harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Pi_{\mu_{1}\cdots\mu_{n}}^{a_{1}\cdotsa_{n},(2)}=\Pi_{\mu_{1}\cdots\mu_{n}}^{a_{1}\cdotsa_{n},(\vec{E})},\qquad\Gamma_{\mathrm{eff}}^{\mathrm{PV},(2)}=\Gamma_{\mathrm{eff}}^{\mathrm{PV},(\vec{E})}
V\rightarrowV-{\frac{1}{2(M_{1}^{2}-M_{2}^{2})}}\int{\frac{d^{2}k}{(2\pi)^{2}}}\left[(m^{2}-M_{2}^{2})\sqrt{k^{2}+M_{1}^{2}}+(M_{1}^{2}-m^{2})\sqrt{k^{2}+M_{2}}\right].
{f^{(12,-12)}(\tau,\bar{\tau})={\frac{1}{2^{12}}}{\frac{\Gamma\left({\frac{27}{2}}\right)}{\Gamma\left({\frac{3}{2}}\right)}}\sum_{(m,n)\ne(0,0)}\tau_{2}^{3/2}{\frac{(m+n\bar{\tau})^{24}}{|m+n\tau|^{27}}}.}
\sum_{\omega}\delta_{\omega}\,\Lambda_{A_{1},\cdots,A_{i-1}A_{\omega(i)}}^{[i]}\,\Lambda_{A_{\omega(i+1)},\cdots,A_{\omega(2i)}}^{[i]}=0,\quad1\leqA_{1}<\cdots<A_{2i}\leqn+1,
S_{gh}[{\bar{c}},c;\theta]\,=\,\intd^{3}x\left[\partial_{\mu}{\bar{c}}\cdotD_{\mu}^{L}c\,+\,\frac{1}{2g}(\partial_{\mu}{\bar{c}}\timesc)^{a}G_{\mu\nu}^{ab}(\theta)(\partial_{\nu}{\bar{c}}\timesc)^{b}\right]\;.
S[\psi,A_{\mu},a_{\mu},b_{\mu}]=\intd^{2}x[\overline{{\psi}}\gamma^{\mu}(-\partial_{\mu}+\frac{i}{2}(q+1)A_{\mu}+\frac{i}{2}(q-1)\gamma_{3}A_{\mu}+ia_{\mu}+i\gamma_{3}b_{\mu})\psi].
\frac{W_{+}(z)W_{+}(z^{\prime})}{e^{i\hbar\gamma^{2}h^{+}(z-z^{\prime})}}-\frac{W_{+}(z^{\prime})W_{+}(z)}{e^{-i\hbar\gamma^{2}h^{-}(z-z^{\prime})}}=0.
S_{inv}[\theta]\;=\;\intd^{3}x\,{\mathcalL}_{inv}(\theta)\;\;,\;\;{\mathcalL}_{inv}(\theta)\;=\;\frac{1}{2}{\tilde{f}}_{\mu}^{a}G_{\mu\nu}^{ab}(\theta){\tilde{f}}_{\nu}^{b}
ds_{E}^{2}=(H_{1}H_{2})^{\frac{3}{4}}[(H_{1}H_{2})^{-1}ds^{2}({\bfE}^{2,1})+H_{2}^{-1}d{\bfx}^{1}\cdotd{\bfx}^{1}+H_{1}^{-1}d{\bfx}^{2}\cdotd{\bfx}^{2}+dz^{2}],
e_{\mu}^{(\lambda)}\gamma_{\mu}=\left(\begin{matrix}{0}&{\alpha_{\lambda}}\\{-\alpha_{\lambda}}&{0}\\\end{matrix}\right),\;\alpha_{\sigma}=\left(\begin{matrix}{0}&{-ie^{-i\varphi_{k}}}\\{ie^{i\varphi_{k}}}&{0}\\\end{matrix}\right),\;\alpha_{\pi}=\left(\begin{matrix}{k_{\perp}}&{-k_{3}e^{-i\varphi_{k}}}\\{-k_{3}e^{i\varphi_{k}}}&{-k_{\perp}}\\\end{matrix}\right){\frac{1}{\omega_{k}}}.
\begin{array}{l}{e^{+2}\nabla\psi_{+2\dot{q}}^{2+}+e^{-2}\nabla\psi_{-2\dot{q}}^{2+}+e^{+}\nabla\chi_{\dot{q}}^{2}+e^{-}\nablah_{-\dot{q}}^{2+}+T^{+2}\psi_{+2\dot{q}}^{2+}+T^{-2}\psi_{-2\dot{q}}^{2+}+T^{+}\chi_{\dot{q}}^{2}+T^{-}h_{-\dot{q}}^{2+}}\\{=\frac12\tilde{\gamma}_{\dot{q}p}^{i}\Omega^{+2i}\pi_{p}^{2-}.}\\\end{array}
W_{\psi_{0}}(\psi_{1}\otimes\psi_{2}\otimes\psi_{3};z_{i})\sim\left(z_{2}-z_{3}\right)^{-\bar{h}t_{23}}\left(z_{1}-z_{3}\right)^{-\bar{h}(t_{12}+t_{13})}W_{\psi_{0}}^{2}(\psi_{1}\otimes\psi_{2}\otimes\psi_{3}).
\begin{array}{l}{R_{12}M_{1}M_{2}=M_{2}M_{1}R_{12}\quad,}\\{R_{12}\tilde{M}_{1}\tilde{M}_{2}=\tilde{M}_{2}\tilde{M}_{1}R_{12}\quad,}\\{R_{12}M_{1}\tilde{M}_{2}=\tilde{M}_{2}M_{1}R_{12}\quad.}\\\end{array}
\tilde{X}^{\mu}(\bar{z})\rightarrowD_{\nu}^{\mu}X^{\nu}(\bar{z})\,\quad\tilde{\psi}^{\mu}(\bar{z})\rightarrowD_{\nu}^{\mu}\psi^{\nu}(\bar{z})\,\quad\tilde{\phi}(\bar{z})\rightarrow\phi(\bar{z})\.
