harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\cal{S}}=\frac{1}{16\pi}\intd^{4}x\sqrt{-\tilde{g}}[\tilde{\Phi}\tilde{R}-\frac{\omega(\tilde{\Phi})}{\tilde{\Phi}}\tilde{g}^{\mu\nu}\partial_{\mu}\tilde{\Phi}\partial_{\nu}\tilde{\Phi}]+{\cal{S}}_{m}[\psi_{m},\tilde{g}_{\mu\nu}]
D_{\alpha}{F}^{\alpha\beta,a}+\frac{\kappa}{2}\epsilon^{\beta\alpha\lambda}{F}_{\alpha\lambda}^{a}=-i\bar{\psi}\gamma^{\beta}{T}^{a}\psi-i\frac{u}{2}\epsilon^{\beta\alpha\lambda}D_{\alpha}(\bar{\psi}\gamma_{\lambda}{T}^{a}\psi).
\frac{1}{2\pii}\oint_{0}\frac{\partial}{\partial\alpha^{j}}H\frac{d\zeta}{\zeta^{m+1}}=\frac{1}{2\pii}\oint_{0}\frac{\partial}{\partial\alpha^{j}}\sum_{i=1}^{k}\frac{\eta_{i}^{2}}{\zeta}\frac{d\zeta}{\zeta^{m+1}}=\frac{1}{2\pii}\oint_{\sum_{i=1}^{k}0_{i}}\frac{\partial}{\partial\alpha^{j}}\frac{\eta^{2}}{\zeta}\frac{d\zeta}{\zeta^{m+1}}
d_{\alpha\beta}^{0}=\frac12\left(\begin{matrix}{-1}&{0}\\{0}&{-1}\\\end{matrix}\right),\qquadd_{\alpha\beta}^{1}=\frac12\left(\begin{matrix}{1}&{0}\\{0}&{-1}\\\end{matrix}\right),\qquadd_{\alpha\beta}^{2}=\frac12\left(\begin{matrix}{0}&{-1}\\{-1}&{0}\\\end{matrix}\right).
\begin{array}{l}{V_{NL}=-\phi(y_{3})e^{2y_{2}}\partial_{x}-2u\phi(y_{3})y_{1}e^{2y_{2}}\partial_{u}+\phi(y_{3})e^{2y_{2}}y_{1}^{2}\partial_{y_{1}}}\\{\qquad\qquad+[f_{2}(y_{3})-y_{1}\phi(y_{3})e^{2y_{2}}]\partial_{y_{2}}+f_{3}(y_{3})\partial_{y_{3}}.}\\\end{array}\tag{3.48}
\begin{aligned}{\eta_{B}(s)}&{{}=2\,\zeta(s-1,\Lambda^{2}/2)-\Lambda^{2}\,\zeta(s,\Lambda^{2}/2)}\\{}&{{}\quad-2s\left(\frac{\Lambda^{2}}{2}\right)f\big((s+1)/2\big)-\frac{2s(s+1)(s+2)}{3!}\left(\frac{\Lambda^{2}}{2}\right)^{3}f\big((s+3)/2\big)+\cdots}\\\end{aligned}
\bar{\Lambda}(\beta;D)=-\frac{16}{\beta}(2\pi\alpha^{\prime})^{-D/2}\sum_{(\sigma,\rho)}\int_{F}\frac{d^{2}\tau}{2\pi\tau_{2}^{2}}\;B(\bar{\tau},\tau;D)A_{\sigma\rho}(\tau;D)D_{\sigma\rho}(\bar{\tau},\tau;\beta)\quad,
{\calL}={\calL}_{0}(C_{\mu\nu\alpha})+{\calL}_{0}(B_{\mu\nu})-2\sqrt{-3\Omega(d-3)}C_{\mu\nu\alpha}D^{\mu}B^{\nu\alpha}-\frac{\Omega(d-3)}{2}C^{\mu\nu\alpha}C_{\mu\nu\alpha}
q^{+}=0\,\,alpha_{n}^{+}=\frac{1}{2}\sqrt{2\alpha^{\prime}}P^{+}\delta_{n,0}=\bar{\alpha}_{n}^{+}\;\alpha_{n}^{-}=\frac{2}{\sqrt{2\alpha^{\prime}}P^{+}}\,L_{n}^{\perp}\,\,bar{\alpha}_{n}^{-}=\frac{2}{\sqrt{2\alpha^{\prime}}P^{+}}\,\bar{L}_{n}^{\perp}\,
R=6C^{2}e^{2\phi_{0}+6\tau}\Bigl(\frac{e^{-3\chi}}{4}+\frac{e^{-3\psi}}{4}+\frac{3}{2}e^{-\chi-\psi}+\frac{e^{-\chi-\psi}}{2}\dot{\chi}\dot{\psi}-e^{-\chi-\psi}\dot{\chi}-e^{-\chi-\psi}\dot{\psi}\Bigr)
(z-w)^{-4}-{\frac{1}{2}}\sum_{n=0}^{\infty}\left({-{\frac{1}{2}}\atopn+1}\right){\frac{n+1}{n+2}}w^{-n-2}(z-w)^{n-2}={\frac{1}{2}}{\frac{z+w}{\sqrt{zw}}}(z-w)^{-4}\,,
\begin{array}{rcl}{Q_{1}}&{=}&{R_{+}C^{+}+R_{-}C^{-}+(\displaystyle\frac{1}{2}(\lambda-8E)L+4\lambdaL^{3})C^{3}+i\bar{C}_{3}C^{+}C^{-}-96\lambdaL\bar{C}_{+}\bar{C}_{-}C^{+}C^{-}C^{3}}\\{}&{}&{+2i(\displaystyle\frac{1}{2}(\lambda-8E)+12\lambdaL^{2})(\bar{C}_{-}C^{-}C^{3}-\bar{C}_{+}C^{+}C^{3})~.}\\\end{array}
\ddot{U}=\dot{U}^{2}\Bigl[{\frac{1}{2U}}+{\frac{1}{U-1}}\Bigr]-{\frac{\dot{U}}{\eta}}+{{\frac{2A^{2}}{\eta^{2}}}}(1-U)^{2}({\frac{1}{U}}-U)+2{\frac{h^{2}-l}{g^{2}\eta^{2}}}{\frac{U(U+1)}{U-1}}.
