathrm{L}} C_{\mathrm{L}}+\frac{\Delta \theta_{\mathrm{TL}}}{\ell} d\left(\Gamma^{\star \alpha} \wedge \vartheta_{\alpha}\right)

\begin{gathered} \chi=\theta_{\mathrm{L}} \varepsilon-\tilde{\theta}_{\mathrm{L}} \tilde{\varepsilon}, \quad \Lambda=\frac{(-1)^{s}}{4 \ell^{2}}\left(\theta_{\mathrm{L}} \varepsilon^{3}-\tilde{\theta}_{\mathrm{L}} \tilde{\varepsilon}^{3}\right), \quad \theta_{\mathrm{T}}=\frac{(-1)^{s}}{2}\left(\theta_{\mathrm{L}} \varepsilon^{2}-\tilde{\theta}_{\mathrm{L}} \tilde{\varepsilon}^{2}\right), \\ \Delta \theta_{\mathrm{L}}=\theta_{\mathrm{L}}-\tilde{\theta}_{\mathrm{L}}, \quad \Delta \theta_{\mathrm{TL}}=\frac{1}{2}\left(\theta_{\mathrm{L}} \varepsilon-\tilde{\theta}_{\mathrm{L}} \tilde{\varepsilon}\right) . \end{gathered}

\theta_{\mathrm{L}} C_{\mathrm{L}}-\tilde{\theta}_{\mathrm{L}} \widetilde{C}_{\mathrm{L}}=L_{\mathrm{MB}}+\frac{\Delta \theta_{\mathrm{TL}}}{\ell} d\left(\Gamma^{\star \alpha} \wedge \vartheta_{\alpha}\right),

c=12 \cdot 4 \pi \theta_{\mathrm{L}}, \quad \tilde{c}=12 \cdot 4 \pi \tilde{\theta}_{\mathrm{L}},

\widetilde{L}=L+\lambda_{\alpha} \wedge T^{\alpha} .

C_{\alpha}-E_{\alpha}=\Sigma_{\alpha}-D^{\{ \}} \mu_{\alpha},

\left.\left.\left.C_{l}=E_{l}^{\alpha} *\left\{D^{\{ \}}\left[2\left(e^{\beta}\right\lrcorner D^{\{ \}} H_{[\alpha \beta]}\right)+\frac{1}{2} \vartheta_{\alpha} \wedge\left(e^{\beta}\right\lrcorner e^{\gamma}\right\lrcorner D^{\{ \}} H_{[\beta \gamma]}\right)\right]\right\}

\left.\left.-E_{\alpha}=-e_{\alpha}\right\rfloor L(\star)-(-1)^{s} \frac{2 \gamma}{\ell}\left(e_{\alpha}\right\rfloor R^{\{ \} \star \beta}\right) \wedge \vartheta_{\beta}=\ell \widetilde{\Sigma}_{\alpha}

G_{\alpha}^{\{ \}}-\tilde{\Lambda} \eta_{\alpha}=\ell \widetilde{\Sigma}_{\alpha},

\tilde{\Lambda}=(-1)^{s}{ }^{*} L(\star)+\frac{1}{3 \ell}\left[4 \gamma+(-1)^{s} \ell\right] *\left(R^{(\{ \} \star \alpha} \wedge \vartheta_{\alpha}\right),

\sigma^{1}=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \sigma^{2}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right), \quad \sigma^{3}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right) .

\left[\sigma^{\alpha}, \sigma^{\beta}\right]=2 i \eta^{\alpha \beta \gamma} \sigma_{\gamma} .

\gamma_{0}=i \sigma^{2}, \quad \gamma_{1}=\sigma^{1}, \quad \gamma_{2}=\sigma^{3} .

