harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
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)\qquad{\mathrm{with}}\qquadT={\binom{1}{z}}\qquad{\mathrm{and}}\qquad\bar{z}\equivz^{\dagger}\quad.
S_{\mathrm{eff}}(\lambda)=S_{\mathrm{eff}}(\lambda=0)+\frac{i}{2}\lambda\left[\mathrm{Tr}V_{1\nu}^{~\mu}(\nabla)-\mathrm{Tr}V_{2\nu}^{~\mu}(\nabla)\right]-\frac{i}{4}\lambda^{2}\mathrm{Tr}[V_{1\nu}^{~\mu}(\nabla)]^{2}+{\calO}(({\calE}^{\mu\nu})^{2}),
{R^{\mu}}_{\nu}={\frac{1}{2}}\,\partial^{\mu}\phi\,\partial_{\nu}\phi+{\frac{1}{2\,4!}}\,e^{\phi/2}\,\left[4\,F^{\mu\alpha_{2}\alpha_{3}\alpha_{4}}\,F_{\nu\alpha_{2}\alpha_{3}\alpha_{4}}\,-\,{\frac{3}{8}}\,{G^{\mu}}_{\nu}\,F^{2}\right]\,,
\frac{B}{A}=\frac{\Gamma(1-|\alpha|)\left(|\alpha|-\frac{m\bar{\lambda}_{\theta}}{2\pi}\right)}{\Gamma(1+|\alpha|)\left(|\alpha|+\frac{m\bar{\lambda}_{\theta}}{2\pi}\right)}\left(\frac{kr_{0}}{2}\right)^{2|\alpha|}.
\Gamma_{kl}=\frac{1}{(4\pi)^{2}}\int\mathrm{d}^{8}z\frac{W^{\alphakl}W_{\alpha}^{kl}\bar{W}_{\dot{\alpha}}^{kl}\bar{W}^{\dot{\alpha}kl}}{(\Phi^{kl})^{2}(\bar{\Phi}^{kl})^{2}}
W[\phi,\pi]=\int\!\left[\frac{d\eta}{2\pi}\right]~\Psi^{\ast}\left[\phi-{\frac{\hbar}{2}\eta}\right]\,e^{-i\intdx\,\eta\left(x\right)\pi\left(x\right)}~\Psi\left[\phi+{\frac{\hbar}{2}\eta}\right],
dx^{\primei}=dx^{i}+i(\theta^{1}\bar{\zeta}^{\dot{1}})(\partial_{\bar{Y}}\psi\sigma^{i}\bar{\zeta})d\bar{Y}+i(\theta^{1}\bar{\zeta}^{\dot{1}})(\psi\sigma^{i}\partial_{Y}\bar{\psi})dY+i\bar{\theta}^{\dot{1}}(\zeta\sigma^{i}\partial_{Y}\bar{\psi})dY,
\Psi(x_{1},\cdots,x_{M}|z_{1},\cdots,z_{N})=\vartheta(x_{1},\cdots,x_{M}|z_{1},\cdots,z_{N})\prod_{\mu=1}^{M}\prod_{j=1}^{N}\psi\Bigl(\frac{x_{\mu}}{z_{j}}\Bigr),
V(H_{1},H_{2})=\frac{1}{8}(g^{2}+g^{\prime\;2})(|H_{1}|^{2}-|H_{2}|^{2})^{2}+\mu_{1}|H_{1}|^{2}+\mu_{2}|H_{2}|^{2}-\mu_{3}(H_{1}H_{2}+h.c),
S^{(n)\prime}(y_{1},\dots,y_{n})=\sum_{i=1}^{n}R_{1}(y_{i})+\sum_{i\nej}^{n}R_{2}(y_{i},y_{j})+\sum_{i\nej\nek}^{n}R_{3}(y_{i},y_{j},y_{k})+\dots.
\langlea_{{\bfq}_{1}}^{\dagger}a_{{\bfq}_{2}}^{\dagger}a_{{\bfp}_{1}}a_{{\bfp}_{2}}\rangle\simeq\langlea_{{\bfq}_{1}}^{\dagger}a_{{\bfp}_{1}}\rangle\langlea_{{\bfq}_{2}}^{\dagger}a_{{\bfp}_{2}}\rangle+\langlea_{{\bfq}_{1}}^{\dagger}a_{{\bfp}_{2}}\rangle\langlea_{{\bfq}_{2}}^{\dagger}a_{{\bfp}_{1}}\rangle,
\begin{array}{cllcll}{{[A_{m}^{i},A_{n}^{j}]}}&{=\;m\delta^{ij}\delta_{m,-n}\;\;\;}&{,\;\;\;}&{[C_{m},C_{n}]}&{=\;m\delta_{m,-n}\;\;\;}&{,}\\{\{B_{r}^{i},B_{s}^{j}\}}&{=\;\delta{ij}\delta_{r,-s}\;\;\;}&{,\;\;\;}&{\{D_{r},D_{s}\}}&{=\;\delta_{r,-s}\;\;\;}&{,}\\\end{array}
H_{I}=\intd^{\nu}x\left[\left(\mu^{2}(\phi_{0})\,\phi_{0}-{\frac{\lambda}{3}}\phi_{0}^{3}\right):\eta(\vec{x}):+{\frac{\lambda}{3!}}\phi_{0}:\eta^{3}(\vec{x}):+{\frac{\lambda}{4!}}:\eta^{4}(\vec{x}):\right].
