harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Gamma\sim\left(\begin{array}{ccc}{3j_{0}+1}&{0}&{-\sqrt{(3j_{0}+1)(3j_{0}+2)}}\\{0}&{3(2j_{0}+1)}&{0}\\{-\sqrt{(3j_{0}+1)(3j_{0}+2)}}&{0}&{3j_{0}+2}\\\end{array}\right).
{\calL}=\frac{1}{24}E^{\mu\nu\alpha\beta,\rho}E_{\mu\nu\alpha\beta,\rho}+\frac{1}{18}E^{\mu\nu\alpha\beta,\rho}E_{\mu\nu\alpha\rho,\beta}-\frac{2}{9}E^{\mu\nu\alpha}E_{\mu\nu\alpha}
I_{2\,0}=q_{1}q_{2}=X^{2}-\frac{q^{2}}{4}={X^{\prime}}^{2}-\frac{{q^{\prime}}^{2}}{4}-\frac{g}{{p^{\prime}}^{2}}={q^{\prime}}_{1}{q^{\prime}}_{2}-\frac{g}{{p^{\prime}}^{2}}\neqI_{2\,0}(g(0)).
\begin{array}{c}{(q^{-{\frac{\sigma^{3}}{2}}}\otimes\sigma^{+}+\sigma^{+}\otimesq^{\frac{\sigma^{3}}{2}})\downarrow\otimes\downarrow=}\\{=q^{1/2}\downarrow\otimes\uparrow+q^{-1/2}\uparrow\otimes\downarrow}\\\end{array}
\bar{\eta}_{\mu\nu}(p_{2},p_{3}-p_{2})=\big[{\frac{i}{\not\!p_{2}-m)}}{\frac{i}{\not\!p_{3}-m}}\big]_{\mu\nu}i{\frac{\lambda_{0}^{2}}{4\pi^{2}}}\intdp_{1}\bar{\eta}_{\alpha\alpha}(p_{1},p_{3}-p_{2})
[n_{1},n_{2},n_{3},n_{5},n_{5}]_{n_{6}}~=~\langle\alpha_{1}+n_{1},\alpha_{2}+n_{2},\alpha_{3}+n_{3},\alpha_{4}+n_{4},\alpha_{5}+n_{5}\rangle\frac{1}{(x^{2})^{n_{6}}}
\left[e^{-t\bar{H}_{s}^{2}}\right]_{\mathrm{diag}}={\frac{1}{(4\pit)^{3/2}}}~~~,~~~\left[e^{-t\bar{H}_{d}^{2}}\right]_{\mathrm{diag}}={\frac{r_{d}}{(4\pit)^{3/2}}}\left(1+\frac12\kappa^{2}t\right)~~~.
\left[l^{\mu\nu},l^{\alpha\beta}\right]=\imathQ\left(g^{\mu\beta}l^{\alpha\nu}-g^{\mu\alpha}l^{\beta\nu}+g^{\nu\alpha}l^{\beta\mu}-g^{\nu\beta}l^{\alpha\mu}\right)+2\sum_{\sigma_{1}\neq\sigma_{2}}\sum_{\sigma_{3},\sigma_{4}}\mathcal{E}_{\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}}^{\mu\nu\alpha\beta}\tag{25}
\begin{array}{lll}{H\left|\Omega(k,k+1)\right\rangle}&{=}&{\left(Q+Q^{-1}\right)\left|\Omega(k,k+1)\right\rangle+\left|\Omega(k-1,k+1)\right\rangle}\\{}&{}&{}\\{}&{+}&{\left|\Omega(k,k+2)\right\rangle}\\\end{array}
M=\left(\begin{array}{cccc}{0}&{-1}&{-1}&{+1}\\{+1}&{0}&{-1}&{-1}\\{+1}&{+1}&{0}&{-1}\\{-1}&{+1}&{+1}&{0}\\\end{array}\right)\;\;\;\;\;\mathrm{with}\;\;\;\;\;M^{-1}=\left(\begin{array}{cccc}{0}&{-1}&{+1}&{-1}\\{+1}&{0}&{+1}&{+1}\\{-1}&{-1}&{0}&{-1}\\{+1}&{-1}&{+1}&{0}\\\end{array}\right)\;.
\Psi_{\underline{{o}}}^{\prime}(\vec{\sigma}_{2o})\sum_{\underline{{n}}}\,c_{\underline{{n}}}\Psi_{\underline{{n}}}^{\prime}(\vec{\sigma}_{1o})=-\Psi_{\underline{{o}}}^{\prime}(\vec{\sigma}_{2o})\frac{R}{\calN}=-\frac{n_{o}(\vec{\sigma}_{2o})}{\calN}.
\widetilde{\calJ}^{\nu}=\eta^{\nu\alpha}[\xi_{\sigma}\nabla_{\alpha}\xi^{\sigma}-\eta_{\sigma}{^{\rho}}\xi^{\sigma}(\nabla_{\alpha}\xi_{\rho}+\nabla_{\rho}\xi_{\alpha})+\eta_{\rho}{^{\sigma}}\xi_{\alpha}\nabla_{\sigma}\xi^{\rho}];
{\frac{\partial\Gamma_{div}^{(n)}}{\partial\kappa}}={\frac{\partial{\calG}^{(n)}}{\partial\kappa}}+\left(R^{(n)},{\frac{\partial\Sigma_{0}}{\partial\kappa}}\right)+\left({\frac{\partialR^{(n)}}{\partial\kappa}},\Sigma_{0}\right),
\frac{\partial^{2}\hat{\phi}_{L}}{\partialt^{2}}+3\Bigl(\frac{\partiala(t)/\partialt}{a(t)}\Bigr)\frac{\partial\hat{\phi}_{L}(t)}{\partialt}+\frac{\deltaV(\hat{\phi}_{L})}{\delta\hat{\phi}_{L}}=0.
