harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\{b(p,s),b^{\dagger}(p^{\prime},s^{\prime})\}=\delta(p^{+}-p^{\prime+)}\delta^{2}(p_{\perp}-p_{\perp}^{\prime})\delta_{s,s^{\prime}}=\delta^{3}(p-p^{\prime})\delta_{s,s^{\prime}}\;,
\langleE_{0}|A_{0}^{(1)}\cdotsA_{0}^{(n)}|E_{0}\rangle=\langle\hat{E}_{0}|\hat{A}_{0}^{(1)}\cdots\hat{A}_{0}^{(n)}|\hat{E}_{0}\rangle\,\langle\check{A}_{0}^{(1)}\cdots\check{A}_{0}^{(n)}\rangle_{G},
G_{0}^{(2)}(p\,,\dot{-}p^{\prime})=\frac{i}{2}\chi_{1}(p_{0})\chi_{1}(-p_{0})\frac{\delta^{(4)}(p\dot{+}(\dot{-}p^{\prime}))+\delta^{(4)}((\dot{-}p^{\prime})\dot{+}p)}{{\mathcalC}_{D}(p)-m^{2}}~.
A^{a}=-\fraci{\psi^{+}\psi}Tr(d\psi\psi^{+}\sigma^{a}-\psid\psi^{+}\sigma^{a})+\fraci{\psi^{+}\psi}Tr(D\psi\psi^{+}\sigma^{a}-\psiD\psi^{+}\sigma^{a}).
32\tilde{V}=\frac{1}{2}M_{\mu}^{(2)}(\rho^{(\mu)})^{2}+\frac{(2-\omega)\tilde{A}^{2}}{16\pi^{2}\omega(1-\omega^{2})}\mu^{4}\left(\frac{\rho^{(\mu)}}{\mu}\right)^{4/(2-\omega)}.
\frac{\partial}{\partial\xi}\frac{\delta^{2}\Gamma}{\delta\psi(y)\delta\bar{\psi}(x)}=+\frac{\delta^{3}\Gamma}{\delta\psi(y)\delta\bar{K}\delta\chi}\frac{\delta^{2}\Gamma}{\delta\bar{\psi}(x)\delta\psi}+\frac{\delta^{2}\Gamma}{\delta\psi(y)\delta\bar{\psi}}\frac{\delta^{3}\Gamma}{\delta\bar{\psi}(x)\deltaK\delta\chi}
\partial^{2}\tilde{\Lambda}(t,\vec{x})=-\dot{A}^{0}(t_{0},x)+{\vec{\nabla}}^{2}f(\vec{x})+\sqrt{2}\kappa\vec{q}\cdot\int_{t_{0}}^{t}d\tau\vec{\nabla}\times\vec{A}(\tau,\vec{x})
F(\beta)=-\pi^{-26}2^{-14}\int_{S}\frac{d^{2}\tau}{\tau_{2}^{2}}\tau_{2}^{-12}|\eta(\tau)|^{-48}\left[\theta_{3}\left(0\left|\frac{i\beta^{2}}{2\pi^{2}\tau_{2}}\right.\right)-1\right]
\deltav^{\ell}(\omega)=-e\left[{\frac{u^{*}\alpha^{\ell}\Lambda^{+}(\vec{p})\alpha^{k}u}{\omega}}-{\frac{u^{*}\alpha^{k}\Lambda^{-}\alpha^{\ell}u}{\omega-2E}}+\omega\leftrightarrow-\omega,\ell\leftrightarrowk\right]\,A_{k}^{e}(\omega)~~.
L_{l\otimes\frac12}(u)=q^{-\beta\sigma^{z}}R_{l\otimes\frac12}(u)\ldotsR_{l\otimes\frac12}(u)=\left(\begin{array}{cc}{q^{-\beta}A_{l}(u)}&{q^{-\beta}B_{l}(u)}\\{q^{+\beta}C_{l}(u)}&{q^{+\beta}D_{l}(u)}\\\end{array}\right).
P_{\mathrm{red}}(i\rightarrowf)=\sum_{m=0}^{+\infty}\sum_{\ell=-\infty}^{+\infty}<r_{f},\theta_{f}|m,\ell>\,e^{-\frac{i}{\hbar}(t_{f}-t_{i})E_{m,\ell}}\,<m,\ell|r_{i},\theta_{i}>\\\.
\frac{\mathrm{d}^{2}\rho(t;x)}{\mathrm{d}t^{2}}=-\frac{(\gamma_{1}-\gamma_{-1})^{2}}{\gamma_{1}\gamma_{-1}}\frac{1}{(2\kappa)^{2}}\big(\mathrm{e}^{-\rho(x-2\kappa)}-2\mathrm{e}^{-\rho(x)}+\mathrm{e}^{-\rho(x+2\kappa)}\big).
A_{1}^{\alpha,i}\equiv\frac{1+\sigma_{3}}{2}\otimes\psi_{-1/2}^{\alpha,i}|0,k\rangle~,~A_{2}^{\alpha,i}\equiv\frac{1-\sigma_{3}}{2}\otimes\psi_{-1/2}^{\alpha,i}|0,k\rangle
\begin{aligned}{[m]\Omega_{+}}&{{}=0}\\{\Omega_{-}}&{{}=-\frac{1}{12}\Theta\left(\Gamma_{+}\Gamma_{-}+1\right)}\\{\Omega_{i}}&{{}=\frac{1}{24}(3\Theta\Gamma_{i}+\Gamma_{i}\Theta)\Gamma_{+}}\\\end{aligned}
\tilde{I}_{\mathrm{DE}}=\frac{\Omega_{\tilde{d}+1}}{16\pi}[(\tilde{d}+1)r_{+}^{\tilde{d}}-r_{-}^{\tilde{d}}]\beta_{H}-\frac{1}{4}\Sigma-\frac{\piQ}{\tilde{d}\Omega_{\tilde{d}+1}r_{+}^{\tilde{d}}}Q\beta_{H}
M=\left(\begin{array}{ccc}{G^{-1}}&{-G^{-1}C}&{-G^{-1}a^{T}}\\{-CG^{-1}}&{G+C^{T}G^{-1}C}&{C^{T}G^{-1}a^{T}+a^{T}}\\{-aG^{-1}}&{aG^{-1}C+a}&{1+aG^{-1}a^{T}}\\\end{array}\right)
\left.-\frac{1}{p\cdotk_{1}\,\,p\cdotk_{2}}p\cdot\varepsilon(k_{1})p\cdot\varepsilon(k_{2})(k_{1}-k_{2})\cdot\varepsilon(k_{3})C_{a_{1}bf}C_{a_{3}fg}C_{a_{2}bg}\right]
S_{2}=\int_{-\infty}^{+\infty}dr_{*}\Phi_{2}(r_{*}){\calD}_{2}\left(\frac{d}{dr_{*}},r_{*}\right)\Phi_{2}(r_{*})=\int_{-\infty}^{+\infty}dr_{*}\Phi_{2}(r_{*}){\calD}_{1}\left(-\frac{d}{dr_{*}},-r_{*}\right)\Phi_{2}(r_{*}).
