harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{rcl}{R_{11}^{\prime}(\tilde{p}^{\prime})}&{=}&{-{p^{\prime}}_{1}{p^{\prime}}_{2}-4{p^{\prime}}_{1}+8{p^{\prime}}_{2}^{2}-16{p^{\prime}}_{2}+64\,,\nonumber}\\{R_{12}^{\prime}(\tilde{p}^{\prime})}&{=}&{-2{p^{\prime}}_{1}+{p^{\prime}}_{2}^{2}+12{p^{\prime}}_{2}\,,\nonumber}\\{R_{22}^{\prime}(\tilde{p}^{\prime})}&{=}&{{p^{\prime}}_{1}+4{p^{\prime}}_{2}+16\,.}\\\end{array}
1\simeqe^{2}(4|eH|)\int\frac{d^{4}\hat{q}}{(2\pi)^{4}}\frac{\mathrm{e}^{-\hat{q}_{\perp}^{2}}}{\hat{q}^{2}}\frac{1}{2|eH|\hat{q}_{\parallel}^{2}+m^{2}},
b_{\sigma_{0}^{(1)}}=\left({\frac{d\rho_{0}}{d\sigma_{0}^{(1)}}}\right)b_{\rho_{0}}=-\left({\frac{d\rho_{0}}{d\sigma_{0}^{(2)}}}\right)b_{\rho_{0}}=-b_{\sigma_{0}^{(2)}},
\left(Q_{2}^{\dagger}\right)^{31}u_{B}^{-}({\bfr})\propto\sum_{n=0}^{\infty}{\frac{(-1)^{n}}{(2n)!!(2n+2\left(2\ell+1-\lambda\right))!!}}\left(\sqrt{E}r\right)^{2n+\left(2\ell+1-\lambda\right)}e^{-i\left(2\ell+1\right)\theta}
\Psi_{0}^{J}[\phi]={\cal{N}}exp\left[-{\frac{1}{4}}\intd^{\nu}x\,d^{\nu}y\,\eta(\vec{x})\,G(\vec{x}-\vec{y})\,\eta(\vec{y})+{\frac{1}{2}}\intd^{\nu}x\,\eta(\vec{x})J(\vec{x})\right]\,,
Z_{E}^{(n)}=\frac{m^{n}}{n!}\int\proddxdy\left\langle\left\{\overline{\psi}_{a}e^{ig\gamma_{5}\phi_{a}}\psi_{a}+\overline{\psi}_{b}e^{ig\gamma_{5}\phi_{b}}\psi_{b}\right\}^{n}\right\rangle.
\langle\deltaj^{\mu}(X)\rangle\simeq\frac{1}{2}\int\frac{d^{\,3}p}{(2\pi)^{3}}\left[\hat{\calP}_{+}^{\mu}Z_{+}\frac{-i\Sigma_{off}^{(+)}}{|Im\Sigma_{R}^{(+)}|}-\hat{\calP}_{-}^{\mu}Z_{-}\frac{-i\Sigma_{off}^{(-)}}{|Im\Sigma_{R}^{(-)}|}\right],
\{\theta_{i}^{~j},t_{k}^{~l}\}=((r_{+})_{ik}^{~sp}+\alpha_{4}P_{ik}^{~sp})\theta_{s}^{~j}t_{p}^{l}-\theta_{i}^{~m}((r_{+})_{mk}^{~jp}+\alpha_{6}P_{mk}^{~jp})t_{p}^{l}+
\Big(\partial_{i}-{\frac{i}{2}}g\sigma^{a}W_{i}^{a}\Big)\Big(\partial_{i}-{\frac{i}{2}}g\sigma^{a}W_{i}^{a}\Big)_{B}^{A}\Phi^{B}(x)=-(E^{2}-m^{2})\Phi^{A}(x)
e^{-2\hat{\phi}}=1+\sum_{1}^{s}{\frac{2\tilde{m}_{k}}{r_{k}}}\,\qquadA_{u}=1+\sum_{1}^{s}{\frac{2\hat{m}_{k}}{r_{k}}}\,\qquadA_{i}=\sum_{1}^{s}{\frac{2(q_{k})_{i}}{r_{k}}}\,\qquadi=4,\dots8.
I\!\!I_{n,k}\left(\xi\right)\;=\;i\,g_{n}\left(\xi\right)\inth_{k}\left(\xi-\xi^{\prime}\right)\left(\vec{q}\left(\xi\right)-\vec{q}\left(\xi^{\prime}\right)\right)\cdot\vec{\lambda}\Phi\left(\xi^{\prime}\right)d^{6}\xi^{\prime}\.
\stackrel{\mathrm{HW}}{S^{\alpha\beta}}=\int\tau_{\mathrm{KG}}^{\alpha\beta0}\,d^{3}x\,=\frac{1}{4mi}\int\left[\left(D^{0}\overline{{\Psi}}\right)\sigma^{\alpha\beta}\Psi-\overline{{\Psi}}\sigma^{\alpha\beta}\left(D^{0}\Psi\right)\right]\,d^{3}x\,.
N=u^{i}{\frac{\partial}{\partialu^{i}}}+\xi_{\alpha}^{*}{\frac{\partial}{\partial\xi_{\alpha}^{*}}}+\left(n-u_{i}^{*}{\frac{\partial}{\partialu_{i}^{*}}}-\xi^{\alpha}{\frac{\partial}{\partial\xi^{\alpha}}}\right),\quadn=n_{+}+n_{-}.
<\xi(x,t)\xi(x^{\prime},t^{\prime})>=N\intD\xi\xi(x,t)\xi(x^{\prime},t^{\prime})e^{-\frac{1}{2}\intdx^{\prime\prime}dt^{\prime\prime}\frac{\xi(x^{\prime\prime},t^{\prime\prime})^{2}}{\theta^{2}}}.