{\dot{\pi}}_{\phi}({\bfk}_{n},t)=i[H,\pi_{\phi}({\bfk}_{n},t)]=-\frac{e^{3Ht}}{L^{3}}[\omega_{n}^{2}(t)\phi({\bfk}_{n},t)+\frac{\deltaV_{F}}{\delta\phi(-{\bfk}_{n},t)}+\alpha\Omega({\bfk}_{n},t)]
\alpha(M_{\alpha}^{-})~=~\frac{\epsilon_{\alpha}}{2\,\sqrt{2}}\;|\alpha|\qquad\mathrm{and}\qquadM_{\alpha}^{-}~=~\frac{\epsilon_{\alpha}}{\sqrt{2}\,|\alpha|}\;(H_{\alpha}^{})_{\mathfrak{a}}^{}~,
\frac{\partial}{\partial\eta^{\prime\prime}{}^{\mu}}\frac{\partial}{\partial\overline{{\psi}}^{\prime\prime}}\Omega(\zeta^{\prime\prime})=\frac{\partial}{\partial\overline{{\psi}}^{\prime\prime}}\frac{\partial}{\partial\eta^{\prime\prime}{}^{\mu}}F(\zeta^{\prime\prime})\,,
S=\intd\tau\{p_{a}\dot{x}^{a}+\frac{ms}{(p,n)}\dot{\varphi}+\varrho\frac{(p,\partial_{\varphi}n)}{(p,n)}\dot{\varphi}-\frac{e(\tau)}{2}(p^{2}+m^{2})\}\,,
{\calL}=\sum_{j=1}^{N}i\bar{\Psi}_{j}\gamma^{\mu}D_{\mu}\Psi_{j}+\frac{G}{2}\sum_{j=1}^{N}\left[(\bar{\Psi}_{j}\Psi_{j})^{2}+(\bar{\Psi}_{j}i\gamma_{5}\Psi_{j})^{2}\right],
\frac{1}{8\pi}\int_{S^{3}(R_{+})}\nabla_{\nu}\tilde{K}_{\mu}^{i}d\Sigma^{\mu\nu}+\frac{1}{8\pi}\int_{S^{3}(R_{c})}\nabla_{\nu}\tilde{K}_{\mu}^{i}d\Sigma^{\mu\nu}=\int_{\Sigma_{t}}T_{\nu\rho}\tilde{K}^{i\rho}d\Sigma^{\nu},
\bar{g}^{\prime}(\alpha)=\kappa^{\prime2}{\frac{i\alpha}{\pi(\xi+1)}}{\frac{\Gamma(1-{\frac{1}{(\xi+1)}}+{\frac{i\alpha}{\pi(\xi+1)}})}{\Gamma({\frac{1}{(\xi+1)}}+{\frac{i\alpha}{\pi(\xi+1)}})}}\.
S_{\mathrm{int}}[\phi,\varphi]=-\,\int_{t_{i}}^{t}\,d^{4}x\left[\textstyle{{\frac{1}{6}}}g^{2}\left(\phi^{3}\varphi+\textstyle{{\frac{3}{2}}}\phi^{2}\varphi^{2}+\phi\varphi^{3}\right)+\textstyle{{\frac{1}{4!}}}g^{2}\varphi^{4}\right].
\Gamma[W,\bar{W}]~=~\mathrm{Im}\int\mathrm{d}^{4}x\mathrm{d}^{4}\theta\,{\calF}(W)~+~\int\mathrm{d}^{4}x\mathrm{d}^{8}\theta\,{\calH}(\bar{W},W)~+~\dots
V_{o}(x)=\frac{(4-2\rho^{2}-16\rho)e^{2x}-4(1-\rho^{2})\rho^{2}e^{-2x}-16\rhox(e^{2x}+\rho^{2}e^{-2x})-64\rho^{2}x-16\rho^{2}}{[e^{2x}-\rho^{2}e^{-2x}+(1-\rho^{2})+4\rhox]^{2}}.
dS^{2}={H_{5}^{\prime}}^{-1/2}[dudv-\mu^{2}\sum_{i=1}^{4}{z^{i}}^{2}du^{2}+\sum_{i=1}^{4}dz^{i}dz^{i}]+{H_{5}^{\prime}}^{1/2}(dy^{2}+y^{2}d{\Omega_{3}^{\prime}}^{2}).
\langleV_{\alpha_{1}}V_{\alpha_{2}}V_{\alpha_{3}}\rangle=\left(C^{even}(\alpha_{1},\alpha_{2},\alpha_{3})+\eta\bar{\eta}C^{odd}(\alpha_{1},\alpha_{2},\alpha_{3})\right)\prod_{i<j}|Z_{i}-Z_{j}|^{\delta_{ij}}
\dot{N}^{\zeta}(t_{0})=\frac{i}{U_{0}^{2}}\left[\mathrm{Re}\,\dot{S}_{0}-2Z\mathrm{Re}\,S_{0}+\left(Z^{2}-{\textstyle{\frac{1}{4}}}{\gamma^{\zeta}}^{2}(t_{0})\right)R_{0}\right],
\langle\varphi_{J}^{a\dagger}\varphi_{b}^{I}\rangle_{\mathrm{FT}}^{\mathrm{1-loop,WZ}}\simg^{2}\delta_{J}^{I}\delta_{b}^{a}p^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{\left(k+\frac{p}{2}\right)^{2}\left(k-\frac{p}{2}\right)^{2}}\,.