\frac{k_{B}T_{\mathrm{cr}}}{m_{0}}=\frac{1}{2\pi}\left[\frac{2\Gamma\left({\displaystyle{\frac{3-D}{2}}}\right)}{\sqrt{\pi}\Gamma\left({\displaystyle{\frac{2-D}{2}}}\right)}(2^{3-D}-1)\zeta(3-D)\right]^{1/(2-D)}\,.
\begin{array}{l}{i\dot{b}_{1}+ic_{1}^{\prime}+m\bar{c}_{2}=0,}\\{i\dot{b}_{2}+ic_{2}^{\prime}+m\bar{c}_{1}=0,}\\{i\dot{c}_{1}+ib_{1}^{\prime}-m\bar{b}_{2}=0,}\\{i\dot{c}_{2}+ib_{2}^{\prime}-m\bar{b}_{1}=0.}\\\end{array}
\biggl[\,G_{ijkl}(x)\frac{\delta}{\deltah_{kl}(x)}\frac{\delta}{\deltah_{ij}(x)}-\frac{1}{(16\piG)^{2}}\,\sqrt{h(x)}\Bigl(\!~R(x)+2\Lambda\Bigr)\biggr]\,\Psi[h]=0\,
V\Big[\begin{array}{cc}{\alpha_{1}}&{\alpha_{2}}\\{\beta_{1}}&{\beta_{2}}\\\end{array}\Big]\stackrel{\mathrm{X}}{\longrightarrow}V\Big[\begin{array}{cc}{\alpha_{1}}&{\alpha_{2}}\\{\alpha_{2}+\beta_{1}}&{\alpha_{1}+\beta_{2}}\\\end{array}\Big].
\Phi_{z\bar{z}}=\partial_{z}x^{i}\deltaE_{ij}\partial_{\bar{z}}x^{j}\mathrm{~where~}\partial_{z}x^{i}=\sum_{n}\alpha_{n}^{i}z^{-n-1}\mathrm{~and~}\partial_{\bar{z}}x^{j}=\sum_{n}\bar{\alpha}_{n}^{j}\bar{z}^{-n-1}.
A_{i}^{(1)}\equivPe^{ig\int_{0}^{x_{0}}dtA_{0}(t,{\bfx})}\left(A_{i}+\frac{1}{ig}\partial_{i}\right)\left(Pe^{ig\int_{0}^{x_{0}}dtA_{0}(t,{\bfx})}\right)^{\dagger}=\mathrm{continuous},
\lambda(p^{\mu})\,\equiv\left(\begin{array}{c}{\left(\zeta_{\lambda}\,\Theta_{[j]}\right)\,\phi_{_L}^{\ast}(p^{\mu})}\\{\phi_{_L}(p^{\mu})}\\\end{array}\right)\,\,,\quad\rho(p^{\mu})\,\equiv\left(\begin{array}{c}{\phi_{_R}(p^{\mu})}\\{\left(\zeta_{\rho}\,\Theta_{[j]}\right)^{\ast}\,\phi_{_R}^{\ast}(p^{\mu})}\\\end{array}\right)\,\,.
\left.\begin{array}{lllll}{\gamma_{1,3,(0,1,\pm1)}}&{=}&{\gamma_{1,3,(0,-1,\pm1)}}&{=}&{{\bf0}}\\{\gamma_{1,{\tilde{7}}_{3},(0)}\gamma_{W4,{\tilde{7}}_{3}}^{\pm1}}&{=}&{-\alpha{\bf1}_{1}}&{}&{}\\{\gamma_{1,7_{4},(\pm1)}}&{=}&{\alpha{\bf1}_{1}}&{}&{}\\{\gamma_{1,7_{5},(\pm1)}}&{=}&{{\bf0}}&{}&{}\\\end{array}\right.
\begin{aligned}{\hat{g}_{\mu\nu}}&{{}=e^{a\phi+b\varphi}g_{\mu\nu}\,,}\\{\hat{g}_{xx}}&{{}=e^{-7(a\phi+b\varphi)}\,,\qquad}\\\end{aligned}\qquad\begin{aligned}{\hat{\phi}}&{{}=c\phi+d\varphi+m_{4}x\,,}\\{\hat{A}_{x}}&{{}=-2e^{-3m_{4}x/4}\chi\,,}\\\end{aligned}
{\calT}_{o}\,=\,\quad\left[\begin{array}{cccc}{1}&{\qquad0}&{\qquad0}&{\qquad0}\\{}&{}&{}&{}\\{1}&{\qquad1}&{\qquad0}&{\qquad0}\\{}&{}&{}&{}\\{-2}&{\qquad0}&{\qquad1}&{\qquad0}\\{}&{}&{}&{}\\{0}&{\qquad0}&{\qquad0}&{\qquadq}\\\end{array}\right]\hphantom{xxxxxxxxxxxxxxxxxx}
L_{r}^{\,\alpha_{0},\,\alpha_{0}}(\Phi):=\intd^{\,4}x\\left(a_{r}^{\,\alpha_{0}}\Phi^{\,2}(x)-b_{r}^{\,\alpha_{0}}\Phi(x)\Box\Phi(x)+c_{r}^{\,\alpha_{0}}\Phi^{\,4}(x)\right)\quad,
H^{2}=\left(\frac{\dot{R}}{R}\right)^{2}=\frac{8\piG_{4}}{3}\,\Lambda_{4}-\frac{k}{R^{2}}+\frac{\mu}{R^{4}}+\frac{8\piG_{4}}{3}\,\frac{\tilde{\rho}}{R^{3}}+\left(\frac{4\piG_{4}}{3\sigma}\,\tilde{\rho}^{2}-q^{2}\right)\,\frac{1}{R^{6}}\,.
v=-\frac{f_{0}}{2m_{0}^{2}}G_{c}^{(2)}-\frac{g_{0}}{6m_{0}^{2}}G_{c}^{(3)}-\frac{g_{0}}{2m_{0}^{2}}G_{c}^{(2)}v-\frac{f_{0}}{2m_{0}^{2}}v^{2}-\frac{g_{0}}{6m_{0}^{2}}v^{3}\,,
\left(\vec{D}\cdot\vec{\gamma}\right)^{2}=-\partial_{3}^{2}-\left(\begin{array}{cccc}{D_{-}D_{+}}&{0}&{0}&{0}\\{0}&{D_{+}D_{-}}&{0}&{0}\\{0}&{0}&{D_{-}D_{+}}&{0}\\{0}&{0}&{0}&{D_{+}D_{-}}\\\end{array}\right)
\Omega_{{M_{1}}...{M_{i}}...{M_{j}}...{M_{p}}}\,=\,-(-)^{{m_{i}}m_{j}}(-)^{(m_{i}+m_{j})(m_{j-1}+...+m_{i})}\,\Omega_{{M_{1}}...{M_{j}}...{M_{i}}...{M_{p}}}.
