harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
0=\phi_{1}\phi_{2}^{\prime}-\phi_{2}\phi_{1}^{\prime}+H_{1}(\phi_{1}^{2}+\phi_{2}^{2})+(im(1-\phi_{1}^{2}-\phi_{2}^{2})+\frac{1}{2}M^{2}r^{2}(H_{1}-\beta^{\prime});
W_{jk}^{\scriptscriptstyle0}=\frac{hm}{4}\left[\left(\frac{4}{(hm)^{2}}+1\right)\delta_{j,k}-\left(\frac{2}{(hm)^{2}}-\frac{1}{2}\right)\left(\delta_{j,k-1}+\delta_{j,k+1}\right)\right].
\delta_{\mathrm{shift}}\int_{\Sigma_{g}}\sqrt{g}=-\int_{\Sigma_{g}}\frac{1}{2}\sqrt{g}\,tr\,(r_{\mu}^{\gamma}g_{\gamma\nu}+r_{\nu}^{\gamma}g_{\gamma\mu})=-\int_{\Sigma_{g}}\sqrt{g}\,r_{\mu}^{\mu}.
Q^{(n)}(w|\zeta)^{\varepsilon}=\prod_{a\inA}\zeta_{a}\left(\prod_{j=1}^{a-1}(xz_{j}-w_{a})\prod_{j=a+1}^{2n}(xw_{a}-z_{j})\right)\prod_{a,b\inA\atopa<b}(x^{-1}w_{b}-xw_{a})^{-1},
\delta_{\kappa}\Gamma=-{\frac{1}{8\pi}}\left[\int_{\Sigma_{2},\xi^{3}=\pi}\widetilde{\iota_{\kappa}\omega^{\prime}}+\int_{\Sigma_{2},\xi^{3}=0}\widetilde{\iota_{\kappa}\omega^{\prime\prime}}\right]+{\frac{1}{2}}\int_{\Sigma_{3}}\delta_{\kappa}X_{3},
G=\left(\begin{array}{llccrr}{R^{2}}&{u}&{v}&{-u}&{-2v-R^{2}}&{-u}\\{u}&{R^{2}}&{u}&{v}&{-u}&{-2v-R^{2}}\\{v}&{u}&{R^{2}}&{u}&{v}&{-u}\\{-u}&{v}&{u}&{R^{2}}&{u}&{v}\\{-2v-R^{2}}&{-u}&{v}&{u}&{R^{2}}&{u}\\{-u}&{-2v-R^{2}}&{-u}&{v}&{u}&{R^{2}}\\\end{array}\right),
S=S_{CS}[A]+\frac{s}{4\pi}\int_{\Sigma}d\phidy\,{\mathrm{Tr}}\,[(A_{\phi})^{2}]-S_{CS}[\bar{A}]+\frac{s}{4\pi}\int_{\Sigma}d\phidy\,{\mathrm{Tr}}\,[({\bar{A}}_{\phi})^{2}]\,.
\frac{SU(1,1)}{U(1)}\times\frac{SO(2,\frac{n_{A}(n_{A}-1)}{2}+\frac{n_{B}(n_{B}-1)}{2})}{SO(2)\timesSO(\frac{n_{A}(n_{A}-1)}{2}+\frac{n_{B}(n_{B}-1)}{2})}\times\frac{SO(4,n_{A}n_{B})}{SO(4)\timesSO(n_{A}n_{B})}
\begin{aligned}{(\vec{\beta}_{\nu}\cdot\vec{\beta}_{\lambda})}&{{}=\pm2\,}&{~\mathrm{for}\\nu=\pm\lambda\;}\\{(\vec{\beta}_{\nu}\cdot\vec{\beta}_{\lambda})}&{{}\in\{-1,0,+1\}\,}&{~\mathrm{for}\\nu\neq\pm\lambda\.}\\\end{aligned}
\not\!\!{n}_{k}\equiv{\bfn}_{k}\cdot\sigma\equivn_{k}^{i}\sigma_{i}=\left(\begin{array}{lr}{n_{k}^{3}}&{n_{k}^{1}+in_{k}^{2}}\\{n_{k}^{1}-in_{k}^{2}}&{-n_{k}^{3}}\\\end{array}\right)
g\left(\begin{array}{cc}{n_{1}}&{n_{2}}\\{m_{1}}&{m_{2}}\\\end{array}\right)=\frac{G}{2^{(n_{1}+m_{1}+n_{2}+m_{2})/2}}\sqrt{\frac{\Gamma(n_{1}+m_{1}+1)\Gamma(n_{2}+m_{2}+1)}{\Gamma(n_{1}+1)\Gamma(m_{1}+1)\Gamma(n_{2}+1)\Gamma(m_{2}+1)}}.
\left(\begin{matrix}{y}\\{y^{\prime}}\\\end{matrix}\right)=-\frac{1}{k^{2}}\left(\begin{matrix}{\frac{s^{\prime}}{s}}&{-1}\\{k^{2}-\left(\frac{s^{\prime}}{s}\right)^{2}}&{\frac{s^{\prime}}{s}}\\\end{matrix}\right)\left(\begin{matrix}{x}\\{x^{\prime}}\\\end{matrix}\right)~,
S=-16(0,606)^{9}1.2\frac{2\pi}{(4\pi^{2}\alpha^{\prime})^{1.2}g}\intdt\left[1+\frac{(2\pi\alpha^{\prime})^{2}}{4}(2\partial_{t}X^{i}\partial_{t}X^{i}+2i\overline{{\theta}}\Gamma^{0}\partial_{t}\theta)\right]
\begin{aligned}{A_{\mu}{}_{\mathrm{bare}}}&{{}=z_{3}^{\frac{1}{2}}A_{\mu},\qquad}&{c_{\mathrm{bare}}}&{{}=z_{c}^{\frac{1}{2}}c,\qquad}&{\overline{c}_{\mathrm{bare}}}&{{}=z_{c}^{\frac{1}{2}}\overline{c},}\\{b_{\mathrm{bare}}}&{{}=z_{3}^{-\frac12}b,\qquad}&{g_{\mathrm{bare}}}&{{}=z_{1}z_{3}^{-\frac32}g,\qquad}&{\xi_{0}{}_{\mathrm{bare}}}&{{}=z_{3}\xi_{0}.}&{}\\\end{aligned}
\left.-\mathrm{Sp}\left(\frac{t-1}{t-c}\right)+\mathrm{Sp}\left(\frac{1-t}{1-\frac{t}{c}}\right)-\mathrm{Sp}\left(\frac{1-t}{1-ct}\right)+\mathrm{Sp}\left(\frac{1-t^{2}}{1-ct}\right)-\mathrm{Sp}\left(\frac{1-t^{2}}{1-\frac{t}{c}}\right)\right]
={\frac{1}{2}}f^{\mu}f_{\mu}-\epsilon^{ij}(f^{0}\partial^{i}A^{j}-f^{i}\partial^{0}A^{j}+f^{i}\partial^{j}A^{0})+{\frac{\mu}{2}}\epsilon^{ij}(2A^{0}\partial^{i}A^{j}-A^{i}\partial^{0}A^{j}).
