harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{l}{\intd^{2}z(\phi-\zetaf_{0})=\mathrm{const}~\zeta\bar{\zeta};\nonumber}\\{\intd^{2}z(\bar{\phi}-\bar{\zeta}f_{0})=\mathrm{const}~\zeta\bar{\zeta};\nonumber}\\{\intd^{2}z(f_{A}-2\zeta\bar{\zeta}f_{0})=0,}\\\end{array}
\frac{\partialK(\Lambda_{1},\Lambda_{2};t)}{\partialt}=\frac{i}{2}\left(\sum_{i=1}^{m}\frac{\partial^{2}}{\partial\lambda_{1i}^{2}}-\sum_{\alpha=1}^{n}\frac{\partial^{2}}{\partial{\bar{\lambda}}_{1\alpha}^{2}}\right)K(\Lambda_{1},\Lambda_{2};t),
P_{j-m}^{(0,\,2m)}(x)=\frac{(-1)^{j-m}}{2^{m}(j-m)!}(1+x)^{-2m}\frac{d^{j-m}}{dx^{j-m}}\left[(1-x)^{j-m}(1+m)^{j+m}\right]
[\delta_{\zeta_{1}},\delta_{\zeta_{2}}]{\tilde{e}}{^a}_{\mu}(e,\psi)=\Xi^{\rho}\partial_{\rho}{\tilde{e}}{^a}_{\mu}(e,\psi)+{\tilde{e}}{^a}_{\rho}(e,\psi)\partial_{\mu}\Xi^{\rho},
W_{(d)}=-\int_{M^{2}}\left({\frac{1}{2}}(\nabla\Phi)^{2}+{\frac{1}{8}}({\frac{d-2}{d-3}})\Phi^{2}R+{\frac{1}{8}}(d-2)^{2}C^{\frac{2}{d-2}}\Phi^{2({\frac{d-4}{d-2}})}\right)~~.
D_{-}^{\alpha}\equiv\partial_{z}+\frac{i}{2}(A_{z}^{1}+\frac{1}{\sqrt{3}}A_{z}^{2}+\cdots+\frac{1}{\sqrt{(\alpha-1)(\alpha-2)/2}}A_{z}^{\alpha-2}+\frac{\alpha}{\sqrt{\alpha(\alpha-1)/2}}A_{z}^{\alpha-1})
\Big\{\frac{1}{2}\Big(x\cdot\Pix+p\cdot\Pip\Big)-\frac{1}{2}\Big(\frac{\hbar}{2}\Big)^{2}\Pi_{ij}\Big(\frac{\partial^{2}}{\partialp_{i}\partialp_{j}}+\frac{\partial^{2}}{\partialx_{i}\partialx_{j}}\Big)-E\Big\}f(x,p)=0,
e^{-1}{\calL}=\frac{1}{4}R-\frac{1}{4}e^{2\Phi}F_{\mu\nu}^{i}F^{\mu\nu\;i}-\frac{1}{4}P_{\mu\;ij}P^{\mu\;ij}-\frac{1}{2}(\partial_{\mu}\Phi)^{2}-\frac{g^{2}}{16}e^{-2\Phi}(T_{ij}T^{ij}-\frac{1}{2}T^{2}),
\{Q^{+},Q\}=2\intd^{3}p\,\vec{p}^{2}\bigl[b_{1}^{+}(\vec{p})b_{1}(\vec{p})+b_{2}^{+}(\vec{p})b_{2}(\vec{p})+c_{1}^{+}(\vec{p})c_{1}(\vec{p})+c_{2}^{+}(\vec{p})c_{2}(\vec{p})\bigr]\quad,
\tilde{\Psi}^{-}=\hat{S}_{0}\bar{\Phi}^{-}=\hat{S}_{0}\sigma_{3}\bar{\Phi}^{+}=\hat{S}_{0}\sigma^{3}\hat{S}_{0}^{-1}\hat{S}_{0}\bar{\Phi}^{+}={\bf\sigma.\hat{e}}_{r}\tilde{\Psi}^{+}
f(a)=-{\frac{1}{3}}\int_{0}^{1}{\frac{db}{b^{\frac{1}{3}}(1-b)^{\frac{1}{3}}}}\int_{\frac{1}{2}}^{a}{\frac{dx}{x^{\frac{2}{3}}(1-x)^{\frac{2}{3}}}}\,,
\{\psi_{R}(x^{-}),\psi_{R}^{\dagger}(y^{-})\}_{D}={\frac{-i}{\sqrt2}}\delta(x^{--}y^{-}),\\\{\stackrel{o\}{A_{-}},\stackrel{o\}{E^{-}}\}_{D}={\frac{1}{2L}}.
\begin{array}{cccc}{\pi_{j}(\hat{H})_{mn}=(j+1-n)\:\delta_{m,n},}&{m,\:n=1,\:2,\:\ldots,\:2j+1.}\\{\pi_{j}(\hat{E})_{mn}=\omega_{m}\:\delta_{m,n-1},}&{\pi_{j}(\hat{F})_{mn}=\omega_{n}\:\delta_{m-1,n}.}\\\end{array}
S={\int}d^{4}x\left[\frac{1}{4}F_{\mu\nu}^{\prime}F_{\mu\nu}^{\prime}+\frac{1}{2}\partial_{\mu}\rho\partial_{\mu}\rho+\frac{1}{2}e^{2}\rho^{2}B_{\mu}B_{\mu}+\frac{1}{2}m^{2}\rho^{2}+\frac{\lambda}{4!}\rho^{4}\right],
a_{A}^{\dagger}(k)\,a_{A}^{\dagger}(k^{\prime})+\sum_{l,l^{\prime}\in{\calI}}{\hat{\calR}}\left(k,k^{\prime};l,l^{\prime}\right)a_{A}^{\dagger}(l)\,a_{A}^{\dagger}(l^{\prime})=0\;,
^{+}{\calS}_{3}=\frac{i}{2}{^+\tau}\int_{M}eF^{\mu\nui}F^{\alpha\betaj}g_{ij}\,\Sigma_{\mu\nu}^{ab}\,\Sigma_{\alpha\beta}^{cd}\,\epsilon_{abcd}-\frac{1}{2}{^+\tau}\int_{M}eF^{\mu\nui}F^{\alpha\betaj}g_{ij}\,\Sigma_{\mu\nu}^{ab}\,\Sigma_{\alpha\beta}^{cd}\,\delta_{ab,cd}.
