harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{-\infty}^{\infty}dk\,\left(\rho_{\parallel}(k)-\rho_{0}(k)\right)\Phi(k^{2};\beta_{T},\mu)=\frac{1}{2\pi}\int_{-\infty}^{\infty}dk\,\Phi(k^{2};\beta_{T},\mu)\frac{d}{dk}\delta_{\parallel}(k)
D(\alpha,{\alpha^{\prime}})={\frac{\sum_{i,j}p_{i}^{\prime\prime}\langle\Phi_{i}|C_{\alpha}|\Psi_{j}\rangle\langle\Phi_{i}|C_{\alpha^{\prime}}|\Psi_{j}\rangle^{*}p_{j}^{\prime}}{\sum_{i,j}p_{i}^{\prime\prime}|\langle\Phi_{i}|C_{u}|\Psi_{j}\rangle|^{2}p_{j}^{\prime}}}.
\left(\partial_{\mu}1\otimes1+\frac{1}{4}\epsilon_{\alpha\beta\delta}\omega_{\mu}^{\alpha\beta}1\otimes\gamma^{\delta}+\frac{i}{2}A_{\mu}^{c}\sigma_{c}\otimes1-\frac{1}{2\ell}e_{\mu\alpha}1\otimes\gamma^{\alpha}\right)\varepsilon_{r}=0\,,
\mathcal{F}_{\mu\nu}^{a}=\left(\begin{array}{ccccccc}{0}&{0}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{\mathcal{F}_{12}^{a}}&{0}&{0}&{0}&{0}\\{0}&{-\mathcal{F}_{12}^{a}}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{\mathcal{F}_{34}^{a}}&{0}&{0}\\{0}&{0}&{0}&{-\mathcal{F}_{34}^{a}}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}&{\mathcal{F}_{56}^{a}}\\{0}&{0}&{0}&{0}&{0}&{-\mathcal{F}_{56}^{a}}&{0}\\\end{array}\right).
W_{l}^{p_{1}\top_{2}}\equiv\intd^{3}\widetilde{\bfk}_{1}\intd^{3}\widetilde{\bfk}_{2}\;\widetilde{\omega}_{l}\frac{d\Gamma^{p_{1}\top_{2}}}{d^{3}\widetilde{\bfk}_{1}d^{3}\widetilde{\bfk}_{2}}\;,
\eta^{IJ}P_{I}^{x}P_{J}^{x}=\eta^{IJ}\big(h_{uv}k_{I}^{u}k_{J}^{v}-m_{I}^{A}m_{J}^{B}\rho_{AB}\big)=\eta^{IJ}h_{uv}k_{I}^{u}k_{J}^{v}\,,
{\calL}_{PV}=\intd^{4}\theta[Q_{+}^{reg~\dagger}e^{V}Q_{+}^{reg}|\Phi|^{2}+Q_{-}^{reg~\dagger}e^{V}Q_{-}^{reg}|\phi|^{2}]+(\intd^{2}\theta\Phi^{3}\Lambda_{UV}Q_{+}^{reg}Q_{-}^{reg}+\mathrm{h.c.}).
\pmi\sqrt{\lambda}\varepsilon^{ijk}\bigg(e_{i}^{a}(\partial_{j}\omega_{ka}-\partial_{k}\omega_{ja})-\varepsilon_{abc}e_{i}^{a}\omega_{j}^{b}\omega_{k}^{c}+{\frac{1}{3}}\lambda\varepsilon_{abc}e_{i}^{a}e_{j}^{b}e_{k}^{c}\bigg),
{\frac{h}{r^{2}}}{\frac{d}{dr}}\left(hr^{2}{\frac{dR}{dr}}\right)~+~\left({\frac{A}{r^{4}}}~+~{\frac{B}{r^{3}}}~+~{\frac{C}{r^{2}}}\right)\omega^{2}R~=~0~.
\begin{array}{lll}{<\frac{1}{4}(1+e_{3}+e_{5}+e_{6}),}&{\frac{1}{4}(1+e_{3}+e_{5}+e_{6}),}&{\pm\frac{1}{4}(-1+e_{3}+e_{5}+e_{6})\pm\frac{1}{2}e_{7}>,}\\{}&{}&{\pm\frac{1}{4}(1-e_{3}+e_{5}+e_{6})\pm\frac{1}{2}e_{2}>,}\\{}&{}&{\pm\frac{1}{4}(1+e_{3}+e_{5}-e_{6})\pm\frac{1}{2}e_{4}>,}\\{}&{}&{\pm\frac{1}{4}(1+e_{3}-e_{5}+e_{6})\pm\frac{1}{2}e_{1}>,}\\\end{array}
\tilde{D}_{\alpha}=e^{i\partial_{i}^{p}D_{i}}e^{-ix_{\mu}p_{\mu}}D_{\alpha}e^{ix_{\nu}p_{\nu}}e^{-i\partial_{j}^{p}D_{j}}\,,\quad(\partial_{i}^{p}=\frac{\partial}{\partialp_{i}}).
G_{\mu\nu}^{0A}(k,\epsilon=0)=-{\frac{1}{k^{2}}}\biggl(g_{\mu\nu}+{\frac{(\lambdak^{2}+\eta^{2})}{(\eta\cdotk)^{2}}}k_{\mu}k_{\nu}-{\frac{k_{[\mu}\eta_{\nu]_{+}}}{\eta\cdotk}}\biggr).
\sigma^{1}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right),\quad\sigma^{2}=\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right),\quad\sigma^{3}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right).
