harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
H^{1}({\mathbbP}^{1}\timesT^{2},{\calO}_{{\mathbbP}^{1}\timesT^{2}}(3\sigma|_{{\mathbbP}^{1}\timesT^{2}}-5F))|_{b_{-5}^{(1)}}\stackrel{m_{1}}{\rightarrow}H^{1}({\mathbbP}^{1}\timesT^{2},{\calO}_{{\mathbbP}^{1}\timesT^{2}}(6\sigma|_{{\mathbbP}^{1}\timesT^{2}}-3F))|_{c_{-3}^{(1)}}.
\begin{array}{lcl}{1\starf(x,p)}&{=}&{f}\\{p\starf(x,p)}&{=}&{fp+\thetaf^{\prime}}\\{p^{2}\starf(x,p)}&{=}&{fp^{2}+2\thetaf^{\prime}p+{\theta}^{2}f^{\prime\prime}}\\{p^{3}\starf(x,p)}&{=}&{fp^{3}+3\thetaf^{\prime}p^{2}+3\theta^{2}f^{\prime\prime}p+{\theta}^{3}f^{\prime\prime\prime}}\\\end{array}
\partial_{p}F_{j}(z|s)={\frac{i}{2}}zF_{j}(z|s)(\delta_{j,3}+\delta_{j,4})-\sum_{l=1}^{4}F_{l}*a_{lj}\bigg|_{z,s}-{\frac{1}{z}}\sum_{l=1}^{4}F_{l}*b_{lj}\bigg|_{z,s}\.
H_{N}(q,J)=\frac{J}{4}\left\{\sum_{j=1}^{N-1}\sigma_{j}^{x}\sigma_{j+1}^{x}+\sigma_{j}^{y}\sigma_{j+1}^{y}+\frac{q+q^{-1}}{2}\sigma_{j}^{z}\sigma_{j+1}^{z}-\frac{q-q^{-1}}{2}(\sigma_{1}^{z}-\sigma_{N}^{z})\right\}
\psi_{1}^{+}=\frac{i}{\sqrt{2R(R-r_{5})}}\left[\begin{array}{c}{r_{3}+ir_{4}}\\{-r_{1}+ir_{2}}\\{R-r_{5}}\\{0}\\\end{array}\right],\psi_{2}^{+}=\frac{i}{\sqrt{2R(R-r_{5})}}\left[\begin{array}{c}{r_{1}+ir_{2}}\\{r_{3}-ir_{4}}\\{0}\\{R-r_{5}}\\\end{array}\right],
{\calG}_{\mathrm{rel}}\simeq\frac{g^{2}}{16\pi}\sum_{A}\lambda_{A}\left(\frac{1}{r_{A}}d{\bfr}_{A}^{2}+r_{A}\,(d\psi_{A}+{\bfw}({\bfr}_{A})\cdotd{\bfr}_{A})^{2}\right),
A_{\mu}^{\alpha}(x)-A_{\mu}^{\alpha}(x)\equiv\deltaA_{\mu}^{\alpha}(x)=iD_{\mu}^{\alpha\beta}c^{\beta}(x)\intd^{4}z\overline{{c}}(z)\frac{\partialF^{\gamma}[A(z);\theta]}{\partial\theta}\mid_{_{\theta}}\delta\theta
\frac{1}{\Box_{1}\Box_{2}}F_{mn}(\Box_{1},\Box_{2})X_{1}X_{2}=-\frac{1}{2}\int_{-\infty}^{0}dq\,q^{m+n}\biggl(\frac{d^{m}}{dq^{m}}\int_{-\infty}^{q}\frac{d{\bar{q}}}{{\bar{q}}}{\calH}_{\bar{q}}X_{1}\biggr)\biggl(\frac{d^{n}}{dq^{n}}\int_{-\infty}^{q}\frac{d{\bar{q}}}{{\bar{q}}}{\calH}_{\bar{q}}X_{2}\biggr)
G=\frac{1}{81\Lambda\bar{\Lambda}-\Box{(\frac{Q^{2}}{16\pi^{2}}\Box+m^{2})}^{2}}\left[\begin{array}{cc}{9\bar{\Lambda}}&{-(\frac{Q^{2}}{16\pi^{2}}\Box+m^{2})\frac{\bar{D}^{2}}{4}}\\{-(\frac{Q^{2}}{16\pi^{2}}\Box+m^{2})\frac{D^{2}}{4}}&{9\Lambda}\\\end{array}\right]
\sum_{A,B}\sum_{|K|=1}\int_{\Omega}E_{A}^{0}(f)I_{AB}^{(K)}E_{B}^{K}(h)=\sum_{A,B}\sum_{|K|=1}\int_{\Omega}D_{K}\biggl(E_{A}^{0}(f)I_{AB}^{(0)}E_{B}^{K}(h)\biggr).
V(y_{i},\theta)=-V_{p+1}(i)^{p}\int_{0}^{\infty}\frac{dt}{t}(8\pi^{2}\alpha^{\prime}\,t)^{-\frac{p+1}{2}}e^{-(\frac{y^{2}}{2\pi\alpha^{\prime}}-\pi)t}t^{4}\eta^{-12}(i/t)\theta_{11}^{4}(1/2,i/t)\,.
Ad\left(\hat{G}_{\mu\nu}\right)\rightarrow2\left(x_{\mu}\frac{\partial}{\partialx_{\nu}}-x_{\nu}\frac{\partial}{\partialx_{\mu}}-w_{\mu\lambda}\frac{\partial}{\partialw_{\lambda\nu}}+w_{\nu\lambda}\frac{\partial}{\partialw_{\lambda\mu}}\right),
e^{-K}\mathcal{O}_{\pm}^{in}(\Omega)e^{K}=\mathcal{O}_{\pm}^{in}(\Omega)+\sum_{N=1}^{\infty}\frac{(-1)^{N}}{N!}\overbrace{[K,[K,\mathcal{O}_{\pm}^{in}(\Omega)],\dots,]]}^{N}\.