{\calP}_{\Lambda}^{0}\,=\,\langle{\bar{V}}\,\vert\,T_{\Lambda}\,V\rangle\,=\,\langle{V}\,\vert\,T_{\Lambda}\,{\bar{V}}\rangle\,=\,\mathrm{exp}\left[{\calK}\right]\,\langle{\bar{\Omega}}\,\vert\,T_{\Lambda}\,{\Omega}\rangle
\frac{\partialf(x;\eta)}{\partial\eta}=\left(\eta^{-1}A(x)+(1-\eta)^{-1}(B_{0}(x)+B_{1}(x)\frac{\partial}{\partialx}+B_{2}(x)\frac{\partial^{2}}{\partialx^{2}})\right)f(x;\eta)+{\calO}(\frac{1}{j_{0}^{3}}).
\renewcommand{\arraystretch}{1.5}\begin{array}{rcl}{f_{1}^{(1)}}&{=}&{{}_{2}F_{1}(\lambda_{3},1-\lambda_{1};\lambda_{2}+\lambda_{3};1-x)\,,}\\{*f_{2}^{(1)}}&{=}&{x^{1-\lambda_{3}-\lambda_{4}}(1-x)^{1-\lambda_{2}-\lambda_{3}}{}_{2}F_{1}(\lambda_{1},1-\lambda_{3};\lambda_{1}+\lambda_{4};1-x)\,,}\\\end{array}
H^{(2)}=\intd^{2}x\left[-\frac{1}{2}\partial_{i}\Phi^{0}\partial^{i}\Phi^{0}+\frac{m}{\sqrt{\kappa}}\Phi^{0}\epsilon_{ij}\partial^{i}\Phi^{j}+\frac{m^{2}}{2\kappa}\Phi^{i}\Phi^{i}-(\frac{\sqrt{\kappa}}{m}\partial_{i}\Phi^{i}+\frac{1}{2}\Phi^{3})\Phi^{3}\right].
\intdU\Upsilon_{\sigma}(U)\Upsilon_{\tau}(U^{\dagger})=\delta_{T_{\sigma},T_{\tau}}C_{\sigma}=\prod_{l}\delta_{\sigma^{(l)},\tau^{(l)}}C_{\sigma^{(l)}}=\prod_{l}\intdU_{l}\Upsilon_{\sigma^{(l)}}(U_{l})\Upsilon_{\tau^{(l)}}(U_{l}^{\dagger}).
\begin{array}{c}{C:\quadu=u_{0},\quad\quadF(u_{0},W)\equiv\int^{W}dv^{\prime}G(u_{0},v^{\prime})=\alpha(\sigma^{+})+\bar{\beta}(\sigma^{-})\,\,\,,}\\{D:\quadv=v_{0},\quad\quad\bar{F}(\bar{W},v_{0})\equiv\int^{\bar{W}}du^{\prime}G(u^{\prime},v_{0})=\bar{\alpha}(\sigma^{-})+\beta(\sigma^{+}),}\\\end{array}
\langleH^{(2)}\rangle_{\Gamma}\equiv{\frac{\int[{\calD}h]\Psi^{*}[h_{ij}]H^{(2)}\left[h_{ij},-i\hbar{\frac{\delta}{\deltah_{ij}}}\right]\Psi[h_{ij}]}{\int[{\calD}h]\Psi^{*}[h_{ij}]\left[h_{ij},-i\hbar{\frac{\delta}{\deltah_{ij}}}\right]\Psi[h_{ij}]}}.
M_{induced}^{2}(X)\simeq\frac{\lambda\,m^{2}}{(2\pi)^{3/2}}\left(\frac{T(X)}{m}\right)^{3/2}e^{-m/T(X)}\left(1+\frac{3}{8}\frac{T(X)}{m}\right)\;\;\;\;\;\;\;\;(T(X)<<m).
\Gamma^{(2)}=\frac{1}{32\pi^{2}}\intd^{4}x\sqrt{g}\left[4RF_{1}(\Box)R+4R_{\mu\nu}F_{2}(\Box)R_{\mu\nu}+\mathrm{Tr}({\calR}_{\mu\nu}F_{3}(\Box){\calR}_{\mu\nu})\right]
\begin{array}{ccl}{{\calY}^{a}}&{=}&{\{{\calY}^{A},{\calY}^{m},{\calY}^{3}\}}\\{{\calY}^{[ab]}}&{=}&{\{{\calY}^{AB},{\calY}^{Am},{\calY}^{mn},{\calY}^{A3},{\calY}^{m3}\}}\\{{\calY}^{[abc]}}&{=}&{\{{\calY}^{ABC},{\calY}^{ABm},{\calY}^{AB3},{\calY}^{Amn},{\calY}^{Am3},{\calY}^{mn3}\}}\\\end{array}
E[f^{(0)}+f^{(1)}]=E[f^{(0)}]+\frac{\deltaE}{\deltaf}[f^{(0)}]\cdotf^{(1)}+\frac{1}{2}\frac{\delta^{2}E}{\deltaf^{2}}[f^{(0)}]\cdot(f^{(1)})^{2}+\cdots.