\left({\begin{array}{ccc}{1}&{2}&{[(2\mu-3)(2\mu-1)+b]}\\{\mu-1}&{-(3\mu-2)-\frac{2\tilde{\lambda}_{g\,1}}{(\mu-1)}}&{\frac{(\mu-1)(2\mu-1+b)}{(2\mu-1)}}\\{1}&{2}&{-\,\frac{8\mu\tilde{\lambda}_{g\,1}}{(2\mu-1)^{2}}+[(2\mu-3)(2\mu-1)+b]}\\\end{array}}\right)
<x|e^{-\beta(\frac{1}{2}K^{ij}v_{i}v_{j}-J^{i}v_{i})}|y>=\left.e^{-\frac{1}{2}\betaK^{ij}u_{i}u_{j}}<x|e^{\varphi^{i}v_{i}}|y>\right|_{\varphi=\betaJ}
\mathbf{E}=\Big[\Phi(x,y,\xi+\varepsilonz)\mathrm{cos}\left(\frac{\pmz}{l(x,y)}+b(x,y)\right);\Phi(x,y,\xi+\varepsilonz)\mathrm{sin}\left(\frac{\pmz}{l(x,y)}+b(x,y)\right);0\Big],
\delta\mathbf{r}^{i}\left(\tau\right)\mathbf{=}\frac12\varepsilon_{MN}\left\{L^{MN}\left(\tau\right),\mathbf{r}^{i}\left(\tau\right)\right\},\quad\delta\mathbf{p}^{i}\left(\tau\right)\mathbf{=}\frac12\varepsilon_{MN}\left\{L^{MN}\left(\tau\right),\mathbf{p}^{i}\left(\tau\right)\right\}.
{\tilde{\Pi}}_{2n}=\intD{\tilde{M}}(1-2g{\tilde{\Pi}}_{2})^{-n}tr({\tilde{M}}^{2n})exp[-\frac{1}{2}tr({\tilde{M}}^{2})-\frac{{\tilde{g}}}{4N}tr({\tilde{M}}^{4})],
J_{\mu}^{ab}=A_{\mu}^{ab}+2~B_{\mu}^{ab}~,~~~~~~~~~~~~~~~~~~A_{\mu}^{ab}=2\;n^{a}\stackrel{\leftrightarrow}{\partial}_{\mu}n^{b}~,{}~~~~~~~~~~~B_{\mu}^{ab}=-i~\overline{{\psi}}^{a}\gamma_{\mu}\psi^{b}~.
\begin{array}{rcl}{s_{p_{1}\wedgek}s_{p_{2}\wedgep_{3}}s_{p_{1}\wedgek}}&{=}&{\displaystyle+1/2~s_{p_{2}\wedgep_{3}}+\cdots,}\\{s_{p_{1}\wedgek}s_{p_{2}\wedgek}s_{p_{3}\wedge(k+p_{1})}}&{=}&{\displaystyle-1/4~s_{p_{3}\wedgep_{1}}+\cdots,}\\{s_{p_{1}\wedgek}s_{k\wedgep_{3}}s_{p_{2}\wedge(k+p_{1})}}&{=}&{\displaystyle-1/4~s_{p_{1}\wedgep_{2}}+\cdots,}\\\end{array}
\begin{array}{c}{P\psi^{\uparrow}=+\psi^{\uparrow}\;,\quad\psi^{\uparrow}=\gamma_{0}\psi_{-}\gamma_{0}-\psi_{-}\;;}\\{P\psi^{\downarrow}=-\psi^{\downarrow}\;,\quad\psi^{\downarrow}=\gamma_{0}\psi_{+}\gamma_{0}+\psi_{+}\;.}\\\end{array}
\widehat{q^{l}p^{m}}={\frac{1}{l+1}}\sum_{k=0}^{l}\widehat{q^{l-k}}\widehat{p^{m}}\widehat{q^{k}}={\frac{1}{l+1}}\sum_{k=0}^{l}{\widehatq}\,{}^{l-k}{\widehatp}\,{}^{m}{\widehatq}\,{}^{k},
T_{\mu\nu}^{\alpha\beta}\sim\epsilon_{~\mu}^{\alpha~\rho\sigma}\epsilon_{~\nu}^{\beta~\lambda\gamma}\big(AF_{\rho\sigma}^{a}F_{\lambda\gamma}^{a}+BF_{\rho\sigma}^{a}\tilde{F}_{\lambda\gamma}^{a}+C\tilde{F}_{\rho\sigma}^{a}\tilde{F}_{\lambda\gamma}^{a}\big)\,,
{\frac{{\tilde{g}}_{Y\!M}^{2}}{(2\pi)^{p-2}}}={\frac{R_{\natural}^{3-p}\ell_{11}^{3(p-2)}}{V_{p}}}={\frac{R_{\star}^{3-p}(\ell_{11}^{\prime})^{3(p-2)}}{V_{p}^{\prime}}}={\tilde{V}}_{p}\left(E_{f}^{\prime}\right)^{3}={\mathrm{finite}}\quad.
D_{\mu\nu}^{a}\equiv{\frac{g_{\mu\nu}-q_{\mu}q_{\nu}/q^{2}}{q^{2}-\mu^{a2}}}={\frac{g_{\mu\nu}-q_{\mu}q_{\nu}/\mu^{a2}}{q^{2}-\mu^{a2}}}+{\frac{1}{\mu^{a2}}}{\frac{q_{\mu}q_{\nu}}{q^{2}}}\.
S=\frac{1}{2\pi}\intd^{2}x\sqrt{-g}\{e^{-2\Phi}[R+4(\nabla\Phi)^{2}+4\lambda^{2}]-(\nabla\tilde{T})^{2}-[(\nabla\Phi)^{2}-\Box\Phi-m_{0}^{2}]\tilde{T}^{2}\}
{\calL}=\frac{1}{g^{2}}\,[\psi_{1}^{\dagger},\psi_{2}^{\dagger}]\,(M^{T}\otimesI)\left[{\psi_{1}\atop\psi_{2}}\right]+\frac{r}{g^{2}}\lambda_{1}+\frac{1}{rg^{2}}\lambda_{2},
=\int_{X}d^{4}x\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{abcd}\bigg(c_{1}{^+}G_{\mu\nu}^{ab}{^+}G_{\alpha\beta}^{cd}+c_{3}{^+}F_{\mu\nu}^{ab}{^+}G_{\alpha\beta}^{cd}\bigg)+\int_{X}d^{4}x\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{abcd}\bigg(c_{2}{^-}G_{\mu\nu}^{ab}{^-}G_{\alpha\beta}^{cd}+c_{4}{^-}F_{\mu\nu}^{ab}{^-}G_{\alpha\beta}^{cd}\bigg).
