harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\calL}=\frac{1}{2}(\partial_{\mu}{\vec{\pi}})^{2}+\frac{1}{2f_{\pi}^{2}}\frac{({\vec{\pi}}\times\partial_{\mu}{\vec{\pi}})^{2}}{(1-\frac{{\vec{\pi}}^{2}}{f_{\pi}^{2}})^{2}}
AH_{0}=c_{H}\,K_{H}\,\int{\frac{d^{2}a_{1}d^{2}a_{2}d^{2}a_{3}\,|a_{45}\,a_{46}\,a_{56}|^{2}}{T^{5}\prod_{i<j}^{6}|a_{ij}|^{2}}}\prod_{l=1}^{4}{\frac{d^{2}z_{l}(r-z_{l})}{y(z_{l})}}I(r)\bar{F}(\bar{a},\bar{z}),
D_{\zeta}^{2s+\tau,2t+\tau^{\prime}}(x,x_{1},\ldots,x_{n})=(-i)^{2s+\tau+1}\sigma_{2s+\tau}^{+}\sum_{p=0}^{t}x^{2s-2p+\tau+1-\zeta}\hat{\sigma}_{2p+\zeta}^{-}\,\,.
x_{n}=\frac{{}_{1}}{{}^{2}}\left[e^{in\theta}A_{n}^{*}+e^{-in\theta}A_{n}\right],\;\;\;x_{n+1}=\frac{{}_{1}}{{}^{2}}\left[e^{i(n+1)\theta}A_{n}^{*}+e^{-i(n+1)\theta}A_{n}\right],
\omega_{\mu}^{~ab}=\frac{1}{2}e^{a\nu}(\partial_{\mu}e_{~\nu}^{b}-\partial_{\nu}e_{~\mu}^{b})+~\frac{1}{4}e^{a\nu}e^{b\lambda}(\partial_{\lambda}e_{~\nu}^{c}-\partial_{\nu}e_{~\lambda}^{c})e_{c\mu}-(a\leftrightarrowb).
V({\bfx},{\bfx}^{\prime})=\int\frac{dk_{1}}{(2\omega_{{\bfk}_{1}})^{1/2}}\frac{d{\bfk}_{2}}{(2\omega_{{\bfk}_{2}})^{1/2}}\frac{1}{2\omega_{({\bfk}_{1}+{\bfk}_{2})/2}}e^{-i{\bfk}_{1}{\bfx}+i{\bfk}_{2}{\bfx}^{\prime}}
B(t)=\frac{C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}}{R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}}\equiv\frac{1}{3}\frac{[xy-x^{2}-\dot{x}+\dot{y}]^{2}}{[x^{4}+2x^{2}y^{2}+3y^{4}+2x^{2}\dot{x}+\dot{x}^{2}+4y^{2}\dot{y}+2\dot{y}^{2}]}
Im(I_{2})={\frac{\pi^{2}}{16a}}\int_{1}^{\lambda-1}{dr{\frac{1}{r}}\left[{\frac{(B^{3}-3\delta^{2}B)}{(B^{2}+\delta^{2})^{3}}}-{\frac{1}{r}}{\frac{B}{(B^{2}+\delta^{2})}}\right]}.
\mathcal{K}^{\left(tr\right)}(p,q)=\underset{s_{1}s_{2}\sigma_{1}\sigma_{2}m_{J}}{-i\sum}C_{Jm_{J}}^{\left(tr\right)s_{1}s_{2}\sigma_{1}\sigma_{2}}\intd\hat{\mathbf{p}}\,d\hat{\mathbf{q}}\,\mathcal{M}
[\mathcal{H}]_{2}=\frac{U}{2(d-1)}\left\{-c_{2}R-\lambda_{IJ}\partial^{\mu}\phi^{I}\partial_{\mu}\phi^{J}+\nabla_{\mu}\left[\left(2(d-1)\partial_{I}\Phi+M_{IJ}\beta^{J}\right)\partial^{\mu}\phi^{I}\right]\right\},
L_{2k}=-\frac{2y_{1}y_{2}}{\zeta}\cdot[R_{k}+\frac{y_{1}^{2}+y_{2}^{2}}{\zetay_{2}}P_{k}-\frac{(y_{1}^{2}-y_{2}^{2})}{2y_{1}y_{2}}\cdot\frac{\partial_{2}P_{k}}{2k}-\frac{\partial_{1}P_{k}}{2k}+\frac{y_{1}^{2}+y_{2}^{2}}{y_{1}y_{2}^{2}}\cdot\frac{P_{k}}{4k}],
G_{11}={\frac{r+P_{2}}{r+P_{1}}},\\G_{22}={\frac{r+Q_{1}}{r+Q_{2}}},\\G_{12}=-B_{12}={\frac{q[r+{\frac{1}{2}}(P_{1}+P_{2})]}{(r+Q_{2})(r+P_{1})}}.
{}~~~~=\omega^{(h-f)(a-b-e+h)}\Bigg\{\sum_{\sigma=0}^{N-1}{\frac{w(x_{3},x_{13},x_{1}|-\sigma-a+c+f-h)}{w(x_{3},x_{23},\omegax_{2}|-\sigma)s(\sigma,-a+b+e-h)}}\Bigg\}_{0}.
H(t_{E})\equiv\frac{\frac{d{a}_{E}}{dt_{E}}}{{a}_{E}}=e^{\Phi-\frac{\sigma_{1}+5\sigma_{2}}{2}}\left(-{\dot{\Phi}}+\frac{{\dot{\sigma}}_{1}+5{\dot{\sigma}}_{2}}{2}+{\dot{b}}\right)
ds_{4}^{2}\stackrel{r\rightarrow\infty}{\longrightarrow}\left(1-{\frac{Q_{6}+\sum_{j}P_{j}}{2r}}\right)(-dt^{2})+\left(1+{\frac{Q_{6}+\sum_{j}P_{j}}{2r}}\right)(dr^{2}+r^{2}d\Omega^{2}).
