harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Q=\frac{m^{2}}{\sqrt{2}\,\lambda^{2}}\int\!\mathrm{d}z\left[\partial_{t}u\;\psi+\partial_{z}u\;\gamma^{0}\gamma^{1}\psi+i\,m\,\partial_{\bar{u}}\overline{{{\calW}}}\;\gamma^{0}\psi^{*}+(u\tov\,,~\psi\to\eta)\right]\;,
\sum_{j}\left(\frac{\partialX^{\mu}}{\partialx_{i}}\frac{\partialX^{\mu}}{\partialx_{j}}+F_{ij}\right)\hat{L}_{jk}\,+\,\left(\frac{\partialX^{\mu}}{\partialx_{i}}\frac{\partialX^{\mu}}{\partialx_{j}}-F_{ij}\right)\hat{L}_{kj}\,=\,2L\delta_{ik}
ds^{2}=Q_{5}\frac{r^{2}}{\alpha^{\prime}}(-dx_{0}^{2}+dx_{1}^{2})+\frac{Q_{5}\alpha^{\prime}}{r^{2}}dr^{2}+Q_{5}\alpha^{\prime}d\Omega_{3}^{2}+dx^{A}dx^{A}~,
(2h)=-\\frac{i}{6(2\pi)^{4}}\tr\intd^{d}k\\frac{\stackrel{\displaystyle(Sk)}{\scriptscriptstyle(-p)\phantom{+}}\stackrel{\displaystyle(S(k+q))}{\scriptscriptstyle(-q)\phantom{+qqqqqqq}}\stackrel{\displaystyle(S(k-p))}{\scriptscriptstyle(p+q)\phantom{+qqqqqqq}}}{k^{2l}(k-p)^{2l}(k+q)^{2l}}
C_{a_{1}\dotsa_{p}}=(-1)^{\frac{\sum{a_{i}}}{h}}(-i)^{p}(\frac{\beta^{2}}{m^{2}h})^{\frac{p}{2}-1}\prod_{k=1}^{p}m_{a_{k}},~~~~~\mathrm{if}~~\sum_{k=1}^{p}a_{k}\equiv0~~(\mathrm{mod}~h).
\begin{array}{rcl}{\psi_{2,1}}&{\cong}&{\rho_{1,1}=L_{-1}\psi_{1,1}}\\{\phi_{2,1}}&{\cong}&{\phi_{1,1}^{\prime}=L_{-1}\phi_{1,1}}\\{\xi_{2,1}}&{\cong}&{\phi_{1,1}=\frac{1}{2}L_{1}\rho_{1,1}\,.}\\\end{array}
H_{\lambda}=\frac{i}{\lambda-{\lambda}^{-1}}\sum_{i=1}^{L}(\sigma_{i}^{X}\sigma_{i+1}^{X}+\sigma_{i}^{Y}\sigma_{i+1}^{Y}+\frac{(\lambda+\lambda^{-1})}{2}(\sigma_{i}^{Z}+\sigma_{i+1}^{Z}))
\begin{array}{l}{\nu(\nu+1)=m_{0}^{2}-\displaystyle\frac{1}{4}(1-n^{2})~,}\\{(\nu^{\prime})^{2}=\displaystyle\frac{1}{4}(n-1)^{2}-(\epsilon^{2}-m_{0}^{2})~.}\\\end{array}
{\bfH}_{\mu\,,\,\nu}^{\nu}-\frac12\,{\bfH}_{,\,\mu}=-l\Box{\bfG_{\xi}}(\Box)\left[\begin{array}{c}{\,\xi_{,\,\mu}^{+}\,}\\{\,\xi_{,\,\mu}^{-}\,}\\\end{array}\right].
Int^{Str}=\frac{1}{8}Str(F_{\mu\nu}F^{\mu\nu})^{2}=\frac{1}{2}Str(F_{x\theta}^{2}g^{xx}g^{\theta\theta}+F_{x\varphi}^{2}g^{xx}g^{\varphi\varphi}+F_{\theta\varphi}^{2}g^{\theta\theta}g^{\varphi\varphi})^{2}
S_{int}[t]=-\frac{1}{g_{o}^{2}\alpha^{\,\prime\,13}}\int\frac{d^{26}k_{1}d^{26}k_{2}d^{26}k_{3}}{(2\pi)^{26}(2\pi)^{26}}\delta(k_{1}+k_{2}+k_{3})\;\frac{1}{3}t(k_{1})t(k_{2})t(k_{3})\,\gamma^{\alpha^{\,\prime}k_{1}^{2}+\alpha^{\,\prime}k_{2}^{2}+\alpha^{\,\prime}k_{3}^{2}-3}.
V_{2}(\mathrm{x)=\frac{1}{32}\mathrm{x^{6}-\frac{\omega}{16\mathrm{g}}\mathrm{x^{4}+(\frac{\omega^{2}}{32\mathrm{g^{2}}}+\frac{3j}{4}+\frac{3}{8})\mathrm{x^{2}-\frac{\omega}{4\mathrm{g}}(5j+\frac{1}{2})}}}}
H=\int\!d^{3}x\,\left(\pi\,{\bar{\pi}}-\frac{1}{2}\,{\bar{\pi}}^{2}+\frac{1}{2}\left(\nabla\phi\right)^{2}+\nabla\phi\cdot\nabla{\bar{\phi}}+\frac{1}{2}\,m^{2}\,\phi^{2}+m^{2}\,\phi\,{\bar{\phi}}\,\right)\,,
\frac{d^{2}v}{dt^{2}}=\biggl(\frac{1}{2v}+\frac{1}{v-1}\biggr)\biggl(\frac{dv}{dt}\biggr)^{2}-\frac{1}{t}\frac{dv}{dt}-\frac{1}{2}\frac{v(v+1)}{v-1}.
