harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\phi^{\mu_{0}\mu_{1}\ldots\mu_{p}}(X)\rightarrow\phi^{\mu_{0}\mu_{1}\ldots\mu_{p}}(X)+\sum_{i=0}^{p}(-1)^{p\cdoti}\frac{\partial}{\partialX_{\mu_{i}}}\Lambda^{\mu_{i+1}\mu_{i+2}\ldots\mu_{p}\mu_{0}\mu_{1}\ldots\mu_{i-1}},
(2\pi)^{3}{\hat{G}}({\hat{q}},{\hat{q}}^{\prime})=D^{-1}({\hat{q}}\intd{\hat{q}}^{\prime\prime}K({\hat{q}},{\hat{q}}^{\prime\prime}){\hat{G}}({\hat{q}}^{\prime\prime},{\hat{q}}^{\prime})
S_{0}=-N\intd\tilde{t}\,STr\left(-\frac{1}{2}\partial_{\tilde{t}}\tilde{\Phi}\partial_{\tilde{t}}\tilde{\Phi}+\frac{1}{4}[\tilde{\Phi},\tilde{\Phi}]^{2}+{\frac{1}{2}}\tilde{\Theta}\dot{\tilde{\Theta}}-{\frac{1}{2}}\tilde{\Theta}\gamma^{i}[\tilde{\Phi}_{i},\tilde{\Theta}]\right)
d\Omega_{(p)\,\alpha_{1}\dots\alpha_{p-1}}^{\prime}=\Pi_{(p-1)\,\alpha_{1}\dots\alpha_{p-1}}^{\prime}\quad;\quad\Pi_{(p)}^{\prime}=\Pi_{(p)\,\alpha_{1}\dots\alpha_{p}}^{\prime}F^{\alpha_{1}}\wedge\dots\wedgeF^{\alpha_{p}}\quad.
4({\frac{15}{64}})^{2}\left[v_{4}^{4}v_{3}^{4}(\nabla_{4}^{i}\nabla_{4}^{j}){\frac{1}{\vert\vec{x}_{4}\vert^{7}}}(\nabla_{3}^{i}\nabla_{3}^{j}){\frac{1}{\vert\vec{x}_{3}\vert^{7}}}\right]{\frac{1}{\vert\vec{x}_{12}\vert^{3}}}
{w}_{\beta}^{~\alpha}=-2d\theta^{\alpha}D_{\beta}\Phi-2d\theta_{\beta}D^{\alpha}\Phi+\Pi^{\gamma\dot{\gamma}}w_{\gamma\dot{\gamma}\beta}^{~~~~\alpha}~,\qquadw_{\dot{\beta}}^{~\dot{\alpha}}=(w_{\beta}^{~\alpha})^{*}~.
\left(\begin{matrix}{\cdot}&{\cdot}&{\cdot}&{\cdot}&{\cdot}&{\\\cdot\\}\\{\cdot}&{\cdot}&{0.0291884}&{0.0321865}&{0.0326601}&{}\\{\cdot}&{0.0271003}&{0.0489937^{*}}&{0.0327393}&{}&{}\\{0.0323937}&{0.0310551}&{0.031359}&{}&{}&{}\\{0.0321239}&{0.0312882}&{}&{}&{}&{}\\{0.0312289}&{}&{}&{}&{}&{}\\\end{matrix}\right),
\langle\,(s_{r}B_{x}^{a})\,\bar{c}_{y}^{b}\,\rangle\,=\,\langle\,s_{r}(B_{x}^{a}\,\bar{c}_{y}^{b})\,\rangle\,-\,\langle\,B_{x}^{a}\,(s_{r}\bar{c}_{y}^{b})\,\rangle\,=\,-\langle\,B_{x}^{a}\,B_{y}^{b}\,\rangle\,+\,\widetildeZ_{1}\,(1-\frac\alpha2)\,\langle\,B_{x}^{a}\,(\bar{c}\timesc)_{y}^{b}\,\rangle\;,
\Omega_{a}^{\hat{\alpha}\hat{\beta}}=\left(\hat{e}_{\mu}^{\hat{\alpha}}\,\partial_{\nu}k_{a}^{\mu}+k_{a}^{\mu}\partial_{\mu}\hat{e}_{\nu}^{\hat{\alpha}}\right)e_{\hat{\lambda}}^{\nu}\eta^{\hat{\lambda}\hat{\beta}}
|\hat{D2}\rangle=\frac{T_{1}R_{9}}{2\sqrt{V\alpha^{\prime}}}|B2\rangle_{\mathrm{NS-NS}}^{U}+\frac{T_{1}}{2\sqrt{2}\pi^{2}\alpha^{\prime}}\left(|T_{1}\rangle_{\mathrm{R-R}}+|T_{2}\rangle_{\mathrm{R-R}}\right).
\Gamma_{\muab}(p,q)=\gamma_{\muab}+\int\frac{d^{3}\ell}{(2\pi)^{3}}\left[S(p+\ell)\Gamma_{\mu}(p+\ell,q+\ell)S(q+\ell)\right]_{dc}K_{cd,ba}(q+\ell,p+\ell,\ell),
A_{0}\left(\overrightarrow{\eta}(\overrightarrow{X},t),t\right)=\hat{A}_{0}(\overrightarrow{X},t)-\frac{e\theta}{2(1+e\theta\widehat{B})}\widehat{A}_{l}(\overrightarrow{X},t)\varepsilon_{kj}\partial_{t}\widehat{A}_{j}(\overrightarrow{X},t)e_{kl}(\overrightarrow{X},t)\,,
<A^{\mu}(x)A^{\nu}(y)>_{MCS}=i({\frac{a}{a^{2}\partial^{2}+4\alpha^{2}}}g^{\mu\nu}\delta(x-y)+{\frac{2\alpha}{\partial^{2}(a^{2}\partial^{2}+4\alpha^{2})}}\epsilon^{\mu\nu\lambda}\partial_{\lambda}\delta(x-y)),
S_{0i}=\intd^{3}x\:\left[\varepsilon_{ijk}\left(F_{0j}+\tilde{F}_{0j}\right)\psi_{k}+\etaD_{i}\phi-\chi_{i}D_{0}\phi-\left(F_{0i}+\tilde{F}_{0i}\right)\psi_{0}+\varepsilon_{ijk}\chi_{j}D_{k}\phi-{\frac{1}{2}}\left[\lambda,\phi\right]\psi_{i}\right].