E=-E_{n}\,,\;n\geq1\;\;\psi_{n,p}^{(-)}=\sqrt{\frac{E_{n}-m}{2E_{n}}}\frac{e^{ipx_{2}}}{\sqrt{2\pi}}e^{-\frac{1}{2}{\zeta}^{2}}\left(\begin{array}{c}{N_{n}H_{n}(\zeta)}\\{-\frac{\sqrt{E_{n}^{2}-m^{2}}}{E_{n}-m}N_{n-1}H_{n-1}(\zeta)}\\\end{array}\right)
\begin{array}{llll}{a_{\mu}^{\prime}=ka_{\mu},}&{B^{\prime3}=lB^{3},}&{C^{\prime3}=mC^{3},}&{\bar{C}^{\prime3}=n\bar{C}^{3},}\\{A_{\mu}^{\primea}=xA_{\mu}^{a},}&{B^{\primea}=yB^{a},}&{C^{\primea}=uC^{a},}&{\bar{C}^{\primea}=v\bar{C}^{a},}\\\end{array}
H_{L}|=\frac{1}{\gamma^{2}}\int_{0}^{2\pi}d\sigma~(f^{{\prime}2}+\alpha^{\prime}\beta^{\prime}),~~~~\omega_{L}|=\frac{1}{\gamma^{2}}\int_{0}^{2\pi}d\sigma~(\mathrm{d}f^{\prime}\wedge\mathrm{d}f+\mathrm{d}\beta^{\prime}\wedge\mathrm{d}\alpha),
\begin{array}{l}{{[}S_{a}^{b},U^{c}{]}=-U^{(bac)}}\\{{[}K_{ab},U^{c}{]}=U_{(acb)}-U_{(bca)}}\\{{[}K^{ab},U_{c}{]}=-U^{(bca)}+U^{(acb)}}\\{{[}S_{a}^{b},S_{c}^{d}{]}=S_{(abc)}^{d}-S_{c}^{(bad)}}\\{{[}S_{a}^{b},K_{cd}{]}=K_{(abc)d}+K_{c(abd)}}\\{{[}S_{a}^{b},K^{cd}{]}=-K^{(bac)d}-K^{c(bad)}}\\{{[}K_{ab},K^{cd}{]}=S_{(acb)}^{d}-S_{(bca)}^{d}-S_{(adb)}^{c}+S_{(bda)}^{c}}\\\end{array}
g=\sum_{l,j=1}^{n-1}\frac{1}{4\pi^{2}}(P^{-1}+X^{-1})_{lj}\sum_{a=1}^{3}dy_{l}^{a}dy_{j}^{a}+\sum_{l,j=1}^{n-1}(P^{-1}+X^{-1})_{lj}^{-1}\alpha^{l}\alpha^{j}
H_{j,j+1}=\mathrm{i}\left.\frac{dR_{j,j+1}(u)}{du}\right|_{u=0}=(1+\Gamma)\sigma_{j}^{x}\sigma_{j+1}^{x}+(1-\Gamma)\sigma_{j}^{y}\sigma_{j+1}^{y}+h(\sigma_{j}^{z}+\sigma_{j+1}^{z})\nonumber
S_{\mathrm{graviton}}[\gamma_{\mu\nu}^{\pm}]=\frac1{16\piG_{4}}\intd^{4}x\,\frac12\,[\,\gamma_{\mu\nu}^{+}\,\,\,\gamma_{\mu\nu}^{-}\,]\,\frac{\bfF(\Box)}{l^{2}}\,\left[\begin{array}{c}{\,\,\gamma_{+}^{\mu\nu}}\\{\,\,\gamma_{-}^{\mu\nu}\,}\\\end{array}\right],
S\,=\,\intd^{3}x\left[\xi_{\mu}(\partial^{\mu}\theta+i{\tilde{\phi}}^{*}\partial^{\mu}{\tilde{\phi}})-\frac{1}{4\rho}\xi_{\mu}\xi^{\mu}+\frac{1}{4\rho}(\partial\rho)^{2}-V(\rho)\right]\;.
{\calL}_{\mathrm{non-pert}}=-{\frac{2V^{(8)}}{2^{10}\pi^{6}}}\sum_{N>0}e^{-2i\piNT_{s}{\calT}_{2}}{\frac{1}{N}}\sum_{N=mn\atop0\leqj<n}[{\calD}{\calA}]\left(\frac{m\,{\calU}+j}{n}\right)\.