{\frac{1}{4}}\,\left\{\,X^{\mu}\,X^{\nu}\,\right\}_{\mathrm{PB}}\left\{\,X_{\mu}\,X_{\nu}\,\right\}_{\mathrm{PB}}\longleftrightarrow{\frac{1}{4}}\,\mathrm{Tr}\mathbf{F}_{\mu\nu}\,\mathbf{F}^{\mu\nu}
\partial_{\mathrm{k}}\Biggl({\frac{f_{\pi}^{2}}{2}}\partial_{\mathrm{k}}UU^{\dagger}+{\frac{1}{8e^{2}}}\biggl[\partial_{\mathrm{i}}UU^{\dagger},\biggl[\partial_{\mathrm{k}}UU^{\dagger},\partial_{\mathrm{i}}UU^{\dagger}\biggr]\biggr]\Biggr)=0
{\calZ}=\sum_{\alpha}\mathrm{Tr}(q^{H_{\overline{{\alphaV}}}^{L}}~{\overline{q}}^{H_{\overline{{\alphaV}}}^{R}}e^{2\pii\alphas}\xi({\overline{{\alphaV}}},{\bfn})\eta({\overline{{\alphaV}}},{\calN}_{\overline{{\alphaV}}},{\bfn}))~.
\left(\begin{array}{c}{z^{\frac{M}{2}}(1-z)^{\frac{1+M}{2}}F(-\frac{1}{2}+M,\frac{3}{2}+M,\frac{3}{2}+M;1-z)}\\{\mpi(\frac{1}{2}-M)\frac{\Gamma(\frac{1}{2}-M)\Gamma(\frac{1}{2}-M)}{\Gamma(\frac{3}{2}-M)\Gamma(-\frac{1}{2}-M)}z^{-\frac{M}{2}}(1-z)^{\frac{M}{2}}F(-1,1,\frac{1}{2}+M;1-z)}\\\end{array}\right).
\langle\zeta\,\zeta\rangle=\frac{\langle\delta\bar{\psi}\,\delta\psi\rangle}{2\bar{\psi}\psi}+\frac{(\bar{\psi}\,\gamma_{a}\psi)\langle\delta\bar{\psi}\,\gamma^{a}\delta\psi\rangle}{2(\bar{\psi}\psi)^{2}}+\cdots.
y_{rr+3}^{(0)}=\frac{1-x_{r+1}x_{r+2}}{1-x_{r}x_{r+1}x_{r+2}}\,\frac{1-x_{r}x_{r+1}x_{r+2}x_{r+3}}{1-x_{r+1}x_{r+2}x_{r+3}}\,.
\psi_{-}=-\gamma^{0}\alpha_{+}{\frac{k\mpim}{\sqrt{k^{2}+m^{2}}}}e^{-i\omegat+i(kx\pm\delta/2)}=-\gamma^{0}\alpha_{+}e^{\pmi\theta/2}e^{-i\omegat+i(kx\pm\delta/2)}\quad\mathrm{~when~}x\to\pm\infty,
N^{NS}=\left(\begin{matrix}{a+c}&{0}&{0}&{0}&{0}&{0}&{a+c-1}&{0}&{0}\\{0}&{a}&{c}&{0}&{0}&{0}&{0}&{a}&{c-1}\\{0}&{c}&{a}&{0}&{0}&{0}&{0}&{c-1}&{a}\\{0}&{0}&{0}&{0}&{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}&{0}&{0}&{0}&{0}\\{a+c-1}&{0}&{0}&{0}&{0}&{0}&{a+c}&{0}&{0}\\{0}&{a}&{c-1}&{0}&{0}&{0}&{0}&{a}&{c}\\{0}&{c-1}&{a}&{0}&{0}&{0}&{0}&{c}&{a}\\\end{matrix}\right).
D_{\mu\nu}^{flow}(x-y)=i\left<0\left|T\bar{A}_{\mu}(x)\bar{A}_{\nu}(y)\right|0\right>+\frac{g_{\mu-}g_{\nu-}}{(1+\alpha)^{2}}\frac{1}{\partial_{-}^{2}}*\delta(x-y)\,
\begin{array}{cc}{\tilde{t}=\frac{1}{2\lambda}[k_{1}(t)+(x^{2}+\lambda^{2})h(t)],}\\{\tilde{w}=\frac{1}{2\lambda}[k_{1}(t)+(x^{2}-\lambda^{2})h(t)],}\\{\tilde{x}^{k}=h(t)x^{k},}\\\end{array}
B_{\mu\nu}(t)=\sum_{\rho=0}^{N}\frac{\eta_{\mu\rho}\eta_{\nu\rho}}{\sqrt{8\omega_{\mu}}}\left[\left(1+\frac{\omega_{\mu}}{\Omega_{\rho}}\right)\mathrm{e}^{i\Omega_{\rho}t}+\left(1-\frac{\omega_{\mu}}{\Omega_{\rho}}\right)\mathrm{e}^{-i\Omega_{\rho}t}\right]\;.
\hbar\dot{\mathrm{C}}_{\mathrm{F}}(t)=-i\mathrm{C}_{\mathrm{F}}(t)\varepsilon_{\mathrm{F}}(t)-\hbar\mathrm{C}_{\mathrm{F}}(t)\langle\mathrm{F},A(t)|\frac{\partial}{\partialt}|\mathrm{F},A(t)\rangle.