\left.\sum_{m=0}^{n}d_{m}(\theta)b_{n-m}(u^{2},w,\theta)u^{2m}\right)s^{n+1}\equiv{\frac{1}{(4\pis)^{d/2}}}\sum_{n=0}^{\infty}a_{\alpha,n+1}s^{n+1}+ES~~~.
\begin{array}{rcl}{\hat{\lambda}_{(p=1)}[\gamma(1)]}&{=}&{\hat{\lambda}_{(p=1)}[\gamma(0)]+1=\left(M_{(p=1)}\hat{\lambda}_{(p=1)}\right)[\gamma(0)]\,,}\\{}&{}&{}\\{M_{(p=1)}}&{=}&{\left(\begin{array}{cc}{1}&{1}\\{0}&{1}\\\end{array}\right)=T\,,}\\\end{array}
{\frac{d^{2}x^{\mu}}{d\tau^{2}}}+\Gamma_{\rho\sigma}^{\mu}{\frac{dx^{\rho}}{d\tau}}{\frac{dx^{\sigma}}{d\tau}}={\frac{F^{\mu}}{m_{0}}}+g_{\rho\sigma}{\frac{F^{\rho}}{m_{0}}}{\frac{dx^{\sigma}}{d\tau}}{\frac{dx^{\mu}}{d\tau}},
L=\sum_{k=1}^{N}(\frac{1}{2}\partial_{\mu}\phi^{k}\partial^{\mu}\phi^{k}-\frac{1}{2}\mu_{0}^{2}\phi^{k}\phi^{k})-\frac{\lambda}{4!N}(\sum_{k=1}^{N}\phi^{k}\phi^{k})^{2},
\frac{V_{af}(\sigma)}{N}=\frac{\sigma^{2}}{g}-\int\frac{d^{2}p}{(2\pi)^{2}}\mathrm{ln}(1+\frac{\sigma^{2}}{p^{2}})=\nonumber\,\frac{\sigma^{2}}{g}-\frac{1}{4\pi}\left[\Lambda^{2}\mathrm{ln}(\frac{\sigma^{2}}{\Lambda^{2}}+1)+\sigma^{2}\mathrm{ln}(\frac{\Lambda^{2}}{\sigma^{2}}+1)\right].
P_{0}=-\frac{BRp}{\alpha^{\prime}}-\frac{Al}{R}+\sqrt{\left(\frac{pR}{\alpha^{\prime}}\right)^{2}+\left(\frac{l}{R}\right)^{2}+{\vec{P}}^{~2}+\frac{2}{\alpha^{\prime}}(N+\tilde{N}-2)}
\langle\psi|\psi^{\prime}\rangle=\int\,d^{m}\eta\,\psi(\eta)^{\ast}\psi^{\prime}(\eta)=\langle\psi|\psi^{\prime}\rangle=\sum_{p}\frac{(-1)^{{\binom{p}{2}}}}{(m-p)!p!}\,\varepsilon^{a_{1}\cdotsa_{p}a_{p+1}\cdotsa_{m}}\psi_{a_{1}\cdotsa_{p}}^{\ast}\psi_{a_{p+1}\cdotsa_{m}}^{\prime}.
\rho(\varphi)\,\mathrm{det\}\left(\frac{\partial\varphi^{r}}{\partial\tilde{\varphi}^{s}}\right)\=\\frac{\lambda\,\cdot\,\tilde{\rho}\,(\tilde{\varphi})}{h\,\left(x\,(\tilde{t},\tilde{\varphi})\right)}
{\frac{1}{m}}{\frac{dm}{d\tilde{\lambda}}}=g_{\mu\nu}A^{\mu}{\frac{dx^{\nu}}{d\tilde{\lambda}}}={\frac{F}{(\sigma+\varrho)a_{0}^{2}}}\left[\left({\frac{dt}{d\tilde{\lambda}}}\right)^{2}-a_{0}^{2}\sum_{i}\left({\frac{dx^{i}}{d\tilde{\lambda}}}\right)^{2}-2\right]\sum_{i}\left({\frac{dx^{i}}{d\tilde{\lambda}}}\right)^{2}.
V=-\frac{T_{0}}{4\lambda^{2}}\sum_{a\neqb}\mathrm{Tr}[X^{a},X^{b}]^{2}-\frac{iT_{0}}{3\lambda}\mathrm{Tr}(X^{i}X^{j}X^{k})F_{0ijk}^{(4)}-\frac{T_{0}}{\lambda}\mathrm{Tr}(\theta^{T}\gamma_{a}[X^{a},\theta])
\left(\frac{sinh\frac{\gamma}{2}(u_{j}+2si)}{sinh\frac{\gamma}{2}(u_{j}-2si)}\right)^{L}=\prod_{k=1,k\neqj}^{M}\frac{sinh\frac{\gamma}{2}(u_{j}-u_{k}+2i)}{sinh\frac{\gamma}{2}(u_{j}-u_{k}-2i)}
\begin{array}{lcr}{{L}_{q}}&{=}&{-\int{\frac{dz}{2i\pi}}J^{q}(z)=-J_{0}^{q};\quadq=0,\pm1}\\{{\bar{L}}_{q}}&{=}&{\int{\frac{d{\bar{z}}}{2i\pi}}\bar{J}^{q}(z)=-{\bar{J}}_{0}^{q};\quadq=0,\pm1.}\\\end{array}
S=\intd^{4}x\Bigl(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2\zeta}\bigl(\partial_{\mu}A^{\mu}\bigr)^{2}+\frac{\sqrt{2}\kappa}{4}\epsilon^{\mu\nu\lambda\sigma}q_{\mu}A_{\nu}F_{\lambda\sigma}\Bigr)
T^{ij}(\lambda){}~=~\delta_{ij}~+~\sum_{n=0}^{\infty}~{\frac{1}{\lambda^{n+1}}}~\big\{~J_{n}^{0}~\delta_{ij}~+~\sum_{a=1}^{N^{2}-1}J_{n}^{a}~(t^{a})_{ij}~\big\}~,
ds^{2}~=~-\left(1-\frac{2GM}{r}+\frac{GQ^{2}}{r^{2}}\right)dt^{2}+\left(1-\frac{2GM}{r}+\frac{GQ^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}~.