Q_{\pm}^{B}=\intdx^{1}\sum_{a,b}\left[f^{B}\right]^{ab}\left[(\partial_{0}\phi^{a}\mp\partial_{1}\phi^{a})\psi_{\pm}^{b}\pm\frac{\partialW^{1}}{\partial\phi^{b}}\psi_{\mp}^{a}\right]
\begin{array}{l}{Y_{2}R_{12}K_{1}R_{21}=R_{12}K_{1}R_{12}^{-1}Y_{2}+q^{2}R_{12}{\calP}\quad,}\\{\tilde{Y}_{2}R_{21}^{-1}K_{1}R_{21}=R_{12}K_{1}R_{21}\tilde{Y}_{2}+q^{-2}R_{12}{\calP}\quad.}\\\end{array}
\int_{0}^{R}\mathrm{d}r\;u_{kl}^{2}\left(r\right)=\frac{1}{2k}\left[u_{kl}^{\prime}\left(R\right)\partial_{k}u_{kl}\left(R\right)-u_{kl}\left(R\right)\partial_{k}u_{kl}^{\prime}\left(R\right)\right]
{\calM}_{T}\to\omega_{T}^{\mathrm{T}}{\calM}_{T}\omega_{T},\\\{\calM}_{U}\to\omega_{U}^{\mathrm{T}}{\calM}_{U}\omega_{U},\\\{\calF}_{\mu\nu}\to(\omega_{T}^{-1}\otimes\omega_{U}^{-1}){\calF}_{\mu\nu},
\chi_{\nu}(\rho)=\frac{1}{N}\left(\begin{array}{c}{\sqrt{E+m}(\epsilon_{n})^{n}J_{\epsilon_{n}\nu}(k\rho)}\\{i\sqrt{E-m}(\epsilon_{n})^{n+1}J_{\epsilon_{n}(\nu+1)}(k\rho)}\\\end{array}\right)
S_{fi}=\overline{{u}}_{R}^{\alpha^{^{\prime}}}(p_{1}^{^{\prime}})\Gamma_{R}^{\mu}(p_{1}^{^{\prime}},p_{1})u_{R}^{\alpha}(p_{1})iD_{R\mu\nu}(k)\overline{{u}}_{R}^{\beta^{^{\prime}}}(p_{2}^{^{\prime}})\Gamma_{R}^{\nu}(p_{2}^{^{\prime}},p_{2})u_{R}^{\beta}(p_{2})
\frac{1}{Z_{0}}\int{\calD}be^{-S^{\prime}[b]}e^{\int_{\calT}d^{2}xJ(x)b(x)}=e^{\frac{1}{2}\int_{\calT}d^{2}x\int_{\calT}d^{2}x^{\prime}J(x)G(x-x^{\prime})J(x^{\prime})}
\sigma_{(1)}(x)=-{\frac{1}{4}}\beta_{H}^{2}R_{(0)}+\beta_{H}^{2}\left({\frac{d(d-3)}{16}}(\nabla_{(0)}\sigma_{(0)})^{2}+{\frac{(d-1)}{4}}\nabla_{(0)}^{2}\sigma_{(0)}+{\frac{(d-2)(d+1)}{4L^{2}}}e^{\sigma_{(0)}}\right)
e^{-1}{\calL}=-{\frac{1}{2}}{\calR}+g_{ab^{*}}\partial_{\mu}z^{a}\partial^{\mu}\bar{z}^{b^{*}}-\mathrm{Im}\,({\calN}_{\Lambda\Sigma}(z,\bar{z}){\calF}_{\mu\nu}^{+\,\Lambda}{\calF}^{+\,\Sigma\,\mu\nu}),
\verta_{\nu}\vert^{2}=4\frac{\langleQ_{0}\rangle+4}{\langleQ_{1}\rangle}\left[\frac{H^{2}e^{\pi\nu}\Gamma(-\sigma)\Gamma(-\sigma^{*})}{2^{5}\pi^{4}m^{2}}\right]=4\frac{\langleQ_{0}\rangle+4}{\langleQ_{1}\rangle}\left[\frac{H^{2}\nu^{2}+H^{2}/4}{2^{4}\pi^{3}m^{2}}\right]\,.
e^{a}e^{b}\hat{\Omega}_{b,a}^{~~i}=0\quad\rightarrow\quad\hat{\Omega}_{b,a}^{~~~i}=\hat{\Omega}_{a,b}^{~i}={\nabla}_{b}E_{a}^{~\underline{b}}u_{\underline{b}}^{~i},
\omega_{\mu}^{\alpha\beta}=\frac{1}{2}e^{\nu\alpha}\left(\partial_{\mu}e_{\nu}^{\beta}-\partial_{\nu}e_{\mu}^{\beta}\right)-\frac{1}{2}e^{\nu\beta}\left(\partial_{\mu}e_{\nu}^{\alpha}-\partial_{\nu}e_{\mu}^{\alpha}\right)-\frac{1}{2}e^{\rho\alpha}e^{\sigma\beta}\left(\partial_{\rho}e_{\sigma\gamma}-\partial_{\sigma}e_{\rho\gamma}\right)e_{\mu}^{\gamma},
Z(\bar{\zeta},\zeta)=\int\!(d\bar{\psi}_{UV}^{\phantom{\rho}})(d\psi_{UV}^{\phantom{\rho}})e^{-\\bar{\psi}_{UVa}^{\phantom{UV}\alpha}\,Delta_{UV\alpha}^{-1\phantom{\alpha}\beta}\\psi_{UV\betaa}^{\phantom{\rho}}}\Z_{\Lambda}[\bar{\psi}_{UV}^{\phantom{\rho}},\psi_{UV}^{\phantom{\rho}},\bar{\zeta},\zeta]
\frac{L^{2}}{4na^{2}}\sum_{N=0}^{\infty}\sum_{\alpha=0}^{N}\frac{(2a)^{8-2n}{\pi}^{\frac{n}{2}}\Gamma(4-n)\Gamma(N-4+\frac{3n}{2})}{\Gamma(N+4-\frac{n}{2})\Gamma(\frac{3n}{2}-4)}Y_{\alpha}^{N}({\eta})\int_{C}d^{n}\eta^{\prime}Y_{\alpha}^{N}(\eta^{\prime})
ds^{2}=\alpha^{\prime}\left\{\frac{U^{2}}{R^{2}}\left((1-U_{T}^{4}/U^{4})dt^{2}+\sum_{i}dx_{i}^{2}\right)+\frac{R^{2}}{U^{2}}\frac{dU^{2}}{1-U_{T}^{4}/U^{4}}+R^{2}d\Omega_{5}^{2}\right\}.
\Lambda_{+}={\frac{1}{2}}\gamma^{0}\gamma^{+}=\left(\begin{array}{cc}{1}&{0}\\{0}&{0}\\\end{array}\right)~~~~~\Lambda_{-}={\frac{1}{2}}\gamma^{0}\gamma^{-}=\left(\begin{array}{cc}{0}&{0}\\{0}&{1}\\\end{array}\right)\;.
\Omega_{\mu}(x)={\frac{g^{2}}{32\pi^{2}}}\,\epsilon_{\mu\alpha\beta\gamma}\,A_{a}^{\alpha}\,(F_{a}^{\beta\gamma}-{\frac{g}{3}}f_{abc}A_{b}^{\beta}A_{c}^{\gamma})\equiv{\frac{g^{2}}{32\pi^{2}}}K_{\mu}\.
