harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{e^{2\pii\left(-\frac{1}{4\tau}\right)P^{2}}}{\eta\left(-\frac{1}{4\tau}\right)}=\intdP^{\prime}e^{2\piiPP^{\prime}}\frac{e^{2\pii\tau{P^{\prime}}^{2}}}{\eta(\tau)}~,
Re\epsilon_{\pm}=\pm\left[\frac12\left\{\left[\left(R^{2}-\frac14\Gamma^{2}\right)^{2}+\left(\vec{R}\cdot\vec{\Gamma}\right)^{2}\right]^{\frac12}+\left(R^{2}-\frac14\Gamma^{2}\right)\right\}\right]^{\frac12},
\int\frac{\mathrm{d}^{3}Q}{|{\bfQ}|}N(|{\bfQ}|)\left\{\frac{\mathrm{d}}{\mathrm{d}Q_{0}}\left(\frac{1}{Q_{0}}\mathrm{ln}(k\cdotQ)\right)\right\}_{Q_{0}=|{\bfQ}|}=\pi\mathrm{ln}(T)\mathrm{ln}(k^{2})+\cdots,
G(x)=2\pix\int_{0}^{1}{\frac{\pi^{2}B_{2}(t)}{(x^{2}+(2\pit)^{2})^{{\frac{3}{2}}}}}dt+2\pi^{3}x\sum_{k=0}^{\infty}{\binom{-{\frac{3}{2}}}{k}}{\frac{x^{2k}}{(2\pi)^{3+2k}}}\int_{0}^{1}B_{2}(t)\zeta(3+2k,t)\,dt\.
{\omega(T)}=t_{\infty}^{-1}{\sum}_{{\lambda}=0}^{\infty}a_{\lambda}t_{\infty}^{\lambda}\;,{a_{o}\neq0},\;\;{\omega(T)}=t_{o}^{-1}{\sum}_{{\lambda}=0}^{\infty}b_{\lambda}t_{o}^{\lambda}\;,{b_{o}\neq0}
R_{\pm}=\frac{\pm\mid\alpha_{1}\mid\mid\alpha_{2}\mid}{\sqrt{(\mid\alpha_{1}\mid^{2}\pm\frac{1}{2}sinh2(\mid\alpha_{1}\mid^{2}+\mid\alpha_{2}\mid^{2})(\mid\alpha_{2}\mid^{2}\pm\frac{1}{2}sinh2(\mid\alpha_{1}\mid^{2}+\mid\alpha_{2}\mid^{2})}}.\nonumber\,
{\frac{1}{\tau}}{\frac{dE(\tau)}{d\tau}}=-{\frac{2e}{\tau^{2}}}\sum_{s=1}^{4}\int[d{\bfk}]({\bfk}_{\perp}^{2}+m^{2})\lambda_{s}\vertf_{{\bfk}s}^{+}\vert^{2},
H_{\mathrm{BPS}}={\frac{1}{2}}\sum_{A}\{Q_{A},Q_{A}^{\dagger}\}=-{\frac{1}{2}}\nabla^{2}+{\frac{1}{2}}(\nablaW)^{2}+{\frac{1}{2}}{\frac{\partial^{2}W}{\partialx^{i}\partialx^{j}}}[\psi^{i\dagger},\psi^{j}]
{\hat{\rho}}^{(1)}=\left(\begin{array}{ccc}{0}&{-\delta_{ij}}&{{\frac{\partial\Omega^{j}}{\partialq_{i}}}}\\{\delta_{ij}}&{0}&{{\frac{\partial\Omega^{j}}{\partialp_{i}}}}\\{-{\frac{\partial\Omega^{i}}{\partialq_{j}}}}&{-{\frac{\partial\Omega^{i}}{\partialp_{j}}}}&{0}\\\end{array}\right)
\displaystyle{\hat{{\calL}}_{l}\,\varphi(s)\equiv\frac{\mu(s)}{\epsilon(s)}\left\{\frac{d^{2}}{ds^{2}}-\frac{l(l+1)}{\left[s-(a_{1}-a_{2})\Theta(s-a_{1})\right]^{2}}\right\}\varphi(s)=-\frac{\omega^{2}}{c^{2}}\varphi(s)},
\left(\begin{array}{c}{t}\\{c}\\{u}\\\end{array}\right);\\\left(\begin{array}{c}{b}\\{s}\\{d}\\\end{array}\right);\\\left(\begin{array}{c}{\tau}\\{\mu}\\{e}\\\end{array}\right);\\\left(\begin{array}{c}{\nu_{1}}\\{\nu_{2}}\\{\nu_{3}}\\\end{array}\right).
e^{2}<A^{\mu}(x)A^{\nu}(y)>=<B^{\mu}(x)B^{\nu}(y)>+{\frac{i}{a^{2}\partial^{2}+4\alpha^{2}}}(ag^{\mu\nu}\delta(x-y)+{\frac{2\alpha}{\partial^{2}}}\epsilon^{\mu\nu\lambda}\partial_{\lambda}\delta(x-y)).
V_{II}(\beta,\varphi_{0})=\frac{1}{\beta}\sum_{s=1}^{\infty}\frac{(-1)^{s+1}}{2s}\biggl(\frac{\lambda}{8\pi^{2}}\biggr)^{s}\biggl(\frac{\varphi_{0}}{\mu}\biggr)^{2s}\int\frac{d^{d}q}{(2\pi)^{d}}A_{1}^{M^{2}}(s,a).