\begin{array}{rcl}{\nabla_{1}F_{01}}&{=}&{-2\lambda^{1/2}/R^{7}\left[Q(r^{3}-3rN^{2})-P(3r^{2}N-N^{3})\right]\,,}\\{}&{}&{}\\{\nabla_{1}{}^{\star}F_{01}}&{=}&{-2\lambda^{1/2}/R^{7}\left[P(r^{3}-3rN^{2})+Q(3r^{2}N-N^{3})\right]\,,}\\\end{array}
\bar{\xi}_{1}^{(i)}(x)\equivT^{-1}(x)\mathrm{exp(}\frac{1}{2}\Phi)|\lambda_{max}^{(i)}\rangle,{}~~~~\bar{\xi}_{2}^{(i)}(x)\equivT^{-1}(x)\mathrm{exp(}\frac{1}{2}\Phi)\mathrm{exp(}\Psi_{-})F_{i}|\lambda_{max}^{(i)}\rangle
{\calL}=-\frac{1}{4}\tilde{F}_{\mu\nu}\tilde{F}_{\mu\nu}+\partial_{\mu}\psi\partial_{\mu}\psi+q^{2}\psi^{2}\tilde{A}_{\mu}\tilde{A}_{\mu}-\mu^{2}\psi^{2}-\frac{\lambda}{4}\psi^{4}
\begin{array}{cc}{T_{n}^{\dagger}=T_{-n}}&{}\\{\}&{\}\\{(G_{s}^{+})^{\dagger}=G_{-s}^{-};}&{(G_{s}^{+K})^{\dagger}=G_{-s}^{-K}}\\{\}&{\}\\{(A_{n}^{+i})^{\dagger}=A_{-n}^{+i};}&{(A_{m}^{-i})^{\dagger}=A_{-m}^{-i}}\\{\}&{\}\\{W_{n}^{\dagger}=W_{-n}}&{}\\{\}&{\}\\{(Q_{r}^{+})^{\dagger}=-Q_{-r}^{-};}&{(Q_{r}^{+K})^{\dagger}=-Q_{-r}^{-K}}\\\end{array}
\sum_{k=0}^{r-1}-(2\pi)^{\,4}\intd^{\,4}q^{\prime}\\tilde{C}_{\alpha}^{\,\alpha_{0}}(q)\tilde{C}_{\alpha}^{\,\alpha_{0}}(q^{\prime})\\hat{\Gamma}_{r-k}^{\,\alpha,\,\alpha_{0}}(q,-q^{\prime},\Phi_{c})\\delta_{\Phi_{c}(p)}\delta_{\Phi_{c}(q^{\prime})}\,\Gamma_{k}^{\,\alpha,\,\alpha_{0}}(\Phi_{c})
{\calH}_{1}=\kappa_{0}^{2}(F_{ij}^{\alpha})^{2}+\frac{9\lambda_{1}^{2}}{2}(\frac{\kappa_{2}}{\kappa_{0}})^{2}(\phi^{4})^{2}(D_{i}\phi^{\alpha})^{2}+\frac{\lambda_{1}^{2}}{2}(D_{i}\phi^{a})^{2}+\frac{1}{2}\kappa_{2}^{2}(D_{[i}\phi^{a}D_{j]}\phi^{b})^{2}
h(X,\Pi,\psi)=\frac{1}{2}\Pi_{a}^{i}\Pi_{a}^{i}+\frac{1}{4}f_{abc}X_{i}^{b}X_{j}^{c}f_{ade}X_{i}^{d}X_{j}^{e}-\frac{i}{2}f_{abc}X_{a}^{j}\psi_{\alpha}^{b}\sigma_{j}^{\alpha\beta}\psi_{\beta}^{c}
\frac{\partial^{2}}{\partialx_{2}^{2}}V(x_{2})\;=\;\phi_{0}(x_{2})\phi_{0}^{\dagger}(x_{2})\int\frac{d^{2}k}{2\pi}\frac{-i\;k_{0}+ie\;V_{0,0}}{(-i\;k_{0}+ie\gamma_{0}V_{0,0})^{2}}\;.
\frac{1}{4}\sum_{a}\intd^{2}\thetaf_{a}(\phi)(W^{\alpha}W_{\alpha})_{a}+h.c=-\frac{1}{4}\{\sum_{a}(Ref)_{a}(F_{\mu\nu}F^{\mu\nu})_{a}-Imf_{a}(F{\tilde{F}})_{a}\},
S_{\sigma}=-\frac{k}{4\pi}\int_{M}d\taud\sigma\;\eta^{\alpha\beta}\mathrm{Tr}[g^{-1}\partial_{\alpha}g\;g^{-1}\partial_{\beta}g]-\frac{k}{6\pi}\int_{B}\mathrm{Tr}[g^{-1}dg\wedgeg^{-1}dg\wedgeg^{-1}dg].
a_{2,3}=ra_{2,3}^{\mathrm{\tiny{scal}}}-{\frac{r}{3!}}\int_{\Sigma_{r}}\bar{h}^{1/2}d^{3}x\left[{\frac{1}{12}}(\bar{R}+12Bm^{2})\bar{F}^{ij}\bar{F}_{ij}-\frac14\bar{F}_{ij}\bar{\nabla}^{2}\bar{F}^{ij}\right]~~,
{{\calL}^{1}}^{\mathrm{eff}}(H,\mu,T)=\frac{eH}{2\pi^{2}}\sum_{n=0}^{\infty}b_{n}\int_{-\infty}^{\infty}d\!p_{z}{}~\frac{p_{z}^{2}}{\varepsilon_{n}(p_{z})}~(f_{+}(T)+f_{-}(T)).