{\calP}_{n}=\left(d+\fracT2+\frac14\right)^{-\frac12}\left(\frac{4d+1-2T}{4d+1+2T}\right)^{\fracn2}P_{n}\left(\frac{4d-1}{\left[\left(4d+1\right)^{2}-4T^{2}\right]^{\frac12}}\right).
f^{1}(u)=f^{i}u_{i}^{1}+f_{k}^{(ij)}u_{i}^{1}u_{j}^{2}u_{2}^{k}+g_{k}^{(ij)}u_{i}^{1}u_{j}^{3}u_{3}^{k}+h_{(ij)}u_{2}^{i}u_{3}^{j}+\ldots
{\frac{d^{2}y}{d\tilde{\lambda}^{2}}}-{\frac{1}{2}}{\frac{{\calW}^{\prime}}{\calW}}\left({\frac{dy}{d\tilde{\lambda}}}\right)^{2}-{\frac{1}{2}}\left[{\calW}-\left({\frac{dy}{d\tilde{\lambda}}}\right)^{2}\right]{\calW}^{-1}\partial_{y}({\calW}g_{\mu\nu}){\frac{dx^{\mu}}{d\tilde{\lambda}}}{\frac{dx^{\nu}}{d\tilde{\lambda}}}=0.
S=-\sum_{I=1}^{K}\,\intd\tau_{I}\,(m_{0}^{I}\,\sqrt{v_{\mu}^{I}\,v_{I}^{\mu}}+\mathrm{tr}\,Z_{I}\lambda_{I}^{-1}{\dot{\lambda}}_{I})-\intd^{4}x\,\mathrm{tr}\,\bigl(j_{\mu}\,A^{\mu}+{\frac{1}{16\pi}}\,F_{\mu\nu}\,F^{\mu\nu}\bigr).
\mu^{2}(k_{1})=\int_{0}^{k_{1}}dk_{2}\frac{{(k_{1}+k_{2})}^{2}}{k_{2}{(k_{1}-k_{2})}^{2}}=\int_{0}^{k_{1}}dk_{2}(\frac{4k_{1}}{(k_{2}-k_{1})^{2}}+\frac{1}{k_{2}}).
j=\frac{r-1}{2}-\frac{t}{u}\,\frac{s-1}{2}\,,\qquadj^{\prime}+\ell+\frac{t}{t-u}\,\frac{1-\theta_{\ell}}{2}=\frac{r^{\prime}-1}{2}-\frac{t}{t-u}\,\frac{s^{\prime}-1}{2}
\frac{\partial}{\partialt}p_{0}(y,t)=\frac{\partial}{\partialy}\left(y\,p_{0}(y,t)\right)+\frac{\langlex^{2}\rangle}{2}\frac{\partial^{2}}{\partialy^{2}}p_{0}(y,t)-\frac{\partial}{\partialy}\intdx\,x\,a(x,y,t)\,.
P^{a}=\pi^{\alpha}\sigma_{\alpha\bar{\beta}}^{a}{\bar{\pi}}^{\bar{\beta}}\quadJ^{a}=\frac{1}{2}\sigma_{\alpha\bar{\beta}}^{a}(\pi^{\alpha}{\bar{\omega}}^{\bar{\beta}}+{\omega}^{\alpha}{\bar{\pi}}^{\bar{\beta}}),
G(z,k)=\frac{V}{(4\pi)^{3/2}}\frac{\Gamma(z-3/2)}{\Gamma(z)}\sum_{j=-\infty}^{+\infty}\omega_{j}^{2k}\left(\omega_{j}^{2}+m^{2}-e^{2}\mu^{2}\right)^{3/2-z}\;.
G_{f}(z,z^{\prime})=G_{(\sigma)}(z,z^{\prime})-\sum_{s=1}^{n}\sum_{p=a,b}\sum_{N_{s}}\int_{C_{p}^{(s)}}G_{(\sigma)}(z,z^{\prime\prime})\frac{d\theta^{\prime\prime}dz^{\prime\prime}}{2\pii}Y_{p,N_{s}}^{(0)}(t^{\prime\prime})\chi_{N_{s}}^{(0)}(z^{\prime})
W_{b}=\int_{r(\tau=-\frac{\tilde{\beta}}{2})=0}^{r(\tau=\frac{\tilde{\beta}}{2})=0}p_{b}dr=\Biggl\lbrace_{\frac{\pi^{2}}2\sigmar_{0}^{3},\qquadfor\quadN=3}^{\frac{4\pi}{3}\sigmar_{0}^{2},\qquadfor\quadN=2}
\delta\Omega^{\underline{{a}}\underline{{b}}}(d)=d\Omega_{~\underline{{b}}}^{\underline{{a}}}(\delta)+\Omega^{\underline{{a}}\underline{{c^{\prime}}}}(\delta)\Omega_{\underline{{c^{\prime}}}}^{~\underline{{b}}}(d)-\Omega_{~\underline{{c^{\prime}}}}^{\underline{{a}}}(d)\Omega^{\underline{{c^{\prime}}}\underline{{b}}}(\delta)\;,
\begin{aligned}{C_{n}}&{{}{}\mapstoC_{n+\theta}\,,}&{\qquad}&{{}A_{n}}&{\mapstoA_{n}+\theta\delta_{n,0}\,,}\\{B_{n}}&{{}{}\mapstoB_{n-\theta}\,,}&{}&{{}\bar{A}_{n}}&{\mapsto\bar{A}_{n}-\frac{\theta}{p}\delta_{n,0}\,.}\\\end{aligned}
R(e_{1})=\left(\begin{matrix}{0}&{0}&{\cdots}&{0}&{1}\\{1}&{0}&{\cdots}&{0}&{0}\\{0}&{1}&{\cdots}&{0}&{0}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}&{\vdots}\\{0}&{0}&{\cdots}&{1}&{0}\\\end{matrix}\right)
\rho_{i\pm}(\tau_{i\pm}(\tau,\vec{\sigma}),\vec{\sigma})=r_{i\pm}(\tau_{i\pm}(\tau,\vec{\sigma}),\vec{\sigma})[1\mp\vec{\beta}_{i}(\tau_{i\pm}(\tau,\vec{\sigma}))\cdot\hat{r}_{i\pm}(\tau_{i\pm}(\tau,\vec{\sigma}
*\wedge\Bigl(\delta\Omega,\Omega\Bigr)=-\frac{1}{4\pi}\Bigl[{\bfF}_{\mu\nu}\nabla_{\sigma}{\bfF}^{\sigma\nu}+(*{\bfF})_{\mu\nu}\nabla_{\sigma}(*{\bfF})^{\sigma\nu}\Bigr]dx^{\mu}\otimes\varepsilon^{1}\wedge\varepsilon^{2}.