\tilde{E}_{n\delta}^{(i)}=\sum_{p_{1}+2p_{2}+\cdots+np_{n}=n}\frac{\left(q^{(\alpha_{i},\alpha_{i})}-q^{-(\alpha_{i},\alpha_{i})}\right)^{\sum_{i}p_{i}-1}}{p_{1}!\;...\;p_{n}!}(E_{\delta}^{(i)})^{p_{1}}(E_{2\delta}^{(i)})^{p_{2}}...(E_{n\delta}^{(i)})^{p_{n}}
L[\phi,\psi_{i}]=-\int_{0}^{\infty}\mathrm{d}r~\left[4\pir^{2}\left\{\frac{1}{2}\left(\frac{\mathrm{d}\phi}{\mathrm{d}r}\right)^{2}+U(\phi)\right\}+\sum_{i}\sum_{KM}\psi_{i}^{\dagger}H_{f}\psi_{i}\right]~~,
Q=\intd^{3}x\left[\sum_{l=1}^{13}\left(\vec{\sigma}\cdot\vec{F}^{+l}\psi^{l}(x)\right)+\left(\sigma^{\mu}\partial_{\mu}\phi^{\dagger}(x)\right)\cdot\psi_{L}(x)\;+{\calW}\psi^{1}\right]
T_{ab}=\frac{1}{16\piG_{10}g_{s}^{2}}\int_{r\to\infty}d\Omega_{8-p}r^{8-p}n^{i}[\eta_{ab}(\partial_{i}h_{\c}^{c}+\partial_{i}h_{\j}^{j}-\partial_{j}h_{\i}^{j})-\partial_{i}h_{ab}],
V_{i}(k_{i},\epsilon_{i},z_{i})=(\epsilon_{i}\cdot\partialX(z_{i})+ik_{i}\cdot\psi(z_{i})\,\epsilon_{i}\cdot\psi(z_{i}))\,\mathrm{e}^{ik_{i}\cdotX(z_{i},\bar{z}_{i})}.
S_{I}^{\prime}=\int_{{\calM}^{p+1}}\Big({\frac{1}{p!}}E^{a_{0}}\wedgee^{a_{1}}\wedge...\wedgee^{a_{p}}\epsilon_{a_{0}...a_{p}}-{\frac{1}{(p+1)!}}e^{a_{0}}\wedgee^{a_{1}}\wedge...\wedgee^{a_{p}}\epsilon_{a_{0}...a_{p}}\Big)
\begin{array}{l}{K_{ij}A^{(i_{1}\ldotsi_{m})}(z;\beta)=-{A}^{(i_{1}\ldotsi_{m})}(z;\beta)\quad\forall\quadi\mathrm{~and~}j\in\{i_{1}\ldotsi_{m}\}\,,}\\{K_{ij}{A}^{(i_{1}\ldotsi_{m})}(z;\beta)=\phantom{-}{A}^{(i_{1}\ldotsi_{m})}(z;\beta)\quad\forall\quadi\mathrm{~and~}j\not\in\{i_{1}\ldotsi_{m}\}\,.}\\\end{array}
\tilde{c}={\frac{3(p-1)V_{p}}{2\piG}}\left[{\frac{(p-3)^{2}+16}{(p-3)^{2}}}\right]^{\frac{p-3}{4}}\bar{\calE}^{-{\frac{p-3}{2}}}\bar{\calP}_{\phi}^{\frac{p-1}{2}},
S=S_{0}+S_{I}=\frac{1}{4\pi\alpha^{\prime}}\int_{M}\!\!d^{2}z\sqrt{g}g^{ab}\partial_{a}x^{\mu}\partial_{b}x^{\nu}G_{\mu\nu}+\frac{1}{2\pi\alpha^{\prime}\mu}\int_{\partialM}dt\;eH_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}
C_{0,1}\;=\;\left\{(J,J_{1},\varepsilon):0\leq\varepsilon\leq\pi/2\;,\;J\geq0\;\mathrm{and}\;\varepsilon=\frac{\pi}{4}+\mathrm{arcsin}\left(\frac{e^{-J+2J_{1}}}{\sqrt{2}}\right)\right\}.
\begin{array}{rcl}{\hat{Q}}&{=}&{S_{+}\hat{C}^{+}+S_{-}\hat{C}^{-}+A_{1}(\hat{S}_{3})\hat{C}^{3}+A_{2}(\hat{S}_{3})\hat{\bar{C}}_{-}\hat{C}^{-}\hat{C}^{3}+A_{3}(\hat{S}_{3})\hat{\bar{C}}_{+}\hat{C}^{+}\hat{C}^{3}}\\{}&{}&{+A_{4}(\hat{S}_{3})\hat{\bar{C}}_{3}\hat{C}^{+}\hat{C}^{-}+A_{5}(\hat{S}_{3})\hat{\bar{C}}_{+}\hat{\bar{C}}_{-}\hat{C}^{+}\hat{C}^{-}\hat{C}^{3}{}~,}\\\end{array}
1-f_{n}^{*}(-q)-f_{n}(q)=1+v_{q}(\frac{1}{2\pi})(\frac{m}{q})ln[\frac{(k_{f}+q/2)^{2}+(\frac{m\mathrm{~}\omega}{q})^{2}}{(k_{f}-q/2)^{2}+(\frac{m\mathrm{~}\omega}{q})^{2}}]=0
S_{\Psi}~=~\int\!dt~\{p_{i}\dot{q}^{i}~+~B_{a}\dot{N}^{a}~+~{\overline{{\calP}}}_{a}\dot{\calC}^{a}~+~{\overline{{\calC}}}_{a}\dot{\calP}^{a}~-~H_{m}~+~i[Q,\Psi]\},
\begin{array}{llll}{\pm(e_{1}+e_{2}),}&{\pm(e_{2}-e_{3}),}&{\pm(e_{3}+e_{4}),}&{{\frac{1}{2}}(\pm(e_{1}+e_{2})\pm(e_{3}+e_{4})\pm(e_{5}-e_{6}+\sqrt2e_{7})),}\\{\pm(e_{1}+e_{3}),}&{\pm(e_{2}+e_{4}),}&{\pm(e_{1}-e_{4}),}&{{\frac{1}{2}}(\pm(e_{1}-e_{4})\mp(e_{2}-e_{3})\pm(e_{5}-e_{6}+\sqrt2e_{7})).}\\\end{array}
Q(x,y)\;=\;\frac{1}{2}\:\sum_{k=1}^{K_{xy}}\frac{\partial{\cal{L}}(\lambda^{xy})}{\partial\lambda_{k}^{xy}}\:F_{k}^{xy}\:P(x,y)\;=\;\frac{1}{2}\:\sum_{k=1}^{K_{xy}}\frac{\partial{\cal{L}}(\lambda^{xy})}{\partial\lambda_{k}^{xy}}\:P(x,y)\:F_{k}^{yx}\;.
