harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\nabla_{i}\left(\begin{matrix}{V}\\{V_{AB}}\\{\bar{V}}\\{\bar{V}^{AB}}\\\end{matrix}\right)=\left(\begin{matrix}{0}&{0}&{0}&{{\frac{1}{2}}P_{CD,i}}\\{0}&{0}&{P_{AB,i}}&{0}\\{0}&{0}&{0}&{0}\\{0}&{{\frac{1}{2}}P_{,i}^{ABCD}}&{0}&{0}\\\end{matrix}\right)\left(\begin{matrix}{V}\\{V_{CD}}\\{\bar{V}}\\{\bar{V}^{CD}}\\\end{matrix}\right)
\begin{array}{l}{\displaystyleT(\xi^{0})=(\xi^{0}\xi^{0})^{-(\mu+\nu)/2}(\xi^{0}\xi^{0}-1)^{\nu/2}F\left(\frac{1}{2}(\mu+\nu)~,\frac{1}{2}(\mu+\nu+1)~,\mu-\frac{n}{2}+1~;(\xi^{0}\xi^{0})^{-1}\right)~,}\\{\displaystyleR(\rho)=\rho^{-\lambda}(\rho^{2}-1)^{\lambda/2}F\left(\frac{1}{2}(\lambda-n-l)+1~,\frac{1}{2}(\lambda+l)~,\frac{1}{2}~;\rho^{-2}\right)~,}\\\end{array}
d\hat{s}_{11}^{2}=e^{\varphi_{2}}(dy_{1}^{2}+dy_{2}^{2})+e^{\varphi_{3}}(dy_{3}^{2}+dy_{4}^{2}+dy_{5}^{2})+e^{-(2\varphi_{2}+3\varphi_{3})/4}ds_{6}^{2},
\mathcal{F}_{b}^{\prime}(0)=\frac{b^{2}\mu_{B}}{2}\sum_{j=0}^{r}e_{j}G_{B}(be_{j});\\mathcal{F}_{b}^{\prime\prime}(0)=b^{2}\mu\sum_{j=0}^{r}e_{j}G_{B}(2be_{j}).
v_{\Lambda\Omega}=\frac{1}{f_{\Lambda}}\mathrm{{sgn}}(\sigma_{R_{ij}^{(\ell)}\Lambda}^{a})\,\frac{v_{\Lambda_{ij}\left(R_{ij}^{(\ell)}\Lambda\right)_{ij}}}{f_{\left(R_{ij}^{(\ell)}\Lambda\right)_{ij}}}f_{\Omega}\,\#(\Omega,\Lambda)\,.
\begin{array}{rcl}{(i_{\hat{k}}{\hat{\Omega}})_{\mu_{1}\dots\mu_{5}y}}&{=}&{-(i_{k}\Omega^{(6)})_{\mu_{1}\dots\mu_{5}}}\\{}&{}&{}\\{(i_{\hat{k}}{\hat{\Sigma}})_{\mu_{1}\dots\mu_{5}y}}&{=}&{-(i_{k}\Sigma^{(6)})_{\mu_{1}\dots\mu_{5}}\,.}\\\end{array}
{\calL}_{\mathrm{hyp}}=\sum_{X}\left\{\!\int\!\!\mathrm{d}^{2}\theta\mathrm{d}^{2}\bar{\theta}\left[\bar{X}\mathrm{e}^{n_{E}^{a}V_{a}}\mathrm{e}^{n_{Ma}V_{D}^{a}}X+\bar{\tilde{X}}\mathrm{e}^{-n_{E}^{a}V_{a}}\mathrm{e}^{-n_{Ma}V_{D}^{a}}\tilde{X}\right]+2\,\mathrm{Re}\!\int\!\!\mathrm{d}^{2}\theta\,\tilde{X}{\calZ}_{X}(A)X\right\},
\bar{Q}_{a}=\bar{\psi}_{a}^{i}R_{i}-2i\bar{\psi}_{a}^{i}m^{j}\frac{\partial^{2}A}{\partial\phi^{i}\partial\phi^{j}}+2i\psi_{a}^{i}n^{j}\frac{\partial^{2}A}{\partial\phi^{i}\partial\phi^{j}}-\frac{1}{2}i\lambda_{ai}^{c}\bar{\psi}_{c}^{i},
\int\frac{dk^{\prime\prime}}{2\pi}\frac{i}{k^{2}+4}\frac{1}{-k^{\prime\prime}-2k_{1}-2i}\left(\frac{1}{k+k^{\prime}+k^{\prime\prime}-2i}+\frac{1}{k+k^{\prime}-k^{\prime\prime}-2i}K_{0}(k^{\prime\prime})\right).
\widehatp_{i}\Psi=-i\frac{\partial\Psi}{\partialq_{i}},\qquad\widehatP_{\alpha}\Psi=-i\frac{\partial\Psi}{\partialQ_{\alpha}},\qquad\widehat\pi_{i}\Psi=-i\frac{\partial\Psi}{\partial\theta_{i}}.
V_{\beta}(\phi)=V_{0}(\phi)+\frac{N}{2}LF_{\beta}(\phi)_{R}-\frac{N}{4!}(\lambda_{0}+\frac{\eta_{0}\phi^{2}}{10N})(F_{\beta}(\phi)_{R})^{2}-\frac{2N\eta(F_{\beta}(\phi)_{R})^{3}}{6!}.
({\calB}_{V})_{cl}=\sum_{{\vec{n}},{\vec{m}}}\sum_{s,s^{\prime}}\left[(a_{\vec{n}}^{s\dagger}\bar{u}_{\vec{n}}^{s}+b_{-{\vec{n}}}^{s}\bar{v}_{-{\vec{n}}}^{s})(a_{\vec{m}}^{s^{\prime}}\gamma^{0}u_{\vec{m}}^{s^{\prime}}+b_{-{\vec{m}}}^{s^{\prime}\dagger}\gamma^{0}v_{-{\vec{m}}}^{s^{\prime}})\right]_{cl}F({\vec{n}}-{\vec{m}},X).
