harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S=\frac{1}{2}\int\left((e^{-\Phi}\starQ_{B}e^{\Phi})(e^{-\Phi}\star\eta_{0}e^{\Phi})-\int_{0}^{1}dt(e^{-t\Phi}\star\partial_{t}e^{t\Phi})\star\left\{(e^{-t\Phi}\starQ_{B}e^{t\Phi}),(e^{-t\Phi}\star\eta_{0}e^{t\Phi})\right\}\right)\,
S=\frac{1}{2\pi}\intd^{2}{\xi}\left(g_{\mu\nu}+2\pi\alpha^{\prime}B_{\mu\nu}\right)\left(\frac{2}{\alpha^{\prime}}\overline{{\partial}}X^{\mu}\partialX^{\nu}{}-\overline{{\partial}}\psi^{\mu}\,\psi^{\nu}+\widetilde{\psi}^{\mu}\partial\widetilde{\psi}^{\nu}\right)~.
\frac{1}{V}<H_{\phi^{3}}^{\theta}>=\frac{1}{4}\int\frac{d^{2}k}{(2\pi)^{2}}[K^{-1}(k)+\frac{16\pi^{2}\theta^{2}}{\eta^{2}}z+(K(k)-\frac{16\pi^{2}\theta^{2}}{\eta^{2}}zK^{2}(k))(k^{2}+\lambda\eta^{2})]
L=\frac{M}{2}{\dot{q}}^{2}-V_{F}(q)+\sum_{\alpha}\{\frac{m_{\alpha}}{2}\dot{y}_{\alpha}^{2}-\frac{m_{\alpha}}{2}\Omega_{\alpha}^{2}y_{\alpha}^{2}+\epsilon_{\alpha}y_{\alpha}q\}+E_{0}.
g_{1}=\frac{2\phi}{R}\eta_{\mu\nu}\Big((\tilde{E}^{\mu}-E^{\mu})\otimesE^{\nu}+E^{\mu}\otimes(\tilde{E}^{\nu}-E^{\nu})\Big)+\frac{R^{2}}{4\phi^{2}}\eta_{ij}\tilde{E}^{i}\otimes\tilde{E}^{j}.
{\frac{d}{dt}}\left(\begin{matrix}{p}\\{q}\\\end{matrix}\right)={\frac{1}{\hbar}}\left(\begin{matrix}{0}&{-{\calH}}\\{{\calH}}&{0}\\\end{matrix}\right)\left(\begin{matrix}{{\frac{\deltaH_{0}}{\deltap}}}\\{{\frac{\deltaH_{0}}{\deltaq}}}\\\end{matrix}\right)
\hat{\gamma}(\hat{y})=e^{-i\omegaz}\hat{f}(\hat{y})=\left(\frac{\alpha+\hat{y}}{\alpha-\hat{y}}\right)^{\frac{-i\zeta\omega}{2\alpha(\alpha^{2}-1)}}\left(\frac{1-\hat{y}}{1+\hat{y}}\right)^{\frac{-i\zeta\omega}{2(\alpha^{2}-1)}}\hat{f}(\hat{y}),
V_{i}\cdotV_{j}=\left(\begin{array}{cc}{-1}&{-{\frac{1}{2}}}\\{-{\frac{1}{2}}}&{(v^{I})^{2}-{\frac{1}{3}}}\\\end{array}\right)~,~~~k_{ij}=\left(\begin{array}{cc}{k_{00}}&{0}\\{{\frac{1}{2}}}&{{\frac{1}{2}}(v^{I})^{2}+{\frac{1}{3}}}\\\end{array}\right)~.
\overline{{\psi^{a}}}_{L}(x)\psi_{R}^{a}(x)\longrightarrow\frac{1}{2a}\frac{1}{2}(\psi_{a,x}^{\dagger}\psi_{a,x}-\psi_{a,x+1}^{\dagger}\psi_{a,x+1}+L_{x}^{(a)}-R_{x}^{(a)})\quad.
\Phi_{1}^{G}=x^{0}-\kappa\tau,\quad\,\Phi_{2}^{G}=\chi,\quad\,\Phi_{3}^{G}=e-\frac{\kappa}{p_{0}},\quad\,\Phi_{4\nu}^{G}=b_{\nu},\quad\,\Phi_{5}^{G}=\xi^{0}.
{\textstyle{\frac{1}{4}}}\,[\,\delta^{Ba}\,\sigma_{ga}^{r}\,D_{r}\,F_{\mathrm{sym}}^{++gH}\,-\,\delta^{Ha}\,\sigma_{ga}^{r}\,D_{r}\,F_{\mathrm{sym}}^{++gB}\,]=-{\textstyle{\frac{1}{4}}}\,\,\epsilon_{\qquadcd}^{BH}\,\,F_{\mathrm{asy}}^{++cd}
C_{1}\left[J_{\lambda}\left(\frac{\sqrt{4\alphac_{\epsilon}}}{\epsilonx^{\frac{\epsilon}{2}}}\right)+\frac{\sqrt{4\alphac_{\epsilon}}}{x^{\frac{\epsilon}{2}}}J_{\lambda}^{\prime}\left(\frac{\sqrt{4\alphac_{\epsilon}}}{\epsilonx^{\frac{\epsilon}{2}}}\right)\right]_{x=a}=0\;\;\;.
\frac{16\,\pi^{2}}{g_{i,\,\mathrm{bare}}^{2}}=\frac{16\,\pi^{2}}{g_{i}^{2}(\mu)}-b_{i}\,(4\pi)^{\epsilon}\int_{0}^{\infty}\frac{dt}{t^{1-\epsilon}}\,e^{-t\frac{\mu^{2}}{M^{2}}}\,.
\sum_{i=0}^{k+1}\left({k+1\atopi}\right)\frac{(-1)^{i}}{(2k-2i+1)}=-\left(\frac{2k+2}{2k+1}\right)\sum_{i=0}^{k}\left({k\atopi}\right)\frac{(-1)^{i}}{(2k-2i-1)}
G_{\mathrm{ac}}^{(n)}({\bf{r}}_{1},\ldots,{\bf{r}}_{n})=\left.\frac{\delta^{(n)}{\cal{G}}_{\mathrm{ac}}\{{\phi}\}}{\delta{\phi}({\bf{r}}_{1})\ldots\delta{\phi}({\bf{r}}_{n})}\right|_{{\phi}=0}\;.