S=\intd^{4}x\sqrt{{g}}e^{-\phi}\Bigl\{{R}+({\nabla}\phi)^{2}+\frac{1}{8}Tr\bigl({\calL}\nabla{\calM}\bigr)^{2}-\frac{1}{4}{\calF}_{\mu\nu}^{T}{\calM}{\calF}^{\mu\nu}-\frac{1}{12}{H}_{\mu\nu\lambda}^{2}\Bigr\}
\frac{{\lambda}^{2}}{16\pi^{2}}\,\left({{\sigma}_{c}}\,{{{\phi}_{c}}}^{2}+\frac{{{\sigma}_{c}}\,{{{\phi}_{c}}}^{2}}{2\,\epsilon}+\frac{{{{\phi}_{c}}}^{4}}{4}+\frac{{{{\phi}_{c}}}^{4}}{4\,\epsilon}+{{{\sigma}_{c}}}^{2}+\frac{{{{\sigma}_{c}}}^{2}}{4\,\epsilon}\right)
{\mathrmRe}\partial_{p^{2}}\Gamma_{+-}(p^{2})\big|_{p^{2}=\mu_{W}^{2}}=1\quad{\mathrmRe}\partial_{p^{2}}\Gamma_{HH}(p^{2})\big|_{p^{2}=\mu_{H}^{2}}=1\quad{\mathrmRe}\partial_{p^{2}}\Gamma_{\chi\chi}(p^{2})\big|_{p^{2}=\mu_{\chi}^{2}}=1
a_{{\bf{k}}}({\bf{q}})=\sum_{i}[a_{{\bf{k}}}({\bf{q}}),b_{i}^{\dagger}({\bf{q}})]b_{i}({\bf{q}})-\sum_{i}[a_{{\bf{k}}}({\bf{q}}),b_{i}(-{\bf{q}})]b_{i}^{\dagger}(-{\bf{q}})
\frac{d^{2}x^{\mu}}{d\lambda^{2}}+\Gamma_{\nu\lambda}^{\mu}\frac{dx^{\nu}}{d\lambda}\frac{dx^{\lambda}}{d\lambda}=-\xi~\kappa{}~\partial_{\nu}\phi~\frac{dx^{\nu}}{d\lambda}\frac{dx^{\mu}}{d\lambda}
\cdot\sum_{J_{1}n_{1}}\sum_{J_{2}n_{2}}\sum_{J_{3}n_{3}}F_{J_{1},n_{1};J_{2},n_{2};J_{3},n_{3}}(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3})A_{J_{1}n_{1}}(\varepsilon_{1})A_{J_{2}n_{2}}(\varepsilon_{2})A_{J_{3}n_{3}}(\varepsilon_{3}),
[x^{I},p^{J}]=i\delta^{IJ}~,~~~[\alpha_{n}^{i},\alpha_{m}^{j}]=\omega_{n}\delta^{ij}\delta_{n+m,0}~,~~~[\alpha_{n}^{i^{\prime}},\alpha_{m}^{j^{\prime}}]=\omega_{n}^{\prime}\delta^{i^{\prime}j^{\prime}}\delta_{n+m,0}~.
\partial_{s}\vec{\sigma}^{\primeQ}=i\hat{\Omega}\left(\Gamma_{s}+i\partial_{s}\right)\hat{\Omega}^{\dagger}\vec{\sigma}^{\primeQ}+i\hat{\Omega}\vec{\Gamma}_{s}Q=i\Gamma_{s}^{\prime}\vec{\sigma}^{\primeQ}+i\vec{\gamma}_{s}^{\prime}Q.
\begin{array}{cc}{\bar{\theta}_{i}=\theta^{i\dagger}\tilde{\gamma}^{0}=(\theta^{j})^{t}\bar{\calE}_{ji}~~~~}&{~~~~\bar{\tilde{\theta}}_{i}=\tilde{\theta}^{i\dagger}\gamma^{0}=(\tilde{\theta}^{j})^{t}\bar{\calE}_{ji}}\\\end{array}
\left\{\left(\frac{g_{0}}{4\pi}\right)^{2}\Phi_{N}\left[\Gamma^{[r,0]}\right]\right\}_{R,\nu}=\left[\Gamma_{N}^{\left[\left.r,0\right)\right.}-\Gamma_{N}^{(0)\mathrm{{pert}}}\right]+0\left(g^{2}(\nu),e(r+1)\right)\,.
H_{B}\to\frac2{3\gamma}t^{-1},\qquadV_{B}\to\left(\frac{\sqrt{3\rho_{0}k_{4}^{2}}\,\gamma}2\,t\right)^{2/\gamma},\qquad\rho_{B}\to\frac4{3k_{4}^{2}\gamma^{2}}\,t^{-2},
{\calE}_{\pm}^{B}=\int\!\!\int_{-\infty}^{\infty}dudv\,A_{\pm}(u,v),\qquad{\calE}_{\pm}^{F}=-\int\!\!\int_{-\infty}^{\infty}dudv\,\sqrt{f^{\prime}(u)g^{\prime}(v)}A_{\pm}(u,v),
\Phi^{\left(i_{1}j_{1}l_{1}\dotsi_{k}j_{k}l_{k}\right)\left(A_{1}B_{1}\dotsA_{k}B_{k}\right)}=\underbrace{U^{i_{1}}U^{j_{1}}U^{l_{1}}V^{A_{1}}V^{B_{1}}\dotsU^{i_{k}}U^{j_{k}}U^{l_{k}}V^{A_{k}}V^{B_{k}}}_{k}.
