harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left(\begin{matrix}{a^{\prime}}\\{b^{\prime}}\\\end{matrix}\right)=I\left(\begin{matrix}{a}\\{b}\\\end{matrix}\right)=\left(\begin{matrix}{\sigma}&{0_{2}}\\{0_{2}}&{-\sigma}\\\end{matrix}\right)\left(\begin{matrix}{a}\\{b}\\\end{matrix}\right)~~~,~~~\sigma\equiv\left(\begin{matrix}{0}&{1}\\{1}&{0}\\\end{matrix}\right)~~~,
{\frac{\partialf_{a}^{\prime}}{\partial\phi^{A}}}{\overline{\delta}}_{a}\phi^{A}\,+\,{\frac{1}{2}}g_{3}^{\prime}{\overline{\delta}}_{2}\,{\overline{\delta}}_{1}\phi^{A}\,-\,{\frac{1}{2}}\epsilon^{ab}{\frac{\partialf_{a}^{\prime}}{\partial\phi^{A}}}{\frac{\partialf_{b}^{\prime}}{\partial{\bar{\phi}}^{A}}}\,\,=\,{\overline{\delta}}_{2}{\overline{\delta}}_{1}{\bf\chi}[\phi^{A}]\,\,,
\delta{\hat{p}_{i}}={\calF}\!L^{*}\{p_{i},\,G^{*}\}-v^{\mu_{0}}{\calF}\!L^{*}(B_{\mu_{0}}^{\nu_{0}}\{p_{i},\,\phi_{\nu_{0}}\})+[L]_{j}{\frac{\partial{\deltaq^{j}}}{\partial{\dot{q}^{i}}}}.
V_{n^{\prime}n}^{r^{\prime}r}=\intV_{1-n^{\prime}\alpha_{-}/2-n\alpha_{+}/2}(z)V_{\alpha_{-}}(z_{0})\cdotsV_{\alpha_{-}}(z_{r^{\prime}})V_{\alpha_{+}}(w_{0})\cdotsV_{\alpha_{+}}(w_{r})\prod_{i=1}^{r^{\prime}}\mathrm{d}z_{i}\prod_{i=1}^{r}\mathrm{d}w_{i},
\renewcommand{\arraystretch}{2}\begin{array}{lcl}{\displaystyle\Omega_{1}=-\frac{1}{4}(F_{2}+F_{3}+F_{4})+\cdots,}&{~}&{\displaystyle\Omega_{2}=-\frac{1}{4}(-F_{1}+F_{3}+F_{4})+\cdots,}\\{\displaystyle\Omega_{3}=-\frac{1}{4}(-F_{1}-F_{2}+F_{4})+\cdots,}&{~}&{\displaystyle\Omega_{4}=-\frac{1}{4}(-F_{1}-F_{2}-F_{3})+\cdots,}\\\end{array}
L=\sum_{i}\left\{2\dot{H}_{i}+4H_{i}^{2}-4\alpha\left[2(\dot{H}_{i}+H_{i}^{2})^{3}-H_{i}^{6}\right]\right\}-4\alpha(\sum_{i}H_{i}^{3})^{2}\to6(\dot{H}+2H^{2})-24\alpha\left[(\dot{H}+H^{2})^{3}+H^{6}\right]
H^{0}(\pi^{*}{\calS},{\calO}_{\pi^{*}{\calS}}(A^{\prime}\sigma|_{\pi^{*}{\calS}}-B^{\prime}F))=H^{0}({\calS},\pi_{*}{\calO}_{\pi^{*}{\calS}}(A^{\prime}\sigma|_{\pi^{*}{\calS}}-B^{\prime}F)).
(\tilde{\Lambda}\psi)_{\alpha}=-4\tilde{\Pi}_{b}P^{-ba}\partial_{a}\theta_{\alpha}+i\varepsilon^{ab}\theta^{\beta}\partial_{a}\theta^{\gamma}\partial_{b}\theta^{\delta}\Gamma_{\alpha(\beta}^{\mu}(C\Gamma^{\mu})_{\gamma\delta)}=0.
f(A)=-{\frac{1}{\partial^{2}+\sigma_{0}}}A_{\mu}A^{\mu}+{\frac{1}{2}}{\frac{1}{\partial^{2}+\sigma_{0}}}(\partial_{\mu}A^{\mu}+A_{\mu}\partial^{\mu}){\frac{1}{\partial^{2}+\sigma_{0}}}(\partial_{\nu}A^{\nu}+A_{\nu}\partial^{\nu}).
\{A,B\}_{D}\,=\,\{A,B\}\,+\,{\frac{1}{2|\phi|^{2}}}\{A,|\phi|^{2}-1\}\{\phi^{a}\pi_{a},B\}\,-\,{\frac{1}{2|\phi|^{2}}}\{A,\phi^{a}\pi_{a}\}\{|\phi|^{2}-1,B\}\,,
\int_{0}^{\infty}\int_{0}^{2\pi}d\taud\phie^{\tauT_{+}}e^{\phiT_{-}}G_{0,+}Q_{0,+}{\bar{\phi}}_{\bar{a}}(1)(\phi_{i}(1)|\phi_{j}(0)\rangle)={\bar{\phi}}_{\bar{a}}(C_{ij}^{k}|k\rangle)
W_{1}(\alpha)=-\frac{1}{2}\int[123]f_{\lambda}\left[\frac{\tilde{\delta}(p_{3\hat{\alpha}}-p_{1\hat{\alpha}2\hat{\alpha}})}{p_{3\hat{\alpha}}^{-}-p_{1\hat{\alpha}2\hat{\alpha}}^{-}}-\frac{\tilde{\delta}(p_{3}-p_{12})}{p_{3}^{-}-p_{12}^{-}}\right]a_{\lambda1}^{\dagger}a_{\lambda2}^{\dagger}a_{\lambda3}-h.c.\;.
