harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
L=\frac{1}{2e}\left(\dot{x}{}^{2}-(J^{2})^{-1}(\dot{x}J)^{2}\right)+sm(J^{2})^{-1}(\dot{x}J)-\frac{1}{2}em^{2}\left(1+s^{2}(J^{2})^{-1}\right)+L_{kin}.
u_{j}(p)=\sqrt{\frac{m_{j}}{2}}\left(\begin{array}{c}{e^{-\theta/2}}\\{e^{\theta/2}}\\\end{array}\right)\quad\quad\mathrm{and\qquad~}v_{j}(p)=i\sqrt{\frac{m_{j}}{2}}\left(\begin{array}{c}{e^{-\theta/2}}\\{-e^{\theta/2}}\\\end{array}\right)\,\,,
\Pi_{+q\atop-\dot{q}}^{m}={\calD}_{+q\atop-\dot{q}}X^{m}-i\left({\calD}_{+q\atop-\dot{q}}\Theta^{\alpha1}\sigma_{\alpha\beta}^{m}\Theta^{\beta1}+{\calD}_{+q\atop-\dot{q}}\Theta^{\alpha2}\sigma_{\alpha\beta}^{m}\Theta^{\beta2}\right)=0,
T^{\mu}\_{\nu}(\eta)=\left(\begin{array}{cc}{H(\eta)c^{-1}(\eta)}&{0}\\{0}&{T_{\beta}^{\beta}-c^{-1}(\eta)H(\eta)\}\\\end{array}\right)+c^{-1}\left(\begin{array}{cc}{\zeta}&{\alpha^{\prime}}\\{-\alpha^{\prime}}&{-\zeta\}\\\end{array}\right)
{f_{\mathrm{inv}}^{(n)}}^{-1}=\left(\begin{array}{c|c}{f^{-1}-f^{-1}AC^{-1}\tilde{A}f^{-1}}&{-f^{-1}AC^{-1}}\\\hline{C^{-1}\tilde{A}f^{-1}}&{C^{-1}}\\\end{array}\right)
m={\textstyle\frac{1}{2}}\left(\begin{array}{cc}{m^{1}}&{m^{2}+m^{3}}\\{}&{}\\{m^{2}-m^{3}}&{-m_{1}}\\\end{array}\right)={\calQ}=\left(\begin{array}{cc}{\frac{\delta}{2}r}&{\betaq}\\{\gammap}&{-\frac{\delta}{2}r}\\\end{array}\right)\,.
S=\frac{1}{(2\pi)^{7}l_{s}^{8}}\intd^{10}x\sqrt{-g}\frac{1}{e^{2\phi}}\left(R+4(\partial\phi)^{2}-\frac{(2\pil_{s})^{4}}{2\cdot3!(2\pi)^{2}}H_{3}^{2}\right).
g_{i}({\bf{q}})=[\sum_{{\bf{k}}}\frac{{\bar{n}}_{{\bf{k}}-{\bf{q}}/2}-{\bar{n}}_{{\bf{k}}+{\bf{q}}/2}}{({\tilde{\omega}}_{i}({\bf{q}})-\frac{{\bf{k.q}}}{m})^{2}}]^{-\frac{1}{2}}
e^{-2\phi}=e^{-2\rho}=C_{1}F(1,\frac{\lambda^{2}}{2}+1;-\frac{T^{2}}{4})+C_{2}(-\frac{T^{2}}{4})^{-\frac{\lambda^{2}}{2}}e^{-\frac{T^{2}}{4}}-\frac{2e^{-\frac{T^{2}}{4}}}{a(T^{2})^{\frac{\lambda^{2}}{2}}}x^{-}
ds^{2}~=~\left(1+{\frac{2Gm}{R}}\right)\left[-dx^{-}~dx^{+}+dx^{2}+dy^{2}\right]+{\frac{4Gm}{R}}\left[{\frac{p_{0}}{m}}dx{-}+{\frac{m}{4p_{0}}}dx^{+}\right]^{2},
I_{ij}^{(1)\mu}={\frac{i}{2}}\Psi^{\lambda}\Psi^{\sigma}(f_{i~\sigma}^{\nu}D_{\nu}f_{j~\lambda}^{\mu}+f_{j~\sigma}^{\nu}D_{\nu}f_{i~\lambda}^{\mu}+{\frac{1}{2}}f_{i}^{\mu\rho}c_{j\lambda\sigma\rho}+f_{j}^{\mu\rho}c_{i\lambda\sigma\rho})
H_{c}=\int(\frac{g^{2}}{64\pi^{2}}G_{ij}^{a}G_{ij}^{a}+\frac{8\pi^{2}}{g^{2}}(\pi_{i}^{a}+\frac{1}{8\pi}\epsilon_{ij}A^{j,a})^{2}-A_{0}^{a}\wedge^{a})d^{2}x
\delta_{\epsilon}A_{\alpha}^{a}=\left(\begin{array}{lll}{\left(D_{\alpha}\right)_{\;\;b}^{a}}&{\left(D_{\alpha}\right)_{\;\;b}^{a}}&{\left(D_{\alpha}\right)_{\;\;b}^{a}}\\\end{array}\right)\left(\begin{array}{l}{\epsilon_{1}^{b}}\\{\epsilon_{2}^{b}}\\{\epsilon_{3}^{b}}\\\end{array}\right)\,,
\nabla\,e_{i}\=\\omega_{i}^{j}\otimese_{j}\=\\omega_{\mui}^{j}\,\gamma^{\mu}\otimese_{j}\\,\in\;\Omega_{D}^{1}({\mathcalA}_{\alpha})\otimes_{{\mathcalA}_{\alpha}}\displaystyle\mathop{{\mathcalH}}^{\circ}\.