F_{8}^{{\mathcal{O}}_{2,0}^{1,0}|++++++--}=\frac{\langle{\mathcal{O}}_{2,0}^{1,0}\rangle(\sigma_{2}^{-})^{1/2}(\sigma_{5}^{+}+\sigma_{3}^{+}\sigma_{2}^{-}+\sigma_{1}^{+}(\sigma_{2}^{-})^{2})/\sigma_{6}^{+}}{2^{6}\,\mathrm{cosh}(\theta_{78}/2)\,\prod_{1\leqi<j\leq6}\mathrm{cosh}(\theta_{ij}/2)}\prod_{i<j}F_{\mathrm{min}}^{\mu_{i}\mu_{j}}(\theta_{ij}),
\frac{1}{(q)_{\infty}}\sum_{n=0}^{\infty}A_{n}(q^{\kappa-\iota})^{kn}q^{kn^{2}}=\chi_{\iota,\kappak+\kappa-1}^{(2\kappa+1,(2\kappa+1)k+2\kappa-1)}(q),
\langlez|e^{-sO_{\Sigma}}|z^{\prime}\rangle\simeq{\frac{\triangle_{\Sigma}^{1/2}(z,z^{\prime})}{(4\pis)}}~e^{-{\frac{\sigma_{\Sigma}^{2}(z,z^{\prime})}{4s}}-m^{2}s}\left(1+a_{\Sigma,1}(z,z^{\prime})s+a_{\Sigma,2}(z,z^{\prime})s^{2}+...\right)~~~,
A_{+}=\frac{1}{2}\left(\begin{array}{cc}{0}&{\sigma_{3}+1}\\{\sigma_{3}-1}&{0}\\\end{array}\right)\;,\;\;A_{-}=\frac{1}{2}\left(\begin{array}{cc}{0}&{\sigma_{3}-1}\\{\sigma_{3}+1}&{0}\\\end{array}\right).
\left(\begin{matrix}{\pi_{ADM}^{0,M}}\\{\pi_{ADM}^{D-1,M}}\\\end{matrix}\right)={\frac{(D-4)\Omega_{D-3}}{2\kappa_{D}^{2}}}\left(\begin{matrix}{Q+Q\dot{F}^{2}+p}&{Q\dot{F}^{i}}&{-Q\dot{F}^{2}-p}\\{-Q\dot{F}^{2}-p}&{-Q\dot{F}^{i}}&{-Q+Q\dot{F}^{2}+p}\\\end{matrix}\right).
\partial_{\mu}T^{\lambda\mu}={\frac{1}{4\pi}}\,\mathrm{tr}\,({\calE}_{\mu}\,F^{\lambda\mu})+\sum_{I=1}^{K}\int\!d\tau_{I}\,\varepsilon_{I}^{\lambda}\,\delta^{4}\bigl[x-z_{I}(\tau_{I})\bigr],
\mathrm{det}U_{x,\mu}=N!\c_{x,+\mu}^{1}c_{x+\hat{\mu},-\mu}^{1\dagger}c_{x,+\mu}^{2}c_{x+\hat{\mu},-\mu}^{2\dagger}...c_{x,+\mu}^{N}c_{x+\hat{\mu},-\mu}^{N\dagger}.
B_{3/2}^{II}=\frac{5\,\alpha\,\sqrt\pi}{64\,R},\quadB_{2}^{II}=\frac{4\,\alpha}{45\,R^{2}},\quadB_{5/2}^{II}=\frac{269\,\alpha\,\sqrt\pi}{2^{13}R^{3}}\,{.}
\widetilde{D}_{0}\pi_{a}+i\varsigma{\frac{\partialE_{a}}{\partial\phi_{i}}}\tau_{i}-i\overline{{p}}^{\prime}{\frac{\partial\overline{{J}}^{a}}{\partial\overline{{\phi}}_{i}}}\overline{{\tau}}_{i}+iE_{a}\beta-i\overline{{J}}^{a}\overline{{\gamma}}=0
{\calA}_{\mathrm{IR}}={\calA}(\infty)-{\calA}(\Lambda)\simeq\int_{0}^{\infty}\frac{dS}{S}S^{n/2-12}(e^{-p\circpS/2\pi\alpha^{\prime}}-e^{-(p\circp+1/\Lambda^{2})S/2\pi\alpha^{\prime}})~,
\left({\frac{x^{-}}{x^{+}}}\right)^{2}F(x^{-},x^{+})={\frac{N}{k}}\sum_{n}{\frac{M_{n}^{4}}{32\pi^{2}}}{\frac{(k-2n)^{2}}{k^{2}}}K_{4}(M_{n}r)
\tilde{A}_{\mu\nu}(k)~\sim~\frac{\tilde{B}_{A}}{(k^{2})^{3/2-\beta_{A}}}\left[\eta_{\mu\nu}~-~(1-\rho)\frac{k_{\mu}k_{\nu}}{k^{2}}~+~\vartheta\frac{\epsilon_{\mu\nu\sigma}k^{\sigma}}{(k^{2})^{1/2}}\right]
\{f,g\}_{1}^{\mathrm{can}}=\sum_{i=1}^{N}\left(\frac{\partial_{r}f}{\partialx^{i}}\frac{\partial_{l}g}{\partial\theta_{i}}+\frac{\partial_{r}f}{\partial\theta_{i}}\frac{\partial_{l}g}{\partialx^{i}}\right).
\zeta_{C}^{(2)}(s)=\sum_{j=-\infty}^{+\infty}\sum_{n}\left[\left(\omega_{j}^{2}+\sigma_{n}+m^{2}-e^{2}\mu^{2}\right)+\frac{4e^{2}\mu^{2}\omega_{j}^{2}}{\left(\omega_{j}^{2}+\sigma_{n}+m^{2}-e^{2}\mu^{2}\right)}\right]^{-s}\;.