\varrho_{1}={\frac{\tilde{\varrho}_{1}\Omega_{R}}{\sqrt{\Omega_{R}^{2}-\dot{R}^{2}/c^{2}}}},\\\\\\\\\\\wp_{1}={\frac{\tilde{\wp}_{1}\sqrt{\Omega_{R}^{2}-\dot{R}^{2}/c^{2}}}{\Omega_{R}}}.
V_{ef}\left(\phi_{c}\right)=\frac{2\alpha^{2}}{4!}\phi_{c}^{4}+\frac{\hbar}{24\pia}\alpha^{3}\phi_{c}^{3}-\frac{\hbar}{16\pi^{2}a^{4}}\sum_{N=1}^{m}\left(\frac{1}{N}\right)^{4}+\frac{\hbar}{16\pi^{2}a^{2}}\alpha^{2}\phi_{c}^{2}\sum_{N=1}^{m}\left(\frac{1}{N}\right)^{2}.
\hat{h}_{\mu\nu}(x,z)=h_{\mu\nu}^{(m)}+h_{\mu\nu}^{(f)}+\frac{1}{k}f_{,\mu\nu}+2ke^{-2k|z|}\eta_{\mu\nu}f-e^{-2k|z|}\eta_{\rho(\nu}F^{\rho}{}_{,\nu)},
Z_{(r,s)}(q)=\sum_{(r^{\prime},s^{\prime}),(r^{\prime\prime},s^{\prime\prime})}N_{r^{\prime}r^{\prime\prime}}^{(A_{g-2})}{}^{r}N_{s^{\prime}s^{\prime\prime}}^{(A_{g-1})}{}^{s}\chi_{r^{\prime},s^{\prime}}(q){\chi_{r^{\prime\prime},s^{\prime\prime}}(q)}^{*}
\left(\frac{m}{e^{2}}\right)_{\mathrm{ren}}=\left(\frac{m}{e^{2}}\right)Z_{m}(0)(\tilde{Z}(0))^{2}=\left(\frac{m}{e^{2}}\right)\left\{1+(\frac{7}{12\pi}-\frac{1}{3\pi})N\frac{e^{2}}{m}\right\}=\left(\frac{m}{e^{2}}\right)+\frac{N}{4\pi}
V_{eff}=tr\\sum_{n,m}\sum_{l=0}^{[m/2]}\frac{1}{4}(_{2l}^{m})\frac{\Gamma(n-l-1/2)}{\Gamma(1/2-l)}\alpha^{2(n-l)-1}K[_{n}^{m}](W)\partial^{m-2l}\frac{1}{(\alpha^{2}W)^{n-l-1/2}}
5[2\pii(\overline{{\omega(\{q_{N_{s}}\},L^{\prime})}}-\omega(\{q_{N_{s}}\},L))^{-1}]_{mn}\sum_{p=a,b}\int_{C_{p}^{(r)}}2\piiD(t)\eta_{m}(t;L)2\pii\eta_{n}(t;L)dtY_{p,N_{r}}(t)
\left[\frac{\Gamma(\frac{n}{2}-1)}{4\pi^{\frac{n}{2}}}\right]^{3}\int_{C}d^{n}\eta^{\prime}|\eta-\eta^{\prime}|^{3(2-n)}=\sum_{N=0}^{\infty}\sum_{\alpha=0}^{N}\frac{(2a)^{6-2n}{\pi}^{\frac{n}{2}}\Gamma(3-n)\Gamma(N-3+\frac{3n}{2})}{\Gamma(N+3-\frac{n}{2})\Gamma(\frac{3n}{2}-3)}Y_{\alpha}^{N}({\eta})\int_{C}d^{n}\eta^{\prime}Y_{\alpha}^{N}(\eta^{\prime})
1\gg\left|{\frac{S_{1}}{S_{0}}}\right|=\left|{\frac{S_{1}^{\prime}}{S_{0}^{\prime}}}\right|=\left|S_{1}^{\prime}\right|\left|\left({\frac{\Omega+S_{0}^{2}}{\mu^{2}+\phi^{2}}}\right)^{1/2}\right|
A_{uv}\!\left(p+\frac{q}{2},\:p-\frac{q}{2}\right)\;=\;\frac{\epsilon(q^{2})}{2\pi^{2}}\:\int_{-\infty}^{\infty}(S_{z}^{\Join}+\Theta(z)\:H_{z})(p+\tauq)\;d\tau
d{}^{*}d\phi+\frac{1}{2}\mathrm{e}^{-\phi}H_{3}\wedge{}^{*}H_{3}-\mathrm{e}^{2\phi}F_{1}\wedge{}^{*}F_{1}-\frac{1}{2}{\tilde{F}}_{3}\wedge{}^{*}{\tilde{F}}_{3}+2\kappa^{2}\frac{\delta{\cal{L}}_{b}}{\delta\phi}=0
\lbrackl_{\dot{K}_{j}}^{\dot{K}_{j}},\sum_{i=1}^{m}\beta_{\dot{K}_{i}}^{\dot{L}_{i}}l_{\dot{L}_{i}}^{\dot{K}_{i}}\rbrack=\beta_{\dot{K}_{j}}^{\dot{L}_{j}}l_{\dot{L}_{j}}^{\dot{K}_{j}}-\beta_{\dot{K}_{j}}^{\dot{L}_{j}}\sum_{\dot{K}_{j1}K_{j2}=\dot{K}_{j}}\delta_{\dot{L}_{j}}^{\dot{K}_{j1}}l_{\dot{K}_{j}}^{\dot{K}_{j}K_{j2}}+\Gamma,
J^{\mu\nu}=X_{o}^{\mu}Q^{\nu}-X_{o}^{\nu}Q^{\mu}-i\sum_{n=1}^{\infty}(a_{n}^{\dagger\mu}a_{n}^{\nu}-a_{n}^{\dagger\nu}a_{n}^{\mu}+\tilde{a}_{n}^{\dagger\mu}\tilde{a}_{n}^{\nu}-\tilde{a}_{n}^{\dagger\nu}\tilde{a}_{n}^{\mu})\;.