S\;=\;\int_{-\infty}^{\infty}\Bigl[\;{\frac{1}{2}}\;(1+{\frac{x^{2}}{\tau^{2}}}\;(1-\varepsilon))\left({\partial_{x}\phi}\right)^{2}+{\frac{\lambda}{4}}{\frac{1}{1+{\frac{x^{2}}{\tau^{2}}}\;(1-\varepsilon)}}\;\phi^{4}\;\Bigr]\;dx,
\begin{array}{c}{m\mapston\,,\quadn\mapsto-m\,,}\\{n^{1}\mapsto-m_{1}\,,\quadn^{2}\mapsto-m_{2}\,,\quadm_{1}\mapsto-n^{1}\,,\quadm_{2}\mapsto-n^{2}\,,}\\{R_{2}\mapstoR_{1}\vartheta\,,\quadR_{1}\mapstoR_{2}\vartheta\,,}\\{\phi\mapsto\phi\vartheta^{2}-\vartheta\,,\quad\lambda^{1}\mapsto\lambda^{2}\,,\quad\lambda^{2}\mapsto-\lambda^{1}\,,\quadg^{2}\mapstog^{2}\vartheta}\\\end{array}
\left(\frac{1}{g_{c}^{\prime2}}+\frac{2t_{2}(A)}{8\pi^{2}}\mathrm{ln}g_{c}^{\prime}\right)=\left(\frac{1}{g_{c}^{2}}+\frac{2t_{2}(A)}{8\pi^{2}}\mathrm{ln}g_{c}\right)-\frac{3t_{2}(A)}{8\pi^{2}}\mathrm{ln}\frac{M}{M^{\prime}}.
\Psi_{E;-n-1}(t,r,\phi)=\frac{1}{N}\left(\begin{array}{c}{\sqrt{mc^{2}+E}K_{1-\eta}(\kappar)}\\{i\sqrt{mc^{2}-E}K_{\eta}(\kappar)e^{i\phi}}\\\end{array}\right)e^{-i(n+1)\phi}e^{-i{\calE}t/\hbar},
\phi^{(in)}(k)\delta_{m}(k)=\theta(k^{0})\frac{A^{in}(\bf{k})}{2\omega_{k}}\delta(k^{0}-\omega_{k})+\theta(-k^{0})\frac{A^{in}(-\bf{k})^{\dagger}}{2\omega_{k}}\delta(k^{0}+\omega_{k}),
M_{\alpha^{\prime}\beta^{\prime}}=\frac{1}{4}\left(\begin{array}{cc}{g_{2}^{2}(v^{2}+w^{2})}&{g_{2}g_{1}(n_{(1)}v^{2}-n_{(2)}w^{2})}\\{g_{2}g_{1}(n_{(1)}v^{2}-n_{(2)}w^{2})}&{g_{1}^{2}(n_{(1)}^{2}v^{2}+n_{(2)}^{2}w^{2})}\\\end{array}\right)
K_{\frac{1}{2}a}(\theta)|B_{a}\rangle=R_{aa}^{\frac{1}{2}}(\theta)K_{\frac{1}{2}a}(-\theta)|B_{a}\rangle+R_{ab}^{\frac{1}{2}}(\theta)K_{\frac{1}{2}b}(-\theta)|B_{b}\rangle\quad,\quadb\neqa\quad,\quada,b=0,1
L=2t\int_{\Sigma}d^{2}z(\frac{1}{2}g_{IJ}\partial_{z}\phi^{I}\partial_{\bar{z}}\phi^{J}+i\psi_{z}^{\bar{i}}D_{\bar{z}}\chi^{i}g_{i\bar{i}}+i\psi_{\bar{z}}^{i}D_{z}\chi^{\bar{i}}g_{\bar{i}i}-R_{i\bar{i}j\bar{j}}\psi_{\bar{z}}^{i}\psi_{z}^{\bar{i}}\chi^{j}\chi_{\bar{j}})
M\Psi_{nlm}(r,\theta,\varphi;\delta_{1},\delta_{2})=\left(l+\frac{\delta_{1}+\delta_{2}}{2}\right)\left(l+\frac{\delta_{1}+\delta_{2}}{2}+1\right)\Psi_{nlm}(r,\theta,\varphi;\delta_{1},\delta_{2}),
\frac{\partial\phantom{iii}}{\partialx_{i}^{a_{i}}}\,X^{a_{1}x_{1}\ldotsa_{i}x_{i}\,\ldots\,a_{n}x_{n}}=\bigl(\,\delta(x_{i}-x_{i-1})-\delta(x_{i}-x_{i+1})\,\bigr)X^{a_{1}x_{1}\ldotsa_{i-1}x_{i-1}\,a_{i+1}x_{i+1}\ldotsa_{n}x_{n}}
g_{ij}={\calZ}_{,i}^{\mu}{\calZ}_{,j}^{\nu}{\calG}_{\mu\nu},\;\;{\calZ}_{,i}^{\mu}{\eta}^{\nu}{\calG}_{\mu\nu}=0,\;\;{\eta}^{\mu}{\eta}^{\nu}{\calG}_{\mu\nu}=\varepsilon
{\calH}(p)=-2i\varepsilon^{\mu\nu\rho}\varepsilon^{\alpha\beta\lambda}g_{\nu\alpha}\frac{\mu^{\epsilon}}{(2\pi)^{d}}\int{\calD}q\frac{1}{(p+q)^{2}q^{2}}I_{\beta\mu\lambda\rho}(q)\;,
\frac{1}{2\pii}\int_{C}ds\,\frac{e^{\rho/s}}{s}F(s^{})=\frac{1}{2\pii}\left(\int_{C_{0}}ds+\int_{C_{1}}ds+\int_{C_{2}}ds\right)\frac{e^{\rho/s}}{s}F(s^{})=0,
