harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{\bar{c}_{1}\bar{d}_{1}}{\bar{a}_{1}\bar{b}_{1}}+\frac{c_{1}d_{1}}{a_{1}b_{1}}=\frac{\bar{c}_{2}\bar{d}_{2}}{\bar{a}_{2}\bar{b}_{2}}+\frac{c_{2}d_{2}}{a_{2}b_{2}}.
\Gamma^{\mu\nu}=(p^{2}+A)\left(\eta^{\mu\nu}-{\frac{p^{\mu}p^{\nu}}{p^{2}}}-{\frac{\bar{\theta}^{\mu}\bar{\theta}^{\nu}}{\bar{\theta}^{2}}}\right)+(p^{2}+A+B)\,{\frac{\bar{\theta}^{\mu}\bar{\theta}^{\nu}}{\bar{\theta}^{2}}}+{\frac{p^{\mu}p^{\nu}}{\xi}}
h_{\mu\nu}^{(\pm)}(x,y)=h_{\mu\nu}(x,y)+l\xi_{,\,\mu\nu}^{\pm}(x)+\frac{2}{l}\eta_{\mu\nu}a^{2}(y)\,\xi^{\pm}(x)+a^{2}(y)\,\xi_{(\mu,\nu)}^{\pm}(x)
D_{\mu\nu}(k)=-i\frac{1}{k^{2}}\left(g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}\right)-id(k_{\perp}^{2},k_{\parallel}^{2})\frac{k_{\mu}^{\parallel}k_{\nu}^{\parallel}}{k^{2}k_{\parallel}^{2}},
\tilde{L}_{n}=\frac{1}{8}\int_{-\pi}^{\pi}d\sigmae^{\pmin\sigma}(\dot{X}\pmX^{\prime})^{2}=\frac{1}{2}\int_{-\pi}^{\pi}d\sigmae^{\pmin\sigma}(P_{c}\pm\frac{1}{2}X^{\prime})^{2},
{\cal{F}}\left\{\left(P-{\mu}^{2}\right)_{Ad}^{-1}\right\}(x)=\pi{\mu}^{\frac{\nu}{2}-1}Q_{-}^{\frac{1}{2}\left(1-\frac{\nu}{2}\right)}{\cal{I}}_{1-\frac{\nu}{2}}\left(\muQ_{-}^{\frac{1}{2}}\right)
{\frac{\partial{\calV}}{\partialt}}={\frac{\partial}{\partial\phi}}\left({\frac{H^{3/2}}{8\pi^{2}}}{\frac{\partial}{\partial\phi}}\left({H^{3/2}}{\calV}\right)+{\frac{V^{\prime}}{3H}}\,{\calV}\right)+3H{\calV}\.
\phi_{b}^{(r)}(x)=\rho_{bA}^{(r)}(\Sigma^{\prime}(x))\zeta^{A}=\rho_{bc}^{(s)}(h(x))\rho_{cA}^{r}(\Sigma(x))\zeta^{A}=\rho_{bc}^{(s)}(h(x))\phi_{c}^{(r)}(x).
f^{*}\Omega=\Omega_{\alpha\beta\gamma}\frac{\partialy^{\alpha}}{\partial\xi^{i}}\frac{\partialy^{\beta}}{\partial\xi^{j}}\frac{\partialy^{\gamma}}{\partial\xi^{k}}\,d\xi^{i}\wedged\xi^{j}\wedged\xi^{k}\equiv6\phi\,d^{3}\xi~,
I_{1}=\int_{T_{\epsilon}^{2}}d^{2}z\partial_{\bar{z}}\overline{{\rho}}\partial_{z}\rho=\frac{1}{2i}\int_{T_{\epsilon}^{2}}\bar{\partial}\overline{{\rho}}\wedge\partial\rho=\frac{1}{2i}\int_{|z|=\epsilon}dz\overline{{\rho}}\partial_{z}\rho,
S_{\mathrm{KKh}}=-T_{\mathrm{KKh}}\intd^{6}\xi\e^{-2{\phi}}k\sqrt{|\mathrm{det}(\\partial_{i}X^{\mu}\partial_{j}{X}^{\nu}g_{\mu\nu}-k^{-2}F_{i}^{\prime}F_{j}^{\prime})|}+\mathrm{WZ}\,.
ds_{11}^{2}=e^{\frac{1}{2\sqrt{3}}\sigma-\frac{1}{6}\phi}(e^{-\frac{2}{\sqrt{3}}\sigma}g_{\mu\nu}dx^{\mu}dx^{\nu}+g_{mn}^{c}dy^{m}dy^{n})+e^{\frac{4}{3}\phi}dz^{2}\quad.
{\calL}_{St}=-{\frac{1}{4}}A_{\mu\nu}A^{\mu\nu}+{\frac{\mu}{4}}\epsilon_{\mu\nu\lambda}A^{\mu\nu}A^{\lambda}+{\frac{m^{2}}{2}}(A_{\mu}-\partial_{\mu}\theta)(A^{\mu}-\partial^{\mu}\theta).
\Phi_{m_{1},n_{1}}\times\Phi_{m_{2},n_{2}}=\sum_{m=|m_{1}-m_{2}|+1}^{m_{1}+m_{2}-1}\,\,\sum_{n=|n_{1}-n_{2}|+1}^{n_{1}+n_{2}-1}\,\,\Phi_{m,n}\,,
f_{B}(y)=\left(\frac{\mu_{1}}{\mu}\right)^{2}f_{PV}(y)=\sum_{n,n_{1}}\frac{Zny(1-y)^{(2n+2n_{1}-1)}}{(2n+2n_{1}-1)!n!n_{1}!}\frac{(g/\mu)^{2n}(g/\mu_{1})^{2n_{1}}}{(16\pi^{2})^{n+n_{1}}}\,,
{\calV}=(U^{\Lambda\Sigma}-3\bar{L}^{\Lambda}L^{\Sigma})P_{\Lambda}^{x}P_{\Sigma}^{x}+\frac13\bar{L}^{\Lambda}L^{\Sigma}h^{uv}\nabla_{u}P_{\Lambda}^{x}\nabla_{v}P_{\Sigma}^{x}\,,
V(\Phi)={\frac{M^{2}}{48\pi^{2}}}(1+ln{\frac{\mu^{2}}{M^{2}}})\Phi^{2}+{\frac{1}{144\pi^{2}}}\Phi^{4}(ln{\frac{\Omega^{2}}{\mu^{2}}}-{\frac{3}{2}})\;.