\mathrm{adj~}\mathrm{E}_{r+1(r+1)}\,=\,\mathrm{adj~}\mathrm{E}_{r(r)}\,\oplus\,\mathrm{adj~}\mathrm{O}(1,1)\,\oplus(\mathbb{D}_{r}^{+}\oplus\mathbb{D}_{r}^{-})
\{\bar{q}_{i}^{\dot{\alpha}},\bar{s}_{\dot{\beta}}^{j}\}\Phi=\left[-\delta_{i}^{j}(\sigma^{\mu\nu})^{\dot{\alpha}}{}_{\dot{\beta}}m_{\mu\nu}-2\delta_{\dot{\beta}}^{\dot{\alpha}}\left(2t_{j}^{i}+\delta_{i}^{j}(\ell+r)\right)\right]\Phi=0\;.
c_{m}^{i}(\xi_{1},\xi_{2})=K_{m}^{i~p_{1},\ldots,p_{\alpha}~q_{1},\ldots,q_{\beta}}\,\xi_{1p_{1}},\ldots,\xi_{1p_{\alpha}}\,\xi_{2q_{1}},\ldots,\xi_{2q_{\beta}}\,.
\overline{{{\calJ}^{ab}}}=\overline{{\overline{{\psi_{A}^{a}}}\,g^{AB}\,\psi_{B}^{b}}}=\overline{{\psi_{B}^{b}}}\,\overline{{g^{AB}}}\,\psi_{A}^{a}=\overline{{\psi_{B}^{b}}}\,g^{BA}\,\psi_{A}^{a}={\calJ}^{ba},
k_{2}{\frac{\delta}{\deltak_{2}}}\bar{L}=\Lambda-\Lambda^{2}\left[{\frac{4c_{1}}{D(D-1)}}+{\frac{2c_{2}}{D}}+2c_{3}\right]-12c_{3}\Lambda[(\dot{H}+2H^{2}+{\frac{k_{1}}{a^{2}}})].
T(u_{E})=\left(\begin{array}{c|c}{a(u_{E}^{0})}&{b(u_{E}^{0})\vec{u}_{E}^{\mathrm{T}}}\\\hline{d(u_{E}^{0})\vec{u_{E}}}&{e(u_{E}^{0})I+(\vec{u}_{E}\otimes\vec{u}_{E}^{\mathrm{T}})f(u_{E}^{0})}\\\end{array}\right),
\begin{aligned}{\Phi(\lambda-\mu,z)\Phi(\mu-\lambda,z)}&{{}=\wp(z)-\wp(\lambda-\mu),}\\{\Phi(\lambda-\nu,z)\Phi(\nu-\mu,z)}&{{}=-\Phi^{\prime}(\lambda-\mu,z)+\Phi(\lambda-\mu,z)\eta(\lambda,\nu,\mu),}\\\end{aligned}
\int\frac{d^{n}k}{(2\pi)^{n}}\frac{e^{ik\wedgep}}{(k^{2}+M^{2})^{\lambda}}=\frac{M^{n/2-\lambda}}{2^{\lambda-1}(2\pi)^{n/2}\Gamma[\lambda]}\frac{K_{n/2-\lambda}(\sqrt{-M^{2}p\circp})}{{(\sqrt{-p\circp})}^{n/2-\lambda}},
E_{\mathrm{cas}}[\sigma]=\sum_{\ell}N_{\ell}\left[\sum_{j}\frac{\omega_{\ellj}}{2}+\int_{0}^{\infty}\frac{dk}{2\pi}\omega(k)\frac{d}{dk}\left[\delta_{\ell}(k)\right]_{N}\right]+E_{\mathrm{FD}}^{N}+E_{\mathrm{CT}}
\begin{aligned}{(Z^{(2|1)},Z^{(2|1)},W^{(8)})}&{{}\rightarrow\tilde{p}=\frac{1}{\tilde{m}-1}\,,}\\{(Z^{(2|1)},Z^{(2|1)},W^{(5)})}&{{}\rightarrow\tilde{q}=\frac{1}{\tilde{n}-1}\,,}\\{(Q,Q,Z^{(2|1)})}&{{}\rightarrowq=\mp1\,,}\\\end{aligned}
{\calL}={\frac{k}{2}}\varepsilon^{\alpha\beta\gamma}A_{\alpha}\partial_{\beta}A_{\gamma}+i\Psi^{\ast}(\partial_{t}+iA_{0})\Psi-{\frac{1}{2}}\left|(\nabla-i{\bfA})\Psi\right|^{2}+{\frac{g}{2}}\left|\Psi\right|^{4}\,.
b_{0}^{n}=\frac{1}{k!n!}(\partial_{x^{i_{1}}}\ldots\partial_{x^{i_{n}}})C_{i_{1}^{\prime}\ldotsi_{2k}^{\prime}a_{1}\ldotsa_{p}}^{0}Str\left(\phi^{i_{1}}\ldots\phi^{i_{n}}\ldots\underbrace{\phi^{i_{2k}^{\prime}}\phi^{i_{2k-1}^{\prime}}}\ldots\right).
\intdwd\alphad\beta\left\{\frac{\partial}{\partial\alpha}\left[\frac{(\alpha-\beta)^{2}}{\beta(c_{\alpha}\alpha-\beta)}\bar{f}\right]-\frac{(N-1)(\alpha-\beta)}{\alpha(c_{\alpha}\alpha-\beta)}\bar{f}\right\}V|k\rangle_{1i}\dots|\Psi\rangle_{N}=0\,
V_{\beta\alpha}\equivV^{\mu}\left(\widetilde{\sigma}_{\mu}\right)_{\beta\alpha}\mathrm{~};\mathrm{~}f_{\alpha\beta}\equiv\left(\tilde{\sigma}_{\mu}\right)_{\alpha\beta}f^{\mu}\mathrm{~};\mathrm{~}f^{\mu}=-\fraci2\varepsilon^{\mu\nu\rho}F_{\nu\rho}\mathrm{~},
\Pi_{\mu_{1}\mu_{2}}^{\mathrm{ln}}=-\left(\frac{13}{3}-\xi\right)\frac{g^{2}C_{G}}{\left(4\pi\right)^{2}}\mathrm{ln}\left(T\right)\left(\eta_{\mu_{1}\mu_{2}}k^{2}-k_{\mu_{1}}k_{\mu_{2}}\right).