\epsilon^{\mu\nu\rho\lambda\sigma}G_{\mu\mu^{\prime}}G_{\nu\nu^{\prime}}G_{\lambda\lambda^{\prime}}G_{\sigma\sigma^{\prime}}=-G_{5}\epsilon_{\mu^{\prime}\nu^{\prime}\rho^{\prime}\lambda^{\prime}\sigma^{\prime}}G_{5}^{\rho\rho^{\prime}},
2M=\frac{a-\chi^{2}/2}{a+\chi^{2}/2}r_{-}+r_{+}\,\Q_{E}=(\frac{r_{+}r_{-}e^{-\chif_{\infty}}}{2a+\chi^{2}})^{1/2}\,\Q_{D}=\frac{\chi^{2}}{a+\chi^{2}/2}r_{-}
{\calA}_{0}\left(t,{\bfx}\right)=\int_{0}^{1}{d\alpha\frac{1}{{1+e^{2}a\left({1-12\alpha^{2}g^{2}\partial^{2}}\right)}}}\frac{{x^{i}\partial_{i}^{\alphax}\left({-J^{0}\left({\alphax}\right)}\right)}}{{\nabla_{\alphax}^{2}-\chi^{2}}}.
[{\mathbb{Q}},{\mathbb{X}}]=[\tilde{\mathbb{Q}},\mathbb{X}]=0\qquad\mathrm{for~any}\quad\mathbb{X}\in\{\mathbb{P}_{m},\mathbb{M}_{mn},\mathbb{D},\mathbb{K}_{m},\mathbb{Q},\tilde{\mathbb{Q}}\}~.
\hat{S}=\intd^{10}\hat{x}\sqrt{|\hat{g}|}\,\left\{e^{-2\hat{\phi}}\left[\hat{R}-4(\partial\hat{\phi})^{2}\right]+{\textstyle\frac{1}{2\cdot7!}}e^{2\hat{\phi}}\hat{\tilde{H}}{}^{2}\right\}\,,
P_{\mu}^{\varepsilon}\:P_{\mu^{\prime}}^{\varepsilon}\;=\;\frac{1}{4}\left(X\:t_{\mu}^{\varepsilon}\:t_{\mu^{\prime}}^{\varepsilon}\:X^{*}+X\:t_{\mu}^{\varepsilon}\:X\:t_{\mu^{\prime}}^{\varepsilon}+t_{\mu}^{\varepsilon}\:X^{*}\:t_{\mu^{\prime}}^{\varepsilon}\:X^{*}\right).
L=\frac{H}{2}\epsilon_{i_{1}j_{1}}Tr[Q_{1}^{i_{1}}D_{t}Q_{1}^{j_{1}}]+\frac{H}{2}\epsilon_{{\tilde{i}}_{1}{\tilde{j}}_{1}}Tr[Q_{1}^{{\tilde{i}}_{1}}D_{t}Q_{1}^{{\tilde{j}}_{1}}]+(2\rightarrow1),
S_{1}=\intdx\left(C^{\alphaa}\Lambda_{\alpha}^{i}\varphi_{ia}^{*}+\varepsilon^{ab}B^{\beta}\Lambda_{\alphab|a}^{*}(-1)^{\varepsilon_{\beta}}+B^{\alpha}\bar{\Lambda}_{\alpha}^{i}\bar{\varphi}_{i}(-1)^{\varepsilon_{i}+\varepsilon_{\alpha}}+\frac{1}{2}\varepsilon^{ab}B^{\alpha}\hat{\Lambda}_{\alpha}^{ij}\varphi_{jb}^{*}\varphi_{ia}^{*}\right),
S_{WZ}^{N(M2_{t},\bar{M2}_{t})}=\mu_{2}\int_{R^{2+1}}\mathrm{Tr}\left(l_{p}^{2}\left[{\hat{k}}^{-2}{\hat{k}}^{(1)}+il_{p}^{2}i_{\hat{\Phi}}i_{\hat{\Phi}}({\hat{C}}-{\hat{k}}^{-2}{\hat{k}}^{(1)}\wedgei_{\hat{k}}{\hat{C}})+\dots\right]\wedge({\hat{F}}^{(2)}-{\hat{F}}^{(2)\prime})+\dots\right)\,.
{\frac{(r^{2}-\mu^{2})^{2}}{r^{4}}}\partial_{r}^{2}h^{\prime\prime}+{\frac{(r^{2}-\mu^{2})(3r^{2}+\mu^{2})}{r^{5}}}\partial_{r}h^{\prime\prime}-{\frac{L(L+2)}{r^{2}}}h^{\prime\prime}=0.
(k-1)U_{0}^{n}Str\left(\ldots\underbrace{\phi^{i_{2j}^{\prime}}\phi^{i_{2j-1}^{\prime}}\phi^{i_{2k-1}^{\prime}}}\ldots\right)+U_{0}^{n}Str\left(\ldots\underbrace{\phi^{i_{2k}^{\prime}}\phi^{i_{2k-1}^{\prime}}}\ldots\right)
{\mathcalA}^{(j)}=\frac{\hbar^{2}}{2M}\frac{\hat{\lambda}}{2}\,\int_{I_{j}}dr\,\delta(r-a)\,{\mathcalE}_{\bfr}\left|\Psi_{\mathrm{(gs)}}^{(j)}({\bfr})\right|^{2}=\frac{\hbar^{2}}{2M}\frac{\hat{\lambda}}{2}\,R^{(j)}(a)\,{\mathcalE}_{r}R^{(j)}(a)\;,
U_{0}=\left(\begin{array}{c}{(2\zeta)^{-1/2}Z}\\{\sqrt{\zeta}\bar{z}_{2}\bar{z}_{1}}\\{-\bar{z}_{2}(Z_{1}+Z_{2}+\theta_{2})}\\{\sqrt{\zeta}\bar{z}_{1}\bar{z}_{1}}\\{-\bar{z}_{1}(Z_{1}+Z_{2}-\theta_{1})}\\\end{array}\right)
S\simeq2\,\pi\,\sqrt{\frac{d}{6}\\overline{N}}\left[1-\frac{3d}{4}\,\frac{1}{\overline{{\calR}}}\right]\simeq2\,\pi\left(\frac{d\,\alpha^{\prime}}{6}\right)^{\frac{1}{2}}\,M\left[1-\frac{3}{4}\,\frac{\ell_{s}^{2}}{R^{2}}\right]\,.
