harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
U_{\alpha}(t)=\frac{1}{\sqrt{2}}\left(\begin{matrix}{f_{\alpha}^{(-)}(t)+f_{\alpha}^{(+)}(t)}\\{f_{\alpha}^{(-)}(t)-f_{\alpha}^{(+)}(t)}\\\end{matrix}\right),~~V_{\alpha}(t)=\frac{1}{\sqrt{2}}\left(\begin{matrix}{g_{\alpha}^{(-)}(t)+g_{\alpha}^{(+)}(t)}\\{g_{\alpha}^{(-)}(t)-g_{\alpha}^{(+)}(t)}\\\end{matrix}\right),
\alpha=\omega\displaystyle\frac{p+1}{p}+\chi_{\infty}\left(\left[\displaystyle\frac{1}{2}+\displaystyle\frac{S}{p+1}+\displaystyle\frac{\omega}{\pi}\right]-\left[\displaystyle\frac{1}{2}+\displaystyle\frac{S}{p+1}-\displaystyle\frac{\omega}{\pi}\right]\right)
\delta_{\vec{u}}(t)<W(\gamma)>=-{\frac{4\pii}{k}}\int_{0}^{1}dsu^{a}(s)({\eta^{\prime}}^{b}(s)+t{u^{\prime}}^{b}(s))\epsilon_{abc}\oint_{\gamma}dy^{c}\delta^{3}(\eta(s,t)-y)\left<\mathrm{Tr}\left(\tau^{l}H(\eta(t)_{x}^{o})H(\gamma_{o}^{y})\tau^{l}H(\gamma_{y}^{o})H(\eta(t)_{o}^{x})\right)\right>.
\begin{array}{ll}{\{Q^{\alpha},Q^{\beta}\}=-2\tilde{\Gamma}^{m\alpha\beta}P_{m},}&{[P_{m},Q^{\alpha}]=-i\tilde{\Gamma}_{m}{}^{\alpha\beta}\Sigma_{\beta},}\\{{}[M_{mn},Q^{\alpha}]=\displaystyle\fraci4Q^{\beta}(\Gamma_{mn})_{\beta}{}^{\alpha},}&{[M_{mn},\Sigma_{\alpha}]=-\displaystyle\fraci4(\Gamma_{mn})_{\alpha}{}^{\beta}\Sigma_{\beta}}\\\end{array}
{\calZ}_{2N}=K(2N)!\int_{\stackrel{\cdots}{R(-\infty,x_{1},\ldots,x_{2N},\infty)}}\int\prod_{i=1}^{2N}e^{-\frac{\beta}{2}\sum_{i=1}^{2N}V(x_{i})}\times[det(x_{j}^{i-1})]_{i,j=1,\ldots,2N}
\delta_{j_{1}0}\delta_{j_{2}0}=\int_{0}^{\frac{2\pi}{l}}dk\int_{0}^{l/A}dqe^{-ij_{1}lk+ij_{2}l\tau_{2}q}|\tilde{c}(k,q)|^{2}\sum_{n=0}^{A-1}|C_{0}(k,q+\frac{ln}{A})|^{2}.
(-m_{\pi}^{2}+\partial_{\bot}^{2})\,\omega_{\pi}^{(1)}=\frac{\lambda}{2L}\int_{-L}^{L}dx^{-}(\varphi_{\pi}^{3}+\varphi_{\pi}\varphi_{\sigma}^{2}+2v\varphi_{\pi}\varphi_{\sigma}),
V_{\mathrm{CMS}}=-\frac{\mathrm{Im}(a^{*}\,a_{D})}{r}\,\frac{\left(n_{E}^{(1)}n_{M}^{(2)}-n_{M}^{(1)}n_{E}^{(2)}\right)^{2}}{\left|\,n_{E}^{(1)}\,a+n_{M}^{(1)}\,a_{D}\right|\left|\,n_{E}^{(2)}\,a+n_{M}^{(2)}\,a_{D}\right|}+{\calO}(\omega^{2})\,.
=\frac{\epsilon^{0ijk5}}{(4!)^{2}}\left\{\partial_{0}\left[\frac{\sigma^{\prime}}{((\sigma^{\prime}/b)^{2}-(\dot{\sigma}/n)^{2})^{2}}\right]-\partial_{5}\left[\frac{\dot{\sigma}}{((\sigma^{\prime}/b)^{2}-(\dot{\sigma}/n)^{2})^{2}}\right]\right\}=0.
D({\tau}_{\pm}^{q})^{-1}\tilde{F}=({\tau}_{\pm}^{q})^{-1}\Big(d+\tau_{\pm}^{q}d({\tau}_{\pm}^{q})^{-1}-iqA_{\pm}\,{\tau}_{\pm}^{q}\sigma_{3}({\tau}_{\pm}^{q})^{-1}\Big)\tilde{F}
n^{\mu}=\left(-{\frac{\dot{a}}{F(a)}},-\sqrt{F(a)+\dot{a}^{2}},0,0,0\right);\qquadn_{\mu}=\left(+\dot{a},-{\frac{\sqrt{F(a)+\dot{a}^{2}}}{F(a)}},0,0,0\right).
\sigma_{1}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right);\sigma_{2}=\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right);\sigma_{3}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)
H^{2}=\frac{\rho_{m}^{2}}{36}+\frac{U_{B}\rho_{m}}{12}-\frac{1}{16a^{4}}\intd\tau\frac{da^{4}}{d\tau}(\dot{\phi}^{2}-2U)-\frac{1}{12a^{4}}\intd\taua^{4}\rho_{m}\frac{dU_{B}}{d\tau},
\intd^{3}\vec{y}\,\,S(\vec{x},\vec{y})S^{-1}(\vec{y}-\vec{z})=\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{\delta_{\mu}^{\rho}}&{0}\\{0}&{0}&{\delta_{j}^{k}}\\\end{array}\right)\,\delta(\vec{x}-\vec{z})\,.