\gamma_{\alpha} \gamma_{\beta}=g_{\alpha \beta} \mathbf{1}+\eta_{\alpha \beta \nu} \gamma^{\nu},

\begin{aligned} L_{\mathrm{D}} & =\frac{i}{2}\left(\bar{\psi}^{*} \gamma \wedge D \psi+D \bar{\psi} \wedge^{*} \gamma \psi\right)-m \bar{\psi} \psi \eta \\ & =\bar{\psi}\left(i^{*} \gamma \wedge D-m \eta\right) \psi+d\left(\frac{i}{2} \bar{\psi}^{*} \gamma \psi\right), \end{aligned}

i^{*} \gamma \wedge D \psi-m \psi \eta-\frac{i}{2}\left(D^{*} \gamma\right) \psi=0

\left.D^{*} \gamma={ }^{*} \gamma \wedge T, \quad T:=e_{\alpha}\right\rfloor T^{\alpha},

\hat{D}:=D-\frac{1}{2} T,

i^{*} \gamma \wedge \hat{D} \psi-m \eta \psi=0,

\hat{R}=R-\frac{1}{2} d T=\frac{i}{4} R^{\alpha \beta} \sigma_{\alpha \beta}-\frac{1}{2} d T

[\hat{D}, \hat{D}]=\hat{R}+T^{\alpha} \hat{D}_{\alpha} .

\hat{D}^{*} \hat{D} \psi-m^{2} \eta \psi+i^{*} \sigma \wedge\left(\hat{R}+T^{\alpha} \hat{D}_{\alpha}\right) \psi=0,

L_{A}=\frac{1}{2}\left(F \wedge{ }^{*} F+m_{\text {photon }} F \wedge A\right) .

d^{*} F-m_{\text {photon }} F=0

\square F-m_{\text {photon }}^{2} F=0,

X:=x_{\alpha} \vartheta^{\alpha}, \quad Y:=y_{\alpha} \vartheta^{\alpha} .

\Sigma_{\alpha}=\varepsilon x_{\alpha} X \wedge Y, \quad \tau_{\alpha}^{\star}=\sigma y_{\alpha} X \wedge Y,

\vartheta^{\alpha} \wedge \Sigma_{\alpha}=0, \quad \vartheta^{\alpha} \wedge \tau_{\alpha}^{\star}=0

R^{\alpha \beta}=\varepsilon \ell^{2} x^{[\alpha} y^{\beta]} X \wedge Y,

R_{\beta}{ }^{\alpha} \wedge \vartheta^{\beta}=\frac{\ell^{2}}{2} \varepsilon\left(x^{\alpha} Y \wedge X \wedge Y-y^{\alpha} X \wedge X \wedge Y\right)=0 .

\begin{aligned} \vartheta^{\hat{0}} & =d t+\ell^{2} \sigma \rho^{* 2}\left[1-\cos \left(\rho / \rho^{*}\right)\right] d \phi \\ \vartheta^{\hat{1}} & =d \rho, \quad \vartheta^{\hat{2}}=\rho^{*} \sin \left(\rho / \rho^{*}\right) d \phi \\ \Gamma^{\hat{1} \hat{2}} & =\cos \left(\rho / \rho^{*}\right) d \phi=-\Gamma^{\hat{2} \hat{1}} \end{aligned}

\mathscr{A}:={ }^{*}\left(\vartheta^{\alpha} \wedge T_{\alpha}\right)=-(-1)^{s} \frac{\varepsilon}{\ell^{2}}

\Psi=\Psi_{i} d x^{i}=\Psi_{\alpha} \vartheta^{\alpha}

\gamma=\gamma_{\alpha} \vartheta^{\alpha} .

\bar{\Psi} \wedge \Psi=0, \quad \bar{\Psi} \wedge \gamma_{5} \gamma^{\alpha} \Psi=0, \quad \bar{\Psi} \wedge \gamma_{5} \Psi=0 .

