harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
n_{1}(t)=C_{2}e^{-\int^{t}\left[\sigma_{12}(t^{\prime})+\sigma_{21}(t^{\prime})\right]\mathrm{d}t^{\prime}}+e^{-\int^{t}\left[\sigma_{12}(t^{\prime})+\sigma_{21}(t^{\prime})\right]\mathrm{d}t^{\prime}}\int^{t}\mathrm{d}t^{\prime}\sigma_{21}(t^{\prime})e^{\int^{t^{\prime}}\left[\sigma_{12}(t^{\prime\prime})+\sigma_{21}(t^{\prime\prime})\right]\mathrm{d}t^{\prime\prime}}.
C_{ijk}\,=\,{\frac{1}{\mu^{2}}}\,{\hat{C}}_{ijk}\,+\,{\calO}\left({\frac{1}{\mu^{3}}}\right)\quad\Longrightarrow\quad{\hat{C}}_{ijk}\,=\,\langle{\partial_{i}\hat{U}}_{j}\vert{\hat{U}}_{k}\rangle
I=\int_{\Sigma}d^{3}\!x\sqrt{\gamma}\left(\!R(\gamma)-{\frac{1}{2}}\!\left(\left(DU\right)^{2}\!-\!e^{-2U}\!\left(DV\right)^{2}\right)-{\frac{1}{2}}\!\left(\left(D\Phi\right)^{2}\!-\!e^{-2\Phi}\!\left(D\Psi\right)^{2}\right)\right)\.
\langle\langle\delta^{(1)}j_{\mu}(x,t)\rangle\rangle=-i\int_{t_{0}}^{\infty}dt^{\prime}\int_{-\infty}^{\infty}dx^{\prime}A_{0}^{cl}(x^{\prime},t^{\prime})\langle\langle[j_{\mu}(x,t),j_{0}(x^{\prime},t^{\prime})]\rangle\rangle\theta(t-t^{\prime}).
\left[Q_{\alpha},T^{\mu\nu}\right]=-{\textstyle\frac{i}{2}}\left\{(\sigma^{\mu\rho})_{\alpha}^{\\beta}\,\partial_{\rho}S_{\beta}^{\nu}+(\sigma^{\nu\rho})_{\alpha}^{\\beta}\,\partial_{\rho}S_{\beta}^{\mu}\right\}\,,
{\calL}^{(6)}=-e^{-2\phi_{6}}\left\{\frac{1}{2}R^{(6)}+2\left(\frac{\partial\omega}{\omega}\right)^{2}-2(\partial\phi_{6})^{2}\right\}-\frac{1}{4}e^{-\phi_{6}}\omega^{2}F^{2}-\frac{1}{16}\omega^{4}(dB-\Omega)^{2}+\dots
\Psi_{+\bfk}^{n}(P)=(-1)^{n}C_{n}^{1/2}\left(\frac{P_{1}-iP_{2}}{P^{0}+m}\right)^{n}\Psi_{+\bfk}^{0}(P),\quad\Psi_{+\bfk}^{0}(P)=\delta(P^{0}-\sqrt{{\bfP}^{2}+m^{2}})\delta({\bfP}-{\bfk})f({\bfk}),
\tilde{\delta}\Theta=-{\frac{1}{2}}\epsilon\left[\bar{F}^{\tilde{m}\tilde{n}}\gamma_{v\tilde{m}\tilde{n}}+\bar{F}^{\tilde{m}\tilde{n}}\partial_{\tilde{m}}X^{I_{1}}\partial_{\tilde{n}}X^{I_{2}}\gamma_{v}\gamma_{I_{1}I_{2}}\right],
G(u_{1},\cdots,u_{2n}|u_{0})=\sum_{k_{0},k_{1},\cdotsk_{2n}}t_{k_{1}}^{k_{0}}(u_{1}-u_{0})\otimes\cdots\otimest_{k_{2n}}^{k_{2n-1}}(u_{2n}-u_{0})F(u_{1},\cdots,u_{2n}|u_{0})^{k_{0}k_{1}\cdotsk_{2n}}.
A(x,p)\starB(x,p)=e^{\kappa(\partial_{x}\partial_{\widetilde{p}}{~}-{~}\partial_{p}\partial_{\widetilde{x}})}A(x,p)B({\widetilde{x}},{\widetilde{p}})\Big|_{{\widetilde{x}=x},{\widetilde{p}=p}}.
(X,Y)_{\lambda,{\bfk}}=\frac{1}{2}\int{\mathrm{Tr}}_{\bfk}\bigg({\frac{\delta_{r}X}{\delta{\calA}}}{\frac{\delta_{l}Y}{\delta{\calA}_{\lambda}^{*}}}-{\frac{\delta_{r}X}{\delta{\calA}_{\lambda}^{*}}}{\frac{\delta_{l}Y}{\delta{\calA}}}\bigg),
T_{\\mu\rho\sigma}^{\mu}=-p^{2}T_{1}\Biggl[\left(p_{\rho}p_{\sigma}-p^{2}g_{\rho\sigma}\right)\mp\frac{1}{4}\left(\varepsilon_{\rho\lambda}p^{\lambda}p_{\sigma}+\varepsilon_{\sigma\lambda}p^{\lambda}p_{\rho}\right)\Biggl].