T_{t}^{t}=T_{z}^{z}=\varphi^{2}e^{-2gM}+\frac{A^{2}\varphi^{2}}{r^{2}}e^{-2gM}+{\calP}M^{2}+\left(\frac{A^{\prime}}{rh}\right)^{2}+{\calR}\left(\frac{G}{gr}\right)^{2}+U.
S[q(\tau)]=\int_{0}^{T}[\frac{1}{2}M(q)\dot{q}^{2}+V(q)+\deltaV(q)]d\tau+\frac{\alpha\Gamma(n+1)}{4\pi}\int_{-\infty}^{\infty}d\tau^{\prime}\int_{0}^{T}d\tau\frac{\{q(\tau)-q(\tau^{\prime})\}^{2}}{\mid\tau-\tau^{\prime}\mid^{n+1}},
0=\delta(y)\bar{\Psi}_{\mu}^{A}\left[2(M+\gamma_{5}\bar{M})_{A}^{\;B}+(\omega+\gamma_{5}\bar{\omega})_{A}^{\;B}\right]\left(\frac{3\mathrm{i}}{2\sqrt{2}}\gamma^{5}F_{\;\;5}^{\mu}\eta_{B}-\gamma^{\mu\nu}\partial_{\nu}\eta_{B}\right)\.
\intd^{D}p\,\,\,\frac{\partial}{\partialp_{\omega}}\left(p_{\lambda}p_{\tau}p_{\kappa}p_{\eta}p_{\xi}p_{\varrho}p_{\vartheta}\,\,\,e^{-\frac{1}{2}p\cdotR\cdotp}\right)=0.
ds^{2}=\left(1+\frac{R^{4}}{r^{4}}\right)^{-1/2}(-dt^{2}+d\vec{x}^{2})+\left(1+\frac{R^{4}}{r^{4}}\right)^{1/2}(dr^{2}+r^{2}d\Omega_{5}^{2})
\left\{\sum_{n=0}^{\infty}\frac{(-1)^{p+1}(p+1)!\\rho_{n}^{(k+1)}(s)}{(t-n-\frac{1}{2})^{p+2}}\right\}_{{s=0}\atop{t=0}}-\left\{\sum_{n=0}^{\infty}\frac{(-1)^{k+1}(k+1)!\r_{n}^{(p+1)}(s)}{(s-n-\frac{1}{2})^{k+2}}\right\}_{{s=0}\atop{t=0}}=0,
\begin{array}{rcll}{\displaystyle\lambdax^{-}}&{=}&{\displaystyle-\alpha^{2}\lambdax^{+}-2\alpha\sqrt{\frac{M}{\lambda}}-{\frac{M}{\lambda}}\qquad}&{\displaystyle\lambdax^{+}\ge1}\\{\displaystyle\lambdax^{-}}&{=}&{\displaystyle-\left(\alpha+\sqrt{\frac{M}{\lambda}}\right)^{2}\lambda{x}^{+}}&{\lambdax^{+}\le1}\\\end{array}
\overleftarrow{\not\!\!D}_{L}\equiv\frac{1}{2}\left(\overleftarrow{\partial}_{\mu}-\frac{1}{2}\omega_{\mu}^{cd}I\sigma_{cd}+\frac{i}{2}\,\tilde{g}\vec{\tau}\tilde{\vec{A}}_{\mu}-\frac{i}{2}\tilde{g}^{\prime}\tilde{B}_{\mu}\right)\gamma^{a}V_{a}^{\mu}
\left[\begin{array}{c}{\,R^{+}\,}\\{\,R^{-}\,}\\\end{array}\right]+\frac6l\left[\begin{array}{cc}{\,1\,\,}&{\,0\,}\\{\,0\,\,}&{\,a^{2}\,}\\\end{array}\right]\Box\left[\begin{array}{c}{\,\,\xi^{+}\,}\\{\,\,\xi^{-}\,}\\\end{array}\right]=0.
\epsilon\equiv{\frac{M_{0}}{\xi^{2}}}=\left({\frac{\Lambda}{\xi}}\right)^{{\frac{3N_{c}-N_{f}}{N_{c}}}}\left[{\frac{m}{M_{P}}}\left({\frac{\xi}{M_{P}}}\right)^{q+\tilde{q}-1}\right]^{{\frac{N_{f}-N_{c}}{N_{c}}}}\;\;\;.
E_{2}=\langle0,0|\hat{h}^{\mathrm{m}}|0,0\rangle=\frac{1}{g_{\mathrm{eff}}^{2}}\left\{\langle0|\varphi^{2}|0\rangle-2\left(\langle0|\varphi|0\rangle\right)^{2}\right\}=\frac{1}{g_{\mathrm{eff}}^{2}}\left(\frac{\pi^{2}}{6}-1\right).
L_{+}^{D}=\left(\begin{array}{ll}{(qt^{-1})^{S^{3}}t^{Z}}&{\OmegaS^{-}}\\{0}&{(qt)^{-S^{3}}t^{-Z}}\\\end{array}\right)\\\,\\\L_{-}^{D}=\left(\begin{array}{ll}{(qt)^{-S^{3}}t^{Z}}&{0}\\{-\OmegaS^{+}}&{(qt^{-1})^{S^{3}}t^{-Z}}\\\end{array}\right)\,
\hat{\cal{L}}_{SD}^{S}=\frac{1}{2}(\hat{f}^{\mu}+\hat{h}^{\mu}-\partial^{\mu}\hat{\phi})(\hat{f}_{\mu}+\hat{h}_{\mu}-\partial_{\mu}\hat{\phi})-\frac{1}{2m}\epsilon^{\alpha\beta\gamma}(\hat{f}_{\alpha}\partial_{\beta}\hat{f}_{\gamma}+2\hat{h}_{\alpha}\partial_{\beta}\hat{f}_{\gamma}).
\frac{1}{\hbar^{2}k^{2}+2\hbark\cdotp}+\frac{1}{\hbar^{2}k^{2}-2\hbark\cdotp}=-\frac{k^{2}}{2(p\cdotk)^{2}}+{\calO}\,(\frac{\,\hbar^{2}k^{2}}{p^{4}})\,.