\left\{F,G\right\}_{PB}=\int_{S}d^{p}\sigma\,\sum_{i}\left[\left(-1\right)^{F\phi^{i}}\frac{\deltaF}{\delta\phi^{i}\left(\sigma\right)}\frac{\deltaG}{\delta\Lambda_{i}\left(\sigma\right)}-\left(-1\right)^{\left(F+1\right)\phi^{i}}\frac{\deltaF}{\delta\Lambda_{i}\left(\sigma\right)}\frac{\deltaG}{\delta\phi^{i}\left(\sigma\right)}\right]\quad,
{\frac{J_{b}^{G/K_{b}}}{J_{f}^{G/H_{f}}}}={\frac{J_{b}^{G/H_{f}}}{J_{f}^{G/H_{f}}}}\,J_{b}^{H_{f}/K_{b}}\,[1+O(({\calM}/M_{\mathrm{Pl}})^{8})]
p_{bj}(z_{1},z_{2})=-\frac{\eta^{4}J_{3}(B_{1},B_{2})}{4\alpha^{4}\pi^{3/2}a^{4}\Gamma(3/2)}=-\frac{\eta^{4}\Lambda^{2}J_{3}(B_{1},B_{2})}{36\pi^{3/2}a^{4}\Gamma(3/2)}.
\begin{array}{l}{\lambda\left(\kappa_{ad}^{S}\epsilon_{dbc}\alpha_{c}+\kappa_{bd}^{S}\epsilon_{dac}\alpha_{c}\right)+R\left(\beta_{a}k_{b}+\beta_{b}k_{a}\right)=0}\\{\\\lambdaR\left(\alpha_{a}k_{b}-\alpha_{b}k_{a}\right)+\kappa_{ad}^{S}\epsilon_{dbc}\beta_{c}-\kappa_{bd}^{S}\epsilon_{dac}\beta_{c}=0.}\\\end{array}
\widetilde{{\calA}}_{r}^{\left(\sigma\right)}=\intd^{2}x\left[\frac1{2\pi\widetilde{b}^{2}}\frac{\partial_{\mu}\chi\partial_{\mu}\overline{{\chi}}}{1+\chi\overline{{\chi}}}+\frac{(\partial_{\mu}\phi)^{2}}{8\pi}+2\widetilde{\mu}\chi\overline{{\chi}}e^{\widetilde{b}e_{r-1}\cdot\phi}+\widetilde{\mu}\sum_{i=1}^{r-1}e^{\widetilde{b}e_{i}\cdot\phi}+\widetilde{U}_{\sigma}\right]
R_{\alpha\beta\mu\nu}=\frac{1}{\Box}\{\nabla_{\mu}\nabla_{\alpha}R_{\nu\beta}+\nabla_{\nu}\nabla_{\beta}R_{\mu\alpha}-\nabla_{\nu}\nabla_{\alpha}R_{\mu\beta}-\nabla_{\mu}\nabla_{\beta}R_{\nu\alpha}\}+O({\calR}^{2})\,\,\,,
V=\frac{1}{2}\left\{\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}g_{n}g_{m}Tr\Phi^{n+m-2}-\frac{1}{N}\sum_{n=0}^{\infty}g_{n}\sum_{p=0}^{n-2}Tr\Phi^{p}Tr\Phi^{k-p-2}\right\}
\begin{array}{l}{\left\{\varphi^{6}~,~~\varphi^{2}\bar{\Psi}\Psi~,~~e^{-1}{\calB}_{\Sigma}({\hat{A}}^{*}A)~,~~e^{-1}{\calB}_{\Sigma}(\bar{\Psi}^{*}\Psi+\bar{\Psi}\Psi^{*})~,~~e^{-1}{\calB}_{\Sigma}(\varphi^{*}\varphi)~\right\}\,,}\\\end{array}
a_{D}=\partial_{a}{\calF}={\frac{\sqrt2}{\pi}}\int_{\Lambda^{2}}^{u}{\frac{dx\sqrt{x-u}}{\sqrt{x^{2}-{\Lambda^{4}}}}},\qquada={\frac{\sqrt2}{\pi}}\int_{-{\Lambda^{2}}}^{\Lambda^{2}}{\frac{dx\sqrt{x-u}}{\sqrt{x^{2}-{\Lambda^{4}}}}},
g=\left(\begin{array}{cccc}{0}&{0}&{-\delta(x-y)}&{0}\\{0}&{0}&{0}&{-\partial_{1}^{2}\delta(x-y)}\\{\delta(x-y)}&{0}&{0}&{0}\\{0}&{\partial_{1}^{2}\delta(x-y)}&{0}&{0}\\\end{array}\right),
\tilde{W}_{tree}=\frac{1}{2\mu}S^{ij}\tilde{Q}_{i}\tilde{Q}_{j}+\frac{1}{2\mu}\Phi^{st}\tilde{q}_{s}\tilde{q}_{t}+\frac{1}{\mu}N^{is}\tilde{Q}_{i}\tilde{q}_{s}.
V_{3}=(0(0~0)^{3}\vert\vert{\tilde{e}}^{2}~0^{2}~\vert({\frac{1}{3}}~{\frac{1}{3}}~{\frac{1}{3}}~{\frac{1}{3}}~{\frac{2}{3}})_{r}^{3}\vert0_{r})~.