\hat{\sigma}_{gh}^{a}=i\hat{z}^{\dagger}\tau^{a}\hat{p}_{z}-i\hat{p}_{z}^{\dagger}\tau^{a}\hat{z}+i\hat{\eta}^{\dagger}\tau^{a}\hat{p}_{\eta}-i\hat{p}_{\eta}^{\dagger}\tau^{a}\hat{\eta}=[\hat{Q},i\hat{z}^{\dagger}\tau^{a}\hat{p}_{\eta}-i\hat{p}_{z}^{\dagger}\tau^{a}\hat{\eta}]\,.
K^{t}{}_{t}=-{\frac{d}{dt}}\left[{\frac{a\dot{\chi}_{*}}{\sqrt{1-\left(a\dot{\chi}_{*}\right)^{2}}}}\right]\,,\qquadK^{i}{}_{j}={\frac{{}-\delta^{i}{}_{j}}{\sqrt{1-\left(a\dot{\chi}_{*}\right)^{2}}}}\left[\dot{a}\dot{\chi}_{*}+{\frac{f^{\prime}(\chi_{*})}{af(\chi_{*})}}\right]\,,\\i,j=1,2\,,
{\ddot{\eta}-{\frac{1}{2}}\ddot{a}=0},\qquad\ddot{\theta}+{\frac{2{p}_{\theta}}{ma^{3}}}\dot{\theta}=0,\qquad{\ddot{\pi}}_{Q}+{\frac{3{{p}_{\theta}}^{2}}{ma^{4}}}\dot{a}+m\ddot{a}=0.
\begin{array}{c}{\omega=d{\bfp}\wedged{\bfx}+cd{\bfe}_{+}\wedged{\bfe}_{1},}\\{{\calH}=\frac{{\tilde{\sigma}}}{2c}\left[c^{2}{\bfe}_{1}^{2}+({\bfp}{\bfe}_{+}-2c)^{2}+{\bfp}^{2}{\bfe}_{+}^{2}\right]}\\\end{array}
\begin{array}{rcl}{\displaystyle{S_{GCS}^{e}}}&{=}&{\displaystyle{\int_{M}\mathrm{Tr}_{\mathrm{\bf~k}}\left(\frac{1}{2}\mathcal{A}Q\mathcal{A}+\frac{1}{3}\mathcal{A}^{3}\right)},}\\{\displaystyle{S_{GCS}^{o}}}&{=}&{\displaystyle{\int_{M}\mathrm{Str}_{\mathrm{\bf~j}}\left(\frac{1}{2}\mathcal{A}Q\mathcal{A}+\frac{1}{3}\mathcal{A}^{3}\right)},}\\\end{array}
\Lambda=1+\sum_{i=2}^{4}\beta_{i}^{2}\vertz_{i}\vert^{2}\quad\mathrm{and}\quadA=\frac{\Lambda^{-1}}{2i}\sum_{i=2}^{4}\beta_{i}\left(\bar{z}_{i}dz_{i}-z_{i}d\bar{z}_{i}\right)~,
{\widehat\Gamma}_{\Lambda_{1}\dots\Lambda_{n}}^{(n)}(k_{1}\dotsk_{n})=(-1)^{n+1}{\frac{\prod_{\{i|\Lambda_{i}=R\}}\Bigl(1+n(k_{i}^{0})\Bigr)}{\prod_{\{i|\Lambda_{i}=A\}}n(k_{i}^{0})}}{\widehat\Gamma}_{{\bar{\Lambda}}_{1}\dots{\bar{\Lambda}}_{n}}^{(n)*}(k_{1}\dotsk_{n})\,,
S_{red}=\intd^{4}x\mathrm{Tr}\left[-{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}-2m^{2}\Phi_{\mu}\Phi^{\mu}+{\frac{1}{4}}\epsilon^{\mu\nu\rho\lambda}E_{\mu\nu}\tilde{F}_{\rho\lambda}\right].
I_{0}=\frac{i\pi}{|\vec{k}|}\left(\frac{1}{4\pi}\right)^{\frac{3-\epsilon}{2}}\frac{\mu^{\epsilon}}{\Gamma\left(\frac{3-\epsilon}{2}\right)}\int_{K_{-}}^{K_{+}}dK\int_{0}^{\infty}d|\vec{q}|\frac{|\vec{q}|^{1-\epsilon}}{|\vec{q}|^{2}+(2\piK)^{2}}\left[\frac12+N\left(\frac{|\vec{q}|}{T}\right)\right],
\mathrm{Im}\;\delta\;=\;\sum_{\mathrm{multiplets}}{\frac{\mathrm{dim}(s)}{2}}\sum_{k\\mathrm{odd}}\mathrm{exp}\left[-{\frac{2\pi\alpha^{\prime}k}{\epsilon}}\left({\frac{b^{2}}{(2\pi\alpha^{\prime})^{2}}}+M(s)^{2}\right)\right]\,
S_{dual}=\intd^{6}x\left[-{\frac16}G_{{\mu}{\nu}{\sigma}}G^{{\mu}{\nu}{\sigma}}+\frac{1}{2({\partial}_{\lambda}a)({\partial}^{\lambda}a)}{\partial}^{\mu}a{\calG}_{{\mu}{\nu}{\sigma}}{\calG}^{{\nu}{\sigma}{\rho}}{\partial}_{\rho}a+A_{{\nu}{\sigma}}{\partial}_{{\mu}}G^{{\mu}{\nu}{\sigma}}\right].