F\equiv\left(\begin{array}{ccc}{f_{1}}&{}&{0}\\{}&{\ddots}&{}\\{0}&{}&{f_{n}}\\\end{array}\right),\quad\Phi\equiv\left(\begin{array}{ccc}{e^{i\phi_{1}}}&{}&{0}\\{}&{\ddots}&{}\\{0}&{}&{e^{i\phi_{n}}}\\\end{array}\right),
{\frac{1}{2}}\sigma_{\mu\nu}{\calF}_{\mu\nu}={\frac{1}{2}}(F_{+}\sigma_{\mu\nu}N_{\mu\nu}^{+}+F_{-}\sigma_{\mu\nu}N_{\mu\nu}^{-})\:\quad\sigma_{\mu\nu}\equiv\frac{[\gamma_{\mu},\gamma_{\nu}]}{2i}\.
\left(\zeta+a_{0}\frac{x_{8}^{4-n}z^{8-n}}{\zeta}\right)\to\zeta+a_{0}\frac{1}{\zeta}z\prod_{i=1}^{N_{s}}(x_{8}-m_{i}^{4})\prod_{j=1}^{N_{f}}(z-m_{j}^{2}).
\psi_{m}(r)=r^{|m|}\,_{1}F_{1}\left(m\left({\frac{1}{2}}+|m|-m{\frac{\delta}{\bar{\delta}}}\right)-{\frac{g}{4}}{\frac{\delta}{\bar{\delta}}},1+2|m|;2\bar{\delta}{\frac{r}{R}}\right)\,e^{-\bar{\delta}{\frac{r}{R}}}\,
\tilde{I}=\int_{X}\varepsilon^{ijk}\bigg(\omega_{i}^{a}(\partial_{j}\omega_{k}^{a}-\partial_{k}\omega_{j}^{a}+{\frac{2}{3}}\varepsilon_{abc}\omega_{j}^{b}\omega_{k}^{c})+\lambdae_{i}^{a}(\partial_{j}e_{k}^{a}-\partial_{k}e_{j}^{a})+2\lambda\varepsilon_{abc}\omega_{i}^{a}e_{j}^{b}e_{k}^{c}\bigg).
S_{10}=\int\left(\frac{1}{4}\,\hat{R}-\frac{1}{2}\,\partial_{M}\hat{\phi}\,\partial^{M}\hat{\phi}-\frac{1}{12}\,e^{-2\hat{\phi}}\,\hat{H}_{MNP}\,\hat{H}^{MNP}\right)\sqrt{-\hat{g}}\,d^{10}\hat{x},
{\cal{G}}^{ij}=\left(\begin{matrix}{.}&{1}&{.}&{.}&{.}&{.}&{.}&{.}\\{1}&{.}&{.}&{.}&{.}&{.}&{.}&{.}\\{.}&{.}&{1}&{.}&{.}&{.}&{.}&{.}\\{.}&{.}&{.}&{.}&{1}&{.}&{.}&{.}\\{.}&{.}&{.}&{1}&{.}&{.}&{.}&{.}\\{.}&{.}&{.}&{.}&{.}&{.}&{1}&{.}\\{.}&{.}&{.}&{.}&{.}&{1}&{.}&{.}\\{.}&{.}&{.}&{.}&{.}&{.}&{.}&{\frac{4}{3}}\\\end{matrix}\right).
I=\intd^{2}x\sqrt{-g}e^{-2\phi}[R+\gammag^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi+\mue^{2\lambda\phi}-\frac{1}{2}g^{\alpha\beta}\partial_{\alpha}f\partial_{\beta}f+\frac{1}{4}e^{\epsilon\phi+\chif}F^{2}],
\begin{aligned}{[b]}&{{}U_{1}:}&{}&{{}\qquad\lambda={\frac{Z_{1}}{Z_{2}}},}&{}&{{}\qquadx={\frac{X}{W}},}&{}&{{}\qquady={\frac{YZ_{2}}{W}},}&{}&{{}\qquadv={\frac{V}{W}}}\\{}&{{}U_{2}:}&{}&{{}\qquad\rho={\frac{Z_{2}}{Z_{1}}},}&{}&{{}\qquadx={\frac{X}{W}},}&{}&{{}\qquadu={\frac{YZ_{1}}{W}},}&{}&{{}\qquadv={\frac{V}{W}}.}\\\end{aligned}
e^{\alpha_{\pm}^{(0)}}\bar{f}_{\pm}=\left.F\pm2\kappa_{5}^{2}\dot{A}e^{-A-\alpha_{\pm}^{(0)}}\bar{\tau}_{\pm(LL)}-\eta^{\mu\nu}k_{\mu}k_{\nu}\frac{3{\alpha^{\prime}}_{\pm}^{(0)}}{2\kappa_{5}^{2}C_{\pm}}\left[F\mp\kappa_{5}^{2}{\alpha^{\prime}}_{\pm}^{(0)}e^{-A-\alpha_{\pm}^{(0)}}\dot{\Psi}^{(0)}\bar{\tau}_{\pm(LL)}\right]\right|_{w=w_{\pm}}.
\left(\psi,A\psi\right)=tr\rhoA\,\,\,\,\,\,with\,\,\,\rho=\left|\alpha\right|^{2}\left|\psi_{-}\right\rangle\left\langle\psi_{-}\right|+\left|\beta\right|^{2}\left|\psi_{+}\right\rangle\left\langle\psi_{+}\right|
\psi_{1}\left(\rho,\varphi\right)=\psi_{n-1,l-1}^{\left(j\right)}\left(\rho,\varphi\right)\,,\;\psi_{2}\left(\rho,\varphi\right)=-i\psi_{n,l}^{\left(j\right)}\left(\rho,\varphi\right)\,.
