harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{8\pi}{3}\rho_{eff}=\frac{(C+E)^{2}g_{S}e^{2\Phi}-|g_{00}|(g_{S}g^{3}+\ell^{2}e^{2\Phi})}{4|g_{00}|g_{rr}g_{S}g^{3}}(\frac{g^{\prime}}{g})^{2}
\frac{i}{\lambda}\int_{0}^{\pi/2}d\phi\left[e^{\lambdae^{i\phi}}e^{-i\phi}-e^{-\lambdae^{i\phi}}e^{-i\phi}\right]-\frac{i}{\lambda}\int_{\pi}^{3\pi/2}d\phi\left[e^{\lambdae^{i\phi}}e^{-i\phi}-e^{-\lambdae^{i\phi}}e^{-i\phi}\right],
h_{j,j+1}=\sum_{n=1}^{N}F(n)X_{j}^{-n}X_{j+1}^{n}+\sum_{n,m=1}^{N}[G_{+}(n,m)X_{j}^{-n}X_{j+1}^{n}Z_{j}^{m}+G_{-}(n,m)X_{j}^{-n}X_{j+1}^{n}Z_{j+1}^{m}]
\frac{1}{2M_{r}^{2}}\left[(\partial_{z}\phi)\,\partial_{z}F-(\partial_{z}^{2}\phi)\,F\right]=\,\frac{1}{M_{r}^{2}}\,{\calW}^{\prime\prime}\left\langle\partial_{z}\chi\left(1-\frac{\partial_{z}^{2}}{M_{r}^{2}}\right)^{-1}\!\partial_{z}\chi\right\rangle\,.
{\calA}^{\mu}(\hat{p},\hat{q})=\int_{-\infty}^{+\infty}dt\int_{0}^{1}dy\,\varepsilon_{\eta\nu}^{dil}\,\varepsilon_{\lambda\sigma}^{dil}\,\langle\,V_{-1}^{\eta}(iy,-\hat{p}-\hat{q})\,V_{-1}^{\nu}(-iy,-\hat{p}-\hat{q})\,V_{0}^{\mu}(t,2\hat{q})\,V_{0}^{\lambda}(i,\hat{p})\,V_{0}^{\sigma}(-i,\hat{p})\,\rangle\quad.
\psi^{3}\left(x\right)=\frac{1}{\sqrt{L}}\sum_{k\geq0}\left[\left(c_{k}e^{ip_{k}x}+c_{k}^{\dagger}e^{-ip_{k}x}\right){\binom{1}{0}}+\left(d_{k}e^{ip_{k}x}+d_{k}^{\dagger}e^{-ip_{k}x}\right){\binom{0}{1}}\right].
\frac{\Phi_{0}^{2}}{2\pi<\Phi_{0},\Phi_{o}>}+\sum_{\omega\neq0}\frac{1}{4\pi}\frac{\Phi_{\omega}(\sigma)\:\Phi_{\omega}(\sigma^{\prime})}{<\Phi_{\omega},\Phi_{\omega}>}=\delta(\sigma-{\sigma^{\prime}})
H_{B}^{2}=\frac{\rho^{2}}{36}+\frac{U_{B}}{12}\rho+\frac{1}{16a^{4}}\intdt\frac{da^{4}}{dt}(2V-\dot{\phi}^{2})-\frac{1}{12a^{4}}\intdt\frac{dU_{B}}{dt}\rho+\frac{C}{a^{4}}
S=\intd^{3}x\left[-\frac{1}{4}F^{a\mu\nu}F_{\mu\nu}^{a}-\frac{1}{4}G^{a\mu\nu}G_{\mu\nu}^{a}+\frac{m}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}^{a}\phi_{\rho}^{a}\right],
|\Psi_{(-)b}>=b_{1}^{\dagger}b_{2}^{\dagger}b_{3}^{\dagger}b_{4}^{\dagger}b_{5}^{\dagger}|0>{\bfN}_{b}+\frac{1}{12}\epsilon^{ijklm}b_{k}^{\dagger}b_{l}^{\dagger}b_{m}^{\dagger}|0>{\bfN}_{bij}+b_{i}^{\dagger}|0>{\bfN}_{b}^{i}
L_{0}=\alpha^{\prime}p^{\mu}g_{\mu\nu}p^{\nu}+\frac{1}{4\pi^{2}\alpha^{\prime}}(x_{2}-x_{1})^{2}+\sum_{n=1}^{\infty}(\eta_{MN}a_{-n}^{M}a_{n}^{N}+n\eta_{MN}c_{-n}^{M}c_{n}^{N})
\{\Psi_{2}^{1},\Psi_{2}^{4}\}\stackrel{\Sigma}{=}\{\Psi_{2}^{2},\Psi_{2}^{3}\}\stackrel{\Sigma}{=}\frac{g}{(q_{2}^{2})^{3/2}}\left(m+\frac{\beta}{\alpha}\sigma\sqrt{m^{2}-p_{1}^{2}}\right),~~~~\{\Psi_{3}^{1},\Psi_{3}^{2}\}=-\alpha\sqrt{g},
\langle\omega_{i_{1}}\cdots\omega_{i_{n}}\rangle_{g,\beta}=(\mathrm{ev}_{1}^{*}(\omega_{i_{1}})\cup\cdots\cup\mathrm{ev}_{n}^{*}(\omega_{i_{n}}))\cap[\overline{{\mathcal{M}}}_{g,n}(Y,\beta)]^{\mathrm{vir}}\,.
\left[\mathrm{Re}\,\Gamma(\varphi,\lambda)\right]_{\lambda-dep}\,=\,\pi\,\sum_{n_{1}}\,\left[0\,+\,\frac{1}{2}\frac{\tau_{2}^{2}\lambda^{2}}{(\lambda^{2}+\tau_{2}^{2}n_{1}^{2})^{\frac{3}{2}}}\,\varphi_{1}^{2}\;+\;{\calO}(\varphi^{4})\right]\;.