S_{\mathrm{boson}}=\frac{8\pi}{e^{2}}\intdtdr\sqrt{A(r)B(r)}\;\:r^{2}\:\:\left\{\frac{1}{\sqrt{B(r)}}\frac{\partiala_{1}(r,t)}{\partialt}-\frac{1}{\sqrt{A(r)}}\frac{\partiala_{0}(r,t)}{\partialr}\right\}^{2}
|\frac{1}{2},\frac{1}{2}\rangle_{0}=\frac{1}{\sqrt{2(q+q^{-1}+1)}}((q^{1/2}+q^{-1/2})|\uparrow\downarrow\uparrow\rangle-q^{-1/2}|\downarrow\uparrow\uparrow\rangle-q^{1/2}|\uparrow\uparrow\downarrow\rangle)
Q={\calC}_{0}^{a}G_{a}+if({\bfx}^{2}){\calC}_{1}^{1}x^{a}{\calP}_{0a}-\frac{1}{2}\varepsilon_{ab}{}^{c}{\calC}_{0}^{a}{\calC}_{0}^{b}{\calP}_{0c}+\pi_{0a}\bar{\calP}_{0}^{a}+\pi_{11}\bar{\calP}_{1}^{1}+\pi_{11}^{1}\bar{\calP}_{1}^{11}\;,
\left(\begin{matrix}{a}&{b}\\{c}&{d}\\\end{matrix}\right)\mapsto\left(\begin{matrix}{K^{\frac{1}{2}}\otimesK^{-\frac{1}{2}}}&{q^{-\frac{1}{2}}\lambdaK^{\frac{1}{2}}\otimesK^{-\frac{1}{2}}E}\\{-q^{\frac{1}{2}}\lambdaFK^{\frac{1}{2}}\otimesK^{-\frac{1}{2}}}&{\quadK^{-\frac{1}{2}}\otimesK^{\frac{1}{2}}-\lambda^{2}FK^{\frac{1}{2}}\otimesK^{-\frac{1}{2}}E}\\\end{matrix}\right).
\Theta=\frac{c}{16\pi^{2}}\,(W_{\mu\nu\rho\sigma})^{2}-\frac{a}{16\pi^{2}}\,\widetilde{R}_{\mu\nu\rho\sigma}\widetilde{R}^{\mu\nu\rho\sigma}+\frac{c}{6\pi^{2}}\,V_{\mu\nu}^{2}~.
\frac{1}{V}\frac{\langle0_{m}|H|0_{m}\rangle}{\langle0_{m}|0_{m}\rangle}=\frac{B^{2}}{2}-\frac{eB}{2\pi}\sum_{n=1}\sqrt{2neB+m^{2}}+m^{2}\frac{eB}{2\pi}\sum_{n=1}^{\infty}\frac{1}{\sqrt{2neB+m^{2}}}
\delta_{\xi}g_{\mu\nu}(p)=i\intd^{4}r\Bigl\{g_{\rho\nu}(p-r)r_{\mu}\xi^{\rho}(r)+g_{\rho\mu}(p-r)r_{\nu}\xi^{\rho}(r)+\xi^{\rho}(p-r)r_{\rho}g_{\mu\nu}(r)\Bigr\},
\left(-1\right)^{F}=2^{4}\,d_{0}^{2}d_{0}^{3}d_{0}^{4}d_{0}^{5}d_{0}^{6}d_{0}^{7}d_{0}^{8}d_{0}^{9}\left(-1\right)^{\sum_{n>0}d_{-n}^{i}d_{n}^{i}},
H_{0}=\int_{(\infty)}\frac{d^{3}\vec{k}}{(2\pi)^{3}2\omega(\vec{k}\,)}\frac{1}{2}\omega(\vec{k}\,)\left[a^{*}(\vec{k}\,)a(\vec{k}\,)+a(\vec{k}\,)a^{*}(\vec{k}\,)\right]\,
\sigma^{1}\,=\,\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\;\;,\;\;\sigma^{2}\,=\,\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right)\;\;,\;\;\sigma^{3}\,=\,\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\;.
g(z)=\frac{(q^{2}z;p,q^{4})_{\infty}(q^{4}z^{-1};p,q^{4})_{\infty}(pz^{-1};p,q^{4})_{\infty}(pq^{2}z;p,q^{4})_{\infty}}{(q^{2}z^{-1};p,q^{4})_{\infty}(q^{4}z;p,q^{4})_{\infty}(pz;p,q^{4})_{\infty}(pq^{2}z^{-1};p,q^{4})_{\infty}}.
U_{i_{1}\,\,i_{2}}^{\,\,j_{1}\,\,j_{2}}V_{j_{1}\,\,j_{2}}^{\,\,p_{1}\,\,p_{2}}=\!\!\!\sum_{\rho\subset\lambda\otimes\mu}\!\!U(\lambda\mu\rho)V(\lambda\mu\rho)\left(\begin{array}{ccc}{\lambda}&{\mu}&{\rho}\\{i_{1}}&{i_{2}}&{m}\\\end{array}\right)^{\ast}\left(\begin{array}{ccc}{\lambda}&{\mu}&{\rho}\\{p_{1}}&{p_{2}}&{m}\\\end{array}\right).
\frac{1}{T_{c}}=\frac{\partial{\calS}_{c}}{\partialE_{c}}=\frac{dN}{dE_{c}}\frac{d{\calS}_{c}}{dN}=\frac{1}{k}\big(2\sqrt{N}-\frac{1}{\sqrt{N}}+{\calO}(N^{-3/2})\big)
\left[E^{2}-P_{z}^{2}-\frac{m^{2}}{4}+4\omega\left(\frac{E}{m}-(N+\left|m\right|+\frac{1}{2})\right)\right]-i\omegaP_{z}b_{-}-i\omega\left(N+\left|m\right|+\frac{1}{2}\right)P_{z}b_{+}=0
\left({\frac{d\tilde{\lambda}}{d\lambda}}\right)^{-1}{\frac{d}{d\tilde{\lambda}}}\left({\frac{d\tilde{\lambda}}{d\lambda}}\right)=-nn^{\prime}\left({\frac{dt}{d\tilde{\lambda}}}\right)^{2}+aa^{\prime}\sum_{j}\left({\frac{dx^{j}}{d\tilde{\lambda}}}\right)^{2}.
