harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\calF}_{{\mu}_{1}{\cdots}{\mu}_{p+1}}^{a}(A)=\frac{(-1)^{p+1}}{(p+1)!}{\Gamma}^{ab}{\epsilon}_{{\mu}_{1}{\cdots}{\mu}_{p+1}{\nu}_{1}{\cdots}{\nu}_{p+1}}{\calF}^{b{\nu}_{1}{\cdots}{\nu}_{p+1}}(A),
J={\frac{ig^{2}f_{acd}f_{bcd}}{(2\pi)^{6}}}{\frac{(1^{-}2^{-})(\theta_{31})^{2}(\theta_{32})^{2}}{\hat{x}_{12}^{2}\check{x}_{31;2}^{2}\check{x}_{32;1}^{2}}}\;.
\begin{array}{lcl}{<u_{k},u_{l}>_{\theta}}&{=}&{\delta_{k+l,1}\int{dzu_{1-k}\staru_{k}},}\\{}&{=}&{\delta_{k+l,1}\int{dzu_{1-k}.u_{k}}}\\{}&{=}&{<u_{k},u_{l}>}\\\end{array}
\langle\kappa^{\prime},s|L_{0}|\kappa,s\rangle=\left[\frac{s}{2}+\frac{i\kappa}{4}\right]\,\delta(\kappa-2i-\kappa^{\prime})+\left[\frac{s}{2}-\frac{i\kappa}{4}\right]\,\delta(\kappa+2i-\kappa^{\prime}).
A_{k}(t,r,\theta,\varphi)=\sum_{l=1}^{\infty}\sum_{m=-l}^{l}\Bigr[c_{l}(r)\partial_{k}Y_{lm}(\theta,\varphi)+d_{l}(r)\varepsilon_{kp}\partial^{p}Y_{lm}(\theta,\varphi)\Bigr]e^{i\omegat}.
P_{j}^{i}h_{i}^{j}(u,p)=\frac{\sqrt{6}}{l^{2}}\left\{[16-6\alpha(\alpha-2)]A_{1}+[4+3\alpha(\alpha-2)]B_{1}+24\frac{L^{2}}{l^{2}}B_{2}\right\}+\mathcal{O}(u)~.
\begin{array}{lll}{\Omega_{J_{0}}^{e}\equivE_{E}}&{\Omega_{\beta_{l}}^{r^{k}}\equivE_{R^{k}}}&{\Omega^{sr^{p}}\equiv\mathrm{gcd}(2m+1,j)(E_{R^{p}}\oplusE_{R^{x-p}})}\\{\Omega_{J_{1}}^{e}\equivE_{E}}&{\Omega^{s_{\gamma_{0}}}\equiv\mathrm{gcd}(2m+1,j)E_{E}}&{~}\\{\Omega_{\alpha_{l}}^{e}\equiv2E_{E}}&{\Omega^{s_{\gamma_{1}}}\equiv\mathrm{gcd}(2m+1,j)E_{E}}&{~}\\\end{array}
\mu\equivm_{H}^{2},~C\equiv\sqrt{1+\frac{Ka_{2}}{2}(\xiA(0)B(0))^{2}\left(1+\frac{\Lambda^{2}}{m_{H}^{2}}\right)^{\frac{K}{2\pi}}}\simeq\sqrt{1+\frac{Ka_{2}}{2}(\xiA(0)B(0))^{2}\left(1+\frac{\Lambda^{2}}{m_{H}^{2}}\frac{K}{2\pi}\right)}.
\frac{E_{0}\left(\ell,\mu,d\right)}{L^{d-1}}\approx-C\left(d\right)\frac{\Gamma\left(\frac{d}{2}+1\right)}{2^{d+3}\pi^{d/2}\Gamma\left(\frac{d}{2}\right)\ell^{d}}\left[\left(\frac{d}{2}+1\right)\mu^{\frac{d}{2}-2}+2\mu^{\frac{d}{2}-1}\right]e^{-2\mu}.
\left(\begin{array}{l}{z_{1}}\\{z_{2}}\\\end{array}\right)=\left(\begin{array}{l}{\left(p_{2}^{14}+p_{2}^{13}\right)-i\left(p_{2}^{11}+p_{2}^{12}\right)}\\{\left(p_{2}^{11}-p_{2}^{12}\right)-i\left(p_{2}^{14}-p_{2}^{13}\right)}\\\end{array}\right)
\{f,g\}_{1}=\sum_{i=1}^{N}\left(\frac{\partial^{R}f}{\partialx^{i}}\frac{\partial^{L}g}{\partial\theta_{i}}+\frac{\partial^{R}f}{\partial\theta_{i}}\frac{\partial^{L}g}{\partialx^{i}}\right)
\int^{\Lambda}\!\frac{d^{D}k}{(2\pi)^{3}}\frac{1}{k^{2}}+\int\!\frac{d^{D}k}{(2\pi)^{3}}\left[\frac{k^{2}+m_{2}^{2}}{(k^{2}+m_{+}^{2})(k^{2}+m_{-}^{2})}-\frac{1}{k^{2}}\right],
L^{*}=\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{abcd}\bigg[\bigg(-{\frac{c_{3}^{2}}{4c_{1}^{+}}}\bigg){^+}F_{\mu\nu}^{ab}{^+}F_{\alpha\beta}^{cd}-\bigg(-{\frac{c_{4}^{2}}{4c_{1}^{-}}}\bigg){^-}F_{\mu\nu}^{ab}{^-}F_{\alpha\beta}^{cd}\bigg].