\int\mid\alpha,\lambda><\alpha,\lambda\mid{\frac{1}{2\pi}}K_{\lambda+\frac{1}{2}}(\mid\alpha\mid)I_{\lambda+\frac{1}{2}}(\mid\alpha\mid)d^{2}\alpha=\sum_{m=0}^{\infty}\midm,\lambda><m,\lambda\mid={\bf1},
L_{b_{2}(k)}^{a_{2}}(\mu)L_{b_{1}(j)}^{a_{1}}(\lambda)=e^{-i\theta(\hat{a}_{1}-\hat{b}_{1})(\hat{a}_{2}-\hat{b}_{2})}L_{b_{1}(j)}^{a_{1}}(\lambda)L_{b_{2}(k)}^{a_{2}}(\mu),\ll{brany}
\left(Pe^{-ig\int_{x_{0}}^{x_{0}+\beta}dt^{\prime}\,A_{0}(t^{\prime},\vec{x})}\right)\rightarrowU^{-1}(x_{0},\vec{x})\left(Pe^{-ig\int_{x_{0}}^{x_{0}+\beta}dt^{\prime}\,A_{0}(t^{\prime},\vec{x})}\right)U(x_{0},\vec{x})
\delta{\calE}_{1}^{2,0}=\frac{K}{4a_{0}^{6}}\left\{\left[4\left(\frac{a_{0}}{\Lambda}\right)^{3}-8\left(\frac{a_{0}}{\Lambda}\right)^{2}+11\left(\frac{a_{0}}{\Lambda}\right)\right]e^{-\Lambda/a_{0}}+11Ei\left(-\frac{\Lambda}{a_{0}}\right)\right\}\,{,}
L\left(\begin{array}{c}{A_{Q}}\\{\bar{A}_{\widetilde{Q}}}\\\end{array}\right)=\left(\begin{array}{cc}{\Box}&{\sqrt{2}\bar{F}_{\Phi}}\\{\sqrt{2}F_{\Phi}}&{\Box}\\\end{array}\right)\left(\begin{array}{c}{A_{Q}}\\{\bar{A}_{\widetilde{Q}}}\\\end{array}\right)
\begin{array}{rcl}{\delta(x_{i})}&{=}&{x_{i}\otimes1+q^{-h_{i}}\otimesx_{i}\,}\\{\delta(y_{i})}&{=}&{y_{i}\otimesq^{h_{i}}+1\otimesy_{i}\,}\\{\delta(h_{i})}&{=}&{h_{i}\otimes1+1\otimesh_{i}\,}\\{\delta(d)}&{=}&{d\otimes1+1\otimesd\,}\\\end{array}
{\calK}_{n}(k)=2\left(\frac{1}{4\pi}\right)^{\frac{D}{2}}\left(\frac{1}{2}\sqrt{\frac{k\circk}{m^{2}}}\right)^{n-\frac{D}{2}}K_{n-\frac{D}{2}}(\sqrt{m^{2}k\circk})~.
\begin{array}{rcl}{\{\theta_{\alpha,n},\theta_{\beta,m}\}}&{=}&{\frac{1}{n}\delta_{n+m,0}\\\\mathrm{for}\\\n\neq0\,,}\\{\{\theta_{\alpha,0},\theta_{\beta,0}\}}&{=}&{\{\xi_{\alpha},\xi_{\beta}\}\\=\\0\,,}\\{\{\xi_{\alpha},\theta_{\beta,0}\}}&{=}&{1\,,}\\\end{array}
\Gamma_{\ell}={\frac{\piJ_{h}^{\ell}}{k^{2}(e^{4\pikr_{0}}-1)}}={\frac{\pi(\Gamma(\ell+1))^{2}}{2^{2\ell+2}(\Gamma(\ell+{\frac{3}{2}}))^{2}(\Gamma(2\ell+1))^{2}}}\;k^{2\ell-1}r_{0}^{2\ell+1}e^{-2\pikr_{0}}|\Gamma(\ell+1+2ikr_{0})|^{2}
{\bf{\calI}}(E)=\left(\begin{array}{cc}{\int\frac{d^{3}k}{(2\pi)^{3}}\frac{1}{E^{+}-\frac{k^{2}}{2\mu}}}&{\int\frac{d^{3}k}{(2\pi)^{3}}\frac{k^{2}}{E^{+}-\frac{k^{2}}{2\mu}}}\\{\int\frac{d^{3}k}{(2\pi)^{3}}\frac{k^{2}}{E^{+}-\frac{k^{2}}{2\mu}}}&{\int\frac{d^{3}k}{(2\pi)^{3}}\frac{k^{4}}{E^{+}-\frac{k^{2}}{2\mu}}}\\\end{array}\right).
{\bar{S}}_{s}^{rs}=\intd^{3}\sigma\{\sigma^{r}[({\vec{\pi}}_{\perp}\times\vec{B})^{s}+{\hat{\pi}}_{\phi^{*}}\partial^{s}{\hat{\phi}}^{*}+{\hat{\pi}}_{\phi}\partial^{s}\hat{\phi}]-(r\leftrightarrows)\}(\tau,\vec{\sigma}).
\delta_{t}~=~{{\frac{1}{2}}}\intd^{4}x\left\{\left[{\frac{\partial^{2}}{\partialt^{2}}}~\varphi(x)\right]~\delta\varphi(x)+\delta\varphi(x)~\left[{\frac{\partial^{2}}{\partialt^{2}}}~\varphi(x)\right]\right\}\quad.