\sum_{n=1}^{\infty}e^{-\betakE_{n}}=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}e^{-\frac{x^{2}}{4}-\frac{1}{x^{2}}{\beta}^{2}k^{2}m^{2}}\frac{1}{e^{\frac{2{\beta}^{2}eBk^{2}}{x^{2}}}-1}\;.
\alpha_{+}={\frac{q_{B}r_{\perp0}^{-(d-1)}\phi_{-}}{V_{a}V_{w}(\phi_{-}\phi_{+}{}^{\prime}-\phi_{+}\phi_{-}{}^{\prime})}},\\\,alpha_{-}={\frac{q_{B}r_{\perp0}^{-(d-1)}\phi_{+}}{V_{a}V_{w}(\phi_{-}\phi_{+}{}^{\prime}-\phi_{+}\phi_{-}{}^{\prime})}}.
\tauT\;\;=\;\;\frac{1}{(p-1)!(q-1)!}\eta^{\mu_{1}\nu_{1}}T_{\mu_{1}...\mu_{p}\nu_{1}...\nu_{q}}dx^{\mu_{2}}\wedge...\wedgedx^{\mu_{p}}\otimesdx^{\nu_{2}}\wedge...\wedgedx^{\nu_{q}}\,\in\,X^{p-1,q-1}
S^{(0)}=\intd^{6}x\sqrt{-g}{\calL}^{(0)}\equiv-\frac{1}{2\lambda_{s}^{4}}\intd^{6}x\sqrt{-g}e^{-\phi}\biggl[R+g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi+\Lambda\biggr],
h_{m\bar{a}\bar{b}}=h_{\bar{m}\bar{a}\bar{b}}+\epsilon_{pac}h_{\bar{c}\bar{b}}+\epsilon_{pbc}h_{\bar{c}\bar{a}}+\frac{3}{10}[\eta_{pa}h_{b}+\eta_{pb}h_{a}-\frac{2}{3}\eta_{ab}h_{p}].
\ddot{a}_{\alpha}^{(m)}({\vec{k}},t)\,+\,(k^{2}\,+\,m^{2})\,a_{\alpha}^{(m)}({\vec{k}},t)\,+\,2\,{\frac{\dot{A}}{A}}\,\dot{a}_{\alpha}^{(m)}({\vec{k}},t)\,=\,0.
\begin{array}{rcl}{\delta\phi_{\mu\nu\rho\sigma}}&{=}&{4\partial_{(\mu}\xi_{\nu\rho\sigma)}}\\{\delta\phi_{\mu\nu}^{\prime}}&{=}&{2\partial_{(\mu}\xi_{\nu)}^{\,\prime}+2\partial^{\rho}\xi_{\mu\nu\rho}}\\{\delta\phi^{\prime\prime}}&{=}&{4\partial^{\mu}\xi_{\mu}^{\,\prime}}\\\end{array}
\begin{array}{lll}{\{x_{1},x_{2}\}=\displaystyle\frac{m}{m^{*}}\,\theta,}\\{\{x_{i},p_{j}\}=\displaystyle\frac{m}{m^{*}}\,\delta_{ij},}\\{\{p_{1},p_{2}\}=\displaystyle\frac{m}{m^{*}}\,B.}\\\end{array}
\left(\stackrel{[0,0]}{z}^{A},\stackrel{[0,0]}{z}^{A_{1}}\right),\;\left(\stackrel{[1,0]}{\calP}_{1a_{0}},\stackrel{[0,1]}{\calP}_{2a_{0}}\right),\;\left(\stackrel{[1,1]}{\lambda}_{a_{0}}\right),
{\bf\nabla}W=iW\int_{0}^{\beta}dt^{\prime}[\nablaA_{0}(t^{\prime},{\bfx})-\partial_{t^{\prime}}{\bfA}(t^{\prime},{\bfx})]=-iW\int_{0}^{\beta}~{\bfE}(t^{\prime},{\bfx})~dt^{\prime},\;\;E_{i}\equivF_{0i}
a_{1}^{0}[\psi]+a_{1}^{0}[\phi]=\int_{0}^{\infty}dr\,\left(-\frac18\delta_{r}^{2}\right)=\frac1{4\pi}\intr^{2}dr\,d\Omega\left(-\frac18\frac{\delta_{r}^{2}}{r^{2}}\right).
\Gamma_{\alpha\beta}^{\mu}=\frac{1}{2}\sum_{\sigma=0}^{3}\left(\frac{\partialg_{\sigma\beta}}{\partialx^{\alpha}}+\frac{\partialg_{\sigma\alpha}}{\partialx^{\beta}}-\frac{\partialg_{\alpha\beta}}{\partialx^{\sigma}}\right)g^{\sigma\mu}.
\Pi(q^{2})=\frac{\alpha}{\pi}\int_{0}^{1}dz\frac{z^{2}}{1-z^{2}}\,\biggl(1-\frac{1}{3}z^{2}\biggr)\frac{q^{2}}{\frac{4m^{2}}{1-z^{2}}-q^{2}}.