\times\sum_{p=0}^{d-2}\sum_{l=0}^{p}(-1)^{p+l}\left(\begin{array}{c}{p}\\{l}\\\end{array}\right)w_{p}\sum_{m=1}^{\infty}\zeta_{R}\left(2z+2k-l,\gammam+{\frac{d-1}{2}}\right)\left(\gammam+{\frac{d-1}{2}}\right)^{p-l}~~~,
S_{\mathrm{pert}}=\intdx^{0}F^{i\overline{{\jmath}}}\left[{\frac{i}{2}}(\phi_{i}\widetilde{D}_{0}\overline{{\phi}}_{j}-\overline{{\phi}}_{j}\widetilde{D}_{0}\phi_{i})+\tau_{i}\overline{{\tau}}_{j}\right]
\begin{aligned}{[m]x^{i}}&{{}=x_{0}^{i}+\eta\,\deltax^{i}(\tau)}\\{r}&{{}=r_{0}+\eta\,\deltar(\tau)\qquad,\qquad\theta=\theta_{0}+\eta\,\delta\theta(\tau)}\\{\varphi}&{{}=\omega_{0}\tau+\varphi_{0}+\eta\,\delta\varphi(\tau)}\\{\psi}&{{}=v_{0}\tau+\psi_{0}+\eta\,\delta\psi(\tau)~.}\\\end{aligned}
\mu_{k}\mu_{k}^{*}{}^{\prime}-\mu_{k}^{\prime}\mu_{k}^{*}=\left[\vertC_{1}(k)\vert^{2}-\vertC_{2}(k)\vert^{2}\right]\frac{\mathrm{e}^{2i\pib}}{2k_{\mathrm{c}}\vert\eta_{0}\vert}{\calF},
T_{a\dot{a}}=\left(\begin{array}{ccc}{T_{2121}}&{T_{2111}-T_{2122}}&{T_{2112}}\\{T_{1121}-T_{2221}}&{h_{1}-h_{2}}&{T_{1112}-T_{2212}}\\{T_{1221}}&{T_{1211}-T_{1222}}&{T_{1212}}\\\end{array}\right).
\chi_{1}^{n}=\frac{t_{3}K^{n}/m}{t_{1}+t_{2}+t_{3}}-M_{1}q^{n},\quad\chi_{2}^{n}=\frac{t_{2}K^{n}/m}{t_{1}+t_{2}+t_{3}}-M_{2}q^{n},\quad\chi_{3}^{n}=\frac{t_{1}K^{n}/m}{t_{1}+t_{2}+t_{3}}+Mq^{n},
{}~~~~~~~~~~~~~\times\Bigg\{\sum_{\sigma\inZ_{N}}\frac{w(x_{3},x_{13},x_{1}|d,h+\sigma)w(x_{4},x_{24},x_{2}|a,g+\sigma)}{w(x_{4},x_{14},x_{1}|e,c+\sigma)w(x_{3}/\omega,x_{23},x_{2}|f,b+\sigma)}\Bigg\}_{0}.
S_{\mathrm{FB}}=y\sum_{x}(-1)^{\sum_{\nu}x_{\nu}}(\bar{\psi}_{Lx}^{(1)}\phi_{x}\psi_{Rx}^{(1)}+\bar{\psi}_{Rx}^{(1)}\phi_{x}^{*}\psi_{Lx}^{(1)}),
\sum_{i}^{n}\!\left<g_{++}\!(\xi_{1})\!\dots\!\delta_{\!f}\!g_{++}\!(\xi_{i})\!\dots\!g_{++}\!(\xi_{n})\right>\!+\!\frac{d\!\!-\!\!28\!\!+\!\!\lambda^{D}}{i48\pi}\!\!\int\!\!d\xi^{2}\!\,\delta\!f\!(\!\xi\!)\left<\partial_{-}^{3}g_{++}\!(\xi)g_{++}\!(\xi_{1})\!\dots\!g_{++}\!(\xi_{n})\right>\!\!=\!\!0.
{\Phi}^{(I)}=-\frac{s\,e^{2}F(1-F)}{2\pi|m|}\left[\frac{1}{6}(F-\frac{1}{2})+\frac{1}{4\pi}\int_{1}^{\infty}\frac{dv}{v{\sqrt{v-1}}}\frac{C_{F}v^{F}-C_{F}^{-1}v^{1-F}}{C_{F}v^{F}+2+C_{F}^{-1}v^{1-F}}\right]
L_{m}=\frac{1}{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{1}{2}m^{2}\varphi^{2}+\frac{1}{2}\xiR\varphi^{2}-\frac{\lambda}{4!}\varphi^{4}+\bar{\psi}\left(i\gamma^{\mu}(x)\nabla_{\mu}-h\varphi\right)\psi,
=\left(1+\frac{2i\varepsilon}{\alpha_{+}}\phi+O(\varepsilon^{2}\phi^{2})\right)\times\frac1{2i\varepsilon}e^{-\frac\phi{\alpha_{+}}}\delta^{(2)}(\gamma-x)+\frac1\pi\left|\gamma-x\right|^{-2-4i\varepsilon}e^{-\frac{1+2i\varepsilon}{\alpha_{+}}\phi}
S_{GB}^{volume}=2\left(1-(e^{U}R^{\prime})^{2}\right)|_{r_{h}}-2(e^{2U})^{\prime}|_{r=r_{h}})^{-1}\left[(e^{2U})^{\prime}\left(1-(e^{U}R^{\prime})^{2}\right)\right]_{r=r_{0}}\.
Q^{(1)}=\frac{1}{\sqrt{2}}(\mathrm{e}^{-i{\frac{\pi}{4}}}Q_{2}+\mathrm{e}^{i{\frac{\pi}{4}}}\overline{{Q}}_{\dot{2}}),\qquadQ^{(2)}=\frac{1}{\sqrt{2}}(\mathrm{e}^{i{\frac{\pi}{4}}}Q_{1}+\mathrm{e}^{-i{\frac{\pi}{4}}}\overline{{Q}}_{\dot{1}}).
Z_{0}(k_{s};l_{1s},l_{2s})=\frac{(-1)^{2l_{1s}+2l_{2s}-1}(1-k_{s})^{2}}{4^{2l_{1s}}k_{s}^{3/2}[1+(-1)^{2l_{2s}}\sqrtk_{s}k_{s}^{l_{1s}}]^{2-2l_{1s}}}.
