harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left.\begin{array}{ccccc}{}&{}&{h^{0,0}}&{}&{}\\{}&{h^{1,0}}&{}&{h^{0,1}}&{}\\{h^{2,0}}&{}&{h^{1,1}}&{}&{h^{0,2}}\\{}&{h^{2,1}}&{}&{h^{1,2}}&{}\\{}&{}&{h^{2,2}}&{}&{}\\\end{array}\right.=\left.\begin{array}{ccccc}{}&{}&{1}&{}&{}\\{}&{0}&{}&{0}&{}\\{1}&{}&{4}&{}&{1}\\{}&{0}&{}&{0}&{}\\{}&{}&{1}&{}&{}\\\end{array}\right.
\left.\frac{\delta^{2}S}{\delta\phi^{a}\delta\phi^{b}}\right|_{\phi=\Phi_{0}}=-\sum_{n}[(\bigtriangledown^{2}-\omega_{n}^{2}+\mu^{2})-\lambdaT(\Phi_{0}^{k}\Phi_{0}^{k}\delta^{ab}+2\Phi_{0}^{a}\Phi_{0}^{b})].
\beta_{ij}^{(k)}\equiv\partial_{x}^{-1}\left({\calL}^{k}\,(\Phi_{j})\Psi_{i}\right)\;\;;\;\;\beta_{ij}^{*\,(k)}\equiv\partial_{x}^{-1}\left(\Phi_{j}\,{\calL}^{*\,k}(\Psi_{i})\right)
x(1+V)^{-1}x=\frac{1}{2}\sum_{r}(x^{(r)},x^{(r)})-\frac{i}{t}\Bigl(x^{(1)}\wedgex^{(2)}+x^{(2)}\wedgex^{(3)}+x^{(3)}\wedgex^{(1)}\Bigr)\,,
M(z)=4=\sum_{m=-\infty}^{m=+\infty}c_{m}^{(3)}z^{m}\\\\mathrm{;}\\\\\,\N(z)=\sum_{m=-\infty}^{m=+\infty}c_{m}^{(2)}z^{m}=0\mathrm{~,}\tag{122}
M_{NSM}^{2}=g^{2}v^{2}\left(\begin{array}{ccccccc}{1}&{-1}&{0}&{\cdots}&{0}&{0}&{0}\\{-1}&{2}&{-1}&{\cdots}&{0}&{0}&{0}\\{0}&{-1}&{2}&{\cdots}&{0}&{0}&{0}\\{}&{}&{}&{\ddots}&{}&{}&{}\\{0}&{0}&{0}&{\cdots}&{-1}&{2}&{-1}\\{0}&{0}&{0}&{\cdots}&{0}&{-1}&{2}\\\end{array}\right).
\delta\lambda_{a}=-{\frac{i}{2}}g_{ab}(\varphi)\Gamma^{\mu}\partial_{\mu}\varphi^{b}\,\varepsilon+{\frac{1}{4}}\left({\frac{3}{4}}\right)^{2/3}t_{\Lambda,a}\Gamma^{\mu\nu}F_{\mu\nu}^{\Lambda}\,\varepsilon.
v=\left(\begin{array}{cccc}{v_{1}}&{0}&{\ldots}&{0}\\{0}&{v_{2}}&{\ldots}&{0}\\{\ldots}&{\ldots}&{\ldots}&{\ldots}\\{0}&{0}&{\ldots}&{v_{N_{f}}}\\{0}&{0}&{\ldots}&{0}\\{\ldots}&{\ldots}&{\ldots}&{\ldots}\\\end{array}\right)
\left(\!\!\begin{array}{ccc}{j_{1}\!}&{\!j_{2}\!}&{\!j_{3}}\\{0\!}&{\!0\!}&{\!0}\\\end{array}\!\!\right)^{2}\simeq\frac{2\theta(\lambda_{E})}{\pi\lambda_{E}(j_{1}^{2},j_{2}^{2},j_{3}^{2})};\qquad\lambda_{E}^{2}\equiv-\lambda^{2},
{V}_{imp}\equiv\intd^{2}z\,\partial_{\alpha}([u_{i}X^{0}]\Theta(X^{0})\partial_{\alpha}X^{i})=\intd\tau\,u_{i}\left(X^{0}\Theta(X^{0})\right)\partial_{n}X^{i})~;\qquadi=1,\ldots,9.
g\left[1\;-\;{\frac{mg}{2}}\int{\frac{d^{3}k}{(2\pi)^{3}}}{\frac{1}{k^{2}-({\bfp}_{1}+{\bfp}_{2})\cdot{\bfk}+{\bfp}_{1}\cdot{\bfp}_{2}-i\epsilon}}\right].
\left[{\frac{d^{2}}{dr^{2}}}+{\frac{2}{r}}{\frac{d}{dr}}+\left(1-{\frac{1}{\alpha}}\right)M^{2}\right]a_{0}+M\left(1-{\frac{1}{\alpha}}\right)\left({\frac{db_{0}}{dr}}+{\frac{2}{r}}b_{0}\right)=M^{2}a_{0},
{\calL}_{\mathrm{eff}}=\frac{1}{8\pi}\mathrm{Im}\!\int\!\!\mathrm{d}^{2}\theta\,\tau_{D}(A_{D})\,W_{D}^{2}-\frac{1}{4\pi}\mathrm{Im}\!\int\!\!\mathrm{d}^{2}\theta\mathrm{d}^{2}\bar{\theta}\,\bar{A}_{D}\,A(A_{D})\,.
\alpha_{2}=\frac{1}{2}\alpha_{ab}\eta^{a}\eta^{b}+\alpha_{\;\;ab}^{\mu\nu}\eta^{a}\eta_{\mu\nu}^{b}+\frac{1}{2}\alpha_{\;\;ab}^{\mu\nu\;\lambda\rho}\eta_{\mu\nu}^{a}\eta_{\lambda\rho}^{b},
\int\int\dots\int_{-\infty}^{\infty}f(x_{i},p_{i})\starg(x_{i},p_{i})\proddx_{j}\proddp_{j}=\int\int\dots\int_{-\infty}^{\infty}f(x_{i},p)g(x_{i},p_{i})\proddx_{j}\proddp_{j}.