\alpha^{*}\left(\mu^{2}\right)={\frac{1}{4yx^{3}}}\left[T\left(\mu^{2}\right)-T\left(\mu^{2}\right]|_{I_{1}=0}\right)|_{W=0}={\frac{\left(\mu^{2}\right)^{2}xI_{1}}{1+xI_{1}-2\mu^{2}I_{1}yx^{2}}}.
R=\left[\begin{array}{llll}{q}&{\,}&{\,}&{\,}\\{\,}&{1}&{0}&{\,}\\{\,}&{\lambda}&{1}&{\,}\\{\,}&{\,}&{\,}&{q}\\\end{array}\right]\quad,\quad{\calP}=\left[\begin{array}{llll}{1}&{\,}&{\,}&{\,}\\{\,}&{0}&{1}&{\,}\\{\,}&{1}&{0}&{\,}\\{\,}&{\,}&{\,}&{1}\\\end{array}\right]\quad.
\partial_{\mu}S^{\mu}=\frac{1}{3}\sum_{i}\gamma_{i}^{\ast}\phi_{i}^{\alpha}\frac{\partialW}{\partial\phi_{i}^{\alpha}}\bigg|+\frac{1}{48\pi^{2}}\left[3T(G)-\sum_{i}T(R_{i})(1-\gamma_{i}^{\ast})\right]F\widetilde{F}=0\,,
{\mathcal{A}}={\mathrm{\bf{j}}}ie\Big(A^{i}T_{i}+\frac{1}{2\sqrt{3}}BY\Big)+{\mathrm{\bf{k}}}\frac{ie}{\sqrt{2}}\Big(\phi_{0}\Sigma_{+}+\phi_{+}\Sigma_{-}+\phi_{0}^{*}\Sigma_{-}^{\prime}+\phi_{+}^{*}\Sigma_{+}^{\prime}\Big),
\psi_{12}(p_{30})=\,{\frac{\Lambda_{12}^{++}}{(P_{0}-S)-(p_{30}-h_{3})+i\epsilon}}\,\left[\,V_{12}\psi_{12}(p_{30})+V_{23}\psi_{23}(h_{1})+V_{31}\psi_{31}(h_{2})\,\right]
\frac{q^{\sum_{\alpha<\beta}2r_{\beta}+\sum_{\alpha}r_{\alpha}}}{\prod_{j\geq1}(q)_{\nu_{j}}}=\frac{q^{\sum_{\beta=1}^{\ell}(2\beta-1)r_{\beta}}}{\prod_{j\geq1}(q)_{\nu_{j}}}=\frac{q^{\sum_{m=1}^{p-2}N_{m}^{2}}}{(q)_{N_{1}-N_{2}}(q)_{N_{2}-N_{3}}\dots},
T_{+p-\dot{p}}^{+q}=-\frac{\chi}{2}\gamma_{q\dot{p}}^{i}\Omega_{+p}^{+2i},\T_{-\dot{q}-\dot{p}}^{+q}=-\frac{\chi}{2}\left(\gamma_{q\dot{q}}^{i}\Omega_{-\dot{p}}^{+2i}+\dot{q}\leftrightarrow\dot{p}\right),
\sum_{k=1}^{n}c\left(\frac{\zeta_{N-1}}{\zeta_{k}}\right)\prod_{j=k+1}^{n}b\left(\frac{\zeta_{N-1}}{\zeta_{j}}\right)\overline{{G}}_{\varepsilon}^{(n-1,l-1)}(\zeta^{\prime})^{-\cdots\stackrel{k}{\check{+}}\cdots-\stackrel{n+1}{\check{+}}\cdots+}.
[a(q,\lambda),a^{\dagger}(q^{\prime},\lambda^{\prime})]=\delta(q^{+}-q^{\prime+})\delta^{2}(q_{\perp}-q_{\perp}^{\prime})\delta_{\lambda,\lambda^{\prime}}=\delta^{3}(q-q^{\prime})\delta_{\lambda,\lambda^{\prime}}\;.
G_{r}^{a}=\oint{\frac{dz}{2\pii}}{\calG}_{r}^{a}(z),\quad\bar{G}_{r}^{a}=\oint{\frac{dz}{2\pii}}\bar{{\calG}}_{r}^{a}(z),\quad(a=1,2,\\r\in{\bfZ}+{\frac{1}{2}})
\mathrm{Tr~}[(D^{2}+B+m^{2})^{-1}-(P^{2}+m^{2})^{-1}]=\int_{0}^{\infty}\,dte^{-tm^{2}}\int\,d^{2}r<\!{\bfr}\,|\,e^{-(D^{2}+B)t}-e^{-P^{2}t}\,|\,{\bfr}\!>\,.
\psi(x,y)\,=\,{\calS}\,\int_{-\infty}^{\infty}dy^{\prime}\bigl[\{H\,\psi\}(x,y^{\prime})\,-i\,\eta\pi\,\mathrm{sign}(y^{\prime}-y)\,\partial_{x}\psi(x,y^{\prime})\bigr].
{\calL}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{\hbar}{2}\bar{\psi}\gamma^{\mu}\left(i\partial_{\mu}-eA_{\mu}\right)\psi-\frac{\hbar}{2}\left[\left(i\partial_{\mu}+eA_{\mu}\right)\bar{\psi}\right]\gamma^{\mu}\psi
\xi_{r}a_{r}({\bfk})=\frac{1}{(2w_{k})^{1/2}}\intd^{3}x\epsilon_{\mur}\left(-i\Pi^{\mu}-w_{k}A^{\mu}\right)e^{i{\bfk}.{\bfx}}+\frac{k^{j}\epsilon_{jr}}{w_{k}}a_{0}({\bfk})
T_{F}^{\mu\nu}=\frac{1}{\Lambda}\left[\{J^{\mu},J^{\nu}\}-g^{\mu\nu}J^{\lambda}J_{\lambda}-:\{J^{\mu},J^{\nu}\}:+g^{\mu\nu}:J^{\lambda}J_{\lambda}:\right]+g^{\mu\nu}\frac{m}{2}\{\bar{\psi},\psi\}
\langleP[\overline{{\psi}},\psi,A,h]\rangle_{0}\;:=\;\frac{1}{Z_{0}}\int{\calD}h{\calD}A{\calD}\overline{{\psi}}{\calD}\psi\;P[\overline{{\psi}},\psi,A,h]\;e^{-S_{G}[A]-S_{h}[h]-S_{F}[\overline{{\psi}},\psi,A,h]}\;.