{\calD}_{Vv}^{\mathrm{sym}}\equiv{\calD}_{Vv}^{(l)}-8r^{(1)}\bigl({\calD}_{g^{2}}^{(l+1)}+l({\calN}_{V}^{(l+1)}-8(l+1)r^{(1)}\delta{\calN}_{V}^{(l+2)})\bigr),
\Pi(p^{2})=-\frac{e^{2}}{2p^{2}}\left(\delta_{\mu\nu}-3\frac{p_{\mu}p_{\nu}}{p^{2}}\right)\int\frac{d^{3}q}{(2\pi)^{3}}\mathrm{tr}\left[\gamma_{\mu}S(q+\mathrm{~\frac{1}{2}~}p)\Gamma_{\nu}(q+\mathrm{~\frac{1}{2}~}p,q-\mathrm{~\frac{1}{2}~}p)S(q-\mathrm{~\frac{1}{2}~}p)\right].
\bar{H}_{\mathrm{gs}}=H_{\mathrm{g}}(q^{r},p^{r})+\lambda_{a}^{(1)r_{a}}\pi_{a}^{r_{a}}+\lambda_{a}^{(2)r_{a}}\psi_{a}^{r_{a}}+\lambda_{a}^{(3)}\theta_{a}+\lambda_{a}^{(4)}\gamma_{a}+\lambda_{A}^{(5)}\omega_{A}.
{\calL}=\overline{{D_{\mu}\phi^{i}}}D^{\mu}\phi^{i}+\sigma(\phi^{i\dagger}\phi^{i}-N\chi)+N(m^{2}\chi+{\frac{\lambda}{2}}\chi^{2}+{\frac{\eta}{3}}\chi^{3})+iN\epsilon_{\mu\rho\nu}A_{\mu}\partial_{\rho}A_{\nu}.
\begin{array}{cccccc}{}&{}&{\S{G_{2}\timesSU(3)}}&{}&{}&{}\\{}&{\diagup}&{}&{\diagdown}&{}&{}\\{\S{SU(3)\timesSU(3)}}&{}&{}&{}&{}&{\S{SO(13)}.}\\{}&{\diagdown}&{}&{\diagup}&{}&{}\\{}&{}&{\S{SU(6)}}&{}&{}&{}\\\end{array}
\Gamma_{\mu}^{\nu\rho}\equiv\Gamma^{\rho}=2ik^{\rho}t\left(P+\xi^{-1}Q\right)-iPP^{,\rho}\left(1-e^{\left(1-\xi^{-1}\right)k^{2}t}\right)-iQP^{,\rho}\left(-1+e^{\left(-1+\xi^{-1}\right)k^{2}t}\right)
\begin{array}{l}{e^{+2}\nabla\psi_{+2q}^{1,2+}\!+\!e^{-2}\nabla\psi_{-2q}^{1,2+}\!+\!e^{+}\nabla\chi_{q}^{1,2}\!+\!e^{-}\nablah_{-q}^{1,2+}\!+\!T^{+2}\psi_{+2q}^{1,2+}\!+\!T^{-2}\psi_{-2q}^{1,2+}\!+\!T^{+}\chi_{q}^{1,2}\!+\!T^{-}h_{-q}^{1,2+}}\\{=-\frac12\gamma_{q\dot{q}}^{i}\Omega^{+2i}\pi_{\dot{q}}^{1(2)-},}\\\end{array}
\tilde{\Pi}^{(2)}=(\frac{4\pi}{\kappa})^{2}[\frac{1}{96\pi^{2}}(1-\gamma)(\frac{4\pi}{\kappa})^{2}R-\frac{1}{96\pi^{2}}(\frac{C_{2}(G)}{2})^{2}(5-4\gamma)]
{\underbrace{\X_{\sigma^{\prime}ij}\}}_{T}=-{\frac{1}{E_{j}-E_{i}}}\left({\frac{d\,}{d\sigma}}f_{\sigmaij}\right)\int_{\sigma}^{\infty}d\sigma^{\prime}X_{\sigma^{\prime}ij}+{\frac{1}{E_{j}-E_{i}}}\left(1-f_{\sigmaij}\right)X_{\sigmaij}.
\xi^{(m)}(u)=u\prod_{k=1}^{N}(1+uq^{2k-N-1})\left\{\sum_{i=1}^{m}\frac{q^{2i-N-1}}{1+uq^{2i-N-1}}-\frac{m}{N}\sum_{i=1}^{N}\frac{q^{2i-N-1}}{1+uq^{2i-N-1}}\right\}.
\left(\eta_{\mu\nu}\partial_{\mu}\partial_{\nu}+m^{2}+\xiR++{\frac{1}{3}}R_{\alpha}^{\nu}y^{\alpha}\partial_{\nu}-{\frac{1}{3}}R_{\\alpha\\beta}^{\mu\\nu}y^{\alpha}y^{\beta}\partial_{\mu}\partial_{\nu}+O(R^{2})\right)\phi=0
\langlej_{\mu}^{l}(x_{1})j_{\nu}^{l}(x_{2})\rangle=\frac{1}{(x_{1}^{2})^{l}}\frac{1}{(x_{2}^{2})^{l}}g_{\mu\rho}(x_{1})g_{\nu\sigma}(x_{2})\langlej_{\rho}(Rx_{1})j_{\sigma}(Rx_{2})\rangle.
\left.e^{\left[ip_{k}(A_{k}^{(1)}+A_{k}^{(2)})(1-2x)-2\vec{A}^{(1)}\cdot\vec{A}^{(2)}(1-\delta(x))/t\right]}\right|_{\stackrel{\!{\scriptstyle\mathrm{linear}\,\mathrm{in}}}{\mathrm{each}\,A_{k}^{(i)}}}=-\left[\delta_{kj}\frac{2(1-\delta(x))}{t}+p_{k}p_{j}\left(1-2x\right)^{2}\right]A_{k}^{(1)}\!A_{j}^{(2)},\nonumber\,
S=T\intd^{2}xd^{N}\theta\left\{\theta^{N}\;(i\;\partial_{-}\Psi^{i}\Psi^{i}+{\frac{1}{N}}\;D_{\alpha}\Psi^{i}D_{\alpha}\Psi^{i})+P^{\{\alpha\beta\}}D_{\alpha}\Psi^{i}D_{\beta}\Psi^{i}\right\}
\varphi_{200}(r,\theta,\phi)=R_{20}(r)Y_{0}^{0}(\theta,\phi)=\frac{2}{(2a_{0})^{\frac{3}{2}}}\left(1-\frac{r}{2a_{0}}\right)e^{-\frac{r}{2a_{0}}}Y_{0}^{0}\,{,}
\kappa^{2}g^{2}\alpha^{\prime}\left(\frac{1}{2}(\frac{11}{3^{2}})^{2}-26\cdot\frac{1}{2}(\frac{5}{3^{2}\sqrt{2}})^{2}\right)\phi^{4}=-\kappa^{2}g^{2}\alpha^{\prime}\left(\frac{34}{27}\right)\phi^{4}.