R(x)e^{2kr_{0}}\left(1+{\calO}(l^{2})\right)\mathrm{Tr}T_{\mu\nu}=\frac{r(x)}{\sqrt{6}M_{Pl}e^{-kr_{0}}}\mathrm{Tr}T_{\mu\nu}\left(1+{\calO}(l^{2})\right),
({\mathcalA}_{3}^{R})_{m}(x_{\perp})=\int\,dx_{-}(R_{i}^{a}(x_{-},x_{\perp})\partial^{i}R_{+}^{b}(x_{-},x_{\perp})-\partial^{i}R_{+}^{a}(x_{-},x_{\perp})R_{i}^{b}(x_{-},x_{\perp}))C_{mba},
\alpha_{0}^{-}=\frac{P^{-}}{2}+\frac{nL}{2\alpha^{\prime}}\,\quad\tilde{\alpha}_{0}^{-}=\frac{P^{-}}{2}-\frac{nL}{2\alpha^{\prime}}\,\quad\alpha_{0}^{i}=\bar{\alpha}_{0}^{i}=\frac{P^{i}}{2}\,\quad
\sum_{j=0}^{a-1}(-1)^{j}\left(\begin{array}{c}{j}\\{a-1}\\\end{array}\right)\mathrm{exp}\left(-\frac{2\pii}{k}\frac{PQ}{a}ej\right)=\left[1-\mathrm{exp}\left(-\frac{2\pii}{k}\frac{Pq}{a}e\right)\right]^{a-1}
\sigma_{i}e_{k_{1}k_{2}\ldotsk_{n}}=\sum_{l_{i},l_{i+1}\in{\calI}}{\hat{\calR}}\left(k_{i},k_{i+1};l_{i},l_{i+1}\right)e_{k_{1}k_{2}\ldotsk_{i-1}l_{i}l_{i+1}k_{i+2}\ldotsk_{n}}\;.
\{g,B,A^{(1)},A^{(2)},k,\phi,T,C_{(0)}^{+},C_{(1)}^{+},C_{(2)}^{+},C_{(3)}^{+},C_{(0)}^{-},C_{(1)}^{-},C_{(2)}^{-},C_{(3)}^{-}\}\,,
T_{+2-2}^{+2}=4i\psi_{+2q}^{2+}\psi_{-2q}^{2+},\T_{\pm2+q}^{+2}=-4i\chi\psi_{\pm2q}^{2+},\T_{+q+p}^{+2}=-4i\delta_{qp}(1+\chi^{2});
S_{B}:\begin{array}{llllll}{\frac{\partial\tilde{m}_{s}}{\partialB_{ij}}}&{=}&{\frac{1}{2}\tilde{m}^{ij}\,\quad}&{\frac{\partial\tilde{m}^{ij}}{\partialB_{kl}}}&{=}&{\frac{1}{2}\tilde{m}^{sijkl}\\quad}\\{\frac{\partial\tilde{m}^{sijkl}}{\partialB_{mn}}}&{=}&{\frac{1}{2}\tilde{m}^{s;sijklmn}\,\quad}&{\frac{\partial\tilde{m}^{s;sijklmn}}{\partialB_{pq}}}&{=}&{0\.}\\\end{array}
\phi(\vec{\epsilon}_{f},t_{f})=\int{}d\epsilon_{i}I_{\vec{\epsilon}_{f}}e^{\vec{\epsilon}_{i}^{t}e^{-M(t_{f}-t_{i})}\vec{\epsilon}_{f}}\phi(\vec{\epsilon}_{i},t_{i})=\phi(e^{-M(t_{f}-t_{i})}\vec{\epsilon}_{f},t_{i}).
\int_{g}dg\,\biggl({}_{\nu\bar{\nu}}\langleO_{2}|\times{}_{\eta\bar{\eta}}\langleO_{2}|\timesf^{*}(Q)\biggl)_{t=0}\,||\,g^{\dagger}(\deltaH+\deltaH^{\dagger})^{n}g\,||\,\biggl(f(Q)\times{|O_{2}\rangle}_{\eta\bar{\eta}}\times{|O_{k}\rangle}_{\nu\bar{\nu}}\biggr)_{t=0}
\left[\begin{array}{cc}{\,r_{-}^{\mu}\,}&{\,\,\;-(Wr_{-})^{\nu}\,}\\{\,r_{+}^{\mu}\,}&{\,\,\;-(Wr_{+})^{\nu}\,}\\\end{array}\right]\left[\begin{array}{cc}{\,(Wv^{+})_{\mu}\,\,\,}&{\,\,(Wv^{-})_{\mu}\,}\\{v_{\nu}^{+}\,}&{\,v_{\nu}^{-}}\\\end{array}\right]=\left[\begin{array}{cc}{\,\theta_{-}^{\;\;+}\;}&{\,0\,}\\{\;0}&{\;\theta_{+}^{\;\;-}\,}\\\end{array}\right],
S_{W}\cong\int\mathcal{A}\starQ_{NS}\mathcal{A}+\frac23\int\mathrm{X}\mathcal{A}\star\mathcal{A}\star\mathcal{A}+\intY(i)\Psi\starQ_{R}\Psi+2\int\Psi\star\mathcal{A}\star\Psi.