S^{ii}(\theta)=\left(\begin{array}{ll}{(2)_{\theta}}&{-(1)_{\theta}}\\{-(1)_{\theta}}&{(2)_{\theta}}\\\end{array}\right)\quad\quadS^{ij}(\theta-\sigma_{ij})=\left(\begin{array}{ll}{\eta_{ij}\,(-1)_{\theta}}&{\eta_{ij}^{2}\,(-2)_{\theta}}\\{\eta_{ij}^{2}\,(-2)_{\theta}}&{\eta_{ij}\,(-1)_{\theta}}\\\end{array}\right)\;.
\left(\begin{array}{cc}{E-m}&{2i\left(\frac{\partial}{\partial_{z}}-ieA_{z}\right)}\\{-2i\left(\frac{\partial}{\partial_{\bar{z}}}-ieA_{\bar{z}}\right)}&{-E-m}\\\end{array}\right)\left(\begin{array}{c}{\chi^{1}}\\{\chi^{2}}\\\end{array}\right)=0
\Pi^{\alpha\beta}(k)\,=\,-e^{2}\int\frac{d^{3}q}{(2\pi)^{3}}\frac{\mathrm{Tr}[\gamma^{\alpha}(\not\!q+\not\!k+m)\gamma^{\beta}(\not\!q+m)]}{[(k+q)^{2}-m^{2}](q^{2}-m^{2})}+(m\rightarrow-m),
\delta^{ab}+C(q^{2})\frac{\delta^{2}\Gamma_{t}[\phi]}{\delta\phi^{a}(\mathbf{-q})\delta\phi^{b}(\mathbf{q})}=\delta^{ab}+q^{2}\left(2\delta^{ab}V^{\prime}+4\phi^{a}\phi^{b}V^{\prime\prime}+q^{2}\delta^{ab}\right)
F_{x_{\perp}\bar{x}_{\perp}}^{21}=-\frac{1}{2\pi|x_{\perp}|}e^{2ix\cdot\omega}\partial_{\bar{x}_{\perp}}(\bar{x}_{\perp}I)\sime^{2ix\cdot\omega}e^{-2|\omega||x_{\perp}|}{\frac{(\omega_{1}+i\omega_{2})\pi^{{\frac{1}{2}}}}{2^{{\frac{3}{2}}}|\omega|^{{\frac{1}{2}}}|x_{\perp}|^{{\frac{3}{2}}}}}
\frac{1}{3\pim_{P}^{2}a}\Bigl(\frac{\partialS_{L(II)}}{\partiala}\Bigr)^{2}+\frac{3\pim_{P}^{2}}{4}V_{g}-\frac{1}{3\pim_{P}^{2}a}V_{q}-\langle\hat{H}_{m}\rangle=0,
m_{\sigma}^{2}=16iN_{c}g^{2}\intd\!\!\!\,{^{-}}^{4}p\left[\frac{m_{q}^{2}}{(p^{2}-m_{q}^{2})^{2}}-\frac{1}{(p^{2}-m_{q}^{2})}\right]=\frac{N_{c}g^{2}m_{q}^{2}}{\pi^{2}}
z^{A}\longrightarrow{\frac{z^{A}+{\frac{2}{3}}(C\,a)^{AB}(d\,zz)_{B}+{\frac{1}{9}}(C\,aa)^{A}(d\,zzz)}{1+a_{B}z^{B}+{\frac{1}{3}}(C\,aa)^{B}(d\,zz)_{B}-{\frac{1}{27}}(C\,aaa)(d\,zzz)}}\,.
\begin{array}{rcl}{{\calR}^{(3)}}&{=}&{3\partiala^{(2)\prime}+{\textstyle\frac{1}{2\pi\alpha^{\prime}}}C^{(3)}-3{\textstyle\frac{m}{2}}(2\pi\alpha^{\prime})b\partialb-3{\textstyle\frac{m}{2}}(2\pi\alpha^{\prime})Bb}\\{}&{}&{}\\{}&{}&{-\left(C^{(1)}-{\textstyle\frac{m}{2}b}\right)\left({\calF}+{\calL}^{(2)}\right)\,,}\\\end{array}
d\Pi_{\mu_{1}\dots\mu_{p-1}\alpha_{1}}=a_{1}\left((C\Gamma_{\nu\mu_{1}\dots\mu_{p-1}})_{\beta\alpha_{1}}\Pi^{\nu}\Pi^{\beta}\right.+{\frac{a_{s}}{a_{0}}}\left.(C\Gamma^{\nu})_{\beta\alpha_{1}}\Pi_{\nu\mu_{1}\dots\mu_{p-1}}\Pi^{\beta})\right)
\begin{array}{lclclclclclclcl}{G_{1111}=1}&{\quad}&{G_{2112}=1}&{\quad}&{G_{3121}=1}&{\quad}&{G_{4122}=1}\\{G_{5211}=1}&{\quad}&{G_{6212}=1}&{\quad}&{G_{7221}=1}&{\quad}&{G_{8222}=1}\\\end{array}
\Pi_{\mathrm{T}}^{(1)(\beta)}(0)=\frac{C(D)e^{2}}{2^{D-2}\pi^{\frac{D-1}{2}}\Gamma(\frac{D-1}{2})}\int_{0}^{\infty}dk\,k^{D-2}\left((D-1)+\frac{k^{2}}{\omega_{k}}\frac{\partial}{\partial\omega_{k}}\right)\left(\frac{n_{F}(\omega_{k})}{\omega_{k}}\right)
d(z)\!\!=\!\!\left(\!\!q^{2j-1}x_{+}z\!-\!2x_{0}skq^{j}\phantom{\frac{q^{j}}{s}}\!\!\!\!\!\!-\!x_{-}k^{2}z^{-1}\!\right)\!\!\left(-q^{2j}y_{+}z\!+\!\frac{2y_{0}}{tk}q^{j}\!\!+qy_{-}k^{-2}z^{-1}\!\right),
Im(I)={\frac{\pi^{2}}{16a}}\int_{-\lambda}^{\lambda}{du{\frac{1}{u+1}}\left[{\frac{(A^{3}-3\delta^{2}A)}{(A^{2}+\delta^{2})^{3}}}+{\frac{1}{u+1}}{\frac{A}{(A^{2}+\delta^{2})}}\right]},
\Sigma_{\mathrm{CS}}=\intd^{3}x\,m\,\varepsilon^{\mu\nu\rho}\left(A_{\mu}^{a}\partial_{\nu}A_{\rho}^{a}+\fracg3f^{abc}A_{\mu}^{a}A_{\nu}^{b}A_{\rho}^{c}\right)\,\,,
\Xi_{[1/2]}=\left(\begin{matrix}{e^{i\phi}}&{0}\\{0}&{e^{-i\phi}}\\\end{matrix}\right),\quad\Xi_{[1]}=\left(\begin{matrix}{e^{i2\phi}}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{e^{-i2\phi}}\\\end{matrix}\right)\quad,
h_{1}^{2}=a^{2}\frac{X_{1}^{2}-X_{2}^{2}}{X_{1}^{2}-1},\h_{2}^{2}=a^{2}\frac{X_{1}^{2}-X_{2}^{2}}{1-X_{2}^{2}},\h_{3}^{2}=a^{2}(X_{1}^{2}-1)(1-X_{2}^{2}).
