harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{-\infty}^{\infty}dk\,\left(\rho(k)-\rho_{0}(k)\right)\Phi(k^{2};\beta_{T},\mu)=-\frac{\mu}{2\pi}\int_{-\infty}^{\infty}dk\,\frac{dk}{\sqrt{1+k^{2}}}+\frac{1}{\beta_{T}}h_{\parallel}(\beta_{T}\mu)
\begin{array}{l}{H=N\intdx^{-}dy^{-}\bigg[(g^{2}/4)M_{--,ij}(x^{-},y^{-})|x^{-}-y^{-}|M_{--,ij}(x^{-},y^{-})}\\{~~~~~~~~~~~~~-(g^{2}/4N)M_{--,ii}(x^{-},x^{-})|x^{-}-y^{-}|M_{--,jj}(y^{-},y^{-})}\\{~~~~~~~~~~~~~-i(m^{2}/4)\;\mathrm{sgn}(x^{-}-y^{-})M_{--,ii}(x^{-},y^{-})\bigg]}\\\end{array}
\bar{M}_{Z}^{2}=m_{1}^{2}\left(1+\left(\frac{\Gamma_{1}}{m_{1}}\right)^{2}\right)^{-1}\quad\mathrm{and}\quad\bar{\Gamma}_{Z}^{2}=\Gamma_{1}^{2}\left(1+\left(\frac{\Gamma_{1}}{m_{1}}\right)^{2}\right)^{-1},
\phi=\frac{1}{2}\Pi^{a}\Pi_{a}+g\left[g^{-1}l_{\alpha\beta}l^{\alpha\beta}+\lambda\sqrt{1+\frac{1}{2}g^{-1}l_{\alpha\beta}l^{\alpha\beta}+\frac{1}{64}g^{-1}g^{\mu\nu}V_{\mu}V_{\nu}}+\frac{\lambda^{2}}{8}+2\right]
A^{NSNS}=-(8\pi^{2})^{-5}\int{\frac{dl}{2}}\left[{\frac{G_{C}}{2}}{\frac{f_{3}(e^{-2\pil})^{8}}{f_{1}(e^{-2\pil})^{8}}}+32G_{M}{\frac{f_{2}(ie^{-2\pil})^{8}}{f_{1}(ie^{-2\pil})^{8}}}+32^{2}{\frac{f_{2}(e^{-2\pil})^{8}}{f_{1}(e^{-2\pil})^{8}}}\right]\,.
{\calV}^{(0)}(\Phi)=-{\frac{\Gamma_{s}^{(0)}[\varphi;\Phi,\delta]}{\int_{x}}}\biggl|_{\varphi_{x}=\Phi}={\frac{1}{2}}f_{xx}-{\frac{1}{2}}I_{0}(M^{2})+{\frac{M^{2}}{4}}I_{1}(M^{2})\;.
\begin{array}{c}{D_{TM}^{(-1)}(t)=\displaystyle{\frac{-\left({n_{0}}^{2}\,R^{2}\,t\,\left(8\,a^{2}+{n_{0}}^{2}\,R^{2}\,t^{2}\right)\right)}{8\,{\left(a^{2}+{n_{0}}^{2}\,R^{2}\,t^{2}\right)}^{2}\,{\sqrt{1+\frac{{n_{0}}^{2}\,R^{2}\,t^{2}}{a^{2}}}}}-\frac{\left(2+{n_{0}}^{2}\,t^{2}\right)\,Z^{2}}{2\,t^{3}\,{\sqrt{1+{n_{0}}^{2}\,t^{2}}}}},}\\\end{array}
{\calL}_{b}=-{\frac{1}{4}}(\partial_{\mu}b_{\nu}^{R}-\partial_{\nu}b_{\mu}^{R})^{2}+{\frac{1}{2}}(m_{b}^{R})^{2}b_{\mu}^{R}b^{\mu}{}^{R}\,-{\frac{1}{4}}\delta_{b}(\partial_{\mu}b_{\nu}^{R}-\partial_{\nu}b_{\mu}^{R})^{2}+{\frac{1}{2}}\delta_{m}b_{\mu}^{R}b^{\mu}{}^{R}.
{\calL}_{inv}=-{\frac{1}{4}}F^{\mu\nu,a}F_{\mu\nu}^{a}+{\frac{m}{2}}\epsilon^{\mu\nu\lambda}A_{\mu}^{a}(\partial_{\nu}A_{\lambda}^{a}+{\frac{g}{3}}f^{abc}A_{\nu}^{b}A_{\lambda}^{c})
D[y(x)]=-2\sqrt{\acute{y}(x)}\frac{d^{2}}{dx^{2}}\left(\frac{1}{\sqrt{\acute{y}(x)}}\right)=\frac{y^{\prime\prime\prime}(x)}{y^{\prime}(x)}-\frac{3}{2}\left(\frac{y^{\prime\prime}(x)}{y^{\prime}(x)}\right)^{2}.
<N\delta_{\beta\rho}\delta^{d}(x-y)F[\sigma]>-<\intdz\sigma_{\alpha\rho}(z,y){\frac{\deltaF}{\delta\sigma_{\alpha\beta}(z,x)}}>-i<\intdz\sigma_{\alpha\rho}(z,y){\frac{\deltaS}{\delta\sigma_{\alpha\beta}(z,x)}}F[\sigma]>=0.
ds^{2}=\alpha^{\prime}\left[\frac{U^{2}}{\sqrt{4\pig_{s}N}}\eta_{ij}dx^{i}dx^{j}+\sqrt{4\pig_{s}N}\frac{dU^{2}}{U^{2}}+\sqrt{4\pig_{s}N}d\Omega_{5}^{2}\right].
q^{(ab)}(r)=\frac{1}{8\pi^{2}a^{2}r^{2}}\sum_{l=1}^{\infty}(2l+1)\sum_{\lambda=0,1}\int_{0}^{\infty}{z\psi_{\mu}(z/a)\Omega_{\lambdal}(z,\eta)F_{\lambdal}^{(q)}(z,zx)dz},\quadx=r/a.
