harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\zeta(s)=\frac{1}{\Gamma(s)}\,\,\int_{0}^{\infty}d\tau\,\tau^{s-1}\sum_{(m,n)\in\mathbb{Z}^{2}}e^{-\tau\pi(n^{2}(1+t)+m^{2}(1+t)+2inmt)}\,[6pt]
\langleX_{1}^{n}\rangle_{pol}=5(5\sqrt{5})^{g-1}\beta^{k}\left[\left({\frac{\sqrt{5}-1}{2}}\right)^{5(k-g+1)}+(-1)^{k}\left({\frac{\sqrt{5}+1}{2}}\right)^{5(k-g+1)}\right]
\Gamma(x_{1},...,x_{m})=\pi^{-\frac{D}{2}}\int_{y_{0}>0}\frac{d^{D+1}y}{y_{0}^{D+1}}\prod_{i=1}^{m}(\frac{y_{0}}{y_{0}^{2}+(\vec{y}-\vec{x_{i}})^{2}})^{\alpha_{i}}
2\kappa^{2}\langleT_{\mu\nu}\rangle=k_{\mu}k_{\nu}\frac{1}{3!}\,\hat{C}^{\mu\nu\rho}\hat{C}_{\mu\nu\rho}^{*}=k_{\mu}k_{\nu}\frac{1}{3!}\,\left[\hat{C}_{ijk}\hat{C}_{ijk}^{*}-3\hat{k}^{l}\hat{C}_{ljk}\hat{k}^{m}\hat{C}_{mjk}^{*}\right]
J\equiv\int_{-L}^{L}dyW^{\frac{n}{2}}\left[\frac{2}{3}(4-n)T_{\mu}^{\mu}+(2n-4)T_{m}^{m}+\frac{1}{6\piG_{6}}\left\{(4-n)\hat{R}+(n-1)\bar{R}W^{-1}\right\}\right]
\gamma^{0}=\left(\begin{array}{cc}{\sigma_{3}}&{0}\\{0}&{-\sigma_{3}}\\\end{array}\right),\\\\\\gamma^{1}=\left(\begin{array}{cc}{i\sigma_{1}}&{0}\\{0}&{-i\sigma_{1}}\\\end{array}\right),\\\\\\gamma^{2}=\left(\begin{array}{cc}{i\sigma_{2}}&{0}\\{0}&{-i\sigma_{2}}\\\end{array}\right)
\Pi_{\mu\nu}(n,k)\;=\;-\frac{e_{N}^{2}}{\beta}\sum_{l=-\infty}^{l=+\infty}\int\frac{d^{2}p}{(2\pi)^{2}}\mathrm{tr}\left[\frac{1}{i(\not\!p+\not\!k)+i\gamma_{0}{\tilde{\omega}}_{l+n}}\gamma_{\mu}\frac{1}{i\not\!p+i\gamma_{0}{\tilde{\omega}}_{l}}\gamma_{\nu}\right]
\Gamma_{\alpha\beta\gamma}^{\bigtriangledown}=\frac{1}{2}\left[\delta_{\beta}g_{\alpha\gamma}+\delta_{\gamma}g_{\beta\alpha}-\delta_{\alpha}g_{\gamma\beta}+g_{\alpha\tau}W_{\gamma\beta}^{\tau}+g_{\beta\tau}W_{\alpha\gamma}^{\tau}-g_{\gamma\tau}W_{\beta\alpha}^{\tau}\right],
\frac{\partialS}{\partial\lambda^{I}}=\frac{K}{2}\int\frac{d\tau}{2\pi}\int\frac{d\tau^{\prime}}{2\pi}\left<\mathcal{O}_{I}(\tau)\{Q_{B},\mathcal{O}(\tau^{\prime})\}\right>_{\mathcal{V}}\,
ds^{2}=\frac{1}{1+u^{2}}(2d{x^{\prime}}^{+}du+d{x^{\prime}}^{2}+{x^{\prime}}^{2}d\Omega_{2}^{2}+\frac{1}{2}(1+\frac{1}{\sqrt{1+u^{2}}})(d{y^{\prime}}^{2}+{y^{\prime}}^{2}d\Omega_{5}^{2}))
\left(\sum_{i=1}^{2}\left(\frac{1}{r_{i}}\frac{\partial}{\partialr_{i}}r_{i}\frac{\partial}{\partialr_{i}}+\frac{1}{r_{i}^{2}}\frac{\partial^{2}}{\partial\psi^{2}}-\frac{r_{i}^{2}}{g^{2}}\right)+\frac{2}{g^{2}}\:E\right)\Psi(r_{1},r_{2},\psi)=0\;.
\gamma_{a\dot{b}}^{i}\tilde{\gamma}_{\dot{b}c}^{j}+\gamma_{a\dot{b}}^{j}\tilde{\gamma}_{\dot{b}c}^{i}=2\delta^{ij}\delta_{ac}\;,\qquad\tilde{\gamma}_{\dot{a}b}^{i}\gamma_{b\dot{c}}^{j}+\tilde{\gamma}_{\dot{a}b}^{j}\gamma_{b\dot{c}}^{i}=2\delta^{ij}\delta_{\dot{a}\dot{c}}\;.
T(a,b,d)=\frac{4}{(2\sqrt\pi)^{d-2}ab}\Gamma(2-\frac{d}{2})\sum_{n,n^{\prime}=1}^{\infty}\frac{1}{{\left(m^{2}+(\frac{n\pi}{a})^{2}+(\frac{n^{\prime}\pi}{b})^{2}\right)^{2-\frac{d}{2}}}}.
\left.\begin{array}{llll}{Tr\,[\,Q_{0}^{3}\,]\,=4\,}&{\,Tr\,[\,Q_{1}^{3}\,]\,=4\,}&{\,Tr\,[\,Q_{2}^{3}\,]\,=-8\,}&{\,Tr\,[\,Q_{3}^{3}\,]\,=0}\\\end{array}\right.
F_{\mu\nu}^{(2,2)}\:\stackrel{??}{=}\:-\frac{1}{2}\{F_{\mu\cdot}^{(1,1)}\stackrel{\theta}{,}F_{\cdot\nu}\}-\frac{1}{2}\{A^{(1,1)}\stackrel{\theta}{,}\frac{\partial+2D}{3}F_{\mu\nu}\}
\partial_{T}\left(\varphi_{,T}^{M}G_{MN}(\varphi)\varphi_{,T}^{N}\right)=2\left({\calD}_{T}\varphi_{,T}^{M}\cdot\varphi_{M,T}\right)+\varphi_{,T}^{M}\varphi_{,T}^{N}{\calD}_{T}G_{MN}(\varphi)
\Psi(n)\;=\;\frac{1}{\tau_{2}}\sum_{k}\,\left\{b_{\pm}(k\midA)u_{\pm}(k\midA)\;+\;d_{\pm}^{\dagger}(k\midA)v_{\pm}(k\midA)\right\}\,e^{i2\pik\cdotn}\;\;,
|\hat{\bboxp}j_{3}({\mathsf{s}}_{R}j_{R})^{-}\rangle=\frac{i}{2\pi}\int_{-\infty_{II}}^{+\infty}d{\mathsf{s}}|\hat{\bboxp}j_{3}({\mathsf{s}}j_{R})^{-}\rangle\frac{1}{{\mathsf{s}}-{\mathsf{s}}_{R}}\,,\quad{\mathsfs}_{R}=\left(M_{R}-i\frac{\Gamma_{R}}{2}\right)^{2}
V(k,k\,^{\prime};\tau)=\frac{m}{\alpha}\,\left(e^{-\alpha(k+k\,^{\prime})\tau}\,\,\delta(k-k\,^{\prime}-i)+\frac{e^{\alpha(k+k\,^{\prime})\tau}}{4\,k\,k\,^{\prime}}\,\,\delta(k-k\,^{\prime}+i)\right).
E(e_{j}-e_{k})=\sigma_{+}^{(j)}\sigma_{-}^{(k)},\quadE(e_{j}+e_{k})=\sigma_{+}^{(j)}\sigma_{+}^{(k)},\quadE(-e_{j}-e_{k})=\sigma_{-}^{(j)}\sigma_{-}^{(k)},
\psi_{j}^{f}(x)=\int\frac{d\theta}{\sqrt{4\pi}}\left(a_{j}(\theta)u_{j}(\theta)e^{-ip_{j}\cdotx}+a_{_{\bar{\jmath}}}^{\dagger}(\theta)v_{j}(\theta)e^{ip_{j}\cdotx}\right)\,,\qquad
\begin{array}{rl}{N}&{even:\;\;\frac{1}{2}-\frac{1}{p_{0}}+\frac{1-v_{s}}{4}<\frac{s}{p_{0}}<\frac{1}{2}+\frac{1-v_{s}}{4}}\\{N}&{odd:\;\;\;\;\frac{1}{2}-\frac{3}{4p_{0}}+\frac{1-v_{s}}{4}<\frac{s}{p_{0}}<\frac{1}{2}-\frac{1}{4p_{0}}+\frac{1-v_{s}}{4}}\\\end{array}
3D_{i\alpha}(W(\theta)\tilde{\gamma}^{A})^{[\beta\gamma]}=\delta_{\alpha}^{~\beta}D_{i\delta}(W(\theta)\tilde{\gamma}^{A})^{[\delta\gamma]}-\delta_{\alpha}^{~\gamma}D_{i\delta}(W(\theta)\tilde{\gamma}^{A})^{[\delta\beta]}
{\hat{V}}_{0}=\left(\begin{array}{cc}{{\bf1}}&{0}\\{0}&{{\bf1}}\\\end{array}\right),\quad{\hat{V}}_{1/2}=\left(\begin{array}{cc}{0}&{V_{1/2}}\\{V_{1/2}^{\prime}}&{0}\\\end{array}\right),\quad{\hat{V}}_{1}=\left(\begin{array}{cc}{V_{1}}&{0}\\{0}&{V_{1}^{\prime}}\\\end{array}\right);
\int_{0}^{\infty}dm\frac{m^{2}~e^{-\epsilon_{m}m}}{e^{-\beta\Omega_{H}(1-x)m}-1}=-\frac{2}{\epsilon_{m}^{3}}-\frac{2\zeta(3)}{[\beta\Omega_{H}(1-x)]^{3}}~,
F(\lambda)=x_{+}^{2\ell-\mu+\lambda}\int_{0}^{\infty}(\rho+1)^{2\ell-\mu}\rho^{\lambda-1}d\rho=\frac{\Gamma(\lambda)\Gamma(-2\ell+\mu-\lambda)}{\Gamma(\mu-2\ell)}\,x_{+}^{2\ell-\mu+\lambda}\,,
\left\{Q_{r},Q_{s}\right\}=Z_{rs}(\Gamma_{\mu}C)_{rs}P^{k}+C\Gamma_{[\mu\nu]}C)_{rs}Z^{[\mu\nu]}+\left(\Gamma_{[\mu_{1}-\mu_{s}]}C\right)_{rs}Z^{[\mu_{1}\ldots\mu_{5}]}\,.
H_{gonihedric}^{3d}=-2\sum_{\vec{r},\vec{\alpha}}\sigma_{\vec{r}}\sigma_{\vec{r}+\vec{\alpha}}+\frac{1}{2}\sum_{\vec{r},\vec{\alpha},\vec{\beta}}\sigma_{\vec{r}}\sigma_{\vec{r}+\vec{\alpha}+\vec{\beta}}
V_{(j,m)}^{\omega}(z,\bar{z})~=~\gamma^{j-m}(z)\bar{\gamma}^{j-m}(\bar{z})e^{\frac2{\alpha_{+}}j\phi(z,\bar{z})}e^{i\sqrt{\frac{2}{k}}mX(z,\bar{z})}e^{i\sqrt{\frac{2}{k}}(m+{\frac{k}{2}}\omega)Y(z,\bar{z})}
\langleA_{\mu}^{a}A_{\nu}^{b}\rangle\sim{\frac{\delta^{ab}}{k^{2}}}\left(g_{\mu\nu}-{\frac{k_{\mu}k_{\nu}}{k^{2}}}\right)+\varepsilon\alpha^{a}\alpha^{b}{\frac{k_{\mu}k_{\nu}}{(k^{2})^{2}}},
\Bigl[-\,\frac{1}{N^{2}}\partial_{i}^{2}-\frac{1}{N^{2}}\sum_{j\neqi}\frac{\partial_{j}-\partial_{i}}{\eta_{j}-\eta_{i}}-\frac{\eta_{i}}{N}\,\partial_{i}-\frac{2\alpha_{\scr\mathrm{KP}}}{N\eta_{i}}\,\partial_{i}+\frac{\alpha_{\scr\mathrm{KP}}}{N\eta_{i}}\sum_{j}\frac{1}{\eta_{j}}-\frac{\alpha_{\scr\mathrm{KP}}^{2}}{\eta_{i}^{2}}\Bigr]{\calZ}_{\scr\mathrm{KP}}=0.
A(0)\simeq1-\frac{K}{2}G_{h}(0)=1-\frac{K}{4\pi}\frac{d\Lambda}{\Lambda},~B(0)\simeq1+\frac{K}{2}G_{\phi}(0)\simeq1+\frac{K}{4\pi}\left(\frac{\Lambda}{m_{H}}\right)^{2}\frac{d\Lambda}{\Lambda},
(d)=-{\frac{{\lambda}^{2}\phi_{0}^{2}}{48(2\pi)^{2}}}\int_{0}^{\frac{\beta}{2}}d\tau\int_{-\infty}^{+\infty}{\frac{dk_{1}\,dk_{2}}{g(k_{1})g(k_{2})g(k_{3})}}\prod_{i=1}^{3}\left[e^{-g(k_{i})\tau}+{\frac{e^{g(k_{i})\tau}+e^{-g(k_{i})\tau}}{e^{\betag(k_{i})}-1}}\right]
ds^{2}=-2dx^{+}dx^{-}-\beta^{2}\sum_{i=1}^{4}(\tilde{x}^{i})^{2}(dx^{+})^{2}+\sum_{i=1}^{4}(d\tilde{x}^{i})^{2}+\sum_{i=5}^{8}(dx^{i})^{2},
|M|^{2}\left(Z^{-1}+\frac{1}{2}\eta^{\mu\nu}\partial_{\mu}x^{m}\partial_{\nu}x^{m}\right)=|M|^{2}\left(Z^{-1}+(2\pi\alpha^{\prime})^{2}\eta^{\mu\nu}\partial_{\mu}\bar{\phi}\partial_{\nu}\phi\right)\,.
\begin{array}{rcl}{{\calQ}_{e}^{I\\prime}}&{=}&{(c\hat{a}_{0}+d){\calQ}_{e}^{I}+ce^{-\hat{\phi}_{0}}{\calQ}_{m}^{I}\,,}\\{}&{}&{}\\{{\calQ}_{m}^{I\\prime}}&{=}&{-ce^{-\hat{\phi}_{0}}{\calQ}_{e}^{I}+(c\hat{a}_{0}+d){\calQ}_{m}^{I}\,.}\\\end{array}
1\otimesr_{\dot{J}}^{\dot{I}}\otimes\Xi_{\lambda_{2}}^{\lambda_{1}}\left(\bar{\phi}^{\rho_{1}}\otimess^{\dot{K}}\otimes\phi^{\rho_{2}}\right)=\delta_{\lambda_{2}}^{\rho_{2}}\sum_{\dot{K}_{1}\dot{K}_{2}=\dot{K}}\delta_{\dot{J}}^{\dot{K}_{2}}\bar{\phi}^{\rho_{1}}\otimess^{\dot{K}_{1}\dot{I}}\otimes\phi^{\lambda_{1}}.
e^{i\hat{\Gamma}[\vec{\varphi}_{1},\vec{\varphi}_{2}]}=\langleT_{C}\mathrm{exp}\left[\frac{i}{\delta}\intd^{\,4}x\sum_{j=1}^{2}\left\{(-)^{j-1}{\calL}^{\prime}(\vec{\phi}_{j}(x)+\vec{\varphi}_{j}(x))-\frac{\delta\hat{\Gamma}[\vec{\varphi}_{1},\vec{\varphi}_{2}]}{\delta\vec{\varphi}_{j}(x)}\cdot\vec{\phi}_{j}(x)\right\}\right]\rangle,
{\calL}_{\mathbf{I}}W_{\mathbf{J}}^{x}-{\calL}_{\mathbf{J}}W_{\mathbf{I}}^{x}+\epsilon^{xyz}W_{\mathbf{I}}^{y}W_{\mathbf{J}}^{z}=f_{\mathbf{I}\mathbf{J}}^{\cdot\cdot\mathbf{L}}W_{\mathbf{L}}^{x}
G_{l+\nu}(r^{\prime\prime},r^{\prime};E|\lambda)=G_{s_{l}}^{(0)}(r^{\prime\prime},r^{\prime};E)=-\frac{2M}{\hbar^{2}}\,\sqrt{r^{\prime}r^{\prime\prime}}\,I_{s_{l}}(\kappar_{<})K_{s_{l}}(\kappar_{>})\;,
\sum_{\begin{array}{l}{n_{1},n_{1}^{\prime},n_{2},n_{2}^{\prime}}\\{n_{1}+n_{1}^{\prime}=1}\\{n_{2}+n_{2}^{\prime}=1}\\\end{array}}[B_{2}^{[n_{1},n_{2}]},B_{1}^{[n_{1}^{\prime},n_{2}^{\prime}]}]=1
1+{\frac{1}{4}}I_{(2)}[\mu^{2}]\int_{-\infty}^{\infty}{\frac{d\alpha}{\sqrt{\pi}}}e^{-\alpha^{2}}V^{(4)}(\alpha\sqrt{2(I_{(1)}[\mu^{2}]-I_{(1)}[M^{2}])}+\Phi)\ge0\;.
\left|{\varepsilon}_{N},2N\right\rangle=\prod_{n=-\infty}^{N-1}a_{n}^{\dagger}|0\rangle\otimes\prod_{m=N}^{\infty}b_{m}^{\dagger}|0\rangle,\quad{\varepsilon}_{N}(c)=\frac{2\pi}{L}\left\{\left(N-\frac{ecL}{2\pi}\right)^{2}-\frac{1}{12}\right\}.
L_{2}^{\epsilon_{1}k}\tilde{Z}_{21}^{-1}L_{1}^{\epsilon_{2}j}=\tilde{Z}_{21}^{-1}L_{1}^{\epsilon_{2}j}\tilde{Z}_{21}L_{2}^{\epsilon_{1}k}\tilde{Z}_{21}^{-1}\ll{zlzl2+}
[D,K](x,y)=a_{N}\Big(\varphi_{N}(x)\quad\varphi_{N-1}(x)\Big)\left(\begin{array}{cc}{\frac{C(x)-C(y)}{x-y}}&{\frac{A(x)-A(y)}{x-y}}\\{\frac{A(x)-A(y)}{x-y}}&{\frac{B_{N}(x)-B_{N}(y)}{x-y}}\\\end{array}\right)\left(\begin{array}{c}{\varphi_{N}(y)}\\{\varphi_{N-1}(y)}\\\end{array}\right).
{\calL}_{\mathrm{spinor}}^{(2)}=\frac{e^{2}}{2}\intd^{4}x^{\prime}\,{\calD}(x-x^{\prime})\,\Bigl[-8\,\langlex|D_{\alpha}G|x^{\prime}\rangle\langlex^{\prime}|D_{\alpha}G|x\rangle+16\,\langlex|G|x^{\prime}\rangle\langlex^{\prime}|D_{\alpha}GD_{\alpha}|x\rangle+16\,\langlex|x^{\prime}\rangle\langlex^{\prime}|G|x\rangle\Bigr].
\Sigma_{A}(\bar{p})\simeq\frac{ie^{2}}{(2\pi)^{3}}|eH|\intdq_{0}dq_{3}\int_{0}^{\infty}dr^{2}\mathrm{e}^{-r^{2}}\frac{-2}{q^{2}}\frac{\Sigma_{A}(\bar{p"})}{\bar{p"}^{2}+\Sigma_{A}(\bar{p"})}
\sum_{Q}m_{Q}^{4}K_{Q}(p)=\tilde{k},\\\sum_{Q}m_{Q}^{4}H_{Q}(p)=\tilde{h},\\,sum_{Q}m_{Q}^{4}S_{Q}(p)=\tilde{s},\\\sum_{Q}m_{Q}^{4}R_{Q}(p)=\tilde{r}.