<\mu|\tilde{\mu}_{I}>=\frac{(1-\tilde{\mu}^{*}\tilde{\mu})^{j+k/2}\;|\mu|^{(2j+k-1)/2}\;e^{\mu^{*}\tilde{\mu}}}{\sqrt{\Gamma(2j+k)I_{2j+k-1}(2|\mu|)}}
\hat{\hat{S}\,}=\intd^{11}\hat{\hat{x}}\sqrt{|\hat{\hat{g}}|}\left[\hat{\hat{R}\,}-{\textstyle\frac{1}{2\cdot4!}}\hat{\hat{G}\,}{}^{2}-{\textstyle\frac{1}{6^{4}}}{\textstyle\frac{1}{\sqrt{|\hat{\hat{g}}|}}}\,\hat{\hat{\epsilon}\,}\partial\hat{\hat{C}\,}\partial\hat{\hat{C}\,}\hat{\hat{C}\,}\right]\,.
\begin{array}{rcl}{L_{-1}}&{L_{0}}&{L_{1}\quadL_{2}\quad\ldots}\\{J_{-2}\quadJ_{-1}}&{J_{0}}&{J_{1}\quadJ_{2}\quad\ldots}\\{\Lambda_{-3}\quad\Lambda_{-2}\quad\Lambda_{-1}}&{\Lambda_{0}}&{\Lambda_{1}\quad\Lambda_{2}\quad\ldots}\\{\ldots\ldots\ldots\ldots\ldots\ldots\ldots}&{\ldots}&{\ldots\ldots\ldots\ldots\ldots\quad.}\\\end{array}
\omega=\frac{1}{4}\lambdaeE\left(\frac{1}{|\pi_{D}(t_{2})|}+\frac{1}{|\pi_{D}(t_{1})|}\right)=\frac{\lambda}{T}=\frac{\lambdaeE}{2p_{0}(t_{2})},\;\;\lambda=\frac{m^{2}+{\bfp}_{\bot}^{2}}{eE}\;.
\gamma_{0}\;=\;\left(\begin{array}{cc}{\sigma_{3}}&{0}\\{0}&{-\sigma_{3}}\\\end{array}\right)\;\;,\;\;\gamma_{1}\;=\;\left(\begin{array}{cc}{\sigma_{1}}&{0}\\{0}&{-\sigma_{1}}\\\end{array}\right)\;\;,\;\;\gamma_{2}\;=\;\left(\begin{array}{cc}{\sigma_{2}}&{0}\\{0}&{-\sigma_{2}}\\\end{array}\right)
\Gamma\left[\bar{\Psi}_{a},\Psi_{a},A_{a}^{\mu}\right]=W\left[\bar{\eta}_{a},\eta_{a},J_{a}^{\mu}\right]-\int\left(\bar{\eta}_{a}\Psi^{a}+\bar{\Psi}_{a}\eta^{a}+J_{a}^{\mu}A_{\mu}^{a}\right)\,,
{\psi}_{\sfGI}({\bf{r}})=V_{\cal{C}}({\bf{r}})\,\psi({\bf{r}})\;\;\;\mathrm{\small~and}\;\;\;{\psi}_{\sfGI}^{\dagger}({\bf{r}})=\psi^{\dagger}({\bf{r}})\,V_{\cal{C}}^{-1}({\bf{r}})\;,
Z_{k}(M)=\mathrm{e}^{i\phi_{\mathrm{fr}}}\sum_{\beta\in{\calF}_{l}}{\frac{\prod_{i=1}^{n}\sum_{\rho+\Lambda_{i}\in{\calF}_{l}}S_{\beta\rho+\Lambda_{i}}{\calU}_{\rho+\Lambda_{i}\,\rho}^{(p_{i},q_{i})}}{S_{\rho\beta}^{n-2}}},
27m_{dyn}^{2}\equiv\bar{\Sigma}_{0}^{2}=\Lambda^{2}e^{2\delta}\left[\left(\frac{1-\omega}{1+\omega}\right)\frac{g-\left(\frac{\omega+1}{2}\right)^{2}}{g-\left(\frac{1-\omega}{2}\right)^{2}}\right]^{1/\omega}.
K_{0}={\frac{x_{0}^{d-4}}{(x_{0}^{2}+{\bfx}^{2})^{d-2}}}dx^{i}dx^{j}-{\frac{x_{0}^{d-5}x^{i}}{(x_{0}^{2}+{\bfx}^{2})^{d-2}}}dx^{0}dx^{j}+{\frac{x_{0}^{d-5}x^{j}}{(x_{0}^{2}+{\bfx}^{2})^{d-2}}}dx^{0}dx^{i},
\delta_{ls}\tilde{E}^{a}=-2i\tilde{E}^{\alpha}\Gamma_{\alpha\beta}^{a}\epsilon^{\beta}(x,\tilde{\theta}(x))\;,\quad\delta_{ls}\tilde{E}^{\alpha}={\calD}\epsilon^{\alpha}(x,\tilde{\theta}(x))+\ldots\;\quad\ldots
V=\left(\begin{array}{cc}{v_{1}}&{-v_{2}^{\ast}}\\{v_{2}}&{v_{1}^{\ast}}\\\end{array}\right),~~~~~~\bar{U}=\left(\begin{array}{cc}{u_{1}}&{-u_{2}^{\ast}}\\{u_{2}}&{u_{1}^{\ast}}\\\end{array}\right)
{\frac{d^{2}x^{\rho}}{d\tilde{\lambda}^{2}}}+\tilde{\Gamma}_{\mu\nu}^{\rho}{\frac{dx^{\mu}}{d\tilde{\lambda}}}{\frac{dx^{\nu}}{d\tilde{\lambda}}}=-\left[{\calW}^{-1}g^{\rho\sigma}+{\frac{1}{2}}{\frac{dx^{\rho}}{d\tilde{\lambda}}}{\frac{dx^{\sigma}}{d\tilde{\lambda}}}\right]{\frac{dy}{d\tilde{\lambda}}}{\frac{dx^{\mu}}{d\tilde{\lambda}}}\partial_{y}\left({\calW}g_{\sigma\mu}\right).
\begin{aligned}{}&{{}\partial_{x_{3}}=y_{2}\partial_{x_{2}},}&{}&{{}\qquad\partial_{y_{3}}=-x_{2}y_{2}\partial_{x_{2}}-y_{2}^{2}\partial_{y_{2}}+y_{2}u_{2}\partial_{u_{2}}+2y_{2}v_{2}\partial_{v_{2}},}\\{}&{{}\partial_{u_{3}}={\frac{1}{y_{2}}}\partial_{u_{2}},}&{}&{{}\qquad\partial_{v_{3}}={\frac{1}{{y_{2}}^{2}}}\partial_{v_{2}}.}\\\end{aligned}
C_{Q}=-\frac{2\pir_{+}^{2}\left[1-\frac{1-a^{2}}{1+a^{2}}\frac{r_{-}}{r_{+}}\right]\left[1-\frac{r_{-}}{r_{+}}\right]^{2a^{2}/(1+a^{2})}}{\left[1-\frac{3-a^{2}}{1+a^{2}}\frac{r_{-}}{r_{+}}\right]}.