G_{\mu\nu}=\left(\frac{\partialG_{\mu\nu}}{\partialF_{\rho\sigma}}(-F_{\rho\sigma}+B_{\rho\sigma}^{N})+2\chi\frac{\partialG_{\mu\nu}}{\partial\chi}+2\chi\frac{\partialG_{\mu\nu}}{\partial\phi}\right)+*B_{\mu\nu}^{R}.
{\calS}_{CJS}=-\intd^{11}x{\frac{\sqrt{-g_{11}}}{2^{.}4!}}F_{M_{1}...M_{4}}^{(4)}F^{(4)M_{1}...M_{4}}+\int{\frac{1}{3}}A^{(3)}\wedgeF^{(4)}\wedgeF^{(4)}
\beta_{Bolt}=\frac{4\pi}{F^{\prime}(r_{b})}=\frac{6\pi\ell^{2}(r_{b}^{2}-N^{2})^{3}}{3r_{b}^{7}-9r_{b}^{5}N^{2}+N^{2}(4\ell^{2}-15N^{2})r_{b}^{3}+9m\ell^{2}r_{b}^{2}+3N^{4}(4\ell^{2}-25N^{2})r_{b}+3m\ell^{2}N^{2}}
L_{eff}={\frac{1}{2}}\dot{\bfa}\cdot\dot{\bfa}\intd^{2}{\bfr}\rho+\intd^{2}{\bfr}\rho{\bfE}\cdot{\bfr}-{\frac{1}{2}}k\intd^{2}{\bfr}\rhor^{2}+\intd^{2}{\bfr}\rho\dot{\bfa}\cdot{\bfA}^{e}({\bfr})
M^{(\pm)}(\nu,0)=4\int_{1}^{\infty}d\alpha(\psi^{\prime\prime}(\alpha)\alpha^{\frac{5}{4}}+\frac{3}{2}\psi^{\prime}(\alpha)\alpha^{\frac{1}{4}})[\alpha^{\frac{\nu}{2}}+\alpha^{\frac{-\nu}{2}}]=2\xi(\frac{1}{2}+\nu).
\left(1+Ne^{2\phi}\right)\left(\dot{\phi}^{\prime}-\dot{\rho}\phi^{\prime}-\rho^{\prime}\dot{\phi}\right)-\dot{\phi}\phi^{\prime}=\frac{N}{12}e^{2\phi}\left[\dot{Z}Z^{\prime}+2\dot{Z}^{\prime}-2\left(\dot{\rho}Z^{\prime}+\rho^{\prime}\dot{Z}\right)\right].
D^{\rho}h^{\mu\nu}=\frac{1}{3}\left(2\frac{\omega_{R}^{\mu\nu}}{\omega_{\rho}^{P}}-\frac{\omega_{R}^{\nu\rho}}{\omega_{\mu}^{P}}-\frac{\omega_{R}^{\rho\mu}}{\omega_{\nu}^{P}}\right)
{\frac{\int_{X}\parallela\rightarrowb\parallel\\parallela\wedgeb\rightarrowc\wedgex_{t}\parallel}{\int_{X}\parallela\rightarrowx_{t}\parallel}}*{\frac{\int_{X}\parallela\wedgeb\rightarrowx_{t}\parallel}{\int_{X}\parallela\wedgeb\rightarrowx_{t}\parallel}}
\left.\begin{array}{ccccc}{}&{}&{h^{0,0}}&{}&{}\\{}&{h^{1,0}}&{}&{h^{0,1}}&{}\\{h^{2,0}}&{}&{h^{1,1}}&{}&{h^{0,2}}\\{}&{h^{2,1}}&{}&{h^{1,2}}&{}\\{}&{}&{h^{2,2}}&{}&{}\\\end{array}\right.=\left.\begin{array}{ccccc}{}&{}&{1}&{}&{}\\{}&{0}&{}&{0}&{}\\{1}&{}&{20}&{}&{1}\\{}&{0}&{}&{0}&{}\\{}&{}&{1}&{}&{.}\\\end{array}\right.
F_{mn}(\Box_{2},\Box_{3})X_{2}X_{3}=-2\int_{-\infty}^{0}dq\,q^{m+n}\Bigl[\Bigl(\frac{d}{dq}\Bigr)^{m+1}{\calH}_{q}X_{2}\Bigr]\Bigl[\Bigl(\frac{d}{dq}\Bigr)^{n+1}{\calH}_{q}X_{3}\Bigr]\quad.
{\sum_{i,j,k,l=1}^{h}\omega_{l}^{2}\partial_{z}\left({\omega_{i}/\omega_{l}}\right)A_{ij}A_{lk}\overline{\omega}_{k}^{2}\partial_{\bar{z}}\left({\overline{\omega}_{j}/\overline{\omega}_{k}}\right)}=\left(\sum_{i,j=1}^{h}\omega_{i}A_{ij}\overline{\omega}_{j}\right)^{3}.
\varpi_{j}(\psi,\phi)=-\frac{1}{6}\sum_{m=1}^{\infty}\frac{(-1)^{m}\alpha^{jm}\Gamma(\frac{m}{6})}{\Gamma(m)\Gamma(1-\frac{m}{2})\Gamma^{2}(1-\frac{m}{6})}(12\psi)^{m}u_{-\frac{m}{6}}((-1)^{j}\phi)
\dot{R}^{2}+k=\frac{4(\beta+2)}{(\beta\ell)^{2}}\,R^{2}\left\{1-\sqrt{1-\left(\frac{\beta\ell}{\beta+2}\right)^{2}\left(\frac{\mu}{R^{4}}-\frac{q^{2}}{R^{6}}\right)}\right\}\,.