S=-\tau_{p}V_{p}-\frac{1}{4g_{\mathrm{YM}}^{2}}\intd^{p+1}\xi\left(F_{\alpha\beta}F^{\alpha\beta}+\frac{2}{(2\pi\alpha^{\prime})^{2}}\partial_{\alpha}X^{a}\partial^{\alpha}X^{a}\right)+\cdots
\overline{{I}}_{2}={\frac{(2ie\pi)^{2}}{2}}\int{\frac{d\overline{{k}}}{(2\pi)^{2}}}\,\epsilon(\overline{{k}})\epsilon(\overline{{k}}+\overline{{p}})\left[n(|\overline{{k}}|)+n(|\overline{{k}}+\overline{{p}}|)-2n(|\overline{{k}}|)n(|\overline{{k}}+\overline{{p}}|)\right]
V(\vec{r})=\frac{{\hat{l}_{1}}^{2}}{r_{1}^{2}}+\frac{{{\hat{l}_{2}}}^{2}}{r_{2}^{2}}+\frac{{{\hat{l}_{3}}}^{2}}{r_{3}^{2}}+{\alpha^{2}}\,(r_{1}^{2}r_{2}^{2}+r_{2}^{2}r_{3}^{2}+r_{1}^{2}r_{3}^{2})
S=\int_{0}^{1}dt\!\![i\chi^{\dagger}\dot{\chi}+\frac{2i\lambda_{3}}{\Omega}\mathrm{tr}(\dot{U}^{\dagger}T_{3}U)-\frac{2i\lambda_{3}}{\Omega}\mathrm{tr}(\dot{h}^{\dagger}T_{3}h)-\frac{2\lambda_{3}}{\Omega}\mathrm{tr}(T_{3}h\alphah^{\dagger})+\chi^{\dagger}U^{\dagger}h\sigma+\sigma^{\dagger}h^{\dagger}U\chi]\,.
\left(\begin{array}{cc}{{\varphi}_{out}^{\prime}}\\{{{\varphi}^{\prime}}_{out}^{*}}\\\end{array}\right)=\left(\begin{array}{cc}{-1}&{0}\\{0}&{-1}\\\end{array}\right)\left(\begin{array}{cc}{\varphi_{in}}\\{\varphi_{in}^{*}}\\\end{array}\right),
E_{00}=\frac{1}{a^{4}}\intd\taua^{4}\left(\partial_{\tau}\left(\frac{3}{16}\dot{\phi}^{2}+\frac{3}{8}U\right)+\frac{3}{2}H\dot{\phi}^{2}+\frac{\dot{U}_{B}}{4}\rho_{m}\right)
A_{nm}^{l}=\frac{2}{2l+\delta_{1}+\delta_{2}}\left[\frac{(l-|m|)(l+|m|+\delta_{1}+\delta_{2})(l+\delta_{1})(l+\delta_{2})(n-l)(n+l+\delta_{1}+\delta_{2})}{(2l+\delta_{1}+\delta_{2}-1)(2l+\delta_{1}+\delta_{2}+1)}\right]^{1/2}.
E_{D4/D0}=T_{4}(2\pi\alpha^{\prime})^{2}(\mathrm{Pf}B+(2\pi)^{2})+\frac{T_{4}}{(\mathrm{Pf}B+(2\pi)^{2})}[\frac{1}{4}|\mathrm{tr}\,\tilde{B}g\tilde{B}g|+(2\pi)^{2}\sqrt{\mathrm{det}g}]+O(\epsilon)\,.
e^{-{\frac{1}{2}}a\phi-(p+1)A}=1+{\frac{\lambda_{1}}{a\sqrt{2}}}{\frac{1}{r^{p+1}}},\\\\e^{+{\frac{1}{2}}a\phi-(p+1)A}=1+{\frac{\lambda_{2}}{a\sqrt{2}}}{\frac{1}{r^{p+1}}},
\begin{array}{rccccc}{\mathrm{\underline{{sector}}}}&{{}}&{\mathrm{\underline{{state}}}}&{{}}&{{}}&{\mathrm{\underline{{~{\cal~R}~}}}}\\{\mathrm{NS}}&{{}}&{\psi_{-{\frac{1}{2}}}^{i}|0\rangle}&{i=2,\cdots,9}&{{}}&{-}\\{\mathrm{R}}&{{}}&{|+,2s_{1},\cdots,2s_{4}\rangle}&{8s_{1}s_{2}s_{3}s_{4}=+1}&{{}}&{+}\\\end{array}
\int\frac{k^{s}d^{N}{\bfk}}{\sqrt{k^{2}+c^{2}}}=c^{N+s-1}\frac{\pi^{(N-1)/2}}{\Gamma(N/2)}\Gamma\left(-\frac{N+s-1}{2}\right)\Gamma\left(\frac{N+s}{2}\right)
N^{(3)}=\frac{\beta^{2}}{8\pi^{3}}\left(\frac{m\beta}{\pi}\right)^{2}\,\left(\sum_{n=0}^{\infty}{\frac{[-4(2n+1)^{2}+(\frac{m\beta}{\pi})^{2}]}{[(2n+1)^{2}+(\frac{m\beta}{\pi})^{2}]^{7/2}}}\right)\,\intdx\,(\theta^{\prime})^{3}
\begin{aligned}{\partial_{+}\partial_{-}X^{r}+\mu^{2}X^{r}}&{{}=0\,,}\\{\partial_{+}\theta^{1}-\muI\theta^{2}}&{{}=0\,,}\\{\partial_{-}\theta^{2}+\muI\theta^{1}}&{{}=0\,.}\\\end{aligned}
\frac{i}{16\pi^{2}}\,{(-1)}^{\ell}e^{-\frac{i}{2}\pi\nu}\,k^{2}\,\left[\frac{(8\pi^{3}\ell^{3}-8\pi^{2}\ell^{2}\Phi-8\pi^{3}\ell+4\pi\ell\Phi^{2}+2\pi^{2}\Phi-\Phi^{3})}{4\nu(\nu^{2}-1)}\right],
G_{l}(v)_{\mu_{1}\ldots\mu_{s}}^{i_{1}\ldotsi_{s}}=2^{s}\left[{\frac{z_{0}}{z_{0}^{2}+\vec{z}^{2}}}\right]^{d-3}\left[\partial_{\mu_{1}}{\frac{z^{i_{1}}}{z_{0}^{2}+\vec{z}^{2}}}\cdots\partial_{\mu_{s}}{\frac{z^{i_{s}}}{z_{0}^{2}+\vec{z}^{2}}}\right]
:c^{+}(\sigma)b_{+}(\tau):=e^{ip_{0}(\sigma-\tau)}e^{\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}(e^{in\sigma}-e^{in\tau})a_{n}^{\dagger}}e^{\sum_{m=1}^{\infty}\frac{1}{\sqrt{m}}(e^{-im\tau}-e^{-im\sigma})a_{m}}\,.