e^{2\Phi}=(ga)^{3}\left(\frac{3}{20}\right)^{3}\!\rho^{3}\left(1-\frac{l^{10/3}}{\rho^{10/3}}\right)^{3/2},\:\:\:\:e^{2h}=ga\frac{27}{400}\rho^{3}\left(1-\frac{l^{10/3}}{\rho^{10/3}}\right)^{1/2},
\Gamma_{3}=\frac{\lambda^{3}}{8\left(2\pi\right)^{5}}\int^{\infty}\frac{\mathrm{d}|{\bfQ}|}{|{\bfQ}|}N\left(|{\bfQ}|\right)\int\mathrm{d}\Omega\left\{\frac{k_{1}^{2}\,k_{2}\cdotk_{3}}{\left(\hat{Q}\cdotk_{1}\right)^{2}\hat{Q}\cdotk_{2}\hat{Q}\cdotk_{3}}+permutations\right\},
{\phi}^{self}(x^{+})=\int_{0}^{L}dy^{-}\int_{0}^{L}dz^{-}\int_{0}^{L}d{\eta}^{-}{\Phi}^{\star,++}(z^{-},x^{+}|y^{-},x^{+})i{\epsilon}(y^{-}-{\eta}^{-}){\Phi}^{++}({\eta}^{-},x^{+}|z^{-},x^{+})
{\frac{1}{2}}F_{\Lambda}^{ab}+{L_{\Lambda}}^{3}G^{-ab}+{L_{\Lambda}}^{i}G_{i}^{+ab}+{(L^{-1})_{3}}^{\Pi}J_{\Pi\Lambda}G^{+ab}-{(L^{-1})_{i}}^{\Pi}J_{\Pi\Lambda}G^{i-ab}=0,
2\eta(\alpha_{j})\sqrt{E_{j}E_{j^{\prime}}}\,\bar{u}^{\alpha_{j}}(\vec{p}_{j})\Gamma_{\mu_{v}}(\eta(\alpha_{j})p_{j},\eta(\alpha_{j^{\prime}})p_{j^{\prime}})u^{\alpha_{j^{\prime}}}(\vec{p}_{j^{\prime}})\,.
{{\mathcal{E}}_{\mu\nu}}=-{{\left({\frac{\tilde{\kappa}}{\kappa}}\right)}^{4}}\left[{\mathcal{U}}\left({u_{\mu}}{u_{\nu}}+{\frac{1}{3}}{h_{\mu\nu}}\right)+{{\mathcal{P}}_{\mu\nu}}+{{\mathcal{Q}}_{\mu}}{u_{\nu}}+{{\mathcal{Q}}_{\nu}}{u_{\mu}}\right],
{\calL}={\frac{V\tau_{p}}{\sqrt{2}}}\sum_{\gamma=0}^{\infty}\left(\begin{matrix}{{\frac{1}{2}}}\\{\gamma}\\\end{matrix}\right){\frac{1}{\gamma!}}(\kappa\partial_{\mu}\chi^{9}\partial^{\mu}\chi^{9})^{\gamma}={\frac{V\tau_{p}}{\sqrt{2}}}L_{\frac{1}{2}}(-\kappa\partial_{\mu}\chi^{9}\partial^{\mu}\chi^{9})\,
mc\frac{dt}{ds}\frac{d}{dt}\left(\frac{dt}{ds}\frac{dx^{\mu}}{dt}\right)+\frac{e}{c}\sum_{k=0}^{N}\sum_{\alpha_{1},...,\alpha_{k}=0}^{3}\eta^{\mu\mu}F_{\mu\alpha_{1}\cdots\alpha_{k}}(x)\frac{dt}{ds}\frac{dx^{\alpha_{1}}}{dt}\cdots\frac{dt}{ds}\frac{dx^{\alpha_{k}}}{dt}=0
K_{lmpp^{\prime}}^{A}({\bfq}_{\bot}^{\prime2})=\frac{2(-1)^{l+p+p^{\prime}}\,(2{\bfq}_{\bot}^{\prime2})^{l}\Gamma(l+m+1)}{(l-p)!p!(l-p^{\prime})!p^{\prime}!\Gamma(m+1)}.
I=\frac{1}{2\kappa^{2}}\intd^{D}z\sqrt{-g}[R-\frac{1}{2}(\partial\phi)^{2}-\frac{e^{-\alpha\phi}}{2(d+1)!}F_{d+1}^{2}-\frac{e^{\beta\phi}}{2(d+1)!}G_{d+1}^{2})
\nabla_{A}W_{BCD}=\nabla_{\underline{{A}}}W_{\underline{{BCD}}}-\frac{1}{4}\varepsilon_{AB}\nabla^{E}W_{ECD}-\frac{1}{4}\varepsilon_{AC}\nabla^{E}W_{EBD}-\frac{1}{4}\varepsilon_{AD}\nabla^{E}W_{EBC}
B(r)^{2}\rightarrow-\frac{4q^{2}r}{(r^{2}+q^{2})^{n}}\lbrace\int^{r}\frac{(s^{2}+q^{2})^{n}}{s^{2}}ds-\int^{iq}\frac{(s^{2}+q^{2})^{n}}{s^{2}}ds-\bar{\alpha}\rbrace,
<q|n>=\left(\frac{m\omega}{\pi\hbar}\right)^{\frac{1}{4}}\,\frac{1}{\sqrt{2^{n}n!}}\,e^{-\frac{m\omega}{2\hbar}q^{2}}\,H_{n}\left(q\sqrt{\frac{m\omega}{\hbar}}\right)\\\,\\\n=0,1,2,\dots\\\,
{\frac{1}{\prod_{i}m_{i}}}\sum_{\beta}C_{\overline{{\betaV}}}^{\overline{{\alphaV}}}g^{-1}({\overline{{\betaV}}})=e^{2\pii\alphas}\eta({\overline{{\alphaV}}},{\calN}_{\overline{{\alphaV}}},{\vec{P}}_{\overline{{\alphaV}}}^{2})~,
{\calL}=a^{3}e^{2\beta\psi}\left[6F_{0}f\left(\frac{\dot{a}}{a}\right)^{2}+6F_{0}(2\betaf+f^{\prime})\dot{\psi}\left(\frac{\dot{a}}{a}\right)+\frac{\gamma}{2}{\varphi_{0}}^{2}\beta^{2}\dot{\psi}^{2}-\frac{V}{\varphi_{0}}^{\alpha}e^{-(2\beta+\alpha)\psi}\right]\.
Z_{C}[J,\zeta,\bar{\zeta}]=\int{\calD}A_{\mu}{\calD}\psi{\calD}\bar{\psi}\delta\left(\partial_{\mu}A^{\mu}-C(x)\right)det{\calM}_{L}e^{iS+i\intdx(J^{\mu}A_{\mu}+\bar{\psi}\zeta+\bar{\zeta}\psi)}.