\left[\frac{d^{2}}{dr^{2}}+\frac{1}{r}\fracd{dr}-\frac1{\alpha^{2}r^{2}}(n^{2}-2neA+e^{2}A^{2})-\beta^{2}\right]g_{n}(r,r^{\prime})=-\frac1{\alphar}\delta(r-r^{\prime})\,
\sum_{j=0}^{n}R_{(k-1)n-j\alpha_{k}}^{\alpha_{k-1}}R_{(k)j+1\alpha_{k+1}}^{\alpha_{k}}\mu_{a}^{\alpha_{k+1}}=-R_{(k-1)n+1\alpha_{k}}^{\alpha_{k-1}}R_{(k)0\alpha_{k+1}}^{\alpha_{k}}\mu_{a}^{\alpha_{k+1}}=0\.
|a_{n}^{l}(\vec{q},\vec{q})|=\left|\sum_{\{m_{i,1}\}}\sum_{\{m_{i,2}\}}\sum_{\{m_{i,3}\}}A_{\{m_{i,\nu}\}}^{nl}V^{(m_{11},m_{12},m_{13})}(\vec{q})\dotsV^{(m_{l1},m_{l2},m_{l3})}(\vec{q})\right|,
S_{B}=\intd^{2}x~\left[-{\frac{1}{4}}\Big(1+{\frac{ae^{2}}{\pi}}\Big)F^{2}+{\frac{e^{2}}{2\pi}}A^{2}+{\frac{1}{2}}\partial_{\mu}\Sigma\partial^{\mu}\Sigma-{\frac{e}{\sqrt{\pi}}}A^{\mu}\partial_{\mu}\Sigma\right].
\frac{\partial^{2}V_{w}^{(1)}}{\partial\mu^{2}}_{\mid\mu=0}=\frac{eH}{2\pi^{2}}\beta^{2}\frac{\partial}{\partial\beta^{2}}\sum_{l=1}^{\infty}\int_{0}^{\infty}\frac{ds}{s}exp(-\frac{m^{2}s}{eH}-\frac{l^{2}\beta^{2}eH}{4s})[\frac{3}{sinh(s)}+4sinh(s)].
\frac{1}{4}\intF^{a\mu\nu}F_{\mu\nu}^{a}\equiv\frac{1}{2}\int\mathrm{tr}(F^{\mu\nu}F_{\mu\nu})\geq\left|\frac{1}{2}\int\mathrm{tr}(F^{\mu\nu}\tilde{F}_{\mu\nu})\right|.
\begin{array}{lll}{k^{1}}&{=}&{m_{1}m^{2}\equiv0}\\{k^{4}}&{=}&{m_{1}m^{5}+m^{2}m^{2}\equiv0}\\{k^{1;6}}&{=}&{m_{1}m^{1;7}+m^{2}m^{5}\equiv0}\\{k^{3;7}}&{=}&{m^{2}m^{1;7}+m^{5}m^{5}\equiv0}\\{k^{6;7}}&{=}&{m^{5}m^{1;7}\equiv0}\\\end{array}
\begin{array}{rcl}{M_{S5_{2}}}&{=}&{{\displaystyle\frac{R_{3}\ldotsR_{7}(R_{8}R_{9})^{2}}{g^{2}\ell_{s}^{10}}}\,,}\\{}&{}&{}\\{M_{F1_{6}}}&{=}&{{\displaystyle\frac{R_{3}(R_{4}\ldotsR_{9})^{2}}{g^{4}\ell_{s}^{14}}}\,.}\\\end{array}
\Lambda_{\mathrm{eff}}=\Lambda+\frac{2GR^{2}}{27\pi^{2}c^{4}}\:\left(\frac{5\alpha^{\prime}\hbar^{3}c}{6}\right)^{\frac{1}{2}}\left\lbrack1+\frac{2}{\pi}\left(\frac{5\alpha^{\prime}\hbar}{4c}R\right)^{\frac{1}{2}}\right\rbrack
J^{-}=-{\partial}_{+}(m\tilde{\Sigma}+\frac{m^{2}}{m^{2}-n_{-}{\partial}_{+}^{2}}{\partial}_{+}^{-1}B),\;\;J^{+}={\partial}_{-}(m\tilde{\Sigma}+\frac{m^{2}}{m^{2}-n_{-}{\partial}_{+}^{2}}{\partial}_{+}^{-1}B)=m{\partial}_{-}\tilde{\Sigma}.
{\calL}_{YM}^{(1)}(D\rightarrow4,2)=\frac{\hbarg^{2}}{32\pi^{2}(4-D)}\left(\frac{11}{3}C+\frac{1}{6}T_{s}-\frac{4}{3}T_{f}\right)F_{\mu\nu}^{a}F_{\mu\nu}^{a}.
{\calL}_{eff}=\chi\,\frac{m}{2}\,\epsilon^{\mu\nu\lambda}\,A_{\mu}\,\partial_{\nu}A_{\lambda}-\frac{m^{2}}{8}\,\lambda\,F_{\mu\nu}\,F^{\mu\nu}-\beta^{2}\,\left(\frac{q-p}{p}\right)\,\left(\lambda\right)^{\fracp{q-p}}\,.
S_{eff}=\int\left(-\frac{1}{4}\sum_{n}(\partial_{\mu}u_{\rho\sigma}^{n}\partial^{\mu}u^{n,\rho\sigma}+{(m_{n}e^{kR})^{2}}u_{\rho\sigma}^{n}u^{n,\rho\sigma})-\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi\right)dz.