G_{bb}^{-2}(x,y)=\left[-\frac{{N^{2}}}{g^{2}}\frac{{4}}{f^{4}({\hat{\phi}})}\left({1-\frac{{3}}{2}\frac{{\hbarG_{aa}(z,z)}}{f({\hat{\phi}})}}\right)\nabla_{x}^{2}+\frac{{1}}{2}\frac{{\hbar}}{f({\hat{\phi}})}G_{aa}^{-1}(z,z)\right]\delta(x-y),
\theta_{+}=\sum_{\alpha,\beta=1}^{2}a^{\alpha\beta}(k)f_{\alpha}(k,\eta)u_{\beta}(k)\,,\qquad\theta_{-}=-{\calC}^{-1}\sum_{\alpha,\beta=1}^{2}a^{*\alpha\beta}(-k)f_{\alpha}^{*}(-k,\eta)u_{\beta}^{*}(-k)\,.
\int_{-i\infty}^{i\infty}\bar{d}\beta\mathrm{exp}\left({\frac{1}{r}}\tilde{f}_{N}(r\beta)\right)\mathrm{e}^{\beta\varepsilon}={\frac{1}{r}}\hat{A}(\varepsilon/r,v/r)\mathrm{e}^{{\beta_{H}\varepsilon}/r}.
\left[\chi^{ab},\chi^{cd}\right]\left(M_{n}^{\,\,\,\,\,m}\right)=\left(1-\Lambda\right)_{efgh}^{abcd}\,\left(\chi^{ef}\ast\chi^{gh}\right)\left(M_{n}^{\,\,\,\,\,m}\right)={\mathbf{C}}_{ef}^{abcd}\chi^{ef}\left(M_{n}^{\,\,\,\,\,m}\right),
B^{\prime}=\int_{0}^{1}d\eta_{1}\int_{0}^{1-\eta_{1}}d\eta_{2}{\frac{1-\eta_{1}-\eta_{2}}{[m^{2}-t\eta_{2}(1-\eta_{1}-\eta_{2})][m^{2}-s\eta_{1}(1-\eta_{1}-\eta_{2})]}}\.
{\calL}=-\frac{1}{4}F_{\mu\nu}^{\prime}{F^{\prime}}^{\mu\nu}+\frac{1}{2}\left[\partial_{\mu}\rho+igA_{\mu}^{\prime}(\rho+v)\right]^{2}-\frac{1}{2}\mu^{2}(\rho+v)^{2}-\frac{1}{4}\lambda(\rho+v)^{4}\.
\left.\frac{\partial^{2}V_{eff}^{(2)}(T=0,\mu,m)}{\partialm^{2}}\right|_{m=m_{1}}=-\frac{1}{\pi}\frac{2\mu-m(0)}{m(0)-\mu}<0,\\mathrm{when}\\frac{m(0)}{2}<\mu<m(0)
\frac{\sqrt{h}}{2}h^{ab}\left[2g_{\mu\nu}(X^{\alpha})\partial_{a}\partial_{b}X^{\mu}+2\partial_{a}X^{\mu}\partial_{b}X^{\rho}\partial_{\rho}g_{\mu\nu}(X^{\alpha})+\partial_{a}X^{\mu}\partial_{b}X^{\eta}\partial_{\nu}g_{\mu\eta}(X^{\alpha})\right]=0
S_{Q2f}^{\prime}(M_{n,i})=\sum_{i=0}^{L-1}\frac{1+\alpha}{N}\mathrm{Tr}\,(M_{1,i}^{2}+M_{2,i}^{2})+\frac{2\alphag}{N^{2}}\mathrm{Tr}\,M_{1,i}^{2}\mathrm{Tr}\,M_{2,i}^{2}
\varepsilon_{0}^{\beta}=\frac{|e|H}{(2\pi)^{2}}\sum_{n=0}^{\infty}\int_{-\infty}^{\infty}dp\;\epsilon_{n}\left\{\frac{-2+\mathrm{e}^{\beta\epsilon_{n}}}{1-\mathrm{e}^{\beta\epsilon_{n}}}\right\}.
\hat{\chi}_{\lambda\phi,1}~=~-\,\frac{4}{\pi^{2}}\left[-\,1+\frac{\rho}{(1+\vartheta^{2})}-\frac{(1-\vartheta^{2})}{(1+\vartheta^{2})^{2}}+2+\frac{2(1-\vartheta^{2})}{(1+\vartheta^{2})^{2}}\right]
ig\left[\delta^{ij}\delta^{kl}\delta_{JI}\delta_{LK}\Gamma_{(n)\,\alpha}^{~~~~~\beta}\Gamma_{~~~\gamma}^{(n)~\delta}~-~\delta^{il}\delta^{kj}\delta_{LI}\delta_{JK}\Gamma_{(n)\,\alpha}^{~~~~~\delta}\Gamma_{~~~\gamma}^{(n)~\beta}\right]
H=1+(\sqrt{\alpha^{\prime}})^{{\frac{x-\tilde{d}}{\beta}}}(g_{s})^{{\frac{2(k-1)}{\beta}}}\left[g_{f}^{2}\,\left({\frac{u}{\calR}}\right)^{\tilde{d}}\,\left({\frac{d_{p}}{c_{p}}}\right)\right]^{-1/\beta}\,.
T_{\alpha\beta}=-T_{\beta\alpha}={\frac{i}{2}}\bigl(\Phi^{\dagger}(\theta)L^{a}e_{a[\alpha}\nabla_{\beta]}\Phi-\bigl(L^{a}e_{a[\alpha}\nabla_{\beta]}\Phi\bigr)^{\dagger}\Phi\bigr)=-{\frac{\kappa}{2G}}\varepsilon_{\alpha\beta}\Phi^{\dagger}\Phi
S_{d3}=-\intd^{4}x\,{\frac{1}{4g_{\mathrm{YM}}^{2}}}\,\mathrm{Tr}\Bigl(F_{ab}F^{ab}+{\frac{2}{(2\pi\alpha^{\prime})^{2}}}D_{a}X^{\mu}D^{a}X_{\mu}+{\frac{1}{(2\pi\alpha^{\prime})^{4}}}[X^{\mu},X^{\nu}]^{2}\Bigr)+\cdots,
\Gamma_{g}^{\alpha\beta\gamma}\;(x,y,z)=\Gamma_{r}^{\alpha\beta\gamma}\;(x,y,z)-{\frac{i}{4\pi^{2}}}\;\varepsilon^{\betai\gammaj}{\frac{\partial}{\partialx_{\alpha}}}{\frac{\partial}{\partialy^{i}}}G(x,y){\frac{\partial}{\partialz^{j}}}\delta(y-z).
