harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde{a}_{L}^{(n)}(\{q\})\sim\rho_{1}^{2}\,,\qquad\tilde{b}_{L}^{(n)}(\{q\})\sim\rho_{1}^{2}\,,\qquad\tilde{\alpha}_{L}^{(n)}(\{q\})\sim\rho_{1}\sqrt{\rho_{1}}
\hat{p}_{L}^{M}=\frac{1}{2}\Bigl(\frac{n_{M}}{R^{M}}+\frac{m_{M}R^{M}}{\alpha^{\prime}}\Bigr),\,\,\,\hat{p}_{R}^{M}=\frac{1}{2}\Bigl(\frac{n_{M}}{R^{M}}-\frac{m_{M}R^{M}}{\alpha^{\prime}}\Bigr)
\pi_{\mu\nu}=-\frac{1}{4}T_{\mu\alpha}T_{\nu}^{\;\alpha}+\frac{1}{12}TT_{\mu\nu}+\frac{1}{8}g_{\mu\nu}T_{\;\beta}^{\alpha}T_{\alpha}^{\;\beta}-\frac{1}{24}g_{\mu\nu}T^{2}
v_{1\epsilon\kappa}\left(r\right)=\sqrt{\frac{\epsilon+m}{2\epsilon}}\,2kr\,j_{l}\left(kr\right)\,,\quadv_{2\epsilon\kappa}\left(r\right)=\sqrt{\frac{\epsilon+m}{2\epsilon}}\,\frac{\kappa_{0}k}{\epsilon+m}\,2kr\,j_{\overline{l}}\left(kr\right)
\mathrm{Im}\Sigma(y)={\hat{g}}^{4}T^{2}\Big\{{\frac{2\pi^{3}}{9}}+7.9y-{\frac{\pi}{2}}y^{2}+{\frac{\pi}{3}}y\mathrm{ln}y^{2}-{\frac{\pi}{3}}(\mathrm{ln}y^{2})^{2}\Big\}+0({\hat{g}}^{5})
\begin{array}{rcl}{\tilde{S}}&{=}&{\intd^{d-1}x\sqrt{|g|}\\left\{k\\left[R-{\textstyle\frac{1}{4}}k^{2}F_{(2)}^{2}+{\textstyle\frac{(-1)^{(d-3)}}{2\cdot(d-2)!}}k^{-2}F_{(d-2)}^{2}-{\textstyle\frac{1}{2}}m^{2}k^{-2}\right]\right.}\\{}&{}&{}\\{}&{}&{\left.+{\textstyle\frac{1}{(d-2)!}}\frac{\epsilon}{\sqrt{|g|}}F_{(d-2)}\left[-mA_{(1)}\right]\right\}\,.}\\\end{array}
H_{\mathrm{free}}=\int\frac{1}{2}\left\{|\pi(\stackrel{\rightharpoonup}{x})|^{2}+|\bigtriangledown\varphi(\stackrel{\rightharpoonup}{x})|^{2}+m|\varphi(\stackrel{\rightharpoonup}{x})|^{2}\right\}d^{3}\stackrel{\rightharpoonup}{x}\\\.
\psi=\left(\begin{array}{c}{-\sqrt{2\zeta}(z_{2}+z_{4})}\\{\sqrt{2\zeta}(\bar{z}_{1}+\bar{z}_{3})}\\{(z_{1}+z_{3})(\bar{z}_{1}+\bar{z}_{3})+(\bar{z}_{2}+\bar{z}_{4})(z_{2}+z_{4})}\\\end{array}\right)\,.
S_{RB}[\Lambda,a_{\mu}]=\intd^{2}x[-\frac{1}{2}(\partial_{\mu}\Lambda-\frac{1}{\sqrt{2\pi}}a_{\mu})(\partial^{\mu}\Lambda-\frac{1}{\sqrt{2\pi}}a^{\mu})+\frac{1}{2\sqrt{2\pi}}\Lambda\epsilon_{\mu\nu}F_{a}^{\mu\nu}]
{\tilde{\langleT_{\mu}^{\mu}\rangle}}=\frac{\hbarH^{3}}{4\pic^{2}}(\bar{m}^{2}\gamma)\left\lbrack(1-6\zeta)-\bar{m}^{2}\gamma\right\rbrack^{\frac{1}{2}}\cdot\mathrm{ctg}\:\pi\left\lbrack(1-6\zeta)-\bar{m}^{2}\gamma\right\rbrack^{\frac{1}{2}}
{\calL}_{q}(x,x^{\prime})=\bar{q}_{\alpha\alpha^{\prime}}(x,x^{\prime})\left(i\hbarc\gamma^{\mu}\partial_{\mu}-mc^{2}\right)_{\alpha\beta}\left(i\hbar{c}^{\prime}\gamma^{\mu^{\prime}}\partial_{\mu^{\prime}}-m{c^{\prime}}^{2}\right)_{\alpha^{\prime}\beta^{\prime}}q_{\beta\beta^{\prime}}(x,x^{\prime})
g_{o}^{2}=\frac{1}{(2\pi\alpha^{\prime})^{2}\tau_{p}};~~~~\tau_{p}=\frac{1}{g_{s}(2\pi)^{p}\sqrt{\alpha^{\prime}}^{p+1}};~~~~\kappa^{2}=\frac{(2\pi)^{7}}{2}g_{s}^{2}\alpha^{4}.
\Omega^{\mu_{1}\nu_{1}...\mu_{n}\nu_{n}}=\frac1{\left(4\pi\right)^{n}}\int\left(\prod_{j=1}^{n}d^{3}x_{j}\frac{(x_{j}-y_{j})^{\mu_{j}}}{\left|x_{j}-y_{j}\right|^{3}}\;\right)<j^{\nu_{1}}(x_{1})...j^{\nu_{n}}(x_{n})>\;.
