harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{rcl}{\delta(i_{k}N^{(8)})}&{=}&{7\left\{\partial(i_{k}\Omega^{(7)})-15\partial(i_{k}{\tilde{\Lambda}})(i_{k}C^{(3)})+30\partial\Lambda^{(2)}(i_{k}C^{(3)})^{2}\right.}\\{}&{}&{}\\{}&{}&{\left.-20C^{(3)}(i_{k}C^{(3)})\partial(i_{k}\Lambda^{(2)})-(i_{k}N^{(7)})\partial\Lambda^{(0)}\right\}\,.}\\\end{array}
{\calR}=(2S_{cl})^{3k}\,\prod_{i=1}^{3}\,\left(S_{i}\right)^{-k}\left(\frac{(S_{i}^{2}+S_{cl}^{2})^{1/2}+S_{cl}}{(S_{i}^{2}+S_{cl}^{2})^{1/2}-S_{cl}}\right)^{-k/2}
M_{pl}^{2}={\frac{M_{X}^{3}}{k}}e^{-2k(y_{3}-y_{2})-2c}\left(-{\frac{1}{2}}e^{-2ky_{1}}+1-e^{-2k(y_{2}-y_{1})}+e^{-2k(2y_{2}-y_{1}-y_{3})}\right)~.~\,
R_{mn}{}^{pq}\rightarrowW_{mn}{}^{pq}-\frac{16}{d-2}\delta_{[m}{}^{[p}(\bar{\psi}_{|r|}\Gamma^{|r|}\psi_{n]}{}^{q]}-\bar{\psi}^{|r|}\Gamma^{q]}\psi_{n]r})\,.
W(J^{a},\sigma)\equiv-{\frac{1}{(2L)^{d-1}}}\left\langle0\right\vertP_{+}\left\vert0\right\rangle=-\left({\frac{Nm_{0}^{2}}{g_{0}}}\sigma-{\frac{N}{2g_{0}}}\sigma^{2}+V_{1-\mathrm{loop}}(\sigma)\right)+{\frac{J^{a}J^{a}}{2\sigma}},
p_{R}=(m_{i}-a_{i}^{I}(p^{I}-{\frac{1}{2}}a_{j}^{I}n^{j})+\xi_{ij}^{*}Q^{j}-{\frac{\xi_{ki}\xi_{kj}^{*}}{2}}n^{j}){\frac{{\bfe}^{*i}}{2R_{i}}}-n^{i}R_{i}{\bfe}_{i}.
2\kappa^{2}S_{\Phi}\equiv(n^{r}\nabla_{r}\Phi|_{r=r_{\mathrm{i}}+\epsilon}-n^{r}\nabla_{r}\Phi|_{r=r_{\mathrm{i}}-\epsilon})={\frac{1}{\sqrt{G_{rr}}}}(\Phi^{\prime}|_{r=r_{\mathrm{i}}+\epsilon}-\Phi^{\prime}|_{r=r_{\mathrm{i}}-\epsilon})\,.
f_{TTT}=-{\frac{2i}{\pi}}\frac{j_{T}(T)}{j(T)-j(U)}\left\{\frac{j(U)}{j(T)}\right\}\left\{\frac{j_{T}(T)}{j_{U}(U)}\right\}\left\{\frac{j(U)-j(i)}{j(T)-j(i)}\right\}\equiv2W(T,U).
E_{q}={\frac{1}{2\pig^{1/2}(L)}}\partial_{\bar{\beta}}\Gamma|_{\bar{\beta}=\beta_{H}}={\frac{1}{96\pi}}\int_{x_{+}}^{L}{\frac{1}{g}}({\frac{4}{\beta_{H}^{2}}}-g^{2}(x))dx-{\frac{1}{16\pig^{1/2}(L)}}g^{\prime}(L).
I=\frac{V}{2^{d}\piG_{d}}\intd^{d_{l}}x\sqrt{-g^{(d_{l})}}e^{d_{t}\psi/2}(e^{a\phi}R^{(d_{l})}+\frac{d_{t}(d_{t}-1)}{4}e^{a\phi}(D\psi)^{2}
0=4a\left(\frac{d\bar{\phi}}{dA}\right)^{2}-\frac{2}{P(A)}\frac{dP(A)}{dA}\frac{d\bar{\phi}}{dA}-\frac{\bar{f}_{0}^{2}}{2P^{2}(A)}+\frac{2\bar{Q}+\mue^{-4a\bar{\phi}}/\bar{Q}}{4P(A)},
\mathrm{D}_{z}^{(s)}(E_{i})=z^{s_{i}}E_{i}\,,~~~\mathrm{D}_{z}^{(s)}(F_{i})=z^{-s_{i}}F_{i}\,,~~~\mathrm{D}_{z}^{(s)}(h_{i})=h_{i}\,,~~~i=0,\cdots,r
\frac{\partial^{2}a_{n}}{\partialx^{2}}=\frac{1}{2}(\frac{1}{a_{n}-1}+\frac{1}{a_{n}})(\frac{\partiala_{n}}{\partialx})^{2}-\frac{1}{x}\frac{\partiala_{n}}{\partialx}-\frac{2}{x}a_{n}(a_{n}-1)+\frac{(n+1)^{2}}{2x^{2}}\frac{a_{n}-1}{a_{n}}.