\begin{array}{rcl}{\phi_{0}}&{=}&{Z_{\phi}^{1/2}\phi,}\\{m_{0}}&{=}&{Z_{m}^{1/2}m,}\\{\lambda_{0}}&{=}&{Z_{\lambda}\lambda,}\\{J_{0}}&{=}&{Z_{\phi}^{-1/2}J.}\\\end{array}
\left[\begin{matrix}{\left(t-\frac{1}{2}\right)}&{z}\\{z^{*}}&{\left(\frac{1}{2}-t\right)}\\\end{matrix}\right]\left[\begin{matrix}{dt}&{dz}\\{dz^{*}}&{-dt}\\\end{matrix}\right]\left[\begin{matrix}{dt}&{dz}\\{dz^{*}}&{-dt}\\\end{matrix}\right]
\begin{array}{r}{D_{\mu}F_{a}^{\mu\nu}=(-\gamma)^{-1/2}\partial_{\mu}\left[(-\gamma)^{1/2}F_{a}^{\mu\nu}\right]+\epsilon_{abc}A_{b\mu}F_{c}^{\mu\nu}=}\\{\frac{g^{2}}{2}\left\{\left(D^{\nu}\Phi\right)^{+}\left(t^{a}\Phi\right)+\left(t^{a}\Phi\right)^{+}\left(D^{\nu}\Phi\right)\right\},}\\\end{array}
\left(\dot{x}\vphantom{x}^{M}G_{MN}\dot{x}\vphantom{x}^{N}\right)+\left(\frac{\gamma}{\alpha^{\prime}}\right)^{2}\left(\acute{x}\vphantom{x}^{M}G_{MN}\acute{x}\vphantom{x}^{N}\right)=0
\hat{\calR}_{\lambda=1}\,=\,Q^{2\left(\hat{\zeta}\otimes\hat{H}+\hat{H}\otimes\hat{\zeta}\right)}\left(1\!\mathrm{l}\otimes1\!\mathrm{l}+\sigmaQ^{\hat{\zeta}}\hat{X}_{+}\otimesQ^{-\hat{\zeta}}\hat{X}_{-}\right)
{\calL}={\frac{1}{2}}\epsilon^{ijk}(\partial_{j}A_{k}^{a})\epsilon_{ab}(\dot{A}_{i}^{b})-{\frac{1}{2}}\epsilon^{ijk}(\partial_{j}A_{k}^{a})\epsilon_{ab}(\partial_{i}A_{0}^{b})-{\frac{1}{4}}F^{a,jk}F_{jk}^{a}
+\intdx^{5}\varepsilon^{\alpha\beta\gamma\delta\sigma}\left[C_{\alpha\beta\gamma\delta\sigma}^{(5)}+{\frac{1}{12}}F_{\alpha\beta\gamma}C_{\delta\sigma}^{(2)}-{\frac{1}{24}}C_{\alpha\beta\gamma}^{(3)}C_{\delta\sigma}^{(2)}\right].
Z^{\ast}(\theta_{2})Z^{\ast}(\theta_{1})\Omega=\frac{1}{\sqrt{2}}(\chi_{21}a^{\ast}(\theta_{2})a^{\ast}(\theta_{1})\Omega+\chi_{12}S(\theta_{2}-\theta_{1})a^{\ast}(\theta_{1})a^{\ast}(\theta_{2})\Omega)
(\Psi,\widetilde\Psi)=\int_{-\infty}^{\infty}dx_{n}^{1}\int_{-\inftyi}^{\inftyi}i\,dx_{n}^{0}\Psi^{*}(x_{n}^{0},x_{n}^{1})\widetilde\Psi(x_{n}^{0},x_{n}^{1})\,
S=\int\!\!d^{D}\!x\,\left(\frac{1}{2}(\partial_{\mu}\phi)^{2}+\frac{1}{2}m_{0}^{2}\phi^{2}+\frac{1}{3!}f_{0}\phi^{3}+\frac{1}{4!}g_{0}\phi^{4}\right)\,.
W_{p}=\sum_{q_{1}=p}^{N}\sum_{q_{2}=p-1}^{q_{1}-1}\cdots\sum_{q_{p}=1}^{q_{p-1}-1}{\calD}^{0}\cdot{\calD}^{1}\cdots{\calD}^{p-1}\cdote^{\theta_{q_{p}-1}},
\vec{y}=\vec{y}_{0}+\left(\vec{v}_{0}-\vec{a}_{0}{\frac{e^{2}}{6\pim}}\right)t+\vec{a}_{0}\left({\frac{e^{2}}{6\pim}}\right)^{2}\,\left(\mathrm{exp}{\frac{6\pimt}{e^{2}}}-1\right)
\left(\begin{array}{ccc}{r(\alpha-1)s(\alpha-1)}&{z\left(1+\frac{MC_{2}(R)}{T(R)}\right)}&{\frac{MC_{2}(R)z}{2T(R)}}\\{-\,4}&{\!2q(\beta)-\Pi^{\prime}\!\left(1-\frac{MC_{2}(R)}{T(R)}\right)}&{\frac{MC_{2}(R)z\Pi^{\prime}}{2T(R)}}\\{0}&{0}&{\!\!\!q(\gamma)-\frac{Mz\Pi^{\prime}(4C_{2}(R)-C_{2}(G))}{4T(R)}}\\\end{array}\right)
{D}^{(1/2)}(R_{w}):={D}^{(1/2)}({\bfp},{\bfn})=\frac{p_{0}-{\bfp}\cdot{\bfn}+m-i{\sigma}\cdot({\bfp}{\times}{\bfn})}{\sqrt{2(p_{0}+m)(p_{0}-{\bfp}\cdot{\bfn})}},\quad{D}^{(1/2)}({\bfp},{\bfn})\,{D}^{\dagger(1/2)}({\bfp},{\bfn})=1.
\sum_{\nu=0}^{3}\eta^{\nu\nu}\left(\frac{\partial}{\partialx^{\nu}}\right)^{2}A_{\mu}(x;y)=\eta_{\mu\mu}4\pim_{y}G\frac{dy^{\mu}(\frac{x^{0}}{c})}{dx^{0}}\delta({\bfx}-{\bfy}(\frac{x^{0}}{c})),
\phi^{i}(x)=\varphi^{i}(x)+\sum_{jk}a_{jk}^{i}\varphi^{j}(x)\varphi^{k}(x)+\sum_{jkl}a_{jkl}^{i}\varphi^{j}(x)\varphi^{k}(x)\varphi^{l}(x)+\cdots.