n={\frac{\alpha}{\pi}}(x_{0}\Omega)^{2}\intdt^{\prime}\bigl[{\frac{-i\pi}{3}}sin\,t^{\prime}-(-\pi)cos\,t^{\prime}\bigr]e^{-it^{\prime}}=\alpha(x_{0}\Omega)^{2}{\frac{T\Omega}{3}}
Q_{Y,i_{1}i_{2}...i_{2n}}^{T,I_{1}I_{2}...I_{n}}=P_{Y}^{i_{1}i_{2}...i_{2l},j_{1}j_{2}...j_{2l}}(\prod_{l=1}^{n}C_{j_{2l-1}j_{2l}}^{I_{l}})
\langle{\calO}\rangle_{\sigma}=G_{\sigma}^{(n)}(\overbrace{x^{-1}\zeta,\cdots,x^{-1}\zeta}^{n},\overbrace{\zeta,\cdots,\zeta}^{n})^{-\varepsilon_{n}\cdots-\varepsilon_{1}\varepsilon_{1}^{\prime}\cdots\varepsilon_{n}^{\prime}}.
\left(f_{\omega}^{R*}\,\tilde{\beta}_{\omega^{\prime\prime}\omega}^{R*}+f_{\omega}^{R}\,\tilde{\alpha}_{\omega^{\prime\prime}\omega}^{R}\right)\left(\tilde{\beta}_{\omega^{\prime\prime}\omega^{\prime}}^{R}\,f_{\omega^{\prime}}^{R}+\tilde{\alpha}_{\omega^{\prime\prime}\omega^{\prime}}^{R*}\,f_{\omega^{\prime}}^{R*}\right)\,,
\left(\begin{matrix}{a^{\prime}}\\{b^{\prime}}\\\end{matrix}\right)=I\left(\begin{matrix}{a}\\{b}\\\end{matrix}\right)=\left(\begin{matrix}{H}&{G}\\{F}&{E}\\\end{matrix}\right)\left(\begin{matrix}{a}\\{b}\\\end{matrix}\right)\RightarrowIJI^{T}=-J.
\Gamma_{k}[\phi]=-W_{k}[J,\varphi_{0}^{k}]+\intJ(\varphi_{0}^{k}+\phi_{k}^{\Lambda})=-{\mathcalW}_{k}[J_{k}^{\Lambda},\varphi_{0}^{k}]+\intJ_{k}^{\Lambda}\phi_{k}^{\Lambda},
\times\delta\varphi_{r}(t_{1})dt_{1}\int_{\hat{C}_{v_{r}}}f_{r}(t_{2})dt_{2}K(t_{2},t^{\prime})+\int_{C_{v_{r}}}K_{(0)}(t,t_{1};L)\varphi_{(0)r}(t_{1})dt_{1}\int_{C_{r}}\deltaf_{r}(t_{2})dt_{2}K(t_{2},t^{\prime})
\left(\Phi^{A},\Phi_{B}^{*(1)}\right)_{1}=\left(\Phi^{A},\Phi_{B}^{*(2)}\right)_{2}=\delta_{\;\;B}^{A},\left(\Phi^{A},\Phi_{B}^{*(1)}\right)_{2}=\left(\Phi^{A},\Phi_{B}^{*(2)}\right)_{1}=0.
i\frac{\partial}{\partialt}\Psi(1,2;t)=-\frac{1}{m}\sum_{p=1,2}\left(\bar{{\bf\bigtriangledown}}_{(p)}{\bf\bigtriangledown}_{(p)}+{\bf\bigtriangledown}_{(p)}\bar{{\bf\bigtriangledown}}_{(p)}\right)\Psi(1,2;t),
U_{\nu_{H}}=\left(\begin{matrix}{1}&{1}&{\ldots}&{1}\\{b_{21}}&{b_{22}}&{\ldots}&{b_{2k}}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{b_{k1}}&{b_{k2}}&{\ldots}&{b_{kk}}\\\end{matrix}\right)
\frac{-\mathbf{q}.(\mathbf{q}+2\mathbf{k})}{k^{2}-m^{2}+i\varepsilon}\frac{1}{(k+q)^{2}-m^{2}+i\varepsilon}+\varepsilonq_{0}\frac{i}{|\mathbf{q}|^{2}+i\varepsilon}\frac{(q+2k)_{0}}{k^{2}-m^{2}+i\varepsilon}\frac{1}{(k+q)^{2}-m^{2}+i\varepsilon}
{\frac{Dp^{\mu}}{d\tilde{\lambda}}}\equiv{\frac{dp^{\mu}}{d\tilde{\lambda}}}+\tilde{\Gamma}_{\rho\sigma}^{\mu}{\frac{dx^{\rho}}{d\tilde{\lambda}}}p^{\sigma}=-\tilde{m}_{0}{\frac{{\calW}^{\prime}}{\calW}}{\frac{dy}{d\tilde{\lambda}}}{\frac{dx^{\mu}}{d\tilde{\lambda}}},
D^{\pm}(x)=\frac{1}{(2\pi)^{4}}\intd^{4}k\:e^{-ikx}\:D^{\pm}(k)=\frac{1}{(2\pi)^{4}}\intd^{4}k\:e^{-ikx}\:2\pi\theta(\pmk^{0})\delta(k^{2})
[a_{m},a_{m}^{\dagger}]-[\psi_{\dot{\alpha}},\;\psi_{\dot{\alpha}}^{\dagger}]_{+}=[\tilde{a}_{m},\tilde{a}_{m}^{\dagger}]-[\tilde{\psi}_{\dot{\alpha}},\;\tilde{\psi}_{\dot{\alpha}}^{\dagger}]_{+}=B-F-1+\mid0\rangle\langle0\mid.