S_{L}[g,\varphi]=\int_{M}\!\!\!\sqrt{g}\,d^{2}z\;\left({\frac{1}{2}}g^{ab}\partial_{a}\varphi\partial_{b}\varphi+R_{g}\varphi+{\frac{\mu}{2}}\mathrm{e}^{\varphi}\right)+\lambda\int_{\partialM}\!\!e\,ds\;\mathrm{e}^{\frac{\widetilde{\varphi}}{2}}\;\;\;,
\frac{\partial}{\partialX^{\mu_{0}}}j_{b}^{\mu_{0}\mu_{1}\ldots\mu_{p}}(X)=-g_{b}\sum_{i=0}^{p}\intd\xi_{b}^{0}d\xi_{b}^{1}\ldots\hat{d\xi_{b}^{i}}\ldotsd\xi_{b}^{p}\left[K_{bi}^{\mu_{1}\mu_{2}\ldots\mu_{p}}\delta^{(D)}(X-X_{b})\right]_{\xi_{b}^{i}=0}^{\xi_{b}^{i}=l_{b}^{i}},
\alpha(\rho,\theta)=\left(\begin{array}{ll}{e^{i\frac{1}{2}\theta}}&{\rho}\\{0}&{e^{-i\frac{1}{2}\theta}}\\\end{array}\right),\,\,p_{R}\sim\left(\begin{array}{ll}{2}&{0}\\{0}&{0}\\\end{array}\right),
F(s)=\left(\frac{ms^{1/2}}{2}\right)^{2}\left(\frac{4}{s^{1/2}}K_{1}^{2}(ms^{1/4})+m^{2}K_{0}^{2}(ms^{1/4})+\frac{4m}{s^{1/4}}K_{0}(ms^{1/4})K_{1}(ms^{1/4})\right).\nonumber
\mathbf{L}^{2}=\left(\begin{array}{cccc}{0}&{}&{}&{}\\{}&{\fbox{\ensuremath{\begin{array}{rr}{0}&{1}\\{0}&{0}\\\end{array}}}}&{}&{}\\{}&{}&{\fbox{\ensuremath{\begin{array}{rr}{0}&{1}\\{0}&{0}\\\end{array}}}}&{}\\{}&{}&{}&{\ddots}\\\end{array}\right).
J.\left[\begin{array}{c}{\xi_{Y}}\\{\xi_{Z}}\\\end{array}\right]=\left[\begin{array}{c}{-Y(Z^{*}\xi_{Y}^{*}+Y^{*}\xi_{Z}^{*})}\\{Z(Z^{*}\xi_{Y}^{*}+Y^{*}\xi_{Z}^{*})}\\\end{array}\right]
P_{\alpha}^{~\beta}=\left(\begin{array}{cc}{\frac{1}{M^{2}}S^{\mu\sigma}P_{\sigma}P^{\nu}}&{S^{\mu\nu}}\\{\frac{1}{M^{2}}P^{\mu}P^{\nu}}&{(g^{\mu\nu}+\frac{1}{M^{2}}P^{\mu}S^{\nu\sigma}P_{\sigma})}\\\end{array}\right).
Z(F_{1},F_{2},L)=C_{1}C_{2}\int_{0}^{\infty}{\frac{dt}{2p^{+}}}{_{(-1)}\langle}B,y_{2}|e^{-(P_{cl}^{-}-p^{-})t}\hat{R}(M^{p_{2}T}M^{p_{1}})|B,y_{1}\rangle_{(-1)},
\curvearrowright(\bar{\psi}_{\alpha_{1}^{\prime}...\alpha_{n}^{\prime}}S_{\alpha_{1}...\alpha_{n}}^{\alpha_{1}^{\prime}...\alpha_{n}^{\prime}})(\theta_{1}+i\pi...\theta_{n}+i\pi)=\psi_{\alpha_{1}...\alpha_{n}}(\theta_{1},...\theta_{n})
(\alpha,\beta)=\int_{\Sigma}^{}\mathrm{tr}(\alpha\wedge{}^{\ast}\beta)=\int_{\Sigma}^{}\,\frac{1}{m!}\alpha_{\mu_{1}\cdots\mu_{m}}^{a}\beta_{a}^{\mu_{1}\cdots\mu_{m}}\sqrt{g}\,d^{3}\!x\;,
h_{2}(\theta)={\frac{-1}{\Gamma(1-\nu)\Gamma(\nu)}}\{\frac{d}{d\theta}}\int_{0}^{\theta}(\theta-\theta^{\prime})^{-\nu}\mathrm{e}^{\mathrm{i}(r+\mu)\theta^{\prime}}\Psi(\theta^{\prime})d\theta^{\prime},\tag{B.36}
S\sim\intd^{10}x\sqrt{g}\partial_{\lambda}h_{\mu\nu}\partial^{\lambda}h_{\mu\nu}+\dots=\intdy~e^{-2A(y)}\sqrt{g(y)}\psi^{2}(y)\cdot\intd^{4}x\partial_{\hat{\lambda}}\hat{h}_{\mu\nu}(x)\partial^{\hat{\lambda}}\hat{h}_{\mu\nu}(x)+\dots
=F[g_{i_{1}}^{1}g_{i_{2}}^{2}...g_{i_{n}}^{n};(g^{1}h^{1}{(g^{1})}^{-1})_{k_{1}}(g^{2}h^{2}{(g^{2})}^{-1})_{k_{2}}...(g^{n}h^{n}{(g^{n})}^{-1})_{k_{n}}]
\{\Phi_{n},\beta\}=\{p_{n},\beta\}=-\frac{1}{\sqrt{L}}\int_{-L/2}^{L/2}dx\left[V^{\prime\prime}(\phi)-V^{\prime\prime}\left(\frac{q_{0}}{\sqrt{L}}\right)\right]e^{-ik_{n}x}\equiv-\frac{\alpha_{n}}{\sqrt{L}}\;,
L_{m}=\frac{1}{2}\left(\partial_{\mu}\Phi_{1}-eA_{\mu}\Phi_{2}\right)^{2}+\frac{1}{2}\left(\partial_{\mu}\Phi_{2}+eA_{\mu}\Phi_{1}\right)^{2}-\frac{1}{2}m^{2}\Phi^{2}+\frac{1}{2}\xiR\Phi^{2}-\frac{1}{4!}\lambda\Phi^{4}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu},