S_{L}=-\frac{\ell}{32\piG}\int_{|r|=\bar{r}}[\frac12\stackrel{(0)}g\strut^{ab}\partial_{a}\phi\partial_{b}\phi-\frac8{\ell^{2}}e^{\phi}+\stackrel{(0)}{\calR}\phi]\,\sqrt{-\stackrel{(0)}{\bfg}}\,dt\,d\varphi\qquad.
\phi_{opt}(t,z,\theta)=\int_{-\infty}^{+\infty}dt^{\prime}\int_{S^{d}}d\mu(\theta^{\prime})e^{\frac{dz^{\prime}}{\beta_{H}}}\left(G\partial_{z^{\prime}}\varphi(z^{\prime},t^{\prime},\theta^{\prime})-\partial_{z^{\prime}}G\varphi(z^{\prime},t^{\prime},\theta^{\prime})\right)_{z^{\prime}=Z}\,
D_{\mu\nu}^{(0)}(p)=-\frac{1}{(p^{2}-m_{+}^{2})(p^{2}-m_{-}^{2})}\left[\eta_{\mu\nu}\,(p^{2}-m^{2})-p_{\mu}p_{\nu}\frac{(1-\xi)(p^{2}-m^{2})+\xi\kappa^{2}}{p^{2}-\xim^{2}}+i\,\kappa\,\epsilon_{\mu\nu\lambda}\,p^{\lambda}\right],
\delta[B_{\mu}\dot{y}^{\mu}+B_{\alpha}\dot{\psi}^{\alpha}+\lambda^{\alpha}\tilde{\chi}_{\alpha}-\tilde{H}]={\frac{d}{dt}}\{[B_{\mu}f^{\mu\nu}{\frac{\partial\tilde{\chi}_{\alpha}}{\partialy^{\nu}}}+B_{\beta}\omega^{\beta\rho}{\frac{\partial\tilde{\chi}_{\alpha}}{\partial\psi^{\rho}}}+\tilde{\chi}_{\alpha}]\epsilon^{\alpha}\}\,,
g_{BTZ}=-\frac{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}{r^{2}}dt^{2}+\frac{r^{2}}{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}dr^{2}+r^{2}(d\phi-\frac{r_{+}r_{-}}{r^{2}}dt)^{2}
\begin{array}{c}{R(u/v)(K_{l}(u)\otimes{\bf1})R(uvq^{-1})({\bf1}\otimesK_{l}(v)=}\\{=({\bf1}\otimesK_{l}(v))R(uvq^{-1})(K_{l}(u)\otimes{\bf1})R(u/v)\,,}\\\end{array}
\hat{\Xi}_{0,p}[\alpha,\beta]^{-1}=e^{i\pi(\alpha_{1}+...+\alpha_{p})}\zeta(\alpha_{1},\alpha_{2},\beta_{3})\prod_{j=3}^{p-2}\zeta(\beta_{j},\alpha_{j},\beta_{j+1})\zeta(\beta_{p-1},\alpha_{p-1},\alpha_{p})\prod_{j=3}^{p-1}{\nu_{1}(d_{\beta})}.
\begin{array}{rll}{\mathrm{(A)}\}&{h_{1}=f_{2}=2\,\h_{2}=f_{1}=0\:\quad}&{N=8\,\g_{\mathrm{s}}=1\,\C=0\;\nonumber}\\{\mathrm{(B)}\}&{h_{1}=2\,\f_{2}=4\,\h_{2}=f_{1}=0\:\quad}&{N=0\,\g_{\mathrm{s}}=\mathrm{~\frac{1}{2}~}\,\C=0\;\nonumber}\\{\mathrm{(C)}\}&{h_{1}=-h_{2}=f_{1}=f_{2}=2\:\quad}&{N=0\,\g_{\mathrm{s}}=1\,\C=0\.}\\\end{array}
\sum_{p_{1},\ldots,p_{i}\inT_{2}}\sum_{s_{1},t_{1},\ldots,s_{G},t_{G}\inS_{n}}N^{n(2-2G)-i}\left[\frac{1}{n!}\delta(p_{1}\cdotsp_{i}\Omega_{n}^{2-2G}\prod_{j}s_{j}t_{j}s_{j}^{-1}t_{j}^{-1})\right].
S^{\mu\nu}\equiv{\frac{\partialL}{\partial\sigma_{\mu\nu}}}=L^{-1}[J^{2}\sigma^{\mu\nu}+{\frac{MJ}{{({\frac{a_{1}a_{2}}{2}}+a_{3}})^{\frac{1}{2}}}}(a_{1}\sigma^{\mu\nu}+(u^{\mu}\sigma^{\nu\lambda}-u^{\nu}\sigma^{\mu\lambda})u_{\lambda})].
1-A_{\pm}(p)-p\,A_{\pm}^{\prime}(p)=-\frac{\alpha_{0}}{\pil}\int_{-\infty}^{\infty}dk\frac{A_{\pm}(k)\,k}{A_{\pm}^{2}(k)\,k^{2}+B_{\pm}^{2}(k)}\,\frac{1}{p-k},
\zeta_{\pm\alpha}^{u}(\rho_{0},x-x_{0})=\frac{1}{\sqrt{\rho_{0}}}\left(\rho_{0}\eta_{\pm\alpha}^{u}+{\sigma^{\mu}}_{\alpha{\dot{\alpha}}}(x_{\mu}-x_{0\,\mu}){\overline{\xi}}_{\pm}^{{\dot{\alpha}\,u}}\right)\,,
L_{3}=\dot{x}p_{x}+\dot{y}p_{y}+\dot{z}p_{z}-\frac{1}{2}p_{x}^{2}-\frac{1}{2}p_{y}^{2}-\frac{1}{2}p_{z}^{2}-\frac{1}{2}\omega^{2}\left(x^{2}+y^{2}\right)-\xi\left[p_{z}+g(xp_{y}-yp_{x})\right]\\\.