\omega^{2}(z)-{\frac{1}{s}}\omega^{\prime}(z)+W^{\prime}(z)\omega(z)+{\frac{1}{4}}R(z)=0,\qquadR(z)\equiv{\frac{4}{s}}\sum_{p=1}^{s}{\frac{W^{\prime}(w_{p})-W^{\prime}(z)}{w_{p}-z}}
\mathsf{E}=(\mathsf{E}^{\mu\nu})\equiv\left|\begin{array}{cccc}{0}&{\mathcal{E}^{1}}&{\mathcal{E}^{2}}&{\mathcal{E}^{3}}\\{-\mathcal{E}^{1}}&{0}&{i\mathcal{E}^{3}}&{-i\mathcal{E}^{2}}\\{-\mathcal{E}^{2}}&{-i\mathcal{E}^{3}}&{0}&{i\mathcal{E}^{1}}\\{-\mathcal{E}^{3}}&{i\mathcal{E}^{2}}&{-i\mathcal{E}^{1}}&{0}\\\end{array}\right|)
\frac{1}{2\kappa^{2}}\left(n_{\mu}\Gamma_{\rho\sigma}^{\mu}-\frac{1}{2}g_{\rho\sigma}n_{\mu}\Gamma_{\alpha\beta}^{\mu}g^{\alpha\beta}-\Gamma_{\alpha(\rho}^{\alpha}n_{\sigma)}+\frac{1}{2}\Gamma_{\beta\alpha}^{\alpha}n^{\beta}g_{\rho\sigma}\right)=T_{\rho\sigma}^{bra}.
({\not\!\!D})^{2}=(D_{\mu})^{2}-\frac{1}{2}\sigma^{\mu\nu}F_{\mu\nu}=\left(\begin{array}{cc}{(D_{\mu})^{2}-\sigma^{\mu\nu}F_{\mu\nu}/2}&{0}\\{0}&{(D_{\mu})^{2}}\\\end{array}\right),
A\equiv2\biggl\vert\frac{\vert\beta_{k}^{\mathrm{(approx)}}\vert-\vert\beta_{k}^{\mathrm{(exact)}}\vert}{\vert\beta_{k}^{\mathrm{(approx)}}\vert+\vert\beta_{k}^{\mathrm{(exact)}}\vert}\biggr\vert
S={\frac{1}{g}}\intd^{3}x\left[-\frac{1}{4}(F_{\mu\nu})^{2}-\frac{1}{2}(\partial_{\mu}\phi_{m})^{2}-\frac{1}{2}(\partial_{\mu}X_{I})^{2}+\frac{i}{2}\bar{\psi}^{(a)}\gamma^{\mu}\partial_{\mu}\psi^{(a)}\right],
{\calC}^{11}(x){\calC}_{11}^{\dagger}(y)=\frac{N^{2}-1}{2(2\pi)^{4}(x-y)^{4}}+\frac{1}{(2\pi)^{2}(x-y)^{2}}\left({\calK}_{\bf{1}}+{\calQ}_{\bf{20}}^{(Y)}+{\calQ}_{\bf{20}}^{(X)}\right)\left(\frac{x+y}{2}\right)+\ldots\;
\frac{1}{m}\sum_{i}\left(\frac{\partial^{2}\Psi}{\partialx_{i}^{2}}+\frac{\partial^{2}\Psi}{\partialy_{i}^{2}}+\frac{\partial^{2}\Psi}{\partialz_{i}^{2}}\right)+\sum_{k}\frac{1}{M_{k}}\left(\frac{\partial^{2}\Psi}{\partialx_{k}^{2}}+\frac{\partial^{2}\Psi}{\partialy_{k}^{2}}+\frac{\partial^{2}\Psi}{\partialz_{k}^{2}}\right)+\frac{8\pi^{2}}{h^{2}}\left(E-V\right)\Psi=0~,
\begin{array}{ll}{{\displaystyleR_{\mathrm{cr}}=0}\,}&{;D=2\,,}\\{{\displaystyleR_{\mathrm{cr}}=8m_{0}^{2}}\,}&{;D=3\,,}\\{{\displaystyleR_{\mathrm{cr}}=12m_{0}^{2}}\,}&{;D=4\,.}\\\end{array}
\begin{array}{c}{\partial_{\pm2}u_{m}^{[+2]}=\Omega_{\pm2}^{(0)}u_{m}^{[+2]}+\Omega_{\pm2}^{[+2](i)}u_{m}^{(i)},}\\{\partial_{\pm2}u_{m}^{[-2]}=-\Omega_{\pm2}^{(0)}u_{m}^{[-2]}+\Omega_{\pm2}^{[-2](i)}u_{m}^{(i)},}\\{\partial_{\pm2}u_{m}^{(i)}=\frac12\Omega_{\pm2}^{[+2](i)}u_{m}^{[-2]}+\frac12\Omega_{\pm2}^{[-2](i)}u_{m}^{[+2]}+\Omega_{\pm2}^{(i)(j)}u_{m}^{(j)}.}\\\end{array}
\frac{1}{8\piG}\sqrt{g}(g^{ab}g^{cd}-g^{ac}g^{bd})\left(\hat{\xi}_{1a}\hat{\xi}_{2|bd}^{t}-\hat{\xi}_{2a}\hat{\xi}_{1|bd}^{t}\right)f^{-1}n_{c}=\frac{1}{8\piG}f\sqrt{\sigma}\sigma^{\alpha\beta}\left(\hat{\xi}_{2}^{r}\hat{\xi}_{1|\alpha\beta}^{t}-\hat{\xi}_{1}^{r}\hat{\xi}_{2|\alpha\beta}^{t}\right)=O(N^{2}),
\langle\Psi|P|\Psi^{\prime}\rangle=\langle\widetilde{\Psi}|\int_{0}^{\infty}dt\;{\eta(t)}\mathrm{e}^{-t(H_{0}^{x}+H_{0}^{\varphi})}|\widetilde{\Psi}^{\prime}\rangle\;\;\;,
\int_{0}^{\infty}dn\,g(n)=\frac{\pi^{4}}{60}\left(\frac{a}{\beta}\right)^{4}-\frac{\zeta(3)}2\left(\frac{a}{\beta}\right)^{3}-\frac{\pi^{2}}{12}\left(\frac{a}{\beta}\right)^{2}+\frac{5}{6}\frac{a}{\beta}-\frac1{3}.
g=d\bar{z}\otimes(h_{,\bar{z}q}dq+h_{,\bar{z}p}dp)+dz\otimes(h_{,zq}dq+h_{,zp}dp)+{\frac{1}{h_{,\bar{z}\bar{z}}}}(h_{,\bar{z}q}dq+h_{,\bar{z}p}dp)^{2}.