I_{\mu_{1}\cdots\mu_{n}}(0)\sim\mathcal{C}_{\left[1+\frac{n}{2}\right]}+p^{2}\mathcal{C}_{\left[\frac{n}{2}\right]}+\cdots+\frac{p^{2\left[\frac{n}{2}\right]}}{\left[\frac{n}{2}\right]!}\mathcal{C}_{1}+\mathrm{const.}
H_{pr}(\tau)=\int{d^{3}\sigma\Big[\frac12\Big(\check{\stackrel{o}{\bar{\psi}}}(\tau,\vec{\sigma})\gamma^{\bar{o}}\partial_{r}\check{\stackrel{o}{\psi}}(\tau,\vec{\sigma})}-\partial_{r}\check{\stackrel{o}{\bar{\psi}}}(\tau,\vec{\sigma})\gamma^{\bar{o}}\check{\stackrel{o}{\psi}}(\tau,\vec{\sigma})\Big)+\Big(\vec{\pi}_{\perp}\wedge\vec{B}\Big)_{r}(\tau,\vec{\sigma})\approx0.
R_{\rho\sigma}n_{2}^{\rho}n_{2}^{\sigma}=R_{\rho\sigma}n_{3}^{\rho}n_{3}^{\sigma},\;\;\;\;\;R_{\rho\sigma}n_{2}^{\rho}n_{3}^{\sigma}=-R_{\rho\sigma}n_{3}^{\rho}n_{2}^{\sigma}=0.
P_{\phi}={\frac{4}{|D|^{2}}}(\tilde{\omega}\tilde{R}_{\frac{\pi}{2}})^{8}={\frac{1}{16}}{\frac{\pi^{2}}{2^{2\sqrt{6}}}}{\frac{(\tilde{\omega}\tilde{R}_{\frac{\pi}{2}})^{2\sqrt{6}+4}}{\Gamma(\sqrt{6})^{2}}}.
\left[S_{+}^{\alpha\beta}(x),S_{-}^{\alpha^{\prime}\beta^{\prime}}(y)\right]=\delta(x-y)\left(\delta^{\beta\alpha^{\prime}}J^{\alpha\beta^{\prime}}(x)+\delta^{\alpha\beta^{\prime}}J^{\beta\alpha^{\prime}}(x)+\delta^{\alpha\alpha^{\prime}}J^{\beta\beta^{\prime}}(x)+\delta^{\beta\beta^{\prime}}J^{\alpha\alpha^{\prime}}(x)\right)~~,
\left[(hr^{3}\partial_{r})^{2}+\omega^{2}r^{6}f-{\frac{8hr^{4}r_{\pm}^{4}}{(r^{2}+r_{\pm}^{2})^{2}}}\left(1+{\frac{r_{0}^{2}}{r_{\pm}^{2}}}\right)\right]\tilde{\phi}_{\pm}=0,
\Pi_{\vec{k}}\rightarrow\widetilde\Pi_{\vec{k}}=k\Psi_{\vec{k}}\,,~~~~~\Psi_{\vec{k}}\rightarrow\widetilde{\Psi}_{\vec{k}}=-k^{-1}\Pi_{\vec{k}}\,,~~~~~S\rightarrow\widetildeS=S^{-1}
\alpha=\sum_{s=1}^{m}\sum_{p+q=s-1}{\frac{(-1)^{m-s}(m-p-1)!(m-q-1)!}{(2m-s)!(m-s)!p!q!}}{\mathrm{sTr}}(\omegaF_{L}^{p}({g^{-1}F_{R}g})^{q}(\omega^{2})^{m-p-q-1})
\begin{array}{c}{^{out}<R_{c}(\beta_{1}^{\prime})\bar{R}_{\underline{d}}(\beta_{2}^{\prime})|R_{a}(\beta_{1})\bar{R}_{\underline{b}}(\beta_{2})>^{in}=\delta(\beta_{1}^{\prime}-\beta_{1})\delta(\beta_{2}^{\prime}-\beta_{2})F_{ab}^{cd}(\beta)}\\{-\delta(\beta_{1}^{\prime}-\beta_{2})\delta(\beta_{2}^{\prime}-\beta_{1})B_{ab}^{dc}(\beta),}\\\end{array}
F_{L}^{\ell}(q)=\sum_{\{\nu\}}\prod_{a=1}^{n-1}\prod_{j=1}^{\ell}q^{{\frac{1}{2}}({\bfh}|{\bfh})}\biggl[\begin{array}{c}{h_{j}^{(a)}+\sum_{b}\sum_{k}G_{ab}^{-1}C_{jk}h_{k}^{(b)}}\\{h_{j}^{(a)}}\\\end{array}\biggr].
f^{(1)}=\left(\begin{array}{cccc}{0}&{-\eta_{ij,kl}}&{0}&{0}\\{\eta_{ij,kl}}&{0}&{\partial_{k,ij}}&{0}\\{0}&{\partial_{i,lk}}&{0}&{\partial_{i}}\\{0}&{0}&{\partial_{k}}&{0}\\\end{array}\right)\,\delta(\vec{x}-\vec{y})
X_{\alpha\beta}(x,y)=e^{2}\left(\begin{array}{cccc}{-1}&{0}&{0}&{0}\\{0}&{-1}&{0}&{0}\\{e^{2}}&{0}&{1}&{0}\\{0}&{e^{2}}&{0}&{-1}\\\end{array}\right)\delta(x-y).