\left|\sum_{\alpha\beta\gamma\delta}\langle\Phi_{\alpha}\Phi_{\beta}\Phi_{\gamma}\Phi_{\delta}\rangle_{c}\left.\right/\sum_{\alpha\beta\gamma\delta}\langle\Phi_{\alpha}\Phi_{\beta}\Phi_{\gamma}\Phi_{\delta}\rangle\right|=\left|\langle\Phi^{4}(x)\rangle_{c}\left.\right/\langle\Phi^{4}(x)\rangle\right|\;(n=4)\;,
\left.\frac{\partial(\varepsilon_{cl\acute{a}}+\varepsilon_{0})}{\partialb^{2}}\right|_{b^{2}=0}=\left.\frac{\mu^{4}}{8e^{2}}-\frac{\sqrt{\pi}}{8\pi^{2}}\mu^{4}B_{2}\c^{-\frac{s+1}{2}}\frac{\Gamma(\frac{s+1}{2})}{\Gamma(s/2)}\right|_{s=-1}.
\Psi_{\kappa}^{(F)}=\frac1{2^{3/2}\pi}\,e^{\pi\kappa/2}\left\{\Gamma(i\kappa)\left(\frac{mx_{+}}2\right)^{-i\kappa}+e^{-\pi\kappa}\Gamma(-i\kappa)\left(\frac{mx_{-}}2\right)^{i\kappa}+...\right\}.
\tilde{{\calL}}^{\prime\prime}=\bar{\psi}(i\!\not{\!\partial}+J)\psi-\frac{1}{2G_{S}}\sigma^{2}-\frac{1}{\sqrt{N}}\sigma\bar{\psi}\psi+\frac{1}{2G_{V}}(A_{\mu}-\sqrt{N}\partial_{\mu}\phi)^{2}+\frac{1}{\sqrt{N}}\bar{\psi}\not{\!\!A}\psi\,
\sum_{n=1}^{N_{\mathrm{TD}}}\int\left[\prod_{i=1}^{n}d{\bfx}_{i}\right]\delta^{2}(\tilde{{\bfP}}-\sum_{i=1}^{n}{\bfx}_{i})|\tilde{\psi}_{n}({\bfx}_{1},{\bfx}_{2},\dots,{\bfx}_{n})|^{2}=1,
\frac{1}{\beta_{0}}\left[\frac{9}{4}u_{1}-\frac{33}{2}x_{1}+\frac{5}{4}x_{3}-2N_{F}\bigg(w_{3}+(w_{1}+w_{2})z_{1}+z_{2}-\frac{1}{3}z_{3}\bigg)\right]\,=\,u_{1}\;,
\langle0\vert[T_{\mu\nu}^{V}(x),T_{\rho\sigma}^{V}(0)]\vert0\rangle=F_{\mu\nu\rho\sigma}(x)-F_{\rho\sigma\mu\nu}(-x)=\int\!d^{2}p\e^{-ipx}\varepsilon(p^{0})G_{\mu\nu\rho\sigma}(p).
\begin{array}{l}{{\calO}[V]=V^{i_{1}\ldotsi_{s}}\sum_{k=0}^{s}{\frac{(-1)^{k}}{k!\left(k+{\frac{D-4}{2}}\right)!(s-k)!\left(s-k+{\frac{D-4}{2}}\right)!}}\partial_{i_{1}}\cdots\partial_{i_{k}}\phi^{*}\;\partial_{i_{k+1}}\cdots\partial_{i_{s}}\phi,}\\{g_{i_{1}i_{2}}V^{i_{1}\ldotsi_{s}}=0}\\\end{array}
E_{\alpha,n}^{\mathrm{TOT}}=E_{\alpha}^{a}+E_{n}^{0}+\langle\alpha,n|V^{a}|\alpha,n\rangle+\sum_{\gamma,m}\frac{\langle\alpha,n|V^{a}|\gamma,m\rangle\langle\gamma,m|V^{a}|\alpha,n\rangle}{E_{\gamma}^{a}+E_{m}^{0}-E_{\alpha}^{a}-E_{n}^{0}}+\cdots
H_{\mathrm{triangle}}(R)=\frac{56}{\sqrt3}\Lambda^{2}L_{3}\left[R^{2}\Lambda^{2}+\frac{1}{7}R^{4}\Lambda^{4}+\frac{4\sqrt3}{315}R^{5}\Lambda^{5}+{\calO}\left(R^{6}\Lambda^{6}\right)\right].
{\calU}=\left(\begin{matrix}{\frac32\delta_{X}{}^{Y}-\frac{1}{W^{2}}{\calJ}_{X}{}^{Z}{\calL}_{Z}{}^{Y}}&{\frac1{W^{2}}{\calJ}_{XZ}\partial^{y}K^{Z}}\\{-\frac1{W^{2}}(\partial_{x}K^{Z}){\calJ}_{Z}{}^{Y}}&{2\delta_{x}{}^{y}}\\\end{matrix}\right)\,,
E_{hg}(\omega_{n})={\frac{1}{2}}\sum_{n=1}^{M-1}\omega_{n}+{\frac{1}{2}}T_{0}^{2}MR_{1}^{2}+g^{2}M\sum_{k=1}^{M}\left[{\frac{a^{2}d}{2}}+R_{k}^{2}\right]^{-d/2}\,.
