harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Gamma_{2}(\phi,G)=\frac{-1}{4!N}\intd^{D}x\left(\lambda_{0}+\frac{\eta_{0}\phi^{2}}{10N}\right)[G_{aa}(x,x)]^{2}-\frac{\eta_{0}}{6!N^{2}}\intd^{D}x[G_{aa}(x,x)]^{3}.
W_{l}\starW_{l^{\prime}}=\sum_{{m=|l-l^{\prime}|+1\atopl+l^{\prime}-m-1\,\equiv\,0\,\mathrm{mod}\,2}}^{l+l^{\prime}-1}{\calS}(l,l^{\prime},m)W_{m}=\sum_{{m=|l-l^{\prime}|+1\atopl+l^{\prime}-m-1\,\equiv\,0\,\mathrm{mod}\,2}}^{l+l^{\prime}-1}{\binom{\frac{1}{2}(l+l^{\prime}-|l-l^{\prime}|-2)}{\frac{1}{2}(m-|l-l^{\prime}|-1)}}W_{m}\,,
\begin{array}{llllllccl}{}&{p}&{s}&{t}&{u}&{v}&{x}&{y}&{w}\\{\lambda\}&{0}&{1}&{1}&{n}&{0}&{2(n+2)}&{3(n+2)}&{0}\\{\mu\}&{0}&{0}&{0}&{1}&{1}&{4}&{6}&{0}\\{\nu\}&{0}&{0}&{0}&{0}&{0}&{2}&{3}&{1}\\{\sigma\}&{1}&{1}&{0}&{q}&{0}&{2(q+2)}&{3(q+2)}&{0}\\\end{array}
\deltaR_{MN}(h)+{\frac{1}{6}}\bar{F}_{\theta\phi}\bar{F}^{\theta\phi}(h_{MN}+2\bar{g}_{MN}{\calF})-(\bar{F}_{M\theta}\bar{F}_{N}^{~\theta}+\bar{F}_{M\phi}\bar{F}_{N}^{~\phi}){\calF}=0,
{\frac{d^{2}V}{dw^{2}}}-\left({\frac{dV}{dw}}\right)^{2}-\left(w+(p-2){\frac{1}{w}}\right){\frac{dV}{dw}}-{\frac{p(p-2)}{4w^{2}}}-{\frac{p+4-2d}{2}}=0.
M_{a}=2i\varrho\frac{\zeta_{a}}{\chi}=-i\varrho\left(\frac{1+z\bar{z}}{z-\bar{z}}\;,\;\frac{1-z\bar{z}}{z-\bar{z}}\;,\;\frac{z+\bar{z}}{z-\bar{z}}\right)\,.
\frac{\partialS_{2}}{\partial\tau^{A}}\delta_{1}\tau^{A}=\left[-\mathscr{F}_{tA}\mathscr{J}^{A}{}_{t}t^{2}-(\mathscr{F}_{tA}\mathscr{J}^{A}{}_{B}+\mathscr{F}_{AB}\mathscr{J}^{A}{}_{t})t\tau^{B}-\mathscr{F}_{AB}\mathscr{J}^{B}{}_{C}\tau^{A}\tau^{C}\right]\delta\lambda
{}^{\star}\left({}^{\star}A_{\mu}\right)=\frac{1}{4\,g^{\prime\,2}}\,\left[\partial_{\mu}\left(\partial_{\nu}A^{\nu}\right)+2\,g^{\prime}\,\partial^{\nu}\left(\frac{1}{2\,g^{\prime}}\right)\,F_{\mu\nu}-\Box\,A_{\mu}\right]\,.
\frac{1}{\beta}\int_{0}^{\beta}d\tau_{1}\cdots\int_{0}^{\beta}d\tau_{n}\,\langle\eta^{k_{1}}(\tau_{1})\cdots\eta^{k_{n}}(\tau_{n})\rangle_{0,c}\propto\beta^{n-1+(k_{1}+\cdots+k_{n})/2}.
T_{D}^{(n)}({\bfr^{\prime\prime}},{\bfr^{\prime}};E)=\sigma^{n}\,\left[G_{D}^{(0)}({\bf0},{\bf0};E)\right]^{n-1}\,\delta^{(D)}({\bfr^{\prime\prime}})\,\delta^{(D)}({\bfr^{\prime}})\;,
\hat{\mathrm{H}}_{\mathrm{phys}}=\mathrm{M}\mathrm{C}^{\dagger}\mathrm{C}+\sum_{p>0}\frac{1}{p}\mathrm{E}_{p}(\overline{{\rho}}_{\mathrm{tot,+}}(-p)\overline{{\rho}}_{\mathrm{tot,+}}(p)+\overline{{\rho}}_{\mathrm{tot,-}}(p)\overline{{\rho}}_{\mathrm{tot,-}}(-p)).
+\frac{1}{2e^{2}\mid\varphi\mid^{2}}\{(\partial_{\nu}\Lambda^{\nu\mu})+\frac{e}{2}(\varphi^{\dagger}\partial_{\mu}\varphi-\partial_{\mu}\varphi^{\dagger}\varphi)\}^{2}+\frac{1}{2}\mid\partial_{\mu}\varphi\mid^{2}.
\mathcal{L}^{\mathrm{(proj)}}=\frac{1}{\sqrt{1-v^{2}}}\,\left(\begin{array}{c|c}{1}&{\mathbf{v}^{T}}\\\hline{\mathbf{v}}&{\sqrt{1-v^{2}}\,{\bf1}\,+\,\frac{\mathbf{v}\mathbf{v}^{T}}{v^{2}}\,\left(1-\sqrt{1-v^{2}}\,\right)\,}\\\end{array}\right)~,
{\calL}=\kappaB\dot{a}+i\psi^{\ast}\dot{\psi}-A_{0}(\kappaB^{\prime}+\rho)-\frac{1}{2m}|(\partial_{x}-ia\mpi\frac{1}{\kappa}B^{\prime})\psi|^{2}-\frac{1}{2m}B^{2}\rho
\begin{array}{lcl}{\{p^{n},f\}_{\theta}}&{=}&{\sum_{s=0}^{n}\theta^{s-1}c_{n}^{s}\{\frac{1-(-)^{s}}{2}\}f^{s}p^{n-s},}\\{\{p^{-n},f\}_{\theta}}&{=}&{\sum_{s=0}^{\infty}\theta^{s-1}c_{s+n-1}^{s}\{\frac{(-)^{s}-1}{2}\}f^{s}p^{-n-s},}\\\end{array}
\Omega=\mathrm{d}\!\left\{S(z,\bar{z},\varphi)\,\mathrm{d}\bar{z}-\bar{S}(z,\bar{z},\varphi)\,\mathrm{d}z\right\}=\mathrm{d}S\wedge\mathrm{d}\bar{z}-\mathrm{d}\bar{S}\wedge\mathrm{d}z\,.