\Big(\left.{\calD}\right|_{\sigma}\left.{\calD}\right|_{\sigma^{\prime}}{\bfG}^{\mu\nu}(\sigma,\sigma^{\prime})\Big)_{\sigma=\sigma^{\prime}}=-\frac1{\sqrt{s}^{3}}(b_{0}^{1}g^{\mu\nu}+sb_{1}^{1}g_{\lambda\rho}z^{\prime\lambda}z^{\prime\rho}g^{\mu\nu}+sb_{2}^{1}z^{\prime\mu}z^{\prime\nu})
\triangle\left(\hat{Q}_{j}\right)=\hat{Q}_{j}\otimesI+q^{T_{j}}\otimes\left(\hat{Q}_{j}-\hat{\epsilon}_{j}I\right)\inU_{q}(\hat{\mathfrak{g}})\otimes\mathcal{B}_{q}^{\epsilon}(\hat{\mathfrak{g}}),
\delta\langleW(\gamma)\rangle=Z^{-1}\frac{-i4\pi}{k}\int\delta(x-z)dx^{\mu}dy^{\nu}dz^{\rho}\epsilon_{\mu\nu\rho}{\calT}r[T^{a}T^{a}U(\gamma_{x}^{x})]e^{iS_{CS}[A]}{\calD}A
z_{i}^{a}{\alpha_{i}^{a}}^{\dagger}=z_{i}^{+}{\alpha_{i}^{+}}^{\dagger}+z_{i}^{-}{\alpha_{i}^{-}}^{\dagger}\\,\\|z|^{2}\equiv|z_{i}^{a}|^{2}=|z_{i}^{+}|^{2}+|z_{i}^{-}|^{2}\\,
H=\frac{V}{2\overline{{g}}}\mathcal{P}^{2}-\frac{V}{72}(\Pi^{(U)ab}\varepsilon_{ab})^{2}+2\Pi^{(U)ab}\partial_{a}U_{0b}+c\varphi_{1}+k_{a}\varphi_{2}^{a}+\lambda^{\mu}\varphi_{3\mu}+\widetilde{\lambda}^{\mu}\varphi_{4\mu}
\mathcal{H}={\frac{1}{2}}\overrightarrow{{\bfp}}^{2}+{\frac{g^{2}}{2}}(x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2})+{\frac{v^{2}}{2}}(x^{2}+y^{2}+z^{2}).
\hat{H}(t)=\sum_{\alpha}\Omega_{\alpha}^{(D)}(t)\Bigl(\hat{a}_{\alpha}^{\dagger}\hat{a}_{\alpha}+\frac{1}{2}\Bigr)+\Omega_{\alpha}(t)\Bigl(\hat{a}_{\alpha}^{\dagger2}+\hat{a}_{\alpha}^{2}\Bigr)\equiv\sum_{\alpha}\hat{H}_{\alpha}(t).
\gamma=\gamma_{\underline{{a}}}^{1}I^{\underline{{a}}}+\gamma_{\underline{{a}}\underline{{b}}}^{1}I^{\underline{{a}}}I^{\underline{{b}}}+...,\mathrm{~and~}Q_{\mu}=q_{\mu,\underline{{a}}}^{1}I^{\underline{{a}}}+q_{\mu,\underline{{a}}\underline{{b}}}^{2}I^{\underline{{a}}}I^{\underline{{b}}}+...
C:=\left(\begin{array}{cc}{[T_{n}^{+},T_{m}^{+}]}&{[T_{n}^{+},T_{m}^{3}]}\\{~[T_{n}^{3},T_{m}^{+}]}&{[T_{n}^{3},T_{m}^{3}]}\\\end{array}\right)=\left(\begin{array}{cc}{0}&{k\delta_{n+m}}\\{-k\delta_{n+m}}&{(kn/2)\delta_{n+m}}\\\end{array}\right)
V_{NS5}(x,b,c)=\frac{A^{2}m^{3}}{18}b^{3/2}\left[\frac{3x^{2}}{3x^{4}+b}\right]\Biggl[\left(x^{4}-1\right)^{2}+c+\frac{3c}{b}x^{4}\Biggr],
\bigg(\langle\langleF_{\mu\nu}(z_{1})F_{\rho\sigma}(z_{1}^{\prime})\rangle\rangle-\langle\langleF_{\mu\nu}(z_{1})\rangle\rangle\,\langle\langleF_{\rho\sigma}(z_{1}^{\prime})\rangle\rangle\bigg)^{SR}=0.
{\calR}_{\mu\nu}^{ab}=\partial_{\mu}\Omega_{\nu}^{ab}-\partial_{\nu}\Omega_{\mu}^{ab}+\frac12[\Omega_{\mu\,c}^{a},\Omega_{\nu}^{cb}]-\frac12[\Omega_{\nu\,c}^{a},\Omega_{\mu}^{cb}]\,,
g=\int_{S_{\infty}^{2}}d\vec{\sigma}\cdot\frac{1}{v}\Phi^{a}\vec{B}_{a}=\frac{1}{v^{2}}\int_{S_{\infty}^{2}}d\vec{\sigma}\cdot\Phi_{a}\Phi^{a}\vec{B}=\int_{S_{\infty}^{2}}d\vec{\sigma}\cdot\vec{B}=\frac{4\pin}{e}\,.