S_{\mathrm{coset}}=\int\!d^{2}x[{\bar{\psi}}\gamma^{\mu}i\partial_{\mu}\psi+\psi_{-}^{\dagger}B_{+}^{a}T^{a}\psi_{-}+\psi_{-}^{\dagger}E_{+}^{b}t^{b}\psi_{-}+\mathrm{ghosts}]\quad,
\{6j\}(\sigma_{E},R)\equiv\left\{\begin{array}{ccc}{J_{1}+R}&{j}&{J_{2}+R}\\{J_{3}+R}&{J_{4}+R}&{J_{5}+R}\\\end{array}\right\}\:\:\:\begin{array}{c}{\longrightarrow}\\{R\gg1}\\\end{array}\:\frac{(-1)^{\Psi}}{2R}\,\:d_{\nu_{2}\nu_{3}}^{j}(\theta),
\theta_{\beta}=\Psi_{\beta\alpha}(z_{\alpha},\theta_{\alpha})=\psi_{\beta\alpha}(z_{\alpha})+\theta_{\alpha}\sqrt{f_{\beta\alpha}^{\prime}(z_{\alpha})+\psi_{\beta\alpha}(z_{\alpha})\psi_{\beta\alpha}^{\prime}(z_{\alpha})}.
\partial_{x}\theta(x,t)=\frac{mv}{\hbar}-{\frac{\lambda}{2}}\rho^{2}(x-vt)\\\,Longrightarrow\\\\theta(x,t)=\frac{mv}{\hbar}x+\theta_{0}(t)-{\frac{\lambda}{2}}\int_{-\infty}^{x-vt}~dy~\rho^{2}(y),
\langleT_{zz}(z,\bar{z})T_{zz}(0,0)\rangle=\frac{\pi}{48}\int_{0}^{\infty}d\mu\,\tilde{c}(\mu)\int\frac{d^{2}p}{(2\pi)^{2}}\frac{e^{\frac{i}{2}(p\bar{z}+\bar{p}z)}}{p\bar{p}+\mu^{2}}\,\bar{p}^{4}\equiv\frac{F(|z|^{2})}{2\,z^{4}}\nonumber
{\calL}={\frac{1}{2}}\partial_{\mu}\phi_{x}\partial^{\mu}\phi_{x}-{\frac{1}{2}}m_{0}^{2}\phi_{x}^{2}-{\frac{m^{2}}{\beta^{2}}}(1-cos(\beta\phi_{x}))\equiv{\frac{1}{2}}\partial_{\mu}\phi_{x}\partial^{\mu}\phi_{x}-U(\phi_{x}),
\left[\partial_{1}-(1-\lambda_{1})\left(\frac{1-\lambda_{2}}{z_{12}}+\frac{1-\lambda_{3}}{z_{13}}+\frac{1-\lambda_{4}}{z_{14}}\right)\right]\langle\mu_{\lambda_{1}}\mu_{\lambda_{2}}\mu_{\lambda_{3}}\mu_{\lambda_{4}}\rangle=0.
\langle\intd^{2}x\left[\lambda(i\partial_{i}+\tilde{a}_{i}^{\prime\prime})(-\sigma_{i\alpha\beta}^{\prime\prime}(\gamma_{3\beta\gamma}^{\prime\prime}\psi^{\gamma})\bar{\psi}^{\alpha})-2\bar{\psi}\lambda\phi\psi\right]_{\star}\rangle_{S}=\intd^{2}x\bigl[4g^{2}\lambda\epsilon_{ij}\partial_{i}a_{j}^{\prime\prime}\bigr]_{\star}
\begin{array}{lcr}{{q^{\prime}}_{j}^{n-2}=-2\delta_{j}^{n-2}+\delta_{j+1}^{n-2}+2\delta_{j-1}^{n-2}-\delta_{j-2}^{n-2}}\\{{q^{\prime}}_{j}^{n-2}=-2\delta_{j}^{n-1}+\delta_{j+1}^{n-1}-\delta_{j-1}^{n-1}+\delta_{j-2}^{n-1}+\delta_{j-3}^{n-1}.}\\\end{array}
D_{ir~k}^{~pj~n}=\{t_{i}^{~p}S(t_{r}^{~j}),t_{k}^{~n}\}=(r_{+})_{ik}^{~ms}t_{m}^{~p}t_{s}^{~n}S(t_{r}^{~j})-t_{i}^{~m}t_{k}^{~s}(r_{+})_{ms}^{~pn}S(t_{r}^{~j})-
\begin{array}{c}{-\frac{1}{2}(k+1)k(k-1)<O_{k-2}O_{k_{1}}...O_{k_{p-1}}P>_{c,g-1}}\\{+4(k+1)\sum_{g_{1}+g_{2}=g}\sum_{i\ink}<O_{i_{1}}...O_{i_{n}}P>_{c,g_{1}}<O_{k}O_{i_{n+1}}...O_{i_{p-1}}P>_{c,g_{2}}}\\{+4t_{2}(k+2)<O_{k+2}O_{k_{1}}...O_{k_{p-1}}P>_{c,g}}\\{+2\sum_{i=1}^{p-1}(2k+k_{i}+2)<O_{k+k_{i}}O_{k_{1}}...\hat{O}_{k_{i}}...O_{k_{p-1}}P>_{c,g}=0.}\\\end{array}
\xi(t;\xi_{f})=\left(\xi_{f}+\frac{b}{a}\right)e^{a(t-t_{f})}-\frac{b}{a}\\,\\\gamma(\xi_{f})=\frac{1}{a}\left(\xi_{f}+\frac{b}{a}\right)\left(1-e^{-a(t_{f}-t_{i})}\right)-\frac{b}{a}(t_{f}-t_{i})\\,