(\bot^{\mu}{_{\lambda}}\widetilde{\nabla}_{\nu}\widetilde{\nabla}^{\nu}-2K_{\lambda}{^{\nu\mu}}\widetilde{\nabla}_{\nu}+\bot^{\mu}{_{\nu}}\eta^{\rho\sigma}{\calR}_{\rho}{^{\nu}}_{\sigma\lambda})\xi^{\lambda}\equiv({\calP}\xi^{\lambda})^{\mu}=0,
H[\varphi,\pi]=a^{d}\sum_{\bfi}\left[\frac{1}{2}\pi_{\bfi}^{2}+\frac{J}{2a^{2}}\sum_{\mu=1}^{d}(\varphi_{{\bfi}+{\bfe}_{\mu}}-\varphi_{\bfi})^{2}-\frac{1}{2}m^{2}\varphi_{\bfi}^{2}+\frac{\lambda}{4}\varphi_{\bfi}^{4}\right].
\left[-{\frac{d^{2}}{dz^{2}}}+\left\{f^{2}\left[V(\alpha_{i};f(z))-E(\alpha_{i})\right]+{\frac{1}{2}}\left({\frac{3f^{2}}{2f^{2}}}-{\frac{f^{\prime\prime\prime}}{f^{\prime}}}\right)\right\}\right]\tilde{\psi}\left(\alpha_{i};z\right)=0.
e^{iW[\vec{H}]}=\int{\calD}\lambda{\calD}{\vec{\Phi}}e^{i\intd^{d}x(\frac{1}{2}(\partial_{\mu}{\vec{\Phi}}(x))^{2}-\frac{1}{2}\lambda(x)({\vec{\Phi}}^{2}(x)-C^{2})+\frac{1}{4U}\lambda^{2}(x)+{\vec{H}}(x)\cdot{\vec{\Phi}}(x)}
S_{II}=k^{2}~\frac{\sigma_{0}}{\pi}\int_{\tau_{0}}^{\tau_{f}}d\tau~\sum_{n\neq0}^{}~|F_{\nu}\left(\frac{2\pi}{\sigma_{0}}|n|\tau,\frac{2\pi}{\sigma_{0}}|n|\tau_{0}\right)|^{2}~|\phi_{n}(\sigma_{0})|^{2}
\hat{T}^{i}=\sum_{\tau=1}^{\nu}\sum_{p,q=1}^{\nu}{a^{\taup}}^{\dagger}T_{pq}^{i}a^{\tauq},\hat{T}_{C}^{i}=\sum_{\tau=1}^{\nu}\sum_{p=1}^{\nu}\lambda_{p}^{i}{a^{\taup}}^{\dagger}a^{\taup}.
G_{R}(x,x^{\prime})=-\int{\frac{d^{4}k}{(2\pi)^{4}}}e^{ik_{\mu}(x^{\mu}-x^{\prime}{}^{\mu})}\Biggl[{\frac{a(y)^{2}a(y^{\prime})^{2}\ell^{-1}}{{\bfk}^{2}-(\omega+i\epsilon)^{2}}}+\int_{0}^{\infty}dm\,{\frac{u_{m}(y)u_{m}(y^{\prime})}{m^{2}+{\bfk}^{2}-(\omega+i\epsilon)^{2}}}\Biggr],
\dot{H}_{\mathrm{h}}(t){+}H_{\mathrm{h}}^{2}(t)={-}\frac{4\pi}{3m_{\mathrm{P}}^{2}}\big(\rho_{\varphi0}{+}\overline{{\rho_{\chi0}}}{+}3p_{\varphi0}{+}3\overline{{p_{\chi0}}}\big)
c=\int_{0}^{1}\frac{df}{d\xi}d\xi\:,\qquad\qquad\int_{0}^{1}\frac{g_{\mu\nu}\dot{X}^{\mu}\dot{X}^{\nu}}{\dot{f}}d\xi=\int_{0}^{c}g_{\mu\nu}\frac{dX^{\mu}}{df}\frac{dX^{\nu}}{df}df\,.\nonumber
S\sim{\frac{1}{G_{p+2}}}\left(G_{p+2}M\right)^{p/(p-1)}={\frac{b^{p}}{G_{p+2}}}\left({\frac{G_{p+2}M}{b^{p-1}}}\right)^{p/(p-1)}\sim{\frac{b^{D-2}}{G_{D}}}\left({\frac{G_{D}M}{b^{D-3}}}\right)^{p/(p-1)}.
Z(\bfx\mit)=\frac{1}{\bfx\mit}+\sum_{\bfq\mit\neq0}\{\;\frac{1}{\bfx-q\mit}+\frac{1}{\bfq\mit}+\frac{1}{\bfq\mit}\bfx\mit\frac{1}{\bfq\mit}+\frac{1}{\bfq\mit}\bfx\mit\frac{1}{\bfq\mit}\bfx\mit\frac{1}{\bfq\mit}+\frac{1}{\bfq\mit}\bfx\mit\frac{1}{\bfq\mit}\bfx\mit\frac{1}{\bfq\mit}\bfx\mit\frac{1}{\bfq\mit}\;\}~~,
S[B,\tilde{B},F]=\intd^{5}x\left\{{\textstyle\frac{1}{2\cdot3!}}H^{2}-{\textstyle\frac{1}{4}}m^{2}B^{2}+{\textstyle\frac{1}{4}}\epsilon\partial\tilde{B}(F+mB)\right\}\,.