{\calH}_{j,j+1}=\sigma_{j}^{+}\sigma_{j+1}^{-}+\sigma_{j}^{-}\sigma_{j+1}^{+}+\Gamma\left(\sigma_{j}^{+}\sigma_{j}^{+}+\sigma_{j}^{-}\sigma_{j+1}^{-}\right)-\frac{h}{2}\left(\sigma_{j}^{z}+\sigma_{j+1}^{z}\right)\;.
J_{p}=\frac{V_{3}N}{\pi^{2}\lambda^{2}}\frac{\sqrt{2m\lambda}(2r_{+}^{2}+l^{2})}{(r_{+}^{2}+l^{2})(l^{2}-2r_{+}^{2})}\frac{l}{\triangle_{*}}\left[\frac{m(l^{2}-2r_{+}^{2})}{r_{+}^{2}(2r_{+}^{2}+3l^{2})}+\frac{u^{2}(u^{2}+l^{2})-m}{u^{2}+l^{2}}-u^{2}\triangle_{*}\right]
[P,R]_{\scriptscriptstyleSN}\;\hat{\longrightarrow}\;\sum_{i=1}^{p}(-1)^{i+1}\widehat{X}_{(1)}\cdots{\widehat{\widehatX}}_{(i)}\cdots\widehat{X}_{(p)}\{-\{\widehat{X}_{(i)},Q\},\widehat{R}\}
\int_{-\infty}^{0}dx_{1}dx_{2}\int_{-\infty}^{\infty}dt_{1}dt_{2}G_{N}(t,x;t_{1},x_{1})~G_{N}(t^{\prime},x^{\prime};t_{2},x_{2})~G_{N}(t_{2},x_{2};t_{1},x_{1})
<:\!e^{ik_{1}X(z_{1},\bar{z}_{1})}\!:\,:\!e^{ik_{2}X(z_{2},\bar{z}_{2})}\!:\,\ldots\,:\!e^{ik_{n}X(z_{n},\bar{z}_{n})}\!:>_{S_{2}}\=2\pi\delta(k_{1}+\ldots+k_{n})\prod_{i<j}|z_{i}-z_{j}|^{2k_{i}k_{j}},
\Box\phi_{0}={\frac{1}{l(1-e^{-2d/l})}}\left[J^{A}+e^{-4d/l}J^{B}\right]+\left[{\frac{l^{2}}{4}}-{\frac{ld}{2}}{\frac{1}{(1-e^{-2d/l})}}\right]\Box^{2}\phi_{0}-{\frac{l}{4}}e^{2d/l}\left[1-e^{-2d/l}\right]\BoxC\.
\overline{\zeta}_{5,S}:=\frac{(d-3)k^{4}}{20(g(k^{2})\pi^{d/2})^{3}}\int\frac{dP_{1}dP_{2}dP_{3}dP_{4}~\delta(k-P_{S})}{P_{1}^{2}P_{2}^{2}P_{3}^{2}P_{4}^{2}(P_{1}-P_{2})^{2}(P_{2}-P_{3})^{2}(P_{3}-P_{4})^{2}(P_{4}-P_{1})^{2}}
F(R)\equiv\int_{\infty}^{R}\frac{K(k(R))^{6}}{q(k(R))^{2}}\frac{R\,dR}{(1+R^{2})^{2}}=\int_{1/\sqrt2}^{k(R)}k\;dk\;{\frac{K(k)^{6}}{q(k)^{2}}}\;.
\hat{\psi}_{n}(t)~=~\hat{\psi}_{n}e^{-i({\frac{2\pi}{L}})^{2}{\frac{\hbar}{2}}n^{2}t}{}~~~~~{\hat{\psi}_{n}}^{\dagger}(t)~=~{\hat{\psi}_{n}}^{i({\frac{2\pi}{L}})^{2}{\frac{\hbar}{2}}n^{2}t}
{\calL}_{WZ}=\frac1{\left(p+1\right)!}\,\epsilon^{\mu_{1}\cdots\mu_{p+1}}\cdot\,\Pi_{,\mu_{1}}^{A_{1}}\cdots\Pi_{,\mu_{p+1}}^{A_{p+1}}\cdot\,\left(WZ\right)_{A_{p+1}\cdotsA_{1}}\left(\phi\right)\quad.
W^{(m,1)}\left(\left.\begin{array}{cc}{c}&{d}\\{b}&{a}\\\end{array}\right|v\right)=\sum_{\sigma\in\mathfrak{S}_{m}}\mathrm{sgn}\,\sigma\prod_{j=1}^{m}W\left(\left.\begin{array}{cc}{b_{j}^{\sigma}}&{a_{j}}\\{b_{j-1}^{\sigma}}&{a_{j-1}}\\\end{array}\right|z-\frac{m+1}{2}+j\right).
Y_{u}=\alpha^{2}\varepsilon\left(\begin{array}{lcl}{(1-e^{-2i\pi/5})\eta\varepsilon^{2}}&{(1-e^{-2i\pi/5})\eta\varepsilon^{3}}&{(1-\etae^{-2i\pi/5})\varepsilon}\\{(1-e^{-2i\pi/5})\eta\varepsilon^{3}}&{0}&{(\eta-e^{-2i\pi/5})\varepsilon^{2}}\\{(\eta-e^{-2i\pi/5})\varepsilon}&{(1-\etae^{-2i\pi/5})\varepsilon^{2}}&{(1-e^{-2i\pi/5})}\\\end{array}\right),
H_{NS}(ND)=\sum_{j}\left(N_{F}^{NS}(ND)+\frac{1}{12}\right)=\sum_{j}\left(\sum_{n=1}^{\infty}r({b_{-n}}^{-j}{b_{n}}_{j}+{b_{-n}}^{j}{b_{n}}_{-j})+\frac{1}{12}\right).