\begin{array}{l}{{[}P_{\mu},P_{\nu}{]}=0}\\{{[}P_{\mu},J_{\nu\rho}{]}=\eta_{\mu\nu}P_{\rho}-\eta_{\mu\rho}P_{\nu}}\\{{[}J_{\mu\nu},J_{\rho\sigma}{]}=\eta_{\mu\sigma}J_{\nu\rho}+\eta_{\nu\rho}J_{\mu\sigma}-\eta_{\mu\rho}J_{\nu\sigma}-\eta_{\nu\sigma}J_{\mu\rho},}\\\end{array}
\tau_{0}=\sum_{\sigma_{1}=0}^{1}\sum_{\sigma_{2}=0}^{1}\sum_{\sigma_{3}=0}^{1}Y_{\sigma_{1}\sigma_{2}\sigma_{3}}^{(0)}(t_{2}Z_{2})^{\sigma_{1}}(t_{4}Z_{4})^{\sigma_{2}}(t_{7}Z_{7})^{\sigma_{3}}
\left(\frac{r}{\beta}\right)^{3}\sum_{n=1}^{\infty}\frac{1}{n^{2}}\left[\frac{dI_{n}\left(\lambda\right)}{d\lambda}\right]_{\lambda=1}=\frac{\pir^{3}\left|m_{eff}^{2}\right|^{1/2}}{2\beta^{2}}\sum_{n=1}^{\infty}\frac{1}{n^{3}}\left\{\frac{d}{d\lambda}\frac{1}{\lambda}\left[H_{1}\left(\alpha\lambda\right)-N_{1}\left(\alpha\lambda\right)\right]\right\}_{\lambda=1},
M^{n}\stackrel{p_{0}^{-1}}{\rightarrow}(M^{2n},M^{n},T_{x}^{*}M^{n},\{,\}_{\omega})\stackrel{p^{-1}}{\rightarrow}(M^{4n},M^{2n},T_{a}^{*}M^{2n},\{,\}_{\Omega})
\left(\begin{array}{cc}{t_{1}}&{}\\{t_{2}}&{}\\{\vdots}&{}\\{t_{k-1}}&{}\\{t_{k}}&{}\\\end{array}\right)=\left(\begin{array}{cc}{\frac{m_{k}}{m_{1}}(t+1)+\frac{m_{k-1}+\cdots+m_{2}}{m_{1}}}&{}\\{\frac{m_{k}}{m_{2}}(t+1)+\frac{m_{k-1}+\cdots+m_{3}}{m_{2}}}&{}\\{\vdots}&{}\\{\frac{m_{k}}{m_{k-1}}(t+1)}&{}\\{t}&{}\\\end{array}\right).
V(\rho)=\frac{1}{G}\rho^{2}-i\mathrm{tr}\int_{0}^{\rho}ds\S(x,x;s)-2i\int_{0}^{\rho}sds\[G_{1}(x,x;s)+G_{2}(x,x;s)]+{\calO}\left(\frac{1}{N}\right),
\xi_{\mathrm{ext}}^{(m)}=\left(\frac{u}{\bar{u}}\right)^{\frac{m}{2}}(u\bar{u})^{\frac{\alpha-1}{2}}\left[a(\rho)J_{\gamma}(2\rho\mup(u\bar{u})^{\frac{\alpha}{2}})+b(\rho)J_{-\gamma}(2\rho\mup(u\bar{u})^{\frac{\alpha}{2}})\right]
{\calL}_{\mathrm{GF}}^{(1)}\longrightarrow{\calL}_{\mathrm{GF}}^{(1)^{\prime}}=\frac{1}{2\alpha\kappa^{2}}\eta_{\mu\nu}\partial_{\rho}\tilde{g}^{\mu\rho}\partial_{\sigma}\tilde{g}^{\nu\sigma}.
{\sfa}=\sqrt{|{\kappa}|\left(\frac{\sqrt{4{\ell}({\ell}+1)+1}-1}{2}\right)\;}\;\;\;\;\;\mathrm{and}\;\;\;\;{\sfb}=\sqrt{|{\kappa}|\left(\frac{\sqrt{4{\ell}({\ell}+1)+1}+1}{2}\right)\;}\;\;\;.
^{0}\theta_{\pm}=\frac{\pm1+x+h}{2h}i\pi\;(\textrm{mod}\,2\pii)\qquad\textrm{and}\qquad^{0}\theta_{\pm}^{B}=\frac{\pm1\mpB-x-h}{2h}i\pi\;(\textrm{mod}\,2\pii).
C^{(1)}+2C^{(2)}r+3C^{(3)}r^{2}+\dots=-\frac{d\psi}{dt}\bigl(1+C^{(1)}r+C^{(2)}r^{2}+\dots\bigr)-i\bigl(\frac{dC^{(1)}}{dt}r+\frac{dC^{(2)}}{dt}r^{2}+\dots\bigr)
\tilde{H}{}_{a}^{0f}=\frac{1}{2}\left[-\frac{d^{2}}{dx^{2}}+W_{-,\nu}^{2}-\nu-3-\frac{1}{2}(W_{-,\nu}^{\prime}-\nu+1)\right]-\frac{1}{4}(W_{-,\nu}^{\prime}-\nu+1)\cdot\sigma_{i}\otimes\sigma_{i}
\left(\eta^{ab}\nabla_{a}E_{b}^{\underline{a}}+{\frac{1}{3!}}\epsilon^{abc}E_{a}^{\underline{d}}E_{b}^{\underline{c}}E_{c}^{\underline{b}}F_{\underline{{bcd}}}^{~~~\underline{a}}\right)u_{\underline{a}}^{~i}=0.