\Phi_{\mu}=(\phi,0,0,0)\\\\\mathrm{and}\\\\\Lambda_{\mu}^{\nu}=\left(\begin{array}{cccc}{0}&{-E_{x}}&{-E_{y}}&{-E_{z}}\\{E_{x}}&{0}&{0}&{0}\\{E_{y}}&{0}&{0}&{0}\\{E_{z}}&{0}&{0}&{0}\\\end{array}\right),
\begin{array}{lll}{U_{k-1}\left|\Omega(k)\right\rangle=\left|\Omega(k-1)\right\rangle}&{}&{U_{k+1}\left|\Omega(k)\right\rangle=\left|\Omega(k+1)\right\rangle}\\{}&{}&{}\\{U_{k}\left|\Omega(m)\right\rangle=0}&{}&{k\neq\{m\pm1,m\}}\\\end{array}
J_{\pm}=\int_{0}^{\verta\vert^{3/2}}\frac{4\pi\imath\vert\beta\vert^{2}\vert\lambda\vert^{2}\verta\vert^{2}(a^{*}\dot{a}-a\dot{a}^{*})}{(1+\vert\beta\vert^{2}\verta\vert^{2}+4\vert\lambda\vert^{2}\verta\vert^{2}\vertb\vert^{2})^{2}}d\vertb\vert^{2}.
\left[\frac{d}{d\eta}(1-\eta^{2})\frac{d}{d\eta}-\frac{(|m|+\delta_{1})^{2}}{2(1+\eta)}-\frac{(|m|+\delta_{2})^{2}}{2(1-\eta)}+2\alphaR\eta+\frac{ER^{2}}{2}(1-\eta^{2})\right]\psi_{2}=-\lambda(R)\psi_{2},
\frac{2|\Lambda|-4\piG_{D}(1-f_{H}^{2})^{2}}{p+d-1}r_{H}^{2}-(8\piG_{D}f_{H}^{2}-d+2)=\pm\frac{p+2}{2}~r_{H}^{\frac{2(d-2)}{p}+1}\left[\frac{4\hat{R}}{p(p+1)}\right]^{(p+1)/p},
{\calL}=\frac{1}{2}(\dot{A}_{i}^{T})^{2}-\frac{1}{4}F_{ij}^{2}(A^{T})+\frac{1}{2}j_{0}\frac{1}{\partial^{2}}j_{0}+j_{i}A_{i}^{T}+{\calL}_{M}
\Phi_{1}^{-}\Phi_{1}^{+}=\Phi_{2}^{-}\Phi_{2}^{+}=\frac{\alpha}{\alpha^{2}-\beta^{2}},~~\Phi_{1}^{-}\Phi_{2}^{+}=\Phi_{2}^{-}\Phi_{1}^{+}=\frac{\beta}{\alpha^{2}-\beta^{2}}.
<T\hat{\eta}(x^{\prime})\hat{\overline{{\eta}}}(0)>=\frac{1}{4\pi^{2}}\hat{\overline{{D}}}{}^{2}D^{2}\delta^{4}(\theta-\theta^{\prime})\frac{1}{(x^{\prime})^{2}}\quad,\quad<T\hat{\eta}(x^{\prime})\hat{\eta}(0)>=\frac{1}{16\pi^{2}}(|\alpha|^{2}+2|\alpha|^{3}\theta^{\prime}\theta)\quad.
M_{\lambda1}^{-2}(\beta)=\frac{i}{2}\int[p_{1}p_{2}p_{3}]f_{\lambda}\left(\frac{\partial}{\partialp_{3}^{2}}\tilde{\delta}\right)a_{1\beta}^{\dagger}a_{2\beta}^{\dagger}a_{3\beta}+h.c.\;.
{\calA}=\left(-\frac{3}{4}(\nabla^{2}X^{\mu}\nabla^{2}X^{\nu}G_{\mu\nu}-4g^{\alpha\beta}P_{\alpha\beta})-\frac{1}{2}R_{(2)}-\frac{1}{6}g^{\alpha\gamma}g^{\beta\delta}W_{\alpha\beta\gamma\delta}\right)\phi-\frac{5}{6}\nabla^{2}X^{\mu}D_{\mu}\phi.
\int_{\Sigma}\Bigl(g_{IJ}\partial_{z}\Phi^{I}\partial_{\bar{z}}\Phi^{J}+i\eta^{\bar{\imath}}(D_{z}\rho_{\bar{z}}^{i}+D_{\bar{z}}\rho_{z}^{i})g_{i\bar{\imath}}+i\theta_{i}(D_{\bar{z}}\rho_{z}^{i}-D_{z}\rho_{\bar{z}}^{i})+R_{i\bar{\imath}j\bar{\jmath}}\rho_{z}^{i}\rho_{\bar{z}}^{j}\eta^{\bar{\imath}}\theta_{k}g^{k\bar{\jmath}}\Bigr)d^{2}z\,
\left.\mathrm{Tr}V_{1\nu}^{~\mu}(\nabla)\right|^{\mathrm{div}}=\frac{2i}{4-d}\,\frac{\kappa^{2}}{24\pi^{2}}\intd^{4}x\sqrt{-g}\left[-5R_{\mu\nu}{\calE}^{\mu\nu}+\frac{5}{2}Rg_{\mu\nu}{\calE}^{\mu\nu}\right].