\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}},
\begin{array}{llllll}{\displaystyleD}&{=3}&{:\quad}&{Sp(4;R)}&{\to}&{OSp(N;1|R)}\\{\displaystyleD}&{=4}&{:\quad}&{SU(2,2)}&{\to}&{SU(2,2|N)}\\{\displaystyleD}&{=6}&{:\quad}&{U_{\alpha}(4;H)}&{\to}&{UU_{\alpha}(N;4|R)}\\\end{array}
\left(\begin{array}{c}{{x^{0}}^{\prime}}\\{\vec{x}^{\prime}}\\{1}\\\end{array}\right)=\left(\begin{array}{c|c}{\Lambda}&{\begin{matrix}{a^{0}}\\{\vec{a}}\\\end{matrix}}\\\hline{0}&{1}\\\end{array}\right)\left(\begin{array}{c}{x^{0}}\\{\vec{x}}\\{1}\\\end{array}\right)
\begin{array}{rcl}{P_{a,b}(\mu,\nu;2\lambda;q)}&{=}&{q^{\lambda(2(a+b)\lambda-a+b)}\displaystyle\left[\begin{array}{c}{\mu+\nu}\\{\mu-(a+b)\lambda}\\\end{array}\right]_{q},}\\{P_{a,b}(\mu,\nu;2\lambda-1;q)}&{=}&{q^{(2\lambda-1)((a+b)\lambda-b)}\displaystyle\left[\begin{array}{c}{\mu+\nu}\\{\mu-(a+b)\lambda+b}\\\end{array}\right]_{q}.}\\\end{array}
\begin{array}{l}{N=4,\quad\alpha=4,2,1,-1}\\{\displaystyleT(f)=f\partial_{t}+\frac{1}{4}f^{\prime}(x\partial_{x}+2y\partial_{y})}\\{\displaystyle\qquad\qquad\qquad-\frac{1}{4}(wf^{\prime}+xyf^{\prime\prime})\partial_{w},}\\{\displaystyleY(g)=g\partial_{y}-\frac{1}{2}xg^{\prime}\partial_{w},}\\{\displaystyleX(h)=h\partial_{x}-yh^{\prime}\partial_{w},\quadW(k)=k\partial_{w}.}\\\end{array}
\mathcal{L}_{NGT}=\left(e\widetilde{e}\right)^{\frac{1}{2}}\left[e_{a}^{\mu}\widetilde{e}_{b}^{\nu}R_{\mu\nu}\,^{ab}\left(^{c}\omega_{\alpha}\right)+\frac{1}{8}e_{b}^{\mu}\widetilde{e}^{\nub}R_{\mu\nu}\,^{a}\,_{a}\left(^{c}\omega_{\alpha}\right)\right].
[(p_{1}^{+}p_{2}^{-}p_{3}^{+}p_{4}^{-}p_{5}^{+}p_{6}^{-})(\epsilon_{1}^{-}\epsilon_{2}^{+})(\epsilon_{3}^{-}\epsilon_{4}^{+}\epsilon_{5}^{-}\epsilon_{6}^{+})]~=~0\,
S=\int\!\!d^{4}\!x\,\Biggl(-\frac{1}{2g^{2}}\sum_{j=1}^{N}\mathrm{tr}\,F_{j}^{2}+f_{s}^{2}\sum_{j=1}^{N}\mathrm{tr}\,\Bigl[(D_{\mu}U_{j,j+1})^{\dagger}D^{\mu}U_{j,j+1}\Bigr]+\dotsb\Biggr)
\left(\begin{array}{cc}{\partial_{1}}&{\partial_{2}}\\{\partial_{4}}&{\partial_{3}}\\\end{array}\right)\mapsto\left(\begin{array}{cc}{\partial_{1}q^{\alphaL_{6}}}&{\partial_{2}}\\{\partial_{4}}&{\partial_{3}q^{-\alphaL_{6}}}\\\end{array}\right)\equiv\tilde{\partial}\quad,
W_{cl}=-{\frac{1}{16\piG}}\int_{M^{4}}^{}R^{(4)}\sqrt{g}d^{4}x+{\frac{1}{16\piG}}\int_{M^{4}}^{}F_{\mu\nu}^{2}\sqrt{g}d^{4}x-{\frac{1}{8\piG}}\int_{\partialM^{4}}^{}K^{(4)}\sqrt{h}d^{3}x,
{\hat{d}}_{2}^{\mu\nu}(k)=\Bigl(g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}\Bigr)\frac{4m^{2}}{2\pi}\frac{1}{k^{2}\sqrt{1-4m^{2}/k^{2}}}{\mathrm{sgn}}(k^{0})\Theta(k^{2}-4m^{2})\quad.
\delta\left(\begin{matrix}{x^{0}}\\{x^{1}}\\\end{matrix}\right)=\left(\begin{matrix}{a}&{b}\\{q^{-1}b^{\mathrm{T}}}&{a^{\mathrm{T}}}\\\end{matrix}\right)\otimes\left(\begin{matrix}{x^{0}}\\{x^{1}}\\\end{matrix}\right),
e^{-2\phi}=\left((1+\sum_{1}^{s}{\frac{2\tilde{m}_{k}}{r_{k}}})(1+\sum_{1}^{s}{\frac{2\hat{m}_{l}}{r_{l}}})-\left[\sum_{1}^{s}{\frac{2(q_{k})_{i}}{r_{k}}}\right]^{2}\right)^{\frac{1}{2}}\.