R_{symmetry}~=~\left(\begin{matrix}{-1}&{0}&{0}&{0}&{0}&{0}\\{0}&{-1}&{0}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{-1}&{0}&{0}\\{0}&{0}&{0}&{0}&{-1}&{0}\\{0}&{0}&{0}&{0}&{0}&{1}\\\end{matrix}\right)
l(l+1)N^{2}f^{a_{1}...a_{l}}=\sum_{i=1}^{l}w^{a_{i}}\left[((l+2)N-\gamma_{c}w^{c})\gamma_{b}f^{a_{1}...{\nota_{i}}...a_{l}b}\right]+d\left[\left(\frac{l(l+3)}{l+2}N-\frac{l}{l+2}\gamma_{c}w^{c}\right)\gamma_{b}j^{a_{1}...a_{l}b}\right]
D_{n}^{i}(\betaR)=-\frac{2}{\Gamma(|\nu|+1)\Gamma(|\nu|)}\left(\frac{g_{i}^{n}-|\nu|f_{i}^{n}}{g_{i}^{n}+|\nu|f_{i}^{n}}\right)\left(\frac{\betaR}{2}\right)^{2|\nu|},
R_{pn}^{mj}\widetilde{R}_{mq}^{in}=\widetilde{R}_{pn}^{mj}R_{mq}^{in}=\delta_{p}^{i}\,\delta_{q}^{j}\,,\\\,overline{R}{}_{pn}^{mj}\widetilde{\overline{{R}}}{}_{mq}^{in}=\widetilde{\overline{{R}}}{}_{pn}^{mj}\overline{{R}}{}_{mq}^{in}=\delta_{p}^{i}\,\delta_{q}^{j}\,.
\frac{\langlep,\bar{z},\bar{\theta},\bar{\chi}|\hat{\calQ}_{\alpha}^{I}|p,\bar{z},\bar{\theta},\bar{\chi}\rangle}{\langlep,\bar{z},\bar{\theta},\bar{\chi}|p,\bar{z},\bar{\theta},\bar{\chi}\rangle}={\calQ}_{\alpha}^{I}+{\calO}(s^{-2})\,.
w_{0}(x,\rho)=\sum_{m_{1},\cdots,m_{k}\geq0}{\frac{\Gamma(-\sum((m_{k}+\rho_{k})l_{0}^{(k)}+1)}{\prod_{1\leqi\leqp}\Gamma(\sum(m_{k}+\rho_{k})l_{i}^{(k)}+1)}}x^{m+\rho}\;\;.
\left|\begin{array}{cccc}{A_{11}}&{A_{12}Q+A_{23}}&{A_{14}Q^{2}+A_{15}Q+A_{16}}&{0}\\{0}&{A_{11}}&{A_{12}Q+A_{13}}&{A_{14}Q^{2}+A_{15}Q+A_{16}}\\{A_{21}}&{A_{22}Q+A_{23}}&{A_{24}Q^{2}+A_{25}Q+A_{26}}&{0}\\{0}&{A_{21}}&{A_{22}Q+A_{23}}&{A_{24}Q^{2}+A_{25}Q+A_{26}}\\\end{array}\right|=0
{\mathbbT}(t)={\bf\Lambda}_{i}(q^{\frac{3}{2}}t)+{\bf\Lambda}_{i}^{-1}(q^{-\frac{1}{2}}t){\overline{{\bf\Lambda}}}_{j}(q^{\frac{1}{2}}t)+{\overline{{\bf\Lambda}}}_{j}^{-1}(q^{-\frac{3}{2}}t),\qquadi\not=j,
({\mathcalL}_{m}^{(r,s)}(u),\tilde{\mathcalL}_{n}^{(\alpha,\beta)}(v))=\delta_{\alpha+s+1,0}\delta_{\beta+r+1,0}Res({\mathcalL}_{m}^{(r,s)}(u)\star\tilde{\mathcalL}_{n}^{(\alpha,\beta)}(v)),
e^{ikX_{<}(z)}\equiv\sum_{m=0}^{\infty}z^{m}\sum_{\sum_{i=1}^{\infty}ir_{i}=m}\frac{(ka_{-1})^{r_{1}+r_{2}+...}(-)^{r_{2}+r_{4}+r_{6}+...}}{1^{r_{1}}r_{1}!2^{r_{2}}r_{2}!...}\quad.
\textstyle{-\frac{1}{4}}(\sigma^{ij})_{\alpha}{}^{\beta}(\sigma_{ij})_{\gamma}{}^{\delta}+\textstyle{\frac{1}{2}}\delta_{\alpha}{}^{\beta}\delta_{\gamma}{}^{\delta}=\delta_{\alpha}{}^{\delta}\delta_{\gamma}{}^{\beta}\,.
\left(\gamma_{\alpha}^{g_{1},g_{2},g_{3}g_{4}}\right)\,\left(\gamma_{\alpha}^{g_{1}g_{2},g_{3},g_{4}}\right)\:=\:\left(\gamma_{\alpha}^{g_{1},g_{2}g_{3},g_{4}}\right)\,\left(\gamma_{\alpha}^{g_{2},g_{3},g_{4}}\right)\,\left(g_{4}^{*}\gamma_{\alpha}^{g_{1},g_{2},g_{3}}\right)
\langle\,\partial_{-}f(\sigma)\partial_{+}f(\sigma^{\prime})\,\rangle\,_{\mathrm{{in}}}\sim-\,\frac{1}{4\pi}\,\frac{\xi\kappae^{-\,\kappa\sigma^{-}}}{(x_{H}-\xi\,e^{-\,\kappa\sigma^{-}}-\sigma^{+})^{2}}\,,
J_{\mu,a}^{R}=i\sum_{s=-\infty}^{\infty}\bar{\psi}_{s}^{+}\gamma_{\mu}T_{a}^{R}\psi_{s}^{+},\qquadJ_{\mu,a}^{L}=i\sum_{s=-\infty}^{\infty}\bar{\psi}_{s}^{-}\gamma_{\mu}T_{a}^{L}\psi_{s}^{-}
\gamma^{\mu}\Psi_{,\mu}=-i\gamma^{\mu}\frac{e^{\gamma^{5}\alpha}}{\rho}\{Im(\overline{{\Psi}}_{,\mu}\Psi)-j_{\mu}-\gamma^{5}[Im(\overline{{\Psi}}_{,\mu}\gamma^{5}\Psi)-g_{\mu}]\}\Psi
p(x)={\frac{1}{2x}}-{\frac{(3-k)x^{2-k}}{2(x^{3-k}+Q_{p})}}+{\frac{(3-k)x^{2-k}-(2-k)\ell^{2}x^{1-k}}{2(x^{3-k}+Q_{p}-\ell^{2}x^{2-k})}}.
