harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1}{2!2}\,\epsilon_{ij}\,\mathrm{tr}\Big(\big(-i[X^{i},X^{j}]\big)\big(-i[\hat{\Phi}^{a},\hat{\Phi}^{b}]\big)-\big(-i[X^{i},\hat{\Phi}^{a}]\big)\big(-i[X^{j},\hat{\Phi}^{b}]\big)\Big)C_{12ab}^{(4)}
{\calL}=\varepsilon^{\mu\nu\alpha}\left(\Gamma_{\mu\beta}^{\lambda}\partial_{\nu}\Gamma_{\alpha\lambda}^{\beta}+\frac{2}{3}\\Gamma_{\mu\sigma}^{\rho}\Gamma_{\nu\varphi}^{\sigma}\Gamma_{\alpha\rho}^{\varphi}\right)
\frac{g^{2}}{R^{2}}\left[-\frac{C_{3}}{f^{2}}+\left(\frac{\dot{f}G}{f}\right)^{2}\right]+R^{2}\left(\frac{\dot{f}}{f}\right)^{2}=-\frac{g^{2}}{R^{2}}[G^{2}+\omega^{2}G^{2}]-R^{2}\left(\frac{g^{\prime}}{g}\right)^{2}.
W=\sum_{\alpha,\alpha^{\prime}}H_{\alpha,\alpha^{\prime}}V_{\alpha+1,\alpha^{\prime}}D_{\alpha+1,\alpha^{\prime}+1}-\sum_{\alpha,\alpha^{\prime}}H_{\alpha,\alpha^{\prime}+1}V_{\alpha,\alpha^{\prime}}D_{\alpha+1,\alpha^{\prime}+1}.
\sum_{\alpha=s}^{\infty}\lambda^{\alpha}d\omega^{k_{s},\alpha}=\sum_{\alpha=s}^{\infty}\lambda^{\alpha}\left[-\frac{1}{2}c_{i_{p}j_{q}}^{k_{s}}\sum_{\beta=0}^{\alpha}\omega^{i_{p},\beta}\wedge\omega^{j_{q},\alpha-\beta}\right]\;,
\frac{1}{T_{\mathrm{DR}}^{\mathrm{on}}(E)}=\frac{1}{C_{\mathrm{DR}}}+\frac{2\mu}{(4\pi)^{w}}(-2\muE)^{w-1}\Gamma(1-w)+i\frac{2\pi\mu}{\Gamma(w)(4\pi)^{w}}(2\muE)^{w-1}.
\left\{{\frac{\partial^{2}}{\partialr^{2}}}+{\frac{5}{r}}{\frac{\partial}{\partialr}}+\tilde{\omega}^{2}\left(1+{\frac{\tilde{R}_{\frac{\pi}{2}}^{4}}{r^{4}}}\right)-{\frac{2R_{\frac{\pi}{2}}^{8}}{r^{10}f^{2}}}\right\}\hat{\phi}=0.
\left\{k^{ij}\left(\vec{x},t\right),s^{mn}\left(\vec{y},t\right)\right\}^{*}=\frac14\left(\eta^{im}\eta^{jn}+\eta^{in}\eta^{jm}-\eta^{ij}\eta^{mn}\right)\delta^{2}\left(\vec{x}-\vec{y}\right)\;\,,
=-\,\delta_{\epsilon,\bar{\epsilon}}\langleX\rangle=\frac{1}{2\pii}\oint_{C}dz\,\epsilon(z)\langleT(z)X\rangle-\frac{1}{2\pii}\oint_{C}d\bar{z}\,\bar{\epsilon}(\bar{z})\langle\bar{T}(\bar{z})X\rangle
E^{i-}=-\frac{\imath}{2}\sum_{n=1}^{\infty}\frac{1}{n}\left(\left[\alpha_{-n}^{i},\frac{1}{p^{+}}\right]_{+}\alpha_{n}^{-}-\alpha_{-n}^{-}\left[\alpha_{n}^{i},\frac{1}{p^{+}}\right]_{+}\right)\tag{53}
f_{5}(\zeta,k,\theta,\phi)=\sum_{i\neqj}^{5}\frac{(\theta_{i}-\theta_{j})\phi_{i}(\zeta_{i}\cdotk_{j})(2\alpha^{\prime})^{11/4}-1/2\phi_{i}\phi_{j}(\zeta_{i}\cdot\zeta_{j})(2\alpha^{\prime})^{9/2}}{x_{i}-x_{j}-\theta_{i}\theta_{j}}\,,
\Gamma_{\mathrm{eff}}^{\mathrm{PV},\,(\vec{E})}=\Gamma_{1,\,\mathrm{eff}}^{\mathrm{PV},\,(\vec{E})}+\Gamma_{2,\,\mathrm{eff}}^{\mathrm{PV},\,(\vec{E})}+\cdots
{\calS}[e,\psi]=\int\,d^{4}x\sqrt{g}\left\{\frac{i}{2}[\overline{{\psi}}\gamma^{\hat{\alpha}}D_{\hat{\alpha}}\psi-(\overline{{D_{\hat{\alpha}}\psi}})\gamma^{\hat{\alpha}}\psi]-m\overline{{\psi}}\psi\right\}
\qquad\qquad\Bigl(\frac1l\sum_{i\in\bfZ/l}q^{-ij}\alpha^{i}\Bigr)f^{(l-1-n)}e^{(l-1-p)}\Bigr(\sum_{i\in\bfZ/l}q^{i(s+p)}k^{-i}\Bigl)\;\quad.
S_{bulk}^{i}=-\frac{1}{2\kappa_{5}^{2}}\intd^{5}x\sqrt{-g_{E}^{(5)}}(R+G_{IJ}\partial_{\mu}X^{I}\partial^{\mu}X^{J}-\frac{1}{2{\calV}^{2}}(\partial{\calV})^{2}-\frac{1}{2{\calV}^{2}}\alpha_{I}^{i}\alpha_{J}^{i}G^{IJ}(X)).