{\calH}_{can}^{mass}=\Pi^{-}\partial_{+}B_{-}+\Pi^{i}\partial_{+}B_{i}-{\calL}_{jWeyl}^{mass}=\frac12(\Pi)^{2}+\frac12(\partial_{i}B_{j})^{2}+\Pi\partial_{i}B_{i}+\frac{m^{2}}{2}B_{i}^{2}-B_{i}j^{i}-B_{-}j^{-},
\left[\prod_{n=M}^{1}ig(t_{G}^{c_{n}})\right]^{ab}\left[\prod_{n=M}^{1}\intd\xi_{n}d\bar{\xi}_{n}\int_{s}^{s^{\prime}}dt_{n}\theta(t_{n}-t_{n-1})\right]exp\left[i\sum_{n=1}^{M}\hat{k}(t_{n})\cdotx(t_{n})\right]+permutations.
(-)^{a}\sum_{j=0}^{a}(-1)^{j}\left(\begin{array}{c}{j}\\{a}\\\end{array}\right)\mathrm{exp}\left[2i\pi\frac{q}{p}(h_{e^{\prime}+ape,n^{\prime}+apn-pj}-h_{e^{\prime}n^{\prime}})\right]|e^{\prime}+ape,n^{\prime}+apn-pj/I>
{\calS}_{\psi}=i\intd^{4}x\intd\phi~[{\itdet}~V]~\bar{\psi}~[\gamma^{a}v_{a}^{\mu}~(\partial_{\mu}-\frac{i}{2}G_{LN}\sigma^{ab}v_{a}^{\nu}\partial_{\mu}v_{b}^{\lambda}\delta_{\nu}^{N}\delta_{\lambda}^{L}-G_{AD}\sigma^{ab}v_{a}^{\beta}v_{b}^{\delta}{\bar{\Gamma}_{MB}}^{A}\delta_{\mu}^{M}\delta_{\beta}^{B}\delta_{\delta}^{D})]~\psi~\delta(\phi-\pi)
\partial_{t}\xi_{2n}=2(n-1)\xi_{2n}+{\frac{k^{2(n-1)}}{Vol}}{\frac{1}{n!}}{\frac{\partial^{n+1}}{\partialR\partial(\phi^{2})^{n}}}\partial_{t}\Gamma_{k}\Big|_{\substack{\phi=0\,R=0}}.
\begin{array}{cccc}{\left\{P_{\phi}^{\tau},X_{\phi}^{\tau}\right\}=1,}&{\left\{P_{\phi}^{r},X_{\phi}^{s}\right\}=-\delta^{rs},}&{\left\{P_{\phi}^{r},X_{\phi}^{\tau}\right\}=0,}&{\left\{P_{\phi}^{\tau},X_{\phi}^{r}\right\}=0,}\\\end{array}
S_{P}[\eta,A]=t\sum_{x}\sum_{\stackrel{\scriptstyle\mu,\nu=1}{\mu<\nu}}^{3}A_{\mu\nu}(x)\Big[\eta_{+\mu}(x,\nu)\eta_{-\mu}(x+\hat{\mu},\nu)+\eta_{+\nu}(x,\mu)\eta_{-\nu}(x+\hat{\nu},\mu)\Big],
Y_{l\,m}(x,\,\phi)=(-1)^{\frac{m+|m|}2}\,i^{l}\,\frac{e^{im\phi}}{2\pi}\,\sqrt{\frac{2l+1}2\,\frac{(l-|m|)!}{(l+|m|)!}}P_{l}^{|m|}(x).
\int_{0}^{2\pi}\frac{d\theta}{2\pi}e^{-|z|^{2}}\,\prod_{\pm}\left\{\prod_{\mu=1}^{M}e^{xe^{\pmi\theta}\left[e^{i\omega_{\mu}}|z_{\mu+}^{\pm}|^{2}+e^{-i\omega_{\mu}}|z_{\mu-}^{\pm}|^{2}\right]}\right\}\\\,
\frac{d^{2}}{d\sigma^{2}}\tilde{\deltax}_{R}+H_{ij}\tilde{n}_{S}^{i}[x^{k}\tilde{\nabla}_{k}(x^{l}\tilde{\nabla}_{l}\tilde{n}_{R}^{j})+2(x^{k}\tilde{\nabla}_{k}\tilde{n}_{R}^{j})\frac{d}{d\sigma}]\tilde{\deltax}^{S}=x^{i}x^{j}\tilde{R}_{iklj}\tilde{n}_{R}^{k}\tilde{n}_{S}^{l}\tilde{\deltax}^{S}.
\partial_{\vec{z}}^{2}H_{F}+H_{3}\partial_{\vec{x}}^{2}H_{F}=q_{F}\delta({\vec{x}-\vec{x}_{0}})\delta(\vec{z}-\vec{z}_{0}),\\\\partial_{\vec{z}}^{2}H_{3}={\frac{q_{3}}{V_{y}}}\delta({\vec{z}-\vec{z}_{0}}),
\mathrm{S}=\intd^{4}xd\tau\left\{\psi^{*}(i\partial_{\tau}+e_{0}a_{5})\psi-\frac{1}{2M}\psi^{*}(p_{\mu}-e_{0}a_{\mu})(p^{\mu}-e_{0}a^{\mu})\psi-\frac{\lambda}{4}f_{\alpha\beta}f^{\alpha\beta}\right\}
\int{\frac{d^{4}q}{(2\pi)^{4}}}\eta_{\mu\nu}\widehatT_{G}^{\mu\nu}(q)=-3e^{2}\int{\frac{d^{4}q}{(2\pi)^{4}}}\hat{h}_{G}(q^{2})=-3e^{2}\left.h_{G}\right|_{x^{2}=0}=0,
\langle(\gamma_{\mu}{\calC})_{\alpha\betaij}\Pi_{\bar{z}}{}^{\mu}(z,\bar{z})\rangle=\left(\begin{matrix}{Z^{ij}}&{|Z|\mathrm{I}}\\{-|Z|\mathrm{I}}&{Z^{*ij}}\\\end{matrix}\right)\equiv{\calZ}\.