K^{\mathrm{(J)}}=\left(\begin{matrix}{1}&{0}&{\dots}&{0}\\{0}&{1}&{\dots}&{0}\\{}&{\dots}&{\dots}&{}\\{0}&{0}&{\dots}&{1}\\\end{matrix}\right)+2m\left(\begin{matrix}{1}&{1}&{\dots}&{1}\\{1}&{1}&{\dots}&{1}\\{}&{\dots}&{\dots}&{}\\{1}&{1}&{\dots}&{1}\\\end{matrix}\right)
{\bar{F}^{\mu\nu}}={\bar{\eta}^{\mu\alpha}}{\bar{\eta}^{\nu\beta}}F_{\alpha\beta},\;\;\;{\bar{\eta}}^{\mu\nu}=\eta^{\mu\nu}-\frac{\partial^{\mu}T\partial^{\nu}T}{1+\partial_{\rho}T\partial^{\rho}T}.
\bar{D}^{\dot{\alpha}}J_{\alpha\dot{\alpha}}^{0}=-\frac{1}{24}D_{\alpha}\left[\frac{3N_{c}-N_{f}}{2\pi^{2}}\mathrm{Tr}\,W^{2}+\gammaZ\bar{D}^{2}\sum_{f}\left({Q}_{f}^{\dagger}e^{V}{Q}_{f}+\bar{Q}_{f}^{\dagger}e^{-V}\bar{Q}_{f}\right)\right]
\delta\alpha_{2}=\frac{1}{2}\left(\partial_{\mu}A_{a}^{*\mu}\right)f_{\;\;bc}^{a}C^{b}C^{c}-A_{a}^{*\mu}f_{\;\;bc}^{a}\left(C^{b}C_{\mu}^{c}+\eta^{b}A_{\mu}^{c}\right).
=\sum_{n}\sum_{\alpha_{n}=0}^{m_{n}-1}\left[x_{n}^{(\alpha_{n}+1)}dp_{n,\alpha_{n}}-dx_{n}^{(\alpha_{n})}\dot{p}_{n,\alpha_{n}}\right]-\sum_{n}dx_{n}\left[\frac{\partialL_{0}}{\partialx_{n}}-\dot{p}_{n,0}\right]\.
+{\frac{1}{2m^{2}}}\Pi_{\theta}^{2}+{\frac{m^{2}}{2}}\partial_{i}\theta\partial_{i}\theta+m^{2}(\partial_{i}A_{i})\theta-A_{0}(\partial_{i}\Pi_{i}+{\frac{\mu}{2}}\epsilon_{ij}\partial_{i}A_{j}+{\frac{m^{2}}{2}}A_{0}),
L{^{\pm}}_{CS}=\int_{X}\varepsilon^{ijk}\bigg({^{\pm}}A_{i}^{AB}\partial_{j}{\^{\pm}}A_{kAB}+{\frac{2}{3}}{^{\pm}}A_{iA}^{B}{^{\pm}}A_{jB}^{C}{^{\pm}}A_{kC}^{A}\bigg),
\frac{\partial}{\partial\bar{T}_{k}}d\Omega_{0}^{*}=\frac{\partial}{\partialX}d\Omega_{k}^{*},\;\;\;\frac{\partial}{\partialT_{k}}d\Omega_{0}^{*}=\frac{\partial}{\partialX}d\tilde{\Omega}_{k}^{*}.
{Z}=\mathrm{Tr}e^{-\beta(\hat{H}-\mu\hat{N})}=\sum_{j=0}^{\mathrm{M}}<j|e^{-\beta(\hat{H}-\mu\hat{N})}|j>=\prod_{\bfp}\sum_{j=0}^{\mathrm{M}}(ze^{-\beta\epsilon})^{j}\;,
e^{-W_{1}[C_{\alpha}]}=Z_{1}[C_{\alpha,\epsilon}]Z[K_{\alpha,\epsilon}]e^{W_{1}^{CAS}(\alpha,\epsilon)}=e^{-(W[C_{\alpha,\epsilon}]+W[K_{\alpha,\epsilon}]-W_{1}^{CAS}(\alpha,\epsilon))}
\delta_{\mathrm{components}}E_{M}{}^{A}=\Xi^{N}\partial_{N}E_{M}{}^{A}+\partial_{M}\Xi^{N}E_{N}{}^{A}+\frac{1}{2}(-)^{BM}(\Lambda^{rs}X_{rs})^{A}{}_{B}E_{M}{}^{B}\,,
I=\intd^{2}x\sqrt{-g}e^{-2\phi}[R+\gammag^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi+\mue^{2\lambda\phi}-\frac{1}{2}e^{-2\phi(\delta-1)}g^{\alpha\beta}\partial_{\alpha}f\partial_{\beta}f+\frac{1}{4}e^{\epsilon\phi}F^{2}],
I\approx2\,\epsilon^{\frac{N}{2}}\,\bigl({\textstyle\frac{\pi}{2}}\bigr)^{\frac{1}{2}}\,\bigl({\textstyle\frac{N-1}{2}}\bigr)^{\frac{N-1}{2}}\,e^{-\frac{N-1}{2}}\,
\Phi_{i}^{(1)a}\equiv\epsilon_{0ijk}\pi^{jka}\approx0,\;\Phi_{i}^{(2)a}\equiv\frac{1}{2}\epsilon_{0ijk}\left(F^{jka}-\left(D^{\left[j\right.}\right)_{\;\;b}^{a}\pi^{\left.k\right]0b}\right)\approx0,
\sigma_{\mathrm{R}}^{0}=\frac{12\pi\tilde{\Gamma}_{e,2}\tilde{\Gamma}_{f,2}}{m_{2}^{2}\Gamma_{2}^{2}}\left[1-(I^{\prime})^{2}+\frac{2G\left(m_{2}^{2}\right)}{m_{2}\Gamma_{2}}\right],
M_{0}=\frac{m_{1}m_{2}}{\mathrm{det}\,(c)},\quadM_{1}=\frac{m_{0}m_{2}}{\mathrm{det}\,(c)},\quadM_{2}=\frac{m_{0}m_{1}}{\mathrm{det}\,(c)},\quadT_{(0,2)}=\frac{m_{0}m_{1}m_{2}}{\mathrm{det}\,(c)}c^{-1}