S=-\frac{1}{2\pi}\intd\taud\sigma\mathrm{Tr}[(D_{\alpha}X^{I})^{2}-i\Theta^{T}\gamma^{\alpha}D_{\alpha}\Theta+{\frac{1}{2g^{2}}}F_{\alpha\beta}^{2}+{\frac{g^{2}}{2}}[X^{I},X^{J}]^{2}-i\Theta^{T}\gamma_{I}[X^{I},\Theta]].
\Omega_{p+1}\,\equiv\int_{\Sigma_{p+1}}d^{p+1}\sigma\,\sqrt{\overline{g}}=\int_{0}^{T}d\tau\,e(\tau)\,\int_{\Sigma_{p}}d^{p}\sigma\,\sqrth\equivV_{p}\,\int_{0}^{T}d\tau\,e(\tau)\.
s_{W}\left(\begin{array}{ll}{A_{\mu}^{a}(p)}&{}\\{\omega_{a}(p)}&{}\\{\bar{\omega}_{a}(p)}&{}\\{\varphi_{a}(p)}&{}\\\end{array}\right)=\left(\begin{array}{ll}{p_{\mu}\omega_{a}(p)}&{}\\{0}&{}\\{-p^{\mu}A_{\mu}^{a}(p)-im_{a}\varphi_{a}(p)}&{}\\{im_{a}\omega_{a}(p)}&{}\\\end{array}\right).
\langle{\bfk}^{\prime}\vertH_{\mathrm{FT}}\vert{\bfk}\rangle=\int{\frac{\mathrm{d}{\bfr}}{(2\pi)^{3}}}e^{i({\bfk}-{\bfk}^{\prime})\cdot{\bfr}}\,\,{\cal{H}}({\bfr},{\frac{{\bfk}^{\prime}+{\bfk}}{2}}),
2\kappa^{2}S_{\Phi}={\frac{1}{\sqrt{G_{rr}}}}(\Phi^{\prime}|_{r=r_{\mathrm{i}}+\epsilon}-\Phi^{\prime}|_{r=r_{\mathrm{i}}-\epsilon})={\frac{1}{\sqrt{G_{rr}}}}\left(-{\frac{3}{4}}{\frac{Z_{6}^{\prime}}{Z_{6}}}+{\frac{1}{4}}{\frac{Z_{2}^{\prime}}{Z_{2}}}\right),
{\calL}_{\mathrm{odd/odd}}^{2}=\epsilon^{r_{1}\cdotsr_{8}mn}\epsilon^{s_{1}\cdotss_{8}}{}_{mn}W_{r_{1}r_{2}s_{1}s_{2}}W_{r_{3}r_{4}s_{3}s_{4}}\,D_{s_{5}}H_{r_{7}r_{8}s_{6}}\,D_{r_{5}}H_{s_{7}s_{8}r_{6}}\,,
A_{1}(s)=\frac{\zeta(2s)}{\pi}\frac{R^{2s-1}}{c_{2}^{2s}\,\Gamma(1-s)}\!\int_{0}^{1}d\eta\,(1-\eta^{2})^{-s}\!\int_{0}^{\infty}\!dk\,k^{-2s+1}\left[\frac{\partial}{\partialk}-\frac{\eta}{k}\frac{\partial}{\partial\eta}\right]D_{1}(k,\eta),
S\vertp,0>=e^{-\overline{{\kappa}}/2}\sum_{j=0}^{\infty}\frac{(-i\sqrt{\overline{{\kappa}}}\,)^{j}}{\sqrt{j!}}\vertp-j\frac{\hbar\omega}{c},j>,\qquad\overline{{\kappa}}=\left(\frac{u_{0}\delta}{\hbarc}\right)^{2},
R_{MN}^{\\\\ab}=E_{M}^{\\\Lambda}E_{N}^{\\\Pi}\left\{\partial_{\Lambda}\Omega_{\Pi}^{\ab}+\Omega_{\Lambda}^{\ac}\Omega_{\Pic}^{\\,b}-\left(-\right)^{\Lambda\Pi}\left(\Lambda\leftrightarrow\Pi\right)\right\}
\frac{Z_{3}}{Z_{\lambda}\lambda}\,\langle\,\partialA_{x}^{a}\,\partialA_{y}^{b}\,\rangle\,=\,\widetildeZ_{3}\,\langle\,i(\partialD_{r}c)_{x}^{a}\,\bar{c}_{y}^{b}\,\rangle\,+\,i\widetildeZ_{1}\,(1-\frac\alpha2)\,\langle\,\partialA_{x}^{a}\,(\bar{c}\timesc)_{y}^{b}\,\rangle\;.
{\calL}_{\rho}=-\frac{1}{4}F_{\mu\nu}^{a}F^{a\mu\nu}-\frac{1}{4}(G_{\mu\nu}^{a}+f_{abc}F_{\mu\nu}^{b}\rho^{c})(G^{a\mu\nu}+f_{abc}F^{b\mu\nu}\rho^{c})+\frac{m}{2}\epsilon^{\alpha\beta\gamma}F_{\alpha\beta}^{a}\phi_{\gamma}^{a}
\kappa(\zeta;r)=\frac{(q^{4}r\zeta^{2};q^{4})_{\infty}(q^{2}r\zeta^{-2};q^{4})_{\infty}(q^{6}\zeta^{4};q^{8})_{\infty}(q^{8}\zeta^{-4};q^{8})_{\infty}}{(q^{4}r\zeta^{-2};q^{4})_{\infty}(q^{2}r\zeta^{2};q^{4})_{\infty}(q^{6}\zeta^{-4};q^{8})_{\infty}(q^{8}\zeta^{4};q^{8})_{\infty}}.
