harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
A_{(a)}^{\mu\nu}=iP^{\alpha,\lambda\kappa}iP^{\gamma\delta,\rho\sigma}i\int{\frac{d^{4}\ell}{(2\pi)^{4}}}{\frac{\tau_{\alpha\beta}(p,p^{\prime}-\ell)\tau_{\gamma\delta}(p^{\prime}-\ell,p^{\prime})\tau_{\rho\sigma,\lambda\kappa}^{\mu\nu}(\ell,q)}{\ell^{2}(\ell-q)^{2}((\ell-p^{\prime})^{2}-m^{2})}}
\mathrm{tr}\,\langleA_{\mu}(x)A_{\nu}(y)\rangle=\mathrm{tr}(t_{a}t_{b})\underbrace{\langleA_{\mu}^{a}(x)A_{\nu}^{b}(y)\rangle^{\mathrm{\scriptsize~singlet}}}_{{\textstyle=:\delta^{ab}D_{\mu\nu}(x,y)}}={\frac{N^{2}-1}{2}}\,D_{\mu\nu}(x,y)
Z=\int{\calD}A^{\mu}{\calD}\pi_{\mu}{\calD}\lambda{\calD}\pi_{\lambda}{\calD}\rho{\calD}\pi_{\rho}\prod_{\alpha,\beta=0}^{3}\delta(\tilde{\Omega}_{\alpha})\delta(\Gamma_{\beta})det\mid\{\tilde{\Omega}_{\alpha},\Gamma_{\beta}\}\mide^{iS^{\prime}},
\frac{\delta\rho}{\rho}(\xi,\theta,\phi)=\sqrt{2}\pi\sum_{l=0}^{\infty}\sum_{m=-l}^{+l}\int_{0}^{+\infty}d\zeta~{\calA}_{lm}(\zeta)~\frac{R_{l}(\xi;\zeta)}{\zeta}~Y_{lm}(\theta,\phi)\nonumber\,
{\frac{ds^{2}}{l_{s}^{2}\sqrt{4\pig_{s}N}}}=\left(\rho^{2}-{\frac{b^{4}}{\rho^{2}}}\right)^{-1}d\rho^{2}+\left(\rho^{2}-{\frac{b^{4}}{\rho^{2}}}\right)d\tau^{2}+\rho^{2}\sum_{i=1}^{3}dx_{i}^{2}+d\Omega_{5}^{2},
q_{c}(a,r)=\frac{1}{16\pi^{2}a^{4}}\sum_{m=-\infty}^{+\infty}\int_{-\infty}^{+\infty}{dk\int_{0}^{\infty}{dz\sum_{\lambda=0,1}\frac{z^{3}\psi_{\mu}(z/a)}{\sqrt{k^{2}+z^{2}/a^{2}}}\frac{f_{\lambdam}^{(q)}(z,zx)}{\bar{J}_{m}^{2}+\bar{Y}_{m}^{2}}}}\,.
{\calL}=\sum_{i}\intd^{2}\thetad^{2}\bar{\theta}Z_{i}\Phi_{i}^{\dagger}e^{2V_{h}}\Phi_{i}+\left(\intd^{2}\theta\widetilde{W}(\tau^{\prime},\lambda_{ijk}^{\prime},m_{ij}^{\prime},c_{i}^{\prime},M^{\prime})+\mathrm{h.c.}\right)
\begin{array}{lll}{B_{11}=F_{12}A_{123}+F_{21}A_{213}}&{,}&{B_{21}=F_{23}A_{123}+F_{32}A_{132}}\\{B_{12}=F_{13}A_{132}+F_{31}A_{231}}&{,}&{B_{22}=F_{13}A_{213}+F_{31}A_{312}}\\{B_{13}=F_{23}A_{312}+F_{23}A_{321}}&{,}&{B_{23}=F_{12}A_{231}+F_{21}A_{321}}\\\end{array}
\mathbf{P}^{n}\equiv\intd^{3}\mathbf{x}\left(\nabla\varphi_{n}^{\alpha}\right)\left\{\frac{\partial\mathcal{F}^{n}}{\partial\varphi_{n}^{\alpha}}-\nabla\mathbf{\cdot}\left(\frac{\partial\mathcal{F}^{n}}{\partial\nabla\varphi_{n}^{\alpha}}\right)\right\}.
iG^{[s]}(x,x^{\prime})_{mn}^{ab}=\int_{0}^{\infty}\,dT\int_{x(0)=x^{\prime},x(T)=x}Dx(t)exp\left({\frac{-1}{4}}\int_{0}^{T}dt\,\dot{x}^{2}\right)\Phi_{mn}^{[s]}[C^{xx^{\prime}}]U[C^{xx^{\prime}}]^{ab},
G_{1}\left(m_{eff}\beta\right)=\left(\frac{\beta}{2\pi}\right)^{2}\frac{d^{2}}{d\lambda^{2}}\sum_{n=1}^{\infty}\frac{1}{n^{2}}\int_{m_{eff}\beta}^{\infty}dy\left(y^{2}-m_{eff}^{2}\beta^{2}\right)^{1/2}e^{-2\pinry\lambda/\beta},
\left(\begin{array}{c}{<W(L_{\mp})>}\\{<W(L_{W\mp})>}\\\end{array}\right)=\left(\begin{array}{cc}{q^{\pm{\frac{1}{4}}}}&{(q^{\mp{\frac{1}{4}}}-q^{\pm{\frac{1}{4}}})}\\{0}&{q^{\mp{\frac{1}{4}}}}\\\end{array}\right)\left(\begin{array}{c}{<W(L_{I})>}\\{<W(L_{W})>}\\\end{array}\right).
E_{i}^{a}\left(x\right)U\left({\bfy^{\prime}},\bfy\right)\left|E\right\rangle=\left({\varepsilon_{i}^{a}\left(x\right)+g\int_{\bfy^{\prime}}^{\bfy}{dz_{i}\delta^{(3)}\left({x-z}\right)T^{a}}}\right)U\left({\bfy^{\prime}},\bfy\right)\left|E\right\rangle.