\begin{array}{ccl}{x(\hat{u},\tau)}&{\rightarrow}&{x(\hat{u}^{M},-\frac{1}{\tau})\equivx^{\prime}(\hat{u},\tau),}\\{y(\tau)}&{\rightarrow}&{y(-\frac{1}{\tau})\equivy^{\prime}(\tau).}\\\end{array}
L_{\mu\nu}^{-1}(0,\tau^{\prime})+\frac{{\calF}_{\mu}^{\alpha}}{2}\int_{0}^{1}L_{\mu\nu}^{-1}(\tau^{\prime\prime},\tau^{\prime})d\tau^{\prime\prime}=\eta_{\mu\nu}\delta(\tau^{\prime})\;,
\hat{D}_{\mu}\hat{\psi}\begin{array}{}{t]{l}=\partial_{\mu}\hat{\psi}+i\hat{B}_{\mu}*\hat{\psi}}\\{=\partial_{\mu}\hat{\psi}-i\hat{\psi}*\hat{B}_{\mu}}\\{=\partial_{\mu}\hat{\psi}+i(\hat{\psi}*\hat{B}_{\mu}-\hat{B}_{\mu}*\hat{\psi})}\\\end{array}
\begin{array}{l}{{\delta}A_{\mu}^{a}=D_{\mu}c^{a},~{\delta}B^{a}=0~,}\\{\displaystyle{\delta}c^{a}=-\frac12f^{abc}c^{b}c^{c},~{\delta}\bar{c}^{a}=\frac{ik}{4\pi}B^{a}~.}\\\end{array}
\stackrel{(3)}{\varphi}_{\mu}\,=\,p_{1\mu}\,-\,m\,\frac{q_{2\mu}}{\sqrt{q_{3}^{2}}}\,-\,\beta\,\frac{\varepsilon_{\mu\nu\lambda}\,q_{2}^{\nu}\,q_{3}^{\lambda}}{q_{2}^{2}\,\sqrt{q_{2}^{2}}}\,=\,0,\quad\mu\,=\,0,\,1,\,2\,{.}
\int_{|k|<\frac{\Lambda}{s}}\frac{d^{D}k_{1}}{(2\pi)^{D}}\int_{|k|<\frac{\Lambda}{s}}\frac{d^{D}k_{2}}{(2\pi)^{D}}\int_{\frac{\Lambda}{s}<|k|<\Lambda}\frac{d^{D}k_{3}}{(2\pi)^{D}}\int_{\frac{\Lambda}{s}<|k|<\Lambda}\frac{d^{D}k_{4}}{(2\pi)^{D}}\phi_{4f}(k_{4})\phi_{3f}(k_{3})\phi_{2s}(k_{2})\phi_{1s}(k_{1}).
\int\frac{dk^{\prime\prime}}{2\pi}\,\,\frac{1}{\sqrt{k^{2}+4}}\,\,\left(\frac{ik^{\prime\prime}+2\sigma_{1}}{ik^{\prime\prime}-2\sigma_{1}}\right)\,\,\frac{1}{(k+k^{\prime}-2k^{\prime\prime}-i\tau)}
S_{(D)}=\frac{1}{2\kappa_{(D+1)}^{2}k}\intd^{D}x\sqrt{-g_{(D)}}\left[\frac{2}{(D-2)}\left(1-e^{-(D-2)\phi}\right)R^{(D)}-2(D-1)e^{-(D-2)\phi}(\nabla\phi)^{2}\right],
{\calL}=-\,\frac{1}{4}\,F_{\mu\nu}F^{\mu\nu}+\frac{1}{12}\,H_{\mu\nu\rho}H^{\mu\nu\rho}+\frac{1}{2}\,m\,\epsilon_{\mu\nu\rho\lambda}\,A^{\mu}\partial^{\nu}B^{\rho\lambda}\,.
E^{j}{\calP}_{jq}^{ip}=E^{p}\delta_{q}^{i}\,,\\\,tilde{\varepsilon}_{i}{\calP}_{jq}^{ip}=\tilde{\varepsilon}_{q}\,\delta_{j}^{p}\,,\\\\varepsilon_{p}\,\tilde{E}^{q}\,{\calP}_{jq}^{ip}=\tilde{E}^{i}\varepsilon_{j}\,,
\langle\psi^{-}|M^{2^{\times}}|\hat{\bboxp}j_{3},{\mathsf{s}}_{R}j^{-}\rangle=\frac{i}{2\pi}\int_{-\infty}^{+\infty}d{\mathsfs}\,{\mathsfs}\langle\psi^{-}|\hat{\bboxp}j_{3}{\mathsfs}j^{-}\rangle\frac{1}{{\mathsfs}-{\mathsfs}_{R}}={\mathsf{s}}_{R}\langle\psi^{-}|\hat{\bboxp}j_{3},{\mathsf{s}}_{R}j^{-}\rangle\quad\mathrm{for~every~}\psi^{-}\in\Phi_{+}\subset{\mathcalH}\subset\Phi_{+}^{\times}.
\delta_{W}S_{L}=-\frac\ell{32\piG}\int_{|r|=\bar{r}}T_{ab}\delta_{W}\stackrel{(0)}g\strut^{ab}\sqrt{\stackrel{(0)}{\bfg}}\,d\varphidt=-\frac\ell{16\piG}\int_{|r|=\bar{r}}\stackrel{(0)}{\calR}\sqrt{\stackrel{(0)}{\bfg}}\,\delta\alpha\,d\varphidt\qquad.
M_{W}(t)/M_{W}(t_{0})\approx1-\frac{m_{B}^{2}}{18k^{2}}{\frac{\rho_{4}(T)}{M_{pl}^{2}k^{2}e^{-4kb_{0}}}}(1+\frac{3P_{4}(T)}{2\rho_{4}(T)})+\frac{\deltab(t)}{b_{0}}.