Tr_{s_{1}}(Q{_{a}}^{2}{(-)^{s_{2}F}}q^{H-11/12}{\bar{q}}^{{\bar{H}}-3/8})_{int}=\frac{1}{|G|}{\displaystyle\mathop{{\sum_{\begin{array}{ccc}{{g,h\;\in\;G}}\\{gh\;=\;hg}\\\end{array}}}}}Tr_{(g,s_{1})}(Q_{a}^{2}(-)^{s_{2}F}q^{L_{o}-11/12}{\bar{q}}^{{\bar{L}}_{o}-3/8}).
{\frac{d^{2}V}{d\tau^{2}}}=V_{ab}{\frac{d\phi^{a}}{d\tau}}{\frac{d\phi^{b}}{d\tau}}+2e^{2U}{\frac{\partialV}{\partial\phi^{a}}}{\frac{\partialV}{\partial\phi^{b}}}G^{ab}\.
S=M^{3}\intd^{4}x\int_{-z_{c}}^{z_{c}}dz\sqrt{g}\left(R-2\Lambda\right)~-\sigma_{0}\int_{z=0}d^{4}x\sqrt{h_{0}}~-\sigma_{c}\int_{z=z_{c}}d^{4}x\sqrt{h_{c}}~.
u^{0}=x^{0}-x^{3},\;u^{1}=q\left(u^{0}\right)r^{2},\;u^{2}=\varphi,\;u^{3}=x^{0}+x^{3}-u^{0}u^{1},\;q\left(u^{0}\right)=\left[\left(u^{0}\right)^{2}+a\right]^{-1},
R^{\prime\prime}+\frac{{w^{\prime}}^{2}-4{R^{\prime}}^{2}}{R}+\frac{4(R^{2}+1)}{\nuR}+\frac{2\nu^{\prime}}{\nu}(R\Phi^{\prime}-R^{\prime})-4R{\Phi^{\prime}}^{2}+10R^{\prime}\Phi^{\prime}-\frac{(w^{2}-1)^{2}}{4\nuR^{3}}=0\,,
{\calG}_{(1)}=\left(\begin{matrix}{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{1}&{-2}&{2}&{1}\\{0}&{0}&{-2}&{1}\\\end{matrix}\right)\;\qquad{\calG}_{(2)}=\left(\begin{matrix}{1}&{-1}&{0}&{1}\\{0}&{0}&{1}&{-1}\\{0}&{0}&{0}&{1}\\{0}&{-1}&{0}&{0}\\\end{matrix}\right)\.
K_{d+1}(s)={\frac{T}{2\pi}}\int_{-\infty}^{\infty}d\omega\sum_{\Lambda(\omega)}e^{-s(-\omega^{2}+\Lambda(\omega))}={\frac{T}{2\pi}}\int_{0}^{\infty}d\omega~e^{s\omega^{2}}~\left[K_{\omega}(s)+K_{-\omega}(s)\right].
T^{-1}=\left(\begin{array}{cc}{a^{-1}+a^{-1}\betad^{-1}\gammaa^{-1}}&{-a^{-1}\betad^{-1}}\\{-d^{-1}\gammaa^{-1}}&{d^{-1}+d^{-1}\gammaa^{-1}\betad^{-1}}\\\end{array}\right)
\langles_{1}\cdot\cdot\cdots_{N}|\Psi\rangle=\left({\frac{B}{4}}\right)^{N/4}e^{-B\sum_{i}s_{i}^{2}/2}e^{-{\frac{1}{4B}}\sum_{i}\left({\frac{\partial}{\partials_{i}}}\right)^{2}}\Psi(\sqrt{2B}s_{1},\cdot\cdot\cdot,\sqrt{2B}s_{N}).
\beta_{\nu}\equiv\frac{\pi^{-\nu/2}}{2^{\nu-1}\Gamma(\nu/2)}\left[\frac{\Gamma(\nu/2)}{4!}\Gamma(4-\nu/2)-\frac{\Gamma(1+\nu/2)}{2!2!}\Gamma(3-\nu/2)+\frac{\Gamma(2+\nu/2)}{4!}\Gamma(2-\nu/2)\right]
{\calS}[\psi]=\int\,d^{5}x\sqrt{g}\,\left\{\frac{i}{2}\left[\overline{{\psi}}\tilde{\gamma}^{\hat{\mu}}\tilde{\nabla}_{\hat{\mu}}\psi-(\overline{{\tilde{\nabla}_{\hat{\mu}}\psi}})\tilde{\gamma}^{\hat{\mu}}\psi\right]-M\overline{{\psi}}\psi\right\}
E=\frac{m_{p}\left(\frac{u}{R}\right)^{7-p}\tilde{f}}{\sqrt{\tilde{f}-\left(\frac{R}{u}\right)^{7-p}\left(\tilde{f}^{-1}\dot{u}^{2}+u^{2}\dot{\phi}^{2}\right)}}-m_{p}\left(\frac{u}{R}\right)^{7-p}\left(1-\frac{u_{0}^{7-p}}{2u^{7-p}}\right).