I_{11}^{B}=\intd^{11}x\left[-\frac{\sqrt{g}}{4\kappa^{2}}R(g)-\frac{\sqrt{g}}{48}F^{2}+\frac{2\kappa}{144^{2}}\epsilon^{1\cdots11}F_{1\cdots}F_{5\cdots}A_{..11}\right],
V_{\omega}^{d-l}(r_{*})=-\omega^{2}{\frac{R^{4}}{r^{4}}}\left(1+{\frac{r_{n}^{2}}{r^{2}}}\right)+K^{2}{\frac{R^{2}}{r^{2}}}h+{\frac{3h}{4r^{2}}}\left(1+{\frac{3r_{0}^{2}}{r^{2}}}\right),
S_{T}=\intd^{2}\xi\phi(\xi)\epsilon^{mn}\left[e_{m}^{--}e_{n}^{++}{\calE}_{--}^{\underline{a}}{\calE}_{++}^{\underline{b}}\eta_{\underline{{ab}}}+B_{mn}^{(2)}+\partial_{m}A_{n}\right],
S_{eff}=\int_{z=z_{c}}d^{4}x\sqrt{\bar{g}}\left[\bar{g}^{\mu\nu}\nabla_{\mu}H^{\dagger}\nabla_{\nu}H-\lambda\left(|H|^{2}-e^{-2kz_{c}}m_{0}^{2}\right)^{2}\right]+\ldots
f^{1}=\left(\begin{array}{c}{f_{1}^{1}}\\{f_{2}^{1}}\\{\vdots}\\{f_{6}^{1}}\\\end{array}\right),~~~~~f^{2}=\left(\begin{array}{c}{f_{1}^{2}}\\{f_{2}^{2}}\\{\vdots}\\{f_{6}^{2}}\\\end{array}\right).
ds_{AH}^{2}={\frac{1}{4}}a^{2}b^{2}c^{2}{\left({\frac{dk}{k{k^{\prime}}^{2}K^{2}}}\right)}^{2}+a^{2}(k){\sigma}_{1}^{2}+b^{2}(k){\sigma}_{2}^{2}+c^{2}(k){\sigma}_{3}^{2},
Z_{ab}=\left(\begin{array}{cccc}{q^{-\frac{1}{2}}}&{0}&{0}&{0}\\{0}&{q^{\frac{1}{2}}}&{0}&{0}\\{0}&{0}&{q^{\frac{1}{2}}}&{0}\\{0}&{0}&{0}&{q^{-\frac{1}{2}}}\\\end{array}\right)\,,
S=\int_{0}^{-1}\frac{(1+u)^{-x}-1}{x(1-u)}\,du=\frac{1}{2}\int_{0}^{1}\frac{1-u^{-x}}{x(1-u/2)}\,du=\sum_{k\ge1}2^{-k}\bigg(\frac{1}{k}-\frac{1}{k-x}\bigg),
e^{\textstyle(J+1)\alpha_{-}\widetilde\Phi(\sigma,\tau)}={\frac{1}{\sqrt\varpi}}{\tilde{c}}_{-J-1}\sum_{M=-J}^{J}(-1)^{J-M}e^{ih(J-M)}\xi_{M}^{(-J-1)}(x_{+})\,{\overline{\xi}_{-M}^{(-J-1)}}(x_{-})\sqrt\varpi.
L=\overline{\psi}\left(p\!\!/+A\!\!\!/+\gamma_{5}A\!\!\!/_{5}\right)\psi-\frac{1}{2}\left(\partial_{\mu}A_{5\nu}\right)\left(\partial_{5}^{\nu}\right)-\frac{1}{2}\left(\partial_{\mu}A_{\nu}\right)\left(\partial^{\mu}A^{\nu}\right)
\left\{F,G\right\}^{(\eta,\Lambda)}=\left\{F^{\prime},G^{\prime}\right\}^{(\eta^{\prime},\Lambda^{\prime})}=\frac{\partial_{r}F^{\prime}}{\partial\eta^{A}}\Lambda^{AB}(\eta^{\prime})\frac{\partial_{l}G^{\prime}}{\partial\eta^{B}}\;,
\zeta_{\pm}^{2}(z)=N_{2}\left(\begin{array}{c}{z^{\frac{1-M}{2}}(1-z)^{-\frac{M}{2}}F(\frac{3}{2}-M,-\frac{1}{2}-M,\frac{3}{2}-M;z)}\\{\mpi(\frac{1}{2}-M)z^{-\frac{M}{2}}(1-z)^{\frac{M}{2}}F(-1,1,\frac{1}{2}-M;z)}\\\end{array}\right)
\langle\hat{\Phi}^{\dagger}(p)\hat{\Phi}(p)\rangle_{t}=\langle\hat{\Phi}^{\dagger}(p)\hat{\Phi}(p)\rangle_{0}\;\Big(\mathrm{ch}^{2}(\omega_{-}(p)t)+{\frac{E^{2}(p)}{\omega_{-}^{2}(p)}}\mathrm{sh}^{2}(\omega_{-}(p)t)\Big)\;,
\partial_{y}^{2}\Big[(1+c\delta_{0})f_{n}^{S}\Big]=\frac{a\delta_{0}^{\prime}}{1+a\delta_{0}}\partial_{y}\Big[(1+c\delta_{0})f_{n}^{S}\Big]-m_{n}^{2}(1+a\delta_{0})f_{n}^{S},
A^{\prime}=\frac{1}{\sqrt{1+\midz^{\prime}\mid^{2}}}\left(\begin{matrix}{1}&{-z^{*}}\\{z^{\prime}}&{1}\\\end{matrix}\right)\left(\begin{matrix}{\frac{\mid\psi_{1}^{\prime}\mid}{\psi_{1}^{*}}}&{0}\\{0}&{\frac{\mid\psi_{1}^{\prime}\mid}{\psi_{1}^{\prime}}}\\\end{matrix}\right)
\lambda_{\alpha_{N}}^{|N|\mathrm{{t}}}\Xi_{N}\stackrel{1}{\Pi}\lambda_{\alpha_{N}^{\prime}}^{|N|}dA_{N}^{\alpha_{N}^{\prime}}(x):=\mathbb{M}_{\alpha_{N}\alpha_{N}^{\prime}}dA_{N}^{\alpha_{N}^{\prime}}(x)=0\,.