\left|\beta_{k}\right|=\frac{\left|3m+i\right|}{\sqrt{\pi}}\frac{2^{n_{s}-3}\left|\Gamma\left(n_{s}\right)\right|}{|n_{1}(1-n_{1})|}z_{S}^{m-3/2}\left(\frac{g_{S}}{g_{1}}\right)^{-l}\left(\frac{\omega}{\omega_{s}}\right)^{2-n_{s}}\left(\frac{M}{M_{s}}\right)_{.}^{-n_{1}/2}
\chi(r)=\frac{\sqrt{2}}{2}\,\left(\begin{array}{c}{1}\\{e^{i\phi}}\\\end{array}\right)\,\delta_{r(+)}\,+\,\frac{\sqrt{2}}{2}\,\left(\begin{array}{c}{-e^{-i\phi}}\\{1}\\\end{array}\right)\,\delta_{r(-)},
\tilde{H}_{\mu\nu\rho}=\frac{1}{6!}\epsilon_{\mu\nu\rho\sigma_{1}\sigma_{2},\cdots,\sigma_{7}}\frac{1}{\sqrt{-X^{2}}}\frac{\delta}{\deltaX_{\sigma_{1}}}B_{(6)}^{\sigma_{2},\cdots,\sigma_{6}},
\frac{\deltaS}{\delta\phi^{A}}\frac{\deltaS}{\delta\phi_{Aa}^{\ast}}+\varepsilon^{ab}\phi_{Ab}^{\ast}\frac{\deltaS}{\delta\overline{{\phi}}_{A}}=i\hbar\frac{\delta^{2}S}{\delta\phi^{A}\delta\phi_{Aa}^{\ast}}
d(l_{0},k_{1}+1;l_{0},k_{1})c(l_{0},k_{1};l_{0},k_{1}+1)=\frac{-(l_{0}+k_{0})a+(l_{0}-k_{0})b}{(k_{1}-l_{0})(k_{1}-l_{0}+1)(k_{1}+l_{0})(k_{1}+l_{0}+1)},
\begin{array}{l}{\rho_{R+}\rho_{L+}\{q_{L3},\,\,e_{Lpq},\,\,ie_{Lp7}q_{Lr}:\,\,r=1,2,\,\,p,q=1,...,6\}\rho_{L+}\rho_{R+}}\\{\,\,\,=\rho_{R+}\rho_{L+}\{q_{L3},\,\,e_{Lpq}\}\rho_{R+}}\\{\,\,\,=\rho_{L+}\rho_{R+}\{q_{L3},\,\,e_{L7}+2e_{R7},\,\,e_{Lpq}-e_{Lmn}:\,p,q,m,n=1,...,6,\,\,e_{p}e_{q}-e_{m}e_{n}=0\}.}\\\end{array}
\hat{\mathrm{P}}_{\mathrm{R}}^{s}=\hat{\mathrm{H}}_{0}^{s}-\frac{e^{2}\mathrm{L}}{2\pi}\hbar\sum_{p>0}{\alpha}_{p}{\alpha}_{-p}-\frac{1}{2}\xi_{\mathrm{R}}\hbar-\frac{1}{2}eb\mathrm{L}\cdot\eta_{\mathrm{R}}\hbar.
{\hat{R}}_{ij^{\star}k\ell^{\star}}\,=\,{\frac{1}{\mu^{2}}}\,({\hat{g}}_{ij^{\star}}{\hat{g}}_{k\ell^{\star}}+{\hat{g}}_{kj^{\star}}{\hat{g}}_{i\ell^{\star}})\,+\,{\hat{C}}_{iks}\,{\hat{C}}_{t^{\star}j^{\star}\ell^{\star}}\,{\hat{g}}^{st^{\star}}
G_{0}=\frac{1}{4\pi^{2}x^{2}}\,\rightarrow\,\frac{\Gamma(\frac{d}{2})}{(d-2)2\pi^{\frac{d}{2}}}\,\biggl(\frac{1}{x^{2}}\biggr)^{(\frac{d}{2}-1)}=\frac{1}{4\pi^{2}}\,\frac{\Gamma(1+\epsilon)}{\pi^{\epsilon}}\,\biggl(\frac{1}{x^{2}}\biggr)^{(1+\epsilon)}
\tilde{\sigma_{5}}=\int{\frac{d^{d}k}{(2\pi)^{d}}}\gamma_{\alpha\beta}^{5}\sigma_{\alpha\beta}(k)=-\tilde{G}_{5}=-\int{\frac{d^{d}k}{(2\pi)^{d}}}Tr[\gamma^{5}G^{(2)}(k)]
{\calL}(\frac{3}{2},\alpha-1)=\bar{\Psi}[\frac{2}{3}L_{\mu}(\partial_{\mu}+ia_{\mu}+iC_{\mu})+M]\Psi-\frac{i}{8\pi(\alpha-1)}\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}\;,
\begin{array}{ccc}{K_{ab}^{-1}=}&{p\left(\begin{array}{cc}{\sqrt{g}}&{{\frac{1}{\sqrt{g}}}}\\{{\frac{1}{\sqrt{g}}}}&{\sqrt{g}}\\\end{array}\right)}&{+(1-p)\left(\begin{array}{cc}{g}&{{\frac{1}{g}}}\\{{\frac{1}{g}}}&{g}\\\end{array}\right)}\\\end{array}
H_{\mathrm{eff}}=-K\sum_{[x,i]}\psi_{A}^{\alpha\dagger}(x+\hat{i})\psi_{B}^{\alpha}(x)\psi_{B}^{\beta\dagger}(x)\psi_{A}^{\beta}(x+\hat{i})=K\sum_{[x,i]}J^{\alpha\beta}(x+\hat{i})J^{\beta\alpha}(x)
\left\langle:\psi_{L}\psi_{R}:\right\rangle_{0}=\left\langle:\psi_{L}^{\dagger}\psi_{R}^{\dagger}:\right\rangle_{0}=\left\langle:\psi_{L}^{\dagger}\psi_{L}:\right\rangle_{0}=\left\langle:\psi_{R}^{\dagger}\psi_{R}:\right\rangle_{0}=0.