I_{\calG}=\int_{-\infty}^{+\infty}\frac{a^{3}\varphi^{2}}{{\calH}^{2}}dw\equiv6\int_{-\infty}^{+\infty}\biggl\{a^{3}\frac{{\calH}^{2}-{\calH}^{\prime}}{{\calH}^{2}}\biggr\}.
\langle2n,0|B|2n,0\rangle={\frac{1}{c_{n}}}\langle0|A^{n}B(A^{\dagger})^{n}|0\rangle={\frac{1}{c_{n}}}\langle0|(2n+B)A^{n}(A^{\dagger})^{n}|0\rangle=2n+3
\frac{d^{2}{\hat{\phi}}_{i}}{d{\hat{t}}^{2}}+3\,{\hat{H}}\,\frac{d{\hat{\phi}}_{i}}{d{\hat{t}}}+\frac{d{\hat{V}}}{d{\hat{\phi}}_{i}}=0\,,\qquad{\hat{\phi}}_{i}={\hat{\phi}},\,{\hat{s}}.
d\Theta_{\underline{\mu}}^{2}=v_{\underline{\mu}q}^{+}\pi_{q}^{2-}+v_{\underline{\mu}\dot{q}}^{-}\pi_{\dot{q}}^{2+},=e^{-2}\psi_{-2\dot{q}}^{2+}v_{\underline{\mu}\dot{q}}^{-}+e_{q}^{-}v_{\underline{\mu}q}^{+},
T^{*}(x,\vec{k}_{\perp};x^{\prime},\vec{k}_{\perp}^{\prime})=\frac{1}{2}\left(\frac{m_{f}^{2}+\vec{k}_{\perp}}{x(1-x)}+\frac{m_{f}^{2}+\vec{k}_{\perp}^{\prime}}{x^{\prime}(1-x^{\prime})}\right),
{\calL}_{\eta}|\Phi_{ph}\rangle=\Bigl(\eta\hat{\partial}+\frac{1}{2}\hat{\partial}^{I}\eta^{J}M^{IJ}+\frac{1}{2}\eta^{I}M^{zI}+\hat{\partial}^{+}\eta^{I}\alpha^{I}(\frac{\partial^{J}}{\partial^{+}}\bar{\alpha}^{J}-\frac{s+d-4}{\hat{\partial}^{+}}\bar{\alpha}^{z})\Bigr)|\Phi_{ph}\rangle\,.
{\mathcalS}_{f}^{reg}(m)=\intd^{D+1}x\,\left[{\bar{\Psi}}^{(+)}\Gamma_{\alpha}D_{\alpha}(m)\Psi^{(+)}+{\bar{\Psi}}^{(-)}\Gamma_{\alpha}D_{\alpha}(m)\Psi^{(-)}\right]
\cdots\mathfrak{L}(x+(2M-12)i)\mathfrak{L}(x+(2M-8)i)v_{0}^{\prime}=\prod_{j=1}^{(M-2)/2}T_{M}(x-(2+4j)i)\left(\begin{array}{c}{T_{0}(x-(M+2)i)}\\{T_{M-1}(x-i)}\\\end{array}\right)
M=\frac{g_{YM}^{2}}{2V{\calN}}(w^{i}+\Theta^{ij}n_{j})^{2}+\frac{V}{4g_{YM}^{2}{\calN}}(m_{ij}+{\calN}\Phi{ij})^{2}+\frac{1}{\calN}\sqrt{\frac{k_{i}^{2}}{\Sigma_{i}^{2}}},
\Psi_{L}^{c}(x)=\frac{m}{2\sqrt{2}L}\sum_{k=0}^{\infty}\int_{-L}^{L}dy^{-}e^{-\frac{i\pi}{L}(k+\frac{1}{2})(x^{-}-y^{-})}\widetilde{\Psi}_{R}^{c}(y;k),
\left[\begin{array}{ccc}{-\zeta_{\rho}\,m^{2}\,\Theta_{[1]}}&{{\quad}}&{{\calO}_{1}}\\{{\calO}_{2}}&{{\quad}}&{-\zeta_{\rho}\,m^{2}\,\Theta_{[1]}}\\\end{array}\right]\rho\{p^{\mu}\}\,=\,0\quad,
\phi_{\alpha\alpha^{\prime}}=g_{\alpha\alpha^{\prime}\beta_{2}\cdots\beta_{p+1}}g_{\alpha\alpha^{\prime}\beta_{1}\beta_{3}\cdots\beta_{p+1}}^{(-1)}\cdotsg_{\alpha\alpha^{\prime}\beta_{1}\cdots\beta_{p}}^{(-1)^{p}}\.
{\calL}_{eff}(g_{\mu\nu})={\calL}_{eff}^{(0)}(g_{\mu\nu})+{\calL}_{eff}^{(2)}(\partial_{\lambda}\partial_{\tau}g_{\mu\nu},(\partial_{\lambda}g_{\mu\nu})^{2})+O[(\partial_{\lambda}g_{\mu\nu})^{4}]\,.
\hat{L}_{i_{2},i_{3}}^{i_{1},i_{4}}f\equivf(q^{i_{1}}x^{1},q^{i_{2}}x^{2},q^{i_{3}}x^{3},q^{i_{4}}x^{4}),\quadi_{1},i_{2},i_{3},i_{4}=1,\dots,4,
{\frac{d\bar{W}}{d\tilde{\phi}}}={\frac{d\tilde{\phi}}{d\tilde{y}}}=\pm\sqrt{\tilde{V}(\tilde{\phi})+3\left(w_{0}+\alpha{\bar{W}}(\tilde{\phi})\right)^{2}},\quad{\frac{dA}{d\tilde{y}}}=-\alphaw_{0}-\alpha^{2}\bar{W}(\tilde{\phi}).
f_{\alpha_{1}}^{N,k}(X)\cdotf_{\alpha_{2}}^{N,k}(X)=h(X)f_{rel}^{N,k}(X)+\langle{\calO}_{e^{\alpha_{1}}}{\calO}_{e^{\alpha_{2}}}{\calO}_{e^{h}}\rangle\eta^{hm}f_{m}^{N,k}(X)
([\hat{A},\hat{B}])_{\mu}^{\nu}=(\hat{A}\hat{B}-\hat{B}\hat{A})_{\mu}^{\nu}=\sum_{\sigma=0}^{3}(\hat{A}_{\mu}^{\sigma}\hat{B}_{\sigma}^{\nu}-\hat{B}_{\mu}^{\sigma}\hat{A}_{\sigma}^{\nu}).