S=\Huge\frac{1}{4\pi\alpha^{\prime}}\int\larged^{\:2}\sigma\left[\sqrt{\gamma}g_{\mu\nu}(X)\partial_{A}X^{\mu}\partial_{B}X^{\nu}\gamma^{AB}+ib_{\mu\nu}(X)\partial_{A}X^{\mu}\partial_{B}X^{\nu}\epsilon^{AB}+\sqrt{\gamma}\R^{(2)}\phi(X)\right],
f_{1}=-\frac{i}{2}\Bigg(\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{2}(k+p)^{2}}-\sum_{i}c_{i}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{(k^{2}-M_{i}^{2})\Big((k+p)^{2}-M_{i}^{2}\Big)}\Bigg)
\deltaR_{\mu\nu}=-{\frac{1}{2}}\bar{\nabla}^{2}h_{\mu\nu}+{\bar{R}}_{\sigma(\nu}h_{~\mu)}^{\sigma}-{\bar{R}}_{\sigma\mu\rho\nu}h^{\sigma\rho}+\bar{\nabla}_{(\nu}\bar{\nabla}_{|\rho|}{\hat{h}}_{~\mu)}^{\rho}.
\left\langleH\right\rangle_{\Omega}=\left\langleH\right\rangle_{0}+\frac{{q^{2}\sqrt\pi}}{{2g}}\sqrt{\frac{a-1}{{a}}}\left({1-e^{-\frac{g}{{\sqrt\pi}}\sqrt{\frac{a}{{a-1}}}|y-y^{\prime}|}}\right),
\hat{x}^{\alpha\dot{\alpha}}=x^{\alpha\dot{\alpha}}+2i\theta^{A\alpha}\bar{\theta}^{B\dot{\alpha}}u_{(A}^{+}u_{B)}^{-},\\\theta^{\pm\alpha}=u_{A}^{\pm}\theta^{A\alpha},\\,bar\theta^{\pm\dot{\alpha}}=u_{A}^{\pm}\bar{\theta}^{A\dot{\alpha}},\\u^{\pm}
\begin{array}{c}{D^{\left[A,B\right]}(\Lambda)u(\Lambda^{-1}p)=u(p)D^{\left(s\right)}(R^{-1}(\Lambda,p))}\\{D^{\left[A,B\right]}(\Lambda)v(\Lambda^{-1}p)=v(p)D^{\left(s\right)*}(R^{-1}(\Lambda,p)),\quadv(p)=u(p)D^{\left(s\right)*}(i\sigma_{2})}\\\end{array}\,\,\,\,\,\,
S_{eff}\sim\intd^{D}X\sqrt{\tilde{G}}\left[\alpha^{\prime}\tilde{R}-\alpha^{\prime}(\partial\tilde{\phi})^{2}+\frac{2}{3}(26-D)e^{\frac{4}{\sqrt{2|D-2|}}\tilde{\phi}}-\alpha^{\prime}\frac{1}{12}H_{\lambda\mu\nu}^{2}e^{-\frac{8}{\sqrt{2|D-2|}}\tilde{\phi}}+...\right]~.
I(n,p)=\int{\frac{d^{n}k}{(2\pi)^{n}}}\,{\frac{e^{ik_{\mu}\theta^{\mu\nu}p_{\nu}}}{(k^{2}+M^{2})}}=\frac{m^{\,\frac{-2+n}{2}}\,(p\circp)^{\,\frac{2-n}{4}}\,K_{\frac{-2+n}{2}}\,(m\,{\sqrt{p\circp}})}{{\left(2\,\pi\right)}^{\frac{n}{2}}}
S_{BH}=2\pi\left(\sqrt{N_{2}^{(1)}}+\sqrt{\bar{N}_{2}^{(1)}}\right)\left(\sqrt{N_{2}^{(2)}}+\sqrt{\bar{N}_{2}^{(2)}}\right)\left(\sqrt{N_{5}^{(1)}}+\sqrt{\bar{N}_{5}^{(1)}}\right)\left(\sqrt{N_{5}^{(2)}}+\sqrt{\bar{N}_{5}^{(2)}}\right)
={\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}
\left[1+{\frac{M_{*}^{N-2}}{c_{1}\,M_{*}^{N-2}\,+\,c_{2}\,\nabla^{N-2}}}\,{\frac{1}{(r_{c}\,\nabla)^{2}}}\,+\,\sum_{n}{\frac{a_{n}}{(r_{c}\,\nabla)^{n}}}\right]\,R\,=\,8\piG_{N}\,T_{\mu}^{\mu}\,,
\begin{array}{l}{-(\mu^{2}-\frac{15}{4}-m_{11}^{2})\sigma^{3}=\muc_{3J}\sigma^{J}+m_{13}^{2}\sigma^{1}-m_{14}^{2}\sigma^{2}}\\{-(\mu^{2}-\frac{15}{4}-m_{33}^{2})\sigma^{1}=\muc_{1J}\sigma^{J}+m_{13}^{2}\sigma^{3}+m_{34}^{2}\sigma^{2}}\\{-(\mu^{2}-\frac{15}{4}-m_{44}^{2})\sigma^{2}=\muc_{2J}\sigma^{J}+m_{34}^{2}\sigma^{1}-m_{14}^{2}\sigma^{3}}\\\end{array}
\lambda=\frac{2g}{N}[\Pi_{\alpha}^{*}\Pi_{\alpha}+\partial_{i}z_{\alpha}^{*}\partial^{i}z^{\alpha}-\frac{4g}{N}(z_{\alpha}^{*}\partial^{i}z_{\alpha})(z_{\beta}\partial_{i}z_{\beta}^{*})]