I_{-1k}\left(\mu,\lambda\right)=\frac{\Gamma\left(k+1/2\right)}{2^{2k+2}\,k^{k+3/2}}\left[\frac{\left(k+1/2\right)}{\mu^{k+1/2}}+\frac{2k}{\mu^{k-1/2}}\right].e^{-2\muk}
q^{H}a_{1}^{\varepsilon}=a_{1}^{\varepsilon}q^{H+1}\,,\,\,q^{H}a_{2}^{\varepsilon}=a_{2}^{\varepsilon}q^{H-1}\,,\,\,q^{\bar{H}}{\bar{a}}_{\varepsilon}^{1}={\bar{a}}_{\varepsilon}^{1}q^{{\bar{H}}-1}\,,\,\,q^{\bar{H}}{\bar{a}}_{\varepsilon}^{2}={\bar{a}}_{\varepsilon}^{2}q^{{\bar{H}}+1}\,,
\frac{T}{M_{pl}}\sim\left(\frac{\rho_{\beta}}{M_{5}^{4}}\right)^{1/4}=\frac{3^{3/4}\sqrt{\beta}}{(2B)^{1/4}(I_{3}M_{5})^{3/4}}\left[\int_{Y=0}^{Y=R}D(-V)^{1/2}dY\right]^{1/2}.
\begin{array}{lcl}{D_{0}\equivC_{0},}&{\quad}&{D_{1}\equivC_{1},}\\{D_{2}\equivC_{2}+B_{2}\wedgeC_{0},}&{}&{D_{3}\equivC_{3}+B_{2}\wedgeC_{1},}\\{D_{4}\equivC_{4}+{\frac{1}{2}}B_{2}\wedgeC_{2}+{\frac{1}{2}}B_{2}\wedgeB_{2}\wedgeC_{0}.}&{}&{}\\\end{array}
\partial_{\tau}W_{\tau}+\frac{1}{2M}(\partial_{\mu}W_{\tau}+\frac{1}{c}A_{\mu})(\partial^{\mu}W_{\tau}+\frac{1}{c}A^{\mu})-\frac{i\hbar}{2M}(\BoxW_{\tau}+\frac{1}{c}\partial_{\mu}A^{\mu})=0
R_{l}(\beta)={\frac{\Gamma({\frac{2l}{\xi}}+{\frac{i\beta}{\pi\xi}})\Gamma(1+{\frac{2l}{\xi}}+{\frac{i\beta}{\pi\xi}})}{\Gamma({\frac{2l+1}{\xi}}+{\frac{i\beta}{\pi\xi}})\Gamma(1+{\frac{2l-1}{\xi}}+{\frac{i\beta}{\pi\xi}})}}\.
\begin{array}{c}{<O_{k}O_{l}P>_{c}=\frac{1}{\rho^{2}}<O_{l}PP><PO_{k}>-\frac{1}{\rho^{3}}<O_{k}P><PPP><PO_{l}>}\\{-\frac{1}{\rho}<O_{l}O_{k}P>+\frac{1}{\rho^{2}}<O_{l}P><PPO_{k}>.}\\\end{array}
\{A,B\}=-\left({\delta^{\mu}}_{\nu}-\frac{k^{\mu}u_{\nu}}{uk}\right)\left(\frac{\partialA}{\partialx^{\mu}}\frac{\partialB}{\partialk_{\nu}}-\frac{\partialB}{\partialx^{\mu}}\frac{\partialA}{\partialk_{\nu}}\right)
\Sigma_{A}(\bar{p})\simeq\frac{ie^{2}}{(2\pi)^{3}}|eH|\intdq_{0}dq_{3}\int_{0}^{\infty}dr^{2}\mathrm{e}^{-r^{2}}\frac{G_{1}}{q^{2}}\frac{\Sigma_{A}(\bar{p"})}{\bar{p"}^{2}+\Sigma_{A}(\bar{p"})}
\begin{array}{rcl}{\vartheta(x_{1},\cdots,x_{n}|z_{1},\cdots,z_{j}\tau^{4},\cdots,z_{N})}&{=}&{\vartheta(x_{1},\cdots,x_{n}|z_{1},\cdots,z_{N})\displaystyle\prod_{\mu=1}^{n}\frac{-z_{j}\tau}{x_{\mu}},}\\{\vartheta(x_{1},\cdots,x_{\mu}\tau^{4},\cdots,x_{n}|z_{1},\cdots,z_{N})}&{=}&{\vartheta(x_{1},\cdots,x_{n}|z_{1},\cdots,z_{N})\displaystyle\prod_{j=1}^{N}\frac{-x_{\mu}\tau}{z_{j}}.}\\\end{array}
S_{D2}=\frac{1}{D-1}\sum_{i\neqj}^{D}\frac{1}{N}\mathrm{Tr}\,M_{i}^{2}M_{j}^{2}\\\\\\\\mathrm{and}\\\\\\\S_{D4}=\frac{1}{D-1}\sum_{i\neqj}^{D}\frac{1}{N}\mathrm{Tr}\,M_{i}^{4}M_{j}^{4}
\Psi_{2s+1}(\xi_{1},\xi_{2})=\sum_{k_{1}<k_{2}}\left\{A_{1}(k_{1},k_{2})\left|\Omega_{1}(k_{1},k_{2})\right\rangle+A_{2}(k_{1},k_{2})\left|\Omega_{2}(k_{1},k_{2})\right\rangle\right\}
F_{\mu\nu}^{4}=\frac{i}{2}\{a_{\mu},a_{\nu}\}-\frac{i}{2}\varepsilon_{\mu\nu\alpha\beta}\{a_{\alpha},a_{\beta}\}+\partial_{\mu}b_{\nu}-\partial_{\nu}b_{\mu}+\frac{i}{2}\{b_{\mu},b_{\nu}\}
\left(-\frac{\delta^{2}\phi^{i}}{\delta\tau^{2}}-R_{\,jkl}^{i}\dot{q}_{b}^{j}\phi^{k}\dot{q}_{b}^{l}+(\nabla^{i}\nabla_{j}V)\phi^{j}\right)\Pi_{i}^{k}(b)=0,~\vec{\phi}(-T/2)=\vec{\phi}(0)={\bf0}.
E_{\mathrm{vac}}=-\frac{1}{2}\sum_{j}(2G_{j}+1)|\epsilon_{j}|-\int_{0}^{\infty}\frac{dk}{2\pi}\,\sqrt{k^{2}+m_{f}^{2}}\,\frac{d}{dk}\,\overline{{\delta}}(k)+E^{(1,2)}+E^{(3,4)}\,,
2(\dot{x}^{AA^{\prime}}\acute{x}^{BB^{\prime}}-\acute{x}^{AA^{\prime}}\dot{x}^{BB^{\prime}})=v\bar{\pi}^{A}\bar{\pi}^{B}\epsilon^{A^{\prime}B^{\prime}}+\bar{v}\pi^{A^{\prime}}\pi^{B^{\prime}}\epsilon^{AB}.