\left\langle\frac{1}{2N}\mathrm{Tr}\,H_{S}^{2}\right\rangle-\left\langle\frac{1}{2N}\mathrm{Tr}\,H_{S}\right\rangle^{2}=\left\langle\frac{1}{N}\mathrm{Tr}\,H_{0}^{2}\right\rangle-\left\langle\frac{1}{N}\mathrm{Tr}\,H_{0}\right\rangle^{2}-\left\langle\frac{1}{2N}\mathrm{Tr}\,(\vec{\sigma}(\vec{A}\times\vec{A})\vec{\sigma}(\vec{A}\times\vec{A}))\right\rangle
\dot{Q}_{B}=-\frac{i}{\hbar}\frac{q^{2}}{1+q^{2}}\Bigl(\;[\;c\;(p^{2}-m^{2}),\dot{\bar{c}}\;\dot{c}\;]_{q}+[\;b\;\dot{c},e\;(p^{2}-m^{2})\;]_{q}\;\Bigr).
d_{C}\alpha=-{\frac{1}{m!}}\Big({\mathrm{sTr}}(F_{L}^{m})-{\mathrm{sTr}}(({g^{-1}F_{R}g})^{m})\Big)=-{\frac{1}{m!}}\Big({\mathrm{sTr}}(F_{L}^{m})-{\mathrm{sTr}}(F_{R}^{m})\Big)\quad.
{\{\pi_{i},\pi_{j}\}}_{D(\Phi)}=gG_{ij}^{a}(x)I^{a}+ig\left(\nabla_{k}^{ab}G_{ij}^{b}(x)\right)I^{a}\xi_{k}({\calP}_{m}\xi_{m})\frac{(b\kappa+a)}{\tilde{\beta}(\omega+\tilde{m})}=gG_{ij}^{a}(q)I_{\varphi}^{a},
\vec{R}_{\nu\mu}\vec{J}^{\mu}=(\vec{\Gamma}_{\nu}\times\vec{\Gamma}_{\mu})[\vec{\Gamma}_{\sigma}\times\vec{R}^{\mu\sigma}]=(\vec{\Gamma}_{\nu}\vec{\Gamma}_{\sigma})(\vec{\Gamma}_{\mu}\vec{R}^{\mu\sigma})-(\vec{\Gamma}_{\mu}\vec{\Gamma}_{\sigma})(\vec{\Gamma}_{\nu}\vec{R}^{\mu\sigma})=0
{\calH}_{1.c_{0}\neq0}=\frac{s}{2c_{1}^{2}c_{0}}\left[2\imathc_{1}^{2}c_{0}{\bfp}({\bfz}-{\bf{\bar{z}}})+(c_{0}^{2}-{\bfp}^{2})({\bfz}{\bar{\bfz}}-c_{1}^{2})\right].
\phi_{1}=\left(\begin{array}{cc}{s-ib_{12}}&{b_{13}-ib_{23}}\\{-b_{13}-ib_{23}}&{s+ib_{12}}\\\end{array}\right),\quad\phi_{2}=\left(\begin{array}{cc}{-b_{03}+ip}&{-b_{01}+ib_{02}}\\{-b_{01}-ib_{02}}&{b_{03}+ip}\\\end{array}\right),
A_{1}=\frac{1}{\sqrt{2}}\left(\begin{matrix}{A_{1}^{0}}&{W_{1}^{1}}&{\cdots}&{W_{1}^{N_{1}}}\\{W_{1}^{1*}}&{}&{}&{}\\{\vdots}&{}&{}&{}\\{W_{1}^{N_{1}*}}&{}&{}&{}\\\end{matrix}\right),\;\;A_{2}=-\frac{1}{\sqrt{2}}\left(\begin{matrix}{A_{2}^{0}}&{W_{2}^{1}}&{\cdots}&{W_{2}^{N_{2}}}\\{W_{2}^{1*}}&{}&{}&{}\\{\vdots}&{}&{}&{}\\{W_{2}^{N_{2}*}}&{}&{}&{}\\\end{matrix}\right)
S_{parent}^{\mathrm{IV}}=-\frac{T_{p}}{2}\intd^{p+1}{\xi}\left[\sqrt{-g}\left({\Psi}^{(p-1)/2}g^{ij}h_{ij}-(p-1){\Psi}^{(p+1)/2}\right)+{\Lambda}\left({\Psi}-{\Phi}\right)\right].
\varphi_{i}(x)=\sum_{n>0}\frac{1}{\sqrt{4n\pi}}\left[a_{n}^{(i)}(x^{+},x^{\bot})e^{-i\frac{\pin}{L}x^{-}}+a_{n}^{(i)\dagger}(x^{+},x^{\bot})e^{i\frac{n\pi}{L}x^{-}}\right]\quad,
\bar{c}_{3}^{N}=\frac{c_{2}^{N}\bar{b}_{1}^{N}}{b_{2}^{N}}\frac{b_{2}^{N}\bar{c}_{1}^{N}-\bar{a}_{1}^{N}d_{2}^{N}}{c_{2}^{N}\bar{b}_{1}^{N}-a_{2}^{N}\bar{d}_{1}^{N}},
\bar{S}_{free}^{(0)}[\phi]=-{\frac{1}{2}}\mathrm{Tr}\left([\hat{x}_{\mu}^{\prime},\phi(\hat{x})][\hat{x}_{\mu}^{\prime},\phi(\hat{x})]\right)={\frac{1}{2}}\mathrm{Tr}\,(\hat{\partial}_{\mu}\phi(\hat{x}))^{2}.