\begin{array}{ccc}{\deltax^{A}=0~~~}&{~~~\delta\theta^{i}=-\theta^{j}T_{j}^{~i}~~~}&{~~~\delta\tilde{\theta}^{\tilde{\imath}}=-\tilde{\theta}^{\tilde{\jmath}}\tilde{T}_{\tilde{\jmath}}^{~\tilde{\imath}}}\\\end{array}
G_{{\bf{k}},{\bf{k}}^{\prime}}^{\Lambda,\Lambda_{0}}=(2\pi)^{D}\delta({\bf{k}}+{\bf{k}}^{\prime})\frac{\theta_{\epsilon}(|{\bf{k}}|,\Lambda)-\theta_{\epsilon}(|{\bf{k}}|,\Lambda_{0})}{{\bf{k}}^{2}+m_{0}^{2}+\Sigma^{\Lambda,\Lambda_{0}}({\bf{k}})}\;.
E^{a}=e^{a}(x)\left(1-{\frac{i}{2R}}\bar{\theta}\theta-{\frac{1}{2\cdot3!R^{2}}}(\bar{\theta}\theta)^{2}\right)-i\theta\gamma^{a}{D}\theta\left(1+{\frac{i}{3!R}}\bar{\theta}\theta\right),
\xi_{even}^{^L}=\phi_{_L}^{\downarrow}+\Theta\phi_{_L}^{\uparrow\,\,\ast}\equiv\left(\begin{matrix}{B^{3}+iB^{0}}\\{B^{1}+iB^{2}}\\\end{matrix}\right)\quad,\quad\xi_{odd}^{^L}=\phi_{_L}^{\downarrow}-\Theta\phi_{_L}^{\uparrow\,\,\ast}\equiv\left(\begin{matrix}{-E^{0}+iE^{3}}\\{-E^{2}+iE^{1}}\\\end{matrix}\right)\quad.
G_{\mathrm{c}}^{(n)}({\bf{r}}_{1},\ldots,{\bf{r}}_{n})\equiv\langle\phi({\bf{r}}_{1})\ldots\phi({\bf{r}}_{n})\rangle_{\mathrm{c}}=\left.\frac{\delta^{(n)}{\cal{G}}_{\mathrm{c}}\{J\}}{\deltaJ({\bf{r}}_{1})\ldots\deltaJ({\bf{r}}_{n})}\right|_{J=0}\;.
S=S^{0}+\phi_{i}^{*}R^{i}{}_{a}c^{a}+(-)^{b}\frac{1}{2}c_{a}^{*}T_{bc}^{a}c^{c}c^{b}+(-)^{i+a}\frac{1}{4}\phi_{i}^{*}\phi_{j}^{*}E_{ab}^{ji}c^{b}c^{a}+...\.
\frac{1}{(2^{8}\cdot3^{2}\cdot7)^{2}}~{\calI}_{\mu\nu\rho,\tau\zeta\varsigma}^{(3)}(x)~{\calI}_{\tau\zeta\varsigma,\alpha\beta\gamma}^{(3)}(x)=\Im_{\mu\nu\rho,\alpha\beta\gamma}^{(3)}.
P=T\frac{1}{T^{\dagger}T}T^{\dagger}=\left(\begin{matrix}{\frac{1}{1+\bar{z}z}}&{\frac{1}{1+\bar{z}z}\bar{z}}\\{z\frac{1}{1+\bar{z}z}}&{z\frac{1}{1+\bar{z}z}\bar{z}}\\\end{matrix}\right)\quad\mathrm{with}\quadT=\left(\begin{matrix}{1}\\{z}\\\end{matrix}\right)\quad\mathrm{and}\quad\bar{z}\equivz^{\dagger}.
\theta=\nabla_{a}n^{a}={\frac{-1}{\sqrt{g}}}{\frac{\partial}{\partialu}}{(\sqrt{g}g^{uv})}={\frac{1}{|g_{uv}|\sqrt{g_{xx}g_{yy}}}}{\frac{\partial}{\partialu}}{(\sqrt{g_{xx}g_{yy}})}\to-\infty
\bar{\partial}_{\dot{\alpha}}^{+}\theta^{\prime}{}^{\gamma}\bar{\partial}_{\dot{\beta}}^{+}\bar{\theta}^{\prime}{}^{\dot{\gamma}}+\bar{\partial}_{\dot{\beta}}^{+}\theta^{\prime}{}^{\gamma}\bar{\partial}_{\dot{\alpha}}^{+}\bar{\theta}^{\prime}{}^{\dot{\gamma}}=0
\psi_{\nu}^{(3)}=\psi_{\nu}^{(1)}+i\psi_{\nu}^{(2)}=\left({\frac{\pi}{2\zeta}}\right)^{1/2}H_{\nu+{\frac{1}{2}}}^{(1)}(\zeta),\quad\psi_{\nu}^{(4)}=\psi_{\nu}^{(1)}-i\psi_{\nu}^{(2)}=\left({\frac{\pi}{2\zeta}}\right)^{1/2}H_{\nu+{\frac{1}{2}}}^{(2)}(\zeta).