N^{(1)}=\frac{1}{\pi}\left(\frac{m\beta}{\pi}\right)^{2}\left(\sum_{n=0}^{\infty}\frac{1}{[(2n+1)^{2}+(\frac{m\beta}{\pi})^{2}]^{3/2}}\right)\,\int_{-\infty}^{\infty}dx\,\theta^{\prime}
\{\hat{\bar{\eta}}_{\dot{\alpha}}^{\prime},\hat{\bar{\eta}}_{\dot{\beta}}^{\prime}\}=\{{\hat{\eta}}_{\alpha}^{\prime},{\hat{\eta}}_{\beta}^{\prime}\}=0,\\\\\\\{\hat{\eta}_{\alpha},\hat{\bar{\eta}}_{\dot{\alpha}}^{\prime}\}=-2\sigma_{\alpha\dot{\alpha}}^{\mu}\hat{p}_{\mu}
ds^{2}=H^{-1/4}\left[\Big(1-\frac{r_{0}^{2}}{r^{2}}\Big)dt^{2}+\sum_{i=1}^{5}(dy^{i})^{2}+H\Big(\Big(1-\frac{r_{0}^{2}}{r^{2}}\Big)^{-1}dr^{2}+r^{2}d\Omega_{3}^{2}\Big)\right]\,
{G}^{\left(4\right)}\left(p,p^{\prime},P_{os}\right)=\frac{-i}{\left(2\pi\right)^{4}}\frac{{\chi}\left(p,P_{os}\right){\bar{\chi}}\left(p^{\prime},P_{os}\right)}{2\omega\left(P^{0}-\omega+i\epsilon\right)}+\mathrm{reg.\,terms},\,\omega:=\sqrt{{\bfP}^{2}+M^{2}}\,
h_{~\nu}^{\mu}=\left(\begin{array}{ccccc}{h_{1}}&{h_{3}}&{0}&{0}&{0}\\{-d^{2}h_{3}/f}&{h_{2}}&{0}&{0}&{0}\\{0}&{0}&{h_{~\chi}^{\chi}}&{h_{~\theta}^{\chi}}&{h_{~\phi}^{\chi}}\\{0}&{0}&{h_{~\chi}^{\theta}}&{h_{~\theta}^{\theta}}&{h_{~\phi}^{\theta}}\\{0}&{0}&{h_{~\chi}^{\phi}}&{h_{~\theta}^{\phi}}&{h_{~\phi}^{\phi}}\\\end{array}\right)
\vec{u}^{2}=\vec{u}^{2}+\left(\frac{\lambdae^{2}}{4\piM}\\frac{1}{(s_{t})^{3}}\right)^{2}-2\frac{\lambdae^{2}}{4\piM}\\frac{1}{(s_{t})^{3}}\\left(\sqrt{s_{t}^{2}-s_{y}^{2}},s_{y},0\right)\cdot(u_{x},0,0)
\epsilon^{\dagger}\hat{\Gamma}_{013}\epsilon=\epsilon^{\dagger}\hat{\Gamma}_{013}(\hat{\Gamma}_{012}\epsilon)=-\epsilon^{\dagger}\hat{\Gamma}_{012}\hat{\Gamma}_{013}\epsilon=-\left(\hat{\Gamma}_{012}\epsilon\right)^{\dagger}\hat{\Gamma}_{013}\epsilon=-\epsilon^{\dagger}\hat{\Gamma}_{013}\epsilon
\displaystyle\frac{1}{\mu-\hat{x}}=\displaystyle\frac{1}{\nu}=\displaystyle\frac{(-1)^{\delta}e^{iZ_{N}(\hat{x}+\nu)}iZ_{N}^{\prime}(\hat{x}+\nu)}{1+(-1)^{\delta}e^{iZ_{N}(\hat{x}+\nu)}}+...
\mathrm{A}_{\mu}^{(n)}(y)={\calL}_{\mu\nu}A^{\nu}({\calL}^{T}y)=-\frac{1}{2}{\calL}_{\mu\nu}F_{\\sigma}^{\nu}{\calL}_{\lambda}^{\\sigma}y^{\lambda}=-\frac{1}{2}({\calL}F{\calL}^{T}y)_{\mu}.
S_{\mathrm{odd}}(a;b_{1},\ldots,b_{k-1})=\sum_{n=1}^{\infty}\frac{1}{(2n-1)^{a}}\prod_{i=1}^{k-1}\sum_{m_{i}=1}^{n-1}\frac{1}{(2m_{i}-1)^{b_{i}}}\,,
<\bar{z}\vert\hat{Q}_{m}\vertz>\equiv<\bar{z}^{\prime}\vert\hat{Q}_{m}\vertz>\Big\vert_{z^{\prime}\rightarrowz}=\hat{Q}_{m}(\bar{z}^{\prime})<z^{\prime}\vertz>\Big\vert_{z^{\prime}=z}=Q_{m}(\bar{z},z)<\bar{z}\vertz>
\renewcommand{\arraystretch}{1.2}\begin{array}{rcl}{d_{\perp}X}&{=}&{0,}&{F}&{=}&{0,}&{d_{\perp}\chi}&{=}&{0,}&{\psi}&{=}&{0,}\\{{}P_{\perp}^{+}}&{=}&{0,}&{\eta_{\parallel}}&{=}&{0,}&{A_{\perp}}&{=}&{0,}&{B_{\parallel}}&{=}&{0,}\\{(*\eta^{+})_{\parallel}}&{=}&{0,}&{(*P)_{\perp}}&{=}&{0,}&{(*B^{+})_{\parallel}}&{=}&{0,}&{(*A^{+})_{\perp}}&{=}&{0.}\\\end{array}
\times\mathrm{exp}\Bigg\{\frac{i}{\hbar}\sum_{\alpha=0}^{N-1}\left[(r_{\alpha+1}-r_{\alpha})p_{r\alpha}+\hbar(\theta_{\alpha+1}-\theta_{\alpha})l_{\alpha}+(z_{\alpha+1}-z_{\alpha})p_{z\alpha}-\epsilonh_{\alpha}-\epsilon\xi_{\alpha}\left(p_{z\alpha}+g\ell_{\alpha}\right)\right]\Bigg\}\\\,
{\calH}_{r}={\frac{1}{2}}(\pi_{\phi}+gA_{1})^{2}+{\frac{1}{2ag^{2}}}(\pi_{1}^{\prime}+g\phi^{\prime})^{2}+{\frac{1}{2}}({\frac{\pi_{1}^{2}}{\tilde{\alpha}}}+\phi^{2})+{\frac{1}{2}}ag^{2}A_{1}^{2}.