\hat{h}\psi_{\varkappa,n}=\epsilon_{\varkappa,n}\psi_{\varkappa,n},\;\psi_{\varkappa,n}=\left(\begin{array}{c}{g^{00}\sqrt{-g}\left(\epsilon_{\varkappa,n}-qA_{0}\right)\varphi_{\varkappa,n}}\\{\varphi_{\varkappa,n}}\\\end{array}\right)\;.
\mathrm{d}\mu=\mathrm{d}\mu_{\,\mathrm{SU}(2)}\,\prod_{R(j)}\left\{(M_{\mathrm{PV}})^{-T(j)}\,e^{-4\pi^{2}\,j\,|v_{j}|^{2}\,\rho_{\mathrm{inv}}^{2}}\,\prod_{i=1}^{2T(j)}\mathrm{d}\,\bar{\eta}_{i}\right\}\,,
\varepsilon=\left(\begin{array}{cccc}{0}&{\varepsilon^{01}}&{\varepsilon^{02}}&{0}\\{-\varepsilon^{01}}&{0}&{\varepsilon^{12}}&{\varepsilon^{01}}\\{-\varepsilon^{02}}&{-\varepsilon^{12}}&{0}&{\varepsilon^{02}}\\{0}&{-\varepsilon^{01}}&{-\varepsilon^{02}}&{0}\\\end{array}\right)
\left\{j_{M}^{0}\left(t,\sigma\right),\widehat{U_{N}^{0}}\left(t,\sigma^{\prime}\right)\right\}=-\delta_{M}U_{N}^{0}\cdot\delta\left(\sigma^{\prime}-\sigma\right)\;+\frac\partial{\partial\sigma^{r}}\left[\delta_{M}\phi^{i}\frac{\partial\widehat{U_{N}^{0}}}{\partial\partial_{r}\phi^{i}}\cdot\delta\left(\sigma^{\prime}-\sigma\right)\right]\quad.
\begin{array}{l}{\theta_{m,8}({\frac{\tau}{2}},{\frac{z}{4}})\stackrel{z\toz+{\frac{\tau}{2}}}{\longrightarrow}q^{-1/6}y^{-1/4}~\theta_{m+2,8}({\frac{\tau}{2}},{\frac{z}{4}})\stackrel{z\toz+{\frac{\tau}{2}}}{\longrightarrow}q^{-1/4}y^{-1/2}~\theta_{m+4,8}({\frac{\tau}{2}},{\frac{z}{4}}),}\\\end{array}
q^{S^{3}}=\left(\begin{array}{cc}{q^{1/2}}&{0}\\{0}&{q^{-1/2}}\\\end{array}\right),\\\S^{+}=\left(\begin{array}{cc}{0}&{q^{1/2}}\\{0}&{0}\\\end{array}\right),\\\S^{-}=\left(\begin{array}{cc}{0}&{0}\\{q^{-1/2}}&{0}\\\end{array}\right),
{\calG}_{WZWC}=\frac{\ell}{16\piG}\int_{|r|=\bar{r}}[\frac{e_{+}^{t}}{e_{-}^{t}}\,k_{-}^{2}+\frac{e_{-}^{t}}{e_{+}^{t}}\,\tilde{k}_{+}^{2}]\,\theta\,d\varphi\,dt\qquad.
\Gamma^{0}=\left(\begin{array}{cc}{I}&{0}\\{0}&{-I}\\\end{array}\right),\quad\Gamma^{i}=\left(\begin{array}{cc}{0}&{\sigma^{i}}\\{-\sigma^{i}}&{0}\\\end{array}\right),\quad\Gamma_{11}=\left(\begin{array}{cc}{0}&{I}\\{I}&{0}\\\end{array}\right),
Q(\xi)=\frac{1}{16\pi}\intd\varphi\left\{\frac{\xi^{\bot}}{lr}\left((g_{\varphi\varphi}-r^{2})-2r^{2}(lg^{-1/2}-1)+4r^{2}\left[\phi^{2}+(1+\alpha)\phi^{4}\right]\right)+2\xi^{\varphi}\pi_{\varphi}^{r}\right\}\,,
\begin{array}{c}{{\calH}=g^{a{\bar{b}}}(\pi_{a}{\bar{\pi}}_{b}+{U}_{a}{\bar{U}}_{\bar{b}})-}\\{-iU_{a;b}\eta_{1}^{a}\eta_{2}^{b}+i{\bar{U}}_{\bar{a};\bar{b}}{\bar{\eta}}_{1}^{\bar{a}}{\bar{\eta}}_{2}^{\bar{b}}-R_{a\bar{b}c\bar{d}}\eta_{1}^{a}\bar{\eta}_{1}^{b}\eta_{2}^{a}\bar{\eta}_{2}^{d},}\\\end{array}
{\calL}_{\mathrm{GNJL}}=-{\frac{1}{4}}F_{\mu\nu}^{2}+\bar{\psi}i\gamma^{\mu}D_{\mu}\psi-\bar{\psi}(\sigma+i\gamma_{5}\pi)\psi-{\frac{1}{2G_{0}}}(\sigma^{2}+\pi^{2})+\frac{m_{0}}{G_{0}}\sigma.
F^{(n)}=dC^{(n-1)}=-(-1)^{\frac{n(n+1)}{2}}d\hat{\star}C^{(11-n)}=(-1)^{\frac{n(n-1)}{2}}\hat{\star}dC^{(9-n)}=(-1)^{\frac{n(n-1)}{2}}\hat{\star}F^{(10-n)},
[\Gamma_{\mu},\Gamma_{\nu}{]_{\lambda}}^{\kappa}\equiv(\Gamma_{\mu}\Gamma_{\nu}-\Gamma_{\nu}\Gamma_{\mu}{)_{\lambda}}^{\kappa}={\Gamma_{\mu\lambda}}^{\sigma}{\Gamma_{\nu\sigma}}^{\kappa}-{\Gamma_{\nu\lambda}}^{\sigma}{\Gamma_{\mu\sigma}}^{\kappa}.