^{(F)}T_{ab}[g]=-{\frac{2}{\sqrt{-g}}}{\frac{\delta}{\deltag^{ab}}}\,S_{anom}^{(F)}[g]=b\,C_{ab}-2b\left[C_{a\b}^{\c\d}R_{cd}\,\varphi+2\,\nabla_{c}\nabla_{d}\left(C_{(a\b)}^{\\c\d}\,\varphi\right)\right]\,,
I_{\mathrm{GS}}=T_{(9)}^{I}\;(\pi\alpha^{\prime})^{2}\;\intd^{10}x\;\left[\;C^{(6)}\wedge\mathrm{X}_{4}+\;(\pi\alpha^{\prime})^{2}\;C^{(2)}\wedge\mathrm{X}_{8}\;\right]\,
B_{1}^{in}=(\int_{M}dva_{1}+\int_{S^{2}}dsc_{1})V^{1/2-s}=\int_{M}(\frac{\lambda}{2}\hat{\phi}^{2})^{3/2-s}dv+\frac{2}{3R}\int_{S^{2}}(\frac{\lambda}{2}\hat{\phi}^{2})^{1/2-s}ds
(\omega^{2}-{\omega^{\prime}}^{2})\int_{\theta}^{2\pi}\!\phi_{\omega}^{L}(\sigma)\:\phi_{\omega^{\prime}}^{L}(\sigma)\,d\sigma+\int_{\theta}^{2\pi}\!\partial_{\sigma}(\phi_{\omega^{\prime}}^{L}(\sigma)\:\partial_{\sigma}\phi_{\omega}^{L}(\sigma)-\phi_{\omega}^{L}(\sigma)\:\partial_{\sigma}\phi_{\omega^{\prime}}^{L}(\sigma))\,d\sigma=0
(\partial_{t}r)^{2}=\left(1+{\frac{r^{2}}{\tilde{L}^{2}}}\right)^{2}\left[1-{\frac{1+{\frac{r^{2}}{\tilde{L}^{2}}}}{\left({\frac{r^{p+1}}{\tilde{L}}}+{\frac{E}{T_{p}V_{p}}}\right)^{2}}}{\frac{T_{p}^{2}V_{p}^{2}L^{2}r^{2p}+\ell^{2}}{T_{p}^{2}V_{p}^{2}L^{2}}}\right],
\Pi_{\delta}(x)=m_{*}^{2}(x)-{\frac{g^{2}}{2}}\int{\frac{d^{4}x^{\prime}d^{4}p}{(2\pi)^{4}}}\;{\frac{d^{3}k}{(2\pi)^{3}E_{k}}}\;{\frac{e^{-ip(x^{\prime}-x)}}{p^{2}-m_{*}^{2}+ip_{0}0^{+}}}\;f_{0}(x^{\prime},{\bfk})\;,
\bar{\epsilon}(\psi_{1+})\Gamma^{\mu}\epsilon(\psi_{2-})=\bar{\psi}_{1+}e^{-\frac{1}{4}x^{+}\theta}\left[\Gamma^{\mu}+\frac{1}{24}x^{i}\Gamma^{\mu}\Gamma^{i}U_{(i)}\Gamma^{+}\right]e^{-\frac{1}{12}x^{+}\theta}\psi_{2-}.
e_{\mu}^{a}=\cfrac{\delta_{\mu}^{a}}{1+\frac{m^{2}x^{2}}{4}},\\\omega_{\mu}^{ab}=\frac{m^{2}}{2}\cfrac{x^{a}\delta_{\mu}^{b}-x^{b}\delta_{\mu}^{a}}{1+\frac{m^{2}x^{2}}{4}}.
m_{s}\Phi_{L_{s}}^{T}(-i\sigma^{2})C_{4}\Phi_{L_{s}}=m_{s}\left[-(\Phi_{L_{s}}^{+})^{T}C_{4}\Phi_{L_{s}}^{-}+(\Phi_{L_{s}}^{-})^{T}C_{4}\Phi_{L_{s}}^{+}\right],
\partial_{\tau}\Phi_{\tau}\equiv-\frac{i}{\hbar}\tilde{H}\Phi_{\tau}=\frac{i\hbar}{2M}\Box\Phi_{\tau}-\frac{1}{M}(\partial^{\mu}W_{\tau}+\frac{1}{c}A_{\tau}^{\mu})\partial_{\mu}\Phi_{\tau}-\frac{1}{2M}\BoxW_{\tau}\Phi_{\tau}
\begin{array}{lll}{SU\left(2\right)^{W}:}&{\varepsilon^{mn}}&{\varepsilon^{12}=-1}\\{SU\left(3\right)^{c}:}&{\varepsilon^{\dot{m}\dot{n}\dot{r}}}&{\varepsilon^{\dot{1}\dot{2}\dot{3}}=1}\\{SU\left(2\right)^{c}:}&{\varepsilon^{\dot{m}\dot{n}}}&{\varepsilon^{\dot{1}\dot{2}}=1}\\{SO\left(7\right)^{c}:}&{\varepsilon^{\alpha\beta\gamma\delta\mu\nu\rho}}&{\varepsilon^{1234567}=-1}\\\end{array}
R_{\mu\nu}=R_{~\mu\lambda\nu}^{\lambda}=\partial_{\nu}\Gamma_{~\mu\lambda}^{\lambda}-\partial_{\lambda}\Gamma_{~\mu\nu}^{\lambda}+\Gamma_{~\tau\nu}^{\lambda}\Gamma_{~\mu\lambda}^{\tau}-\Gamma_{~\tau\lambda}^{\lambda}\Gamma_{~\mu\nu}^{\tau}.