\Gamma^{0}=\left(\begin{matrix}{\gamma^{0}}&{0}\\{0}&{-\gamma^{0}}\\\end{matrix}\right)\,\;\;\;\Gamma^{1}=\left(\begin{matrix}{\gamma^{1}}&{0}\\{0}&{-\gamma^{1}}\\\end{matrix}\right)\,\;\;\;\Gamma^{2}=\left(\begin{matrix}{\gamma^{2}}&{0}\\{0}&{-\gamma^{2}}\\\end{matrix}\right)\,
\tilde{H}=\intd{\bfr}\left[\{\textstyle\frac{1}{2}}\Pi_{i}^{a}({\bfr})\Pi_{i}^{a}({\bfr})+{\textstyle\frac{1}{4}}F_{ij}^{a}({\bfr})F_{ij}^{a}({\bfr})+{\psi^{\dagger}}({\bfr})\left(\betam-i\alpha_{i}\partial_{i}\right)\psi({\bfr})\right]+\tilde{H}^{\prime}\,.
{\frac{\partial}{\partial\tau}}G^{(N=2)}\sim\intd^{4}x_{L0}d^{4}\theta_{0}\langle\mathrm{Tr}\;W^{2}(0)|\widetildeq^{+}\widetildeq^{+}|q^{+}q^{+}|\widetildeq^{+}\widetildeq^{+}|q^{+}q^{+}\rangle\;.
f\star_{\lambda}g=f\g+\sum_{m=1}^{\infty}C_{m}(\lambda)\J^{a_{1}b_{1}}J^{a_{2}b_{2}}\cdotsJ^{a_{m}b_{m}}\\partial_{a_{1}}\cdots\partial_{a_{m}}f\\partial_{b_{1}}\cdots\partial_{b_{m}}g.
{\bfP}_{\eta}=diag(1\mathbf{,\eta,}\eta^{2},\eta^{3},\eta^{4}),\quad{\bfQ}=\left(\begin{array}{ccccc}{0}&{0}&{0}&{0}&{1}\\{1}&{0}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}\\\end{array}\right)
H_{1}=1+\sum_{a=1}^{l_{1}}\frac{Q_{a}^{(1)}}{|x-x_{a}^{(1)}|};~~H_{2}=1+\sum_{b=1}^{l_{2}}\frac{Q_{b}^{(2)}}{|x-x_{b}^{(2)}|},~~|x-x_{a}|=(\sum_{i=1}^{3}|x_{_{i}}-x_{a_{i}}|^{2})^{1/2}.
Q=\left(\begin{array}{cccccc}{0}&{0}&{0}&{0}&{0}&{-1}\\{1}&{0}&{0}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}&{0}&{-1}\\{0}&{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}&{-1}\\{0}&{0}&{0}&{0}&{1}&{0}\\\end{array}\right).
\hat{R}_{\phi}=\left(\begin{array}{cccc}{0}&{-1}&{0}&{0}\\{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\\end{array}\right);\qquad\hat{R}_{r}=\hat{R}_{z}=\hat{R}_{t}=0.
I(g,\mathrm{A})={\frac{1}{4\pi}}\mathrm{Tr}\int\mathrm{d}\tau~\mathrm{d}\theta~\left(\mathrm{A}_{\theta}\partial_{\tau}gg^{-1}-\mathrm{A}_{\tau}g^{-1}\partial_{\theta}g+\mathrm{A}_{\theta}g\mathrm{A}_{\tau}g^{-1}-\mathrm{A}_{\tau}\mathrm{A}_{\theta}\right).
\frac{1}{2}\left(\begin{array}{c}{1}\\{1}\\{0}\\{0}\\{-\sqrt{2}}\\{0}\\\end{array}\right),\quad\frac{1}{2}\left(\begin{array}{c}{0}\\{0}\\{1}\\{1}\\{0}\\{-\sqrt{2}}\\\end{array}\right).
\int_{SO(4)}dg\D_{\alpha_{1}\beta_{1}}^{(N_{1})}(g)D_{\alpha_{2}\beta_{2}}^{(N_{2})}(g){D_{\alpha\beta}^{(\Lambda)}(g)}=C_{\alpha_{1}\alpha_{2}\alpha}^{N_{1}N_{2}\,\Lambda}\\{C}_{\beta_{1}\beta_{2}\beta}^{N_{1}N_{2}\,\Lambda}.
(1-\tilde{P})\left\{\partial_{z}\tilde{P}-(\partial_{z}P)\tilde{P}\right\}=0\qquad\mathrm{and}\qquad(1-\tilde{P})\left\{\frac{i}{2}\partial_{t}\tilde{P}+(\partial_{\bar{z}}P)\tilde{P}\right\}=0.
\left<\left<{\mathcalL}_{m}^{(r,s)}(u),\tilde{\mathcalL}_{n}^{(\alpha,\beta)}(v)\right>\right>=\delta_{m+n,0}\delta_{\alpha+s+1,0}\delta_{\beta+r+1,0}\int{dx{\widehatRes}\left[{\mathcalL}_{m}^{(r,s)}(u)\star\tilde{\mathcalL}_{n}^{(\alpha,\beta)}(v)\right]}
\tilde{g}_{k}(\eta)=\sum_{n=1}^{\infty}\left(\frac{i}{2\Omega_{k0}}\right)^{n}\int_{\eta_{0}}^{\eta}d\eta_{1}K_{k}(\eta,\eta_{1})\cdots\int_{\eta_{0}}^{\eta_{n-1}}d\eta_{n}K_{k}(\eta_{n-1},\eta_{n})\,.