\left.\frac{1}{(q+k)^{2}}\right|_{q^{2}=0}=\frac12\frac{1}{{q}\cdotk}-\frac14\frac{k^{2}}{\left({q}\cdotk\right)^{2}}+\frac18\frac{\left(k^{2}\right)^{2}}{\left({q}\cdotk\right)^{3}}+\cdots,
M(p)=\frac{8}{\pi^{2}N}\left[\int_{0}^{p}dq\;\left(\frac{1}{p}+\frac{\xi\alpha}{2p^{2}}\right)\frac{q^{2}M(q)}{q^{2}+M^{2}(q)}+\int_{p}^{\alpha}dq\;\left(q+\frac{\xi\alpha}{2}\right)\frac{M(q)}{q^{2}+M^{2}(q)}\right].
h_{\mu\nu}^{(TT)}=\hat{h}_{\mu\nu}^{(1)}(x^{\rho})a^{2}(y)+\hat{h}_{\mu\nu}^{(2)}(x^{\rho})a^{2}(y)\int^{y}{\frac{dy^{\prime}}{a^{4}(y^{\prime})}},
{\bar{U}}(\phi,\chi)=\frac{1}{2}\mu^{2}r^{2}(\phi^{2}-a^{2})^{2}-\mu^{2}r(\phi^{2}-a^{2})\chi^{2}+2\mu^{2}\phi^{2}\chi^{2}+\frac{1}{2}\mu^{2}\chi^{4}.
\vartheta_{m,m^{\prime}}\left(\frac{\sigma}{\tau},-\frac{1}{\tau}\right)=e^{-i\pim^{\prime}\sigma^{2}/\tau}\sqrt{\frac{-i\tau}{2m^{\prime}}}\,\sum_{r=0}^{2m^{\prime}-1}e^{-i\pirm/m^{\prime}}\vartheta_{r,m^{\prime}}(\sigma,\tau)
K_{\bfa}^{(1)}(x,x^{\prime},t)={\frac{1}{8(\pit)^{3/2}}}{\frac{1}{|x||x^{\prime}|}}e^{{\frac{4\pi}{\bfa}}(|x|+|x^{\prime}|)}\int_{|x|+|x^{\prime}|}^{\infty}dzze^{-{\frac{4\pi}{\bfa}}z}e^{-{\frac{z^{2}}{4t}}}~~.
S=-\frac{1}{8\pi}\intd^{2}\zeta\sqrt{\hat{g}}({\hat{g}}^{\alpha\beta}\partial_{\alpha}X_{\mu}\partial_{\beta}X_{\mu}~+~i\hat{R}Q_{\mu}X_{\mu}~-~\frac{i}{2}\psi_{\mu}\gamma^{\alpha}D_{\alpha}\psi_{\mu}~+~ghosts)
{\calH}={\frac{1}{2}}\rhov_{i}v_{i}+R(\rho)~,~~\{v_{i}(x),\rho(y)\}={\frac{\partial}{\partialx^{i}}}\delta(x-y)~,~~\{v_{i}(x),v_{j}(y)\}=-{\frac{\partial_{i}v_{j}-\partial_{j}v_{i}}{\rho}}\delta(x-y).
\Lambda_{C}=\left(\begin{matrix}{1\!\!1}&{0}\\{C}&{1\!\!1}\\\end{matrix}\right),\quad\Lambda_{R}=\left(\begin{matrix}{(R^{T})^{-1}}&{0}\\{0}&{R}\\\end{matrix}\right),\quad\Lambda_{i}=\left(\begin{matrix}{-1\!\!1+e_{i}}&{e_{i}}\\{e_{i}}&{-1\!\!1+e_{i}}\\\end{matrix}\right),\quad(e_{i})_{jk}=\delta_{ij}\delta_{jk},
{\calS}=\intd^{5}x\sqrt{-g}\left[{\frac{\calR}{16\piG_{N}}}-{\frac{1}{2}}\partial_{\mu}\Phi^{a}\partial^{\mu}\Phi^{a}-{\frac{\lambda}{4}}(\Phi^{a}\Phi^{a}-\eta^{2})^{2}\right]-\int_{i}d^{4}x\sqrt{-h}\sigma_{i}(\Phi^{a})\,,
\frac{kn}{2kn+2}\,r_{n,k}=\frac{1}{2\pii}\int_{\Gamma}\,\frac{1}{z^{(k-1)n+3}}\,f_{n+1}(z^{-1})\,\left(z^{n}\,l_{n}(z^{-1})\right)^{k-1+\frac{1}{n}}\,dz
=L_{0}(q_{n,0},q_{n,1},\cdots,q_{n,m_{n}-1},\dot{q}_{n,m_{n}-1})+\sum_{n}\sum_{i_{n}=1}^{m_{n}-1}\left(q_{n,i_{n}}-\dot{q}_{n,i_{n}-1}\right)\mu_{n,i_{n}}\.