S_{mm^{\prime},nn^{\prime}}^{R}=\frac{1}{u}(-1)^{G^{\prime}+m+n+u(u-n-1)}e^{-i\pi(u+1)m(u-n-1)/u}e^{-i\pi(u-1)(m-2m^{\prime}+u)(n-2n^{\prime}+u)/u};
\frac{de}{d\omega}=\frac{\gamma^{3}}{2\pi^{3}R_{0}^{3}}\;\frac{\omega^{2}+(\gammav\omega_{0})^{2}}{\omega^{2}}\;\frac{v^{3}w^{2}}{w^{2}+(2\gammav)^{2}}\;h_{\gamma}(w),
d\Pi^{\mu_{1}\dots\mu_{p}}\equiv{\frac{1}{2}}(C\Gamma^{\mu_{1}\dots\mu_{p}})_{\alpha\beta}\Pi^{\alpha}\Pi^{\beta}\,,\qquad(\Gamma^{\mu_{1}\dots\mu_{p}}=\Gamma^{[\mu_{1}}\Gamma^{\mu_{2}}\cdots\Gamma^{\mu_{p}]}\equiv{\frac{1}{p!}}\epsilon_{\nu_{1}\dots\nu_{p}}^{\mu_{1}\dots\mu_{p}}\Gamma^{\nu_{1}}\dots\Gamma^{\nu_{p}})\,,
\left[a_{\eta^{\prime}}(p^{\prime\,\mu}),\,a_{\eta}^{\dagger}(p^{\mu})\right]_{\pm}\,=\,(2\,\pi)^{3}\,2\,p_{0}\,\delta^{3}({\bfp}\,-\,{\bfp}^{\prime})\,\delta_{\eta,\,-\,\eta^{\prime}}\quad.
\begin{array}{rcl}{u_{\lambda,l,m}(r,\theta,\phi)}&{=}&{N_{\lambdal}\v_{1,\lambda,l,m}^{(\circ)}=}\\{}&{=}&{N_{\lambdal}\r^{l}\_{2}F_{1}\left({\frac{l+1+i\Lambda}{2}},{\frac{l+1-i\Lambda}{2}};l+{\frac{3}{2}};-r^{2}\right)Y_{l,m}(\theta,\phi),}\\\end{array}
L_{1}=\int_{0}^{1}dx_{3}x_{3}^{2\alpha^{\prime}\rho-1}(1-x_{3})^{2\alpha^{\prime}\alpha_{34}}\int_{0}^{1}duu^{2\alpha^{\prime}\alpha_{12}-1}(1-u)^{2\alpha^{\prime}\alpha_{23}}(1-ux_{3})^{2\alpha^{\prime}\alpha_{24}-1}\,,
{\calL}_{mat}=-m\,\beta_{ab}\,{\calJ}^{ab}-\frac{1}{2}\,\mathrm{Im}\left\{\beta_{ab}\,{(\gamma^{\mu})^{b}}_{c}\,(\partial_{\mu}{\calJ}^{ac})+\beta_{ab}\,{(\gamma^{\mu})^{b}}_{c}\,C_{\mu}^{ac}\right\}.
b_{n}^{L}=\frac{\sqrt{2}}{\pi}\frac{(-1)^{(n-1)}}{(2n-1)^{3/2}}p_{0}+\frac{1}{\sqrt{2}}a_{2n-1}+\frac{1}{\sqrt{2}}\sum_{m=1}^{\infty}M_{1\,m,n}a_{2m}-M_{2\,m,n}a_{2m}^{\dagger},
\frac{D_{2}}{D_{1}}=e^{-2i\pi\mu}e^{4aL\mu}\frac{\Gamma(2\mu)\Gamma(1-\mu-\nu-\lambda)\Gamma(-\mu-\nu+\lambda)}{\Gamma(-2\mu)\Gamma(1+\mu-\nu-\lambda)\Gamma(\mu-\nu+\lambda)}
R_{\quadkl}^{ij}=\delta_{\k}^{i}\delta_{\l}^{j}\bigl(\delta^{ij}+\Theta^{ji}q_{ij}^{-1}+\Theta^{ij}q_{ji}r^{-2}\bigr)+\delta_{\l}^{i}\delta_{\k}^{j}\Theta^{ij}\bigl(1-r^{-2}\bigr),
F_{\pm}(s)=\frac{V}{8\pi\sqrt{\pi}}\frac{\Gamma(s-3/2)}{\Gamma(s)}\left(\frac{2\pi}{\beta}\right)^{3-2s}\sum_{j=1}^{\infty}\left[(j\pmie\bar{\mu})^{2}+\bar{m}^{2}\right]^{3/2-s}\;,
{\calK}_{0}^{b}({\bfr})=\sum_{n=0}^{\infty}i^{n}{\epsilon}_{(n)}^{{\vec{\delta}}ba}L_{(n)}^{\vec{\delta}}{\Phi}_{(n)}^{a}({{\bfr}})=-j_{0}^{b}({\bfr})+\sum_{n=1}^{\infty}i^{n}{\epsilon}_{(n)}^{{\vec{\delta}}ba}L_{(n)}^{\vec{\delta}}{\Phi}_{(n)}^{a}({{\bfr}})
\begin{array}{c}{\Gamma_{\rho\nu\lambda}^{\left(2\right)}(p_{1,}k_{2},q_{2})=g_{\rho\nu}(p_{1}+k_{2})_{\lambda}+g_{\nu\lambda}(q_{2}-k_{2})_{\rho}-g_{\lambda\rho}(q_{2}+p_{1})_{\nu}}\\{\Gamma_{\sigma\mu\lambda}^{(2)}(p_{2},k_{1},q_{2})=g_{\mu\sigma}(p_{2}+k_{1})_{\lambda}-g_{\mu\lambda}(k_{1}+q_{2})_{\sigma}+g_{\lambda\sigma}(q_{2}-p_{2})_{\mu}}\\\end{array}