\zeta\left(s\left|\frac{-\partial_{d}^{2}+\xiR+m^{2}}{\sigma^{2}}\right.\right)=\sigma^{2s}\sum_{n=-\infty}^{\infty}\sum_{\ell=0}^{\infty}\left[\left(\frac{2\pin}{\beta}\right)^{2}+\frac{M_{\ell}^{2}\left(\mu,\chi\right)}{r^{2}}\right]^{-s},
I=\frac{1}{2\kappa^{2}}\intd^{D}X\sqrt{|g|}\left(R-\frac{1}{2}(\nabla\vec{\phi})^{2}-\sum_{I=1}^{k}\frac{s_{I}e^{-\vec{\alpha}^{(I)}\vec{\phi}}}{2(d_{I}+1)!}F_{d_{I}+1}^{(I)2}\right),
f^{(2)}=\left(\begin{array}{cccccc}{0}&{-\eta_{ij,kl}}&{0}&{0}&{\partial_{k,ij}}&{0}\\{\eta_{ij,kl}}&{0}&{\partial_{k,ij}}&{0}&{0}&{0}\\{0}&{\partial_{i,lk}}&{0}&{\partial_{i}}&{0}&{0}\\{0}&{0}&{\partial_{k}}&{0}&{0}&{0}\\{\partial_{i,lk}}&{0}&{0}&{0}&{0}&{\partial_{i}}\\{0}&{0}&{0}&{0}&{\partial_{k}}&{0}\\\end{array}\right)\,\delta(\vec{x}-\vec{y})
A^{\dot{c}d}=-\frac{i}{q}\left[\frac{\alpha(v^{d}\partial^{\dot{c}e}u_{e}+u^{d}\partial^{\dot{c}e}v_{e})-\beta(\bar{u}^{\dot{c}}\partial^{\dot{e}d}\bar{v}_{\dot{e}}-\bar{v}^{\dot{c}}\partial^{\dot{e}d}\bar{u}_{\dot{e}})-2im(\alphau^{d}\bar{u}^{\dot{c}}+\betav^{d}\bar{v}^{\dot{c}})}{(\alphau^{a}v_{a}+\beta\bar{u}^{\dot{a}}\bar{v}_{\dot{a}})}\right].
\Pi_{\mu\nu}=i\int\frac{d^{3}k}{(2\pi)^{3}}\,\frac{tr(\gamma_{\mu}(k\!\!\!/+m)\gamma_{\nu}(k\!\!\!/-p\!\!\!/+m))}{[(k-p)^{2}-m^{2}][k^{2}-m^{2}]}
\begin{array}{c}{(a^{\dagger})^{m}\rightarrow\frac{z^{m}}{w-z}\;\theta(|w|-|z|),}\\{(a)^{n}\rightarrow(\frac{D^{n}}{Dz^{n}}\frac1{w-z})\;\theta(|w|-|z|).}\\\end{array}
e^{-2\lambda_{5}}=\frac{\left|{\calM}{\calQ}\right|^{2}e^{-2\phi_{\infty}}}{\left[Im\left({\calM}z_{\infty}{\calQ}\right)\right]^{2}},\quad\lambda_{7}=\frac{Im\left({\calM}{\calQ}\right)}{Im\left({\calM}z_{\infty}{\calQ}\right)},
S_{\sigma}={\frac{-1}{4\pi\alpha^{\prime}}}\intd^{2}\xi\left(\sqrt{\gamma}\gamma^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}g_{\mu\nu}+\epsilon^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}B_{\mu\nu}-{\frac{1}{2}}\alpha^{\prime}\sqrt{\gamma}R^{(2)}\Phi\right),
g=\sum_{PQRSUV}\left(\frac{(U+2I+1)!(V+2I+1)!}{P!Q!R!S!U!V!(2I+1)}\right)^{(1/2)}p^{P}q^{Q}r^{R}s^{S}u^{U}v^{V}\vertPQRSUV>\,.
{\dot{\pi}}_{\phi}({\bfk}_{n},t)=i[H_{T},\pi_{\phi}({\bfk}_{n},t)]=-\frac{e^{3Ht}}{L^{3}}\left[(e^{-2Ht}{\bfk}_{n}^{2}+m^{2})\phi({\bfk}_{n},t)+\frac{\deltaV_{F}}{\delta\phi(-{\bfk}_{n},t)}+\sum_{i}\alpha_{i}\eta_{i}({\bfk}_{n},t)\right],
(\Omega_{2})_{\alpha\beta}=\left(\begin{matrix}{0}&{i}&{0}&{0}\\{-i}&{0}&{0}&{0}\\{0}&{0}&{0}&{-i}\\{0}&{0}&{i}&{0}\\\end{matrix}\right)\qquad(\Omega_{3})_{\alpha\beta}=\left(\begin{matrix}{0}&{1}&{0}&{0}\\{-1}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}\\{0}&{0}&{-1}&{0}\\\end{matrix}\right)
e_{1}=\frac{4}{A}\bigg(\frac{A^{\prime}}{A^{2}}\bigg)^{\prime}\bigg[1-\frac{12\alpha}{M^{2}}\frac{1}{A^{2}}\bigg(\bigg(\frac{A^{\prime}}{A}\bigg)^{2}+\bigg(\frac{\dot{A}}{A}\bigg)^{2}\bigg)\bigg],
e_{x^{+}}(\lambda_{\alpha})^{-1}e_{x^{-}}(\lambda_{\alpha})e_{1}(\lambda_{\alpha})g_{1}(\lambda_{\alpha})e_{q_{1}}(\lambda_{\alpha})^{N}e_{q_{2}}(\lambda_{\alpha})^{N}=-\prod_{\beta=1}^{M}e_{2}(\lambda_{\alpha}-\lambda_{\beta})e_{2}(\lambda_{\alpha}+\lambda_{\beta})
[Im(\tau{})^{\frac{1}{2}(B_{0}-B_{1})}{\cal{Z}}(\tau{})]={\calN}\tau^{-\frac{1}{4}(\chi{}-\sigma{})}\bar{\tau}^{-\frac{1}{4}(\chi{}+\sigma{})}[Im(-\frac{1}{\tau})^{\frac{1}{2}(B_{0}-B_{1})}{\cal{Z}}(-\frac{1}{\tau})
\begin{array}{l}{{\displaystyle{\frac{dg_{t}^{2}}{dt}}=-(D-2)g_{t}^{2}+{\frac{1}{4\pi}}{\frac{g_{t}^{4}}{2\lambda_{2,1}(t)+2N\lambda_{1,2}(t)}}}\}\\{{\displaystyle{\bigg[(12N+12)\lambda_{3,1}(t)+24N\lambda_{1,3}(t)+4(N^{2}+N+4)\lambda_{2,1;1,1}(t)}}}\\{{\displaystyle{+4(2N+1)\lambda_{2,1}(t)+4(N^{2}+2)\lambda_{1,2}(t)\bigg]}}\.}\\\end{array}
{\tt\frac{1}{z-z_{a}}}=\sum_{m=0}^{\infty}{\ttz_{a}}^{m}\left({\tt\frac{1}{z}}\right)^{m+1}\,,\qquad{\tt\frac{1}{\bar{z}-\bar{z}_{a}}}=\sum_{m=0}^{\infty}{\tt\bar{z}_{a}}^{m}\left({\tt\frac{1}{\bar{z}}}\right)^{m+1}\,.