G^{(2,2,2)}(1\vert2\vert3)=\langle\mathrm{Tr}\left(\tilde{q}^{+}(1)\right)^{2}\mathrm{Tr}\left(\tilde{q}^{+}(2)q^{+}(2)\right)\mathrm{Tr}\left(q^{+}(3)\right)^{2}\rangle
\widetildeM_{\lambda}(j_{p},j_{q})=-e{\frac{1}{2\sqrt{2\omega_{k}E_{q}E_{p}}}}e^{(i{\frac{\pi}{2}}(|l|-|n|)}\delta(E_{p}-E_{q}-\omega_{k})\delta(p_{3}-q_{3}-k_{3})\;m_{\lambda}\,,
p_{1}\,=\,-\,\frac{\partialL}{\partial\dot{x}}\,-\,\dot{p}_{2},\quadp_{2}\,=\,-\,\frac{\partialL}{\partial\ddot{x}}\,-\,\dot{p}_{3},\quadp_{3}\,=\,-\,\frac{\partialL}{\partial\stackrel{\ldots}{x}}\,{,}
\breve{\omega}_{i}dx^{i}\simeq\sum_{n=-\infty}^{\infty}{\frac{m_{0}a\{(x^{1}-n)dx^{2}-x^{2}dx^{1}\}}{\{\breve{R^{2}}+(x^{1}-n)^{2}\}^{\frac{5}{2}}}}\,
\begin{array}{l}{\delta\lambda=-\frac{1}{2}H\!\!\!\!/\sigma^{3}\epsilon+\frac{1}{4}e^{\phi}F^{(4)}\!\!\!\!\!\!\!\!\!/\,\,\,\,\,\sigma^{1}\epsilon=}\\{\qquad=\mu\left(-\Gamma^{+12}\sigma^{3}+\Gamma^{+234}\sigma^{1}\right)\epsilon=\frac{1}{2}\mu\Gamma_{-}\Gamma_{12}\left(1+\Gamma_{134}(i\sigma^{2})\right)\epsilon}\\\end{array}
{\cal{L}}_{m}=-{\scriptstyle{\frac{1}{2}}}(m^{2}+e\xi)({{\tilde{A}}_{1}}^{2}+{{\tilde{B}}_{1}}^{2})-{\scriptstyle{\frac{1}{2}}}(m^{2}-e\xi)({{\tilde{A}}_{2}}^{2}+{{\tilde{B}}_{2}}^{2})-im\bar{\psi}\psi
c_{E}^{2}\;=\;\frac{T}{U}\;=\;1-\frac{\tilde{\chi}(1-\frac{\tilde{\chi}}{m_{*}^{2}})^{-2}}{m^{2}-\frac{\tilde{\chi}}{2}(1-\frac{{\tilde{\chi}}}{m_{*}^{2}})^{-1}+{\tilde{\chi}}(1-\frac{{\tilde{\chi}}}{m_{*}^{2}})^{-2}}\;\;,
+\frac{\gamma}{2N}\left[\sum_{\{p_{\sigma^{\prime}}\mu^{\prime}\nu^{\prime}\}}Tr(U_{p_{\sigma}^{\prime}}+U_{p_{\sigma}^{\prime}}^{+})+\sum_{\{p_{\tau^{\prime}}\mu^{\prime}d\}}Tr(U_{p_{\tau}^{\prime}}+U_{p_{\tau}^{\prime}}^{+})\right],
\left[{\frac{\partial^{2}}{\partialu^{2}}}+{\frac{u^{2}+3\Lambda^{4}}{4(u^{2}-\Lambda^{4})^{2}}}\right]\sqrt{\Lambda^{4}-u^{2}}{\frac{\partiala_{D}}{\partialu}}=0=\left[{\frac{\partial^{2}}{\partialu^{2}}}+{\frac{u^{2}+3\Lambda^{4}}{4(u^{2}-\Lambda^{4})^{2}}}\right]\sqrt{\Lambda^{4}-u^{2}}{\frac{\partiala}{\partialu}}.
\kappa{\calS}^{-}[w]=\kappa{\calS}^{-}[\tilde{g}]+\kappa{\calS}^{-}[e^{\displaystyle\,\gamma}]+\frac{1}{\pix}\int\mathrm{str}\left\{\bar{\partial}e^{\displaystyle-\gamma}\cdote^{\displaystyle\,\gamma}\tilde{J}\right\}\,\,.
\begin{array}{rcl}{\mathrm{Dilaton\scalar}}&{\phi}&{\eta^{\mu\nu}\bar{\partial}X_{\mu}\partialX_{\nu}\qquad(\mu,\nu=1,...,D)}\\{\mathrm{Graviton}}&{g_{\mu\nu}}&{\bar{\partial}X_{\mu}\partialX_{\nu}}\\{\mathrm{Gauge\field}}&{A_{\mu}^{a}}&{\bar{\partial}X_{\mu}\partialX^{a}}\\{\mathrm{Higgs\field}}&{\Phi^{ab}}&{\bar{\partial}X^{a}\partialX^{b}}\\\end{array}
{\frac{\partial^{2}\widetilde{\calF}(\tilde{a})}{\partial\tilde{a}^{2}}}=\left[-\left({\frac{\partial\tilde{a}}{\partiala}}\right)^{-3}{\frac{\partial^{2}\tilde{a}}{\partiala^{2}}}{\frac{\partial}{\partiala}}+\left({\frac{\partial\tilde{a}}{\partiala}}\right)^{-2}{\frac{\partial^{2}}{\partiala^{2}}}\right]\widetilde{\calF}(\tilde{a}).
\Psi^{\mu}(z)\Psi^{\mu}(w)=\frac{\alpha^{\prime}}{2}\frac{1}{z-w}+\cdots\,,\quad\tilde{\Psi}^{\mu}(z)\tilde{\Psi}^{\mu}(w)=\frac{\alpha^{\prime}}{2}\frac{1}{\bar{z}-\bar{w}}+\cdots\,,
k_{\varepsilon_{2n}}(\zeta_{2n})\overline{{G}}_{b}^{(n)}(\zeta^{\prime},\zeta_{2n})^{\varepsilon^{\prime}\,\varepsilon_{2n}}=\overline{{G}}_{b}^{(n)}(\zeta^{\prime},\zeta_{2n}^{-1})^{\varepsilon^{\prime}\,\varepsilon_{2n}},
\left.\begin{array}{lll}{W}&{=}&{\lambda\wp(z-a),}\\{E}&{=}&{8|\lambda|^{2}\frac{|\wp(z-a)||\wp^{2}(z-a)-\wp^{2}(L/2)|}{[1+|\lambda\wp(z-a)|^{2}]^{2}}.}\\\end{array}\right.