\begin{array}{rcl}{Q^{(n)}(w|\zeta)^{kk_{1}\cdotsk_{2n-1}k}}&{=}&{\displaystyle\prod_{a\inA}\{v_{a}-u_{a}+\frac{1}{2}-k_{a}\}\left(\prod_{j=1}^{a-1}[v_{a}-u_{j}-\frac{1}{2}]\prod_{j=a+1}^{2n}[u_{j}-v_{a}-\frac{1}{2}]\right)}\\{}&{\times}&{\displaystyle\prod_{j=1}^{2n-1}\{k_{j}\}^{-1}\prod_{a,b\inA\atopa<b}[v_{a}-v_{b}+1]^{-1},}\\\end{array}
Z_{a\otimes_{T}b}^{(\kappa)}(q):=\sum_{c\inT_{\kappa}}\hat{\chi}_{a\,c}(q)\left(\hat{\chi}_{\zeta(c)\,b}(q)\right)^{*}=\sum_{c\inT_{\kappa},d,e\inH}{\hat{N}}_{a\,c}{}^{d}{\hat{N}}_{b\,\zeta(c)}{}^{e}\hat{\chi}_{d}(q)\left(\hat{\chi}_{e}(q)\right)^{*}
\begin{array}{rcl}{\Leftrightarrowx^{(1)}{}_{(2)}\otimesx^{(2)^{\prime}}\widehat{x^{(1)}{}_{(1)}}(a)}&{=}&{x_{(2)}{}^{(1)}\otimesx_{(1)}(a)x_{(2)}{}^{(2)^{\prime}}}\\{\Leftrightarrowx^{(1)}{}_{(2)}\otimesx^{(2)^{\prime}}<x^{(1)}{}_{(1)},a_{(1)}>a_{(2)}}&{=}&{x_{(2)}{}^{(1)}\otimesx_{(1)}(a)x_{(2)}{}^{(2)^{\prime}}}\\\end{array}
a_{0}=-\frac{6}{g}\Sigma_{3}+\frac{6}{g^{2}}\left(2\Sigma_{2}\Sigma_{3}+2\Sigma_{3}\Sigma_{2}+\sum_{k=1}^{M}\frac{a_{k}\Sigma_{3}a_{k}^{\dagger}+a_{k}^{\dagger}\Sigma_{3}a_{k}-\Sigma_{3}}{k}\right)+O(1/g^{3})\;.
\frac{1}{24}(\hat{z}+\frac{b}{\hat{z}}+2)x_{0}^{12}+\frac{1}{12}x_{3}^{12}+\frac{1}{3}x_{4}^{3}+\frac{1}{2}x_{5}^{2}+\frac{1}{6\sqrt{c}}(x_{0}x_{3})^{6}+\left(\frac{a}{\sqrt{c}}\right)^{1/6}x_{0}x_{3}x_{4}x_{5}=0,
{\frac{d\tilde{m}_{0}}{d\tilde{\lambda}}}={\frac{\tilde{m}_{0}}{2}}\partial_{y}({\calW}g_{\mu\nu}){\frac{dx^{\mu}}{d\tilde{\lambda}}}{\frac{dx^{\nu}}{d\tilde{\lambda}}}{\frac{dy}{d\tilde{\lambda}}}.
ds^{2}={\frac{U^{2}}{\sqrt{4\pigN}}}\left[-\left(1-{\frac{U_{0}^{4}}{U^{4}}}\right)dt^{2}+dy_{i}dy^{i}\right]+{\frac{\sqrt{4\pigN}}{U^{2}}}\left(1-{\frac{U_{0}^{4}}{U^{4}}}\right)^{-1}dU^{2}+\sqrt{4\pigN}d\Omega_{5}.
\left[\partial_{t}+D-4+\sum_{i=1}^{4}{\bf{q}}_{i}\cdot\nabla_{{\bf{q}}_{i}}\right]\tilde{\Gamma}_{t}^{(4)}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4})=g_{t}^{2}I^{(4)}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4})\;,
=\;\frac{e}{2}\,\{x_{a}^{k}\,,\,x_{b}^{j}\}\,(\partial_{j}A_{k}(x_{b})-\partial_{k}A_{j}(x_{b}))\;=\;-\frac{1}{2\theta}\,\epsilon_{jk}\,\{x_{a}^{k}\,,\,x_{b}^{j}\}
\rho_{P}(\lambda)=\frac{2}{a^{2}\pi}\mathrm{Im}\,\left(\frac{1}{G_{Q_{\lambda}}(0)}-\frac{a^{2}}{4}G_{Q_{\lambda}}(0)\right)=-\frac{2}{a^{2}\pi}\mathrm{Im}\,G_{Q_{\lambda}}(0)\left(\frac{1}{(\mathrm{Re}\,G_{Q_{\lambda}}(0))^{2}+(\mathrm{Im}\,G_{Q_{\lambda}}(0))^{2}}+\frac{a^{2}}{4}\right)
\rho_{\nu}(x)=\rho_{\nu^{\prime}}(x)\frac{q^{C(\nu)/2}+\epsilon(\nu)\epsilon(\nu^{\prime})xq^{C(\nu^{\prime})/2}}{xq^{C(\nu)/2}+\epsilon(\nu)\epsilon(\nu^{\prime})q^{C(\nu^{\prime})/2}}\;,~~~~\forall\nu\neq\nu^{\prime}.