Z=\sum_{h_{ij}}\prod_{ni}\int_{-\infty}^{\infty}\!dA_{i}(n)\int_{-\infty}^{\infty}de_{i}(n)\;exp\left(-\sum_{ni}\:[e_{i}(n)]^{2}+\frac{2i}{\kappa}\sum_{nij}\epsilon_{ijk}\:e_{k}(n)[\bigtriangleup_{i}A_{j}(n)+\frac{1}{2}h_{ij}(n)]\right).
{\tilde{\calL}}_{B}={\tilde{\calL}}_{W}-4(D-2)^{2}\intdre^{(D-3)A}\left(\left({\frac{\partialW}{\partial\phi}}\right)^{2}-\kappa^{D-2}\left({\frac{D-3}{D-2}}\right)W^{2}\right)\eta^{\mu\nu}\partial_{\mu}\tilde{r}\partial_{\nu}\tilde{r}\,
\mathrm{~\psi(x)~}\rightarrow\psi^{\prime}(x)=C\overline{{\psi}}(x)^{\mathrm{T}},\;\;\;\mathrm{~\overline{{\psi}}(x)~}\rightarrow\overline{{\psi}}^{\prime}(x)=-\mathrm{~\psi(x)~}^{\mathrm{T}}C^{\dagger},
\Sigma^{ab}=\frac{1}{4}(\gamma^{a}\gamma^{b}-\gamma^{b}\gamma^{a})=\frac{1}{4}\left(\begin{array}{cc}{\sigma^{a}\bar{\sigma}^{b}-\sigma^{b}\bar{\sigma}^{a}}&{0}\\{0}&{\bar{\sigma}^{a}\sigma^{b}-\bar{\sigma}^{b}\sigma^{a}}\\\end{array}\right).
{\calT}_{S}\,=\,\left[\begin{array}{cc}{1}&{\qquad0}\\{}&{}\\{1}&{\qquad1}\\\end{array}\right]\qquad{\calS}_{S}\,=\,q\gamma_{q}\left[\begin{array}{cc}{0}&{\qquad-1}\\{}&{}\\{1}&{\qquad0}\\\end{array}\right]
\begin{array}{c}{\widehat{L}^{n}\left(\lambda\right)=-\left(\begin{array}{cc}{iA_{0}^{n}}&{\lambdaA_{1}^{n}+\frac1\lambdaA_{-1}^{n}}\\{-\lambdaA_{1}^{n^{*}}-\frac1\lambdaA_{-1}^{n^{*}}}&{iA_{0}^{n^{*}}}\\\end{array}\right),}\\{G^{n}=\left(\begin{array}{cc}{a_{n}}&{0}\\{0}&{a_{n}^{*}}\\\end{array}\right),}\\\end{array}
U(p)=i\int\!d^{2}x\e^{ipx}\langle0\vertT\lbrackT_{++}(x)T_{++}(0)\rbrack\vert0\rangle=\frac{1}{24\pi}\frac{p_{+}^{3}}{p_{-}}=\frac{1}{12\pip^{2}}p_{+}^{4}\,\,.
{\calF}(\{t_{k}\})=\sum_{\{k_{i}\},\{d_{i}\},s}\langle\sigma_{k_{1}}^{d_{1}}\cdots\sigma_{k_{s}}^{d_{s}}\rangle\frac{t_{k_{1}}^{d_{1}}\cdotst_{k_{s}}^{d_{s}}}{d_{1}!\cdotsd_{s}!},
\begin{array}{c}{(\xi_{i}-1)a_{i}^{2}=0}\\{a_{i}a_{j}=e^{i(\lambda_{ji}-\lambda_{ij})}a_{j}a_{i}\qquadi\neqj}\\{a_{i}a_{j}^{+}=e^{i(\lambda_{ij}-\lambda_{ji})}a_{j}^{+}a_{i},\qquadi\neqj}\\{a_{i}a_{i}^{+}=1+\xi_{i}a_{i}^{+}a_{i}}\\\end{array}
\tilde{\calL}=\frac{1}{2}\,\partial_{\mu}\phi^{A}\partial^{\mu}\phi^{A}+\frac{1}{2}\,\lambda\,\bigl(\phi^{A}\phi^{A}-\mathrm{e}^{\eta^{1}}\bigr)-\frac{1}{8}\,\mathrm{e}^{\eta^{1}}\partial_{\mu}\eta^{1}\partial^{\mu}\eta^{1}
P^{-}=P_{\mathrm{free}}^{-}-{\frac{\pi}{2L}}C_{0}^{2}+{\frac{e^{2}L}{2\pi^{2}}}\sum_{N=-\infty}^{\infty}\!\!\!\!{}^{\stackrel{\prime}{}}\;\left({\frac{1}{N^{2}}}\right)C_{N}C_{-N}\;,
\left({\frac{d\tilde{\lambda}}{d\lambda}}\right)^{-1}{\frac{d}{d\tilde{\lambda}}}\left({\frac{d\tilde{\lambda}}{d\lambda}}\right)=-\sqrt{h}\left[{\frac{h^{\prime}}{2}}\left({\frac{dt}{d\tilde{\lambda}}}\right)^{2}-y\sum_{j}\left({\frac{dx^{j}}{d\tilde{\lambda}}}\right)^{2}\right].
