harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
x^{\mu}-y^{\mu}+\frac{2i}{(1^{1}2_{4})}\xi_{2/4}\sigma^{\mu}\bar{\theta}^{1}\,,\qquad\xi_{2/4}\equiv[(1^{2}2_{4})\theta_{2}+(1^{3}2_{4})\theta_{3}+(1^{4}2_{4})\theta_{4}]
\left({\calD}^{k}|_{\sigma}{\bfG}^{\mu\nu}(\sigma,\sigma^{\prime})\right)_{\sigma=\sigma^{\prime}}=-\left({\calD}^{k}|_{\sigma^{\prime}}{\bfG}^{\mu\nu}(\sigma,\sigma^{\prime})\right)_{\sigma=\sigma^{\prime}}+O(\sqrt{s})
e^{\alphax(t)}=e^{\alphat}(1+\alpha(\sum_{f=1}^{\infty}e^{ft}\sum_{\sigma_{f}\inP_{f}}(-1)^{l(\sigma_{f})}(\alpha+f)^{l(\sigma_{f})-1}S(\sigma_{f})\prod_{j=1}^{l(\sigma_{f})}\frac{b_{d_{j}}}{d_{j}}))
i\int_{0}^{\infty}\frac{f\left(iy\right)-f\left(-iy\right)}{e^{2\piy/h}-1}=-2\left(\frac{r}{\beta}\right)^{3}\,\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n-1}}{\left(2n-1\right)!}\,\zeta\left(2-2n\right)G_{n}\left(m_{eff}\beta\right),
S=S_{0}+S_{1}+\intd^{2}x\left[c^{\ast}\left(c^{\prime}c+\phi^{\prime\,\,2}l^{\prime}l\right)+l^{\ast}\left(l^{\prime}c+2c^{\prime}l\right)+2\phi^{\ast}h^{\ast}\phi^{\prime}l^{\prime}l\right],
\frac{c_{2}^{-\nu}}{c_{0}^{-\nu}}=\frac{1}{\displaystyleh^{-2}[s^{2}-(-\nu+2)^{2}]-\frac{1}{\displaystyleh^{-2}[s^{2}-(-\nu+4)^{2}]-\frac{1}{\displaystyleh^{-2}[s^{2}-(-\nu+6)^{2}]\cdot\cdot\cdot}}}
\int_{C}{d^{n}}{\eta}=a^{n}\int_{0}^{2\pi}d\theta_{1}\int_{0}^{\pi}d\theta_{2}sin\theta_{2}\int_{0}^{\pi}d\theta_{3}sin^{2}\theta_{3}...\int_{0}^{\theta_{n}^{0}}d\theta_{n}sin^{n-1}\theta_{n}
\tilde{A}_{x}=2\frac{y}{z_{0}}\frac{H_{40}f_{1}}{\sqrt{c_{0}^{2}z_{0}^{2}\bar{z}^{2}+f_{1}^{2}(x^{2}+y^{2})}}\,\\\tilde{A}_{y}=-2\frac{x}{z_{0}}\frac{H_{40}f_{1}}{\sqrt{c_{0}^{2}z_{0}^{2}\bar{z}^{2}+f_{1}^{2}(x^{2}+y^{2})}}\.
A_{\mu}=\mp\,{\frac{2i}{g}}\sum_{I=1}^{K}\,[\,H_{I}\,{\frac{v_{\mu}^{I}}{\rho_{I}}}+g\,\kappa\,E_{I\,K+1}^{\pm}\,R_{\mu}^{I}\,\prod_{I=1}^{K-1}\delta(R^{K}\cdotR^{I})\,].
G_{N}\geG_{N}^{crit,lin}\equiv\frac{\kappa^{2}}{8\piV_{1}x_{\mathrm{lin}}^{11}}=\frac{\alpha_{1}^{2}}{16\pi^{2}}\left|\int_{CY_{3}}\omega\wedge\left(\frac{\mathrm{tr}F^{2}-\frac{1}{2}\mathrm{tr}R^{2}}{8\pi^{2}}\right)\right|\;.
T=\frac{i\gamma}{2}\sum_{j=1}^{2}\int_{U_{j}}\left(2i(\partial_{\bar{z}}\zeta_{z}-\partial_{z}\zeta_{\bar{z}})+\partial_{t}(\partial_{z}h\partial_{\bar{z}}\chi+\partial_{\bar{z}}h\partial_{z}\chi)-\dot{\chi}\Omega^{2}\right)dz\wedged\bar{z}
\tilde{\Omega}_{\lambdal}(z,\sigma)\equiv\frac{s_{l}^{(\lambda)}(z\sigma)/s_{l}^{(\lambda)}(z)}{e_{l}^{(\lambda)}(z\sigma)s_{l}^{(\lambda)}(z)-e_{l}^{(\lambda)}(z)s_{l}^{(\lambda)}(z\sigma)}.
|R_{\ell}|^{2}\sim{\frac{1}{v}}\;{\frac{|\Gamma(\ell+1+i\eta)|^{2}}{(\Gamma(2\ell+1))^{2}}}\;(2kr)^{2\ell}e^{-\pi\eta}={\frac{m}{\hbar}}\;{\frac{2^{2\ell}}{(\Gamma(2\ell+1))^{2}}}\;e^{-\pi\eta}k^{2\ell-1}r^{2\ell}|\Gamma(\ell+1+i\eta)|^{2}
\left[F,\pi_{a_{2k+1}}\right]=\left[F,\nu_{a_{2k+1}}^{a_{2j}}\right]\gamma_{a_{2j}}+\left[F,\gamma_{a_{2j}}\right]\nu_{a_{2k+1}}^{a_{2j}}\approx0,\;k=0,\cdots,\Lambda,
{\mathcalL}_{+}=h^{\mu}\partial_{\mu}+\lambda_{\mathbfi}^{\alpha}D_{\alpha}^{\mathbfi}\,\qquad\qquad{\mathcalL}_{-}={\bar{h}}^{\mu}\partial_{\mu}+{\bar{\lambda}}_{\dot{\alpha}}^{\mathbfi}{\bar{D}}_{\mathbfi}^{\dot{\alpha}}\.