\beta^{2}=\frac{c\lambda\,r\,L^{2}}{(r^{2}-L^{2})^{n}}\left(\frac{L^{2n-1}}{\sqrt{\pi}}\Gamma(n+1)\,\Gamma(\frac{1}{2}-n)+\left(\frac{1}{r^{2}}\right)^{\frac{1}{2}-n}\,\frac{1}{2n-1}\,\,_{2}F_{1}[\frac{1}{2}-n,-n,\frac{3}{2}-n,\frac{L^{2}}{r^{2}}]\right).
\{\phi^{i},p_{j}\}=\delta_{j}^{i},\quad\{\psi^{ai},p_{(\psi),bj}\}=-\delta_{b}^{a}\delta_{j}^{i},\quad\{\bar{\psi}_{a}^{i},p_{(\bar{\psi}),j}^{b}\}=-\delta_{b}^{a}\delta_{j}^{i},
\sum_{\mathrm{\scriptsize{\it~\sum_{i=1}^sd_i=d~}}}\,<\prod_{i=1}^{s}\sigma_{d_{i}}>_{n}\prod_{i=1}^{s}{\frac{(2d_{i})!}{d_{i}!\lambda_{i}^{2d_{i}+1}}}=\sum_{\Gamma\in{G}_{g,s,n}}{\frac{1}{S(\Gamma)}}\int_{{\bfR}_{+}^{L(\Gamma)}}\rho_{g,n,s}\,\mathrm{e}^{-\sum_{i=1}^{s}\lambda_{i}p_{i}}\prod_{e\inL(\Gamma)}dl(e),
(\lambda-\lambda_{0})[\frac{1}{u}-i\int\frac{d^{d}{k}}{(2\pi)^{d}}(\frac{1}{(k^{2}-\lambda)(k^{2}-\lambda_{0})}-\frac{1}{(k^{2}-\mu)^{2}})]=(\vec{\phi}^{2}-v^{2})
\partial_{i}{\tilde{\Pi}}_{i}^{a}({\bfr})+gf^{abc}\,A_{i}^{b}({\bfr})\,{\tilde{\Pi}}_{i}^{c}({\bfr})+{\tilde{j}}_{0}^{a}({\bfr})=\partial_{i}\Pi_{i}^{a}({\bfr})+gf^{abc}\,A_{i}^{b}({\bfr})\,\Pi_{i}^{c}({\bfr})\,.
E=\int{\left\{\frac{1}{4}F_{ij}^{a}F_{ij}^{a}+\frac{1}{2}F_{i0}^{a}F_{i0}^{a}+\frac{1}{2}D_{i}\Phi^{a}D_{i}\Phi^{a}+\frac{1}{2}D_{0}\Phi^{a}D_{0}\Phi^{a}+\frac{\lambda}{4}(\Phi^{a}\Phi^{a}-\eta^{2})^{2}\right\}d^{3}r}\.
G(x,x^{\prime})=\theta(t-t^{\prime})\left(\begin{array}{cc}{H(x,x^{\prime})}&{-H(x^{\prime},x)}\\{-H(x,x^{\prime})}&{H(x^{\prime},x)}\\\end{array}\right)+\theta(t^{\prime}-t)\left(\begin{array}{cc}{H(x^{\prime},x)}&{-H(x^{\prime},x)}\\{-H(x,x^{\prime})}&{H(x,x^{\prime})}\\\end{array}\right),
e^{+2\sqrt{\pi}\sqrt{\frac{\pi}{\pi+gN}}U_{1a}\big(\delta(x)-\delta(y),C_{m_{d}}\alpha^{(1)}\big)}\prod_{I=2}^{N}e^{+2\sqrt{\pi}U_{Ia}\big(\delta(x)-\delta(y),C_{0}\alpha^{(I)}\big)}\;\rho(x-y)\;,
\begin{array}{ll}{T_{N}^{\mu\nu}}&{=\displaystyle{\frac{\partial{\calL}}{\partialA_{\lambda,\mu}^{a}}A_{\lambda}^{a,\nu}-\eta^{\mu\nu}{\calL}}}\\{{}}&{{}}\\{{}}&{=-\Pi^{a\mu\lambda}A_{\lambda}^{a,\nu}-\eta^{\mu\nu}{\calL}}\\\end{array}
\left(y^{(1)a_{1}},\pi_{a_{1}}^{(1)}\right)\rightarrow\left(\varphi_{a}^{(1)},\pi^{(1)a}\right),\;\left(y^{(2)a_{1}},\pi_{a_{1}}^{(2)}\right)\rightarrow\left(\varphi_{a}^{(2)},\pi^{(2)a}\right),
\nu_{\alpha_{3}}=Z_{\;\;\alpha_{3}}^{\alpha_{2}}Z_{\;\;\alpha_{2}}^{\alpha_{1}}\eta_{\alpha_{1}}^{*}+\frac12C_{\alpha_{2}}^{\alpha_{0}\beta_{0}}Z_{\;\;\alpha_{3}}^{\alpha_{2}}\Phi_{\alpha_{0}}^{*}\Phi_{\beta_{0}}^{*}+A_{\beta_{1}}^{\;\;\alpha_{2}}C_{\alpha_{3}}^{\beta_{1}\beta_{0}}\Phi_{\alpha_{2}}^{*}\Phi_{\beta_{0}}^{*},
\int_{1}^{\infty}\!\!dy\,e^{-\pi(mR)^{2}/y}y^{D/2-1}=\left[\pi(mR)^{2}\right]^{D/2}\Gamma(-D/2)-\int_{1}^{\infty}\!\!dy\,e^{-\piy(mR)^{2}}y^{-D/2-1}\quad.