\frac{d^{2}f}{d\rho^{2}}+\left[a+1+\frac{a\rho_{0}^{a}}{(\rho^{a}-\rho_{0}^{a})}+\frac{a(\omegaR)^{a}(4-a)}{2(\rho^{a}+(\omegaR)^{a})}\right]\frac{1}{\rho}\frac{df}{d\rho}-\frac{\rho^{a-2}l(l+a)}{\rho^{a}-\rho_{0}^{a}}f+\frac{\rho^{a}(\rho^{a}+(\omegaR)^{a})}{(\rho^{a}-\rho_{0}^{a})^{2}}f=0\,,
u=gh^{-n}\intd^{n}pH_{0}({\bfp})\left[\frac{\lambdae^{-\betaH_{0}({\bfp})}}{1-\lambdae^{-\betaH_{0}({\bfp})}}-(q+1)\frac{\lambda^{q+1}e^{-\beta(q+1)H_{0}({\bfp})}}{1-\lambda^{q+1}e^{-\beta(q+1)H_{0}({\bfp})}}\right].
ds^{2}=-4dx^{+}dx^{-}-{\frac{\mu^{2}}{R^{4}}}\bigg[\sum_{a=1}^{4}x_{a}^{2}+(1+{\frac{MR^{2}}{r^{4}}})\sum_{b=5}^{7}x_{b}^{2}+(1-3{\frac{MR^{2}}{r^{4}}})x_{8}^{2}\bigg](dx^{+})^{2}+\sum_{i=1}^{8}dx_{i}^{2}
\Omega^{ij}\;=\;\left(\begin{array}{cccc}{k_{1}}&{0}&{0}&{-k_{2}}\\{0}&{0}&{k_{2}}&{0}\\{0}&{k_{2}}&{0}&{0}\\{-k_{2}}&{0}&{0}&{0}\\\end{array}\right)\,,\,\Omega_{ij}\;=\;\left(\begin{array}{cccc}{0}&{0}&{0}&{-q_{2}}\\{0}&{0}&{q_{2}}&{0}\\{0}&{q_{2}}&{0}&{0}\\{-q_{2}}&{0}&{0}&{q_{1}}\\\end{array}\right)\,,
\partial_{\mu}\pi_{i}^{j}\to\partial_{\mu}\pi_{i}^{j}-\left(\begin{array}{cc}{O}&{-m_{1}A_{\mu}+m_{2}\gamma_{5}B_{\mu}}\\{m_{1}A_{\mu}+m_{2}\gamma_{5}B_{\mu}}&{O}\\\end{array}\right)_{i}^{j}
\mathcalL=-\frac{\kappa}{2}\sqrt{|\eta|}\,h_{\mu\nu}(\xi)\,\eta^{ij}\,\partial_{i}\xi^{\mu}\partial_{j}\xi^{\nu}-\frac{\kappa}{2}\,\lambda_{\mu\nu}(\xi)\,\epsilon^{ij}\,\partial_{i}\xi^{\mu}\partial_{j}\xi^{\nu}-\kappa\sqrt{|\eta|}\,V(\xi).
B(t)=\left[\left(\frac{k^{2}B_{0}^{4}}{\epsilon_{0}^{2}b_{0}^{2}}+\dot{b}_{0}^{2}\right)t^{2}+2b_{0}\dot{b}_{0}t+b_{0}^{2}\right]^{1/2}+{\calO}(e^{-t/T_{0}}),
S=\intd^{4}x\sqrt{-\tilde{g}}\left[\tilde{R}-\frac{1}{2}\left(\tilde{\nabla}\Phi\right)^{2}-\frac{1}{2}\left(\tilde{\nabla}y\right)^{2}-\frac{1}{2}Q^{2}e^{2\Phi-\sqrt{3}y}-\frac{1}{2}m^{2}e^{2\Phi+\sqrt{3}y}\right],
D_{n_{1}n_{2}n_{3}}=\prod_{i=1}^{3}\frac{1}{n_{i}!}\left(\frac{\partial}{\partial\bar{z}_{i}^{\prime\prime}}\right)^{n_{i}}G(\bar{z}^{\prime\prime},z^{\prime};0)\Big\vert_{\bar{z}_{1}^{\prime\prime}=\bar{z}_{2}^{\prime\prime}=\bar{z}_{3}^{\prime\prime}=0}
\langle\alpha|\delta\rangle\,\theta_{\delta}(0)^{-1}\langle\prod_{i=1}^{4}p_{i}\!\cdot\!\psi(z_{i})\rangle\langle\epsilon_{1}\!\cdot\!\psi(z_{1})\epsilon_{2}\!\cdot\!\psi(z_{2})\rangle\langle\psi(x_{1})\!\cdot\!\psi(x_{2})\epsilon_{3}\!\cdot\!\psi(z_{3})\epsilon_{4}\!\cdot\!\psi(z_{4})\rangle
\begin{array}{ccc}{\left[B_{a},B_{b}\right]}&{=}&{C_{ab}^{c}B_{c}}\\{}&{}&{}\\{\left[B_{a},Q_{\beta}\right]}&{=}&{C_{a\beta}^{\gamma}Q_{\gamma}}\\{}&{}&{}\\{\left\{Q_{\alpha},Q_{\beta}\right\}}&{=}&{C_{\alpha\beta}^{c}B_{c}}\\\end{array}
\partial_{t}^{2}\overrightarrow{\phi}=[-(\partial_{t}\overrightarrow{\phi})^{2}+(\partial_{x}\overrightarrow{\phi})^{2}+(\partial_{y}\overrightarrow{\phi})^{2}]\overrightarrow{\phi}+\partial_{x}^{2}\overrightarrow{\phi}+\partial_{y}^{2}\overrightarrow{\phi}
\begin{array}{l}{{\calX}^{+k}\vertn,z\rangle_{+k}=q^{2n}z\vertn,z\rangle_{+k}~,}\\{{\calD}_{+k}\vertn,z\rangle_{+k}=q^{1-2n}\lambda^{-1}z^{-1}\vertn-1,z\rangle_{+k}~,}\\{\Lambda_{+k}\vertn,z\rangle_{+k}=\vertn-1,z\rangle_{+k}~.}\\\end{array}