\frac{Fl^{4}}{E^{2}(x-y)^{4}}\left[\frac{1}{G}(\frac{\mathrm{d}x}{\mathrm{d}\tau})^{2}+\frac{1}{F}(\frac{\mathrm{d}y}{\mathrm{d}\tau})^{2}\right]=\frac{E^{2}-V^{2}}{E^{2}}
{\calM}_{ab}^{V}=O^{T}(\theta)\left(\begin{array}{cccc}{0}&{M_{+-}}&{0}&{0}\\{M_{+-}}&{0}&{0}&{0}\\{0}&{0}&{M_{ZZ}}&{0}\\{0}&{0}&{0}&{M_{AA}}\\\end{array}\right)O(\theta)\quad\mathrm{with}\quad\begin{array}{lcl}{M_{+-}}&{=}&{M_{W}^{2}+\deltaM_{W}^{2}}\\{M_{ZZ}}&{=}&{M_{Z}^{2}+\deltaM_{Z}^{2}}\\{M_{AA}}&{=}&{0}\\{\theta}&{=}&{0+\delta\theta}\\\end{array}
\alpha_{i}^{a}=\sqrt{\frac{\omega}{2\hbarg}}\left[\hat{q}_{i}^{a}+i\frac{g}{\omega}\hat{p}_{i}^{a}\right]\\,\\{\alpha_{i}^{a}}^{\dagger}=\sqrt{\frac{\omega}{2\hbarg}}\left[\hat{q}_{i}^{a}-i\frac{g}{\omega}\hat{p}_{i}^{a}\right]\\,
I=\frac{1}{\pi}\intd^{2}x[-\frac{1}{2}\bar{\partial}\chi^{\alpha}\partial\chi^{\alpha}+\frac{1}{2}h\partial\chi^{\alpha}\partial\chi^{\alpha}+\frac{1}{3}bd^{\alpha\beta\gamma}\partial\chi^{\alpha}\partial\chi^{\beta}\partial\chi^{\gamma}],
d_{abc}\hat{\xi}_{a}\hat{\xi}_{b}=2t_{\alpha\beta}^{a}\left(\frac{n}{6}+\frac{1}{4}\right)a_{\alpha}^{\dagger}a_{\beta}=\frac{\left(\frac{n}{3}+\frac{1}{2}\right)}{\sqrt{\frac{1}{3}n^{2}+n}}\hat{\xi}_{a}.
2(\partial\Phi)^{2}-\nabla^{2}\Phi={\frac{(p-3)}{4(p+2)!}}e^{2\Phi}G^{2}+\left({\frac{\Lambda\rho_{\perp}}{\sqrt{-g}}}\right)e^{\Phi}V(T)\sqrt{-A}\left[1-{\frac{1}{4}}(A^{-1})^{\lambda\sigma}g_{\lambda\sigma}\right]\,.
\psi_{\pi\varphi\ldots}(\stackrel{\rightharpoonup}{x_{1}},\ldots,\stackrel{\rightharpoonup}{x_{N}})\rightarrow(-1)^{\#\pi-\mathrm{indices}}\psi_{\pi\varphi\ldots}(-\stackrel{\rightharpoonup}{x_{1}},\ldots,-\stackrel{\rightharpoonup}{x_{N}})
\left|Q^{(1)}\vertE,p^{3}\rangle\right|^{2}=\langleE,p^{3}\vert\left(Q^{(1)}\right)^{2}\vertE,p^{3}\rangle=\langleE,p^{3}\vertH^{\prime}-P^{3}\vertE,p^{3}\rangle=E-p^{3}=0.
I=-{\frac{1}{16\piG_{5}}}\intd^{5}x\sqrt{-g^{5}}\left[R+{\frac{12}{l^{2}}}-{l^{2}}F^{2}-{\frac{l^{3}}{6\sqrt{3}}}\epsilon^{\mu\alpha\beta\gamma\delta}A_{\mu}F_{\alpha\beta}F_{\gamma\delta}\right].
\bar{V}_{i}=(\bar{a}_{4}^{i}-\bar{a}_{5}^{i})\,\bar{Q}^{1/2}\Big[\begin{array}{cc}{0}&{0}\\{1}&{0}\\\end{array}\Big]\,\bar{Q}^{1/2}\Big[\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\Big],
Q_{1}~=~{\frac{1}{2}}~\sigma_{1}~p~+~{\frac{1}{2}}~\sigma_{2}~W^{\prime}(x)~,~~Q_{2}~=~{\frac{1}{2}}~\sigma_{2}~p~-~{\frac{1}{2}}~\sigma_{1}~W^{\prime}(x)~.
{\calN}_{1}=\left(A_{1}^{+\;f,g}-z_{1}^{\ast}\right)\left(A_{1}^{f,g}-z_{1}\right),\quad{\calN}_{2}=\left(A_{2}^{+\;\alpha,\beta}-z_{2}^{\ast}\right)\left(A_{2}^{\alpha,\beta}-z_{2}\right).
f^{(1)}(x,y)=\left(\begin{array}{cccccc}{0}&{-1}&{0}&{0}&{0}&{q\partial_{y}}\\{1}&{0}&{0}&{0}&{0}&{q}\\{0}&{0}&{0}&{0}&{0}&{q^{2}(a-1)}\\{0}&{0}&{0}&{0}&{-1}&{q^{2}}\\{0}&{0}&{0}&{1}&{0}&{\partial_{y}}\\{-q\partial_{x}}&{-q}&{-q^{2}(a-1)}&{-q^{2}}&{-\partial_{x}}&{0}\\\end{array}\right)\delta(x-y),
P_{-}(X^{0})=P_{-}=N\int_{\nu/H}^{\mu/H}dr\frac{(H^{2}r^{2}-1)^{2}+H^{2}L^{2}}{(1-H^{2}r^{2})^{3/2}\sqrt{1-H^{2}r^{2}-(L/r)^{2}}}.
\hat{\phi}(x^{\mu})=\int_{(\infty)}\frac{d^{3}\vec{k}}{(2\pi)^{3}2\omega(\vec{k}\,)}\left[a(\vec{k}\,)e^{-ik\cdotx}+a^{\dagger}(\vec{k}\,)e^{ik\cdotx}\right]\.
\pi={\calP}+\frac{i}{W}\frac{\partialW}{\partialz}(\psi\overline{{\psi}}-\frac{3}{2}),\overline{{\pi}}=\overline{{{\calP}}}-\frac{i}{W}\frac{\partialW}{\partial\overline{{z}}}(\psi\overline{{\psi}}-\frac{1}{2}),
\{f,g\}={\frac{\partialf}{\partial\Pi}}\cdot{\frac{\partialg}{\partial\nabla^{2}X}}+{\frac{\partialf}{\partialP}}\cdot{\frac{\partialg}{\partial\dot{X}}}+{\frac{\partialf}{\partialp}}\cdot{\frac{\partialg}{\partialX}}-(f\leftrightarrowg)\,.