T_{\mu\nu}^{eff}=\frac{\zeta+b_{k}}{\zeta+b_{g}}\varphi_{,\mu}\varphi_{,\nu}-K\tilde{g}_{\mu\nu}+\frac{b_{g}M^{4}}{(\zeta+b_{g})^{2}}e^{-2\alpha\phi/M_{p}}\tilde{g}_{\mu\nu}+\frac{\zeta}{\zeta+b_{g}}T_{\mu\nu}^{(f,canonical)}-\frac{mF(\zeta)}{\sqrt{\zeta+b_{g}}}\overline{{\Psi}}^{\prime}\Psi^{\prime}\tilde{g}_{\mu\nu},
\left(\begin{matrix}{-\frac{d^{2}}{dx^{2}}+\tilde{V}_{\ell}(x)}&{0}\\{0}&{-\frac{d^{2}}{dx^{2}}+V_{\ell}(x)}\\\end{matrix}\right)\psi_{k}(x)=k^{2}\psi_{k}(x)
\Omega=i\frac{4\piT}{e^{2}LN^{2}}\left(\begin{matrix}{2}&{1}&{\ldots}&{1}\\{1}&{2}&{\ldots}&{1}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{1}&{1}&{\ldots}&{2}\\\end{matrix}\right)\equivig_{ij}\\.
y^{2}=x^{3}+\sum_{k=-4}^{4}\sum_{l=0}^{8}f_{kl}z^{\primel}z^{4+k}p^{8-l-k}x+\sum_{k=-6}^{6}\sum_{l=0}^{12}g_{kl}z^{\primel}z^{6+k}p^{12-l-k}
\vec{\sigma}\cdot\vec{p}\,\,\tilde{\xi}_{\lambda}(\vec{p})=2p\lambda\,\tilde{\xi}_{\lambda}(\vec{p})\,,\quad\vec{\sigma}\cdot\vec{p}\,\,\tilde{\eta}_{\lambda}(\vec{p})=-2p\lambda\,\tilde{\eta}_{\lambda}(\vec{p})\,.
\bar{F}_{6}=\frac{l\sqrt{Q_{4}Q_{4}^{\prime}}}{(Q_{4}Q_{4}^{\prime}-l^{2})^{3/4}(x^{-})^{3/2}}dx^{-}\wedgedx^{1}\wedgedx^{2}\wedge(dx^{3}\wedgedx^{4}+dx^{5}\wedgedx^{6})\wedgedz.
\left(\begin{matrix}{\phi_{R}}\\{\xi_{R}}\\{\phi_{L}}\\{\xi_{L}}\\\end{matrix}\right)=\left(\begin{matrix}{r}&{t}&{-s}&{-t}\\{t}&{-r}&{t}&{-s}\\{-s}&{-t}&{r}&{t}\\{t}&{-s}&{t}&{-r}\\\end{matrix}\right)\left(\begin{matrix}{u_{R}}\\{v_{R}}\\{u_{L}}\\{v_{L}}\\\end{matrix}\right)
{\calT}=\frac{1}{4}\left\{|Q_{o}+Q_{v}|^{2}\Lambda^{4}+|Q_{o}-Q_{v}|^{2}\left|\frac{2\eta}{\vartheta_{2}}\right|^{4}+16|Q_{s}+Q_{c}|^{2}\left|\frac{\eta}{\vartheta_{4}}\right|^{4}+16|Q_{s}-Q_{c}|^{2}\left|\frac{\eta}{\vartheta_{3}}\right|^{4}\right\}\,
{\calL}_{\mathrm{eff}}=-\frac{N}{\pi}\epsilon^{\mu\nu\lambda}\left(B_{\mu}^{a}\partial_{\nu}B_{\lambda}^{a}+\frac{8e\sqrt{N/\kappa_{r}}}{3}\epsilon^{abc}B_{\mu}^{a}B_{\nu}^{b}B_{\lambda}^{c}\right)+\frac{N}{\pi}M^{ab}B_{\mu}^{a}B^{b\mu}+O(\partial^{2},v^{4})
_{+}^{-}{\calD}_{nn^{\prime}}=\int{}\{}_{+}^{-}\phi_{p_{-},n,+1,r}^{\dagger}(x)\{}_{+}^{-}\phi_{p_{-},n^{\prime},+1,r}(x){}dx^{2}dx^{4}...dx^{2\left[(d-1)/2\right]},\;\mp\pi_{-}<0.
\hat{H}=\frac{\hbar}{2}\Omega_{G}\Bigl(\hat{a}^{\dagger}\hat{a}+\hat{a}\hat{a}^{\dagger}\Bigr)-\frac{3\lambda{\hbar}^{2}}{16\Omega_{G}^{2}}+\frac{\lambda{\hbar}^{2}}{16\Omega_{G}^{2}}\sum_{k=0}^{4}{\binom{4}{k}}\hat{a}^{\daggerk}\hat{a}^{4-k}.
f_{s}(t)=-\int_{C(z)}f_{s}(t_{1})dt_{1}K_{s}^{(1)}(t_{1},t)\quad\mathrm{and}\quadK_{s}^{(1)}(t,t^{\prime})=\int_{C(z)}K_{s}^{(1)}(t,t_{1})dt_{1}K_{s}^{(1)}(t_{1},t^{\prime})
{\calS}[\psi]=\int\,d^{5}x\sqrt{g}\left\{\frac{i}{2}\,\left[\overline{{\psi}}\tilde{\gamma}^{\hat{\mu}}\tilde{\nabla}_{\hat{\mu}}\psi-(\overline{{\tilde{\nabla}_{\hat{\mu}}\psi}})\tilde{\gamma}^{\hat{\mu}}\psi\right]-M\overline{{\psi}}\psi\right\}
S[x^{\mu},p_{\mu};\lambda]=\int_{\tau_{i}}^{\tau_{f}}d\tau\left[\dot{x}^{\mu}P_{\mu}-\lambda\phi\right]=\int_{\tau_{i}}^{\tau_{f}}d\tau\left[\dot{x}^{\mu}P_{\mu}-\frac{1}{2}\lambda\left[P^{2}+m^{2}\right]\right]\.
\begin{array}{l}{i\partialH_{1}^{\prime}=\Psi^{+}\Psi^{-}~~~(\mathrm{as~defined~before})}\\{i\partialH_{2}=\Psi^{0}\Psi^{2}}\\{-i\sqrt{3}\partialH_{3}=J_{{\calN}/U(1)}-Qi\partialX^{2}~~.}\\\end{array}
\hat{A}={\frac{p-3m^{2}M_{P}^{2}}{\rho+3m^{2}M_{P}^{2}}}\,+\,\gamma_{0}{\frac{2m^{\prime}a^{-1}M_{P}^{2}}{\rho+3m^{2}M_{P}^{2}}}=A_{1}+\gamma_{0}A_{2}\,.