\varepsilon_{\mu\nu\lambda\rho}\partial_{\lambda}\partial_{\rho}\theta_{i}^{\mathrm{sing}}(x)=2\pi\Sigma_{\mu\nu}^{i}(x)\equiv2\pi\int_{\Sigma_{i}}^{}d\sigma_{\mu\nu}\left(x^{(i)}(\xi)\right)\delta\left(x-x^{(i)}(\xi)\right).
\nabla\left(\begin{matrix}{V_{AB}}\\{V_{I}}\\{\bar{V}^{AB}}\\{\bar{V}^{I}}\\\end{matrix}\right)=\left(\begin{matrix}{0}&{0}&{\frac{1}{2}P_{ABCD}}&{P_{ABJ}}\\{0}&{0}&{\frac{1}{2}P_{ICD}}&{P_{IJ}}\\{\frac{1}{2}P^{ABCD}}&{P^{ABJ}}&{0}&{0}\\{\frac{1}{2}P^{ICD}}&{P^{IJ}}&{0}&{0}\\\end{matrix}\right)\left(\begin{matrix}{V_{CD}}\\{V_{J}}\\{\bar{V}^{CD}}\\{\bar{V}^{J}}\\\end{matrix}\right)
V(\varphi,\sigma)=\frac{\lambda_{\varphi}}{4}(\varphi^{2}-\eta^{2})^{2}+f_{\varphi\sigma}\varphi^{2}\sigma^{2}+\frac{\lambda_{\sigma}}{4}\sigma^{4}-\frac{m_{\sigma}^{2}}{2}\sigma^{2},
S=\int(\frac{1}{2}G_{0}^{+}\Phi\tilde{G}_{0}^{+}\Phi-\frac{1}{6}\Phi\{G_{0}^{+}\Phi,\tilde{G}_{0}^{+}\Phi\}-\frac{1}{24}[\Phi,G_{0}^{+}\Phi][\Phi,\tilde{G}_{0}^{+}\Phi]+\cdots).
\frac{1}{\mathrm{dim~}F}\sum_{i\in{\bfZ},\alpha\in|\Sigma|}\,i\,\mathrm{dim~}F^{[\alpha]}(i)\,\left(D_{\alpha}\cupJ^{n-1}\right)<\frac{1}{\mathrm{dim~}E}\sum_{i\in{\bfZ},\alpha\in|\Sigma|}\,i\,\mathrm{dim~}E^{[\alpha]}(i)\,\left(D_{\alpha}\cupJ^{n-1}\right)
\omega_{\mathrm{\,CS}}[A_{1},A_{2},A_{3}\,]\equiv\,\frac{1}{16\,\pi^{2}}\;\epsilon^{klm}\;\mathrm{tr}\left(A_{kl}\,A_{m}-{\textstyle\frac{2}{3}}\,A_{k}\,A_{l}\,A_{m}\right)\;,
{\calL}^{\star}=-\frac{1}{2}\left(\frac{m}{e}\right)^{2}\left(V^{a,\mu}-\frac{e}{m}j^{a,\mu}\right)M_{\mu\nu}^{ab}\left(V^{b,\nu}-\frac{e}{m}j^{b,\nu}\right)-\frac{1}{4}m^{2}(W_{\mu\nu}^{a})^{2}+\cdots.
\delta\Psi=\epsilon^{2}(1+i\Psi\partial_{\tau}\Psi),\quad\delta\Phi=-i\epsilon^{2}(2\eta-\Psi\partial_{\tau}\Phi)=-i\epsilon^{2}(2\eta-L),\quadL=\Psi\partial_{\tau}\Phi={\frac{\partial_{\tau}\PhiD\Phi}{1+\sqrt{1-(\partial_{\tau}\Phi)^{2}}}},
z^{\alpha}K_{\alpha}(z)=2^{\alpha-1}\Gamma(\alpha)\left[1+\sum_{j=1}^{\infty}\frac1{j!(1-\alpha)_{j}}\left(\frac{z}2\right)^{2j}-\frac{\Gamma(1-\alpha)}{\Gamma(1+\alpha)}\left(\frac{z}2\right)^{2\alpha}\sum_{j=0}^{\infty}\frac1{j!(1+\alpha)_{j}}\left(\frac{z}2\right)^{2j}\right],
{\calL}_{HD}=\frac{1}{2m^{2}}A^{\mu}\Box(\Box+m^{2})\theta_{\mu\nu}A^{\nu}+\frac{\zeta^{2}}{2M^{2}}A^{\mu}\Box(\Box+M^{2})\omega_{\mu\nu}A^{\nu}-j_{\mu}A^{\mu}
\left.I_{\mathrm{\scriptsize~grav}}\right|_{\mathrm{\scriptsize~sing}}=\frac{2^{10}}{35\pi^{6}}\frac{1}{x_{13}^{8}x_{24}^{8}}\left[s(7t^{2}+6t^{4})+s^{2}(-7+3t^{2})-8s^{3}\right].
+\frac{ie\phi_{0}\phi_{1}^{\dagger}}{(\lambda_{1}^{2}-\widehat\partial^{2})\widehat\partial^{2}}[\widehat{/\!\!\!\partial}\gamma^{0}V_{0,1}\widehat{/\!\!\!\partial}]P_{L}+\frac{ie\phi_{0}\phi_{0}^{\dagger}}{(\widehat\partial^{2})^{2}}[\widehat{/\!\!\!\partial}\gamma^{0}V_{0,0}\widehat{/\!\!\!\partial}]P_{L}.
{\calL}_{eff}=\chi\,\frac{m}{2}\,\epsilon^{\mu\nu\lambda}\,A_{\mu}\,\partial_{\nu}A_{\lambda}-\lambda\,\left[\frac{m^{2}}{8}\,F_{\mu\nu}\,F^{\mu\nu}-\frac{\beta^{2}}{4\,m^{2}}\right]+\frac{\beta^{2}\,m^{2}}{\lambda}\,,
\psi^{\prime\prime}\left(r\right)+\frac{1}{r}\psi^{\prime}\left(r\right)+R\left(r\right)\psi\left(r\right)+0\,,\;R\left(r\right)=\left(k_{0}+f_{1}\right)^{2}-\left(k_{3}+f_{2}\right)^{2}-\frac{\left(\bar{l}+A\right)^{2}}{r^{2}}-m^{2}\,.