\deltag_{ij}\,=\,\deltax_{,i}^{\mu}\,x_{,j}^{\mu}\,+\,x_{,i}^{\mu}\deltax_{,j}^{\mu}\,=\,f(n_{,i}^{\mu}x_{,j}^{\mu}\,+\,n_{,j}^{\mu}x_{,i}^{\mu}).
W^{\mu\nu}=\frac{\partial^{2}L}{\partial\dot{x}_{\mu}\partial\dot{x}_{\nu}}=-\frac{m}{\sqrt{\dot{x}^{2}}}\left(g^{\mu\nu}-\frac{\dot{x}^{\mu}\dot{x}^{\nu}}{\dot{x}^{2}}\right)=-m\left(g^{\mu\nu}-\dot{x}^{\mu}\dot{x}^{\nu}\right)\;,
{\frac{\ddot{a}}{a}}=-{\frac{1}{L^{2}}}\left[p(\bar{E}^{2}-\bar{P}_{\phi}^{2})\left({\frac{L}{a}}\right)^{2(p+1)}-(p-1)\bar{E}^{2}\left({\frac{L}{a}}\right)^{2p}+(p-1)\bar{P}_{\phi}\left({\frac{L}{a}}\right)^{p+1}\right],
E={\frac{1}{2i}}\int_{-\infty}^{\infty}{\frac{d\omega}{2\pi}}e^{-i\omega\tau}\sum_{l=1}^{\infty}(2l+1)\int_{0}^{\infty}r^{2}\,dr\,2k^{2}[F_{l}(r,r)+G_{l}(r,r)].
\Phi_{\alpha_{1}\ldots\alpha_{k}\dot{\alpha}_{1}\ldots\dot{\alpha}_{k}}=(-i)^{k}(\sigma^{\mu_{1}})_{\alpha_{1}\dot{\alpha}_{1}}\ldots(\sigma^{\mu_{k}})_{\alpha_{k}\dot{\alpha}_{k}}\nabla_{\mu\{k\}}\phi
p_{1}\equivz_{6}^{4}(z_{1}^{8}+z_{2}^{8}-2\phiz_{1}^{4}z_{2}^{4})+(z_{3}^{2}-\psiz_{2}^{4}z_{6}^{2})^{2}+z_{4}^{4}+z_{5}^{4}=0\;,
\frac{dN(\omega)}{d\sigma}=\frac{1}{2\pi}\frac{e^{2}}{\hbarc^{3}}(2a_{0}\omega_{0})^{3}\langlex^{2}\rangle\int_{-\infty}^{+\infty}ds\,e^{-is\omega/a_{0}\omega_{0}}\mathrm{csch}^{4}(s-i\epsilon)~,
\tilde{S}_{SU(3)}(\theta,\sigma_{12})=\left(\begin{array}{ccc}{-1}&{-(\sigma_{12},2)}&{-((1-\nu)\sigma_{12},3)}\\{(\sigma_{21},2)}&{-1}&{-(\nu\sigma_{21},1)}\\{-((\nu-1)\sigma_{12},1)}&{-(\nu\sigma_{12},3)}&{-1}\\\end{array}\right)\,.
\partial^{\mu}\left(\partial^{\nu}\tilde{\partial}_{+}\bar{\phi}\,\,\varepsilon^{\prime}\sigma_{\nu\mu}\tilde{\partial}_{-}\psi\,\,\right)+\tilde{\partial}_{+}\left(\tilde{\partial}_{-}\bar{V}_{+}\,\bar{\varepsilon}_{\dot{\alpha}}\bar{\sigma}_{\mu}^{\dot{\alpha}\beta}\partial^{\mu}\tilde{\partial}_{-}\psi_{\beta}\right).
S_{\mathrm{grav}}^{(0)}=\frac{1}{2}\intd^{4}x[\partial_{\sigma}\gamma_{\lambda\rho}\partial^{\sigma}\gamma^{\lambda\rho}-\partial_{\lambda}\gamma^{\rho\kappa}\partial_{\kappa}\gamma_{\rho}^{\lambda}-\frac{1}{4}\partial_{\rho}\partial^{\rho}\gamma-\frac{1}{\alpha}\partial_{\rho}\gamma_{\lambda}^{\rho}\partial_{\kappa}\gamma^{\kappa\lambda}],
\gamma_{\scriptscriptstyle{l}}=\Phi/2\pi-l+\frac{a^{2l}}{\int_{0}^{a}dr\,r^{2l-1}\,e^{2\phi(r)}}-k^{2}\,a\chi_{\scriptscriptstyle{l}}^{\prime}(a)+\mathrm{O}(k^{4})\,.