\delta\phi^{i}\left(\frac{\partial}{\partial\phi^{i}}f^{(p)}\right)\left[\psi^{p}\right]_{\mathrm{max}}=\delta\phi^{i}\phi^{i}\left(\frac{d}{\phid\phi}f^{(p)}\right)\left[\psi^{p}\right]_{\mathrm{max}}=-f^{(p+2)}\delta_{\psi}\left[\psi^{p+2}\right]_{\mathrm{max}},
\pi_{k}^{a}(x^{0},{\bfx})\equiv\frac{\deltaL^{\prime}}{\delta\dot{A}_{a}^{k}(x^{0},{\bfx})}=\left(\nabla_{k}(A)A_{0}\left\{{\bfA}\right\}\right)^{a}-\dot{A}_{k}^{a}-\left(\nabla_{k}(A)C\right)^{a},
Z=\sum_{1}^{n-1}(t^{I}q_{I})+t^{n}(t^{1},\dotst^{n-1})q_{n}\qquad\hat{Z}=\sum_{1}^{n-1}(t^{I}q_{I})+\hat{t}^{n}(t^{1},\dotst^{n-1})q_{n}
ds^{2}=\alpha^{\prime}\left[\frac{u^{2}}{R^{2}}[(-\tilde{f}d\tilde{x}_{0}^{2}+d\tilde{x}_{1}^{2})+\tilde{h}(d\tilde{x}_{2}^{2}+d\tilde{x}_{3}^{2})]+\frac{R^{2}}{u^{2}}[f^{-1}du^{2}+u^{2}d\Omega_{5}^{2}]\right],
\delta_{\epsilon}A_{\mu}^{a}=\left(D_{\mu}\right)_{\;\;b}^{a}\epsilon^{b},\;\;\delta_{\epsilon}B_{\mu\nu}=\partial_{\mu}\epsilon_{\nu}-\partial_{\nu}\epsilon_{\mu}\equiv\partial_{\left[\mu\right.}\epsilon_{\left.\nu\right]},
{\bfZ({\bfx})}=\frac{1}{{\bfx}}+\sum_{q\neq0}\{\frac{1}{{\bfx-q}}+\frac{1}{{\bfq}}+\frac{1}{{\bfq}}{\bfx}\frac{1}{{\bfq}}+\frac{1}{{\bfq}}{\bfx}\frac{1}{{\bfq}}{\bfx}\frac{1}{{\bfq}}+\frac{1}{{\bfq}}{\bfx}\frac{1}{{\bfq}}{\bfx}\frac{1}{{\bfq}}{\bfx}\frac{1}{{\bfq}}\},
B(p)=\frac{\alpha}{\pi^{2}N}\frac{1}{p}\int_{0}^{\infty}dk\frac{kB(k)G(p^{2},k^{2})}{k^{2}A(k)^{2}+B(k)^{2}}\{4ln\frac{p+k+\alpha}{|p-k|+\alpha}\}
V=\epsilon_{\mu\nu}J_{\+}^{\mu}(z)U_{+}(z)U_{-}(\bar{z})J_{\-}^{\nu}(\bar{z})=\epsilon_{\mu\nu}J_{\+}^{\mu}(z)e^{2ikX(z,\bar{z})}J_{\-}^{\nu}(\bar{z}),
\frac{\partial^{2}}{\partialx_{2}^{2}}V(x_{2})=\phi_{0}(x_{2})\phi_{0}^{\dagger}(x_{2})\frac{e^{2}\;V_{1,1}}{2}+\phi_{1}(x_{2})\phi_{1}^{\dagger}(x_{2})\frac{e^{2}\;V_{0,0}}{2}.
\Gamma_{\ell}={\frac{\pi(\Gamma(\ell+1))^{2}}{2^{2\ell+2}(\Gamma(\ell+{\frac{3}{2}}))^{2}(\Gamma(2\ell+1))^{2}}}\;k^{2\ell-1}(T_{H}A)^{2\ell+1}e^{-k/(2T_{H})}|\Gamma(\ell+1+ik/(2\piT_{H})|^{2}
H_{\mathrm{mass}}=\left(E+\frac{m_{1}^{2}+m_{2}^{2}}{4E}\right)\left[\begin{array}{cc}{1}&{\phantom{+}0}\\{0}&{\phantom{+}1}\\\end{array}\right]+\frac{\deltam^{2}}{4E}\left[\begin{array}{cc}{-1}&{\phantom{+}0}\\{\phantom{+}0}&{\phantom{+}1}\\\end{array}\right]\,.
\frac{1+g}{1-g}=\lambda_{1}^{2}coth\frac{1}{2}k(r+it)+\lambda_{2}^{2}coth\frac{1}{2}k(r+it-ick^{-1})+\lambda_{3}^{2}coth\frac{1}{2}k(r+it+ick^{-1})
\varepsilon_{c}(t)\equiv\varepsilon(d,T)-\varepsilon(\infty,T)=-4t^{3}\sum_{m=1}^{\infty}\frac{1}{\mu}\mathrm{coth}(t\mu)\mathrm{csch}^{2}(t\mu),\;\;\;t=\pi\frac{T}{T_{c}}.
\left(\begin{matrix}{a}&{b}\\{c}&{d}\\\end{matrix}\right)={\frac{1}{2}}\left(\begin{matrix}{O_{L}+O_{R}}&{O_{L}-O_{R}}\\{O_{L}-O_{R}}&{O_{L}+O_{R}}\\\end{matrix}\right)\\\\mathrm{s.t.}\\\O_{L},O_{R}\inO(d,{\bfZ}).
\nabla_{\mu}\left(-\Box\psi_{(\lambda)}^{\mu}\right)+{\frac{1}{\alpha}}\left[({\mathrm{Tr}}\;K)n_{\sigma}\psi_{(\lambda)}^{\sigma}+n^{\mu}K_{\mu\sigma}\psi_{(\lambda)}^{\sigma}+n^{\mu}n_{\sigma}\nabla_{\mu}\psi_{(\lambda)}^{\sigma}\right]=\lambda\nabla_{\mu}\psi_{(\lambda)}^{\mu}\;,
{\calL}^{(1)}=-\partial_{\mu}\overline{\psi}_{1}\gamma_{\mu\nu}\partial_{\nu}\psi_{1}-\partial_{\mu}\overline{\psi}_{2}\gamma_{\mu\nu}\partial_{\nu}\psi_{2}-m^{2}\overline{\psi}_{1}\psi_{1}+m^{2}\overline{\psi}_{2}\psi_{2}.