\frac{P_{N}^{2}(x)-4\Lambda^{2N}}{4\Lambda^{2N}}=\left(T_{N}^{2}(\frac{x-\phi}{2\eta\Lambda})-1\right)=\left[\left(\frac{x-\phi}{2\eta\Lambda}\right)^{2}-1\right]U_{N-1}^{2}\left(\frac{x-\phi}{2\eta\Lambda}\right).
{\frac{m}{M}}={\frac{\sqrt{3\pi}(P_{k}^{S})^{1/4}}{\sqrt{L}}}\left[{\frac{(1+u^{2})^{7/8}u^{1/2}}{(1+3u^{2})^{1/8}}}\right]\approx1.3\times10^{-3}\left(10^{10}P_{k}^{S}\right)^{1/4}\left({\frac{60}{L}}\right)^{1/2}\left[{\frac{(1+u^{2})^{7/8}u^{1/2}}{(1+3u^{2})^{1/8}}}\right].
V(R)\,=\,M_{0}^{2}\,R\,+\,\frac{D-2}{R}\,\left[v^{ren}(q)\,-\,\frac{\pi}{4\sqrt{\epsilon}}\,-\,\frac{1}{2\sqrt{\epsilon}}\,\sum_{n=1}^{\infty}\,\frac{1}{n}\,K_{1}\left(\frac{2\pin}{\sqrt{\epsilon}}\right)\right]\,{,}
e^{iW}=\intdg_{\mu\nu}~d\bar{\Gamma}_{~\mu\nu}^{\sigma}~d\bar{c}^{\mu}~dc^{\nu}~d\bar{\chi}^{\mu}~d\chi^{\nu}~d\bar{\eta}^{\mu}~d\eta^{\sigma\lambda\nu}e^{iS_{quan}}\left(det\omega_{\mu\nu}\right)^{\frac{1}{2}}\left(det\zeta_{\mu\nu}\right)^{\frac{1}{2}}\left(det\varsigma_{\mu\nu}\right)^{\frac{1}{2}}
u^{\nu}*\frac{\partialW}{\partialx^{\nu}}=u^{\nu}*\left(\frac{\partialW}{\partial\left(E/B\right)_{i}}*\frac{\partial\left(E/B\right)_{i}}{\partialx^{\nu}}+\frac{\partialW}{\partialV}*\frac{\partialV}{\partialx^{\nu}}\right)=0
J_{a}=\left(\underbrace{\Gamma_{a}{\otimes}\mathbf{1}{\otimes}\cdots{\otimes}\mathbf{1}}_{L-times}+\mathbf{1}{\otimes}\Gamma_{a}{\otimes}\cdots{\otimes}\mathbf{1}+\cdots+\mathbf{1}{\otimes}\mathbf{1}{\otimes}\cdots{\otimes}\Gamma_{a}\right)_{sym}.
F(\beta)=-\frac{1}{2^{6}\pi^{10}}\int_{S}\frac{d^{2}\tau}{\tau_{2}^{2}}\tau_{2}^{-4}\frac{\theta_{2}^{4}}{\eta^{12}}\frac{\bar{\Theta}_{\Gamma}}{\bar{\eta}^{24}}\theta_{2}\left(0\left|\frac{2i\beta^{2}}{\pi^{2}\tau_{2}}\right.\right)
\begin{array}{c}{\varphi^{*}(x^{i})=x^{i}\,\quad\varphi^{*}(x_{i}^{*})=x_{i}^{*}\,\quad\varphi^{*}(\bar{c}_{*}^{i})=\bar{c}_{*}^{i}\,\quad\varphi^{*}(\bar{c}_{i})=\bar{c}_{i}\,}\\{\varphi^{*}(y^{i})=b_{*}^{i},\quad\varphi^{*}(b_{*}^{i})=-y^{i},\quad\varphi^{*}(b_{i})=y_{i}^{*},\quad\varphi^{*}(y_{i}^{*})=-b_{i}\,.}\\\end{array}
L={\frac{1}{2}}\dot{X}_{i}^{2}+{\frac{1}{2}}(w_{--}w_{+})\epsilon_{ij}X_{i}\dot{X}_{j}-{\frac{1}{2}}w_{+}w_{-}X_{i}^{2};~~X_{i}={\sqrt{\frac{w_{+}w_{-}}{w_{+}^{2}+w_{-}^{2}}}}(x_{i}-y_{i}).
D_{\mu}({\calA})j_{L}^{\mu}(x,t)=\frac{1}{48\pi^{2}}\epsilon^{\lambda\mu\rho\sigma}\partial_{\lambda}\left[{\calA}_{\mu}\partial_{\rho}{\calA}_{\sigma}+\frac{1}{2}{\calA}_{\mu}{\calA}_{\rho}{\calA}_{\sigma}\right]\,
\left(\frac{\dot{a}}{a}\right)^{2}=\left(\frac{\Lambda_{b}}{6M^{3}}+\frac{\Lambda_{i}^{2}}{36M^{6}}\right)+\frac{\Lambda_{i}}{18M^{6}}\rho_{i}+\frac{1}{36M^{6}}\rho_{i}^{2}.