U^{T}:=\left(c^{(1)}\!\left[\begin{array}{c}{1}\\{1}\\{.}\\{.}\\{.}\\{.}\\{1}\\\end{array}\right],c^{(2)}\!\left[\begin{array}{c}{1}\\{.}\\{.}\\{.}\\{.}\\{1}\\{-(N\!-\!1)}\\\end{array}\right],c^{(3)}\!\left[\begin{array}{c}{1}\\{.}\\{.}\\{.}\\{1}\\{-(N\!-\!2)}\\{0}\\\end{array}\right]\;.\;.\;.\;c^{(N)}\!\left[\begin{array}{c}{\;1}\\{-1}\\{0}\\{.}\\{.}\\{0}\\{0}\\\end{array}\right]\right).
\begin{array}{rcl}{D^{\prime}(z)}&{=}&{\displaystyle\sum_{\alpha}\int\frac{d^{3}\vec{p}}{(2\pi)^{3}2E_{\vec{p},\alpha}}\left|\langle\Omega|\varphi(0)|\vec{p},\alpha\rangle\right|^{2}e^{-ip_{+}\cdotz}}\\{}&{=}&{\displaystyle\sum_{\alpha}\int\frac{d^{4}p}{(2\pi)^{3}}\tau(p_{0})\delta(p^{2}-m_{\alpha}^{2})\left|\langle\Omega|\varphi(0)|\vec{p},\alpha\rangle\right|^{2}e^{-ip\cdotz}.}\\\end{array}
g_{ab}=\left(\begin{array}{ccccc}{0}&{-\omega_{1}}&{}&{}&{}\\{\omega_{1}}&{0}&{}&{}&{}\\{}&{}&{\ddots}&{}&{}\\{}&{}&{}&{0}&{-\omega_{\frac{p+1}{2}}}\\{}&{}&{}&{\omega_{\frac{p+1}{2}}}&{0}\\\end{array}\right).
V={x^{p-2}}\,{y^{p-2}}-{x^{p-4}}\,{y^{p-4}}\,\left({x^{3}}+{y^{3}}\right)+{x^{p-6}}\,{y^{p-6}}\,\left({x^{6}}+{y^{6}}\right)+\cdots+\left({x^{3\,{\frac{p-2}{2}}}}+{y^{3\,{\frac{p-2}{2}}}}\right)
{\cal{T}}_{p}\{\prod_{n}\{\hat{\varphi}}^{a_{n}}(x_{n})\\}=\tilde{{\cal{T}}}\{\prod_{a_{n}=2}{\hat{\varphi}}^{a_{n}}(x_{n})\}{\cal{T}}\{\prod_{a_{k}=1}{\hat{\varphi}}^{a_{k}}(x_{k})\}\.
{\frac{\overline{\alpha}(Q)}{Q^{2}}}\Longrightarrow{\frac{1}{2}}\sqrt{\overline{\alpha}(Q_{q})\overline{\alpha}(Q_{\bar{q}})}\left({\frac{1}{Q_{q}^{2}}}+{\frac{1}{Q_{\bar{q}}^{2}}}\right)\,,
\eta_{\mathrm{\tiny~GL~{}_{q}(2)~}}=-\left(\begin{array}{cccc}{q^{-3}}&{0}&{0}&{0}\\{0}&{0}&{q^{-1}}&{0}\\{0}&{q^{-3}}&{0}&{0}\\{0}&{0}&{0}&{q^{-1}}\\\end{array}\right).
\stackrel{\left(1\right)}{\omega}_{1}=a_{\;\;ij}^{ab}\eta_{2a}^{i}\eta_{2b}^{j}+a_{\;\;ijk}^{ab}\eta_{2a}^{i}\partial^{\left(j\right.}\eta_{2b}^{\left.k\right)}+a_{\;\;ijkl}^{ab}\partial^{\left(i\right.}\eta_{2a}^{\left.j\right)}\partial^{\left(k\right.}\eta_{2b}^{\left.l\right)},
\begin{array}{c}{\left(D(\beta,L)-i\mu\gamma^{0}\right)K_{bc}(t,x;t^{\prime},x^{\prime})=\delta(x-x^{\prime})\delta(t-t^{\prime})}\\{BK_{bc}(t,x;t^{\prime},x^{\prime})=0,\mathrm{~for~}x=0,1,}\\{K_{bc}(1,x;t^{\prime},x^{\prime})=-K_{bc}(0,x;t^{\prime},x^{\prime}).}\\\end{array}
bigh\left(\rho_{(1)}^{A}\right)=\left(-gh_{1}\Phi_{A}^{*(2)}-1,-gh_{2}\Phi_{A}^{*(2)}\right),\;\epsilon\left(\rho_{(1)}^{A}\right)=\epsilon\left(\Phi_{A}^{*(2)}\right)+1\;\mathrm{mod}\;2.
{\calL}=\left\{\frac{1}{8\,e^{2}}\int\!\mathrm{d}^{2}\theta\,W^{2}+\mathrm{H.c.}\right\}+\frac{1}{4}\int\!\mathrm{d}^{4}\theta\left(\bar{S}e^{V}S+\bar{T}e^{-V}T\right)-\frac{\xi}{4}\int\!\mathrm{d}^{2}\theta\mathrm{d}^{2}\bar{\theta}\,V(\!x,\theta,\bar{\theta})\,,
\begin{array}{cllcll}{{[a_{m}^{i},a_{n}^{j}]}}&{=\;m\delta^{ij}\delta_{m,-n}\;\;\;}&{,\;\;\;}&{[c_{m},c_{n}]}&{=\;m\delta_{m,-n}\;\;\;}&{,}\\{\{b_{r}^{i},b_{s}^{j}\}}&{=\;\delta^{ij}\delta_{r,-s}\;\;\;}&{,\;\;\;}&{\{d_{r},d_{s}\}}&{=\;\delta_{r,-s}\;\;\;}&{,}\\\end{array}
\frac{\ddot{b}}{b}-\frac{\dot{b}}{b}\dot{\phi}=\frac{\beta^{\prime\prime}}{\beta},~~~~\Biggl[\frac{d}{dt}\left(\frac{\dot{c}}{c}\right)-\frac{\dot{b}}{b}\frac{\dot{c}}{c}+\frac{\dot{c}}{c}\dot{\phi}\Biggr]=\Biggl[\frac{d}{dx}\left(\frac{\kappa^{\prime}}{\kappa}\right)+\frac{\beta^{\prime}}{\beta}\frac{\kappa^{\prime}}{\kappa}\Biggr].
{\frac{{g(\tau,\vec{\sigma})}}{{\gamma(\tau,\vec{\sigma})}}}=g_{\tau\tau}(\tau,\vec{\sigma})-\gamma^{{\check{r}}{\check{s}}}(\tau,\vec{\sigma})g_{\tau{\check{r}}}(\tau,\vec{\sigma})g_{\tau{\check{s}}}(\tau,\vec{\sigma}).