\Gamma_{S}^{Q^{\prime}Q^{\prime\prime}}(-i\Omega_{m},\stackrel{\rightharpoonup}{p})[\delta_{Q^{\prime\prime}Q}-N_{Q^{\prime\prime}}(-i\Omega_{m},\stackrel{\rightharpoonup}{p})g_{Q^{\prime\prime}Q}]=g_{Q^{\prime}Q}/2,
V=\frac{1}{2}\omega^{2}\phi^{2},\;\;\;\;\;\;\phi=\frac{1}{\sqrt{2\omega}}(e^{i\omegat}a^{\dagger}+e^{-i\omegat}a),\;\;\;\;\;\;\pi=i\sqrt\frac{\omega}{2}(e^{i\omegat}a^{\dagger}-e^{-i\omegat}a)
\int\,d\mu_{B}=\int\,\frac{d^{3}X}{(2\pi)^{\frac{3}{2}}}({\calJ}_{X}^{})^{\frac{3}{2}}\int_{0}^{2\pi}\,\frac{d\theta}{(2\pi)^{\frac{1}{2}}}({\calJ}_{\theta})^{\frac{1}{2}}
F_{n}\left({\vec{z}}\right)=\sum_{{\vec{c}}}D_{{\vec{c}},n}\theta\left[\begin{array}{c}{{\vec{0}}}\\{\frac{{\vec{c}}}{\ell}}\\\end{array}\right]\left({\vec{z}}-\frac{\tau}{\omega\kappa}\left(n+\frac{\beta}{2\pi}\right){\hat{e}}+\frac{\alpha}{2\pi\omega\kappa}{\hat{e}},\Omega\right)
{\widetildeT}_{\mu\nu}=T_{\mu\nu}-{\widehatM}_{P}^{D-d-2}\biggl[-\partial^{\lambda}\partial_{\lambda}H_{\mu\nu}+\partial_{\mu}\partial_{\nu}\chi-{\frac{d-1}{d}}\eta_{\mu\nu}\partial^{\lambda}\partial_{\lambda}\chi\biggr]~.
\frac{F}{V}=-\sum_{\tau\in{\calF}}\frac{\nu^{-2}e^{-\frac{\beta^{2}|n\tau-m|^{2}}{4\pi\alpha^{\prime}\tau_{2}}}}{n^{2}+\nu^{-2}r^{2}}\left(\frac{1}{4\pi\alpha^{\prime}\tau_{2}}\right)^{13}\vert\eta(\tau)\vert^{-48}
\nonumber\begin{array}{ll}{SU(n)^{3}:}&{n-4+k=0}\\{SU(2)^{2}U(1)_{R}:}&{(n+2k-4)(n-2)+(k-2)k=0}\\{{U(1)_{R}}^{3}:}&{n(n-1)(n+2k-4)^{3}+2kn(k-2)^{3}-k(k-1)n^{3}=0}\\\end{array}
{\frac{1}{2}}\intd^{3}\!x({\bfD}\varphi)^{2}=-{\frac{1}{2}}\intd^{3}\!x\,\varphi\,{\bfD}\cdot{\bfD}\,\varphi=-{\frac{1}{2}}\intd^{3}\!x\,\varphiD^{2}\varphi
\tilde{C}\left(l_{1}^{\mu}\left[\sigma\right],r_{2}^{\mu}\left(\sigma\right)\right)=\int\left[dz\right]\,\tilde{A}\left(l_{1}^{\mu}\left[\sigma\right],z^{\mu}\left[\sigma\right]\right)\,\tilde{B}\left(z^{\mu}\left[\sigma\right],r_{2}^{\mu}\left[\sigma\right]\right),
F_{\mu\nu\rho\lambda}=\partial_{\mu}A_{\nu\rho\lambda}-\partial_{\nu}A_{\mu\rho\lambda}+\partial_{\rho}A_{\lambda\mu\nu}-\partial_{\lambda}A_{\rho\mu\nu}\equiv\partial_{\left[\mu\right.}A_{\left.\nu\rho\lambda\right]},
\Omega=-H\left(t,q,p=\psi(t,q)\right){\mathrm{d}}t+\psi(t,q)\,\mathrm{d}q\stackrel{\textstyle!}{=}\mathrm{d}S(t,q)=\partial_{t}S(t,q)\,\mathrm{d}t+\partial_{q}S(t,q)\,\mathrm{d}q\,.
V(\phi,\chi)=\frac{1}{2}\mu^{2}r^{2}(\phi^{2}-a^{2})^{2}+\mu^{2}\phi^{2}\chi^{2}-\mu^{2}a^{2}s^{2}\chi^{2}+\frac{1}{2}\mu^{2}\frac{s^{4}}{r^{2}}\chi^{4}.
\left[\pi_{l}^{\left(N\right)}\left(\bar{x}\right),\,\pi_{k}^{\left(N\right)}(\bar{y})\right]_{x^{0}=y^{0}}\,=\,-i\,\theta\,\left(-1\right)^{N-1}\left(b+a\nabla^{2}\right)\epsilon_{ik}\partial_{l}\partial^{i}\delta^{2}\left(\bar{x}-\bar{y}\right),
b_{\scriptscriptstyle{L,L}}/b_{\scriptscriptstyle{L-1,L}}=\frac{ik\psi_{\scriptscriptstyle{L+1,L}}^{>}(k,a)}{(\gamma_{\scriptscriptstyle{L,L}}-\frac{1}{2})\,\psi_{\scriptscriptstyle{L,L}}^{>}(k,a)}\,(1+\mathrm{O}(ka)^{2})\,.
{\calJ}_{i}^{0}\equiv-\frac\partial{\partialx}\psi^{i}-2\varepsilon^{ijk}\psi^{j}\pi_{k}=-\sqrt{1-{\bf\varphi}^{2}}\;\varpi_{i}-\varepsilon^{ijk}\varphi^{j}\varpi_{k}\equivJ_{i}^{0}.