f_{exp}^{(3)}=\left(\begin{matrix}{e^{-i\beta}}&{0}&{0}\\{0}&{f_{11}e^{-i\frac{\beta}{2}}}&{f_{12}e^{-i\frac{\beta}{2}}}\\{0}&{f_{21}e^{-i\frac{\beta}{2}}}&{f_{22}e^{-i\frac{\beta}{2}}}\\\end{matrix}\right)\inSU^{loc}(2)\otimesU^{loc}(1),
\begin{array}{rclrcl}{\left[L_{1},L_{-2}\right]}&{=}&{L_{-1},}&{\quad\left[L_{1},L_{2}\right]}&{=}&{0,}\\{\left[L_{2},L_{-2}\right]}&{=}&{N+\frac{D+1}{2},}&{\quad\left[L_{0},L_{n}\right]}&{=}&{0,}\\{\left[L_{1},L_{-1}\right]}&{=}&{L_{0},}&{\quad\left[N,L_{n}\right]}&{=}&{{}-nL_{n},\quadn=0,\pm1,\pm2,}\\\end{array}
\tilde{E}_{n\delta}=\sum_{\begin{array}{c}{k_{1}p_{1}+...+k_{m}p_{m}=n}\\{0<k_{1}<...<k_{m}}\\\end{array}}\frac{\left(q^{(\alpha,\alpha)}-q^{-(\alpha,\alpha)}\right)^{\sum_{i}p_{i}-1}}{p_{1}!\;...\;p_{m}!}(E_{k_{1}\delta})^{p_{1}}...(E_{k_{m}\delta})^{p_{m}}
\frac{\partial(2\piD)}{\partialz}=0,\;\;2\piD\equiv\frac{2\piB(1-{\dot{R}}^{2})}{\sqrt{R^{2}(1-{\dot{R}}^{2}+{R^{\prime}}^{2}-E_{z}^{2})+B^{2}(1-{\dot{R}}^{2})}}.
\left\langle\partial_{z}\chi\left(1-\frac{\partial_{z}^{2}}{M_{r}^{2}}\right)^{-1}\partial_{z}\chi\right\rangle=i\int\!\frac{\mathrm{d}^{2}p}{(2\pi)^{2}}\,p_{z}^{2}\,\frac{1}{p^{2}\left[1+(p_{z}^{2}/M_{r}^{2})\right]^{2}}=\frac{M_{r}^{2}}{4\pi}\,.
\Omega=2iJ\left[\frac{d{\bar{\xi}}\wedged\xi}{1+\mid\xi\mid^{2}}-\frac{(\xid{\bar{\xi}})\wedge({\bar{\xi}}d\xi)}{(1+\mid\xi\mid^{2})^{2}}\right],\\\\mid\xi\mid^{2}=\sum_{p}\mid\xi_{p}\mid^{2}.
\left(\lambda^{\alpha,A}\right)^{\dagger}=-\lambda^{\beta,B}\epsilon_{\beta\alpha}\eta_{BA}^{1},\qquad\left(\bar{\lambda}_{\alpha^{\prime},A}\right)^{\dagger}=-\bar{\lambda}_{\beta^{\prime},B}\epsilon^{\beta^{\prime}\alpha^{\prime}}\eta^{1,BA}.
W_{\bfa}^{ren}[M^{4}]=\kappa_{0}^{ren}A(\Sigma)S_{2}(e^{\frac{2\pi}{{\bfa}_{ren}}})+B(\Sigma)(\kappa_{1}^{ren}+S_{0}(e^{\frac{2\pi}{{\bfa}_{ren}}}))~~.
D_{l}=[s_{l}(x^{\prime})e_{l}^{\prime}(x)-s_{l}^{\prime}(x^{\prime})e_{l}(x)]^{2}-\xi^{2}[s_{l}(x^{\prime})e_{l}^{\prime}(x)+s_{l}^{\prime}(x^{\prime})e_{l}(x)]^{2},
\phi_{m}=\frac{1}{\sqrt{2\omega_{m}}}(e^{i\omega_{m}t}a_{m}^{\dagger}+e^{-i\omega_{m}t}a_{m}),\;\;\;\;\;\;\pi_{m}=i\sqrt\frac{\omega_{m}}{2}(e^{i\omega_{m}t}a_{m}^{\dagger}-e^{-i\omega_{m}t}a_{m})
\varepsilon_{(10)}^{s_{1}\cdotss_{8}lm}\,\Gamma_{l}\,\psi_{s_{7}s_{8}}=\Gamma_{[9]}^{s_{1}\cdotss_{8}m}\,\psi_{s_{7}s_{8}}=\widetilde{\calE}(\psi)-42\,\Gamma_{[5]}^{[s_{1}\cdotss_{5}}\,\psi_{\phantom{[5]}}^{s_{6}m]}\,,
[J^{x\bar{x}},Q^{\pmi}]=\pm\frac{1}{2}Q^{\pmi}\,,\quad[J^{x\bar{x}},Q_{i}^{\pm}]=\mp\frac{1}{2}Q_{i}^{\pm}\,,\quad[J^{x\bar{x}},S_{i}^{\pm}]=\pm\frac{1}{2}S_{i}^{\pm}\,,\quad[J^{x\bar{x}},S^{\pmi}]=\mp\frac{1}{2}S^{\pmi}\,.
\left.\begin{array}{l}{(A)(D_{2})(\gamma_{2})(\overline{{D_{1}}})=[S_{2}],}\\{(B)(D_{3})(\gamma_{3})(\overline{{D_{2}}})=[S_{3}],}\\{(\overline{{A}})(\gamma_{4})(\overline{{D_{3}}})=[S_{4}],}\\\end{array}\right.
\begin{array}{cccc}{\mathrm{NS-NS}}&{{\bf1+28+35_{v}},}&{\mathrm{NS-R}}&{{\bf8_{s}+56_{s}},}\\{\mathrm{R-R}}&{{\bf1+28+35_{c}},}&{\mathrm{R-NS}}&{{\bf8_{s}+56_{s}}.}\\\end{array}
F_{8}^{{\mathcal{O}}_{2,0}^{1,0}|++++----}=\frac{\langle{\mathcal{O}}_{2,0}^{1,0}\rangle(\sigma_{4}^{-})^{1/2}((\sigma_{3}^{+})^{2}+\sigma_{1}^{+}\sigma_{3}^{+}\sigma_{2}^{-}+\sigma_{4}^{-}(\sigma_{1}^{+})^{2})/(\sigma_{4}^{+})^{2}}{2^{8}\,\prod_{1\leqi<j\leq4}\mathrm{cosh}(\theta_{ij}/2)\prod_{5\leqi<j\leq8}\mathrm{cosh}(\theta_{ij}/2)}\prod_{i<j}F_{\mathrm{min}}^{\mu_{i}\mu_{j}}(\theta_{ij}),
<T_{--}^{\mathrm{f}}(\sigma^{+},\sigma^{-})>|_{\sigma^{+}\rightarrow\infty}=\frac{\lambda^{2}}{48}\left[1-\frac{1}{(1+\frac{m}{\lambda}e^{\lambda(\sigma^{-}-\sigma_{0}^{+})})^{2}}\right]
\begin{array}{ll}{L_{n}^{\pm}=}&{\sum_{m}:\alpha_{-m}^{-}\alpha_{n+m}^{+}:}\\{L_{n}^{S}=}&{\frac1{k-2}\left(\sum_{m}\,:s_{-m}s_{n+m}:\,+ins_{n}+\frac14\delta_{n,0}\right)}\\\end{array}
I_{A}^{P}={\frac{g}{4\pi}}\intds\intds^{\prime}({\frac{\partial{\hat{u}}}{\partials}}\times{\frac{\partial{\hat{u}}}{\partials^{\prime}}})\cdot{\hat{u}}(1+\muR(s,s^{\prime}))e^{-\muR(s,s^{\prime})}\;.
