harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\frac{dn(\omega)}{d\omega}}=\int_{\Sigma_{r}}\sqrt{h}d^{D-1}x\sum_{k}\left[2\omega|\phi_{\omega,k}|^{2}+i\phi_{\omega,k}^{*}a^{i}(\nabla_{i}+i\omegaa_{i})\phi_{\omega,k}-i(\nabla_{i}-i\omegaa_{i})\phi_{\omega,k}^{*}a^{i}\phi_{\omega,k}\right]~~~,
S_{4}\,[\,g_{\mu\nu}^{\pm},\xi^{\pm}]=S_{\mathrm{grav}}[\,g_{\mu\nu}^{\pm}]+S_{\mathrm{n-m}}[\,g_{\mu\nu}^{\pm},\xi^{\pm}]+S_{\mathrm{rad}}\,[\,g_{\mu\nu}^{\pm},\xi^{\pm}].
B\left({\frac{1}{3}},\,{\frac{1}{6}}\right)={\frac{\Gamma\left({\frac{1}{3}}\right)\Gamma\left({\frac{1}{6}}\right)}{\Gamma\left({\frac{1}{2}}\right)}}=2^{\frac{2}{3}}{\frac{\Gamma^{2}\left({\frac{1}{3}}\right)}{\Gamma\left({\frac{2}{3}}\right)}}\simeq8.413
S=\frac{\beta}{4}\intd^{2}x\,\epsilon^{\mu\nu}\left[\phi_{ab}\star\left({\calR}_{\mu\nu}^{ab}-2\Lambdae_{\left[\mu\right.}^{a}\stare_{\left.\nu\right]}^{b}\right)-2\phi_{a}\starT_{\mu\nu}^{a}\right]\,,
{\hat{M}}=-\frac{1}{2\pigP}\frac{1}{R^{\prime}}\frac{d}{dz}\frac{R{R^{\prime}}^{2}(1+D^{2}{R^{\prime}}^{2})}{(1+{R^{\prime}}^{2})^{3/2}}\frac{d}{dz}\frac{1}{R^{\prime}}
\frac{\partialF_{j}}{\partialx^{1}}\mid_{(\vec{z}_{i},t^{*})}=0,\,quad\quad\frac{\partialF_{j}}{\partialx^{2}}\mid_{(\vec{z}_{i},t^{*})}=0,\,quad\quad\frac{\partialF_{j}}{\partialt}\mid_{(\vec{z}_{i},t^{*})}=0,\,quad\quadj=1,2,
\begin{array}{l}{\,s\;\omega_{D}^{g}\;+\;d\;\omega_{D-1}^{g+1}\;=\;0\;,}\\{s\;\omega_{D-1}^{g+1}\;+\;d\;\omega_{D-2}^{g+2}\;=\;0\;,}\\{\;\;\;\;\;\;\;\;\;\;.....}\\{\;\;\;\;\;\;\;\;\;\;.....}\\{s\;\omega_{1}^{g+D-1}\;+\;d\;\omega_{0}^{g+D}=\;0\;,}\\{s\;\omega_{0}^{g+D}=\;0\;,}\\\end{array}
L=-8\pi\hat{\beta}_{R}^{2}s(\hat{\beta}_{R}^{2})(\hat{\beta}_{R}^{2}-1){\frac{\Gamma^{2}(1-\hat{\beta}_{R}^{2})\Gamma(\hat{\beta}_{R}^{2}-1)\Gamma(2-\hat{\beta}_{R}^{2})}{\Gamma^{2}(3-\hat{\beta}_{R}^{2})}}.
({\calB}_{V})_{cl}^{\it{approx}}=\sum_{{\vec{n}},{\vec{m}}}\sum_{s,s^{\prime}}\left[([a_{\vec{n}}^{s}]_{cl}^{*}\bar{u}_{\vec{n}}^{s}+[b_{-{\vec{n}}}^{s\dagger}]_{cl}^{*}\bar{v}_{-{\vec{n}}}^{s})([a_{\vec{m}}^{s^{\prime}}]_{cl}\gamma^{0}u_{\vec{m}}^{s^{\prime}}+[b_{-{\vec{m}}}^{s^{\prime}\dagger}]_{cl}\gamma^{0}v_{-{\vec{m}}}^{s^{\prime}})\right]F({\vec{n}}-{\vec{m}},X).