(\frac{1}{gf^{2}}-\frac{1}{g^{2}f})\tilde{\gamma}^{k0i}\partial_{0}(\psi_{i}^{-})_{R}\phi^{-}+\frac{3}{2}(\frac{g^{\prime}}{f}-\frac{f^{\prime}}{g})\phi^{+}\gamma^{ki}(\psi_{i}^{+})_{L}=0.
\begin{array}{rclcl}{\dot{\varphi}^{0}}&{=}&{[\varphi^{0},H]}&{=}&{\varphi_{1}^{0},}\\{\dot{\varphi}^{m}}&{=}&{[\varphi^{m},H]}&{=}&{2\varphi_{1}^{m},}\\{\dot{\varphi}_{1}^{0}}&{=}&{[\varphi_{1}^{0},H]}&{=}&{-\partial_{m}\varphi_{1}^{m},}\\{\dot{\varphi}_{1}^{m}}&{=}&{[\varphi_{1}^{m},H]}&{=}&{0.}\\\end{array}
\begin{array}{cc}{\{\tilde{Q}^{\tilde{\imath}},(\tilde{Q}^{\tilde{\jmath}})^{t}\}=2\tilde{{\calE}}{}^{\tilde{\imath}\tilde{\jmath}}\tilde{\gamma}^{A}P_{A}~~~}&{~~~\tilde{\imath},\tilde{\jmath}=1,2,\cdots,2\tilde{N}}\\\end{array}
|W\rangle=(:e^{\frac{i}{\sqrt{2}}X^{25}(1)}:\otimes\chi_{-}-:e^{-\frac{i}{\sqrt{2}}X^{25}(1)}:\otimes\chi_{+})(:e^{\frac{i}{\sqrt{2}}X^{24}(1)}:\otimes\chi_{-}^{\prime}-:e^{-\frac{i}{\sqrt{2}}X^{24}(1)}:\otimes\chi_{+}^{\prime})|w\rangle
S_{thermo}=2\pi\left[{\frac{4}{(D-3)^{2}}}Q_{1}^{(1)}Q_{1}^{(2)}(2m)^{\frac{2}{D-3}}-{\frac{2}{(D-3)}}\sum_{i=1}^{[{\frac{D-1}{2}}]}J_{i}^{2}\right]^{\frac{1}{2}}.
S=S_{cm}-{\frac{1}{2}}\,\int_{\Sigma_{p+1}}d^{p+1}\sigma\,\sqrt{\,\overline{g}\,}\,\left[\,\overline{{g}}^{\,mn}\,\partial_{m}\,Y^{\mu}\,\partial_{n}\,Y_{\mu}-m_{p+1}\,p\,\right]\.
C^{A}\equiv\left\{B^{\alpha};\Pi_{\ast}^{\alpha}(-1)^{\varepsilon_{\alpha}+1}\right\},\quad\bar{{\calP}}_{A}\equiv\left\{\Pi_{\alpha};B_{\alpha}^{\ast}\right\},\quad[C^{A},\bar{{\calP}}_{B}]=i\hbar\delta_{B}^{A}.
[K,\tilde{a}_{n}^{in}]=\sum_{m}c[(\tilde{b}_{m}^{in})^{2},\tilde{a}_{n}^{in}]=-2c\tilde{b}_{m}^{in}\,[K,[K,\tilde{a}^{in}]]=-4c\overline{{c}}\tilde{a}_{n}^{in}\
R(u)=\left(\begin{array}{ccc|ccc|ccc}{c}&{}&{}&{}&{}&{}&{}&{}&{}\\{}&{b}&{}&{e}&{}&{}&{}&{}&{}\\{}&{}&{d}&{}&{g}&{}&{f}&{}&{}\\\hline{}&{\bar{e}}&{}&{b}&{}&{}&{}&{}&{}\\{}&{}&{\bar{g}}&{}&{a}&{}&{g}&{}&{}\\{}&{}&{}&{}&{}&{b}&{}&{e}&{}\\\hline{}&{}&{}&{}&{}&{}&{}&{}&{}\\{}&{}&{\bar{f}}&{}&{\bar{g}}&{}&{d}&{}&{}\\{}&{}&{}&{}&{}&{\bar{e}}&{}&{b}&{}\\{}&{}&{}&{}&{}&{}&{}&{}&{c}\\\end{array}\right),
\epsilon_{L}(\omega,{\bfk})=1+{\frac{m^{2}}{\bfk}}[1+{\frac{\omega}{2|{\bfk}|}}\mathrm{ln}|{\frac{\omega-|{\bfk}|}{\omega+|{\bfk}|}}|+{\frac{i\pi\omega}{2|{\bfk}|}}\Theta(|{\bfk}|-\omega)],
\frac{1}{\rho}k(\rho\mu)^{2}=\left.\frac{1}{\rho}\frac{a(\rho)c_{1}(\rho\mu)^{2}+b(\rho)c_{2}(\rho\mu)^{-2\gamma}}{a(\rho)c_{3}+c_{4}b(\rho)(\rho\mu)^{-2\gamma}}\right.
J^{i}=\partial\theta_{L}^{i}+\Sigma_{ji}\theta_{R}^{j}+{\frac{1}{2}}\Gamma_{ai}^{L}\partialX^{a},\\\bar{J}^{i}=\bar{\partial}\theta_{R}^{i}+\Sigma_{ij}\bar{\theta}_{L}^{j}+{\frac{1}{2}}\Gamma_{ia}^{R}\bar{\partial}X^{a}.
\betaV\cdot{\calN}_{\overline{{\alphaV}}}\equiv\beta{\vec{U}}\cdot{\vec{P}}_{\overline{{\alphaV}}}+\sum_{r}({\betaV})^{r}N_{\overline{{\alphaV}}}^{r}\epsilon^{r}+\sum_{\ell}({\betaW})^{\ell}J_{\overline{{\alphaV}}}^{\ell}\epsilon^{\ell}~.