[\{\chi_{i},\chi_{j}\}]=\left(\begin{array}{cccccc}{0}&{-iKe}&{-e}&{0}&{0}&{0}\\{iKe}&{0}&{0}&{-e}&{0}&{0}\\{e}&{0}&{0}&{0}&{-e}&{0}\\{0}&{e}&{0}&{0}&{0}&{-e}\\{0}&{0}&{e}&{0}&{0}&{iKe}\\{0}&{0}&{0}&{e}&{-iKe}&{0}\\\end{array}\right)
\Big\langle\prod_{i=1}^{n}{\calO}_{+}^{a}(\{x_{i}\})\prod_{i=1}^{m}{\calO}_{-}^{a}(\{y_{i}\})\Big\rangle\;\;\Big\langle\prod_{i=1}^{n^{\prime}}{\calO}_{+}^{a}(\{x_{i}^{\prime}\})\prod_{i=1}^{m^{\prime}}{\calO}_{-}^{a}(\{y_{i}^{\prime}\})\Big\rangle\;,
\partial_{\mu}e_{\nu}^{\a}-\omega_{\mub}^{\\a}e_{\nu}^{\b}-\Gamma_{\mu\nu}^{\\\rho}e_{\rho}^{\a}=0,\quad\omega_{\mu}^{\ab}=-\omega_{\mu}^{\ba},\quad\Gamma_{\mu\nu}^{\\\rho}=\Gamma_{\nu\mu}^{\\\rho},
C_{i,j}(x,y)=\left(\begin{array}{ccc}{a(\vec{x},\vec{y})}&{0}&{b(\vec{x},y^{\bot})}\\{0}&{0}&{c(x^{\bot},y^{\bot})}\\{-b(\vec{y},x^{\bot})}&{-c(y^{\bot},x^{\bot})}&{0}\\\end{array}\right),
c_{3/2}=\frac{\sqrt{\pi}}{2}((\frac{-1}{6}\hat{R}-\frac{1}{4}R_{nn}^{0}+\frac{3}{32}K^{2}-\frac{1}{16}K_{ij}K^{ij}+Q)+\frac{5}{16}Kf^{(1)}-\frac{1}{4}f^{(2)}),
\mathrm{tr}\left[\widehat{D}(p)\,\delta\Pi(p)\right]^{n}=2\sum_{m=0}^{[n/2]}{}_{n}C_{2m}\frac{\left[R(p)\right]^{n-2m}\left[{\bfS}^{2}(p)\right]^{m}}{\left[Q(p)\right]^{n}},
A^{(3)}=i\bar{\Theta}\Gamma_{\underline{{ab}}}d\Theta({E}^{\underline{a}}{E}^{\underline{b}}-i{E}^{\underline{a}}\bar{\Theta}\Gamma^{\underline{{b}}}d\Theta-{\frac{1}{3}}\bar{\Theta}\Gamma^{\underline{{a}}}d\Theta\bar{\Theta}\Gamma^{\underline{{b}}}d\Theta)
H^{1}({\mathbbP}^{1}\timesT^{2},{\calO}_{{\mathbbP}^{1}\timesT^{2}}(6\sigma|_{{\mathbbP}^{1}\timesT^{2}}-3F))|_{c_{-3}^{(1)}}=H^{0}({\calS},{\calO}_{{\calS}}(1))^{*}.
{\calL}_{m}=\left(\begin{matrix}{\psi_{R}^{*}}&{\psi_{L}^{*}}\\\end{matrix}\right)\left(\begin{matrix}{i(\sqrt{1+s}\partial_{+}-\sqrt{1-s}\partial_{-})}&{im}\\{-im}&{i(\sqrt{1-s}\partial_{+}-\sqrt{1+s}\partial_{-})}\\\end{matrix}\right)\left(\begin{matrix}{\psi_{R}}\\{\psi_{L}}\\\end{matrix}\right),
\left(\begin{matrix}{t^{{\vec{\Lambda}}_{1}}=g^{{\vec{\Lambda}}_{1}}+\mathrm{i}\,e_{{\vec{\Lambda}}_{1}}\,}\\{{\bar{t}}_{{\vec{\Lambda}}_{1}}=g^{{\vec{\Lambda}}_{1}}-\mathrm{i}\,e_{{\vec{\Lambda}}_{1}}\,}\\\end{matrix}\right)
\Upsilon_{\mu\nu\alpha\beta}^{0}\equivg_{\{\mu\nu}g_{\alpha\beta\}}\int_{k}\frac{1}{(k^{2}-m^{2})^{2}}-24\int_{k}\frac{k_{\mu}k_{\nu}k_{\alpha}k_{\beta}}{(k^{2}-m^{2})^{4}},
\int_{0}^{\infty}\prod_{\alpha=1}^{N}d\tau_{\alpha}{\frac{1}{n!}}\left(t\sum_{\alpha=1}^{N}e^{-i\mu_{\alpha}T}\tau_{\alpha}\right)^{n}e^{-s\sum_{\alpha=1}^{N}\tau_{\alpha}}={\frac{t^{n}}{s^{n+N}}}\sum_{\|m_{\alpha}\|=n}\prod_{\alpha=1}^{N}e^{-im_{\alpha}\mu_{\alpha}T}\,
b=\frac{1}{4\pi}\sum_{j,m}\left(2j+1\right)\left(\begin{array}{c}{b_{+}\left(j,m;r\right)D_{+1,m}^{j}\left(\theta,\varphi\right)}\\{b_{0}\left(j,m;r\right)D_{0,m}^{j}\left(\theta,\varphi\right)}\\{b_{\tilde{o}}\left(j,m;r\right)D_{\tilde{o},m}^{j}\left(\theta,\varphi\right)}\\{b_{-}\left(j,m;r\right)D_{-1,m}^{j}\left(\theta,\varphi\right)}\\\end{array}\right)
