harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{d\alpha^{i}(\tau,\vec{\Sigma}(\tau,\vec{\sigma}_{o}))}{d\tau}=\frac{\partial\alpha^{i}(\tau,\vec{\Sigma}(\tau,\vec{\sigma}_{o}))}{\partial\tau}+[I(\tau,\vec{\Sigma}(\tau,\vec{\sigma}_{o}))]_{\;\check{r}}^{i}\,\frac{\partial\Sigma^{\check{r}}(\tau,\vec{\sigma}_{o})}{\partial\tau}=0,
L^{+4}(x,\theta^{+},u)=\mathrm{Tr}\;(A^{++}\partial^{+\alpha}A_{\alpha}^{+}-{\frac{1}{2}}A^{+\alpha}D^{++}A_{\alpha}^{+}+A^{++}A^{+\alpha}A_{\alpha}^{+})\.
\frac{1}{R}\sum_{n,m}\frac{1}{(\frac{1}{2\sqrt{\alpha^{\prime}}}(2n+1)i+\frac{(2m+1)i}{2R}-t_{0})^{2}}=\frac{\frac{1}{R}\frac{\partial}{\partialt_{0}}}{e^{\frac{i}{2R}\frac{\partial}{\partialt_{0}}}-e^{-\frac{i}{2R}\frac{\partial}{\partialt_{0}}}}\sum_{n}\frac{1}{\frac{1}{2\sqrt{\alpha^{\prime}}}(2n+1)i-t_{0}}.
{\tilde{f}}^{(0)}=\left(\begin{array}{cccccc}{0}&{-1}&{0}&{0}&{0}&{q\partial_{y}}\\{1}&{0}&{0}&{0}&{0}&{q}\\{0}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{-1}&{q^{2}}\\{0}&{0}&{0}&{1}&{0}&{\partial_{y}}\\{-q\partial_{x}}&{-q}&{0}&{-q^{2}}&{-\partial_{x}}&{0}\\\end{array}\right)\delta(x-y)
a_{j}({\mathsf{s}})=\frac{R_{Z}}{{\mathsf{s}}-\left(M_{R}-i\frac{\Gamma_{R}}{2}\right)^{2}}=\frac{R_{Z}}{{\mathsf{s}}-{\mathsf{s}}_{R}}\,,\quad-\infty_{II}<{\mathsfs}<+\infty\,.
\frac{6\ddot{a}}{a}+\dot{\phi}^{2}=\frac{2\ddot{a}}{a}+\frac{4(\dot{a}^{2}+k)}{a^{2}}=-\frac{1}{\alpha}\left(\ddot{\phi}+\frac{3\dot{a}\dot{\phi}}{a}\right)=\Lambdae^{\alpha\phi}\;.
\deltaS\vert_{\hat{E}^{I}=0=\tilde{E}^{i}}={\frac{1}{2}}\int_{{\calM}^{1+1}}\left(\hat{M}_{2}^{I}U_{\underline{{m}}}^{I}+j_{1}\wedge\left(\hat{E}^{++}U_{\underline{{m}}}^{--}-\hat{E}^{--}U_{\underline{{m}}}^{++}-2mf^{i}u_{\underline{{m}}}^{i}\right)\right)\delta\hat{X}^{\underline{{m}}}
Tr\,e^{-isH}={\frac{T\,2a\ell^{2}}{4\pi^{2}i}}{\frac{e^{-ism^{2}}}{s^{2}}}[1+2\sum_{n=1}^{\infty}(-1)^{n}e^{i(an)^{2}/s}][1+iseB\,L(iseB)]\,,
ds^{2}={\frac{1}{g_{s}}}\left(\mu_{6}VZ_{2}-\mu_{2}Z_{6}\right)dx_{\perp}^{2}={\frac{\alpha^{-3/2}}{(2\pi)^{2}g}}\Biggl(\frac{V}{V_{*}}-1-\frac{g_{s}N\alpha^{1/2}}{r}\Biggr)(dr^{2}+r^{2}d\Omega_{2}^{2})\.
{\calL}_{2}\equiv{\calL}_{3}\left(\varphi,\partial_{\mu}\varphi,\partial_{\mu\nu}\varphi;B_{\rho},F^{(1)},F^{(2)},C^{(1)},C^{(2)};B_{\rho\sigma},D^{(1)},D^{(2)},G^{(2)},G^{(3)}\right).
V(\phi,g)=g\int{\mathrmd}^{4}x\left\{\frac{\mu^{(1)}}{2}\phi(x)^{2}+\frac{\zeta^{(1)}}{2}\phi(x)(-\bigtriangleup)\phi(x)+\frac{1}{4!}\phi(x)^{4}\right\}+O(g^{2}).
L=\frac1{2\lambda}\int_{M}Tr\left(X^{-1}\partial_{a}X\cdotX^{-1}\partial^{a}X\right)d^{2}M+\frack{24\pi}\int_{B}\epsilon^{abc}Tr\left(X^{-1}\partial_{a}X\cdotX^{-1}\partial_{b}X\cdotX^{-1}\partial_{c}X\right)d^{3}B,
\frac{d^{2}}{d\xi^{2}}\left(e^{-\frac{\xi}{2}}\xi^{\frac{k}{2}}F(\xi)\right)=e^{-\frac{\xi}{2}}\xi^{\frac{k}{2}}\left[\frac{d^{2}F}{d\xi^{2}}+\left(\frac{k}{\xi}-1\right)\frac{dF}{d\xi}+\left(\frac{1}{4}-\frac{k}{2\xi}+\frac{\frac{k}{2}\left(\frac{k}{2}-1\right)}{\xi^{2}}\right)\right].
\begin{array}{rll}{\displaystyle\int_{0}^{\infty}\psi_{\ellj}(r)\psi_{\ellj^{\prime}}(r)\,dr}&{\displaystyle\delta_{jj^{\prime}}}&{\mathrm{for~the~bound~states~and}}\\{\displaystyle\int_{0}^{\infty}\psi_{\ell}^{*}(k,r)\psi_{\ell}(k^{\prime},r)\,dr}&{\displaystyle\frac{\pi}{2}\delta(k-k^{\prime})}&{\mathrm{for~the~continuum~states.}}\\\end{array}
\{\theta_{1},\theta_{2}\}_{\mathrm{c.b.}}=\beta_{1}\left(P-\frac{1}{N}I\right)+\left\{\theta_{1}[\theta_{2},r]\right\}+\beta_{2}\left(\theta_{1}P\theta_{2}+\theta_{2}P\theta_{1}-\frac{2}{N}\theta_{1}\theta_{1}-\frac{2}{N}\theta_{2}\theta_{2}\right).
S_{\mathrm{eff}}={\frac{1}{2}}\intd^{4}x\sqrt{-g}\,e^{-\sigma}\left[\ell_{\mathrm{st}}^{-2}(R+\partial_{\mu}\sigma\partial^{\mu}\sigma)+R^{2}+\mathrm{matter~terms}+O(e^{\sigma})\right]\,.
e_{\mu}^{(0)}=\frac{p_{\mu}}{\sqrt{-p^{2}}},\quade_{\mu}^{(\alpha)}\eta_{\alpha\beta}e_{\nu}^{(\beta)}=\eta_{\mu\nu},\quad\varepsilon_{\mu\nu\lambda}e^{(0)\mu}e^{(i)\nu}e^{(j)\lambda}=\varepsilon^{0ij}.
K=\mathrm{Tr}\left(\frac{1}{2}\bar{W}W+\tilde{q}^{+}\nabla^{--}q^{+}-q^{+}\nabla^{--}\tilde{q}^{+}\right)=\mathrm{Tr}\left(\bar{\phi}\phi+\bar{q}^{i}q_{i}\right)+\theta\\mathrm{terms}\.
\widehat{\varrho}{}^{g}=\widehat{\calP}\gamma\otimes1+m\cdot1\otimes\sigma_{3},\quad2\widehat{\phi}{}^{g}=\widehat{\calP}{}^{2}+m^{2}+\frac{1}{4}q\epsilon_{\mu\nu\lambda}{\calF}^{\mu\nu}\gamma^{\lambda}\otimes1,
V=Tr{\frac{1}{4}}[{\bfA}^{i}\,,{\bfA}^{j}][{\bfA}^{i}\,,{\bfA}^{j}]={\frac{1}{4}}\sum_{i,j;m,n}\langlem\vert[{\bfA}^{i}\,,{\bfA}^{j}]\vertn\rangle\langlen\vert[{\bfA}^{i}\,,{\bfA}^{j}]\vertm\rangle\,.
\begin{array}{c}{a^{\pm}=\frac1{\sqrt{2}}\left(\mp\fracd{dx}+\frac{d\rho\left(x\right)}{dx}\right)=}\\{\mp\frac1{\sqrt{2}}e^{\mp\rho\left(x\right)}\fracd{dx}e^{\pm\rho\left(x\right)}}\\\end{array}
H_{f}=-{\frac{i}{2}}\sum_{\vec{x},j}(-1)^{\sum_{p=1}^{j-1}x_{p}}\left(\psi^{\dagger}(\vec{x}+\vec{j})\psi(\vec{x})-\psi^{\dagger}(\vec{x})\psi(\vec{x}+\vec{j})\right)
e^{\frac43\phi}\simeq\frac{\left(3\,r_{3}^{2}+10\,r_{3}\,r_{4}+3\,r_{4}^{2}\right)^{2}\,\left(5\,r_{3}^{2}+6\,r_{3}\,r_{4}+5\,r_{4}^{2}\right)}{128\,\left(r_{3}+r_{4}\right)}\,\,\sqrt{u_{0}+2\,\kappa\,|u|^{2}}\;\kappa\,x+O(x^{2})\,,
\begin{array}{cccc}{{\hat{\delta}}D_{i}=\partial_{i}d_{1}}&{{\hat{\delta}}{\overline{D}}_{i}=-E_{i}+\partial_{i}d_{2}}&{{\hat{\delta}}E_{i}=\partial_{i}d_{3}}&{{\hat{\delta}}{\overline{d}}_{\overline{a}}=-b_{\overline{a}}}\\{{\hat{\delta}}d_{2}=d_{3}}&{{\hat{\delta}}d={\dot{d}}_{1}}&{{\hat{\delta}}{\overline{d}}=-b}&{}\\\end{array}\;.
(\:{\tilde{\Gamma}}_{{\bar{j}}{\bar{i}}}^{\bar{l}}\:G_{k{\bar{l}}}+\partial_{\bar{i}}G_{k{\bar{j}}}\:)\:|\sigma_{1}\rangle+d\:G_{k{\bar{j}}}\:|{\hat{\phi}}_{\bar{i}}\rangle=(\:{\tilde{\Gamma}}_{{\bar{i}}{\bar{j}}}^{\bar{l}}\:G_{k{\bar{l}}}+\partial_{\bar{j}}G_{k{\bar{i}}}\:)\:|\sigma_{1}\rangle+d\:G_{k{\bar{i}}}\:|{\hat{\phi}}_{\bar{j}}\rangle\,
P_{0}(l)=\frac{4\pi^{2}}{l!^{2}(l+1)!^{2}}\left(\frac{\lambda\omegar_{0}}{4}\right)^{2l+2}=\frac{\pi^{2}\omega^{4}r_{1}^{2}r_{5}^{2}}{4}\frac{(\omega^{2}r_{1}r_{5})^{2l}}{2^{4l}l!^{2}(l+1)!^{2}}\,,
\left\langle{\calO}_{\alpha_{1}}\ldots{\calO}_{\alpha_{n}}\cdot\widetilde{\calO}_{\widetilde{\beta}_{1}}\ldots\widetilde{\calO}_{\widetilde{\beta}_{s}}\right\rangle=\int_{{\bfR}{\calS}}\iota^{\ast}\Phi^{\ast}\alpha_{1}\wedge\ldots\wedge\iota^{\ast}\Phi^{\ast}\alpha_{n}\wedge\Phi^{\ast}\widetilde{\beta}_{1}\wedge\ldots\wedge\Phi^{\ast}\widetilde{\beta}_{s}.
\Omega(q)=\frac1{N!}\sum_{PP^{\prime}}\intdx^{+}e^{iq_{PP^{\prime}}^{-}x^{+}}\rho(x^{+})l_{c}^{3}(x^{+})e^{-4(q_{PP^{\prime}}^{+})^{2}l_{c}^{2}(x^{+})}
\begin{array}{lcll}{\mathrm{floor}}&{\mathrm{SU(2/1)}}&{\mathrm{field}}&{\mathrm{helicity}}\\{\mid\mathrm{gnd}>}&{(0~+\frac{1}{2})}&{\phi}&{~0}\\{\mid\mathrm{1st}>}&{(0~+\frac{1}{2})}&{2\lambda}&{-\frac{1}{2}}\\{\mid\mathrm{2nd}>}&{(0~+\frac{3}{2})}&{A_{\mu}}&{-1.}\\\end{array}
{\itL}_{eff}=-\frac{1}{2}\gamma_{~\mu\nu}^{\sigma}\tilde{F}_{~\sigma~~\lambda}^{~\mu\nu~\alpha\beta}\gamma_{~\alpha\beta}^{\lambda}-\frac{1}{2}h^{\alpha\beta}~X_{\alpha\beta\mu\nu}~h^{\mu\nu}-\gamma_{~\alpha\beta}^{\lambda}B_{\lambda~~~\mu\nu}^{~\alpha\beta\sigma}\nabla_{\sigma}h^{\mu\nu}+\gamma_{~\mu\nu}^{\sigma}~H_{\sigma~~\alpha\beta}^{~\mu\nu}{}~h^{\alpha\beta}
U(z,\bar{z})=\frac12J_{\alpha\beta}\psi^{\alpha}(z)\bar{\psi}^{\beta}(\bar{z}),\qquadV^{a}(z,\bar{z})=\frac12d_{\alpha\beta}^{a}\psi^{\alpha}(z)\bar{\psi}^{\beta}(\bar{z}).
C\left(a^{\ast},k\right)=\sum_{n=0}^{\infty}C\left(n,k\right)\frac{\left(a^{\ast}\right)^{n}}{\sqrt{n!}},\;\;C^{+}\left(a,k\right)=\sum_{n=0}^{\infty}C^{+}\left(n,k\right)\frac{a^{n}}{\sqrt{n!}}.
<x,T|{\calT}\left[\hat{x}_{H}(t_{1})\cdot\frac{\hat{\gamma}_{H}(t_{2})}{2}\right]|y,0>\equiv<x|e^{-i\hat{\calH}T}{\calT}\left[\hat{x}_{H}(t_{1})\cdot\frac{\hat{\gamma}_{H}(t_{2})}{2}\right]|y>=\left<x(t_{1})\cdot\zeta(t_{2})\right>_{S}
I=\frac{(4\pi)^{(n-2)/2}\beta}{8\pi\ell^{2}}\left[m\ell^{2}-(n-1)\sum_{i=0}^{(n-2)/2}\left({\frac{(n-2)}{2}\atopi}\right)(-1)^{i}N^{2i}\frac{r_{+}^{(n-2i-1)}}{(n-2i-1)}\right]
\vec{\Phi}_{\perp}=\frac{1}{\sqrt{2}}\left(\vec{A}_{\perp}^{1}-i\vec{A}_{\perp}^{2}\right)\quad,\quad\vec{\Pi}_{\perp}=\frac{1}{\sqrt{2}}\left(\vec{E}_{\perp}^{1}+i\vec{E}_{\perp}^{2}\right).
\tilde{Z}=\left(\begin{array}{cccc}{\tilde{z}_{++}}&{0}&{0}&{0}\\{0}&{\tilde{z}_{--}}&{0}&{0}\\{0}&{0}&{\tilde{z}_{ZZ}}&{\tilde{z}_{ZA}}\\{0}&{0}&{\tilde{z}_{AZ}}&{\tilde{z}_{AA}}\\\end{array}\right)
S_{q,g}^{ext}={\frac{1}{4\sqrt{\pi}}}\int_{\epsilon^{2}/l^{2}}^{\infty}{\frac{ds}{s^{3/2}}}k_{H}(s)\Theta_{\Sigma_{g}}(s{\frac{l^{2}}{l_{1}^{2}}})e^{-{s/4}}~~
\Gamma_{\mu}(p^{\prime},p)=\left\langlep^{\prime}\left|j_{\mu}(0)\right|p\right\rangle,\quadW_{\mu\nu}(p,x)=\left\langlep\left|j_{\mu}(\frac{x}{2})j_{\nu}(\frac{-x}{2})\right|p\right\rangle_{conn.}
={(\frac{\partial^{2}A}{\partial\phi^{p}\partial\phi^{q}})}^{-1}\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}}(\bar{\psi}_{a}^{i}\psi^{aj}\bar{\psi}_{b}^{l}\psi^{bk}+\bar{\psi}_{a}^{i}\psi^{ak}\bar{\psi}_{b}^{l}\psi^{bj})
\left(Q_{\alpha_{1}i_{1}}\cdotsQ_{\alpha_{(N-4)/2}i_{(N-4)/2}}\right)\left(\left(\Phi_{\beta_{1}}Q_{\gamma_{1}j_{1}}\right)\cdots\left(\Phi_{\beta_{(N-4)/2}}Q_{\gamma_{(N-4)/2}j_{(N-4)/2}}\right)\right)\epsilon_{i_{1}\dotsi_{(N-4)/2}j_{1}\dotsj_{(N-4)/2}}~.
\frac{\partialf_{\mu}}{\partial\dot{x}^{\nu}}+\frac{\partialf_{\nu}}{\partial\dot{x}^{\mu}}=0\qquad\Longrightarrow\qquad\frac{\partial^{2}f_{\mu}}{\partial\dot{x}^{\nu}\partial\dot{x}^{\lambda}}+\frac{\partial^{2}f_{\nu}}{\partial\dot{x}^{\mu}\partial\dot{x}^{\lambda}}=0\.
\left(\frac{\partial}{\partialu}\right)^{n_{1}}a_{\mu}^{m_{1}}\left(\frac{\partial}{\partial\bar{u}}\right)^{n_{2}}\bar{a}_{\mu}^{m_{2}}=\left(\frac{\partial}{\partialu}\right)^{n_{1}}\left(\frac{\partial}{\partial\bar{u}}\right)^{n_{2}}a_{\mu}^{m_{1}}\bar{a}_{\mu}^{m_{2}},
{\calH}_{D}=-\tilde{K}\cdot\left({\calF}\frac{\delta}{\delta\tilde{K}}\right)-\tilde{K}\cdot\left(\frac{\delta}{\delta\tilde{K}}\vee\frac{\delta}{\delta\tilde{K}}\right)-g_{\mathrm{st}}^{2}\tilde{K}\cdot\left(\tilde{K}\cdot\left(\wedge\frac{\delta}{\delta\tilde{K}}\right)\right),
D_{n}=\int_{0}^{\infty}r^{2}~dr\{\frac{sin2nF}{nr}[1+4(F^{2}+\frac{sin^{2}nF}{n^{2}r^{2}})+F^{\prime}(1+\frac{8sin^{2}nF}{n^{2}r^{2}})\}
\intd^{4}x\left\{\frac{i}{4}\bar{\psi}_{1-}\gamma_{-}\partial_{+}\psi_{1-}+\frac{i}{4}\bar{\psi}_{-}\gamma_{-}\frac{\Box}{\partial_{-}}\psi_{-}+i\bar{\tilde{\psi}}_{+}\hat{\partial}_{\perp}\tilde{\psi}_{-}+i\bar{\tilde{\psi}}_{-}\hat{\partial}_{\perp}\tilde{\psi}_{+}\right\}
\begin{array}{rcl}{}&{}&{\mathbb{H}_{m}=3\partialB_{m}+3V_{mn}F_{n}\equivH_{m}+3V_{mn}F_{n}\,,}\\{}&{}&{}\\{}&{}&{F_{m}=2\partialA_{m}-mB_{m}\,,}\\{}&{}&{}\\{}&{}&{{\calF}_{mn}=2\partialV_{mn}+2mV_{mp}V_{np}\,.}\\{}&{}&{}\\\end{array}
J^{(N)}(n;1,\ldots,1)=\frac{1}{\Lambda^{(N)}}\;\left(\prodm_{i}^{2}\right)\;\sum_{i=1}^{N}\frac{1}{m_{i}^{2}}\;F_{i}^{(N)}\;J_{i}^{(N)}(n;1,\ldots,1).
{\frac{\partialW_{0}}{\partialm^{2}}}=-(i^{\nu+1}){\frac{AT}{2}}\Gamma(\nu+1)\sum_{n=1}^{\infty}\int\int{\frac{dk_{1}dk_{2}}{(2\pi)^{2}}}\int{\frac{dk_{0}}{2\pi}}[k_{0}^{2}-(k_{1}^{2}+k_{2}^{2}+{\frac{n^{2}\pi^{2}}{a^{2}}}+m^{2})]^{-(\nu+1)}\,.
F_{0,1}=\frac{1}{\pi^{3}}\intdz_{1}dz_{2}dz_{3}\frac{\rho_{0,1}(z_{1},z_{2},z_{3},p_{1}^{2},p_{2}^{2},p_{3}^{2})}{(z_{1}-p_{1}^{2})(z_{2}-p_{2}^{2})(z_{3}-p_{3}^{2})},
\begin{array}{rl}{(b,\lambda,p_{2})\longleftrightarrow(c,\mu,p_{3});}&{P\leftrightarrowR,s\leftrightarrowu\,;}\\{(b,\lambda,p_{2})\longleftrightarrow(d,\nu,p_{4});}&{P\leftrightarrowQ,s\leftrightarrowt\,.}\\\end{array}
\tilde{L}_{0}=\pi\sum_{n}\left[(|n+mH\alpha^{\prime}|-n)^{2}A_{1n}A_{1n}^{\dagger}+(|n-mH\alpha^{\prime}|-n)^{2}A_{2n}A_{2n}^{\dagger}\right]-\frac{\pi}{2}m^{2}\alpha^{2}.
\delta_{\varepsilon}\Phi_{a_{I}}^{I}=\varepsilon^{i}{(C_{i}^{I})_{a_{I}}}^{a_{I+1}}\Phi_{a_{I+1}}^{I+1}+\varepsilon^{i}{(\tilde{C}_{i}^{I})_{a_{I}}}^{a_{I-1}}\frac{d}{d\tau}{\Phi}_{a_{I-1}}^{I-1},
\frac{\partialQ_{s}}{\partialt}=\frac{\partial}{\partialt}\left[\int_{-\infty}^{0}(T_{s+1}-\Theta_{s-1})\right]=[T_{s+1}+\Theta_{s-1}]_{-\infty}^{0}=(T_{s+1}+\Theta_{s-1})|_{x=0}\,.
H_{{\bfF}0}:=E\:,\:\:\:\:\:D_{\bfF}:=-i\left(\frac{1}{2}+E\frac{d\:}{dE}\right)\:,\:\:\:\:\:C_{\bfF}:=-\frac{d\:}{dE}E\frac{d\:}{dE}+\frac{(k-\frac{1}{2})^{2}}{E}\:,
+\frac{b_{0}^{(3)}}{L_{0}^{(3)}}Q^{(3)}|\Lambda_{a_{1}a_{2}}^{e}\rangle_{3}+\frac{b_{0}^{(3)}}{L_{0}^{(3)}}[\delta\varepsilon_{1}|\Omega_{1a_{1}a_{2}}\rangle+\delta\varepsilon_{2}|\Omega_{2a_{1}a_{2}}\rangle]
N=\int_{t_{N}}^{t_{end}}Hdt=\int_{\phi_{N}}^{\phi_{end}}H\frac{d\phi}{\dot{\phi}}=-\int_{\phi_{N}}^{\phi_{end}}\frac{3H^{2}Zd\phi}{V^{\prime}}=-\frac{1}{M_{P}^{2}}\int_{\phi_{N}}^{\phi_{end}}\frac{V}{V^{\prime}}Zd\phi
W(x_{B},Q;{\mathcalT})=\left(\begin{array}{cccc}{W^{00}}&{0}&{W^{02}}&{0}\\{0}&{W^{11}}&{0}&{0}\\{W^{20}}&{0}&{W^{22}}&{0}\\{0}&{0}&{0}&{0}\\\end{array}\right)
\lambda_{r}^{-1}(q^{2})=\frac{3Nm_{F}}{4\pi}\left(1+\frac{m_{F}}{\sqrt{q^{2}}}\biggl(1+\frac{q^{2}}{4m_{F}^{2}}\biggr)\mathrm{arcsin}\sqrt{\frac{q^{2}}{4m_{F}^{2}+q^{2}}}\;\right)\,.
\begin{array}{l}{{\calD}_{+k}{\calX}^{+k}-{\calX}^{+k}{\calD}_{+k}=\mu_{+k}~,~~~~~~\mu_{+k}{\calX}^{+k}=q^{2}{\calX}^{+k}\mu_{+k}~,}\\{\mu_{+k}{\calD}_{+k}=q^{-2}{\calD}_{+k}\mu_{+k}~,~~~~~~\mu_{+k}\equiv1+(q^{2}-1){\calX}^{+k}{\calD}_{+k}~,}\\\end{array}
u^{0}=1,\quad\frac{du^{1}}{dt}=-\frac{2\dot{a}}{a}u^{1},\quad\frac{du^{2}}{dt}=0,\quad\frac{du^{3}}{dt}=\frac{2\dot{a}}{a}\Omega+\dot{\Omega}-\frac{2\dot{a}}{a}u^{3}.
\begin{array}{}{C_{2}C_{3}}&{=}&{6C_{3}}\\{{C_{3}}^{2}}&{=}&{\displaystyle-\frac{9}{4}C_{2}(C_{2}-3)(C_{2}-8)}\\{C_{2}W}&{=}&{3W\displaystyle-\frac{1}{2}C_{2}(C_{2}-3)(C_{2}-8)}\\{W^{2}}&{=}&{\displaystyle5W+\frac{2}{27}{C_{3}}^{2}\,\,.}\\\end{array}
S_{\mathrm{cosmo}}=\intd^{4}x\,\Lambda\epsilon^{\mu\nu\rho\sigma}(B_{\mu\nu}^{a}-\nabla_{\mu}G_{\nu}^{a}+\nabla_{\nu}G_{\mu}^{a})(B_{\rho\sigma}^{a}-\nabla_{\rho}G_{\sigma}^{a}+\nabla_{\sigma}G_{\rho}^{a})\.
\begin{array}{rl}{{\calL}=}&{\frac{1}{g^{2}}\int\left(-\frac{1}{4}F_{mn}F^{mn}+\frac{1}{2}D_{m}\Phi^{ab}D^{m}\Phi_{ab}+\frac{1}{4}\{\Phi^{ab},\Phi^{cd}\}_{*}\{\Phi_{ab},\Phi_{cd}\}_{*}\right.+}\\{}&{\left.+i\lambda_{a}\sigma^{m}D_{m}\bar{\lambda}^{a}+i\{\lambda_{a},\lambda_{b}\}_{*}\Phi^{ab}+i\{\bar{\lambda}^{a},\bar{\lambda}^{b}\}_{*}\Phi_{ab}\right)\,.}\\\end{array}
A^{\mu}={\frac{2ic}{s}}\;{\frac{(12)(13)(23)}{x_{12}^{4}x_{34}^{4}}}\;X_{2}^{\mu}+[2isX_{2}^{\mu}\partial_{4}^{--}\Pi+S_{44}^{\mu}\Pi]\cdotf_{s}+[-2itX_{3}^{\mu}\partial_{4}^{--}\Pi+T_{44}^{\mu}\Pi]\cdotf_{t}\;.
\left(\begin{array}{c}{h_{+}}\\{h_{-}}\\{h_{3}}\\{1}\\\end{array}\right)=\left(\begin{array}{cclc}{N^{-1/2}}&{}&{}&{}\\{}&{N^{-1/2}}&{}&{}\\{}&{}&{1}&{N/2}\\{}&{}&{}&{1}\\\end{array}\right)\left(\begin{array}{c}{N\Sigma_{+}^{(N)}}\\{N\Sigma_{-}^{(N)}}\\{N\Sigma_{3}^{(N)}/2}\\{{\bf1}^{(N)}}\\\end{array}\right),
\mathrm{tr}\,\left\{\left(U^{-1}\partial_{\nu}U\right)\left(U^{-1}\partial_{\rho}U\right)\left(U^{-1}\partial_{\sigma}U\right)\right\}=\frac{\partial\xi^{i}}{\partialx_{\nu}}\frac{\partial\xi^{j}}{\partialx_{\rho}}\frac{\partial\xi^{k}}{\partialx_{\sigma}}\mathrm{tr}\,\left\{\left(U^{-1}\partial_{i}U\right)\left(U^{-1}\partial_{j}U\right)\left(U^{-1}\partial_{k}U\right)\right\}\.
{\calL}=i\psi^{+}D_{\circ}\psi-\frac{1}{2}(D_{i}\psi)^{+}(D_{i}\psi)-k\epsilon_{\alpha\beta\gamma}tr(A^{\alpha}\partial^{\beta}A^{\gamma}+\frac{2}{3}A^{\alpha}A^{\beta}A^{\gamma})
f_{4,m}^{(8)}=m^{6}|m|^{1/2}\frac{1}{g_{\mathrm{YM}}^{28}}\left(\frac{g_{\mathrm{YM}}^{2}}{\phi}\right)^{13/2}K_{13/2}(|m|\phi/g_{\mathrm{YM}}^{2})e^{im\phi^{8}/g_{\mathrm{YM}}^{2}}~,
S_{0}=\displaystyle\intd\tau\left\{\pi_{{\bar{a}}\mu}\dot{x}_{\bar{a}}^{\mu}-\frac12\sum_{\bar{a}=0}^{D-1}\phi_{\bar{a}}(\pi_{{\bar{a}}\mu}\pi_{\bar{a}}^{\mu}+c_{\bar{a}}^{2})\right\},
h^{\beta}(a^{2})={\frac{\partialh}{\partiala^{2}}}={\frac{1}{2\pi{\hat{\beta}}^{2}}}\int_{0}^{\infty}{\frac{dt}{{\sqrt{t^{2}+a^{2}}(e^{\sqrt{t^{2}+a^{2}}}-1})}}\;.
S_{duality}~=~\left(\begin{matrix}{0}&{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{0}&{0}&{-1}\\{-1}&{0}&{0}&{0}&{0}&{0}\\{0}&{-1}&{0}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}&{0}\\\end{matrix}\right)
I=-R\int_{0}^{\rho}sds\int\frac{d^{4}p}{(2\pi)^{4}}\int_{0}^{\infty}tdt\e^{-t(p^{2}+s^{2})}=\frac{R}{2(4\pi)^{2}}\int_{0}^{\infty}dt\frac{1}{t^{2}}\left(e^{-t\rho^{2}}-1\right).
(\Omega_{12}-V_{c})\psi_{12}=\{[(m_{1}+\frac{1}{2}~V_{s})^{2}-\nabla^{2}]^{\frac{1}{2}}+[(m_{2}+\frac{1}{2}~V_{s})^{2}-\nabla^{2})]^{\frac{1}{2}}\}\psi_{12}
Z_{1/R}(1,0^{15})\to\int\frac{d^{2}\tau}{\tau_{2}}\tau_{2}^{-5}|\eta(\tau)|^{-16}\left(Z_{0}^{0}(\tau)^{16}{Z_{0}^{0}(\tau)^{*}}^{4}-Z_{1}^{0}(\tau)^{16}{Z_{1}^{0}(\tau)^{*}}^{4}-Z_{0}^{1}(\tau)^{16}{Z_{0}^{1}(\tau)^{*}}^{4}\right)\,
ig^{2}\epsilon^{mq}\epsilon^{pn}{\overline{{C}}}^{m}C^{n}A^{\mup}A_{\mu}^{q}\rightarrow\frac{1}{2}g^{2}\langlei{\overline{{C}}}^{m}C^{m}\rangle\langleA_{\mu}^{n}A^{\mun}\rangle=\frac{1}{2}g^{2}\left(-\frac{v}{16\pi}\right)\left(-8{\calG}\right)=\frac{vg^{2}}{20\pi}\varphi^{*}\varphi
{\cal{G}}_{\mu\nu\rho}\;\;:=\;\;\square\phi_{\mu\nu\rho}-3\partial^{\alpha}\partial_{(\mu}\phi_{\nu\rho)\alpha}+\partial_{(\mu}\partial_{\nu}{\phi}_{\rho)}^{\prime}+\frac{1}{\square}\left(2\partial^{\alpha}\partial^{\beta}\partial_{(\mu}\partial_{\nu}\phi_{\rho)\alpha\beta}-\partial_{\mu}\partial_{\nu}\partial_{\rho}\partial^{\alpha}\phi_{\alpha}^{\prime}\right)\;\;=\;\;0
u\circv=\sum_{s=0}^{\infty}{\frac{\kappa^{2s}}{2^{2s}(2s+1)!}}\omega^{i_{1}j_{1}}\ldots\omega^{i_{2s}j_{2s}}{\frac{\partial^{2s}u}{\partialx^{i_{1}}\ldots\partialx^{i_{2s}}}}{\frac{\partial^{2s}v}{\partialx^{j_{1}}\ldots\partialx^{j_{2s}}}}\,.
g^{\mu\nu}g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}\deltag_{\mu\nu}-\nabla^{\mu}\nabla^{\nu}\deltag_{\mu\nu}=\nabla_{\mu}(g^{\alpha\beta}\nabla^{\mu}\deltag_{\alpha\beta}-g^{\mu\alpha}\partial^{\beta}\deltag_{\alpha\beta})
Z_{\mathrm{F}}[A]=\int\!D{\bar{\psi}}D\psie^{-\int\!d^{2}x\,[\psi_{1}^{\dagger}i\partial\psi_{1}+\psi_{2}^{\dagger}i{\bar{\partial}}\psi_{2}-\psi_{1}^{\dagger}A\psi_{1}-\psi_{2}^{\dagger}{\bar{A}}\psi_{2}]}\;,
Q_{n}^{\mathcal{O}|{\mathfrak{M}}(l_{+},l_{-})}({\mathfrak{X}}_{+},{\mathfrakX}_{-})=\prod_{\mu_{i}\in\mathfrak{M}_{+}}Q_{2s_{i}+\tau_{i},2\bar{s}_{i}+\bar{\tau}_{i}}^{\nu_{i},\varsigma_{i}}({\mathfrak{X}}_{i},I_{ij}{\mathfrak{\hat{X}}}_{j}),
s_{gauge}=\int_{0}^{T}\sum_{ll^{\prime}}\left(\frac1S\frac{dS_{l}(t)}{dt}-(r_{b}-r_{l})\right)\frac{A_{ll^{\prime}}(t)}{2}\left(\frac1S\frac{dS_{l^{\prime}}(t)}{dt}-(r_{b}-r_{l^{\prime}})\right)dt\.
{L}^{(quarks)}=\sum_{i}\,[\intd^{4}\theta\,\{Q_{i}^{\dagger}e^{V}Q_{i}+{\tilde{Q}_{i}}^{\dagger}e^{{\tilde{V}}}{\tilde{Q}}_{i}\}+\intd^{2}\theta\,\{\sqrt{2}{\tilde{Q}}_{i}\PhiQ^{i}+m_{i}{\tilde{Q}}_{i}Q^{i}\}]
f=\left(1-\frac{\rho_{0}}{\rho}\right)^{\frac{1-k^{2}}{1+k^{2}}}\;,\;\;r^{2}=\rho^{2}\left(1-\frac{\rho_{0}}{\rho}\right)^{\frac{2k^{2}}{1+k^{2}}}\;,\;\;e^{\phi-\phi_{0}}=\left(1-\frac{\rho_{0}}{\rho}\right)^{\frac{2l}{1+k^{2}}}
L(\hat{A}_{\mu}^{tot}(\vec{X},t),\vec{X},\dot{\vec{X}},\vec{P},\dot{\vec{P}})=L_{DH}(A_{\mu}^{CS}(\vec{\eta},t)+A_{\mu}^{ext}(\vec{\eta},t)\,\vec{\eta},\dot{\vec{\eta}},\vec{\cal{P}},\dot{\vec{\cal{P}}})
\langle\delta^{2}k_{R}\rangle=\langlek_{R}^{2}\rangle-\langlek_{R}\rangle^{2}=\lambda^{2}\left(\frac{n+2}{Nn}\right)^{2}\left\{\left\langle\left(\sum_{\bfi}\Vert\varphi_{\bfi}\Vert^{2}\right)^{2}\right\rangle-\left(\sum_{\bfi}\langle\Vert\varphi_{\bfi}\Vert^{2}\rangle\right)^{2}\right\}
\begin{array}{lll}{A_{1}^{IJ}}&{=}&{\frac{8}{7}\,\theta\delta_{IJ}+\frac{1}{7}T_{IK\,JK}}\\{A_{2}^{I\dot{A}}}&{=}&{-\frac{1}{7}\Gamma_{A\dot{A}}^{J}T_{IJ\,A}}\\{A_{3}^{\dot{A}\dot{B}}}&{=}&{2\theta\delta_{\dot{A}\dot{B}}+\frac{1}{48}\Gamma_{\dot{A}\dot{B}}^{IJKL}T_{IJ\,KL}.}\\\end{array}
{\frac{1}{\sqrt{-g}}}{\frac{\delta}{\delta\bar{g}^{ab}}}\int\,d^{4}x\,\sqrt{-\bar{g}}\,\overline{F}_{4}\,\sigma=2\,\overline{C}_{a\b}^{\c\d}\overline{R}_{cd}\\sigma+4\,\overline{\nabla}_{c}\overline{\nabla}_{d}\left(\overline{C}_{(a\b)}^{\\c\d}\\sigma\right)\,.
S[q]=S[\bar{q}]+{\frac{1}{2!}}\delta^{2}S[\bar{q}]=S[\bar{q}]+{\frac{1}{2}}\int_{t^{\prime}}^{t^{\prime\prime}}\bigg(y{\frac{\partial}{\partialq}}+\dot{y}{\frac{\partial}{\partial\dot{q}}}\bigg)^{(2)}L(q,\dot{q},t)dt.
\frac{\partial\tilde{N}_{\pm}(x;\pm{\bfp})}{\partialx_{0}}\pm\frac{{\bfp}}{\omega_{\pm}}\cdot\frac{\partial\tilde{N}_{\pm}}{\partial{\bfx}}\simeq\frac{-i\Sigma_{off}^{(bare)}(x;\omega_{\pm},{\bfp})}{2\omega_{\pm}}.
L_{\psi^{\prime}\psi^{\prime}\psi\psi}=g^{2}\sum_{i,j=L,R}\sum_{q=u,d}\frac{\epsilon_{ij}}{(\Lambda_{ij}^{\epsilon})^{2}}\,(\bar{\psi}_{i}^{\prime}\gamma_{\mu}\psi_{i}^{\prime})\,(\bar{\psi}_{j}\gamma^{\mu}\psi_{j})
\tilde{\Pi}_{np}^{(1)}=\frac{4\pi}{\kappa}\frac{m}{p^{2}}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{k^{2}p^{2}-(k\cdotp)^{2}}{k^{2}(k^{2}+m^{2})(k+p)^{2}}e^{i\thetak\wedgep}.
\begin{array}{l}{\left.X_{1}^{+}\right.^{*}=-q^{2}X_{1}^{-}~,~~~~~\left.X_{1}^{-}\right.^{*}=-q^{-2}X_{1}^{+}~,~~~~~H_{1}^{*}=H_{1}~,}\\{\left.X_{2}^{+}\right.^{*}=q^{3/2}X_{2}^{-}~,~~~~~\left.X_{2}^{-}\right.^{*}=q^{-3/2}X_{2}^{+}~,~~~~~H_{2}^{*}=H_{2}~.}\\\end{array}
S_{E}=V_{(1)}V_{(2)}...V_{(T)}\left\{-\left[\frac{\partial\beta}{\partial\xi}\right]_{\xi_{S}}^{\xi_{N}}-\frac{1}{n_{(1)}+n_{(2)}+...+n_{(T)}-1}\int\mathrm{d}\xi\beta(\xi)\left(2{\calV}-\frac{(d_{f}-1)(M^{2}-E^{2})f(\phi)}{\prod_{i=lit}a_{i}^{2n_{i}}}\right)\right\}.
h_{\mu\nu}(t,{\bfx})=-8\pi{\calG}\frac{1}{(2\pir)^{(D-2)/2}}\partial_{t}^{(\frac{D-4}{2})}\left[\intd^{D-1}{\bfx^{\prime}}S_{\mu\nu}(t-|{\bfx-x^{\prime}}|,{\bfx^{\prime}})\right]\,,
\lambda_{A}^{i}=\epsilon^{AB}\,\mathcal{C}\,\left(\overline{{\lambda}}^{iB}\right)^{T}\,\quad;\quad\overline{{\lambda}}^{iB}=\left(\lambda_{B}^{i}\right)^{\dagger}\gamma_{0}\quad;\quad~~~A,B=1,\dots,2\,.