F_{\mu\nu}^{pq}(x)=\int_{\tilde{T}^{4}}d^{4}z\int_{\tilde{T}^{4}}d^{4}z^{\prime}\,\psi^{p}{}^{\dagger}(z;x)\sigma_{\mu}~(D_{x}^{\dagger}D_{x})^{-1}(z,z^{\prime})~\sigma_{\nu}^{\dagger}\psi^{q}(z^{\prime};x)-[\mu\leftrightarrow\nu],
\sigma_{\mathrm{abs}}=\frac{A_{H}}{1-Re\Psi(\alpha,\beta)}\left(\left|\frac{\gamma-1}{\alpha}\right|^{2}\right)\left|\frac{\Gamma(\gamma)\Gamma(\gamma-\alpha-\beta)}{\Gamma(\gamma-\alpha)\Gamma(\gamma-\beta)}\right|^{-2}
\vec{\psi}=\vec{\psi}^{T}+\left(\frac{1}{2}\vec{\gamma}-\frac{1}{2\vec{k}^{2}}{\vec{k}}({\vec{k}}\cdot\vec{\gamma})\right)\theta+\left(\frac{3}{2\vec{k}^{2}}\vec{k}-\frac{1}{2\vec{k}^{2}}\vec{\gamma}({\vec{k}}\cdot\vec{\gamma})\right){\vec{k}}\cdot\vec{\psi}\,,
\Sigma_{n_{i}=0}^{\infty}\frac{H_{n_{i}}(\sqrt{\omega}x_{1i})H_{n_{i}}(\sqrt{\omega}x_{2i})(\frac{z}{2})^{n_{i}}}{n_{i}!}=\frac{1}{\sqrt{1-z^{2}}}e^{\omega\left(\frac{2x_{1i}x_{2i}z-x_{1i}^{2}z^{2}-x_{2i}^{2}z^{2}}{1-z^{2}}\right)}
\int{\calD}z\mu[z]{\calP}_{\mu_{1}\nu_{1}}[z]\cdots{\calP}_{\mu_{2n}\nu_{2n}}[z]=\frac{\left(a^{2}\right)^{2n}}{(2n-1)!!}\left[\hat{1}_{\mu_{1}\nu_{1},\mu_{2}\nu_{2}}\cdots\hat{1}_{\mu_{2n-1}\nu_{2n-1},\mu_{2n}\nu_{2n}}+{\,}\mathrm{permutations}{\,}\right].
K_{\nu}(z)=\sqrt{\frac{\pi}{2z}}e^{-z}\left(1+\sum_{k=1}^{\infty}\frac{1}{(2z)^{k}}\frac{\Gamma\left(\nu+k+\frac{1}{2}\right)}{k!\Gamma\left(\nu-k+\frac{1}{2}\right)}\right)\.
F_{a}^{\mu\nu}=\frac{1}{a}\phi_{a}F^{\mu\nu}=\frac{1}{a}\phi_{a}\,\left(\frac{1}{a^{2}g}\phi\cdot(\partial^{\mu}\phi\wedge\partial^{\nu}\phi)+\partial^{\mu}f^{\nu}-\partial^{\nu}f^{\mu}\right),
\Theta_{\alpha}^{A}\equiv\epsilon^{ABC}\,\left(\,\frac{3}{16}\,\tau_{\alpha\mu\nu\rho}\,\eta^{B}\,D_{\mu}\,T_{\nu\rho}^{C}+\frac{9}{2}\,\tau_{\mu}\,\eta^{B}\,T_{\alpha\mu}^{C}\right)
Z(t)={\frac{1}{(2\pi\hbar)^{2}}}\left({\frac{2\pi}{t}}\right)^{3/2}\sqrt{{\frac{2}{tv^{4}}}}\int_{0}^{\infty}dse^{-s}s^{-1/2}\left(1+{\frac{s}{2z}}\right)^{-1/2},
\zeta_{\mu\nu}^{(1)}\zeta^{(2)\mu\nu}k_{2}^{2}{\frac{\Gamma(k_{1}.k_{2})\Gamma(k_{2}^{2})}{\Gamma(k_{1}.k_{2}+k_{2}^{2})}}-\zeta_{\nu\mu}^{(1)}\zeta^{(2)\mu\nu}k_{2}^{2}{\frac{\Gamma(k_{1}.k_{2}+1)\Gamma(k_{2}^{2})}{\Gamma(k_{1}.k_{2}+1+k_{2}^{2})}}.
U^{\phi}(\tau,\sigma)=\frac{2E}{m(Mm^{2}+E^{2})\;\tau^{4}}\sum_{n}\sum_{p}(A_{p\parallel}+\tilde{A}_{p\parallel})(A_{n-p\perp}+\tilde{A}_{n-p\perp})e^{-in\sigma}+{\calO}(\frac{1}{\tau^{3}}),
W_{f}=h^{4}\sum_{\bfm,m^{\prime}}\psi_{\overline{{\bfm}}}^{\dagger}\left({\frac{2}{h}}\bbox{\Gamma\cdot{\calD}}+i\mu\right)_{\bfm,m^{\prime}}\psi_{\overline{{\bfm}}^{\prime}}^{\vphantom\dagger},\quad\bbox{\Gamma}=(\gamma^{0}\gamma_{k},\gamma^{0}).
b(y)^{2}=\frac{16}{3}\frac{(1-y)^{4}}{(1-y/2)^{2}(1-3y/4)^{2}}\left(\frac{\Gamma(1+y/2)}{\Gamma(1-y/2)}\right)^{4}\left(\frac{\Gamma(1-y/4)}{\Gamma(1+y/4)}\right)^{3}\left(\frac{\Gamma(1-y)}{\Gamma(1+y)}\right)^{2}\left(\frac{\Gamma(1+3y/4)}{\Gamma(1-3y/4)}\right)=\frac{16}{3}+O(y).\nonumber
\frac{\partial^{2}\tilde{\phi}}{\partial\tilde{t}^{2}}-\frac{\partial^{2}\tilde{\phi}}{\partial\tilde{x}^{2}}+\tilde{\eta}_{a}\frac{\partial\tilde{\phi}}{\partial\tilde{t}}=-\tilde{\phi}+\tilde{\alpha}\tilde{\phi}^{2}-\tilde{\lambda}\tilde{\phi}^{3}+\tilde{\xi}_{a}.
\eta(1-k\eta)\frac{d^{2}A(\eta)}{d\eta^{2}}+\left(\frac52-3k\eta\right)\frac{dA(\eta)}{d\eta}-A(\eta)(k-a\ddot{a}+\dot{a}^{2})=-\omega_{0}a^{3}(k-a\ddot{a}+\dot{a}^{2}).
g^{-1}=\left(\begin{array}{cccc}{0}&{0}&{\delta(x-y)}&{0}\\{0}&{0}&{0}&{\frac{1}{\partial_{1}^{2}}}\\{-\delta(x-y)}&{0}&{0}&{0}\\{0}&{-\frac{1}{\partial_{1}^{2}}}&{0}&{0}\\\end{array}\right).
\frac{1}{\sqrt{N^{q}}}\mathrm{tr}[\Gamma_{\alpha_{1}}^{\dagger}(k_{1}^{+})\cdots\Gamma_{\alpha_{q}}^{\dagger}(k_{q}^{+})]\cdot\frac{1}{\sqrt{N^{s}}}\mathrm{tr}[\Gamma_{\beta_{1}}^{\dagger}({\tilde{k}}_{1}^{+})\cdots\Gamma_{\beta_{s}}^{\dagger}({\tilde{k}}_{s}^{+})]\vert0\rangle,
\begin{array}{l}{P^{+}(z)\times\,:X^{-}(w)\,P^{+}(w)X^{-}(w):}\\{\quad=\,:P^{+}(z)\,X^{-}(w)\,P^{+}(w)X^{-}(w):}\\{\quad\quad+2<P^{+}(z)\,X^{-}(w)>\,:X^{-}(w)\,P^{+}(w):}\\{\quad\rightarrow\frac{-2\,i}{z-w}(:X^{-}(w)\,P^{+}(w):)+\cdots}\\\end{array}
\begin{array}{rl}{}&{1234=1234(2)=1342=1342(1)=3421=3421(4)=3214=3214(3)=2143}\\{}&{=2143(1)=2431=2431(2)=4312=4312(3)=4123=4123(4)=1234=\cdot\cdot\cdot}\\\end{array}
\begin{array}{rcl}{}&{}&{e_{i}^{2}=\left[2\right]e_{i}~,}\\{}&{}&{e_{i}e_{i+1}e_{i}=e_{i}~,}\\{}&{}&{e_{i}e_{j}=e_{j}e_{i}~,~~~~\mathrm{if}~|i-j|\geq2~.}\\\end{array}
{\lambda_{(0)\alpha}^{A}\equiv\delta\lambda_{\alpha}^{A}={\frac{1}{2}}F_{(0)\mu\nu}^{-}\sigma_{\alpha}^{\mu\nu\,\beta}{\frac{1}{\sqrt{\rho_{0}}}}\left(\rho_{0}\eta_{\beta}^{A}+(x-x_{0})_{\mu}\sigma_{\beta\dot{\beta}}^{\mu}\bar{\xi}^{\dot{\beta}A}\right)\,.}
\overline{{R}}^{\primea}{}_{b}{}^{\mu\nu}=\overline{{X}}^{\prime\mu}\overline{{C}}^{a}{}_{b}{}^{\nu}-\overline{{X}}^{\prime\nu}\overline{{C}}^{a}{}_{b}{}^{\mu}-\overline{{C}}^{a}{}_{e}{}^{\mu}\overline{{C}}^{e}{}_{b}{}^{\nu}+\overline{{C}}^{a}{}_{e}{}^{\nu}\overline{{C}}^{e}{}_{b}{}^{\mu}.
S_{\mathrm{WZ}}^{NM10}=\mu_{9}\int_{R^{9+1}}\mathrm{Tr}\left(\left[l_{p}^{4}i_{\hat{k}}{\hat{\tilde{C}}}\wedge{\hat{\calF}}\wedge{\hat{\calF}}+l_{p}^{6}{\hat{C}}_{{\hat{\mu}}{\hat{\nu}}{\hat{\rho}}}D{\hat{X}}^{\hat{\mu}}D{\hat{X}}^{\hat{\nu}}D{\hat{X}}^{\hat{\rho}}\wedge{\hat{\calF}}\wedge{\hat{\calF}}\wedge{\hat{\calF}}+\dots\right]\wedge{\calD}{\hat{T}}\right)\,,
f=1+\frac{\tilde{R}^{4}}{r^{4}\triangle},\\\triangle=\tilde{\triangle}=1+\frac{l^{2}}{r^{2}},\\h=1-\frac{2m}{r^{4}\triangle}=\tilde{h}=\frac{1}{\triangle}\left(1+\frac{l^{2}}{r^{2}}-\frac{2m}{r^{4}}\right),
Z(u)={\frac{\left(-1\pmi{e^{\eta}}+{e^{2\eta}}\right)\left({e^{\eta}}\mpi{e^{u}}\right)\left(-1+{e^{u}}\right)\left(1+{e^{u}}\right){z_{1}}}{2{e^{u}}\left(\pmi+{e^{3\eta+u}}\right)}},
\intd{\mathbfx}\int_{0}^{a}\frac{d\xi}{\xi}A_{\alpha}^{(\sigma)\mu}(x)A_{\alpha^{\prime}\mu}^{(\sigma^{\prime})*}(x)=-\frac{2\pi}{\omega}\delta_{\alpha\alpha^{\prime}}\delta_{\sigma\sigma^{\prime}},\quad\alpha=({\mathbfk},\omega).
L^{-2}g_{ind}^{(0)}=\left(\begin{matrix}{A^{2}b^{2}}&{A^{2}b^{2}x}&{0}&{0}\\{A^{2}b^{2}x}&{b^{2}(1-x^{2})+A^{2}b^{2}x^{2}}&{0}&{0}\\{0}&{0}&{\frac{b^{2}}{1-x^{2}}}&{0}\\{0}&{0}&{0}&{-1}\\\end{matrix}\right)
S=\int_{0}^{1}\left[-\frac{\dot{x}^{2}}{2e}-e\frac{m^{2}}{2}-g\dot{x}^{\alpha}A_{\alpha}+igeF_{\alpha\beta}\psi^{\alpha}\psi^{\beta}+i\left(\frac{\dot{x}_{\alpha}\psi^{\alpha}}{e}-m\psi^{5}\right)\chi-i\psi_{n}\dot{\psi}^{n}\right]d\,\tau\,,
\begin{aligned}{[m]ds^{2}}&{{}=2dx^{-}dx^{+}-4\lambda^{2}\sum_{i=1}^{8}(x^{i})^{2}(dx^{-})^{2}+\sum_{i=1}^{8}(dx^{i})^{2}}\\{F_{5}}&{{}=\lambdadx^{-}\wedge\left(dx^{1}\wedgedx^{2}\wedgedx^{3}\wedgedx^{4}+dx^{5}\wedgedx^{6}\wedgedx^{7}\wedgedx^{7}\right)~}\\\end{aligned}
0=2(n_{c}-\tilde{n}_{c})\sum_{k=1}^{n_{c}-\tilde{n}_{c}}\gamma_{k}-2\frac{1}{2}(n_{c}-\tilde{n}_{c})\sum_{k=1}^{n_{c}-\tilde{n}_{c}}\gamma_{k}-(n_{c}-\tilde{n}_{c})\sum_{i=1}^{n_{f}}m_{i},
S_{\Phi}=i\intd^{2}xd^{4}\theta_{+}du_{1}du_{2}\;{\frac{1}{u_{1}^{+}u_{2}^{+}}}\Phi_{+}^{+Y^{\prime}}(1)\partial_{++}\Phi_{+Y^{\prime}}^{+}(2)+{\frac{m^{2}}{2}}\intd^{2}xdud^{2}\theta_{+}^{+}\;\Phi_{+}^{+Y^{\prime}}\Phi_{+Y^{\prime}}^{+}\;.
X\Psi_{n_{1}n_{2}m}(\mu,\nu,\varphi;\delta_{1},\delta_{2})=-\alpha\frac{n_{1}-n_{2}+\frac{\delta_{1}-\delta_{2}}{2}}{n+\frac{\delta_{1}+\delta_{2}}{2}}\,\Psi_{n_{1}n_{2}m}(\mu,\nu,\varphi;\delta_{1},\delta_{2}),
C_{J}=\left(\frac{\partialE}{\partialT}\right)_{J}=12\pi^{11/4}\kappa^{-3/2}N^{1/2}m^{3/2}r_{+}^{-1}\frac{2r_{+}^{2}+l^{2}}{2r_{+}^{4}+5l^{2}r_{+}^{2}-l^{4}},
\bar{X}(B^{\dagger})-\bar{X}^{(0)}(B^{\dagger})=\sum_{k=1}^{\infty}\lambda^{k}\bar{X}^{(k)}(B^{\dagger})=\sum_{k=1}^{\infty}\lambda^{k}\sum_{w}X_{\bar{w}}^{(k)}B^{\dagger\;w}
\frac{1}{\sum_{n=0}^{\infty}C_{2n}p^{2n}}=\frac{1}{C_{0}}-\frac{C_{2}}{C_{0}^{2}}p^{2}+\left[\left(\frac{C_{2}}{C_{0}^{2}}\right)^{2}-\frac{C_{4}}{C_{0}^{2}}\right]p^{4}+\ldots.
\frac{1}{\sqrt{3!}}\varepsilon_{b_{1}b_{2}b_{3}}\overline{{\psi}}_{1d_{1}}(y_{1})\overline{{\psi}}_{2d_{2}}(y_{2})\overline{{\psi}}_{3d_{3}}(y_{3})U^{d_{1}b_{1}}(y_{1},y_{M})U^{d_{2}b_{2}}(y_{2},y_{M})U^{d_{3}b_{3}}(y_{3},y_{M})|0\rangle
\delta_{a}\lambda_{\alpha_{1}}^{*(b)}=\left(-\right)^{\epsilon_{\alpha_{1}}+1}\delta_{ab}\left(\pi_{d\alpha_{1}}^{*(d)}+\bar{\eta}_{d\alpha_{1}}^{(d)}-\tilde{\Phi}_{\alpha_{0}}Z_{\;\;\alpha_{1}}^{\alpha_{0}}\right).