f\,\sim\,\sqrt{\frac{\gamma_{-}(k)-b(k)+\mid\lambda\mid}{\gamma_{-}(k+\frac{A}{2\pi})-b(k+\frac{A}{2\pi})+\mid\lambda\mid}}\,\sim\,\frac{\mid\tau\,C_{1}(ak)-C_{2}(ak)\mid}{\mid\tauC_{1}(a(k+\frac{A}{2\pi}))-C_{2}(a(k+\frac{A}{2\pi}))\mid}\;,
\chi=\left[-\frac{rs+\lambda\sqrt{(1+r^{2})(1+s^{2})}}{2\sqrt{(1+r^{2})(1+s^{2})}}+\left\{\frac{\left(rs+\lambda\sqrt{(1+r^{2})(1+s^{2})}\right)^{2}-1}{4(1+r^{2})(1+s^{2})}\right\}^{\frac{1}{2}}\right]^{-\frac{1}{2}}
\delta_{D}S_{CS}=-\frac\ell{16\piG}\int_{|r|=\bar{r}}Tr[2\,\xi^{r}\,A_{r}(A_{[\varphi}A_{t]}+\tilde{A}_{[\varphi}\tilde{A}_{t]})]\sqrt{\stackrel{(0)}{\bfg}}\,\,d\varphidt\qquad,
0=(m^{2}-A^{2}-B^{2}-2A\cdotB\gamma_{5})\hat{W}+\frac{\kappa^{2}}{16}e_{a}^{\mu}e_{d}^{\rho}\gamma^{a}R_{\mu\nu}^{bc}R_{\rho\sigma}^{ef}\frac{\partial^{2}}{\partialp_{\nu}\partialp_{\sigma}}\left\{\sigma_{bc},\gamma^{d}\left\{\sigma_{ef},\hat{W}\right\}\right\}
\alpha_{2}=-\varepsilon^{\lambda\mu\nu\rho}\left(\eta_{\;\;\;(\lambda)}^{*\alpha}A_{\alpha(\mu)}+\varepsilon^{\alpha\beta\gamma\delta}\frac{1}{4}B_{\alpha\beta(\lambda)}^{*}B_{\gamma\delta(\mu)}^{*}\right)C_{(\nu)}C_{(\rho)}+\xiC_{(\lambda)}^{*}\eta^{(\lambda)},
\alpha_{\omega}=\frac{-\pi}{2i}[4\omega^{2}b^{2}K_{o}(2\omegab)+\frac{e^{-2\omegab}}{8\omegab}(\frac{3}{4}+\frac{3}{2}\omegab-\frac{15}{8}\omega^{2}b^{2})](\frac{r_{g}}{b})^{2},
{\calE}^{\underline{a}}=d\xi^{m}{\calE}_{m}^{\underline{a}}=d\xi^{m}\left(\partial_{m}x^{\underline{m}}-i\partial_{m}\bar{\theta}^{I}\Gamma^{\underline{m}}\theta^{I}\right)\delta_{\underline{m}}^{\underline{a}}=e^{a}\left(D_{a}x^{\underline{m}}-iD_{a}\bar{\theta}^{I}\Gamma^{\underline{m}}\theta^{I}\right)\delta_{\underline{m}}^{\underline{a}},
K_{i}\equiv\frac{i}{2}\partial_{k}g_{i\bar{j}}\bar{\chi}^{\bar{j}\beta}\chi_{\beta}^{k}\qquad\qquad\bar{K}_{\bar{j}}\equiv-\frac{i}{2}\partial_{\bar{l}}g_{i\bar{j}}\bar{\chi}^{\bar{l}\beta}\chi_{\beta}^{i}
P_{a_{1},a_{2},\cdots,a_{m}}^{a_{1},d_{2}^{\prime},\cdots,d_{m}^{\prime}}\delta_{c_{1}^{\prime}}^{c_{1}}=\sum_{\scriptstyle{\begin{array}{l}{b_{2},\cdots,b_{m}}\\{c_{2},\cdots,c_{m}}\\\end{array}}}D_{m}(a,b,c,d^{\prime};z,z^{\prime})P_{c_{1}^{\prime},b_{2},\cdotsb_{m}}^{c_{1},c_{2},\cdots,c_{m}},
\sum_{\nu=1}^{4p}\beta_{\nu}=0\:\:,\:\:\sum_{\nu=1}^{4p}(2\alpha_{\nu}\beta_{\nu}+1-\alpha_{\nu}^{2})=0\:\:,\:\:\sum_{\nu=1}^{4p}[\beta_{\nu}\alpha_{\nu}^{2}+\alpha_{\nu}(1-\alpha_{\nu}^{2})]=0~~.
g_{\mathrm{eff}}^{2}=g_{f}^{2}u^{5},\quad\tau_{D}=\left({\frac{1}{4\pi}}\left[g_{f}^{2}\left({\frac{u}{\calR}}\right)^{5}\left({\frac{16}{\pi}}\right)\right]^{1/6}\right)^{-2},
\hat{g}=\left(\begin{array}{ccccc}{0}&{1}&{0}&{\cdots}&{0}\\{0}&{0}&{1}&{\ddots}&{\vdots}\\{\vdots}&{0}&{0}&{\ddots}&{0}\\{0}&{\vdots}&{\ddots}&{\ddots}&{1}\\{1}&{0}&{\cdots}&{0}&{0}\\\end{array}\right).
+\nabla_{m}(\pi^{ij}N^{m})-\pi^{jm}\nabla_{m}N^{i}-\pi^{im}\nabla_{m}N^{j}+N\sum_{n}\delta^{2}(x-q_{n})\frac{P_{na}P_{nb}g^{ai}g^{bj}}{2\sqrt{m_{n}^{2}+P_{na}P_{nb}g^{ab}}}
{\calG}_{2}^{(1)}(t_{1},t_{2})=-4\tauR(\Omega-1)x^{2(t_{2}-t_{1})}x^{2(T-1)}(1+x^{2T})^{\Omega-2}\left[(t_{2}-t_{1})+\frac{T}{2}(\Omega-2)\frac{x^{2T}}{1+x^{2T}}\right]\.