\left(\frac{1}{2}f^{\prime}\,+\,e^{f}\,\frac{1}{12\sqrt{2}}\bar{g}\left(2e^{-\varphi}+e^{2\varphi}\right)\,+\,\frac{1}{6}\sqrt{\frac{1}{2}}e^{f}e^{\varphi}\frac{1}{2\bar{g}}\gamma^{\nu\rho}R_{\nu\rho}^{ij}\gamma_{ij}\right)\gamma_{0}\,\varepsilon_{a}=0,
\left(\begin{array}{c}{B^{N}}\\{B^{R}}\\\end{array}\right)\to(\Lambda^{-1})^{T}\left(\begin{array}{c}{B^{N}}\\{B^{R}}\\\end{array}\right)=\left(\begin{array}{cc}{s}&{-r}\\{-q}&{p}\\\end{array}\right)\left(\begin{array}{c}{B^{N}}\\{B^{R}}\\\end{array}\right)
q_{k}\rightarrow\mathrm{e}^{-i\pi\widetilde{T}_{1}}\ast\left(\mathrm{e}^{i\piT_{1}}\astq_{k}\right),\qquadp_{k}\rightarrow\mathrm{e}^{-i\pi\widetilde{T}_{1}}\ast\left(\mathrm{e}^{i\piT_{1}}\astp_{k}\right)
{\calL}_{ooc}=\phi(\partial_{i}\partial_{j}\varphi*^{\prime}{\bar{\partial}}_{\bar{i}}{\bar{\partial}}_{{\bar{j}}}\varphi-\partial_{i}{\bar{\partial}}_{{\bar{i}}}\varphi*^{\prime}\partial_{j}{\bar{\partial}}_{{\bar{j}}}\varphi)(G^{-1}-\Theta)^{i{\bar{j}}}(G^{-1}+\Theta)^{j{\bar{i}}},
\sum_{\alpha_{1}\,,\alpha_{2}}z_{\alpha_{1}\alpha_{2}}\Pi(\lambda)_{\alpha_{1}}^{\alpha_{1}}\,z_{\alpha_{1}\alpha_{2}}^{-1}\Pi(\lambda^{\prime})_{\alpha_{2}}^{\alpha_{2}}=\sum_{\alpha_{1}\,,\alpha_{2}}z_{\alpha_{1}\alpha_{2}}\Pi(\lambda^{\prime})_{\alpha_{2}}^{\alpha_{2}}\,z_{\alpha_{1}\alpha_{2}}^{-1}\Pi(\lambda)_{\alpha_{1}}^{\alpha_{1}}\,,\nonumber
\int_{-i\infty}^{i\infty}ds{\frac{sin(\pi[a+c+1])sin(\pi[d+s+\lambda])}{sin(\pi[a+c+d+s+\lambda+1])}}{\frac{\Gamma(a+c+2)\Gamma(d+s+\lambda+1)}{\Gamma(a+c+d+s+\lambda+3)}},
1-c=\frac{-2\hat{u}^{3/2}+3\sqrt{3}\hat{v}}{2(\hat{u}-4)\sqrt{\hat{u}-1}+3\sqrt{3}\hat{v}},\:\:\:\:1-\frac{c}{(1-a)^{2}}=\frac{2\hat{u}^{3/2}+3\sqrt{3}\hat{v}}{2(\hat{u}-4)\sqrt{\hat{u}-1}-3\sqrt{3}\hat{v}}.
\tilde{\Pi}^{(1)}(p)=\frac{4\pi}{\kappa}\frac{C_{2}(G)}{2p^{2}}\mu\int\frac{d^{3}k}{(2\pi)^{3}}\frac{(k.p)^{2}-k^{2}p^{2}}{k^{2}(k+p)^{2}(k^{2}+\mu^{2})}
[\bar{\delta}_{f},\bar{\delta}_{g}]A_{\mu}=\bar{\delta}_{[f,g]}A_{\mu}+D_{\mu}X\quad\mathrm{and}\quad[\bar{\delta}_{f},\bar{\delta}_{g}]F_{\mu\nu}=\bar{\delta}_{[f,g]}F_{\mu\nu}+D_{\mu}X_{\nu}\;,
{\calL}_{I}=-\frac{2}{9}\Omega^{\rho,\mu\nu\alpha\beta}\Omega_{\mu,\rho\nu\alpha\beta}+\frac{2}{9}\Omega^{\mu\nu\alpha}\Omega_{\mu\nu\alpha}-\frac{1}{9}\Omega^{\rho,\mu\nu\alpha\beta}E_{\mu\nu\alpha\beta,\rho}-\frac{4}{9}\Omega^{\mu\nu\alpha}E_{\mu\nu\alpha}
\tilde{\phi}=\phi+{\frac{\alpha(1+\alpha^{2}l^{-2})}{(r_{+}^{2}-\alpha^{2})}}\chi,\quad\tilde{\psi}=\psi+{\frac{\alpha(1+\alpha^{2}l^{-2})}{(r_{+}^{2}-\alpha^{2})}}\chi\,.
A^{11}=\frac{b^{\prime}r-b}{2r^{3}}+\frac{1}{r\left(e^{2\psi}\alpha^{2}-\beta^{2}r^{2}\right)}\left[e^{2\psi}\psi^{\prime}\left(1-\frac{b}{r}\right)\alpha^{2}-\beta^{2}b\right]
\phi(r,\theta,\varphi)=\int_{r_{H}}^{\infty}dr_{o}\int_{-1}^{1}dcos{\theta}_{o}\int_{0}^{2\pi}d{\varphi}_{o}(r_{o}^{2}+A^{2}{cos}^{2}\theta_{o})G(r,\theta,\varphi;r_{o},\theta_{o},\varphi_{o}){\cal{J}}(r_{o},\theta_{o},\varphi_{o})
M=\left(\begin{array}{cc}{Q_{nm}+\frac{\pi}{2}A_{n}(Im\,\Omega)^{-1}A_{m}}&{-\frac{\pi}{2}A_{n}(Im\,\Omega)^{-1}\bar{A}_{m}}\\{-\frac{\pi}{2}\bar{A}_{n}(Im\,\Omega)^{-1}A_{m}}&{\bar{Q}_{nm}+\frac{\pi}{2}\bar{A}_{n}(Im\,\Omega)^{-1}\bar{A}_{m}}\\\end{array}\right)\;,
\hat{P}=-\frac{2\lambda^{2}}{4!}\left(\begin{array}{cc}{3{\varphi_{1}}^{2}+{\varphi_{2}}^{2}}&{2{\varphi_{1}}{\varphi_{2}}}\\{2{\varphi_{1}}{\varphi_{2}}}&{3{\varphi_{2}}^{2}+{\varphi_{1}}^{2}}\\\end{array}\right),
U_{OM,JJ}\propto\frac{e^{-J\beta}}{JV}\bigg(\Big[\frac{V_{J}}{V}+\frac{1}{J}+\beta\Big]^{2}+\frac{1}{J^{2}}+\Big(\frac{V_{J}}{V}\Big)^{2}-\frac{V_{JJ}}{V}\bigg)
g_{\hat{M}\hat{N}}=\left(\begin{matrix}{-\delta_{\hat{0}\hat{0}}+h_{\hat{0}\hat{0}}(x^{p})}&{h_{\hat{0}\hat{m}}(x^{p})}\\{h_{\hat{n}\hat{0}}(x^{p})}&{\delta_{\hat{m}\hat{n}}+h_{\hat{m}\hat{n}}(x^{p})}\\\end{matrix}\right)
\left(\frac{d}{d\tau}\frac{\partialL}{\partial{\dot{x}}_{\mu}}-\frac{d^{2}}{d\tau^{2}}\frac{\partialL}{\partial{\ddot{x}}_{\mu}}+\frac{d^{3}}{d\tau^{3}}\frac{\partialL}{\partial{\stackrel{\ldots}{x}}_{\mu}}\right){\dot{x}}_{\mu}=0.