|n\rangle\langlen+\ell|=2(-1)^{n}\sqrt{\frac{n!}{(n+\ell)!}}\left(\frac{2r^{2}}{\theta}\right)^{\ell/2}L_{n}^{\ell}(2r^{2}/\theta)e^{-r^{2}/\theta}e^{i\ell\varphi}\,,
B_{mod}^{(n)}\left[\left(\prod_{s}\partial_{\mu_{s}}\partial_{\bar{\mu}_{s}}\partial_{\nu_{s}}\partial_{\bar{\nu}_{s}}\right)Z_{L,L^{\prime}}^{(n)}K_{L,L^{\prime}}^{(n,m)}I_{LL^{\prime}}^{(n)}\right]_{\{k_{s},u_{s},v_{s}+\mu_{s}\nu_{s}\}}+B_{mod}^{(n)}\sum_{\hat{q}}\partial_{\hat{q}}H_{\hat{q}}^{(n,m)}+R
[\tau(\lambda)Q(\lambda)\psi]({\vec{x}})=\intd{\vec{y}}\prod_{i=N}^{1}w_{i}Tr\left({\tilde{L}}(\lambda|x_{N},\partial_{x_{N}})\ldots{\tilde{L}}(\lambda|x_{1},\partial_{x_{1}})\right)\psi({\vec{y}})\;.
2\sqrt{2}\hat{P}^{0}=(1+x_{I}^{2})(p^{+}-P^{-})+2\partial_{p^{+}}D-\frac{1}{p^{+}}l^{IJ}M^{IJ}+\frac{x^{I}}{zp^{+}}\{M^{zJ},M^{JI}\}\,.
\hat{A}(M)=1+\frac{1}{(4\pi)^{2}}\frac{1}{12}\,tr\,R^{2}+\frac{1}{(4\pi)^{4}}\left[\frac{1}{288}(tr\,R^{2})^{2}+\frac{1}{360}\,tr\,R^{4}\right]+\ldots
\partial^{{\breve{\alpha}}+}\partial_{\breve{\alpha}}^{-}e^{-4}+{\frac{1}{2}}\partial^{{\breve{\alpha}}+}\partial^{{\breve{\beta}}+}e^{-4}\partial_{\breve{\alpha}}^{+}\partial_{\breve{\beta}}^{+}e^{-4}=0\,
{\cal{H}}_{\eta_{o}}(u,v)=\left(\frac{v}{u}\right)^{\frac{1}{2}\eta_{o}}\sum_{n=0}^{\infty}\frac{v^{n}}{n!}\frac{({\textstyle{\frac{1}{2}}}\eta_{o})_{n}^{4}}{(\eta_{o})_{2n}(\eta_{o}+1-\mu)_{n}}{}_{2}F_{1}({\textstyle{\frac{1}{2}}}\eta_{o}+n,{\textstyle{\frac{1}{2}}}\eta_{o}+n;\eta_{o}+2n;1-\frac{v}{u}),
Z=\int{\calD}A^{\mu}{\calD}\pi_{\mu}{\calD}\phi{\calD}\pi_{\phi}{\calD}\theta{\calD}\pi_{\theta}\prod_{i,j=1}^{2}\delta(\widetilde{\Omega}_{i})\delta(\Gamma_{j})\mathrm{det}\mid\{\widetilde{\Omega}_{i},\Gamma_{j}\}\mide^{iS^{\prime}},
X=\frac{2}{\mu+\nu}\left[\mu\frac{\partial}{\partial\nu}\left(\nu\frac{\partial}{\partial\nu}\right)-\nu\frac{\partial}{\partial\mu}\left(\mu\frac{\partial}{\partial\mu}\right)\right]+\frac{\mu-\nu}{2\mu\nu}\frac{\partial^{2}}{\partial\varphi^{2}}+\frac{2c_{1}\nu}{\mu(\mu+\nu)}-\frac{2c_{2}\mu}{\nu(\mu+\nu)}-\alpha\frac{\mu-\nu}{\mu+\nu}.
\Gamma=-\intd^{2}x\left\{\partial_{\mu}\pi^{+}\partial_{\mu}\pi^{-}+\frac{1}{2}(\partial_{\mu}\pi^{0})^{2}+\frac{1}{2}(\partial_{\mu}\eta)^{2}+m_{\pi}^{2}(\pi^{+}\pi^{-}+\frac{1}{2}\pi^{0}\pi^{0})+\frac{1}{2}m_{\eta}^{2}\eta^{2}\right\}+...,
{\frac{1}{2}}|{\calD}_{\mu}\phi|^{2}={\frac{1}{2}}\left[e^{2}v^{2}\tilde{A}_{\mu}\tilde{A}^{\mu}+{\frac{g^{2}v^{2}}{4}}F_{\mu}F^{\mu}+egv^{2}\tilde{A}_{\mu}F^{\mu}\right]\,.