\tilde{\psi}(t,x,y,\zeta)=\chi(t,x,y,\zeta)\psi(t,x,y,\zeta)\quad\mathrm{with}\quad\chi=1+\sum_{\alpha=1}^{s}\sum_{k=1}^{m}\frac{R_{\alphak}}{(\zeta-\mu_{k})^{\alpha}},
S_{CSM}~=~\intd^{2}x~\left[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+eA_{\nu}(\eta^{\mu\nu}-\epsilon^{\mu\nu})\partial_{\mu}\phi+\frac{1}{2}ae^{2}A_{\mu}A^{\mu}~\right],
L=(\vec{\dot{A_{1}}}\cdot\vec{A_{2}}-\vec{\dot{A_{2}}}\cdot\vec{A_{1}})+\vec{\phi}^{2}-[(\vec{A_{1}^{2}}+\vec{A_{2}^{2}})\vec{\phi}^{2}-(\vec{A_{1}}\cdot\vec{\phi})^{2}-(\vec{A_{2}}\cdot\vec{\phi})^{2}]-\frac{\kappa}{4}(\vec{\phi}^{2}-1)^{2}
0=<\Omega{\calO}>_{J}=<{\frac{\partial_{r}{\calO}}{\partial\Phi^{A}}}{\frac{\partial_{l}\Sigma}{\partialK_{A}}}-{\frac{\partial_{r}{\calO}}{\partialK_{A}}}{\frac{\partial_{l}\Sigma}{\partial\Phi^{A}}}-i\hbar(-1)^{\varepsilon_{A}+1}{\frac{\partial_{r}}{\partial\Phi^{A}}}{\frac{\partial_{r}{\calO}}{\partialK_{A}}}>_{J}.
\hat{D}_{\bar{z}^{\bar{1}}}\hat{\phi}_{1}-\hat{D}_{z^{2}}\hat{\phi}_{2}=0\qquad\mathrm{and}\qquad\hat{D}_{\bar{z}^{\bar{2}}}\hat{\phi}_{1}+\hat{D}_{z^{1}}\hat{\phi}_{2}=0,
\Lambda_{M}(u)=\prod_{i=1}^{p_{0}}\frac{1}{a(u-u_{i}^{(0)})}\sum_{j=1}^{n}m_{j}\prod_{l=1}^{p_{j-1}}a(u-u_{l}^{(j-1)})\prod_{m=1}^{p_{j}}a(u_{m}^{(j)}-u)
M_{2}^{2}=-^{2}R+\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)E^{\mu\tau}E_{\tau}^{\nu}+\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)E^{\mu\sigma}E_{\sigma}^{\nu}
W=\frac{eE}{\beta^{2}(2\pi)^{2}}\frac{2}{\sqrt{\pi}}e^{-\bar{m}^{2}\pi}\sum_{k=1}^{\infty}\frac{\left(\bar{t}_{f}\bar{\beta}\right)^{2k-2}\left(-1\right)^{k+1}}{\left(2k-1\right)\left(k-1\right)!}\int_{0}^{\infty}\frac{dzz^{(2k-5)/2}}{\frac{z}{4}+\frac{\pi}{\bar{\beta}^{2}}}e^{-z((\bar{\beta}\bar{m})/2)^{2}-\frac{1}{z}}.
F=\left[\left\{\sum_{i=1}^{2k}\left(\frac{dT_{i}}{dx^{i}}\right)^{2}-m^{2}T_{i}^{2}+\lambdaT_{i}^{4}+\frac{m^{2}}{4\lambda}\right\}+\left\{\partial_{\mu}\mathcal{T}\partial^{\mu}\mathcal{T}-m^{2}\mathcal{T}^{2}+\lambda\mathcal{T}^{4}+f(\mathcal{T})\overline{{\theta}}_{R}\theta_{L}\right\}\right]
\langle0|\varphi^{2}(x)|0\rangle=\langle0_{{\mathrm{m}}}|\varphi^{2}(x)|0_{{\mathrm{m}}}\rangle+\langle\varphi^{2}(x)\rangle_{b}^{(a)}+\langle\varphi^{2}(x)\rangle_{b}^{(b)}+\langle\varphi^{2}(x)\rangle^{(ab)},
Z_{t}^{\mu}=\sum_{\mathrm{\tiny~full~cuts~C~of~t~}}(-1)^{n_{C}}\left[\prod_{i}B_{t_{i}^{C}}\mu_{t_{i}^{C}}^{-\#(t_{i}^{C})x}\right]B_{t_{R}}\mu_{t_{R}}^{-\#(t_{R})x},
g_{AB}(x,\phi)=\left(\begin{array}{cc}{g_{\mu\nu}(x)}&{\sum_{a=1}^{dimG}A_{\mu}^{a}(x)\xi_{i}^{a}(\phi)}\\{\sum_{a=1}^{dimG}A_{\mu}^{a}(x)\xi_{i}^{a}(\phi)}&{\gamma_{ij}(\phi)}\\\end{array}\right)
S=\int_{0}^{1}\left(-\frac{1}{2e}(\dot{x}_{\mu}-i\psi_{\mu}\chi-\epsilon_{\mu\nu\lambda\rho}b^{\nu}\psi^{\lambda}\psi^{\rho}+i\frac{\alpha}{2}\theta^{+}\theta^{-}b_{\mu})^{2}-i\psi_{\mu}\dot{\psi}{}^{\mu}+L_{kin}^{\theta}\right)d\tau,
g_{ij}^{(11)}=\left(\begin{matrix}{e^{-{\frac{2}{3}}\phi}(g_{\alpha\beta}+e^{2\phi}C_{\alpha}C_{\beta})}&{\nue^{{\frac{4}{3}}\phi}C_{\alpha}}\\{\nue^{{\frac{4}{3}}\phi}C_{\beta}}&{\nu^{2}e^{{\frac{4}{3}}\phi}}\\\end{matrix}\right)\,
\sigma^{(1)}=\frac{m^{2}}{8\sqrt{3}\pi}\eta^{-1/2}\int_{0}^{1}d{\tilde{h}}\;({\tilde{h}}-1)(17+18{\tilde{h}})=-\frac{23\;m^{2}}{16\sqrt{3}\pi}\eta^{-1/2}.
N_{{\bfp},r}=e^{-\pi\lambda}\left[1+O\left(\left[\frac{1+\lambda}{\xi_{1}}\right]^{3}\right)+O\left(\left[\frac{1+\lambda}{\xi_{2}}\right]^{3}\right)\right],\;\;-\sqrt{eE}\frac{T}{2}\leq\xi_{1}\leq-K.