\tilde{E}_{n\delta}=\sum_{p_{1}+2p_{2}+\cdots+np_{n}=n}\frac{\left(q^{(\alpha,\alpha)}-q^{-(\alpha,\alpha)}\right)^{\sum_{i}p_{i}-1}}{p_{1}!\;...\;p_{n}!}(E_{\delta})^{p_{1}}(E_{2\delta})^{p_{2}}...(E_{n\delta})^{p_{n}}
\sqrt{-G}\partial_{y}\Phi\Bigr|_{y=0^{+}}=-\sqrt{-g^{0}}\frac{V_{0}}{2}\,\qquad\sqrt{-G}\partial_{y}\Phi\Bigr|_{y=\pir_{c}^{-}}=\sqrt{-g^{\pir_{c}}}\frac{V_{\pi}}{2}\.
\sum_{l=0}^{\infty}(2l+1)[s_{l}^{\prime}(\lambdar)e_{l}(\lambda\rho)]^{2}=\frac{1}{2r\rho}\int_{r-\rho}^{r+\rho}\left(\frac{1}{\lambda}\,\frac{\partial{\calD}}{\partialr}\right)^{2}R\,dR\,.
{\calL}_{\mathrm{F}}=\frac{1}{2}i\,\bar{\chi}^{aj}\gamma^{\mu}\nabla\!_{\mu}\chi_{aj}-\bar{\chi}^{aj}(\alpha_{\,jk}^{I}A^{I}+i\gamma_{5}\beta_{\,jk}^{I}B^{I})_{ab}\chi^{bk},
{\calL}=-{\frac{1}{4}}\mathrm{Re}\;S\sum_{a}k_{a}F^{a\mu\nu}F_{\mu\nu}^{a}+{\frac{1}{4}}\mathrm{Im}\;S\sum_{a}k_{a}F^{a\mu\nu}\tilde{F}_{\mu\nu}^{a},
L_{0}^{\parallel}=-\sum_{n=0}^{\infty}(n+i\epsilon)(a_{n}^{+})^{*}a_{n}^{-}-\sum_{n=1}^{\infty}(n-i\epsilon)(a_{n}^{-})^{*}a_{n}^{+}+{\frac{1}{2}}i\epsilon(1-i\epsilon).
ds^{2}={\frac{u^{2}}{\tilde{L}^{2}}}[-dt^{2}+dx_{1}^{2}+\cdots+dx_{m-2}^{2}]+\tilde{L}^{2}{\frac{du^{2}}{u^{2}}}+L^{2}d\Omega_{p+2}^{2},
\dot{u}_{0}=-\frac{16}{3}\left[v_{0}^{\prime\prime}+2u_{0}^{\prime}v_{0}^{\prime}\right]\quad,\quad\dot{v}_{0}=u_{0}^{\prime\prime}-\left(u_{0}^{\prime}\right)^{2}+\frac{16}{3}\left(v_{0}^{\prime}\right)^{2}\;,
\varepsilon_{p}^{D}(L)=\frac{L^{p-D}}{2^{D+1}}\pi^{-D/2}\int_{0}^{\infty}d\tau\left(\sqrt{\tau}\right)^{D-2}\left\{\bigg[1-\left[\frac{\pi}{\tau}\right]^{\frac{1}{2}}\bigg]^{p}-\bigg[\theta_{3}\left(0,e^{-\tau}\right)-\left[\frac{\pi}{\tau}\right]^{\frac{1}{2}}\bigg]^{p}\right\}
{\calF}_{\mu\nu}^{ab}=F_{\mu\nu}^{ab}+\Sigma_{\mu\nu}^{ab}+\Theta_{\mu\nu}^{ab},\\\{\calF}_{\mu\nu}^{i}=F_{\mu\nu}^{i}+\Sigma_{\mu\nu}^{i},\\\{\calF}_{\mu\nu}^{a}=F_{\mu\nu}^{a}+\Sigma_{\mu\nu}^{a},
ds_{t\to0}^{2}\sim\mathrm{e}^{-\frac{2pf_{0}}{q-1}}(-du^{2}+u^{2}d\Sigma_{k,-1}^{2}+d\vec{y}_{q-k}^{2})+\mathrm{e}^{2f_{0}}d\vec{x}_{p}^{2},
2iA_{i}=\epsilon_{iak}\tau_{a}\hat{x}_{k}(\frac{\phi_{1}(r)-1}{r})-(\tau_{i}-\hat{x}_{i}\hat{x}\cdot\vec{\tau})\frac{\phi_{2}(r)}{r}+\hat{x}_{i}\hat{x}\cdot\vec{\tau}H_{1}(r),
\frac{1}{2}\intdt\,d^{3}xh_{\mu}^{\nu}T_{\nu}^{\mu}=\frac{1}{2}\intdt\,d^{3}x\left(H_{tt}^{0}T^{tt}+\frac{1}{2}H_{aa}^{0}T^{aa}+H_{zz}^{0}T^{zz}+2H_{tz}^{0}T^{tz}\right)\,.
L_{GB}={\frac{4}{R^{2}}}\,\left([(e^{2U})^{\prime}\,R^{\prime}]^{2}-(e^{2U})^{\prime\prime}+(e^{2U})\,(R^{\prime})^{2}\,(e^{2U})^{\prime\prime}+2\,(e^{2U})\,(e^{2U})^{\prime}\,R^{\prime}(r)\,R^{\prime\prime}\right)\.
H=\intd^{2}x[{\frac{1}{2}}(P_{1}^{2}+\partial_{i}Q_{1}\partial_{i}Q_{1}+M_{1}^{2}Q_{1}^{2})+{\frac{1}{2}}(P_{2}^{2}+\partial_{i}Q_{2}\partial_{i}Q_{2}+M_{2}^{2}Q_{2}^{2})],
u=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}}\,\,\,\,\,\,\mathrm{and}\,\,\,\,\,v=\frac{x_{12}^{2}x_{34}^{2}}{x_{14}^{2}x_{23}^{2}}.