\begin{array}{rl}{}&{<q_{f}|\hat{U}(t_{f},t_{i})|q_{i}>=}\\{}&{}\\{}&{=\int_{-\infty}^{\infty}\prod_{\alpha=1}^{N-1}dq_{\alpha}\,\prod_{\alpha=0}^{N-1}dp_{\alpha}\,\prod_{\alpha=0}^{N-1}\left[<q_{\alpha+1}|p_{\alpha}><p_{\alpha}|e^{-\frac{i}{\hbar}\epsilon\hat{H}_{0}}|q_{\alpha}>\right]\.}\\\end{array}
S_{0}^{\mathrm{NA}}={\frac{\kappa}{2}}\intd^{3}x\,\epsilon^{\mu\nu\rho}\,\left(A_{\mu}^{i}\,\partial_{\nu}\,A_{\rho}^{i}+{\frac{1}{3}}\epsilon_{ijk}\,A_{\mu}^{i}A_{\nu}^{j}A_{\rho}^{k}\right)\,.
\left[-\gamma_{44}E^{2}+2iE\gamma_{4i}\vec{p}_{i}+\gamma_{ij}\vec{p}_{i}\vec{p}_{j}+m^{2}\right]u_{1}^{\sigma}(\vec{p})=0\quad(\mathrm{or}\,\,v_{1}^{\sigma}(\vec{p}))\quad,
\omega=\omega_{+}\equiv\sum_{j}\frac{(r^{2}-r_{j}^{2})dr^{3}-(r^{3}-r_{j}^{3})dr^{2}}{|\vec{r}-\vec{r_{j}}|(-r^{1}+r_{j}^{1}+|\vec{r}-\vec{r_{j}}|)}~,
S_{p}=-T_{p}V_{p}\intdt\,r^{p}\sqrt{1+{\frac{r^{2}}{\tilde{L}^{2}}}-{\frac{(\partial_{t}r)^{2}}{1+{\frac{r^{2}}{\tilde{L}^{2}}}}}-L^{2}h_{ij}\partial_{t}\varphi^{i}\partial_{t}\varphi^{j}}+T_{p}V_{p}\intdt\,{\frac{r^{p+1}}{\tilde{L}}},
\frac{\partialL^{*}(x,{\dot{x}};{\xi})}{\partial{\dot{x}}^{\mu\nu}}\Big\vert_{x=x({\xi})}=0\,\rightarrow\Pi_{\mu\nu}({\xi})=\partial_{\,[\mu}S\partial_{\nu]}U\,\rightarrowL^{*}(x,{\dot{x}};{\xi})\Big\vert_{x=x({\xi})}=0\.
y^{\alpha_{1}}\longleftrightarrow\psi_{\;A}^{\alpha},\;z_{\alpha_{1}}\longleftrightarrow\bar{\psi}_{\alpha}^{\;A},\;\sigma_{\;\;\beta_{1}}^{\alpha_{1}}\longleftrightarrow-\mathrm{i}\left(\gamma^{0}\right)_{\;\;\beta}^{\alpha}\delta_{A}^{\;\;B},
\left(\begin{array}{l}{f_{n}(r)}\\{g_{n}(r)}\\\end{array}\right)=\left(\begin{array}{c}{C_{n}^{(1)}(E)J_{n-\Phi^{(0)}}(kr)+C_{n}^{(2)}(E)Y_{n-\Phi^{(0)}}(kr)}\\{{\frac{ik}{E+m}}\bigl[C_{n}^{(1)}(E)J_{n+1-\Phi^{(0)}}(kr)+C_{n}^{(2)}(E)Y_{n+1-\Phi^{(0)}}(kr)\bigr]}\\\end{array}\right),
\begin{array}{l}{\beta^{T}=-\frac{1}{2}\alpha^{\prime}D^{2}T\,}\\{\beta_{\mu\nu}^{G}=\alpha^{\prime}R_{\mu\nu}+\frac{1}{2}\alpha^{2}R_{\mu\alpha\beta\lambda}R_{\nu}^{\\alpha\beta\lambda}\,}\\{\beta^{\phi}=\frac{1}{6}(D-25)-\frac{1}{2}\alpha^{\prime}D^{2}\phi+\frac{1}{16}\alpha^{2}(R_{\mu\nu\alpha\lambda})^{2}\.}\\\end{array}
D_{\nu}(z)=\sqrt{{\frac{2}{\pi}}}2^{\nu}e^{-{\frac{\pi}{2}}\nui}e^{{\frac{z^{2}}{4}}}\int_{-\infty}^{\infty}x^{\nu}e^{-2x^{2}+2ixz}dx,\;\;\mathrm{Re}\,\nu>-1.
{4M^{2}}\,\left[D\epsilon^{\beta\gamma\kappa\lambda}\partial_{\nu}{\partial_{\beta}}T_{(\kappa\lambda)}{}^{\nu}+2a\partial_{\nu}{\partial_{\beta}}T^{(\beta\gamma)\nu}\right]=e^{2}\,\partial_{\nu}\left({\partial}^{\gamma}\lambda^{\nu}-{\partial}^{\nu}\lambda^{\gamma}\right).
\begin{array}{c}{e_{L13}=\frac{1}{2}(-e_{R13}+e_{R7}+e_{R26}+e_{R45}),}\\{e_{L26}=\frac{1}{2}(+e_{R13}+e_{R7}-e_{R26}+e_{R45}),}\\{e_{L45}=\frac{1}{2}(+e_{R13}+e_{R7}+e_{R26}-e_{R45}).}\\\end{array}
G_{(2,1)}^{c}:=\{G_{2}^{c},\,G_{1}^{c}\}-\left({\dot{\lambda}}^{\nu_{0}}(B_{1\nu_{0}}^{\mu_{0}}B_{2\mu_{0}}^{\sigma_{0}}-B_{2\nu_{0}}^{\mu_{0}}B_{1\mu_{0}}^{\sigma_{0}})+...\right)\phi_{\sigma_{0}}.