\deltaS^{spin}=-\frac{i}{2}\omega_{ab}\sum_{i,j=1}^{n}\int_{\Lambda}(D_{c}-\frac{4x_{c}}{R^{2}})[\frac{\partialL}{\partialu_{i,c}}(\Gamma^{ab})_{j}^{i}u^{j}(x)]d\Omega(x)
{\frac{d^{2}x^{i}}{d\lambda^{2}}}+2{\frac{\dot{a}}{a}}{\frac{dt}{d\lambda}}{\frac{dx^{i}}{d\lambda}}+2{\frac{a^{\prime}}{a}}{\frac{dy}{d\lambda}}{\frac{dx^{i}}{d\lambda}}=0,
\begin{array}{ll}{a_{10}=1,}&{}\\{a_{20}=\sqrt{6},}&{}\\{a_{30}=\sqrt{10+\sqrt{73}},}&{a_{31}=\sqrt{10-\sqrt{73}},}\\{a_{40}=\sqrt{25+3\sqrt{33}},}&{a_{41}=\sqrt{25-3\sqrt{33}}.}\\\end{array}
S=\intf_{\mathrm{kin}}(x_{i},y_{j})\,d^{4}\thetad^{2}z+t\intf_{\mathrm{FI}}(y_{j})\,d\theta^{+}d\bar{\theta}^{-}d^{2}z+\intW(x_{i})\,d^{2}\theta^{+}d^{2}z+\mathrm{h.c.},
{\calA}_{\mathrm{M}}=V_{p+1}\int\frac{d^{p+1}k}{(2\pi)^{p+1}}\int_{0}^{\infty}{\frac{dt}{2t}}\sum_{I}{\frac{\Omega_{i}}{2}}e^{-2\pi{\alpha^{\prime}}t(p^{2}+M_{I}^{2})},
V_{ef}\left(\phi_{c}\right)=\frac{2\alpha^{2}}{4!}\phi_{c}^{4}+\frac{\hbar}{24\pia}\alpha^{3}\phi_{c}^{3}-\frac{\hbar\pi^{2}}{1440a^{4}}+\frac{\hbar\alpha^{2}\phi_{c}^{2}}{96a^{2}}.
\frac{\mathrm{d}^{2}\rho_{n}(r)}{\mathrm{d}r^{2}}=-(\gamma_{1}-\gamma_{-1})^{2}\frac{1}{(2\kappa)^{2}}\big(\mathrm{e}^{-\rho_{n-2}(r)}-2\mathrm{e}^{-\rho_{n}(r)}+\mathrm{e}^{-\rho_{n+2}(r)}\big),
\frac{{q^{2}+\frac{{l(m-l)}}{{R^{2}}}}}{{(q^{2}+\frac{l^{2}}{R^{2}})(q^{2}+\frac{(m-l)^{2}}{{R^{2}}})}}=\frac{R}{m}\left(\frac{\frac{l}{R}}{{q^{2}+\frac{l^{2}}{R^{2}}}}-\frac{\frac{{l-m}}{R}}{{q^{2}+\frac{(m-l)^{2}}{{R^{2}}}}}\right).
V_{1-\mathrm{loop}}(\sigma)=-{\frac{N}{2L}}\sum_{n}p_{n+}=-{\frac{N}{2L}}\sum_{n}{\frac{1}{c}}\left[-s{\frac{n\pi}{L}}+\sqrt{\left({\frac{n\pi}{L}}\right)^{2}+c\sigma^{2}}\right].
\vec{v}^{(1)}=\left(\begin{matrix}{3}\\{4}\\{3}\\\end{matrix}\right),\qquad\vec{v}^{(2)}=\left(\begin{matrix}{1}\\{0}\\{-1}\\\end{matrix}\right),\qquad\vec{v}^{(3)}=\left(\begin{matrix}{1}\\{-2}\\{1}\\\end{matrix}\right),
X_{0}^{+}\tilde{p}^{-\mu}=2p^{-\mu}+\{Q,m^{+\mu}\}+\oint\!\frac{dz_{2}}{2\pii}\;(c\partial\xi^{+}e^{-\varphi^{-}}i\psi^{-\mu}-\xi^{+}\partial(ce^{-\varphi^{-}}i\psi^{-\mu}))(z_{2}).
\displaystyle{\cal\mathcal{L}}_{mTh}=\int_{-\infty}^{+\infty}\left(\bar{\Psi}(i\gamma_{\nu}\partial^{\nu}+m_{0})\Psi-\frac{g}{2}\bar{\Psi}\gamma^{\nu}\Psi\bar{\Psi}\gamma_{\nu}\Psi\right)dx\,,
{{A}_{i}^{\gamma}}({\bf{r}})=\frac{1}{gr}\left\{\left({\delta}_{{i}\,{\gamma}}-\frac{r_{i}r_{\gamma}}{r^{2}}\right)\,{\calN}+\frac{r_{i}\,r_{\gamma}}{r}\frac{d\,{\calN}}{dr}\right\}+{\delta}_{{i}\,{\gamma}}\,{\calT}_{A}+\frac{r_{i}\,r_{\gamma}}{r^{2}}\,{\calT}_{B}+{\epsilon}_{i{\gamma}n}\frac{r_{n}}{r}\,{\calT}_{C}\,,
\begin{array}{lll}{\xi=P_{0}(B^{0}r+\frac{C^{0}}{r}+O(\frac{1}{r^{2}}))}&{+}&{J_{0}(\frac{B^{2}}{l}r+\frac{D^{0}}{r}+O(\frac{1}{r^{2}}))}\\{\quad+P_{1}(\beta^{1}+O(\frac{1}{r^{2}}))}&{+}&{J_{1}(\gamma^{1}+O(\frac{1}{r^{2}}))}\\{\quad+P_{2}(B^{2}r+\frac{C^{2}}{r}+O(\frac{1}{r^{2}}))}&{+}&{J_{2}(\frac{B^{0}}{l}r+\frac{D^{2}}{r}+O(\frac{1}{r^{2}})),}\\\end{array}
\begin{array}{l}{S_{+}\rightarrowd^{2}S_{+}+2cdS_{0}-c^{2}S_{-}\,,}\\{S_{0}\rightarrowbdS_{+}+(1+2bc)S_{0}-acS_{-}\,,}\\{S_{-}\rightarrow-b^{2}S_{+}-2abS_{0}+a^{2}S_{-}\,.}\\\end{array}
{\calA}_{\mathrm{NS}}=\frac{T}{2\pi}{\frac{1}{\sqrt{8\pi^{2}\alpha^{\prime}}}}\int_{0}^{\infty}{dt}\left(\frac{\pi}{t}\right)^{\frac{1}{2}}\,\mathrm{e}^{-b^{2}/(2\alpha^{\prime}t)}\,\left[\left({\frac{f_{4}}{f_{2}}}\right)^{8}-\left({\frac{f_{3}}{f_{2}}}\right)^{8}\right]=-{\frac{T}{4\pi\alpha^{\prime}}}|b|~.