\left\langle\xi(x)\xi(x^{\prime})\right\rangle\equivN(x,x^{\prime})=\left(\frac18\right)\left[\left\langle\left\{\Phi^{2}(x),\Phi^{2}(x^{\prime})\right\}\right\rangle_{0}-2<\Phi^{2}>_{0}^{2}\right]
\chi[{\calM}](y,\bar{y})=\sum_{q_{\mathrm{L}},q_{\mathrm{R}}=0}^{\hat{c}}(-1)^{q_{\mathrm{L}}+q_{\mathrm{R}}}h_{q_{\mathrm{L}},\hat{c}-q_{\mathrm{R}}}y^{q_{\mathrm{L}}}\bar{y}^{q_{\mathrm{R}}}\,,
H=\frac{R}{2\hbar}\Pi_{\lambda}^{k}\Pi_{\lambda}^{k}-\frac{R}{16\pi\alpha^{3}}F_{\rho}^{\mu\nu}F_{\rho}^{\tau\epsilon}Y_{\mu}^{i}Y_{\tau}^{i}Y_{\nu}^{j}Y_{\epsilon}^{j}-\frac{iR}{{2\pi}\left(\hbar\alpha^{\prime}\right)^{3/2}}\pi^{\mu}Y_{\sigma}^{i}F_{\mu}^{\sigma\nu}\gamma_{i}\theta_{\nu}
\int_{-\epsilon}^{\infty}dl\:\mathrm{e}^{-l\zeta}\int_{-\epsilon}^{\infty}dl^{\prime}\mathrm{e}^{-l^{\prime}\zeta}ll^{\prime}{\frac{l^{\prime}-l}{l+l^{\prime}}}\{3\,\delta^{\prime\prime}(l)-{\frac{3}{4}}t\,\delta(l)\}=0.
\Gamma_{PC}[A]=\sqrt{\pi}(-1\pm\mathrm{sign}(m))\sum_{n=0}^{(d-1)/2}\frac{(-2)^{n}}{(2n+1)!!}(m^{2})^{n+\frac{1}{2}}H_{d-1-2n}+\frac{1}{2}\sum_{n=\frac{d+1}{2}}^{\infty}\frac{\Gamma\left(n-\displaystyle{\frac{d}{2}}\right)}{(m^{2})^{n-d/2}}H_{2n},
W^{(m,m^{\prime})}\left(\left.\begin{array}{cc}{c}&{d}\\{b}&{a}\\\end{array}\right|z\right)=\sum_{\sigma\in\mathfrak{S}_{m^{\prime}}}\mathrm{sgn}\,\sigma\prod_{j=1}^{m^{\prime}}W^{(m,1)}\left(\left.\begin{array}{cc}{d_{j}}&{d_{j-1}}\\{a_{j}^{\sigma}}&{a_{j-1}^{\sigma}}\\\end{array}\right|z+\frac{m^{\prime}+1}{2}-j\right),
{\bfe}_{1}=\left(\begin{matrix}{1}\\{0}\\{0}\\{\vdots}\\{0}\\\end{matrix}\right),\quad{\bfe}_{2}=\left(\begin{matrix}{0}\\{1}\\{0}\\{\vdots}\\{0}\\\end{matrix}\right),\quad\ldots\quad,{\bfe}_{N}=\left(\begin{matrix}{0}\\{0}\\{0}\\{\vdots}\\{1}\\\end{matrix}\right).
\begin{array}{ccl}{(det_{q}K)P_{-\;ij,kl}}&{\propto}&{\epsilon_{ij}^{q}\epsilon_{ab}^{q}(X_{a}Z_{c}^{\dagger}+Z_{a}X_{c}^{\dagger})R_{bc,mn}^{(3)}(X_{m}Z_{p}^{\dagger}+Z_{m}X_{p}^{\dagger})\epsilon_{pn}^{q}\epsilon_{kl}^{q}}\\{\,}&{=}&{\epsilon_{ij}^{q}[(X^{t}\epsilon^{q}Z)(X^{t}\epsilon^{q}Z)^{\dagger}+(Z^{t}\epsilon^{q}X)(Z^{t}\epsilon^{q}X)^{\dagger}]\epsilon_{kl}^{q}\;\neq0}\\\end{array}
\begin{array}{ll}{}&{R_{2}^{\mathrm{in}}=e^{-i\omegav_{*}}\,,}\\{}&{R_{2}^{\mathrm{out}}=e^{-i\omegav_{*}}e^{2[i(\omega-\omega_{0})-C]r_{*}}\,,~~~~~~~(r>r_{H})\,.}\\\end{array}
D={\left(\begin{array}{cccc}{0}&{-\frac{c}{eB}}&{\frac{1}{2}}&{0}\\{\frac{c}{eB}}&{0}&{0}&{\frac{1}{2}}\\{-\frac{1}{2}}&{0}&{0}&{-\frac{eB}{4c}}\\{0}&{-\frac{1}{2}}&{\frac{eB}{4c}}&{0}\\\end{array}\right)}\;.
{\calN}_{IJ}={\bar{F}}_{IJ}+2i\frac{\mathrm{Im}(F_{IK})\mathrm{Im}(F_{JL})\,X^{K}X^{L}}{\mathrm{Im}(F_{KL})\,X^{K}X^{L}},\;\;F_{IJ}=\frac{\partial^{2}F}{{\partialX_{I}}{\partialX_{J}}},\;F_{I}={\calN}_{IJ}X^{J}.