{\smallg_{\alpha\beta}=\left[\begin{array}{ccccc}{1}&{0}&{0}&{0}&{0}\\{0}&{g_{1}+w_{1}^{\2}h_{3}+n_{1}^{\2}h_{4}}&{w_{1}w_{2}h_{3}+n_{1}n_{2}h_{4}}&{w_{1}h_{3}}&{n_{1}h_{4}}\\{0}&{w_{2}w_{1}h_{3}+n_{1}n_{2}h_{4}}&{g_{2}+w_{2}^{\2}h_{3}+n_{2}^{\2}h_{4}}&{w_{2}h_{3}}&{n_{2}h_{4}}\\{0}&{w_{1}h_{3}}&{w_{2}h_{3}}&{h_{3}}&{0}\\{0}&{n_{1}h_{4}}&{n_{2}h_{4}}&{0}&{h_{4}}\\\end{array}\right],}
(\Psi_{1}\star\Psi_{2})[z(\sigma)]\equiv\int\Psi_{1}[x(\sigma)]\Psi_{2}[y(\sigma)]\prod_{\pi/2\leq\sigma\leq\pi}\delta\lbrackx(\sigma)-y(\pi-\sigma)]\,\,dx(\sigma)\,dy(\pi-\sigma)\,\,,
\phi_{G}^{\#}((f,g)^{a})=G_{b}^{a}(\phi_{G}^{\#}(f),\phi_{G}^{\#}(g))^{b}\,,\qquad\phi_{G}^{\#}(V^{a}f)=G_{b}^{a}V^{b}(\phi_{G}^{\#}(f))\,,
\begin{array}{c}{c(1)={4l(1)+6l(2)+s(1)}}\\{c(2)={4l(1)+2l(5)+s(2)}}\\{c(3)={6l(2)+2l(5)+s(3)}}\\{c(4)={4l(5)+s(4)}\qquad\quad\;}\\{c(5)={4l(4)+2l(5)+s(5)}}\\{c(6)={4l(4)+2l(5)+s(6)}}\\{c(7)={2l(1)+2l(4)+s(7)}}\\{c(8)={6l(3)+2l(4)+s(8)}}\\{c(9)={2l(1)+6l(3)+s(9)}}\\\end{array}
K^{(4)}=K_{IK\bar{J}\bar{L}}\bigl[{\frac{2}{3}}\,\Phi^{I}\Phi^{K}\bar{\Phi}^{\bar{J}}\bar{\Phi}^{\bar{L}}\,+\,{\frac{1}{3}}\,\Phi^{I}\bar{\Phi}^{\bar{J}}\Phi^{K}\bar{\Phi}^{\bar{L}}\bigr].
\left<\left(\begin{array}{l}{\varphi_{\varphi}}\\{\varphi_{\pi}}\\\end{array}\right)_{1}\Bigg|\left(\begin{array}{l}{\varphi_{\varphi}}\\{\varphi_{\pi}}\\\end{array}\right)_{2}\right>=-i\int(\varphi_{\pi1}^{*}\varphi_{\varphi2}-\varphi_{\varphi1}^{*}\varphi_{\pi2})\d^{3}\stackrel{\rightharpoonup}{x}
\begin{array}{|c|cccccc|}\hline{\mathrm{NS~State}}&{\epsilon_{R_{1}}}&{\epsilon_{R_{2}}}&{\epsilon_{\Omega,9}}&{\epsilon_{\Omega,5_{1}}}&{\epsilon_{\Omega,5_{2}}}&{\epsilon_{\Omega,1}}\\\hline{\psi_{-{\frac{1}{2}}}^{2,3,4,5}}&{+}&{-}&{-}&{-}&{+}&{+}\\{\psi_{-{\frac{1}{2}}}^{6,7,8,9}}&{-}&{+}&{-}&{+}&{-}&{+}\\\hline{}\\\end{array}
D_{0}(x)=\frac{1}{2\pi}\delta(x^{2})\,\mathrm{sgn}(x^{0})=\frac{i}{(2\pi)^{3}}\int\!\!d^{4}p\,\delta(p^{2})\,\mathrm{sgn}(p^{0})\,e^{-i\,p\cdotx}\,.
W_{\tiny\mathrm{div}}[g,\beta,D]=-{\frac{\eta}{2}}\int_{0}^{\infty}{\frac{ds}{s}}e^{-m^{2}s}{\frac{1}{(4\pis)^{D/2}}}\left(B_{0}+sB_{1}+s^{2}B_{2}\right)~~~.
\phi^{I}={\frac{1}{\sqrt{2}}}\left(\varphi^{I}+i\varphi^{I+3}\right)\quad\quad\phi_{I}^{\dagger}={\frac{1}{\sqrt{2}}}\left(\varphi^{I}-i\varphi^{I+3}\right)\qquadI=1,2,3\;,
Z_{\mathrm{1-loop}}^{(c)}(M)={\frac{(2\pix)^{{\frac{1}{2}}(\mathrm{dim}\,H_{A^{(c)}}^{0}-\mathrm{dim}\,H_{A^{(c)}}^{1})}}{\mathrm{vol}(H_{c})}}\mathrm{e}^{-{\frac{1}{x}}S_{\mathrm{CS}}(A^{(c)})-{\frac{i\pi}{4}}\varphi}{\sqrt{|\tau_{R}^{(c)}|}},
\tilde{V}(\tilde{q})=\tilde{V}_{0}(\tilde{q})+\epsilong^{2}\tilde{V}_{1}(\tilde{q}),\quad\tilde{V}_{0}(\tilde{q})=\frac{1}{2}\tilde{q}^{2}(1-\tilde{q})^{2},\quad\tilde{V}_{1}(\tilde{q})=-\tilde{q}.
\tilde{\Psi}_{\mathrm{far}}(r)={\frac{1}{r}}\left[\left(A_{\mathrm{in}}+A_{\mathrm{out}}\right)e^{\sqrt{1-s}}+i\left(A_{\mathrm{in}}-A_{\mathrm{out}}\right)e^{-\sqrt{1-s}}\right].