\Pi_{Ti}^{a}({\bf{r}})=\sum_{{\bf{k}};s=1,2}-i\epsilon_{i}^{s}({\bf{k}})\sqrt{k/2}\,[a_{s}^{a}({\bf{k}})e^{+i{\bf{k}}\cdot{\bf{r}}}-a_{s}^{a\dagger}({\bf{k}})e^{-i{\bf{k}}\cdot{\bf{r}}}]\;,
\delta^{lm}\partial_{l}\partial_{m}H\,+\,\frac{1}{4}\sum_{i}\left|\partial_{z_{i}}\gamma_{i}\right|^{2}\delta_{i}^{4}(x)\,+\,4\pi^{3}Q\,f_{0}(N_{I})\delta(x^{4})\dots\delta(x^{9})=0
\deltaS_{\mathrm{matter}}=\intd^{5}x\sqrt{-g}~\delta(g^{B}\,_{A})T^{(2)A}\,_{B}=\intd^{4}x\intdy~b_{0}\sqrt{-g_{4}}~\delta(\delta^{B}\,_{A})T^{(2)A}\,_{B}~,
\left(P^{2}-m^{2}-\frac{e}{2}\sigma^{\mu\nu}F_{\mu\nu}\right)\phi(x)=0\;,\;\;\;F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\;,\;\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]\;.
{\frac{\partial}{\partialt_{2}}}J=\left[D,{\overline{D}}\right]J^{\prime}+2JJ^{\prime}-2(\Phi{\overline{\Phi}})^{\prime}\;,{\frac{\partial}{\partialt_{2}}}\Phi=\Phi^{\prime\prime}-2D{\overline{D}}(J\Phi)\;.
\eta_{1}(z)=-4\sum_{n=1}^{\infty}\left[{\frac{1}{nz}}\int_{0}^{1}{\mathrmd}y\,{\frac{y^{2}}{\sqrt{1-y^{2}}}}\,K_{1}(\frac{nzy}{2})-2{\frac{1}{n^{2}z^{2}}}\right].
K(T)_{SPA}=e^{\frac{i}{\hbar}S(\phi_{cl})}\int{\cal{D}}[\eta_{a}(x)]\int_{\eta_{a},0}^{\eta_{a},T}{\cal{D}}[\eta(x,t)]\;e^{-\frac{i}{\hbar}\frac{1}{2}\int_{L\textrm{x}T}dtdx\;\eta(\square+U^{\prime\prime})\eta+\dots}.
\delta_{a}\eta_{c\alpha_{1}}^{*(b)}=\left.\left(\eta_{c\alpha_{1}}^{*(b)},S\right)_{a}\right|_{\mathrm{ghosts}=0}=\left(-\right)^{\epsilon_{\alpha_{1}}}\delta_{ab}\Phi_{\alpha_{0}}^{*(c)}Z_{\;\;\alpha_{1}}^{\alpha_{0}},
D_{i\alpha}\lambda^{j\beta}(z)=\delta_{i}^{~j}(\textstyle{\frac{1}{2}}\hat{\lambda}(z)\delta_{\alpha}^{~\beta}-\tilde{\hat{\omega}}_{\alpha}{}^{\beta}(z))-\hat{T}_{i}^{~j}(z)\delta_{\alpha}^{~\beta}
\frac{S\left(\beta\right)}{2\pir\widetilde{E}\left(\beta\right)}\approx\frac{\frac{\delta^{-3}}{180}+\frac{\left|\mu_{eff}^{2}\right|}{24}\delta^{-1}}{\left[\frac{1}{240}\left(\delta^{-4}-1\right)+\frac{\left|\mu_{eff}^{2}\right|}{48}\left(\delta^{-2}-9\right)\right]},\,\,\,\,\delta\ll1.
b_{-\omega+1}=\frac{1}{2^{2\omega}\pi^{\omega}}(2\omega-[1-\alpha^{\omega-1}])\{{\frac{\Gamma(\omega)-\frac{1}{\omega}\Gamma(\omega+1)}{\Gamma[\omega]}}\}N_{c}A_{\mu}^{a}(x)A_{\mu}^{a}(x)=0
\Gamma^{\alpha\beta}=\frac{\pi^{2}T^{4}\kappa}{180}\left(4\,\delta_{0}^{\alpha}\delta_{0}^{\beta}-\eta^{\alpha\beta}\right)+\frac{m^{2}T^{2}\kappa}{48}\left(\eta^{\alpha\beta}-2\,\delta_{0}^{\alpha}\delta_{0}^{\beta}\right)+\cdots.
(\kappa_{4})^{4}U_{OM}>\frac{3|h|^{2}}{4dJ^{2}}\Bigg\{\beta\Big[e^{-J\betax}-e^{-J\beta(1-x)}\Big]^{2}+\frac{2}{3}\frac{J}{V}\beta^{2}\Big[e^{-J\betax}x-e^{-J\beta(1-x)}(1-x)\Big]^{2}\Bigg\}\;.
I_{4}=\frac{\Gamma(2-\frac{D}{2})}{(4\pi)^{\frac{D}{2}}}\int_{0}^{1}dx(1-x)\sum_{n}\left[(3-D)\frac{1}{[k_{0}^{2}+m^{2}]^{2-\frac{D}{2}}}+(D-4)\frac{m^{2}}{[k_{0}^{2}+m^{2}]^{3-\frac{D}{2}}}\right].