\mathrm{\hat{H}|\Psi>=\left[\tilde{R}^{2}\frac{\partial^{2}}{\partial\tilde{R}^{2}}+p\tilde{R}\frac{\partial}{\partial\tilde{R}}-6\frac{\partial^{2}}{\partial\tilde{\varphi}^{2}}-36k\tilde{R}^{4}+12\lambda\,\tilde{R}^{6}(\tilde{\varphi}^{2}-\tilde{a}^{2})^{2}\right]|\Psi>=0.}
L(z,u_{a},D)\equiv\int_{0}^{\infty}d\hat{T}\,J(z,u_{a},\hat{T},D)=L_{02}(z,u_{a},D)+g(z,D)G_{Bab}^{1-{\frac{D}{2}}}+O\bigl(z^{4},G_{Bab}^{2-{\frac{D}{2}}}\bigr)\nonumber\,
\left(\partial_{n}r\right)_{\rho_{0}=R}={\frac{R^{2}+r^{2}-\rho^{2}}{2Rr}}~~,~~\left(\partial_{n}r_{1}\right)_{\rho_{0}=R}={\frac{\rho^{2}+r^{2}-R^{2}}{2\rhor}}
\begin{array}{ll}{\mathrm{~\tilde{~}a_i~=~a_i~\\for~i~\in~I_+~;}}\\{\mathrm{~\tilde{~}a_j=-a_j+\sum_{i~\leftrightarrow~j}~a_i~\\~for~j\in~I_-~;}}\\\end{array}\qquad\begin{array}{ll}{\mathrm{~\tilde{~}c_i~=~a_i-c_i~\\~for~i~\in~I_+~;}}\\{\mathrm{~\tilde{~}c_j=~c_j~\\~for~j\in~I_-~.}}\\\end{array}
\tilde{\calM}^{(\pm)}=\left(\bigcup_{n,m\in{\bfZ}}{\calM}_{S}^{(\pm)nm}\right)\cup\left(\bigcup_{n,m\in{\bfZ}}{\calM}_{N}^{(\pm)nm}\right)\cup\tilde{\calM}_{T}^{(\pm)}\cup\left(\bigcup_{n,m\in{\bfZ}}{\calM}_{0}^{(\pm)nm}\right).
H_{R}^{(\sigma)}=\int_{0}^{\infty}d\Omega\,\intd^{n-2}k\,\Omega\left[b_{k}^{(\sigma)\dagger}b_{k}^{(\sigma)}\,+\,\bar{b}_{k}^{(\sigma)}\bar{b}_{k}^{(\sigma)\dagger}\right].
\lambda^{(1)}=\left(\begin{matrix}{A_{1}}&{A_{2}}&{S}&{A_{3}}\\{A_{2}}&{-A_{1}}&{-A_{3}}&{S}\\{-S}&{-A_{3}}&{A_{1}}&{A_{2}}\\{A_{3}}&{S}&{A_{2}}&{-A_{1}}\\\end{matrix}\right);\;\;\;\lambda^{(2)}=\left(\begin{matrix}{A_{3}}&{S}&{A_{1}}&{A_{2}}\\{-S}&{A_{3}}&{A_{2}}&{-A_{1}}\\{A_{1}}&{A_{2}}&{-A_{3}}&{-S}\\{A_{2}}&{-A_{1}}&{S}&{-A_{3}}\\\end{matrix}\right);
{{\psi}_{k}^{in}}^{\prime}S^{2}=\frac{n_{s}\sqrt{k_{s}}}{\sqrt{\pi}}2^{-n_{s}}\Gamma\left(-n_{s}\right)a_{s}^{m}e^{l\phi_{s}/2}\left(\frack{k_{s}}\right)^{n_{s}}=const.
S(\mathbf{r},B)=-m\int_{0}^{\infty}\frac{dE}{2\pi}\mathrm{tr}\left<\mathbf{r}|((\mathbf{P}_{\perp}-\mathbf{A}_{\perp})^{2}-\sigma_{3}B+m^{2}+E^{2})^{-1}|\mathbf{r}\right>.
\begin{array}{c}{C:\quadu=c_{1}\quad\quad\int^{v(\sigma^{+},\sigma^{-})}dv^{\prime}G(c_{1},v^{\prime})=\alpha(\sigma^{+})+\beta(\sigma^{-})\,\,\,,}\\{D:\quadv=c_{2,}\quad\quad\int^{u(\sigma^{+},\sigma^{-})}du^{\prime}G(u^{\prime},c_{2})=\tilde{\alpha}(\sigma^{+})+\tilde{\beta}(\sigma^{-}),}\\\end{array}
\mathcal{E}_{\star}=16\frac{2r^{4}+4r^{2}+4t^{2}(1+2r^{2})-4t(x^{2}-y^{2})+1}{[2r^{4}+2r^{2}+4t(x^{2}-y^{2})+4t^{2}+1]^{2}}[1+O(\theta/r^{2})],
{{\frac{1}{2}}{\frac{{\partial}^{2}V}{{\partialx}^{2}}}=P(x,T,T^{*})\left[\widetilde{E}+{\frac{{\left(\alpha+\beta+2x\alpha\beta\right)}^{2}-\left(1+\alpha+\beta+2x\alpha\beta\right)}{1+\alpha+\beta+2x\alpha\beta-{\frac{1}{4x^{2}}}}}\right]}
I_{3}^{s}=\int_{C^{s}}(x_{i}^{s}n_{i}^{s}+\epsilon)(\frac{4}{\epsilon^{2}}+\frac{4}{\epsilon}b_{j}^{s}n_{j}^{s}+b_{j}^{s}b_{j}^{s})\,dl=8\pi+4\pib_{j}^{s}x_{j}^{s}.
\frac{1}{2}f^{\prime\prime}(3-D)-\frac{1}{2}f\frac{s^{\prime\prime}}{s}+f^{\prime}\frac{s^{\prime}}{s}\frac{(3-D)}{4}+\frac{1}{4}f\left(\frac{s^{\prime}}{s}\right)^{2}-f\left(\frac{f^{\prime}}{f}\right)^{2}\left(\frac{D^{2}-9D+18}{8}\right)=-\alpha^{2}f\frac{(D-2)^{2}+(D-2)}{8}
S_{\mathrm{general}}=\frac{1}{2}\sum_{n}\intd^{4}x\Bigl[\hat{\Phi}_{n}^{\dagger}\hat{\Phi}_{n}\Bigr]_{D}+\intd^{4}x\Bigl[f(\hat{\Phi})\Bigr]_{F}+\intd^{4}x\Bigl[f(\hat{\Phi})\Bigr]_{F}^{*}.