{\calL}=-{\frac{1}{4}}\mathrm{F}_{\mu\nu}\mathrm{F}^{\mu\nu}+\bar{\psi}i{\hbar}{\gamma}^{\mu}{\partial}_{\mu}\psi+e_{+}\bar{\psi}_{+}{\gamma}^{\mu}{\psi_{+}}A_{\mu}+e_{-}\bar{\psi}_{-}{\gamma}^{\mu}{\psi_{-}}A_{\mu},
Z\equiv\left(\begin{array}{cc}{vx^{1}}&{u^{-1}x^{4}}\\{ux^{2}}&{v^{-1}x^{3}}\\\end{array}\right)\quad,\quadX=\left(\begin{array}{cc}{x^{1}}&{qx^{4}}\\{qx^{2}}&{x^{3}}\\\end{array}\right)
\tilde{\calA}\vphantom{\calA}_{n,TT}^{z}+\frac{4\alpha}{1+\alpha}(T+\tilde{\Lambda})^{-1}\tilde{\calA}_{n,T}^{z}+\left[n^{2}-\frac{2\alpha(1-\alpha)}{(1+\alpha)^{2}}(T+\tilde{\Lambda})^{-2}\right]\tilde{\calA}_{n}^{z}=0,
{\calE}=\frac{\sqrt{1-\lambda_{8}^{2}}}{\lambda_{8}}\left(v+iu\right)=\frac{(1-\lambda_{8}^{2}){\calE}_{0}}{1-\lambda_{8}^{2}Re{\calE}_{0}},\;\;\;z=i\left(1-\lambda_{8}^{2}{\calE}_{0}\right).
{\calE}_{-}=\frac{Lm^{3}\lambda}{8\pi}\sum_{j=0}^{\infty}\frac{1}{j!}(2eB\lambda^{2})^{(-j)}\sum_{k=1}^{\infty}\frac{(2k+j-1)!}{(2k)!}\frac{{\calB}_{2k+2j}}{(2k+j-1/2)(2k+j-3/2)}\left(\frac{2eB}{m^{2}}\right)^{2k+j}
y^{2}=x^{3}+\sum_{k=-4}^{4}\sum_{l=0}^{8-nk}f_{kl}z^{\primel}z^{4+k}p^{8-l-qk}x+\sum_{k=-6}^{6}\sum_{l=0}^{12-nk}g_{kl}z^{\primel}z^{6+k}p^{12-l-qk}
\displaystyle\frac{\vartheta_{\sigma}^{(n)}(w^{\prime},x^{-1}z_{2n-1}|\zeta^{\prime\prime},\zeta_{2n-1},\epsilonx^{-1}\zeta_{2n-1})}{\vartheta_{\epsilon\sigma}^{(n-1)}(w^{\prime}|\zeta^{\prime\prime})}=\displaystylex^{2n}z_{2n-1}^{-2n}\prod_{j=1}^{2n-2}z_{j}^{-1}\prod_{a\inA\atopa\neqa_{2n}}w_{a}^{-1}\Theta_{x^{2}}(xw_{a}/z_{2n-1}).
{\calL}^{D+M}=-{\frac{1}{2}}\sum_{l^{\prime},\,l=e,\mu,\tau\ldots}\overline{{(\nu_{l^{\prime}\,L})^{c}}}M_{l^{\prime}l}^{L}\nu_{l\,L}\,-\sum_{l^{\prime},\,l=e,\mu,\tau\ldots}\bar{\nu}_{l^{\prime}\,R}M_{l^{\prime}l}^{D}\nu_{l\,L}\,-\,{\frac{1}{2}}\sum_{l^{\prime},\,l=e,\mu,\tau\ldots}\overline{{\nu_{l^{\prime}\,R}}}M_{l^{\prime}l}^{R}(\nu_{l\,R})^{c}+h.c.
G_{\mu\rho}^{ac}(\theta)\,M_{cb}^{\rho\nu}(\theta)\;=\;\delta_{\mu}^{\nu}\delta_{b}^{a}\;\;,\;\;M_{ab}^{\mu\nu}\,=\,g^{\mu\nu}\delta_{ab}+g^{-\frac{1}{2}}\epsilon^{\mu\lambda\nu}f^{acb}\theta_{\lambda}^{c}\;.
=S\left(\frac{Z_{1}}{Z_{3}}A,\frac{Z_{1}}{Z_{3}^{2}}B,\phi,\tilde{Z}_{3}\psi,\frac{Z_{1}^{2}}{Z_{3}^{3}}g^{2},Z_{3}\alpha\right)+C\left(\frac{Z_{1}}{Z_{3}}A,\frac{Z_{1}}{Z_{3}^{2}}B,\phi,\tilde{Z}_{3}\psi,\frac{Z_{1}^{2}}{Z_{3}^{3}}g^{2},Z_{3}\alpha,R\right)
G^{\mathrm{ret}}(t,{\bfx})=\frac{1}{16\pi^{3}}\left(\frac{\delta^{\prime\prime}(t-r)}{r^{3}}+3\frac{\delta^{\prime}(t-r)}{r^{4}}+3\frac{\delta(t-r)}{r^{5}}\right)\,\,,\,\,\,\,\,D=8\,.
\Gamma_{g.f.}+\Gamma_{\phi\pi}\longrightarrow\Gamma_{g.f.}+\Gamma_{\phi\pi}+\chi\int\left\{\frac12\bar{c}B+(\partial_{\xi}\xi_{A})m\bar{c}\varphi_{2}-(\partial_{\xi}\hat{\xi}_{A})m\bar{c}\hat{\varphi}_{2}\right\}\;.
F_{\mu\nu}=\mathrm{tr}\,\Psi^{\dagger}\left(\partial_{\left[\mu\right.}\mathcal{D}_{z}^{\dagger}\frac{1}{\mathcal{D}_{z}\mathcal{D}_{z}^{\dagger}}\partial_{\left.\nu\right]}\mathcal{D}_{z}\right)\Psi\,.