S_{B}={\frac{(p-1)V_{p}}{128G}}\left[(p-3)^{2}\bar{\calE}^{2}\bar{\calP}_{\phi}^{-2{\frac{p-1}{p-3}}}a^{\frac{p^{2}-1}{p-3}}-(p+1)(p-7)a^{p-1}\right].
T^{R}(p^{2})=T(p^{2})-T(0)-p^{2}\left.\frac{d}{dp^{2}}T(p^{2})\right|_{p^{2}=0}=\frac{p^{4}}{\pi}\int_{4m^{2}}^{\infty}\!\!\frac{dt}{t-p^{2}}\frac{1}{t^{2}}ImT(t)
\begin{array}{c}{\omega=d{\bfp}\wedged{\bfx}+\frac{\imath}{c}\sum_{i}d{\bfz}_{i}\wedged{\bar{\bfz}}_{i},}\\{\;\;}\\{{\calH}=\frac{s}{2c}\left[\imath{\sqrt{2}}{\bfp}({\bf{\bar{z}}}_{1}-{\bf{z}}_{1})+\imath\sum_{a=1}^{N-1}{k}_{a}({\bfz}_{a}{\bf{\bar{z}}}_{a+1}-{\bfz}_{a+1}{\bf{\bar{z}}}_{a})+{k}_{N}({\bfz}_{N}{\bf{\bar{z}}}_{N}-c^{2})\right],}\\\end{array}
\frac{[u+1]}{\{k-1\}}\frac{\{v+\frac{5}{2}-k\}}{[v+\frac{3}{2}]}[v-u-\frac{1}{2}]+\frac{[u+2]}{\{k\}}\frac{\{v+\frac{3}{2}-k\}}{[v+\frac{3}{2}]}[u-v-\frac{1}{2}]=-\frac{[1]\{u+2-k\}\{v-u+\frac{1}{2}-k\}}{\{k\}\{k-1\}},
\begin{array}{l}{1=[0,0,0]=0\\\,\3=[0,0,1]=x\\\,\{\overline{3}}=[0,0,2]=y\\\,\6=[0,1,1]=x^{2}-y}\\{{\overline{6}}=[0,2,2]=y^{2}-x\\,\8=[0,1,2]=xy-1\,\10=[1,1,1]=x^{3}-2xy+1}\\{{\overline{{10}}}=[2,2,2]=y^{3}-2xy+1\\,\15=[1,1,2]=x^{2}y-x-y^{2}\\,\{\overline{{15}}}=[1,2,2]=xy^{2}-x^{2}-y}\\\end{array}
F_{j_{1}m_{1},j_{2}m_{2}}^{++~j_{3}m_{3}}\langle{\calO}_{j_{3}m_{3}}^{r_{1}+r_{2}~+}{\calO}_{j_{3}~{-m_{3}}}^{r_{3}~-}\rangle=F_{j_{2}m_{2},j_{3}~{-m_{3}}}^{+-~j_{1}{-m_{1}}}\langle{\calO}_{j_{1}m_{1}}^{r_{1}+}{\calO}_{j_{1}~{-m_{1}}}^{r_{2}+r_{3}-}\rangle\nonumber
S=\frac{1}{2\pi}\intd^{2}z(\eta_{\mu\nu}+B_{\mu\nu})(\bar{\partial}x^{\mu}\partialx^{\nu}-\bar{\partial}\varphi_{1}^{\\mu}\varphi_{1}^{\\nu}+\varphi_{2}^{\\mu}\partial\varphi_{2}^{\\nu}).
\begin{array}{rcl}{\check{S}^{K,L}\check{S}^{L,K}}&{=}&{\mathrm{id.},}\\{(\check{S}^{V_{z_{1}},V_{z_{2}}^{*}})_{jl^{*}}^{i^{*}k}}&{=}&{(\check{S}^{V_{z_{2}},V_{z_{1}}})_{ij}^{kl},}\\{(\check{S}^{V_{z_{1}}^{*},V_{z_{2}}})_{i^{*}j}^{kl^{*}}}&{=}&{(\check{S}^{V_{z_{2}},V_{z_{1}+nw}})_{jl}^{ik}.}\\\end{array}
\alpha_{n}^{\left(\pm\right)0^{\prime}}=\frac1{m_{2}}\left(-p_{2}^{+}\cdot\hat{\alpha}_{n}^{\left(\pm\right)-}-p_{2}^{-}p_{1}^{+}\delta_{n,0}+p_{2}^{i}\alpha_{n}^{\left(\pm\right)i}\right)
\frac{k_{\mu}k^{\lambda}l_{\,\,\,\,nu\lambda}^{\mu\nu}}{4(k^{2})^{3/2}(\sqrt{k^{2}}-\lambda)^{3}}-\frac{k_{\mu}k_{\lambda}l_{\,\,\nu}^{\nu\,\,\,\,\mu\lambda}}{4(k^{2})^{3/2}(\sqrt{k^{2}}-\lambda)^{3}},
Q^{(n)}(w|\zeta)^{\varepsilon}=\prod_{a\inA}\zeta_{a}\left(\prod_{j=1}^{a-1}(xz_{j}-w_{a})\prod_{j=a+1}^{2n}(xw_{a}-z_{j})\right)\left/\prod_{a,b\inA\atopa<b}(x^{-1}w_{b}-xw_{a}).\right.