W^{\Omega_{z_{1}},\Omega_{z_{2}}^{*}}\left(\begin{array}{cc}{c}&{d}\\{b}&{a}\\\end{array}\right)=\frac{G_{b}}{G_{c}}W\left(\left.\begin{array}{cc}{d}&{a}\\{c}&{b}\\\end{array}\right|z_{2}-z_{1}-\frac{n}{2}\right),
U=\left(\begin{matrix}{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}\\{\vdots}&{\vdots}&{\vdots}&{\vdots}&{\ddots}\\{1}&{0}&{0}&{0}&{\cdots}\\\end{matrix}\right),
k_{2}^{\nu}T_{\rho\sigma\mu\nu}^{(3)abcd}(p_{1},p_{2};k_{1},k_{2})=C_{3}k_{1\mu}S_{\rho\sigma}^{(3)}+C_{3}\Gamma_{\rho\sigma\mu}^{(3)}(p_{1},p_{2},q_{3})
\begin{array}{ll}{S\sim}&{\intd^{4}x\sqrt{\hat{g}}\left\{\hat{R}+2\frac{\partialU\partial\bar{U}}{\left(U-\bar{U}\right)^{2}}+2\frac{\partialS\partial\bar{S}}{\left(S-\bar{S}\right)^{2}}+2\frac{\partialT\partial\bar{T}}{\left(T-\bar{T}\right)^{2}}\right.}\\{}&{\left.-\frac{\alpha^{\prime}}{8}\sum_{\alpha}\left(v_{\alpha}S_{2}+\tilde{v}_{\alpha}T_{2}\right)trF_{\alpha}^{2}\right\}.}\\\end{array}
H=H_{c}=\int_{-\infty}^{\infty}dx\left[\frac{1}{2}(3\lambda\omega^{2}-m^{2})\varphi^{2}+\lambda(\omega\varphi^{3}+\frac{1}{4}\varphi^{4})+\frac{1}{4\lambda}(\lambda\omega^{2}-m^{2})^{2}\right],
\sum_{e}\sum_{m,n=0}^{1}C_{e}(m,n)\theta^{4}[e]F_{R,tws}^{m,n}=E(z,0)^{-2}\sum_{i=2}^{3}C_{i}\theta_{i}^{12}(0|\tau)\frac{\theta_{i}^{2}(z|\tau)}{\theta_{i}^{2}(0|\tau)},
\left(1-\frac{1}{x^{2}}+\frac{A^{\prime}}{AB}\right)+\frac{y}{x^{2}}=\pm\sqrt{\left(1-\frac{1}{x^{2}}+\frac{A^{\prime}}{AB}\right)^{2}+\frac{2}{xB^{2}}\left(\frac{A^{\prime}}{A}+\frac{B^{\prime}}{B}\right)}.
S^{ij}(\theta-\sigma_{ij})=\left(\begin{array}{ccc}{\eta_{ij}(-1)_{\theta}}&{\eta_{ij}^{2}(-2)_{\theta}}&{\eta_{ij}^{3}(-3)_{\theta}}\\{\eta_{ij}^{2}(-2)_{\theta}}&{-(-3)_{\theta}(-1)_{\theta}}&{\eta_{ij}^{2}(-2)_{\theta}}\\{\eta_{ij}^{3}(-3)_{\theta}}&{\eta_{ij}^{2}(-2)_{\theta}}&{\eta_{ij}(-1)_{\theta}}\\\end{array}\right)
{\calG}=\frac{4\pi^{2}l}{e^{2}}\,\left(C_{AB}d\vec{x}_{A}\cdotd\vec{x}_{B}+(C^{-1})_{AB}(d\psi_{A}+\vec{\omega}_{AC}\cdotd\vec{x}_{C})(d\psi_{B}+\vec{\omega}_{BD}\cdotd\vec{x}_{D})\right).
\int_{M}\{e^{-\phi}(\kappa\hat{{\calR}}\star1-\frac{\alpha}{2}d\phi\wedge\stard\phi+\frac{1}{2}dB\wedge\stardB)+\frac{1}{2}e^{-2\phi}dA\wedge\stardA+\lambdaA\wedgedB\}.
{\bfG}_{4}=\sqrt{\frac{\alpha}{\sqrt{2}}}\left(\begin{array}{ccc}{0}&{0}&{1}\\{0}&{0}&{0}\\{0}&{1}&{0}\\\end{array}\right),\quad{\bfG}_{5}=\sqrt{\frac{\alpha}{\sqrt{2}}}\left(\begin{array}{ccc}{0}&{0}&{0}\\{0}&{0}&{1}\\{-1}&{0}&{0}\\\end{array}\right),\quad{\bfe}=\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{0}\\\end{array}\right),
A^{p}(M,\wedge^{q}T^{1,0}M\niV=V_{{\bar{i}}_{1}{\bar{i}}_{2}\cdots{\bar{i}}_{p}}^{j_{1}j_{2}\cdotsj_{q}}dz^{{\bar{i}}_{1}}dz^{{\bar{i}}_{2}}\cdotsdz^{{\bar{i}}_{p}}\partial_{j_{1}}\partial_{j_{2}}\cdots\partial_{j_{q}}
{\bfD}_{\beta}{\bfF}^{\beta\alpha}=g{\bfJ}^{\alpha}\,\mathrm{with}\quad{\bfJ}^{\alpha}=-i\bigl[\Phi,{\bfD}^{\alpha}\Phi\bigr]\,\quad{\mathrm{and}}\quad{\bfD}^{\alpha}{\bfD}_{\alpha}\Phi=0\.