\nabla_{\dot{A}a}\overline{{W}}^{2}=2W^{\dot{C}\dot{D}}\nabla_{\dot{A}a}W_{\dot{C}\dot{D}}=4iW^{\dot{C}\dot{D}}W_{\dot{C}\dot{D}\dot{A}a}-\frac{4}{3}W_{\dot{A}\dot{B}}\Lambda_{a}^{\dot{B}}
\Sigma_{\mu\nu}^{(2)}(p)=g^{2}T\sum_{n}\int\frac{d^{2}q}{(2\pi)^{2}}\epsilon_{\mu\sigma\eta}G_{\sigma\lambda}^{0}(p+q)\epsilon_{\lambda\tau\nu}D_{\tau\eta}^{0}(q)\;,
T(z_{1},\theta_{1})\cdotT(z_{2},\theta_{2})=\frac{\frac{1}{4}\hat{c}}{z_{12}^{3}}+\frac{3\theta_{12}}{2z_{12}^{2}}~T(z_{2},\theta_{2})+\frac{\frac{1}{2}}{z_{12}}~D_{2}T+\frac{\theta_{12}}{z_{12}}~\partial_{2}T+\dots~,
\Bigg\lbrace\frac{\partial}{\partial\mu}-\intdV_{x}\overline{\beta}_{\phi}\frac{\partial}{\partial\overline{\phi}}-\overline{\beta}_{m}\frac{\partial}{\partial\overline{m}^{2}}-\overline{\beta}_{\lambda}\frac{\partial}{\partial\overline{\lambda}}\Bigg\rbrace\Gamma=\frac{\partial\Lambda(\overline{m}^{2},\mu)}{\partial\mu}
\begin{array}{l}{\Omega=\displaystyle{\sum_{l<k}}{\bfF}_{lk}^{f}\,dx_{f}^{l}\wedgedx_{f}^{k},\quad{\bfB}={\bfB}_{l}^{f}dx_{f}^{l},}\\{{\bfF}_{lk}^{f}=\partial_{l}^{f}{\bfB}_{k}-\partial_{k}^{f}{\bfB}_{l}-ig[{\bfB}_{l},{\bfB}_{k}].}\\\end{array}
<0\vert(a_{n})^{r_{n}}\cdot...\cdot(a_{1})^{r_{1}}(a_{1}^{\dagger})^{r_{1}}\cdot...\cdot(a_{n}^{\dagger})^{r_{n}}\vert0>\quad=\prod_{i=1}^{n}[r_{i}]_{q}!
\rho=\frac{1}{2}\delta_{ij}\lbrack\Phi^{i},_{z}+\frac{\partialW}{\partial\Phi^{j}}\rbrack\lbrack\Phi^{j},_{z}+\frac{\partialW}{\partial\Phi^{i}}\rbrack-\frac{\partialW}{\partial\Phi^{i}}\Phi^{i},_{z},
{\calL}=\frac12\epsilon^{\mu\nu\lambda\rho}\partial_{\mu}A_{\lambda\rho}-\frac14\left(B_{\lambda\rho}+\partial_{\rho}-\partial_{\rho}C_{\lambda}\right)\left(B^{\lambda\rho}+\partial^{\rho}-\partial^{\rho}C^{\lambda}\right),
\frac{Q^{1}(Q^{2})^{*}}{|Q^{1}||Q^{2}|}=\mathrm{e}^{-i\omega_{Q}}\,,\qquad\frac{(a_{1})^{*}a_{2}}{|a_{1}||a_{2}|}=\mathrm{e}^{i\omega_{V}}\,,
M_{1}=\epsilon_{1}\,(\frac{\Lambda_{1}^{4}m_{0}}{m_{1}})^{\frac{1}{2}}+\epsilon_{2}\,(\frac{\Lambda_{2}^{4}m_{2}}{m_{1}})^{\frac{1}{2}},\quad\mathrm{det}\,T_{(0,\,2)}=0,\quadA=\epsilon_{1}\epsilon_{2}\,(\frac{m_{0}m_{2}}{\Lambda_{1}^{4}\Lambda_{2}^{4}})^{\frac{1}{2}}.