S_{\mathrm{cl}}=\mathrm{tr}\left(\bar{\sigma}^{\dot{\alpha}}\bar{D}_{\dot{\alpha}}D_{\alpha}\sigma^{\alpha}\right)\longrightarrow\mathrm{tr}\left(\bar{\sigma}^{\dot{\alpha}}\bar{\nabla}_{\dot{\alpha}}\nabla_{\alpha}\sigma^{\alpha}\right)=\mathrm{tr}\left(\bar{\sigma}^{\dot{\alpha}}e^{\bar{W}}\bar{D}_{\dot{\alpha}}e^{-V}D_{\alpha}e^{W}\sigma^{\alpha}\right)\,.
E_{0}=\frac{\pi}{p^{+}}\left(\left(\frac{p^{I_{N}}}{\pi}\right)^{2}+\left(\frac{m}{3}\right)^{2}(x^{i_{N}})^{2}+\left(\frac{m}{6}\right)^{2}(x^{i_{N}^{\prime}})^{2}+\frac{i}{\pi}\frac{m}{3}\psi_{0}\gamma^{4}\Omega\psi_{0}-\frac{i}{\pi}\frac{m}{6}\psi_{0}^{\prime}\gamma^{4}\Omega\psi_{0}^{\prime}\right)~.
\frac{\deltaS}{\delta\theta^{\alpha\beta}}=\frac{\beta}{2}\intd^{2}x{\mathrm{Tr}}\left[-\fraci4\{F_{\alpha\beta},\Phi\}\starF_{01}-\fraci4\Phi\star\{F_{0\alpha},F_{\beta1}\}+\fraci4\Phi\star\{F_{0\beta},F_{\alpha1}\}\right]\,.
{\nabla}^{2}\frac{1}{{\phi}(r)}\left({\nabla}^{2}\frac{1}{{\phi}(r)}\left[{\calK}_{0}^{b}({\bfr})+j_{0}^{b}({\bfr})\right]\right)+{\calL}_{(1)}^{bs}{\nabla}^{2}\frac{1}{{\phi}(r)}j_{0}^{s}({\bfr})=-{\calL}_{(2)}^{bs}{\calK}_{0}^{s}({\bfr})\;\;\;\mathrm{and}\;\;\;
\nabla_{\hat{\mu}}\hat{F}^{\hat{\mu}\hat{\mu}_{1}\hat{\mu}_{2}\hat{\mu}_{3}\hat{\mu}_{4}}={\textstyle\frac{3}{5!4}}\epsilon^{ij}\epsilon^{\hat{\mu}_{1}\ldots\hat{\mu}_{10}}\hat{\calH}_{\hat{\mu}_{5}\hat{\mu}_{6}\hat{\mu}_{7}}^{(i)}\hat{\calH}_{\hat{\mu}_{8}\hat{\mu}_{9}\hat{\mu}_{10}}^{(j)}\,.
\langle\sigma^{z}\rangle=-\frac{2(q^{4};q^{4})_{\infty}^{4}(p^{2};p^{2})_{\infty}^{4}}{(q^{2};q^{2})_{\infty}^{8}(-p;p)_{\infty}^{8}+(-q^{2};q^{2})_{\infty}^{8}(p;p)_{\infty}^{8}}.
P^{+}=\sum_{N>0}\left({\frac{N\pi}{L}}\right)a{_N^{\dagger}}a_{N}+\sum_{n>0}\left({\frac{n\pi}{L}}\right)\left[b{_n^{\dagger}}b_{n}+d{_n^{\dagger}}d_{n}\right]-{\frac{\pi}{2L}}(D_{0}^{3})^{2}
\begin{array}{lcl}{0}&{=}&{{{\mathcalL}_{n}}(u)\starK[a]-K[a]\star{{\mathcalL}_{n}}(w)}\\{}&{=}&{(u_{3}-w_{3}+2n\thetaa_{2}^{\prime})p^{n-3}}\\{}&{+}&{(u_{4}-w_{4}+(n-3)\theta(w_{3}^{\prime}-u_{3}^{\prime})+2n\thetaa_{3}^{\prime}+4n\theta^{2}a_{2}^{\prime\prime})p^{n-4}}\\{}&{}&{\vdots}\\\end{array}
C(\omega)=\left(\begin{matrix}{\vec{p}^{~2}-\omega^{2}}&{-\mathrm{i}\sqrt{2}\kappaq\omega}&{~0~}\\{{}~0~}&{\vec{p}^{~2}-\omega^{2}+2\kappa^{2}q^{2}\frac{p_{1}^{2}}{\vec{p}^{~2}}}&{~0~}\\{{}~0~}&{~0~}&{\vec{p}^{~2}-\omega^{2}}\\\end{matrix}\right)
Q_{\mathrm{CS}}^{-}(T)=2^{-1/4}\mathrm{i}\sum_{|n^{\perp}|\leT}\int_{0}^{\infty}dk\frac{1}{\sqrt{k}}\left[a_{ij}^{\dagger}(k,n^{\perp})b_{ij}(k,n^{\perp})+b_{ij}^{\dagger}(k,n^{\perp})a_{ij}(k,n^{\perp})\right]\,,
\Omega=\int_{0}^{2\pi}d\sigma\;\biggl(L(\sigma)\eta(\sigma)+\bar{L}(\sigma^{\prime})\bar{\eta}(\sigma^{\prime})-{\calP}(\sigma)\eta(\sigma)\eta^{\prime}(\sigma)+\bar{\calP}(\sigma^{\prime})\bar{\eta}(\sigma^{\prime})\bar{\eta}^{\prime}(\sigma^{\prime})\biggr).