<<i\varepsilon\intd^{3}\mathbf{x}a_{\nu0}\frac{\partial}{\partialt}A_{\nu}^{\alpha}(\mathbf{x},t)>>=\varepsilon\intd^{3}\mathbf{x}a_{\nu0}\frac{\partial}{\partialt}\frac{\deltaW}{\deltaJ_{\nu}^{\alpha}(\mathbf{x},t)}
\frac12\int_{s_{0}}^{s_{1}}[\frac12\left(\frac{dA_{1}}{ds}-\frac{dA_{2}}{ds}\right)^{*}\left(\frac{dA_{1}}{ds}-\frac{dA_{2}}{ds}\right)+\sum_{i=1}^{2}\left(\frac{dZ}{ds}-ieA_{i}Z\right)^{*}\left(\frac{dZ}{ds}-ieA_{i}Z\right)]ds
[D_{\alpha}^{\mathbfi},{\mathcalL}]=\frac{3}{{\mathcalN}(4-{\mathcalN})}\left(({\mathcalN}-2)\sigma+2{\bar{\sigma}}\right)\D_{\alpha}^{\mathbfi}-i\Lambda^{\mathbfi}{}_{\mathbfj}\D_{\alpha}^{\mathbfj}-\Omega_{\alpha}{}^{\beta}\D_{\beta}^{\mathbfi}\,
(\partial_{I}^{\,I}-\partial_{N}^{\,N})f_{L_{1}\ldotsL_{r}}^{K_{1}\ldotsK_{q}}(u)=(\delta_{K_{1}I}-\delta_{K_{1}N}-\delta_{L_{1}I}+\delta_{L_{1}N}+\ldots)f_{L_{1}\ldotsL_{r}}^{K_{1}\ldotsK_{q}}(u)\;.
\ddot{x}^{a}=-\kappa^{-2}\dot{\kappa}\bar{\pi}{}^{A}\pi^{A^{\prime}}+\kappa^{-1}(\dot{\bar{\pi}}{}^{A}\pi^{A^{\prime}}+\bar{\pi}{}^{A}\dot{\pi}^{A^{\prime}}),\ddot{x}{}^{2}=-2\kappa^{-2}.
T_{\alpha},\quadG_{a}=\frac{1}{12}\sigma_{a}^{\alpha\dot{\beta}}(R_{\,\,\dot{\beta}\alpha\gamma}^{\gamma}+R_{\,\,\alpha\dot{\beta}\dot{\gamma}}^{\dot{\gamma}}),\quadW_{\alpha\beta\gamma}=\sigma_{(\alpha\dot{\delta}}^{a}\sigma_{\beta}^{b\dot{\delta}}T_{ab\gamma)}
\begin{array}{lcl}{{\calC}_{\infty}}&{\equiv}&{\{\tau=i\infty\},}\\{{\calC}_{0}}&{\equiv}&{\{\hat{u}(\tau)=\frac{3}{2}e_{1}(\tau)\},}\\{{\calC}_{c}^{(1)}}&{\equiv}&{\{\hat{u}(\tau)=(e_{3}+\frac{1}{2}e_{1})(\tau)\},}\\{{\calC}_{c}^{(2)}}&{\equiv}&{\{\hat{u}(\tau)=(e_{2}+\frac{1}{2}e_{1})(\tau)\},}\\{{\calC}_{1}}&{\equiv}&{\{{\tau=0}\}.}\\\end{array}
\gammaA_{i}^{a}=0,\;\gammaB_{a}^{0i}=\epsilon^{0ijk}\partial_{j}\eta_{ka}^{\left(2\right)},\;\gammaB_{a}^{ij}=\epsilon^{0ijk}\eta_{ka}^{\left(1\right)},\;\gamma\pi_{ij}^{a}=0,
{\frac{1}{n^{2}}}\left[\ddot{\Omega}-\left(3{\frac{\dot{a}}{a}}+{\frac{\dot{n}}{n}}\right)\dot{\Omega}\right]-\left[\Omega^{\prime\prime}-\left(3{\frac{a^{\prime}}{a}}-{\frac{n^{\prime}}{n}}\right)\Omega^{\prime}\right]+{\frac{k^{2}}{a^{2}}}\Omega=0\,.