V(\tilde{\varphi})=\!\sum_{m=0}^{\infty}\frac{1}{m!}\partial_{\tilde{\varphi}}^{m}V(\tilde{\varphi})|_{\tilde{\varphi}=\tilde{\varphi}_{0}}(\tilde{\varphi}-\tilde{\varphi}_{0})^{m}=\!\sum_{m=0}^{\infty}\frac{1}{m!}\partial_{\tilde{\varphi}}^{m}V(\tilde{\varphi})|_{\tilde{\varphi}=\tilde{\varphi}_{0}}u^{m}\left\{\sum_{n=0}^{\infty}\!\frac{1}{n!}\Lambda_{n}(z,\bar{z})u^{n}\right\}^{m}.
u=\frac{t}{\ell}+\phi,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,v=\frac{t}{\ell}-\phi,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,e^{2\rho}=\frac{r^{2}-r_{ex}^{2}}{\ell^{2}},
\delta_{\mu}^{\tau}H_{\;\;\nu\rho}^{\sigma}+\delta_{\mu\;\;\rho}^{\;\;\sigma}+\delta_{\rho}^{\tau}H_{\mu\nu}^{\;\;\;\;\sigma}=\delta_{\mu}^{\sigma}H_{\;\;\nu\rho}^{\tau}+\delta_{\mu\;\;\rho}^{\;\;\tau}+\delta_{\rho}^{\sigma}H_{\mu\nu}^{\;\;\;\;\tau},
\tilde{C}=n!~\alpha^{n(n-1)/2}\prod_{p=1}^{n-1}p!~~\mathrm{and}~~C=n!~((\alpha)^{3/2}/2\sqrt{\beta})^{n(n-1)/2}(\pi/\beta)^{n/2}\prod_{p=1}^{n-1}p!\:.
S=\intd^{4}x\int_{0}^{L}dy\sqrt{-G}\left(M^{3}R+{\scrL}\right)+\intd^{4}x\sqrt{-g^{0}}L^{0}\Bigr|_{y=0}+\intd^{4}x\sqrt{-g^{L}}L^{L}\Bigr|_{y=L}\,
\sum_{j^{\prime},\epsilon^{\prime}}\intA({\bfv},j,\epsilon;{\bfv}^{\prime},j^{\prime},\epsilon^{\prime})\varphi({\bfv}^{\prime},j^{\prime},\epsilon^{\prime})d\rho({\bfv}^{\prime})=\psi({\bfv},j,\epsilon)
M\frac{d^{2}\xi_{\mu}^{(Q)}}{ds^{2}}=(F_{\mu\nu}(\xi^{(cl)}+\xi^{(Q)})-F_{\mu\nu}^{cl}(\xi^{(cl)}))\frac{d\xi^{(cl)\nu}}{ds}+F_{\mu\nu}(\xi^{cl}+\xi^{Q})\frac{d\xi^{(Q)\nu}}{ds}
\begin{array}{rlllll}{{[c_{m},c_{n}]}}&{=}&{m\delta_{m,-n}\,}&{}\\{{[\widetilde{c}_{m},\widetilde{c}_{n}]}}&{=}&{m\delta_{m,-n}\,}&{\;\;\;m,n\in\mathbb{Z}\setminus\{0\}}\\\end{array}
\Theta=\left\{\begin{array}{lc}{\hphantom{-}\Theta_{C}(\mathrm{mod}2\pi),}&{0<F<\frac{1}{2}}\\{}&{}\\{\hphantom{-}0_{\hphantom{C}}(\mathrm{mod}2\pi),}&{F=\frac{1}{2}}\\{}&{}\\{-\Theta_{C}(\mathrm{mod}2\pi),}&{\frac{1}{2}<F<1}\\\end{array}\right\},-\pi<\Theta_{C}\leq\pi.
\psi_{\chi}(q,\alpha)=\sum_{IM}C_{M\chi}^{I}\psi_{IM\chi}(q,\alpha)=\sum_{IM,\;k/\chi_{k}=\chi,\;\mu}C_{M\chik}^{I}\varphi_{(\chi\mu)}(q)D_{M(\chi_{k},\mu)}^{I}(\alpha),
\begin{array}{c}{H=\dot{m}_{j}\frac{\partialL}{\partial\dot{m}_{j}}(m,\dot{m})-L(m,\dot{m})+v_{j}P_{j}(m,p)}\\{\equivH_{0}(m,p)+v_{j}P_{j}(m,p)\qquad.}\\\end{array}
u_{+}(n,k;\rho,\varphi)=\frac{1}{\sqrt{2\bar{\omega}_{k}(M+\bar{\omega}_{k})}}\left(\begin{array}{c}{ikJ_{n}(k\rho)e^{in\varphi}}\\{(M+\bar{\omega}_{k})J_{n+1}(k\rho)e^{i(n+1)\varphi}}\\\end{array}\right)
Q_{+}=\intdx\left\{\bar{\psi}_{-}^{\bar{j}}(x)\left(\frac{\partial}{\partial\bar{\Phi}^{\bar{j}}}(x)+\frac{1}{2}G_{i\bar{j}}\frac{\partial\Phi^{i}(x)}{\partialx}\right)+\psi_{+}^{j}(x)\Omega_{j}(\Phi(x))-\psi_{+}^{l}\bar{\psi}_{-}^{\bar{i}}\bar{\psi}_{+}^{\bar{j}}\frac{\partialG_{l\bar{j}}}{\partial\bar{\Phi}^{\bar{i}}}\right\}\,.
m_{0}^{2}+\frac{4\lambda\mu^{4-D}}{(\pi)^{D/2}}\Gamma(\frac{D}{2}-1)\left[(\beta_{c}^{2-D}+L_{c}^{2-D})\zeta(D-2)+2E_{2}((D-2)/2;\beta_{c},L_{c})\right]=0.
\intdx\Bigl[\frac{\delta\Gamma}{\deltaA^{\mu}}\partial_{\mu}C+\frac{1}{\xie^{2}}\frac{\delta\Gamma}{\delta\bar{C}}\partial_{\mu}A^{\mu}+i\frac{\delta\Gamma}{\delta\Phi}\mathcal{E}_{\Phi}^{2}(C+c)(\Phi+\phi)\Bigr]=0.