E_{\mathrm{S}}=mc^{2}[1-\frac{1}{2}\frac{m_{N}}{(m+m_{N})}(\frac{Z\alpha}{n})^{2}-\frac{1}{8}\frac{mm_{N}^{2}}{(m+m_{N})^{3}}(\frac{Z\alpha}{n})^{4}-\cdots]
L=mi\frac{z_{1}\dot{\bar{z}_{1}}+z_{2}\dot{\bar{z}_{2}}}{L_{1}}+ni\frac{z_{3}\dot{\bar{z}_{3}}+(z_{2}-z_{1}z_{3})(\dot{\bar{z}_{2}}-\dot{\bar{z}_{1}}\bar{z}_{3}-\bar{z}_{1}\dot{\bar{z}_{3}})}{L_{2}}-H_{t}
{\calS}_{eff}[\phi_{a}]={\calS}_{+}^{(0)}[\varphi_{a}^{(+)}]+{\calS}_{-}^{(0)}[\varphi_{a}^{(-)}]+\intd^{4}k\frac1{8\omega^{2}}\left\{J_{a}^{(-)}[\varphi_{a}^{(+)}]+J_{a}^{(+)}[\varphi_{a}^{(-)}]\right\}^{2}
\intd^{4}x\,\mathcal{O}(x)\to\mathrm{Tr}_{\mathcal{H}}\mathcal{O}(x)\equiv\left(\frac{\pi\zeta}{2}\right)\sum_{(n_{1},n_{2})}\langlen_{1}:n_{2}|\mathcal{O}(x)|n_{1}:n_{2}\rangle\,,
\delta\dot{\bfA}_{a}^{M}=-\bbox\nabla\delta\dot{\Lambda}_{a}^{M}-gf_{ab}^{c}\left(\dot{\bfA}_{c}^{M}\overline{{\delta\Lambda}}_{b}^{M}+\overline{{\bfA}}_{c}^{M}\delta\dot{\Lambda}_{b}^{M}\right),
\Omega_{i}=\frac{1}{\epsilon}\left[(2\pi\alpha^{\prime})^{2}B_{ij}P_{0}^{j}-(2\pi\alpha^{\prime})^{2}B_{ij}B^{jk}(X_{1k}-X_{0k})+g_{ij}(X_{1}^{j}-X_{0}^{j})\right]\approx0.
\int{\frac{d^{d}q}{(2\pi)^{d}}}{\frac{1}{q^{2}+m_{1}^{2}}}{\frac{1}{(q-k)^{2}+m_{2}^{2}}}={\frac{1}{(4\pi)^{d}}}\big\{{\frac{2}{\epsilon}}-\gamma-b(k^{2},m_{1}^{2},m_{2}^{2})+{\calO}(\epsilon)\big\}
\begin{array}{rc}{\mathrm{Sector~\qquad~}}&{SO(4)\mathrm{~rep.}}\\{\mathrm{Untwisted~NSNS:~}}&{({\bf3},{\bf3})+11({\bf1},{\bf1})}\\{\mathrm{RR:~}}&{({\bf3},{\bf1})+({\bf1},{\bf3})+6({\bf1},{\bf1})}\\{\mathrm{Twisted~NSNS:~}}&{48({\bf1},{\bf1})}\\{\mathrm{RR:~}}&{16({\bf1},{\bf1})}\\\end{array}
\Gamma(\Omega,\Omega^{\prime})=-\frac{p_{0}^{4}}{M(1+n_{0})^{3/2}(1+n_{0}^{\prime})^{3/2}}\hat{U}^{-1}(n)G(\vec{p},\vec{p}^{~\!\prime})\\hat{U}(n^{\prime})
K_{ab}^{}~=\;-\,\frac{1}{2}\left(E_{aa}\,+\,E_{\theta(a)\,\theta(a)}\right)+\,\frac{1}{2n}\,\mathbf{1}_{2n}^{}~~,~~M_{ab}^{}~=\;-\,\frac{1}{2}\left(E_{aa}\,-\,E_{\theta(a)\,\theta(a)}\right)
\partial_{m}e_{n}^{\\\underline{p}}-\partial_{n}e_{n}^{\\\underline{p}}+\omega_{m\underline{q}}^{\\\\underline{p}}e_{n}^{\\\underline{q}}-\omega_{n\underline{q}}^{\\\\underline{p}}e_{m}^{\\\underline{q}}=0\,
\psi_{+,m}^{(n)}(x)=\frac{1}{{\calN}_{m,+}^{(n)}}\left(\begin{array}{c}{(d_{+})^{m}\chi_{1}^{(n)}(z,\bar{z})}\\{i\sqrt{m}(d_{+})^{m-1}\chi_{1}^{(n)}(z,\bar{z})}\\\end{array}\right)
{\calL}=\sqrt{-\hat{G}}\,[{\calR}_{\hat{G}}+\hat{G}^{MN}\partial_{M}\Phi\partial_{N}\Phi-{\textstyle{\frac{1}{12}}}\hat{H}_{MNP}\hat{H}^{MNP}-{\textstyle{\frac{1}{4}}}\hat{F}_{MN}^{I}\hat{F}^{I\,MN}],
F_{j_{1},m_{1},j_{2},m_{2}}^{j_{1}+j_{2}-1,m_{1}+m_{2}}=\frac{N(j_{3},m_{3})(j_{3}-1/2)!\sqrt{j_{3}}}{N(j_{1},m_{1})(j_{1}-1)!\sqrt{j_{1}}N(j_{2},m_{2})(j_{2}-1/2)!\sqrt{j_{2}}}(j_{2}m_{1}-j_{1}m_{2}).