\partial_{t}^{2}F_{0}(s,t)=\int_{0}^{\infty}\left[{\frac{e^{-u-t-s}}{1-e^{-u-t-s}}}-{\frac{e^{-u-t}}{1-e^{-u-t}}}\right]du=\sum_{n\geq1}\left[{\frac{e^{-n(t+s)}}{n}}-{\frac{e^{-nt}}{n}}\right].
\begin{array}{ll}{\dot{p}^{\alpha}=0,\qquadp^{\alpha}=mu^{\alpha}+\displaystyle\frac{1}{c^{2}}S^{\alpha\beta}\dot{u}_{\beta},}\\{\dot{S}^{\alpha\beta}=p^{\alpha}u^{\beta}-p^{\beta}u^{\alpha},}\\{S^{\alpha\sigma}u_{\sigma}=0,}\\\end{array}
R_{\alpha}^{j}\frac{\deltaR_{\beta}^{i}}{\delta\phi^{j}}-R_{\beta}^{j}\frac{\deltaR_{\alpha}^{i}}{\delta\phi^{j}}=C_{\alpha\beta}^{\gamma}R_{\gamma}^{i}+M_{\alpha\beta}^{ij}\frac{\deltaI}{\delta\phi^{i}},\;\;M_{\alpha\beta}^{ij}=-M_{\alpha\beta}^{ji}
\int_{\Sigma}\sqrt{W}[-\overline{{\Psi}}\Gamma_{-}\Gamma_{r}\mathcal{D}_{r}\Psi+\overline{{\Psi}}\Gamma_{-}\Gamma_{m}\{X^{m},\Psi\}+\Lambda\{\overline{{\Psi}}\Gamma_{-},\Psi\}],
R_{+}\rightarrowq^{-1/2}R_{+}=\left(\begin{array}{cccc}{q^{1/2}}&{0}&{0}&{0}\\{0}&{q^{-1/2}}&{(q-q^{-1})q^{-1/2}}&{0}\\{0}&{0}&{q^{-1/2}}&{0}\\{0}&{0}&{0}&{q^{1/2}}\\\end{array}\right),
\Gamma_{+}^{2}=\frac{1}{(2\pi)^{2}}\intdt~d^{2}{\bfx}\intd^{2}{\bfp}~\left[\frac{-e^{2}}{2}\pi\delta(p_{1},p_{2})\epsilon_{ij}a_{j}\dot{a}_{i}\right]~e^{i{\bfp}\cdot{\bfx}},
\widehatG(p;\pi,T)=\frac{c}{2}p_{3}+\frac{1}{4}(A_{0}p_{3}^{2}+A_{1}p_{2})+\frac{1}{6}(B_{0}p_{3}^{3}+B_{1}p_{3}p_{2}+B_{2}p_{1})\;.
I=\int_{\Sigma}d^{3}\!x\sqrt{\gamma}\left(\!R(\gamma)\!-\!{\frac{1}{2}}\!\left(\!DU\right)^{2}\!-\!{\frac{1}{2}}e^{2U}V_{i}V^{i}\!+\!V\!D_{i}V^{i}\!-\!{\frac{1}{2}}\!\left(\left(\!D\Phi\!\right)^{2}\!-\!e^{-2\Phi}\!\left(\!D\Psi\!\right)^{2}\right)\right)\.
[\hat{E},\hat{x}]=-i\frac{\hbar}{m}\hat{p},\,\,\,\,\,\,[\hat{E},\hat{p}]=im\omega^{2}\hbar\hat{x},\,\,\,\,\,\,[\hat{x},\hat{p}]=i\hbar(1+\frac{1}{mc^{2}}\hat{E})\,,
\hat{\calL}_{0}\equiv\sum_{j=1}^{2}(-)^{j-1}{\calL}_{0}(\vec{\phi}_{j}(x),\,\vec{\varphi}_{j}(x))\;\;(\in\hat{\calL}\equiv\sum_{j=1}^{2}(-)^{j-1}{\calL}(\vec{\phi}_{j}(x)+\vec{\varphi}_{j}(x))),
\Psi=\left[\begin{array}{c}{0}\\{\mathrm{L}_{N}^{\left|m\right|}F_{m}}\\{c_{3}\mathrm{L}_{N+1}^{\left|m\right|-1}F_{m-1}}\\{d_{3}\mathrm{L}_{N}^{\left|m\right|}F_{m}}\\\end{array}\right]
\delta_{\lambda}x_{\bar{a}}^{\mu}=\lambda_{\bar{a}\bar{b}}\pi_{\bar{b}}^{\mu},\qquad\delta_{\lambda}x_{\bar{b}}^{\mu}=\lambda_{\bar{b}\bar{a}}\pi_{\bar{a}}^{\mu},\qquad\lambda_{\bar{a}\bar{b}}\equiv\lambda_{\bar{b}\bar{a}},
\left[\Box_{o}+\frac{1}{4}\left(\frac{1}{e}e_{\mu}^{m}(\partial_{m}(ee_{n}^{\mu}))\right)^{2}-\frac{1}{2}\partial^{n}\left(\frac{1}{e}e_{\mu}^{m}\partial_{m}(ee_{n}^{\mu})\right)+\xiR\right]\tilde{G}(x,x^{\prime};\sigma)=-\partial_{\sigma}\tilde{G}(x,x^{\prime};\sigma)
\begin{array}{l}{1^{0}\quadE_{j}(s){\neq}E_{k}(s)\quad\foralls\in[0,1],\quadj{\neq}k,}\\{2^{0}\quad\forallj\P_{j}(s)}\\{mbox{isdoublecontinuously}}\\{\qquad\qquad\quad\mathrm{differentiable~function~of}\s\in[0,1],}\\\end{array}