\frac{\partial\deltaN_{\pm}(X;\pm\omega_{\pm},\pm\hat{\bfp})}{\partialX_{0}}+{\bfv}_{\pm}\cdot\frac{\partial\deltaN_{\pm}}{\partial{\bfX}}\simeqZ_{\pm}\left[\Gamma_{\pm}^{(p)}+\frac{\left\{f_{B},\;Re\Sigma_{R}\right\}}{2\omega_{\pm}}\right]_{p_{0}=\pm\omega_{\pm}}.
\iota_{\xi}\,dz^{M}\,=\,\xi^{M}\\\\Rightarrow\\\,iota_{\xi}\,\Omega_{p}\,=\,\frac{1}{(p-1)!}dz^{M_{1}}...dz^{M_{p-1}}\xi^{M_{p}}\,\Omega_{M_{p}...M_{1}}.
V_{CKM}=\left(\begin{array}{ccc}{c_{12}c_{13}}&{s_{12}c_{13}}&{s_{13}e^{-i\delta_{13}}}\\{-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta_{13}}}&{c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta_{13}}}&{s_{23}c_{13}}\\{s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{13}}}&{-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta_{13}}}&{c_{23}c_{13}}\\\end{array}\right);
T_{h}=\left(\begin{matrix}{n}&{0}&{\cdots}&{0}&{0}\\{0}&{n-1}&{\cdots}&{0}&{0}\\{\vdots}&{\ddots}&{\ddots}&{\vdots}&{\vdots}\\{0}&{0}&{\cdots}&{-n+2}&{0}\\{0}&{0}&{\cdots}&{0}&{-n}\\\end{matrix}\right),\quad[T_{a},T_{b}]=f_{ab}^{c}T_{c}\,.
\Sigma_{\mathrm{BFYM}}=\intd^{n}x\,\,\left\{\varepsilon^{\mu_{1}\ldots\mu_{n}}B_{\mu_{1}\ldots\mu_{n-2}}^{a}F_{\mu_{n-1}\mu_{n}}^{a}+\left(B_{\mu_{1}\ldots\mu_{n-2}}^{a}+D_{[\mu_{1}}\eta_{\mu_{2}\ldots\mu_{n-2}]}^{a}\right)^{2}\right\}\,\,,
e^{-\frac{\bar{z}\starz+\hbar}{\hbar}}=1+(-\frac{\bar{z}\starz+\hbar}{\hbar})+\frac{1}{2!}(-\frac{\bar{z}\starz+\hbar}{\hbar})(-\frac{\bar{z}\starz+\hbar}{\hbar})+\cdots.
\begin{array}{rcl}{\psi^{\prime}(t,x)}&{=}&{e^{-ie\epsilon_{0}(t,x)}\psi(t,x)\,}\\{{A^{1}}^{\prime}(t,x)}&{=}&{A^{1}(t,x)-\partial_{1}\epsilon_{0}(t,x)\,}\\{\pi_{1}^{\prime}(t,x)}&{=}&{\pi_{1}(t,x)\,\}\\{{A^{0}}^{\prime}(t,x)}&{=}&{A^{0}(t,x)+\partial_{0}\epsilon_{0}(t,x)\,}\\\end{array}
(\kappa)={\frac{1}{\sqrt{1+\vert\nu_{A}\vert^{2}}}}\,\left(\begin{array}{cc}{\nu_{A}^{c}\,\kappa_{A}\,\nu_{A}}&{\nu_{A}^{c}\,\kappa_{A}}\\{\kappa_{A}\,\nu_{A}}&{\kappa_{A}}\\\end{array}\right)\,{\frac{1}{\sqrt{1+\vert\nu_{A}\vert^{2}}}}\;,
u_{E,q,j,m_{j}}^{\pm}=N_{E,q,j}^{\pm}\frac{1}{r}\left(\begin{array}{r}{-i(E+M)\,f_{E,q,j}^{\pm}\,\Psi_{q,j,m_{j}}^{\pm}}\\{\sqrt{V}(h_{E,q,j}^{\pm}\,\Psi_{q,j,m_{j}}^{\pm}+g_{E,q,j}^{\pm}\,\Psi_{q,j,m_{j}}^{\mp})\,,}\\\end{array}\right)
\frac{\partialP_{c}(\phi,t|\chi)}{\partialt}=\frac{1}{2}\frac{H^{3/2}(\chi)}{2\pi}\,\frac{\partial}{\partial\chi}\left(\frac{H^{3/2}(\chi)}{2\pi}\,\frac{\partial}{\partial\chi}P_{c}(\phi,t|\chi)\right)-\frac{V^{\prime}(\chi)}{3H(\chi)}\frac{\partial}{\partial\chi}P_{c}(\phi,t|\chi)\.
\zeta+a_{0}\frac{1}{\zeta}x_{1}^{4-n}z^{6-n}\to\zeta+a_{0}\frac{1}{\zeta}\prod_{i=1}^{N_{s}}(x_{1}-m_{i}^{4})\prod_{j=1}^{N_{f}}(z-m_{j}^{2}),
{\calL}_{c}^{M}(V_{c})=\frac{1}{16}\intd^{2}\theta\left(\frac{1}{g_{c}^{2}}+i\frac{\theta}{8\pi^{2}}\right)W^{a}(g_{c}V_{c})W^{a}(g_{c}V_{c})+\mathrm{h.c.}
\mathrm{d}s_{*}^{2}=r_{0}^{2}\left[\mathcal{R}^{2}\left\{u^{2}\mathrm{d}x_{d}^{2}+\left({\frac{\mathrm{d}u}{u}}\right)^{2}\right\}+\mathrm{d}\Omega_{\tilde{d}+1}^{2}\right]\,.