\frac{1}{2<\psi_{AB}|\psi_{AB}>}\left[<\psi_{AB}|z_{A}\frac{\partial}{\partialz_{A}}\psi_{AB}>+<z_{A}\frac{\partial}{\partialz_{A}}\psi_{AB}|\psi_{AB}>\right]=\frac{1}{2}(1-\frac{N}{2})
h(w)=-\frac{2v}{\delta_{b}R\mathrm{cosh}^{2}\frac{w}{\delta_{b}}}+\frac{1}{2\piR^{4}}\sum_{i}\sum_{KM}\frac{KG_{i}}{{\calN}_{i}}\int_{w}^{\infty}\mathrm{d}w^{\prime}~\frac{w^{\prime}}{\mathrm{cosh}^{2\gamma_{i}}\frac{w^{\prime}}{\delta_{b}}}~~,
S_{\mathrm{kin}}^{(4)}=\intd^{5}\xie^{-\phi}\sqrt{g}\left\{1+{\textstyle\frac{1}{2}}{\calF}^{2}+{\textstyle\frac{1}{8}}({\calF}^{2})^{2}-{\textstyle\frac{1}{4}}{\calF}^{4}\right\}^{1/2}\,,
*R*^{\mu\nu}{}{}_{\rho\sigma}=-R^{\mu\nu}{}{}_{\rho\sigma}+\delta_{\sigma}^{\mu}R_{\rho}^{\nu}+\delta_{\rho}^{\mu}R_{\sigma}^{\nu}-\delta_{\sigma}^{\mu}R_{\rho}^{\nu}-\delta_{\rho}^{\nu}R_{\sigma}^{\mu}+\frac{1}{2}R(\delta_{\rho}^{\nu}\delta_{\sigma}^{\mu}-\delta_{\rho}^{\mu}\delta_{\sigma}^{\nu}).
S=\frac{1}{2}\intd^{4}x\epsilon^{\mu\nu\alpha\beta}\epsilon_{abcd}\bigg[({^+}\tau-{^-}\tau)F_{\mu\nu}^{~~ab}F_{\alpha\beta}^{~~cd}+({^+}\tau+{^-}\tau)F_{\mu\nu}^{~~ab}\tilde{F}_{\alpha\beta}^{~~cd}\bigg].
\frac12\Omega_{+q}^{+2i}+2i\gamma_{q\dot{q}}^{i}\psi_{-2\dot{q}}^{1-}=0,\-\frac12\Omega_{-q}^{+2i}+2i\gamma_{q\dot{q}}^{i}\psi_{-2\dot{q}}^{2+}=0.
W_{ij}^{(0)}(x,y)=\left(\begin{array}{ccccc}{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\\end{array}\right)\delta^{2}(x-y)\equiv\widetilde{W}_{ij}^{(0)}\delta^{2}(x-y),
\tilde{\Phi}_{\lambdam-1}^{\muV^{(l)}}(z)=\frac{1}{[l-m]}\left\{\tilde{\Phi}_{\lambdam}^{\muV^{(l)}}(z)f_{1}-q^{2m-l}f_{1}\tilde{\Phi}_{\lambdam}^{\muV^{(l)}}(z)\right\},\qquadm=1,2,\ldots,l.
S=-\intd^{11}x~\sqrt{-g_{11}}[{\frac{1}{4^{.}4!}}F_{M_{1}...M_{4}}^{(4)}F^{(4)M_{1}...M_{4}}+{\frac{1}{4^{.}7!}}F_{M_{1}...M_{7}}^{(7)}F^{(7)M_{1}...M_{7}}\big]+\int{\frac{1}{6}}F^{(7)}\wedgeF^{(4)}
ds^{2}=\left(1-\frac{a^{2n+2}}{r^{2n+2}}\right)^{-1}dr^{2}+r^{2}\left(1-\frac{a^{2n+2}}{r^{2n+2}}\right)(d\psi+A)^{2}+r^{2}ds_{M}^{2}.
\frac{da_{1}}{dT}=\frac{4(a_{1}-x_{0})}{(a_{1}-c)(a_{1}-b_{1})(a_{1}-a_{2})(a_{1}-b_{2})}\,,\qquad\frac{db_{2}}{dT}=\frac{4(b_{2}-x_{0})}{(b_{2}-c)(b_{2}-a_{1})(b_{2}-b_{1})(b_{2}-a_{2})}\,,
K=\left(\begin{array}{ll}{-\sigma_{1}\bar{\beta}_{11}}&{\\-\sigma_{1}\bar{\beta}_{12}}\\{-\sigma_{2}\bar{\beta}_{12}}&{\\-\sigma_{2}\bar{\beta}_{22}}\\\end{array}\right)\equiv\left(\begin{array}{ll}{K_{11}}&{\\K_{12}}\\{K_{21}}&{\\K_{22}}\\\end{array}\right).
\langle\bar{\psi}_{f}\psi_{f}\rangle_{A}={\frac{m_{f}}{\sqrt{m_{f}^{2}+h_{f}^{2}}}}\int_{-\infty}^{\infty}d\lambda\rho(\lambda){\frac{1}{\sqrt{(m_{f}^{2}+h_{f}^{2})}+i\lambda}},
s\Gamma^{\alpha}{}_{\beta\gamma}(u)=\Gamma^{\alpha}{}_{\beta\gamma}(u)-\left(\begin{array}{cc}{0}&{L^{i}{}_{jc}}\\{C^{a}{}_{bi}}&{0}\\\end{array}\right)=\left(\begin{array}{cc}{L^{i}{}_{jk}}&{0}\\{0}&{C^{a}{}_{bc}}\\\end{array}\right)
\frac{m}{\lambdaM}=\frac{1}{\mu^{2}}\left(\begin{matrix}{m_{\pi}^{2}}\\{m_{\sigma}^{2}}\\\end{matrix}\right)+\left(\begin{matrix}{m_{\pi}^{2}{\calF}_{\mathrm{EPI}}(m_{\pi}^{2},M^{2})}\\{(m_{\sigma}^{2}-4M^{2}){\calF}_{\mathrm{EPI}}(m_{\sigma}^{2},M^{2})}\\\end{matrix}\right).
={\frac{1}{(d-2)!}}\sum_{k=0}^{\infty}C_{k}(z)\sigma^{k}\sum_{n=0}^{\infty}\left(n+{\frac{d+1}{2}}\right)^{-2z-2k}\left[d^{2}+d(n-3)+4-n\right]\sum_{p=0}^{d-3}n^{p}w_{p}^{\prime}~~~,
\begin{aligned}{\nabla_{M}\eta+\frac{1}{8}H_{MNR}\Gamma^{NR}\eta}&{{}=0}\\{\Gamma^{M}\partial_{M}\phi\eta+\frac{1}{12}H_{MNR}\Gamma^{MNR}\eta}&{{}=0}\\{F_{MN}\Gamma^{MN}\eta}&{{}=0~,}\\\end{aligned}
q=\frac{1}{8\pi^{2}}\sum_{l=1}^{\infty}\left\{2\int_{0}^{\infty}{\omega^{3}\psi_{\mu}(\omega)D_{l}^{(q)}(\omegar)d\omega}-\frac{1}{a^{4}}\int_{0}^{\infty}{\chi_{\mu}(z/a)F_{l}^{(q)}(z,x)dz}\right\},\quadq=\varepsilon,p_{\bot},p,
(W_{0}(n)\overline{{P}}_{\beta}^{k}W_{0}(n)^{-1})_{-}{}~=~\overline{{B}}_{k\beta}^{KP}(n)|_{t=(x,0,\cdots)}~~~~\mathrm{for}~^{\forall}k\in\mathrm{~\bf{Z}~}_{\geq1}.
\langlek_{\mu}(x)k_{\nu}(y)\rangle=g_{\mu\nu}\int{\frac{d^{4}p}{(2\pi)^{4}i}}\left({\frac{1}{p^{2}}}-{\frac{1}{\eta^{2}}}\right)^{-1}e^{ip(x-y)}=\eta^{2}g_{\mu\nu}\int{\frac{d^{4}p}{(2\pi)^{4}i}}{\frac{p^{2}}{\eta^{2}-p^{2}}}e^{ip(x-y)}.
{\cal{L}}_{quad}=\frac{1}{4}h_{\mu\nu,\rho}h_{\mu\nu,\rho}-\frac{1}{8}h_{,\rho}h_{,\rho}-\frac{1}{2}(h_{\mu\rho,\mu}-\frac{1}{2}h_{,\rho})(h_{\rho\nu,\nu}-\frac{1}{2}h_{,\rho})-\frac{1}{2}(h_{\mu\nu}^{2}-\frac{1}{2}h^{2})\Lambda\,
\rho_{\mathrm{S}}(\zeta)=\left(\pi\rho(0)\right)^{2}|\zeta|\left(J_{\alpha}^{2}(2\pi\rho(0)\zeta)-J_{\alpha+1}(2\pi\rho(0)\zeta)J_{\alpha-1}(2\pi\rho(0)\zeta)\right)
\begin{array}{rclrcl}{Z_{1}}&{=}&{\partial_{\mu}Z_{\nu}-\partial_{\nu}Z_{\mu},}&{Z_{2}}&{=}&{[Z_{\mu},Z_{\nu}],}\\{A_{1}}&{=}&{\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},}&{A_{2}}&{=}&{[A_{\mu},A_{\nu}],}\\{M}&{=}&{[Z_{\mu},A_{\nu}]-[Z_{\nu},A_{\mu}].}&{}&{}&{}\\\end{array}
Q_{f}=f_{\mu_{1}\dots\mu_{r}}\Pi^{\mu_{1}}\psi^{\mu_{2}}\dots\psi^{\mu_{r}}+\frac{i}{r+1}(-1)^{r+1}f_{[\mu_{1}\dots\mu_{r};\mu_{r+1}]}\psi^{\mu_{1}}\dots\psi^{\mu_{r+1}}
J_{-}=\left(\begin{array}{ccccc}{0}&{}&{}&{}&{*}\\{1}&{0}&{}&{}&{*}\\{}&{1}&{\ddots}&{}&{\vdots}\\{}&{}&{\ddots}&{0}&{*}\\{}&{}&{}&{1}&{*}\\\end{array}\right),{}~~~~J_{+}=\left(\begin{array}{ccccccc}{}&{0}&{0}&{1}&{}&{}&{}\\{}&{}&{0}&{0}&{1}&{}&{}\\{}&{}&{}&{\ddots}&{\ddots}&{\ddots}&{}\\{}&{}&{}&{}&{0}&{0}&{1}\\{}&{*}&{*}&{...}&{*}&{*}&{*}\\{}&{*}&{*}&{...}&{*}&{*}&{*}\\\end{array}\right),
S_{2}=-T_{2}\intd^{2}\zeta\left({\frac{1}{2}}\sqrt{-\gamma}\gamma^{ij}\partial_{i}X^{M}\partial_{j}X^{N}g_{MN}e^{\frac{\phi}{2}}+{\frac{1}{2}}\epsilon^{ij}\partial_{i}X^{M}\partial_{j}X^{N}B_{MN}\right),
{\calF}^{(0)}={\frac{1}{a^{3}}}\int_{-\infty}^{\infty}{\frac{d\omega}{2\pi}}e^{-i\omega\tau}\sum_{l=1}^{\infty}{\frac{2l+1}{4\pi}}\left\{x^{\prime}[\tilde{s}_{l}^{\prime}(x^{\prime})\tilde{e}_{l}^{\prime}(x^{\prime})-\tilde{e}_{l}(x^{\prime})\tilde{s}_{l}^{\prime\prime}(x^{\prime})]-x[\tilde{s}_{l}^{\prime}(x)\tilde{e}_{l}^{\prime}(x)-\tilde{e}_{l}(x)\tilde{s}_{l}^{\prime\prime}(x)]\right\}.
+\imath(\alpha+\imath\beta)(\alpha^{2}+\beta^{2})[-h^{\prime}-\frac{2}{r}h+2g(k^{\prime}+\frac{2k}{r})+2kg^{\prime}]e^{-\imath\theta}+\imath(\alpha+\imath\beta)^{3}fce^{-5\imath\theta}+\ldots
D(x,x^{\prime})=\sum_{neven}\int{\frac{d^{4}k}{(2\pi)^{4}}}\int_{0}^{\infty}{\frac{qdq}{\pi}}{\frac{e^{ik_{\mu}(x^{\mu}-x^{\mu})}e^{in(\theta-\theta^{\prime})}}{k_{4}^{2}+q^{2}}}J_{n}(qr)J_{n}(qr^{\prime}).
\kappa^{2}g^{2}\alpha^{\prime}\left(\frac{1}{2}(\frac{2^{4}}{3^{2}})^{2}\right)(A^{\mu}A_{\mu})^{2}=\kappa^{2}g^{2}\alpha^{\prime}\left(\frac{128}{81}\right)(A^{\mu}A_{\mu})^{2}
A_{p}\!=\!e^{\frac{\phi}{2}}[z_{1}\,\gamma_{p}\otimes1\!\!1+z_{2}\Pi^{(1)j}\,h_{jp}\otimes\sigma_{3}+z_{3}\Pi^{(2)j}\,h_{jp}+z_{4}\Pi_{p}^{(3)}\otimes\sigma_{1}+z_{5}{\tilde{\gamma}}^{j}h_{jp}\otimes\sigma_{2}]
\hat{H}_{\mathrm{D}}(q^{1},q^{1(s)},p^{n})=p_{a}^{n_{a}}\dot{q}_{a}^{n_{a}}(q^{1},q^{1(s)},p^{n})-L_{\mathrm{q}}\left(q^{1},q^{1(s)},\dot{q}^{n}(q^{1},q^{1(s)},p^{n})\right).
\left(\begin{array}{cc}{0}&{f}\\{-f}&{0}\\\end{array}\right)\star\left(\begin{array}{cc}{0}&{g}\\{-g}&{0}\\\end{array}\right)-\left(\begin{array}{cc}{0}&{g}\\{-g}&{0}\\\end{array}\right)\star\left(\begin{array}{cc}{0}&{f}\\{-f}&{0}\\\end{array}\right)=\left(\begin{array}{cc}{\left[g,f\right]_{\star}}&{0}\\{0}&{\left[g,f\right]_{\star}}\\\end{array}\right)
\begin{array}{cc}{\left\{\,b(\sigma,\tau),c({\sigma}^{\prime},\tau)\right\}\,=2\pi\delta(\sigma-{\sigma}^{\prime})}&{}\\{\left[\beta(\sigma,\tau),\gamma({\sigma}^{\prime},\tau)\right]=2\pi\delta(\sigma-{\sigma}^{\prime})}&{}\\\end{array}
P^{-}=\frac{1}{2}\intdx^{-}d^{2}x^{\perp}:\left[-\phi\partial^{\perp2}\phi+\sum_{i=1}^{2}\left(\bar{\psi}_{m}^{(i)}\frac{-\partial^{\perp2}+m^{2}}{i\partial^{+}}\gamma^{+}\psi_{m}^{(i)}+2g\phi\bar{\psi}_{m}^{(i)}\psi_{m}^{(i)}+g^{2}\bar{\psi}_{m}^{(i)}\phi\frac{\gamma^{+}}{i\partial^{+}}\phi\psi_{m}^{(i)}\right)\right]:\;.
{\not\!\nabla}^{2}=\nabla^{\mu}\nabla_{\mu}-e_{a}^{\mu}e_{b}^{\nu}e_{c}^{\lambda}J^{ab}T_{\mu\nu}^{c}\nabla_{\lambda}+\frac{1}{2}e_{a}^{\mu}e_{b}^{\nu}J^{ab}J^{cd}R_{cd\mu\nu},
\left\{P_{\mu}\left(\sigma,\tau\right),M_{f}\left(\tau\right)\right\}=-\partial_{\sigma}\left[f\left(\sigma,\tau\right)\left(\frac{h^{\tau\sigma}\left(\sigma,\tau\right)}{h^{\tau\tau}\left(\sigma,\tau\right)}P_{\mu}\left(\sigma,\tau\right)+\fracT{2h^{\tau\tau}\left(\sigma,\tau\right)\sqrt{h\left(\sigma,\tau\right)}}\partial_{\mu}\left(\sigma,\tau\right)\right)\right]
\left[\,{\frac{1}{4\piR^{2}}}\,{\frac{\partial}{\partialR}}\,{\frac{1}{R^{2}}}\,{\frac{\partial}{\partialR}}-{\frac{pM_{0}^{2}}{l_{Pl}^{2}}}\,\right]\,\psi\left(\,R\,\right)=0\.
x_{\bar{1}2}^{\mu}=x_{1-}^{\mu}-x_{2+}^{\mu}-2i\theta_{2}^{a}\sigma^{\mu}\bar{\theta}_{1a}=x_{12}^{\mu}+i\theta_{12}^{a}\sigma^{\mu}\bar{\theta}_{12a}=(x_{12}^{\mu})_{-}=-(x_{21}^{\mu})_{+}
{\calA}_{{\bfk}_{1}{\bfk}_{2}}^{\sigma_{1}\sigma_{2}}=\;\langlep_{2}\vert\otimes\langle{f_{1}}_{{\bfk}_{1}\sigma_{1}},\bar{f_{2}}_{{\bfk}_{2}\sigma_{2}}\vert\;\hat{S}_{I}\;\vert0\rangle\otimes\vertp_{1}\rangle\;,
<\overline{{\psi}}_{L}^{(2)}\overline{{\psi}}_{L}^{(1)}\psi_{R}^{(1)}\psi_{R}^{(2)}>=(\frac{e^{\gamma}}{4\pi})^{2}\frac{2}{\pi}e_{c}^{2}=(\frac{e^{\gamma}}{4\pi})^{2}m_{S}^{2}\quad.
\psi_{1\xi}^{\dagger}(\epsilon)\psi_{2\eta}(-\epsilon)=ie^{\frac{i}{2}[-\theta(\eta)-\theta(\xi)+\frac{1}{2}R_{2}(\epsilon,\xi,\eta)]}e^{-i\pi\intd^{2}y\frac{1}{e}\partial_{i}E_{i}(y)}e^{iF_{2}(A,E)}
G_{D}^{(n)}({\bfr^{\prime\prime}},{\bfr^{\prime}};E)=\sigma^{n}\,\left[G_{D}^{(0)}({\bf0},{\bf0};E)\right]^{n-1}\,G_{D}^{(0)}({\bfr}^{\prime\prime},{\bf0};E)\,G_{D}^{(0)}({\bf0},{\bfr}^{\prime};E)\;,
{\calI}(a,b)\equiv\int_{a}^{\infty}ds\,\frac{(s^{2}-a^{2})^{D/2}\left(\frac{2s}{b}+\frac{2}{b}+1\right)e^{-2s}}{\left(\frac{2s}{b}+1\right)\left[\left(\frac{2s}{b}+1\right)^{2}-e^{-2s}\right]}.
{\calL}_{HE}^{(1)}=\sum_{k=2}^{\infty}{\frac{(-1)^{k}}{8\pi^{2}}}{\frac{2^{2k}\vertB_{2k}\vert\;m^{4}}{2k(2k-1)(2k-2)}}{\frac{B^{2k}}{B_{\mathrm{cr}}^{2k}}}\,,
N^{Ii}\equiv\frac{\gamma^{\tau\tau}\Pi_{\tau}^{i}+(\gamma^{\tau\sigma}\mp1/\sqrt{-\gamma})\Pi_{\sigma}^{i}}{\gamma^{\tau\tau}\Pi_{\tau}^{+}+(\gamma^{\tau\sigma}\mp1/\sqrt{-\gamma})\Pi_{\sigma}^{+}}~,
d\Omega_{D}=\frac{i}{6}\bar{L}\hat{L}^{3}\tau_{3}^{\prime}\tau_{1}^{\prime}L+i\bar{L}\tau_{1}^{\prime}(\frac{e^{\frac{\phi}{2}}}{\sqrt{1+e^{2\phi}C_{0}^{2}}}\tilde{F}-b_{2}^{\prime})\hat{L}L+\frac{1}{30}(\varepsilon^{a_{1}...a_{5}}L^{a_{1}}\wedge\cdots\wedgeL^{a_{5}}+\varepsilon^{a_{1}^{\prime}...a_{5}^{\prime}}L^{a_{1}^{\prime}}\wedge\cdots\wedgeL^{a_{5}^{\prime}})\,.
K_{S}^{(1)}([\omega_{1}],[\omega_{2}])=Res\sum_{A=1}^{d}\frac{1/2(P_{2}{\frac{\partial}{\partialX^{A}}}P_{1}-P_{1}{\frac{\partial}{\partialX^{A}}}P_{2})dX^{1}\ldotsdX^{d}}{\frac{\partialW}{\partialX^{A}}\prod_{B=1}^{d}\frac{\partialW}{\partialX^{B}}}
\begin{array}{l}{\{n^{a}(x),n^{b}(y)\}=0\;\;\;\;\;,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\{{\pi}^{a}(x),n^{b}(y)\}=(\delta^{ab}-n^{a}(x)n^{b}(x))\delta(x-y)}\\{\{{\pi}^{a}(x),{\pi}^{b}(y)\}=({\pi}^{b}(x)n^{a}(x)-{\pi}^{a}(x)n^{b}(x))\;\delta(x-y)}\\\end{array}
{\calL}_{3Leading}[A,H,V]=\frac{i\theta_{CS}}{2}\epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}A_{\rho}+\frac{1}{2G}V_{\mu}V^{\mu}+\frac{1}{2}\epsilon^{\mu\nu\rho}H_{\mu}F_{\nu\rho}[A+V].
\mathbf{E}(\mathbf{x})=\frac{1}{4\pi}\int\rho_{e}(\mathbf{x}^{\prime})\frac{\mathbf{r}}{r^{3}}\,d^{3}x^{\prime}\,-\frac{1}{4\pi}\int\mathbf{J}_{m}(\mathbf{x}^{\prime})\,{\bf\times}\,\frac{\mathbf{r}}{r^{3}}\,d^{3}x^{\prime}
\begin{aligned}{[b]|X_{0}|^{2}-|X_{2}|^{2}+|X_{3}|^{2}-|X_{4}|^{2}}&{{}=}&{\mathrm{Re}(s)}\\{-2|X_{0}|^{2}+|X_{1}|^{2}+|X_{4}|^{2}}&{{}=}&{\mathrm{Re}(t)}\\\end{aligned}
W_{cr}=({\frac{g^{2}}{4\pi}})^{2}\,2\,C_{F}\,C_{A}({\calA})^{2}\int_{0}^{1}d\,\xi\int_{0}^{1}d\,\eta\,log{\frac{|sin\rho(\xi-\eta)|}{|sin\rho\xi|}}log{\frac{|sin\rho(\xi-\eta)|}{|sin\rho\eta|}},
\mathcal{E}\equiv\left|\begin{array}{c}{\overrightarrow{\mathcal{E}}}\\{\mathcal{E}^{0}}\\\end{array}\right|=\mathrm{column}\left|E^{1}-iH^{1},E^{2}-iH^{2},E^{3}-iH^{3},E^{0}-iH^{0}\right|,
S=(-1)^{s}\int\left(-\nabla^{\mu}h^{\mu_{1}\ldots\mu_{s}}\nabla_{\mu}h_{\mu_{1}\ldots\mu_{s}}+[s^{2}+(d-6)s-2(d-3)]h^{\mu_{1}\ldots\mu_{s}}h_{\mu_{1}\ldots\mu_{s}}\right)