H\,=\,\frac{\rho_{0}l}{2\pi}\int_{0}^{\pi}d\sigma\left[\left(\dot{{\bfx}}_{1}^{2}\,+\,a^{2}{\bfx}_{1}^{2}\right)\,-\,\left(\dot{{\bfx}}_{2}^{2}\,-\,a^{2}{\bfx}_{2}^{2}\,-\,\frac{a^{2}}{\epsilon}{\bfx}_{2}^{2}\,-\,2{\bfx}_{2}\ddot{{\bfx}}_{2}\right)\right].
S_{eff}=\int\left(-\frac{1}{4}\sum_{n}(e^{-2kR}\partial_{\mu}b_{\rho\sigma}^{\primen}\partial^{\mu}b^{\primen,\rho\sigma}+{m_{n}^{2}}b_{\rho\sigma}^{\primen}b^{\primen,\rho\sigma})-\frac{1}{2}e^{2kR}\partial_{\mu}\phi^{\prime}\partial^{\mu}\phi^{\prime}\right)dz,
ds_{{\calM}_{\mathrm{IR}}}^{2}={\frac{1}{8\pi^{2}g_{\mathrm{YM}}^{2}}}\left[{\frac{3}{4}}du^{2}+u^{2}\left({\frac{4}{3}}\sigma_{3}^{2}+{\sigma_{1}^{2}+\sigma_{2}^{2}}\right)\right]\.
I_{\bigtriangledown}\sim\int_{Z}\mathrm{Tr}(\alpha_{1}(\vertF_{A}\vert^{2}+\vertF_{A^{\prime}}\vert^{2})+\alpha_{2}\vertD\phi\vert^{2}+\alpha_{3}(\vert\phi+1\vert^{2}-1)^{2}+\cdots)
Z_{one-loop}=\int{\frac{d^{2}\tau}{(\mathrm{Im}\tau)^{{\frac{D}{2}}+1}|\eta|^{24}}}\sum_{\alpha,\beta}C\Big[\begin{array}{c}{\alpha}\\{\beta}\\\end{array}\Big]\,\prod_{f\inF^{\prime}}\Theta^{1/2}\Big[\begin{array}{c}{\alpha(f)}\\{\beta(f)}\\\end{array}\Big](\tau).
\omega_{\sigma}^{2}(q)=q^{2}\frac{1+\frac{2}{\pi}V_{(0)}^{11}+\frac{1}{2\pi^{2}}\{(U_{(0)}^{11})^{2}-2U_{(0)}^{11}U_{(1)}^{11}\}}{1+\frac{2}{\pi}V_{(1)}^{11}-\frac{1}{2\pi^{2}}(U_{(1)}^{11})^{2}},
\begin{array}{l}{J_{+}^{\prime}=\displaystyle\sqrt{\frac{[J_{3}+J_{0}]_{q}[J_{3}-1-J_{0}]_{q}}{(J_{3}+J_{0})(J_{3}-1-J_{0})}}J_{+}~,}\\{J_{-}^{\prime}=\displaystyleJ_{-}\sqrt{\frac{[J_{3}+J_{0}]_{q}[J_{3}-1-J_{0}]_{q}}{(J_{3}+J_{0})(J_{3}-1-J_{0})}}~,}\\{J_{3}^{\prime}=J_{3}~.}\\\end{array}
ds^{2}=\frac{1}{4}\left(\frac{b^{2}}{K^{2}}dK^{2}+a^{2}\sigma_{1}^{2}+b^{2}\sigma_{2}^{2}+c^{2}\sigma_{3}^{2}\right)+\frac{1}{2}\frac{EK}{M+mEK}\left(d\phi-\frac{k^{\prime}K}{E}\sigma_{1}\right)^{2},
n_{(r,s,l),(r_{1},s_{1},l_{1})}^{(r_{2},s_{2},l_{2})}=\sum_{r^{\prime}=1}^{m-1}\sum_{s^{\prime}=1}^{m+1}\sum_{l^{\prime}=0}^{2}\frac{S_{(r,s,l)}^{(r^{\prime},s^{\prime},l^{\prime})}S_{(r_{1},s_{1},l_{1})}^{(r^{\prime},s^{\prime},l^{\prime})}(S^{-1})_{(r_{2},s_{2},l_{2})}^{(r^{\prime},s^{\prime},l^{\prime})}}{S_{(1,1,0)}^{(r^{\prime},s^{\prime},l^{\prime})}}\in\mathbb{N}_{0}.
z_{\alpha\beta}=m\left(\begin{matrix}{\gamma_{ab}^{0}\delta_{qp}+{\calP}_{aq~bp}^{(-)}}&{0}\\{0}&{{\calP}^{(-)}{}_{\dot{q}\dot{p}}^{ab}}\\\end{matrix}\right)=\Sigma_{i=1}^{24}\lambda_{\alpha}^{i}\lambda_{\beta}^{i}\;,
I_{o}=\intd^{3}x[-\mu\epsilon^{mnp}A_{m}^{i}\partial_{n}a_{p}^{i}-\frac{1}{2}\mu^{2}a_{m}^{i}a^{mi}-\mu\epsilon^{mnp}A_{m}^{i}\partial_{n}v_{p}^{i}+\frac{1}{2}\mu\epsilon^{mnp}v_{m}^{i}\partial_{n}v_{p}^{i}].