S_{R}=-\intd^{2}x\,\sqrt{g}\,\left\{\frac{1}{2}Zg^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi+CR+V+\frac{1}{\epsilon}\left(-\frac{V^{\prime}}{C^{\prime}}+\frac{ZV}{2{C^{\prime}}^{2}}\right)\right\}.
R(u)=\left(\begin{array}{ccc|ccc|ccc}{a}&{}&{}&{}&{}&{}&{}&{}&{}\\{}&{b}&{}&{d}&{}&{}&{}&{}&{}\\{}&{}&{c}&{}&{e}&{}&{f}&{}&{}\\\hline{}&{d}&{}&{b}&{}&{}&{}&{}&{}\\{}&{}&{e}&{}&{g}&{}&{e}&{}&{}\\{}&{}&{}&{}&{}&{b}&{}&{d}&{}\\\hline{}&{}&{f}&{}&{e}&{}&{c}&{}&{}\\{}&{}&{}&{}&{}&{d}&{}&{b}&{}\\{}&{}&{}&{}&{}&{}&{}&{}&{a}\\\end{array}\right),
{\hat{\calC}}^{a}|Q_{i}^{\prime},{\calC}^{\primea},\bar{\calC}_{a}^{\prime},{\pi}_{a}^{\prime}\rangle=\hat{\bar{\calC}}_{a}|Q_{i}^{\prime},{\calC}^{\primea},\bar{\calC}_{a}^{\prime},{\pi}_{a}^{\prime}\rangle={\hat{\pi}}_{a}|Q_{i}^{\prime},{\calC}^{\primea},\bar{\calC}_{a}^{\prime},{\pi}_{a}^{\prime}\rangle=0,
\Phi_{k}\left(\frac{a\tau+b}{c\tau+d},\frac{\boldsymbol{z}}{c\tau+d}\right)=(c\tau+d)^{k}{\bfe}\left[{\frac{c\langle\boldsymbol{z},\boldsymbol{z}\rangle}{2(c\tau+d)}}\right]\Phi_{k}(\tau,\boldsymbol{z})\,.
\begin{array}{c}{\nabla_{\rho}\nabla^{\rho}(\nabla_{\sigma}\Lambda^{\sigma\mu_{3}\ldots\mu_{s}})+\tilde{m}^{2}\nabla_{\sigma}\Lambda^{\sigma\mu_{3}\ldots\mu_{s}}=0}\\{\mathrm{with}\;\;\tilde{m}^{2}=s^{2}+(d-2)s-2}\\\end{array}
\left(\begin{array}{cccccc}{\mu_{1}+1/r_{N1}+1/r_{12}}&{-1/r_{12}}&{0}&{\cdots}&{0}&{-1/r_{N1}}\\{-1/r_{12}}&{\mu_{2}+1/r_{12}+1/r_{23}}&{-1/r_{23}}&{\cdots}&{0}&{0}\\{0}&{-1/r_{23}}&{\mu_{3}+1/r_{23}+1/r_{34}}&{\cdots}&{0}&{0}\\{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}\\{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}\\\end{array}\right)
W_{D}S^{(0)}[\phi]=-\intdx\bigg[\partial^{\rho}\left(x^{\mu}*T_{\rho\mu}^{I}\right)+\underbrace{{\frac{\lambda}{4!}}x^{\mu}*\left(4\partial_{\mu}\phi*(\phi)_{*}^{3}-\partial_{\mu}(\phi)_{*}^{4}\right)}_{=:B}\bigg].
\{p_{\alpha}^{+}{}^{\beta},\chi_{\gamma}\}=0,\qquad\{p_{\alpha}^{+}{}^{\beta},p_{\gamma}^{-}{}^{\delta}\}=0,\qquad\{p_{\alpha}^{+}{}^{\beta},p_{\gamma}^{+}{}^{\delta}\}=0;
q=\left(\begin{array}{ccc}{\frac{m_{2}+2m_{3}}{m_{1}+m_{2}+m_{3}}I_{m_{1}}}&{0}&{0}\\{0}&{\frac{-m_{1}+m_{3}}{m_{1}+m_{2}+m_{3}}I_{m_{2}}}&{0}\\{0}&{0}&{\frac{-2m_{1}-m_{2}}{m_{1}+m_{2}+m_{3}}I_{m_{3}}}\\\end{array}\right),
\Gamma_{\mu}(\partial,\stackrel{\leftarrow}{\partial}^{\dagger})=a_{\mu}^{0}+\sum_{l=1}^{N-1}\sum_{k=0}^{l}\overline{{[\mu_{1},...,\mu_{k}]}}a_{\mu\mu_{1}...\mu_{l}}^{k}[\mu_{k+1},...,\mu_{l}]
V_{\mu_{1}\mu_{2}\mu_{3}}^{a_{1}a_{2}a_{3}}(p,q,r)=-\frac{is}{3!4\pi}(2\pi)^{3}f^{a_{1}a_{2}a_{3}}\epsilon^{\mu_{1}\mu_{2}\mu_{3}}\delta^{(3)}(p+q+r)
\simeq\frac{-\lambda_{1}(a^{+}\Gamma_{1}\Lambda_{3})+i\lambda_{2}(\Lambda_{3}\Gamma_{2}\Gamma_{3}a^{+})}{\sqrt{2}}u_{R}(\lambda_{1},\lambda_{2})\mid0\,\rangle\mid\psi(R,\Lambda_{3})\,\rangle.
D_{z}^{\dagger}=\left(\begin{array}{ccc}{\bar{z}_{2}+a\bar{z}_{4}}&{-z_{1}-az_{3}}&{\sqrt{\zeta(1+a^{2})}}\\{\bar{z}_{1}+a\bar{z}_{3}}&{z_{2}+az_{4}}&{0}\\\end{array}\right)\,.