G(p)=\int\frac{d^{3}q}{(2\pi)^{3}}\,\frac{1}{(p-q)^{2}}\frac{\left[2\left(\frac{\bf{p.q}}{\left|{\bfp}\right|\left|{\bfq}\right|}\right)^{2}-1\right]\,\,G(q)}{\left|-\frac{1}{2}\delta+iq_{3}+\frac{\left|{\bfq}\right|^{2}}{2m}+\Sigma_{\pm}(q_{3},\left|{\bfq}\right|)\right|^{2}}\,.
\begin{aligned}{U(1)_{a}}&{{}:\,(\Phi_{1},\Phi_{2},\Phi_{3},\Phi_{4})\sim(e^{i\theta_{a}}\Phi_{1},e^{i\theta_{a}}\Phi_{2},e^{i\theta_{a}}\Phi_{3},\Phi_{4})\,,}\\{U(1)_{b}}&{{}:\,(\Phi_{1},\Phi_{2},\Phi_{3},\Phi_{4})\sim(e^{i\theta_{b}}\Phi_{1},\Phi_{2},\Phi_{3},e^{i\theta_{b}}\Phi_{4})}\\\end{aligned}
\begin{aligned}{\frac{\partial\psi_{ik}(u,\lambda)}{\partialu_{j}}}&{{}=\beta_{ij}(u)\psi_{jk}(u,\lambda),\quadi\nej}\\{\sum_{k=1}^{n}\frac{\partial\psi_{ij}(u,\lambda)}{\partialu_{k}}}&{{}=\lambda\psi_{ij}(u,\lambda),}\\\end{aligned}
\frac{\mathrm{d}}{\mathrm{d}z}p+\frac{\mathrm{d}}{\mathrm{d}\bar{z}}\bar{p}=\frac{\mathrm{d}}{\mathrm{d}z}\frac{\partial{\calL}}{\partialv}+\frac{\mathrm{d}}{\mathrm{d}\bar{z}}\frac{\partial{\calL}}{\partial\bar{v}}=\partial_{\varphi}{\calL}\,.
\bar{\psi}^{2C}|0>\sim\left(\varepsilon_{\alpha_{1}\cdots\alpha_{C}}\bar{\psi}^{a_{1}\alpha_{1}}\cdots\bar{\psi}^{a_{C}\alpha_{C}}\right)\left(\varepsilon_{\beta_{1}\cdots\beta_{C}}\bar{\psi}^{b_{1}\beta_{1}}\cdots\bar{\psi}^{b_{C}\beta_{C}}\right)|0>.
\frac{d^{2}F(0)}{d^{2}t}=-K_{L}(\mathcal{I})\star[Q,K]_{L}(\mathcal{I})+[Q,K]_{L}(\mathcal{I})\starK_{L}(\mathcal{I})=-[[Q,K],K]_{L}(\mathcal{I})=E_{L}(\mathcal{I})
S_{inv}^{A}=\int\!\!d^{3}x~\left(\frac1{12}H_{\mu\nu\alpha}H^{\mu\nu\alpha}+\frac12\partial_{\mu}\varphi\partial^{\mu}\varphi+\fracm2\epsilon^{\mu\nu\alpha}B_{\mu\nu}\partial_{\alpha}\varphi\right),
I_{\mu\nu}^{ab}(p)\;=\;\frac{g}{2^{4}}\;\delta^{ab}\;p\;\{\frac{1}{2}\;[3\,\frac{p_{\mu}p_{\nu}}{p^{2}}\;-\;\delta_{\mu\nu}]\;+\;(\frac{\lambda-1}{\lambda})\;[\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}]\}\;.
\left.\begin{array}{l}{f(\phi)=e^{K_{\mathrm{dil}}(S,S^{*})}\left(|K_{S}|^{2}K^{S\,S^{*}}-3\right),\}\\{widehatV=0,}\\{V_{0}(\tau_{1,2})=\left(e^{K_{\mathrm{matt}}}|W|^{2}\right)_{T_{1,2}}.}\\\end{array}\right.
\left(\begin{array}{c}{\psi_{1}}\\{\psi_{2}}\\\end{array}\right)\longrightarrow\left(\begin{array}{c}{\widetilde\psi_{1}}\\{\widetilde\psi_{2}}\\\end{array}\right)=\left(\begin{array}{c}{a}\\{c}\\\end{array}\begin{array}{cc}{b}\\{d}\\\end{array}\right)\left(\begin{array}{c}{\psi_{1}}\\{\psi_{2}}\\\end{array}\right),
\begin{array}{rl}{g_{\mu\nu}}&{\rightarrow\(3,3)+58(1,1)\nonumber}\\{\phi,\chi}&{\rightarrow\2(1,1)\nonumber}\\{B_{\mu\nu}^{(1)},B_{\mu\nu}^{(2)}}&{\rightarrow\2(3,1)+2(1,3)+44(1,1)\nonumber}\\{A_{\mu\nu\rho\lambda}^{+}}&{\rightarrow\3(1,3)+19(3,1)+(1,1).}\\\end{array}
\bar{\mu}^{\dot{A}}=\left(x^{B\dot{A}}-i\Theta^{B}\bar{\Theta}^{\dot{A}}\right)\lambda_{B}+2\bar{z}^{\dot{A}\dot{B}}\bar{\lambda}_{\dot{B}}-i\bar{\Theta}^{\dot{A}}\bar{\Theta}^{\dot{B}}\bar{\lambda}_{\dot{B}}
G(x,x^{\prime})=\int\frac{d\omega}{2\pi}\,e^{-i\omega(\tau-\tau^{\prime})}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}\,e^{i{\bfk}\cdot({\bfr}-{\bfr}^{\prime})}\,{\calG}(\omega,{\bfk};z,z^{\prime}).
I_{i}=\frac{h_{1}^{i}+h_{2}^{i}}{\alpha^{\prime}\pi}\left[\sum_{n=1}^{\infty}\left(\overline{{b}}_{n}^{i}b_{n}^{i}-\overline{{a}}_{n}^{i}a_{n}^{i}\right)-\frac{\left|\epsilon^{i}\right|}{\epsilon^{i}}\overline{{a}}_{0}^{i}a_{0}^{i}\right],
<\psi_{1},\psi_{2}>=\int_{\Sigma}\sqrt{h}dx^{D-1}\bar{\psi}_{1}\gamma^{t}\psi_{2}~~~,~~~(\psi_{1},\psi_{2})=\int_{\Sigma}\sqrt{h}dx^{D-1}\bar{\psi}_{1}\tilde{\gamma}^{t}\psi_{2}~~~.