L_{n}=n\sum_{l_{1}=0}^{\left[\frac{n}{2}\right]}\cdots\sum_{l_{k-1}=0}^{\left[\frac{n}{k}\right]}\frac{1}{l_{1}!\cdotsk_{k-1}!}\frac{\left(n-1-\sum_{j=1}^{k-1}jl_{j}\right)!}{\left(n-\sum_{j=2}^{k}jl_{j-1}\right)!}
\hat{\phi}(x^{0}\,,\,x^{1})\,=\,\frac{1}{\sqrt{2\pi}}\left(\frac{\pi}{R}\right)\sum_{n\geq0}\frac{1}{\sqrt{\omega_{n+\frac{1}{2}}}}\left[\hat{\Lambda}_{n}e^{-i\omega_{n+\frac{1}{2}}(x^{0}+x^{1})}\,+\,\hat{\Lambda}_{n}^{\dagger}e^{i\omega_{n+\frac{1}{2}}(x^{0}+x^{1})}\right],
\tau_{0}=\frac12\left(\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}\right)\,,\quad\tau_{1}=-\frac12\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\,,\quad\tau_{2}=\frac12\left(\begin{array}{cc}{-1}&{0}\\{0}&{1}\\\end{array}\right)\,
H_{F}=1+\sum_{i}{\frac{Q_{i}}{[|\vec{x}-\vec{x}_{0\,i}|^{2}+{\frac{4Q_{p}}{(p-4)^{2}}}|\vec{z}-\vec{z}_{0}|^{p-4}]^{\frac{(p-3)^{2}+3}{2(p-4)}}}},\\\\H_{p}={\frac{Q_{p}}{|\vec{z}-\vec{z}_{0}|^{6-p}}}.
\begin{array}{c}{\deltaX^{I}=2i\alpha^{\dagger}\Gamma^{I}S+2i\epsilon^{ab}\partial_{a}X^{I}\alpha^{\dagger}\intd\tau\partial_{b}S}\\{\deltaS=-{\frac{1}{2\sqrt{2}P^{+}}}\left(\dot{X}^{I}\Gamma_{I}-{\frac{1}{2}}\epsilon^{ab}\partial_{a}X^{I}\partial_{b}X^{J}\Gamma_{IJ}\right)\alpha+2i\epsilon^{ab}\partial_{a}S\alpha^{\dagger}\intd\tau\partial_{b}S+\beta}\\\end{array}
\langleR^{\prime}\|{\hat{\calT}}^{R}\|R^{\prime\prime}\rangle=[D(R^{\prime\prime})]^{-1}\left\{\begin{array}{lll}{R^{\prime}}&{R}&{R^{\prime\prime}}\\{R^{1\prime}}&{R^{1}}&{R^{1\prime\prime}}\\{R^{2\prime}}&{R^{2}}&{R^{2\prime\prime}}\\\end{array}\right\}\langleR^{1\prime}\|{\hat{\calT}}^{R^{1}}(1)\|R^{1\prime\prime}\rangle\langleR^{2\prime}\|{\hat{\calT}}^{R^{2}}(2)\|R^{2\prime\prime}\rangle,
e^{-\eta}=e^{-\Phi}\sqrt{\mathrm{det}\,G_{mn}}=e^{-(\Phi+\sigma)},\\\\\epsilon^{\mu\nu\rho\sigma}\partial_{\sigma}a=\sqrt{-g}e^{-\eta}g^{\mu\sigma}g^{\nu\lambda}g^{\rho\tau}H_{\sigma\lambda\tau},
S\approx\intd^{10}x\sqrt{|g|}\left[e^{-\tilde{\Phi}}\left(R+|\nabla\tilde{\Phi}|^{2}-|dB_{2}^{(2)}|^{2}\right)-|dB_{2}^{(1)}|^{2}-|dD_{4}|^{2}\right]-\ldots
\overline{{\Pi}}_{2}(k^{2})=\frac{\overline{\beta}_{2}\overline{{a}}}{2(d-1)k^{4}}\left\{1+\left(\frac{2\overline{\beta}_{3}}{\overline{\beta}_{2}}\right)\overline{{a}}+\left(\frac{3\overline{\beta}_{4}}{\overline{\beta}_{2}}\right)\overline{{a}}^{2}+O(\overline{{a}}^{3})\right\}
\Omega=\Omega_{0}-\sum_{i=1}^{\infty}(-1)^{i}\int_{0}^{\beta}d\beta_{1}\int_{0}^{\beta_{1}}d\beta_{2}\cdots\int_{0}^{\beta_{i-1}}d\beta_{i}\langle{\calV}(\beta_{1}){\calV}(\beta_{2})\cdots{\calV}(\beta_{i})\rangle^{c},
K^{-1}K^{\prime}(N^{2}{L^{2}}^{\prime}-L^{2}{N^{2}}^{\prime})+2[(K^{-1}K^{\prime})^{\prime}+(K^{-1}K^{\prime})^{2}]N^{2}L^{2}=-16{\pi}GN^{2}B_{\hat{~}}^{2}.
\dot{\rho}+3\left(\rho+{\frac{p}{c_{\mathrm{grav}}^{2}}}\right){\frac{\dot{a}}{a}}={\frac{3}{4\piG}}{\frac{\dot{c}_{\mathrm{grav}}}{c_{\mathrm{grav}}}}\left({\frac{\dot{a}}{a}}\right)^{2}-{\frac{c_{\mathrm{grav}}\dot{c}_{\mathrm{grav}}}{4\piG}}\Lambda+{\frac{3}{4\piG}}{\frac{kc_{\mathrm{grav}}\dot{c}_{\mathrm{grav}}}{a^{2}}}.
\begin{array}{lclclcl}{\delta_{1234}\lambda_{i}}&{=}&{\eta_{0}\,\rho_{i}}&{,}&{\delta_{1234}\rho_{i}}&{=}&{-\eta_{0}\,\lambda_{i},}\\{\delta_{1234}\lambda_{ijk}}&{=}&{\eta_{0}\,\rho_{ijk}}&{,}&{\delta_{1234}\rho_{ijk}}&{=}&{-\eta_{0}\,\lambda_{ijk}.}\\\end{array}
m=\frac{\kappa\hat{e}_{m}}{\sqrt{2}\pi}\left(\frac{N_{2}^{(2)}\bar{N}_{2}^{(2)}N_{5}^{(1)}\bar{N}_{5}^{(1)}N_{5}^{(2)}\bar{N}_{5}^{(2)}}{N_{2}^{(1)}\bar{N}_{2}^{(1)}}\right)^{\frac14}(l_{1}l_{2})^{-1}
L=\frac{1}{2\pi}\int_{\Sigma}\left\{\frac{1}{2}\left[T_{\mu\nu}^{a}u^{a}-\left(e_{\mu}^{a}D_{\nu}u^{a}-e_{\nu}^{a}D_{\mu}u^{a}\right)\right]\right\}dx^{\mu}\wedgedx^{\nu}.
|11\rangle=\sqrt{\frac{3}{10}}\left|\frac32\,,-\frac12\right>\left|\frac32\,\frac32\right>+\sqrt{\frac{3}{10}}\left|\frac32\,\frac32\right>\left|\frac32\,,-\frac12\right>-\sqrt{\frac{2}{5}}\left|\frac32\,\frac12\right>\left|\frac32\,\frac12\right>.