=-\left[(5\,h_{0}+1)\,h_{0}\,\kappa\left(\frac{2}{3}h+\frac{2}{15}-\frac{22+5c}{240}\right)+18\,q\,q_{0}\,(h_{0}+1)\right]\;\zeta^{-1}\;\left\langle\phi^{\prime},\left(\phi\otimes\phi_{0}\right)\right\rangle,
s_{Q}\Upsilon_{\mathrm{T}}=\intd^{4}x\,\mathrm{tr}\Bigr\{-A_{\mu}^{*}D^{\mu}C-\psi_{\mu}^{*}(D^{\mu}\phi+\{C,\psi^{\mu}\})+C^{*}C^{2}+\phi^{*}[C,\phi]\Bigr\}.
det\left|\begin{array}{ccc}{Q_{1}(\lambda-i)}&{Q_{1}(\lambda)}&{Q_{1}(\lambda+i)}\\{P_{1}(\lambda-i)}&{P_{1}(\lambda)}&{P_{1}(\lambda+i)}\\{R_{1}(\lambda-i)}&{R_{1}(\lambda)}&{R_{1}(\lambda+i)}\\\end{array}\right|=\lambda^{N}.
\begin{array}{c}{S=\frac1{16\piG_{p+1}}\intd^{p+1}y\sqrt{-\tilde{g}}e^{-x^{0}}\{-\partial^{A}x^{0}\partial_{A}x^{0}+a^{i}a^{j}\partial^{A}a_{k}\partial_{A}a_{l}\eta_{ij}^{kl}-2\Lambda\}}\\{-\frac1{16\piG_{p+1}}\intd^{p+1}y\sqrt{-\tilde{g}}e^{-x^{0}}\tilde{R},}\\\end{array}
\sum_{\mu,\nu=0}^{3}\left(\hat{A}_{\mu}\hat{A}_{\nu}\right)\left(\hat{A}^{\mu}\hat{A}^{\nu}\right)-\sum_{\nu=0}^{3}\left(\left(\hat{A}_{\mu}\hat{A}_{\nu}\right)\hat{A}^{\mu}\right)\hat{A}^{\nu}=3\lambda\sum_{\nu=0}\hat{A}_{\nu}\hat{A}^{\nu}
\left.{\frac{\partial_{r}<{\calO}>_{J}}{\partialK_{A}}}\right|_{J}=\left.{\frac{\partial_{r}<{\calO}>_{J}}{\partialK_{A}}}\right|_{\Phi}-\left.{\frac{\partial_{r}<{\calO}>_{J}}{\partialJ_{B}}}\right|_{K}\left.{\frac{\partial_{r}J_{B}}{\partialK_{A}}}\right|_{\Phi},
\frac{1}{2^{n-l}}\frac{(n-1)!}{(2n-1)!}\left|\sum_{\{m_{i}\}}\frac{(2(n-l))!}{m_{1}!\dotsm_{l}!}V^{(m_{1},0,0)}(q)\dotsV^{(m_{l},0,0)}(q)\right|.
\langle{\calE}\rangle={\frac{1}{\Omega}}\int_{\Omega}dt\,d^{3}\!k\,{\calE}_{\bfk}(t)={\frac{1}{\Omega}}\sum_{M\in\Omega}\deltat\int_{\Omega}d^{3}\!k\,{\calE}_{\bfk}^{M},
M=\left(\begin{array}{ccccccc}{h(0+1)}&{\gamma_{-}}&{0}&{\cdots}&{0}&{0}&{0}\\{\gamma_{+}}&{h(1+2)}&{2\gamma_{-}}&{\cdots}&{0}&{0}&{0}\\{0}&{2\gamma_{+}}&{h(2+3)}&{\cdots}&{0}&{0}&{0}\\{\vdots}&{\vdots}&{\vdots}&{\ddots}&{\vdots}&{\vdots}&{\vdots}\\{0}&{0}&{0}&{\cdots}&{(N-2)\gamma_{+}}&{h(N-1+N-2)}&{(N-1)\gamma_{-}}\\{0}&{0}&{0}&{\cdots}&{0}&{(N-1)\gamma_{+}}&{h(N-1)}\\\end{array}\right)\nonumber
f[u_{0}]=\frac{\partialG_{2}[\xi]}{\partial\xi}\bigg|_{\xi=u_{0}}\left[\hat{h}^{-1}+\frac{1}{2}[\hat{h}-l(2\omega_{sph})]^{-1}\right]G_{2}[u_{0}]-\frac{3}{2}G_{3}[u_{0}].
S=\intd^{2}x\left[{\frac12}{\partial}_{\mu}{\phi}{\partial}^{\mu}{\phi}+{\frac12}{\lambda}_{{\mu}{\nu}}\left({\partial}^{\mu}{\phi}-{\epsilon}^{{\mu}{\sigma}}{\partial}_{\sigma}{\phi}\right)\left({\partial}^{\nu}{\phi}-{\epsilon}^{{\nu}{\rho}}{\partial}_{\rho}{\phi}\right)\right],
\langle\,f\,|\,\left(R^{\dagger}-1\right)\left(E_{i}-H+i\epsilon\,\right)^{-1}H_{\mathrm{\scriptsize~I}}|i\,\rangle=(i\epsilon)^{-1}\langle\,f\,|\,\left(R^{\dagger}-1\right)T\,|i\,\rangle
\frac{1}{M_{6}^{4}}f_{0}(\rho)=-\frac{5}{8M_{6}^{4}}{\Lambda}+\frac{3}{8M_{6}^{4}}f_{\theta}(\rho)-\frac{{\left(\sigma^{3/4}\gamma^{1/2}\right)}^{\prime\prime}}{\sigma^{3/4}\gamma^{1/2}}\.