S=\frac{1}{16\pi}\intd^{2}\!x\,(-g)^{1/2}e^{-2\phi}\left[R+2(\nabla\phi)^{2}+2e^{2\phi}-2\Lambda-2Q^{2}e^{4\phi}-\frac{1}{2}\sum_{i=1}^{N}(\nablaf_{i})^{2}\right],
\left[\delta^{ab}+C.\frac{\delta^{2}\Gamma[\phi]}{\delta\phi^{a}\delta\phi^{b}}\right]^{-1}(\mathbf{q},\mathbf{-q})=\left[\delta^{ab}+2q^{2}\left\{\delta^{ab}V^{\prime}+2\phi^{a}\phi^{b}V^{\prime\prime}+q^{2}\delta^{ab}\right\}\right]^{-1}
ds^{2}=\left(1+\frac{\tilde{k}_{3}}{\hat{r}^{5}}\right)^{-2/3}dx^{\mu}dx^{\nu}\eta_{\mu\nu}+\left(1+\frac{\tilde{k}_{3}}{\hat{r}^{5}}\right)^{1/3}(dz^{2}+dy^{\tilde{m}}dy^{\tilde{m}}).
\begin{array}{ll}{L_{+}=\epsilon^{\mu\nu\rho\sigma}\bigg(AR_{\mu\nu}^{+ab}R_{\rho\sigmaab}^{+}+B\Sigma_{\mu\nu}^{+ab}R_{\rho\sigmaab}^{+}+C\Sigma_{\mu\nu}^{+ab}\Sigma_{\rho\sigmaab}^{+}+EG_{\mu\nu}^{+ab}}\\{\big(R_{\rho\sigmaab}^{+}-\partial_{\rho}\omega_{\sigmaab}^{+}+\partial_{\sigma}\omega_{\rhoab}^{+}+\frac{1}{2}f_{[ab][cd][ef]}\omega_{\rho}^{+cd}\omega_{\sigma}^{+ef})\bigg),}\\\end{array}
{\calJ}=\frac{1}{4\pi}\int\left(\frac{(|R_{z}|^{2}+|R_{{\bar{z}}}|^{2})(1+|z|^{2})^{2}}{(1+|R|^{2})^{2}}\right)\frac{2i~\mathrm{d}z\mathrm{d}{\bar{z}}}{(1+|z|^{2})^{2}}
S_{c}^{(2)}=\intd^{4}x\sqrt{-g}\left\{vE+xC^{2}+y\left[\frac{R}{3}-\frac{{\Box}\zeta}{\zeta}+\frac{({\nabla}\zeta)^{2}}{2\zeta^{2}}\right]^{2}+\frac{3z\zeta}{\gamma}\left[\frac{R}{3}-\frac{{\Box}\zeta}{\zeta}+\frac{(\nabla\zeta)^{2}}{2\zeta^{2}}\right]-\frac{u\zeta^{2}}{\gamma^{2}}\right\},
(U-\overline{U})_{rr}=6i\Biggl[{\frac{(n+1)^{2}g_{n+1}^{2}-n^{2}g_{n}^{2}}{2n+1}}+{\frac{1}{3}}\sum_{l=0}^{n}(-1)^{l}g_{n-l}^{2}\Biggr].
ds^{2}=\frac{4a\bar{f}_{0}^{2}}{-\mu\chi^{2}}\frac{dt^{2}}{r^{2a/(1-a)}}-\frac{a+\chi^{2}/2}{-\mu(1-a)^{2}}\frac{dr^{2}}{(r^{*}/r)^{2(a+\chi^{2}/2)/a}-1}-r^{2}d\Omega
\psi_{i}=\frac{1}{\sqrt{2}}\sum_{n=-\infty}^{i(n+v_{i})(\tau-\sigma)}\psi_{i,n+v_{i}},\quad\bar{\psi}_{i}=\frac{1}{\sqrt{2}}\sum_{n=-\infty}^{i(n-v_{i})(\tau-\sigma)}\bar{\psi}_{i,n-v_{i}},
\{F,G\}=\frac{\partialF}{\partialx_{l}}\frac{\partialG}{\partialp_{l}}-\frac{\partialF}{\partialp_{l}}\frac{\partialG}{\partialx_{l}}+i(-1)^{F}\frac{\partialF}{\partial\psi_{l}}\frac{\partialG}{\partial\psi_{l}}\.
V_{GEP}(\phi_{0})-V_{GEP}(0)={\frac{m^{2}}{2}}\phi_{0}^{2}+\frac{\lambda}{4!}\phi_{0}^{4}+I_{1}(\mu)-I_{1}(\mu_{0})-\frac{\lambda}{8}[I_{0}^{2}(\mu)-I_{0}^{2}(\mu_{0})]\;,
A(E,\nabla)=\left(\mu_{2}(E)-\frac{i}{2\pi}\frac{\left(\mu_{1}(E)^{+}+\left(\iota(\theta)\mu_{2}(E)\right)^{+}\right)^{2}}{\mu_{0}(E)+\iota(\theta)\mu_{1}(E)+\frac{1}{2}\iota(\theta)^{2}\mu_{2}(E)}\right)[T],
\Omega_{2}=\int\mathrm{d}^{3}{\vec{x}}\,v^{\lambda}(\vec{x})\partial_{i}J_{i}(\vec{x})=-\int\mathrm{d}^{3}{\vec{x}}\,v^{\lambda}(\vec{x})\frac{\partial}{\partialt}\nabla^{2}\phi(\vec{x}).
W(u)_{aebcfd}=\sum_{\alpha,\beta,\gamma,\delta}S_{ae\alpha}S_{be\beta}\left(\delta_{\alpha\gamma}\delta_{ef}\delta_{\beta\delta}+Q^{-1/2}f_{0}\delta_{\alpha\beta}\delta_{\gamma\delta}+f_{1}\delta_{\alpha\beta\gamma\delta}+Q^{-1/2}f_{2}\delta_{\alpha\gamma}\delta_{\beta\delta}\right)S_{\gammafc}S_{\deltafd}