E_{0}\left(\ell,\mu\gg1,eB\right)\approx-\frac{eBL^{2}}{8\pi^{2}\ell}\,\sum_{k=1}^{\infty}\,\frac{\left(-1\right)^{k+1}}{2^{2k}k^{k*5/2}}\Gamma\left(k+1/2\right)\left[\frac{\left(k+1/2\right)}{\mu^{k+1/2}}+\frac{2k}{\mu^{k-1/2}}\right]e^{-2\muk},
L_{M}=\frac{\alpha}{4}(f_{\mu}f^{\mu})^{\frac{\delta}{2\delta-1}}-\beta\epsilon^{\mu\nu\rho}f_{\mu}\partial_{\nu}A_{\rho}+\frac{m}{2}\epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}A_{\rho}
\hat{{\bfr}}^{(j+1)}\simeq\hat{{\bfq}}^{(j+1)}+\frac{1}{q^{(j+1)}}\left[{\bfp}-({\bfp}\cdot\hat{{\bfq}}^{(j+1)})\,\hat{{\bfq}}^{(j+1)}\right]\,.
D=\left(\begin{array}{cc}{v^{-1}\partial_{1}}&{u^{-1}\partial_{2}}\\{u\partial_{4}}&{v\partial_{3}}\\\end{array}\right)\;,\quad\bar{D}=\hat{V}^{\epsilon\,-1}D=\left(\begin{array}{cc}{v\partial_{3}}&{-q^{-2}u^{-1}\partial_{2}}\\{-q^{-2}u\partial_{4}}&{v^{-1}\partial_{1}}\\\end{array}\right)\;,
\zeta(-\frac{1}{2},a_{D})_{\mida_{D}\rightarrow\infty}=-\frac{2}{3}a_{D}^{3/2}+\frac{1}{2}a_{D}^{1/2}-\frac{1}{48}a_{D}^{-1/2}+O(a_{D}^{-3/2})
H=\frac{1}{V_{1}(Q_{1})+V_{2}(Q_{2})}\{\frac{1}{2mW_{1}(Q_{1})}P_{1}^{2}+\frac{1}{2mW_{2}(Q_{2})}P_{2}^{2}+U_{1}(Q_{1})+U_{2}(Q_{2})\}
\mathrm{det}\left(1\!\!1\,\pm\,\hat{F}\right)=1\,-\,\frac{1}{2}Tr(\hat{F}^{2})\,+\,\frac{1}{8}\left[Tr(\hat{F}^{2})\right]^{2}-\frac{1}{4}\,Tr(\hat{F}^{4})\,+\mathrm{det}\,(\hat{F})
Z_{\Lambda}^{\mathrm{ren}}(\phi)=\frac{1}{192\pi^{2}}\frac{\lambda_{R}^{2}\phi^{2}}{m_{R}^{2}+\frac{1}{2}\lambda_{R}\phi^{2}}\left[1-\frac{\Lambda^{3}}{(\Lambda^{2}+m_{R}^{2}+\frac{1}{2}\lambda_{R}\phi^{2})^{3/2}}\right].
S_{++}(qt,p_{1}t_{1})=p_{1i}\stackrel{\approx}{q}_{1}^{i}+\int_{\gamma_{2}}(\tilde{\tilde{p}}_{i}d\stackrel{\approx}{q}^{i}(\tau)-H(\stackrel{\approx}{q}(\tau),\stackrel{\approx}{p}(\tau)))d\tau
E_{\mu\nu}={\tilde{E}}_{\mu\nu}-\frac{1}{D-2}q_{\mu\nu}{}^{(D)}R_{\alpha\beta}n^{\alpha}n^{\beta}-\frac{1}{D-2}q_{\mu}^{\alpha}q_{\nu}^{\beta}{}^{(D)}R_{\alpha\beta}+\frac{1}{(D-1)(D-2)}q_{\mu\nu}{}^{(D)}R,
\tilde{G}_{4}^{*}(p,q,r)=\tilde{G}_{2}^{*}(p)\tilde{G}_{2}^{*}(q){\frac{B}{(p+q)^{2}+\mu^{\prime2}}}\tilde{G}_{2}^{*}(r)\tilde{G}_{2}^{*}(p+q-r).
\begin{array}{c}{{\calL}(A,B)=\int_{M_{4}}<(H^{A}\otimesI_{E})\wedge(I_{E}\otimesK^{B})>-\frac12<(I_{E}\otimesK^{B})^{2}>}\\{+(A\leftrightarrowB,H^{A}\leftrightarrowK^{B})}\\\end{array}
r,s\approx\sigma_{\mp}\tilde{\epsilon}^{-1/2}\;\qquad\sigma_{\mp}=\frac{\sqrt{2\,\tilde{u}_{0}^{6}+\tilde{u}_{1}-(\zeta+\frac{1}{\zeta})}\mp\sqrt{\tilde{u}_{1}-(\zeta+\frac{1}{\zeta})}}{2\,\tilde{u}_{0}^{6}}\.