\begin{array}{rcl}{{\calL}_{2G}}&{=}&{\displaystyle-\frac{1}{4}(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})^{2}-\frac{1}{4}(\partial_{\mu}Z_{\nu}-\partial_{\nu}Z_{\mu})^{2}}\\{}&{}&{\displaystyle-\frac{1}{2}(\partial_{\mu}W_{\nu}^{+}-\partial_{\nu}W_{\mu}^{+})(\partial^{\mu}W^{-\nu}-\partial^{\nu}W^{-\mu}),}\\\end{array}
S_{BV}\,=\,\,S_{BV}\,[\,\phi^{A},\phi_{A}^{\ast}={\frac{\partial\Psi}{\delta\phi^{A}}}\,]\,=\,S_{0}[\phi^{i}]\,+\,{\frac{\partial\Psi}{\partialA^{m}}}R_{\alpha}^{m}c^{\alpha}-{\frac{\partial\Psi}{\partialc^{\alpha}}}T_{\beta\,\gamma}^{\alpha}\,c^{\gamma}\,c^{\beta}\,
A^{(\kappa)}=\frac{V_{0}^{(\kappa)}}{\sqrt{b}}\frac{\nu(\kappa)-1}{(\overline{{W}},\,v^{(\kappa)})}\left[V_{00}+\frac{b}{2}\frac{\nu(\kappa)+1}{\nu(\kappa)-1}-(\overline{{W}},\,\mathscr{P}\frac{1}{C^{\prime}U^{\prime}-\nu(\kappa)}\,W)\right]
\Omega_{ij}^{ab}(\vec{x},\vec{y})=\vert\vert\{\Omega_{i}^{a}(\vec{x}),\Omega_{j}^{b}(\vec{y})\}\vert\vert=-\left(\begin{array}{cc}{0}&{\partial_{1}^{x}}\\{\partial_{1}^{x}}&{0}\\\end{array}\right)\delta(\vec{x}-\vec{y})g^{ab}
\overline{{\hat{\epsilon}}}\,\hat{\Gamma}{}^{\hat{a}_{1}\cdots\hat{a}_{n}}\,\hat{\psi}=(-1)^{n+\left[n/2\right]}\,\bar{\hat{\psi}}\,\hat{\Gamma}{}^{\hat{a}_{1}\cdots\hat{a}_{n}}\,\hat{\epsilon}\,,
H_{E}\left(q_{n,\alpha_{n}},p_{n,\alpha_{n}}\right)=\overline{{H}}_{0}\left(q_{n,\alpha_{n}},p_{n,m_{n}-1}\right)+\sum_{n}\sum_{i_{n}=1}^{m_{n}-1}q_{n,i_{n}}p_{n,i_{n}-1}\.
\begin{array}{l}{\left[:X^{-}\,P^{+}X^{-}:\,-ik\partial_{z}X^{-}+2X^{-}S\right](z)\,\times\frac{T^{\prime}(w)}{P^{+}(w)}}\\{\rightarrow\,\quad\,\frac{-2iT^{\prime}(w)}{\left(z-w\right)}\left(:X^{-}\frac1{P^{+}}:\,+S\right)(w)}\\{\quad\quad\quad+\frac{k-2}{(z-w)^{2}}\frac{T^{\prime}(w)}{\left(P^{+}(w)\right)^{2}}+\cdots}\\\end{array}
S=\frac{1}{16\pi}\int\left\{-R+\frac{1}{3}e^{-4\phi}H_{\mu\nu\lambda}H^{\mu\nu\lambda}+2\partial_{\mu}\phi\partial^{\mu}\phi-e^{-2\phi}F_{\mu\nu}F^{\mu\nu}\right\}\sqrt{-g}d^{4}x,
H^{2}={\frac{\kappa^{2}\rho_{0}}{3a^{3\gamma}}}+{\frac{\kappa^{2}\rho_{0}^{2}}{6\lambdaa^{6\gamma}}}-{\frac{k}{a^{2}}}+\frac{1}{3}\Lambda+{\frac{\Sigma^{2}}{a^{6}}}+{\frac{{\calU}_{0}}{a^{4}}}.
M_{\nu,\rho}^{ab,cd}=-\frac{i}{4}\left(\delta_{\nu\rho}\delta_{c^{\prime}d^{\prime}}^{ab}\eta^{c^{\prime}c}\eta^{d^{\prime}d}+\left(\delta_{\nu\alpha}^{ab}\delta_{\rho\beta}^{cd}-\frac{2}{N-2}\delta_{\rho\alpha}^{ab}\delta_{\nu\beta}^{cd}\right)\eta^{\alpha\beta}\right)\delta^{N}(x-y)
\mathrm{~H_{nm}^{\{{\bf~R}\}}(0,0)~}=\left(\frac{R_{11}^{n}R_{22}^{m}}{2^{n+m}}\right)^{\frac12}\sum_{l=0}^{\left[\frac{\mu_{nm}}2\right]}\frac{(-1)^{\frac{n+m}2}n!m!}{l!\left(\mu_{nm}-2l\right)!\left(\frac{|n-m|}2+l\right)!}(2r)^{\mu_{nm}-2l},
(1.g)=-\\frac{i}{6(2\pi)^{4}}\mathrm{tr}\intd^{d}k\\frac{\stackrel{\displaystyle(Sk)}{\scriptscriptstyle(-p)\phantom{+}}\stackrel{\displaystyle(S(k+q))}{\scriptscriptstyle(-q)\phantom{+qqqqqqq}}\stackrel{\displaystyle(S(k-p))}{\scriptscriptstyle(p+q)\phantom{+qqqqqqq}}}{k^{2l}(k-p)^{2l}(k+q)^{2l}}
\mu^{\prime}=\alpha^{2}\left(e^{2\gamma\psi}N(K^{\prime})^{2}+\frac{1}{2}Nx^{2}(H^{\prime})^{2}+\frac{1}{2x^{2}}(K^{2}-1)^{2}e^{2\gamma\psi}+K^{2}H^{2}\right.\,\nonumber
I_{\alpha\rho}=-\int_{0}^{1}dh\int_{0}^{h}dt\frac{\partialc_{t}^{\mu}}{\partialx^{\rho}}\frac{\partialc_{t}^{\nu}}{\partialt}\frac{\partialc_{h}^{\delta}}{\partialx^{\alpha}}\frac{\partialc_{h}^{\beta}}{\partialh}[G_{\beta\delta}(c_{h}),G_{\mu\nu}(c_{t})]
\vec{\psi}_{\mp}(x)\left(\frac{d\vec{\phi}}{dx}\pm\vec{\nabla}W\right)|\vec{\phi}(x),\vec{\psi}(x)\rangle=0\qquad,\,\vec{\psi}_{\pm}(x)=\vec{\psi}_{1}(x)\pm\vec{\psi}_{2}(x)
\frac{1}{2}\int\prod_{i=1}^{4}dg_{i}~~~\phi^{2}(g_{1},\ldots,g_{4})=\frac{1}{2}\sum_{N_{1}\ldotsN_{4}}\sum_{\Lambda}\phi_{\calA}^{N_{1}\ldotsN_{4}\,\Lambda}\phi_{\calA}^{N_{1}\ldotsN_{4}\,\Lambda}.