V_{B}=-\frac{T_{0}}{4\lambda^{2}}\sum_{a\neqb}\mathrm{Tr}[X^{a},X^{b}]^{2}+\frac{2ifT_{0}}{3\lambda^{2}}\mathrm{Tr}(X^{i}X^{j}X^{k})\epsilon_{ijk}+\frac{T_{0}M^{2}}{2}\sum_{i=1,2,3}\mathrm{Tr}(X^{i})^{2}.
\left(V,a,C\right)=\left(\begin{array}{ccc}{\frac{a+\frac1a}2+\frac{V^{2}}{2a}}&{\frac{a-\frac1a}2-\frac{V^{2}}{2a}}&{-V^{T}C}\\{\frac{a-\frac1a}2+\frac{V^{2}}{2a}}&{\frac{a+\frac1a}2-\frac{V^{2}}{2a}}&{-V^{T}C}\\{-\fracVa}&{\fracVa}&{{\bf1}_{n-2}}\\\end{array}\right)\quad.
\hat{\omega}_{\hat{i}~\tilde{I}}^{~\tilde{i}}=\frac{1}{3}h^{-2/3}\partial_{\hat{I}}h~,\\hat{\omega}_{\hat{I}}^{~\tilde{I}\tilde{J}}=-\frac{1}{6}h^{-2/3}\partial_{\hat{J}}h~,
{\calL}_{M}=-\frac14(F_{\mu\nu}^{a})^{2}-\frac12g^{\mu\nu}(D_{\mu}\phi^{a})(D_{\nu}\phi^{a})-\frac12g^{\mu\nu}(\partial_{\mu}\chi^{a})(\partial_{\nu}\chi^{a})-V(\phi^{a},\chi^{a})\,
\left[\frac{\partial^{2}}{\partial\xi^{2}}+2\,\kappa\,\delta\,\frac{s}{c}\frac{\partial}{\partial\xi}-\left\{\frac{1}{\kappa^{2}}-\kappa^{2}\delta(1+\delta)\right\}\frac{s^{2}}{c^{2}}+(\varepsilon+\kappa^{2}\delta)\right]\Psi(\xi)=0\;,
ds^{2}=\frac{1}{2}\sum_{A}f^{2}(r^{A})dr^{A}dr^{A}+a^{2}(r^{A})(\sigma_{1}^{A})^{2}+b^{2}(r^{A})(\sigma_{2}^{A})^{2}+c^{2}(r^{A};r^{B})(\sigma_{3}^{A})^{2}
<B|T_{\nu}^{\mu}(r)|B>=\left(\begin{array}{cc}{T_{\alpha}^{\alpha}-\Omega^{-1}(r)H(r)}&{0}\\{0}&{\Omega^{-1}H(r)\}\\\end{array}\right)+\Omega^{-1}(\frac{-\pi}{24a^{2}}+\frac{1}{384\pim^{2}})\left(\begin{array}{cc}{1}&{0}\\{0}&{-1\}\\\end{array}\right).
\int\frac{dk^{\prime\prime}}{2\pi}\,\,\frac{1}{\sqrt{k^{2}+4}}\,\,\frac{2ik^{\prime\prime}\epsilon}{(ik^{\prime\prime}-2\sigma_{1})^{2}}\,\,\frac{1}{(k+k^{\prime}-2k^{\prime\prime}-i\tau)}.
T_{+2-2}^{-q}=\frac12\gamma_{q\dot{q}}^{i}\Omega_{-2}^{-2i}\psi_{+2\dot{q}}^{2+}-\frac12\gamma_{q\dot{q}}^{i}\Omega_{+2}^{-2i}\psi_{-2\dot{q}}^{1-},
\begin{array}{c}{F_{12}=F_{43}=F_{65}=\pmF_{78}\,,}\\{{}}\\{F_{13}=F_{24}=F_{75}=\pmF_{86}\,,}\\{{}}\\{F_{14}=F_{32}=F_{76}=\pmF_{58}\,,}\\{{}}\\{F_{ab}=0~~~\mathrm{for}~~~{a\in\{1,2,3,4\}}~~~{b\in\{5,6,7,8\}}\,.}\\\end{array}
B_{\Sigma}=\intd^{D}x\,\left(\frac{\delta\Sigma}{\delta\Phi^{i}}\frac\delta{\delta\Phi^{i*}}+\frac{\delta\Sigma}{\delta\Phi^{i*}}\frac\delta{\delta\Phi^{i}}\right)\;,\;\;\;\;\;\;B_{\Sigma}B_{\Sigma}=0\;.