J_{D}(x)=\left(\begin{array}{cccc}{\phi_{1}(x)}&{k}&{0}&{\cdots}\\{0}&{\ddots}&{\ddots}&{0}\\{\vdots}&{}&{\ddots}&{k}\\{0}&{\cdots}&{0}&{\phi_{N}(x)}\\\end{array}\right)\;,\;\sum_{i=1}^{N}\phi_{i}(x)=0.
q^{+}(\zeta,u)=f^{i}(x)u_{i}^{+}+\theta^{+\alpha}\psi_{\alpha}(x)+\bar{\theta}_{\dot{\alpha}}^{+}\bar{\kappa}^{\dot{\alpha}}(x)+2i\theta^{+}/\!\!\!\partial\bar{\theta}^{+}f^{i}(x)u_{i}^{-},
g_{n}({\sigma}_{0})=\sum_{\{{\sigma}_{i}\ne{\sigma}_{0}\}}exp\left\{J_{3}\sum_{\triangle}{\sigma}_{0}{\sigma}_{1}{\sigma}_{1}+h\sum{\sigma}_{1}+J_{3}\sum_{\triangle}{\sigma}_{i}{\sigma}_{j}{\sigma}_{k}+h\sum_{i}{\sigma}_{i}\right\}.
{\calL}=\frac{3}{4}\Omega^{\mu,\nu\alpha\beta}\Omega_{\nu,\mu\alpha\beta}-\frac{3}{4}\Omega^{\alpha\beta}\Omega_{\alpha\beta}-\frac{1}{2}\Omega^{\mu,\nu\alpha\beta}T_{\nu\alpha\beta,\mu}+\frac{3}{2}\Omega^{\alpha\beta}T_{\alpha\beta}
L_{i_{1}}^{(1)}(x)=\intd^{3}y\left[W_{i_{1}j}^{(1)}(x,y)\ddot{\varphi}^{j}(y)+\alpha_{i_{1}}^{(1)}(y)\delta^{3}(x-y)\right]=0,~~~i_{1}=1,\cdot,6,
\intd^{\mathrm{d}}zd^{\mathrm{d}}x^{\prime}d^{\mathrm{d}}y^{\prime}\gamma_{\mu}D_{\mu\nu}^{\mathrm{T}}(x-z)S(x-x^{\prime})\Gamma_{\nu}(z;x^{\prime},y^{\prime})S(y^{\prime}-y)=0~;
\frac{d^{2}\psi(\zeta)}{d\zeta^{2}}+\frac{2\zeta-1}{\zeta(\zeta-1)}\frac{d\psi(\zeta)}{d\zeta}-\frac{q+r(1-2\zeta)^{2}}{\zeta^{2}(\zeta-1)^{2}}\psi(\zeta)=0
m_{\pm}^{2}=\frac{1}{2Z^{2}}\left(m_{r}^{2}+(1+6\xi\gamma^{2})m_{h}^{2}\pm((m_{r}^{2}-m_{h}^{2}(1+6\xi\gamma^{2}))^{2}+144\gamma^{2}\xi^{2}m_{r}^{2}m_{h}^{2})^{1/2}\right)
S\,=\,S_{0}+\phi_{A}^{\ast\,a}\,s_{a}\phi^{A}+{\frac{1}{2}}{\bar{\phi}}_{A}s_{2}s_{1}\phi^{A}\,+\,{\frac{1}{2}}\epsilon^{ab}\phi_{A\,a}^{\ast}\,\pi_{b}^{A},
\deltaR_{\theta\theta}=g_{\theta\theta}e^{-2b_{0}}\lbracec_{0}^{\prime}(\deltab^{\prime}-\deltaa^{\prime})+2(c_{0}^{\prime}(a_{0}^{\prime}-b_{0}^{\prime}+3c_{0}^{\prime})+c_{0}^{\prime\prime})\deltab\rbrace,
(S^{-})^{\dagger}=\int_{M}d^{4}xe\left[-i(\partial_{\mu}{\overline{\Psi}}_{L})\gamma^{\mu}\Psi_{L}+{\overline{\Psi}}_{L}(\frac{i}{2}A_{\muAB}\sigma^{AB}+W_{\mua}T^{a})\gamma^{\mu}\Psi_{L}\right].
{\calA}_{\mathrm{vac},+}^{(k_{+},q_{+})}(x,t)=\overline{{\langle\mathrm{vac};A;+|}}a_{1}a_{2}\cdot...\cdota_{k_{+}}\cdoti\frac{\delta}{\deltaA_{1}(x,t)}\cdota_{q_{+}}^{\dagger}\cdot...\cdota_{2}^{\dagger}a_{1}^{\dagger}\overline{{|\mathrm{vac};A;+\rangle}}.
\mathcal{L}=\frac{4}{9}F_{(\alpha\beta\gamma)\nu}\,F^{(\alpha\beta\gamma)\nu}+\frac{2}{3}F_{(\alpha\beta\gamma)\nu}\,F^{(\alpha\beta\nu)\gamma}-F_{(\alpha\beta\mu)}{}^{\mu}\,F^{(\alpha\beta\nu)}{}_{\nu}\,
\left(\left(D^{i}\right)_{\;\;b}^{a}+f_{bc}^{a}\pi^{0ic}\right)\Phi_{i}^{(2)b}=-\epsilon^{0ijk}f_{bc}^{a}\chi_{i}^{(1)b}\left(D_{j}\right)_{\;\;d}^{c}\chi_{k}^{(1)d}\approx0.