l(k)=\left(\frac{k^{2}\alpha_{2}+(E+m)^{2}\beta_{1}+ik(E+m)(\alpha_{1}-\beta_{2})}{k^{2}\alpha_{2}-(E+m)^{2}\beta_{1}-ik(E+m)(\alpha_{1}+\beta_{2})}\right)e^{-2ik\xi}
{\gamma}_{\mathrm{F}}^{\mathrm{Berry}}\equiv\int_{0}^{T}dt\int_{-\mathrm{L}/2}^{\mathrm{L}/2}dx\dot{A_{1}}(x,t)\langle\mathrm{F},A(t)|i\frac{\delta}{\deltaA_{1}(x,t)}|\mathrm{F},A(t)\rangle
\frac{1}{G(y)}\partial_{t}\partial_{t}\tilde{\phi}-\partial_{y}[G(y)\partial_{y}\tilde{\phi}]+\partial_{x}[G(x)\partial_{x}\tilde{\phi}]+\frac{1}{G(x)}\partial_{\varphi}\partial_{\varphi}\tilde{\phi}+\frac{1}{6}[\partial_{x}^{2}G(x)-\partial_{y}^{2}G(y)]\tilde{\phi}=0,
\hat{G}_{a}=V^{-{\frac{1}{2}}}f^{\frac{1}{2}}\xi_{a}^{i}\hat{p}_{i}f^{-{\frac{1}{2}}}V^{\frac{1}{2}}=\xi_{a}^{i}\hat{p}_{i}-{\frac{i}{2}}\xi_{a,i}^{i}+{\frac{i}{2}}C_{ab}^{b}+{\frac{i}{2}}C_{ao}^{o},
\left(P_{b}^{i_{1}\ldotsi_{p-2k-1}},b_{i_{1}\ldotsi_{p-2k-1}}\right),\;\left(P_{b^{1}}^{i_{1}\ldotsi_{p-2k-1}},b_{i_{1}\ldotsi_{p-2k-1}}^{1}\right),\;k=0,\ldots,a,
\sum_{k=1}^{\infty}\frac{T_{\nu}(\lambda_{\nu,k})}{\lambda_{\nu,k}^{\sigma}}J_{\sigma+\nu+1}(\lambda_{\nu,k})J_{\nu}(\lambda_{\nu,k}t)=\frac{1}{2}\int_{0}^{\infty}{J_{\sigma+\nu+1}(z)J_{\nu}(zt)\frac{dz}{z^{\sigma}}}=\frac{\left(1-t^{2}\right)^{\sigma}t^{\nu}}{2^{\sigma+1}\Gamma(\sigma+1)}
\left\{Q^{\alpha},Q^{\beta}\right\}=c\left(\Gamma^{a}{\calC}^{-1}\right)^{\alpha\beta}P_{a}+\sum_{n=2,5,6,9,10}{\textstyle\frac{c_{n}}{n!}}\left(\Gamma^{a_{1}\cdotsa_{n}}{\calC}^{-1}\right)^{\alpha\beta}{\calZ}_{a_{1}\cdotsa_{n}}^{(n)}\,.
2E-E_{c}=\frac{r_{+}l}{2R}\left[\left(2-\frac{l^{2}}{r_{+}^{2}}k\right)\left(k+\frac{r_{+}^{2}}{l^{2}}\right)^{m-1}-(2m+1)k+\frac{l^{2}}{r_{+}^{2}}k^{2}\right].
S_{\mathrm{eff.}}=\sigma\intd^{2}\xi\sqrt{g}+\kappa\intd^{2}\xi\sqrt{g}{\calR}+\frac{1}{\alpha_{0}}\intd^{2}\xi\sqrt{g}g^{ab}(\partial_{a}t_{\mu\nu})(\partial_{b}t_{\mu\nu})+{\calO}\left(\frac{T_{g}^{6}\alpha_{s}\left<\left(F_{\mu\nu}^{a}(0)\right)^{2}\right>}{R^{2}}\right),
y^{2}D_{\infty}^{\prime\prime}+2yD_{\infty}^{\prime}+\left(\delta^{2}\,y^{2}+\delta\tilde{\beta}\,y-\frac{1}{4}\left(2\tilde{\beta}+\delta^{2}+(\tilde{\beta}+\delta)(\tilde{\beta}+3\delta)\right)\right)D_{\infty}=0,
A_{\sigma}=A_{\sigma}^{(s)}+A_{\sigma}^{(m)}\;\;,\;\;\psi_{-}=\psi_{-}^{(s)}+\psi^{(m)}\;\;,\;\;\bar{\psi}_{-}=\bar{\psi}_{-}^{(s)}+\bar{\psi}_{-}^{(m)}.
\intd^{25-p}q_{\bot}\,\left(\frac{kW^{2}}{1+W^{2}}\right)^{q_{\bot}^{2}}\frac{2}{W^{2}}=\mathrm{Const}\times\int_{0}^{\infty}dr\,r^{25-p-1}\left(\frac{kW^{2}}{1+W^{2}}\right)^{r^{2}}\frac{2}{W^{2}}.
\bar{A}_{2}^{T}[S_{\beta}^{d}]=A_{2}^{(1)}[\tilde{S}_{\beta}^{d}]-A_{2}^{(0)}[\tilde{S}_{\beta}^{d}]+(4\pi)^{d/2}\delta_{d,2n}+O((1-\gamma)^{2})~~~.
K(\hat{\beta})={\frac{\lambda^{2}|z|^{4-2\hat{\beta}^{2}}}{z^{\hat{\beta}^{2}}}}\intd^{2}w_{1}d^{2}w_{2}w_{1}^{\hat{\beta}^{2}}w_{2}^{-\hat{\beta}^{2}}(w_{1}-1)^{-\hat{\beta}^{2}}(w_{2}-1)^{\hat{\beta}^{2}}|w_{1}-w_{2}|^{-2\hat{\beta}^{2}}.
T\sum_{n=-\infty}^{\infty}\frac{1}{(((2n+1)\piT)^{2}+m^{2})^{p}}\sim\frac{m^{1-2p}}{2\sqrt{\pi}}\frac{\Gamma(p-\frac{1}{2})}{\Gamma(p)}-\frac{(2mT)^{1-p}}{m\Gamma(p)}e^{-\frac{m}{T}}+\cdots
{\frac{\bar{t}}{\bar{t}_{*}}}=\left({\frac{9-4r^{2}}{12}}\right)\left[{\frac{2}{3-2r}}\left|{\frac{\eta}{\eta_{*}}}\right|^{(3/2)-r}+{\frac{2}{3+2r}}\left|{\frac{\eta}{\eta_{*}}}\right|^{(3/2)+r}\right]
W_{i_{1}j}^{(1)}(x,y)=\left(\begin{array}{cc}{\widetilde{W}_{ij}^{(0)}}\\\hline{}\\{\begin{array}{ccccc}{0}&{-\partial_{x}^{1}}&{-\partial_{x}^{2}}&{-\partial_{x}^{3}}&{-m^{2}}\\\end{array}}\\\end{array}\right)\delta^{3}(x-y),
S_{*}=\intd^{4}x\sqrt{|g|}e^{-\varphi}\left[R+\left(\nabla\varphi\right)^{2}-\frac{1}{2}\left(\nabla\beta\right)^{2}-\frac{1}{2}e^{2\varphi}\left(\nabla\sigma\right)^{2}\right]\,,
(2)(f+h.c)=i(T-{\bar{T}})^{2}(U-{\bar{U}})^{2}\partial_{\bar{U}}\partial_{\bar{T}}\int_{\calF}\frac{d^{2}\tau}{{\tau}_{2}^{2}}\partial_{\bar{\tau}}(\tau_{2}Z){\bar{F}}({\bar{\tau}}).