H^{0}(\pi^{*}{\calS},{\calO}_{\pi^{*}{\calS}}(A^{\prime}\sigma|_{\pi^{*}{\calS}}-B^{\prime}F))=H^{0}({\calS},{\calO}_{{\calS}}(-B^{\prime})\oplus\bigoplus_{i=2}^{A^{\prime}}(-B^{\prime}+i(r-2)).
J(X_{\tau})=\int_{X}d^{m}z_{i}d^{n}z_{\hat{i}}\prod_{i=1}^{m}z_{i}^{\alpha}(1-z_{i})^{\beta}\prod_{\hat{i}=\hat{1}}^{n}z_{\hat{i}}^{\alpha^{\prime}}(1-z_{\hat{i}})^{\beta^{\prime}}\left[\prodz_{ij}\prodz_{\hat{i}\hat{j}}\prodz_{i\hat{j}}^{2\sigma}\right]_{\tau}
\mu_{0}c\rho_{\mathrm{em}}=\frac{1}{\lambda}\frac{\Phi_{0}}{2\pi\lambda^{2}}\,J^{0}\\,\\\mu_{0}\vec{J}_{\mathrm{em}}=\frac{1}{\lambda}\frac{\Phi_{0}}{2\pi\lambda^{2}}\,\vec{J}.
ds^{2}=-dt^{2}+k[dx^{2}+\frac{sin^{2}x}{cos^{2}x+R^{2}sin^{2}x}d\theta_{1}^{2}+\frac{R^{2}cos^{2}x}{cos^{2}x+R^{2}sin^{2}x}d\theta_{2}^{2}]
\prod_{k=1}^{n_{c}-\tilde{n}_{c}}(\Lambda\omega^{2\ell-1}+\delta_{\ell}-\Lambda\omega^{2k}-\gamma_{k})^{2}=\prod_{k=1}^{n_{c}-\tilde{n}_{c}}(\Lambda\omega^{2\ell-1}-\Lambda\omega^{2k})^{2}\left[1+2\sum_{k=1}^{n_{c}-\tilde{n}_{c}}\frac{\delta_{\ell}-\gamma_{k}}{\Lambda\omega^{2\ell-1}-\Lambda\omega^{2k}}\right].
G_{r}^{M}=\sum_{n=-\infty}^{\infty}\alpha_{-n}\cdot(b_{r+n}+\eta_{\phi}b_{r+n}^{\prime})=\sum_{n=-\infty}^{\infty}\alpha_{-n}\cdot(b_{r+n}+i\;b_{r+n}^{\prime})=\sum_{n=\infty}\alpha_{n}\cdotB_{r+n}
i{\calG}_{2}(p)=-{\frac{2ie^{2}B^{2}p_{\bot}^{2}}{(p^{2}-m^{2})^{4}}}D+2e^{3}B^{3}\left[{\frac{1}{(p^{2}-m^{2})^{4}}}+{\frac{4p_{\bot}^{2}}{(p^{2}-m^{2})^{5}}}\right]\bar{D},
\mathcal{L}^{(0)}=p_{i}\dot{q}_{i}+\epsilon\hat{\bar{p}}_{i}\dot{\bar{q}\,}_{i}+\pi\dot{\phi}+\epsilon\hat{p}_{i}^{b}\dot{b}_{i}+p_{i}^{c}\dot{c}_{i}-\mathcal{H}_{ext}\,.
{\calL}^{(1)}=-\partial_{\mu}A^{*}\partial^{\mu}A^{\prime}-\left|{\frac{d{\calW}_{\mathrm{np}}^{(1)}}{dA^{\prime}}}\right|^{2}=-\partial_{\mu}A^{*}\partial^{\mu}A^{\prime}-\left|m^{2}\left(1-\left({\frac{A^{\prime}}{m}}\right)^{2n}\right)\right|^{2}.
\delta_{\xi}\psi^{i}=\sqrt{2}\left[(i\sigma^{1}\bar{\xi}-\Omega_{-}^{*}\xi)\partial_{1}A_{\mathrm{cl}}^{i}+(i\sigma^{2}\bar{\xi}+i\Omega_{-}^{*}\xi)\partial_{2}A_{\mathrm{cl}}^{i}\right].
{\calM}_{n}({\bfu}_{1},{\bfu}_{2}){_{\displaystyle\longrightarrow}\atop{\scriptstyle\omega_{n}\rightarrow\infty}}\,\,{\frac{g({\bfu}_{1})}{\pi\omega_{n}}}~\delta(|{\bfx}({\bfu}_{1})-{\bfx}({\bfu}_{2})|^{2})={\frac{\sqrt{g({\bfu}_{1})}}{2\pi\omega_{n}}}~{\frac{\delta(|{\bfu}_{1}-{\bfu}_{2}|)}{|{\bfu}_{1}-{\bfu}_{2}|}}\,.
Z=\int\prod_{n=1}^{\mathcal{N}}D[P_{n}]D[\pi^{ij}]D[h_{ij}]\delta(\chi)\delta(\chi^{1})\delta(\chi^{2})|\mathrm{Det}\{\chi^{\mu},H_{\nu}\}|\delta(\frac{H_{i}}{\sqrt{h}})\delta(\frac{H}{\sqrt{h}})e^{iS}.
\varphi_{n_{1}}^{\,\,m_{1}}\star\left(\varphi^{\dagger}\right)_{m_{2}}^{\,\,n_{2}}\star\varphi_{n_{3}}^{\,\,m_{3}}=\delta_{\,\,m_{2}}^{m_{1}}\,\delta_{\,\,n_{3}}^{n_{2}}\,\varphi_{n_{1}}^{\,\,m_{3}},
{e_{c}}^{({{\partial}_{\eta}}_{i}{{\partial}_{\bar{\eta}}}^{i})}:=\sum_{r}\frac{{({{\partial}_{\eta}}_{i}{{\partial}_{\bar{\eta}}}^{i})}^{r}}{[r]_{c}!}\mathrm{\qquad~with~\qquad}[r]_{c}:=\frac{c^{2r}-1}{c^{2}-1}.