\langle{\bfp}\,\vert{\bfx},t\rangle=\langle{\bfp}\,\vertU(I,({\bfx},t)\vert{\bf0},0\rangle=e^{i{\bfp}\cdot{\bfx}-i\omega_{m}({\bfp}\,)t}\langle{\bfp}\,\vert{\bf0},0\rangle
P_{N}=\sum_{N=n_{1}+n_{2}+n_{3}}\frac{1}{n_{1}!\,n_{2}!}\,(a_{1}^{\dagger})^{n_{1}}(a_{2}^{\dagger})^{n_{2}}(b^{\dagger})^{n_{3}}|0><0|(b)^{n_{3}}(a_{2})^{n_{2}}(a_{1})^{n_{1}}\,.
\sum_{I_{1}I_{2}\cdotsI_{n}=(I)}X^{I_{1},I_{2},\ldots,I_{n}}\equiv\sum_{I_{1},I_{2},\ldots,I_{n}}\delta_{(I)}^{I_{1}I_{2}\cdotsI_{n}}X^{I_{1},I_{2},\ldots,I_{n}}.
S={\frac{1}{2\kappa_{5}^{2}}}\intd^{5}x\sqrt{-g}\left\{R-{\frac{4}{3}}(\nabla\lambda)^{2}-4(\nabla\nu)^{2}-{\frac{1}{4}}e^{{\frac{8}{3}}\lambda}F^{(K)2}-{\frac{1}{4}}e^{-{\frac{4}{3}}\lambda+4\nu}F^{2}-{\frac{1}{4}}e^{-{\frac{4}{3}}\lambda-4\nu}H^{2}\right\},
\begin{array}{l}{w_{2i+1}=x_{i}^{-1}x_{i+1},~~i=0,1,2,\cdots,N-1,}\\{w_{2j}=y_{j},~~j=1,2,\cdots,N-1,}\\{w_{2N}=\prod_{l=1}^{N-1}y_{l}^{-1},}\\\end{array}
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).
\cdot\left[\frac{\Gamma(n_{1}+|m|+\delta_{1}+1)\Gamma(n_{2}+|m|+\delta_{2}+1)\Gamma(n+l+\delta_{1}+\delta_{2}+1)}{(n_{1})!(n_{2})!(n-l-1)!\Gamma(l+\delta_{2}+1)\Gamma(l+|m|+\delta_{1}+\delta_{2}+1)}\right]^{1/2}
\Gamma_{0}^{M}(k)=i\int{\frac{d^{4}u}{(2\pi)^{4}}}\hat{I}_{ab}\big(k-u,{\frac{k+u}{2}},{\frac{k+u}{2}}\big)\sigma^{a}\hat{H}(u)\Gamma_{0}^{M}(u)\hat{H}(u)\sigma^{b}\.
{\itL}_{gh}-\left(\bar{c}^{\mu}~\bar{\chi}^{\mu}\right)\left(\begin{array}{cc}{\left(g_{\mu\nu}\nabla^{2}+R_{\mu\nu}\right)}&{0}\\{0}&{(f_{1}+f_{2}+4f_{3})g_{\mu\nu}(-g)^{\alpha}}\\\end{array}\right)\left(\begin{array}{c}{c^{\nu}}\\{\tilde{\chi}^{\nu}}\\\end{array}\right)
TrK_{B_{3}^{\alpha}}=\left({\frac{V_{B_{3}^{\alpha}}}{l^{3}}}+{\frac{A_{+}}{l}}(2\pi\alpha)c_{2}(\alpha)~\bar{s}~\right){\frac{e^{-\mu\bar{s}}}{(4\pi\bar{s})^{3/2}}}~+~ES~~,
\begin{array}{ll}{l=(l_{12}+l_{13}+l_{23})/(2N),}&{l^{\prime}=(l_{14}+l_{15}+l_{45})/(2N)}\\{l^{\prime\prime}=(l_{24}+l_{26}+l_{46})/(2N),}&{l^{\prime\prime\prime}=(l_{35}+l_{36}+l_{56})/(2N)}\\\end{array}
\begin{aligned}{ds^{2}}&{{}=2dx^{+}dx^{-}+H(x^{i},x^{-})(dx^{-})^{2}+\sum_{i=1}^{9}(dx^{i})^{2}}\\{F_{4}}&{{}=dx^{-}\wedge\xi(x^{i},x^{-})}\\\end{aligned}
S=S_{\mathrm{SUGRA}}(h,A,\psi)+\intd^{11}x(h_{IJ}T^{IJ}+A_{IJK}J^{IJK}+A_{IJKLMN}^{D}M^{IJKLMN}+\bar{\psi}_{\alphaI}S_{\alpha}^{I}+\bar{S}_{\alpha}^{I}\psi_{\alphaI})
e^{-Y_{6}}=e^{{\frac{t}{24}}}e^{-({\frac{Y_{1}}{24}}+{\frac{Y_{2}}{24}}+{\frac{Y_{3}}{12}}+{\frac{Y_{4}}{3}}+{\frac{Y_{5}}{2}})}=e^{{\frac{t}{24}}}X_{1}X_{2}X_{3}X_{4}X_{5}.
\frac{\delta\mathcal{E}_{\mathrm{vac}}}{M_{\beta}^{2}}=\frac{\epsilon_{\mathrm{vac}}\left(l_{0}\right)}{l_{0}}\left(\frac{M}{M_{\beta}}\right)^{2}-\frac{\mathcal{E}_{\mathrm{vac}}^{(0)}}{M_{\beta}^{2}}
M^{-}\approx2r\left(\begin{matrix}{c_{R}^{2}}&{c_{R}\,d_{R}}&{0}&{0}\\{c_{R}\,d_{R}}&{d_{R}^{2}}&{0}&{0}\\{0}&{0}&{c_{L}^{2}}&{c_{L}\,d_{L}}\\{0}&{0}&{c_{L}\,d_{L}}&{d_{L}^{2}}\\\end{matrix}\right)
\begin{array}{rl}{\vec{b}_{N}^{\,\prime}(v,\theta_{i}^{\prime},\phi_{i}^{\prime})}&{=\vec{b}_{p,N}^{\,\prime}(v,\theta_{i}^{\prime},\phi_{i}^{\prime})+\vec{\beta}_{N}(\theta_{i}^{\prime},\phi_{i}^{\prime})}\\{\}&{=\vec{a}_{N}^{\,\prime}(v,\theta_{i}^{\prime},\phi_{i}^{\prime})|_{\hat{x}\hat{y}\hat{z}\rightarrow\hat{x}^{\prime}\hat{y}^{\prime}\hat{z}^{\prime}}.}\\\end{array}
\zeta_{i}=\frac{\partialL}{\partial\dot{\eta}^{i}}=i\,h_{ij}\,\eta^{j}\,,\qquad\{\zeta_{i}\,,\eta^{j}\}=i\,\delta_{i}^{j}\,,\qquad\{\eta^{i}\,,\eta^{j}\}=h^{ij}\,.