\left(\hat{A}_{\mu}^{a}\hat{A}_{\nu}^{a}\right)\left(\hat{A}^{a\mu}\hat{A}^{a\nu}\right)-\left(\left(\hat{A}_{\mu}^{a}\hat{A}_{\nu}^{a}\right)\hat{A}^{a\mu}\right)\hat{A}^{a\nu}=\lambda\hat{A}_{\nu}^{a}\hat{A^{a\nu}}.
F=\frac{g^{2}}{m^{2}-p^{2}}+\frac{g^{2}}{m^{2}-p^{2}}(\deltam^{2}+g^{2}\Sigma(p^{2}))\frac{g^{2}}{m^{2}-p^{2}}+\ldots=\frac{g^{2}}{m^{2}-p^{2}-\deltam^{2}-g^{2}\Sigma(p^{2})}
\sqrt{-g}g^{\mu_{1}\nu_{1}}\cdotsg^{\mu_{D-p-1}\nu_{D-p-1}}\tilde{f}_{\nu_{1}\dots\nu_{D-p-1}}={\frac{1}{p!}}\epsilon^{\mu_{1}\dots\mu_{D-p-1}\nu_{1}\dots\nu_{p}}f_{\nu_{1}\dots\nu_{p}}.
\begin{array}{rl}{a^{-1}=}&{(a_{0}^{-1}+F_{0}^{-1})\,\sqrt{1+F_{0}\theta}-F_{0}^{-1}-\theta}\\{=}&{(a_{0}^{-1}+F_{0}^{-1}-F_{0}^{-1}\sqrt{1+F_{0}\theta})\,\sqrt{1+F_{0}\theta}\,.}\\\end{array}
z^{l}F^{\prime}(\frac{1}{z})-{z^{l+1}}[\mu+\frac{l+2}{1-z}]F(\frac{1}{z})=-\frac{1}{\mu}z(1-z)F^{\prime\prime}(z)-\frac{1}{\mu}(1+zl)F^{\prime}(z)+\frac{1}{\muz}F(z)-
\left(\hat{A}_{\mu}^{a}\hat{A}_{\mu}^{a}\right)\left(\hat{A}^{a\mu}\hat{A}^{a\mu}\right)-\left(\left(\hat{A}_{\mu}^{a}\hat{A}_{\mu}^{a}\right)\hat{A}^{a\mu}\right)\hat{A}^{a\mu}=0.
K_{\beta_{H}}\equiv\beta_{H}\left(\frac{\partialQ}{\partial\varphi}\right)_{\beta_{H}}=\frac{\tilde{d}\Omega_{\tilde{d}+1}(r_{+}r_{-})^{\tilde{d}/2}}{\pi}\frac{[1-(\tilde{d}-1)(r_{-}/r_{+})^{\tilde{d}}]}{[1-(r_{-}/r_{+})^{\tilde{d}}]}.
\psi_{\alpha}^{i}(x^{0},x^{1},x^{2},x^{3})\!\!=\!\!\sum_{n}\left(\begin{matrix}{b_{n}(x^{0},x^{3})\psi_{n1}^{i}(x^{1},x^{2})}\\{c_{n}(x^{0},x^{3})\psi_{n2}^{i}(x^{1},x^{2})}\\\end{matrix}\right)
\hat{\varphi}^{(1)}(x)=\mathrm{i}\,\intd^{6}y\,\,\Big<\hat{\varphi}(x)\,J(y)\,\hat{\varphi}(y)\Big>=\frac{\kappa_{orb}\,M}{2}\,\int\frac{d^{5}k}{(2\pi)^{5}}\frac{\mathrm{e}^{\mathrm{i}k\cdotx}}{k^{2}}~~.
P=\frac{2\pi\lambda}{[\nu!(\nu-1)!]^{2}}\frac{|\Gamma\left(\nu/2+\eta_{+}(\nu)+i\lambda/2\right)\Gamma\left(\nu/2+\eta_{-}(\nu)+i\lambda/2\right)|^{2}}{|\Gamma(1+i\lambda)|^{2}}\left(\frac{\omegar_{0}}{2}\right)^{2\nu}
I_{\mathrm{matter}}=\frac{m^{2}}{4}\intd^{2}x\left\langleq\Biggl|\epsilon^{\mu\nu}[D_{\mu}\Phi,D_{\nu}\Phi]+\frac{m^{2}}{2}e(q)\Bigl[\Phi,[\Phi,q]\Bigr]\right\rangle\,,
{\frac{dQ(\lambda)}{d\lambda}}\,=\,\left.\frac{\partialG}{\partialp}\right|_{q=Q(\lambda),p=P(\lambda)},\;{\frac{dP(\lambda)}{d\lambda}}\,=\,-\,\left.\frac{\partialG}{\partialq}\right|_{q=Q(\lambda),p=P(\lambda)}.
{\tilde{G}}[{\tilde{x}},{\tilde{x}^{\prime}}]=\sum_{i}\left[u_{i}({\tilde{x}})u_{i}^{*}({\tilde{x}^{\prime}})\right]\;\sum_{j}\left[v_{j}^{*}({\tilde{x}})\,v_{j}({\tilde{x}^{\prime}})\right].
\epsilon_{ab_{1}\cdotsb_{d-1}}\beta_{0}\bar{R}_{\beta_{1}}^{b_{1}b_{2}}\cdot\cdot\cdot\bar{R}_{\beta_{k}}^{b_{2k-1}b_{2k}}e^{b_{2k+1}}\!\cdot\cdot\cdote^{b_{d-1}}=0
T_{\nu}^{AB}(\lambda,z)=z\left\{\frac{\bar{J}_{\nu}^{2}(z)}{\bar{J}_{\nu}^{2}(\lambdaz)}\left[A^{2}+B^{2}(\lambda^{2}z^{2}-\nu^{2})\right]-A^{2}-B^{2}(z^{2}-\nu^{2})\right\}^{-1}.
\gamma^{0}=\left(\begin{array}{cc}{\sigma_{3}}&{0}\\{0}&{\sigma_{3}}\\\end{array}\right)\quad\gamma^{1}=\left(\begin{array}{cc}{-i\sigma_{2}}&{0}\\{0}&{-i\sigma_{2}}\\\end{array}\right)\quad\gamma^{2}=\left(\begin{array}{cc}{i\sigma_{1}}&{0}\\{0}&{-i\sigma_{1}}\\\end{array}\right)\quad\gamma^{3}=\left(\begin{array}{cc}{0}&{i\sigma_{1}}\\{i\sigma_{1}}&{0}\\\end{array}\right),
\overline{{\varphi}}_{\mu_{j}}^{a}=\sum_{\mathrm{\scriptsize{part}}\atop\alpha_{j}\inI_{a}}\prod_{b=1}^{N}\sum_{\mathrm{\scriptsize{perm}}}\frac{E_{-\alpha_{1_{a}}}^{b}E_{-\alpha_{2_{a}}}^{b}\ldotsE_{-\alpha_{q_{a}}}^{b}|\lambda^{b}>}{(\mu_{1_{a}}-\mu_{2_{a}})(\mu_{2_{a}}-\mu_{3_{a}})\ldots(\mu_{q_{a}}-Z_{q})}.
\begin{array}{l}{v_{k+1}=W\left[\left(2F(u_{k},v_{k})-F(u_{k},v_{k-1})\right)\,,u_{k}\right],}\\{u_{k+1}=\bar{W}\left[\left(2\bar{F}(u_{k},v_{k})-\bar{F}(u_{k-1},v_{k})\right)\,,v_{k}\right].}\\\end{array}
\begin{array}{rl}{{\calL}=}&{-\frac{\mu^{2}}{4}F_{\mu\nu}F^{\mu\nu}+\frac{\kappa}{2}\varepsilon^{\mu\nu\lambda}F_{\mu\nu}A_{\lambda}}\\{}&{+\frac{\kappa\mu}{1+\phi^{3}}\varepsilon^{\alpha\beta}\phi^{\alpha}D_{0}\phi^{\beta}-\frac{1}{2m}|D_{i}\phi^{a}|^{2}-\frac{\mu^{2}}{2}(1-\phi^{3})^{2}.}\\\end{array}
\begin{array}{ccccc}{\{L=(00000),M=5\}}&{+}&{\{L=(00000),M=7\}}&{=}&{\{L=(00001),M=6\}\nonumber}\\{(1,0,0,0)}&{+}&{(-4,-1,-8,5)}&{=}&{(-3,-1,-8,5)}\\\end{array}
{\calL}_{\mathrm{\tiny{4D}}}={\calL}_{\mathrm{\tiny{CFT}}}+{\frac{1}{4g^{2}}}F_{\mu\nu}F^{\mu\nu}+A_{\mu}J_{\mathrm{\tiny{CFT}}}^{\mu}++|D_{\mu}\phi|^{2}+c\left(\phi{\calO}_{4}+\mathrm{h.c.}\right).
\Gamma_{kmn}(\Box,\Box_{2},\Box_{3})X_{2}X_{3}(x)=\frac{4(-1)^{k+m+n}}{(k+m+n)!}\int_{-\infty}^{0}dq\,\Biggl(q^{k}\frac{d^{k}}{dq^{k}}K_{0}\left(\sqrt{2q\Box}\right)\Biggr){\calF}(q,x)
\sum_{N=0}^{\infty}\frac{e^{-\frac{N\beta}{2R^{+}}}}{N}\sum_{\stackrel{r|N}{r\odd}}\sum_{s=0}^{N/r-1}\longrightarrow\frac{\betaR^{+}}{\sqrt{2}2\pi\alpha^{\prime}}\int_{0}^{1}d\tau_{1}\int_{0}^{\infty}\frac{d\tau_{2}}{\tau_{2}^{2}}\sum_{\stackrel{r=1}{r\odd}}^{\infty}e^{-\frac{r^{2}\beta^{2}}{4\pi\alpha^{\prime}\tau_{2}}}
\delta_{b}x^{\mu}=b^{\mu}{}_{\nu}n^{\nu},\qquad\delta_{b}z^{\mu}=-\omega^{\mu}{}_{\nu}(\frac1{e}\Pi^{\nu}-B^{\nu}),\qquad\delta_{b}B^{\mu}=\frac1{e}\omega^{\mu}{}_{\nu}\dot{n}^{\nu},
\left[K_{\quad\quad\nu}^{(1)\mu}-\delta_{\nu}^{\mu}K^{(1)}\right]|_{y=0}={\frac{l}{2}}\left(R_{\\nu}^{\mu}-{\frac{1}{2}}\delta_{\nu}^{\mu}R\right)+\chi_{\\nu}^{\mu}={\frac{\kappa^{2}}{2}}T_{\\nu}^{\mu}\.
\int_{0}^{R}\mathrm{d}r\,\left[u_{1\epsilon\kappa}\left(r\right)u_{1\epsilon^{\prime}\kappa}\left(r\right)+u_{2\epsilon\kappa}\left(r\right)u_{2\epsilon^{\prime}\kappa}\left(r\right)\right]=\frac{u_{1\epsilon\kappa}\left(R\right)u_{2\epsilon^{\prime}\kappa}\left(R\right)-u_{1\epsilon^{\prime}\kappa}\left(R\right)u_{2\epsilon\kappa}\left(R\right)}{\epsilon-\epsilon^{\prime}}\,.
H_{n}={}-\frac12\left(\partial_{r}^{2}+r^{-1}\partial_{r}-r^{-2}\left(A_{\varphi}^{2}(r)-2iA_{\varphi}(r)\partial_{\varphi}-\partial_{\varphi}^{2}\right)\right)+\fracn2B(r)\sigma_{3},
V_{\mathrm{eff.}}^{T}(\phi_{c},\mu)=\frac{m^{2}}{2}\phi_{c}^{2}+\frac{\lambda}{4}\phi_{c}^{4}+(I_{1}+I_{1}^{\beta})-\frac{3\lambda}{2}(I_{0}+I_{0}^{\beta})^{2},
(T_{i}\otimesT_{j})_{{\calH}_{\{\alpha_{k},{\alpha\prime}_{k}\}}}=\prod_{k\nei,j}^{r}\delta_{\alpha_{k},{\alpha\prime}_{k}}(T^{a})_{\alpha_{i}{\alpha\prime}_{i}}(T^{a})_{\alpha_{j}{\alpha\prime}_{j}}
\underline{{\chi}}^{(0)}=1+\left(\frac14-\frac{4}{3r_{0}^{2}}\right)r^{2}+\frac{9}{32}\frac{\kappa^{2}}{r_{0}^{2}}r^{4}+\frac{3}{16}\frac{\kappa^{2}}{r_{0}^{2}}\left(\frac18-\frac{1}{r_{0}^{2}}\right)r^{6}.