\left(\begin{array}{cc}{1}&{0}\\{-2i\theta^{a}(x_{+}{\cdot\tilde{\sigma}})^{-1}}&{1}\\\end{array}\right){\calF}(z)\left(\begin{array}{cc}{1}&{-2i(x_{+}{\cdot\tilde{\sigma}})^{-1}\bar{\theta}_{b}}\\{0}&{1}\\\end{array}\right)=\left(\begin{array}{cc}{-ix_{+}{\cdot\tilde{\sigma}}}&{0}\\{0}&{v_{~b}^{a}(-z)}\\\end{array}\right)
{\calL}_{\mathrm{gf}}+{\calL}_{\mathrm{ghost}}=-\frac{1}{2\xi}(\partial_{\mu}A^{\mu})\star(\partial_{\nu}A^{\nu})+\partial^{\mu}\bar{c}\star(\partial_{\mu}c-ie\left[A_{\mu},c\right]_{\mathrm{MB}})
N_{lm}(\delta_{1},\delta_{2})=\frac{(-1)^{\frac{m-|m|}{2}}}{2^{|m|}}\sqrt{\frac{(2l+\delta_{1}+\delta_{2}+1)(l-|m|)!\Gamma(l+|m|+\delta_{1}+\delta_{2}+1)}{2^{\delta_{1}+\delta_{2}+2}{\pi}\Gamma(l+\delta_{1}+1)\Gamma(l+\delta_{2}+1)}}.
{\calC}_{{\bfq}_{1}{\bfq}_{2};{\bfp}_{1}{\bfp}_{2}}\equiv\langlea_{{\bfq}_{1}}^{\dagger}a_{{\bfq}_{2}}^{\dagger}a_{{\bfp}_{1}}a_{{\bfp}_{2}}\rangle-\langlea_{{\bfq}_{1}}^{\dagger}a_{{\bfp}_{1}}\rangle\langlea_{{\bfq}_{2}}^{\dagger}a_{{\bfp}_{2}}\rangle-\langlea_{{\bfq}_{2}}^{\dagger}a_{{\bfp}_{1}}\rangle\langlea_{{\bfq}_{1}}^{\dagger}a_{{\bfp}_{2}}\rangle.
\bar{f}_{1}^{^{\prime}}(\theta_{1}^{\prime})\bar{f}_{2}^{^{\prime}}(\theta_{2}^{\prime})\left\langle0\right|a(\theta_{2}^{\prime})a(\theta_{1}^{\prime})Sa^{\ast}(\theta_{2})a^{\ast}(\theta_{1}\left|0\right\ranglef_{2}(\theta_{2})f_{1}(\theta_{1})
\begin{array}{lll}{d=10:}&{\Gamma_{M}}&{M=1,\cdots,10\,,}\\{d=11:}&{{\tilde{\Gamma}}_{\tilde{M}}}&{{\tilde{M}}=1,\cdots,10,11\,,}\\{d=12:}&{{\hat{\Gamma}}_{\hat{M}}}&{{\hat{M}}=1,\cdots,10,11,12\,.}\\\end{array}
\psi_{1}=\intd^{2}we^{i{\vec{k}.\vec{w}}}a_{1}K_{L}(kx_{0})\;\;\;\psi_{2}=\intd^{2}we^{i\vec{k}.\vec{w}}\frac{i{\gamma.\vec{k}}}{k}K_{L+1}(kx_{0})
{\mathbbP}H^{0}(dP_{9},{\calO}_{dP_{9}}(3\sigma|_{dP_{9}}+5F))\simeq{\mathbbP}H^{0}({\calS},{\calO}_{{\calS}}(5)\oplus{\calO}_{{\calS}}(3)\oplus{\calO}_{{\calS}}(2))\simeq{\mathbbP}^{12}.
W[Y]=\frac{1}{2}\intd^{4}x\int\frac{d^{4}p}{(2\pi)^{4}}\int_{0}^{\infty}\frac{dt}{t^{3}}\rho(t,\Lambda^{2})e^{-p^{2}}\mathrm{tr}\left(e^{-t(m^{2}+A)}\right)\cdot1,
f[u_{0}]=-\frac{1}{2}\frac{\partialG_{2}}{\partial\eta}\bigg|_{\eta=u_{0}}\left[\hat{h}^{-1}+\frac{1}{2}[\hat{h}-l(2\omega_{s})]^{-1}\right]G_{2}[u_{0}]+\frac{3}{4}G_{3}[u_{0}]
\dot{\tilde{g}}(t)=\frac{4}{m+1}\tilde{g}(t)-\pi\frac{m}{m+1}\tilde{C}_{\Phi\Phi}^{\Phi}(\tilde{g}(t),t)\tilde{g}(t)^{2}=y\tilde{g}(t)-\pi\tilde{C}_{\Phi\Phi}^{\Phi}(\tilde{g}(t),t)\tilde{g}(t)^{2}\frac{m}{m+1}.
D_{M}\left(\left(v_{\alpha}e^{-\phi}+\tilde{v}_{\alpha}e^{\phi}\right)F_{\alpha}^{MN}\right)-\frac{1}{2}v_{\alpha}e^{-2\phi}H_{PQ}^{N}F_{\alpha}^{PQ}-\frac{1}{2}\tilde{v}_{\alpha}\mathrm{}^{*}H_{PQ}^{N}F_{\alpha}^{PQ}=0,
\rho_{0}={\frac{V_{0}}{2|z_{+}-z_{-}|}}=\mp{\frac{A}{2|z_{+}-z_{-}|^{5}}};\qquad\qquad\quadA\equiv{\frac{\pi^{2}}{32}}\zeta_{R}^{\prime}(-4)\approx2.46\cdot10^{-3}.