{\calL}=-\sqrt{-G}\sqrt{1+{\frac{\tilde{Y}M\tilde{Y}}{G(\partiala)^{2}}}+\left({\frac{\tilde{Y}L\tilde{Y}}{2G(\partiala)^{2}}}\right)^{2}}-{\frac{\tilde{Y}L\partialY\cdot\partiala}{2(\partiala)^{2}}},
\begin{array}{rl}{e_{\ell}e_{m}}&{=-\delta_{\ellm}e_{0}+\epsilon_{\ellmn}e_{n}}\\{e_{\ell}\hat{e}_{m}}&{=-\delta_{\ellm}e_{7}-\epsilon_{\ellmn}\hat{e}_{n}}\\{\hat{e}_{\ell}\hat{e}_{m}}&{=-\delta_{\ellm}e_{0}-\epsilon_{\ellmn}e_{n}}\\\end{array}
K=\sigma_{\mu}x^{\mu}=\left[\begin{array}{cc}{q(x^{0}-x^{3})}&{-\left[2\right]^{1/2}q^{-1/2}x^{+}}\\{-\left[2\right]^{1/2}q^{1/2}x^{-}}&{qx^{0}+q^{-1}x^{3}}\\\end{array}\right]\equiv\left[\begin{array}{cc}{qD}&{B}\\{A}&{C/q}\\\end{array}\right]\;,
(x_{13}^{2})^{-2r}(x_{24}^{2})^{-2}(\frac{v}{u})^{2}\sum_{D\in\{{\bf{84}},{\bf{105}},{\bf{175}}\}}\sum_{\substack{a,b;\,s\inS_{r}\,(a>b)}}\frac{\sigma_{D}}{N^{2}}P_{D}^{I_{a}I_{b},J_{s(a)}J_{r+1}}\delta^{I_{r+1}J_{s(b)}}\prod_{\substack{i=1\,i\notin\{a,b\}}}^{r}\delta^{I_{i}J_{s(i)}}
\mathcal{H}_{\lambda}=\bigoplus_{\mu\in\mathcal{C}_{2N}}\mathcal{H}_{\lambda,\mu}=\bigoplus_{\mu\in\mathcal{C}_{2N}}\mathcal{V}_{\lambda}^{\mu}\otimes\bar{\mathcal{V}}_{\lambda^{*}}^{\mu^{*}}.
\Psi\equiv\left(\begin{array}{c}{\psi_{1}}\\{\psi_{2}}\\{\psi_{3}}\\{\psi_{4}}\\\end{array}\right)\equiv\left(\begin{array}{c}{\varphi_{1}+i\eta_{1}}\\{\varphi_{2}+i\eta_{2}}\\{\varphi_{3}+i\eta_{3}}\\{\varphi_{4}+i\eta_{4}}\\\end{array}\right)~~~~~~~\varphi_{m},\eta_{m}\in{\calR},~~~m=1,2,3,4~,
{{\calG}^{-1}}_{\mu\nu}^{\alpha\beta}\eta_{\alpha\beta}^{\mathrm{TT}}=\left(k^{2}-\Lambda_{l}+\frac{1}{2r^{2}}N(N-1)-\frac{N+d}{N+d-2}\Lambda\right)\eta_{\mu\nu}^{\mathrm{TT}}.
{\calS}_{quad}=tr\intd^{4}x\left(\frac{g^{2}}8\omega^{2}\chi^{*\mu\nu}\chi_{\mu\nu}^{*}-\frac{g^{2}}4\omega\chi^{*\mu\nu}\varepsilon_{\mu}\psi_{v}^{*}-\frac{g^{2}}{32}\varepsilon^{\mu}\varepsilon^{\nu}\psi_{\mu}^{*}\psi_{v}^{*}+\frac{g^{2}}{32}\varepsilon^{2}\psi^{*\mu}\psi_{\mu}^{*}\right)\;.
M_{\alpha\beta}^{\mathrm{higher}}(\kappa_{1}\tilde{x},\kappa_{2}\tilde{x})=M_{\alpha\beta,a}^{\mathrm{tw3}}(\kappa_{1}\tilde{x},\kappa_{2}\tilde{x})+M_{\alpha\beta,a}^{\mathrm{tw4}}(\kappa_{1}\tilde{x},\kappa_{2}\tilde{x})
\begin{array}{rcl}{\psi_{\uparrow}^{(+)}}&{=}&{\sqrt{\rho}e^{-\gamma_{21}mt}\,;}\\{\psi_{\downarrow}^{(+)}}&{=}&{\sqrt{\rho}\gamma_{31}e^{-\gamma_{21}mt}\,;}\\{\psi_{\uparrow}^{(-)}}&{=}&{\sqrt{\rho}\gamma_{5}(\gamma_{12}e^{\gamma_{21}mt})\,;}\\{\psi_{\downarrow}^{(-)}}&{=}&{\sqrt{\rho}\gamma_{5}\gamma_{31}(\gamma_{12}e^{\gamma_{21}mt})\,.}\\\end{array}
\frac{1}{2}(S,S)^{a}+\intdx\varepsilon^{ab}\Phi_{Ab}^{*}\frac{\delta}{\delta{\bar{\Phi}}_{A}}S=0,\quad\varepsilon^{ab}=-\varepsilon^{ba},\quad\varepsilon_{ab}\varepsilon^{bc}=\delta_{a}^{c},\quad\varepsilon_{12}=\varepsilon^{21}=1,
\begin{aligned}{l_{1}}&{{}\mapstoH-E_{1}\,,}\\{l_{2}}&{{}\mapstoH-E_{2}\,,}\\{e_{1}}&{{}\mapstoH-E_{1}-E_{2}\,,}\\{e_{a}}&{{}\mapstoE_{a+1}\,,\,\,\,\,a=2,\cdots,k-1\,.}\\\end{aligned}
{\mathrm{Tr}}\Omega_{\;\nu}^{\mu}\Omega_{\;\rho}^{\nu}\Omega_{\;\mu}^{\rho}=\left[{\frac{1}{2}}({\mathrm{Tr}}K)^{3}+3({\mathrm{Tr}}K)({\mathrm{Tr}}K^{2})\right]({\mathrm{Tr}}1)={\frac{81}{2}}({\mathrm{Tr}}1)\;,
{\calG}_{k}(U,U^{\prime})=-\frac{1}{N^{2}}\frac{1+\gamma}{2\gammak^{2}}\frac{1}{UU^{\prime}}\left(\frac{U_{*}}{U_{>}}\right)^{-\gamma}\left[(\gamma-1)\left(\frac{U_{*}}{U_{<}}\right)^{\gamma}+(\gamma+1)\left(\frac{U_{*}}{U_{<}}\right)^{-\gamma}\right]~.