W=\frac{1}{8\sqrt{6}V}\left\{\left[(\alpha+\beta)\left(1+\sigma^{2}+(V+\xi^{2})^{2}\right)+2(3\alpha-\beta)(V-\xi^{2})\right]^{2}+128\alpha(\alpha-\beta)\xi^{2}V\right\}^{1/2}\,.
p_{4}\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_{7}^{2}(z_{4}^{4}+z_{5}^{4})=0\;,
\begin{array}{rl}{*W_{2}*}&{=*(s_{2}\overline{{s}}_{2}+\overline{{s}}_{2}s_{2})*=(-1)^{(q-1)(r-q+1)}*(s_{2}**\overline{{s}}_{2}+\overline{{s}}_{2}**s_{2})*}\\{}&{=(-1)^{(q-1)(r-q+1)+r(q-1)+1+rq}(\overline{{s}}_{2}s_{2}+s_{2}\overline{{s}}_{2})=(-1)^{q(r-q)}W_{2}\quad,}\\\end{array}
V_{0}={\frac{(6-n)H_{0}^{2}}{16\piG}}\left[\frac{\Omega_{\mathrm{NR}}^{n}}{(1-\Omega_{\mathrm{NR}})^{3}}\right]^{\frac{1}{n-3}};\quad\alpha=(3-n)(2\piG/n)^{1/2}
\{\circ,\circ\}_{D}\equiv\{\circ,\circ\}-\{\circ,\phi_{i}^{\alpha}\}\frac{1}{\{\phi_{i}^{\alpha},\bar{\phi}_{\bar{j}\beta}\}}\{\bar{\phi}_{\bar{j}\beta},\circ\}-\{\circ,\bar{\phi}_{\bar{j}\beta}\}\frac{1}{\{\bar{\phi}_{\bar{j}\beta},\phi_{i}^{\alpha}\}}\{\phi_{i}^{\alpha},\circ\}
{\calL}_{quad}=\frac{1}{2}h_{\mu\nu}\gamma^{\mu\nu,\rho\sigma}\left[\delta_{\rho\sigma}^{\alpha\beta}(-\Box)+2\delta_{(\rho}^{(\alpha}\nabla_{\sigma)}\nabla^{\beta)}-g^{\alpha\beta}\nabla_{(\rho}\nabla_{\sigma)}-{K_{\rho\sigma}}^{\alpha\beta}\right]h_{\alpha\beta}\,
\gamma_{(3)}\epsilon=-iG_{z\overline{{z}}}^{-1}(\partial_{z}Z^{m})(\overline{{\partial_{z}Z^{n}}})\hat{\Gamma}_{0m\overline{{n}}}\epsilon=G_{z\overline{{z}}}^{-1}(\partial_{z}Z^{m})(\overline{{\partial_{z}Z^{n}}})\delta_{m\overline{{n}}}\epsilon=\epsilon
=g_{2}^{2}g^{4}{\frac{\Lambda^{2}}{M^{4}}}\left(\lambda_{1}^{2}-\lambda_{2}^{2}\right)-g^{6}{\frac{m_{0}^{2}}{M^{4}}}\left(\lambda_{1}^{2}-\lambda_{2}^{2}\right)-g^{2}ln{\frac{m_{0}^{2}+\lambda_{1}^{2}}{m_{0}^{2}+\lambda_{2}^{2}}}+...
\left(\begin{matrix}{x_{1}}\\{x_{2}}\\\end{matrix}\right)=\left(\begin{matrix}{(m_{2}/m_{1})^{1/4}}&{0}\\{0}&{(m_{1}/m_{2})^{1/4}}\\\end{matrix}\right)\left(\begin{matrix}{y_{1}}\\{y_{2}}\\\end{matrix}\right).
\nabla_{-}x^{i}-\frac12\Omega_{-}^{+2i}x^{-2}-\frac12\Omega_{-}^{-2i}x^{+2}-ih_{-q}^{1+}\gamma_{q\dot{q}}^{i}\theta_{\dot{q}}^{1-}-i\chi_{\dot{q}}^{1}\tilde{\gamma}_{\dot{q}q}^{i}\theta_{q}^{1+}-(1\leftrightarrow2)=0.
D_{F}^{N}(x)=\frac{-i}{(2\pi)^{4}}\int\frac{S_{\vec{k}}(N+1)^{2}e^{i\vec{k}.\vec{r}-ik_{0}\,h(N_{\vec{k}})\,t}\,\,d^{4}k}{k^{2}+m^{2}}-(N\rightarrowN-1)\,,
\mathcal{L}_{B}^{kin}=\frac{1}{4}\sqrt{-g}\,\epsilon^{\mu\nu\rho\sigma\lambda}B_{\mu\nu}^{\mathcal{M}}\,\mathcal{D}_{\rho}B_{\sigma\lambda}^{\mathcal{N}}\,\Omega_{\mathcal{MN}}
R_{\r}^{r}=F^{-2}\left((\phi^{\prime})^{2}+\frac{1}{4}(\eta^{\prime})^{2}+\frac{1}{2}\frac{q_{2}^{2}}{q_{1}^{2}}e^{-\eta}(\hat{b}^{\prime})^{2}\right)+\frac{p-3}{8}Y\,,
\{{\calF},{\calG}\}_{p}=\intd^{2p+1}x\left[\frac{\delta{\calF}}{\deltaD^{i_{1}...i_{p}}}\\epsilon^{i_{1}...i_{p}kj_{1}...j_{p}}\,nb_{k}\frac{\delta{\calG}}{\deltaB^{j_{1}...j_{p}}}-({\calF}\rightleftharpoons{\calG})\right]\,
H(\{q_{N_{s}}\})=g^{2n}(u_{1}-u_{2})(v_{1}-u_{2})\left[1-\frac{\mu_{1}\mu_{2}}{2(u_{1}-u_{2})}-\frac{\nu_{1}\mu_{2}}{2(v_{1}-u_{2})}\right]\prod_{s=1}^{n}(u_{s}-v_{s}-\mu_{s}\nu_{s})^{-1}
\sum_{k=0}^{1-a_{ij}}(-1)^{k}q_{i}^{-k(1-a_{ij}-k)}\left\vert\left[\begin{array}{c}{1-a_{ij}}\\{k}\\\end{array}\right]\right._{q_{i}^{-1}}(X_{i}^{-})^{k}X_{j}^{-}(X_{i}^{-})^{1-a_{ij}-k}=0~,
Q=\tilde{Y}=\left(\begin{array}{ccc}{0}&{0}&{0}\\{0}&{-1}&{0}\\{0}&{0}&{-1}\\\end{array}\right)\:,\:\tilde{T}_{3}=\left(\begin{array}{ccc}{-1}&{0}&{0}\\{0}&{0}&{0}\\{0}&{0}&{-1}\\\end{array}\right)
G(p_{1},p_{2},p_{3},p_{4})=\intd^{4}k\frac{1}{(\rho^{2}+k^{2})(\rho^{2}+(k-p_{1})^{2})(\rho^{2}+(k-p_{1}-p_{2})^{2})(\rho^{2}+(k+p_{4})^{2})}
{\mathbf{v}}=({\mathbf{q}}_{inv})\sqrt{\frac{2}{\alpha^{\prime\prime}}},\qquadv_{iL,R}=\frac{\tilde{n}_{i}}{R_{i}^{\prime}}\pm\frac{w_{i}R_{i}^{\prime}}{\alpha^{\prime\prime}},\quadi=1,2\qquadv_{3L,R}=\frac{n_{3}}{R_{3}}\pm\frac{mw_{3}R_{3}}{\alpha^{\prime\prime}}.
