harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\phi_{3}^{(brst)}=\frac{1}{2}(p^{2}+\gamma)+\gamma\gamma^{ij}[\partial_{j}c^{3}\mu_{i}]+(\partial_{i}c^{i})\mu_{3}+\frac{3}{2}\partial_{i}c^{3}\partial_{j}c^{3}\mu_{j}\mu_{i}+\partial_{i}(c^{i}\mu_{3}).
\rho_{L+}^{(2)}\rho_{R+}^{(2)}\rho_{L+}^{(3)}\rho_{R+}^{(3)}\lambda_{L+}^{(2)}\lambda_{R+}^{(2)}\lambda_{L+}^{(3)}\lambda_{R+}^{(3)}[X]\equivP[X]=\rho_{+}^{(2)}\rho_{+}^{(3)}\lambda_{+}^{(2)}\lambda_{+}^{(3)}X^{\sim}\equivpX^{\sim},
\left\{\psi_{L}\left(\eta\right),\psi_{L}^{\dagger}\left(\eta^{\prime}\right)\right\}=\delta\left(\eta-\eta^{\prime}\right),\quad\left\{\psi_{R}\left(\eta\right),\psi_{R}^{\dagger}\left(\eta^{\prime}\right)\right\}=\delta\left(\eta-\eta^{\prime}\right).
\zeta_{r}(s)={\frac{1}{3^{s}}}-{\frac{3}{\sqrt{\pi}}}{\frac{1}{2^{2s+1}}}{\frac{\Gamma\left(s+{\frac{1}{2}}\right)}{\Gamma(s+1)}}-{\frac{3}{\sqrt{\pi}}}{\frac{1}{2^{2s+3}}}{\frac{\Gamma\left(s+{\frac{3}{2}}\right)}{\Gamma(s+2)}}\F\left(1,s+{\frac{3}{2}},s+2,{\frac{3}{4}}\right)
R_{n+1}=-p^{2}(-\pi^{2}g^{2}\hbar)^{n+1}\Big((1{-}\alpha)^{n+1}+O\Big(\sum_{j=1}^{n+1}P_{r,j}^{n+1}(\alpha)\Big(\frac{m_{P}^{2}}{p^{2}}\Big)^{\!jr}\,\Big)\Big)~.
S_{k}^{(1)}=\left(\begin{matrix}{\tau_{k}^{(1)},}&{1,}&{0,}&{0,}&{\cdots,}&{0}\\{\tau_{k}^{(2)},}&{0,}&{1,}&{0,}&{\cdots,}&{0}\\{\vdots}&{}&{}&{}&{}&{\vdots}\\{\tau_{k}^{(n)},}&{0,}&{0,}&{0,}&{\cdots,}&{1}\\{\tau_{k}^{(n+1)},}&{0,}&{0,}&{0,}&{\cdots,}&{0}\\\end{matrix}\right).
S_{0}^{\prime}=\intd\tau\left[\frac{1}{2}g_{st}\left(\dot{q}^{s}+\sum_{a,b}\dot{\alpha}_{a}\zeta_{ab}\delta_{b}q^{s}\right)\left(\dot{q}^{t}+\sum_{c,d}\dot{\alpha}_{c}\zeta_{cd}\delta_{d}q^{t}\right)+V[q]\right],
C_{Jm_{J}}^{\left(tr\right)s_{1}s_{2}\sigma_{1}\sigma_{2}}=C_{Jm_{J}}^{\left(tr\right)Jm_{\sigma}}C_{Jm_{J}}^{\left(tr\right)Jm_{s}}/\sum_{\nu_{1}\nu_{2}m_{J}}\left(C_{Jm_{J}}^{\left(tr\right)Jm_{\nu}}\right)^{2}.
[\,A_{{\sfGI}\,i}^{{\gamma}}({\bf{r}})\,{\textstyle\frac{{\lambda}^{\gamma}}{2}}\,]=V_{\cal{C}}({\bf{r}})\,[\,A_{i}^{\gamma}({\bf{r}})\,{\textstyle\frac{\lambda^{\gamma}}{2}}\,]\,V_{\cal{C}}^{-1}({\bf{r}})+{\textstyle\frac{i}{g}}\,V_{\cal{C}}({\bf{r}})\,\partial_{i}V_{\cal{C}}^{-1}({\bf{r}})\,,
{\frac{1}{r_{*}^{2}}}\intd^{3}x^{\prime}{\frac{R+\ell}{R^{2}}}\left[{\calO}_{r_{1}}^{[1]}{\calO}_{r_{3}}^{[2]}v(r_{1},r_{3})\right]_{r_{1}=r_{3}=r^{\prime}},
\delta\theta_{\dot{A}}^{1-}=-\frac{i}{c(\alpha^{\prime})^{1/2}}\theta_{A}^{1+}\gamma_{A\dot{A}}^{i}e^{\mu[-2]}w_{\mu}^{(i)},\\delta\theta_{A}^{1+}=\kappa_{A}^{+}+\frac{i}{c(\alpha^{\prime})^{1/2}}\theta_{\dot{A}}^{1-}\tilde{\gamma}_{\dot{A}A}^{i}e^{\mu[+2]}w_{\mu}^{(i)},
\begin{array}{cll}{}&{\{<U,0>,\;<0,V>:\;U,V\inE_{8}^{+}\}}&{(2\times240=480\mathrm{~elements)~}}\\{\cup}&{\{<U,V>:\;U,V\inE_{8}^{-},\;UV^{\dagger}=\pme_{a}\}}&{(16\times240=3840\mathrm{~elements)}}\\\end{array}