L~=~\frac{1}{2}(\partial_{\mu}\phi^{ai})^{2}~+~\frac{1}{2}\sigma\phi^{ai}\phi^{ai}~+~\frac{1}{2}T^{ab}\phi^{ai}\phi^{bi}~-~\frac{3\sigma^{2}}{2w}~-~\frac{3}{2v}T^{ab}T^{ab}
(\partial_{\tau}^{2}-\partial_{\sigma}^{2})\hat{X}^{i}(\tau,\sigma)=\left(\partial_{\sigma}\hat{X}^{i}-{\calP}_{i}^{0}\right)\delta(\sigma)-\left(\partial_{\sigma}\hat{X}^{i}-{\calP}_{i}^{\pi}\right)\delta(\sigma-\pi).
\begin{array}{cl}{}&{\displaystyle\sum_{s=\pm1}t_{k^{\prime}+s}^{k^{\prime}}(u_{2n}-u_{0}+2)F_{\sigma}^{(n)}(\zeta_{1},\cdots,\zeta_{2n-1},x^{2}\zeta_{2n})^{kk_{1}\cdotsk_{2n-2}k^{\prime}k^{\prime}+s}}\\{=}&{\displaystyle\sum_{s=\pm1}t_{k}^{k-s}(u_{2n}-u_{0})F_{\sigma}^{(n)}(\zeta_{2n},\zeta_{1},\cdots,\zeta_{2n-1})^{k-skk_{1}\cdotsk_{2n-2}k^{\prime}}.}\\\end{array}
T_{A}^{B}=-\frac{1}{2}\frac{Q^{2}}{\gamma^{2}}\delta_{A}^{B},\,\,\,\,T_{\rho}^{\rho}=-\frac{1}{2}\frac{Q^{2}}{\gamma^{2}},\,\,\,\,T_{\vartheta}^{\vartheta}=T_{\varphi}^{\varphi}=\frac{1}{2}\frac{Q^{2}}{\gamma^{2}},
\hat{\Theta}^{\hat{a}\hat{b}\hat{c}}=-\,\ell_{p}^{3}\,z^{-2}\,\hat{g}^{\hat{a}\hat{k}}\hat{\calH}_{\hat{k}\hat{l}\hat{m}}\,(\hat{G}_{OM})^{\hat{b}\hat{l}}\,(\hat{G}_{OM})^{\hat{c}\hat{m}}=-\,\ell_{p}^{3}\,\left((1+{\frac{1}{6}}\hat{\calH}^{2})\hat{\calH}^{\hat{a}\hat{b}\hat{c}}-{\frac{1}{2}}(\hat{\calH}^{3})^{\hat{a}\hat{b}\hat{c}}\right)\,.
x_{0}^{\mu}={\frac{i}{2}}\sqrt{2\alpha^{\prime}}(a_{0}^{\mu}-a_{0}^{\dagger\mu}),\p_{0\mu}={\frac{1}{\sqrt{2\alpha^{\prime}}}}\eta_{\mu\nu}(a_{0}^{\nu}+a_{0}^{\dagger\nu}),\\[x_{0}^{\mu},p_{0\nu}]=i\delta_{\nu}^{\mu},\[a_{0}^{\mu},a_{0}^{\dagger\nu}]=\eta^{\mu\nu}.
\Sigma\;=\intd^{4}x\;\left(-\frac1{4g^{2}}F_{\mu\nu}^{a}F^{a\mu\nu}\;+\;A^{*a\mu}(D_{\mu}c)^{a}-\frac12\;f_{abc}C^{*a}c^{b}c^{c}\;\right)\;\;,
\frac{\deltaW}{\deltaA_{i}^{a}}=0\quad\Longrightarrow\quad\left[\delta_{ij}D_{k}^{2}\,(A)-D_{j}\,(A)~D_{i}\,(A)\right]^{ac}A_{j}^{c}=D^{ac}\,_{0}\,(A)\left[\partial_{0}A_{i}^{c}-{D(A)}_{i}^{cb}A_{0}^{b}\right]
e^{2\Phi_{D}}=\frac{W\bar{W}}{\Lambda^{2}}G^{1/2}\bar{G}^{1/2}\left[\frac{1+G^{1/2}\bar{G}^{1/2}}{G^{1/2}+\bar{G}^{1/2}}-{\frac{1}{g_{YM}^{2}(W)}}+{\frac{2K}{g_{YM}^{2}(W)}}\left(\frac{2}{W\bar{W}(G^{1/2}+\bar{G}^{1/2})}\right)\right],
({\calE}_{\tau}^{\underline{{D-1}}})^{2}+\sum_{i=1}^{i=D-2}({\calE}_{\tau}^{i})^{2}=({\calE}_{\tau}^{\underline{0}})^{2}\equiv(\partial_{\tau}X^{\underline{0}}-i\partial_{\tau}\bar{\Theta}\Gamma^{\underline{0}}\Theta)^{2},
F_{(e)}^{\mu\nu}=\left(\begin{array}{cccc}{0}&{-E_{x}}&{-E_{y}}&{-E_{z}}\\{E_{x}}&{0}&{-B_{z}}&{B_{y}}\\{E_{y}}&{B_{z}}&{0}&{-B_{x}}\\{E_{z}}&{-B_{y}}&{B_{x}}&{0}\\\end{array}\right).
\stackrel{.}{a_{1}}=\Gamma^{\rho}\sum_{i,j}T_{i}X_{0,\rho}T_{j}\,e^{\left(m_{i}^{2}-m_{j}^{2}\right)t}-\sum_{i,j}T_{i}X_{1}T_{j}\,e^{\left(m_{i}^{2}-m_{j}^{2}\right)t}
S(x,\alpha)=\frac{(1+\alpha)^{2}m^{2}[(1+\alpha)(1+4x-3\alpha)m^{2}+\{\alpha(2-\alpha)+(2\alpha-1)x\}(\alpha-x)M^{2}]}{\alphaM^{2}[(\alpha-x)(1+x)M^{2}-(1+\alpha)^{2}m^{2}][x(\alpha-x)M^{2}+(1+\alpha)m^{2}]},
\frac{(3\gamma-2)\kappa_{4)}^{2}\rho_{0}}{6}a^{-3\gamma}+\frac{(3\gamma-1)\kappa_{(4)}^{2}\rho_{0}^{2}}a^{-6\gamma}+\frac{2}{3}\sigma_{0}a^{-6}-\frac{\Lambda_{(4)}}{3}a+\frac{2{\calU}_{0}}{\lambda\kappa_{(4)}^{2}}a^{-4}<0.