e^{-1}{\calL}=-\frac{1}{2}R-\frac{1}{4}G_{IJ}(\phi)F_{\mu\nu}^{I}F^{\mu\nu{J}}-\frac{1}{2}g_{\mu\nu}\bar{g}_{xy}(\phi)\partial^{\mu}\phi^{x}\partial^{\nu}\phi^{y}+\frac{e^{1}}{48}\epsilon^{\mu\nu\rho\sigma\lambda}C_{IJK}F_{\mu\nu}^{I}F_{\rho\sigma}^{J}A_{\lambda}^{K}
\begin{array}{l}{z_{1}^{(3)}=z_{1}^{(4)}\tau^{2}=\cdots=z_{1}^{(n)}\tau^{2(n-3)}=z_{0}\tau^{2(n-2)},}\\{z_{i}^{(2)}=z_{i}^{(3)}\tau^{2}=\cdots=z_{i}^{(n)}\tau^{2(n-2)},~~2\leqi\leqm-1,}\\{z_{m}^{(2)}=z_{m}^{(3)}\tau^{2}=\cdots=z_{m}^{(n-1)}\tau^{2(n-3)},}\\{z_{1}^{(1)}=z_{1}^{(2)}\tau^{2},}\\\end{array}
s_{\rho\sigma}^{TS}=\left|\begin{array}{cc}{s_{\rho\sigma}^{T}}&{0}\\{0}&{0}\\\end{array}\right|\in(1,0)\oplus(0,0),\quads_{\rho\sigma}^{T}=-s_{\sigma\rho}^{T}:\quads_{mn}^{T}=-i\varepsilon^{mnj}S^{j},\quads_{0j}^{T}=S^{j},
W_{\mu}^{\pm}=\frac{B_{\mu}^{1}\mpiB_{\mu}^{2}}{\sqrt{2}},\quadZ_{\mu}=\frac{-g^{\prime}{\calA}_{\mu}+gB_{\mu}^{3}}{\sqrt{g^{2}+g^{2}}}\quad\mathrm{and}\quad{A}_{\mu}=\frac{g{\calA}_{\mu}+g^{\prime}B_{\mu}^{3}}{\sqrt{g^{2}+g^{2}}}.
\langle\Psi,\Phi\rangle\;=\;\sum_{k=0}^{K}\\frac{1}{k!}\,sum_{\alpha_{1}\cdots\alpha_{k}\atopA_{1}\cdotsA_{k}}\int\prod_{j,A}dX_{j}^{A}\\overline{{\psi_{A_{1}\cdotsA_{k}}^{\alpha_{1}\cdots\alpha_{k}}(X)}}\\times\\phi_{A_{1}\cdotsA_{k}}^{\alpha_{1}\cdots\alpha_{k}}(X).
\varepsilon^{\mu}(\mathrm{k},\lambda)=\frac{(\mathrm{p}_{A}\varepsilon)}{(\mathrm{p}_{A}\mathrm{p}_{B})}\mathrm{p}_{B}^{\mu}+\frac{(\mathrm{p}_{B}\varepsilon)}{(\mathrm{p}_{A}\mathrm{p}_{B})}\mathrm{p}_{A}^{\mu}+\epsilon_{\bot}^{\mu}(\mathrm{k},\lambda)
\sigma_{V}\,\longleftrightarrow\,(-1)^{\sumJ}\:\:\left\{\begin{array}{ccc}{J_{1}}&{J_{2}}&{J_{3}}\\{J_{4}}&{J_{5}}&{J_{6}}\\\end{array}\right\}\,\doteq\,(-1)^{\sumJ}\:\{6j\}(\sigma_{V}),
34\tilde{V}=\frac{\tilde{A}}{16\pi^{2}\omega(1-\omega^{2})}\bar{\Sigma}_{0}^{4}\left[(2-\omega)\left(\frac{\rho_{r}}{\bar{\Sigma}_{0}}\right)^{4/(2-\omega)}-2\left(\frac{\rho_{r}}{\bar{\Sigma}_{0}}\right)^{2}\right],
\left(-{\frac{1}{r}}{\frac{\partial}{\partialr}}r{\frac{\partial}{\partialr}}+{\frac{\left(m-\delta\;\Theta(R-r)\right)^{2}}{r^{2}}}-{\frac{g}{2}}\delta{\frac{1}{R}}\delta(r-R)\right)\psi_{m}(r)=\epsilon\psi_{m}(r)\,.
\varepsilon_{remain}\left(X,P\right)=\sum_{s=0}^{\infty}\sum_{k=0}^{s}\varepsilon_{s,k}^{\mu_{1}\cdots\mu_{k}}\,\left(\kappap_{\kappa}-wp_{w}-\frac{w}{\kappa}B\right)^{s-k}\,(p_{\mu_{1}}+A_{\mu_{1}})\cdots(p_{\mu_{k}}+A_{\mu_{k}}).