\begin{array}{rcl}{J_{\mathrm{vac},3,t_{0}}(t_{1})}&{=}&{\displaystyle\frac{em^{3}}{2\pi^{2}}\int_{-\infty}^{\infty}\mathrm{d}p_{z}\\int_{0}^{\infty}\mathrm{d}p\p\\left\{\frac{E_{\tau_{1}}+p_{z}+a(\tau_{1}))}{E_{\tau_{1}}}\right.}\\{}&{-}&{\displaystyle\left.\frac{E_{\tau_{0}}+p_{z}+a(\tau_{0}))}{E_{\tau_{0}}}|f(\tau_{1})|^{2}\right\}}\\\end{array}
[\bar{q}q]=Z\bar{q}q-{\frac{N_{c}m^{3}\mu^{-2\epsilon}}{4\pi^{2}}}\biggl(\hat{\frac{1}{\epsilon}}+{\frac{C_{F}g^{2}}{8\pi^{2}}}\Bigl(3\hat{\frac{1}{\epsilon}}^{2}-\hat{\frac{1}{\epsilon}}\Bigl)\biggl),
R_{1}^{\prime\prime}+{\frac{1}{\rho}}R_{1}^{\prime}-\left[\left({\frac{l}{\rho}}-eA_{\varphi}\right)^{2}-e\left(A_{\varphi}^{\prime}+{\frac{1}{\rho}}A_{\varphi}\right)-p_{\perp}^{2}\right]R_{1}=0;
\left\{\left(\begin{array}{c}{f_{i}}\\{0}\\\end{array}\right),\left(\begin{array}{c}{0}\\{f_{j}^{\ast}}\\\end{array}\right)\right\}=\delta_{ij}g_{j},\\\\\\left\{\left(\begin{array}{c}{f_{i}}\\{0}\\\end{array}\right),\left(\begin{array}{c}{f_{j}}\\{0}\\\end{array}\right)\right\}=0
M^{[\,A_{\mu}\,A_{\mu}\,]}=1\,\,;\,\,M^{[\,c\,\overline{c}\,]}=1\,\,;\,\,M^{[\,\overline{c}\,c\,]}=-1\,\,;\,\,M^{[\,G\,G\,]}=1\,\,;\,\,
\langle\Phi_{E}^{(p)},\Phi_{E^{\prime}}^{(p^{\prime})}\rangle=\delta_{n_{1}n_{1}^{\prime}}\delta_{n_{2}n_{2}^{\prime}}...\delta_{n_{D}n_{D}^{\prime}}\delta_{p_{1}p_{1}^{\prime}}\delta_{p_{2}p_{2}^{\prime}}...\delta_{p_{D}p_{D}^{\prime}}.
\frac{\partial^{2}}{\partialt^{2}}\delta\phi=\frac{\partial^{2}}{\partial\rho^{2}}\delta\phi+\frac{1}{\rho}\frac{\partial}{\partial\rho}\delta\phi+\frac{1}{\rho^{2}}\frac{\partial^{2}}{\partial\theta^{2}}\delta\phi-(3\phi_{0}^{2}-1)\delta\phi.
L=\frac{1}{2kz}\left[A_{\mu}(\partial^{2}\eta^{\mu\nu}-z\partial_{z}(\frac{1}{z}\partial_{z})\eta^{\mu\nu}-\partial^{\mu}\partial^{\nu})A_{\nu}+2A_{5}\partial_{z}\partial^{\mu}A_{\mu}-A_{5}\partial^{2}A_{5}\right].
A^{\mu\nu}=g^{\mu\lambda}A_{\lambda}^{\nu}=\frac{k^{\mu}}{k^{2}}\left(k_{\lambda}A^{\lambda\nu}\right)-\frac{\varepsilon^{\mu\alpha}k_{\alpha}}{k^{2}}\left(\varepsilon_{\lambda\beta}k^{\beta}A^{\lambda\nu}\right)\;.
S_{E}(q,A_{\mu})=\sum_{k=1}^{\infty}\left\{\frac{1}{2}\frac{(2\pik)^{2}}{\tau}|q_{\mathrm{time}~k}|^{2}+\left[\frac{\gamma}{2}\frac{(2\pik)^{2}}{\tau}+\eta2\pik\right]|{\bfq}_{k}|^{2}-F_{\mu\nu}(2\pik)q_{\muk}^{*}q_{\nuk}\right\}+{\calE}_{0}^{2}\tau.
E=-E_{n}\,,\;n\geq1\;\;\psi_{2}^{(-)}=\sqrt{\frac{E_{n}+m}{2E_{n}}}{e^{-ipx_{2}-\frac{1}{2}{\xi}^{2}}}\left(\begin{array}{c}{0}\\{0}\\{N_{n}H_{n}(\xi)}\\{\frac{\sqrt{E_{n}^{2}-m^{2}}}{E_{n}+m}N_{n-1}H_{n-1}(\xi)}\\\end{array}\right)
{\bfX}^{\mu}({\bfz},\overline{{\bfz}})=\sqrt{\frac{2}{\alpha^{\prime}}}X^{\mu}(z,\overline{{z}})+i\theta\psi^{\mu}(z,\overline{{z}})+i\overline{{\theta}}\widetilde{\psi}^{\mu}(z,\overline{{z}})+i\theta\overline{{\theta}}F^{\mu}(z,\overline{{z}})~.
\mathrm{Tr}\left(T^{a_{1}}T^{a_{2}}T^{a_{3}}\right)\left(\frac{e^{\frac{i}{2}(p_{1}\timesp_{2}+p_{1}\timesp_{3}+p_{2}\timesp_{3})}}{-(k\timesp_{3})(k\times(p_{2}+p_{3}))}+(\mathrm{cyclic})\right)+(1\leftrightarrow2)~,
\tilde{\calI}(j_{1},j_{2},j_{3},k)=\tilde{\mu}^{s_{-}}\Gamma(-s_{-})\int\prod_{i=1}^{s_{-}}d^{2}w_{i}|w_{i}|^{4j_{1}+2\tilde{\rho}}|1-w_{i}|^{4j_{2}}\prod_{i<j}|w_{i}-w_{j}|^{4\tilde{\rho}}
\intd^{10}\hat{x}\sqrt{|\hat{\jmath}|}\,\left\{{\textstyle\frac{1}{4\cdot5!}}\left(\hat{G}_{(5)}^{+}\right)^{2}+{\textstyle\frac{1}{4\cdot5!}}\left(\hat{G}_{(5)}^{-}\right)^{2}+{\textstyle\frac{1}{4!\cdot5!}}\hat{\epsilon}\left[\hat{G}_{(5)}^{+}\hat{\calH}\hat{C}_{(2)}^{-}+\hat{G}_{(5)}^{-}\hat{\calH}\hat{C}_{(2)}^{+}\right]\right\}\,.