L_{\mathrm{RS}}=\frac{i}{4}(\bar{\Psi} \wedge D \Psi-\Psi \wedge \overline{D \Psi})+\frac{i}{4} m \bar{\Psi} \wedge \gamma \wedge \Psi,

D \Psi=d \Psi-\frac{1}{2} \gamma_{\alpha} \Gamma^{\star \alpha} \wedge \Psi,

\Sigma_{\alpha}:=\frac{\delta L_{\Psi}}{\delta \vartheta^{\alpha}}=\frac{\partial L_{\Psi}}{\partial \vartheta_{\alpha}}+D \frac{\partial L_{\Psi}}{\partial T^{\alpha}},

\begin{aligned} \Sigma_{\alpha} & \left.\left.:=e_{\alpha}\right\rfloor L_{\Psi}-\left(e_{\alpha}\right\rfloor \Psi\right) \wedge \frac{\partial L_{\Psi}}{\partial \Psi}-\left(e_{\alpha} \downharpoonleft \bar{\Psi}\right) \wedge \frac{\partial L_{\Psi}}{\partial \bar{\Psi}} \\ & \left.-\left(e_{\alpha}\right\rfloor D \Psi\right) \wedge \frac{\partial L_{\Psi}}{\partial D \Psi}-\left(e_{\alpha} \downharpoonleft D \bar{\Psi}\right) \wedge \frac{\partial L_{\Psi}}{\partial D \bar{\Psi}} ; \end{aligned}

\Sigma_{\alpha}=-\frac{i}{4} m \bar{\Psi} \wedge \gamma_{\alpha} \Psi=-(-1)^{s} 2 m \tau_{\alpha}^{\star}

\tau_{\alpha}^{\star}:=\frac{1}{2} \eta_{\alpha \beta \gamma} \tau^{\beta \gamma}=\frac{(-1)^{s}}{2} \frac{\delta L_{\Psi}}{\delta \Gamma_{\alpha}^{\star}}=\frac{(-1)^{s}}{2} \frac{i}{4} \bar{\Psi} \wedge \gamma_{\alpha} \Psi .

L=L_{\infty}\left(\vartheta^{\alpha}, \Gamma_{\alpha}^{\star}, \Psi\right)=L_{\mathrm{MB}}+L_{\Psi}

\delta L=\delta \vartheta^{\alpha} \wedge \frac{\delta L}{\delta \vartheta^{\alpha}}+\delta \Gamma_{\alpha}^{\star} \wedge \frac{\delta L}{\delta \Gamma_{\alpha}^{\star}}+\delta \bar{\Psi} \wedge \frac{\delta L}{\delta \bar{\Psi}},

\begin{aligned} \delta_{\text {susy }} \vartheta^{\alpha} & \left.=i \bar{\sigma} \Psi \gamma^{\alpha}, \quad \delta_{\text {susy }} \Gamma_{\alpha}^{\star}=i \bar{\sigma} \gamma_{\alpha}^{*} D \Psi+i c \bar{\sigma}\left(\gamma_{\alpha} \Psi+e_{\alpha}\right\rfloor^{*} \Psi\right) \\ \delta_{\text {susy }} \Psi & =2 D \sigma+c \gamma \sigma, \end{aligned}

\delta_{\mathrm{susy}} L=i \bar{\sigma} \Psi \gamma^{\alpha} \wedge \frac{\delta L}{\delta \vartheta^{\alpha}}+\delta_{\mathrm{susy}} \Gamma_{\alpha}^{\star} \wedge \frac{\delta L}{\delta \Gamma_{\alpha}^{\star}}+(2 \overline{D \sigma}+c \bar{\sigma} \gamma) \wedge \frac{\delta L}{\delta \bar{\Psi}},

\delta_{\text {susy }} L \cong \bar{\sigma}\left(i \gamma^{\alpha} \Psi \wedge \frac{\delta L}{\delta \vartheta^{\alpha}}-2 D \frac{\delta L}{\delta \bar{\Psi}}+c \gamma \wedge \frac{\delta L}{\delta \bar{\Psi}}\right)+2 d\left(\bar{\sigma} \wedge \frac{\delta L}{\delta \Psi}\right) .

\frac{2}{i} \frac{\delta L}{\delta \Psi}=D \Psi+\frac{1}{2} m \gamma \wedge \Psi \cong 0