X^{a}(\tau,\rho)=\sqrt{\frac{\alpha^{\prime}}{2}}\sum_{n=-\infty,n\neq0}^{\infty}\rho^{n}X_{n}^{a}e^{in\tau},\,partial_{n}X^{a}(\tau,\rho=1)=\sqrt{\frac{\alpha^{\prime}}{2}}\sum_{n=-\infty,n\neq0}^{\infty}nX_{n}^{a}e^{in\tau}\.
\psi_{n}=e^{-{\frac{i}{2}}\gamma_{0}\phi_{n}(\vec{x})}\tilde{\psi}_{n},\qquad\overline{{\psi}}_{n}=\overline{{\tilde{\psi}}}_{n}e^{-{\frac{i}{2}}\gamma_{0}\phi_{n}(\vec{x})}
Q(\xi)=\frac{1}{16\piG}\intd\varphi\left\{\xi^{\perp}\left[\frac{l}{r}+\frac{r^{3}}{l^{3}}\left(g_{rr}-\frac{l^{2}}{r^{2}}\right)\right]+\frac{1}{lr}\left(\xi^{\perp}+r\xi^{\perp},_{r}\right)\left[g_{\varphi\varphi}-r^{2}\right]+2\xi^{\varphi}\pi_{\varphi}^{r}\right\}.
\tilde{I}_{\mathrm{sub}}(M^{2}/\mu^{2},\mu_{i}^{2}/\mu^{2},\Lambda^{2}/\mu^{2})\equiv\tilde{I}(\mu^{2},M^{2},\Lambda^{2})+\sum_{i}\frac{\alpha_{i}}{\alpha}\tilde{I}(\mu_{i}^{2},M^{2},\Lambda^{2})\,.
Z_{DD}={\frac{1}{N}}\sum_{p=0}^{N-1}\chi_{p}^{f}(q){\frac{1}{\eta(q)}}\int_{-\infty}^{\infty}dxq^{x^{2}/2N}\sum_{m=-\infty}^{\infty}[|a_{m}|^{2}e^{2\piimx}+(-1)^{p}|b_{m}|^{2}e^{2\pii(m+1/2)x}]
\eta_{ij}^{(2)}=\frac{\pi^{2}}{h\,H}\,B(2-B)A_{ij}^{-1}\,t_{j}=\frac{\pi^{2}}{h\,h^{\vee}}\,B(2-B)\,\left\langle\lambda_{i},\lambda_{j}\right\rangle\,\,\,\,.
\bar{N}_{00}^{rs}=\delta^{rs}+{\frac{\sqrt{\alpha_{r}\alpha_{s}}}{\alpha_{3}}}-\mu\sqrt{\alpha_{r}\alpha_{s}}\,\epsilon^{rt}\epsilon^{su}\alpha_{t}\alpha_{u}B^{T}\Gamma_{+}^{-1}B\quadr,s,t,u=1,2
\begin{array}{rcr}{\{\eta,c\}}&{=}&{-c}\\{\{\eta,c^{*}\}}&{=}&{c^{*}}\\{\{\eta,z\}}&{=}&{-2z}\\{\{\eta,z^{*}\}}&{=}&{2z^{*}}\\{\{c,z^{*}\}}&{=}&{\frac{1}{\sqrt{2}}c^{*}}\\\end{array}\;\;\;\;\begin{array}{rcr}{\{c^{*},c\}}&{=}&{1}\\{\{z^{*},z\}}&{=}&{-\frac{1}{2}\eta}\\{\{c,z\}}&{=}&{0}\\{\{c^{*},z^{*}\}}&{=}&{0}\\{\{c^{*},z\}}&{=}&{\frac{1}{\sqrt{2}}c}\\\end{array}
\overline{{F}}^{a}{}_{\mu\nu}=g\left[\partial_{\mu}\overline{{A}}^{a}{}_{\nu}-\partial_{\nu}\overline{{A}}^{a}{}_{\mu}+g{}\overline{{f}}^{a}{}_{bc}\overline{{A}}^{b}{}_{\mu}\overline{{A}}^{c}{}_{\nu}\right].
\begin{aligned}{\frac{2}{n}\epsilon^{sr}\lambda_{rA}\lambda_{sB}d_{AB}^{C}=}&{{}\quadg_{AB}^{C}\qquad\textrm{with}}\\{d_{AB}^{C}=}&{{}\intd^{2}\sigma\sqrt{W}Y_{A}Y_{B}\overline{{Y}}_{C}}\\\end{aligned}
\psi_{L}^{a}(x^{+},x^{-})=\frac{1}{2i}~\frac{1}{2}\int_{-L}^{L}dy^{-}~\epsilon(x^{-}-y^{-})\{\phi_{0}(x^{+})+\varphi(x^{+},y^{-})\}~\psi_{R}^{a}(x^{+},y^{-})~,
\pi_{j}=-i\sqrtj{\frac{\partial}{\partial\eta_{j}}},~~~\pi_{j}^{*}=-i\sqrtj{\frac{\partial}{\partial\bar{\eta}_{j}}},~~~\lambda_{j}={\frac{\eta_{j}}{\sqrtj}}.~~~\lambda_{j}^{*}={\frac{\bar{\eta}_{j}}{\sqrtj}}.