{M}{=}\left(\begin{array}{ll}{p}&{-p^{\prime}}\\{q}&{-q^{\prime}}\\{-n{^\prime}}&{n}\\{-m^{\prime}}&{m}\\\end{array}\right){,}\quad{m}{p}{+}{m}{q}{-}{m^{\prime}q^{\prime}}{-}{n^{\prime}p{^\prime}}={N}
\begin{array}{c}{c_{j}^{(i)}=\frac{1}{(\frac{d\varphi_{i}}{dN_{i}})_{0}}\delta_{ij}}\\{c_{ij}^{(i)}=-\frac{1}{2}(c_{ij}+c_{ji}^{*})c_{i}^{(i)}c_{j}^{(j)}}\\{c_{jk}^{(i)}=0,\quadj\neqi,\quadk\neqi}\\\end{array}
\begin{array}{lcr}{u=\prod_{j=0}^{n}X_{j}^{j}}&{;\qquadN^{+{\frac{n(n+1)}{2}}}}&{=\prod_{j=0}^{n}(\eta_{j}^{+})^{n-j}}\\{v=\prod_{j=0}^{n}X_{j}^{n-j}}&{;\qquadM^{+{\frac{n(n+1)}{2}}}}&{=\prod_{j=0}^{n}(\eta_{j}^{+})^{j}}\\{z=\prod_{j=0}^{n}X_{j}}&{;\qquadS^{+{(n+1)}}}&{=\prod_{j=0}^{n}\eta_{j}^{+}.}\\\end{array}
{\calL}={\calL_{0}}+\frac{1}{2}(D_{\mu}\pi)^{2}+\frac{g_{4}^{(1)}}{2}(D_{\mu}\pi\cdotD^{\mu}\pi)^{2}+\frac{g_{4}^{(2)}}{2}(D_{\mu}\pi\cdotD_{\nu}\pi)(D_{\mu}\pi\cdotD_{\nu}\pi)+\cdots
e_{\alpha}^{\;\;\mu}e_{\beta}^{\;\;\nu}\left.\frac{\partial\Gamma_{R}}{\partialb^{\mu\nu}}\right|_{g_{\mu\nu}}={\bar{e}}_{\alpha}^{\;\;i}{\bar{e}}_{\beta}^{\;\;j}\frac{\partial\Gamma_{R}}{\partialb^{ij}}.
{\frac{1}{6}}\big(d\bar{\theta}\Gamma_{MNPQR}d\thetad\bar{\theta}\Gamma^{R}+d\bar{\theta}\Gamma^{R}d\thetad\bar{\theta}\Gamma_{MNPQR}\big)=d\bar{\theta}\Gamma_{[MN}d\thetad\bar{\theta}\Gamma_{PQ]}.
\ddot{Z}^{1}=\partial_{\sigma}\partial_{\sigma}Z^{1}+\partial_{\sigma}\partial_{\rho}Z^{2},\ddot{Z}^{2}=\partial_{\rho}\partial_{\rho}Z^{2}+\partial_{\sigma}\partial_{\rho}Z^{1},\ddot{Z}^{I}=\partial_{\sigma}\partial_{\sigma}Z^{I}+\partial_{\rho}\partial_{\rho}Z^{I},
({\partial_{r}}^{2}+{\partial_{t}}^{2})\omega={\frac{1}{2}}({\partial_{r}}^{2}+{\partial_{t}}^{2})(e^{2\zeta}-2\zeta)={\frac{1}{2}}({\partial_{r}}^{2}+{\partial_{t}}^{2})e^{2\zeta}+{\frac{1}{r^{2}}}(1-e^{2\zeta})
L_{1/2}=\frac{1}{2}h_{ij}(y)\dot{y}^{i}\dot{y}^{j}+\frac{1}{2}L_{\alpha\beta}(y)(\dot{\psi}^{\alpha}+w_{i}^{\alpha}(y)\dot{y}^{i})(\dot{\psi}^{\beta}+w_{j}^{\beta}(y)\dot{y}^{j}).
B_{\mathbf{p,j}}[\mathbf{f}(\mathbf{u}(\mathbf{y}))]=\sum_{|\mathbf{j}|\le|\mathbf{m}|\le|\mathbf{p}|}B_{\mathbf{p,m}}[\mathbf{u}]B_{\mathbf{m,j}}[\mathbf{f}],
P^{J_{1}J_{2}\ldotsJ_{M}}|_{C_{J_{1}J_{2}\ldotsJ_{I}}}=P^{J_{1}J_{2}\ldotsJ_{I}}|_{C_{J_{1}J_{2}\ldotsJ_{I}}}P^{J_{I+1}\ldotsJ_{M}}|_{C_{J_{1}J_{2}\ldotsJ_{I}}},\;I<M,
\left\{{\frac{\partial^{2}}{\partialr^{2}}}+{\frac{5}{r}}{\frac{\partial}{\partialr}}+\tilde{\omega}^{2}\left(1+{\frac{\tilde{R}_{\frac{\pi}{2}}^{4}}{r^{4}}}\right)\right\}\hat{\phi}-{\frac{4R_{\frac{\pi}{2}}^{8}(h_{~2}^{2}+h_{~3}^{3})}{r^{10}f^{2}}}=0,
K=i\int_{0}^{\infty}d\lambda^{\prime}\;se^{i\lambda^{\prime}(\pi^{\prime\prime}-\pi^{\prime})}e^{-i[s\lambda^{\prime}(\gamma^{\mu}P_{\mu}^{\prime}-m)+{\frac{i}{s}}({\bar{\eta}}^{\prime\prime}-{\bar{\eta}}^{\prime})(\eta^{\prime\prime}-\eta^{\prime})]}\delta^{4}(P^{\prime\prime}-P^{\prime})\,.