W^{iAB}=\mathrm{i}g^{i\bar{\jmath}}\left(\begin{array}{cc}{-{\nabla_{\bar{\jmath}}}\bar{P}^{1}}&{{\nabla_{\bar{\jmath}}}\bar{P}^{3}}\\{{\nabla_{\bar{\jmath}}}\bar{P}^{3}}&{{\nabla_{\bar{\jmath}}}\bar{P}^{1}}\\\end{array}\right)^{AB},
\langle\Psi^{(p)}(z^{i},dz^{j})|\Xi^{(q)}(z^{k},dz^{l})\rangle={\frac{1}{2^{p}\pi^{D}}}\int_{{\bfC}^{D}}e^{-z^{m}\bar{z}_{m}}\Psi^{(p)}(z^{i},dz^{j})*\Xi^{(q)}(\bar{z}^{k},d\bar{z}^{l}),
\varepsilon_{10}^{\mu\kappar_{1}\cdotsr_{8}}\varepsilon_{10}^{\nu\lambdas_{1}\cdotss_{8}}\times\eta_{\mu\nu}\times\zeta_{\kappa\lambda}^{(5)}\times\prod_{i=1}^{4}\zeta_{r_{2i-1}s_{2i-1}}^{(i)}k_{r_{2i}}^{(i)}k_{s_{2i}}^{(i)}\.
{\calL}_{eff}=-\frac{1}{4}G_{\mu\nu}^{a}G_{\mu\nu}^{a}+\frac{m}{4}\varepsilon^{\alpha\beta\gamma}\left(G_{\alpha\beta}^{a}A_{\gamma}^{a}-\frac{g}{3}\varepsilon_{abc}A_{\alpha}^{a}A_{\beta}^{b}A_{\gamma}^{c}\right)+\frac{1}{2}A_{0}^{a}\Pi_{00}(T)\delta^{ab}A_{0}^{b},
\frac{(n-l)!}{(n-l-m_{i})!m_{i}!}\frac{(n-l-m_{i})!}{(n-l-2m_{i})!m_{i}!}\dots\frac{(n-l-(\frac{l}{2}-1)m_{i})!}{(n-l-\frac{l}{2}m_{i})!m_{i}!}=\frac{(n-l)!}{(m_{i}!)^{l/2}}.
L=\frac{-Tr}{4g^{2}}\left[4g^{\mu\nu}\partial_{+}A_{\mu}\partial_{-}A_{\nu}-g^{\mu\nu}g^{\rho\kappa}[A_{\rho},A_{\mu}][A_{\kappa},A_{\nu}]-2\bar{\psi}\gamma^{\mu}[A_{\mu},\psi]\right],
+{\frac{i}{e^{2}}}\{(g+{\frac{a}{a^{2}\partial^{2}+4\alpha^{2}}})g^{\mu\nu}\delta(x-y)-{\frac{2\alpha}{\partial^{2}(a^{2}\partial^{2}+4\alpha^{2})}}\epsilon^{\mu\nu\lambda}\partial_{\lambda}\delta(x-y)\},
P^{(N_{f}=1,\nu=1)}(\lambda,m)=\frac{\lambda}{2}e^{-\lambda^{2}/4}\frac{1}{mI_{1}(m)}\left[tI_{2}(\lambda)I_{3}(t)-\lambdaI_{2}(t)I_{3}(\lambda)\right]_{t=\sqrt{\lambda^{2}+m^{2}}}\:.
S_{UV\psi}=N\mathrm{tr}_{N}\psi_{i}^{\dagger}\psi_{i}+N\frac{1}{2}\mathrm{tr}_{N}(U^{2}+\mathrm{e}^{2\beta}V^{2})-Ng\mathrm{tr}_{N}(\sum_{i}\psi_{i}U\psi_{i}^{\dagger}+\sum_{i,j}L_{ij}\psi_{j}V\psi_{i}^{\dagger}).
R^{(+)}\left(L^{\left(\varepsilon_{1}\right)}\otimes1\!\mathrm{l}\right)\left(1\!\mathrm{l}\otimesL^{\left(\varepsilon_{2}\right)}\right)\,=\,\left(1\!\mathrm{l}\otimesL^{\left(\varepsilon_{2}\right)}\right)\left(L^{\left(\varepsilon_{1}\right)}\otimes1\!\mathrm{l}\right)R^{(+)}\,,
\Xi^{\nu\rho}(x)=\intd^{3}x_{2}....d^{3}x_{n}F_{\mu_{2}\nu_{2}}(x_{2})....F_{\mu_{n}\nu_{n}}(x_{n})\Omega^{\nu\rho\mu_{2}\nu_{2}...\mu_{n}\nu_{n}}(x,x_{2},...,x_{n})\;.
F_{a}[f]=a^{-\frac12}\int_{0}^{\infty}e^{-t^{2}}f(t\sqrt{a})dt=\sum_{n=0}^{\infty}a^{\frac{n-1}{2}}\frac{f^{(n)}(0)}{n!}\int_{0}^{\infty}e^{-t^{2}}t^{n}dt=
(\frac{gH_{min}+\frac{R}{4}}{T^{2}})^{\frac{1}{2}}=\frac{\frac{11g^{2}}{15\pi^{2}}+\sqrt{(\frac{11g^{2}}{15\pi^{2}})^{2}+\frac{R}{5\pi^{2}}}}{2}
\hat{x}_{++}=x_{++}+i\theta_{+}^{AA^{\prime}}\theta_{+A^{\prime}}^{B}u_{(A}^{+}u_{B)}^{-},\\x_{--},\\\theta_{+}^{\pmA^{\prime}}=u_{A}^{\pm}\theta_{+}^{AA^{\prime}},\\u^{\pm}\;.
\Lambda_{0}I\Lambda_{0}^{T}=\Lambda_{0}\Lambda_{0}^{T}={\calM}_{0}=e^{\phi_{0}}\left(\begin{array}{cc}{e^{-2\phi_{0}}+\chi_{0}^{2}\qquad}&{\chi_{0}}\\{\chi_{0}\qquad}&{1}\\\end{array}\right).
\left[\begin{array}{c}{A_{jm}^{(1)}}\\{A_{jm}^{(2)}}\\{A_{jm}^{(3)}}\\\end{array}\right]=R\cdot\left[\begin{array}{c}{a_{jm}^{(1)}}\\{a_{jm}^{(2)}}\\{a_{jm}^{(3)}}\\\end{array}\right],
\left(\begin{array}{c}{\delta(\bar{\psi}\psi)}\\{\delta(\bar{\psi}i\tau^{1}\psi)}\\{\delta(\bar{\psi}i\gamma^{5}\psi)}\\\end{array}\right)=2\left(\begin{array}{ccc}{0}&{-\theta^{1}}&{-\theta^{2}}\\{\theta^{1}}&{0}&{-\theta^{3}}\\{\theta^{2}}&{\theta^{3}}&{0}\\\end{array}\right)\left(\begin{array}{c}{\bar{\psi}\psi}\\{\bar{\psi}i\tau^{1}\psi}\\{\bar{\psi}i\gamma^{5}\psi}\\\end{array}\right).
\Psi_{\mu_{1}}^{(i,1-i)}(\xi_{1})\Psi_{\mu_{2}}^{*(1-i,i)}(\xi_{2})={\frac{g\delta_{\mu_{1}\mu_{2}}}{1-\xi_{2}^{2}/\xi_{1}^{2}}}\left({\frac{\xi_{2}}{\xi_{1}}}\right)^{i+{\frac{1-\mu_{1}}{2}}}+\cdots\qquad(\xi_{1}\rightarrow\pm\xi_{2}),
c^{\prime}=\prod_{t_{1},\mu_{1}}(1+\omega^{-1}e^{2\pii\mu_{1}/t_{1}})^{t_{1}h_{t_{1}}},~~c^{\prime\prime}=\prod_{t_{1},\mu_{1}}(\omega+e^{2\pii\mu_{1}/t_{1}})^{-t_{1}h_{t_{1}}}\.
\mathrm{det}\{\sigma_{\alpha}^{\mathrm{(s)}},\sigma_{\beta}^{\mathrm{(s)}}\}_{\mathrm{P}}=\mathrm{det}^{2}\left(\frac{\partial\Xi_{a}}{\partialv_{a}}\right)\mathrm{det}\{\gamma_{A},\gamma_{B}\}_{\mathrm{P}}.
\Pi_{\mu\nu}^{ij}(p)=-\int\!\frac{d^{3}k}{i(2\pi)^{3}}\frac{(2k+p)_{\mu}(2k+p)_{\nu}}{(k^{2}-m_{i}^{2})[(k+p)^{2}-m_{j}^{2}]}+\int\!\frac{d^{3}k}{i(2\pi)^{3}}\frac{2g_{\mu\nu}}{k^{2}-m_{i}^{2}}.
\left(\begin{array}{cc}{{\bf1}}&{Q}\\{{\bf0}}&{{\bf1}}\\\end{array}\right)\quad\mathrm{with}\quadQ=\left(\begin{array}{ccc}{0}&{\delta^{\prime}}&{\epsilon^{\prime}}\\{\delta^{\prime}}&{\alpha^{\prime}}&{\beta^{\prime}}\\{\epsilon^{\prime}}&{\beta^{\prime}}&{\gamma^{\prime}}\\\end{array}\right)
\frac{1}{i\hbar}[x^{\alpha},[p_{\beta},p_{\mu}]]\equiv0=\frac{1}{i\hbar}\left([[x^{\alpha},p_{\beta}],p_{\mu}]-[[x^{\alpha},p_{\mu}],p_{\beta}]\right)=i\hbar\partial_{\mu}B_{\beta}^{\alpha}\,.
\left(\begin{array}{c}{\tilde{A}}\\{\tilde{B}}\\\end{array}\right)=S(k)\left(\begin{array}{c}{A}\\{B}\\\end{array}\right)\quad\Rightarrow\quadS(k)=\left(\begin{array}{cc}{t(k)}&{r(k)}\\{l(k)}&{t(k)}\\\end{array}\right)
D=1+\chi_{1}^{2}\,e^{2\varphi_{1}}+\chi_{2}^{2}\,e^{2\varphi_{2}}+\chi_{3}^{2}\,e^{2\varphi_{3}}-2i\,\chi_{1}\,\chi_{2}\,\chi_{3}\,e^{\varphi_{1}+\varphi_{2}+\varphi_{3}}\.
{\bfA}=\left(\begin{matrix}{N^{-1/2}}&{0}&{0}\\{0}&{(LM)^{1/2}}&{0}\\{0}&{0}&{0}\\\end{matrix}\right)\left(\begin{matrix}{{\bfa}^{(1)}}&{{\bfa}^{(3)}}&{0}\\{{{\bfa}^{(3)}}^{\dagger}}&{{\bfa}^{(2)}}&{0}\\{0}&{0}&{0}\\\end{matrix}\right)\left(\begin{matrix}{N^{-1/2}}&{0}&{0}\\{0}&{(ML)^{1/2}}&{0}\\{0}&{0}&{0}\\\end{matrix}\right)
n\:C\;(\alpha_{1}+\gamma)^{1-p}\:\beta_{1}^{1-q}\left(\prod_{j=2}^{p}(\alpha_{1}+\gamma)^{a_{j}-\alpha_{j}}\:\alpha_{j}^{-a_{j}}\right)\left(\prod_{k=2}^{q}\:\beta_{1}^{b_{k}-\beta_{k}}\:\beta_{k}^{-b_{k}}\right)\;,
\begin{aligned}{\big[M^{(8)},Q\big]}&{{}=A\,\GammaQ\,,}&{\big[M^{(5)},Q\big]}&{{}=B\,\GammaQ\,,}\\{\big[W^{(8)},Q\big]}&{{}=A_{z}\,\GammaQ\,,}&{\big[W^{(5)},Q\big]}&{{}=B_{z}\,\GammaQ\,,}\\{\big[Z^{(2|1)},Q\big]}&{{}=C\,\GammaQ\,,}&{\big[Z^{(4|2)},Q\big]}&{{}=D\,\GammaQ\,.}\\\end{aligned}
\gamma_{{\itk}^{2}}=\mu^{2}\frac{\partial\overline{{{\itk}^{2}}}}{\partial\mu^{2}}=-\frac{\overline{{\lambda}}}{32\pi^{2}}\overline{{{\itk}}}^{2}\left(t^{2}-\frac{8}{3}t+\frac{4}{3}+\frac{9}{t^{2}}\right)
\breve{\omega}_{1}\simeq-\int_{-\infty}^{\infty}du{\frac{m_{0}ax^{2}}{\{\breve{R^{2}}+(x^{1}-u)^{2}\}^{\frac{5}{2}}}}\,\\,\qquad\breve{\omega}_{2}\simeq\int_{-\infty}^{\infty}du{\frac{m_{0}a(x^{1}-u)}{\{\breve{R^{2}}+(x^{1}-u)^{2}\}^{\frac{5}{2}}}}\,.