\begin{array}{ll}{[h_{i},h_{j}]=0,}&{[e_{i},f_{j}]=\delta_{ij}h_{i},}\\{\mathrm{~[~}h_{i},e_{j}\mathrm{~]~}=A_{ij}e_{j},}&{\mathrm{~[~}h_{i},f_{j}\mathrm{~]~}=-A_{ij}f_{j},}\\\end{array}
\rho(z_{1};\sigma_{1}|z_{2};\sigma_{2})=\langle(\Psi_{\sigma_{1}}^{\dagger}(z_{1})\Psi_{\sigma_{2}}^{\dagger}(z_{2}))*(\Psi_{\sigma_{2}}(z_{2})\Psi_{\sigma_{1}}(z_{1}))\rangle\,.
\stackrel{\displaystyle\Gamma^{M}\Gamma^{N}=\Gamma^{M}\Gamma^{N}+\Gamma^{M}\Gamma^{N}}{\rule{0.4pt}{1.5mm}\rule{6mm}{1pt}\rule{0.4pt}{1.5mm}\phantom{{}^{N}=\Gamma}\rule{0.4pt}{1.5mm}\rule{6mm}{1pt}\rule{0.4pt}{1.5mm}~{}_{S}\phantom{{}+\Gamma}\rule{0.4pt}{1.5mm}\rule{6mm}{1pt}\rule{0.4pt}{1.5mm}~{}_{A}}
{\calP}_{(\alpha_{1},\alpha_{2})}^{(2)}(z,\bar{z})=\frac{e^{-|z-\alpha_{1}|^{2}}+e^{-|z-\alpha_{2}|^{2}}}{1-e^{-|\alpha_{1}-\alpha_{2}|^{2}}}-\frac{e^{-(z-\alpha_{1})(\bar{z}-\bar{\alpha}_{2})}+e^{-(z-\alpha_{2})(\bar{z}-\bar{\alpha_{1}})}}{e^{|\alpha_{1}-\alpha_{2}|^{2}}-1}.
T_{+2-2}^{-2}=4i\psi_{+2\dot{q}}^{1-}\psi_{-2\dot{q}}^{1-},\T_{\pm2-\dot{q}}^{-2}=4i\chi\psi_{\pm2\dot{q}}^{1-},\T_{-\dot{q}-\dot{p}}^{-2}=-4i\delta_{\dot{q}\dot{p}}(1+\chi^{2}).
i\langle\bar{\psi}_{f}\gamma_{5}\psi_{f}\rangle_{A}={\frac{h_{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}},
\left.\int\,e^{iqA_{b}}\,G(A)\,d\mu_{I_{\beta}}\,(A)=e^{-\tilde{E}_{\beta}(a,q)}\,\int\,e^{iqA_{b}}\,\left[\prod_{p:b\in\partialp}\,i_{\beta}\,(\epsilon(b,p)a,dA)\right]\,G(A)\,d\mu_{I_{\beta}}\,(A)\,\right.
\langleJ_{\mu}(x)\rangle=\left.\frac{e^{2}}{2\pi^{2}}\Phi(x,x^{\prime})\frac{m^{2}K_{1}\left(\left[-m^{2}(x-x^{\prime})^{2}\right]^{1/2}\right)}{\left[-m^{2}(x-x^{\prime})^{2}\right]^{1/2}}(x-x^{\prime})_{\mu}(x-x^{\prime})_{\nu}\widetilde{F}^{\rho\nu}\right|_{x^{\prime}\rightarrowx}\,
F\epsilon^{ij}={\frac{1}{2}}E^{{\alpha}i}\wedgeE^{{\beta}j}\epsilon_{{\alpha}{\beta}}+{\frac{1}{2}}\bar{E}^{\dot{\alpha}i}\wedge\bar{E}^{\dot{\beta}j}\epsilon_{\dot{\alpha}\dot{\beta}}+{\frac{1}{2}}E^{a}\wedgeE^{b}f_{ab}\epsilon^{ij}\;,\quaddF\equiv0\;.
\Gamma^{(1)}=\intd^{8}z\big(\phi\bar{\phi}+\Phi\bar{\Phi}+\hbarK^{(1)}[\phi,\bar{\phi},\Phi,\bar{\Phi}]\big)+\big(\intd^{6}z(\frac{M}{2}\Phi^{2}+\frac{\lambda}{2}\Phi\phi^{2}+\frac{g}{3!}\phi^{3})+h.c.\big)\,
\begin{array}{c|ccccccc}{\,}&{}&{-1}&{0}&{1}&{2}&{3}&{\,}\\\hline{\,}&{}&{}&{}&{\vdots}&{}&{\,}\\{3}&{}&{0}&{0}&{0}&{0}&{0}&{\,}\\{2}&{}&{1}&{1}&{1^{*}}&{0}&{0}&{\,}\\{1}&{\cdots}&{2}&{2}&{2}&{1^{*}}&{0}&{\cdots}\\{0}&{}&{2}&{2}&{2}&{1}&{0}&{\,}\\{-1}&{}&{2}&{2}&{2}&{1}&{0}&{\,}\\{\,}&{}&{}&{}&{\vdots}&{}&{\,}\\\end{array}
\{A,B\}=-i\hbar\int_{-\infty}^{+\infty}d\,x\left({\frac{\deltaA}{\delta{\phi}(x)}}{\frac{\deltaB}{\delta{\phi}^{*}(x)}}-{\frac{\deltaA}{\delta{\phi}^{*}(x)}}{\frac{\deltaB}{\delta{\phi}(x)}}\right).
\underbrace{\mathcal{O}(M^{4})}_{\mathrm{\small~tree-level}}+\underbrace{\mathcal{O}(T_{br})}_{\mathrm{\small~SM~loops}}+\underbrace{\mathcal{O}(T_{br}^{2}M^{-4})+\ldots}_{\mathrm{\small~higher~order}}
X_{z^{a}}=\theta_{a\bar{a}}S^{N}\bar{z}^{\bar{a}}(S^{\dagger})^{N}+\sum_{n=0}^{N-1}\lambda_{a,n}|n\rangle\langlen|\otimesP_{0}^{(2)}\qquad\mathrm{and}\qquad\phi=\gammaS^{N}
Z_{\,\overline{\Psi}}\,[J_{A}]\,=\,\int[d\varphi^{I}]exp{\frac{i}{\hbar}}\Big(S_{_\Sigma}\,\,+\,\hbarM_{1}[\phi^{A},\chi^{a}]\,\,+\,\,J_{A}\,\phi^{A}\Big)
L_{eff}^{(2)}=\frac{1}{2}\bar{M}[c_{1}^{2}\sum_{p_{1}=1}^{n_{1}}\vec{V}_{p_{1}}^{2}+c_{2}^{2}\sum_{p_{2}=1}^{n_{2}}\vec{V}_{p_{2}}^{2}]+\deltaL_{eff}\;\;,
S_{gauge-fixed}^{conf}\sim\intd^{4}x\sqrt{g}\left(-\frac{M_{P}^{2}}{2}R-\frac{1}{4}F_{\mu\nu}g^{\mu\rho}g^{\nu\sigma}F_{\rho\sigma}+\frac{1}{2}\bar{\lambda}\not\!\!D\lambda\right)\,.
S[\psi]=-\intd^{D}x\left\{\frac{1}{2}\sum_{j=1}^{N}\bar{\psi}^{j}\gamma\cdot\partial\psi^{j}+\frac{\lambda}{8N}\sum_{ij=1}^{N}\bar{\psi}^{i}*\psi^{i}*\bar{\psi}^{j}*\psi^{j}\right\}
w_{j}^{2}=\sum_{i=1}^{l}\frac{h(p_{i})\mathrm{det}(p_{i}^{2}-m^{2})^{1/2}}{\prod_{k\nei}(p_{k}^{2}-p_{i}^{2})(p_{i}^{2}-m_{j}^{2})}\Lambda_{N=2}^{2N_{c}-2-N_{f}}.
\left(\frac{d}{dw}\right)^{2m+1}\bigg(aw^{2}+bw+c\bigg)^{m+n+1}=\bigg({\,frac{d}{dw}}\bigg)^{2m+1}F(Q(w))=\bigg({\frac{d}{dw}}\bigg)^{2m+1}f(w).
{\calL}=\frac{1}{2}\partial_{\mu}\rho\partial^{\mu}\rho+\frac{1}{2}\left(1+\frac{1}{v}\rho\right)^{2}\partial_{\mu}\xi\partial^{\mu}\xi-\frac{1}{2}\mu^{2}(\rho+v)^{2}-\frac{1}{4}\lambda(\rho+v)^{4}\.
{\calA}_{D2}={\frac{z_{12}z_{13}z_{14}}{2\prod_{i=1}^{4}y(z_{i})}}\left(\sum_{i=1}^{6}{\frac{1}{z_{1}-a_{i}}}-2\sum_{i=1}^{3}{\frac{1}{z_{1}-b_{i}}}\right)\langle\prod_{i=1}^{4}\mathrm{e}^{ik_{i}\cdotX(z_{i},\bar{z}_{i})}\rangle.
i\hbar{\frac{\partial\Phi}{\partialt}}(1,\cdots,N)={\frac{1}{2m}}\sum_{p=1}^{N}\left\{i\hbar\partial_{i}^{(p)}-{\frac{e}{c}}\sum_{r=1(\nep)}^{N}{\calA}_{i}(p,r)\right\}^{2}\Phi(1,\cdots,N)\,.
\mathrm{\large~\frac{1}{2}~}(W_{m}^{(0)},W_{m}^{(0)})^{a}-V_{m}^{a}W_{m}^{(0)}=0,\qquad\mathrm{\large~\frac{1}{2}~}\{W_{m}^{(0)},W_{m}^{(0)}\}_{A}-V_{A}W_{m}^{(0)}=0,
H\left(m_{eff}\beta\right)=\frac{r}{4\pi^{2}\beta}\sum_{n=1}^{\infty}\frac{1}{n^{2}}\left[\frac{d^{2}}{d\lambda^{2}}\int_{\left|m_{eff}^{2}\right|^{1/2}\beta}^{\infty}\,dy\,\left(y^{2}+\left|m_{eff}^{2}\right|\beta^{2}\right)^{1/2}e^{-2\pirny\lambda/\beta}\right]_{\lambda=1}.