\widetilde{q}_{p}\widetilde\phi_{p^{\prime}}+\widetildeq_{p^{\prime}}\widetilde\phi_{p}=\pm\left[\frac{q_{p}^{1}q_{p^{\prime}}^{1}}{|\kappa_{1}|}-\frac{q_{p}^{2}q_{p^{\prime}}^{2}}{|\kappa_{2}|}\right]\,,
A_{\mu}=\pm\,[\,{\frac{i}{g}}\,(\,H_{2}-H_{1})\,{\frac{v_{\mu}^{1}}{\rho_{1}}}+\kappa\,E_{1\,2}^{\pm}\,R_{\mu}^{1}]\pm\,[\,{\frac{i}{g}}\,(H_{4}-H_{3})\,{\frac{v_{\mu}^{3}}{\rho_{3}}}+\kappa\,E_{3\,4}^{\pm}\,R_{\mu}^{3}\,].
W=\frac{1}{\sqrt{1+\mid\xi\mid^{2}}}~\left(\begin{array}{cc}{1}&{-\zeta^{\ast}}\\{\zeta}&{1}\\\end{array}\right)~\left(\begin{array}{cc}{e^{i\frac{\alpha}{2}}}&{0}\\{0}&{e^{-i\frac{\alpha}{2}}}\\\end{array}\right)
{\frac{d\sigma}{d\varphi}}={\frac{\vec{\rho}\;\vec{\jmath}}{j_{p}}}=|F(\varphi-\varphi_{p})|^{2}\;{\frac{\psi^{\dagger}(\vec{p})\;G^{\dagger}(\varphi-\varphi_{p})\;\alpha_{\rho}(\varphi)\;G(\varphi-\varphi_{p})\;\psi(\vec{p})}{\psi^{\dagger}(\vec{p})\;\alpha_{\vec{p}}(\varphi)\;\psi(\vec{p})}}
{\calL}_{d}[h]=-{\frac{1}{4}}(h_{\mu\nu}{}^{i})^{2}+{\frac{1}{12\eta^{2}}}(H_{\mu\nu\lambda}{}^{i})^{2}+{\frac{1}{4\gamma^{4}}}(\partial^{\lambda}H_{\lambda\mu\nu}{}^{i})^{2}
h_{\mu\nu}=-\frac{\kappa_{5}^{2}}{\int_{0}^{r_{c}}dza^{2}}\frac{1}{\Box^{(4)}}(T_{\mu\nu}^{+}-\frac{1}{3}T^{+}\eta_{\mu\nu})-\left.\frac{U_{B}}{2}\right|_{0}\xi^{+}\eta_{\mu\nu},
(\frac{\alpha^{\prime}}{2})^{\frac{(r+s)}{2}}\frac{\sqrt{\alpha^{\prime}}}{\pi}\mu^{\frac{(r+s)}{2}}|\mathrm{ln}\mu|\sum_{k=0}^{r}\left(\begin{array}{c}{r}\\{k}\\\end{array}\right)\left(\begin{array}{c}{s}\\{{\frac{(s+r)}{2}}-k}\\\end{array}\right)\frac{4({\frac{r}{2}}-k)^{2}}{\alpha^{\prime}p^{2}-4({\frac{r}{2}}-k)^{2}},
{}_{I}\langle\ell,s|\tilde{q}^{H^{(c)}}|\ell^{\prime},s^{\prime}\rangle_{I}=\delta_{\ell,\ell^{\prime}}\delta_{s,s^{\prime}}(-1)^{s}\chi_{\ell}^{(N-2)}(\tilde{q})\chi_{s}^{SO(8)}(\tilde{q})~,
J_{N}^{2}=\left|\begin{array}{ccccccc}{N}&{s_{1}}&{s_{2}}&{s_{3}}&{\ldots}&{}&{s_{N-1}}\\{s_{1}}&{s_{2}}&{s_{3}}&{}&{}&{}&{s_{N}}\\{s_{2}}&{s_{3}}&{}&{}&{}&{}&{}\\{s_{3}}&{}&{}&{}&{}&{}&{}\\{\vdots}&{}&{}&{}&{}&{}&{}\\{s_{N-1}}&{s_{N}}&{}&{}&{}&{}&{s_{2N-2}}\\\end{array}\right|
(1+2c^{-2}U-c^{-2}\sum_{j=1}^{3}\left(\frac{dx^{j}}{dt}\right)^{2})^{1/2}\frac{d}{dt}\left((1+2c^{-2}U-c^{-2}\sum_{j=1}^{3}\left(\frac{dx^{j}}{dt}\right)^{2})^{-1/2}\frac{dx^{i}}{dt}\right)+\frac{\partialU}{\partialx^{i}}=0
\omega=-\frac{i\kappa}{2}\sum_{r,s=1}^{N}\left(\frac{2R^{2}\delta_{rs}}{\left(1+|z_{r}|^{2}\right)^{2}}+\frac{\partial\bar{b}_{s}}{\partialz_{r}}\right)dz_{r}\wedged\bar{z}_{s}.
Tr(K(s)-K_{0}(s))={\frac{1}{(4\pis)^{\omega}}}\intd^{2\omega}x(e^{-\Phi^{2}(x)s}(1-s^{2}({\frac{1}{6}}\Box\Phi^{2}(x)-{\frac{1}{12}}(\nabla\Phi^{2}(x))^{2}+...)-1)
\begin{array}{lcr}{{V^{\prime}}_{a}=V_{a},\quada=0,\ldots,n-2}\\{{V^{\prime}}_{n-1}={\frac{1}{2}}(V_{n-1}+V_{n})}\\{\tilde{V}_{l}=\tilde{V}_{l},\quadl=n,\ldots,n+3}\\{{q^{\prime}}_{j}^{a}=q_{j}^{a},\quada=0,\ldots,n-3.}\\\end{array}
{\calX}=a{\calX}_{1}+b{\calX}_{2}=a\left(\begin{array}{ccc}{0}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{-1}\\\end{array}\right)+b\left(\begin{array}{ccc}{0}&{0}&{0}\\{0}&{0}&{1}\\{0}&{1}&{0}\\\end{array}\right).