\begin{array}{ll}{\eta_{1}\eta_{1}^{\prime}=B\eta_{1}^{\prime}\eta_{1},~~~~~\eta_{1}\eta_{2}^{\prime}=(2-B)\eta_{2}^{\prime}\eta_{1}+(B-2A+AB)\eta_{1}^{\prime}\eta_{2},}\\{\eta_{2}\eta_{2}^{\prime}=B\eta_{2}^{\prime}\eta_{2},~~~~~\eta_{2}\eta_{1}^{\prime}=(2-B)A(2-A)\eta_{1}^{\prime}\eta_{2}-(4-3B-2A+AB)\eta_{2}^{\prime}\eta_{1}.}\\\end{array}
={\frac{1}{4G}}A_{\Sigma}-\int_{\Sigma}\left(8\pic_{1}R+4\pic_{2}R_{\mu\nu}n_{i}^{\mu}n_{i}^{\nu}+8\pic_{3}R_{\mu\nu\lambda\rho}n_{i}^{\mu}n_{i}^{\lambda}n_{j}^{\nu}n_{j}^{\rho}\right)~{}~~
\Psi\in\Gamma(K^{1/2}\otimes_{\mathbbR}X^{*}(T{\mathbbR}^{4}))\oplus\Gamma(K^{-1/2}\otimes_{\mathbbR}X^{*}(T{\mathbbR}^{4}))\\\mathrm{and}\\\partial\!\!\!/\Psi=0\,
S_{*}=\intd^{4}x\sqrt{|\tilde{g}|}\left[\tilde{R}-\frac{1}{2}\left(\tilde{\nabla}\varphi\right)^{2}-\frac{1}{2}\left(\tilde{\nabla}\beta\right)^{2}-\frac{1}{2}e^{2\varphi}\left(\tilde{\nabla}\sigma\right)^{2}-\frac{1}{2}Q^{2}e^{2\varphi-\sqrt{3}\beta}\right]
H={\frac{\nuT}{4}}\intd^{2}\sigma\left(\dot{X}^{i}\dot{X}^{i}+{\frac{2}{\nu^{2}}}\{X^{i},X^{j}\}\{X^{i},X^{j}\}-\frac{2}{\nu}\theta^{T}\gamma_{i}\{X^{i},\theta\}\right)
\dot{x}^{AA^{\prime}}=o^{A}\bar{o}^{A^{\prime}},\,\,\acute{x}^{AA^{\prime}}=o^{A}\bar{\iota}^{A^{\prime}}+\iota^{A}\bar{o}^{A^{\prime}},\,\,(\rho^{\tau}o^{A}\bar{o}^{A^{\prime}})\dot{}={\calF}^{AA^{\prime}},
\{\theta\,,\,\bar{\theta}\}^{\ast}=\frac{i}{s}\zeta\left(1+\frac{1}{2}\frac{z\bar{z}\theta\bar{\theta}}{\zeta}\right)\qquad\{z\,,\,\bar{\theta}\}^{\ast}=\frac{i\zeta}{2s}z\bar{\theta}\qquad\{\bar{z}\,,\,\theta\}^{\ast}=-\frac{i\zeta}{2s}\bar{z}\theta\,.
{\calL}=-\frac{1}{4}F_{ij}^{2}+\frac{1}{2}{\dot{A}_{i}}^{2}+\frac{1}{2}\partial_{i}{\dot{A}_{i}}\frac{1}{\partial^{2}-m^{2}}\partial_{j}{\dot{A}_{j}}-\frac{m^{2}}{2}A_{i}^{2}\nonumber
V_{II}(\beta,\phi)=\mu^{D}\sum_{s=1}^{\infty}f(D,s)g^{s}\phi^{2s}\sqrt{\pi}\biggl(\frac{m}{2\pi\mu}\biggr)^{D-2s}\frac{1}{\Gamma(s-\frac{d}{2})}\biggl(\Gamma(s-\frac{D}{2})+4\sum_{n=1}^{\infty}\biggl(\frac{mn\beta}{2}\biggr)^{s-\frac{D}{2}}K_{\frac{D}{2}-s}(mn\beta)\biggr)
\begin{aligned}{}&{{}U_{2}:}&{}&{{}\qquadx_{2}(t)=0,}&{}&{{}\qquady_{2}(t)=t,}&{}&{{}\qquadu_{2}(t)=0,}&{}&{{}\qquadv_{2}(t)={\frac{\mu}{t}}}\\{}&{{}U_{3}:}&{}&{{}\qquadx_{3}(t^{\prime})=0,}&{}&{{}\qquady_{3}(t^{\prime})=t^{\prime},}&{}&{{}\qquadu_{3}(t^{\prime})=0,}&{}&{{}\qquadv_{3}(t^{\prime})={\frac{\mu}{t^{\prime}}}.}\\\end{aligned}
\{x_{3}^{-},x_{3}^{+}\}_{p/q}={\frac{\left({\frac{q}{p}}\right)^{H_{2}}(k_{3}-k_{3}^{-1})}{(pq)^{1/2}-(pq)^{-1/2}}},\quad\mathrm{where}\quadk_{3}=k_{1}k_{2}
\underline{{{\mathcalS}}}^{\prime}\:=\{\mathcalS}^{\prime}\otimes\Lambda\bigl((s\mathrm{Vir}_{\mathrm{even}})^{*}\bigr)\otimesS\bigl((s\mathrm{Vir}_{\mathrm{odd}})^{*}\bigr)\otimes\Lambda\bigl(\overline{\mathrm{Vir}}^{\,*}\bigr)\.
\left(\begin{array}{cccc}{\left(E-m\right)}&{0}&{-P_{z}}&{-\pi_{-}}\\{0}&{\left(E-m\right)}&{-\pi_{+}}&{P_{z}}\\{-P_{z}}&{-\pi_{-}}&{\left(E-m\right)}&{0}\\{-\pi_{+}}&{P_{z}}&{0}&{\left(E-m\right)}\\\end{array}\right)\left(\begin{array}{c}{\varphi_{+}}\\{\varphi_{-}}\\{\chi_{+}}\\{\chi_{-}}\\\end{array}\right)=0