X^{\mu}=x^{\mu}+w^{\mu}\sigma+\taug^{\mu\nu}(p_{\nu}-B_{\nu\rho}w^{\rho})+{\frac{i}{\sqrt{2}}}\sum_{n\neq0}\left[{\frac{\alpha_{n}^{\mu}}{n}}e^{-in(\tau-\sigma)}+{\frac{\tilde{\alpha}_{n}^{\mu}}{n}}e^{-in(\tau+\sigma)}\right],
S=-\frac1{4\pi\alpha^{\prime}}\intd^{2}\sigma\{\sqrt{h}h^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}g_{\mu\nu}(X^{\rho})+\epsilon^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}B_{\mu\nu}(X^{\rho})+\alpha^{\prime}\sqrt{h}\phi(X^{\rho})R^{(2)}\}
\begin{array}{ll}{Z_{23}=}&{\frac{1}{s}(\sum_{l=1}^{g(s)}e^{2\pii(l-1)/ns}+e^{-2\pii/n}\sum_{l=g(s)+1}^{s}e^{2\pii(l-1)/ns})}\\{}&{\times(\sum_{p=0}^{p_{max}}e^{-4\piip/n}\sum_{u(p)>j\geql(p)}e^{4\pii(j-1)/M}),}\\\end{array}
\left.\mathrm{Res}\right|_{s\rightarrow-\frac12}\zeta_{\mathrmint}(s)+\left.\mathrm{Res}\right|_{s\rightarrow-\frac12}\zeta_{\mathrmext}(s)=\frac1{R^{2}}\left[-\frac{z^{3}}{3\pi}-\frac1{15}\frac{z}{\pi}-\frac12\frac{z^{2}}{\pi}\left(-\alpha+\left|\alpha\right|\right)\right]
\frac1G={\frac{1}{12\pi}}\left[\sum_{s}m_{s}^{D-2}(1-6\xi_{s}){}~\Gamma\left(1-D/2,m_{s}^{2}\delta\right)+2\sum_{d}m_{d}^{D-2}\Gamma\left(1-D/2,m_{d}^{2}\delta\right)\right]
\langleF_{\mu}^{i}[\xi|s]F_{\mu^{\prime}}^{i^{\prime}}[\xi^{\prime}|s^{\prime}]\rangle=-{\frac{3i}{4a_{\xi^{\prime}}(s^{\prime})}}\delta^{ii^{\prime}}g_{\mu\mu^{\prime}}\,\delta(s-s^{\prime})\prod_{\bar{s}=0}^{2\pi}\,\delta^{4}(\xi(\bar{s})-\xi^{\prime}(\bar{s})),
{\frac{{\calF}_{c}}{N}}=-{\frac{m_{L}^{3}}{2\pi}}({\frac{1}{6}}+\sum_{n=1}^{\infty}\Bigl({\frac{e^{-nLm_{L}}}{(nLm_{L})^{3}}}+{\frac{e^{-nLm_{L}}}{(nLm_{L})^{2}}}\Bigr)).
W=\left(\zeta+a_{0}\frac{x_{8}^{4-n}z^{8-n}}{\zeta}\right)+x^{2}+z^{5}-zy^{2}+c_{5}z^{4}+c_{4}z^{3}+c_{3}z^{2}+c_{2}z+c_{1}+\cdots,
\Sigma=m+2g^{2}\Sigma\int^{\Lambda}\frac{d^{d}{\bfq}}{(2\pi)^{d}}\frac{1}{\omega}\left(\frac{1}{e^{i\theta}e^{\beta\omega}+1}-\frac{1}{e^{i\theta}e^{-\beta\omega}+1}\right),
\Pi_{\mu\nu}=-\frac{1}{4}T_{\mu\alpha}T_{\nu}^{\alpha}+\frac{1}{12}T_{\alpha}^{\alpha}T_{\mu\nu}+\frac{1}{24}(3T_{\alpha\beta}T^{\alpha\beta}-(T_{\alpha}^{\alpha})^{2})g_{\mu\nu},
V_{j}^{-1}V_{F}c^{F}\left(V_{0F}^{-1}V_{F}\Gamma{\calP}^{-1}\xi_{E}\right)V_{F}^{-1}V_{j}=\left(\frac{dV_{0j}^{-1}V_{F}\Gamma{\calP}^{-1}\xi_{E}}{dV_{0F}^{-1}V_{F}\Gamma{\calP}^{-1}\xi_{E}}\right)^{-1}c^{j}\left(V_{j}^{-1}V_{F}\Gamma{\calP}^{-1}\xi_{E}\right)\.
U_{3}=T_{3}^{\prime}=({\vec{0}}\vert\vert0\vert\sqrt{{\frac{2}{3}}})({\bf0}^{\prime}\vert\vert({\frac{\sqrt{2}}{3}})^{3}\vert0)(({\frac{1}{3}}{\frac{1}{3}}{\frac{1}{3}}{\frac{1}{3}}{\frac{2}{3}})^{3}\vert0)
\frac{\partial}{\partialu_{k}}\Big(F+F_{D}\Big)=\frac{\partialF}{\partiala_{i}}\frac{\partiala_{i}}{\partialu_{k}}+\frac{\partialF_{D}}{\partiala_{D}^{i}}\frac{\partiala_{D}^{i}}{\partialu_{k}}=a_{D}^{i}\frac{\partiala_{i}}{\partialu_{k}}-a_{i}\frac{\partiala_{D}^{i}}{\partialu_{k}}=\frac{in}{\pi}\delta_{k1}.
B_{3/2}^{in}=\int_{S^{2}}dsc_{3/2}V^{-s}=\frac{-\pi^{3/2}}{6}\int_{S^{2}}ds(\frac{\lambda}{2}\hat{\phi}^{2})^{-s}+\frac{\sqrt{\pi}}{2}\int_{s^{2}}(\frac{\lambda}{2}\hat{\phi}^{2})^{1-s}ds,
U^{\prime}=\left(\begin{array}{cc}{\begin{array}{ccc}{{\bf1}_{L_{0}}}&{}&{}\\{}&{{\bf1}_{\scriptstyle\frac{L_{1}}{2}}\otimesr(\alpha_{1})}&{}\\{}&{}&{{\bf1}_{\scriptstyle\frac{L_{2}}{2}}\otimesr(\alpha_{2})}\\\end{array}}&{0}\\{0}&{\begin{array}{ccc}{\ddots}&{}&{}\\{}&{{\bf1}_{\scriptstyle\frac{L_{M-1}}{2}}\otimesr(\alpha_{M-1})}&{}\\{}&{}&{-{\bf1}_{L_{M}}}\\\end{array}}\\\end{array}\right),
\widetilde{\phi}=e\Omega^{2}\delta(\omega-\overrightarrow{k}\cdot\overrightarrow{v})\left(\omega^{2}-v_{s}^{2}k^{2}\right)/2\pi\varepsilonk^{2}\left(v_{s}^{2}k^{2}-\omega^{2}-\Omega^{2}\right)
D_{{\vec{c}},n-1}=D_{{\vec{c}},n}e^{-i2\piN_{\Phi}c_{\gamma}/\ell}e^{\frac{i\pi\tau}{\omega\kappa^{2}}}e^{\frac{-i2\pi\tauN_{A}}{\omega\kappa^{2}}\left(n+\beta/2\pi\right)}e^{i\alphaN_{A}/\omega\kappa^{2}}