\alpha=\frac{\lambda}{e^{2}}=\frac{1}{2}\left(\frac{m_{H}}{m_{W}}\right)^{2}\,,\quad\beta=\frac{\eta^{2}}{M_{7}^{5}}\,,\quad\gamma=\frac{\Lambda_{7}}{e^{2}\eta^{4}}\,,\quad\epsilon_{i}=\frac{f_{i}}{e^{2}\eta^{4}}\,,
\deltaS=\frac{1}{4\pi}\,\mathrm{Im}\,\,\left\{-\frac{i}{24}\int\!\!d^{12}z\D^{\alpha{\mathbfj}}L_{{\mathbfi}{\mathbfj}}D_{\alpha}^{\mathbfi}{\calF}-3\int\!\!d^{8}z_{+}\sigma{\calA}{\calA}_{D}-\frac{i}{48}\int\!\!d^{12}z\(H-{\bar{H}}){\bar{\calA}}{\calA}_{D}\right\}.
\rho(z)\;=\;{\frac{z}{\pi}}\sqrt{\frac{b-z}{z-a}}\left[{\frac{1}{4\beta}}-{\frac{d-2}{2}}\int_{a}^{b}{\frac{dy}{y+z}}\sqrt{\frac{y+a}{y+b}}{\frac{\rho(y)}{y}}\right]\;\;\;\;\;\;\;\;\;\;\mathrm{for}\;\;\;\;\beta<\beta_{c}\;.
S_{{\mathrm{\scriptsize~{dual}}}}=\frac12\int_{M}d^{2}x\left(G_{jk}\partial^{a}X^{j}\partial_{a}X^{k}+G^{\alpha\beta}\partial^{a}Y_{\alpha}^{+}\partial_{a}Y_{\beta}^{+}+G_{\alpha\beta}\partial^{a}X_{-}^{\alpha}\partial_{a}X_{-}^{\beta}\right)\,,
S_{W}=-\intd^{2}x\left(F_{i}{\frac{\partialW}{\partial\phi_{i}}}~+~{\frac{\partial^{2}W}{\partial\phi_{i}\partial\phi_{j}}}\psi_{-i}\psi_{+j}~+~{\overline{F}}_{i}{\frac{\partial\overline{{W}}}{\partial\overline{{\phi}}_{i}}}~-~{\frac{\partial^{2}\overline{{W}}}{\partial\overline{{\phi}}_{i}\partial\overline{{\phi}}_{j}}}\overline{{\psi}}_{-i}\overline{{\psi}}_{+j}\right)\.
\sigma_{1}=\left(\begin{matrix}{0}&{1}\\{1}&{0}\\\end{matrix}\right),\qquad\sigma_{2}=\left(\begin{matrix}{0}&{-i}\\{i}&{0}\\\end{matrix}\right)\qquad\sigma_{3}=\left(\begin{matrix}{1}&{0}\\{0}&{-1}\\\end{matrix}\right)
{{\calH}}_{\mathrm{coll}}=\frac{1}{2}{\calI}^{\alpha\beta}I_{\alpha}I_{\beta}-{\calI}^{\alpha\beta}I_{\alpha}\overline{{J_{\beta}^{(2)}}}+\frac{1}{2}{\calI}^{\alpha\beta}\overline{{J_{\alpha}^{(2)}J_{\beta}^{(2)}}}.
\begin{array}{llllllllll}{\mathrm{~k~odd}\;\;\;}&{M_{0}}&{>}&{M_{k-1}}&{>}&{M_{2}}&{>}&{M_{k-3}}&{>\cdots>}&{M_{(k-1)/2}\;,}\\{\mathrm{~k~even}\;\;\;}&{M_{k}}&{>}&{M_{0}}&{>}&{M_{k-2}}&{>}&{M_{2}}&{>\cdots>}&{M_{k/2}\;.}\\\end{array}
{\hat{G}}({\hat{\xi}}{\hat{\eta}};{\hat{\xi}}^{\prime}{\hat{\eta}}^{\prime})=\sum_{n}{\phi}_{n}({\hat{\xi}}{\hat{\eta}};P){\phi}_{n}^{*}({\hat{\xi}}^{\prime}{\hat{\eta}}^{\prime};P)/(P^{2}+M^{2})
\Gamma_{\mu\nu\lambda}^{kln}(p,q,r)=-(2\pi)^{4}\left(\frac{2}{g^{\prime}}C_{kl}^{n}\right)\left[\delta_{\mu\lambda}(r+q-p)_{\nu}+\delta_{\nu\lambda}(q-r)_{\mu}+\delta_{\mu\nu}(p-q-r)_{\lambda}\right]\,.
\begin{array}{c}{U(\Lambda)\left|p,s_{3}\right\rangle=\sum_{s_{3}^{^{\prime}}}\left|\Lambdap,s_{3}^{^{\prime}}\right\rangleD_{s_{3}^{^{\prime}}\,s_{3}}(R(\Lambda,p))\,\,\,\,\,\mathrm{and~for~translations}\,:}\\{\,U(a)\,\left|p,s_{3}\right\rangle=e^{ipa}\,\,\left|p,s_{3}\right\rangle\,}\\\end{array}
f^{(0)}(x,y)=\left(\begin{array}{ccccc}{0}&{-1}&{0}&{0}&{0}\\{1}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{-1}\\{0}&{0}&{0}&{1}&{0}\\\end{array}\right)\delta(x-y).