N_{1}N_{2}N_{3}N_{4}{\frac{45}{256}}v_{4}^{4}(\vec{v}_{3}\cdot\vec{\nabla}_{4})^{2}{\frac{1}{x_{4}^{7}}}\nabla_{3}^{2}{\frac{1}{x_{3}^{5}}}{\frac{1}{x_{12}}}.
\psi:=(\sum_{i<j}\sum_{l=1}^{5}\lambda_{ijl}a_{-l}^{(i)}a_{-1}^{(j)}+\sum_{i}\sum_{\tinyl=1,3,5}\mu_{il}a_{-l}^{(i)}a_{-1}^{(i)}+\mu_{17}a_{-7}^{(1)}a_{-1}^{(1)})\vert0>
z_{1}\equiv\frac{\mathrm{tr}(\tilde{G}\tilde{{\calH}}\tilde{G}\tilde{{\calH}})}{2(-G)(1+C_{1}^{2})}=\frac{\tilde{{\calH}}^{\mu\nu}\tilde{{\calH}}_{\nu\mu}-2y^{\mu}y_{\mu}}{2(1+C_{1}^{2})},
S=\intd^{4}x\sqrt{|g_{4}|}\left[e^{-\Phi}\left(R+\left(\nabla\Phi\right)^{2}-6\left(\nabla\beta\right)^{2}-\frac{1}{2}e^{2\Phi}\left(\nabla\sigma\right)^{2}\right)-\frac{1}{2}Q^{2}e^{-6\beta}\right].
j_{\mu}=mN\mathrm{Tr}\left\{\gamma_{\mu}\frac{1}{-\partial^{2}+m^{2}}\left(m\hat{\partial}V^{(d+1)}\frac{1}{-\partial^{2}+m^{2}}\right)^{d}(V^{(d+1)})^{\dagger}\right\},
L_{soft}=\frac{1}{2}\sum_{\alpha}M_{\alpha}{\tilde{\lambda}}{\tilde{\lambda}}-\sum_{i}m_{i}^{2}|{\tilde{\phi}}|^{2}-(A_{ijk}{\tilde{Y}}_{ijk}{\tilde{\phi}}_{i}{\tilde{\phi}}_{j}{\tilde{\phi}}_{k}+B_{ij}{\tilde{\mu}}_{ij}{\tilde{\phi}}_{i}{\tilde{\phi}}_{j}+h.c),
f^{(0)}=\left(\begin{array}{ccc}{0}&{0}&{0}\\{0}&{0}&{-{\frac{1}{2}}\bigl(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk}\bigr)}\\{0}&{{\frac{1}{2}}\bigl(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk}\bigr)}&{0}\\\end{array}\right)\,\delta^{(3)}(\vec{x}-\vec{y}\,)\,.
L_{0}=(2n+1){\frac{|h|}{2}}-|h|\sum_{k=1/2}^{\infty}\left(d_{k}^{\dagger}d_{k}-\tilde{d}_{k}^{\dagger}\tilde{d}_{k}+a_{k}^{\dagger}a_{k}-b_{k}^{\dagger}b_{k}\right)+L_{0}^{free}.
I=\frac{1}{16\piG_{N}}\intd^{4}x\sqrt{-g}\left[R+\tilde{a}_{1}R^{2}+\tilde{a}_{2}R_{\nu}^{\mu}R_{\mu}^{\nu}+\tilde{a}_{3}R_{\\\rho\sigma}^{\mu\nu}R_{\\\mu\nu}^{\rho\sigma}\right],
(-i)\tilde{S}(p)={\frac{p^{2}+b^{2}-m^{2}+2\left(b\cdotp+m\not\!b\right)\gamma_{5}-[\not\!b,\not\!{\hat{p}}]\gamma_{5}}{\left(p^{2}+b^{2}-m^{2}+i\varepsilon\right)^{2}-4\left[(b\cdotp)^{2}-b^{2}\left(m^{2}-\hat{p}^{2}\right)\right]}}\left(\not\!p+m+\not\!b\gamma_{5}\right)\,
{\calL}=\intd^{4}xe^{-2\alpha}\left(\partial_{\mu}\bar{\phi}\partial^{\mu}\phi+\bar{\psi}{\not\!\partial}\psi-\bar{F}F\right)+\intd^{4}xe^{-2\alpha}m\left(\frac{1}{2}\psi\psi+\phiF\right)+\mathrm{h.c.}
K={\frac{f_{1}^{2}(\alpha)}{f_{2}(\alpha)}}{\frac{g^{2}C_{2}\sigma^{2}}{2\pi^{2}}}\left({\frac{\eta^{2}}{\gamma^{2}}}\right)^{2}={\frac{f_{1}^{2}(\alpha)}{f_{2}(\alpha)}}{\frac{g^{2}C_{2}\sigma^{2}}{2\pi^{2}}}{\frac{m_{1}^{2}m_{2}^{2}}{(m_{1}^{2}+m_{2}^{2})^{2}}},
F_{IJ}=\left(\begin{array}{ccccccc}{0}&{-a_{1}}&{}&{}&{}&{}&{}\\{a_{1}}&{0}&{}&{}&{}&{}&{}\\{}&{}&{\ddots}&{}&{}&{}&{}\\{}&{}&{}&{0}&{-a_{4}}&{}&{}\\{}&{}&{}&{a_{4}}&{0}&{}&{}\\{}&{}&{}&{}&{}&{0}&{-a_{5}(=0)}\\{}&{}&{}&{}&{}&{a_{5}(=0)}&{0}\\\end{array}\right),
f(x^{2})={\frac{8i\left(m^{2}\right)^{4}}{\pi^{3}}}\,\int_{0}^{\infty}d\kappa_{1}\,\int_{0}^{\infty}d\kappa_{2}\,\kappa_{1}\kappa_{2}\,e^{-i(\kappa_{1}+\kappa_{2})(m^{2}x^{2}-i\delta)}e^{-i(1-i\epsilon)(\kappa_{1}+\kappa_{2})/(4\kappa_{1}\kappa_{2})}.