\lambda(p^{\mu})=\left(\begin{matrix}{(\zeta_{\lambda}\Theta_{[j]})\phi_{_L}^{\ast}(p^{\mu})}\\{\phi_{_L}(p^{\mu})}\\\end{matrix}\right)\quad,\quad\rho(p^{\mu})=\left(\begin{matrix}{\phi_{_R}(p^{\mu})}\\{(\zeta_{\rho}\Theta_{[j]})^{\ast}\phi_{_R}^{\ast}(p^{\mu})}\\\end{matrix}\right)\,\,,
d\left(\frac{k^{2}}{\lambda^{2}},e_{2}^{2}\right)=\frac{d_{C}(k^{2}/m^{2},e_{1}^{2})}{d_{C}(\lambda^{2}/m^{2},e_{1}^{2})}~,~~e_{2}^{2}=e_{1}^{2}d_{C}(\lambda^{2}/m^{2},e_{1}^{2})~,
S=k<x_{1},x_{2},x_{3}>\left/\left(\begin{array}{r}{x_{2}x_{1}-x_{1}x_{2}+p_{12}x_{2}x_{2}=0}\\{x_{3}x_{2}-x_{2}x_{3}=0}\\{x_{3}x_{1}-x_{1}x_{3}+p_{13}x_{2}x_{2}=0}\\\end{array}\right)\right.
L(H_{\nu\sigma\tau})=\frac{1}{2}\left(\epsilon^{\mu\nu\sigma\tau}\partial_{\mu}H_{\nu\sigma\tau}\right)^{2}+3m^{2}H_{\nu\sigma\tau}H^{\nu\sigma\tau}-J\epsilon^{\mu\nu\sigma\tau}\partial_{\mu}H_{\nu\sigma\tau}+\frac{1}{2}J^{2}\,,
ds_{p}^{2}=\frac{i}{2}\,\partial\bar{\partial}\,\mathrm{~log~}\vert{\calR}_{p}\vert^{2}=\frac{i}{2}\,\prod_{q}e^{2k_{pq}{\calQ}_{q}}\,dZ\wedged\bar{Z};\quadR_{p}\,=\,\sum_{q}k_{pq}ds_{q}^{2},
S=\frac{1}{2}\intd^{d+1}x\,e\,(\bar{\psi}_{\mu}\Gamma^{\mu\nu\rho}D_{\nu}\psi_{\rho}+m_{1}\bar{\psi}_{\mu}\psi^{\mu}+m_{2}\bar{\psi}_{\mu}\Gamma^{\mu\nu}\psi_{\nu})
D_{\nu}^{ij}[A(x)]F^{\nu\mu,j}[A(x)]=-g^{2}C_{A}\frac1{(4\pi)^{2}}\frac{11}{3}\intd^{4}yH(x-y)(g^{\mu\nu}\partial^{2}-\partial^{\mu}\partial^{\nu})A_{\nu}^{a}(y)
\delta~\psi_{1}=\xi^{z}\partial_{z}\psi_{1}-(\partial_{z}\xi^{z})\psi_{1}+\xi^{zz}(\partial_{z}^{2}\psi_{1})-{\frac{1}{2}}(\partial_{z}\xi^{zz})(\partial_{z}\psi_{1})+(({\frac{1}{6}}\partial_{z}^{2}-{\frac{2}{3}}\tilde{u}_{2})\xi^{zz})\psi_{1}\quad,
\beta={\frac{2\pir_{+}({r_{+}}^{2}+a^{2})({r_{+}}^{2}+b^{2})l^{2}}{2r_{+}^{6}+r_{+}^{4}(l^{2}+b^{2}+a^{2})-a^{2}b^{2}l^{2}}}\.
\delta_{tot}|\phi\rangle=\Bigl(\xi\partial+\frac{d-2}{2z}\xi^{z}+\frac{1}{2}\partial^{I}\xi^{J}M^{IJ}+M^{IJ}\partial^{+}\xi^{I}\frac{\partial^{J}}{\partial^{+}}-\frac{d-4}{2z\partial^{+}}\partial^{+}\xi^{I}R^{zI}-\frac{\partial^{+}\xi^{z}}{z\partial^{+}}\Bigr)|\phi\rangle
\frac1v=gh^{-n}\intd^{n}p\left[\frac{\lambdae^{-\betaH_{0}({\bfp})}}{1-\lambdae^{-\betaH_{0}({\bfp})}}-(q+1)\frac{\lambda^{q+1}e^{-\beta(q+1)H_{0}({\bfp})}}{1-\lambda^{q+1}e^{-\beta(q+1)H_{0}({\bfp})}}\right]
\left(\begin{matrix}{z_{1}}\\{z_{2}}\\\end{matrix}\right)\rightarrowU_{L}\,\left(\begin{matrix}{z_{1}}\\{z_{2}}\\\end{matrix}\right)~~~~~;~~~~~\left(\begin{matrix}{z_{1}}\\{\bar{z}_{2}}\\\end{matrix}\right)\rightarrowU_{R}\,\left(\begin{matrix}{z_{1}}\\{\bar{z}_{2}}\\\end{matrix}\right)
\hat{U}^{\dagger}(t_{2},t_{1})=\hat{U}^{-1}(t_{2},t_{1})=\hat{U}(t_{1},t_{2})\\,\\\hat{U}(t_{3},t_{2})\,\hat{U}(t_{2},t_{1})=\hat{U}(t_{3},t_{1})\,
w_{ij}(\eta,\bar{\eta})=\frac{e^{-\frac{\alpha_{ij}(D)}{k}(\eta-\bar{\eta})}}{2\pii}\frac{\theta_{11}(\eta-\frac{\alpha_{ij}(D)}{\pik}\mathrm{Im}\tau)\theta_{11}^{\prime}(0)}{\theta_{11}(\eta)\theta_{11}(\frac{\alpha_{ij}(D)}{\pik}\mathrm{Im}\tau)},
\psi_{1}\left(r\right)=\bar{n}^{3/4}\psi_{\lambda-\frac{1}{2},1-\mu}\left(\rho\right)\,,\;\psi_{2}\left(r\right)=\bar{n}^{1/4}\psi_{\lambda,\mu}\left(\rho\right)\,,\;2\bar{n}=2\lambda+\mu-1\,.