\mathrm{Tr}\,\Big(\hat{\partial}_{\rho}\Big[{\frac{1}{2}}\big(\hat{\partial}_{\rho}\phi(\hat{x})\hat{\partial}_{\mu}\phi(\hat{x})+\hat{\partial}_{\mu}\phi(\hat{x})\hat{\partial}_{\rho}\phi(\hat{x})\big)-\delta_{\rho\mu}\bar{\mathcal{L}}(\phi(\hat{x}))\Big]\Big)=\mathrm{Tr}\,\Big(\hat{\partial}_{\rho}T_{\rho\mu}\Big)=0.
\frac{dT}{dk}=\frac{4}{\sqrt{-K}}\;\frac{E(k)}{1-k^{2}},\;\;\;\;\;\frac{dS^{\mathrm{cl}}}{dk}=\frac{8}{K\alpha^{\prime}}\;\frac{kE(k)}{(1-k^{2})^{3/2}}.
T_{ECS}^{\mu\nu}=-m^{-1}\left[\left(\varepsilon^{\mu\alpha\beta}f^{\nu}+\varepsilon^{\nu\alpha\beta}f^{\mu}\right)\partial_{\alpha}f_{\beta}-g^{\mu\nu}\varepsilon^{\alpha\beta\gamma}f_{\alpha}\partial_{\beta}f_{\gamma}\right]\,
\left.|\rho_{b}^{\Omega_{\tau}}+D_{b}|=|\rho_{b}^{\Omega_{\tau}}|\,|\left(1+\frac{D_{b}}{\rho_{b}^{\Omega_{\tau}}}\right)|\geq|\rho_{b}^{\Omega_{\tau}}|\,\frac{1}{2}\,\right.
N(t)^{2}=\left({\frac{\xi_{-}}{\xi_{+}}}\right)^{2}{\frac{\xi_{+}^{2}+R^{{\frac{2}{3}}(k^{2}-3)}\dot{R}^{2}}{\xi_{-}^{2}+R^{{\frac{2}{3}}(k^{2}-3)}\dot{R}^{2}}}\,.
\left(\frac12,\frac12\right)\otimes\left[\left(0,1\right)\oplus\left(1,0\right)\right]=\left(\frac12,\frac12\right)\oplus\left(\frac12,\frac32\right)\oplus\left(\frac32,\frac12\right)\oplus\left(\frac12,\frac12\right)
M_{N}=Nm_{0}\,\Xi\left(\sigma,\frac{\mu_{0}}{m_{0}}\right)\,,\quad\sigma_{0}\geq\sigma\geq\sigma_{1}=\frac{1}{\sqrt{2}}\frac{\mu_{0}}{m_{0}}+\frac{\alpha_{1}}{8\alpha_{0}}\left(\frac{\mu_{0}}{m_{0}}\right)^{2}+O\left(\left(\frac{\mu_{0}}{m_{0}}\right)^{2}\right)\,,
\tilde{\Pi}_{R{(2)}}^{(\beta)}(\lambda,p)=-\frac{8e^{4}}{p^{2}}m^{2}I(T/\lambda)\intd^{2}q\,n_{F}(|q^{0}|)\left(q^{2}+m^{2}\right)\delta^{\prime\prime}(q^{2}-m^{2}).
ds_{D}^{2}=H^{-\frac{d-2}{D-2}}\left[-fdt^{2}+\sum_{i=1}^{p}(dx^{i})^{2}+H\Big(f^{-1}dr^{2}+dz^{2}+r^{2}d\Omega_{d-2}^{2}\Big)\right]
\left(\partial_{\alpha}-\widehat{\calA}_{\alpha}\right){\calV}^{\prime}=0\qquad\mathrm{with}\qquad\widehat{\calA}_{\alpha}=\left(\begin{matrix}{\hat{\Gamma}_{\alpha\beta}^{\gamma}}&{C_{\alpha\beta\gamma}}\\{0}&{-\hat{\Gamma}_{\alpha\gamma}^{\beta}}\\\end{matrix}\right)\,
c_{n-1}=\frac{(4(n-\frac{1}{2})(n-\frac{3}{2})+2(n-\frac{1}{2}))c_{n-4}-m_{0}^{2}c_{n-2}}{4(n+\frac{5}{2})(n+\frac{3}{2})-10(n+\frac{5}{2})}.
g^{2}\,\Phi_{i}^{2}={\frac{m(z=h_{i};\mu)}{a^{\prime}(z=h_{i})}}={\frac{\displaystyle\prod_{j=1}^{N-1}(h_{i}-\mu_{j})}{\displaystyle\prod_{j=1,j\neqi}^{N}(h_{i}-h_{j})}}\,\cdotp
\overline{{{\cal{A}}_{i}^{\gamma}}}({\bf{r}})=\left({\delta}_{i\,{\gamma}}-\frac{r_{i}\,r_{\gamma}}{r^{2}}\right)\left(\frac{\overline{{\cal{N}}}}{gr}+{\varphi}_{A}\right)+{\epsilon}_{i{\gamma}n}\frac{r_{n}}{r}\,{\varphi}_{C}\,.
G_{F}=\left(\begin{array}{cc}{\left(1+\frac{1}{2}P\Xi^{R}\overline{{\Xi}}_{R}Q\right)_{\alpha}^{.\beta}}&{(P\Xi)_{\alphaN}}\\{(\overline{{\Xi}}Q)^{\betaM}}&{1}\\\end{array}\right)\left(\begin{array}{cc}{\left(1+\frac{1}{2}Q\Xi^{R}\overline{{\Xi}}_{R}P\right)_{\alpha}^{.\beta}}&{(Q\Xi)_{\alphaN}}\\{(\overline{{\Xi}}P)^{\betaM}}&{1}\\\end{array}\right)
\mathcal{S}=\intd^{4}x\left(|W(z)|^{2}+\frac{\partial^{2}K}{\partialz^{i}\partialz^{\bar{j}}}Dz^{i}Dz^{\bar{j}}-\frac{1}{4g_{ij}^{2}}F^{(i)}\wedge*F^{(j)}+\theta_{ij}F^{(i)}\wedgeF^{(j)}\right)