\vec{A}\:(\vec{r}\,)\;=\;-\lambda\:\hat{n}\:{\times}\:{\int}\frac{\vec{r}-\vec{r}\,^{\prime}}{{\mid\vec{r}-\vec{r}\,^{\prime}\mid}^{2}}\;\rho(\vec{r}\,^{\prime}\,)\;d^{2}r^{\prime}\;,
T^{00}=\frac{1}{2}B^{2}+\frac{1}{2}E^{2}+\frac{a}{e^{2}\Lambda}(\partial_{i}E_{i})^{2}+b\frac{e^{2}}{4}\LambdaA^{2}+\frac{1}{\Lambda}\left[c:(A\cdot\tilde{E})^{2}:+d:E^{2}A^{2}:\right]+fA\cdot\tilde{E}
\kappa_{11}^{2}\,\deltaS=-\frac{1}{72}\Big(\bar{\psi}_{\lambda}\Gamma^{\mu\nu}{}_{\sigma_{1}\cdots\sigma_{4}}\epsilon+24\,\bar{\psi}_{\lambda}\Gamma^{\sigma_{1}\sigma_{2}}\delta_{\sigma_{3}}{}^{\mu}\delta_{\sigma_{4}}{}^{\nu}\epsilon\Big)\,\Big(\frac{1}{2}T_{\mu\nu}{}^{\lambda}-T_{\mu\kappa}{}^{\kappa}\delta_{\nu}{}^{\lambda}\Big)\,.
{\overline{\phi}}(p,z)=-\int{\frac{d^{2}k}{(2\pi)^{2}}}e^{-ik\cdotz}\left[bL^{2}{\frac{{\calO}_{4}(p)}{k^{2}+p^{2}}}+(2\pi)^{4}cL^{-2}{\frac{\delta^{(4)}(p)}{k^{2}}}\right]~.
ds^{2}~\sim~\widetilde\Omega^{\frac{1}{4}}~\Big[2^{\frac{2}{3}}\,3^{\frac{1}{3}}\,e^{2C_{1}}\,\chi^{\frac{2}{3}}\,\big(\eta_{\mu\nu}dx^{\mu}dx^{\nu}\big)-d\chi^{2}\Big]~-~\widetilde\Omega^{-{\frac{3}{4}}}~\Big[~{\frac{1}{3\chi^{2}}}\,dw_{2}^{2}~+~d\tilde{s}_{4}^{2}\Big]\,,
L(\phi,\Lambda)=\sum_{m=1}^{\infty}{\frac{1}{(2m)!}}\int{\frac{dp_{1}...dp_{2m}}{(2\pi)^{8m-4}}}L_{2m}(p_{1},...,p_{2m};\Lambda)\delta^{4}(p_{1}+...+p_{2m})\phi(p_{1})...\phi(p_{2m})
B^{A_{1}...A_{p}}=\epsilon^{A_{1}...A_{p}rB_{1}...B_{p}}\\partial_{r}A_{B_{1}...B_{p}}+p\,\epsilon^{A_{1}...A_{p}B_{1}rB_{2}...B_{p}}\A_{rB_{2}...B_{p}||B_{1}}\
\Pi(q^{2})\longrightarrow-i\frac{Ng^{2}}{2}\left[8-\left(\frac{n-2}{n-1}\right)\right]\mu^{2\epsilon}\int\frac{d^{n}\ell}{(2\pi)^{n}}\frac{1}{\ell^{2}(\ell+q)^{2}}
\chi_{\mu}=\left(\begin{matrix}{A_{\mu}}&{\bar{B}_{\mu}}\\{B_{\mu}}&{A_{\mu}}\\\end{matrix}\right)\to\left(\begin{matrix}{A_{\mu}-C_{\mu}}&{\bar{B}_{\mu}}\\{B_{\mu}}&{A_{\mu}+C_{\mu}}\\\end{matrix}\right).
\omega^{i_{p}}(g,\lambda)=\sum_{\alpha=0}^{\infty}\lambda^{\alpha}\omega^{i_{p},\alpha}(g)=\omega^{i_{p},0}(g)+\lambda\omega^{i_{p},1}(g)+\lambda^{2}\omega^{i_{p},2}(g)+\ldots\;;\quadp=0,1\;,
0\rightarrow{\calO}_{\pi^{*}{\calS}}\rightarrow{\calO}_{\pi^{*}{\calS}}(\sigma|_{\pi^{*}{\calS}})\rightarrow{\calO}_{\pi^{*}{\calS}}(\sigma|_{\pi^{*}{\calS}})|_{\sigma|_{\pi^{*}{\calS}}}\rightarrow0.
D^{1}(\psi)=\left(\begin{matrix}{\psi_{1}^{2}}&{-\sqrt{2}\psi_{1}\psi_{2}^{*}}&{\psi_{2}^{*}}\\{\sqrt{2}\psi_{1}\psi_{2}}&{\mid\psi_{1}\mid^{2}-\mid\psi_{2}\mid^{2}}&{-\sqrt{2}\psi_{1}^{*}\psi_{2}^{*}}\\{\psi_{2}^{2}}&{\sqrt{2}\psi_{2}\psi_{1}^{*}}&{\psi_{1}^{*2}}\\\end{matrix}\right)=e^{i\Sigma_{3}\alpha}~e^{i\Sigma_{2}\beta}~e^{i\Sigma_{3}\gamma}
\begin{array}{l}{\deltaa=-i\bar{\zeta}\psi}\\{\deltab=\bar{\zeta}\gamma_{W}\psi}\\{\delta\psi=(f+i\gamma_{W}g)+i\gamma^{\mu}\partial_{\mu}(a+i\gamma_{W}b)\zeta}\\{\deltaf=\bar{\zeta}\not\!\partial\psi}\\{\deltag=i\bar{\zeta}\gamma_{W}\not\!\partial\psi,}\\\end{array}
{\calL}[E_{\mu}{}^{\bar{\Lambda}}=\partial_{\mu}Z^{\Lambda}E_{\Lambda}{}^{\bar{\Lambda}}(Z),A_{\mu_{0}\dots\mu_{p}}=\partial_{\mu_{0}}Z^{\Lambda_{0}}\dots\partial_{\mu_{p}}Z^{\Lambda_{p}}A_{\Lambda_{0}\dots\Lambda_{p}}(Z)]
\langle\zeta^{a}\left(z\right)\zeta^{b}\left(w\right)\rangle=\frac{i}{\pi}\theta^{ab}\mathcal{A}\left(z,w\right)-\frac{1}{\pi}G^{ab}\mathcal{B}\left(z,w\right)+\frac{1}{2\pi}g^{ab}\mathcal{C}\left(z,w\right),
V(u)=E\left\{1-\frac{1}{2}\left(\frac{u}{u_{*}}\right)^{7-p}\left[1-\frac{1}{\left(1+(u_{*}/u)^{7-p}\right)^{2}}\right]\right\}+\frac{Eu_{**}^{2}}{2u^{2}\left(1+(u_{*}/u)^{7-p}\right)^{2}},
F^{(1)}=-i\int_{x_{0}}^{x}dx_{j}A_{j}^{(1)}-\frac{i}{2}\int_{x_{0}}^{x}dx_{j}\,\epsilon^{ml}k_{m}\,\partial_{l}A_{j}^{(0)}+\frac{i}{2}\int_{x_{0}}^{x}dx_{j}\,\epsilon^{ml}A_{m}^{(0)}\,\partial_{l}A_{j}^{(0)}.