\begin{array}{lll}{q}&{=}&{(+\frac{1}{2},\,\,\vartheta_{1}-\frac{1}{2},\vartheta_{2}-\frac{1}{2},\vartheta_{3}-\frac{1}{2})}\\{q}&{=}&{(\pm\frac{1}{2},\pm\frac{1}{2},\,\,\vartheta_{2}-\frac{1}{2},\vartheta_{3}-\frac{1}{2})}\\{q}&{=}&{(\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},\,\,\vartheta_{3}-\frac{1}{2})}\\\end{array}
X_{L}^{25}(\sigma^{+})=X_{0L}^{25}+\sqrt{\frac{\alpha^{\prime}}{2}}P_{L}^{25}(\tau+\sigma)+i\sqrt{\frac{\alpha^{\prime}}{2}}\sum_{n\neq0}\frac{1}{l}\alpha_{L,l}^{25}exp\bigg(-il(\tau+\sigma)\bigg),
S_{I}[x,\vec{\alpha}]=\int_{0}^{t}{dt^{\prime}}\left\{\frac{1}{2}\left(\vec{\alpha}\cdot\frac{d\vec{\alpha}^{*}}{dt^{\prime}}-\vec{\alpha}^{*}\cdot\frac{d\vec{\alpha}}{dt^{\prime}}\right)-\frac{i}{\hbar}(H_{R}-\dot{x}h_{I})\right\}
g_{{\mathcalI}\bar{\mathcalJ}}W^{{\mathcalI}11}W_{11}^{\bar{\mathcalJ}}=\left(P_{X}^{1}+\mathrm{i}P_{X}^{2}\right)\left(P_{Y}^{1}-\mathrm{i}P_{Y}^{2}\right)U^{XY}=4g^{k\bar{l}}\nabla_{k}L\nabla_{\bar{l}}\bar{L}
\left(\begin{array}{c}{\ddot{\hat{\phi}}_{1n}}\\{\ddot{\hat{\phi}}_{2n}}\\\end{array}\right)+\left(\begin{array}{cc}{n^{2}+m^{2}H^{2}\alpha^{2}}&{2inmH\alpha^{\prime}}\\{-2inmH\alpha^{\prime}}&{n^{2}+m^{2}H^{2}\alpha^{2}}\\\end{array}\right)\left(\begin{array}{c}{\hat{\phi}_{1n}}\\{\hat{\phi}_{2n}}\\\end{array}\right)=\left(\begin{array}{c}{0}\\{0}\\\end{array}\right),
\sum_{j}\;{\frac{t_{j}^{k}\,dt_{j}}{s_{j}}}=4\,d\tau\,u(\lambda)\,\sum_{j}\;(-1)^{N-1-j}{\frac{t_{j}^{k}}{t_{j}-\lambda}}{\frac{D_{j}}{D}}=-4\,{\frac{\lambda^{k}}{\prod_{l}\,(\lambda-a_{l})}}\,d\tau
V_{0}(y)=4G^{55}\epsilon(y)\bar{\Phi}^{\prime}(|y|)\delta(y)+\frac{4}{\sqrt{-G}}\left(\sqrt{-G}G^{55}\right)^{\prime}\bar{\Phi}(|y|)\delta(y)\+4G^{55}\bar{\Phi}(|y|)\delta^{\prime}(y)
U_{k}^{ij}=\sqrt{-g}\left[-g^{pi}(-\Gamma_{kp}^{j}+\frac{1}{2}g_{k}^{j}\Gamma_{ap}^{a}+\frac{1}{2}g_{p}^{j}\Gamma_{ak}^{a})+\frac{1}{2}g_{k}^{i}g^{pm}(-\Gamma_{pm}^{j}+\frac{1}{2}g_{p}^{j}\Gamma_{am}^{a}+\frac{1}{2}g_{m}^{j}\Gamma_{ap}^{a})\right],
{\calF}(x^{a},\varphi,\tilde{p}_{a},\tilde{p}_{\varphi})=\varrho\int_{\varphi_{0}}^{\varphi}\frac{(\tilde{p},\partial_{\varphi}n)}{(\tilde{p},n)}d\varphi+\varphi\tilde{p}_{\varphi}+x^{a}\tilde{p}_{a}\,
\beta(\alpha)=\sum_{k=1}^{\infty}\beta_{k}\left(\frac{\alpha}{\pi}\right)^{k}=\frac13\left(\frac{\alpha}{\pi}\right)+\frac14\left(\frac{\alpha}{\pi}\right)^{2}-\frac{121}{288}\left(\frac{\alpha}{\pi}\right)^{3}+\ldots
E_{n\ell}\approx\hbar\omega\left[\left(n+\frac{D}{2}\right)+\frac{1}{2}\left\{(k^{2}+{k^{\prime}}^{2})\left(n+\frac{D}{2}\right)^{2}+(k^{2}-{k^{\prime}}^{2})\left(L^{2}+\frac{D^{2}}{4}\right)+{k^{\prime}}^{2}\frac{D}{2}\right\}\right]\;.
\int_{-\infty}^{\infty}d\theta\gamma_{i}(\theta)\,e^{it\theta}=2\pi\sum_{m=0}^{\infty}(-i)^{m}\nu_{i}^{(m)}t^{m}=\pi\sum_{m=0}^{\infty}\frac{E_{2m}}{(2m)!}(\pi\eta_{i})^{2m}\,t^{2m}\,\,.
\displaystyle\frac{1}{(q)_{\infty}}\sum_{r=-\infty}^{\infty}(-1)^{r}q^{r((2k+1)r+2k-2i+1)/2}=\displaystyle\sum_{n_{1}\geq\cdots\geqn_{k-1}\geq0}\frac{q^{n_{1}^{2}+\cdots+n_{k-1}^{2}+n_{i}+\cdots+n_{k-1}}}{(q)_{n_{1}-n_{2}}\cdots(q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}}},
\begin{array}{rl}{F_{ij}^{\mu\nu}dx_{\mu}dx_{\nu}\;\equiv}&{-\frac{1}{2}\left[\nabla^{\mu},\nabla^{\nu}\right]_{ij}dx_{\mu}dx_{\nu}=}\\{=\frac{1}{2}\{\left(\partial^{\mu}A^{I\nu}-\partial^{\nu}A^{I\mu}\right)\otimes\sigma_{ij}^{I}\;+}&{(A^{I\mu}A^{J\nu}-A^{I\nu}A^{J\mu})\otimes\sigma_{ik}^{I}\sigma_{kj}^{J}\}dx_{\mu}dx_{\nu}\;,}\\\end{array}
\begin{array}{l}{H=\displaystyle\sum_{n=1}^{N}\frac{{\sigma}_{n}^{z}}{2},\;\;\;\;q=e^{i{\eta}},}\\{X_{0}=\displaystyle\sum_{n=1}^{N}q^{-\frac12({\sigma}_{1}^{z}+...+{\sigma}_{n-1}^{z})}{\sigma}_{n}^{-}q^{-\frac12({\sigma}_{n+1}^{z}+...+{\sigma}_{N}^{z})}.}\\\end{array}
{\calD}\Psi_{D}=\left(\begin{array}{c}{-B(\bar{Z}e^{2}+\bar{Y},_{3}e^{3})}\\{dB+B[h,_{4}+(Z-\bar{Z})h/2]e^{3}}\\{dC+C[h,_{4}-(Z-\bar{Z})h/2]e^{3}}\\{C(Ze^{1}+Y,_{3}e^{3})}\\\end{array}\right).