J^{-}=\frac{k}{2\pi}\mu,\,\,\,\,\,\,\,\,\,\,F_{-\alpha}^{>}=0,\,\,\,\,\,\,\,\,\,\,\widetilde{J}^{+}=\frac{k}{2\pi}\nu,\,\,\,\,\,\,\,\,\,\,\widetilde{F}_{+\alpha}^{>}=0,
L_{\mu\nu}^{(1)}=\int{\frac{d^{D}p}{(2\pi)^{D}}}{\frac{dz}{(kz)^{3}}}{\frac{dz^{\prime}}{(kz^{\prime})^{3}}}D_{q}(1/k,z)S_{q+p}(z,z^{\prime})(2p+q)_{\mu}(2p+q)_{\nu}S_{p}(z,z^{\prime}),D_{q}(z^{\prime},1/k)
\mathrm{with}\;\;\overline{{{\cal{Y}}^{a}}}({\bf{r}})={\textstyle\frac{\partial_{j}}{\partial^{2}}\overline{{{\calA}_{j}^{\alpha}}}({\bfr})}=r_{a}\,\frac{\overline{{\cal{N}}}}{gr}\;\;\mathrm{and}\;\;{\cal{X}}^{a}({\bf{r}})={\textstyle\frac{\partial_{i}}{\partial^{2}}}A_{i}^{\alpha}({\bf{r}})=r_{a}\,\frac{\cal{N}}{gr}\,.
\begin{array}{lcll}{\mathrm{floor}}&{\mathrm{SU(3/1)}}&{\mathrm{field}}&{\mathrm{helicity}}\\{\mid\mathrm{gnd}>}&{(0~0~-\frac{3}{2})}&{A_{\mu}}&{+1}\\{\mid\mathrm{1st}>}&{(0~1~-\frac{3}{2})}&{\overline{3}\lambda}&{+\frac{1}{2}}\\{\mid\mathrm{2nd}>}&{(1~0~-\frac{1}{2})}&{3\phi}&{~0}\\{\mid\mathrm{3rd}>}&{(0~0~+\frac{1}{2})}&{\chi}&{-\frac{1}{2},}\\\end{array}
\left[\omega_{\mu_{1}\nu}^{0}\right]_{4}\;=\frac1{g^{2}}\;\left(F_{\mu_{1}\sigma}^{a}F_{\nu}^{a\;\sigma}\;-\;\frac14g_{\mu_{1}\nu}F_{\rho\sigma}^{a}F^{a\rho\sigma}\right)+\mathcal{B}_{\Sigma}\left(A_{\mu_{1}}^{*a}A_{\nu}^{a}\right)\;\;,\;
\begin{array}{lll}{l_{q}}&{=}&{q^{2}[(x^{0})^{2}-q^{-1}x^{+}x^{-}-qx^{-}x^{+}-(x^{3})^{2}]}\\{\,}&{=}&{q^{2}[(x_{0})^{2}-qx_{+}x_{-}-q^{-1}x_{-}x_{+}-(x_{3})^{2}]\;;}\\{\,}&{\,}&{\,}\\{l_{q}}&{=}&{CD-q^{2}AB\quad.}\\\end{array}
{\delta}{\lambda}_{4}(1235)=\frac{8}{3}g_{4}^{4}\int_{{\Lambda}}\frac{d^{4}\vec{p}_{4}}{(2{\pi})^{4}}\frac{{\Phi}(4;1235)}{(\vec{p}_{4}^{2}+m^{2})((\vec{p}_{1}+\vec{p_{2}}+\vec{p}_{4})^{2}+m^{2})},
{\calA}_{b}={\frac{e\rho\,(Z_{1}^{2}-Z_{-}^{2})}{2\sqrt{1-e^{2}\rho^{2}}}}+{\frac{1}{2}}\int_{1}^{{\frac{1}{e\rho}}}\!{\frac{dx}{(x^{2}-1)^{3/2}}}\left(W_{1}^{2}(\rhox)-W_{-}^{2}(\rhox)-x(Z_{1}^{2}-Z_{-}^{2})\right)\,\cdotp
(-g)\,{\epsilon}^{{\delta}_{(1)}ba}\langle{A_{{\sfGI}\,i}^{{\delta}_{(1)}}({\bf{r}})\rangle\partial_{i}\int\frac{d{\bfx}_{1}}{4{\pi}|{\bfr}-{\bfx}_{1}|}\,j_{0}^{a}({{\bfx}_{1}}})=ig\,{\epsilon}^{{\delta}_{(1)}ba}{\phi}(r)L^{{\delta}_{(1)}}{\int\frac{d{\bfx}_{1}}{4{\pi}|{\bfr}-{\bfx}_{1}|}\,j_{0}^{a}({{\bfx}_{1}}}),
\langlep_{1}p_{2}|S|p_{1}^{\prime}p_{2}^{\prime}\rangle=\int\prod_{i=1}^{2}\frac{e^{i\omega(\vec{p}_{i})t-i\vec{x}\vec{p}_{i}}}{(2\pi)^{2})\sqrt{2\omega(p_{i})}}\,\cdot\prod_{j=1}^{2}\frac{e^{-i\omega(p_{j}^{\prime})t+i\tilde{y}\vec{p}_{j}}}{(2\pi)^{2}\sqrt{2\omega(p_{j})}}\cdotK(x,y)dxdy
\langle0|T^{+}A_{-}({x})A_{\nu}({y})|0\rangle=i\int\frac{d^{d+2}k}{(2\pi)^{d+2}}\frac{e^{-i{k}\cdot({x}-{y})}}{k^{2}+i\epsilon}\frac{(k_{-}n_{\nu}^{Weyl}+k_{-}^{Weyl})}{k_{+}+i\\epsilon^{\prime}\\mathrm{sgn}(k_{-})}.
{\bfL}_{n}=\int{\frac{dz}{2i\pi}}\lbracka_{0}{{\gamma}^{n}}J_{sl(2)}^{0}-{\frac{a_{-}}{2}}{{\gamma}^{n+1}}J_{sl(2)}^{-}+{\frac{a_{+}}{2}}{{\gamma}^{n-1}}J_{sl(2)}^{+}\rbrack,
\Gamma_{I}^{\phi_{S}^{0}}(-i\Omega_{m},\stackrel{\rightharpoonup}{p})=-G/\sum_{Q}g_{QQ}(-\Omega_{m}^{2}-{\stackrel{\rightharpoonup}{p}}^{2}-4m_{Q}^{2})K_{Q}^{T}(-i\Omega_{m},\stackrel{\rightharpoonup}{p}).