E=\frac{1}{2}\intd^{3}x\{~Tr(~E_{i}^{2}+B_{i}^{2}+(D_{i}X)^{2}+(D_{t}X)^{2}~)+\frac{\lambda}{2}(~(x^{1})^{2}+(x^{2})^{2}+(x_{3})^{2}-1)~\}
\Gamma_{\mu\rho,\nu\sigma}^{(2)}({p})=\frac{p^{2}}{2}\left[1+\frac{\kappa^{2}\mu^{-1}}{512}\left(-p^{2}\right)^{\frac{1}{2}}\right]P_{\mu\rho,\nu\sigma}^{(2)}+\cdots.
\dot{\vec{x}}(\tau^{*}+\epsilon)-\dot{\vec{x}}(\tau^{*}-\epsilon)=\frac{\lambdae^{2}}{4\piM}\nabla\int_{\tau^{*}-\epsilon}^{\tau^{*}+\epsilon}d\tau\frac1R\delta(t-\tau-R/c)=-\frac{\lambdae^{2}}{4\piM}\frac{1}{(R^{*})^{2}}\hat{x}^{*}\.
\Gamma_{R(\tau)}^{(N,L)}(k_{i(\tau)},p_{i(\tau)},g_{\tau},\kappa_{\tau})=Z_{\phi(\tau)}^{\frac{N}{2}}Z_{\phi^{2}(\tau)}^{L}\Gamma_{(\tau)}^{(N,L)}(k_{i(\tau)},p_{i(\tau)},\lambda_{\tau},\Lambda_{\tau}),
\mathrm{Res}_{\bar{z}^{\prime}=\bar{z}}^{\Psi}\bar{\partial}^{\Psi}\!:\!\psi(z^{\prime})\!:\,\bar{\partial}^{\Psi}\!:\!\phi(z)\!:=-:\!(\phi_{;\mu\eta}-iC_{\eta\nu\mu}\phi_{;\nu})\psi^{;\eta}\bar{\partial}^{\Psi}\!X^{\mu}\!:+O(\epsilon^{5}).
\sigma(x,x^{\prime})\BoxF(x,x^{\prime})=-\frac{1}{24\pi}g_{\gamma\delta}R\sigma^{;\gamma}\sigma^{;\delta}+\frac{1}{96\pi}g_{\gamma\delta}R_{;\rho}\sigma^{;\gamma}\sigma^{;\delta}\sigma^{;\rho}+O(\sigma^{2})~.
\stackrel{(0)}{S_{0}}+\stackrel{(1)}{S_{0}}=\intd^{3}x({\frac{1}{2}}\varepsilon^{ijk}k_{ab}A_{i}^{a}F_{jk}^{b}+{\frac{2}{3}}\varepsilon^{ijk}f_{abc}A_{i}^{a}A_{j}^{b}A_{k}^{c})
F_{n+1}=\sum_{l_{1}=0}^{\left[\frac{n}{2}\right]}\cdots\sum_{l_{k-1}=0}^{\left[\frac{n}{k}\right]}\frac{1}{l_{1}!\cdotsl_{k-1}!}\frac{\left(n-\sum_{j=1}^{k-1}jl_{j}\right)!}{\left(n-\sum_{j=2}^{k}jl_{j-1}\right)!}\,,
P(s,l)\;=\;\gamma^{j}\:P_{j}(s,l)\;\;\;\;{\mathrm{with}}\;\;\;\;P_{j}(s,l)\;:=\;\int_{-\infty}^{\infty}du\int_{-\infty}^{\infty}dv\;h_{j}(u,v)\:e^{-i(us+vl)}\;.
{\calO}^{\epsilon\alpha}\equiv-\epsilon^{\epsilonk\alpha}\partial_{k}-{\frac{\Box}{M}}\left(\eta^{\epsilon\alpha}-{\frac{\partial^{\epsilon}\partial^{\alpha}}{\Box}}\right)+{\frac{\Box}{\hat{\alpha}}}\left({\frac{\partial^{\epsilon}\partial^{\alpha}}{\Box}}\right)\;,
\begin{array}{ccccccc}{}&{}&{}&{1}&{}&{}&{}\\{}&{}&{0\!}&{}&{\!0}&{}&{}\\{}&{0\!}&{}&{h^{1,1}}&{}&{\!0}&{}\\{1}&{}&{h^{1,2}\!}&{}&{\!h^{1,2}}&{}&{1}\\{}&{0\!}&{}&{h^{1,1}}&{}&{\!0}&{}\\{}&{}&{0\!}&{}&{\!0}&{}&{}\\{}&{}&{}&{1}&{}&{}&{}\\\end{array}\,
\left(\begin{array}{ccc}{j_{1}}&{j_{2}}&{j_{3}}\\{-m_{1}}&{-m_{2}}&{-m_{3}}\\\end{array}\right)_{q}=\left(\begin{array}{ccc}{j_{2}}&{j_{1}}&{j_{3}}\\{m_{2}}&{m_{1}}&{m_{3}}\\\end{array}\right)_{q}=(-)^{j_{1}+j_{2}+j_{3}}\left(\begin{array}{ccc}{j_{1}}&{j_{2}}&{j_{3}}\\{m_{1}}&{m_{2}}&{m_{3}}\\\end{array}\right)_{q^{-1}}
\Phi\simB\left[\frac{(\sqrt{2\muk}u)^{\nu+1/2}}{2^{\nu}}\frac{(\sqrt{E_{b}})^{\nu+\frac{1}{2}}e^{i\frac{\nu\pi}{2}}}{\Gamma(1+\nu)}-\frac{(\sqrt{2\muk}u)^{-\nu+1/2}}{2^{-\nu}}\frac{(\sqrt{E_{b}})^{-\nu+\frac{1}{2}}e^{i\frac{\nu\pi}{2}}}{\Gamma(1-\nu)}\right].