\left.\begin{array}{lcl}{T_{\;\;\eta}^{\eta}}&{=}&{V_{\mathrm{c}}(\phi)+{{{{{\dot{\phi}}}^{2}}}{{{e}^{-2\,a}}}}}\\{T_{\;\,i}^{i}}&{=}&{V_{\mathrm{c}}(\phi)-{{{{{\dot{\phi}}}^{2}}}{{{e}^{-2\,a}}}},}\\\end{array}\right.
\begin{array}{l}{{\bfM}_{{\calN}}(E)\equiv}\\{\left(\renewcommand{\arraystretch}{1.3}\begin{array}{cc}{\begin{array}{cccc}{2E+{\calN}-1}&{0}&{2}&{0}\\{2g(1-{\calN})}&{2E+{\calN}-3}&{0}&{6}\\{0}&{2g(2-{\calN})}&{2E+{\calN}-5}&{0}\\{0}&{0}&{2g(3-{\calN})}&{2E+{\calN}-7}\\\end{array}}&{\cdots}\\{\vdots}&{\ddots}\\\end{array}\right)}\\\end{array}
{\calC}^{22}(x){\calC}_{22}^{\dagger}(y)=\frac{N^{2}-1}{2(2\pi)^{4}(x-y)^{4}}+\frac{1}{(2\pi)^{2}(x-y)^{2}}\left({\calK}_{\bf{1}}+{\calQ}_{\bf{20}}^{(Y)}-{\calQ}_{\bf{20}}^{(X)}\right)\left(\frac{x+y}{2}\right)+\ldots\;,
\begin{array}{lcl}{[X_{\underline{{a}}\underline{{b}}ij\cdots}Y_{\underline{{c}}\underline{{d}}kl\cdots}]_{s}}&{\equiv}&{~~X_{abij\cdots}Y_{cdkl\cdots}+X_{cdij\cdots}Y_{abkl\cdots}}\\{}&{}&{+X_{acij\cdots}Y_{bdkl\cdots}+X_{bdij\cdots}Y_{ackl\cdots}}\\{}&{}&{+X_{adij\cdots}Y_{bckl\cdots}+X_{bcij\cdots}Y_{adkl\cdots}\.}\\\end{array}
\begin{array}{rcl}{J_{1}}&{=}&{\{0,1,\cdots,m\}\nij_{1},}\\{J_{2}(j_{1})}&{=}&{\{j_{1},m+1,\cdots,2m\}\nij_{2},}\\{}&{\vdots}&{}\\{J_{n-2}(j_{1},\cdots,j_{n-3})}&{=}&{\{j_{n-3},(n-3)m+1,\cdots,(n-2)m\}\nij_{n-2}}\\{J_{n-1}(j_{1},\cdots,j_{n-2})}&{=}&{\{j_{n-2},(n-2)m+1,\cdots,(n-1)m\}\nij_{n-1}}\\\end{array}
\Psi=\int\frac{d^{3}{\bfp}}{(2\pi)^{3}}\frac{1}{2p_{0}}\sum_{h}\left[(\psi_{h}^{D}(p^{\mu}))^{c}a_{h}exp(-ip\cdotx)+\psi_{h}^{D}(p^{\mu})b_{h}^{\dagger}exp(ip\cdotx)\right]\quad.
E_{1}=A_{1}^{(\alpha_{1}+\alpha_{2},\alpha_{1}-\alpha_{2})}=\frac{1}{2\pii}\oint_{C_{0}}dz:\left(\frac{d}{dz}-\frac{d}{d\xi}\right)U^{\alpha_{1}}(z)U^{\alpha_{2}}(\xi):\mid_{z=\xi}
h_{\mu\nu}(x,y)\rightarrowh_{\mu\nu}^{\prime}(x,y)=h_{\mu\nu}(x,y)-\frac{1}{k}\partial_{\mu}\partial_{\nu}\xi^{5}(x)+e^{-2ky}[2\partial_{(\mu}\hat{\xi}_{\nu)}(x)-2k\xi^{5}(x)\eta_{\mu\nu}].
\begin{aligned}{}&{{}U_{1}:}&{}&{{}\qquadx_{1}(t)=t,}&{}&{{}\qquady_{1}(t)=0,}&{}&{{}\qquadu_{1}(t)=\mu,}&{}&{{}\qquadv_{1}(t)=0}\\{}&{{}U_{2}:}&{}&{{}\qquadx_{2}(t^{\prime})=t^{\prime},}&{}&{{}\qquady_{2}(t^{\prime})=0,}&{}&{{}\qquadu_{2}(t^{\prime})={\frac{\mu}{t^{\prime}}},}&{}&{{}\qquadv_{2}(t^{\prime})=0.}\\\end{aligned}
p^{2}{\calM}(p)=\int{\frac{d^{3}k}{(2\pi)^{3}}}\left[{\frac{2p^{2}{\calM}(k)}{{\calM}^{2}(k)+k^{2}}}-{\frac{2(p.q)(q.k)}{q^{2}}}{\frac{{\calM}(p)}{{\calM}^{2}(k)+k^{2}}}\right]\left({\frac{1}{q^{2}+\alphaq/8}}\right).