S={\frac{1}{16\piG_{10}}}\intd^{10}x\sqrt{-G}[e^{-2\phi}({\calR}+4\partial_{M}\phi\partial^{M}\phi-{\frac{1}{12}}|H_{3}|^{3})-{\frac{1}{2(p+2)!}}|F_{p+2}|^{2}],
{\calV}_{1,1}^{(i)}\geq0,\;\;{\calV}_{2,2}^{(i)}\geq0,\;\;{\calV}_{1,1}^{(i)}\geq|{\calV}_{1,2}^{(i)}|,\;\;{\calV}_{2,2}^{(i)}\geq|{\calV}_{2,1}^{(i)}|.
\int\frac{d\xi}{\xi}\phi_{\omega}^{(1)}(\xi)\phi_{\omega^{\prime}}^{(1)}(\xi)=\frac{\xi}{\omega^{2}-\omega^{2}}\left[\phi_{\omega}^{(1)}(\xi)\frac{d\phi_{\omega^{\prime}}^{(2)}(\xi)}{d\xi}-\phi_{\omega^{\prime}}^{(2)}(\xi)\frac{d\phi_{\omega}^{(1)}(\xi)}{d\xi}\right].
\Sigma_{\mu\nu}\Phi_{\mu\nu}=\frac{1}{i}\left(b_{i}b_{j}\Phi_{\overline{c}_{i}\overline{c}_{j}}+b_{i}^{\dagger}b_{j}^{\dagger}\Phi_{c_{i}c_{j}}+2b_{i}^{\dagger}b_{j}\Phi_{c_{i}\overline{c}_{j}}-\Phi_{c_{n}\overline{c}_{n}}\right).
\left({\frac{dx}{d\lambda}}\right)^{2}+{\frac{g_{zz}}{g_{xx}}}\left({\frac{dz}{d\lambda}}\right)^{2}+{\frac{J_{m}^{2}}{g_{xx}g_{\phi_{m}\phi_{m}}}}+{\frac{E^{2}}{g_{xx}g_{tt}}}+{\frac{\epsilon}{g_{xx}}}=0,
\tilde{L}=\sum_{i}\tilde{\alpha}_{i}^{\prime2}-\tilde{\alpha}_{i}^{\prime}\sum_{j}\tilde{\alpha}_{j}^{\prime}+{\frac{1}{D-2}}\varphi^{\prime2}+\sum_{a}\alpha_{a}^{\prime2}-{\frac{P^{2}}{2}}e^{\tilde{\Phi}}
I\left(a\right)=\frac{\int_{{\calF}_{a}}\left[{\calD}\Psi{\calD}\overline{{\Psi}}\right]e^{-{\calA}_{MTM}}}{\int_{{\calF}_{0}}\left[{\calD}\Psi{\calD}\overline{{\Psi}}\right]e^{-{\calA}_{MTM}}},
\begin{array}{rcl}{\eta_{ji}}&{=}&{-<S\chi_{j},b_{i}>}\\{}&{=}&{-<S\chi_{j},b_{k}>ST^{k}{}_{l}T^{l}{}_{i}}\\{}&{=}&{-<\chi_{k},Sb_{l}>T^{k}{}_{j}T^{l}{}_{i}}\\{}&{=}&{\eta_{kl}T^{k}{}_{j}T^{l}{}_{i},}\\\end{array}
\delta_{L}\psi(x)=-{\frac{i}{2}}\epsilon_{ab}\sigma^{ab}\psi,\quad\delta_{L}{e^{a}}_{\mu}(x)=\epsilon{^a}_{b}e{^b}_{\mu}+{\frac{\kappa^{4}}{4}}\varepsilon^{abcd}\bar{\psi}{^j}\gamma_{5}\gamma_{d}\psi{^j}(\partial_{\mu}\epsilon_{bc}).
{\bf{\tilde{V}}}=V^{a}(z)\frac{\partial}{\partialz^{a}}-V_{,b}^{a}\pi_{a}\frac{\partial}{\partial\pi_{a}}+{\bar{V}}^{a}(\bar{z})\frac{\partial}{\partial\bar{z}^{a}}-{\bar{V}}_{,\bar{b}}^{a}\bar{\pi}_{a}\frac{\partial}{\partial\bar{\pi}_{a}}\quad.
\begin{array}{lcl}{F_{1}\,=\,dD_{0},}&{\quad}&{F_{2}\,=\,dD_{1},}\\{F_{3}\,=\,dD_{2}-B_{2}\wedgedD_{0},}&{}&{F_{4}\,=\,dD_{3}-B_{2}\wedgedD_{1},}\\{F_{5}\,=\,dD_{4}-B_{2}\wedgedD_{2}+{\frac{1}{2}}B_{2}\wedgeB_{2}\wedgedD_{0}.}&{}&{~}\\\end{array}
a^{2}=r^{2}(1-\frac{2}{r})-4r^{2}e^{-r}+...\quadb^{2}=r^{2}(1-\frac{2}{r})+4r^{2}e^{-r}+...\quadc^{2}=4(1-\frac{2}{r})^{-1}+...\.