\sum_{i=1}^{N_{1}}\sum_{r=N_{1}+1}^{N}{\bfk}_{i}\circ{\bar{\bfk}}_{r}=-{\bfk}\circ{\bar{\bfk}}={\frac{1}{8\pi\alpha^{\prime}}}\sum_{\mu,\nu=0}^{p}\Theta^{\mu\sigma}({\calM})_{\sigma\lambda}\Theta^{\lambda\nu}~p_{\mu}^{i}p_{\nu}^{j}\quad,
\tilde{{\calL}}_{\mathrm{Bose}}=-g_{A\bar{B}}\partial_{\mu}\lambda^{A}\partial^{\mu}\bar{\lambda}^{B}-g^{A\bar{B}}\frac{\partial{\calW}}{\partial\lambda^{A}}\frac{\partial\bar{{\calW}}}{\partial\bar{\lambda}^{B}}
\begin{array}{l}{<\chi^{0}\parallel\hat{\Phi}(\zeta)\parallel\chi>=\S_{\lambda\mu=1}^{2}\S_{r^{\lambda\mu}=1}^{N_{\lambda\mu}}\bar{c}(r^{\lambda\mu})F_{r^{\lambda\mu}}(\zeta),}\\{<\chi\parallel\bar{\hat{\Phi}}(\zeta)\parallel\chi^{0}>=\S_{\lambda\mu=1}^{2}\S_{r^{\lambda\mu}=1}^{N_{\lambda\mu}}{\bar{c}}^{*}(r^{\lambda\mu})F^{r^{\lambda\mu}}(\zeta).}\\\end{array}
{\mathcal{K}}_{B}(\hat{X_{B}})|_{X_{B}\to\infty}=(2\pi)^{n_{b}/2}\prod_{l=1}^{n_{b}}\left[X_{B}^{(l)}\right]^{-2N}\prod_{l=1}^{n_{b}}\Gamma\left[2(N-n_{b}+l)-1/2\right]
\imath\delta_{\b}^{a}=\imathG^{ca}\circG_{(0)\ab}^{-1}-\frac{\lambda}{2}h_{abef}\circG^{ca}G^{ef}-\frac{\imath\lambda}{3!}G^{ca}h_{beld}\circ\alpha_{afgh}\circG^{fl}G^{gd}G^{he}.
ds_{dual}^{2}\simeq\underbrace{\left(k\right)^{-\frac{5-p}{7-p}}r^{5-p}\,\,dx^{\mu}\otimesdx^{\nu}\,\eta_{\mu\nu}+\left(k\right)^{\frac{2}{7-p}}\,\frac{dr^{2}}{r^{2}}}_{\mathrm{~\mathrm{AdS}~-metric}}+\,\left(k\right)^{\frac{2}{7-p}}\,ds_{S^{8-p}}^{2}
\langle\partial_{\mu}j_{R}^{\mu}\rangle=-\frac{1}{16\pi^{2}}\mathrm{Im}\\mathrm{tr}\intd^{2}\theta_{1}d^{2}\theta_{2}\left[\Phi^{2}+\frac{\Lambda^{4}}{2\Phi^{2}}+O(\Lambda^{8})\right],
\Gamma_{I}^{\phi^{-}}(p)=-i/\left\{(p^{2}+i\varepsilon)\left[K_{UD}(p)+H_{UD}(p)\right]+E_{UD}(p)-i(p^{2}-\bar{M}^{2}+i\varepsilon)S_{UD}^{I}(p)\right\},
\displaystyle\mathop{Z_{(\Theta^{4},1)}^{torus}(\tau,\bar{\tau},T,U)={\frac{1}{4}}{\sum_{\stackrel{m_{1},m_{2}\inZ}{n^{1},n^{2}\inZ}}}e^{2\pii\tau(m_{1}\frac{n^{1}}{2}+m_{2}\frac{n^{2}}{2})}e^{{\frac{-\pi\tau_{2}}{T_{2}U_{2}}}|TU{\frac{n^{2}}{2}}+T{\frac{n^{1}}{2}}-Um_{1}+m_{2}|^{2}}}
S_{free}=\frac{1}{4\pi\alpha^{\prime}}\intd^{2}z\partial^{\alpha}\xi_{\mu}\partial_{\alpha}\xi^{\mu}+\frac{i}{4\pi\alpha^{\prime}}\int_{z^{2}=0}dz^{1}F_{\mu\nu}\xi^{\nu}\partial_{1}\xi^{\mu}.
\{\rho,{\calQ}\}^{(3)}=\Omega_{s^{\prime}}V_{-s^{\prime}}-\Omega_{s^{\prime}}v_{-s^{\prime}}^{(2)}+\omega_{s^{\prime}}^{2}\eta_{-t}\bar{\eta}_{-s^{\prime}}[G_{s^{\prime}},v_{t}^{(2)}]+iC_{s^{\prime}t}^{u}\Omega_{-s^{\prime}}\eta_{-t}\pi_{u},
(x_{\mu}^{i}-x_{\mu}^{j})S_{(ij)}^{\mu\nu}=(\bar{\xi^{i}}-\bar{\xi^{j}})\Gamma^{\mu\alpha\nu}(\xi^{i}-\xi^{j})\frac{(x_{\mu}^{i}-x_{\mu}^{j})(x_{\alpha}^{i}-x_{\alpha}^{j})}{(x^{i}-x^{j})^{4}}=0.