\begin{array}{llll}{\mathrm{for~}\phi=+1}&{~:~}&{(i^{k},1,0,0,0,1)}&{,(k=0,...,3)}\\{\mathrm{for~}\phi=-1}&{~:~}&{(i^{k}e^{i\pi/4},1,0,0,0,1)}&{,(k=0,...,3)\;.}\\\end{array}
\Psi_{2}(z_{1},z_{2})=(z_{1}z_{2})^{\frac{C_{2}([\tilde{m}]_{n})+C_{2}([m]_{n})-2C_{2}}{4(k+n)}}(T_{1}\chi_{1}^{+}(z)+T_{2}\chi_{2}^{+}(z))
\sim\langleV_{\vec{\lambda}_{1},\vec{m}_{1}}(z_{1})Q_{i}(z_{a},z_{b})\cdotsQ_{j}(z_{c},z_{d})V_{\vec{\lambda}_{2},\vec{m}_{2}}(z_{2})\cdotsQ_{k}(z_{e},z_{f})V_{\vec{\lambda}_{N},\vec{m}_{N}}\rangle~,
z={\frac{\hat{q}_{1}^{(e)}+i\hat{q}_{0}^{(e)}}{\hat{q}_{(m)}^{0}-i\hat{q}_{(m)}^{1}}},\\\\\\M^{2}=|Z|^{2}=|\hat{q}_{(m)}^{0}\hat{q}_{1}^{(e)}-\hat{q}_{0}^{(e)}\hat{q}_{(m)}^{1}|.
V_{o}^{\mathrm{int}}=\intd\etae^{\biggl(i\sqrt{{\frac{\alpha^{\prime}}{2}}}(k\cdot{\bar{X}}+i{\bar{k}}\cdotX)-\eta(D_{L}+D_{R})(k\cdotX-{\bar{k}}\cdotX)\biggr)}|_{\theta_{L}=\theta_{R}},
G^{\pm}(x,t;x^{\prime},t^{\prime})=\int\frac{d\omega}{2\pi}\int\frac{dk}{2\pi}\;\frac{ie^{-i\omega(t-t^{\prime})}}{\omega{}^{2}-k{}^{2}-m^{2}+i\epsilon}\left[e^{ik(x-x^{\prime})}+K_{b}^{\pm}(k)\;e^{-ik(x+x^{\prime})}\right].
a-{\frac{bc}{2\pi^{2}}}\int_{k^{2}=p^{2}}{\frac{d^{2}k}{k^{2}}}+{\frac{Nd^{2}}{\pi^{2}}}\int_{k^{2}=0}{\frac{d^{2}k}{k^{2}+p^{2}}}=a~.
ds^{2}=-\frac{t^{2}}{(t^{2}+r_{-}^{2})(t^{2}-r_{+}^{2})}dt^{2}+t^{2}(d\phi+\frac{r_{+}r_{-}}{t^{2}}dr)^{2}+\frac{(t^{2}+r_{-}^{2})(t^{2}-r_{+}^{2})}{t^{2}}dr^{2}.
\hat{\psi}_{\alpha}(x)=\sum_{s}\int\frac{d^{3}k}{\sqrt{(2\pi)^{3}2\omega}}[\hat{a}_{k}(s)e^{-ikx}u_{\alpha}(k,s)+\hat{b}_{k}^{\dagger}(s)e^{ikx}v_{\alpha}(k,s)]
\partial_{\mu}\partial^{\mu}\,\hat{\xi}\,^{5}(x)=\frac{(1+\gamma)^{-1}}{6\,M_{*}^{3}}\,\Big[\frac{S_{\lambda}^{\lambda}(x)}{2}+\ell\,T_{55}(0)-\frac{1+\gamma}{1-\gamma}\,\ell\,\int_{0}^{y_{m}}\,dy\,\partial^{\mu}T_{\mu5}\,\Big]\,.
{\frac{R_{N}^{\mathrm{out}}(\omega)}{A_{N}(\omega)}}=-{\frac{R_{N}^{\mathrm{in}}(-\omega)}{A_{N}(-\omega)}},~~{\frac{R_{N}^{\mathrm{out}}(-\omega)}{A_{N}(-\omega)}}=-{\frac{R_{N}^{\mathrm{in}}(\omega)}{A_{N}(\omega)}},
D_{+}^{I}=\frac{\partial}{\partial\theta_{I}^{+}}+i\eta^{IJ}\theta_{J}^{+}\frac{\partial}{\partial\sigma^{+}},\\\\D_{-}^{I^{\prime}}=\frac{\partial}{\partial\theta_{I^{\prime}}^{-}}+i\eta^{I^{\prime}J^{\prime}}\theta_{J^{\prime}}^{-}\frac{\partial}{\partial\sigma^{+}},
dQ^{a}=2J(d\xi\cdot\frac{\partial}{\partial\xi}+d\bar{\xi}\cdot\frac{\partial}{\partial\bar{\xi}})\left[\frac{(T^{a})_{00}+(T^{a})_{0q}\xi_{q}+\bar{\xi}_{p}(T^{a})_{po}+\bar{\xi}_{p}(T^{a})_{pq}\xi_{q}}{1+\mid\xi\mid^{2}}\right].
\left[\rho^{^{\prime}}(z)\right]^{2}=\sum_{n=1}^{\infty}(n+1)^{2}(n+2)^{2}G_{2(n+3)}\mathrm{~}z^{2n}-4\sum_{n=1}^{\infty}(n+4)(n+5)G_{n+6}z^{n}\mathrm{~.}\tag{155}
\bar{M}_{Z}=M_{R}\left(1-\frac{1}{4}\left(\frac{\Gamma_{R}}{M_{R}}\right)^{2}\right)^{\frac{1}{2}}\quad\mathrm{and}\quad\bar{\Gamma}_{Z}=\Gamma_{R}\left(1-\frac{1}{4}\left(\frac{\Gamma_{R}}{M_{R}}\right)^{2}\right)^{-\frac{1}{2}}.