{\bfn}_{L\widetildeL}=g^{{\frac{|L-\widetildeL|}{2}}}+g^{{\frac{|L-\widetildeL|}{2}}+1}+\dots+g^{{\frac{L+\widetildeL}{2}}}-g^{-1-{\frac{|L-\widetildeL|}{2}}}-\dots-g^{-1-{\frac{L+\widetildeL}{2}}},
\Psi(\theta)=\left(\begin{array}{lr}{\psi_{n,m-1}(\theta)}\\{\psi_{n,m}(\theta)}\\{\psi_{n,m-1}(\theta)}\\{\psi_{n,m}(\theta)}\\{\,\,\,\,\,\,\,\vdots}\\{\psi_{n,m-1}(\theta)}\\\end{array}\right)_{(2k+1)\times1}
\hat{\calM}_{\mathrm{F}}=\gamma^{\mu}\left(\partial_{\mu}+\frac{2i}{\rho_{\mathrm{cl}}}\mathrm{Im}(\bar{w}_{\mathrm{cl}}\partial_{\mu}w_{\mathrm{cl}})\right)+\left(1-\frac{2|w_{\mathrm{cl}}|^{2}}{\rho_{\mathrm{cl}}^{2}}\right)\left[\mathrm{Re}(m)I+i\mathrm{Im}(m)\gamma^{5}\right]\Psi
ds^{2}=e^{2A(x)}\sum_{\mu,\nu=0}^{q-1}\eta_{\mu\nu}dy^{\mu}dy^{\nu}+\sum_{i=q}^{D-s-3}e^{2F_{i}(x)}dy^{i}dy^{i}+e^{2B(x)}\sum_{\gamma=D-s-2}^{D-1}dx^{\gamma}dx^{\gamma},
\deltaL_{0}=i\Lambda^{a}D_{-}\chi_{a}(\varphi,\tilde{\varphi})+\Lambda^{a}A_{-}^{b}\left(\partial_{\alpha}\chi_{a}\xi_{b}^{\alpha}+\partial_{\overline{{\alpha}}}\chi_{a}A_{-}^{b}\tilde{\xi}_{b}^{\overline{{\alpha}}}\right)(\varphi,\tilde{\varphi}).
\ell_{2n+1}(\lambda):=\left(\begin{array}{cc}{-(\partial_{1}+2v)\rho_{n}(\lambda)/2}&{\rho_{n}(\lambda)}\\{\lambda\rho_{n}(\lambda)-\partial_{1}(\partial_{1}+2v)(\rho_{n}(\lambda)/2-R_{n})}&{(\partial_{1}+2v)\rho_{n}(\lambda)/2}\\\end{array}\right).
G_{0}^{(2)}(\lambda,\vec{r}_{1},z_{1},\vec{r}_{3},z_{3})=\frac{\lambda}{2}\intd^{d-1}r_{2}\int_{0}^{L}dz_{2}G_{0}^{(2)}(\vec{r}_{1},z_{1};\vec{r}_{2},z_{2})G_{0}^{(2)}(\vec{r}_{2},z_{2};\vec{r}_{2},z_{2})G_{0}^{(2)}(\vec{r}_{2},z_{2};\vec{r}_{3},z_{3}).
\deltav^{\ell}(\omega)=-\left(\Phi_{p},\left[\int{\frac{L^{\ell}(\vec{x},0)d\vec{x}}{<\rho>/e_{o}}}\,\,{\frac{1}{\omega-(H-E)}}{e_{0}L^{k}(\vec{y},0)\atop{\phantom{1}}}+\mathrm{crossed~term}\right]\Phi_{p}\right)A_{k}^{e,u}
\epsilon=\left(\begin{matrix}{0}&{1}\\{-1}&{0}\\\end{matrix}\right),\qquad\tau_{1}=\left(\begin{matrix}{0}&{1}\\{1}&{0}\\\end{matrix}\right),\qquad\tau_{3}=\left(\begin{matrix}{1}&{0}\\{0}&{-1}\\\end{matrix}\right).
\frac{1}{2}\int_{B_{1}}\left[\kappa_{1}\,b_{\mu\nu}^{(0)}(x)T^{(1)\mu\nu}+\kappa_{2}\sum_{n=1}^{\infty}\omega_{n}^{(1)}b_{\mu\nu}^{(n)}(x)T^{(1)\mu\nu}-\frac{\kappa_{2}}{\sqrt{3}}\varphi\,T_{\mu}^{(1)\mu}\right]dx.
B^{(2)}=idx^{\underline{a}}(d\bar{\theta}^{1}\Gamma_{\underline{a}}\theta^{1}-d\bar{\theta}^{2}\Gamma_{\underline{a}}\theta^{2})+d\bar{\theta}^{1}\Gamma_{\underline{a}}\theta^{1}d\bar{\theta}^{2}\Gamma_{\underline{a}}\theta^{2}.
\begin{array}{c}{V(\phi,\beta)=\mu^{3}\sum_{s=1}^{\infty}a^{\frac{3}{2}-s}h(3,s)(\gamma+g\phi)^{2s}\frac{2}{\Gamma(-\frac{3}{2}+s+1)}}\\{\left(\int_{1}^{\infty}dt\,t^{2s-3}\frac{e^{\frac{t}{2}}}{e^{t}-1}+\sum_{n=0}^{\infty}\frac{B_{n}(\frac{1}{2})}{n!}\frac{1}{2s-3+n}\right)}\\\end{array}
\Gamma_{\alpha\beta}^{\gamma}=g^{\gamma\bar{\delta}}\partial_{\beta}g_{\alpha\bar{\delta}}\;\qquad{\calD}_{\alpha}U_{\beta}=\partial_{\alpha}U_{\beta}-\Gamma_{\alpha\beta}^{\gamma}U_{\gamma}\;\qquad{\calD}_{\bar{\alpha}}U_{\beta}=\partial_{\bar{\alpha}}U_{\beta}\.