{\frac{1}{2}}\delta\Lambda_{I}=-\gamma^{\mu}\partial_{\mu}\phi\,\varepsilon_{I}+{\frac{1}{\sqrt{2}}}\sigma^{\mu\nu}(e^{-\phi}F_{\mu\nu}\alpha_{IJ}-e^{\phi}\tilde{G}_{\mu\nu}\beta_{IJ})^{-}\,\varepsilon^{J}.
\begin{array}{ll}{V_{\phi}(1,\cdots,N)=}&{exp\left\{\frac{1}{2}\sum_{r\neqs}\phi_{0}^{(r)}\,\tilde{N}_{0,0}^{r,s}\phi_{0}^{(s)}+\right.}\\{}&{\left.+\sum_{r,s}\sum_{n}\phi_{0}^{(r)}\tilde{N}_{0,n}^{r,s}\phi_{n}^{(s)}+\frac{1}{2}\sum_{r,s}\sum_{n,m}\phi_{0}^{(r)}\tilde{N}_{n,m}^{r,s}\phi_{m}^{(s)}\right\}.}\\\end{array}
\sum_{ij}\left\langle\frac{1}{x_{i}-\phi_{a}}\right\rangle_{N}\left\langle\frac{1}{y_{j}-\chi_{b}}\right\rangle_{N}=N^{2}\left\langle\frac{1}{x_{1}-\phi_{a}}\right\rangle_{N}\left\langle\frac{1}{y_{1}-\chi_{b}}\right\rangle_{N}
{\calL}=\left(\partial_{\mu}\phi\right)^{\dagger}\left(\partial^{\mu}\phi\right)-(m^{2}-k^{2}X^{2})\phi^{\dagger}\phi-ikX_{\mu}[\phi\partial^{\mu}\phi^{\dagger}-\phi^{\dagger}\partial^{\mu}\phi]-\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho},
h_{l}(z)=-\pii-\frac{e^{-\frac{\pii}{N_{l}}}\Gamma(\frac1{N_{l}})}{\displaystyle{\prod_{u\in\bar{U}_{l}}}\Gamma(1-\frac{N_{u}}{N_{s}})}\,\psi_{l}+O(\psi_{l}^{2}),
\left\langle\Omega(\psi_{n},\bar{\psi}_{n};p_{0})\right|=\left\langle-p_{0N}\right|\otimes\prod_{n=1}^{\infty}\left(-\frac{\sqrt{\alpha^{\prime}}}{2}\psi_{n},-\frac{\sqrt{\alpha^{\prime}}}{2}\bar{\psi}_{n}\right|~,
\frac{\partialX^{M}}{\partial\sigma^{a}}\nabla_{M}\Bigl(\frac{\partialX^{N}}{\partial\sigma^{b}}\partial_{N}\Bigr)=\tilde{\Gamma}_{ab}^{c}\Bigl(\frac{\partialX^{N}}{\partial\sigma^{c}}\partial_{N}\Bigr)+K_{ab}^{i}n_{i}~.
S=\frac{N^{2}}{16\piG_{10}}\intd^{10}x\sqrt{-g}(Ne^{\phi})^{\lambda}\left[R+\frac{4(p-1)(p-4)}{(7-p)^{2}}(\partial\phi)^{2}-\frac{1}{2N^{2}(8-p)!}F_{8-p}^{2}\right],
\deltaP_{\mu(1)}^{A}=\frac{1}{2\pi\alpha^{\prime}}\sqrt{-\bar{G}}\left((\bar{D}_{ij}\;^{A}\Phi_{(1)}^{j})\bar{n}_{\mu}^{i}+(\bar{\Omega}_{i}\;^{AB}-\bar{G}^{AB}\bar{\Omega}_{iC}\;^{C})\Phi_{(1)}^{i}\bar{x}_{\mu,B}\right)
(D_{\mu})_{i}{}^{j}=D_{\mu}^{G}\delta_{i}^{j}\pm\frac14\varepsilon^{abc}(S_{\mu}^{-}-P_{\mu})_{ab}\tau^{c}{}_{i}{}^{j}+e^{\varphi}Q_{\mu}^{a+}\tau^{a}{}_{i}{}^{j}.
\frac{\partial^{2}\varphi_{L(I)}(s)}{\partials^{2}}+3\Bigl(\frac{\partiala/\partials}{a}\Bigr)\frac{\partial\varphi_{L(I)}(s)}{\partials}+\frac{\delta^{2}V(\hat{\phi})}{\delta\hat{\phi}^{2}}\varphi_{L(I)}(s)=0.
\dot{x}{}^{a}=\kappa^{-1}\bar{\pi}{}^{A}\pi^{A^{\prime}},\,\,\,\,\acute{x}{}^{a}=i\zeta(\dot{\bar{\pi}}{}^{A}\pi^{A^{\prime}}-\bar{\pi}{}^{A}\dot{\pi}{}^{A^{\prime}}).
G[\chi_{2},\chi_{1};\psi_{2},\psi_{1}]~=\int_{0}^{\infty}\prod_{x}dN(x)\int{\calD}\psi{\calD}\chie^{-\intd^{2}x[{\frac{1}{2N}}{({\dot{\psi}}^{2}+{\frac{1}{4}}{{\dot{\chi}}}^{2})+{\frac{1}{8}}\mu^{2}Ne^{{\alpha\psi+\beta\chi}}}]}.