\bar{\alpha}^{-}|\Phi_{s}\rangle=\Bigl(-\frac{\partial^{I}}{\partial^{+}}\bar{\alpha}^{I}+\frac{s+d-2}{\hat{\partial}^{+}}\bar{\alpha}^{z}-\frac{2(\hat{\partial}^{+}-\alpha^{+}\bar{\alpha}^{z})}{\hat{\partial}^{+}(\hat{\partial}^{+}-2\alpha^{+}\bar{\alpha}^{z})}\bar{\alpha}^{z}\Bigr)|\Phi_{s}\rangle\,.
v=\left[{\frac{z_{0}^{2}+z_{i}^{2}+1}{2z_{0}}},{\frac{z_{0}^{2}+z_{i}^{2}-1}{2z_{0}}},{\frac{z_{i}}{z_{0}}}\right],\;\;\;,l=[1,-1,\vec{0}],\;\;\;(v\cdotl)={\frac{z_{0}^{2}+z_{i}^{2}}{z_{0}}}
f\Big(\Lambda/q\Big)=\sum_{i}c_{i}\frac{2\pi^{2}}{\Big(1+q^{2n}/\Lambda^{2n}\Big)^{2}}\sqrt{1+\frac{4M_{i}^{2}}{q^{2}}}\,\mathrm{arctanh}\sqrt{\frac{q^{2}}{4M_{i}^{2}+q^{2}}},
\sum_{b=0}^{N-1}\phi_{j}^{(0)}(b|u,v,w)L(a,b|u,v,w)\,{\bfp}_{j+1}=\phi_{j}^{(1)}(a|u,v,w)\,{\bfp}_{j}\quad\mathrm{for}\quada\in{\bfZ}/N{\bfZ}.
\rho_{II}\cdot\left(\begin{array}{c}{x_{1}+\mathrm{i}y_{1}}\\{x_{2}+\mathrm{i}y_{2}}\\{x_{3}+\mathrm{i}y_{3}}\\{x_{4}+\mathrm{i}y_{4}}\\\end{array}\right)=\left(\begin{array}{c}{-x_{2}+\mathrm{i}y_{2}}\\{x_{1}-\mathrm{i}y_{1}}\\{x_{4}-\mathrm{i}y_{4}}\\{-x_{3}+\mathrm{i}y_{3}}\\\end{array}\right),
V(\varphi,\sigma,\Lambda)=\frac{\lambda_{\varphi}}{4}(\varphi^{2}-\eta^{2})^{2}+f_{\varphi\sigma}\varphi^{2}\sigma^{2}+\frac{\lambda_{\sigma}}{4}\sigma^{4}-\frac{m_{\sigma}^{2}}{2}\sigma^{2}+l^{2}\sigma^{2}\Lambda^{2},
\int{\calD}\beta{\calD}\gamma\,\delta(\beta(q_{1}))\,\delta(\beta(q_{2}))\,\mathrm{e}^{iS_{\beta\gamma}}=(\mathrm{det}^{\prime}\bar{\partial}_{3/2})^{-1}\,\int\mathrm{d}\beta_{1}\mathrm{d}\beta_{2}\,\delta(\beta_{0}(q_{1}))\delta_{0}(\beta(q_{2})),
\widehatI=\mathrm{Tr}\\int_{X}d^{4}x\\varepsilon^{\mu\nu\rho\sigma}\left[{\calR}_{\mu\nu}{\calR}_{\rho\sigma}+2\theta^{\tau\vartheta}{\calR}_{\mu\nu}{\calR}_{\rho\sigma\tau\vartheta}^{(1)}+\theta^{\tau\theta}\theta^{\vartheta\zeta}\left(2{\calR}_{\mu\nu}{\calR}_{\rho\sigma\tau\theta\vartheta\zeta}^{(2)}+{\calR}_{\mu\nu\tau\theta}^{(1)}{\calR}_{\rho\sigma\vartheta\zeta}^{(1)}\right)\right].
(\nablav_{\alphaq}^{\underline{\alpha}})v_{\underline{\alpha}}^{~\betar}={\frac{1}{4}}\delta_{q}^{r}\hat{\Omega}^{ab}(\gamma_{ab})_{\alpha}^{~\beta}+{\frac{1}{4}}\delta_{\alpha}^{~\beta}\hat{\Omega}^{ij}(\gamma_{ij})_{q}^{~r},
M^{2}=\frac{\lambdaT^{2}}{24}\left[\frac{1-\frac{3}{2}\left(\frac{\lambda}{24\pi^{2}}\right)^{\frac{1}{2}}}{1+\frac{3}{2}\left(\frac{\lambda}{24\pi^{2}}\right)^{\frac{1}{2}}}\right].
B_{2}=\sum_{i=1}^{3}b^{i}\omega_{2}^{i};\,\,\,C_{2}=\sum_{i=1}^{3}c^{i}\omega_{2}^{i}\,;\,\,C_{6}=\sum_{i=1}^{3}A_{4}^{i}\wedge\omega_{2}^{i}\.
A=\frac{C_{l-1}}{2\pi}\int\frac{dt}{t}e^{-(\frac{b^{2}t}{2\pi\alpha^{\prime}})}(8\pi^{2}\alpha^{\prime}t)^{-(\sharpNN-1)/2}\Theta_{3}(0,8i\pi^{2}\alpha^{\prime}t/L^{2})B\timesJ.
\langle\hat{\rho}(x)\hat{\rho}(y)\rangle_{c}=-{\frac{1}{\pi^{2}N^{2}\beta}}{\frac{1}{(x-y)^{2}}}{\frac{a^{2}-xy}{[(a^{2}-x^{2})(a^{2}-y^{2})]^{1/2}}}.
[\textstyle{\frac{1}{2}}t_{i}^{~j}T_{1j}{}^{i},\textstyle{\frac{1}{2}}t_{k}^{~l}T_{2l}{}^{k}]=\textstyle{\frac{1}{2}}t_{i}^{~j}[T_{1},T_{2}]_{j}^{~i}
q^{i}[A,E;\sigma]=\int_{\Sigma}d^{2}x\bigl[\epsilon^{ab}A_{a}^{i}\partial_{b}\sigma^{i}-2\phi_{n}\epsilon_{\j}^{i}E^{bj}\partial_{b}({\frac{\sigma^{i}\Pi_{n}}{\mathrm{det}E}})\bigr]
U_{M,N}(a)={\frac{2\pi}{L}}\left(M-{\frac{gLa}{2\pi}}\right)^{2}-{\frac{2\pi}{L}}\left(N-{\frac{1}{2}}-{\frac{gLa}{2\pi}}\right)^{2}-{\frac{\pi}{4L}}+V^{[N=2]}.