\left(\begin{matrix}{P^{I}}\\{iQ_{I}}\\\end{matrix}\right)\rightarrow\Gamma\left(\begin{matrix}{P^{I}}\\{iQ_{I}}\\\end{matrix}\right)=\left(\begin{matrix}{U}&{Z}\\{W}&{V}\\\end{matrix}\right)\left(\begin{matrix}{P^{I}}\\{iQ_{I}}\\\end{matrix}\right)
S_{nr}^{(k)}=\sum_{\ell=1}^{n-r}(-1)^{n+\ell-r}(1-\frac{n}{n+\ell-1}\delta_{r2})\frac{(n+\ell-1)!}{n!}\frac{K_{n-r,\ell}}{(n+\ell-r)!}(1+b(0)^{2})^{-\ell}.
\begin{array}{lcl}{\mathrm{Tr}\gamma_{k,6}}&{=}&{\mathrm{Tr}\gamma_{k,6^{\prime}}=0,\mathrm{~}k=1,3,4,5,7}\\{\mathrm{Tr}\gamma_{2,6}}&{=}&{\mathrm{Tr}\gamma_{2,6^{\prime}}=16\sqrt{2}}\\{\mathrm{Tr}\gamma_{6,6}}&{=}&{\mathrm{Tr}\gamma_{6,6^{\prime}}=-16\sqrt{2}}\\\end{array}
S=-\frac{1}{4\pi\alpha^{\prime}}\intd^{2}\xi\left[\sqrt{-g\:}g^{\alpha\beta}\:\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\:G_{\mu\nu}+(-4\pi\alpha^{\prime})\left(e_{0}\:\delta(\sigma)+e_{\pi}\:\delta(\pi-\sigma)\right)a_{\mu\:}\partial_{0}X^{\mu}\right]
L={\frac{1}{2}}(\dot{x}^{2}+\dot{y}^{2})+{\frac{1}{2a^{2}}}(x\dot{y}+y\dot{x})^{2}-{\frac{a^{2}(x^{2}+y^{2})}{2x^{2}y^{2}(a^{2}+x^{2}+y^{2})}}{\bfJ}^{2}-{\frac{2(x^{2}+y^{2})}{(x^{2}-y^{2})^{2}}}Q^{2}
F_{m}(y)\equiv\left|\begin{matrix}{y_{1}^{m_{1}}}&{\cdots}&{y_{k}^{m_{1}}}\\{\vdots}&{\ddots}&{\vdots}\\{y_{1}^{m_{k}}}&{\cdots}&{y_{k}^{m_{k}}}\\\end{matrix}\right|,\quadm=(m_{1},\dots,m_{k}).
\omega={R}\,\mathrm{d}X^{I}\wedge{\calW}_{I}={R}\,\mathrm{d}X^{I}\wedge\mathrm{d}Y_{I}={R}\Big(\mathrm{d}X^{I}\wedge\mathrm{d}B_{I}-\mathrm{d}F_{I}\wedge\mathrm{d}A^{I}\Big)\.
C_{ijk}^{a}\eta^{(2)b},\eta_{a}^{ijk}\eta_{b}^{i^{\prime}j^{\prime}k^{\prime}},\eta_{a}^{ijk}\eta^{(2)b}\eta^{(2)c},\eta^{(2)a}\eta^{(2)b}\eta^{(2)c}\eta^{(2)d}.
\begin{array}{rcll}{{\calC}_{+}}&{=}&{\sigma_{2}\otimes\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{1}\otimes\ldots\qquad}&{\eta=+1\nonumber}\\{{\calC}_{-}}&{=}&{\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{1}\otimes\sigma_{2}\otimes\ldots\propto{\calC}_{+}\Gamma_{*}\qquad}&{\eta=-1\,.}\\\end{array}
A\,=\,\left(\begin{array}{ccc}{0}&{{\,\,}}&{\Theta_{[1]}}\\{\Theta_{[1]}}&{{\,\,}}&{0}\\\end{array}\right)\,,\quad\Theta_{[1]}\,=\,\left(\begin{array}{ccccc}{0}&{{\,\,}}&{0}&{{\,\,}}&{1}\\{0}&{{\,\,}}&{-1}&{{\,\,}}&{0}\\{1}&{{\,\,}}&{0}&{{\,\,}}&{0}\\\end{array}\right)\quad,
H_{0}^{c}=\frac{1}{2}\sum_{a=1}^{D}\left[(p_{a}^{\prime})^{2}+\left(\partial_{a}^{2}W^{\prime}(x^{(i)})\right)^{2}\left(x_{a}^{\prime}-x_{a}^{(i)}\right)^{2}+\hbar\partial_{a}^{2}W^{\prime}(x^{(i)})\sigma_{a}^{(i)}\right]
[T^{2},\Phi^{2}]=2iT\frac{dT}{d\hat{x}_{1}}\,[T^{4},\Phi^{2}]=4iT^{3}\frac{dT}{d\hat{x}_{1}},\\dots,\[T^{2n},\Phi^{2}]=i2nT^{2n-1}\frac{dT}{d\hat{x}_{1}}\.
H(t)=\sum_{\alpha}\left[\frac{1}{2}e^{-3H_{0}t}\pi_{\alpha}^{2}+\frac{1}{2}e^{3H_{0}t}\left(m^{2}+12\xiH_{0}^{2}+{\bfk}^{2}e^{-2H_{0}t}\right)\phi_{\alpha}^{2}\right].
\Phi_{0}\star\Phi_{0}=\mathcal{T}_{L}(\mathcal{I})\star\mathcal{T}_{L}(\mathcal{I})=-\mathcal{T}_{L}(\mathcal{I})\star\mathcal{T}_{R}(\mathcal{I})=-\mathcal{T}_{L}(\mathcal{T}_{L}(\mathcal{I}))=-\frac{1}{2}\left\{\mathcal{T}_{L},\mathcal{T}_{L}\right\}(\mathcal{I})\.
\intd^{3}x\,\dot{X}_{\gamma}^{ax}\frac{1}{\nabla^{2}}\dot{X}_{\gamma}^{ax}\longrightarrow\frac{a^{3}}{\tau^{2}}\sum_{\vec{n}}\sum_{\vec{n}^{\prime}}f(\vec{n}-\vec{n}^{\prime})X_{\gamma^{\prime}-\gamma}^{a}(\vec{n})X_{\gamma^{\prime}-\gamma}^{a}(\vec{n}^{\prime})
H=\int{d^{2}x\left\{{-\frac{1}{2}F_{i0}F^{i0}+\frac{1}{4}F_{ij}F^{ij}+c^{\prime}\left(x\right)\left[{\partial_{i}\Pi^{i}+\frac{\chi}{2}\varepsilon^{ij}\partial_{i}A_{j}-J^{0}}\right]}\right\}},
\begin{array}{l}{{}[L_{i},L_{j}]=[\Lambda_{i},\Lambda_{j}]=0;\;\;}\\{[L_{i},\Lambda_{i}]=H;\;\;[H,L_{i}]=2L_{i};\;\;[H,\Lambda_{i}]=-2\Lambda_{i}}\\{{}K_{ij}=-K_{ji},[K_{ij},K_{jk}]=2K_{ik},[K_{ij},H]=0}\\{{}[K_{ij}L_{j}]=2L_{i};\;\;[K_{ij}\Lambda_{j}]=2\Lambda_{i}}\\{{}[K_{ij},L_{k}]=[K_{ij},\Lambda_{k}]=0\;\;(k\neqi,j)}\\\end{array}
E=\intd^{2}x\left({\frac{\kappa^{2}-e^{2}|\phi|^{2}}{2e^{2}|\phi|^{2}}}B^{2}+{\frac{1}{2}}\left[|\partial_{i}\phi|^{2}+{\frac{\kappa^{2}e^{2}|\phi|^{2}}{\kappa^{2}-e^{2}|\phi|^{2}}}\tilde{A}_{i}^{2}\right]+V(\phi)\right)\,.
L_{\hphantom{\mu}\nu}^{\kappa}=\left(\begin{array}{cc}{\frac{p^{0}}{m}}&{-\frac{p_{n}}{m}}\\{\frac{p^{\kappa}}{m}}&{\delta_{n}^{\kappa}\!-\!\frac{\frac{p^{\kappa}}{m}\frac{p_{n}}{m}}{1+\frac{p^{0}}{m}}}\\\end{array}\right)\,.
[Q_{g},T_{n}\Bigl(W_{1}f_{1}\otimes...\otimesW_{n}f_{n}\Bigr)]=\Bigl(\sum_{m=1}^{n}g(W_{m})\Bigr)T_{n}\Bigl(W_{1}f_{1}\otimes...\otimesW_{n}f_{n}\Bigr)
\sum_{\nu=0}^{N}\omega_{\nu}^{2}\eta_{\nu\rho}\eta_{\nu\sigma}+\sum_{k=1}^{N}\lambda_{k}\left(\eta_{k\rho}\eta_{0\sigma}+\eta_{0\rho}\eta_{k\sigma}\right)=\delta_{\rho\sigma}\Omega_{\sigma}^{2}\;,
\hat{Z}_{88}^{10}=\hat{Z}_{88}^{\overline{{10}}}=\frac{(det\Im\Omega)^{2}}{4}\biggl\vert\Theta\left[\begin{array}{ll}{1/2}&{1/2}\\{1/2}&{1/2}\\\end{array}\right](0\vert\Omega)\biggr\vert^{8}
\langleC|{\tilde{J}}_{0}^{\mu^{\prime}}z^{L_{0}^{\mathrm{WZW}}\vert_{B}}{\bar{z}}^{{\tilde{L}}_{0}^{\mathrm{WZW}}\vert_{B}}|C\rangle=-\sum_{l,N}S_{0}{}^{l}|z|^{2L_{0}(l,N)}\sum_{j}\langlel;N,j|J_{0}^{\mu^{\prime}}|l;N,j\rangle=0,
f^{(0)}=\left(\begin{array}{ccc}{f_{ik}^{(0)AA}}&{f_{ikl}^{(0)AA}}&{f_{ikl}^{(0)A\pi}}\\{f_{ijk}^{(0)AA}}&{f_{ijkl}^{(0)AA}}&{f_{ijkl}^{(0)A\pi}}\\{f_{ijk}^{(0)\piA}}&{f_{ijkl}^{(0)\piA}}&{f_{ijkl}^{(0)\pi\pi}}\\\end{array}\right)
\left[{\Gamma}^{0}P^{0}-{\Gamma}^{1}P^{1}-k^{1}p^{1}+\frac{e_{1}e_{2}}{c}({\gamma}^{0}\otimes{\gamma}^{0})D_{-}(x_{-})-m_{1}cI\otimes{\gamma}^{0}-m_{2}c{\gamma}^{0}\otimesI\right]{\Phi}(x_{-},t|x_{+},t)=0.
g^{2}\left(\begin{array}{c|c}{H_{\sigma\nu}+H_{\sigma\tau}\bar{D}_{\tau}\left(\frac{1-\beta}{(1-\beta)\tilde{E}+\beta}\right)\bar{D}_{\alpha\nu}}&{H_{\sigma\tau}\bar{D}_{\tau}\left(\frac{i}{(1-\beta)\tilde{E}+\beta}\right)}\\{\left(\frac{-i}{(1-\beta)\tilde{E}+\beta}\right)\bar{D}_{\alpha\nu}}&{\left(\frac{1-\tilde{E}}{(1-\beta)\tilde{E}+\beta}\right)}\\\end{array}\right)
A_{+}(f^{+}(x^{+}))={\frac{aA_{+}(x^{+})+b}{cA_{+}(x^{+})+d}},\\\\\A_{-}(f^{-}(x^{-}))={\frac{aA_{-}(x^{-})+\epsilonb}{\epsiloncA_{-}(x^{-})+d}}.