s(g_{1})s(g_{2})=c(g_{1},g_{2})s(g_{1}g_{2})\quad\rightarrow\quads(g_{1})s(g_{2})=\frac{c(g_{1}g_{2})}{c(g_{1})c(g_{2})}c(g_{1},g_{2})s(g_{1}g_{2})\,.
\psi_{1}^{(\pm)}=\hat{\psi_{1}}\pm\varepsilon\Biggl(\frac{1}{2}tan\frac{\chi_{0}}{2}-\frac{\mida_{0}\mid^{2}sin\chi_{0}}{1+\mida_{0}\mid^{4}+2\mida_{0}\mid^{2}sin\chi_{0}}\Biggr)
E(x)=const\,exp(-\frac{im\tau}{2})f(r)\left\|\begin{array}{c}{-exp(i(\varphi-\tau))r_{\bot}/\sqrt{R^{2}+r^{2}}}\\{iexp(i(\varphi-\tau))r_{\bot}/\sqrt{R^{2}+r^{2}}}\\{1}\\{i}\\{0}\\\end{array}\right\|
a^{-1}(sin{\theta_{n}^{0}})^{m_{1}+l_{1}}C_{N-m_{1}}^{m_{1}+\frac{n-1}{2}}(cos{\theta_{n}^{0}})C_{N^{\prime}-l_{1}}^{l_{1}+\frac{n-1}{2}}(cos{\theta_{n}^{0}})\delta^{m_{1}l_{1}}\delta_{aa^{\prime}}
S=\intd^{2}xd^{4}\thetaK(\Phi,\overline{{\Phi}})-\intd^{2}xd\theta^{+}d\theta^{-}W(\Phi)-\intd^{2}xd\overline{{\theta}}^{+}d\overline{{\theta}}^{-}\overline{{W}}(\overline{{\Phi}})\quad,
J_{T}^{\mu}=H^{a\mu\nu\lambda}F_{\nu\lambda}^{a}+m\epsilon^{\mu\nu\lambda\rho}\big(A_{\nu}^{a}F_{\lambda\rho}^{a}-{\frac{2}{3}}f^{abc}A_{\nu}^{a}A_{\lambda}^{b}A_{\rho}^{c}\big)\,.
V=\left(\begin{matrix}{1}&{w}\\{0}&{1}\\\end{matrix}\right),\quadg=\left(\begin{matrix}{a}&{b}\\{c}&{d}\\\end{matrix}\right),\quadV_{g}=\left(\begin{matrix}{1}&{{\frac{aw+b}{cw+d}}}\\{0}&{1}\\\end{matrix}\right).
\sigma_{n_{j}n_{k}}=\frac12\mathrm{T}\mathrm{r}\left[\Lambda_{j}{\bfS}\Lambda_{k}{\bfS}\right]+\frac12{\bfy}{\bfR}{\bfS}\Lambda_{j}{\bfS}\Lambda_{k}{\bfS}\Sigma_{x}{\bfR}{\bfy},\qquadj\neqk
\etaT\;\;=\;\;\frac{1}{(p+1)!(q+1)!}\eta_{\mu_{1}\nu_{1}}T_{\mu_{2}...\mu_{p+1}\nu_{2}...\nu_{q+1}}dx^{\mu_{1}}\wedge...\wedgedx^{\mu_{p}}\wedgedx^{\mu_{p+1}}\otimesdx^{\nu_{1}}\wedge...\wedgedx^{\nu_{q}}\wedgedx^{\nu_{q+1}}
I=\left(\begin{array}{llll}{0}&{1}&{1}&{1}\\{1}&{0}&{0}&{0}\\{1}&{0}&{0}&{0}\\{1}&{0}&{0}&{0}\\\end{array}\right),\quad\quad\hat{\tau}(\vec{\alpha})=\left(\begin{array}{l}{q^{2}\alpha_{1}}\\{q^{2}\alpha_{2}}\\{\alpha_{3}}\\{\alpha_{4}}\\\end{array}\right),\quad\quad\omega(\vec{\alpha})=\left(\begin{array}{l}{\alpha_{1}}\\{\alpha_{4}}\\{\alpha_{2}}\\{\alpha_{3}}\\\end{array}\right)\,.