\mathrm{I}m\:\frac{d}{ds}\zeta_{H}(-1;ix)=-\frac{1}{4\pi}\sum_{n=1}^{+\infty}\frac{e^{-2n\pix}}{n^{2}}-\frac{x}{2}\mathrm{ln}\:x+\frac{\pi}{24}-\frac{\pix^{2}}{4}\:.
M_{k}^{2}=\sum_{i=1}^{n}\left\{(2m_{i}+1)({\calF}_{2i-1,2i}^{(1)}-{\calF}_{2i-1,2i}^{(2)})\right\}\mp2({\calF}_{2k-1,2k}^{(1)}-{\calF}_{2k-1,2k}^{(2)})\,
\begin{array}{ll}{[\alpha_{m-qR},\alpha_{n-qR}^{\dagger}]=(m-qR)\delta_{mn},}&{[\alpha_{m+qR},\alpha_{n+qR}^{\dagger}]=(m+qR)\delta_{mn},}\\{\{b_{m-qR},b_{n-qR}^{\dagger}\}=\delta_{mn},}&{\{b_{m+qR},b_{n+qR}^{\dagger}\}=\delta_{mn},}\\\end{array}
\left(\ell_{1},\dot{\ell}_{1},h_{1})(\ell_{2},\dot{\ell}_{2},h_{2})=(\ell_{1}+h_{1}\ell_{2}h_{1}^{-1},\dot{\ell}_{1}+[\ell_{1},h_{1}\ell_{2}h_{1}^{-1}]+h_{1}\dot{\ell}_{2}h_{1}^{-1},h_{1}h_{2}\right)\.
\mathbf{M}\,[\phi(\alpha,s)]=\int_{1}^{\infty}\Big[\sum_{k}b_{k}\tau^{\lambda_{k}}+\alpha\,\tau^{h}\,\phi(\tau)\Big]\tau^{-s-1}\,d\tau+\int_{1}^{\infty}\phi(\tau)\tau^{s-1}\,d\tau\,[6pt]
T_{B}^{A}=\frac{\delta(y)}{b_{0}}\mathrm{diag}(V_{\star}+\rho_{\star},V_{\star}-p_{\star},V_{\star}-p_{\star},V_{\star}-p_{\star},0)+\frac{\delta(1-y)}{b_{0}}\mathrm{diag}(-V+\rho,-V-p,V-p,V-p,0),
M^{05}(n_{1}^{\prime}n_{2}^{\prime}n^{\prime}k^{\prime}n_{1}n_{2}nk)=(m+2n+2n_{1}+2n_{2})\delta_{n_{1}n_{1}^{\prime}}\delta_{n_{2}n_{2}^{\prime}}\delta_{nn^{\prime}}\delta_{kk^{\prime}}
B(z,\bar{z})=w_{+}e^{\omegaz+\bar{\omega}\bar{z}}+w_{-}e^{-\left(\omegaz+\bar{\omega}\bar{z}\right)}+we^{\omegaz-\bar{\omega}\bar{z}}+\bar{w}e^{-\left(\omegaz-\bar{\omega}\bar{z}\right)},
\Psi(\omega;\lambda)=\Phi(\omega;\lambda)-{\frac{1}{4\lambda}}\sum_{k}\left[(\phi_{\omega,k}^{(\lambda)},\partial_{\lambda}H^{2}(\lambda)\phi_{\omega,k}^{(\lambda)})+(\phi_{\omega,k}^{(\lambda)},\partial_{\lambda}H^{2}(\lambda)\phi_{\omega,k}^{(\lambda)})^{*}\right],
{\sum_{j}}^{\prime}c_{j}({M_{j}^{\prime}}^{2})^{2}\int_{0}^{\frac{\Lambda^{2}}{{M_{j}^{\prime}}^{2}}}ydy{\frac{1}{(y+1)^{s}}}\left(\alphay^{s}+{\frac{\beta(p_{i})}{{M_{j}^{\prime}}^{2}}}y^{s-1}+{\frac{\gamma(p_{i})}{{M_{j}^{\prime}}^{4}}}y^{s-2}+\ldots\right).
\Lambda\mathrm{~\binom{i_1~i_2~\dots~i_m}{j_1~j_2~\dots~j_n}~}=(-)^{n+m}\sum_{r=1}^{n}(-)^{r+1}\Lambda_{j_{r}i_{m}}\Lambda\mathrm{~\binom{i_1~i_2~\dots~i_{m-1}}{j_1~\dots~\check{j}_r~\dots~j_n}~}
\begin{array}{l}{S_{m}=-i\displaystyle\frack{4\pi}\displaystyle\int_{x}{\epsilon}^{\mu\nu\rho}\left(\displaystyle\frac12A_{\mu}^{a}{\partial}_{\nu}A_{\rho}^{a}+\frac1{3!}f^{abc}A_{\mu}^{a}A_{\nu}^{b}A_{\rho}^{c}\right)+\displaystyle\frac{|k|}{16m\pi^{2}}\int_{x}F_{\mu\nu}^{a}F^{{\mu\nu}a}\,,}\\\end{array}
R^{(1)}M_{1}K_{1}\tilde{M}_{1}^{-1}R^{(2)}M_{2}K_{2}\tilde{M}_{2}^{-1}=M_{2}K_{2}\tilde{M}_{2}^{-1}R^{(3)}M_{1}K_{1}\tilde{M}_{1}^{-1}R^{(4)}
V^{\dagger}\,\Upsilon_{l\,m}^{\pm}=\frac1{\sqrt2}\left(\begin{array}{c}{\sqrt{\frac{l+m}{2l}}\,Y_{l^{-}m^{-}}\pm\sqrt{\frac{l-m+1}{2l+2}}\,Y_{l^{+}m^{-}}}\\{\sqrt{\frac{l-m}{2l}}\,Y_{l^{-}m^{+}}\mp\sqrt{\frac{l+m+1}{2l+2}}\,Y_{l^{+}m^{+}}}\\\end{array}\right)=\frac1{\sqrt2}\left(\Omega_{l,\,l^{-},\,m}\mp\Omega_{l,\,l^{+},\,m}\right).