\sum_{s}\sum_{n=1}^{\infty}N_{m_{1},n}^{rs}nN_{n,m_{2}}^{st}=\delta^{rt}\frac{1}{m_{1}}\oint\frac{dy}{2\pii}\xi_{t}(y)^{-m_{1}}\frac{1}{m_{2}}\oint\frac{dz}{2\pii}\xi_{t}(z)^{-m_{2}}\frac{\xi_{t}^{\prime}(z)\bar{\xi}_{t}^{\prime}(y)}{(1-\xi_{t}(z)\bar{\xi}_{t}(y))^{2}}
\ddot{f}f\;-\frac{2p}{t}f\dot{f}^{3}+(\frac{6p^{2}}{t^{2}}f^{2}-1)\dot{f}^{2}+\frac{3p}{t}(1-\frac{2p^{2}}{t^{2}}f^{2})\dot{f}f+\frac{2p^{4}}{t^{4}}f^{4}+p(1-3p)\frac{f^{2}}{t^{2}}+1=0
\left\{\begin{array}{ccc}{l_{1}}&{l_{2}}&{l}\\{{}}&{{}}&{}\\{{\frac{L}{2}}}&{{\frac{L}{2}}}&{{\frac{L}{2}}}\\\end{array}\right\}\sim{\frac{(-1)^{l_{1}+l_{2}+l}}{\sqrt{L+1}}}\left(\begin{array}{ccc}{l_{1}}&{l_{2}}&{l}\\{{}}&{{}}&{}\\{0}&{0}&{0}\\\end{array}\right)+O(L^{-3/2})
N_{n_{1},n_{2}}^{d}(w_{1},w_{2})=k^{2(n_{1}+n_{2}-1)-d}\int_{0}^{\infty}dxx^{\frac{d}{2}}\frac{\partial}{\partial{t}}\left\{\dot{P}(P+w_{1})^{-n_{1}}(\tilde{P}+w_{2})^{-n_{2}}\right\}
\begin{array}{ccc}{\qquad\nu_{e}\qquad}&{\qquad\nu_{\mu}\qquad}&{\qquad\nu_{\tau}\qquad}\\{\qquade\qquad}&{\qquad\mu\qquad}&{\qquad\tau\qquad}\\{\qquadu\qquad}&{\qquadc\qquad}&{\qquadt\qquad}\\{\qquadd\qquad}&{\qquads\qquad}&{\qquadb\qquad}\\\end{array}
\begin{array}{rl}{A\circ_{1,Z}B}&{=A(Z^{\dagger}B)}\\{}&{=(AZ^{-2/3}Z^{2/3})(Z^{-2/3}Z^{-1/3}B)}\\{}&{=Z^{2/3}((Z^{-2/3}AZ^{-2/3})(Z^{-1/3}B))}\\{}&{=Z^{2/3}((Z^{-2/3}AZ^{-1/3}Z^{-1/3})(Z^{1/3}Z^{-2/3}B))}\\{}&{=Z^{2/3}((Z^{-2/3}AZ^{-1/3})(Z^{-2/3}BZ^{-1/3}))Z^{1/3}}\\\end{array}
I_{B}=\frac{1}{C}(-1)^{\frac{S-{\overline{S}}}2}\sum_{m_{j^{\prime}}}\delta_{\frac{M-M^{\prime}}{2}+\sum\frac{K^{\prime}}{2k_{j}+4}(m_{j^{\prime}}+1)}^{(K^{\prime})}\prod_{j=1}^{r}N_{L_{j},\overline{{L}}_{j}}^{m_{j^{\prime}}-1}
K(z,u_{a},D)\equiv\int_{0}^{\infty}d\hat{T}\,I(z,u_{a},\hat{T},D)=K_{02}(z,u_{a},D)+f(z,D)G_{Bab}^{1-{\frac{D}{2}}}+O\bigl(z^{4},G_{Bab}^{2-{\frac{D}{2}}}\bigr)\nonumber\,
a_{2m,n}={\frac{2^{2m}}{(2m)!}}\sum_{p=2m}^{n}{\frac{\varrho^{2p-2m}}{(p-2m)!}}\bar{a}_{n-p},~~~b_{2m,n}={\frac{2^{2m}}{(2m)!}}\sum_{p=2m}^{n}{\frac{\varrho^{2p-2m}}{(p-2m)!}}\bar{b}_{n-p}
S_{\bar{\psi}_{r}^{+}\psi_{r}^{+}}=S_{\bar{\psi}_{r}^{-}\psi_{r}^{-}}=S_{\bar{\Phi}_{r}^{+}\Phi_{r}^{+}}=-S_{\bar{\Phi}_{r}^{-}\Phi_{r}^{-}}=\frac{\hat{P}}{P^{2}+M_{r}^{2}},
\frac{\partial}{\partialZ_{a}}(\chi\varphi)=h_{a}\chi\varphi+\sum_{j=1}^{m}\chif_{\mu_{j}}^{a}\varphi_{\mu_{j}}^{a}-\sum_{j=1}^{m}\frac{\partial}{\partial\mu_{j}}(\chi\varphi_{\mu_{j}}^{a})
S_{\mathrm{bulk}}=\intd^{d+1}x\left[-\frac{1}{2(p+1)!}\sqrt{-\hat{g}}\hat{g}^{M_{1}N_{1}}\hat{g}^{M_{2}N_{2}}\cdots\hat{g}^{M_{p+1}N_{p+1}}\hat{F}_{M_{1}M_{2}\ldotsM_{p+1}}\hat{F}_{N_{1}N_{2}\ldotsN_{p+1}}\right].