\left(-{\frac{1}{r}}{\frac{\partial}{\partialr}}r{\frac{\partial}{\partialr}}+{\frac{\left(m-\delta{\frac{r^{2}}{R^{2}}}\right)^{2}}{r^{2}}}-{\frac{g}{2}}\delta{\frac{2}{R^{2}}}\right)\psi_{m}(r)=\epsilon\psi_{m}(r)\,.
S=\frac{1}{16\pi}\intd^{4}x\sqrt{-g}\biggl\{-R+\frac{1}{2}\frac{\partial_{\mu}\lambda\partial^{\mu}\overline{{\lambda}}}{({\mathrm{Im}}\;\lambda)^{2}}-\sum_{n=1}^{N}F_{\mu\nu}^{(n)}{}^{\star}\tilde{F}^{(n)\mu\nu}\biggr\}\;.
\left(\begin{matrix}{e^{i\lambda_{1}}}&{0}\\{0}&{e^{i\lambda_{2}}}\\\end{matrix}\right)=e^{-i(\lambda_{1}+\lambda_{2})/2}\left(\begin{matrix}{e^{-i\lambda/2}}&{0}\\{0}&{e^{i\lambda/2}}\\\end{matrix}\right),
L_{ij}=\left(\begin{array}{cccc}{0}&{k_{1}-k_{3}}&{-k_{1}+k_{2}}&{k_{3}-k_{2}}\\{-k_{1}+k_{3}}&{0}&{k_{1}}&{-k_{3}}\\{k_{1}-k_{2}}&{-k_{1}}&{0}&{k_{2}}\\{-k_{3}+k_{2}}&{k_{3}}&{-k_{2}}&{0}\\\end{array}\right).
e_{\mu}^{a}(x)=-\frac{4\piG}{\lambda}p^{a}(\xi)\epsilon_{\mu\nu}{\frac{\dot{X}^{\nu}(\xi)}{\dot{X}^{0}(\xi)}}\delta(\sigma-X^{1}(\xi))=-\frac{2\piG}{\lambda}p^{a}(\xi)\partial_{\mu}\epsilon(\sigma-X^{1}(\xi))
(x-x^{\prime})^{2}\,\int\!\mathrm{d}\rho^{2}\mathrm{d}^{4}x_{0}\,\frac{\rho^{8}}{\left[(x-x_{0})^{2}+\rho^{2}\right]^{4}\left[(x^{\prime}-x_{0})^{2}+\rho^{2}\right]^{4}}\,.
w(r)=\frac{g^{2}}{2\pi^{2}Lr^{6}}\sum_{q_{3}}\frac{q_{3}^{2}e^{-2\omega_{3}r}}{\omega_{3}^{4}}\left(3+6(\omega_{3}r)+5(\omega_{3}r)^{2}+2(\omega_{3}r)^{3}+(\omega_{3}r)^{4}\right)
T_{1}=\frac{-N_{f}/2+N_{c}}{2\pii},\\T_{0}=-\frac{1}{4\pii}\sum_{i=1}^{N_{f}}m_{i},\\\frac{\partialF}{\partialT_{1}}=2u-\frac{1}{2}\sum_{i=1}^{N_{f}}m_{i}^{2}
\sum\frac{f(z_{n})}{s^{\prime}(z_{n})}=w+\int_{0}^{\infty}{f(x)dx}+\int_{0}^{\infty}{\left[\frac{f(ix)}{e^{-2\piis(ix)}-1}-\frac{f(-ix)}{e^{2\piis(-ix)}-1}\right]dx},
\mathrm{d}s^{2}=g_{tt}(r)\;\mathrm{d}t^{2}+g_{rr}(r)\;\mathrm{d}r^{2}+2\;g_{t\phi}(r)\;\mathrm{d}t\,\mathrm{d}\phi+g_{\phi\phi}(r)\;\mathrm{d}\phi^{2}\,.
\tilde{F}(\tilde{R})=\frac{\pi^{2}\beta^{2}}{2\mu^{4}}\left(\tilde{R}^{2}\,\langle\phi\phi\rangle-4\tilde{R}\int^{\tilde{R}}\!d\tilde{R}^{\prime}\,\langle\phi\phi\rangle+6\int^{\tilde{R}}\!d\tilde{R}^{\prime}\int^{\tilde{R}^{\prime}}\!d\tilde{R}^{\prime\prime}\,\langle\phi\phi\rangle\right)+\alpha_{1}+\alpha_{2}\tilde{R},\nonumber
{\mathcalJ_{l+\nu}}(\tilde{\xi};\tilde{k})=\frac{1}{2}\left[w_{l+\nu}(\tilde{\xi};\tilde{k})\right]^{2}{\mathcalM}_{l+\nu}^{(<)}(\tilde{\xi};\tilde{k})+{\mathcalU}_{l+\nu}(\tilde{\xi};\tilde{k})\;,
\underline{{hs_{\bfR}(2,2)}}:\;\;\;\;f=\alpha_{i\bar{j}}a^{i}\bar{a}^{\bar{j}}+\alpha_{i_{1}i_{2}i_{3}\bar{j}_{1}\bar{j}_{2}\bar{j}_{3}}a^{i_{1}}a^{i_{2}}a^{i_{3}}\bar{a}^{\bar{j}_{1}}\bar{a}^{\bar{j}_{2}}\bar{a}^{\bar{j}_{3}}+\ldots
0=\frac{\partial}{\partial{\bfr}_{C}}U_{\mathrm{eff}}=-\frac{\lambda_{C}{\bfr}_{C}}{r_{C}^{3}}\left((\tilde{q}_{C})^{2}-\sum_{I=1}^{5}\sum_{A,B=1}^{r-1}C_{CA}C_{CB}\tilde{a}_{I}^{A}\tilde{a}_{I}^{B}\right)
d{}^{*}\!F\rightarrow{\calD}{}^{*}\!F=\frac{1}{3!}\left(\partial_{\alpha}{}^{*}\!F_{\beta\gamma}+\Gamma_{\rho\alpha}^{\rho}{}^{*}\!F_{\beta\gamma}\right)\delta_{\mu\nu\omega}^{\alpha\beta\gamma}\,dx^{\mu}\wedgedx^{\nu}\wedgedx^{\omega},
{\calH}=\frac{s}{2c}\left[\imath{\sqrt{2}}{\bfp}({\bf{\bar{z}}}_{1}-{\bf{z}}_{1})+\imath\sum_{a=1}^{N-1}{\tilde{k}}_{a}({\bfz}_{a}{\bf{\bar{z}}}_{a+1}-{\bfz}_{a+1}{\bf{\bar{z}}}_{a})+{\tilde{k}}_{N}({\bfz}_{N}{\bf{\bar{z}}}_{N}-c^{2})\right].