\left\langle0\right|T\left[\bar{\psi}^{\uparrow}(t,\theta)\psi^{\downarrow}(0)\right]\left|0\right\rangle=\frac{e^{i\theta}}{2\pi}\left\langle0\right|T\left[\bar{\psi}_{1}^{\uparrow}(t)\psi_{0}^{\downarrow}(0)\right]\left|0\right\rangle,
\langle\hat{t}(\frac{p_{1}}{\delta},\ldots,\frac{q}{\delta}),\stackrel{\vee}{\phi}\rangle=\delta^{3-3l}\intd^{3r-3}\tilde{p}d^{3v-3}\tilde{q}\hat{t}_{1}^{[\mu]}(\frac{\tilde{p}}{\delta})\hat{t}_{2}^{[\nu]}(\frac{\tilde{q}}{\delta})\psi_{[\mu\nu]}^{0}(\tilde{p},\tilde{q}).
l_{c}=\intd\varphi\sqrt{g_{\varphi\varphi}}=2\pi\rho_{c}\left(1-{\frac{\sum_{i}{\hat{q}}_{i}^{2}}{2\rho_{c}^{2}}}-{\frac{{\calF}(1)^{2}}{2A^{2}\rho_{c}^{2}}}\right),
T_{~~A;C}^{(\phi)C}=\left\{\Box_{5}\phi(w,t,y)-V^{\prime}[\phi(w,t,y)]\right\}\phi_{,A}=\left[\frac{\Gamma_{D}^{w}}{2k}\delta(w)\frac1N\delta_{t}\phi+\frac{\Gamma_{D}^{y}}{2k_{y}}\delta(y)\frac1N\delta_{t}\phi+\Gamma_{B}\frac1N\delta_{t}\phi\right]\delta_{A}\phi.
\nabla\Pi^{(\underline{m})}=-i\pi^{1(\underline{\alpha})}\sigma_{(\underline{\alpha})(\underline{\beta})}^{(\underline{m})}\pi^{1(\underline{\beta})}-i\pi_{(\underline{\alpha})}^{2}\tilde{\sigma}^{(\underline{m})(\underline{\alpha})(\underline{\beta})}\pi_{(\underline{\beta})}^{2},\qquad(IIA)
M_{\lambda1}^{-2}(\beta)=-i\frac{d}{d\beta}\int[k_{1}k_{2}k_{3}]\frac{f_{\lambda}}{k_{3\hat{\beta}}^{-}-k_{1\hat{\beta}2\hat{\beta}}^{-}}\tilde{\delta}(k_{3\hat{\beta}}-k_{1\hat{\beta}2\hat{\beta}})a_{\lambda1}^{\dagger}a_{\lambda2}^{\dagger}a_{\lambda3}+h.c.\;,
W\left(\begin{array}{cc}{(d_{1},d_{2})}&{(c_{1},c_{2})}\\{(a_{1},a_{2})}&{(b_{1},b_{2})}\\\end{array}\right)=W\left(\begin{array}{cc}{(\tilde{d}_{1},\tilde{d}_{2})}&{(\tilde{c}_{1},\tilde{c}_{2})}\\{(\tilde{a}_{1},\tilde{a}_{2})}&{(\tilde{b}_{1},\tilde{b}_{2})}\\\end{array}\right).
V(y_{i})=\frac{-k\,\pi^{\frac{D-2}{2}}}{\Gamma(\frac{D-2}{2})V_{T}}\int_{0}^{\infty}dx\,\prod_{i=1}^{D}\theta_{3}\Big(\frac{y_{i}}{L_{i}},\frac{ix}{L_{i}^{2}}\Big)\,,
\widetilde{\calH}_{\scriptscriptstyleBFV}=\intd^{3}x\{\Lambda_{A}\omega^{AB}\vec{\delta}_{B}H_{\scriptscriptstyleBFV}(\xi)+i\bar{\Gamma}_{A}\omega^{AC}\stackrel{\rightarrow}{\delta}_{C}H_{\scriptscriptstyleBFV}(\xi)\stackrel{\leftarrow}{\delta}_{B}\Gamma^{B}\}\nonumber\tag{18}
a_{\mathrm{correlation}}(x,\tau)=\frac{ec^{2}}{2\pi^{2}}\frac{\partial}{\partialR^{2}}\\intdu\\frac{\theta\Bigl(R^{2}-c^{2}u^{2}-c^{2}(\alpha-u)^{2}\Bigr)}{\sqrt{R^{2}-c^{2}u^{2}-c^{2}(\alpha-u)^{2}}}\.
{\frac{1}{2\pi}}\sum_{l=1/2}^{\infty}2le^{-{\frac{l^{2}}{\rho^{2}}}t}={\frac{\rho^{2}}{4\pit}}+{\frac{\rho^{2}}{(4\pit)^{\frac{3}{2}}}}\mathrm{P}\int_{-\infty}^{\infty}dx({\frac{x}{2\rho}}\mathrm{cosec}{\frac{x}{2\rho}}-1)e^{-{\frac{x^{2}}{4t}}},
{\calA}=mH\alpha^{\prime}\left(\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}\right),\;\;\;\;\;\;\;\;{\calB}=\left(\begin{array}{cc}{n^{2}}&{2inmH\alpha^{\prime}}\\{-2inmH\alpha^{\prime}}&{n^{2}}\\\end{array}\right).
\begin{array}{lllllll}{\mathbb{C}}&{}&{x_{0}^{2}+x_{1}^{2}}&{=1}&{defines}&{a\;unit}&{S^{1}}\\{\mathbb{H}}&{}&{x_{0}^{1}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}&{=1}&{defines}&{a\;unit}&{S^{3}}\\{\mathbb{O}}&{}&{x_{0}^{1}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}}&{=1}&{defines}&{a\;unit}&{S^{7}}\\\end{array}