\begin{array}{lll}{\deltae^{a}=-i\left(\bar{\epsilon}\Gamma^{a}\psi-\bar{\psi}\Gamma^{a}\epsilon\right)}&{;}&{\delta\psi=D\epsilon}\\{\deltac=\left(\bar{\epsilon}\psi-\bar{\psi}\epsilon\right)}&{;}&{\delta\bar{\psi}=D\bar{\epsilon}}\\\end{array}
A_{a}^{\pm}=\frac{1}{2R(R\mpr_{5})}\left[\begin{array}{c}{r_{4}\sigma_{1}+r_{3}\sigma_{2}-r_{2}\sigma_{3}}\\{-r_{3}\sigma_{1}+r_{4}\sigma_{2}+r_{1}\sigma_{3}}\\{r_{2}\sigma_{1}-r_{1}\sigma_{2}+r_{4}\sigma_{3}}\\{-r_{1}\sigma_{1}-r_{2}\sigma_{2}-r_{3}\sigma_{3}}\\{0}\\\end{array}\right]
\hat{T}={\frac{1}{y_{14}^{2}y_{23}^{2}}}\left(\hat{y}_{13}^{2}\hat{y}_{24}^{2}-16i\theta_{13}\cdot\bar{\theta}_{24}\cdot(\hat{y}_{13}y_{12}\hat{y}_{24})-16y_{12}^{2}(\theta_{13}\cdot\bar{\theta}_{24})^{2}\right)
\bigg\lbrack\Gamma_{\mu}(z;x,y)\bigg\rbrack_{a}^{b}=\bigg\lbrackF_{\mu}(y-z;x-z)\bigg\rbrack_{aa^{\prime}}^{bb^{\prime}}\bigg\lbrackS_{F}^{-1}(x,y)\bigg\rbrack_{b^{\prime}}^{a^{\prime}}\;\;,
E_{\mathrm{vdW}}=-BN^{2}{\frac{L^{D-2}}{a^{\gamma-D-2}}}{\frac{2^{D-\gamma}\pi^{D/2+1/2}\Gamma(\gamma/2-D/2+1)\Gamma(D/2-\gamma/2+1/2)}{\Gamma(\gamma/2)\Gamma(D/2-\gamma/2+2)(D-\gamma)}}.
q_{c}(a,r)=\frac{1}{8\pi}\sum_{m=-\infty}^{+\infty}\int_{-\infty}^{+\infty}{dk\sum_{\lambda,\gamma}\beta_{\lambdam}^{2}f_{m}^{(q)}(\gammar)},\quadq_{c}=\varepsilon,p_{1},p_{2}
\begin{array}{rl}{\Im(2\bar{C})}&{=p^{0}\,,}\\{\Im(2\bar{C}\,\tau^{ij})}&{=P^{ij}\,,}\\{\Im(2\bar{C}\,\mathrm{Cof}\,\tau_{ij})}&{=-Q_{ij}\,,}\\{\Im(2\bar{C}\,\mathrm{det}\,\tau)}&{=q^{0}\,.}\\\end{array}
\begin{array}{c}{u(\sigma^{+},\sigma^{-})=u_{0}+\frac{p^{+}}2\left[(\sigma^{+}+\left|\sigma^{+}\right|_{per})+(\sigma^{-}-\left|\sigma^{-}\right|_{per})\right]}\\{v(\sigma^{+},\sigma^{-})=v_{0}+\frac{p^{-}}2\left[(\sigma^{+}-\left|\sigma^{+}\right|_{per})+(\sigma^{-}+\left|\sigma^{-}\right|_{per})\right]}\\\end{array}
S_{3}=-\frac{m^{2}}{2}\intd^{4}\thetad^{d}xe^{\sigma+\bar{\sigma}}\int\frac{d^{d}k}{{(2\pi)}^{d}}\frac{-A^{2}k^{2}+m^{2}}{k^{2}{(-A^{2}k^{2}+m^{2})}^{2}-81\Lambda\bar{\Lambda}}
\Phi_{c}(p,J)=\frac{1}{i}(2\pi)^{\,4}\delta_{J(-p)}\,W_{c}^{\,\alpha,\,\alpha_{0}}(J)=\tilde{C}_{\alpha}^{\,\alpha_{0}}(p)\,\Bigl\{\,(2\pi)^{\,4}\,\delta_{\Phi(-p)}\,i\,L^{\,\alpha,\,\alpha_{0}}(\Phi)\,\vert_{\,\Phi=i\tilde{C}_{\alpha}^{\,\alpha_{0}}\!J}\,+i\,J(p)\,\Bigr\}\.
\left[{\frac{\partial^{2}}{\partialr^{2}}}+{\frac{5}{r}}{\frac{\partial}{\partialr}}-{\frac{l(l+4)}{r^{2}}}+\omega^{2}\left(1+{\frac{R^{4}}{r^{4}}}\right)\right]\phi^{(l)}(r)=0.