\frac{W_{\mathrm{c.p.1}}^{3}}{W_{\mathrm{c.p.2}}^{3}}=\frac{27\rho_{\mathrm{cr}}^{6}}{(2+\rho_{\mathrm{cr}}^{6})^{3}}=\frac{27(1-\beta)(1-2\gamma)^{2}}{4(2\gamma-\beta)^{3}}\,.
\tilde{V}_{t}(r)=-\omega^{2}(f-1)+h\left\{{\frac{3}{4r^{2}}}\left(1+{\frac{3r_{0}^{2}}{r^{2}}}\right)+{\frac{8R^{4}}{r^{2}(r^{2}+R^{2})^{2}}}-{\frac{2}{(1+r_{0}^{2}/r^{2})^{2/3}(1+R^{2}/r^{2})^{1/3}}}\right\},
\int_{\mu}^{\infty}{\frac{d\tilde{n}^{(\lambda)}(\omega)}{d\omega}}e^{-t\omega^{2}}d\omega=\mathrm{Tr}\left[\left(1-{\frac{1}{2\lambda}}\partial_{\lambda}H^{2}(\lambda)\right)e^{-tH^{2}(\lambda)}\right]~~~,
{\frac{[(\xi_{1}+i\xi_{3})\kappa_{2}+i\lambda-\xi_{1}(1-\lambda)][\kappa_{1}(1+i\xi_{1}-i\xi_{2})-1]+(\xi\rightarrow-\xi_{1})}{[\kappa_{1}(1+i\xi_{1}-i\xi_{2})-1][\kappa_{2}(1-i\xi_{1}-i\xi_{2})-1]}}.
\begin{aligned}{[m]ds^{2}}&{{}=2dx^{+}dx^{-}+\sum_{i,j}A_{ij}x^{i}x^{j}(dx^{-})^{2}+\sum_{i}dx^{i}dx^{i}}\\{F}&{{}=\mu\,dx^{-}\wedgedx^{1}\wedgedx^{2}\wedgedx^{3}~,}\\\end{aligned}
{}F_{\bar{y}\bar{z}}=\left(\begin{matrix}{0}&{X}\\{0}&{0}\\\end{matrix}\right)\quad,\qquad{}F_{yz}=\left(\begin{matrix}{0}&{0}\\{-X}&{0}\\\end{matrix}\right)\quad,\qquad{}F_{y\bar{y}}+F_{z\bar{z}}=\left(\begin{matrix}{X}&{0}\\{0}&{-X}\\\end{matrix}\right)\,
G_{55}=3\left[\frac{a^{\prime}}{a}\left(\frac{a^{\prime}}{a}+\frac{n^{\prime}}{n}\right)-\frac{b^{2}}{n^{2}}\left(\frac{\dot{a}}{a}\left(\frac{\dot{a}}{a}-\frac{\dot{n}}{n}\right)-\frac{\ddot{a}}{a}\right)\right].
V^{\mu}=\left(-{\frac{\sqrt{\dot{a}^{2}-|F(a)|}}{|F(a)|}},\dot{a},0,0\right);\qquadn^{\mu}=\pm\left(-{\frac{\dot{a}}{|F(a)|}},\sqrt{\dot{a}^{2}-|F(a)|},0,0\right);
{\bf\tilde{\Omega}}_{g}={\bf\Omega}_{g}+\frac{1}{2}\left[\begin{array}{cc}{0}&{I}\\{I}&{0}\\\end{array}\right]=\frac{i\beta^{2}}{2\pi^{2}}\left[\begin{array}{cc}{\tau_{1}\tau_{2}^{-1}\tau_{1}+\tau_{2}}&{-\tau_{1}\tau_{2}^{-1}}\\{-\tau_{2}^{-1}\tau_{1}}&{\tau_{2}^{-1}}\\\end{array}\right]+\frac{1}{2}\left[\begin{array}{cc}{0}&{I}\\{I}&{0}\\\end{array}\right]
T_{+p\pm2}^{+q}=\frac12\gamma_{q\dot{q}}^{i}\Omega_{+p}^{+2i}\psi_{\pm2\dot{q}}^{1-},\quadT_{-p\pm2}^{+q}=\frac12\gamma_{q\dot{q}}^{i}\Omega_{-p}^{+2i}\psi_{\pm2\dot{q}}^{1-};
R(\theta)=\left(\begin{matrix}{R_{12}(\theta_{1})}&{}&{}&{}&{}\\{}&{R_{34}(\theta_{2})}&{}&{}&{}\\{}&{}&{R_{56}(\theta_{3})}&{}&{}\\{}&{}&{}&{R_{78}(\theta_{4})}&{}\\{}&{}&{}&{}&{R_{9\natural}(\theta_{5})}\\\end{matrix}\right)~,
D_{++}^{+}(x)=\frac{2}{(2{\pi})^{3}}\intd^{3}k_{+}\frac{k_{+}}{k_{\bot}^{2}}{\theta}(k_{+})\{{\theta}(x^{+})(\mathrm{e}^{-ik\cdotx}-\mathrm{e}^{-ik\cdotx}|_{x^{-}=0})+{\theta}(-x^{+})(\mathrm{e}^{ik\cdotx}-\mathrm{e}^{ik\cdotx}|_{x^{-}=0})\}
\rho^{2}(x-vt)=\rho_{0}^{2}\left[1-\left(1-{\frac{mv}{\lambda\hbar\rho_{0}^{2}}}\right)\mathrm{sech}^{2}\left(\sqrt{{\frac{mv\lambda\rho_{0}^{2}}{\hbar}}\left(1-{\frac{mv}{\hbar\lambda\rho_{0}^{2}}}\right)}(x-vt)\right)\right].
e^{-B}\eta^{\rho\sigma}\partial_{\rho}\partial_{\sigma}h_{\mu\nu}+e^{-{\frac{p+1}{2}}B}{\frac{1}{\sqrt{g}}}\partial_{i}\left[e^{{\frac{p+1}{2}}B}\sqrt{g}g^{ij}\partial_{j}h_{\mu\nu}\right]=0,
\Bigl[\theta_{3}^{2}\theta_{3}^{\prime}\theta_{3}^{\prime}-\theta_{4}^{2}\theta_{4}^{\prime}\theta_{4}^{\prime}-\theta_{2}^{2}\theta_{2}^{\prime}\theta_{2}^{\prime}\Bigr]/\eta^{12}=-{\frac{\pi^{2}}{4}}\.