\begin{aligned}{\mathrm{if~d=4~and~p=4~:}}&{{}}&{\quadc}&{{}\neq0,}&{\quadk}&{{}=c}\\{\mathrm{otherwise:}}&{{}}&{\quadc}&{{}=0,}&{\quadk}&{{}=\frac{d(2d+2p-dp)}{8p}}\\\end{aligned}
U=\left(\begin{array}{cccccc}{0}&{0}&{0}&{i}&{0}&{0}\\{0}&{0}&{0}&{0}&{i}&{0}\\{0}&{0}&{0}&{0}&{0}&{i}\\{i}&{0}&{0}&{0}&{0}&{0}\\{0}&{i}&{0}&{0}&{0}&{0}\\{0}&{0}&{i}&{0}&{0}&{0}\\\end{array}\right),
f(x)=f(z=x^{1}-ix^{2},\bar{z}=x^{1}+ix^{2})\longrightarrow\hat{f}(a,a^{\dagger})=\int\frac{d^{2}xd^{2}y}{(2\pi)^{2}}f(x)e^{i[\bar{p}(\sqrt{2\theta}a-z)+p(\sqrt{2\theta}a^{\dagger}-\bar{z})]},
F^{n}\left[\eta\right]\equivF^{n}+\frac{{}_{1}}{{}^{2}}T\left\{\bar{\eta}_{n}\psi_{n}+\bar{\eta}_{n+1}\psi_{n+1}+\bar{\psi}_{n}\eta_{n}+\bar{\psi}_{n+1}\eta_{n+1}\right\}
\Psi(\lambda)\equiv{\left[\begin{array}{l}{\psi_{1}(\lambda)}\\{\psi_{2}(\lambda)}\\{\psi_{3}(\lambda)}\\\end{array}\right]}=\sum_{n=-\infty}^{+\infty}\lambda^{n}\,{\left[\begin{array}{l}{q_{1}^{(n)}}\\{q_{2}^{(n)}}\\{q_{3}^{(n)}}\\\end{array}\right]}.
\begin{array}{l}{<1,1\mid\{{\hat{b}}_{r}^{11},{\hat{b}}_{11}^{r^{\prime}}\}\mid1,1>=<2,2\mid\{{\hat{b}}_{r}^{22},{\hat{b}}_{22}^{r^{\prime}}\}\mid2,2>=}\\{=<1,2\mid\{{\hat{b}}_{r}^{12},{\hat{b}}_{12}^{r^{\prime}}\}\mid1,2>=<2,1\mid\{{\hat{b}}_{r}^{21},{\hat{b}}_{(21}^{r^{\prime}}\}\mid2,1>=\delta_{r}^{r^{\prime}}.}\\\end{array}
\alpha=16\zeta\int_{0}^{\infty}\mathrm{d}\xi\;\xi^{2}\,\Psi_{\sigma}^{2}\left(\xi\right)\,,\quad\Psi_{\sigma}\left(\xi\right)=\mathrm{e}^{-\xi}\int_{0}^{\infty}\mathrm{d}s\;\mathrm{e}^{-2\xis}\,R_{\sigma}\left(\eta,\zeta,s\right)\,.