\left({\frac{V_{2}}{V_{1}}}\right)^{2}={\frac{n_{1}\bar{n}_{1}}{n_{2}\bar{n}_{2}}}\,\qquad\left({\frac{V_{3}}{V_{2}}}\right)^{2}={\frac{n_{2}\bar{n}_{2}}{n_{3}\bar{n}_{3}}}\,\qquad\left({\frac{V_{4}}{V_{3}}}\right)^{2}={\frac{n_{3}\bar{n}_{3}}{n_{4}\bar{n}_{4}}}\,
R_{n}(x)=-\int_{-\infty}^{\infty}d^{2}y\,G_{R}(x,y)\,\left[\frac{3\omega_{0}}{2}\,\sum_{i+j=n}R_{i}(y)R_{j}(y)+\frac{1}{2}\,\sum_{i+j+k=n+1}R_{i}(y)R_{j}(y)R_{k}(y)\right]\,;\;\;\;n\ge2\,.
F_{8}^{{\mathcal{O}}_{0,2}^{0,1}|++++----}=\frac{\langle{\mathcal{O}}_{0,2}^{0,1}\rangle((\sigma_{3}^{-})^{2}+\sigma_{1}^{-}\sigma_{3}^{-}\sigma_{2}^{+}+\sigma_{4}^{+}(\sigma_{1}^{-})^{2})/(\sigma_{4}^{-})^{3/2}}{2^{8}\,\prod_{1\leqi<j\leq4}\mathrm{cosh}(\theta_{ij}/2)\prod_{5\leqi<j\leq8}\mathrm{cosh}(\theta_{ij}/2)}\prod_{i<j}F_{\mathrm{min}}^{\mu_{i}\mu_{j}}(\theta_{ij}),
S=\intd^{4}x\epsilon^{\mu\nu\alpha\beta}\epsilon_{abcd}\bigg({^+}\tau{^+}F_{\mu\nu}^{~~ab}{^+}F_{\alpha\beta}^{~~cd}-{^-}\tau{^-}F_{\mu\nu}^{~~ab}{^-}F_{\alpha\beta}^{~~cd}\bigg),
\tilde{m}_{sb}={\frac{{\tilde{m}_{f}}^{2}}{2\tilde{m}_{s}}}[{\frac{1}{\sqrt{1-\tilde{m}_{sb}^{2}}}}tan^{-1}\sqrt{{\frac{1+\tilde{m}_{sb}^{2}}{1-\tilde{m}_{sb}^{2}}}}-{\frac{\pi}{4}}]
L=\left(\frac{1}{2}{\dot{x}}^{2}+\frac{1}{2}i{\dot{\gamma}}_{i}\cdot\gamma_{i}\right)-\left(\frac{1}{2}\lambdax^{2}+i\lambda_{i}\gamma_{i}\cdotx+\frac{1}{2}i\lambda_{ij}\gamma_{i}\cdot\gamma_{j}\right)
S=\intd^{10}x\sqrt{-g_{10}}\left\{e^{-\varsigma}\left[R_{10}+\left(\nabla\varsigma\right)^{2}\right]-\frac{1}{4}F_{AB}F^{AB}-\frac{1}{48}F_{ABCD}F^{ABCD}-\frac{1}{2}m^{2}\right\},
E=-{\frac{1}{2}}\sum_{i,j=q^{1}}^{q^{2}}\sum_{\alpha=1}^{M_{j}}Z_{ij}(\lambda_{\alpha}^{i}+\Omega)-{\frac{1}{2}}\sum_{i,j=q^{1}}^{q^{2}}\sum_{\alpha=1}^{M_{j}}Z_{ij}(\lambda_{\alpha}^{i}-\Omega)\,.
{\frac{-i\mu^{2}}{8H_{5}\piV}}(2\pi)^{3}\delta^{3}(\vec{k}+\vec{k}^{\prime})\intdt|t|\int\intdudu^{\prime}e^{i(u-u^{\prime}){\frac{k_{y}}{H_{5}}}}e^{-i(\mathrm{cosh}u+\mathrm{cosh}u^{\prime})mt}e^{-\epsilon|t|},
0=\langle0\midH\mid0\rangle=\langle0\mid\pi_{m}\pi_{m}-\frac{\lambda^{2}}{2}\;[\phi_{m},\phi_{n}][\phi_{m},\phi_{n}]+\lambda[\Lambda_{\alpha},\;\Lambda_{\beta}]_{+}\Gamma_{\alpha\beta}^{m}\;\phi_{m}\mid0\rangle
\langle\langle\Omega_{0}({\bfk}_{n},t)\Omega_{0}({\bfk}_{n^{\prime}},t^{\prime})\rangle\rangle_{T}=\delta_{n,-n^{\prime}}\langle\langle\Omega_{0}({\bfk}_{n},t)\Omega_{0}(-{\bfk}_{n},t)\rangle\rangle_{T}.
\begin{array}{cc}{(x_{2-}{\cdot\tilde{\sigma}})^{-1}{}x_{\bar{2}1}{\cdot\tilde{\sigma}}(x_{1+}{\cdot\tilde{\sigma}})^{-1}=-x_{\bar{1}2}^{\prime}{\cdot\sigma}~~~~}&{~~~~~\displaystyle{x_{\bar{1}2}^{\prime2}=\frac{x_{\bar{2}1}^{2}}{x_{2-}^{2}\,x_{1+}^{2}}}}\\\end{array}
S_{M}^{d+1}=\int_{M_{d+1}}\Bigl[{\frac{1}{8}}b_{d-1}\wedge{}^{\star}b_{d-1}+{\frac{\mu^{2}}{2}}a\wedge{}^{\star}a-{\frac{1}{4}}a\wedgeH+{\frac{1}{2}}(b_{d-1}-B_{d-1})\wedgeF\Bigl]\;.
j\overline{{\Psi}}_{L}\Psi_{R}=j\overline{{\psi}}_{L}j\psi_{L}^{\prime}-j\overline{{\psi}}_{R}j\psi_{R}^{\prime}+j(\overline{{\psi}}_{L}\psi_{R}+\overline{{\psi}}_{R}^{\prime}\psi_{L}^{\prime}).
G^{(n)}(t,t^{\prime})=G^{(n-1)}(t,t^{\prime})-{\frac{1}{G^{(n-1)}(s_{n},0)}}[G^{(n-1)}(t,0)-G^{(n-1)}(t,s_{n})-G^{(n-1)}(t^{\prime},0)+G^{(n)}(t^{\prime},s_{n})]^{2},