T_{\nu}^{ab}(\eta,z)=z\left\{\frac{\bar{J}_{\nu}^{(a)2}(z)}{\bar{J}_{\nu}^{(b)2}(\etaz)}\left[A_{b}^{2}+B_{b}^{2}(\eta^{2}z^{2}-\nu^{2})\right]-A_{a}^{2}-B_{a}^{2}(z^{2}-\nu^{2})\right\}^{-1},\quad\eta=\frac{b}{a}.
C_{\mu}=\frac{h_{\mu}^{2}}{2}\{d-l\}C_{\mu}+\sum_{\mu^{\prime}\not=\mu}\frac{h_{\mu^{\prime}}^{2}}{2}\{I-l\}C_{\mu^{\prime}}-\frac{h_{\varepsilon}^{2}}{2}\{I-l\}C_{\varepsilon}.
I_{N-n}\equiv\frac{1}{n!}\biggl\{\underbrace{\sum_{i=1}^{N}q_{i},\dots\biggl\{\sum_{i=1}^{N}q_{i}}_{\mathrm{\small~n~times}},I_{N}\biggr\}\dots\biggr\},\quadn=0,1,\dots,N-1.
L=\frac{1}{2}f_{\mu}^{A}f^{\mu\,A}-\frac{1}{4m}\varepsilon_{\mu\nu\rho}f^{\mu\nu\,A}f^{\rho\,A}+\frac{1}{12m}\varepsilon_{\mu\nu\rho}c^{ABC}f^{\mu\,A}f^{\nu\,B}f^{\rho\,C}.
d\pi_{\dot{q}}^{1-}=-\frac12\Omega^{(0)}\pi_{\dot{q}}^{1-}-\frac14\Omega_{\dot{q}\dot{p}}\pi_{\dot{p}}^{1-}-\frac12\tilde{\gamma}_{\dot{q}p}^{i}\Omega^{-2i}\pi_{p}^{1+},
\tilde{\tau}={\frac{\tau}{2r_{+}}}~~,~~0\leq\tilde{\tau}\leq2\pi~~,~~x=\left({\frac{r-r_{+}}{r_{B}-r_{+}}}\right)^{\frac12}e^{\frac{r-r_{B}}{2r_{+}}}~~,~~0\leqx\leq1~,
S_{\sigma}(t_{s}^{b},t^{\prime})=Q_{\Gamma_{b,s}}^{-2}(t)\left(S_{\sigma}(t,t^{\prime})+\sum_{N_{s}}\hat{Y}_{\sigma,N_{s}}^{(1)}(t)\Psi_{\sigma,N_{s}}(t^{\prime})\right)
E_{0}=\frac{1}{2L}\sum_{n=-\infty}^{\infty}\int\frac{dk_{2}}{2\pi}\left\{\Big((n+\mathrm{~\frac{1}{2}~})\mathrm{~\frac{\pi}{L}~}\Big)^{2}+k_{2}^{2}+m^{2}\right\}^{1/2}
\mathrm{e}^{i\omega}=\frac{n_{E}^{(1)}+n_{M}^{(1)}\,(a_{D}^{*}/a^{*})}{\left|n_{E}^{(1)}+n_{M}^{(1)}\,(a_{D}/a)\right|}\;\frac{n_{E}^{(2)}+n_{M}^{(2)}\,(a_{D}/a)}{\left|n_{E}^{(2)}+n_{M}^{(2)}\,(a_{D}/a)\right|}\,.
G_{f}^{(1)}(z,z^{\prime};\hat{m};l_{1},l_{2})\to\frac{1}{z-z^{\prime}}-\frac{a_{gh}(k;l_{1},l_{2})(u-v)^{4}}{(z-u)(z^{\prime}-u)^{3}}\left[\frac{3}{z^{\prime}-u}-\frac{1}{z-u}\right]
F_{ik}=\frac{i}{32\pi^{2}p^{2}}\int_{0}^{\Lambda}d\bar{q}^{2}\frac{\sqrt{c^{2}+4\bar{p}^{2}\bar{q}^{2}}-c}{\bar{q}^{2}+m_{k}^{2}+\Sigma_{k}(-\bar{q}^{2})}\,,
\intdx^{4}dy\sqrt{G}\,i\bar{\Psi}\Gamma^{M}D_{M}\Psi=\int_{-1/K}^{1/K}dy\,e^{3A}f_{0}^{2}\intdx^{4}\,i\bar{\psi}_{0}\gamma^{\alpha}\partial_{\alpha}\psi_{0}.
F_{v}=-\frac{1}{180},,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,F_{x}=\frac{1}{60},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,F_{y}=-\frac{5}{4},
\varphi_{2}^{\prime}(N)=\left(\frac{j(N-j+1)}{(2j+1)(N+2)}\right)^{1/2}\varphi_{+}^{\prime}(N)-\left(\frac{(j+1)(N+j+2)}{(2j+1)(N+2)}\right)^{1/2}\varphi_{-}^{\prime}(N).