J_{top}^{\mu}=\frac1{8\pi}\varepsilon^{\mu\nu\alpha}\pmb{\phi}\cdot\left[D_{\nu}\pmb{\phi}\timesD_{\alpha}\pmb{\phi}-\frace2F_{\nu\alpha}\left(1-{\bf{n}}\cdot\pmb{\phi}\right)\pmb{\phi}\right]{\,}
\mathrm{Tr}~(\lambda^{a}\,\lambda^{b})\,\quad\mathrm{Tr}~(\bar{\lambda}^{\bar{a}}\,\bar{\lambda}^{\bar{b}})\,\quad\mathrm{Tr}~(X^{A}\,X^{B})~-~{\frac{1}{6}}~\delta^{AB}~\mathrm{Tr}~(X^{C}\,X^{C})\.
\begin{array}{l}{\displaystyle\tilde{\calQ}_{\alpha}^{1}=ip_{\alpha\beta}(in^{\beta}{}_{\gamma}\theta^{\gamma}+b\chi^{\beta})-m(\theta_{\alpha}-ibn_{\alpha\beta}\chi^{\beta})}\\{\displaystyle\tilde{\calQ}_{\alpha}^{2}=ip_{\alpha\beta}(in^{\beta}{}_{\gamma}\chi^{\gamma}-b\theta^{\beta})-m(\chi_{\alpha}+ibn_{\alpha\beta}\theta^{\beta})\,.}\\\end{array}
G_{o}(K;\vec{x}-\vec{x^{\prime}};\phi-\phi^{\prime})=-\frac{1}{8\pi^{2}}\sum_{n=-\infty}^{+\infty}\frac{e^{i\sqrt{K^{2}-\frac{n^{2}}{R^{2}}}|\vec{x}-\vec{x}^{\prime}|}}{|\vec{x}-\vec{x}^{\prime}|}e^{in(\phi-\phi^{\prime})},
\begin{array}{lll}{\Lambda_{24}^{2}}&{=}&{\{<A,B,0>,\;<0,A,B>,\;<B,0,A>:\;\;A,B\in\frac{1}{\sqrt{2}}{\calE}_{8}^{odd},}\\{}&{}&{AB^{\dagger}=\pm\frac{1}{2}e_{a},\;\;a\in\{0,...,7\}\},}\\\end{array}
\left\{\psi_{\alpha\vec{x}},\psi_{\beta\vec{y}}\right\}=\left\{\psi_{\alpha\vec{x}}^{\dagger},\psi_{\beta\vec{y}}^{\dagger}\right\}=0\quad,\quad\left\{\psi_{\alpha\vec{x}}^{\dagger},\psi_{\beta\vec{y}}\right\}=\frac{1}{a^{3}}\delta_{\alpha\beta}\delta_{\vec{x}\vec{y}}\quad
M_{\mathrm{ultra-ext}}=\left|\sqrt{(\breve{P}_{1}^{(1)})^{2}+\beta^{2}}-\sqrt{(\breve{P}_{1}^{(2)})^{2}+\beta^{2}}\right|+\left|\sqrt{(\breve{Q}_{2}^{(1)})^{2}+\beta^{2}}-\sqrt{(\breve{Q}_{2}^{(2)})^{2}+\beta^{2}}\right|
\{X_{i}^{1},X_{i}^{2}\}=-r_{-}X_{i}^{1}X_{i}^{2}-X_{i}^{1}X_{i}^{2}r_{+}+X_{i}^{1}r_{-}X_{i}^{2}+X_{i}^{2}r_{+}X_{i}^{1},
\psi_{\mathrm{canonical}}(\vec{p})={\frac{1}{\sqrt2}}\left(\begin{array}{cc}{I}&{I}\\{I}&{-I}\\\end{array}\right)\psi(\vec{p})={\frac{1}{\sqrt2}}\left(\begin{array}{c}{\phi_{R}(\vec{0})+\phi_{L}(\vec{0})}\\{\phi_{R}(\vec{0})-\phi_{L}(\vec{0})}\\\end{array}\right).
H[\varphi,\pi]=a^{d}\sum_{\alpha}\sum_{\bfi}\left[\frac{1}{2}(\pi_{\bfi}^{\alpha})^{2}+\frac{J}{2a^{2}}\sum_{\mu=1}^{d}(\varphi_{{\bfi}+{\bfe}_{\mu}}^{\alpha}-\varphi_{\bfi}^{\alpha})^{2}-\frac{1}{2}m^{2}(\varphi_{\bfi}^{\alpha})^{2}\right]+\frac{\lambda}{4}\sum_{\bfi}\left[\sum_{\alpha}(\varphi_{\bfi}^{\alpha})^{2}\right]^{2}.
\left(L_{n}^{1/2}(x)\right)_{ik}K_{n}(y)\left(K(y-x+\frac{i}{2})\right)_{kj}=\left(K(y-x+\frac{i}{2})\right)_{ik}K_{n}(y)\left(L_{n}^{1/2}(x)\right)_{kj},
V(\rho)=E\left[1-\frac{1+\frac{\rho_{*}^{4}}{2(\rho^{2}-l^{2})^{2}}}{\left(1+\frac{\rho_{*}^{4}}{(\rho^{2}-l^{2})^{2}}\right)^{2}}\right]+\frac{L^{2}}{2m_{p}\rho^{2}}\frac{1}{\left(1+\frac{\rho_{*}^{4}}{(\rho^{2}-l^{2})^{2}}\right)^{2}},
\psi=\left(\begin{matrix}{{\psi_{1}}(\rho,z){e}^{i(m_{z}-{\frac{1}{2}})\phi}}\\{{\psi_{2}}(\rho,z){e}^{i(m_{z}+{\frac{1}{2}})\phi}}\\{-i{\psi_{3}}(\rho,z){e}^{i(m_{z}-{\frac{1}{2}})\phi}}\\{-i{\psi_{4}}(\rho,z){e}^{i(m_{z}+{\frac{1}{2}})\phi}}\\\end{matrix}\right)
0=\omega_{c\rightarrowc^{,}}^{-1}f_{\rho}^{c}(c^{,}(x,x_{0},t))\omega_{c\rightarrowc^{,}}\frac{\partialc^{\rho^{,}}}{\partialt}-\frac{i}{g}\omega_{c\rightarrowc^{,}}^{-1}\frac{\partialc^{,}}{\partialt}\frac{\partial}{\partialc^{,}}\omega_{c}
\left(\begin{array}{cc}{n_{1}^{\prime}}&{n_{2}^{\prime}}\\{l_{1}^{\prime}}&{l_{2}^{\prime}}\\\end{array}\right)=\left(\begin{array}{cc}{a}&{c/2}\\{2b}&{d}\\\end{array}\right)\left(\begin{array}{cc}{n_{1}}&{n_{2}}\\{l_{1}}&{l_{2}}\\\end{array}\right)
\delta\chi=\frac12F_{ab}\Gamma^{ab}\eta=\frac12({{P_{1}}^{ab}}_{cd}+{{P_{2}}^{ab}}_{cd})F_{ab}\Gamma^{cd}\eta=\frac23F_{ab}G^{ab}\eta+\frac16(F_{ab}-\frac12{C_{ab}}^{cd}F_{cd})\Gamma^{ab}\eta
{\frac{\hbar^{2}}{16m^{2}}}{\frac{|p_{1,n}|^{2}}{E_{1,n}-E_{+}}}={\frac{\hbar^{2}}{8m}}{\frac{E_{1,n}-U_{1}(0)}{E_{1,n}-E_{+}}}<{\frac{\hbar^{2}}{8m}},
{\calR}\equiv\left(\begin{array}{cccc}{R^{(1)}}&{0}&{0}&{0}\\{0}&{R^{(3)}}&{0}&{0}\\{0}&{0}&{R^{(2)\;-1}}&{0}\\{0}&{0}&{0}&{{\calP}R^{(4)}{\calP}}\\\end{array}\right)\quad.\qquad
\frac{Z_{n}^{(N_{f}+1)}(M,M_{N_{f}+1})}{C_{n}^{(N_{f}+1)}(M,M_{N_{f}+1})}\=\\frac{Z_{n}^{(N_{f})}(M)}{C_{n}^{(N_{f})}(M)}\,\frac{(n+N_{f})!}{(N\Sigma^{2})^{n+N_{f}}}\,\frac{1}{\prod_{f=1}^{N_{f}}(M_{f}^{2}-M_{N_{f}+1}^{2})}
D_{+2}\theta_{A}^{+}=0,\qquadD_{-2}\theta_{A}^{+}=e_{-2}\theta_{A}^{+},\qquad\tilde{D}_{+2}\theta_{\dot{A}}^{-}=e_{+2}\theta_{\dot{A}}^{-},\qquad\tilde{D}_{-2}\theta_{\dot{A}}^{-}=0.
S(z,\bar{z},\varphi)=\sum_{n=0}^{\infty}\frac{1}{n!}A_{n}(z,\bar{z})y^{n}\,,\quad\bar{S}(z,\bar{z},\varphi)=\sum_{n=0}^{\infty}\frac{1}{n!}\bar{A}_{n}(z,\bar{z})y^{n}\,.
=2\pi\hbar\sum_{j}<\psi_{j}|\bigg\{\frac{-1}{3i\hbar}\hat{\theta}[\partial_{x}\hat{\theta},\partial_{y}\hat{\theta}]+\frac{1}{2}\big((\partial_{x}\hat{\theta})(\partial_{\tilde{x}}\hat{\theta})+(\partial_{y}\hat{\theta})(\partial_{\tilde{y}}\hat{\theta})\big)\bigg\}|\psi_{j}>,
\Big(\partial_{t}-{\frac{1}{2}}(e^{2U}f\partial_{r}f+f^{2}e^{U}\partial_{r}(e^{U}))\gamma_{0}\gamma_{1}+{\frac{f}{4\bar{Z}}}e^{3U}\partial_{r}e^{-2U}\gamma_{1}-ige^{2U}{\kappa}_{I}Y^{I}+{\frac{i}{2}}gf{\kappa}
\Big(\phi_{1}\star\phi_{2}\Big)(x):=\int\frac{d^{4}k}{(2\pi)^{4}}\intd^{4}y\phi_{1}(x+\frac12\tilde{k})\phi_{2}(x+y)e^{iky},\quad\tilde{k}_{\mu}=\theta_{\mu\nu}k_{\nu}.
{\frac{d\bar{w}}{dt}}\,\Pi^{T}\,{\frac{dw}{dt}}={\frac{1}{\kappa^{2}}}\sum_{r=3}^{N}\left[{\frac{t_{1r}\bar{t}_{1r}}{b+r}}+{\frac{t_{2r}\bar{t}_{2r}}{b-r}}\right]\,\,.
{\calL}_{0}^{\prime}=\frac{1}{2}\sum_{i\geqj=1}^{N}\{n_{ij}((\partial_{\mu})^{2}+h^{2}M_{ij}^{I}(q))n_{ij}^{\ast}+m_{ij}((\partial_{\mu})^{2}+h^{2}M_{ij}^{II}(q))m_{ij}^{\ast}\}
{\hat{H}}_{BRST}=i\{{\hat{\Psi}},\{\hat{\Omega}}\},\quad{\hat{\Psi}}={\hat{\bar{\calP}}}{\hat{\lambda}},\quad{\hat{\Omega}}=-i{\hat{\calP}}{\hat{\pi}}+{\hat{\calC}}\left(\frac{{\hat{p}}_{x}^{2}}{2m}+{\hat{p}}_{t}\right).
\alpha^{T}\|c\|\alpha\equiv\sum_{j=1}^{N}\sum_{l=1}^{N}c_{jl}\alpha_{j}\alpha_{l}=\sum\alpha_{i}^{2}+2\!\begin{array}{c}{{}}\\{{\sum\sum}}\\{{}_{j<l}}\\\end{array}\alpha_{j}\alpha_{l}c_{jl}.
\begin{aligned}{S_{WZ}=-i\int\!\mathrm{d}^{2}\sigma\,\epsilon^{ij}\Big[e_{i}^{r}\big(\Theta^{T}{\calC}\Gamma_{r}{\calD}_{j}\Theta\big)-\frac{1}{2}i\big(\bar{\Theta}}&{{}\Gamma^{r}{\calD}_{i}\Theta\big)\big(\Theta^{T}{\calC}\Gamma_{r}{\calD}_{j}\Theta\big)}\\{}&{{}+\frac{1}{12}e_{i}^{r}\big(\Theta^{T}{\calC}\Gamma_{r}{\calM}^{2}{\calD}_{j}\Theta\big)\Big]\,,}\\\end{aligned}
H^{2}=\frac{8\piG}{3}\left(\rho_{1}+\rho_{2}+V_{\mathrm{eff}}+W_{\mathrm{eff}}\right)+\frac{2\alpha^{2}}{1+2\alpha^{2}}\dot{\phi}^{2}+\frac{1}{1+2\alpha^{2}}\dot{R}^{2}.