\int_{0}^{1}\left\{\mathrm{i}(r+\mu)\Phi(\tau,\zeta)+{\frac{1}{2}}{\frac{\partial\Phi}{\partial\zeta}}(\tau,\zeta)\right\}_{|\zeta={\frac{t(\theta-\tau)}{2}}}\mathrm{e}^{\mathrm{i}(r+\mu)t(\theta-\tau)}(1-t)^{-\nu}t^{\nu}dt.\tag{B.35}
\beta_{\lambda}(\lambda,e^{2})\frac{\partial}{\partial\lambda}\zeta(\lambda,e^{2})+\beta_{e^{2}}(\lambda,e^{2})\frac{\partial}{\partiale^{2}}\zeta(\lambda,e^{2})=2\gamma_{J}(\lambda,e^{2})\zeta(\lambda,e^{2})+\delta(\lambda,e^{2})
S=\frac{1}{G}\intdxdt\:\sqrt{-\bar{g}}\:e^{-2\bar{\phi}}(\bar{R}+4(\bar{\nabla}\bar{\phi})^{2}+4\lambda^{2})-\frac{1}{2}\intdxdt\:\sqrt{-\bar{g}}\:\sum_{i=1}^{N}(\bar{\nabla}f_{i})^{2},
D(p)=\left\{\frac{p^{2}}{4\pin\kappa}[\zetaF(p)-4\kappa^{2}z]-(z\zeta-\frac{p^{4}}{4\pi^{2}n^{2}})\sqrt{F(p)}\right\}\left[\zeta^{2}F(p)-\frac{\kappa^{2}p^{4}}{\pi^{2}n^{2}}\right]^{-1}
\rho_{p\rightarrow1+2+\cdots+j+(j+1)\cdots+N}=\int\rho_{p\rightarrowA+B}\,\rho_{A\rightarrow1+2+\cdotsj}\,\rho_{B\rightarrow(j+1)+\cdots+N}\,\bar{d}m_{A}^{2}\,\bar{d}m_{B}^{2}.
{\calL}=-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}+\frac{1}{2}A_{\mu}A^{\mu}+\frac{1}{4m}\epsilon_{\mu\nu\rho\sigma}B^{\mu\nu}\partial^{\sigma}-\frac{1}{4m}\epsilon_{\mu\nu\rho\sigma}\partial^{\nu\rho}A^{\sigma},
\frac{\partial^{2}\tau}{\partialx_{i}\partialx_{j}}=0,\;(\theta_{i}-\theta_{j})\frac{\partial^{2}\alpha^{0}}{\partialx_{i}\partialx_{j}}=2[\frac{\partial\tau}{\partialx_{i}}\frac{\partial\alpha^{0}}{\partialx_{j}}-\frac{\partial\tau}{\partialx_{j}}\frac{\partial\alpha^{0}}{\partialx_{i}}].
J_{\mu}^{reg}=J_{\mu}^{f}-{\frac{1}{\sqrt\pi}}[a{\frac{\epsilon_{\mu}\epsilon_{\nu}-\tilde{\epsilon}_{\mu}\tilde{\epsilon}_{\nu}}{\epsilon_{2}}}\tilde{\partial}_{\nu}(\sigma+\eta)+2\alpha{\frac{\epsilon_{\mu}\epsilon_{\nu}}{\epsilon^{2}}}\tilde{\partial}_{\nu}\Box\sigma],
I\sim\frac{(2|\omega|)^{\frac{1}{2}}e^{-2|\omega||x_{\perp}|}}{2(\omega_{1}-i\omega_{2})}\int_{R^{2}}d^{2}z(y_{1}\bar{y}_{1})^{-\frac{1}{4}}e^{-|x_{\perp}|\sqrt{y_{1}\bar{y}_{1}}}=\frac{(2|\omega|)^{\frac{1}{2}}e^{-2|\omega||x_{\perp}|}}{2(\omega_{1}-i\omega_{2})}\cdot\frac{\pi^{\frac{3}{2}}}{|x_{\perp}|^{\frac{3}{2}}}.
\begin{array}{rclcccc}{\mathrm{\underline{{sector}}}}&{{}}&{\mathrm{\underline{{state}}}}&{{}}&{{}}&{\mathrm{\underline{{~R_1~}}}}&{\mathrm{\underline{{~R_2~}}}}\\{\mathrm{NS}}&{{}}&{|2s_{3},2s_{4}\rangle,}&{s_{3}=s_{4}}&{{}}&{+}&{+}\\{\mathrm{R}}&{{}}&{|-,2s_{1},2s_{2}\rangle,}&{s_{1}=s_{2}}&{{}}&{+}&{+}\\\end{array}
L_{F}=2\\frac{{\bfE}\Gammar_{0}^{4}}{{\bfl}^{3}}\int_{R^{*}/{\bfl}}^{\infty}\frac{u^{2}+u_{R}^{2}}{(u^{2}-u_{R}^{2})^{3}}\frac{u}{(u^{2}-1)^{1/2}}du
\Pi_{\mu\nu}^{\mathrm{(fermion)}}(p)=\left.\frac{\deltaj_{\mu}(p)}{\deltaA^{\nu}(-p)}\right|_{A_{\mu}=0}=-\frac{8\,e^{2}}{(2\pi)^{3}}\int\frac{d^{3}k}{|\vec{k}|}\,n_{F}(|\vec{k}|)\,G_{\mu\nu}({k};p).