\hat{e}_{n}{}^{A}=e^{mz}\left(\begin{array}{cc}{e^{\frac{1}{12}\phi}e_{\mu}{}^{a}}&{e^{-\frac{2}{3}\phi}A_{\mu}}\\{0}&{e^{-\frac{2}{3}\phi}}\\\end{array}\right)\quad\hat{e}_{A}{}^{n}=e^{-mz}\left(\begin{array}{cc}{e^{-\frac{1}{12}\phi}e_{a}{}^{\mu}}&{-e^{-\frac{1}{12}\phi}A_{a}}\\{0}&{e^{\frac{2}{3}\phi}}\\\end{array}\right)\;,
\begin{array}{ll}{\delta\Psi_{\alpha}=\displaystyle\frac18\omega^{mn}(\Gamma_{mn})_{\alpha}{}^{\beta}\Psi_{\beta},}&{\Gamma_{mn}\equiv\Gamma_{m}\tilde{\Gamma}_{n}-\Gamma_{n}\tilde{\Gamma}_{m},}\\{\delta\chi^{\alpha}=\displaystyle\frac18\omega^{mn}(\tilde{\Gamma}_{mn})^{\alpha}{}_{\beta}\chi^{\beta},}&{\tilde{\Gamma}_{mn}\equiv\tilde{\Gamma}_{m}\Gamma_{n}-\tilde{\Gamma}_{n}\Gamma_{m},}\\\end{array}
\left({\frac{d}{dx}}+{\frac{g}{x}}-x\right)e^{-{\frac{1}{2}}x^{2}}\,x^{g+1}\,M(-n+1,g+{\frac{3}{2}},x^{2})=(2g+1)\,e^{-{\frac{1}{2}}x^{2}}\,x^{g}\,M(-n,g+{\frac{1}{2}},x^{2})
\langle\oint_{K_{\ell}}dx^{\mu}A_{\mu}(x)\oint_{K_{m}}dy^{\nu}A_{\nu}(y)\rangle,~~K_{\ell}\neK_{m};~~~\mathrm{and}~~~~\langle\oint_{K}dx^{\mu}A_{\mu}(x)\oint_{K}dy^{\nu}A_{\nu}(y)\rangle
\left(i\gamma^{0}\partial_{t}+i\gamma^{i}\left(\frac{1+{\tilde{R}}^{2}}{2}\partial_{{\tilde{X}}_{i}}-{\tilde{X}}_{i}\right)-gU^{\gamma_{5}}\right){\tilde{\psi}}\big({\tilde{X}}_{i},t\big)=0,
\begin{array}{rcl}{U_{b}/m^{2}}&{=}&{\displaystyle+[9/2D_{m}^{2}(k_{1})D_{m}(k_{2})D_{m}(k_{1}+k_{2})+1/2D_{m}^{2}(k_{1})D(k_{2})D(k_{1}+k_{2})}\\{}&{}&{\displaystyle+D^{2}(k_{1})D_{m}(k_{2})D(k_{1}+k_{2})](1+K_{12})}\\{}&{}&{\displaystyle+2(N-1)D_{m}^{2}(k_{1})D(k_{2})D(k_{1}+k_{2})}\\{}&{}&{\displaystyle+4ND^{2}(k_{1})D_{m}(k_{2})D(k_{1}+k_{2}),}\\\end{array}
{\tilde{\nu}}_{\tilde{\alpha}}^{(1)}\frac{\partialV_{(n-1)}^{(1)}(a_{i},p_{i},\theta)}{\partial{\tilde{\xi}}_{\tilde{\alpha}}^{(1)}}-\frac{\partial{\calG}^{(n)}(a_{i},p_{i},\theta)}{\partial\theta}=0,
D_{i}{}^{j}=\left(\begin{array}{ccc}{d_{AA}}&{d_{Ac}}&{d_{A\bar{c}}}\\{d_{cA}}&{d_{cc}}&{d_{c\bar{c}}}\\{d_{\bar{c}A}}&{d_{\bar{c}c}}&{d_{\bar{c}\bar{c}}}\\\end{array}\right)
J_{x}=J_{(2)}={\frac{1}{\sqrt{2}}}({\bar{\alpha}}_{A}{\bar{\alpha}}^{B}-{\bar{\beta}}_{A}{\bar{\beta}}^{B}){\bar{\pi}}_{B}\omega^{A}+{\frac{1}{\sqrt{2}}}(\alpha_{A^{\prime}}\alpha^{B^{\prime}}-\beta_{A^{\prime}}\beta^{B^{\prime}})\pi_{B^{\prime}}{\bar{\omega}}^{A^{\prime}},
\hat{N}_{(a_{1},m_{1})\;(a_{2},m_{2})}{}^{(a_{3},m_{3})}=\sum_{(a^{\prime},m)}\frac{\Psi_{(a_{1},m_{1})}{}^{(a^{\prime},m)}\Psi_{(a_{2},m_{2})}{}^{(a^{\prime},m)}{\Psi_{(a_{3},m_{3})}{}^{(a^{\prime},m)}}^{*}}{\Psi_{(1,0)}{}^{(a^{\prime},m)}}
G_{MN}^{(I)}=e^{-{\frac{\Phi^{(I)}}{2}}}G_{MN}^{(H)},\\\Phi^{(I)}=-\Phi^{(H)},\\B_{MN}^{(I)}=B_{MN}^{(H)},\\A_{M}^{(I)}=A_{M}^{(H)}.
\mathbf{B}^{\prime}(\mathbf{x})=\frac{1}{4\pi}\int\rho_{m}^{\prime}(\mathbf{x}^{\prime})\frac{\mathbf{r}}{r^{3}}\,d^{3}x^{\prime}\,+\frac{1}{4\pi}\int\mathbf{J}_{e}^{\prime}(\mathbf{x}^{\prime})\,{\bf\times}\,\frac{\mathbf{r}}{r^{3}}\,d^{3}x^{\prime}.
\mathcal{L}=\partial_{\mu}\phi^{*}\,\partial^{\mu}\phi-\phi^{*}m^{2}\phi+\sum_{i=1}^{2}\left[\frac{1}{2}\,\partial_{\mu}\varphi_{i}\,\partial^{\mu}\varphi_{i}-\phi^{*}U_{i}(\varphi_{i})\phi\right],
{\frac{d{e}_{1}(s)}{ds}}=-{e}_{2}^{*}(s)e_{3}(s),\quad{\frac{d{e}_{2}(s)}{ds}}=-{e}_{1}^{*}(s){e}_{3}(s),\quad{\frac{d{e}_{3}(s)}{ds}}=-{e}_{1}(s)e_{2}(s).