F_{~~~~\mu\nu~\alpha\beta}^{-1\sigma~~\rho}F_{\rho~~\omega}^{~\alpha\beta~\eta\kappa}=\frac{1}{2}\delta_{\omega}^{\sigma}\left(\delta_{\mu}^{\eta}\delta_{\nu}^{\kappa}-\delta_{\mu}^{\kappa}\delta_{\nu}^{\eta}\right)
{\calL}=\bar{\Psi}_{a}(i\partial\!\!\!/-m_{0})\Psi_{a}+\frac{g^{2}}{2}\left\{(\bar{\Psi}_{a}\Psi_{a})^{2}+\lambda(\bar{\Psi}_{a}i\gamma_{5}\Psi_{a})^{2}\right\},\\\a=1,\ldots,N.
22<0\vert\bar{\psi}\psi\vert0>_{J_{1}}=\frac{\tilde{A}\Sigma_{0}^{2}\Lambda}{12\pi\alpha\omega}\left[e^{\omega\delta}(\omega-1)\left(\frac{\Lambda}{\Sigma_{0}}\right)^{\omega}+e^{-\omega\delta}(\omega+1)\left(\frac{\Sigma_{0}}{\Lambda}\right)^{\omega}\right].
\mathcal{S}=\Sigma+\displaystyle\intd^{4}x\left(b^{a}\partialA^{a}+h_{\nu}^{a}\partial_{\mu}B^{a\mu\nu}+\omega^{a}\partial\xi^{a}+h^{a\mu}\partial_{\mu}e^{a}+\omega^{a}\lambda^{a}+-\lambda^{a}\partial\overline{{\xi}}^{a}\right)\;,
\left(D_{Z_{1}Z_{2}}^{\mu}\partial_{\mu}-m_{Z_{1}Z_{2}}\right)\phi_{Z_{2}}({\bfr},t)=\sum_{h}U_{Z_{1}\{Z_{2}Z_{3}Z_{4}\}_{as}}^{h}\phi_{Z_{2}}({\bfr},t)\phi_{Z_{3}}({\bfr},t)\phi_{Z_{4}}({\bfr},t)
V_{eff}=\frac{1}{2}m_{10}^{2}(\beta)v^{2}+\frac{1}{4}\frac{\lambda_{10}(\beta)}{\beta}v^{4}+\frac{28}{9}\frac{\tilde{\lambda}^{3}(\beta)\zeta(3)}{2^{9}\pi^{4}}v^{6}.
\prod_{a=1}^{N}\frac{1}{(2n_{a})!}\;\left\langle\prod_{b=1}^{N}\left[\chi_{+}^{(b)}[Q^{0,S}]\chi_{-}^{(b)}[Q^{0,S}]\right]^{n_{b}}\right\rangle_{Q^{0,S}}\;\leq\;\prod_{a=1}^{N}\left(\xi\;8^{[1-\frac{g}{\pi+gN}]}\;2^{\frac{1}{q}}\right)^{2n_{a}}\;.
\begin{array}{c}{A_{0}^{n}=\left(1-q^{-1}V_{1}^{n^{-2}}\right)U_{1}^{n}U_{2}^{n},}\\{A_{1}^{n}=V_{1}^{n}U_{2}^{n},A_{-1}^{n}=V_{1}^{n^{-1}}U_{2}^{n},}\\{a_{n}=V_{1}^{n^{-\frac12}}V_{2}^{n^{\frac14}}V_{2}^{n+1^{-\frac14}}.}\\\end{array}
L=\frac12M\dot{x}^{\mu}\dot{x}_{\mu}+\frac{e_{0}}{c}\dot{x}^{\mu}a_{\mu}+e_{0}\phi-\frac{\lambda}{4c}f^{\alpha\beta}f_{\alpha\beta}=\frac12M\dot{x}^{\mu}\dot{x}_{\mu}+\frac{e_{0}}{c}\dot{x}^{\alpha}a_{\alpha}-\frac{\lambda}{4c}f^{\alpha\beta}f_{\alpha\beta}\.
S_{k}=\kappa_{B}\frac{2k\pi\Sigma_{d-2}}{\Omega_{d-2}G_{k}}\int_{r_{\mathrm{min}}}^{r_{+}}\,r^{(d-2k-1)}\left(\gamma+\frac{r^{2}}{l^{2}}\right)^{k-1}dr\;,
\begin{array}{l}{\alpha_{i}\alpha_{j}^{*}(b_{i}b_{j}^{\dagger}-q_{ij}b_{j}^{\dagger}b_{i})+\beta_{i}^{*}\alpha_{j}^{*}(b_{i}^{\dagger}b_{j}^{\dagger}-q_{ij}b_{j}^{\dagger}b_{i}^{\dagger})+\alpha_{i}\beta_{j}(b_{i}b_{j}-q_{ij}b_{j}b_{i})}\\{+\beta_{i}^{*}\beta_{j}(b_{i}^{\dagger}b_{j}-q_{ij}b_{j}b_{i}^{\dagger})=\delta_{ij}.}\\\end{array}
{\frac{8}{D-2}}\left[{\frac{A}{B}}\,C^{D-2}\,\phi^{\prime}\right]^{\prime}-{\frac{\alpha_{e}\,A\,B\,\kappa^{2}\,Q^{2}\,e^{\alpha_{e}\phi}}{2\,C^{D-2}\,A_{D-2}^{2}}}=0
I_{s}=<-\frac{1}{3!}m^{2}(C_{mnp}-L_{mnp})(C^{mnp}-L^{mnp})-\frac{1}{2}m^{2}(\phi-\omega)^{2}+\frac{1}{4!}m\epsilon^{mnpq}(\phi-\omega)G_{mnpq}>.
I(p,m,\theta,\varepsilon)=i\int\frac{d^{4}k}{(\theta^{2}k_{0}^{2}-|\mathbf{k}|^{2}-m^{2}+i\varepsilon)[\theta^{2}(k+p)_{0}^{2}-|\mathbf{k}+\mathbf{p}|^{2}-m^{2}+i\varepsilon]}
\phi(\langle\Gamma_{q}\rangle,.)=\sum_{\langle\Gamma_{q_{0}}\rangle\in\pi_{1}(Q)}\psi_{S}^{m}(\langle\Gamma_{q_{0}}\cup\Gamma_{q}\rangle)\chi_{m}^{F}(\langle\Gamma_{q_{0}}\cup\Gamma_{q}\rangle,.).