S=\int{\calL}=-\frac{1}{8\pi}\intd\taud\sigma\sqrt{-\mathrm{det}(h_{\alpha\beta})}h^{\alpha\beta}\eta_{\mu\nu}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\sim-\frac{1}{8\pi}\intd\taud\sigma[(\partial_{\sigma}X)^{2}-(\partial_{\tau}X)^{2}]
B=\left(\frac{z^{*}}{|z|}\right)^{3}e^{\hat{K}/2}\left[-m_{3/2}-\hat{K}^{S}F_{S}-\hat{K}^{T}F_{T}\left(1+\sum_{i=1}^{2}\frac{n_{i}}{3}-\frac{T+T^{*}}{3}\frac{\mu_{12}^{\prime}(T)}{\mu_{12}(T)}\right)\right]
\Gamma\,[A]=-\,i\,N_{f}\,\mathrm{Tr}\,\biggl[-\,\frac{1}{p-m+i\epsilon}\,A-\frac{1}{2}\,\frac{1}{p-m+i\epsilon}\,A\,\frac{1}{p-m+i\epsilon}\,A+\cdots\biggr]
I=\int_{R^{2}}d^{2}z\frac{(y_{2}\bar{y}_{2})^{\frac{1}{4}}}{\bar{y}_{2}}(y_{1}\bar{y}_{1})^{-\frac{1}{4}}e^{-|x_{\perp}|(\sqrt{y_{1}\bar{y}_{1}}+\sqrt{y_{2}\bar{y}_{2}})}
\mathrm{Res}_{z^{\prime}=z}^{\Psi}\partial^{\Psi}\!:\!\psi(z^{\prime})\!:\,\bar{\partial}^{\Psi}\!:\!\phi(z)\!:=-:\!(\phi_{;\mu\eta}-iC_{\eta\nu\mu}\phi_{;\nu})\psi^{;\eta}\bar{\partial}^{\Psi}\!X^{\mu}\!:+O(\epsilon^{5})
\begin{array}{lclclcl}{s_{1}^{*}\,e_{4n-3}}&{=}&{e_{2n-1},}&{\quad}&{s_{1}^{*}\,e_{4n-2}}&{=}&{0,}\\{s_{1}^{*}\,e_{4n-1}}&{=}&{0,}&{\quad}&{s_{1}^{*}\,e_{4n}}&{=}&{e_{2n},}\\{s_{2}^{*}\,e_{4n-3}}&{=}&{0,}&{\quad}&{s_{2}^{*}\,e_{4n-2}}&{=}&{-e_{2n-1},}\\{s_{2}^{*}\,e_{4n-1}}&{=}&{-e_{2n},}&{\quad}&{s_{1}^{*}\,e_{4n}}&{=}&{0,}\\\end{array}\qquadn=1,\,2,\,\ldots.
{\calL}_{int}^{NR}=\frac{v_{1}}4\left(\phi^{\dagger}\frac{(\nabla^{2}\phi^{\dagger})}{m^{2}}\phi^{2}+\frac{(\vec{\nabla}\phi^{\dagger})^{2}}{m^{2}}\phi^{2}\right)+\frac{v_{2}}4\left(\phi^{\dagger}\frac{(\nabla^{2}\phi^{\dagger})}{m^{2}}\phi^{2}-\frac{(\vec{\nabla}\phi^{\dagger})^{2}}{m^{2}}\phi^{2}\right)
Z_{0}^{K3}(\tau)=\frac{N_{1}}{4}G(2\tau){\tilde{Z}}_{a}(\tau)+N\left(Z_{even}^{S_{0}}(\tau){\tilde{Z}}_{2}(\tau){\tilde{Z}}_{-}(\tau)+Z_{odd}^{S_{0}}(\tau){\tilde{Z}}_{2}(\tau){\tilde{Z}}_{+}(\tau)\right),
{\calL}_{0}=\intd^{4}\theta\mathrm{\,tr}\{\hat{Q}^{\dagger}e^{2g\hat{V}}\hat{Q}+\hat{\bar{Q}}e^{-2g\hat{V}}\hat{\bar{Q}}^{\dagger}\}+\frac{1}{2}\intd^{2}\theta~\mathrm{\,tr}WW+\frac{1}{2}\intd^{2}\bar{\theta}~\mathrm{\,tr}\bar{W}\bar{W}
{\calL}_{HD}[\hat{A},\Box\hat{A}]=\frac{1}{2}\hat{A}\Box\hat{A}+\frac{1}{2}\hat{A}\Box\Box(\frac{\theta}{m^{2}}+\frac{\omega}{M^{2}})\hat{A}-j(\theta+\frac{\omega}{\zeta})\hat{A}.
({\widehatA}f)(\bar{\mit\Gamma})=\int_{{\calL}^{1|2}}A({\mit\Gamma_{1}},\bar{\mit\Gamma})f(\bar{\mit\Gamma}_{1})L_{s,b}({\mit\Gamma}_{1},\bar{\mit\Gamma})\mathrm{e}^{-\Phi({\mit\Gamma}_{1},\bar{\mit\Gamma}_{1})}d\mu({\mit\Gamma}_{1},\bar{\mit\Gamma}_{1})\,.
\sum_{m=0}{\textstyle\frac{g^{m}}{m!}}(-1)^{m}f_{(m)}^{\vec{\mu}ac}\,{\calR}_{(m)}^{\vec{\mu}}({\bfr})\,\sum_{m^{\prime}=0}{\textstyle\frac{g^{m^{\prime}}}{m^{\prime}!}}(-1)^{m^{\prime}}f_{(m^{\prime})}^{\vec{\mu}^{\prime}ac^{\prime}}\,{\calR}_{(m^{\prime})}^{\vec{\mu}^{\prime}}({\bfr})=\delta_{c,c^{\prime}}\,.
R_{\mathrm{sing.}}=\frac{4}{C^{2}}\left(\frac{A^{\prime}}{A}+3\frac{B^{\prime}}{B^{\prime}}\right)\delta(r-r_{0})=\left(2h^{\prime}(r_{0})+12\frac{h(r_{0})}{r_{0}}\right)\delta(r-r_{0}).
p^{2}-i\alpha\hbar\left(\frac{1-z^{2}}{z^{2}}\right)\frac{dp}{dz}=E-A^{2}-\frac{B^{2}}{A^{2}}-A(A-\alpha\hbar)\left(\frac{1-z^{2}}{z^{2}}\right)+\frac{2B}{z}.