P_{i}T(P_{i_{1}}\cdotsP_{i_{n}})V_{\vec{\lambda},\vec{m}}(z)=\sum_{k=0}^{n}q_{i}^{(a_{ii_{1}}+\cdots+a_{ii_{k}})/2}T(P_{i_{1}}P_{i_{2}}\cdotsP_{i_{k}}P_{i}P_{i_{k+1}}\cdotsP_{i_{n}}V_{\vec{\lambda},\vec{m}}(z)~,
\begin{array}{llll}{{\bfG}^{(N)}}&{=}&{({\bf1}+{\bfK})[({\bf1}+{\bf\tilde{K}})({\bf1}+{\bfK})]^{\frac{N-1}{2}}-{\bf1}\;\;\;}&{\;N\in\mathrm{odd}\;,}\\{\tilde{{\bfG}}^{(N)}}&{=}&{[({\bf1}+{\bfK})({\bf1}+{\bf\tilde{K}})]^{\frac{N}{2}}-{\bf1}\;\;\;}&{\;N\in\mathrm{even}\;.}\\\end{array}
\tilde{L}_{\omega}=\int_{0}^{2\pi}\!\{\Theta^{+}[\tilde{f}_{\omega}^{E}(+)T_{++}^{E}+\tilde{f}_{-\omega}^{E}(-)T_{--}^{E}]+\Theta^{-}[\tilde{f}_{\omega}^{L}(+)T_{++}^{L}+\tilde{f}_{-\omega}^{L}(-)T_{--}^{L}]\}d\sigma
T_{s}=\frac{1}{4\pi}\frac{n-2}{\sqrt{\left({\frac{4(n-1)}{{n}^{2}}}\right)^{\frac{2}{\left(n-2\right)}}-\left({\frac{4(n-1)}{{n}^{2}}}\right)^{\frac{n}{\left(n-2\right)}}}}\,\frac{1}{\alpha}.
L=2t\int_{\Sigma}d^{2}z(\frac{1}{2}g_{i\bar{j}}(\partial_{z}\phi^{i}\partial_{\bar{z}}\phi^{\bar{j}}+\partial_{z}\phi^{\bar{j}}\partial_{\bar{z}}\phi^{i})+i\psi_{z}^{\bar{i}}D_{\bar{z}}\chi^{i}g_{i\bar{i}}+i\psi_{\bar{z}}^{i}D_{z}\chi^{\bar{i}}g_{i\bar{i}}-R_{i\bar{i}j\bar{j}}\psi_{\bar{z}}^{i}\psi_{z}^{\bar{i}}\chi^{j}\chi^{\bar{j}})
g^{-1}dg=\sum_{n=-1}^{+\infty}\omega_{\pm}^{n}L_{n}^{\pm}+\sum_{n=-2}^{+\infty}\Omega_{\pm}^{n}J_{n}^{\pm}+\ldots=g_{red}^{-1}dg_{red}\in\widetilde{\calG}\quad,
\langle\tilde{D}_{-}\tilde{A}_{k}|\dot{\tilde{A}}_{k}\rangle=i\frac{1}{2}\sum_{p>0}\big[(a_{k}^{p})^{\dagger}\dot{a}_{k}^{p}-(\dot{a}_{k}^{p})^{\dagger}a_{k}^{p}\big],
\langle\bar{\psi}\psi\rangle_{0\mu}~\simeq~-~\frac{\verteH\vert}{\pi^{2}}m_{\mu}\left[\theta(m_{\mu}-\mu)\int_{0}^{\sqrt{\verteH\vert}}\frac{dp_{3}}{\sqrt{p_{3}^{2}+m_{\mu}^{2}}}+\theta(\mu-m_{\mu})\int_{\sqrt{\mu^{2}-m_{\mu}^{2}}}^{\sqrt{\verteH\vert}}\frac{dp_{3}}{\sqrt{p_{3}^{2}+m_{\mu}^{2}}}\right]_{.}
{\frac{2}{\rho}}\,\gamma_{,\rho}=2\,\psi_{,\rho}^{2}-2\,\psi_{,z}^{2}+4\,\phi_{,\rho}^{2}-4\,\phi_{,z}^{2}-\kappa^{2}\,e^{2\,\psi-a\,\phi}\,(A_{,\rho}^{2}-\,A_{,z}^{2})
\Gamma[\bar{\Phi},\Phi]=\intd^{8}z{\calL}_{eff}(\Phi,D_{A}\Phi,D_{A}D_{B}\Phi;\bar{\Phi},D_{A}\bar{\Phi},D_{A}D_{B}\bar{\Phi})+(\intd^{6}z{\calL}_{eff}^{(c)}(\Phi)+h.c.)+\ldots
H^{2}\simeq\left(\frac{8\,\pi\,\rho_{r}}{3\,M_{p}^{2}}+\omega_{4}\,M\right)\frac{1}{a^{4}}-\frac{3\,\omega_{4}^{2}\,Q^{2}}{16}\,\frac{1}{a^{6}}+\frac{4\,\pi\,\rho_{r}^{2}}{3\,M_{p}^{2}\,\rho_{0}}\,\frac{1}{a^{8}}\,\,.