\intd^{3}x\psi^{\dagger}(j,x)\psi(j^{\prime},x)=\delta_{j,j^{\prime}}={\delta_{s,s^{\prime}}\delta_{l,l^{\prime}}}{\delta(p_{3}-p_{3^{\prime}})}{\frac{\delta(p_{\perp}-p_{\perp}^{\prime})}{\sqrt{p_{\perp}p_{\perp}^{\prime}}}},
W_{12}=\overline{{V}}_{21}=\left(\begin{array}{cccc}{R}&{\cdot}&{\cdot}&{\cdot}\\{\cdot}&{R}&{\lambda}&{\cdot}\\{\cdot}&{\cdot}&{\overline{{R}}}&{\cdot}\\{\cdot}&{\cdot}&{\cdot}&{-\overline{{R}}}\\\end{array}\right)\,.
\begin{aligned}{{}[\Gamma^{rs},\Gamma^{t}]}&{{}=4\,\Gamma^{[r}\delta_{t}^{s]}\,,}\\{{}[\Gamma^{rs},\Gamma_{tu}]}&{{}=8\,\Gamma^{[r}{}_{[u}\delta^{s]}{}_{t]}\,,}\\{{}[\Gamma^{rs},\Gamma_{tuv}]}&{{}=12\,\Gamma^{[r}{}_{[tu}\delta^{s]}{}_{v]}\,.}\\\end{aligned}
\begin{array}{rcl}{S_{\mathrm{twisted}}}&{\sim}&{\intd^{6}y\sqrt{|\mathrm{det}g|}\left(1+({\calH}^{(3)})^{2}+\dots\right)-\intd^{6}y\sqrt{|\mathrm{det}g|}=}\\{}&{}&{}\\{}&{=}&{\intd^{6}y\sqrt{|\mathrm{det}g|}({\calH}^{(3)})^{2}+\dots\,.}\\\end{array}
\Gamma_{\varphi}^{\rho}(p,p^{\prime})=\frac{ie^{3}}{\theta}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\gamma^{\mu}(\not\!p^{\prime}-\not\!k-m)\gamma^{\rho}(\not\!p-\not\!k-m)\gamma^{\alpha}\epsilon_{\alpha\mu\nu}{\bar{k}}^{\nu}}{[(p^{\prime}-k)^{2}-m^{2}+i\epsilon][(p-k)^{2}-m^{2}+i\epsilon](-{\vec{k}}^{2})}+ie\gamma^{\rho}
\Big\langleJ_{\mu_{1}}^{(1)}(x_{1})J_{\mu_{2}}^{(1)}(x_{2}).....J_{\mu_{n}}^{(1)}(x_{n})\Big\rangle\;\;\Big\langleJ_{\nu_{1}}^{(l_{1})}(y_{1})J_{\nu_{2}}^{(l_{2})}(y_{2}).....J_{\nu_{k}}^{(l_{k})}(y_{k})\Big\rangle\;.
{\frac{d^{2}\tilde{\varphi}}{dz^{2}}}+\left({\frac{1}{z}}-{\frac{1}{1-z}}\right){\frac{d\tilde{\varphi}}{dz}}-\left({\frac{C_{-}}{z}}-{\frac{C_{+}}{1-z}}+{\frac{m_{\varphi}^{2}\ell^{2}}{4}}\right){\frac{\tilde{\varphi}}{z(1-z)}}=0,
S_{E}(p_{\mu},x^{\mu},\eta,{\calP},\rho,\bar{C},N,\Pi)=\int_{\tau^{\prime}}^{\tau^{\prime\prime}}d\tau(p_{\mu}\dot{x}^{\mu}+\dot{\eta}{\calP}+\dot{N}\Pi+\dot{\bar{C}}\rho-\{K,\Omega\}).
L_{0}+\tilde{L}_{0}=\frac{1}{2}k^{2}+N+\tilde{N}+\frac{1}{2}\Pi_{i}G^{ij}\Pi_{j}+\frac{1}{2}L^{i}(G-BG^{-1}B)_{ij}L^{j}+L^{i}B_{ij}G^{jk}\Pi_{k}.
\delta_{{\hat{\xi}}_{2}}L[{\hat{\xi}}_{1}]=\left\{L[{\hat{\xi}}_{2}],L[{\hat{\xi}}_{1}]\right\}=L[\{{\hat{\xi}}_{1},{\hat{\xi}}_{2}\}_{\mathrm{\scriptsize~SD}}]+K[{\hat{\xi}}_{1},{\hat{\xi}}_{2}]
f^{2}\tilde{\psi}^{\prime\prime}+(f^{2})^{\prime}\tilde{\psi}^{\prime}+\left[{\frac{f^{2}}{4\rho^{2}}}-{\frac{(f^{2})^{\prime}}{2\rho}}+{\frac{\omega^{2}}{f^{2}}}+{\frac{\mu}{R^{2}}}\right]\tilde{\psi}=0,
7{\frac{R^{6}}{N}}(\partial_{\rho}R)^{2}-2\partial_{\rho}({\frac{R^{7}\partial_{\rho}R}{N}})+7NR^{6}-{\frac{9}{144}}{\frac{R^{8}}{N}}(\partial_{\rho}\phi)^{2}+9({\frac{Q}{\pi^{5}}})^{2}{\frac{Ne^{-2\phi}}{R^{10}}}=0.
H_{c}^{\prime}=\frac{1}{2}\pi^{i}\pi_{i}^{\prime}+\frac{1}{4}\tilde{F}^{ij}\tilde{F}_{ij}+\frac{1}{2}\left[(\partial_{i}\pi^{i})^{2}+(\partial_{i}\pi)^{2}-A^{2}\right]-A_{i}^{\prime}\partial_{i}\pi
X_{1}=\int_{0}^{\sqrt{\lambda^{2}-4}}{dx{\frac{2}{\left(\lambda^{2}-1-x^{2}\right)}}\left[{\frac{(x^{4}-3\delta^{2}x^{2})}{(x^{2}+\delta^{2})^{3}}}-{\frac{2}{\left(\lambda^{2}-1-x^{2}\right)}}{\frac{x^{2}}{(x^{2}+\delta^{2})}}\right]},