[m{\bf1}-p_{\mu}\gamma^{\mu},W^{(1)}]_{\pm}+\frac{ig}{4c}\partial_{\mu}[\gamma^{\mu},W^{(0)}]_{\mp}=\frac{ig}{2c}F_{\mu\nu}(x)\partial_{p}^{\nu}[\gamma^{\mu},W^{(0)}]_{\mp}
S_{\Sigma}=\frac{1}{4\pi\alpha^{\prime}}\int_{\Sigma}d^{2}\sigma\sqrtg[g^{ab}G_{\mu\nu}(X)\partial_{a}X^{\mu}\partial_{b}X^{\nu}+i\epsilon^{ab}B_{\mu\nu}(X)\partial_{a}X^{\mu}\partial_{b}X^{\nu}+\alpha^{\prime}R\Phi(X)]
\begin{aligned}{|\Phi_{1}|^{2}+|\Phi_{2}|^{2}+|\Phi_{3}|^{2}}&{{}=r_{1}\,,}\\{|\Phi_{1}|^{2}+|\Phi_{4}|^{2}}&{{}=r_{2}\,,\,\,r_{1}\geqr_{2}\geq0\,,}\\\end{aligned}
V_{F}-V_{0}=\frac{l}{2\pi\alpha^{\prime}}\int_{r_{*}/{\bfl}}^{\infty}du[\frac{u}{(u^{2}-1)^{1/2}}-1]\equiv\frac{1}{2\pi\alpha^{\prime}}\left({r^{*}}-\sqrt{{r^{*}}^{2}-{\bfl}^{2}}\right)
\rho(x)=\frac{1}{x^{2}}+\sum_{q\neq0}\{\frac{1}{(x-q)^{2}}-\frac{1}{q^{2}}-\frac{2x\cdotq}{q^{4}}-\frac{1}{q^{6}}(4(q\cdotx)^{2}-q^{2}x^{2})\},
{\frac{3}{b^{2}}}{\frac{a^{\prime}}{a}}\left({\frac{a^{\prime}}{a}}+{\frac{n^{\prime}}{n}}\right)-{\frac{3}{c^{2}n^{2}}}\left[{\frac{\dot{a}}{a}}\left({\frac{\dot{a}}{a}}-{\frac{\dot{n}}{n}}\right)+{\frac{\ddot{a}}{a}}\right]-{\frac{3k}{a^{2}}}=-{\frac{8\piG_{5}}{c^{4}}}\Lambda.
L_{\theta}dt=\left[-\frac{3e^{2}\hbar}{8mc^{2}}K_{1}(|\mu|\varrho)|\mu|\varrho-\frac{\mu}{|\mu|}\frac{e^{2}\hbar}{2c}K_{1}(\mu\varrho)\varrho+\frac{e^{2}\hbar}{2\muc}\right]d\phi~~,
{\calO}_{2}{}^{i}{}_{j}\equiv\left[\begin{matrix}{-{\frac{\partial^{2}{\calW}}{\partialA_{\mathrm{cl}}^{i}\partialA_{\mathrm{cl}}^{j}}}}&{-i\left(\partial_{1}-i\partial_{2}\right)\delta_{j}^{i}}\\{-i\left(-\partial_{1}-i\partial_{2}\right)\delta_{j}^{i}}&{-{\frac{\partial^{2}{\calW}^{*}}{\partialA_{\mathrm{cl}}^{*i}\partialA_{\mathrm{cl}}^{*j}}}}\\\end{matrix}\right]
D_{3}({\vec{B}}_{3}\times{\vec{B}}_{I})\cdot{\vec{e}}_{I}+({\vec{B}}_{I}\times{\vec{B}}_{3})\cdot(D_{I}{\vec{B}}_{3})\beta-({\vec{B}}_{I}\times{\vec{B}}_{3})\cdotD_{I}[\frac{1}{|{\vec{B}}_{3}|}({\vec{B}}_{J}\times{\vec{e}}_{J})]=0
\overline{{\not\!\partial}}=\gamma^{a}\delta_{a}^{\mu}\partial_{\mu}\,,\qquad\not\!\partial=\gamma^{\mu}\partial_{\mu}=a^{-1}\overline{{\not\!\partial}}\,,\qquad\overline{{\gamma}}_{\mu}=\delta_{\mu}^{a}\gamma_{a}\,,\qquad\overline{{\gamma}}^{\mu}=\delta_{a}^{\mu}\gamma^{a}\,.
(36)T_{a}(\zeta,\bar{z})=\kappa(\zeta)_{a}^{\alpha}\Lambda_{\alpha}(\zeta,\bar{z})-\varphi_{a}(\zeta),\;\;\;dK(\zeta,\bar{z})=\Lambda_{\alpha}d\zeta^{\alpha}+\Lambda_{\bar{\beta}}d\bar{z}^{\beta}.
\left[{\bfD}_{\mu\nu}^{(0)}(k)\right]^{-1}=\left[-k^{2}g_{\mu\nu}+(1-\xi)k_{\mu}k_{\nu}\right]\sigma_{3}-i\epsilong_{\mu\nu}\left[{\bfU}^{-1}(k_{0})\right]^{2}\,,
H=\intd^{3}{x}\,\biggl[\,\frac{1}{2}\,\left(P_{ai}-\frac{\theta}{8\pi^{2}}\,B_{ai}^{(+)}(S)\right)^{2}+\left(P_{a}-\frac{\theta}{8\pi^{2}}\,B_{a}^{(-)}(S)\right)^{2}+\frac{1}{2}\,V(S)\,\biggr]\,.
A^{\prime}=-ie^{2}\beta^{\prime}A\;,\;\;\;\;B^{\prime}=ie^{2}\beta^{\prime}B\;,\;\;\;\;C^{\prime}=ie^{2}\beta^{\prime}C\;,\;\;\;\;D^{\prime}=-\left[\frac{1}{x^{2}}+ie^{2}\beta^{\prime}\right]D\;,
\tilde{f}(q,\phi)\equivf(\tilde{q})=f(q)+\Sigma_{n=1}^{\infty}\tilde{f}(q,\phi)^{(n)}~~;~~\tilde{f}^{(1)}=-\phi_{\mu}^{\alpha}\omega_{\alpha\beta}^{\mu\nu}X_{\nu\lambda}^{\beta\delta}\{\bar{\Theta}_{\delta}^{\lambda},f\}_{(q)}