B^{r}=\dot{x}^{+}(e_{+}^{r}-\frac{1}{4}\bar{\theta}\Gamma^{rst}\theta\omega_{+st})+\dot{x}^{i}(e_{i}^{r}-\frac{1}{4}\bar{\theta}\Gamma^{rst}\theta\omega_{ist})+\bar{\theta}\Gamma^{r}\dot{\theta}~,
\left(\Gamma_{eff}^{PV(1)}\right)_{3}=i\intd^{2}x\,B\left[c_{1}(\nabla^{2}\hat{A}_{0}){\frac{\partial}{\partiala}}+c_{2}\nabla^{2}\right]{\frac{\partial}{\partialM^{2}}}\left({\frac{\Gamma(a+\beta\hat{A}_{0},M)}{M}}\right)
\left(-\partial_{z_{i}}^{2}+\frac{D(D-2)}{4\big(|z_{i}|+\ell\big)^{2}}\,\sum_{i=1}^{D-4}sgn(z_{i})^{2}-\frac{(D-2)}{\left(|z_{i}+\ell|\right)}\,\delta(z_{i})\right)\psi(z_{i})=m^{2}\psi(z_{i})\,.
\begin{array}{rcl}{\partial_{\mu}\eta^{\alpha\beta}}&{=}&{0,}\\{\partial_{\mu}\eta_{\alpha\nu}}&{=}&{\eta_{\alpha\sigma}\Lambda_{\nu\mu}^{\sigma}-\eta_{\nu\sigma}\Lambda_{\alpha\mu,}^{\sigma}}\\{\partial_{\mu}\eta_{\alpha\beta}}&{=}&{\eta_{\alpha\nu}\Lambda_{\beta\mu}^{\nu}+\eta_{\beta\nu}\Lambda_{\alpha\mu}^{\nu}.}\\\end{array}
h^{\prime\prime}-\frac{1}{r}h^{\prime}+\left[\omega^{2}-k^{2}-\kappa^{2}(F^{\mathrm{ANO}}(r))^{2}\right]h=2\kappa^{2}F^{\mathrm{ANO}}(r)\chi^{\mathrm{ANO}}(r)f(r,k,\omega).
V^{a{\mu}_{1}{\cdots}{\mu}_{q+1}}=\frac{(-1)^{p}}{(2p-q+1)!}{\Gamma}^{ab}{\epsilon}^{{\mu}_{1}{\cdots}{\mu}_{q+1}{\nu}_{1}{\cdots}{\nu}_{2p-q+1}}{\partial}_{[{\nu}_{1}}Z_{{\nu}_{2}{\cdots}{\nu}_{2p-q+1}]}^{b},
\eta_{+}^{A}=\phi_{+}(y)\left(\begin{array}{c}{\hat{\eta}_{R}}\\{-\hat{\eta}_{L}}\\\end{array}\right)^{A}\,\qquad\eta_{-}^{A}=\phi_{-}(y)\epsilon(y)\left(\begin{array}{c}{\hat{\eta}_{L}}\\{\hat{\eta}_{R}}\\\end{array}\right)^{A}\,
\left(\begin{array}{l}{f_{n}(r)}\\{g_{n}(r)}\\\end{array}\right)=\left(\begin{array}{c}{C_{n}^{(1)}(E)J_{n-\Phi^{(0)}}(|E|r)+C_{n}^{(2)}(E)Y_{n-\Phi^{(0)}}(|E|r)}\\{i({E/|E|})\bigl[C_{n}^{(1)}(E)J_{n+1-\Phi^{(0)}}(|E|r)+C_{n}^{(2)}(E)Y_{n+1-\Phi^{(0)}}(|E|r)\bigr]}\\\end{array}\right),
\frac{d}{dr}\left\{\frac{w^{\prime}}{2(V^{2}-K^{2})}\left(\frac{K^{2}\sqrt{1+V^{2}}}{1+K^{2}}-\frac{(2V^{2}-K^{2})\mathcal{A}}{\sqrt{V^{2}-K^{2}}}\right)\right\}=\frac{w\,V(K^{2}\mathcal{A}-V^{2})}{(V^{2}-K^{2})\sqrt{1+V^{2}}}.
\int{\calD}A{\calD}c{\calD}{\bar{c}}e^{iS_{\mathrm{eff}}^{L}[A,c,{\bar{c}}]}=\int{\calD}A^{\prime}{\calD}c^{\prime}{\calD}{\bar{c}}^{iS_{\mathrm{eff}}^{A}[A^{\prime},c^{\prime},{\bar{c}}^{\prime}]}.
\langle0_{M}|O^{(+)}|0_{M}\rangle=\mathrm{Tr}\left[e^{-\frac{2\pi}{a}aH_{R}^{(+)}}O^{(+)}\right]/\mathrm{Tr}\left[e^{-\frac{2\pi}{a}aH_{R}^{(+)}}\right],
\Gamma^{(1)}=N\intd^{3}x\left(f^{(0)}u^{4}+f^{(2)}u^{3}\left[\psi^{2}\right]+f^{(4)}u^{2}\left[\psi^{4}\right]+f^{(6)}u\left[\psi^{6}\right]+f^{(8)}\left[\psi^{8}\right]\right),
\int\mathrm{d}z_{1}\ldots\int\mathrm{d}z_{n}\frac{e^{-\frac{\left(x-z_{1}\right)^{2}}{4\sigma}}}{\sqrt{4\pi\sigma}}\frac{e^{-\frac{\left(z_{1}-z_{2}\right)^{2}}{4\sigma}}}{\sqrt{4\pi\sigma}}\ldots\frac{e^{-\frac{\left(z_{n}-y\right)^{2}}{4\sigma}}}{\sqrt{4\pi\sigma}}=\frac{e^{-\frac{\left(x-y\right)^{2}}{4\sigma(n+1)}}}{\sqrt{4\pi\sigma(n+1)}}
\partial_{\mu}T\left(j_{A\alpha}^{\mu}(x)\bar{j}_{B\dot{\beta}}^{\nu}(y)\right)=2(\sigma^{\rho})_{\alpha\dot{\beta}}T_{\rho}^{\nu}\delta^{4}(x-y)\delta_{AB}+2(\sigma^{\nu})_{\alpha\dot{\beta}}C_{AB}\delta^{4}(x-y)