S_{eff}^{(o)}=\intd^{4}x\sqrt{-g}\left({\frac{1}{2}}R+{\frac{|DS|^{2}}{(S+\overline{{S}})^{2}}}+3{\frac{|DT|^{2}}{(T+\overline{{T}})^{2}}}+{\frac{\alpha^{\prime}}{8}}(ReS){\cal{R}}_{GB}^{2}+{\frac{\alpha^{\prime}}{8}}(ImS){\cal{R}}{\tilde{\cal{R}}}\right)
\mathbb{D}=\left(\begin{array}{ccccc}{\partial}&{-2e^{2}q_{1}^{\dagger}}&{-2e^{2}q_{2}^{\dagger}}&{\cdots}&{-2e^{2}q_{N}^{\dagger}}\\{-q_{1}}&{\partial-(\xi-m_{1})}&{0}&{\cdots}&{0}\\{-q_{2}}&{0}&{\partial-(\xi-m_{2})}&{}&{}\\{\vdots}&{\vdots}&{}&{\ddots}&{}\\{-q_{N}}&{0}&{}&{}&{\partial-(\xi-m_{N})}\\\end{array}\right),
\partial^{\alpha\dot{\alpha}}{\calJ}_{\alpha\dot{\alpha}}^{f}=i\,D^{2}\left\{\frac12\,Q_{f}\frac{\partial{\calW}}{\partialQ_{f}}+\frac{T(R_{f})}{16\pi^{2}}\,\mathrm{Tr}\,W^{2}\right\}+\mathrm{H.c.}\,,
\ll1\gg_{\calQ}=G^{(0)}(q,g_{\calQ}(q))=\gamma_{(0)}(g_{\calQ}(q))\int_{\calM}d\rho\,{\calQ}(\rho,q)=\gamma_{(0)}(g_{\calQ}(q))<\prod_{i}{\calO}_{\gamma_{i}}>,
G_{E}^{+}\left(x,x^{\prime}\right)=\left\langleE\right|\phi\left(x\right)\phi\left(x^{\prime}\right)\left|E\right\rangle=\frac{\Gamma\left(h_{+}\right)\Gamma\left(h_{-}\right)}{16\pi^{2}}F\left(h_{+},h_{-};2;\frac{1+P}{2}\right),
\partial_{t}^{2}(\delta\tilde{T})-A^{2}{(r)}\partial_{r}^{2}(\delta\tilde{T})-\frac{M}{\lambda}e^{-r}A{(r)}\partial_{r}(\delta\tilde{T})+\frac{1}{4\lambda}A{(r)}(\frac{\lambda^{2}-2}{\lambda}+Me^{-r})(\delta\tilde{T})=0
{\calW}=\sum_{I=1}^{N-R}\left(\sum_{b=1}^{N^{\prime}/2}q_{I}^{b}T^{I}H_{b}\tilde{H}_{b}+\frac{1}{2}\sum_{c=N^{\prime}+1}^{N^{\prime}+N^{\prime\prime}}q_{I}^{c}T^{I}H_{c}H_{c}\right)\;,
\langle\prod_{j=1}^{m}{\calO}_{A_{j}}(z_{j})\rangle=\oint_{C_{\infty}}\cdots\oint_{C_{\infty}}\frac{\prod_{j=1}^{m}f_{A_{j}}(X)}{\prod_{i=1}^{N}(\partial_{X_{i}}W_{q}^{Gr(N,N+M)}(X))}dX_{1}\cdotsdX_{N}
\begin{array}{rcl}{\mathsf{E}}&{=}&{\aleph\,+\,\aleph^{2}a^{2}g^{2}\,+\,6g^{2}N^{2}\,,}\\{}&{}&{}\\{\mathsf{N}}&{=}&{N\left(\aleph\,-\,\aleph^{2}a^{2}g^{2}\,+\,4g^{2}N^{2}\right)\;,}\\{}&{}&{}\\{\alpha}&{=}&{a^{2}\left(1+\aleph-\aleph^{2}\right)\,-\,N^{2}\left(\aleph-3\aleph^{2}a^{2}g^{2}+3g^{2}N^{2}\right)+P^{2}\;,}\\\end{array}
F_{\mathrm{M5}}=-\frac{2^{6}\pi^{3}}{3^{7}}N^{3}V_{5}T^{6}\left[1+\frac{9}{8\pi^{2}}\sum_{i=1}^{2}\omega_{i}^{2}+\frac{27}{128\pi^{4}}\left(\sum_{i=1}^{2}\omega_{i}^{2}\right)^{2}+\frac{81}{256\pi^{4}}\sum_{i=1}^{2}\omega_{i}^{4}+\ldots\right]
m\equiv\left(\partial_{\underline{{y}}}\Lambda\right)\Lambda^{-1}={\textstyle\frac{1}{2}}m^{i}T_{i}={\textstyle\frac{1}{2}}\left(\begin{array}{cc}{m^{1}}&{m^{2}+m^{3}}\\{}&{}\\{m^{2}-m^{3}}&{-m_{1}}\\\end{array}\right)\,.
\tilde{J}=\frac{\kappa}{2}\partial\tilde{g}\cdot\tilde{g}^{-1}\mapsto\frac{\kappa}{2}\partiale^{\displaystyle\gamma}\tilde{g}\cdot\tilde{g}^{-1}e^{\displaystyle-\gamma}\,[2mm]=\frac{\kappa}{2}e_{-}+W(\tilde{J}),
\int\!d\mu_{\tilde{C}}[\varphi]G_{\beta_{0}\alpha_{\pi(1)}}(x_{0},x_{\pi(1)};\varphi)G_{\beta_{\pi(1)}\alpha_{\pi(2)}}(x_{\pi(1)},x_{\pi(2)};\varphi).....G_{\beta_{\pi(n)}\alpha_{0}}(x_{\pi(n)},x_{0};\varphi)
\frac{\partial^{2}V_{ferm.}^{(1)}}{\partial\mu^{2}}=\frac{eH}{\pi^{2}}\beta^{2}\frac{\partial}{\beta^{2}}\sum_{l=1}^{\infty}(-1)^{l+1}\int_{0}^{\infty}\frac{ds}{s}exp(-m^{2}s-\beta^{2}l^{2}/4s)coth(eHs).
\epsilon\equiv\frac{M_{P}^{2}}{2}\left(\frac{V^{\prime}}{V}\right)^{2}\sim\frac{M_{P}^{2}}{f^{2}}\;,\quad\eta\equivM_{P}^{2}\frac{V^{\prime\prime}}{V}\sim\frac{M_{P}^{2}}{f^{2}}
C_{2}=\frac{kq^{2}}{2}+k^{3}q^{4}\left(\frac{b_{1}}{\epsilon}+b_{2}+b_{3}(\epsilon)\right)+k^{5}q^{6}\left(\frac{b_{4}}{\epsilon^{2}}+\frac{b_{5}}{\epsilon}+b_{6}+b_{7}(\epsilon)\right)+\cdots
Y=(y_{1}+iy_{2})/2=\sum_{n}\left[{\frac{1}{\sqrt{4p^{+}\omega_{n}}}}\,\alpha_{n}^{+}e^{-i\omega_{n}\tau+in\sigma/\alpha^{\prime}p^{+}}+{\frac{1}{\sqrt{4p^{+}\omega_{-n}}}}\left(\alpha_{n}^{-}\right)^{\dagger}e^{i\omega_{-n}\tau-in\sigma/\alpha^{\prime}p^{+}}\right]\.
\bar{u}_{1}u_{2}=\sum_{n=1}^{N}\left(\!\begin{array}{cc}{\bar{u}_{n1,1}u_{2,n1}}&{\bar{u}_{n1,1}u_{2,n2}}\\{\bar{u}_{n2,1}u_{2,n1}}&{\bar{u}_{n2,1}u_{2,n2}}\\\end{array}\!\right)\equiv\left(\!\begin{array}{cc}{U_{x}}&{U_{y}}\\{U_{z}}&{U_{t}}\\\end{array}\!\right),
\phi_{\calA}^{N_{1}\ldotsN_{4}\,\Lambda}:=\phi_{\alpha_{1}\ldots\alpha_{4}}^{N_{1}\ldotsN_{4}}\C_{\alpha_{1}\ldots\alpha_{4}}^{N_{1}\ldotsN_{4}\,\Lambda}\frac{1}{\sqrt{\mathrm{dim}_{N_{1}}\ldots\mathrm{dim}_{N_{4}}}},
\Phi^{(2)}\rightarrow-\delta_{\alpha\beta}(k^{2}\delta_{\mu\nu}-k_{\mu}k_{\nu})\delta_{\mu\nu}G_{F}^{2}(\tau_{1},\tau_{2})=-\delta_{\alpha\beta}(D-1)k^{2}G_{F}^{2}(\tau_{1},\tau_{2}).
\Big(\left(\left.{\calD}\right|_{\sigma}+\left.{\calD}\right|_{\sigma^{\prime}}\right){\bfG}^{\mu\nu}(\sigma,\sigma^{\prime})\Big)_{\sigma=\sigma^{\prime}}={\calD}\left(\frac{b_{0}^{0}}{\sqrt{s}}g^{\mu\nu}\right)=0
S_{BF}[A,\phi]=\intd^{2}x\,tr\left(\phi^{0}F^{0}+\phi^{a}F^{b}\right)=\intd^{2}x\,\left(-\phi^{0}F_{\alpha\beta}^{0}\epsilon^{\alpha\beta}+\Lambda\delta_{ab}\phi^{a}F_{\alpha\beta}^{b}\epsilon^{\alpha\beta}\right).
S_{\mathrm{topol}}=-{\frac{\gamma}{32\pi^{2}}}\intd^{4}x\sqrtg\,({\calR}_{\mu\nu\lambda\delta}{\calR}^{\mu\nu\lambda\delta}-4{\calR}_{\mu\nu}{\calR}^{\mu\nu}+{\calR}^{2})={\frac{3\gamma}{4\pi^{2}}}\intd^{4}x{\frac{R^{\prime\prime}(r)(1-R^{2}(r))}{R^{3}(r)}}\.
B^{(N_{f})}(\lambda,m)=\left|\begin{array}{cccc}{J_{0}(\lambda)}&{I_{0}(-m_{1})}&{\cdots}&{I_{0}(-m_{N_{f}})}\\{\lambdaJ_{1}(\lambda)}&{m_{1}I_{1}(-m_{1})}&{\cdots}&{m_{N_{f}}I_{1}(-m_{N_{f}})}\\{\vdots}&{\vdots}&{}&{\vdots}\\{\lambda^{N_{f}}J_{N_{f}}(\lambda)}&{m_{1}^{N_{f}}I_{N_{f}}(-m_{1})}&{\cdots}&{m_{N_{f}}^{N_{f}}I_{N_{f}}(-m_{N_{f}})}\\\end{array}\right|\.
\hat{\mathbf{J}}^{2}\,\Upsilon_{lm}=(\hat{\mathbf{L}}_{C}+\hat{\mathbf{S}}_{C})^{2}\,\Upsilon_{lm}=l(l+1)\,\Upsilon_{lm}\qquad\mathrm{and}\qquad\hat{J}_{z}\,\Upsilon_{lm}=m\,\Upsilon_{lm};
\intd\alpha\left|G_{\mathrm{B}}\Bigl(\alpha(u_{\mathrm{B}}-u_{\mathrm{T}})\Bigr)\right|^{2}=\frac{1}{(u_{\mathrm{B}}^{3}-u_{\mathrm{T}}^{3})}\intd\xi\left|G_{\mathrm{B}}\Bigl(\frac{\xi}{v},0,0,\xi\Bigr)\right|^{2}
\langle\phi_{k_{1}}\phi_{k_{2}}\cdots\phi_{k_{2n}}\rangle=\sum_{\mathrm{pairings}}\alpha(P)\langle\phi_{k_{P_{1}}}\phi_{k_{P_{2}}}\rangle\cdots\langle\phi_{k_{P_{2n-1}}}\phi_{k_{P_{2n}}}\rangle,
W_{\mu\nu}^{a}\rightarrowW_{\mu\nu}^{a}+\nabla_{\mu}^{ab}\Lambda_{\nu}^{b}-\nabla_{\nu}^{ab}\Lambda_{\mu}^{b},\enspace\nabla_{\mu}^{ab}=\delta^{ab}\partial_{\mu}+\frac{m}{e}\epsilon^{acb}P_{\mu}^{c}.
V_{\parallel}=\frac{1-2(\dot{f}-Hf)^{2}}{1-(\dot{f}-Hf)^{2}}\dot{H}+\frac{2}{f^{2}}\frac{(1+Hf(\dot{f}-Hf))^{2}}{1-(\dot{f}-Hf)^{2}}+2H^{2},
e=\frac{i}{2}\left\{d\tau+\frac{S}{2i}\left({\tt\frac{1}{z}}dz-{\tt\frac{1}{\bar{z}}}d\bar{z}\right)\right\}\left(\begin{array}{lr}{1}&{0}\\{0}&{-1}\\\end{array}\right)+\frac{i}{2}\left(\begin{array}{cc}{0}&{dz}\\{d\bar{z}}&{0}\\\end{array}\right)\,,
J\epsilon_{ab3}\epsilon_{ij}\int_{\tau_{0}}^{\tau_{0}-i\beta}dz^{0}\,\left[S_{c}(z^{0}-x^{0},\vec{p})\right]_{\rho\mu}\left(\gamma_{5}C\right)_{\rho\sigma}\left[S_{c}(z^{0}-y^{0},-\vec{p})\right]_{\sigma\nu}.
E_{2}^{\prime}(\rho)=\frac{1}{(c\rho+d)^{4}}E_{2}^{\prime}\left(\tilde{\rho}\right)-\frac{2c}{(c\rho+d)^{3}}E_{2}\left(\tilde{\rho}\right)+\frac{6}{\pii}\frac{c^{2}}{(c\rho+d)^{2}}
\pi\left(\begin{array}{c}{a}\\{b}\\{c}\\{d}\\\end{array}\mathrm{~}\right)r=\frac{r}{\sqrt{2}}\left(\begin{array}{c}{a+d}\\{b+c}\\{c-b}\\{d-a}\\\end{array}\right)=\left(\begin{array}{c}{\Psi_{A}}\\{\Psi_{B}}\\{\Psi_{C}}\\{\Psi_{D}}\\\end{array}\right)
A^{(k)}=\left(\begin{array}{cccc}{1}&{2}&{\cdots}&{k-1}\\{0}&{1}&{\cdots}&{k-2}\\{\vdots}&{\ddots}&{\ddots}&{\vdots}\\{0}&{\cdots}&{0}&{1}\\{0}&{\cdots}&{0}&{0}\\\end{array}\right).
m_{\mathbf{n},\mathbf{\delta}}=\alpha_{\mathbf{n}}-\alpha_{\mathbf{n}-\mathbf{\delta}}+\tilde{n}_{\tilde{\mathbf{\delta}}}(\eta_{\tilde{\mathbf{n}}+\tilde{\mathbf{\delta}}}-\eta_{\tilde{\mathbf{n}}}),
\begin{array}{c}{^{(1)}H_{\mu\nu}=2R_{;\mu\nu}-2g_{\mu\nu}\squareR-\frac{1}{2}g_{\mu\nu}R^{2}+2RR_{\mu\nu}}\\{^{(3)}H_{\mu\nu}=\frac{1}{12}R^{2}g_{\mu\nu}-R^{\rho\sigma}R_{\rho\mu\sigma\nu}}\\\end{array}
ds^{2}=M_{ij}d\vec{x}_{i}\cdotd\vec{x}_{j}+M_{ij}^{-1}\left(d\theta_{i}+\sum_{k}\vec{W}_{ik}\cdotd\vec{x}_{k}\right)\left(d\theta_{j}+\sum_{l}\vec{W}_{jl}\cdotd\vec{x}_{l}\right)
:e^{ip_{i}x^{i}\left(\tau\right)}::e^{iq_{i}x^{i}\left(0\right)}:\,=e^{-\frac{i}{2}\theta^{ij}p_{i}q_{j}\epsilon\left(\tau\right)}:e^{ip_{i}X^{i}\left(\tau\right)+iq_{i}X^{i}\left(0\right)}:,
\delta_{0}:f\rightarrow\left(\begin{array}{c}{-zf}\\{2\left(\frac{\partial}{\partial\bar{s}}+\frac{1}{2}\chi\right)f}\\\end{array}\right),\\\delta_{1}:\left(\begin{array}{c}{g_{0}}\\{g_{1}}\\\end{array}\right)\rightarrow\left(-2\left(\frac{\partial}{\partial\bar{s}}+\frac{1}{2}\chi\right)g_{0}-zg_{1}\right).
\begin{array}{cccc}{z:}&{\left(x^{2,3},x^{4,5},x^{6,7},x^{8,9}\right)}&{\to}&{\left(x^{2,3},x^{4,5},e^{-2\piiz}x^{6,7},e^{2\piiz}x^{8,9}\right)}\\{y:}&{\left(x^{2,3},x^{4,5},x^{6,7},x^{8,9}\right)}&{\to}&{\left(e^{-2\piiy}x^{2,3},e^{2\piiy}x^{4,5},x^{6,7},x^{8,9}\right)}\\\end{array}
\langle\bar{G}_{1}\bar{G}_{2}\bar{G}_{3}\bar{G}_{4}\rangle=\prod_{i}\langlee^{iW_{i}^{1}\cdotJ_{i}}(1)e^{iW_{i}^{2}\cdotJ_{i}}(2)e^{iW_{i}^{3}\cdotJ_{i}}(3)e^{iW_{i}^{4}\cdotJ_{i}}(4)\rangle=\prod_{k<l}({\bar{z}}_{kl})^{\sum_{i}W_{i}^{k}\cdotW_{i}^{l}}\prod_{i}C_{i}^{1234}
\dot{G}_{\pm}+\frac{i}{2}\phi^{\pm}D_{\pm}^{2}G_{\pm}+\frac{i}{4}D_{\pm}\phi^{\pm}D_{\pm}G_{\pm}+\frac{3}{4}iD_{\pm}^{2}\phi^{\pm}G_{\pm}=\frac{\kappa_{0}}{2}D_{\pm}^{5}\phi^{\pm}~.
L_{n}=\frac12\sum:\alpha_{n+p}^{\mu}\alpha_{-p}^{\nu}:\eta_{\mu\nu}\,\,\,\,\,\,\mathrm{and}\,\,\,\,\,\,\widetilde{L}_{n}=\frac12\sum:\widetilde{\alpha}_{n+p}^{\mu}\widetilde{\alpha}_{-p}^{\nu}:\eta_{\mu\nu}\,\,,
\biggl(\frac{{\dot{a}}_{4}}{a_{4}}\biggr)^{2}+\frac{k_{4}}{a_{4}^{2}}-\frac{7}{6}\frac{{\ddot{a}}_{7}}{a_{7}}+\frac{7}{2}\frac{{\dot{a}}_{7}{\dot{a}}_{4}}{a_{4}a_{7}}=\frac{1}{3}\lambda_{4}.
S_{(a)}[J^{\mu},\varphi,\theta,{\tilde{\alpha}}^{r},\beta_{r};{}^{4}g_{\mu\nu}]=\intd^{4}x\Big[|J|a({\frac{|J|}{\sqrt{{}^{4}g}}},\partial_{\mu}\thetaJ^{\mu})-J^{\mu}(\partial_{\mu}\varphi+\beta_{r}\partial_{\mu}{\tilde{\alpha}}^{r})\Big]
\frac{2r_{n}+t}{r_{n}+t}f^{(n,0)}(r_{1},\dots,r_{n-1},r_{n}+t)-\frac{2r_{n-1}+t}{r_{n-1}+t}f^{(n,0)}(r_{1},\dots,r_{n-1}+t,r_{n})=0.
\mu^{2}\frac{\partial}{\partial\mu^{2}}\Gamma_{m}(0|\phi,\phi_{a}^{*},\bar{\phi},\eta)=\bigr(\beta_{g}g\frac{\partial}{\partialg}+\gamma_{Q}\mathbf{\tilde{N}}_{Q}+\gamma_{B}\mathbf{\tilde{N}}_{B}\bigr)\Gamma_{m}(0|\phi,\phi_{a}^{*},\bar{\phi},\eta),
\delta^{New}B_{\mu_{1}\ldots\mu_{p_{a}}}^{a}=\partial_{[\mu_{1}}\Lambda_{\mu_{2}\ldots\mu_{p}]}^{a}+g_{b}p_{a}!p_{b}k_{b\mu_{1}\ldots\mu_{p_{a}}}^{a\nu_{2}\ldots\nu_{p_{b}}}\Lambda_{\nu_{2}\ldots\nu_{p_{b}}}^{b}.
{\calL}=-\sum_{j=1}^{N}\left(\partial^{\nu}\Psi_{j}^{\ast}\partial_{\nu}\Psi_{j}+m_{0}^{2}\Psi_{j}^{\ast}\Psi_{j}-g\Phi^{p}\left(\Psi_{j}^{\ast}\Psi_{j}\right)^{q}\right)-\frac{1}{2}\left(\partial^{\nu}\Phi\partial_{\nu}\Phi+\mu_{0}^{2}\Phi^{2}\right)
\hat{\bfV}_{1}^{\dagger}\hat{\bfV}_{1}=\hat{\bfV}_{2}^{\dagger}\hat{\bfV}_{2}=1,\quad\hat{\bfV}_{1}^{n}=\hat{\bfV}_{2}^{n}=1,\quad\hat{\bfV}_{1}\hat{\bfV}_{2}=\hat{\bfV}_{2}\hat{\bfV}_{1}e^{2\pii\theta_{n}},
W_{z}\left(\begin{array}{cc}{a+N\bar{\epsilon}_{j}}&{b+N\bar{\epsilon}_{j}}\\{d+N\bar{\epsilon}_{j}}&{c+N\bar{\epsilon}_{j}}\\\end{array}\right)=W_{z}\left(\begin{array}{cc}{a}&{b}\\{d}&{c}\\\end{array}\right).
\psi_{L}\left(z,\overline{{z}}\right)\rightarrow\left(\frac{\partial\xi}{\partialz}\right)^{\frac12}\psi_{L}\left(\eta,\theta\right);\\psi_{R}\left(z,\overline{{z}}\right)\rightarrow\left(\frac{\partial\overline{{\xi}}}{\partial\overline{{z}}}\right)^{\frac12}\psi_{R}\left(\eta,\theta\right),
P_{Y_{(2r,2)}}^{j_{1}j_{2}...j_{2r}j_{2r+1}j_{2r+2},i_{1}i_{2}...i_{2r}i_{2r+1}i_{2r+2}}P_{Y_{(4)}}^{i_{1}i_{2}i_{2r+1}i_{2r+2},j_{1}j_{2}j_{2r+1}j_{2r+2}}\prod_{a=2}^{r}P_{Y_{(2)}}^{i_{2a-1}i_{2a},j_{2a-1}j_{2a}}=0
\langlex_{0N}^{\prime}|x_{0N}\rangle=\prod_{\mu=0}^{p}\delta\left(x_{0}^{\prime\mu}-x_{0}^{\mu}\right)~,\quad\langlex_{0D}^{\prime}|x_{0D}\rangle=\prod_{i=p+1}^{D-1}\delta\left(x_{0}^{\primei}-x_{0}^{i}\right)~.
\pi_{i}^{(3)a}=-\,\frac{1}{3m^{2}}\,\eta^{1b}\,G_{1i}^{(2)ba}+\frac{g}{6m^{2}}\,\bar{\eta}^{bc}\,(D_{j}\pi^{j})^{b}\,G_{1i}^{(2)ca}-\frac{1}{3}\,\eta^{2b}\,G_{2i}^{(2)ba}\,,
L=i\left(w_{\alpha}\nabla_{-}\varphi^{\alpha}-\tilde{w}_{\overline{{\alpha}}}\nabla_{-}\tilde{\varphi}^{\overline{{\alpha}}}\right)(\varphi,\tilde{\varphi})-qA_{-}^{a}{W}_{a}(\varphi,\tilde{\varphi})
\begin{array}{ll}{A:\qquadu=u(\sigma^{+}),}&{\quadv=v(\sigma^{-})}\\{B:\qquadu=\bar{u}(\sigma^{-}),}&{\quadv=\bar{v}(\sigma_{\}^{+})}\\{C:\qquadu=c_{1},}&{\quadv=v(a(\sigma^{-}),b(\sigma^{+}))}\\{D:\qquadu=u(\bar{a}(\sigma^{+}),\bar{b}(\sigma^{-})),}&{\quadv=c_{2}\,\,\,,}\\\end{array}