P\phi_{>}(x)^{2}{\frac{Q}{H^{(0)}}}\phi_{>}(y)^{2}P=\langle0|\phi_{>}(x)^{2}Q\phi_{>}(y)^{2}|0\rangle\cdot{\frac{P}{2\omega_{\Lambda}+H^{(0)}}}\quad;
S={\frac{1}{16\pi\alpha^{(n-1)/2}}}\intd^{n+1}x\sqrt{|g|}e^{-\varphi}\left[R+(\nabla\varphi)^{2}+{\frac{k\alpha^{\prime}}{4}}\left(R_{GB}^{2}-(\nabla\varphi)^{4}\right)\right].
{\frac{\partial}{\partialx^{\alpha}}}\;\Gamma_{r}^{\alpha\beta\gamma}\(x,y,z)=\{\frac{i}{4\pi^{2}}}\;\varepsilon^{\betai\gammaj}{\frac{\partial}{\partialy^{i}}}{\frac{\partial}{\partialz^{j}}}\delta(x-y)\delta(x-z).
{e}_{\hat{0}}^{\;\;0}{\bar{e}}_{\alpha}^{\;\;i}\left[-\frac{1}{2}\frac{\partial\delta\Gamma^{(1)}}{\partialv^{i}}\right]={e}_{\hat{0}}^{\;\;0}{\bar{e}}_{\alpha}^{\;\;i}\left[\deltaB_{i}^{v(2)}\right]+{e}_{\alpha}^{\;\;\mu}{e}_{\hat{0}}^{\;\;\nu}X_{\mu\nu}^{v},
H\alpha^{\prime}P=\frac{(H^{4}r^{4}-2H^{2}r^{2}-H^{2}b\alpha^{2})H^{2}r\dot{r}+(3H^{4}r^{4}-2H^{2}r^{2}+H^{2}b\alpha^{2})H\sqrt{b}\alpha^{\prime}}{2(1-H^{2}r^{2})(H^{4}r^{4}+H^{2}b\alpha^{2})}.
\ddot{F}+\left[2\dot{A}-\frac{(e^{A})^{\cdot\cdot\cdot}}{(e^{A})^{\cdot\cdot}}\right]\dot{F}-\left[2e^{-A}(e^{A})^{\cdot\cdot}+\eta^{\mu\nu}k_{\mu}k_{\nu}\right]F=0.
{\bf1}=\left(\begin{matrix}{1}&{0}\\{0}&{1}\\\end{matrix}\right);~~~~\gamma_{0}=\left(\begin{matrix}{0}&{1}\\{-1}&{0}\\\end{matrix}\right);~~~~\gamma_{1}=\left(\begin{matrix}{0}&{1}\\{1}&{0}\\\end{matrix}\right);~~~~\gamma_{2}=\left(\begin{matrix}{1}&{0}\\{0}&{-1}\\\end{matrix}\right).
\langle\Psi_{\{0\}}|\sum_{\vec{b}}h_{\vec{b}}^{\mathrm{m}}|\Psi_{\{0\}}\rangle=\frac{1}{g_{\mathrm{eff}}^{2}}\left(\frac{\pi^{2}}{6}-1\right)\cdot(2N_{\bot}^{2})\,.
J_{i}={\frac{1}{2}}\intd^{3}x\langlex|\mathrm{tr}\,\gamma_{5}\gamma_{i}{\frac{1}{\gamma\cdotD+M}}|x\rangle=-{\frac{1}{2}}\intd^{3}x\langlex|\mathrm{tr}\,\gamma_{5}\gamma_{i}(\gamma\cdotD){\frac{1}{-(\gamma\cdotD)^{2}+M^{2}}}|x\rangle
\sum_{\gamma=\beta-\alpha}^{\beta}c_{\beta-\gamma\\gamma}\left({\frac{1}{2\alpha^{\prime}}}\right)^{\gamma+\alpha-\beta}\left(\begin{matrix}{\gamma}\\{\beta-\alpha}\\\end{matrix}\right)B(2\beta-\alpha-\gamma+\frac{1}{2},\gamma+\alpha-\beta+\frac{1}{2})=0\.
(-)^{{\bar{\alpha}}+{\bar{\beta}}+\bar{\alpha}\bar{\beta}}\frac{{\bar{\vartheta}}[_{\bar{\beta}}^{\bar{\alpha}}]}{{\bar{\eta}}}\rightarrow\frac{{\bar{\vartheta}}[_{\bar{\beta}}^{\bar{\alpha}}]^{5}}{{\bar{\eta}}^{5}}{\frac{1}{2}}\sum_{\gamma,\delta}\frac{{\bar{\vartheta}[_{\delta}^{\gamma}]}^{8}}{{\bar{\eta}}^{8}}
\frac{E_{0}\left(\ell,\mu,d\right)}{L^{d-1}}\approx-\frac{C\left(d\right)\,\left(2^{d}-1\right)\Gamma\left(d\right)\zeta_{R}\left(d+1\right)}{2^{3d-1}\pi^{d/2}\Gamma\left(\frac{d}{2}\right)\,\ell^{d}}.
\left[{\frac{d^{2}}{dr^{2}}}+\left({\frac{d^{\prime}}{d}}+{\frac{3}{r}}\right){\frac{d}{dr}}\right]\delta\tilde{\phi}+\left[{\frac{\omega^{2}f}{d^{2}}}-{\frac{f_{5}^{2}}{f_{n}d}}K_{5}^{2}-{\frac{f_{5}}{d}}K_{i}^{2}-{\frac{l(l+2)}{r^{2}d}}-{\frac{8R^{4}}{r^{6}f_{5}^{2}d}}\right]\delta\tilde{\phi}=0.