\langle\,a_{\omega}^{R}a_{\omega^{\prime}}^{R}\,\rangle\,_{\mathrm{{in}}}=\frac{1}{2\pi}\,\int_{0}^{\infty}\,d\omega^{\prime\prime}\:\tilde{\alpha}_{\omega^{\prime\prime}\omega}^{R}\,\tilde{\beta}_{\omega^{\prime\prime}\omega^{\prime}}^{R}\:\propto\:\frac{1}{\sqrt{\omega\omega^{\prime}}}\,\delta(\omega+\omega^{\prime})\,,
L(spin-1)=-\frac{1}{2\kappa^{2}}\left(\partial_{\mu}h_{ir}\partial^{\mu}h^{ir}+\partial_{r}h_{ir}\partial^{r}h^{ir}\right)+\frac{1}{4\kappa^{2}}h_{ri}h^{ri}{A^{\prime}}^{2}-\frac{1}{4}h_{ri}h^{ri}\Phi^{\prime}{^2}~
\frac{M_{H}^{2}}{4}=\intdxx\frac{\Sigma^{3}(x)\frac{d\Sigma(x)}{d\sigma}\langle\sigma\rangle}{(x+\Sigma^{2}(x))^{2}}\left/\intdxx\frac{\Sigma^{2}(x)-\frac{x}{4}\frac{d\Sigma^{2}(x)}{dx}}{(x+\Sigma^{2}(x))^{2}}\right.,
(\;p_{L}\;\vert\;p_{R}\;)=\left(\;{\frac{\tilde{m}}{r_{11}}}-{\frac{wr_{11}}{\alpha_{h}^{\prime}}}\;\Big\vert\;{\frac{\tilde{m}}{r_{11}}}+{\frac{wr_{11}}{\alpha_{h}^{\prime}}}\;;\;\sqrt{\frac{2}{\alpha_{h}^{\prime}}}({\bfQ}+w{\bfA})\;\right)
0=ga_{0}+a_{0}^{3}+\sum_{n\neq0}{\frac{1}{|n|}}\left(a_{0}a_{n}a_{-n}+a_{n}a_{-n}a_{0}+a_{n}a_{0}a_{-n}-{\frac{3a_{0}}{2}}\right)+6\Sigma_{3}\;
S=\int{dx^{0}}{dx^{1}}\sqrt{-g}e^{-\overline{{\phi}}}\left[R+\partial_{\alpha}\overline{{\phi}}\partial^{\alpha}\overline{{\phi}}+\frac{1}{8}\mathrm{Tr}\left(\partial_{\alpha}M^{-1}\partial^{\alpha}M\right)\right],
TrK_{\bfa}^{(1)}={\frac{1}{2\imath}}\int_{0}^{\infty}{\frac{dy}{e^{\piy}+1}}\left({\frac{(\sqrt{t}k_{0})^{-\imathy}}{\Gamma(1-\imath{\frac{y}{2}})}}-{\frac{(\sqrt{t}k_{0})^{\imathy}}{\Gamma(1+\imath{\frac{y}{2}})}}\right)~~,
\left[-\left(1-\frac{2M}{r}\right)\frac{d^{2}}{dr^{2}}-\frac{2(r-4M)(2r-3M)}{r^{2}(r-3M)}\frac{d}{dr}+\frac{8M}{r^{2}(r-3M)}\right]H_{1}(r)=\lambdaH_{1}(r).
\underline{{g}}_{\mu\nu}(u)=\underline{{g}}_{\mu\nu}(u_{0})+\ell\partial_{\alpha}\underline{{g}}_{\mu\nu}(u_{0})u_{[1]}^{\alpha}+\frac{1}{2}\ell^{2}\partial_{\alpha}\partial_{\beta}\underline{{g}}_{\mu\nu}(u_{0})u_{[1]}^{\alpha}u_{[1]}^{\beta}+...
\kappa^{4}\intEG^{2}\left(\overline{{\nabla}}^{2}R\right)\left(\nabla^{2}\overline{{R}}\right)d^{4}\theta=-\frac{3}{2}\kappa^{4}\int\epsilon\left(\overline{{\nabla}}^{2}+\frac{1}{3}\overline{{R}}\right)\left[G^{2}\left(\overline{{\nabla}}^{2}R\right)\left(\nabla^{2}\overline{{R}}\right)\right]d^{2}\theta+\mathrm{h.c.}
{\frac{1}{\psi}}{\frac{d^{2}\psi}{dv^{2}}}\approx-{\frac{1/4}{v^{2}}}-{\frac{1/c}{v(v_{\infty}{-}v)}}-{\frac{(c{-}1)/c^{2}}{(v_{\infty}{-}v)^{2}}}~,
C=i_{\hat{z}}\hat{C}+(dy+i_{\hat{z}}\hat{B}_{2})\wedge\left(\hat{C}^{(-)}-\,{\frac{\hat{E}^{{\#}(-)}}{\hat{E}_{\hat{z}}^{{\#}}}}\,\wedgei_{\hat{z}}\hat{C}\right)\,,
Q(\zeta)=\frac{1}{4}\left(\frac{1-\mu_{1}^{2}}{\zeta^{2}}+\frac{1-\mu_{2}^{2}}{(\zeta-1)^{2}}+\frac{1-\mu_{1}^{2}-\mu_{2}^{2}+\mu_{\infty}^{2}}{\zeta(1-\zeta)}\right).
\imath\delta_{a}^{\c}=\imathG_{(0)\ab}^{-1}\circG^{bc}-\frac{\lambda}{2}h_{abef}\circG^{ed}G^{bc}-\frac{\imath\lambda}{3!}h_{beld}\circ\alpha_{afgh}\circG^{fl}G^{gd}G^{he}G^{bc}.