\sum_{m,I_{m}}[(\partial_{0}\tilde{P}_{I_{m}}^{(m)}-\tilde{d}\tilde{Q}_{I_{m}}^{(m)})\omega_{(m)}^{I_{m}}+\tilde{P}_{I_{m}}^{(m)}\partial_{0}\omega_{(m)}^{I_{m}}]=\gamma\beta,
{}~{\displaystyle\sum_{\{k_{i}\},\atopi=1,\cdots,6}}R(u_{3},u_{5},u_{6})_{i_{3},i_{5},i_{6}}^{k_{3},k_{5},k_{6}}R(u_{2},u_{4},u_{6})_{i_{2}i_{4}k_{6}}^{k_{2}k_{4}j_{6}}R(u_{1},u_{4},u_{5})_{i_{1}k_{4}k_{5}}^{k_{1}j_{4}j_{5}}R(u_{1},u_{2},u_{3})_{k_{1}k_{2}k_{3}}^{j_{1}j_{2}j_{3}}
\sum_{k=1}^{\infty}\frac{J_{\nu}(\lambda_{\nu,k}t)J_{\nu}(\lambda_{\nu,k}x)}{\left(\lambda_{\nu,k}^{2}-a^{2}\right)J_{\nu+1}^{2}(\lambda_{\nu,k})}=\frac{\pi}{4}\frac{J_{\nu}(ax)}{J_{\nu}(a)}\left[J_{\nu}(at)Y_{\nu}(a)-Y_{\nu}(at)J_{\nu}(a)\right],\quad0\leqx\leqt\leq1.
\delta\left(\eta_{\alpha_{L-3}}^{*}Z_{\;\;\beta_{L-2}}^{\alpha_{L-3}}+\alpha_{4}\Phi_{\alpha_{L-4}}^{*}\Phi_{\alpha_{0}}^{*}C_{\;\;\beta_{L-2}}^{\alpha_{L-4}\alpha_{0}}\right)=\Phi_{\alpha_{L-2}}^{*}D_{\;\;\beta_{L-2}}^{\alpha_{L-2}},
{\frac{dA^{(0)}}{d\tilde{y}}}=0,\quad{\frac{dA^{(1)}}{d\tilde{y}}}=-\alphaw_{0},\quad{\frac{dA^{(2)}}{d\tilde{y}}}=-\alpha^{2}\bar{W}^{(0)}(\tilde{\phi}),\cdots.
A_{i}^{\prime}(\vec{x},t)=e^{-it\tilde{A}_{0}}\left[A_{i}^{\prime}(\vec{x},0)+\int_{0}^{t}dt^{\prime}\,e^{it^{\prime}\tilde{A}_{0}}(\partial_{i}\tilde{A}_{0})e^{-it^{\prime}\tilde{A}_{0}}\right]e^{it\tilde{A}_{0}}
{\calO}=\mathrm{exp}\left\{\sum_{k=1}^{\tau}\intd^{2}z[\xi_{k}^{1}\,(\Phi\tilde{\Phi})_{k}+\xi_{k}^{2}\,(\Phi\tilde{\Pi})_{k}+\xi_{k}^{3}\,(\Pi\tilde{\Phi})_{k}+\xi_{k}^{4}\,(\Pi\tilde{\Pi})_{k}]\right\},
\begin{array}{c}{I=0,\,\alpha_{i},\,\alpha_{i_{1}}\alpha_{i_{2}},\,\ldots,\,\alpha_{i_{1}}\!\cdots\alpha_{i_{k}},\,\ldots\,,\quadi_{1}<i_{2}<\cdots<i_{k},}\\{e^{0}=1,\quade^{\alpha_{i_{1}}\alpha_{i_{2}}\cdots\alpha_{i_{k}}}=\xi^{\alpha_{i_{1}}}\xi^{\alpha_{i_{2}}}\cdots\xi^{\alpha_{i_{k}}},\quadk=0,1,\ldots,n.}\\\end{array}
\begin{array}{rcllcccc}{\mathrm{\underline{{sector}}}}&{{}}&{\mathrm{\underline{{state}}}}&{{}}&{{}}&{\mathrm{\underline{{~R_1~}}}}&{\mathrm{\underline{{~R_2~}}}}&{\mathrm{\underline{{~R_3~}}}}\\{\mathrm{NS}}&{{}}&{|2s_{3},2s_{4}\rangle,}&{s_{3}=s_{4}}&{{}}&{+}&{+}&{-2is_{3}}\\{\mathrm{R}}&{{}}&{|-,2s_{1},2s_{2}\rangle,}&{s_{1}=s_{2}}&{{}}&{+}&{+}&{+2is_{1}.}\\\end{array}
J_{nn^{\prime\prime}}(\hat{q}_{\perp})\equiv\sum_{m=0}^{\mathrm{min}(n,n^{\prime\prime})}\frac{n!n^{\prime\prime}!}{m!(n-m)!(n^{\prime\prime}-m)!}[i\mathrm{sgn}(eH)\hat{q}_{\perp}]^{n+n^{\prime\prime}-2m}.
+\sum_{B\neqA}\frac{\partial\beta_{B}}{\partialM}\frac{1}{z_{A}-z_{B}}\left[\frac{\prod_{C\neqA}(z_{A}-z_{C})}{{\calP}(z_{A})}+\frac{\prod_{C\neqB}(z_{B}-z_{C})}{{\calP}(z_{B})}\right].