{\underbrace{h_{\beta\alpha}~h_{\delta\gamma}}}=\left[\frac{1}{2}\left(\eta_{\beta\delta}\eta_{\alpha\gamma}+\eta_{\alpha\delta}\eta_{\beta\gamma}\right)-\frac{1}{8}\,\eta_{\alpha\beta}\eta_{\gamma\delta}\right]\,\frac{1}{k_{\bot}^{2}}
{\calK}_{{\bfC}^{n}}(z,\bar{z})=N\,\mathrm{exp}\left(\mathrm{i}\sum_{j=1}^{n}\bar{z}^{j}z^{j}\right)\mathrm{d}z^{1}\wedge\ldots\wedge\mathrm{d}z^{n}\wedge\mathrm{d}\bar{z}^{1}\wedge\ldots\wedge\mathrm{d}\bar{z}^{n},
\begin{array}{cl}{\Xi_{2}^{h}=}&{\{(\pme_{a}\pme_{b}\pme_{c}\pme_{d})/2:a,b,c,d\mathrm{~distinct},}\\{}&{e_{a}(e_{b}(e_{c}e_{d}))=\pme_{h}\mathrm{~if~exactly~one~of~}a,b,c,d=0\mathrm{~or~}h,}\\{}&{e_{a}(e_{b}(e_{c}e_{d}))=\pm1\mathrm{~otherwise~}\}.}\\\end{array}
\begin{array}{l}{\partial_{0}x^{i}=-(p^{i}+(pP_{y})P_{y}^{i}),\qquad\partial_{0}p^{i}=-\partial_{1}\partial_{1}x^{i};}\\{(\partial_{0}-\partial_{1})\Gamma^{-}\theta^{+}=0,\qquad(\partial_{0}+\partial_{1})\Gamma^{-}\theta^{-}=0.}\\\end{array}
=-\bigg[\frac{(D-2)\beta+4\gamma}{4(D-3)}\bigg]E+\bigg(\frac{D-2}{D-3}\bigg)(\beta/4+\gamma)C^{2}+\bigg[\frac{4(D-1)\alpha+D\beta+4\gamma}{4(D-1)}\bigg]R^{2}\,
\begin{array}{c}{\tilde{R}_{\mu\nu}-\frac{1}{2}\tilde{G}_{\mu\nu}\tilde{R}=T_{\mu\nu}^{matter}~,}\\{3\sqrt{2|D-2|}\alpha^{\prime}D^{2}\tilde{\phi}+4(26-D)e^{\frac{4}{\sqrt{2|D-2|}}\tilde{\phi}}+\alpha^{\prime}H_{\lambda\mu\nu}^{2}e^{-\frac{8}{\sqrt{2|D-2|}}\tilde{\phi}}=0~,}\\{D_{\lambda}[e^{-\frac{8}{\sqrt{2|D-2|}}\tilde{\phi}}H_{~\mu\nu}^{\lambda}]=0}\\\end{array}
\langle0_{D}|T_{i}^{k}|0_{D}\rangle=\delta_{i}^{k}\piA_{d}\int_{0}^{\infty}dkk^{d}\sum_{n=1}^{\infty}\frac{I_{i\omega}(k\xi_{1})}{I_{i\omega}(k\xi_{2})\frac{\partialD_{i\omega}(k\xi_{1},k\xi_{2})}{\partial\omega}}
ds^{2}=-dy^{2}+dy^{s}dy_{s}+\left[1-\left(\frac{r_{H}}{r}\right)^{7-p}\right]dw^{2}+\frac{dr^{2}}{1-\left(\frac{r_{H}}{r}\right)^{7-p}}+r^{2}d\Omega_{8-p}^{2}.
\begin{array}{c}{\displaystyle(p^{2}+m^{2})F(p,\bar{z},\bar{\theta})=0}\\{\displaystyle(4(p,\hat{J})+m\hat{{\calK}})F(p,\bar{z},\bar{\theta})=m(4s+1)F(p,\bar{z},\bar{\theta})\,.}\\\end{array}
f^{(2)}=\left(\begin{array}{cccc}{0}&{-\delta_{ij}}&{0}&{-\partial_{i}^{x}}\\{\delta_{ij}}&{0}&{-\partial_{i}^{x}}&{0}\\{0}&{-\partial_{j}^{x}}&{0}&{0}\\{-\partial_{j}^{x}}&{0}&{0}&{0}\\\end{array}\right)\delta(\vec{x}-\vec{y}).
\gamma(-\Box)=(\Box+\mu^{2})^{n}\int_{0}^{\infty}\frac{dm^{2}}{m^{2}-\Box}\frac{w(m^{2})}{(m^{2}+\mu^{2})^{n}}+\sum_{k=0}^{n-1}\frac{(-1)^{k}}{k!}(\Box+\mu^{2})^{k}{\left(\frac{\partial}{\partial\mu^{2}}\right)}^{k}\gamma(\mu^{2})
[N_{i},{\calP}_{j}]=i\,\delta_{ij}\left({\frac{\kappa}{2}}\left(e^{2{{\calP}_{0}/\kappa}}-1\right)-{\frac{1}{2\kappa}}\vec{{\calP}}\,{}^{2}\,\right)+i\,{\frac{1}{\kappa}}{\calP}_{i}{\calP}_{j},\quad[N_{i},{\calP}_{0}]=i{\calP}_{i}.
(-1)^{n/2}(n+1)\epsilon_{\mu\mu_{1}\dots\mu_{2n}}X^{\mu_{1}}X^{\mu_{2}}\dotsX^{\mu_{2n}}+(-1)^{(n-1)/2}\frac{n+1}{2}\epsilon_{\mu\mu_{1}\mu_{2}\dots\mu_{2n}}\theta^{\mu_{1}\mu_{2}}X^{\mu_{3}}\dotsX^{\mu_{2n}}=0\.
\psi^{z}(T,\sigma)=\kappa(T+\tilde{\Lambda})^{{\frac{1-\alpha}{1+\alpha}}-1}\sum_{n=-\infty}^{\infty}\left[\tilde{\alpha}\vphantom{\alpha}_{n}^{0}e^{in(\sigma-T)}+\tilde{\beta}\vphantom{\beta}_{n}^{0}e^{in(\sigma+T)}\right]+\kappa\tilde{b}\vphantom{b}_{0}^{0}(T+\tilde{\Lambda})^{\frac{1-\alpha}{1+\alpha}}
{\mathcal{L}}(\xi,\xi^{\prime})=2^{(D-1)/2}\pim^{D-1}\Gamma\left(\frac{D-1}{2}\right)\int_{0}^{\infty}dy\,\frac{\pi^{2}-3y^{2}}{(\pi^{2}+y^{2})^{3}}\frac{K_{(D-1)/2}(m\gamma)}{(m\gamma)^{(D-1)/2}},