\delta{\calE}=D(p,q,r){\calE}D^{T}(p,q,r)-{\calE}=\left(\begin{array}{cccc}{0}&{(rb-qc)}&{(pc-ra)}&{(qa-pb)}\\{-(rb-qc)}&{0}&{0}&{0}\\{-(pc-ra)}&{0}&{0}&{0}\\{-(qa-pb)}&{0}&{0}&{0}\\\end{array}\right)
\pi(d\alpha)=\left(\begin{matrix}{-g^{\mu\nu}\partial_{\mu}a_{i1}\partial_{\nu}b_{i1}}&{-2\gamma^{\mu\nu}\gamma^{5}V_{\nu}a_{i1}\partial_{\mu}b_{i2}}\\{2\gamma^{\mu\nu}\gamma^{5}V_{\nu}a_{i2}\partial_{\mu}b_{i1}}&{-g^{\mu\nu}\partial_{\mu}a_{i2}\partial_{\nu}b_{i2}}\\\end{matrix}\right)
\delta\Phi=\frac{1}{48}{\bar{D}}^{4}(L_{{\mathbfi}{\mathbfj}}F^{{\mathbfi}{\mathbfj}})-\frac{i}{48}{\bar{D}}^{4}D^{{\mathbfi}{\mathbfj}}(L_{{\mathbfi}{\mathbfj}}\Phi-{\bar{L}}_{{\mathbfi}{\mathbfj}}{\bar{\Phi}})\,
{\calL}=\sum_{j=1}^{N}\left[i\bar{\Psi}_{j}\gamma^{\mu}D_{\mu}\Psi_{j}-\bar{\Psi}_{j}(\sigma_{j}+i\gamma_{5}\pi_{j})\Psi_{j}-\frac{1}{2G}\left(\sigma_{j}^{2}+\pi_{j}^{2}\right)\right]
A_{0}^{a}(x^{0},{\bfx})=\Lambda^{a}-\frac{1}{4\pi}\int_{{\bfR}^{3}}d^{3}{\bfy}\frac{1}{\mid{\bfx}-{\bfy}\mid}\left\{U_{~~~b}^{ka}(x^{0},{\bfy})\frac{\partialA_{0}^{b}(x^{0},{\bfy})}{\partialy^{k}}+V_{~~b}^{a}(x^{0},{\bfy})A_{0}^{b}(x^{0},{\bfy})\right\}.
\Phi_{1}=p_{0}+qA_{0}+\zeta\tilde{\omega}\,,\;\Phi_{2}=\phi_{2}^{G}\,,\;\Phi_{3}=\phi_{3,1}^{(1)}\,,\;\Phi_{4}=\phi_{3,2}^{(1)}\,,\;\;\,\Phi_{5}=\phi_{3,3\;}^{(1)}\;.
V(k,k\,^{\prime};\tau)=\frac{m}{\alpha}\,\left(e^{-\alpha(k+k\,^{\prime})\tau}\,\,\delta(k-k\,^{\prime}-i)+\frac{e^{\alpha(k+k\,^{\prime})\tau}}{4\,k\,k\,^{\prime}}\,\,\delta(k-k\,^{\prime}+i),\right).
{\calL}_{\mathrm{int}}=\frac{g^{2}}{2}\left((\bar{\psi}\psi)^{2}+(\mathrm{i}\bar{\psi}\gamma_{5}\psi)^{2}\right)=\frac{g^{2}}{4}\left(\left(\varphi^{\dagger}\sigma_{1}\chi+\chi^{\dagger}\sigma_{1}\varphi\right)^{2}+\left(\varphi^{\dagger}\sigma_{2}\chi+\chi^{\dagger}\sigma_{2}\varphi\right)^{2}\right)\,
\frac{1}{(2^{14}\cdot3^{4})^{2}}~{\calI}_{\mu\nu\rho\sigma,\zeta\varsigma\xi\kappa}^{(4)}(x)~{\calI}_{\zeta\varsigma\xi\kappa,\alpha\beta\gamma\delta}^{(4)}(x)=\Im_{\mu\nu\rho\sigma,\alpha\beta\gamma\delta}^{(4)}
\left[D_{i,\pm^{\prime}}\Phi\left(X^{0},X^{+^{\prime}},X^{-^{\prime}},X^{i}\right)\right]_{X^{0}=\pm\sqrt{X^{i}X_{i}-2X^{+^{\prime}}X^{-^{\prime}}}}=\partial_{i,\pm^{\prime}}\Phi|_{X^{0}}
R_{ij,kl}=\frac{1}{4}{(\frac{\partial^{2}A}{\partial\phi^{p}\partial\phi^{q}})}^{-1}(\frac{\partial^{3}A}{\partial\phi^{i}\partial\phi^{l}\partial\phi^{p}}\frac{\partial^{3}A}{\partial\phi^{q}\partial\phi^{j}\partial\phi^{k}}-\frac{\partial^{3}A}{\partial\phi^{i}\partial\phi^{k}\partial\phi^{p}}\frac{\partial^{3}A}{\partial\phi^{q}\partial\phi^{j}\partial\phi^{l}}).
\begin{array}{c}{u=u_{0}+\frac{p^{+}}2\left[(\sigma^{+}+f(\sigma^{+}))+(\sigma^{-}-g(\sigma^{-}))\right]}\\{v=v_{0}+\frac{p^{-}}2\left[(\sigma^{+}-f(\sigma^{+}))+(\sigma^{-}+g(\sigma^{-}))\right]}\\\end{array}
|i_{1}i_{2}...i_{n}\rangle_{PW}=|i_{1}^{\prime}i_{2}^{\prime}...i_{n}^{\prime}\rangle\langlei_{1}^{\prime}i_{2}^{\prime}...i_{n}^{\prime}|i_{1}i_{2}...i_{n}\rangleC_{i_{1}i_{2}...i_{n}}
\displaystyle{\tilde{\sigma}_{2}^{1}=\,\{-{\frac{1}{3}}\,{\itc^{zz}}\,\left(\,\partial^{5}\mu_{\bar{z}}^{zz}-4\,(\,\rho_{zz}^{2}\,\partial\mu_{\bar{z}}^{zz}+\rho_{zz}\,\partial\rho_{zz}\,\mu_{\bar{z}}^{zz}\,)\,\right)+4\,{\itc^{z}}\,\partial^{3}\mu_{\bar{z}}^{z}\}dzd\bar{z}}
G_{E}^{\iji^{\prime}j^{\prime}}(\mu,X,X^{\prime})=4\kappa\sum_{p=3i}^{i\infty}\frac{i}{2p}\frac{\Psi_{p}^{r}(X)\Psi_{p}^{l}(X^{\prime})}{(1+p^{2})}W_{(p)}^{\iji^{\prime}j^{\prime}}(\mu).