S^{(2)}={\frac{4}{\rho^{2}}}\mathrm{tr}_{2k}\,(\tilde{a}_{n}^{\prime})^{2}+4\rho^{2}\mathrm{tr}_{2k}\,(\tilde{\chi}_{a}^{\perp})^{2}+\mathrm{tr}_{2k}\,\varphi^{2},
\begin{aligned}{\mathcal{A}_{1}=}&{{}\mathcal{A}_{1}^{(a_{1},0)}T_{(a_{1},0)},}\\{\mathcal{A}_{2}=}&{{}\mathcal{A}_{2}^{(a_{1},a_{2})}T_{(a_{1},a_{2})}\quad\textrm{with}\quada_{2}\neq0}\\\end{aligned}
{\bfP}_{\alpha}=diag(1,\alpha,\alpha^{2},\alpha^{3},\alpha^{4}),\quad{\bfQ}=\left(\begin{array}{ccccc}{0}&{0}&{0}&{0}&{1}\\{1}&{0}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}\\\end{array}\right)
L_{\mathrm{int}}^{\prime\prime}=-{\frac{g}{2}}\Bigl(\psi_{1}^{\dagger}(t)\psi_{2}(t)e^{2\sqrt{2}\piip_{X}}(-1)^{F}\-\\psi_{1}^{\dagger}(l+t)\psi_{2}(l+t)\Bigr)\+\\mathrm{h.c.},
\deltaL=\frac{d}{dt}(G)-\dot{p}^{\mu}\circ\deltaq^{\mu}+\frac{1}{2}\dot{q}^{\mu}(\deltaq^{\alpha}\partial_{\alpha}g_{\mu\nu})\dot{q}^{\nu}-\deltaq^{\mu}\partial_{\mu}U_{q}+\frac{1}{2}[\deltaq^{\mu},\dot{g}_{\mu}]
V_{C}^{\dagger}J_{i}^{B}(x)V_{C}=iV_{C}^{\dagger}(\Phi^{*}\partial_{i}\Phi-\partial_{i}\Phi^{*}\Phi)V_{C}=J_{i}^{B}(x)+{\frac{2\pi}{N}}n_{i}^{C}(x)\delta(x\inC)\Phi^{*}\Phi(x)
\begin{aligned}{s_{i\,y}-s_{i\,xx}=s_{i}}&{{}\biggl(\lambda_{i\,x}^{2}\wp(z)+\sum_{j\nei}\bigl[a_{i}\wp(\lambda_{i}-\lambda_{j})+b_{j}\zeta(\lambda_{i}-\lambda_{j})\bigr]+c\biggr)+{}}\\{{}}&{{}+\sum_{j\nei}s_{j}\bigl(a_{i}\Phi^{\prime}(\lambda_{i}-\lambda_{j},z)+b_{j}\Phi(\lambda_{i}-\lambda_{j},z)\bigr).}\\\end{aligned}
{L}_{0}^{(2)}({\bf{r}}_{1},{\bf{r}}_{2})=G_{0}^{-1}({\bf{r}}_{1}-{\bf{r}}_{2})=\delta({\bf{r}}_{1}-{\bf{r}}_{2})\left[-\nabla_{{\bf{r}}_{2}}^{2}+m_{0}^{2}\right]\;.
T^{00}=\sum_{s,t=0}^{n}\{\Box^{s}\frac{\partial{\calL}}{\partial\phi^{(s+t+1)}}\ddot{\phi}^{(t)}-\partial^{0}\Box^{s}\frac{\partial{\calL}}{\partial\phi^{(s+t+1)}}\dot{\phi}^{(t)}\}-{\calL}
\begin{array}{c}{\displaystyle\alpha_{j}^{n}(x;z)=\int_{\vert\xi\vert\leq1}\int_{0}^{\infty}t^{n}D_{-1-j}(x,t;\xi,t;z)\\dt\{\frac{{d\xi}}{{(2\pi)^{\nu-1}}}}}\\{\displaystyle\beta_{j}^{n}(x;z)=\int_{\vert\xi\vert=1}\int_{0}^{\infty}t^{n}D_{-1-j}(x,t;\xi,t;z)\\dt\{\frac{{d\xi}}{{(2\pi)^{\nu-1}}}}}\\\end{array}
C_{\mu}^{i}=J_{\mu}^{i}+K_{\mu}^{i}~,~~~~~J_{\mu}^{i}=-v_{a}^{~i}\partial_{\mu}\varphi^{a}~,~~~~~K_{\mu}^{i}=\fraci2~\varepsilon^{ijk}v_{a}^{~j}v_{b}^{~k}\overline{{\psi}}^{a}\gamma_{\mu}\psi^{b}~.
\deltaG^{AB}(\partial_{A}\partial_{B}\varphi-\overline{{\Gamma}}_{AB}^{C}\partial_{C}\varphi)+\overline{{G}}^{AB}(\partial_{A}\partial_{B}\chi-\overline{{\Gamma}}_{AB}^{C}\partial_{C}\chi-\delta\Gamma_{AB}^{C}\partial_{C}\varphi)+\frac{\partialV}{\partial\varphi}\chi=0,
\mathcal{H}=\left(a_{i}^{\dagger},\,a_{j}\right)\left(\begin{matrix}{\mathcal{E}_{il}}&{\mathcal{F}_{jl}^{\dagger}}\\{\mathcal{F}_{im}}&{\mathcal{E}_{jm}^{T}}\\\end{matrix}\right)\left(\begin{matrix}{a_{l}}\\{\a_{m}^{\dagger}}\\\end{matrix}\right).
i\frac{\partial}{\partialt}\langleq^{\prime},t\midq,0\rangle_{l}=-\frac{1}{2}\frac{\partial^{2}}{\partialq^{2}}\langleq^{\prime},t\midq,0\rangle_{l}+\left(\frac{l(l+1)}{2}\frac{1}{q^{2}}+V(q^{\prime})\right)\langleq^{\prime},t\midq,0\rangle_{l},
T(B,A;\tau)\equiv\int_{0}^{2\pi}d\sigmaW^{-1}B_{\mu\nu}W\frac{\partialx^{\mu}}{\partial\sigma}\frac{\partialx^{\nu}}{\partial\tau}=\frac12\int_{0}^{2\pi}d\sigmaW^{-1}B_{\mu\nu}W\frac{\partial(x^{\mu},x^{\nu})}{\partial(\sigma,\tau)}
A=\int\frac{d^{3}p}{(2\pi)^{3}}\\overline{{\Psi}}(p)S^{-1}(p)\Psi(p)\+\int\frac{d^{3}p\d^{3}q}{(2\pi)^{3}}\\overline{{\Psi}}(p+\frac{\1}{\2}\q)\Gamma(p,q)\Psi(p-\frac{\1}{\2}\q),
C\left[\begin{array}{c}{0}\\{0}\\\end{array}\right]=-C\left[\begin{array}{c}{1}\\{0}\\\end{array}\right]=-C\left[\begin{array}{c}{0}\\{1}\\\end{array}\right]=\pmC\left[\begin{array}{c}{1}\\{1}\\\end{array}\right]={\frac{1}{2}}.