O_{F}=\left(\begin{array}{cc}{-\frac{d^{2}}{dz^{2}}+\frac{\partial^{2}}{\partial\phi^{2}}V}&{\frac{\partial^{2}}{\partial\chi\partial\phi}V}\\{\frac{\partial^{2}}{\partial\phi\partial\chi}V}&{-\frac{d^{2}}{dz^{2}}+\frac{\partial^{2}}{\partial\chi^{2}}V}\\\end{array}\right)_{\vert\phi=\phi(z),\chi=\chi(z)}
2\sum_{a=1}^{\tilde{n}_{c}}(\delta_{\ell}-\phi_{a})=2\tilde{n}_{c}\delta_{\ell}-2\sum_{a=1}^{\tilde{n}_{c}}\phi_{a}=2\tilde{n}_{c}\delta_{\ell}+2\sum_{k=1}^{n_{c}-\tilde{n}_{c}}\gamma_{k},
\overline{{G}}_{\sigma}^{(n)}(\zeta^{\prime},x^{2}\zeta_{2n})^{\varepsilon^{\prime}\,\varepsilon_{2n}}=\sigma\overline{{G}}_{\sigma}^{(n)}(\zeta_{2n},\zeta^{\prime})^{\varepsilon_{2n}\,\varepsilon^{\prime}}\prod_{j=1}^{2n-1}\frac{\zeta_{2n}}{\zeta_{j}}.
\left(\kappa\frac{\partial}{\partialz}-\frac{\beta^{2}}{2z}\right)G^{\epsilon}\left(z,\bar{z}\right)=\frac{\beta\beta_{1}}{z}\quadG^{-\epsilon}\left(z,\bar{z}\right)-\frac{1}{\sqrt{z(1-z)}}H^{\epsilon}(z,\bar{z}),
\frac{H}{p}ww^{\prime}\Bigg|_{r_{1}}^{r_{2}}=\int_{r_{1}}^{r_{2}}dr\,\Bigg\{\frac{H}{p}(w^{\prime})^{2}-\frac{p}{H}u^{2}w^{2}-\frac{1}{pr^{2}}w^{2}(1-w^{2})\Bigg\}~~.
d_{j_{1},j_{2}}^{j_{3}}=\frac{a_{j_{1}}a_{j_{2}}}{a_{j_{3}}}\;\prod_{l=1}^{j_{1}+j_{2}-j_{3}}\frac{\Gamma(\frac{l}{k+2})}{\Gamma(-\frac{l}{k+2})}\;\prod_{l=0}^{j_{1}+j_{2}-j_{3}-1}\frac{\Gamma(-\frac{2j_{1}-l}{k+2})\;\Gamma(-\frac{2j_{2}-l}{k+2})\;\Gamma(\frac{2j_{3}+2+l}{k+2})}{\Gamma(\frac{2j_{1}-l}{k+2})\;\Gamma(\frac{2j_{2}-l}{k+2})\;\Gamma(-\frac{2j_{3}+2+l}{k+2})}
\int_{\calM}\,\left\langle\xi(z)\cdot\prod_{\alpha}^{3g-3}\,\int\mu^{\alpha}b\cdot\prod_{\beta=1}^{2g-2}\,X(z_{\beta})\cdot\prod_{\gamma=1}^{g}\,\delta_{P_{\gamma}\,,\,p_{\gamma}}\,\oint_{{\calA}_{\gamma}}\eta\cdot\prod_{l=1}^{n}\,\intd^{2}z_{l}\,V_{i_{l}}(z_{l})\,\right\rangle\.
I_{B}=\frac{1}{C}(-1)^{\frac{S-\widetilde{S}}{2}}\sum_{\{m_{a}\}}\delta_{\frac{M+\widetilde{M}}{2}+\sum_{j=1}^{r}\frac{K^{\prime}}{2k_{j}+4}(m_{j}+1)}^{(K^{\prime})}\prod_{j=1}^{r}N_{L_{j},\widetilde{L}_{j}}^{m_{j}-1}
{\calL}_{F}^{(2)}=\frac{\mathrm{i}}{2}(\theta_{i}\tilde{{\calD}}_{0}\theta^{i}+\eta_{i}\tilde{{\calD}}_{0}\eta^{i}-2\mathrm{i}\eta_{i}e_{0}^{i}{}_{j}\eta^{j})+e^{2\phi}\eta^{i}C_{ij}^{\prime}\tilde{{\calD}}_{1}\theta^{j}+h.c.\,
K_{x\mu\nu}^{R\rho\sigma}=\frac{1}{4\beta}(x_{\mu}x^{\rho}\delta_{\nu}^{\sigma}+x_{\nu}x^{\sigma}\delta_{\mu}^{\rho}+x_{\mu}x^{\sigma}\delta_{\nu}^{\rho}+x_{\nu}x^{\rho}\delta_{\mu}^{\sigma}).
\left(Z^{(B)\alpha\beta\lambda}\right)^{\rho}=\frac{g}{2}\left(g^{\alpha\rho}g^{\beta\lambda}-g^{\alpha\lambda}g^{\beta\rho}\right)+g\left(\varepsilon^{\alpha\beta\lambda\rho}+\varepsilon^{\lambda\mu\nu\rho}\varepsilon^{\alpha\beta\gamma\delta}A_{\gamma(\mu)}A_{\delta(\nu)}\right).