\begin{array}{c}{\left\{L_{1}^{n},L_{2}^{n}\right\}=-AL_{1}^{n}L_{2}^{n}+L_{1}^{n}L_{2}^{n}D,}\\{\left\{L_{1}^{n},L_{2}^{n+1}\right\}=L_{1}^{n}B^{*}L_{2}^{n+1},}\\{\left\{L_{1}^{n},L_{2}^{m}\right\}=0,\left|n-m\right|\geq2.}\\\end{array}
\begin{array}{rcl}{Q}&{=}&{\int_{0}^{L}dx\,\psi^{\dagger}\psi=-\lambda(p_{+}-p_{-})\,}\\{Q_{5}}&{=}&{\int_{0}^{L}dx\psi^{\dagger}\gamma_{5}\psi=-\frac{i\lambda\sqrt{2\mu}}{2\sqrt{\pi}\sqrt{L}}\,\left[\varphi_{0}-\varphi_{0}^{\dagger}\right]\.}\\\end{array}
\mathrm{e}^{\phi^{i}(x)}=\xi_{1}^{(i)}(x)\bar{\xi}_{1}^{(i)}(x),{}~~~~\psi_{+}^{i}(x)=\frac{\xi_{1}^{(i)}(x)\bar{\xi}_{2}^{(i)}(x)}{\xi_{1}^{(i)}(x)\bar{\xi}_{1}^{(i)}(x)},{}~~~~\psi_{-}^{i}(x)=\frac{\xi_{2}^{(i)}(x)\bar{\xi}_{1}^{(i)}(x)}{\xi_{1}^{(i)}(x)\bar{\xi}_{1}^{(i)}(x)}.
\hat{\hat{G}}\;=\;d\hat{\hat{C}}\;-\;{\textstyle\frac{1}{2}}\left(i_{\hat{\hat{k}}_{\scriptscriptstyle(n)}}\hat{\hat{C}}\right)\,Q^{nm}\,\left(i_{\hat{\hat{k}}_{\scriptscriptstyle(n)}}\hat{\hat{C}}\right)\;.
\Phi_{\alpha\beta\sigma}=\frac{e}{3\sqrt{2}}\left(\partial_{\alpha}h_{\beta\sigma}+\partial_{\beta}h_{\sigma\alpha}+\partial_{\sigma}h_{\alpha\beta}\right)-\frac{q}{12}\left(g_{\sigma\beta}K_{\alpha}+g_{\beta\alpha}K_{\sigma}+g_{\alpha\sigma}K_{\beta}\right)\,.
Tr\,e^{-isH}=2\frac{eB{\ell}^{2}}{2\pi}\sum_{n=0}^{\infty}e^{-iseB(2n+1)}\sum_{n_{1}=-\infty}^{\infty}e^{-is(({2n_{1}\pi-\theta})/{a})^{2}}\sum_{n_{2}=-\infty}^{\infty}e^{is(2n_{2}{\pi}i/\beta)^{2}}e^{-ism^{2}}
\int\mathrm{d}\mu_{f}=\Big(\frac{e^{2}}{2\pi}\Big)^{kN_{f}}\int\,\Pi_{i=1}^{N_{f}}\Pi_{A=1}^{k}\,\mathrm{d}\rho_{i}^{A}\mathrm{d}{\tilde{\rho}}_{i}^{A}\.
\Gamma\lbrackA\rbrack=\int_{0}^{\infty}{\frac{dT}{T}}\,\mathrm{e}^{-m^{2}T}\int{\calD}x\,\mathrm{exp}\left[-\int_{0}^{T}\!\!\!d\tau\left({\frac{1}{4}}{\dot{x}}^{2}+ieA_{\mu}\dot{x}^{\mu}\right)\right]
\intd^{d}x\,\sqrt{g}e^{-2\Phi}\Bigl(R_{g}+|\nabla\Phi|^{2}+|dB|^{2}\Bigr)+\intd^{d}x\,\sqrt{g}g^{\mu\nu}G_{ij}\partial_{\mu}\phi^{i}\partial_{\nu}\phi^{j}+\ldots,
\vert\beta_{\omega^{\prime}\omega}^{B}\vert^{2}=\frac1{e^{2}}\vertj_{\alpha}(k_{+},k_{-})\vert^{2},\qquad\vert\beta_{\omega^{\prime}\omega}^{F}\vert^{2}=\frac1{e^{2}}\vert\rho(k_{+},k_{-})\vert^{2}.
\{\omega,\varphi(x)\}_{D}=-\left(\frac{3\lambda}{4}\right)lim_{L\rightarrow\infty}\frac{\int_{-\infty}^{\infty}dy\in(x-y)[2\omega\varphi(y)+\varphi(y)^{2}]}{(3\lambda\omega^{2}-m^{2})L+3\lambda\int_{-\infty}^{\infty}dx\varphi(x)^{2}}
\gamma_{\mu}=\left(\begin{array}{cc}{\sigma_{\mu}}&{0}\\{0}&{-\sigma_{\mu}}\\\end{array}\right)\,;\mu=1\sim3\,,\quad\gamma_{4}=\left(\begin{array}{cc}{0}&{{\bf1}}\\{{\bf1}}&{0}\\\end{array}\right)\,,\quad\gamma_{5}=\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}=\left(\begin{array}{cc}{0}&{i{\bf1}}\\{-i{\bf1}}&{0}\\\end{array}\right)\.