\dot{\xi}_{\underline{{c}}}-{h}_{\\underline{{c}}}^{i}\,{h}_{\\underline{{b}}}^{j}\,T_{ij}^{\underline{{a}}}\,\xi_{\underline{{a}}}\,\xi^{\underline{{b}}}-{h}_{\\underline{{c}}}^{i}\,T_{i0}^{\underline{{a}}}\,\xi_{{a}}=0,
\lambda_{H}=\left(\textstyle{\frac{6}{\mu}E_{0}+\frac{1}{2}N,\,{\frac{1}{2}j_{z}}\,,\,{\frac{1}{2}(j_{w}+2j_{v}+j_{u})}\,,\,{\frac{1}{2}(j_{w}+j_{u})}\,,\,{\frac{1}{2}(j_{w}-j_{u})}}\right)\,,
\overline{{\Phi}}_{2}\left(p\right)_{a_{1}...a_{m}b_{1}...b_{n}}=\epsilon_{a_{1}b_{1}c_{1}d_{1}}...\epsilon_{a_{n}b_{n}c_{n}d_{n}}p_{a_{n+1}c_{n+1}}...p_{a_{m}c_{m}}
P_{\mathrm{min}}^{\alpha}({\bfm})={\frac{\pi}{M}}[(({\bfB}-1){\bfm})_{\alpha}-A_{\alpha}+1],~~~~P_{\mathrm{max}}^{\alpha}({\bfm})=-P_{\mathrm{min}}^{\alpha}({\bfm})+{\frac{2\pi}{M}}({\frac{\bfu}{2}}-{\bfA})_{\alpha}
\alpha=\sum_{p=1}^{m}\alpha_{m}(p)\Omega_{(p)}^{\prime}\quad,\quad\alpha_{m}(p)=\prod_{r=1}^{p-1}{\frac{2m-r}{r}}\quad,\quad\Omega_{(p)}^{\prime}=\Omega_{(p)\,\alpha_{1}\dots\alpha_{p-1}}^{\prime}F^{\alpha_{1}}\wedge\dots\wedgeF^{\alpha_{p-1}}\quad.
\deltaM-\bar{\cal{E}}_{(m)}=-\delta\left[\int_{\Sigma_{t}}\left(\theta_{(g)}^{\mu}(t)-t^{\mu}L_{(g)}\right)d\Sigma_{\mu}\right]+\delta\left({\frac{\kappa}{2\pi}}S_{(g)}\right)+\delta\left[\int_{C_{t}}\sqrt{\gamma}d^{2}yBN\right]~~~.
\Omega(N,T,V)={\frac{V^{N}}{N!}}\Bigl\{1+{\frac{N^{2}}{2V}}\int_{0}^{2\pi}d\theta\int_{0}^{\infty}dr\,r\Bigl(\mathrm{e}^{-{\frac{\phi({\bfr}_{ij})}{kT}}}-1\Bigr)+\cdots\Bigr\}\.
\psi^{\prime\prime}+\gamma^{\rho\lambda}\psi_{|\rho\lambda}+\left(3\mathcal{H}-2\frac{\varphi_{0}^{\prime\prime}}{\varphi_{0}^{\prime}}\right)\psi^{\prime}+\left(4\mathcal{H}^{\prime}-4\mathcal{H}\frac{\varphi_{0}^{\prime\prime}}{\varphi_{0}^{\prime}}+6K\right)\psi=0.
{\calW}g_{\mu\nu}{\frac{dx^{\mu}}{d\tilde{\lambda}}}{\frac{dx^{\nu}}{d\tilde{\lambda}}}=-\left[\epsilon_{D}\left({\frac{d\lambda}{d\tilde{\lambda}}}\right)^{2}+\left({\frac{dy}{d\tilde{\lambda}}}\right)^{2}\right].
\mathcal{O}:={\frac{1}{\Omega^{(D)}}}\intd^{D}x\\mathrm{tr}\left[{\frac{1}{2}}\mathscr{A}_{\mu}(x)\cdot\mathscr{A}_{\mu}(x)-\alphai\mathscr{C}(x)\cdot\bar{\mathscr{C}}(x)\right],
W^{\Omega_{z_{1}}^{*},\Omega_{z_{2}}}\left(\begin{array}{cc}{c}&{d}\\{b}&{a}\\\end{array}\right)=\frac{G_{b}}{G_{a}}W\left(\left.\begin{array}{cc}{b}&{c}\\{a}&{d}\\\end{array}\right|z_{2}-z_{1}-\frac{n}{2}\right).
T=2\pi\xi\,n+\int\!\mathrm{d}^{2}x\left\{\frac{1}{2g^{2}}\left[F_{12}+g^{2}\left(|\varphi|^{2}-\xi\right)\right]^{2}+\left|\left({\calD}_{1}+i{\calD}_{2}\right)\varphi\right|^{2}\right\}.
\left(\partial_{\mu}1\otimes1+\frac{1}{4}\epsilon_{\alpha\beta\delta}\omega_{\mu}^{\alpha\beta}1\otimes\gamma^{\delta}+\frac{i}{2\ell}A_{\mu}^{I}(-\frac{1}{\ell}N_{I}^{c}\mpK_{I}^{c})\sigma_{c}\otimes1\pm\frac{1}{2\ell}e_{\mu\alpha}1\otimes\gamma^{\alpha}\right)\varepsilon_{r}=0\,
{\bar{\psi}}^{a}=\left(\begin{array}{c}{{\bar{\phi}}^{\dot{L}}}\\{{\bar{\varphi}}_{L}}\\\end{array}\right)\equiv\left(\begin{array}{c}{{\bar{\phi}}^{\dot{1}}}\\{{\bar{\phi}}^{\dot{2}}}\\{{\bar{\varphi}}_{1}}\\{{\bar{\varphi}}_{2}}\\\end{array}\right)\,,
\deltaW=-{\frac{\alpha(1+\alpha)^{2}}{1536}}{\frac{\kappa^{4}}{\lambda^{6}}}\int_{M^{10}}d^{10}x\epsilon^{A_{1}A_{2}\cdotsA_{10}}\mathrm{tr}\epsilonF_{A_{1}A_{2}}\mathrm{tr}F_{A_{3}A_{4}}F_{A_{5}A_{6}}\mathrm{tr}F_{A_{7}A_{8}}F_{A_{9}A_{10}}.
I_{i}(r)\prod_{l=1}^{4}{\frac{(a_{i}-z_{l})}{y(z_{l})}}{\frac{1}{T^{5}\prod_{k<l}^{6}a_{kl}}}={\frac{1}{4}}{\frac{\partial}{\partiala_{i}}}\left\{{\frac{1}{r-a_{i}}}{\frac{1}{T^{5}\prod_{k<l}^{6}a_{kl}}}\prod_{l=1}^{4}{\frac{(a_{i}-z_{l})}{y(z_{l})}}\right\},
\begin{array}{lll}{b_{i}}&{=}&{q^{-2}-(q^{2}+q^{-2})P_{1}(i,i+1)+q(q^{3}-q^{-3})P_{0}(i,i+1)\;,}\\{e_{i}}&{=}&{(q^{2}+1+q^{-2})P_{0}(i,i+1)}\\\end{array}
\frac{F(3,\frac{i}{4}(k+k^{\prime}-2k^{\prime\prime})+2,\frac{i}{4}(k+k^{\prime}-2k^{\prime\prime})+3,e^{-4x_{0}})}{k+k^{\prime}-2k^{\prime\prime}-8i}=\frac{1}{2!}\sum_{n=2}^{\infty}\frac{n(n-1)e^{-4(n-2)x_{0}}}{k+k^{\prime}-2k^{\prime\prime}-4ni},
S=\intd^{4}\!x\,\sqrt{-g}\biggl(\frac{1}{2\kappa_{4}^{2}}R-\lambda_{\mathrm{bare}}-\frac{1}{2\cdot4!}\sum_{i=1}^{N_{3}+1}{Z_{i}}F_{4,i}^{2}\biggr)+S_{\mathrm{branes}}\.
\delta_{Q+S}\hat{s}=0\\Rightarrow\\left[4is(\epsilon_{4}^{-\alpha}+x_{4}^{\alpha\dot{\beta}}\bar{\eta}_{4\dot{\beta}}^{-})\left({\frac{x_{14}}{x_{14}^{2}}}-{\frac{x_{34}}{x_{34}^{2}}}\right)_{\alpha\dot{\alpha}}+\sum_{a=1}^{4}(\epsilon_{a}^{+\alpha}+x_{a}^{\alpha\dot{\beta}}\bar{\eta}_{a\dot{\beta}}^{+})S_{a4\;\alpha\dot{\alpha}}\right]\bar{\theta}_{4}^{+\dot{\alpha}}=0\;.
\begin{array}{l}{{\calX}^{+}\vertn,z\rangle_{+}=q^{2n}z\vertn,z\rangle_{+}~,}\\{{\calD}_{+}\vertn,z\rangle_{+}=q^{1-2n}\lambda^{-1}z^{-1}\vertn-1,z\rangle_{+}~,}\\{\mu_{+}\vertn,z\rangle_{+}=\vertn-1,z\rangle_{+}~.}\\\end{array}
m\simeq\frac{\alpha}{\pi^{2}}\intdq_{3}dq_{4}\int_{0}^{\infty}dr^{2}\mathrm{e}^{-r^{2}}\frac{1}{2r^{2}+L^{2}(q_{3}^{2}+q_{4}^{2})}~\frac{m}{m^{2}+(q_{3}^{2}+q_{4}^{2})}
\tilde{X}=\tilde{\chi}+\biggl(\frac{M}{L}\tilde{C}-\frac{M^{2}}{L^{2}}\tilde{E}^{\prime}\biggr)\varphi^{\prime}+\biggl[\frac{M}{L}\tilde{{\calP}}-\frac{M^{2}}{L^{2}}\dot{\tilde{E}}\biggr]\dot{\varphi},
\begin{array}{l}{{\displaystyle|F_{\nu}\left(\frac{2\pi}{\sigma_{0}}|n|\tau,\frac{2\pi}{\sigma_{0}}|n|\tau_{0}\right)|^{2}\sim}}\\{\sim{\displaystyle\frac{2^{2\nu}}{\Gamma^{2}(1-\nu)}\left(\frac{2\pi|n|}{\sigma_{0}}\right)^{2-2\nu}(-\tau_{0})~J_{\nu}^{2}\left(-\frac{2\pi}{\sigma_{0}}|n|\tau_{0}\right)(-\tau)^{1-2\nu}}}\\\end{array}
\left\langle\prod_{X_{i}\in({\itG_{1}})_{X_{i^{\prime}}^{\prime}}}exp(\eta_{1}\sum_{\stackrel{i_{\alpha_{1}}<j_{\alpha_{1}}<k_{\alpha_{1}}}{\left\{i_{\alpha_{1}},j_{\alpha_{1}},k_{\alpha_{1}}\right\}\inJ_{X_{i}}}}t_{i_{\alpha_{1}}}t_{j_{\alpha_{1}}}t_{k_{\alpha_{1}}}{\bf1}_{(w(t_{i_{\alpha_{1}}})=w(t_{j_{\alpha_{1}}})=w(t_{k_{\alpha_{1}}}))}+\right.
\left|\left\langle\gamma_{(0)}^{j_{1}-m_{1}}\gamma_{(1)}^{j_{2}-m_{2}}\prod_{i=1}^{s_{-}}\beta_{(w_{i})}^{k-2}\right\rangle\right|^{2}=(-)^{-\tilde{\rho}s_{-}}\triangle(1+j_{1}-m_{1})\triangle(1+j_{2}-m_{2})\prod_{i=1}^{s_{-}}|w_{i}|^{2\tilde{\rho}}|1-w_{i}|^{2\tilde{\rho}},
L=m(\Pi,n)+mb(\theta_{\alpha}\dot{\chi}^{\alpha}-\chi_{\alpha}\dot{\theta}^{\alpha})-mb\theta^{\alpha}n_{\alpha\gamma}\dot{n}^{\gamma}{}_{\beta}\chi^{\beta}+is\frac{\bar{z}{\dot{z}}-z\dot{\bar{z}}}{1-z\bar{z}}\,,
\rho_{m_{0}\rightarrowm_{1}+m_{2}}^{D}=\frac{1}{4\piR^{2}}\sum_{j}(2j+1)\rho_{m_{0}\rightarrowm_{1j}+m_{2j}}^{D-2};\qquadm_{ij}^{2}=m_{i}^{2}+\left(\frac{j}{R}\right)^{2}.
S^{ab}(x)S^{Tbc}(x)=S^{ab}(x)S^{cb}(x)=\frac{1}{4}\tau_{ij}^{b}\tau_{kl}^{b}(U(x)\tau^{a}U^{\dagger}(x))_{ij}(U(x)\tau^{c}U^{\dagger}(x))_{kl}=\frac{1}{2}tr[\tau^{a}\tau^{c}]=\delta^{ac}
G_{\ell}(r,r^{\prime},k)=-\frac{2}{\pi}\int_{0}^{\infty}dq\frac{\psi_{\ell}^{*}(q,r)\psi_{\ell}(q,r^{\prime})}{(k+i\epsilon)^{2}-q^{2}}-\sum_{j}\frac{\psi_{\ellj}(r)\psi_{\ellj}(r^{\prime})}{k^{2}+\kappa_{j}^{2}}\,.
\int_{\calI}F(t,\star;\rho,w,\rho^{\prime\prime},w^{\prime\prime},\bullet,\bullet^{\prime\prime})R_{F}(t,\star;\rho^{\prime\prime},w^{\prime\prime},\rho^{\prime},w^{\prime},\bullet^{\prime\prime},\bullet^{\prime};\alpha)G[t;\rho^{\prime\prime},w^{\prime\prime}]{\rho^{\prime\prime}}^{d-1}d\rho^{\prime\prime}dw^{\prime\prime}(d\bullet^{\prime\prime});\notag
\begin{array}{l}{G=KH=\left(\begin{array}{ccc}{1}&{\Xi}&{1/2\Xi\Xi^{+}+iT}\\{0}&{1}&{\Xi^{+}}\\{0}&{0}&{1}\\\end{array}\right)\times\left(\begin{array}{ccc}{L}&{0}&{0}\\{0}&{U}&{0}\\{0}&{0}&{L^{+-1}}\\\end{array}\right)=}\\{\phantom{G=KH}=\left(\begin{array}{ccc}{L}&{\XiU}&{\left(1/2\Xi\Xi^{+}+iT\right)L^{+-1}}\\{0}&{U}&{\Xi^{+}L^{+-1}}\\{0}&{0}&{L^{+-1}}\\\end{array}\right)}\\\end{array}
Y(R,R)=\int_{{\calF}_{1}}\frac{d\tau_{1}\,d\tau_{2}}{\tau_{2}}\vartheta_{3}^{2}\left(i\frac{\tau_{2}}{R^{2}}\right)\,\overline{{\rho}}+\int_{1}^{\infty}\frac{d\tau_{2}}{\tau_{2}}\vartheta_{3}^{2}\left(i\frac{\tau_{2}}{R^{2}}\right)\,\int_{-\frac{1}{2}}^{+\frac{1}{2}}d\tau_{1}\,\overline{{\rho}}+{\calO}\left(e^{-\piR^{2}}\right)\,
\{\rho(x),\rho(y)\}=0~,~~\{v_{i}(x),\rho(y)\}={\frac{\partial}{\partialx^{i}}}\delta(x-y)~,~~\{v_{i}(x),v_{j}(y)\}=-{\frac{\partial_{i}v_{j}-\partial_{j}v_{i}}{\rho}}\delta(x-y).
\epsilon^{kj}(p_{i}+\frac{1}{2}i\partial_{i})e_{k}^{\alpha}R_{\alpha\beta}\frac{\partialW_{j}}{\partialp_{\beta}}=\epsilon_{i}^{j}e_{j}^{\alpha}R_{\alpha\beta}(p^{k}+\frac{1}{2}i\partial^{k})\frac{\partialW_{j}}{\partialp_{\beta}}
\Big(\frac{\delta}{\deltaB_{\alphaa}^{*}}+\mathrm{\large~\frac{1}{2}~}\frac{\delta}{\delta\bar{C}_{\alphaa}}\Big)W_{m}^{(0)}=-m^{2}\frac{\delta}{\deltaF_{\alphaa}}W_{m}^{(0)}\quad\mathrm{and}\quad\frac{\delta}{\delta\bar{B}_{\alpha}}W_{m}^{(0)}=\mathrm{\large~\frac{1}{2}~}m^{2}\epsilon_{ab}\frac{\delta}{\deltaC_{\alphaba}^{*}}W_{m}^{(0)}.
\frac{d^{2}h_{\infty}(z)}{dz^{2}}+\frac{1}{z}\frac{dh_{\infty}(z)}{dz}+(1-\frac{1}{z^{2}})h_{\infty}(z)=-\frac{\kappa^{2}}{k_{1}^{2}}\frac{c_{1}^{2}}{\alpha_{0}^{2}}K_{1}(\frac{\kappa}{k_{1}}z)K_{0}^{2}(\frac{k_{0}}{k_{1}}z).
I_{1}=\int_{M_{(p+1)}}\Big({\frac{1}{(p+1)!}}E^{a_{0}}\wedgeE^{a_{1}}\wedge...\wedgeE^{a_{p}}\epsilon_{a_{0}a_{1}...a_{p}}e^{-{\frac{p-3}{2}}\phi}\sqrt{-det(\eta_{ab}+{\calF}_{ab})}
T_{+2-2}^{+q}=\frac12\gamma_{q\dot{q}}^{i}\Omega_{+2}^{+2i}\psi_{-2\dot{q}}^{1-}-\frac12\gamma_{q\dot{q}}^{i}\Omega_{-2}^{+2i}\psi_{+2\dot{q}}^{1-},
\langleJ(x)J(0)\rangle=\frac{N^{2}}{\lambda^{(2-\alpha)}}\frac{1}{|x|^{2\alpha+p-1}}=\frac{N^{\alpha}}{g_{YM}^{2(2-\alpha)}}\frac{1}{|x|^{2\alpha+p-1}}\,~~~~~~~\alpha=\frac{7-p}{5-p}\,
[\delta_{\chi}^{(1)},\delta_{\xi}^{(2)}]C_{MN}\,=\,-\frac{i}{16}(\chi^{\top}C\Gamma_{MN}\xi)\;,\qquad[\delta_{\chi}^{(1)},\delta_{\xi}^{(2)}]Z_{M_{1}\ldotsM_{6}}^{+}=-\frac{i}{16}(\chi^{\top}C\Gamma_{M_{1}\ldotsM_{6}}\xi)\;.
\begin{array}{lcr}{\{X^{\mu},X^{\nu}\}=\sum_{\stackrel{m_{1},n_{1}}{m_{2},n_{2}}}X_{m_{1}n_{1}}^{\mu}(\sigma)X_{m_{2}n_{2}}^{\nu}(\sigma)\{L_{m_{1}n_{1}},L_{m_{2}n_{2}}\},}\\\end{array}
a=a_{M}\left(\left[F_{\mu\nu}^{a}\right],\left[\partial^{\mu}H_{\mu}^{a}\right]\left[\varphi_{a}\right],\left[B_{a}^{\mu\nu}\right],\left[\Phi_{\alpha_{0}}^{*}\right],\left[\eta_{\alpha_{1}}^{*}\right]\right)e^{M}\left(\eta^{\alpha_{1}}\right)+\gammab,
{\mathcalM}\times{\mathcalG}_{0}\rightarrow{\mathcalA}_{0}:\quad(r_{j},\varphi^{a})\mapstoA_{i}[r_{j},\varphi^{a}]:=\,^{U^{-1}[\varphi^{a}]}\!\tilde{A}_{i}[r_{j}]\;,
{\calL}=\mu\int_{S^{1}\timesR^{3+1}}\,tr\left\{-\frac{1}{2}{\calF}_{MN}{\calF}^{MN}+\sum_{P}{\calD}_{M}\Phi_{P}{\calD}^{M}\Phi_{P}+\frac{1}{2}\sum_{P,Q}[\Phi_{P},\Phi_{Q}]^{2}\right\}
\left\{\left(\frac{g_{0}}{4\pi}\right)^{2}\Phi_{N}\left[\Gamma^{(0)\mathrm{{pert}}}\right]\right\}_{R,\nu}=\left[\frac{g(\nu)}{4\pi}\right]^{2}\Gamma_{N}^{(1){\mathrm{pert}}}+0(g^{4})\,.
p_{L}=\frac{i\sqrt{2}}{T-{\bar{T}}}\left(n_{1}+n_{2}{\bar{T}}^{2}+2m{\bar{T}}\right),\;\;p_{R}=\frac{i\sqrt{2}}{T-{\bar{T}}}\left(n_{1}+n_{2}T{\bar{T}}+m(T+{\bar{T}})\right).
\sum_{\nu=0}^{1-a_{ij}}(-1)^{\nu}q_{i}^{\nu(\nu-1+a_{ij})}\left[\begin{array}{c}{1-a_{ij}}\\{\nu}\\\end{array}\right]_{d_{i}}E^{1-a_{ij}-\nu}E_{j}E_{i}^{\nu}=0\;\;(i\neqj),
J_{r}={\frac{en}{4}}(J_{2}+{\frac{r_{1}^{2}}{\sigma^{2}J_{2}}}-{\frac{2r_{1}}{\sigma}})={\frac{en}{4}}(J_{r}+J_{1}+{\frac{r_{1}^{2}}{\sigma^{2}J_{2}}}-{\frac{2r_{1}}{\sigma}})~.
V(a,\phi_{I})=\frac{1}{g^{2}}\left[\frac{1}{2}(\sum_{I=1}^{N_{f}}|\phi_{I}|^{2}-\mua+\xi)^{2}+\sum_{I=1}^{N_{f}}(a-M_{I})^{2}|\phi_{I}|^{2}\right].
\begin{array}{lcl}{[U_{\alpha}(X),U_{\alpha}(Y)]}&{=}&{Z_{\alpha}^{-1}[U(\Phi_{\alpha}Y),U(\Phi_{\alpha}Y)]Z_{\alpha}}\\{}&{}&{}\\{}&{=}&{Z_{\alpha}^{-1}U([\Phi_{\alpha}X,\Phi_{\alpha}Y])Z_{\alpha}}\\{}&{}&{}\\{}&{=}&{Z_{\alpha}^{-1}U(\Phi_{\alpha}[X,Y]_{\alpha})Z_{\alpha}=U_{\alpha}([X,Y]_{\alpha})}\\\end{array}
M_{n}={\frac{2\sqrt{2}\pi^{2}}{\kappa_{(7)}}}\;{\frac{\vertS_{n0}\vert}{\sqrt{S_{00}}}}\;=\;{\frac{(2\pi)^{3/2}(2k+4)^{1/4}}{\kappa_{(7)}}}\;\mathrm{sin}\left({\frac{n\pi}{k+2}}\right)\,
C_{\tau\alpha}\lambda_{\tau_{1}\alpha_{1}}^{a_{1}}\lambda_{\tau_{2}\alpha_{2}}^{a_{2}}\lambda_{\tau_{3}\alpha_{3}}^{a_{3}}=f_{\tau\alpha,\tau_{1}\alpha_{1},\tau_{2}\alpha_{2},\tau_{3}\alpha_{3},a_{1},a_{2},a_{3}}\Psi.
S=\sum_{x}\sum_{i=1}^{2}\left[\frac{1}{2}\sum_{\hat{\mu}}\left(\Phi_{i}(x+\hat{\mu})-\Phi_{i}(x)\right)^{2}+\frac{1}{2}r_{0}\left(\Phi_{i}(x)\Phi_{i}(x)\right)+\frac{\lambda_{0}}{4}\left(\Phi_{i}(x)\Phi_{i}(x)\right)^{2}-J_{i}\Phi_{i}(x)\right]
(\psi,\psi\,^{\prime})=\int_{0}^{2\pi}d\phi\int_{0}^{\infty}drrf\psi^{\ast}\psi\,^{\prime}=\int_{0}^{2\pi}d\phi\int_{0}^{\infty}d\rho\gamma\psi^{\ast}\psi\,^{\prime}\;.
V(\vec{r})=\frac{1}{r}+\sum_{}\sum_{\{\vec{q}\}\neq0}\sum_{}\{\frac{1}{|\vec{r}-\vec{q}\,|}-\frac{1}{|\vec{q}\,|}[1+\frac{\vec{q}\cdot\vec{r}}{q^{2}}+\frac{1}{2q^{4}}(3(\vec{q}\cdot\vec{r})^{2}-q^{2}r^{2})]\}~~.
L^{MN}\left(\Gamma_{MN}\xi\right)=\varepsilon^{kj}X_{k}^{M}X_{j}^{N}X_{i}^{R}\left(\Gamma_{MN}\Gamma_{R}\kappa^{i}\right)=2\varepsilon^{kj}X_{k}^{M}\left(\Gamma_{M}\kappa^{i}\right)X_{j}\cdotX_{i}
\delta\gamma_{\mu\nu}(\rho)=-\zeta^{\lambda}\diamondsuit\partial_{\lambda}\gamma_{\mu\nu}+\partial^{\rho}\zeta_{\mu}\diamondsuit\gamma_{\rho\nu}+\partial^{\rho}\zeta_{\nu}\diamondsuit\gamma_{\mu\rho}-\partial^{\rho}\zeta_{\rho}\diamondsuit\gamma_{\mu\nu}+\frac{1}{\kappa}(\partial_{\nu}\zeta_{\mu}+\partial_{\mu}\zeta_{\nu}-\partial^{\rho}\zeta_{\rho}\eta_{\mu\nu}).
{\calL}_{L}={\frac{\alpha}{32\pi^{2}}}\{R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}-2R^{\alpha\beta}R_{\alpha\beta}+R^{2}\}+{\frac{\beta}{8\pi^{2}}}\{F^{\mu\nu}\tilde{F}_{\mu\nu}\}.
\left({\frac{\partial^{2}\phi}{\partialt^{2}}}-{\frac{1}{4M^{2}}}{\frac{\partial^{2}\phi}{\partialx^{2}}}\right)+e^{x}\left(-{\frac{1}{4M^{2}}}\nabla_{(2)}^{2}\phi+m^{2}\phi\right)=0
\frac{\partial}{\partialk_{\nu}}\left(\frac{k_{\mu}}{\left(k^{2}-m^{2}\right)^{\alpha}}\right)=\frac{g_{\mu\nu}}{\left(k^{2}-m^{2}\right)^{\alpha}}-2\alpha\frac{k_{\mu}k_{\nu}}{\left(k^{2}-m^{2}\right)^{\alpha+1}}.
\begin{array}{rccclcrcccl}{\left(\frac{G}{H}\right)_{7}}&{=}&{\frac{SO(8)}{SO(7)}}&{\equiv}&{S^{7}}&{;}&{\left(\frac{G}{H}\right)_{4}}&{=}&{\frac{SO(5)}{SO(4)}}&{\equiv}&{S^{4}}\\\end{array}
\frac{\bar{F}_{b}(a,b)}{B}=\frac{\bar{F}(b)}{B}+\bar{P}_{b}(a,b,b)=\frac{\eta^{2}}{24\pib^{4}}(c_{1}\Lambda^{\prime}+c_{2}\Lambda)+\frac{\eta^{2}\Lambda^{\prime}}{3}P_{b}(a,b,b).
{\calL}_{\infty}=-\frac{1}{6}\theta\frac{\partial^{2}\theta}{\partialp^{A}\partial\tilde{q}^{B}}\frac{\partial^{2}\theta}{\partialp_{A}\partial\tilde{q}_{B}}-\frac{1}{2}\frac{\partial\theta}{\partialp^{A}}\frac{\partial\theta}{\partial\tilde{q}_{A}}.
S[{\hat{\phi}}_{c}]=\intd^{4}x[\frac{1}{2}Z_{3}{\partial}_{\mu}{\hat{\phi}}_{c}{\partial}^{\mu}{\hat{\phi}}_{c}-\frac{m^{2}+{\delta}m^{2}}{2}{\hat{\phi}}_{c}^{2}-\frac{g^{4}}{4!}Z_{1}{\hat{\phi}}_{c}^{4}],
{\hat{F}}[T,-1]=\alpha~\sum_{m,\ell,R}{\frac{(-)^{6\epsilon_{\ell}-3\epsilon_{R}}{S_{0\Lambda_{\ell}}~S_{0\Lambda_{R}}~S_{\Lambda_{\ell}\Lambda_{m}}~S_{\Lambda_{\ell}\Lambda_{m}}~S_{\Lambda_{R}\Lambda_{m}}^{*}~q^{-7C_{\ell}+{\frac{3}{2}}C_{R}}}}{S_{00}~S_{0\Lambda_{m}}}}~.
(x-a)^{2}=\frac{\mathcal{A}^{2}}{3\mathcal{X}},\qquad(x-b)^{2}=\frac{\mathcal{B}^{2}}{3\mathcal{X}},\qquad(x-c)^{2}=\frac{\mathcal{C}^{2}}{3\mathcal{X}}
Ns_{\mathrm{eff}}={\frac{1}{4}}\int\!d\tau\,\Biggl[\Bigl({\frac{dx}{d\tau}}\Bigr)^{2}+\beta(x^{2}-2\kappa^{-2/3})-{\frac{\beta^{2}}{4}}\Biggr]+{\calO}(1/N^{2/3})\,.
a_{0,1}=\int_{{\calM}_{3}}h^{1/2}d^{3}x~\mathrm{Tr}\left[{\frac{1}{6}}\bar{R}-V\right]=-\int_{{\calM}_{3}}h^{1/2}d^{3}x~\left[{\frac{1}{6}}\bar{R}+{\frac{1}{16}}F^{jk}F_{jk}\right],