<J^{a}(z)V_{1}(z_{1},\bar{z}_{1})\ldotsV_{n}(z_{n},\bar{z}_{n})>=\sum_{i=1}^{n}\frac{T_{i}^{a}}{z-z_{i}}<V_{1}(z_{1},\bar{z}_{1})\ldotsV_{n}(z_{n},\bar{z}_{n})>
S[A_{(p+1)},B_{(p+2)}]=\intd^{d}x\\sqrt{|g|}\\left[{\textstyle\frac{(-1)^{(p+1)}}{2\cdot(p+2)!}\frac{1}{e_{(p)}^{2}}}F_{(p+2)}^{2}+{\textstyle\frac{(-1)^{(p+2)}}{2\cdot(p+3)!}\frac{1}{e_{(p+1)}^{2}}}H_{(p+3)}^{2}\right]\,,
R_{H}=\prod_{i=1}^{N_{f}}\left[\frac{\sqrt{|m_{i}|^{2}+({\tilde{m}}_{i}+\frac{1}{2}M_{W})^{2}}+{\tilde{m}}_{i}+\frac{1}{2}M_{W}}{\sqrt{|m_{i}|^{2}+({\tilde{m}}_{i}-\frac{1}{2}M_{W})^{2}}+{\tilde{m}}_{i}-\frac{1}{2}M_{W}}\right]^{-k/2}\.
\deltae_{\mu}^{a}-\delta_{D}e_{\mu}^{a}=\frac{1}{e_{G}}v^{\alpha}[{\calD}_{\mu}^{ab}{[}f{]}e_{\alphab}-{\calD}_{\alpha}^{ab}{[}f{]}e_{\mub}-e_{G}E_{\mu\alpha}\psi^{a}v(\Phi)]
L_{g}=\frac{1}{2}g_{\mu\nu}{\dot{q}}^{\mu}{\dot{q}}^{\nu}+\frac{i}{2}\psi_{a}({\dot{\psi}}^{a}+{\stackrel{\circ}{\omega^{a}}}_{b\rho}\psi^{b}{\dot{q}}^{\rho}),
\tilde{W}\equiv\left|\begin{matrix}{f}&{f_{1}}&{\cdots}&{f_{n}}\\{Df}&{Df_{1}}&{\cdots}&{Df_{n}}\\{\vdots}&{\vdots}&{}&{\vdots}\\{D^{n}f}&{D^{n}f_{1}}&{\cdots}&{D^{n}f_{n}}\\\end{matrix}\right|=0\quad.
\begin{array}{rcl}{{\hat{u}}_{t}}&{=}&{{\hat{u}}({\hat{v}}_{x}+{\hat{w}}_{x})\,,}\\{{\hat{v}}_{t}}&{=}&{{\hat{v}}({\hat{w}}_{x}+{\hat{u}}_{x})\,,}\\{{\hat{w}}_{t}}&{=}&{{\hat{w}}({\hat{u}}_{x}+{\hat{v}}_{x})\,,}\\\end{array}
\vartheta^{N}({\alpha}^{i},\beta)\Omega_{p}(p^{+},\overline{p},x^{+})=\vartheta^{N}({\alpha}^{i},\beta)\mathrm{e}^{{\frac{ix^{+}}{2p^{+}}}(\overline{{p}}^{2}+2\alpha(N-\alpha(0)))}\Omega_{p}(p^{+},\overline{p},0)\;\;\;.
h_{ij}(m)=\!\int\!\mathrm{d}z\,g_{ab}(\phi_{\mathrm{sol}})\,\,\frac{\partial\phi_{\mathrm{sol}}^{a}}{\partialm^{i}}\,\,\frac{\partial\phi_{\mathrm{sol}}^{b}}{\partialm^{j}}\,,\qquadi,j=0,1,2,...,n\,,
\bar{S}_{D2}(M_{i})=2x_{D2}\frac{1}{N}\mathrm{Tr}\,M_{i}^{2}\\\\\\\\mathrm{or}\\\\\\\,bar{S}_{D4}(M_{i})=2x_{D4}\frac{1}{N}\mathrm{Tr}\,M_{i}^{4}
H^{(1)}=-\frac{1}{e^{2}}\intdx~[\Phi^{1}(e^{2}\Omega_{3}+G_{4}^{(0)})-\Phi^{2}(e^{2}\Omega_{2}+\Omega_{4}-e^{2}\partial_{1}\Omega_{1})-\Phi^{3}\Omega_{3}-\Phi^{4}\Omega_{2}].
v_{particle}={\frac{dE}{dp}}={\frac{\sqrt{E^{2}-(1-\lambdaE)^{2}m^{2}}}{E-\lambda^{2}m^{2}E+\lambdam^{2}}}={\frac{v}{1+\lambdam\sqrt{1-v^{2}}}}=\frac{p-\lambdaEp}{E-\lambdap^{2}}
\Phi({\bfr}_{1},\alpha_{1},\cdots,{\bfr}_{N},\alpha_{N};t)=\langle0|\frac{1}{\sqrt{N!}}\psi_{\alpha_{1}}({\bfr}_{1},t)\cdots\psi_{\alpha_{N}}({\bfr}_{N},t)|\Phi\rangle\;,
\left.+\;\delta_{M}\phi^{K}\cdot\left\{\Lambda_{K},\delta_{N}A_{\nu_{2}\ldots\nu_{q}}\right\}\;-\;\delta_{N}\phi^{K}\cdot\left\{\Lambda_{K},\delta_{M}A_{\nu_{2}\ldots\nu_{q}}\right\}\right]\cdot\Lambda^{\nu_{2}\ldots\nu_{q}}\quad.