D_{a}=\frac{\partial}{\partial\theta^{a}}-\frac{i}{2}\overline{{\theta}}_{a}\frac{\partial}{\partial\tau},\quad\overline{{D}}^{a}=\frac{\partial}{\partial\overline{{\theta}}_{a}}-\frac{i}{2}\theta^{a}\frac{\partial}{\partial\tau},
d\Pi^{\underline{m}}=-id\Theta^{1}\sigma^{\underline{m}}d\Theta^{1}-id\Theta^{2}\tilde{\sigma}^{\underline{m}}d\Theta^{2},dd\Theta^{1\underline{\mu}}=0,dd\Theta_{\underline{\mu}}^{2}=0,\qquad(IIA)
\frac{1}{2}\left\{T_{L},T_{L}\right\}=\frac{1}{2}\left\{\frac{1}{2\pii}\int_{C_{1}}dzc(z)t(X^{0}(z)),\frac{1}{2\pii}\int_{C_{2}}dwc(w)t(X^{0}(w))\right\}\,
S^{I}(-c_{0}^{-})^{I}S_{I}=\hbar\bigl({\frac{\delta}{\delta\Phi^{I}}}(-c_{0}^{-})^{I}{\frac{\delta}{\delta\Phi_{I}}}\bigr)S=\hbarS^{IJ}(-c_{0}^{-})^{J}R_{JI}.
\deltax^{\mu}={\dot{x}}^{\mu}\xi\;,\;\;\deltae=\frac{d}{d\tau}\left(e\xi\right)\;,\;\;\deltab^{\mu}=\frac{d}{d\tau}\left(b^{\mu}\xi\right)\;,\;\;\delta\psi^{\mu}=\dot{\psi}^{\mu}\xi\;,\;\;\delta\chi=\frac{d}{d\tau}\left(\chi\xi\right)\;,
A_{cc}(p_{1},p_{2})\sim\int_{H^{+}}d^{2}z\,d^{2}w\langleV_{(-1,0)}(z,p_{1}/2)V_{(-1,0)}(\bar{z},p_{1}/2)V_{(0,-1)}(w,p_{1}/2)V_{(0,-1)}(\bar{w},p_{1}/2)\rangle.
{\calL}=-{\frac{1}{4}}E^{(11)}[{\calR}^{(11)}+{\frac{1}{12}}F_{MNPQ}^{(11)}F^{(11)\MNPQ}-{\frac{8}{12^{4}}}\varepsilon^{M_{1}\cdotsM_{11}}F_{M_{1}\cdotsM_{4}}F_{M_{5}\cdotsM_{8}}A_{M_{9}M_{10}M_{11}}],
{\cal{L}}=\frac{1}{16\piG_{B}}(R-2\Lambda_{B})+\frac{1}{4\pi}\left[a_{B}R^{2}+b_{B}R_{\rho\sigma}R^{\rho\sigma}+c_{B}R_{\rho\sigma\tau\lambda}R^{\rho\sigma\tau\lambda}\right]
\rho_{1}[\beta;\phi,\phi]=-\frac{\lambda}{4!}\rho_{0}\,\left[\int_{1}\widehat\varphi^{4}(1)+6\int_{1}G_{\eta}(1;1)\widehat\varphi^{2}(1)+\int_{1}\langle\eta^{4}(1)\rangle\right]\;.
\delta_{0}=\mathrm{\large~\frac{1}{4}~}(\delta_{1}\cdot\overline{{\delta_{1}}}+\overline{{\delta_{1}}}\cdot\delta_{1})+\mathrm{\large~\frac{1}{4}~}(\delta_{2}\cdot\overline{{\delta_{2}}}+\overline{{\delta_{2}}}\cdot\delta_{2}).
{\calD}_{1}^{\mathrm{full}}\left(x^{2}\right)=\frac{m_{D}^{2}}{\pi^{2}M^{2}|x|^{4}}+\frac{m^{2}}{2\pi^{2}Mx^{2}}\left[\frac{K_{1}(M|x|)}{|x|}+\frac{M}{2}\left(K_{0}(M|x|)+K_{2}(M|x|)\right)\right].
I_{2}=(-1)\int\,\delta^{3}\!\left({\vec{r}_{1}+\vec{r}_{2}-\vec{r}}\right)\,\delta(r_{1}+r_{2}-t)\,\frac{d^{3}r_{1}}{4\pi\,r_{1}}\,\frac{d^{3}r_{2}}{4\pi\,r_{2}\,{\ell_{0}}^{2}}
S^{+}=\left(\begin{array}{ccc}{0}&{a}&{0}\\{0}&{0}&{a}\\{0}&{0}&{0}\\\end{array}\right)\,,\quadS^{-}=(S^{+})^{t},\quadS^{3}=\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{0}&{0}\\{0}&{0}&{-1}\\\end{array}\right)\,,
V(a)=\int_{0}^{a}\left[\frac{(3\gamma(a)-2)\kappa_{(4)}^{2}}{6}\rho+\frac{(3\gamma(a)-1)\kappa_{(4)}^{2}}{6\lambda}\rho^{2}+\frac{2}{3}\sigma^{2}-\frac{\Lambda_{(4)}}{3}+\frac{2{\calU}_{0}}{\lambda\kappa_{(4)}^{2}}a^{-4}\right]ada,
S=\frac{1}{4}\int\!\mathrm{d}^{4}x\,\mathrm{d}^{4}\theta\,{\calK}(\Phi_{i},\bar{\Phi}_{j})+\left\{\frac{1}{2}\int\!\mathrm{d}^{4}x\,\mathrm{d}^{2}\theta\,{\calW}(\Phi_{i})+\mathrm{H.c.}\right\}\,,
T_{\mathrm{HK}}=\frac{1}{4\pir_{\mathrm{BH}}}\left((n-1)-(n+1)\frac{r_{\mathrm{BH}}^{2}}{l_{n+2}^{2}}\right),\\\\S=\frac{r_{\mathrm{BH}}^{n}\Omega_{n}}{4G_{n+2}}.
Z^{\calN}\,=\,\left(\begin{array}{clcl}{Z_{1}\,\epsilon}&{0}&{0}&{0}\\{0}&{Z_{2}\,\epsilon}&{0}&{0}\\{0}&{0}&{Z_{3}\,\epsilon}&{0}\\{0}&{0}&{0}&{Z_{4}\,\epsilon}\\\end{array}\right)\qquad,\qquad\epsilon\,=\,\left(\begin{array}{cl}{0}&{1}\\{-1}&{0}\\\end{array}\right)
Q_{\alphaa}=\frac\partial{\partial\theta^{\alphaa}}-\frac12\sigma_{\alpha\dot{\beta}}^{\mu}\gamma_{a\dot{b}}^{m}\bar{\theta}^{\dot{\beta}\dot{b}}\partial_{\mu}\tilde{\partial}_{m},\quad\bar{Q}_{\dot{\beta}\dot{b}}=\frac\partial{\partial\bar{\theta}^{\dot{\beta}\dot{b}}}-\frac12\theta^{\alphaa}\sigma_{\alpha\dot{\beta}}^{\mu}\gamma_{a\dot{b}}^{m}\partial_{\mu}\tilde{\partial}_{m}
ds^{2}=-e^{\frac{\Phi-\xi}{2}}dt^{2}+e^{\frac{\Phi-\xi}{2}}d\vec{x}^{2}+e^{\frac{\Phi+\xi}{2}-5\eta}d\rho^{2}+e^{\frac{\Phi+\xi}{2}-\eta}d\Omega_{5}^{2}
\sum_{r,k}{\frac{c_{A}}{4\omega_{k}\omega_{p}}}{\frac{\left[H_{1},A_{i}^{c}(r)\right]\left[H_{1},A_{\alpha}^{c}(-r)\right]}{(\omega_{k}+\omega_{p})^{3}}}\left(k_{i}k_{\alpha}-k_{i}p_{\alpha}\right)P_{j\gamma}(k)P_{j\gamma}(p)
\prod_{\stackrel{j=0}{j\neqa}}^{l-1}\!\!\bigl(X-Q_{j}\bigr)\;=\;\sum_{j=0}^{l-1}\prod_{s=j+1}^{l-1}\!\!\bigl(Q_{a}-Q_{s}\bigr)\,E^{j}F^{j}\;.\hphantom{xxxxxxxxxxxxxxx}
S_{WZNW}(g,k)=-{\frac{k}{4\pi}}~\int\left[\mathrm{Tr}|g^{-1}\mathrm{d}g|^{2}~+~{\frac{i}{3}}\mathrm{d}^{-1}\mathrm{Tr}(g^{-1}\mathrm{d}g)^{3}\right],
F(\beta)=-\frac{1}{2^{6}\pi^{10}}\int_{F(\Gamma_{0}(2))}\frac{d^{2}\tau}{\tau_{2}^{2}}\tau_{2}^{-4}\frac{\bar{\Theta}_{\Gamma}}{\bar{\eta}^{24}}\frac{\theta_{2}^{4}}{\eta^{12}}\theta\left[\begin{array}{cc}{0}&{\frac{1}{2}}\\{0}&{0}\\\end{array}\right](0|4\Omega)
\mu^{\ast}=\left(\begin{array}{c}{\mu_{c}^{\ast}}\\{\mu_{b}^{\ast}}\\\end{array}\right)\,,~\varepsilon=\left(\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}\right)\,,~\mathcal{X}^{rs}=\left(\begin{array}{cc}{\bar{X}^{sr}}&{0}\\{0}&{X^{rs}}\\\end{array}\right)\,,~C=\left(\begin{array}{cc}{C}&{0}\\{0}&{C}\\\end{array}\right)\,,~C_{nm}=(-1)^{n}\delta_{n,m}\,.
\int_{0}^{1}dx_{3}\int_{0}^{x_{3}}dx_{2}x_{3}^{2\alpha^{\prime}\alpha_{13}}(1-x_{3})^{2\alpha^{\prime}\alpha_{34}}{x_{2}}^{2\alpha^{\prime}\alpha_{12}}(1-x_{2})^{2\alpha^{\prime}\alpha_{24}}(x_{3}-x_{2})^{2\alpha^{\prime}\alpha_{23}}\cdot\left\{{\kappa(x_{2},x_{3})\atop\lambda(x_{2},x_{3})}\right\},
|\mathcal{T}(T)\rangle_{\alpha_{-1}\cdot\tilde{\alpha}_{-1}\tilde{c}_{1}c_{1}|0\rangle}\sim\frac{1}{T^{2}}\,\,U_{f(v)}^{\dagger}c(i)\partialX^{\mu}(i)\,\tilde{c}(i)\bar{\partial}X_{\mu}(i)|0\rangle,
\sigma_{1}=\left(\begin{matrix}{0}&{1}\\{1}&{0}\\\end{matrix}\right)\,,\qquad\sigma_{2}=\left(\begin{matrix}{0}&{-i}\\{i}&{0}\\\end{matrix}\right)\,,\qquad\sigma_{3}=\left(\begin{matrix}{1}&{0}\\{0}&{-1}\\\end{matrix}\right)\,.
\sum_{k\nej}^{r}{\frac{1}{\bar{x}_{j}-\bar{x}_{k}}}+{\frac{g_{S}+g_{L}}{2g_{M}}}{\frac{1}{\bar{x}_{j}-1}}+{\frac{g_{L}}{2g_{M}}}{\frac{1}{\bar{x}_{j}+1}}=0,\quadj=1,\ldots,r,
\Gamma_{g.f.}=\int\left\{\frac{1}{2}\xiB^{2}+B\partialA-eB\Bigl((\hat{\varphi}_{1}-\xi_{A}\fracme)\varphi_{2}-\hat{\varphi}_{2}(\varphi_{1}-\hat{\xi}_{A}\fracme)\Bigr)\right\}
{\calB}=\left(\begin{matrix}{H_{ij}}&{}&{0}\\{0}&{}&{-k^{-2}}\\\end{matrix}\right)\left(\begin{matrix}{\delta_{kj}}&{}&{H_{kl}^{-1}U_{l}^{-}}\\{k^{2}U_{j}^{+}}&{}&{1}\\\end{matrix}\right)\,,
\sigma_{S}(p^{2})=\frac{1}{m}\frac{1+\frac{\Sigma_{B}}{m}}{\epsilon\left(1+\frac{\Sigma_{A}}{m}\right)^{2}+2\left(1+\frac{\Sigma_{A}+\Sigma_{B}}{2m}\right)\frac{\Sigma_{B}-\Sigma_{A}}{m}},
\sum_{\{\delta\}}\int_{{\calC}/\Gamma}dx~k(x,\gamma[x])=\sum_{\delta\neq{\bf1}}\sum_{n=1}^{\infty}\sum_{\beta\in\Gamma/\Gamma^{(\delta)}}\int_{\beta[{\calC}/\Gamma]}dx~k_{0}(x,\delta^{n}[x])~+~\int_{{\calC}/\Gamma}dx~k_{0}(x,x),
{\calL}(J)=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}++{\calD}_{\mu}\phi{\calD}^{\mu}\phi^{*}+\frac{\lambda}{4}(\phi\phi^{*})^{2}+J(\phi\phi^{*})-\zeta/2J^{2}+{\calL}^{\mathrm{CT}}+{\calL}^{\mathrm{gauge}}
\partial_{\mu}(eF^{\mu\nu})=+\partial_{\mu}(e{\calJ}_{e}^{\nu\mu})+\partial_{\mu}(e{\calJ}_{m}^{\nu\mu})+{\textstyle\frac{ig}{2}}\epsilon^{\nu\lambda\rho\sigma}\bar{\psi}_{\lambda}\gamma_{5}\gamma_{\rho}\sigma^{2}\psi_{\sigma}\,.
\left(i\frac{\partial}{\partialt}+\frac{\nabla^{2}}{2m}\right)G_{\alpha\beta}(x,x^{\prime})+i\lambda\langleT\left(\hat{\psi}_{\gamma}^{+}(x)\hat{\psi}_{\gamma}(x)\hat{\psi}_{\alpha}(x)\hat{\psi}_{\beta}^{+}(x^{\prime})\right)\rangle=\delta(x-x^{\prime}).
{\frac{1}{2}}\sqrt{-\eta}\;\eta^{\mu\nu}={\frac{1}{1-\lambda^{2}}}\;\left(\begin{array}{cc}{{\lambda_{--}}}&{{{\frac{1+\lambda^{2}}{2}}}}\\{{{\frac{1+\lambda^{2}}{2}}}}&{{\lambda_{++}}}\\\end{array}\right)
\mu^{l}=f_{\;bc}^{a}f_{\;de}^{c}\left(\epsilon^{0ijl}A_{j}^{e}\eta_{i}^{\left(2\right)d}-{\calP}^{\left(2\right)le}\eta^{d}\right)\left(B_{a}^{0k}A_{k}^{b}+\eta_{a}^{\left(2\right)k}{\calP}_{k}^{\left(2\right)b}+\eta_{a}{\calP}^{b}\right).
{\calH}=\vec{\Lambda}\cdot(H^{1},\cdots,H^{N-1})=\sum_{i=1}^{N-1}\Lambda^{i}H^{i}={\frac{1}{2}}\mathrm{diag}\left(\frac{N-1}{N},-\frac{1}{N},\cdots,-\frac{1}{N}\right),
\begin{array}{c}{P=P(q_{\bot},q_{\bot}^{\prime})}\\{q_{\bot}^{m}=q^{m}-\frac{(qd)d^{m}}{\Box};}\\{{q^{\prime}}_{\bot}^{m}={q^{\prime}}^{m}-\frac{(q^{\prime}d)d^{m}}{\Box}.}\\\end{array}
\begin{array}{ccc}{\big[L_{i},A^{\prime}\big]=-\frac{\tau_{i}}{2}A^{\prime},}&{}&{\big[J_{i},A^{\prime}\big]=\frac{\tau_{i}}{2}A^{\prime},}\\{\big[I_{i},A\big]=-\frac{\tau_{i}}{2}A,}&{}&{\big[K_{i},A\big]=\frac{\tau_{i}}{2}A.}\\\end{array}
{\calH}_{-}={\frac{1}{2}}g_{i}^{2}+{\frac{1}{2m_{-}^{2}}}(\epsilon_{ij}\partial_{i}g_{j})^{2}={\frac{1}{2}}(\pi_{g}^{2}+m_{-}^{2}g^{2})+{\frac{1}{2m_{-}^{2}}}(\epsilon_{ij}\partial_{i}g_{j})^{2}.
\begin{array}{c}{tr_{q}M\left(\lambda\right)=qA\left(\lambda\right)+q^{-1}D\left(\lambda\right),}\\{\mathrm{where~}M\left(\lambda\right)=\left(\begin{array}{cc}{A\left(\lambda\right)}&{B\left(\lambda\right)}\\{C\left(\lambda\right)}&{D\left(\lambda\right)}\\\end{array}\right)}\\{\mathrm{and~}\left[tr_{q}M\left(\lambda\right),tr_{q}M\left(\mu\right)\right]=0.}\\\end{array}
F_{\mu\nu}(p)={\frac{|\Lambda|}{2}}\int{\frac{d^{2}q}{2(\pi)^{2}}}{\frac{\displaystyleq_{\mu}q_{\nu}}{\displaystyle(q^{2}+1)^{3/2}}}[1-{\frac{3}{8}}{\frac{p^{2}}{\Lambda^{2}((q^{2}+1)}}+{\frac{5}{8}}{\frac{(q.p)^{2}}{\Lambda^{2}(q^{2}+1)^{2}}}+...]
L_{G}(u)_{j}^{i}=E_{0}+\sum_{\alpha\neq(0,0)}E_{\alpha}(I_{\alpha})_{j}^{i}=\sum_{k}\phi(u,z)_{k}^{i}\phi^{-1}(u,z)_{j}^{k}\partial_{u}-l\sum_{k}\partial_{u}\phi(u,z)_{k}^{i}\phi^{-1}(u,z)_{j}^{k},
\partial_{y}\Big[\big(\partial_{y}+2\,\ell^{-1}\big)\,h+\frac{(1-\gamma)^{-1}\,\ell}{3\,M_{*}^{3}}\,e^{-2y/\ell}\,T_{55}(y)\Big]=\frac{(1-\gamma)^{-1}\,\ell}{3\,M_{*}^{3}}\partial_{\mu}T^{\mu5}\,.
S=\intd\tau\left\{-p^{0}(\dot{x}^{0}+i\theta^{1}\dot{\theta}^{1})+p_{1}\dot{x}^{1}-{\frac{e(\tau)}{2}}(p_{M}p^{M}+m^{2})+im\,\theta^{2}\dot{\theta}^{2}\right\}.
H_{a}\equivC_{a}+\frac{1}{4\lambda}\hat{V}_{Q}(m)\epsilon_{~b}^{a}G^{b}=p^{a}+\frac{1}{4}\hat{V}_{Q}(m)[\epsilon_{~b}^{a}\rho^{\primeb}+\frac{2\pi\kappa}{\lambda}(p^{a}+2\lambda^{2}\rho^{a}\Pi_{M})]\,,
\alpha_{k}^{(\sigma)}=\frac{1}{\sqrt{2}}(d_{k}^{(\sigma)}+\bar{d}_{\tilde{k}}^{(-\sigma)}),\quad\beta_{k}^{(\sigma)}=\frac{1}{\sqrt{2}}(d_{k}^{(\sigma)}-\bar{d}_{\tilde{k}}^{(-\sigma)})\,{.}
{\frac{3}{b^{2}}}{\frac{a^{\prime}}{a}}\left({\frac{a^{\prime}}{a}}+{\frac{n^{\prime}}{n}}\right)-{\frac{3}{n^{2}c^{2}}}\left[{\frac{\dot{a}}{a}}\left({\frac{\dot{a}}{a}}-{\frac{\dot{n}}{n}}\right)+{\frac{\ddot{a}}{a}}\right]-{\frac{3k}{a^{2}}}=-{\frac{8\piG_{5}}{c^{4}}}\Lambda,
\begin{array}{c}{D^{B}H^{A}=0,\qquad\qquadD^{B}D^{A}\Phi_{A}=[H^{A},\Phi_{B}],}\\{D^{A}K^{B}=0,\qquad\qquadD^{A}D^{B}\Phi_{B}=[K^{B},\Phi_{A}].}\\\end{array}
Z=\int\!\!\!\int_{0}^{2}\frac{d\zeta_{1}}{2\pi}\frac{d\zeta_{2}}{2\pi}\sum_{m_{1},m_{2}\in{\bfZ}}q^{(\zeta_{+}+2m_{+})^{2}+(\lambda+2m_{-})^{2}-\frac{1}{12}}\prod_{n=1}^{\infty}(1-q^{n})^{-2},
\begin{array}{ccc}{s_{j}^{(a)}}&{=}&{(Q_{j}^{(a)}-Q_{j-1}^{(a)})-(Q_{j+1}^{(a)}-Q_{j}^{(a)})}\\{}&{=}&{LG_{a1}^{-1}\delta_{j1}-\sum_{b}\sum_{k}G_{ab}^{-1}C_{jk}h_{k}^{(b)},}\\\end{array}
g=Z_{g}g_{\mathrm{R}},\quad\bar{A}_{\mu}^{i}=Z_{A^{i}}^{1/2}(\bar{A}_{\mathrm{R}})_{\mu}^{i},\quad\bar{A}_{\mu}^{a}=Z_{A^{a}}^{1/2}(\bar{A}_{\mathrm{R}})_{\mu}^{a},
\begin{array}{l}{<p_{1},...,p_{n}|S|q_{1},....,q_{m}>=\intdx_{1}.....dx_{n}dy_{1}....dy_{m}\overline{{f_{p_{1}}}}(x_{1})....\overline{{f_{p_{n}}}}(x_{n})f_{q_{1}}(y_{1})..}\\{......f_{q_{m}}(y_{m})(\Box_{x_{1}}-M^{2}).....(\Box_{y_{m}}-M^{2})<\phi_{\tau}(x_{1}).......\phi_{\tau}(x_{n})\phi_{\tau}(y_{1})...\phi_{\tau}(y_{m})>}\\\end{array}
k={\frac{u_{1}}{2}}\,\Big[\,{\frac{{\calE}_{1}+\sqrt{{\calE}_{1}^{2}+\mu^{2}}}{K+\sqrt{K^{2}+\mu^{2}}}}-{\frac{K+\sqrt{K^{2}+\mu^{2}}}{{\calE}_{1}+\sqrt{{\calE}_{1}^{2}+\mu^{2}}}}\,\Big]\,.
\underbrace{2s-2}_{1},\underbrace{2s-{3}/{2}}_{8},\underbrace{2s-1}_{28},\cdots,\underbrace{2s}_{35^{+}+35^{-}},\cdots,\underbrace{2s+1}_{28},\underbrace{2s+{3}/{2}}_{8},\underbrace{2s+2}_{1}
\begin{array}{c}{\Bigl(a^{+}(P^{0}-P_{2})+a^{-}\left(iP_{1}-P_{3}\right)\Bigr)|\psi\rangle=0,}\\{\Bigl(a^{-}(P^{0}+P_{2})-a^{+}\left(iP_{1}+P_{3}\right)\Bigr)|\psi\rangle=0.}\\\end{array}
\frac{4\pi}{g_{SU(N+M)}^{2}}=\frac{\theta_{\mathrm{B}}}{2\pig_{\mathrm{s}}}\,\quad\frac{4\pi}{g_{SU(N)}^{2}}=\frac{2\pi-\theta_{\mathrm{B}}}{2\pig_{\mathrm{s}}}\.
\Phi_{\left(k\right)}\left(\tau,\theta^{+},\theta^{-}\right)=q_{0\left(k\right)}\left(\tau\right)+\theta^{+}\lambda_{-}W^{-k}+\theta^{-}\lambda_{+}W^{k}+\theta^{+}\theta^{-}\left(-W+\nu\frac{W^{\prime}}{2W}\right).
S=\frac{1}{4g_{YM}^{2}}\mathrm{Tr}\left(V\sum_{\alpha\beta}(F_{\alpha\beta}+\phi_{\alpha\beta}\cdot{\bf1})g^{\alpha\mu}g^{\beta\nu}(F_{\mu\nu}+\phi_{\mu\nu}\cdot{\bf1})\right)\,.
\theta_{m,k}(q)~\theta_{m^{\prime},k^{\prime}}(q)=\sum_{\ell=1}^{k+k^{\prime}}\theta_{mk^{\prime}-m^{\prime}k+2\ellkk^{\prime},kk^{\prime}(k+k^{\prime})}(q)~\theta_{m+m^{\prime}+2\ellk,k+k^{\prime}}(q).
\Theta\left[\begin{array}{c}{r/k}\\{0}\\\end{array}\right]\Big(x\Big|-\frac{k}{\tau}\Big)=\frac{1}{k}\,\left(-ik\tau\right)^{1/2}\,e^{i\pi\taur^{2}/k}\,\sum_{r^{\prime}=0}^{k-1}\,e^{2i\pirr^{\prime}/k}\,\Theta\left[\begin{array}{c}{r^{\prime}/k}\\{0}\\\end{array}\right]\Big(-\taux\Big|k\tau\Big)\\\.
\begin{array}{ll}{\dot{A_{0}}=-\partial_{i}A^{i},}&{\dot{A}^{i}=\pi_{i}+\partial^{i}A^{0},}\\{\dot{\pi}_{0}=\partial^{i}\pi_{i}+m^{2}A^{0},}&{\dot{\pi}_{i}=\partial_{j}F^{ij}-m^{2}A^{i},}\\\end{array}
M_{r}^{i-}\;=\;\Gamma_{r}^{i}(\tau)_{r}+\widetilde\Gamma_{r}^{i}(\tau)+{\frac{iP_{r}^{i}}{\sqrt\alpha}}{\frac{\!\partial}{\partial\alpha_{r}}}+{\frac{iP_{r}^{i}}{\sqrt\alpha}}\left(Q_{r}(\tau)+\widetildeQ_{r}(\tau)\right)\,
W=u_{1}^{\prime}(x)u_{2}(x)-u_{1}(x)u_{2}^{\prime}(x)=\frac{2}{\lambda}\frac{\Gamma(1+2\alpha)\Gamma(1+2\beta)}{\Gamma(\alpha+\beta\mpB\lambda^{2})\Gamma(\alpha+\beta+1\pmB\lambda^{2})}
+(1-t_{1}^{2}){\frac{\partial}{\partialt_{1}}}\Big(\lambda((g^{\dagger})^{-1}\eta_{2}^{\dagger}-g\eta_{2}g^{-1})\Big)-{t_{2}}\Big((g^{\dagger})^{-1}\eta_{2}^{\dagger}g^{\dagger}-g\eta_{2}g^{-1})\Big)g
\frac{1}{2m}-\frac{1}{32\pim}\frac{g}{m^{2}}\frac{\Lambda}{m}+\frac{1}{128\pim}\frac{g}{m^{2}}\frac{\Lambda^{3}}{m^{3}}+\frac{1}{m}O(\frac{g^{2}}{m^{4}}\frac{\Lambda^{5}}{m^{5}})
\begin{array}{cc}{\frac{\Sigma_{3}}{2}D^{\pm}\Psi_{ls}=}&{\mp\frac{1}{2}D^{\pm}\Psi_{ls}}\\{L_{3}D^{\pm}\Psi_{ls}=}&{\left(\begin{array}{c}{l+1}\\{l}\\\end{array}\right)D^{\pm}\Psi_{ls}\.}\\\end{array}
V_{\mathrm{Z}}(r)=\left(1-{\frac{2M}{r}}\right)\left\{\frac{2\lambda^{2}(\lambda+1)r^{3}+6\lambda^{2}Mr^{2}+18\lambdaM^{2}r+18M^{3}}{r^{3}(\lambdar+3M)^{2}}\right\}
\left[\hat{q},\frac{d\hat{q}}{dt}\right]=\left[\hat{r}+\hat{\lambda},\frac{\,\hat{p}+2\alpha^{3}}{3\alpha^{2}-k}\right]=\fraci{3\alpha^{2}-k}\left[1+\frac{6\alpha}{\left(3\alpha^{2}-k\right)^{2}}\left(\hat{p}+\alpha\left(5\alpha^{2}-k\right)\right)\right]\neqi\;.
\chi^{\mathrm{disc}}(m_{v})=\frac{1}{N}\left\langle\sum_{k,l=1}^{N}\frac{1}{(i\lambda_{k}+m_{v})(i\lambda_{l}+m_{v})}\right\rangle-\frac{1}{N}\left\langle\sum_{k=1}^{N}\frac{1}{i\lambda_{k}+m_{v}}\right\rangle^{2}\:.
\mathrm{S=\int\left[-\frac{3}{\kappa^{2}}\frac{R\,{\dot{R}}^{2}}{N}+\frac{R^{3}}{2N}{\dot{\varphi}}^{2}+\frac{3}{\kappa^{2}}kNR-NR^{3}V(\varphi)\right]\,dt\,,}
P_{\alphaV}=(H_{0},\cdots,H_{3},{\bfQ}_{1}^{R},{\bfQ}_{2}^{R},{\bfQ}_{3}^{R}\vert{\bfQ}_{1}^{L},{\bfQ}_{2}^{L},{\bfQ}_{3}^{L},Q_{4},\cdots,Q_{19})
\nabla^{\mu}\partial_{\mu}\Phi^{a}=-{\frac{\Phi^{b}(\nabla^{\mu}\partial_{\mu}\Phi^{b})}{\eta^{2}}}\Phi^{a}={\frac{(\partial^{\mu}\Phi^{b})(\partial_{\mu}\Phi^{b})}{\eta^{2}}}\Phi^{a}\,.
{\mathrm{}^{*}\mathcal{L}}_{eff}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\beta^{2}\sqrt{1+\frac{1}{6m^{2}\beta^{2}}H_{\mu\nu\rho}H^{\mu\nu\rho}}-\frac{\chi}{2\theta}\varepsilon^{\mu\nu\rho\lambda}B_{\nu\rho}\partial_{\mu}A_{\lambda}\;.
A_{parent}^{\mathrm{V}}=-\frac{T_{p}}{2}\intd^{p+1}{\xi}\left[\sqrt{-G}\left(-{\Phi}^{(p+3)/2}G_{ij}h^{ij}+(p+3){\Phi}^{(p+1)/2}\right)+{\Lambda}^{ij}\left(G_{ij}-g_{ij}\right)\right],
S_{CS}=\frac{i}{32\pi^{2}}\int_{0}^{1}\!\!dt\int\!\!\!d^{4}\!x\tr\left\{{\calA}_{0}G_{\mu\nu}({\calA})G_{\rho\sigma}({\calA})\epsilon^{\mu\nu\rho\sigma}+\dot{{\calA}}_{\mu}P^{\mu}({\calA})-\partial_{\mu}\left({\calA}_{0}P^{\mu}({\calA})\right)\right\}
\tilde{A}=e^{\lambda_{0}+\tilde{\lambda}_{0}}\int_{{\calM}(\tau)}d^{2}\omega\prod_{j,i=1}^{N}e^{2\pi(\alpha_{i}-1)G(\omega-\omega_{i}|\tau)}e^{2\pi(\tilde{\alpha}_{i}-1)G(\omega-\tilde{\omega_{i}}|\tau)}
{\calL}_{Coul}=\frac{ig}{2}[(\partial\partial^{*}A_{-}^{a})(\frac{1}{\partial_{+}}A_{+})T^{a}A_{+}+(\partial\partial^{*}A_{+}^{a})(\frac{1}{\partial_{-}}A_{-})T^{a}A_{-}]\;\;\;.
L_{gf}=\biggl(\frac{1}{2}F^{\mu}\omega_{\mu\nu}F^{\nu}+\frac{1}{2}f^{\mu}\zeta_{\mu\nu}f^{\nu}+\frac{1}{2}\rho^{\mu}\varsigma_{\mu\nu}\rho^{\nu}+\frac{1}{2}f^{\mu\nu}\varrho_{\mu\nu\alpha\beta}f^{\alpha\beta}\biggr)e
\begin{array}{l}{Y^{(1/2)}=-2D_{2}\Phi;}\\{Y^{(n+1/2)}=\partial_{-}Y^{(n-1/2)}-2i\partial_{-}\Phi\cdot\sum_{l=1}^{n-1}D_{2}Y^{(l-1/2)}\cdotY^{(n-l-1/2)};}\\{n=1,2,3,\cdots}\\\end{array}
F\rightarrowF+\frac{\Gamma(-1/2)}{(4\pi)^{3/2}R}[\int_{B^{3}}(\frac{\lambda}{2}\hat{\phi}^{2})^{3/2}dv+\frac{2}{3R}\int_{S^{2}}(\frac{\lambda}{2}\hat{\phi}^{2})^{1/2}ds]
\zeta(s,r)=\frac{1}{2\pi\alphar^{3-2s}}\frac{\Gamma(s-1)}{\Gamma(s)}\left[G_{2\alpha}(s-1/2)-G_{2\pi}(s-1/2)\right]=\frac{1}{2\sqrt\pi\alphar^{3-2s}}\frac{I_{2\alpha}(s-1/2)}{\Gamma(s)}{,}
\nu_{\omega}=\left(\begin{matrix}{0.131\omega^{3}+1.661\omega^{2}-0.804\omega+4.655}\\{0.464\omega^{3}+0.687\omega^{2}-0.532\omega+2.276}\\{-0.249\omega^{3}-0.127\omega^{2}+0.298\omega-1.189}\\{1}\\\end{matrix}\right).
S=\intd^{4}x\sqrt{|\tilde{g}|}\left[\tilde{R}-\frac{1}{2}\left(\tilde{\nabla}\varphi\right)^{2}-\frac{1}{2}\left(\tilde{\nabla}\beta\right)^{2}-\frac{1}{12}e^{-2\varphi}\tilde{H}_{\mu\nu\lambda}\tilde{H}^{\mu\nu\lambda}-\frac{1}{48}e^{\sqrt{3}\beta-2\varphi}\tilde{F}_{\mu\nu\lambda\kappa}\tilde{F}^{\mu\nu\lambda\kappa}\right]
\nu_{2}\nu_{3}dJ_{\nu_{1}~\nu_{2}+1~\nu_{3}+1}^{(d+2)}+2\nu_{1}m_{1}^{2}~J_{\nu_{1}+1~\nu_{2}\nu_{3}}^{(d)}-(d-2\nu_{1})J_{\nu_{1}\nu_{2}\nu_{3}}^{(d)}\equiv0.
\begin{array}{lll}{\left[a^{-},a^{+}\right]=\Pi+\nuR,\qquad\qquad}&{\{R,a^{\pm}\}=0,\qquad\qquad}&{R^{2}=\Pi,}\\{\left[\bar{a}^{-},\bar{a}^{+}\right]=\bar{\Pi}+\nu\bar{R},}&{\{\bar{R},\bar{a}^{\pm}\}=0,}&{\bar{R}^{2}=\bar{\Pi}.}\\\end{array}
\sigma^{\mathrm{reg}}(\xi)=-\,{\frac{|\xi|}{\pi}}\sum_{n=1}^{\infty}{\frac{K_{1}(2n\pi\,|\xi|)}{n}}+{\frac{1}{2\sqrt{\pi}}}\int_{0}^{|\xi|}x\,dx\int_{0}^{\infty}t^{-1/2}\Big(\sqrt{{\frac{\pi}{t}}}-1\Big)e^{-x^{2}t}\,dt\,