\begin{array}{rcl}{\psi_{\uparrow}^{(+)}}&{=}&{\sqrt{\rho}e^{-\gamma_{21}mt}\,;}\\{\psi_{\downarrow}^{(+)}}&{=}&{\sqrt{\rho}\gamma_{31}e^{-\gamma_{21}mt}\,;}\\{\psi_{\uparrow}^{(-)}}&{=}&{\sqrt{\rho}\gamma_{5}e^{-\gamma_{21}mt}\,;}\\{\psi_{\downarrow}^{(-)}}&{=}&{\sqrt{\rho}\gamma_{5}\gamma_{31}e^{-\gamma_{21}mt}\,.}\\\end{array}
u(\rho)=\left(C_{1}F\left(-\frac{e^{2}+1}{4},-\frac{1}{2},\rho^{2}\right)+C_{2}\rho^{3}F\left(-\frac{e^{2}-5}{4},\frac{5}{2},\rho^{2}\right)\right)e^{-\frac{\rho^{2}}{2}}
e^{\gamma\Phi}=\frac{\tau\bar{\tau}}{4}\left[\bar{\omega}e^{\fraci2(\sqrt{\tau}z+\sqrt{\bar{\tau}}\bar{z})}+\omegae^{-\fraci2(\sqrt{\tau}z+\sqrt{\bar{\tau}}\bar{z})}-ue^{\fraci2(\sqrt{\tau}z-\sqrt{\bar{\tau}}\bar{z})}-ve^{-\fraci2(\sqrt{\tau}z-\sqrt{\bar{\tau}}\bar{z})}\right]^{-2}\,,
\varphi_{n,\{\lambda_{i}\}}^{(s=0)}\left(\left\{z_{j}\right\};0\right)=z_{12}z_{23}...z_{n1}B^{(s=-1)}(\lambda_{1})...B^{(s=-1)}(\lambda_{p})\frac{1}{z_{1}^{2}...z_{n}^{2}}
\begin{array}{cll}{(3,6)\otimes(2,\overline{{4}})}&{=}&{(2\oplus4,\,4\oplus\overline{{20}})\,,}\\{{}}&{{}}&{{}}\\{(1,10)\otimes(2,\overline{{4}})}&{=}&{(2,\,4\oplus36)\,,}\\\end{array}
\sqrt{-G}W_{\mu\nu}^{10}\Big|_{y=0}=-\sqrt{-g^{0}}\frac{1}{2}V_{0}(\Phi)g_{\mu\nu}^{0}\Bigr|_{y=0}\,\quad\sqrt{-G}W_{\mu\nu}^{10}\Big|_{y=\infty}=\sqrt{-g^{\infty}}\frac{1}{2}V_{\infty}(\Phi)g_{\mu\nu}^{\infty}\Bigr|_{y=\infty}\.
\begin{array}{ll}{\{A^{0}(x),A^{j}(y)\}_{D}=\frac{1}{m^{2}}\partial_{x}^{j}\delta(x-y),}&{\{A^{0}(x),A^{0}(y)\}_{D}=0,}\\{\{A^{j}(x),A^{k}(y)\}_{D}=0,}&{\{\pi_{\mu}(x),\pi_{\nu}(y)\}_{D}=0,}\\{\{A^{i}(x),\pi_{j}(y)\}_{D}=\delta_{j}^{i}\delta(x-y),}&{\{A^{0}(x),\pi_{\nu}(y)\}_{D}=0,}\\{\{A^{i}(x),\pi_{0}(y)\}_{D}=0.}&{}\\\end{array}
m_{0}^{2}={\frac{4\kappa}{3}}\nu_{1}^{2}\varphi_{(-)}^{2}e^{-2d/\ell}\left[{\frac{\sqrt{1+(M^{2}\ell^{2}/4)}}{e^{4\sqrt{1+(M^{2}\ell^{2}/4)}d/\ell}-1}}\right].
\sum_{j,j^{\prime};\mu,\mu^{\prime}}z_{j}^{(\mu)}\{J_{(\mu)}^{j,-j},J_{(\mu^{\prime})}^{j^{\prime},-j^{\prime}}\}^{*}(y)t_{j^{\prime},-j^{\prime}}^{(\mu^{\prime})}=[R(\vec{z};y),y]
\frac{\dot{\sigma}}{R^{2}-1}=\frac{\dot{\sigma}G^{2}}{R^{2}W^{2}}=-\frac{\dot{\sigma}\eta_{G}^{2}\eta_{U}}{2\sigma_{0}R^{2}W^{2}}\frac{1}{U}\approx-\frac{\dot{\sigma}\eta_{G}^{2}\eta_{U}}{2\sigma_{0}}\frac{1}{\tilde{U}}.
h_{\alphar,\betaq^{\prime}}=\epsilon_{\alpha\beta}(h_{j}\tilde{\gamma}_{q^{\prime}r}^{i}+h_{j_{1}j_{2}j_{3}}\gamma_{q^{\prime}r}^{j_{1}j_{2}j_{3}})+\gamma_{\alpha\beta}^{a}(h_{aj}\tilde{\gamma}_{q^{\prime}r}^{j}+h_{aj_{1}j_{2}j_{3}}\gamma_{q^{\prime}r}^{j_{1}j_{2}j_{3}}),
S=\intd^{3}x\left[\frac{1}{4}F_{ij}F_{ij}+\frac{1}{2}(D_{i}\Phi)^{\dagger}(D_{i}\Phi)+V(\Phi)+{\calL}_{\mathrm{g.f.}}+{\calL}_{\mathrm{gh}}+{\calL}_{\mathrm{c.t.}}\right],
\mathrm{Tr}\left(e^{ik\cdotB}\right)\longleftrightarrow\intd^{2p}xe^{ik\cdotx}\mathrm{tr}_{N}\left(P_{*}e^{i\int_{0}^{1}d\sigmak\cdot\theta\cdotA(x+\sigmak\cdot\theta)}\right).
D_{\mu\nu}^{prop}=\left[-g_{\mu\nu}^{\perp}+\frac{n_{\mu}\partial_{\nu}^{\perp}+n_{\nu}\partial_{\mu}^{\perp}}{(n\partial)}-\frac{n_{\mu}n_{\nu}\partial_{\perp}^{2}}{(n\partial)^{2}}\right]\frac{1}{\partial_{\mu}^{2}+i\varepsilon}.
{\dot{\pi}}_{A^{\prime}(k)}=\{H,\\pi_{A^{\prime}(k)}\}=-i{\frac{\partialg}{\partiala^{(j)(k)}}}\pi_{A^{\prime}(j)},\qquad\qquad{\dot{\calP}}^{a}=\{H,\{\calP}^{a}\}=0,
S_{\Phi}=-\frac{1}{2}\intd^{6}x\sqrt{^6g}g^{AB}\partial_{A}\Phi\partial_{B}\Phi\simeq-\frac{1}{2}\epsilon^{2}\int_{1}^{a}\phi^{2}d\phi\intd^{4}x\sqrt{-\eta}\eta^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi.