harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(a_{\alpha},a_{\beta})=(b_{\alpha},b_{\beta})~~,~~(a_{\alpha},b_{\beta})=-(b_{\alpha},a_{\beta})=\delta_{\alpha\beta}~~\rightarrowJ=\left(\begin{matrix}{0_{2}}&{1_{2}}\\{-1_{2}}&{0_{2}}\\\end{matrix}\right)~.
\deltaI=\frac{\kappa}{l^{n}}\intd^{n+1}x\sqrt{g}\left[-H_{j}^{i}+H\delta_{j}^{i}+\frac{n}{l}\delta_{j}^{i}+\frac{l}{n-1}\left(-R_{j}^{i}+\frac12R\delta_{j}^{i}\right)+\cdots\right]h_{i}^{j}~,
={\frac{t_{1}t_{2}}{1-t_{1}t_{2}}}(\delta(\epsilon_{12}^{\prime},t_{1})-\delta(\epsilon_{12}^{\prime},{t_{2}^{-1}}))g={\frac{t_{1}t_{2}}{1-t_{1}t_{2}}}(\delta(\epsilon_{12}^{\prime},t_{1})+\tilde{\delta}(g_{0}\epsilon_{12}^{\prime}g_{0}^{-1},t_{2}))g,
\tilde{S}_{2}[\deltab_{i}]=\frac{1}{2\pi^{2}g_{o}^{2}\alpha^{\,\prime\,\frac{p+1}{2}}}\int\frac{d^{p+1}k}{(2\pi)^{p+1}}\;\frac{k_{\alpha}^{2}}{2}\deltab_{i}(-k)\deltab^{i}(k)
\tau^{-1}=\Gamma=\frac{1}{2E^{0}}\sum_{n\geq2}\int\prod_{i=1}^{n}\frac{d^{3}p^{i}}{(2\pi)^{3}2E^{i}}\,\delta^{4}\left(p^{0}-\sum_{i=1}^{n}p^{i}\right)|T_{n}|^{2}
{\calL}_{0}(x)=-i\overline{{\xi}}(x)(\partial\!\!\!/\partial\!\!\!/^{\dagger})^{N}\partial\!\!\!/\xi(x)+m\overline{{\xi}}(x)(\partial\!\!\!/\partial\!\!\!/)^{N}\xi(x),\,\,\,\,\,\,\,\,\,\,(N>0),
e^{-\mu}(\partial_{1}+iA_{1}+i\partial_{2}-A_{2}+{\frac{1}{2}}\partial_{1}\mu+{\frac{i}{2}}\partial_{2}\mu)\psi_{1}+e^{-\nu}(-\partial_{3}-iA_{3}+i\partial_{4}-A_{4}-{\frac{1}{2}}\partial_{3}\nu+{\frac{i}{2}}\partial_{4}\nu)\psi_{2}=0.
\begin{array}{rcl}{\delta_{\epsilon}A^{a}}&{=}&{\{A^{a},Q_{\epsilon}\}=gf^{abc}\epsilon^{b}A^{c}\\\,}\\{\delta_{\epsilon}\pi^{a}}&{=}&{\{\pi^{a},Q_{\epsilon}\}=gf^{abc}\epsilon^{b}\pi^{c}\\\,}\\{\delta_{\epsilon}\phi^{a}}&{=}&{-\dot{\epsilon}^{a}+gf^{abc}\epsilon^{b}\phi^{c}\\\,}\\\end{array}
B_{b}^{ai}={(\frac{\partial^{2}A}{\partial\phi^{i}\partial\phi^{j}})}^{-1}(\frac{1}{4}\lambda_{bj}^{a}-\frac{1}{2}\frac{\partial^{3}A}{\partial\phi^{j}\partial\phi^{k}\partial\phi^{p}}(\bar{\psi}_{a}^{k}\psi^{bp}+\bar{\psi}^{bk}\psi_{a}^{p})),
S_{\mathrm{free}}\!=\!\frac{1}{2}\int\!\!d^{D}\!x[(\partial_{x}\Phi)^{2}\!-\!m^{2}\Phi^{2}]=\sum_{j,k}\!\frac{1}{2}\int\!\!d^{D-2}\!{\bfx}[(\partial_{\bfx}\phi_{jk}^{*})(\partial_{\bfx}\phi_{jk})\!-\!m_{j}^{2}\phi_{jk}^{*}\phi_{jk}],
B=\left(\begin{array}{ccccccccr}{0}&{x}&{z}&{y}&{0}&{-y}\\{-x}&{0}&{x}&{z}&{y}&{0}\\{-z}&{-x}&{0}&{x}&{z}&{y}\\{-y}&{-z}&{-x}&{0}&{x}&{z}\\{0}&{-y}&{-z}&{-x}&{0}&{x}\\{y}&{0}&{-y}&{-z}&{-x}&{0}\\\end{array}\right)\;,
H^{1}({\mathbbP}^{1}\timesT^{2},{\calO}_{{\mathbbP}^{1}\timesT^{2}}(3\sigma|_{{\mathbbP}^{1}\timesT^{2}}-5F))|_{b_{-5}^{(1)}}=H^{0}({\calS},{\calO}_{{\calS}}(3))^{*}
\Lambda=\sum\nu^{3}\left\{e^{2\nu[\eta(z)-\eta(za/b)]}-1\right\}^{-1}\approx\frac{b^{4}}{16h^{4}(1+z^{2})^{2}}\int_{0}^{\infty}{\frac{s^{3}ds}{e^{s}-1}}=\frac{\pi^{4}b^{4}}{240h^{4}(1+z^{2})^{2}}.
F\partial_{\beta}\partial^{\beta}\vartheta+2\partial_{\beta}F\partial^{\beta}\vartheta-2F^{\prime}\vartheta^{\prime}-F\left(\vartheta^{\prime\prime}+\frac{1}{r}\vartheta^{\prime}\right)+2A^{\beta}\partial_{\beta}F+\left(\partial_{\beta}A^{\beta}\right)F=0,
\frac{f_{\lambdam}^{(q)}(z,zx)}{\bar{J}_{m}^{2}+\bar{Y}_{m}^{2}}-f_{m}^{(q)}(zx)=-\frac{1}{2}\sum_{n=1,2}\Omega_{\lambdam}^{(n)}(z)f_{m}^{(nq)}(zx),
V^{g^{4}}(r)=-\frac{3g^{4}C_{F}C_{A}}{(4\pi)^{3}}\int\!d^{3}z\int\!d^{3}w\frac1{\mod{\boldsymbolz-\boldsymbolw}}\left(\partial_{j}^{z}\frac1{\mod{\boldsymbolz-\boldsymboly}}\right)\left(\partial_{k}^{w}\frac1{\mod{\boldsymbolw-\boldsymbol{y}^{\prime}}}\right)\langleA_{k}^{T}(w)A_{j}^{T}(z)\rangle\,.
H={\frac{\alpha^{\prime}}{2}}\left[-{\frac{1}{\rho}}{\frac{\partial}{\partial\rho}}\left(\rho{\frac{\partial}{\partial\rho}}\right)-{\frac{1}{\rho^{2}}}{\frac{\partial^{2}}{\partial\phi^{2}}}-{\frac{\partial^{2}}{\partialy^{2}}}+{\frac{1}{R^{2}}}\left(k+\mathrm{i}\gamma{\frac{\partial}{\partial\phi}}\right)^{2}\right]\,.
{\cal{L}}_{YM}=\frac{1}{4}{F_{\mu\nu}}^{a}{F_{a}}^{\mu\nu}-\frac{1}{2}(\partial_{\mu}{A_{\nu}}^{c}-\partial_{\nu}{A_{\mu}}^{c}+{\varphi_{ab}}^{c}{A_{\mu}}^{a}{A_{\nu}}^{b}){F_{c}}^{\mu\nu}~,
H_{G}=\frac{z_{A}(z_{A}-1)(z_{A}-z_{3})}{z_{3}(z_{3}-1)}\left\{b_{A}^{2}-(\frac{\mu_{1}}{z_{A}}+\frac{\mu_{2}}{z_{A}-1}+\frac{\mu_{3}-1}{z_{A}-z_{3}})b_{A}+\frac{\kappa}{z_{A}(z_{A}-1)}\right\}
A(l,\omega^{\prime})\approx\frac{1}{2}\frac{\eta}{{\calR}^{4}}\left\{-\pi^{2}(l+\omega^{\prime})^{2}{\overline{\Lambda}}^{2}+\frac{1}{2}{\overline{\Lambda}}^{4}-\frac{1}{3}\frac{{\overline{\Lambda}}^{6}}{\pi^{2}(l+\omega^{\prime})^{2}}+\cdots\right\}
{\calI}=-\frac{1}{(2\pi)}\sum_{\begin{array}{c}{k>0}\\{l\inZ}\\\end{array}}\sum_{p>0}\delta_{n,kl}(\frac{2kl}{p}+\frac{l}{\piT_{2}p^{2}}+\frac{k}{\piU_{2}p^{2}}+\frac{1}{2\pi^{2}T_{2}U_{2}p^{3}})x^{p}+h.c,
\lambda_{1}^{2}=\dot{\lambda}+\lambda\frac{q_{2}q_{3}}{q_{2}^{2}},~~~~\lambda_{1}^{1}=\ddot{\lambda}+\frac{\lambda}{q_{2}^{2}}\left[q_{2}{\dot{q}}_{3}-\frac{q_{2}q_{3}}{q_{2}^{2}}(2q_{2}{\dot{q}}_{2}+q_{2}q_{3})\right].
L_{MS}={\frac{1}{2}}\intd^{3}r\mathrm{Tr}\left[{\dot{A}_{i}}^{2}+{\dot{\Phi}}^{2}+B_{i}^{2}+D_{i}\Phi^{2}\right]={\frac{1}{2}}g_{ij}(z)\dot{z}_{i}\dot{z}_{j}+M
\left(V_{\zeta_{1}}^{cl}\otimes\mho_{j_{1}}\right)\otimes\left(V_{\zeta_{2}}^{cl}\otimes\mho_{j_{2}}\right)=\left(\bigoplus_{\zeta_{1}+\zeta_{2}\le\zeta_{3}}V_{\zeta_{3}}^{cl}\right)\otimes\left(\bigoplus_{j_{3}=\vertj_{1}-j_{2}\vert}^{\mathrm{min}\,\{j_{1}+j_{2},N-2-J_{1}-j_{2}\}}\mho_{j_{3}}\right).
\begin{array}{ll}{\hat{q}^{\prime}}&{=\frac{2}{(k^{+}+1)}\ell_{3}^{+}}\\{}&{}\\{\hat{h}^{\prime}}&{=\frac{1}{(k^{+}+1)}\{\ell^{+}(\ell^{+}+1)-(\ell_{3}^{+})^{2}\}}\\\end{array}
e^{2\Phi^{\prime}}={\frac{(r+P_{1})(r+P_{2})}{\bigg[(r+Q_{1})(r+Q_{2})(r+P_{1})(r+P_{2})-q^{2}[r+{\frac{1}{2}}(P_{1}+P_{2})]^{2}\bigg]^{\frac{1}{2}}}}\,
<\widetilde{h}_{\mu\rho}(p)\widetilde{h}_{\nu\sigma}(-p)>=i\left[\frac{2P_{\mu\rho,\nu\sigma}^{(2)}}{p^{2}+~\frac{\kappa^{2}}{512}(-p^{2})^{\frac{3}{2}}}+~\frac{2P_{\mu\rho,\nu\sigma}^{(0-\mathrm{s})}}{-p^{2}+~\frac{\kappa^{2}}{256}(-p^{2})^{\frac{3}{2}}}\right].
S_{bulk}=\int\,d\rho\,d^{d}\,x\,\rho^{-{\frac{d}{2}}-1}\sqrt{g}\,B(x,\rho)\equiv\int\,d\rho\,d^{d}\,x\,\rho^{-{\frac{d}{2}}-1}\sqrt{g^{(0)}}\,b(x,\rho)\,,
={\frac{1}{4\pi}}\int_{-\infty}^{\infty}d\omega~\varepsilon(\omega)\int_{C}dz~e^{-\omega^{2}(t-iz)}\int_{\mu}^{\infty}d\lambdae^{-iz\lambda}\left(2\omega\varphi(\lambda,\omega)+{\frac{1}{iz}}\partial_{\omega}\partial_{\lambda}N(\lambda,\omega)\right).
{\calL}_{\mathrm{CT}}(J)=\frac{\deltaZ}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{\deltaZ_{2}J}{2}\phi^{2}+\frac{\deltaZ_{4}\lambda\mu^{2\varepsilon}}{4!}\phi^{4}-\mu^{-2\varepsilon}\frac{\delta\zeta}{2}\;J^{2}.
f_{e}^{(2)}=\frac{g_{1}^{2}g_{2}^{2}}{(8\pi^{2})^{2}}\int_{D_{e}}\frac{d\alpha_{1}d\beta_{1}d\alpha_{2}d\beta_{2}}{\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}}\Theta(\alpha_{1}\beta_{1}-\lambda_{1}^{2})~\Theta(\alpha_{2}\beta_{2}-\lambda_{2}^{2}),
\left.\begin{array}{ll}{\mathrm{nilpotence}}&{Q_{B}^{2}=0,}\\{\mathrm{odd~derivation}}&{Q_{B}(A*B)=(Q_{B}A)*B+(-1)^{A}A*(Q_{B}B),}\\{\mathrm{partial~integrability}}&{\intQ_{B}(\ldots)=0,}\\{}&{\intA*B=(-1)^{AB}\intB*A,}\\\end{array}\right.
m^{2}=\sum_{m_{1},m_{2}\atopn^{1},n^{2}}\frac{2}{Y}|-TU^{\prime}n^{2}+iTn^{1}-iU^{\prime}m_{1}+3m_{2}|_{U^{\prime}=U-2i}^{2}={|{\calM}^{pert}|^{2}}/{(Y/2)},
\left[-i\left(\rho_{1}\partial_{0}+i\rho_{2}\partial_{3}\right)-m_{n}^{(1)}{\frac{1+\rho_{3}}{2}}-m_{n}^{(2)}{\frac{1-\rho_{3}}{2}}\right]\left[\begin{matrix}{c_{n}(x^{0},x^{3})}\\{ib_{n}(x^{0},x^{3})}\\\end{matrix}\right]=0,
\sqrt{-h}e^{-2{\bar{\varphi}}}h^{\bar{\mu}\bar{\mu}^{\prime}}h^{\bar{\nu}\bar{\nu}^{\prime}}(\bar{M}\bar{L})_{\bar{i}\bar{j}}\bar{{\calF}}_{\bar{\mu}^{\prime}\bar{\nu}^{\prime}}^{\bar{j}}={\textstyle{\frac{1}{2}}}\epsilon^{\bar{\mu}\bar{\nu}\bar{\rho}}\partial_{\bar{\rho}}\psi^{\bar{i}}.
\begin{array}{l}{\lambda=(\frac{2A^{3}}{27}-\frac{B}{3}+C)^{\frac{1}{3}}-\frac{A}{3},}\\{\lambda_{3}=-2(\frac{2A^{3}}{27}-\frac{B}{3}+C)^{\frac{1}{3}}-\frac{A}{3}.}\\\end{array}
\sum_{m_{1}}R_{U}(J,J^{\prime};\varpi)_{m_{\phantom{2}}m^{\prime}}^{m_{2}m_{1}}\prod_{t=1}^{J^{\prime}+m_{2}}(y^{\prime}q^{2J}+yq^{\varpi+4J-2J^{\prime}+2t-1})\prod_{s=1}^{J+m_{1}}(y^{\prime}+yq^{-(\varpi-2J+2m_{2}+2s-1)})
K^{x}\,=\,\frac{1}{2}tr(E^{t}\wedgeJ^{x}E)\quad;\quad\omega^{x}\,=\,-\frac{1}{2}tr(\thetaJ^{x})\quad;\quad{\calU}^{A\alpha}\,=\,\frac{1}{\sqrt{2}}E^{at}(e_{a})_{B}^{A}
f(\hat{x})=\displaystyle\oint_{\Gamma_{\hat{x}}}\displaystyle\frac{d\mu}{2\pii}\displaystyle\frac{f(\mu)}{\mu-\hat{x}}=\displaystyle\oint_{\Gamma_{\hat{x}}}\displaystyle\frac{d\mu}{2\pii}f(\mu)\displaystyle\frac{(-1)^{\delta}e^{iZ_{N}(\mu)}iZ_{N}^{\prime}(\mu)}{1+(-1)^{\delta}e^{iZ_{N}(\mu)}}
\tilde{D}_{2}=\frac{\kappa_{5}^{2}l^{2}}{B_{2}(w_{+})-B_{2}(w_{-})}\left[e^{-2A_{+}-\alpha_{+}^{(0)}}\bar{\tau}_{+(T)}^{[1]}+e^{-2A_{--}\alpha_{-}^{(0)}}\bar{\tau}_{-(T)}^{[1]}\right]-\frac{B_{1}(w_{+})-B_{1}(w_{-})}{B_{2}(w_{+})-B_{2}(w_{-})}\tilde{D}_{1}.
\begin{array}{rcl}{\left(\begin{array}{c}{Z}\\{A}\\\end{array}\right)}&{=}&{\left(\begin{array}{lr}{c}&{-s}\\{s}&{c}\\\end{array}\right)\left(\begin{array}{c}{G_{1}}\\{G_{2}}\\\end{array}\right),}\\{g}&{=}&{\sqrt{g_{1}^{2}+g_{2}^{2}},~c=g_{1}/g,~s=g_{2}/g.}\\\end{array}
{\calK}_{i}=\sum_{a}\sum_{s=0}^{\infty}D^{s}[-\varphi_{a}\delta_{i}^{j}+v_{a}^{j}(x_{ai}-\varphi_{a}\varphi_{ai})\varphi_{at}^{-1}]\frac{\partial}{\partialx_{a}^{j(s)}}.
{\calE}_{\nu}(\eta,\eta^{\prime})=(rr^{\prime})^{-\frac{d-1}{2}-i\nu}{W}_{\nu}^{\infty}({\omega},{\omega}^{\prime})=\frac{\Gamma\left(\frac{d-1}{2}+i\nu\right)\Gamma(-i\nu)}{4\pi^{\frac{d+1}{2}}}\\\left[-(\eta-\eta^{\prime})^{2}\right]^{-i\nu-\frac{d-1}{2}}
{\frac{d^{2}u}{dl^{2}}}+\left[{\frac{d(A^{-1/2})}{dr}}+{\frac{8\piGrK}{\sqrt{A}}}\right]{\frac{du}{dl}}={\frac{u(u^{2}-1)}{r^{2}}}+e^{2}v^{2}uh^{2}\.
Q=4N_{00}^{11}\left(\begin{array}{cc}{1}&{1}\\{1}&{1}\\\end{array}\right)+\left(\begin{array}{cc}{2T_{2}+2T_{3}}&{T_{2}}\\{T_{2}}&{2T_{1}+2T_{2}}\\\end{array}\right)+D^{T}\frac{1}{1-S\tilde{N}}SD
S_{\mathrm{bulk}}=\intd^{4}x\sqrt{-g}\left[\frac{1}{2\kappa_{4}^{2}}\;^{4}{\calR}(g)\right]-\frac{1}{4.5!\kappa_{10}^{2}}\intd^{4}x\sqrt{-g}\left(\int_{\calM}drd\Omega_{5}\sqrt{-\bar{G}}F^{2}\right)
\textrm{E}=4\pi\int_{0}^{\infty}drr^{2}[{(\omega+eA_{0})}^{2}{\sigma}^{2}e^{-\nu}+U+e^{-\lambda}{\sigma^{\prime}}^{2}+(1/2){A_{0}^{\prime}}^{2}e^{-\nu-\lambda}].
{\calL}=-\frac{3}{8}\left\{\phantom{|}_{abcd}^{\mu\nu\alpha\beta}\right\}\Omega_{\mu\nu}{}^{abe}\Omega_{\alpha\beta}{}^{cde}+\frac{1}{12}\left\{\phantom{|}_{abcde}^{\mu\nu\alpha\beta\gamma}\right\}\Omega_{\mu\nu}{}^{abc}T_{\alpha\beta\gamma}{}^{de}
\frac{R(1+{\dot{R}}^{2})}{\sqrt{(R^{2}+D^{2})(1+{\dot{R}}^{2})}}-\frac{\partial}{\partial\tau}\frac{{\dot{R}}(R^{2}+D^{2})}{\sqrt{(R^{2}+D^{2})(1+{\dot{R}}^{2})}}-hR=0.
\begin{array}{l}{\chi_{i}^{NS}(q)=\sum(S_{NS}^{NS})_{i}^{j}\chi_{j}^{NS}(\tilde{q}),}\\{\chi_{i}^{(-)NS}(q)=\sum(S_{R}^{NS})_{i}^{j}\chi_{j}^{R}(\tilde{q}),}\\{\chi_{i}^{R}(q)=\sum(S_{NS}^{R})_{i}^{j}\chi_{j}^{(-)NS}(\tilde{q})}\\\end{array}
g^{(2)}(y)=\frac{\hbarc\xi^{2}}{4\pi}\sum_{n=-\infty}^{\infty}\,y^{2}\partial_{y}\left[y{{\partial}_{y}}(I_{n}(ay)K_{n}(ay))\right]^{2}=\frac{2\hbarc\xi^{2}}{\pi^{2}}\int_{0}^{1}du\,\frac{u^{2}}{\sqrt{1-u^{2}}}y^{2}{{\partial}_{y}}(ayK_{1}(2ayu))^{2},
\left\langle{^0t},{(l^{\alpha\beta})^{\rho}}_{\mu}x^{\mu}\partial_{\rho}wx^{\alpha_{1}}x^{\alpha_{2}}\right\rangle={(l^{\alpha\beta})^{\alpha_{1}}}_{\sigma}c^{\sigma\alpha_{2}}+{(l^{\alpha\beta})^{\alpha_{2}}}_{\sigma}c^{\sigma\alpha_{1}}.
{\calL}_{g+m}=\frac{1}{4\piG}\left(\eta_{a}\dot{e}_{1}^{a}+\eta_{2}\dot{\omega}_{1}+\eta_{3}\dot{a}_{1}\right)+p_{a}\dot{q}^{a}+\Pi\dot{\varphi}+e_{0}^{a}G_{a}+\omega_{0}G_{2}+a_{0}G_{3}-u{\calE}-v{\calP}\,,
\begin{aligned}{|F|^{2}=}&{{}\;m^{2}|\phi_{1}|^{2}+m^{2}|\phi_{2}|^{2}+mg|\phi_{1}|^{2}\left(\phi_{1}+\phi_{1}^{*}\right)}\\{}&{{}\qquad+2mg|\phi_{2}|^{2}\left(\phi_{1}+\phi_{1}^{*}\right)+4g^{2}|\phi_{1}|^{2}|\phi_{2}|^{2}+g^{2}\left(|\phi_{1}|^{2}\right)^{2}}\\\end{aligned}
V(x_{o})\,V(\tilde{x_{o}})=c^{2}\,e^{-i\,Q\,[\phi(x_{o})-\phi(\tilde{x_{o}})]}\,(-8\pi^{2}\,r_{o}^{4})\bar{\psi}(x_{o})\psi(x_{o})\,\bar{\psi}(\tilde{x}_{o})\psi(\tilde{x}_{o})
S=\intd\tau\left[\hat{\lambda}\sigma_{i}\hat{\bar{\lambda}}\partial_{\tau}x^{i}+\hat{\lambda}\sigma_{3}\hat{\bar{\lambda}}\partial_{\tau}\hat{x}^{3}\right]=\intd\tau\hat{\lambda}\sigma_{m}\hat{\bar{\lambda}}\partial_{\tau}{\hat{x}}^{m},
{\calA}=\left(\begin{matrix}{\omega-{\frac{e^{+-}}{\sqrt{3}}}}&{(\omega^{-}+e^{-})}&{\sqrt{2}e^{--}}\\{(\omega^{+}-e^{+})}&{{\frac{2e^{+-}}{\sqrt{3}}}}&{(-\omega^{-}+e^{-})}\\{\sqrt{2}e^{++}}&{(-\omega^{+}-e^{+})}&{-\omega-{\frac{e^{+-}}{\sqrt{3}}}}\\\end{matrix}\right)
\omega=s\left(\frac23\left(f_{abc}A_{\mu}^{a}A_{\nu}^{b}A_{\rho}^{c}+A_{\mu}^{a}F_{a\nu\rho}+A_{\rho}^{a}F_{a\mu\nu}+A_{\nu}^{a}F_{a\rho\mu}\right)^{2}\right),
L=-\frac{1}{4g^{2}}(F_{\mu\nu}^{3})^{2}-\frac{1}{2}(\partial_{\mu}\phi^{3})^{2}-\frac{\lambda}{2}((\phi^{3})^{2}-\eta^{2})^{2}-\frac{\theta}{\eta}\epsilon_{\mu\nu\lambda}\partial_{\mu}\phi^{3}F_{\nu\lambda}^{3}
m_{0}^{2}\approx{\frac{2\kappa}{3}}\left.\left[\int_{y^{(+)}}^{y^{(-)}}{\frac{dy}{a^{2}}}\right]\right/\left[\int_{y^{(+)}}^{y^{(-)}}{\frac{dy}{a^{4}\dot{\varphi}_{0}^{2}}}\right].
\delta_{\sigma}W_{fin}=-{\frac{1}{16\piG}}{\frac{2d}{l}}\intd^{d}x\sqrt{g_{(0)}}\sigma(x)\left[\rho_{cr}^{-d/2}(x)\left(\sum_{k\neq{\frac{d}{2}}}a_{k}(x)l^{2k}\rho_{cr}^{k}(x)\right)~~+\nabla_{i}J^{i}\right],
R^{\prime\prime}+\frac{{w^{\prime}}^{2}-4{R^{\prime}}^{2}}{R}+\frac{4(R^{2}+1)}{R}-4R{\Phi^{\prime}}^{2}+10R^{\prime}\Phi^{\prime}-\frac{(w^{2}-1)^{2}}{4R^{3}}=0\,,
\times\prod_{a<b}^{N}\left[\int_{\Lambda}\left[\prod_{i=1}^{n_{a}}d^{2}x_{i}^{(a)}d^{2}y_{i}^{(a)}\prod_{j=1}^{n_{b}}d^{2}x_{j}^{(b)}d^{2}y_{j}^{(b)}\right]\Big|B(a,b)\Big|^{\frac{g}{\pi+gN}q(N-1)}\right]^{\frac{1}{q(N-1)}}\;.
f_{R}(q,\lambda)=\sum_{g\ge0}\sum_{Q,R^{\prime},R^{\prime\prime}}C_{RR^{\prime}R^{\prime\prime}}N_{R^{\prime},g,Q}S_{R^{\prime\prime}}(q)(q^{{\frac{1}{2}}}-q^{-{\frac{1}{2}}})^{2g-1}\lambda^{Q},
\begin{array}{l}{A_{1}^{(n)}=A^{(1)}\otimesI\otimes\cdots\otimesI,}\\{\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots}\\{A_{n}^{(n)}=I\otimesI\otimes\cdots\otimesA^{(1)}.}\\\end{array}
{n_{i}}^{\mu}(\Xi)\Pi_{\mu}=\bar{\Pi}_{i}+{\frac{{l_{B}}^{2}}{8}}\varepsilon^{mn}{\rho_{i}}^{k},_{m}{\rho_{j}}^{l},_{n}\varepsilon^{jh}(\bar{\Pi}_{k}\bar{\Pi}_{h}\bar{\Pi}_{l}+\bar{\Pi}_{l}\bar{\Pi}_{h}\bar{\Pi}_{k})+{\calO}({l_{B}}^{4})
\begin{array}{rcl}{\left[L_{n},L_{m}\right]}&{=}&{(n-m)L_{n+m}+\frac{1}{12}D(n^{3}-n)\delta_{n+m,0}\,}\\{}&{}&{}\\{\left[\bar{L}_{n},\bar{L}_{m}\right]}&{=}&{(n-m)\bar{L}_{n+m}+\frac{1}{12}D(n^{3}-n)\delta_{n+m,0}\.}\\\end{array}
\left(\begin{array}{c}{A_{0}}\\{A_{1}}\\\end{array}\right)\rightarrow\,\left(\begin{array}{cc}{cosh\,\varphi}&{sinh\,\varphi}\\{sinh\,\varphi}&{cosh\,\varphi}\\\end{array}\right)\,\left(\begin{array}{c}{\tilde{A_{0}}}\\{\tilde{A_{1}}}\\\end{array}\right)\;\;,
\bar{M}-\bar{\cal{E}}_{(m)}=-\int_{\Sigma_{t}}\left(\bar{\theta}_{(g)}^{\mu}(t)-t^{\mu}\bar{L}_{(g)}\right)d\Sigma_{\mu}+{\frac{\bar{\kappa}}{2\pi}}\bar{S}_{(g)}+\int_{C_{\infty}}\sqrt{\gamma}d^{2}y\bar{B}\bar{N}~~~.
F[\psi,\varphi]=\int_{-\infty}^{+\infty}dx~\psi^{i}\;\left(\sqrt{1-{\bf\varphi}^{2}}\frac\partial{\partialx}\varphi^{i}-\varphi^{i}\frac\partial{\partialx}\sqrt{1-{\bf\varphi}^{2}}+\varepsilon^{ijk}\varphi^{j}\frac\partial{\partialx}\varphi^{k}\right).
W_{m(2)}=-\mathrm{~\frac{1}{2}~}E_{\gammaab}^{*}f_{\alpha\beta}^{\gamma}C^{\alphaa}C^{\betab}+\bar{E}_{\gammaa}f_{\alpha\beta}^{\gamma}B^{\alpha}C^{\betaa}+\mathrm{~\frac{1}{2}~}\epsilon_{ab}\bar{Q}_{i}R_{\alpha,j}^{i}R_{\beta}^{j}(A+Q)C^{\betab}C^{\alphaa}.
\eta_{1}=\zeta(\omega_{1})=i\pi\left({\frac{1}{12}}-2\sum_{n=1}^{\infty}{\frac{nq^{2n}}{1-q^{2n}}}\right){{}\atop{{\longrightarrow}\atop{\omega_{3}\to+\infty}}}i{\frac{\pi}{12}}.
I={\frac{igf_{acb}}{(2\pi)^{6}}}\intdu_{4}d^{4}x_{4_{A}}d^{2}\theta_{4}^{+}d^{2}\bar{\theta}_{4}^{+}\{\frac{(14)}{\hat{x}_{14}^{2}}}\;{\frac{(42)}{\hat{x}_{42}^{2}}}\;{\frac{(\theta_{34})^{2}}{\tilde{x}_{34}^{2}}}\;.
R=\left(\begin{array}{cccccccc}{a}&{0}&{0}&{0}&{0}&{0}&{0}&{p}\\{0}&{a}&{0}&{0}&{0}&{0}&{q}&{0}\\{0}&{0}&{\pmb}&{0}&{0}&{q}&{0}&{0}\\{0}&{0}&{0}&{a}&{r}&{0}&{0}&{p}\\{0}&{0}&{0}&{q}&{b}&{0}&{0}&{0}\\{0}&{0}&{r}&{0}&{0}&{\pma}&{0}&{0}\\{0}&{r}&{0}&{0}&{0}&{0}&{a}&{0}\\{qrp^{-1}}&{0}&{0}&{0}&{0}&{0}&{0}&{b}\\\end{array}\right),
\psi(x,\tau)=\int\frac{d^{4}k}{(2\pi)^{4}}b(k)e^{i(k\cdotx-\kappa\tau)}\qquad\psi^{*}(x,\tau)=\int\frac{d^{4}k}{(2\pi)^{4}}b^{*}(k)e^{-i(k\cdotx-\kappa\tau)}
{\calL}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{\kappa}{4}\epsilon^{\mu\nu\rho}A_{\mu}F_{\nu\rho}+i\phi^{*}D_{t}\phi-\frac{1}{2m}|D_{i}\phi|^{2}+\frac{g}{2}(\phi^{*}\phi)^{2},
S_{Tr.}\,=\,S_{0}+\phi_{A}^{\ast\,a}\,{\overline{\delta}}_{a}\phi^{A}+{\frac{1}{2}}{\bar{\phi}}_{A}{\overline{\delta}}_{2}{\overline{\delta}}_{1}\phi^{A}\,+\,{\frac{1}{2}}\epsilon^{ab}\phi_{A}^{\ast\,a}\,\pi_{b}^{A}
J_{\mathrm{asym}}=-\int_{R_{*}^{2}}d^{2}x\;\epsilon_{ij}x^{i}\,\biggl\{\dot{f}\partial_{j}f-\kappa\bar{A}_{j}\epsilon_{kl}\partial_{k}\bar{A}_{l}\biggr\}-n\pi\rho_{e}|\vec{q}|^{2}.
G^{ij}=H_{L}^{ik}(-l^{k\underline{{m}}}r_{\underline{{m}}}^{l}+{\frac{2(r^{(++)\underline{{n}}}l_{\underline{{n}}}^{k})(l^{(--)\underline{{n^{\prime}}}}r_{\underline{{n^{\prime}}}}^{l})}{(l^{(--)\underline{{n^{\prime\prime}}}}r_{\underline{{n^{\prime\prime}}}}^{(++)})}})H_{R}^{jl}~.
{(\gamma^{0})^{a}}_{b}\,=\,\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right)\;,\;{(\gamma^{1})^{a}}_{b}\,=\,\left(\begin{array}{cc}{0}&{-i}\\{-i}&{0}\\\end{array}\right)\;,\;{(\gamma^{2})^{a}}_{b}\,=\,\left(\begin{array}{cc}{-i}&{0}\\{0}&{i}\\\end{array}\right)\,,
(-\frac{d^{2}}{dt^{2}}-q^{2})g_{ab}\bar{\phi}_{b}({\vec{q}},t)+\intdt^{\prime}\frac{\partial^{2}S_{int}}{\partial\varphi_{a}(-{\vec{q}},t)\partial\varphi_{b}({\vec{q}},t^{\prime})}\bar{\phi}_{b}({\vec{q}},t^{\prime})=0
\Psi_{2n}-\Phi_{2n}=-T_{(2n)}(p)(\vec{p}\cdot\xi)\vec{\xi}\cdot\left[\frac{\partial\vec{A}(x)}{\partialx^{0}}+\vec{\partial}A_{0}(x)\right]=+T_{(2n)}(p)(\vec{p}\cdot\xi)\vec{\xi}\cdot\vec{E}(x).
{\calS}\supset\intd^{5}x\intd^{4}\theta\left(c_{1}{\frac{\Sigma^{\dagger}\SigmaBB^{\dagger}}{M_{Pl}^{2}}}\delta(x_{5}-r_{c})+c_{2}{\frac{QQ^{\dagger}BB^{\dagger}}{M_{Pl}^{2}}}\delta(x_{5})\right)
q(r)=\frac{1}{(2\pi)^{N+1}a^{3}}\sum_{{\bf\alpha}}\frac{\lambda_{n,l}^{3}T_{n}(\lambda_{n,l})}{\sqrt{\lambda_{n,l}^{2}+k^{2}a^{2}+m^{2}a^{2}}}f_{n}^{(q)}[J_{n}(\lambda_{n,l}r/a)],\quadq=\varepsilon,p_{i},
{\calN}_{\{l_{i}\},\{\bar{l}_{i}\}}^{\{k_{i}\}}=\prod_{i\inS_{2}}\delta_{l_{i},\bar{l}_{i}}\prod_{j\inS_{1}}(\delta_{l_{j},\bar{l}_{j}}+\delta_{l_{j},k_{j}-\bar{l}_{j}})
S=-\tau_{0}\intd\xi\left\{(1-v^{2})^{1/2}(1-\phi)-{\frac{1}{2}}(1-v^{2})^{-1/2}(h_{00}^{IIA}+2h_{0i}v^{i}+h_{ij}v^{i}v^{j})-C_{0}-C_{i}v^{i}\right\}
\kappa_{5}^{2}\langleT^{\mathrm{GW}}{}_{5}^{5}\rangle=-\frac{3}{2a^{2}}\left(\dot{H}_{T}^{2}-k^{2}H_{T}^{2}\right)+G_{5}^{5}H_{T}^{2}+\frac{2}{a^{2}}(\dot{A}+\dot{B})H_{T}\dot{H}_{T}-\frac{2}{b^{2}}H_{T}H_{T}^{\prime\prime},
f^{\left(1\right)}(\theta)=\frac{1}{\left(2\piik\right)^{\frac{1}{2}}}\sum_{l}^{{}}e^{il\theta}\chi^{\dagger\left(s^{\prime}\right)}\left[-i\pi\alpha\mathrm{sgn}\left(l\right)\left(1-\delta_{l,0}\right)-i\pi\alpha\delta_{l,0}\sigma_{3}\right]\chi^{\left(s\right)}.
\begin{array}{c}{\displaystyle\sum_{s=1}^{m}\sum_{p+q=s-1}{\frac{(-1)^{m-s}}{(2m-s)!(m-s)!}}{\frac{(m-p-1)!(m-q-1)!}{p!q!}}{\mathrm{sTr}}(\omegaF_{L}^{p}({g^{-1}F_{R}g})^{q}(\omega^{2})^{m-p-q-1})}\\{\displaystyle=\int_{0}^{1}dt\,\mathrm{Tr}\Big(\omega\big(tF_{L}+(1-t){g^{-1}F_{R}g}+t(t-1)\omega^{2}\big)^{m-1}\Big)\quad,}\\\end{array}
{\frac{1}{2}}{\frac{\delta^{2}}{\deltaC^{\mu\nu}\deltaC_{\mu\nu}}}\equiv{\frac{1}{2}}\left(\oint_{C}dl(s)\right)^{-1}\oint_{C}dl(s){\frac{\delta^{2}}{\delta\sigma^{\mu\nu}(s)\delta\sigma_{\mu\nu}(s)}}.
\int{\calD}V=\int\prod_{n}\left(dV_{n}^{1}dV_{n}^{2}dV_{n}^{3}d\lambda_{n}^{1}d\lambda_{n}^{2}d\bar{\lambda}_{n}^{1}d\bar{\lambda}_{n}^{2}dD_{n}\right)\prod_{i}^{2n_{0}}dV_{0}^{i}\prod_{i}^{n_{0}}d\lambda_{0}^{i}
\delta_{\alpha}D_{\beta}^{\left(2q\right)}+\delta_{\alpha}^{\left(2q\right)}=\frac{-2i}{\left(2q\right)!}\Pi^{m_{2q}}\cdots\Pi^{m_{1}}\cdot\left(S\Gamma_{m_{1}\ldotsm_{2q}}\right)_{\alpha\beta}\quad=
{\calG}=\int\left[{\calD}_{\mu}c\frac{\delta}{\deltaA_{\mu}}-\frac{1}{2}g_{0}fcc\frac{\delta}{\deltac}+\frac{\delta\tilde{S}}{\deltaA_{\mu}^{\alpha}}\frac{\delta}{\delta\kappa_{\mu}^{\alpha}}+\frac{\delta\tilde{S}}{\deltac^{\alpha}}\frac{\delta}{\deltal^{\alpha}}\right]
A_{3}=\frac{1}{\sqrt{2}}f_{L}^{\bar{A}\bar{B}\bar{C}}f_{R}^{ABC}\Phi^{\bar{A}A}\Phi^{\bar{B}B}\Phi^{\bar{C}C}=\frac{1}{\sqrt{2}}\mathrm{Tr\,}\{[\Phi,\Phi]\Phi\},
H_{nm}^{\{{\bfR}\}}(0,0)=\mu_{mn}!(-1)^{\frac{m+n}2}R_{11}^{n/2}R_{22}^{m/2}\left(r^{2}-1\right)^{\frac{m+n}4}P_{(m+n)/2}^{|m-n|/2}\left(\fracr{\sqrt{r^{2}-1}}\right),
C_{1}b_{n}^{i}C_{1}^{-1}=\oint_{\xi_{i}=0}d\xi_{i}\\left(V_{0i}^{-1}V_{i}\xi_{i}\right)^{n+1}\left(\frac{d}{d\xi_{i}}V_{0i}^{-1}V_{i}\xi_{i}\right)^{-1}b^{i}(\xi_{i})\.
H_{\mathrm{vis,\pm}}^{2}=\frac{\rho_{\mathrm{vis}}(\rho_{\mathrm{vis}}+2\Lambda_{2}^{\pm})}{36M^{6}(1+4\alpha\Lambda_{b}/3M^{5})}=\frac{\pm\rho_{\mathrm{vis}}}{3M_{P}^{2}\sqrt{1+4\alpha\Lambda_{b}/3M^{5}}}\left[1+\frac{\rho_{\mathrm{vis}}}{2\Lambda_{2}^{\pm}}\right]~~.
G_{\sigma}(x)=\sigma{\pi}_{\sigma}{\pi}_{\phi}-{\frac12}{\sigma}^{2}{\pi}_{\sigma}^{2}+{\frac12}{{\phi}^{\prime}}^{2}-\frac{{\sigma}^{\prime}{\phi}^{\prime}}{\sigma}+\frac{2({\phi}^{\prime}\sigma)^{\prime}}{\sigma}-\frac{c}{\sigma}\approx0
\Pi=\int\mathrm{d}\mu_{\mathrm{SU}(2)}\,\mathrm{Tr}\,W_{\mathrm{inst}}^{2}(x_{L},\theta=0)\,\mathrm{Tr}\,W_{\mathrm{inst}}^{2}(x_{L}^{\prime},\theta^{\prime}=0)\,,
\partial_{0}^{2}h_{j}^{i}+\Boxh_{j}^{i}+\frac{1-d}{x_{0}}\partial_{0}h_{j}^{i}=\delta_{j}^{i}b-\partial^{i}\partial_{j}a+\partial_{j}a^{i}+\partial^{i}a_{j}+\frac12(x_{0}^{2}-\epsilon^{2})\partial^{i}\partial_{j}b.
T^{ab}:={\frac{1}{\sqrt{-h}}}{\frac{\deltaS_{P}}{\deltah_{ab}}}={\frac{1}{4\pi{\alpha}^{\prime}}}\bigg({\partial}^{a}X^{\mu}{\partial}^{b}X_{\mu}-{\frac{1}{2}}{h}^{ab}h^{cd}{\partial}_{c}X^{\mu}{\partial}_{d}X_{\mu}\bigg)=0.
(x^{2}AN\psi^{\prime})^{\prime}=2\gammaA(e^{2\gamma\psi}(N(K^{\prime})^{2}+\frac{(K^{2}-1)^{2}}{2x^{2}})-e^{-2\gamma\psi}\frac{\beta^{2}x^{2}}{4}(H^{2}-1)^{2})\,
\frac12\left(N^{\prime}r+N-1\right)+\frac{r^{2}}{2\sigma^{2}}u^{\prime2}+\frac1{N\sigma^{2}}u^{2}w^{2}+Nw^{\prime2}+\frac1{2r^{2}}\left(w^{2}-1\right)^{2}=0,\tag{4.8}
Q_{ABC}(z)\equiv\textstyle{\frac{1}{4}}F_{IJK}\big(X(z)\big)\,{\displaystyle{\frac{\partialX^{I}(z)}{\partialz^{A}}}{\frac{\partialX^{J}(z)}{\partialz^{B}}}{\frac{\partialX^{K}(z)}{\partialz^{C}}}},\quad\bar{Q}^{ABC}=g^{\bar{D}A}g^{\bar{E}B}g^{\bar{F}C}\bar{Q}_{DEF}\,.
\left\{i\omega\gamma^{0}+i\vec{\gamma}\left(\vec{\partial}_{\vec{r}^{\prime}}+ie\vec{A}(\vec{r}^{\prime})\right)-m+\gamma^{0}\mu\right\}\widetilde{G}_{\omega}(\vec{r},\vec{r}^{\prime})=\delta(\vec{r}-\vec{r}^{\prime}).
L[W]=\int\,dx\,dy\,\frac{1}{(1+|W|^{2})^{2}}\left(\frac{|\dot{W}|^{2}}{(1+r^{2})^{2}}-\left|\frac{\partialW}{\partialx}\right|^{2}-\left|\frac{\partialW}{\partialy}\right|^{2}\right).
f(q_{a},z_{i})={\frac{1}{T}}\int{\frac{y(z_{i})}{y(q_{a})}}\,{\frac{1}{q_{a}-z_{i}}}{\frac{(q_{a}-u)(q_{a}-v)}{(z_{i}-u)(z_{i}-v)}}\,\left|\frac{u-v}{y(u)y(v)}\right|^{2}\mathrm{d}^{2}u\mathrm{d}^{2}v.
I_{eff}=\intdtd^{2}{\bfr}(-\sum_{m}\rho\partial_{t}(\dot{\bfa}^{m}\cdot{\bfr}G^{m})-\rho\partial_{t}\omega-{\frac{1}{2}}\sum_{m}\rho(\dot{\bfa}^{m}G^{m})\cdot(\dot{\bfa}^{m}G^{m}))
AB^{\prime\prime}={\frac{[v_{\gamma}(p)^{2}-V^{2}]v_{\gamma}(p_{0})}{v_{\gamma}(p)\sqrt{v_{\gamma}(p^{\prime})^{2}-V^{2}}}}N{\frac{\tau_{0}}{2}}={\frac{v_{\gamma}(p)^{2}-V^{2}}{v_{\gamma}(p)\sqrt{v_{\gamma}(p^{\prime})^{2}-V^{2}}}}AB^{\prime}~,
\overline{{\delta}}(k)=\sum_{\sigma\in\{+,-\}}\sum_{G=0}^{\infty}(2G+1)\left(\delta_{G,\sigma}(k)-\delta_{G,\sigma}^{(1)}(k)-\delta_{G,\sigma}^{(2)}(k)\right)+\delta_{\mathrm{lim}}(k)\,.
\left[{\mathcalJ}_{\ell}^{0}\left(t,\underline{{x}}\right),~{\mathcalJ}_{\ell}^{0}\left(t,\underline{{y}}\right)\right]\;=\;\frac{i\,\sigma_{H}}{2\pi}\;\delta^{\prime}\left(\underline{{x}}-\underline{{y}}\right)~,
\left[\widehat\Lambda\stackrel{\ast}{,}\widehat{A}_{\mu}\right]=\left\{\widehat{\Lambda}^{I}\stackrel{\ast}{,}\widehat{A}_{\mu}^{J}\right\}\left[T_{I},T_{J}\right]+\left[\widehat\Lambda^{I}\stackrel{\ast}{,}\widehat{A}_{\mu}^{J}\right]\left\{T_{I},T_{J}\right\},