\partial_{x}^{\mu}\partial_{y}^{\nu}\partial_{z}^{\lambda}G_{\mu\nu\lambda}^{abc}(x,y,z)=\alpha\mu^{2}\{\partial_{y}^{\nu}G_{~~\nu}^{cab}(z,x,y)+\partial_{z}^{\lambda}G_{~~\lambda}^{bac}(y,x,z)\}
D_{0}=\frac{1}{2}\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\;,\;\;\;D_{+}=\left(\begin{array}{cc}{0}&{\sigma_{1}}\\{0}&{0}\\\end{array}\right)\;,\;\;\;D_{-}=\left(\begin{array}{cc}{0}&{0}\\{\sigma_{1}}&{0}\\\end{array}\right);
ds^{2}=-\left[\frac{2m_{0}}{t}+\frac{1}{3}\Lambdat^{2}-1\right]^{-1}dt^{2}+\left[\frac{2m_{0}}{t}+\frac{1}{3}\Lambdat^{2}-1\right]dx^{2}+t^{2}d\Omega^{2}.
\langle\\underline{\alpha}\|\\underline{j},m\\rangle=\sum_{m_{2},\dots,m_{N-1}}\left|\begin{matrix}{j_{N}}&{1/2}&{j_{N-1}}\\{m}&{\alpha_{N}}&{m_{N-1}}\\\end{matrix}\right|_{q}\cdots\left|\begin{matrix}{j_{3}}&{1/2}&{j_{2}}\\{m_{3}}&{\alpha_{3}}&{m_{2}}\\\end{matrix}\right|_{q}\left|\begin{matrix}{j_{2}}&{1/2}&{1/2}\\{m_{2}}&{\alpha_{2}}&{\alpha_{1}}\\\end{matrix}\right|_{q}
l_{\bar{q}}^{\mu}=\left(k_{g}+k_{\bar{q}}^{\prime}-k_{\bar{q}}\right)^{\mu}={\frac{\vec{k}_{g_{\!\perp}}^{\,2}}{2k_{g}^{+}}}\\eta^{\mu}\quad\mathrm{and}\quadl_{q}^{\mu}=\left(k_{g}+k_{q}-k_{q}^{\prime}\right)^{\mu}={\frac{\vec{k}_{g_{\!\perp}}^{\,2}}{2k_{g}^{+}}}\\eta^{\mu}
W_{l}^{(j)}\starW_{l^{\prime}}^{(j^{\prime})}=\sum_{{m=|l-l^{\prime}|+1\atopl+l^{\prime}-m-1\,\equiv\,0\,\mathrm{mod}\,2}}^{l+l^{\prime}-1}W_{m}^{(j+j^{\prime}-r)}\,,
e^{2i\pin\alpha_{0}^{-}\alpha_{0}^{+}}\prod_{n,m,k=1}^{\infty}\left(\alpha_{-n}^{+}\right)^{a_{n}}\left(\alpha_{-m}^{-}\right)^{b_{m}}\left(s_{-k}\right)^{c_{k}}\,|p^{+}p^{-}s_{0}>
\tilde{\Gamma}_{\mathrm{div}}^{n}=\intd^{4}x\left[\c_{1}{\frac{1}{4}}F_{\mu\nu}^{2}+\sigma\left(K^{a\mu}{c_{2,\mu}}^{\nu}A_{\nu}^{a}+c_{3}L^{a}C^{a}\right)\\right],
I=\int_{\Sigma}d^{3}\!x\sqrt{\gamma}\left(\!R(\gamma)-{\frac{1}{2}}\!\left(DU\right)^{2}-{\frac{1}{4}}e^{2U}V_{ij}V^{ij}-{\frac{1}{2}}\!\left(\left(D\Phi\right)^{2}\!-\!e^{-2\Phi}\!\left(D\Psi\right)^{2}\right)\right)\,
\zeta\left(\begin{array}{rcr}{s_{1}}&{\ldots}&{s_{k}}\\{\lambda^{p_{1}}}&{\ldots}&{\lambda^{p_{k}}}\\\end{array}\right):=\sum_{n_{j}>n_{j+1}>0}\quad\prod_{j=1}^{k}\frac{\lambda^{p_{j}n_{j}}}{n_{j}^{s_{j}}}
\frac{\delta}{\deltaC(t)}\left(\intdt\left(\frac1C\frac{\partialC}{\partialt}+\frac1C\frac{\sigma^{2}}{2}S^{2}\frac{\partial^{2}C}{\partial^{2}S}-r_{b}(1-\fracSC\frac{\partialC}{\partialS})\right)^{2}-\frac{1}{\beta_{2}}\deltas_{g}\right)=0
c(|x|)=|x|^{4}\inte^{-ikx}\frac{{\calF}(k^{2})}{k^{4}}d^{4}k=4\pi^{2}|x|^{3}\int_{0}^{\infty}\frac{J_{1}(k|x|)}{k^{2}}{\calF}(k^{2})dk\,,