S^{\prime}=\intd^{2}\sigma\Big(-e^{-\phi}\sqrt{-\mathrm{det}\,(G_{\mu\nu}+{\calF}_{\mu\nu})}+\frac{1}{2}\tilde{H}^{\mu\nu}(F_{\mu\nu}-2\partial_{\mu}A_{\nu})+\frac{1}{2}e^{-\phi}\epsilon^{\mu\nu}C_{\mu\nu}-\frac{1}{2}C_{0}\epsilon^{\mu\nu}F_{\mu\nu}\Big)
\mathscr{C}^{A}\rightarrow\pm\bar{\mathscr{C}}^{A},\quad\bar{\mathscr{C}}^{A}\rightarrow\mp\mathscr{C}^{A},\quad\mathscr{B}^{A}\rightarrow-\bar{\mathscr{B}}^{A},\quad\bar{\mathscr{B}}^{A}\rightarrow-\mathscr{B}^{A}\quad(\mathscr{A}_{\mu}^{A}\rightarrow\mathscr{A}_{\mu}^{A}),
{\mathcalD}_{0}^{-1}\simeq\widetilde{\phi}_{1}\phi_{1}^{\dagger}\lambda_{1}P_{L}-\widetilde{\phi}_{1}\widetilde{\phi}_{1}^{\dagger}\widehat{/\!\!\!\partial}P_{R}+\phi_{1}\widetilde{\phi}_{1}^{\dagger}\lambda_{1}P_{R}-\phi_{1}\phi_{1}^{\dagger}\widehat{/\!\!\!\partial}P_{L}-\phi_{0}\phi_{0}^{\dagger}\widehat{/\!\!\!\partial}P_{L}.
M=\left(\begin{matrix}{A}&{B}&{\Phi_{1}}\\{F}&{-A^{\top}}&{\Phi_{2}}\\{-i\Phi_{2}^{\top}}&{i\Phi_{1}^{\top}}&{0}\\\end{matrix}\right)=\left(\begin{matrix}{m}&{\Psi}\\{-i\Psi^{\top}C}&{0}\\\end{matrix}\right)\;,
\Pi_{00}^{(\mathrm{rising\sun})}=-4e^{2}m^{2}\,\frac{1}{\beta}\sum_{n}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{(k_{0}^{2}+\omega^{2})+\frac{k_{0}^{2}\,(k_{0}^{2}+\omega^{2}+\kappa^{2})}{m^{2}}}{((k_{0}+p_{0})^{2}+\omega_{\sigma}^{2})(k_{0}^{2}+\omega_{+}^{2})(k_{0}^{2}+\omega_{-}^{2})},
W_{even}=\sum_{k=1}^{N_{c}-2}\mu_{2k}\mathrm{Tr}(\Phi_{cl}^{2k})+\mu_{2(N_{c}-1)}s_{2(N_{c}-1)}^{cl}+\lambda\mathrm{Pf}\Phi_{cl}\pm\sum_{i=1}^{k}j_{i}\nu_{i}\Lambda_{i}^{3}
\delta\int_{T}^{t}d\tauL=\int_{T}^{t}d\tau~\deltaq({\frac{\deltaL}{\deltaq}}-{\frac{d}{dt}}{\frac{\deltaL}{\delta\dot{q}}})+{\frac{\deltaL_{t}}{\delta\dot{q}_{t}}}\deltaq_{t}-{\frac{\deltaL_{T}}{\delta\dot{q}_{T}}}\deltaq_{T}~.
\partial_{i}(\partial_{0}\deltaA_{i}-\partial_{i}\deltaA_{0})=\frac{E_{c}^{2}}{1+E_{c}^{2}}\triangle\eta+\frac{2E_{c}^{\prime}}{E_{c}(1+E_{c}^{2})}\frac{x_{i}}{r}\left[(\partial_{0}\deltaA_{i}-\partial_{i}\deltaA_{0})-C_{i}\right]+\partial_{i}C_{i}
t_{1}(x)=\frac{2}{3}W_{1}^{\prime}(x)+\frac{1}{3}W_{2}^{\prime}(x),\quadt_{2}(x)=-\frac{1}{3}W_{1}^{\prime}(x)+\frac{1}{3}W_{2}^{\prime}(x),\quadt_{3}(x)=-\frac{1}{3}W_{1}^{\prime}(x)-\frac{2}{3}W_{2}^{\prime}(x).
\begin{array}{rcl}{\tilde{e}_{n_{+},n_{-}}^{\alpha}(z)}&{=}&{(P_{+})^{n_{+}}(P_{-})^{n_{-}}V_{\alpha}(z)}\\{}&{=}&{\displaystyle\int_{\infty}^{z}dt_{1}J_{+}(t_{1})\cdots\int_{\infty}^{z}dt_{n_{+}}J_{+}(t_{n_{+}})\int_{\infty}^{z}dt_{1}^{\prime}J_{-}(t_{1}^{\prime})\cdots\int_{\infty}^{z}dt_{n_{-}}^{\prime}J_{-}(t_{n_{-}}^{\prime})V_{\alpha}(z),}\\\end{array}
\left\langleH\right\rangle_{\Omega}=\left\langle\Omega\right|\int{dx_{1}}\left({-\frac{1}{2}\Pi_{1}\left({1-\frac{{g^{2}}}{\pi}\frac{a}{{a-1}}\frac{1}{{\partial_{1}^{2}}}}\right)^{-1}\Pi^{1}}\right)\left|\Omega\right\rangle.
\deltaA_{\mu}=i\bar{\epsilon}\gamma_{5}\gamma_{\mu}\Psi,\quad\delta\phi=-\bar{\epsilon}\Psi,\quad\delta\Psi=-\frac{1}{2}\epsilon\epsilon^{\mu\nu}F_{\mu\nu}+i\gamma^{\mu}\epsilonD_{\mu}\phi.\quad(\epsilon^{01}=-\epsilon_{01}=1)
T_{+p\pm2}^{-\dot{q}}=-\frac{\chi}{2}\tilde{\gamma}_{\dot{q}p}^{i}\Omega_{\pm2}^{-2i}+\frac12\tilde{\gamma}_{\dot{q}q}^{i}\Omega_{+p}^{-2i}\psi_{\pm2q}^{2+},
(A_{M})_{rs}=\left(\begin{array}{c}{\frac{i}{\sqrt2}\pi_{rs}^{-}}\\{\psi_{rs}}\\\end{array}\right),\;\;\;\;\;\;(A_{M}^{\dagger})_{rs}=\left(\begin{array}{c}{-\frac{i}{\sqrt2}\pi_{rs}^{+}}\\{\psi_{rs}^{\dagger}}\\\end{array}\right)