\int{\calD}x_{d}^{I}{\calD}\psi_{\alpha}{\calD}\bar{\psi}_{\dot{\alpha}}(\prod_{i=1}^{2g_{S}-2+p}V_{i})(\prod_{a=1}^{p}\phi_{a})e^{-S(x_{d}^{I},\psi_{\alpha},\bar{\psi}_{\dot{\alpha}})}=<\prod_{a=1}^{p}\phi_{a}(p_{a})>
T_{fg}\equiv\frac{1}{2}(Q_{0})_{f\beta_{0}}(Q_{1})_{\alpha_{1}\beta_{1}}(Q_{2})_{\alpha_{2}\beta_{2}}\cdots(Q_{r})_{\alpha_{r}g}\epsilon^{\beta_{0}\alpha_{1}}\epsilon^{\beta_{1}\alpha_{2}}\cdots\epsilon^{\beta_{r-1}\alpha_{r}}.
\sqrt{3}\hat{T}_{I}={\frac{{\frac{\sqrt3}{2}}(f_{1}-f_{2})}{f_{1}+f_{2}+{\frac{1}{f_{1}f_{2}}}}}={\frac{{\frac{\sqrt3}{2}}f_{1}f_{2}(f_{1}-f_{2})}{f_{1}f_{2}(f_{1}+f_{2})+1}}.
{\calL}_{2}=-\frac{1}{2\alpha}\\\{({\partial}_{\mu}W_{1}^{\mu}-{\sqrt{2}}k{\phi}_{4})^{2}+({\partial}_{\mu}W_{2}^{\mu}-{\sqrt{2}}k{\phi}_{3})^{2}+({\partial}_{\mu}Z^{\mu}-{\frac{2{\sqrt{2}}}{{\sqrt{3}}}}k{\phi}_{2})^{2}+({\partial}_{\mu}A^{\mu})^{2}\}.
A=\frac{1}{\sqrt{1+\midz\mid^{2}}}\left(\begin{matrix}{1}&{-z^{*}}\\{z}&{1}\\\end{matrix}\right)\left(\begin{matrix}{\frac{\mid\psi_{1}\mid}{\psi_{1}^{*}}}&{0}\\{0}&{\frac{\mid\psi_{1}\mid}{\psi_{1}}}\\\end{matrix}\right)
M(\gamma)=|Z(\Gamma)|=\frac{|\int_{\Gamma}{\Omega(z)}|}{(\int_{Y}{i\,\Omega(z)\wedge{\overline{\Omega}(z)}})^{1/2}}~~=\frac{|q^{i}\int_{\Gamma_{i}}{\Omega}|}{(\int_{Y}{i\,\Omega(z)\wedge{\overline{\Omega}(z)}})^{1/2}}~~.
e_{n}H_{n-1+{\frac{D}{2}}}^{(1)\prime}(|\omega|r^{\prime})+f_{n}H_{n-1+{\frac{D}{2}}}^{(2)\prime}(|\omega|r^{\prime})-d_{n}H_{n-1+{\frac{D}{2}}}^{(1)\prime}(|\omega|r^{\prime})={\frac{(2n+D-2)\Gamma\left({\frac{D-2}{2}}\right)}{4(\pir^{\prime})^{\frac{D}{2}}|\omega|}}.
S=\intd^{D}x\sqrt{-g}\biggl\{i\overline{{\psi}}\gamma^{\mu}(x)D_{\mu}\psi+{\frac{\lambda}{2N}}\biggl[(\overline{{\psi}}\psi)^{2}+(\overline{{\psi}}i\gamma_{5}\psi)^{2}\biggr]\biggr\},
{\calL}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m_{0})\psi+\frac{1}{2}g(\bar{\psi}\gamma^{\mu}\psi)^{2}+\bar{\psi^{\prime}}(i\gamma^{\mu}\partial_{\mu})\psi^{\prime}+\frac{1}{2}g(\bar{\psi^{\prime}}\gamma^{\mu}\psi^{\prime})^{2}\;.
\begin{array}{rl}{{\calP}^{\mu}(q_{1},q_{2})=-(q_{1}+q_{2})_{\perp}^{\mu}}&{-\alpha_{1}p_{A}^{\mu}\left(1-2{\frac{|q_{1\perp}|^{2}+m^{2}}{|q_{1\perp}-q_{2\perp}|^{2}+m^{2}}}\right)}\\{}&{+\beta_{2}p_{B}^{\mu}\left(1-2{\frac{|q_{2\perp}|^{2}+m^{2}}{|q_{1\perp}-q_{2\perp}|^{2}+m^{2}}}\right)}\\\end{array}
D_{12}^{(l)}\stackrel{1}{\widehat{F}^{(l)}}\stackrel{2}{\widehat{F}^{(1)}}D_{12}^{(l)\;-1}=\left(\begin{array}{cc}{\widehat{F}^{(l+1)}}&{0}\\{0}&{-\alpha\beta\widehat{F}^{(l-1)}}\\\end{array}\right),
{\calM}_{CY,J}^{\epsilon}=S\,\omegaM_{CY}^{\epsilon}\,S^{-1}=\left(\begin{array}{rrrrrrrr}{\omega}&{}&{}&{}&{}&{}&{}&{}\\{}&{\omega}&{}&{}&{}&{}&{}&{}\\{}&{}&{\omega}&{}&{}&{}&{}&{}\\{}&{}&{}&{\omega}&{}&{}&{}&{}\\{}&{}&{}&{}&{1}&{0}&{}&{}\\{}&{}&{}&{}&{-2}&{1}&{}&{}\\{}&{}&{}&{}&{}&{}&{\omega^{2}}&{0}\\{}&{}&{}&{}&{}&{}&{-2\omega^{2}}&{\omega^{2}}\\\end{array}\right)
\timesD_{M+K+L-I}\biggl(E_{C}^{K}(f)E_{D}^{L}(g)\biggr)+D_{M+K+L-I}I_{CD}E_{A}^{M}\biggl(E_{C}^{K}(f)E_{D}^{L}(g)\biggr)\biggr]\Biggr).
{\calM}_{b}^{uu}=\frac{ig^{4}}{2}[T^{a}T^{b}\otimesT_{a}T_{b}]\int\frac{d^{2}k}{(2\pi)^{2}}\left[\frac{T(k,p_{1})T^{*}(k,p_{3})}{w_{k}-w_{p}}\right]
\phi_{4}=\frac{\lambda_{1}\phi_{2}(\beta_{2}\psi_{1}-\delta_{1}\phi_{2})}{\beta_{2}\mu_{2}\psi_{1}-\lambda_{1}\delta_{1}\phi_{2}},\qquad\psi_{3}=\frac{\mu_{2}\psi_{1}(\beta_{2}\psi_{1}-\delta_{1}\phi_{2})}{\beta_{2}\mu_{2}\psi_{1}-\lambda_{1}\delta_{1}\phi_{2}}.
e^{-2\phi^{\prime}}F^{2}=\frac{Q^{2}}{\rho^{4}}\Bigl(1-\frac{2GM}{\rho}\Bigr)^{-1}\rightarrow\frac{2G^{2}M^{2}}{\rho^{3}(2GM-\rho)}\propto\frac{1}{x}
\begin{array}{lcl}{Z_{m}^{2m}}&{=}&{1~,}\\{Z_{m}^{2m+1}}&{=}&{g_{2m+1}(1)~,}\\{Z_{m}^{2m+2}}&{=}&{Z_{m}^{2m+1}g_{2m+2}(2)Z_{m}^{2m+1}~,}\\{\vdots}&{\vdots}&{\vdots~,}\\{Z_{m}^{2m+l}}&{=}&{Z_{m}^{2m+l-1}g_{2m+l}(l)Z_{m}^{2m+l-1}~,}\\{\vdots}&{\vdots}&{\vdots~,}\\{Z_{m}^{n}}&{=}&{Z_{m}^{n-1}g_{n}(n-2m)Z_{m}^{n}~,}\\\end{array}
{\calL}^{\prime}=\left(\frac{\partial\{x_{0},x_{1},....,x_{p}\}}{\partial\{x_{0}^{\prime},x_{1}^{\prime},....,x_{p}^{\prime}\}}\right){\calL}\\Rightarrow\S^{\prime}=\intd^{p+1}x^{\prime}\{\calL}^{\prime}=\intd^{p+1}x^{\prime}J^{-1}\{\calL}=\intd^{p+1}x\{\calL}=S.
\left\langleq^{\prime\prime}|A|q^{\prime}\right\rangle=\int_{\mathcalJ}dz^{\prime\prime}\int_{\mathcalJ}dz^{\prime}\,\Psi_{z^{\prime\prime}}(q^{\prime\prime})\,\left\langle\Psi_{z^{\prime\prime}}|A|\Psi_{z^{\prime}}\right\rangle\,\Psi_{z^{\prime}}^{*}(q^{\prime})\;
h_{ij}dq^{i}dq^{j}=\frac{1}{4V^{2}}dV^{2}+\frac{1}{4V^{2}}\left[d\sigma+\mathrm{i}(\bar{\xi}d\xi-\xid\bar{\xi})\right]^{2}+\frac{1}{V}d\xid\bar{\xi}\.
e^{-i\vec{\xi}_{N}\cdot{^5\!\vec{N}}}\,:\quad\begin{array}{lcl}{Z^{0}}&{\to}&{Z^{0}-\vec{\xi}_{N}\cdot\vec{Z}+\frac{1}{2}\,{{\xi}_{N}}^{2}\,(Z^{0}-Z^{5})}\\{Z^{5}}&{\to}&{Z^{5}-\vec{\xi}_{N}\cdot\vec{Z}+\frac{1}{2}\,{{\xi}_{N}}^{2}\,(Z^{0}-Z^{5})}\\{Z^{i}}&{\to}&{Z^{i}-\xi_{N}^{i}\,(Z^{0}-Z^{5})}\\\end{array}
Y(s)=\frac{(a^{0}+\varepsilon\frac{\vec{a}\cdot\vec{P}}{\mid\vec{P}\mid})e^{\psi(0)}-(a^{0}-\varepsilon\frac{\vec{a}\cdot\vec{P}}{\mid\vec{P}\mid})(\frac{P^{0}+\varepsilon\mid\vec{P}\mid}{m})}{e^{\psi(0)}+\frac{P^{0}-\varepsilon\mid\vec{P}\mid}{m}}e^{sa\cdotP}.
+\left(\frac{1}{2}\bar{C}_{\gamma}^{\dagger}U_{\alpha\beta}^{\dagger\gamma}C^{\alpha}C^{\beta}+\frac{1}{2}C^{\dagger\beta}C^{\dagger\alpha}U_{\alpha\beta}^{\gamma}\bar{C}_{\gamma}+C^{\dagger\alpha}\bar{U}_{\alpha\beta}^{\gamma}\bar{C}_{\gamma}C^{\beta}+C^{\dagger\beta}\bar{C}_{\gamma}^{\dagger}\bar{U}_{\alpha\beta}^{\dagger\gamma}C^{\alpha}\right)(-1)^{\varepsilon_{\beta}}.
ds_{deS}^{2}=-{\frac{\ell^{2}}{4}}\left({\frac{d\rho}{\rho}}\right)^{2}+{\frac{1}{\rho}}\left[\left(1+{\frac{\ell^{2}\rho}{4}}\right)^{2}d\vec{n}\cdotd\vec{n}\right]=-{\frac{\ell^{2}}{4}}{\frac{d\rho^{2}}{\rho^{2}}}+{\frac{1}{\rho}}\,g_{ab}(x,\rho)\,dx^{a}dx^{b}\,,
\sigma_{i}F^{(n)}=\sum_{k_{1},k_{2},\ldots,k_{n}\in{\calI}}e_{k_{1}k_{2}\ldotsk_{n}}\sum_{l_{i},l_{i+1}\in{\calI}}{\hat{\calR}}\left(l_{i},l_{i+1};k_{i},k_{i+1}\right)F^{(n)}\left(k_{1},k_{2},\ldots,k_{i-1},l_{i},l_{i+1},k_{i+2},\ldots,k_{n}\right)\;.