L=\frac{1}{2h^{2}}g^{\mu\nu}\partial_{\mu}\sigma\partial_{\nu}\sigma-\frac{N}{2\lambda_{1}h^{2}}\sigma^{2}+\frac{N}{2\xi_{1}h^{2}}R\sigma^{2}-\frac{N}{4!\lambda_{2}h^{4}}\sigma^{4}+\bar{\psi}(i\gamma^{\mu}(x)\nabla_{\mu}-\sigma)\psi,
t_{\mu\nu}^{(\pm)}(p,k)=\mathrm{tr}(I)\sum_{n=-\infty}^{+\infty}\,\frac{-p_{\mu}(p+k)_{\nu}-p_{\nu}(p+k)_{\mu}+[p\cdot(p+k)+\omega_{n}^{(\pm)}]\delta_{\mu\nu}}{(p^{2}+(\omega_{n}^{(\pm)})^{2})[(p+k)^{2}+(\omega_{n}^{(\pm)})^{2}]}
\begin{array}{l}{\delta\theta^{\alpha}=\epsilon^{\alpha},\qquad\delta\bar{\theta}_{\alpha}=\bar{\epsilon}_{\alpha},}\\{\deltax^{\bar{\mu}}=-i\bar{\epsilon}_{\alpha}\tilde{\Gamma}^{{\bar{\mu}}\alpha\beta}\bar{\theta}_{\beta}+i\epsilon^{\alpha}\Gamma_{\alpha\beta}^{\bar{\mu}}\theta^{\beta},\qquad\deltax^{11}=0,}\\\end{array}
\psi(x)=Z^{a+\frac{1}{2}}\,\,\,r^{-(l+1)}\,\,\,(y-1)^{\frac{(\lambda-s)}{2}}\,\,\,(y+1)^{\frac{-(\lambda+s)}{2}}\,\,\,P_{n}^{(\lambda-s-\frac{1}{2}),(-\lambda-s-\frac{1}{2})}(y)\,\,\,P_{k}(x)
Z_{ab}^{S}=\left(\begin{array}{cccc}{0}&{z_{+}^{2}}&{0}&{0}\\{z_{+}^{2}}&{0}&{0}&{0}\\{0}&{0}&{z_{H}^{2}}&{0}\\{0}&{0}&{0}&{z_{\chi}^{2}}\\\end{array}\right)\qquad\begin{array}{ccc}{Z_{f_{i}}^{R}}&{=}&{\tilde{z}_{f_{i}}^{2}}\\{Z_{f_{i}}^{L}}&{=}&{z_{f_{i}}^{2}}\\\end{array}
\lambda_{SW}=-{\frac{1}{2\pii}}{\frac{1}{\sqrt{-\mu}}}\sum_{n=0}^{\infty}{\frac{(2n)!}{2^{2n}(n!)^{2}}}{\frac{W_{G}(x,\tilde{t})^{{\frac{2n+1}{2}}}dx}{(2n+1)(4\mu)^{n}}},
a^{\mu}(x,\tau)=\sum_{s=\mathrm{polarizations}}\int\frac{d^{4}k}{2\kappa}\left[\varepsilon_{s}^{\mu}a(k,s)e^{i(k\cdotx+\sigma\kappa\tau)/\hbar}+\varepsilon_{s}^{\mu*}a^{*}(k,s)e^{-i(k\cdotx+\sigma\kappa\tau)/\hbar}\right]
A_{\mu}(x)=\int_{(\infty)}\frac{d^{3}\vec{k}}{(2\pi)^{3}2|\vec{k}\,|}\sum_{\lambda=\pm}\left[e^{-ik\cdotx}\epsilon_{\mu}(\vec{k},\lambda)a(\vec{k},\lambda)\+\e^{ik\cdotx}\epsilon_{\mu}^{*}(\vec{k},\lambda)a^{\dagger}(\vec{k},\lambda)\right]\,
\left(-\frac{\sqrt{30}}{3r_{0}^{\2}}\,\right)_{1}\left(\frac{c^{3}}{16\piG}\right)_{2}\left(c\right)_{3}\left(\frac{16\piG}{c^{4}}\right)_{4}^{\1/2}\left(\frac{128\hbarGr_{0}}{\sqrt3\,c^{3}}\right)_{5}\left(-\frac12\right)_{6}\left(\frac12\right)_{7},
{\widetildeH}_{BF}=H+{\displaystyle\int}d^{2}x\left\{\phi\phi^{*}(\Phi^{1})^{2}+\left(\frac{2\pi}{\alpha}\right)^{2}\phi\phi^{*}(\Phi^{2})^{2}-2\phi\phi^{*}\left[\Phi^{1}{\widetildeQ}_{1}-\Phi^{2}{\widetildeQ}_{2}\right]\right\}
U_{0}=\left(\begin{array}{cc}{Z_{1}+Z_{2}}&{0}\\{0}&{Z_{1}+Z_{2}+\zeta}\\{-\sqrt{\zeta+\rho^{2}}\bar{z}_{2}}&{\rhoz_{1}}\\{-\sqrt{\zeta+\rho^{2}}\bar{z}_{1}}&{-\rhoz_{2}}\\\end{array}\right)
e^{-2\phi}=\left((e^{-2\hat{\phi}_{0}}+{\frac{2\tilde{m}_{k}}{r}})((A_{u})_{0}+{\frac{2\hat{m}_{k}}{r}})-\left[(A_{i})_{0}+{\frac{2(q_{k})_{i}}{r}}\right]^{2}\right)^{\frac{1}{2}}\.
R+\frac{\gamma}{4}(\frac{(D\Omega)^{2}}{\Omega^{2}}-2\frac{D\cdotD\Omega}{\Omega})+(1-\lambda)\mu\Omega^{-\lambda}-\frac{1}{4}(1-\frac{\epsilon}{2})\Omega^{-\epsilon/2}F^{2}=T_{\Omega}^{X},
\sum_{k_{1},k_{2}=-\infty}^{+\infty}\,<z|\hat{U}(k_{1},k_{2})|z>=\sum_{k_{1},k_{2}=-\infty}^{+\infty}\<z|e^{i\pikk_{1}k_{2}}\,e^{i\sqrt{\frac{\pik}{\tau_{2}}}\left\{[\bar{\tau}k_{1}-k_{2}]\alpha+[\tauk_{1}-k_{2}]\alpha^{\dagger}\right\}}|z>\\\.
\mathbf{s}_{m}^{a}B^{\alpha_{0}}=-m^{2}C^{\alpha_{0}a}+\mathrm{~\frac{1}{2}~}F_{\beta_{0}\gamma_{0}}^{\alpha_{0}}B^{\beta_{0}}C^{\gamma_{0}a}+\mathrm{~\frac{1}{12}~}\epsilon_{cd}(F_{\eta_{0}\beta_{0}}^{\alpha_{0}}F_{\gamma_{0}\delta_{0}}^{\eta_{0}}+2R_{\beta_{0}}^{i}F_{\gamma_{0}\delta_{0},i}^{\alpha_{0}})C^{\gamma_{0}a}C^{\delta_{0}c}C^{\beta_{0}d}.
\bar{I}^{(d-2)}=\bar{I}^{(d)}-\left(\frac{m^{2}}{4m^{2}-q^{2}}\right)^{\frac{d}{2}}\frac{\Gamma(d-3)\Gamma(2-\frac{d}{2})}{i^{d}\Gamma(\frac{d}{2}-1)}.
\nabla^{2}\vec{E}-\frac{\partial^{2}\vec{E}}{\partialt^{2}}=\vec{\nabla}(\vec{\nabla}h\cdot\vec{E})-\frac{\partial\left(\displaystyle{\frac{\partialh}{\partialt}}\vec{E}\right)}{\partialt}-\vec{\nabla}h\times(\vec{\nabla}\times\vec{E})\.
J_{ij}^{(0)}=-{\frac{1}{4}}\Psi^{\lambda}\Psi^{\sigma}\Psi^{\rho}\Psi^{\tau}(R_{\mu\nu\lambda\sigma}f_{i~\rho}^{\mu}f_{j~\tau}^{\nu}+{\frac{1}{2}}c_{i\lambda\sigma}^{~~~\pi}c_{j\rho\tau\pi})
U_{\gamma}=(\alpha_{\gamma})\otimes\left(\begin{array}{ccc}{\beta_{\gamma}}&{}&{}\\{}&{\beta_{\gamma}}&{}\\{}&{}&{\beta_{\gamma}^{-2}}\\\end{array}\right)\otimes\left(\begin{array}{ccc}{\delta_{\gamma}}&{}&{}\\{}&{\delta_{\gamma}^{-1}}&{}\\{}&{}&{}\\\end{array}\right)
\begin{array}{l}{\mu_{+}=1+q\lambdax^{+}\partial_{+}~,}\\{\Lambda=1+q\lambda\displaystyle\sum_{j=0,\pm}x^{j}\partial_{j}+q^{3}\lambda^{2}\left(q^{-1/2}x^{-}x^{+}+\frac{q}{1+q}x^{0}x^{0}\right)\left(q^{-1/2}\partial_{+}\partial_{-}+\frac{q}{1+q}\partial_{0}\partial_{0}\right)~.}\\\end{array}
\begin{array}{l}{[4(3,\bar{3},1)]_{1}+}\\{+[2(1,\bar{3},3)+(1,1,1+8)]_{2}+}\\{+[2(1,1,1+8)+(1,\bar{3},3)]_{3}+}\\{+[2(1,\bar{3},3)+(\bar{3},1,3)+(1,1,1+8)]_{4}+}\\{+[4(1,3,\bar{3})]_{5}+}\\{+[3(1,3,\bar{3})+(\bar{3},3,1)]_{6}+}\\{+[2(\bar{3},1,3)+(1,1,1+8)]_{7}+}\\{+[3(1,3,\bar{3})+(1,1+8,1)]_{8}+}\\{+[3(1,1,1+8)+(1,\bar{3},3)]_{9}+}\\{+[3(1,1,1+8)+(1,\bar{3},3)]_{10}}\\\end{array}
L:=\frac{1}{2}\dot{x}^{2}-\frac{1}{2}\Phi^{2}(x)+\frac{\mathrm{i}}{2}\left(\overline{{\psi}}\dot{\psi}-\dot{\overline{{\psi}}}\psi\right)-\Phi^{\prime}(x)\overline{{\psi}}\psi.
{\times}\int{d^{3}x\:\mathrm{tr}\Bigl[(\partial_{j}A_{k}+A_{j}A_{k})(\partial_{i}f\,g-\partial_{i}g\,f)+\partial_{i}(A_{j}A_{k})\Bigl(\frac{1}{1+\mu}\,fg-\frac{\mu}{1+\mu}\,gf\Bigr)\Bigr]}.
T=\left(\begin{array}{lcccccccc}{}&{n_{1}}&{n_{2}}&{n_{3}}&{n_{4}}&{n_{5}}&{n_{6}}&{n_{7}}&{n_{8}}\\{u_{1}}&{1}&{0}&{1}&{1}&{0}&{0}&{1}&{0}\\{u_{2}}&{0}&{1}&{0}&{0}&{1}&{1}&{0}&{1}\\{u_{3}}&{0}&{0}&{1}&{0}&{1}&{0}&{1}&{1}\\{u_{4}}&{0}&{0}&{0}&{1}&{0}&{1}&{1}&{1}\\\end{array}\right).
\intD{\calB}_{+}det\left(\begin{array}{cc}{0}&{i\partial_{t}+{\calB}_{+}}\\{i\partial_{-}}&{0}\\\end{array}\right)=det\left(\begin{array}{cc}{0}&{i\partial_{t}}\\{i\partial_{-}}&{0}\\\end{array}\right)~\intDU~e^{-(1+2N)\dot{W}[U]},
\theta_{A_{i}}=\theta_{B_{i}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\xi_{A_{i}}^{\ell}(M_{A_{i}})^{\ell}=\xi_{B_{i}}^{\ell}(M_{B_{i}})^{\ell},
\left(\delta_{\mu_{1}\nu_{1}}-{\frac{p_{1\mu_{1}}p_{1\nu_{1}}}{p_{1}^{2}}}\right)\cdots\left(\delta_{\mu_{n}\nu_{n}}-{\frac{p_{n\mu_{n}}p_{n\nu_{n}}}{kp_{n}^{2}}}\right)<A_{\nu_{1}}(p_{1})\cdotsA_{\nu_{n}}(p_{n})>\Big|_{\mathrm{one\,loop}}
\Psi_{{\bfp}}^{(-)}({\bfx},t)=\mathrm{e}^{+iw_{{\bfp}}t}\,\Psi_{{\bfp}}^{(-)}({\bfx})=\frac{1}{{2\pi}}\frac{w_{{\bfp}}+m}{\sqrt{4mw_{{\bfp}}}}\left(\begin{array}{c}{\frac{m-w_{{\bfp}}}{m+w_{{\bfp}}}}\\{1}\\\end{array}\right)\mathrm{e}^{\,-i{\bfp}\cdot{\bfx}+iw_{{\bfp}}t}\;,
\rho_{p\rightarrow1+2}=\frac{\pi^{1-\ell}\Gamma(\ell-1)\lambda^{2\ell-3}}{2^{2\ell-1}(p^{2})^{\ell-1}\Gamma(2\ell-2)};\quad\lambda\equiv\sqrt{p^{4}+m_{1}^{4}+m_{2}^{4}-2m_{1}^{2}m_{2}^{2}-2p^{2}m_{1}^{2}-2p^{2}m_{2}^{2}}.
\begin{array}{l}{-\frac{1}{2}\alpha^{\prime}D^{2}\tilde{T}-\left(\frac{D-2}{12}-\frac{1}{8}\sqrt{\alpha^{\prime}}R-\frac{1}{32}\alpha^{\prime}R^{2}+...\right)\tilde{T}=0~,}\\{-\frac{1}{2}\alpha^{\prime}D^{2}f+\left(\frac{26-D}{6}+\frac{1}{16}\alpha^{2}R^{2}+...\right)f=0~.}\\\end{array}
j=\left(\begin{array}{cccccc}{\theta_{1}}&{}&{}&{}&{}&{}\\{}&{\ddots}&{}&{}&{}&{}\\{}&{}&{\theta_{q+1}}&{}&{}&{b}\\{}&{}&{}&{\ddots}&{}&{}\\{}&{}&{}&{}&{\theta_{p}}&{}\\{}&{}&{c^{t}}&{}&{}&{0}\\\end{array}\right).
\begin{array}{c}{\left({\frac{D}{Dx^{\mu}}}-i^{\frac{p}{2}}{\frac{\partialf}{\partialR}}\Gamma^{\mu}\prod_{I=1}^{p-1}\Gamma^{I}\right)\Psi_{++}=0\;\;\;(\mu=1,\ldots,p-1)}\\{\left({\frac{D}{Dy^{\mu}}}-i^{\frac{q}{2}}{\frac{\partialf}{\partialR}}\Gamma^{p+\mu}\prod_{I=1}^{q-1}\Gamma^{p+I}\right)\Psi_{++}=0\;\;\;(\mu=1,\ldots,q-1)}\\\end{array}
\chi^{\pm}(z)=\sum_{n\in\mathbb{Z}}\chi_{-n\mp\lambda}^{\pm}z^{n-1\pm\lambda}\,,\qquad\bar{\chi}^{\pm}(\bar{z})=\sum_{n\in\mathbb{Z}}\bar{\chi}_{-n\pm\lambda}^{\pm}\bar{z}^{n-1\mp\lambda}\,
\left(\begin{array}{cccc}{0}&{}&{}&{}\\{}&{\fbox{\ensuremath{\begin{array}{rr}{0}&{2}\\{0}&{0}\\\end{array}}}}&{}&{}\\{}&{}&{\fbox{\ensuremath{\begin{array}{rr}{0}&{4}\\{0}&{0}\\\end{array}}}}&{}\\{}&{}&{}&{\ddots}\\\end{array}\right).
(ds)^{2}=-\left(1-\frac{2GM}{x}+\frac{{Q}^{2}G^{2}}{x^{2}}\right)d^{2}t+\left(1-\frac{2GM}{x}+\frac{Q^{2}G^{2}}{x^{2}}\right)^{-1}d^{2}x,
I=\frac{1}{16\pi(8\pi^{6})}\intd^{10}x\sqrt{-g^{(10)}}\left(e^{-2\phi^{(10)}}R^{(10)}+4e^{-2\phi^{(10)}}(D\phi^{(10)})^{2}-\frac{1}{12}H^{(10)2}\right),
H={\frac{1}{2}}\left(p-i{\frac{\partialW}{\partialq}}\right)\left(p+i{\frac{\partialW}{\partialq}}\right),\qquadH={\frac{1}{2}}\sum_{j=1}^{r}\left(p_{j}-i{\frac{\partialW}{\partialq_{j}}}\right)\left(p_{j}+i{\frac{\partialW}{\partialq_{j}}}\right),
\Gamma^{(2)}=\frac{n_{6}}{g_{s}(2\pi)^{6}}V_{6}\intdt\left(-\frac{3\epsilon^{2}M^{2}}{4r^{2}}v^{2}-\frac{\sqrt{3}\epsilonM^{2}}{r^{2}}v^{4}-\frac{M^{2}}{r^{2}}v^{6}\right).
S=\bar{S}+\intdx\,dt\,\lbrace\pi_{\xi}\delta\dot{\xi}-[{\frac{1}{2}}{\frac{\bar{p}}{x}}(\pi_{\xi}^{2}+(\partial_{\phi}\delta\xi)^{2})+{\frac{1}{6x^{2}}}(\deltap_{+}^{3}-\deltap_{-}^{3})]\rbrace.
A_{N}=(\prod_{i=1}^{N-3}\intd\tau_{i})<\prod_{r=1}^{N}O^{(r)}(w_{r})\prod_{i=1}^{N-3}\ointdw_{i}^{\prime}b(w_{i}^{\prime})\prod_{j}{\calZ}(w_{j}^{\prime\prime})>_{R_{\tau}},
X=\left(\begin{matrix}{\alpha\dot{\beta}}&{-\alpha\dot{\alpha}}\\{\beta\dot{\beta}}&{-\beta\dot{\alpha}}\\\end{matrix}\right)=\left(\begin{matrix}{\alpha}\\{\beta}\\\end{matrix}\right)\left(\begin{matrix}{\dot{\beta}}&{-\dot{\alpha}}\\\end{matrix}\right).
I_{n}(\mu)=-\left(\frac{\mu}{2}\right)^{4-D}\frac{M}{(4\pi)^{3/2}}\frac{\Gamma(\frac{2n+D-1}{2})}{\Gamma(\frac{3}{2})}(-ME)^{(2n+D-3)/2}\Gamma\left(\frac{3-D-2n}{2}\right),
{\hat{\lambda}}={\frac{\nu}{\xia^{1-\nu}(1-(a/\ell)^{2\nu})}}-{\frac{\pi\nu}{(1-(a/\ell)^{2\nu})}}-{\frac{C\nu^{2}a^{2\nu}}{(1-(a/\ell)^{2\nu})^{2}}}
a=\frac{\Lambda}{3A^{2}}-\frac{27\,m^{4}-36\,b\,q^{2}\,m^{2}+8\,b^{2}\,q^{4}}{32\,q^{6}\,A^{2}}-\frac{m\,\sqrt{(9\,m^{2}-8\,q^{2}\,b)^{3}}}{32\,q^{6}\,A^{2}}
\Gamma_{\mu}=\left(\begin{array}{cc}{0}&{\gamma_{\mu}}\\{\gamma_{\mu}}&{0}\\\end{array}\right),\quad\mu=1,2\cdots,p+q-1,\quad\Gamma_{p+q}=\left(\begin{array}{cc}{0}&{I}\\{-I}&{0}\\\end{array}\right),
P_{\{\mu,\,n,\varsigma\}}(q,q^{\prime})=(\rho\rho^{\prime})^{-\gamma}J_{2n+\gamma+\varsigma}(\mu\rho)J_{2n+\gamma+\varsigma}(\mu\rho^{\prime})\sum_{k=0}^{n}(\frac{\tau}{\rho\rho^{\prime}})^{2k+\varsigma}r_{2k+\varsigma,2n+\gamma+\varsigma}
ds^{2}=\left(1-\alpha\frac{e^{-\betar}}{r}\right)c^{2}dt^{2}-\left(1-\alpha\frac{e^{-\betar}}{r}\right)^{-1}dr^{2}-r^{2}\left(d\theta^{2}+sin^{2}\thetad\phi^{2}\right),
(f^{1}\otimes\cdots\otimesf^{n},g^{1}\otimes\cdots\otimesg^{n})=\int\cdots\int\prod_{i=1}^{n}(\sum_{\mu_{i}}\bar{f}_{\mu_{i}}^{i}(p_{i}))\prod_{i=1}^{n}(\sum_{\nu_{i}}g_{\nu_{i}}^{i}(p_{i}))dp_{1}\ldotsdp_{n}
=\frac{1}{2}+\sum_{k=1}^{m}\frac{1-\omega^{2k+1}+1-\omega^{-(2k+1)}}{1-\omega^{2k+1}-\omega^{-(2k+1)}+1}=\frac{1}{2}+m=\frac{1}{2}(n_{c}-\tilde{n}_{c}).
\langle0|{\hat{f}}_{2}^{(0)}(0)|\epsilon,\epsilon^{\prime}\rangle=-\pi^{2}{\frac{\hbar}{a^{3}}}{\frac{ll^{\prime}}{\sqrt{\omega_{\epsilon}\omega_{\epsilon^{\prime}}}}}(-1)^{l+l^{\prime}}\delta_{n_{2},-n_{2}^{\prime}}...\delta_{n_{D},-n_{D}^{\prime}}.
T_{1}=igm:[A^{\mu}A_{\mu}\phi+u\tilde{u}\phi-\frac{1}{m}A_{\mu}(\phi\partial^{\mu}\Phi-\Phi\partial^{\mu}\phi)-\frac{m_{H}^{2}}{2m^{2}}\phi\Phi^{2}+\delta_{4}\phi^{3}]:
{p_{L}}=\frac{1}{\sqrt{2ImTImU}}(m_{1}+m_{2}{\bar{U}}+n_{1}{\bar{T}}+n_{2}{\bar{U}}{\bar{T}}),\;p_{R}=\frac{1}{\sqrt{2ImTImU}}(m_{1}+m_{2}{\bar{U}}+n_{1}T+n_{2}T{\bar{U}}),
\mu_{k}(\eta)\simeq\frac{\alpha(k)}{\sqrt{2\omega(k,\eta)}}e^{-i\int^{\eta}\omega(k,\eta^{\prime})\mathrm{d}\eta^{\prime}}+\frac{\beta(k)}{\sqrt{2\omega(k,\eta)}}e^{i\int^{\eta}\omega(k,\eta^{\prime})\mathrm{d}\eta^{\prime}},
\mathrm{I=\int\frac{d^{4}q}{(2\pi)^{4}}\frac{{\itq}_{i}}{q^{2}(q-k)^{2}[(p+q)^{2}+m^{2}]\vec{q}^{2}(\vec{q}-\vec{k})^{2}}~,k\equivp^{\prime}-p,~i=1,2,3~,}
V(\psi\otimes\bar{v},z)\;V(\chi\otimes\bar{w},\zeta)\;(\varphi\otimes\bar{u})=\sum_{\bar{w}^{\prime},\bar{v}^{\prime},\bar{u}^{\prime}}\left(\Phi_{213}\;R_{12}\Phi^{-1}\right)_{\bar{v}^{\prime}\;\bar{w}^{\prime}\;\bar{u}^{\prime}}^{\bar{v}\;\bar{w}\;\bar{u}}\;\;V(\chi\otimes\bar{w}^{\prime},z)\;V(\psi\otimes\bar{v}^{\prime},\zeta)\;(\varphi\otimes\bar{u}^{\prime}),
S_{GB}=S_{GB}^{volume}+S_{EGH}^{bound}={\frac{1}{4\pi^{2}}}\left(\int_{\partialV}-\int_{\partialM}\right)\omega^{01}\wedgeR^{23}=-{\frac{1}{4\pi^{2}}}\left(\int\omega^{01}\wedgeR^{23}\right)_{r=r_{h}}\.
\{Q_{\alpha\alpha^{\prime}\alpha^{\prime\prime}}^{I},Q_{\beta\beta^{\prime}\beta^{\prime\prime}}^{J}\}=-4\tau_{3IJ}(iC_{\alpha^{\prime}\beta^{\prime}}(C\gamma^{a})_{\alpha\beta}J_{a}^{J}+C_{\alpha\beta}(C^{\prime}\gamma^{a^{\prime}})_{\alpha^{\prime}\beta^{\prime}}J_{a^{\prime}}^{J})\varepsilon_{\alpha^{\prime\prime}\beta^{\prime\prime}}.