\frac{1}{\frac{1}{2}(1-x)(({p_{2}}_{0}-{p_{1}}_{0})^{2}+({p_{4}}_{0}-{p_{3}}_{0})^{2}))-x({p_{2}}_{0}-{p_{1}}_{0})({p_{4}}_{0}-{p_{3}}_{0})-{\vec{q}}^{2}}
T_{a,b}^{(s,s)}\left(\theta\right)=L_{a,a_{1}}^{(s,s)}\left(\theta\right)\otimesL_{a_{1},a_{2}}^{(s,s)}\left(\theta\right)\otimes\cdots\otimesL_{a_{N-1},b}^{(s,s)}\left(\theta\right),
\hat{\hat{S}}=\intd^{11}x\,sqrt{|\hat{\hat{g}}|}\\left[\hat{\hat{R}}-{\textstyle\frac{1}{2\cdot4!}}\hat{\hat{G}}{}^{2}-{\textstyle\frac{1}{6^{4}}}\frac{1}{\sqrt{|\hat{\hat{g}}}|}\hat{\hat{\epsilon}}\partial\hat{\hat{C}}\partial\hat{\hat{C}}\hat{\hat{C}}\right]\,.
\left[\left({\frac{1}{\kappa}}+{\frac{p^{2}}{\omega^{2}}}\right)I_{\mu\nu\alpha\beta}+\left(-{\frac{1}{\kappa^{2}}}+{\frac{p^{2}}{\eta^{2}}}+{\frac{p^{4}}{\gamma^{4}}}\right)P_{\mu\nu\alpha\beta}\right]\tilde{B}_{\alpha\beta}(p)=-i\tilde{\mathcal{J}}_{\mu\nu}(p).
\begin{array}{rcl}{F_{\sigma}^{(n)}(\zeta^{\prime\prime},\zeta_{2n-1},sx^{-1}\zeta_{2n-1})^{k\cdotsk^{\prime}k^{\prime}\pm1k_{2n}}}&{=}&{\delta_{k^{\prime},k_{2n}}\frac{d_{s}}{\{k_{2n}\}}F_{\sigma}^{(n-1)}(\zeta^{\prime\prime})^{k\cdotsk^{\prime}}\;\;\;\;(s=\pm),}\\\end{array}
{\calL}=-\frac{1}{2}\varepsilon^{\mu\nu\alpha\beta}[3\varepsilon_{abcd}f_{\mu}{}^{a}f_{\nu}{}^{b}\delta_{\alpha}{}^{c}\delta_{\beta}{}^{d}-f_{\mu}{}^{a}T_{\nu\alpha\beta}{}^{a}]
I=\frac{1}{2}\intd^{2}x\sqrt{-g}e^{-2\phi}\Big(R+4(D\phi)^{2}+4\Lambda-(D\psi)^{2}-\frac{1}{4}(D\psi_{1})^{2}-\frac{1}{4}e^{-\psi_{1}}F_{2}^{2}-\frac{1}{4}e^{\psi_{1}}F^{2}\Big)
\tilde{\Gamma}_{\mu}^{5}(p+q,p)=\intd^{4}x\,d^{4}y\,e^{-iqx-ipy}\left<0\right|\mathrm{T}\left(A_{\mu}^{5}(x)\psi(y)\bar{\psi}(0)\right)\left|0\right>_{\mathrm{amp}},
\begin{array}{cccccccc}{I_{0}}\\{U_{14}I_{0}}&{U_{24}I_{0}}&{U_{34}I_{0}}\\{U_{14}U_{24}I_{0}}&{U_{14}U_{34}I_{0}}&{U_{24}U_{34}I_{0}}\\{U_{4.}U_{14}U_{24}I_{0}}&{U_{4.}U_{14}U_{34}I_{0}}&{U_{4.}U_{24}U_{34}I_{0}}&{U_{14}U_{24}U_{34}I_{0}}\\{U_{34}U_{4.}U_{14}U_{24}I_{0}}&{U_{24}U_{4.}U_{14}U_{34}I_{0}}&{U_{14}U_{4.}U_{24}U_{34}I_{0}}&{U_{4.}U_{14}U_{24}U_{34}I_{0}}\\\end{array}
i\hbar\partial_{t}\psi=\frac{\hbar^{2}}{2m}\Bigl\{\Bigl(\frac{1}{i}\partial_{x_{1}}\pm\varkappa^{2}\delta(x_{1}-x_{2})\Bigr)^{2}+\Bigl(\frac{1}{i}\partial_{x_{2}}\pm\varkappa^{2}\delta(x_{2}-x_{1})\Bigr)^{2}\Bigr\}\psi.