S=\frac{1}{16\piG}\intd^{4}x\sqrt{-g}\left[R-2(\partial\phi)^{2}+2\Lambdae^{2b\phi}-e^{-2a\phi}F_{\mu\nu}F^{\mu\nu}\right]-\frac{1}{8\piG}\intd^{3}x\sqrt{-h}K,
{\calL}=\frac{1}{2}\left(\partial_{\mu}\chi-\frac{e}{\sqrt{\pi}}A_{\mu}\right)^{2}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+(D_{\mu}\phi)(D^{\mu}\phi)^{*}-\lambda(|\phi|^{2}-c^{2})^{2}
{\frac{1}{4\piG_{4}}}\int_{\partial\Sigma}dS_{\mu\nu}e\hat{E}^{\mu\nu}=\bar{\varepsilon}_{\infty}\left[P_{\mu}^{ADM}\gamma^{\mu}+{\frac{1}{2\sqrt{2}G_{4}}}e^{-\varphi_{\infty}/2}\{V_{R\,\infty}L(\vec{Q}-i\gamma^{5}\vec{P})\}^{a}\Gamma^{a}\right]\varepsilon_{\infty},
E_{\mathrm{in}}^{\mathrm{(vol)}}=E_{\mathrm{in}}-\left(2\xi-\frac{1}{2}\right)\left(\frac{L}{2\pi}\right)^{N}\frac{1}{a}\sum_{\alpha}\frac{\lambda_{n,l}^{2}T_{n}(\lambda_{n,l})}{\sqrt{\lambda_{n,l}^{2}+k^{2}a^{2}+m^{2}a^{2}}}J_{n}(\lambda_{n,l})J_{n}^{\prime}(\lambda_{n,l}),
\tilde{b}^{\rho}={\frac{k^{2}b^{\rho}-(b\cdotk)k^{\rho}}{b^{2}k^{2}-(b\cdotk)^{2}}}\,\quad\tilde{b}^{\rho}k_{\rho}=0\,\quad\tilde{b}^{\rho}b_{\rho}=1\.
M=\frac{1}{\sqrt{1+{\bf\bar{w}}\cdot{\bfw}-{\frac{1}{4}}({\bfw}\times{\bf{\bar{w}}})^{2}}}\left(\begin{array}{cc}{1-{\frac{i}{2}}({\bfw}\times{\bf\bar{w}})\cdot{\bf\sigma}}&{-{\bfw}\cdot{\bf\sigma}}\\{{\bf\bar{w}}\cdot{\bf\sigma}}&{1+{\frac{i}{2}}({\bfw}\times{\bf\bar{w}})\cdot{\bf\sigma}}\\\end{array}\right)
{\calM}\,=\,\left(\begin{array}{ccccc}{E^{2}\,+\,p_{z}^{2}}&{{\;\;}}&{\sqrt{2}\,p_{-}\,p_{z}}&{{\;\;}}&{p_{-}^{2}}\\{\sqrt{2}\,p_{+}\,p_{z}}&{{\;\;}}&{E^{2}\,+\,p_{-}\,p_{+}\,-\,p_{z}^{2}}&{{\;\;}}&{-\,\sqrt{2}\,p_{-}\,p_{z}}\\{p_{+}^{2}}&{{\;\;}}&{-\,\sqrt{2}\,p_{+}\,p_{z}}&{{\;\;}}&{E^{2}\,+\,p_{z}^{2}}\\\end{array}\right)\quad.
(\nabla_{a}E_{\alphaq}^{\underline{\alpha}})E_{\underline{\alpha},\betar}={\frac{1}{4}}\hat{\Omega}_{a,b}^{~~i}m_{b}^{~c}\gamma_{c\alpha\beta}(\gamma_{i})_{qr}+{\calD}_{a}h_{\alpha\beta}C_{qr}.
{\overline{{V_{0}+V_{2}}}}=(-{\frac{1}{2}}(-{\frac{1}{2}}~0)(0~{\frac{1}{2}})^{2}\vert\vert{\frac{e_{1}}{2}}~({\frac{1}{2}})^{2}\vert0^{5}~0_{r}^{5}~0_{r})\,
\begin{array}{c}{\underline{{g}}^{\infty}(\Lambda^{\infty})=\intd^{3}x\Lambda^{\infty}[\partial_{i}E_{i}+J_{0}]\approx0\,\quad\Lambda^{\infty}\inT^{\infty},}\\{\underline{{g}}^{\infty}(\Lambda^{\infty})\mid\cdot>=0}\\\end{array}
\mathrm{Det^{\prime}}_{P}^{-{\frac{1}{2}}}\Bigl[-{\partial}_{\tau}^{2}+2ieF{\partial}_{\tau}\Bigr]={\lbrack4\piT\rbrack}^{-{\frac{D}{2}}}\mathrm{Det^{\prime}}_{P}^{-{\frac{1}{2}}}\Bigl[{\bfI}-2ieF{{\partial}_{\tau}}^{-1}\Bigr]\quad
U_{1}(\phi_{0},a_{1},...,a_{d})=\mu^{D}\sqrt{a_{1}\cdotsa_{d}}\sum_{s=1}^{\infty}\frac{(-1)^{s}}{2s}g^{s}\phi_{0}^{2s}\sum_{n_{1},...,n_{d}=-\infty}^{+\infty}\int\frac{d^{D-d}q}{(a_{1}n_{1}^{2}+\cdots+a_{d}n_{d}^{2}+c^{2}+{\bfq}^{2})^{s}}.
R_{x}^{0}=\frac{1}{2~A}\Biggl\{\frac{\dot{B}}{B}\frac{A^{\prime}}{A}+\frac{\dot{A}}{A}\frac{B^{\prime}}{B}+\frac{B^{\prime}}{B}\frac{\dot{B}}{B}-\frac{\dot{C}}{C}\frac{C^{\prime}}{C}-2\frac{{\dot{B}}^{\prime}}{B}\Biggr\},