|0\rangle,\,\,A_{\vec{k}}^{\dagger}|0\rangle,\,\,A_{\vec{k}}^{\dagger}A_{\vec{k}^{\prime}}^{\dagger}|0\rangle\,\,\mathrm{for}\,\,\vec{k}\not=\vec{k}^{\prime},\,\,(A_{\vec{k}}^{\dagger})^{2}|0\rangle,\,\,\cdots\,\,,
\frac{\delta\rho}{\rho}=\frac{16\sqrt{6\pi}}{5}\\\frac{V^{3/2}}{{\frac{\partialV}{\partial\phi}}}~=~\frac{16\sqrt{6\pi}\left({\frac{M^{4}}{4\lambda}}+{\frac{m^{2}\phi^{2}}{2}}\right)^{3/2}}{5{m^{2}\phi}}\.
D_{\nu}K_{0}^{\nu+}-D_{0}K^{+}=\kappa_{5}^{2}\left[{\frac{\Gamma_{D}}{{4k}}(}\delta_{t}\phi)^{2}(0,t,0)+{\frac{\Gamma_{D}}{{4}k_{y}}(}\partial_{y}\phi)^{2}(0,t,0)\right],
\left[\frac{d}{d\xi}(\xi^{2}-1)\frac{d}{d\xi}+\frac{(|m|+\delta_{1})^{2}}{2(\xi+1)}-\frac{(|m|+\delta_{2})^{2}}{2(\xi-1)}-2\alphaR\xi+\frac{ER^{2}}{2}(\xi^{2}-1)\right]\psi_{1}=+\lambda(R)\psi_{1},
{\calQ}_{A}^{B}=\frac{1}{2}\,\delta_{A}^{B}\,{\calR}_{\mathrm{EGB}}^{2}-2\,R\,R_{A}^{B}+4\,R_{A\,C}\,R^{C\,B}+4\,R_{CD}\,R_{A}^{~~C\,B\,D}-\,R_{A\,C\,D\,E}\,R^{B\,C\,D\,E},
S_{F}=-\frac{1}{2(p+1)!}\intd^{d+1}X\sqrt{-\hat{g}}\left(\hat{g}^{M_{1}N_{1}}\cdots\hat{g}^{M_{p+1}N_{p+1}}\hat{F}_{M_{1}\cdotsM_{p+1}}\hat{F}_{N_{1}\cdotsN_{p+1}}\right)~.
\{\tilde{L}_{\alpha},\tilde{\theta}^{\beta}\}=k_{2}{p^{+}}_{\alpha}{}^{\beta}+O^{2}(\tilde{\theta},\tilde{L})+O^{1}(T){p^{-}}_{\alpha}{}^{\beta}+O^{1}(T){p^{+}}_{\alpha}{}^{\beta}.
{\calL}=-\frac{{pe^{2}}}{4}F_{\mu\nu}F^{\mu\nu}+\frac{{qe^{2}}}{2}\varepsilon_{\mu\nu\lambda}A^{\mu}F^{\nu\lambda}+\overline{\psi}i\gamma^{\mu}\left({\partial_{\mu}-ieA_{\mu}}\right)\psi\,-m\overline{\psi}\psi+\frac{g}{2}|\overline{\psi}\gamma^{\mu}\psi|^{2}+J_{0}A^{0}\,,
T_{\mu\nu}=g_{\mu\nu}T-F_{\mu\rho}\mathrm{}^{a}F_{\nu}\mathrm{}^{\rhoa}+D_{\mu}\phi^{a}D_{\nu}\phi^{a}-\frac{1}{2}\\phi^{2}\left(R_{\mu\nu}-\frac{1}{2}\g_{\mu\nu}R\right)
h_{+}^{+}=-h_{rr}+\frac{e^{A}}{2m^{2}}\left[3h_{rr}({A^{\prime}}^{2}-A^{\prime\prime})+3A^{\prime}h^{\prime}{rr}+2\phi(\Phi^{\prime\prime}-A^{\prime}\Phi^{\prime})-2\phi^{\prime}\Phi^{\prime}\right]~,
\widetilde{A}_{\mu}(x)=4\pi\int\delta\xids\,\epsilon_{\mu\nu\rho\sigma}{\Omega}_{\xi}(s,0)T^{i}{\Omega}_{\xi}^{-1}(s,0)L_{i}^{\rho\sigma}[\xi|s]\dot{\xi}^{\nu}(s)\delta^{4}(x-\xi(s)).
U_{i_{1}\,\,i_{2}}^{\,\,j_{1}\,\,j_{2}}V_{j_{1}\,\,j_{2}}^{\,\,p_{1}\,\,p_{2}}=V_{i_{1}\,\,i_{2}}^{\,\,j_{1}\,\,j_{2}}U_{j_{1}\,\,j_{2}}^{\,\,p_{1}\,\,p_{2}}
H^{(2)}=\intdx\left[\frac{1}{2}e^{4}(\Phi^{1})^{2}+e^{2}\Phi^{1}\Phi^{3}+e^{2}(\partial_{1}\Phi^{1})^{2}-e^{2}\Phi^{1}\partial_{1}\Phi^{2}-\Phi^{2}\Phi^{4}+\frac{1}{2}(\Phi^{3})^{2}\right].
S\left[\vec{h}_{\mu\nu}\right]=\intd^{4}x\Biggl(\frac{1}{4}\vec{h}_{\mu\nu}^{2}-\frac{i\thetag^{2}}{32\pi^{2}}\vec{h}_{\mu\nu}\tilde{\vec{h}}_{\mu\nu}\Biggr)+V\left[\vec{h}_{\mu\nu}\right].