\dot{\alpha}_{0}^{2}=\frac{1}{l^{2}}\left(\frac{f(t/l)^{\prime}-g(t/l)^{\prime}}{f(t/l)-g(t/l)}\right)^{2}=\alpha_{1}^{2}-1/l^{2}=\frac{\kappa^{2}}{3l}\rho+\frac{\kappa^{4}\rho^{2}}{36}\.
i\hbar\,\left\langle\frac{\partialA}{\partialt}\right\rangle_{\scriptstyle\!\!\Psi}=\left\langle[A,H]_{\mathrm{extra}}\right\rangle_{\scriptstyle\!\Psi}=-\left\langle\;[A,H]_{\mathrm{regular}}\right\rangle_{\scriptstyle\!\Psi}\;,
\begin{array}{ccc}{R}&{=}&{-2a^{i}D^{A}\partial_{A}a_{i}-a^{i}a^{j}\partial_{A}a_{i}\partial^{A}a_{j}}\\{}&{}&{}\\{}&{}&{+a^{i}a^{j}\partial_{A}a_{k}\partial^{A}a_{l}\eta_{ij}^{kl}+\tilde{R}.}\\\end{array}
S^{\star}=\sum_{C}\left\{-\frac{1}{\oint_{C}dx_{t}}\oint_{C}dx_{t}\biggl|\left(\frac{\delta}{\delta\sigma^{\mut}}-i\frac{2\pi}{e}W_{\mut}\right)\Phi[C]\biggr|^{2}-M_{0}^{2}\biggl|\Phi[C]\biggr|^{2}\right\},
\Phi_{0}^{(E)}=\int_{S}{\bfE}_{0}({\bfr}-{\bfr}_{s}(t))\,d{\bfS}\qquad\mathrm{or}\qquad\Phi_{0}^{(B)}=\int_{S}{\bfB}_{0}({\bfr}-{\bfr}_{s}(t))\,d{\bfS}
\epsilon^{i_{1}...i_{N}j_{1}...j_{M}}\epsilon_{i_{1}...i_{N}l_{1}...l_{M}}=N!\\delta_{[l_{1}...l_{M}]}^{j_{1}...j_{M}}:=N!\\delta_{[l_{1}}^{j_{1}}...\delta_{l_{M}]}^{j_{M}}\
\begin{array}{lcl}{\{p^{n},f\}_{\theta}}&{=}&{\sum_{s=0}^{n}\theta^{s-1}c_{n}^{s}\{\frac{1-(-)^{s}}{2}\}f^{s}p^{n-s},}\\{\{p^{-n},f\}_{\theta}}&{=}&{\sum_{s=0}^{\infty}\theta^{s-1}c_{s+n-1}^{s}\{\frac{(-)^{s}-1}{2}\}f^{s}p^{-n-s}.}\\\end{array}
\underset{d\rightarrow4}{\mathrm{lim}}F_{B}\cdotF_{C}=\frac{(\frac{1}{2}(\lambda-2))_{\frac{1}{2}(l-M)}(\frac{1}{2}(\lambda-2))_{\frac{1}{2}(l-M)+r+t}}{(\lambda-2)_{l-M+r+t}}.
\begin{array}{c}{\left(V_{1},C_{1},V_{2},C_{2},a\right)=\left(V_{1},C_{1},0,1,1\right)\left(0,1,V_{2},C_{2},1\right)\left(0,1,0,1,a\right)\quad,}\\{\left(V_{1},C_{1},0,1,1\right)=\left(V_{1},1,0,1,1\right)\left(0,C_{1},0,1,1\right)\quad,}\\{\left(0,1,V_{2},C_{2},1\right)=\left(0,1,V_{2},1,1\right)\left(0,1,0,C_{2},1\right)\quad.}\\\end{array}
\left\langle\alpha_{1},\alpha_{2},...,\alpha_{2N},t\right|U_{L}\left|\alpha_{1}^{\prime},\alpha_{2}^{\prime},...,\alpha_{2N}^{\prime},t\right\rangle=R_{\alpha_{1}\alpha_{2}}^{\alpha_{2}^{\prime}\alpha_{3}^{\prime}}\:R_{\alpha_{3}\alpha_{4}}^{\alpha_{4}^{\prime}\alpha_{5}^{\prime}}\:...\:R_{\alpha_{2N-1}\alpha_{2N}}^{\alpha_{2N}^{\prime}\alpha_{1}^{\prime}}
(\stackrel{(2,1)}{\pi^{\alpha_{1}}}-\stackrel{(1,2)}{\pi^{\alpha_{1}}})=-{\frac{1}{2}}\tilde{\pi}^{\alpha_{1}}\qquad\mathrm{and}\qquad(\stackrel{(2,1)}{\pi^{\alpha_{1}}}+\stackrel{(1,2)}{\pi^{\alpha_{1}}})=-{\frac{1}{2}}\bar{\pi}^{\alpha_{1}}.