\Omega=\left(\begin{matrix}{0}&{-1}&{-4}&{-1}&{0}&{-1}\\{1}&{0}&{1}&{1}&{0}&{0}\\{4}&{-1}&{0}&{0}&{1}&{0}\\{1}&{-1}&{0}&{0}&{0}&{0}\\{0}&{0}&{-1}&{0}&{0}&{0}\\{1}&{0}&{0}&{0}&{0}&{0}\\\end{matrix}\right)
\tau^{*}\Omega^{(1)}=-\overline{{\Omega^{(1)}}},\;\;\;\;\tau^{*}\Omega^{(2)}=\overline{{\Omega^{(4)}}},\;\;\;\;\tau^{*}\Omega^{(3)}=-\overline{{\Omega^{(3)}}},\;\;\;\;\tau^{*}\Omega^{(4)}=\overline{{\Omega^{(2)}}}.
\overline{{W}}\left(\phi,\phi_{a}^{*},\overline{{\phi}},\pi_{a},\varphi,\varphi_{a}^{*},\overline{{\varphi}},\pi_{\varphia}\right)=F\left(\left(\epsilon^{-1}\right)\phi\right)-\frac12\varphi^{A}O_{AB}^{-1}\varphi^{B}+I\left(\phi,\phi_{a}^{*},\overline{{\phi}},\pi_{a},\varphi,\varphi_{a}^{*},\overline{{\varphi}},\pi_{\varphia}\right)
\tilde{S}[\bar{\Phi}]\!=\!2\frac{u}{N}\int\bar{\Phi}^{2}(x)\left\langle\phi^{2}(x)\right\rangle\!-\!2\left(\frac{u}{N}\right)^{2}\int\!\!\!\!\int\bar{\Phi}^{2}(x)\bar{\Phi}^{2}(y)\left(\left\langle\phi^{2}(x)\phi^{2}(y)\right\rangle\!-\!\left\langle\phi^{2}(x)\right\rangle\left\langle\phi^{2}(y)\right\rangle\right).
\begin{array}{rl}{\varrho=}&{\varrho_{0}+\frac{1}{4\pi}\varepsilon_{ij}\partial_{i}[(\phi^{3}-1)A_{j}]}\\{}&{=\frac{1}{8\pi}\varepsilon_{ij}\varepsilon^{abc}D_{i}\phi^{a}D_{j}\phi^{b}\phi^{c}+\frac{1}{4\pi}B(\phi^{3}-1).}\\\end{array}
\intd^{2}x\displaystyle\sum_{n=1}^{\infty}(-1)^{n}(\pi_{(n)}^{2}+\varphi_{(n)}^{2})\leq\nonumber\,\frac{1}{2}\intd^{2}x\displaystyle\sum_{n=1}^{\infty}(-1)^{n}\frac{1}{n^{2}}(h^{2}+f^{2})=\frac{\pi^{2}}{24}\intd^{2}x(h^{2}+f^{2}).
e^{a}e^{b}\hat{\Omega}_{b,a}^{~~i}=0\quad\rightarrow\quad\hat{\Omega}_{b,a}^{~~i}=\hat{\Omega}_{a,b}^{~i}={\nabla}_{b}E_{a}^{~\underline{b}}u_{\underline{b}}^{~i},
I_{M}\equiv\beta\left(\frac{\partialJ}{\partial\mu}\right)_{M}=\left[\frac{1}{2r_{+}^{2}}+\frac{J^{2}}{8Mr_{+}^{4}\triangle}+\frac{J^{2}}{2M^{2}l^{2}r_{+}^{2}\triangle^{2}}\right]^{-1}.
\frac{d}{dr}\biggl(r^{2}\biggl(1-\frac{2M}{r}\biggr)\frac{dR}{dr}\biggr)+\biggl(-r^{2}\biggl[k^{2}+\biggl(1-\frac{2M}{r}\biggr)^{-1}\frac{n^{2}}{16M^{2}}\biggr]+p^{2}\biggr)R=0,
\mathrm{K.B}:~~~~\mathrm{\bf~Tr}_{{\calH}_{c}}\left[\frac{\Omega}{2}(-1)^{F_{s}}\,e^{-2\pit\,\mathrm{H}_{c}}\right]-\mathrm{\bf~Tr}_{\tilde{{\calH}}_{c}}\left[\frac{\Omega}{2}(-1)^{F_{s}}\,e^{-2\pit\,\mathrm{H}_{c}}\right]
L_{eff}=\intd^{2}x\;[(\partial_{t}\mid\phi\mid)^{2}+(\partial_{t}\mid\chi\mid)^{2}]+\deltaL_{eff}+\kappa\intd^{2}x\;[\frac{B_{2}}{q_{1}}\dot{\omega}_{1}+\frac{B_{1}}{q_{2}}\dot{\omega}_{2}]\;\;.
\Bigl(\frac{1}{16\piG}\Bigr)^{2}S_{1}[h]=\Bigl(\frac{1}{G\mu^{2}}\Bigr)^{2}\intd^{3}x\sqrt{h(x)}\mu^{3}\biggl\{c_{1}+c_{2}\frac{R(x)}{\mu^{2}}\biggr\}~~~,
\langle0|\partial^{\mu}D_{\mu}|0\rangle=\langle0|\theta_{\mu}^{\mu}|0\rangle=4V_{\mathrm{np}}(\Sigma_{0})\bigg|_{\Sigma_{0}=\bar{\Sigma}_{0}}=-\frac{A^{2}N_{f}^{2}}{6}\bar{\Sigma}_{0}^{3}\neq0,
L_{F}=\bar{\psi}i\gamma^{\mu}(\partial_{\mu}-ieA_{\mu})\psi-m\bar{\psi}\psi+{\frac{g}{2}}\mid\bar{\psi}\gamma^{\mu}\psi\mid^{2}-{\frac{pe^{2}}{4}}\midA_{\mu\nu}\mid^{2}+{\frac{qe^{2}}{2}}\epsilon_{\mu\nu\lambda}A^{\mu}A^{\nu\lambda}+J_{\mu}A^{\mu}.
H^{1}(dP_{9},{\calO}_{dP_{9}}(6\sigma|_{dP_{9}}-5F))|_{b_{-8}}\stackrel{-m_{1}g_{2}}{\rightarrow}H^{1}(dP_{9},{\calO}_{dP_{9}}(9\sigma|_{dP_{9}}-F))|_{c_{-3}}.
E_{0}=-\frac{\mu_{eff}^{2}}{\left(2\pi\right)^{2}}\sum_{r\,n=1}^{\infty}\frac{1}{n^{2}}K_{2}\left(2\pin\mu_{eff}\right)+\frac{\mu_{eff}^{3}}{2\pi}\sum_{n=1}^{\infty}\frac{1}{n}K_{3}\left(2\pin\mu_{eff}\right).
\Pi_{00}^{\Sigma}={\frac{4\alpha}{\beta}}\int_{0}^{1}dx\int{\frac{d|{\bfl}|}{(2\pi)}}\left[|{\bfl}|S_{1}^{\Sigma}+(|{\bfl}|^{2}+\Sigma^{2}){\frac{\partialS_{1}^{\Sigma}}{\partial|{\bfl}|}}+{\frac{p_{0}^{2}}{p}}{\frac{\partialS_{1}^{\Sigma}}{\partialp}}|{\bfl}|-(1-2x){\frac{\betap_{0b}}{2}}{\frac{\partialS_{1}}{\partial\omega}}\right]
{f^{\mathrm{odd}}}(z,z^{\prime})=\left[\begin{array}{cc}{{S^{\mathrm{odd}}}(z,z^{\prime})}&{{I^{\mathrm{odd}}}(z,z^{\prime})}\\{{D^{\mathrm{odd}}}(z,z^{\prime})}&{{S^{\mathrm{odd}}}(z^{\prime},z)}\\\end{array}\right]
(L_{m})_{ij}=\frac{1}{2}\sum_{n=-\infty}^{\infty}g_{IJ}(\alpha_{m-n}^{I}\alpha_{n}^{J})_{ij}\,(L_{0})_{ij}=\frac{1}{2}g_{IJ}(\alpha_{0}^{I}\alpha_{0}^{J})_{ij}+g_{IJ}\sum_{n=1}^{\infty}(\alpha_{-n}^{I}\alpha_{n}^{J})_{ij}\
G_{(1+2)}^{\mathrm{D}}(\tau)=360\tau^{5}(e^{\Omega}-1)\int_{0}^{\infty}d\Omega^{\prime}\int_{0}^{\Omega^{\prime}}dK\,K\,\frac{\sqrt{(\Omega^{2}-K^{2})\left[(\Omega+\Omega^{\prime})^{2}-K^{2}\right]}\,e^{\Omega^{\prime}}}{(e^{\Omega^{\prime}}-1)(e^{\Omega+\Omega^{\prime}}-1)}.
\begin{array}{l}{h_{0}(p)=1}\\{h_{1}(p)=\frac{-1}{24}(2p-1)}\\{h_{2}(p)=\frac{1}{5760}(20p^{2}-48p+7)}\\{h_{3}(p)=\frac{-1}{2903040}(280p^{3}-1596p^{2}+1874p-93),}\\\end{array}
\hat{l}\left(\begin{array}{c}{u}\\{v}\\\end{array}\right)=\hat{h}\left(\begin{array}{c}{u}\\{v}\\\end{array}\right)+\left(\begin{array}{c}{G_{2}^{u}(u,v)}\\{G_{2}^{v}(u,v)}\\\end{array}\right)+\left(\begin{array}{c}{G_{3}^{u}(u,v)}\\{G_{3}^{v}(u,v)}\\\end{array}\right)+\cdots
{\calF}=\frac{H^{2}}{1+H^{2}r^{2}}\left(\begin{array}{ccc}{0}&{2E\dot{r}}&{0}\\{0}&{2E^{2}-m^{2}\alpha^{2}+H^{2}m^{2}r^{2}}&{0}\\{0}&{0}&{(1+H^{2}r^{2})m^{2}\alpha^{2}}\\\end{array}\right).
N(\theta)=\quad\frac{\textrm{sn}\left[\frac{\textbf{K}}{i\pi}\left(\theta-i\frac{\pi}{2}a\right)\right]}{\textrm{sn}\left[\frac{\textbf{K}}{i\pi}\left(\theta+i\frac{\pi}{2}a\right)\right]}\frac{\textrm{sn}\left[\frac{\textbf{K}}{i\pi}\left(\theta+i\frac{\pi}{2}(1+a)\right)\right]}{\textrm{sn}\left[\frac{\textbf{K}}{i\pi}\left(\theta-i\frac{\pi}{2}(1+a)\right)\right]}\,\,\,,
\begin{array}{lclclcl}{\pi^{0}}&{=}&{\sqrt{1/2}(u\bar{u}-d\bar{d})}&{}&{K^{0}}&{=}&{d\bar{s}}\\{\pi^{-}}&{=}&{d\bar{u}}&{}&{K^{+}}&{=}&{u\bar{s}}\\{\pi^{+}}&{=}&{u\bar{d}}&{}&{K^{-}}&{=}&{s\bar{u}}\\{\eta}&{=}&{\sqrt{1/3}(u\bar{u}+d\bar{d}+s\bar{s})}&{}&{\bar{K}^{0}}&{=}&{s\bar{d}}\\\end{array}
\int{\frac{d^{d}k}{(2\pi)^{d}}}\,e^{i{\bfk}\cdot{\bfx}}{\frac{k_{i}k_{j}}{|{\bfk}|^{2}}}|{\bfk}|^{\delta}=-{\frac{1}{\delta}}\left\{\delta_{ij}-(\delta+d){\frac{x_{i}x_{j}}{|{\bfx}|^{2}}}\right\}{\frac{C_{d,\delta}}{|{\bfx}|^{d+\delta}}}.
\frac{1}{q^{2}-m^{2}+i{\varepsilon}}{\delta}^{\prime}(q_{-})=-\frac{1}{m^{2}+n_{-}q_{+}^{\;\;2}}{\delta}^{\prime}(q_{-})+\frac{2n_{+}q_{+}}{(m^{2}+n_{-}q_{+}^{\;\;2})^{2}}{\delta}(q_{-}).
\langleV_{3}|L_{-k}^{(2)}=\langleV_{3}|\left(A^{k}\cdotc+\sum_{n\ge0}a_{n}^{k}L_{n}^{(1)}+\sum_{n\ge0}b_{n}^{k}L_{n}^{(2)}+\sum_{n\ge0}c_{n}^{k}L_{n}^{(3)}\right)
S_{cl}=\frac{1}{2\pi}\intd^{2}x\;[e^{-2\phi}(2\partial_{+}\partial_{-}\rho-4\partial_{+}\phi\partial_{-}\phi+\lambda^{2}e^{2\rho})+\frac{1}{2}\sum_{i=1}^{N}\partial_{+}f_{i}\partial_{-}f_{i}]
S_{E}=V_{(1)}V_{(2)}...V_{(T)}\left\{-\left[\frac{\partial\beta}{\partial\xi}\right]_{\xi_{S}}^{\xi_{N}}-\frac{1}{n_{(1)}+n_{(2)}+...+n_{(T)}-1}\int\mathrm{d}\xi\beta(\xi)\left(2{\calV(\phi)}-\frac{(d_{f}-1)(M^{2}-E^{2})}{\prod_{i=lit}a_{i}^{2n_{i}}}\right)\right\}
\frac{M_{i}}{4(k^{2}-M_{i}^{2}+i0)}\,\bar{D}^{2}\delta^{4}(\theta_{1}-\theta_{2})\quad\mathrm{and}\quad\frac{M_{i}}{4(k^{2}-M_{i}^{2}+i0)}\,D^{2}\delta^{4}(\theta_{1}-\theta_{2})
Q(r)=Q(\bar{r})+\left[{\frac{dQ}{d\mu^{2}}}\right]_{\bar{r}}\left(\mu^{2}(r)-\mu^{2}(\bar{r})\right)+\left[{\frac{dQ}{d(\Gamma^{-2})}}\right]_{\bar{r}}\left(\Gamma^{-2}(r)-\Gamma^{-2}(\bar{r})\right)+...
E_{\pm}=Ae^{-\left(Q^{X}+\frac{2}{Q^{X}+Q^{T}}\right)X+\left(Q^{T}-\frac{2}{Q^{X}+Q^{T}}\right)T}\pmBe^{-\left(Q^{X}+\frac{2}{Q^{X}-Q^{T}}\right)X+\left(Q^{T}+\frac{2}{Q^{X}-Q^{T}}\right)T}\,
\sqrt{2}V_{L}^{\vphantom{T}}L\partial_{i}V_{R}^{T}=-e^{-(A+\eta/2)}E_{i}^{(L)}\hat{n}-e^{A-\eta/2}{\frac{\hat{g}_{ij}}{\sqrt{\hat{g}}}}(B_{j}^{(L)}+\epsilon_{jkl}E_{k}^{(L)}C_{l})\hat{m}\.
V_{i}\cdotV_{j}=\left(\begin{array}{ccc}{3}&{4}&{2}\\{4}&{4}&{2}\\{2}&{2}&{2}\\\end{array}\right)~,\quad\quadk_{ij}=\left(\begin{array}{ccc}{k_{00}}&{k_{\theta^{2}0}}&{k_{r0}}\\{k_{\theta^{2}0}}&{k_{\theta^{2}0}}&{k_{r\theta^{2}}}\\{k_{r0}}&{k_{r\theta^{2}}}&{k_{r0}}\\\end{array}\right)~,
\begin{array}{rcl}{a^{-2}\left[\nabla^{2}\hat{h}_{\mu\nu}+2R^{\rho}{}_{(\mu\nu)}{}^{\sigma}\hat{h}_{\rho\sigma}\right]-\hat{h}_{\mu\nu}^{\prime\prime}}&{}&{}\\{}&{}&{}\\{-(\hat{d}-5)a^{-1}a^{\prime}\hat{h}_{\mu\nu}^{\prime}+2[(\hat{d}-4)a^{-2}(a^{\prime})^{2}+a^{-1}a^{\prime\prime}]\hat{h}_{\mu\nu}}&{=}&{0\,.}\\\end{array}
\begin{array}{c}{E_{1}=Ze_{1}=e_{1}Z^{\dagger};\;\;\;\;E_{5}=Ze_{5}=e_{5}Z^{\dagger};}\\{E_{2}=Ze_{2}=e_{2}Z^{\dagger};\;\;\;\;E_{3}=Ze_{3}=e_{3}Z^{\dagger};}\\{E_{4}=Z^{\dagger}e_{4}=e_{4}Z;\;\;\;\;E_{6}=Z^{\dagger}e_{6}=e_{6}Z;}\\{E_{7}=Ze_{7}=e_{7}Z.}\\\end{array}
{\hat{Y}}_{l}^{I}{\hat{Y}}_{m}^{J}=\sum_{K=0}^{N}(-)^{N+K}\sqrt{(2I+1)(2J+1)}\Bigl\{\begin{array}{ccc}{I}&{J}&{K}\\{{\frac{N}{2}}}&{{\frac{N}{2}}}&{{\frac{N}{2}}\}\\\end{array}\Bigr\}C_{lmn}^{IJK}{\hat{Y}}_{n}^{K}
S=\intd^{4}x\sqrt{|g_{4}|}\left[e^{-\Phi}\left(R+\left(\nabla\Phi\right)^{2}-6\left(\nabla\beta\right)^{2}-\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}\right)-\frac{1}{48}e^{6\beta}F_{\mu\nu\lambda\kappa}F^{\mu\nu\lambda\kappa}\right],
{\calL}=\frac{1}{2}\left\{\partial_{m}\phi_{i}^{\prime}\partial^{m}\phi_{i}^{\prime}+\imath\bar{\psi}_{i}\gamma^{m}\partial_{m}\psi_{i}-{\calW}_{i}{\calW}_{ikl}\phi_{k}^{\prime}\phi_{l}^{\prime}-{\calW}_{ik}{\calW}_{il}\phi_{k}^{\prime}\phi_{l}^{\prime}-{\calW}_{ij}\bar{\psi}_{i}\psi_{j}\right\},
L^{-1}:=\overline{{L}}=\{\overline{{L}}_{M}^{A}\}=\left(\begin{array}{cc}{\overline{{L}}_{\alpha}^{a}}&{\overline{{L}}_{\alpha}^{i}}\\{0}&{\overline{{L}}_{\alpha}^{i}}\\\end{array}\right)\,,\quad\overline{{L}}_{\mu}^{a}=0\,,
a_{S}^{\dagger}(k)\,a_{S}^{\dagger}(k^{\prime})-q^{-2}\sum_{l,l^{\prime}\in{\calI}}{\hat{\calR}}\left(k,k^{\prime};l,l^{\prime}\right)a_{S}^{\dagger}(l)\,a_{S}^{\dagger}(l^{\prime})=0\;,
\begin{array}{ccccc}{\delta_{j}}&{\Longleftrightarrow}&{({\mathbbE}_{j})_{\mu\nu}}&{=}&{\delta_{0\mu}\delta_{j\nu}-\delta_{0\nu}\delta_{j\mu}-f_{j\mu\nu}\quad,}\\{1|\delta_{j}}&{\Longleftrightarrow}&{(1|{\mathbbE}_{j})_{\mu\nu}}&{=}&{\delta_{0\mu}\delta_{j\nu}-\delta_{0\nu}\delta_{j\mu}+f_{j\mu\nu}\quad,}\\\end{array}
\int_{0}^{\ell}dy\;\{\xi_{\alphaA}(x),\xi_{\betaB}(y)\}\;\;\Omega\Big|_{\big({\frac{\delta}{\delta\xi_{\betaB}(y)}},\;\chi\big)}=\int_{0}^{\ell}dy\;P_{(\alphaA)(\gammaD)}(x,y)\;\chi_{\gammaD}(y)\quad,