L_{fk}=\frac{e^{\frac{1}{2}\alpha\phi/M_{p}}}{\zeta+k}\left[(\zeta+h_{E})f_{E}\overline{{L_{L}}}\,H\,e_{R}+(\zeta+h_{U})f_{U}\overline{{Q_{L}}}\,\tilde{H}\,U_{R}+(\zeta+h_{D})f_{D}\overline{{Q_{L}}}\,H\,D_{R}+H.c.\right]
Z=\int{\calD}A^{\mu}{\calD}\pi_{\mu}{\calD}\rho{\calD}\pi_{\rho}\prod_{i,j=1}^{2}\delta(\widetilde{\Omega}_{i})\delta(\Gamma_{j})\mathrm{det}\mid\{\widetilde{\Omega}_{i},\Gamma_{j}\}\mide^{iS},
H^{(2)}=\intd^{2}x\left[-\frac{1}{2}\partial_{i}\Phi^{0}\partial^{i}\Phi^{0}+\sqrt{m}\Phi^{0}\epsilon_{ij}\partial^{i}\Phi^{j}+\frac{1}{2}m\Phi^{i}\Phi^{i}-(\frac{1}{\sqrt{m}}\partial_{i}\Phi^{i}+\frac{1}{2}\Phi^{3})\Phi^{3}\right].
{\calL}_{\mathrm{eff}}[Q_{i}^{A},Q_{i}^{3},B_{\mu}^{3}]={\calL}_{\mathrm{YM}}[Q_{\mu},B_{\mu}]|_{Q_{4}=0}-\frac{1}{2}M^{2}(T)Q_{i}^{A}Q_{i}^{A},
\left[\mathrm{Tr}e^{-\bar{H}_{s}^{2}t}\right]_{\mathrm{div}}={\frac{\Gamma\left(1-\fracD2\right)}{(4\pi)^{(D-1)/2}}}{\frac{m^{D-4}}{2\kappat^{3/2}}}\int_{\Sigma}\left[\left(m^{2}-\left(\frac16-\xi\right)R\right)t-{\frac{P}{4\kappa^{2}}}\right]~~~,
\frac{\partial}{\partialz_{m}}\int_{X_{\epsilon}}F~\frac{i}{2}dz\wedged\bar{z}=\int_{X_{\epsilon}}\frac{\partialF}{\partialz_{m}}\frac{i}{2}dz\wedged\bar{z}-\oint_{\partialD_{\epsilon}^{m}}F~\frac{i}{2}d\bar{z}
Z_{n}^{F}={\frac{1}{n!}}\left[\omega_{WP}^{(n)}\right]^{n-4}\cap\left[D^{F}\right]={\frac{\pi^{2}}{(n-1)n!}}\sum_{k=1}^{[n/2]-1}F(n,k)k(n-k-2)\left[\omega_{WP}^{(n)}\right]^{n-4}\cap\left[D_{k}\right],
M_{ADM}^{2}=(8G_{N})^{-1}\left({}{\calM}_{\infty\,ab}({\vec{v}}^{a\,T}\mu_{+}{\vec{v}}^{b})+\left[{2}{\calL}_{ac}{\calL}_{bd}({\vec{v}}^{a\,T}\mu_{+}{\vec{v}}^{b})({\vec{v}}^{c\,T}\mu_{+}{\vec{v}}^{d})\right]^{\frac{1}{2}}\right),
\left({\frac{d\tilde{\lambda}}{d\lambda}}\right)^{-1}{\frac{d}{d\tilde{\lambda}}}\left({\frac{d\tilde{\lambda}}{d\lambda}}\right)=-{\frac{1}{2}}{\calW}^{-{\frac{1}{2}}}{\calW}^{\prime}+{\frac{1}{2}}{\calW}^{-{\frac{1}{2}}}\partial_{y}({\calW}g_{\mu\nu}){\frac{dx^{\mu}}{d\tilde{\lambda}}}{\frac{dx^{\nu}}{d\tilde{\lambda}}}.
\phi(z,{\bfb})=\sum_{k}a_{k}u_{k}(z,{\bfb})+a_{k}^{\dagger}u_{k}^{*}(z,{\bfb})=\sum_{k}\bar{a_{k}}\bar{u_{k}}(z,{\bfb})+\bar{a_{k}}^{\dagger}\bar{u_{k}}^{*}(z,{\bfb}),
e^{a}\rightarrow\tilde{e}^{a}=e^{a}+ih^{a},\;\;\;\;\;\stackrel{2}{\omega}{^{ab}}\rightarrow\stackrel{2}{\omega}{^{ab}}+i\left(\frac{1}{2}C_{\;\cdotc}^{ab}h^{c}+K^{ab}+S^{ab}\right),
\langleA\rangle=v\left(\begin{array}{ccccc}{{\mathbf1}}&{}&{}&{\ldots}&{}\\{}&{\omega\,{\mathbf1}}&{}&{\ldots}&{}\\{}&{}&{\omega^{2}\,{\mathbf1}}&{\ldots}&{}\\{\vdots}&{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{}&{}&{}&{\ldots}&{\omega^{\Gamma-1}\,{\mathbf1}}\\\end{array}\right),
{\mathcalH}=\frac{g^{2}}{2}\sum_{i,a=1}^{2}e^{ia}e^{ia}-a_{0}\,{\mathcalG}+{\mathcalV}_{\scriptscriptstyle\mathrm{YM}}^{2\!\times\!2}\;,\quad{\mathcalG}=\epsilon^{ab}\,a_{i}^{a}\,e^{ib}\;,
ds^{2}=\left(1-\frac{2M}{r}\right)\left(1+\epsilonH_{0}(r)\right)d\tau^{2}+\frac{1+\epsilonH_{1}(r)}{1-\frac{2M}{r}}dr^{2}+r^{2}\left[1-\frac{1}{2}\epsilon(H_{0}(r)+H_{1}(r))\right]d\Omega^{2},
\left.\frac{d}{dT}\left(\frac{(z-x_{0})}{\sqrt{z^{2}-2mz+m^{2}-d^{2}}}\right)\right|_{T=T_{c}^{-}}=-\frac{1-c_{1}z+c_{1}^{2}}{s_{1}^{4}(z-2c_{1})^{2}}\,,
\frac{i}{2}\mathrm{Tr}\left(\frac{1}{-\partial^{2}-m^{2}}\hat{T}_{x}^{(1)}\right)=\frac{i}{2}\sigma(x)\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{k^{2}-m^{2}}\,.