\left(\Sigma^{\mu\nu0\lambda}\right)^{ij\alpha\beta}=\left(\Sigma^{\mu\nu}\right)^{ij\alpha\beta}\eta^{\lambda0}-\frac12\left(\eta^{\mu0}\eta^{\nu\lambda}-\eta^{\nu0}\eta^{\mu\lambda}\right)\left(\eta^{i\alpha}\eta^{j\beta}+\eta^{i\beta}\eta^{j\alpha}\right)
G_{MN}^{(H)}dx^{M}dx^{N}=-(1-{\frac{2m}{r^{2}}})dt^{2}+dx_{1}^{2}+\cdots+dx_{5}^{2}+(1-{\frac{2m}{r^{2}}})^{-1}dr^{2}+r^{2}d\Omega_{3}^{2},
\begin{aligned}{\deltaX^{C}=}&{{}\sum_{A,B}g_{AB}^{C}\epsilon^{A}X^{B}}\\{\delta\mathcal{A}_{r}^{C}=}&{{}-\lambda_{rA}^{C}\epsilon^{A}-\sum_{A,B}g_{AB}^{C}\mathcal{A}_{r}^{A}\epsilon^{B}}\\\end{aligned}
J_{\mu\nu}[\xi|s]=\tilde{g}\epsilon_{\mu\nu\rho\sigma}[\bar{\psi}(\xi(s))\gamma^{\rho}t^{i}\psi(\xi(s)))]\Omega_{\xi}^{-1}(s,0)t_{i}\Omega_{\xi}(s,0)\dot{\xi}^{\sigma}(s),
e^{{\cal{G}}_{\mathrm{c}}\{J\}}=\frac{{\cal{Z}}\{J\}}{{\cal{Z}}_{0}}=\frac{\int{\cal{D}}\{\phi\}e^{-S\{\phi\}+(J,\phi)}}{\int{\cal{D}}\{\phi\}e^{-S_{0}\{\phi\}}}\;.
\left\|\left(f_{0}\underline{{1}}+\sum_{n=1}^{N}\idotsintf_{n}^{(\lambda_{1}....\lambda_{n})}(x_{1}....x_{n})A^{(\lambda_{1})}(x_{1})....A^{(\lambda_{n})}(x_{n})\right)\left|0\right\rangle\right\|^{2}\geq0
{\frac{\partialS_{J}}{\partial\Omega_{p+1}}}={\frac{1}{2m_{p+1}\,(p+1)!}}\,P^{0)}{}_{\mu\mu_{2}\dots\mu_{p+1}}P^{0)}{}^{\mu\mu_{2}\dots\mu_{p+1}}-\frac{m_{p+1}}{2}\,p\.
\begin{array}{ccccc}{\left\{\pi_{\mathrm{ab}},H_{\mathrm{C}}\right\}\approx0,}&{}&{\left\{\chi_{1},H_{\mathrm{C}}\right\}\approx0,}&{}&{\left\{\chi_{2},H_{\mathrm{C}}\right\}\approx0.}\\\end{array}
\frac{\ell^{3}E_{0}\left(t,\mu\right)}{L^{2}}=-\frac{t\mu^{3}}{\pi^{2}}\sum_{p=-1}^{\infty\,\,\,\,\,\,^{\prime}}\,\sum_{k=1}^{\infty}\,\frac{\left(-1\right)^{k+1}}{k}J_{pk}\left(t,\mu\right),
f=-4\pi(T-{\bar{T}})(U-{\bar{U}})\int\frac{d^{2}\tau}{\tau_{2}^{2}}{\bar{F}}({\bar{\tau}})\partial_{{\bar{\tau}}}[\tau_{2}^{2}\partial_{{\bar{\tau}}}(\tau_{2}\sum_{P_{L},P_{R}}{\bar{P}}_{L}{\bar{P}}_{L}e^{i\pi\tau|P_{L}|^{2}}e^{-i\pi{\bar{\tau}}|P_{R}|^{2}})],
\begin{array}{lll}{}&{ds_{N}^{2}}&{=G_{\alpha\beta}dy^{\alpha}dy^{\beta}}\\{}&{}&{=[\phi(x)]^{-(N-3)/(N-2)}\,g_{\mu\nu}(x)\,dx^{\mu}dx^{\nu}+[\gamma\phi(x)]^{2/(N-2)}\,d\Omega_{N-2}^{2}\,,~~~~~\phi>0\,.}\\\end{array}
ds^{2}=-\left(dX^{0}\right)^{2}+\left(dX^{1}\right)^{2}+\left(dX^{2}\right)^{2}+\left(dX^{3}\right)^{2}+\left(dX^{4}\right)^{2}-\left(dX^{5}\right)^{2}\,
\mathrm{Tr}~e^{-(L_{i}+m_{i}^{2}g_{\tau\tau})t}\simeq{\frac{n_{i}}{(4\pit)^{{\frac{D-1}{2}}}}}\int_{\widetilde{\calM}^{\prime}}d\tilde{V}^{\prime}e^{-m_{i}^{2}g_{\tau\tau}t}~~~,
M^{6}r_{c}^{4}\sim\frac{M_{s}^{2}}{T_{2}}E_{2}(U,\bar{U})=\frac{M_{s}^{2}}{T_{2}}\sum_{(m,n)\in(Z,Z)-(0,0)}\frac{U_{2}^{2}}{|m+nU|^{4}}\,.
\left(\begin{array}{c}{\psi_{r}}\\{\psi_{-r}}\\\end{array}\right)=\left(\begin{array}{cc}{M_{rs}}&{{\widetildeM}_{rs}}\\{-{\widetildeM}_{rs}}&{-M_{rs}}\\\end{array}\right)\left(\begin{array}{c}{\psi_{r}}\\{\psi_{-r}}\\\end{array}\right),
\alpha(Q^{2})\equiv{\frac{B^{*}(Q^{2})-b_{0}(Q^{2})g^{3}-g^{5}B_{2}(Q^{2},Q^{2})-g^{7}B_{5}(Q^{2},Q^{2})-...}{B_{1}(Q^{2},Q^{2})g^{3}}}
I=\int_{\Omega}\left(\frac{1}{2}\epsilon_{\mu\nu\lambda}B_{\mu}^{*}(\partial_{\nu}-iga_{\nu})B_{\lambda}+MB_{\mu}^{*}B^{\mu}+\frac{1}{2}\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}\right)\;.