{{\widehatT}_{a}}^{b}={\frac{M}{8\pil^{3}}}\left[4u_{a}u^{b}+{{\delta}_{a}}^{b}\right]-{\frac{l^{3}}{64\piG}}\left[{\frac{1}{12}}{{\delta}_{a}}^{b}R^{2}-R^{cd}{R_{cad}}^{b}\right],
L_{\hphantom{\mu}\nu}^{\mu}=\left(\begin{array}{cc}{\frac{p^{0}}{w}}&{-\frac{p_{n}}{w}}\\{\frac{p^{k}}{w}}&{\delta_{n}^{k}\!-\!\frac{\frac{p^{k}}{w}\frac{p_{n}}{w}}{1+\frac{p^{0}}{w}}}\\\end{array}\right),\,\,\,\,L(\hat{p})\left(\begin{array}{c}{1}\\{0}\\{0}\\{0}\\\end{array}\right)=\hat{p}.
\langle\Phi(x)\Phi(0)\rangle\simA_{1}\frac{e^{-m_{1}r}}{\sqrt{m_{1}r}}+A_{2}\frac{e^{-m_{2}r}}{\sqrt{m_{2}r}}+A_{3}\frac{e^{-m_{3}r}}{\sqrt{m_{3}r}}\cdots
\partial_{zz}^{2}f_{k_{+},k_{-}}+{\frac{1}{z}}\partial_{z}f_{k_{+},k_{-}}+\left[{\frac{l^{2}k_{+}^{2}}{4z^{2}(1-z)}}-{\frac{l^{2}k_{-}^{2}}{4z(1-z)}}-{\frac{m^{2}l^{2}}{4z(1-z)^{2}}}\right]f_{k_{+},k_{-}}=0\,.
S_{5}=-\frac{1}{2\kappa_{5}^{2}}\left[\int_{\mathcal{M}_{5}}\sqrt{-g}\left[R+\frac{1}{2}\partial_{\alpha}\phi\partial^{\alpha}\phi+V(\phi)\right]+\sum_{i=1}^{2}\int_{\mathcal{M}_{4}^{i}}\sqrt{-g}\left[\frac{1}{2}\partial_{\mu}\phi_{i}\partial^{\mu}\phi_{i}+V_{i}(\phi_{i},\phi_{i})\right]\right].
{\frac{\xi_{j}-\xi_{j-1}}{1-\xi_{j}^{*}\xi_{j}}}={\frac{-i\epsilonh\xi_{j-1}}{1-\xi_{j}^{*}\xi_{j-1}}}\,\quad{\frac{\xi_{j}^{*}-\xi_{j+1}^{*}}{1-\xi_{j}^{*}\xi_{j}}}={\frac{-i\epsilonh\xi_{j+1}}{1-\xi_{j+1}^{*}\xi_{j}}}\quad\left({1\lej\leN}\right)\quad.
T_{B}^{(Z,\overline{{Z}})}(z)=-\displaystyle\frac{\varepsilon\,\delta_{I\overline{{J}}}}{\alpha^{\prime}}:\partialZ^{I}\partial\overline{{Z}}^{\overline{{J}}}(z):~,\quadT_{B}^{(\Psi,\overline{{\Psi}})}(z)=-\frac{\varepsilon\,\delta_{I\overline{{J}}}}{4}\left[:\Psi^{I}\partial\overline{{\Psi}}^{\overline{{J}}}(z):{}-:\partial\Psi^{I}\,\overline{{\Psi}}^{\overline{{J}}}(z):\right]~.
\hat{I}=-\frac{1}{4}\intd^{4}x\;[F^{\mu\nu}F_{\mu\nu}-\frac{1}{2}\theta^{\alpha\beta}F_{\alpha\beta}F^{\mu\nu}F_{\mu\nu}+2\theta^{\alpha\beta}F_{\alpha\mu}F_{\beta\nu}F^{\mu\nu}]\;,
\langle{\mathrm{T}r}\left(\lambda\left[L_{i}+\rhoa_{i},-\psi_{\beta}L_{j}\bar{\psi}_{\alpha}(\sigma_{i}\sigma_{j})^{\alpha\beta}\right]\right)-\frac{2\rho^{2}}{\alpha}{\mathrm{T}r}\left(\bar{\psi}\lambda\phi\psi\right)\rangle_{S}=I.
\xi_{1,2}(a_{i}^{\dagger})=\xi_{1}(a_{i}^{\dagger}){\frac{\partial}{\partiala_{i}^{\dagger}}}\xi_{2}(a_{i}^{\dagger})-\xi_{2}(a_{i}^{\dagger}){\frac{\partial}{\partiala_{i}^{\dagger}}}\xi_{1}(a_{i}^{\dagger})\,.
\gamma_{s}=\frac{\left(-1\right)^{s}}2\int_{\Sigma_{s}}\int_{\partial\Sigma_{s}={\bfc}}\frac{\left(\vec{R}-i\frac12\vec{\Gamma}\right)\cdotd\vec{S}}{\left[\left(\vec{R}-i\frac12\vec{\Gamma}\right)^{2}\right]^{\frac32}},