H_{\lambda}=-\sum_{j=1}^{L}\sum_{n=1}^{N^{\prime}-1}\left(\frac{1}{sin\gamman}\left[\left(\Sigma_{j}^{+}(\lambda)\Sigma_{j+1}^{-}(\lambda)\right)^{n}+\left(\Sigma_{j}^{-}(\lambda)\Sigma_{j+1}^{+}(\lambda)\right)^{n}\right]+\mu_{j}(\lambda)+\mu_{j+1}(\lambda)\right)
G_{\phi_{2}}(x-x^{\prime})=i\int\frac{d^{3}q}{(2\pi)^{3}}e^{i{\bfq}.({\bfx}-{\bfx}^{\prime})}\left(\begin{array}{ll}{G_{\phi_{2}}^{++}({\bfq},t-t^{\prime})}&{\:\:G_{\phi_{2}}^{+-}({\bfq},t-t^{\prime})}\\{G_{\phi_{2}}^{-+}({\bfq},t-t^{\prime})}&{\:\:G_{\phi_{2}}^{--}({\bfq},t-t^{\prime})}\\\end{array}\right)\:,
\begin{array}{c}{{\bfP}_{A}=\bar{p}_{A}\,\quad{\bfM}^{AB}=x^{A}\bar{p}^{B}-x^{B}\bar{p}^{A}+S^{AB}\,}\\{S^{AB}=2i\{Z_{i}^{A}\partial^{iB}-\overline{{Z}}_{i}^{A}\overline{{\partial}}^{iB}-(\Sigma^{AB})_{b}^{\a}(\psi_{a}\partial^{b}+\overline{{\psi}}_{a}\overline{{\partial}}^{b})\}\Phi}\\\end{array}
\beta_{p,q}^{\ast}=-i\epsilon\int_{-\infty}^{\infty}dy\left(\phi_{p,\epsilon}^{\ast}(y)\partial_{t}\bar{\phi}_{q,\epsilon}^{\ast}(\bar{y})-\partial_{t}\phi_{p,\epsilon}^{\ast}(y)\bar{\phi}_{q,\epsilon}^{\ast}(\bar{y})\right)
e^{-2\hat{\phi}}=e^{-2\hat{\phi}_{0}}+{\frac{2\tilde{m}_{k}}{r}}\,\qquadA_{u}=(A_{u})_{0}+{\frac{2\hat{m}_{k}}{r}}\,\qquadA_{i}=(A_{i})_{0}+{\frac{2(q_{k})_{i}}{r}}\,\qquadi=4,\dots9.
\lambda_{m}(\alpha)=\sum_{l=0}^{m}\left({\frac{i\alpha}{2}}\right)^{l}{\frac{(m+l)!}{l!(m-l)!}}\times\left\{\begin{array}{cl}{-\lambda_{2N}(\alpha)}&{\mathrm{~for~odd~m+l~}}\\{\lambda_{0}(\alpha)}&{\mathrm{~for~even~m+l~}}\\\end{array}\right\}
W{}={}\frac{\alpha^{2}\,{\overline{{T}}}^{0i}\,k_{i}{}\pm{}\sqrt{\left(\alpha^{2}\,{\overline{{T}}}^{0i}\,k_{i}\right)^{2}{}+{}\left(1{}+{}\alpha^{2}\,{\overline{{T}}}^{00}\right)\left({\bfk}^{2}{}-{}\alpha^{2}\,{\overline{{T}}}^{ij}\,k_{i}\,k_{j}\right)}}{1{}+{}\alpha^{2}\,{\overline{{T}}}^{00}}\quad.
L_{10}=\sqrt{-g}\biggl[\phi\biggl(R+3\frac{\phi_{;\rho}\phi^{;\rho}}{\phi^{2}}-\frac{1}{12}F_{\mu\nu\lambda}F^{\mu\nu\lambda}\biggl)-\frac{1}{8}F_{\mu\nu}F^{\mu\nu}\biggr].
\partial_{u}\xi_{t}^{(1)}+{\frac{1+u^{2}}{uf}}\xi_{t}^{(1)}={\frac{fC_{t}}{u(\piTR)^{2}}}H_{tt}^{\prime}+{\frac{fC_{z}}{u(\piTR)^{2}}}\left({\frac{H_{tz}}{f}}\right)^{\prime}\,.
\frac{x_{12}^{2}x_{34}^{2}}{x_{24}^{4}}=(uv)^{\frac{1}{2}}+O(|x_{12}|,|x_{34}|)\,\,\,,\,\,\,\left(\frac{u}{v}\right)^{k}=1+O(|x_{12}|,|x_{34}|)
\begin{array}{l}{f_{(+)}(\theta_{+k})=Ce^{\chi_{R}(\theta_{+k})-i\chi_{J}(\theta_{+k})},}\\{f_{(-)}(\theta_{-k})=\left.f_{(+)}^{*}(\theta_{+k})\right|_{\theta_{+k}=\theta_{-k}}=Ce^{\chi_{R}(\theta_{-k})+i\chi_{J}(\theta_{-k})}.}\\\end{array}
\dot{\mathcal{S}}=\int\left[\frac{\delta\mathcal{S}}{\delta\phi(x)}\dot{\phi}(t,x)+\frac{\delta\mathcal{S}}{\delta\tilde{\phi}(x)}\dot{\tilde{\phi}}(t,x)+\frac{\delta\mathcal{S}}{\delta\gamma_{ij}(x)}\dot{\gamma_{ij}}(t,x)+\frac{\delta\mathcal{S}}{\delta\tilde{\gamma}_{ij}(x)}\dot{\tilde{\gamma}_{ij}}(t,x)\right]d^{3}x~~~,