\langle\prod_{i=1}^{k}{\calO}_{i}\rangle=\int{\calD}\phi{\calD}\psi{\calD}\chie^{-L}\prod_{i=1}^{k}{\calO}_{i}=\sum_{n=0}^{\infty}e^{-nt}\int_{B_{n}}{\calD}\phi{\calD}\psi{\calD}\chie^{-it\int_{\Sigma}d^{2}z\\{Q,V\}}\prod_{i=1}^{k}{\calO}_{i}
mm^{\dagger}=\rho^{2}\left(\begin{array}{ccc}{x^{2}}&{xy}&{xz}\\{yx}&{y^{2}}&{yz}\\{zx}&{zy}&{z^{2}}\\\end{array}\right)=\rho^{2}\left(\begin{array}{c}{x}\\{y}\\{z}\\\end{array}\right)(x,y,z),
\nabla_{a}\,\nabla^{2}\,=\,i\,{\nabla_{a}}^{b}\,\nabla_{b}\,\pm\,i\,W_{a}\;\;\;\mathrm{e}\;\;\;(\nabla^{2})^{2}\,=\,\Box\,\mp\,i\,W^{a}\,\nabla_{a}\,,
\frac{F^{2}\psi^{4}X^{l}}{r^{9+l}},\;\;\;\;\;\frac{F\psi^{6}X^{l}}{r^{10+l}},\;\;\;\;\;\frac{\psi^{8}X^{l}}{r^{11+l}},\;\;\;\;\;\frac{(\psiD\psi)F\psi^{2}X^{l}}{r^{8+l}},\;\;\;\;\;\frac{(\psiD\psi)\psi^{4}X^{l}}{r^{9+l}}
I=\int_{Y}d^{4}y\sqrtg[e_{a}^{\mu}e_{b}^{\nu}(R_{1\mu\nu}^{ab}+R_{2\mu\nu}^{ab})+g^{\mu\nu}g^{\alpha\beta}e_{\mu}^{a}\frac{V_{\nu}V_{\alpha}}{\sqrt{V^{2}}}(H_{\beta}^{a5}+\tilde{H}_{\beta}^{a5}-H_{\beta}^{5a}-\tilde{H}_{\beta}^{5a})]
E_{\alpha}=(-1)^{\alpha_{1}}\sigma_{\alpha}(0)\sum_{j}\prod_{k\neqj}\frac{\sigma_{\alpha}(z_{jk})}{\sigma_{0}(z_{jk})}\left[\frac{l}{n}\sum_{i\neqj}\frac{\sigma_{\alpha}^{\prime}(z_{ji})}{\sigma_{\alpha}(z_{ji})}-\partial_{j}\right],
{\tilde{A}}_{2}(p)+\tilde{C}(p)=2\gamma\,g^{2}\delta_{a}^{b}\delta_{J}^{I}\left[\Phi_{I}^{a\dagger}(p,\theta,{\overline{\theta}}){\Phi}_{b}^{J}(-p,\theta,{\overline{\theta}})\right]\left[I_{1}(p)+I_{2}(p)\right]\,,
\epsilon({\bf{q}},\omega)=\frac{U_{ext}({\bf{q}},t)}{U_{eff}({\bf{q}},t)}=1+\frac{v_{{\bf{q}}}}{V}\sum_{{\bf{k}}}\frac{n_{F}({\bf{k+q/2}})-n_{F}({\bf{k-q/2}})}{\omega-\frac{{\bf{k.q}}}{m}}
a_{1}=x+v,\,\,\,\,\,\,\,\,\,\,a_{2}=2\,x-4\,v,\,\,\,\,\,\,\,\,\,\,a_{3}=-\frac{1}{3}\,x+v+\frac{y}{9},\,\,\,\,\,\,\,\,\,\,a_{4}=z\zeta.
L={\bar{\psi}}_{\alpha}^{j}\gamma^{\mu}i\partial_{\mu}\psi_{\alpha}^{j}+\eta\psi_{+\;\alpha}^{\daggerj}\psi_{-\;\alpha}^{j}+{\bar{\eta}}\psi_{-\;\alpha}^{\daggerj}\psi_{+\;\alpha}^{j}\quad,
{\bf16}\rightarrow({\bf2},{\bf1},{\bf2},{\bf1})_{1}^{s}\oplus({\bf2},{\bf1},{\bf1},{\bf2})_{1}^{a}\oplus({\bf1},{\bf2},{\bf2},{\bf1})_{-1}^{s}\oplus({\bf1},{\bf2},{\bf1},{\bf2})_{-1}^{a}
D=(D\mu)\frac{\partial}{\partial\mu}+(D\widetilde{\lambda}_{i})\frac{\partial}{\partial\widetilde{\lambda}_{i}}+(D\widetilde{\Phi})\frac{\partial}{\partial\widetilde{\Phi}}=\mu\frac{\partial}{\partial\mu}+\widetilde{\beta}_{\lambda_{i}}\frac{\partial}{\partial\widetilde{\lambda}_{i}}-\widetilde{\gamma}\widetilde{\Phi}\frac{\partial}{\partial\widetilde{\Phi}},
V=-\frac{48|A|}{\left[|A|(h+h^{*})+B+B^{*}\right]^{2}\left[h+Ah^{\prime}+h^{*}+A^{*}{h^{\prime}}^{*}\right]}\left|B+\left|A\right|A^{*}{h^{\prime}}^{*}\right|^{2},
{{F_{ij}}^{\alpha}}_{\beta}=\partial_{i}{{A_{j}}^{\alpha}}_{\beta}-\partial_{j}{{A_{i}}^{\alpha}}_{\beta}+{{A_{i}}^{\alpha}}_{\gamma}{{A_{j}}^{\gamma}}_{\beta}-{{A_{j}}^{\alpha}}_{\gamma}{{A_{i}}^{\gamma}}_{\beta}\quad.
\begin{array}{rclrcl}{Solv_{3}}&{\equiv}&{Solv\left(SO^{\star}(12)/U(6)\right)}&{Solv_{4}}&{\equiv}&{Solv\left(E_{6(2)}/SU(2)\timesSU(6)\right)}\\{\mathrm{rank~}\,Solv_{3}}&{=}&{3}&{\mathrm{rank~}\,Solv_{4}}&{=}&{4}\\{\mathrm{dim~}\,Solv_{3}}&{=}&{30}&{\mathrm{dim~}\,Solv_{4}}&{=}&{40}\\\end{array}
\mathrm{Trace}\;(\gamma_{A}\gamma^{\overline{{B}}})=\mathrm{Trace(I)}\;\delta_{A}{}^{B},\;\;\;\sum_{A}(\gamma_{A})_{\alpha\beta}(\gamma^{\overline{{A}}})_{\gamma\delta}=\mathrm{Trace(I)}\;\delta_{\alpha\gamma}\;\delta_{\beta\delta}