\deltaL=\partial_{\tau}\left(p_{\mu}^{i}\varepsilon_{ij}e^{jk}\dot{x}_{k}^{\mu}-\frac12p_{\mu}^{i}\,p_{\nu}^{j}\,\eta^{\mu\nu}\varepsilon_{ij}-\frac12\left(m^{2}\right)^{ij}\varepsilon_{ij}\right).
D_{J}^{\,I}=\partial_{J}^{\,I}-i\theta_{J}^{\alpha}\bar{\theta}^{I\;\dot{\alpha}}\partial_{\alpha\dot{\alpha}}-\theta_{J}\partial^{I}+\bar{\theta}^{I}\bar{\partial}_{J}\;,\quadI=1,\ldots,k,\J=k+1,\ldots,N
R_{\mu\nu}=\frac{k^{2}q^{2}}{r^{6}}\left(\begin{array}{cccc}{r^{2}}&{0}&{0}&{0}\\{0}&{x^{2}}&{xy}&{xz}\\{0}&{xy}&{y^{2}}&{yz}\\{0}&{xz}&{yz}&{z^{2}}\\\end{array}\right).
G_{2}(k)=-\frac{1}{A^{2}(k^{2}+a_{2})(k^{2}+a_{3})}+\frac{1}{A^{2}}\frac{\frac{m^{2}}{A^{2}}+a_{1}}{(k^{2}+a_{1})(k^{2}+a_{2})(k^{2}+a_{3})}
\frac{\partial^{2}\phi(x,t)}{\partialt^{2}}-\frac{\partial^{2}\phi(x,t)}{\partialx^{2}}+\eta\phi^{2}(x,t)\frac{\partial\phi(x,t)}{\partialt}=-V^{\prime}(\phi)+\phi(x,t)\tilde{\xi}(x,t,T).
\langle\mathrm{out}|f(\phi)|\mathrm{in}\rangle={\frac{e^{-i\pi/4}}{\sqrt{2\pip}}}\left({\frac{2}{1-e^{2i\phi}}}\right)\langle\mathrm{out}|{\calR}^{\phi/\pi}\left({\calR}^{-1}-{\calR}\right)|\mathrm{in}\rangle\;,
0\:=\:-8m+{\calM}_{\nu}^{\mu}{\mathbfa}_{\nu}^{\dagger}{\mathbfa}^{\mu}+{\calN}_{\nu}^{\mu}{\mathbfb}^{\dagger\nu}{\mathbfb}_{\mu}+4m\,{\calW}_{\nu\rho}^{\mu\sigma}{\mathbfa}^{\nu}{\mathbfb}_{\mu}{\mathbfa}^{\rho}{\mathbfb}_{\sigma}\:\equiv\:{\mathbfR}
dv_{\underline{{\mu}}}^{~A}\equiv{\frac{1}{4}}\Omega^{\underline{{b}}\underline{{a}}}v_{\underline{{\mu}}}^{~B}(\sigma_{\underline{{b}}\underline{{a}}})_{B}^{~A}={\frac{1}{2}}v_{\underline{{\mu}}}^{~B}f^{i}\Gamma_{BA}^{i}-{\frac{1}{4}}A^{ij}v_{\underline{{\mu}}}^{~B}\Gamma_{BA}^{ij}
Z=\int_{-\infty}^{\infty}\frac{dk_{Y}}{2\pi}\int_{0}^{2}\frac{d\zeta_{-}}{\sqrt{2}\pi}\sum_{m_{-}=-\infty}^{\infty}q^{\frac{1}{2}k_{Y}^{2}+(\lambda+2m_{-})^{2}-\frac{1}{12}}\prod_{n=1}^{\infty}(1-q^{n})^{-2}.
\begin{array}{cc}{H_{\circ}\Psi_{ls}=}&{E\Psi_{ls}}\\{J_{3}\Psi_{ls}=}&{\left(l+\frac{1}{2}\right)\Psi_{ls}}\\{p_{3}\Psi_{ls}=}&{p_{3}\Psi_{ls}}\\{S_{3}\Psi_{ls}=}&{\pms\Psi_{ls}}\\\end{array}
I=\int_{X}\varepsilon^{ijk}\bigg(e_{ia}(\partial_{j}\omega_{k}^{a}-\partial_{k}\omega_{j}^{a})+\varepsilon_{abc}e_{i}^{a}\omega_{j}^{b}\omega_{k}^{c}+{\frac{1}{3}}\lambda\varepsilon_{abc}e_{i}^{a}e_{j}^{b}e_{k}^{c}\bigg).
L_{L1}={\frac{K\,v^{2}}{2\,R}}~\left[1+\sum_{L=1}^{\infty}\sum_{q=0}^{L-1}\,d_{LL}(0,q)~K^{q}~N^{-q}\left({\frac{N\,v^{2}}{R^{2}M_{p}^{9}V_{T}~b^{7-p}}}\right)^{L}~(1+S_{L1})\right]
D_{\alpha}^{i}={\frac{\partial}{\partial\theta_{i}^{\alpha}}}+i\bar{\theta}^{\dot{\alpha}i}(\sigma^{\mu})_{\alpha\dot{\alpha}}\partial_{\mu}\;,\qquad\bar{D}_{\dot{\alpha}i}=-{\frac{\partial}{\partial\bar{\theta}^{\dot{\alpha}i}}}-i\theta_{i}^{\alpha}(\sigma^{\mu})_{\alpha\dot{\alpha}}\partial_{\mu}\;.
\hat{G}_{RS2}[a,b;E]=\hat{G}_{0}^{\xi}[a,b;E]+\frac{\hat{G}_{0}^{\xi}[a,c;E]\hat{G}_{0}^{\xi}[c,b;E]}{\frac{1}{v}-\hat{G}_{0}^{\xi}[c^{+},c;E]}