\begin{array}{c}{a_{1}^{+}|jm>=q^{-\frac{1}{2}n_{2}}\sqrt{[n_{1}+1]_{q}}\\|j+\frac{1}{2},m+\frac{1}{2}>}\\{a_{2}^{+}|jm>=q^{\frac{1}{2}n_{1}}\sqrt{[n_{2}+1]_{q}}\\|j+\frac{1}{2},m-\frac{1}{2}>}\\{a_{1}|jm>=q^{-\frac{1}{2}n_{2}}\sqrt{[n_{1}]_{q}}\\|j-\frac{1}{2},m-\frac{1}{2}>}\\{a_{2}|jm>=q^{\frac{1}{2}n_{1}}\sqrt{[n_{2}]_{q}}\\|j-\frac{1}{2},m+\frac{1}{2}>.}\\\end{array}
{}^{i}\hat{O}_{\lambda,\mu}^{r_{1}r_{2}}={}^{i}\hat{O}_{\lambda}^{r_{1}}\otimes{}^{i}\hat{O}_{\mu}^{r_{2}}\equiv{}^{i}\hat{O}_{\lambda,\mu}^{r}={}^{i}O_{\lambda,\mu}^{r}(\alpha_{\lambda}\otimes\alpha_{\mu}),
\begin{array}{rll}{\delta(D_{\mu}^{ij}[A]Q^{\mu,j}-a^{i})}&{=}&{\delta(D_{\mu}^{ij}[A]Q^{\mu,j}+D_{\mu}^{ij}[A]\deltaQ_{\mu}^{j}-a^{i}-\deltaa^{i})}\\{}&{=}&{\delta(D_{\mu}^{ij}[A]Q^{\mu,j}-a^{i})}\\\end{array}
E(x)=\frac{1}{{\partial}_{-}^{2}}D(x)-\frac{1}{{\partial}_{\bot}^{2}+n^{2}{\partial}_{+}^{2}}D_{s}(x)-\frac{\mathrm{sin}2{\theta}{\partial}_{+}+{\partial}^{-}}{({\partial}_{\bot}^{2}+n^{2}{\partial}_{+}^{2})^{2}}{\partial}_{-}D_{s}(x)
\tau={\frac{2^{4/3}g_{c}(1+2\sqrt{7})^{2}(1+(-1+\sqrt{7})e^{2h}+(2+\sqrt{7})e^{4h})Z}{15e^{2h}(1+e^{2h})(\Phi(Z,T)+(4T)^{2/3})}}\epsilon^{-1/3}+{\calO}(\epsilon^{0})\.
\rho_{r}(x,x^{\prime};t)=\intdx_{i}dx_{i}^{\prime}\int_{x_{i}}^{x}{\calD}x\int_{x_{i}^{\prime}}^{x^{\prime}}{\calD}x^{\prime}\;e^{{\frac{i}{\hbar}}(S_{x}[x]-S_{x}[x^{\prime}])}\;e^{{\frac{i}{\hbar}}S_{\mathrm{IF}}[x,x^{\prime}]}\rho_{r}(x_{i},x_{i}^{\prime};t_{i})
\begin{array}{cr}{-\frac{1}{2}\frac{\alpha^{\prime}}{a^{2}}\ddot{U}+\frac{\sqrt{\alpha^{\prime}}}{a}Q\dot{U}-\lambdaU=0\,}&{}\\{L(1-L)H^{\prime\prime}(L)+(\frac{D}{2}-DL)H^{\prime}(L)+2(2-\lambda)H(L)=0\}&{(\H^{\prime}=\frac{d}{dL}H\)\.}\\\end{array}
P_{0}=\int\!d\,^{3}xT_{e^{+}e^{-}}^{00}=\int\!d\,^{3}x[\overline{{\Psi}}(i\gamma_{i}\partial_{i}-e\gamma_{i}A_{i}+m)\Psi+e\overline{{\Psi}}\gamma_{0}A_{0}\Psi]=\int\!d\,^{3}x\overline{{\Psi}}i\gamma_{0}\partial_{0}\Psi\,
{\calS}\,=\,{\frac{1}{2}}\int\,dx\,dt\,\left[\frac{\dot{\phi}^{a}\dot{\phi}^{a}\,+\,\phi^{a}\partial_{i}\partial_{i}\phi^{a}}{\phi^{a}\,\phi^{a}}\,-\,\frac{(\,\dot{\phi}^{a}\dot{\phi}^{a}\,)^{2}}{(\,\phi^{a}\,\phi^{a}\,)^{2}}\right]\;\;,
\Omega_{\gamma}\approx\frac{1}{2}\int_{\gamma}dx_{\mu}^{\prime}\,\varepsilon_{\mu\nu}\left(\frac{x_{\nu}^{\prime}-x_{\nu}}{|x^{\prime}-x|^{2}}-\frac{x_{\nu}^{\prime}-y_{\nu}}{|x^{\prime}-y|^{2}}\right).
\begin{array}{c}{2\pi{\calE}=\int_{M_{3}}d(\mathrm{tr}(\Phi_{A}H^{A}))+\int_{M_{3}}d(\mathrm{tr}(\Phi_{B}K^{B}))}\\{=\int_{S^{2}}\mathrm{tr}(\Phi_{A}H^{A})+\int_{S^{2}}\mathrm{tr}(\Phi_{B}K^{B})}\\\end{array}
\left<\xi_{i}(\sigma){\calO}\left[\vec{\xi}{\,}\right]\right>_{\vec{\xi}}=\int_{0}^{1}d\sigma^{\prime}G\left(\sigma-\sigma^{\prime}\right)\left<\frac{\delta{\calO}\left[\vec{\xi}{\,}\right]}{\delta\xi_{i}\left(\sigma^{\prime}\right)}\right>_{\vec{\xi}},
n(\beta,\mu)=\frac{\sum_{l=0}^{n-1}\sum_{p=0}^{n-1}q^{-pl}\sum_{k_{l}^{m}}f_{q^{p}}(\beta(\epsilon^{l}(k_{l}^{m})-\mu))\prod_{k_{l}^{m}}(1+q^{p}e^{\beta(\mu-\epsilon^{l}(k_{l}^{m}))})}{M\sum_{l=0}^{n-1}\sum_{p=0}^{n-1}q^{-pl}\prod_{k_{l}^{m}}(1+q^{p}e^{\beta(\mu-\epsilon^{l}(k_{l}^{m}))})}\.
\hat{H}_{\mathrm{rel}}^{\prime}=-\frac{1}{2\mu}\left[\frac{\partial^{2}}{\partialr^{2}}+\frac{1}{r}\frac{\partial}{\partialr}+\frac{1}{r^{2}}\left(\frac{\partial}{\partial\theta}+i\omega_{j}\right)^{2}\right].
{\frac{-4}{\sigma_{0}}x_{1}x_{3}+x_{2}^{~2}+x_{3}^{~2}\frac{1}{\sigma_{0}}\left(\epsilonk_{0}\ell^{2}-\frac{12}{\sigma_{0}}+\frac{2}{3}C_{\rho}+\epsilon\frac{1}{2}C_{adj}\right)}=0
\intd{\bar{\eta}}d{\eta}\bar{\phi}e_{c}^{({\partial}_{{\eta}^{i}}{\partial}_{\bar{\eta}}^{i})}\psi:={(\bar{\phi}{e_{c}}^{({{\partial}_{\eta}}_{i}{{\partial}_{\bar{\eta}}}^{i})}\psi)}_{\mathrm{\small~evaluated~at~}\eta=0=\bar{\eta}\mathrm{}}
D=\frac{(1-\eta)^{1+\frac{1}{x_{\alpha}x_{\beta}}}}{\kappa_{\beta}^{\frac{1}{x_{\beta}}}}+\frac{\eta^{1+\frac{1}{x_{\alpha}x_{\beta}}}}{\kappa_{\alpha}^{\frac{1}{x_{\alpha}}}}\,,
L^{(-)}=\left(\begin{array}{cc}{q^{\left(\begin{array}{cc}{1}&{0}\\{0}&{0}\\\end{array}\right)}}&{0}\\{(q^{-1}-q)Z^{-}}&{(-1)^{F}q^{-\left(\begin{array}{cc}{0}&{0}\\{0}&{1}\\\end{array}\right)}}\\\end{array}\right)
\left[G_{i}^{(1)a},G_{j}^{(1)b}\right]^{*}=0,\;\left[G_{i}^{(1)a},G_{j}^{(2)b}\right]^{*}=0,\;\left[G_{i}^{(2)a},G_{j}^{(2)b}\right]^{*}=0,
C_{V}=\sum_{n,w}\int_{0}^{\infty}[dt]e^{-y}\left[4t^{2}({\frac{w^{2}\beta^{2}}{4\pi^{2}\alpha^{\prime}}}-{\frac{\alpha^{\prime}\pi^{2}n^{2}}{\beta^{2}}}))^{2}-2t({\frac{w^{2}\beta^{2}}{4\pi^{2}\alpha^{\prime}}}+3{\frac{\alpha^{\prime}\pi^{2}n^{2}}{\beta^{2}}})\right].
\int|x|\y^{2}\\Gamma_{x,y,0}^{\prime}\\mathrm{d}^{3}x\mathrm{d}^{3}y-2\int|x|\y\cdotz\\Gamma_{x,y,z,0}^{\prime}\\mathrm{d}^{3}x\mathrm{d}^{3}y\mathrm{d}
D(x,x^{\prime})=\int{\frac{d^{6}k}{(2\pi)^{6}}}{\frac{1}{k^{2}}}e^{ik_{\mu}(x-x^{\prime})^{\mu}}\left(e^{i{\vec{k}}\cdot({\vec{x}}-{\vec{x}}^{\prime})}+e^{i{\vec{k}}\cdot({\vec{x}}+{\vec{x}}^{\prime})}\right).
{\tilde{T}}_{R}(k)=a^{k^{2}-1}\left({\tilde{T}}_{R}(k)+\intdk_{1}\intdk_{2}\frac{{\tilde{T}}_{R}(k_{1}){\tilde{T}}_{R}(k_{2})}{2k_{1}\cdotk_{2}+1}\delta^{(26)}(k_{1}+k_{2}-k)\right)
\left\langleF(f_{1^{^{\prime}}})F(f_{2^{^{\prime}}})F(f_{2})F(f_{1})\right\rangle_{therm}=\left\langleF(f_{2^{^{\prime}}})F(f_{2})F(f_{1})F(f_{1^{^{\prime}}}^{2\pii})\right\rangle_{therm}
\overline{{\Gamma}}_{5}=-\varepsilon^{\widetilde{0},0}-\varepsilon^{0,\widetilde{0}}-\varepsilon^{\widetilde{\mu},\mu}-\varepsilon^{{\mu},\widetilde{\mu}}-\frac12e_{\mu\nu\rho\omega}\varepsilon^{[\mu\nu],[\rho\omega]}.
0\le\left(\frac{1}{2}(1-\Gamma)\epsilon\right)^{\dagger}\left(\frac{1}{2}(1-\Gamma)\epsilon\right)=\epsilon^{\dagger}\frac{1}{2}(1-\Gamma)\frac{1}{2}(1-\Gamma)\epsilon=\epsilon^{\dagger}\frac{1}{2}(1-\Gamma)\epsilon
H_{\mathrm{loc}}=\int_{\Sigma}d^{2}z~\left(i\chi{\calP}\frac{\delta}{\delta\xi}-\frac{4\pi}{k^{\prime}}\xi\frac{\partialJ^{0}}{\partialt}\right)~~~~~~,~~~~~~H_{\mathrm{top}}=i\left(4\pi\bar{j}_{l}\gamma^{l}+\frac{1}{k^{\prime}}j^{l}\frac{\partial}{\partial\gamma^{l}}\right)
\Lambda_{IR}^{+}(p,p)=4\lambdap^{+}e^{3}\int_{\alpha}^{1}{\frac{dx}{x}}\int{\frac{d^{2}k_{\perp}}{(2\pi)^{3}}}{\frac{1}{k_{\perp}^{2}+m^{2}x^{2}}}-4\lambdap^{+}e^{3}\int_{\alpha}^{1}dx\int{\frac{d^{2}k_{\perp}}{(2\pi)^{3}}}{\frac{m^{2}x}{\bigl(k_{\perp}^{2}+m^{2}x^{2}\bigr)^{2}}}\;,
H_{\gamma}=\frac{v^{3}w^{2}}{w^{2}+(2\gammav)^{2}}\;h_{\gamma}(w),\quadK_{\gamma}=4v^{4}\;k_{\gamma}(w),\quadJ_{\gamma}=\frac{v^{3}w^{2}}{w^{2}+(2\gammav)^{2}}\;j_{\gamma}(w),
[P_{a},P_{b}]=-\betaM_{ab},\;\;[M_{aa^{\prime}},P_{b}]=2\delta_{[a}^{c}g_{a^{\prime}]b}P_{c},\;\;[M_{aa^{\prime}},M_{bb^{\prime}}]=4\delta_{[a}^{c}g_{a^{\prime}][b}\delta_{b^{\prime}]}^{c^{\prime}}M_{cc^{\prime}},
\left\{z(1-z)\partial_{z}^{2}+\left[1-\frac{iw_{in}}{\rho}-z(2+2\frac{iw_{-}}{\rho})\right]\partial_{z}-\frac{iw_{-}}{\rho}+\frac{w_{-}^{2}}{\rho^{2}}\right\}\widetilde{\chi}(z,k)=0
H(\omega,0)=\frac{\phi_{1}f_{2}\partial_{r}(\phi_{2}^{(-)}/A_{-})-(\phi_{2}^{(-)}/A_{-})f_{1}\partial_{r}\phi_{1}}{\phi_{1}f_{2}\partial_{r}(\phi_{2}^{(+)}/A_{+})-(\phi_{2}^{(+)}/A_{+})f_{1}\partial_{r}\phi_{1}}\.
{\calL}=\Psi^{\dagger}\left[i\partial_{0}-eA_{0}+\mu-\frac{1}{2m}(i{\bf\nabla}+e{\bfA})^{2}\right]\Psi+\frac{e}{m}B^{a}\Psi^{\dagger}\frac{\sigma^{a}}{2}\Psi.
i\overline{{\psi}}\gamma^{\mu}\psi\longleftrightarrow-\frac{1}{\sqrt{\pi}}\epsilon^{\mu\nu}\partial_{\nu}\Lambda\;\;\;,\;\;\;i\overline{{\psi}}\gamma^{\mu}\gamma_{3}\psi\longleftrightarrow\frac{1}{\sqrt{\pi}}\partial^{\mu}\Lambda.
N_{\frac{1}{4}}-N_{\frac{1}{4}}^{(0)}=\frac{1}{2\pii}\oint_{C_{\frac{1}{4}}}dk\{\frac{[M^{(-)}(\nu,k)]^{\prime}}{M^{(-)}(\nu,k)}-\frac{[M_{o}^{(-)}(k)]^{\prime}}{M_{o}^{-}(k)}\},
S=\intd^{d+1-n}x\sqrt{{g}}e^{-2\phi}\Bigl\{{R}+4({\nabla}\phi)^{2}+\frac{1}{8}Tr\bigl({\calL}\nabla{\calM}\bigr)^{2}-\frac{1}{4}{\calF}_{\mu\nu}^{T}{\calLML}{\calF}^{\mu\nu}-\frac{1}{12}{H}_{\mu\nu\lambda}^{2}\Bigr\}
U\phi_{\ell}=2\alpha\left({\tt\frac{1}{\bar{z}}}\partial_{z}\phi_{\ell}-{\tt\frac{1}{z}}\partial_{\bar{z}}\phi_{\ell}\right)+\frac{\alpha^{2}}{2}\left({\tt\frac{1}{z}{\tt\frac{1}{\bar{z}}}}+{\tt\frac{1}{\bar{z}}}{\tt\frac{1}{z}}\right)\phi_{\ell}\,,
F_{\mathrm{Yukawa}}(k)=-\int\left[d^{4}p\right]\mathrm{tr}\left[\left(\begin{matrix}{i\gamma_{5}}\\{1}\\\end{matrix}\right){\overline{S}}(p)\left(\begin{matrix}{i\gamma_{5}}\\{1}\\\end{matrix}\right){\overline{S}}(p-k)\right]
G_{\mu\nu}(p,\hat{k})=\left({\frac{p^{2}\hat{k}_{\mu}\hat{k}_{\nu}}{(\hat{k}\cdotp)^{2}}}-{\frac{p_{\mu}\hat{k}_{\nu}+p_{\nu}\hat{k}_{\mu}}{\hat{k}\cdotp}}+\eta_{\mu\nu}\right)
\mathbf{d}\widetilde\Omega=\widetilde{d\Omega}+A^{\alpha}\widetilde{L_{\alpha}\Omega}-F^{\alpha}\widetilde{i_{\alpha}\Omega}=\widetilde{d\Omega}+\widetilde{A^{\alpha}L_{\alpha}\Omega}-\widetilde{F^{\alpha}i_{\alpha}\Omega}
M(p^{2})=\alpha\nu^{2\epsilon}c_{\epsilon}\left[{\frac{1}{p^{2}}}\int_{0}^{p^{2}}\frac{dk^{2}(k^{2})^{1-\epsilon}M(k^{2})}{k^{2}+M^{2}(k^{2})}+\int_{p^{2}}^{\infty}\frac{dk^{2}(k^{2})^{-\epsilon}M(k^{2})}{k^{2}+M^{2}(k^{2})}\right]\;\;\;,
\tilde{\Phi}^{1}=c\partialc,~\tilde{\Phi}^{2}=\frac{1}{3}c\partialcT_{B},~\tilde{\Phi}^{3}=\frac12\partialc\partial^{2}c,~\tilde{\Phi}^{4}=\frac16\partial^{3}cc.
c_{1}=(\frac{r_{m}^{3}}{l^{2}}-\frac{Mr_{m}}{2})^{-1},\;\;\;\;c_{2}=\frac{1}{12}(\frac{6r_{m}^{2}}{l^{2}}-M)(\frac{r_{m}^{3}}{l^{2}}-\frac{Mr_{m}}{2})^{-1},
\tilde{H}^{(s),(a)}(\zeta,\zeta^{\prime};t,\lambda_{t})={\frac{{\calN}_{\tilde{B}^{(s),(a)}}(\zeta,\zeta^{\prime};t,\lambda_{t})}{{\calD}_{\tilde{B}^{(s),(a)}}(t,\lambda_{t})}}.\tag{4.1}
|\frac{1}{2},\frac{1}{2}\rangle_{1}=\frac{1}{\sqrt{2(q+q^{-1}-1)}}(q^{1/2}|\uparrow\uparrow\downarrow\rangle+(q^{1/2}-q^{-1/2})|\uparrow\downarrow\uparrow\rangle-q^{-1/2}|\downarrow\uparrow\uparrow\rangle)
\renewcommand{\arraystretch}{1.5}\begin{array}{rcl}{f_{1}^{(\infty)}}&{=}&{x^{-\lambda_{3}}{}_{2}F_{1}(\lambda_{3},1-\lambda_{4};\lambda_{1}+\lambda_{3};1/x)\,,}\\{*f_{2}^{(\infty)}}&{=}&{x^{-\lambda_{4}}(1-x)^{1-\lambda_{2}-\lambda_{3}}{}_{2}F_{1}(\lambda_{4},1-\lambda_{3};\lambda_{2}+\lambda_{4};1/x)\,.}\\\end{array}
S^{\prime}(\phi^{\prime})=\sum_{x}\left[\frac{a^{d-2}}{2}\sum_{\mu=1}^{d}(\phi_{x+\hat{\mu}}^{\prime}-\phi_{x}^{\prime})^{2}+a^{d}\left(\frac{m^{2}}{2}{\phi^{\prime}}_{x}^{2}+\frac{1}{4}{\phi^{\prime}}_{x}^{4}\right)\right]
G_{\alpha}^{(n)}\equiv\sum_{m=0}^{n}\{\tau_{\alpha}^{(n-m)},\tilde{H}^{(m)}\}+\sum_{m=0}^{n-2}\{\tau_{\alpha}^{(n-m)},\tilde{H}^{(m+2)}\}_{(\Phi)}+\{\tau_{\alpha}^{(n+1)},\tilde{H}^{(1)}\}_{(\Phi)},~~n\geq2
F_{ij}^{\alpha}\equiv\partial_{i}\bar{A}_{j}^{\alpha}(\bar{x})-\bar{A}_{i}^{\alpha}(\bar{x})\stackrel{\leftarrow}\partial_{j}+gF^{\alpha\beta\gamma}\bar{A}_{i}^{\beta}(\bar{x})\bar{A}^{\gamma}(\bar{x})
\begin{array}{c}{U_{i}^{n}V_{i}^{n}=qV_{i}^{n}U_{i}^{n},i=1,2;n=1,\ldots,N,}\\{U_{i}^{n^{*}}=U_{i}^{n^{-1}},V_{i}^{n^{*}}=V_{i}^{n^{-1}}.}\\\end{array}
\deltaS_{2}^{(F)}=\frac{b}{2}\intd^{4}x\sqrt{-g}\,h^{ab}\Biggl(2R^{cd}C_{acbd}\psi+4\nabla^{c}\nabla^{d}(C_{acbd}\psi)+\frac{1}{2}g_{ab}F_{4}\psi\Biggr)\,,
{\calH}_{0}^{\prime}=-i\overline{{\psi}}^{1}\gamma^{1}\partial_{1}\psi^{1}-i\overline{{\psi}}^{2}\gamma^{1}\partial_{1}\psi^{2}+i\overline{{\psi}}^{3}\gamma^{1}\partial_{1}\psi^{3}+m\overline{{\psi}}^{1}\psi^{1}.
\sum_{ij}S_{i^{\prime}i}Z_{ij}S_{~jj^{\prime}}^{\dagger}=Z_{{i^{\prime}}{j^{\prime}}}~,~~\sum_{ij}T_{i^{\prime}i}Z_{ij}T_{~jj^{\prime}}^{\dagger}=Z_{{i^{\prime}}{j^{\prime}}}~.
d=\frac{1}{4\pi}\sum_{j,m}\left(2j+1\right)\left(\begin{array}{c}{d_{+}\left(j,m;r\right)D_{+1,m}^{j}\left(\theta,\varphi\right)}\\{d_{0}\left(j,m;r\right)D_{0,m}^{j}\left(\theta,\varphi\right)}\\{d_{\tilde{o}}\left(j,m;r\right)D_{\tilde{o},m}^{j}\left(\theta,\varphi\right)}\\{d_{-}\left(j,m;r\right)D_{-1,m}^{j}\left(\theta,\varphi\right)}\\\end{array}\right)
\langleP[\{\Phi^{(I)}\}]\rangle\;=\;\frac{1}{Z}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\;\langle\;P[\{\Phi^{(I)}\}]\;S_{int}[\{\Phi^{(I)}\}]^{n}\;\rangle_{Q}\;.
\delta_{k}\equiv\frac{{\calP}_{\calR}^{\eta_{k}}-{\calP}_{\calR}^{-\infty}}{{\calP}_{\calR}^{-\infty}}\simeq2\Re\left[\frac{\beta_{k}^{\eta_{k}}}{\alpha_{k}^{\eta_{k}}}\,\frac{({\calH}_{1/2-q})^{\ast}}{{\calH}_{1/2-q}}\right]
M_{fermion\,\pm}^{2}~=~M_{\pm}^{2}~-~{\frac{3}{4}}\,\alpha\gamma\,\mu\,\widetilde{m}_{\lambda}\Bigg(1\pm\sqrt{1+x}~\Bigg)\Bigg(3~\pm~{\frac{1}{\sqrt{1+x}}}~\Bigg),
C_{\mu}^{ab}=2ig\,\overline{{\psi_{A}^{a}}}\,g^{AB}\psi_{C}^{b}\,{A_{\muB}}^{C}+\left(\,\overline{{\psi_{A}^{a}}}\,g^{AB}\,(\partial_{\mu}\psi_{B}^{b})+\psi_{B}^{b}\,g^{AB}\,(\partial_{\mu}\overline{{\psi_{A}^{a}}})\,\right),