{\frac{3}{b^{2}}}{\frac{a^{\prime}}{a}}\left({\frac{a^{\prime}}{a}}+{\frac{n^{\prime}}{n}}\right)-{\frac{3}{c^{2}n^{2}}}\left[{\frac{\dot{a}}{a}}\left({\frac{\dot{a}}{a}}-{\frac{\dot{n}}{n}}\right)+{\frac{\ddot{a}}{a}}\right]-{\frac{3k}{a^{2}}}=-{\frac{8\piG_{5}}{c^{4}}}\Lambda,
C_{ab}^{cc^{\prime}}=\frac{1}{8}\left(C_{ab}^{d}C_{\;\cdotd}^{cc^{\prime}}-\delta_{a}^{c}\delta_{b}^{c^{\prime}}+\delta_{b}^{c}\delta_{a}^{c^{\prime}}\right),\;\;\;\;\;C_{aa^{\prime}\,b}^{cc^{\prime}}=\delta_{[a}^{[c}C_{a^{\prime}]b}^{c^{\prime}]}+\frac{1}{2}C_{\;\cdot[a}^{cc^{\prime}}g_{a^{\prime}]b}.
\delta\left[\begin{array}{c}{\phi}\\{-i\psi_{2}}\\\end{array}\right]=\varepsilon_{1}\left[\begin{array}{c}{-i\psi_{2}}\\{-i\dot{\phi}}\\\end{array}\right]\,,\quad\quad\delta\left[\begin{array}{c}{\psi_{1}}\\{\partial_{z}\phi+F}\\\end{array}\right]=\varepsilon_{1}\left[\begin{array}{c}{\partial_{z}\phi+F}\\{-i\dot{\psi}_{1}}\\\end{array}\right]\,.
V_{eff}(z)=\frac{15}{4\big(|z|+\ell\big)^{2}}-\frac{3}{\ell}\,\delta(z)\bigg[1-\frac{2\ell\gamma}{9}\left(\partial_{z}\psi(0_{+})+\frac{3}{2\ell}\,\psi(0_{+})\right)\bigg]\,,
\psi^{\mu_{0}\mu_{1}\ldots\mu_{p}}(X)\equiv\phi^{\mu_{0}\mu_{1}\ldots\mu_{p}}(X)+\frac{e}{g}\sum_{i=0}^{p}(-1)^{p\cdoti}\frac{\partial}{\partialX_{\mu_{i}}}B^{\mu_{i+1}\mu_{i+2}\ldots\mu_{p}\mu_{0}\mu_{1}\ldots\mu_{i-1}}(X).
{\calL}_{+}=\epsilon^{\mu\nu\rho\sigma}\bigg(AR_{\mu\nu}^{+ab}R_{\rho\sigmaab}^{+}+B\Sigma_{\mu\nu}^{+ab}R_{\rho\sigmaab}^{+}+C\Sigma_{\mu\nu}^{+ab}\Sigma_{\rho\sigmaab}^{+}\bigg),
G_{a}^{(n)}=\sum_{m=0}^{n}\bigl\{T_{a}^{(n-m)},\,A^{(m)}\bigr\}_{(q,p)}+\sum_{m=0}^{n-2}\bigl\{T_{a}^{(n-m)},\,A^{(m+2)}\bigr\}_{(\eta)}+\bigl\{T_{a}^{(n+1)},\,A^{(1)}\bigr\}_{(\eta)}\,.
{\calK}_{{\hat{\imath}}_{1}\dots{\hat{\imath}_{4}}}^{(4)}={\frac{1}{2!\sqrt{\mathrm{det\Pi^{(A)}}}\sqrt{1+e^{2\phi}k^{-2}(i_{k}C^{(1)})^{2}}}}\Pi_{{\hat{\imath}}_{1}{\hat{\jmath}}_{1}}^{(A)}\dots\Pi_{{\hat{\imath}}_{4}{\hat{\jmath}}_{4}}^{(A)}\epsilon^{{\hat{\jmath}}_{1}\dots{\hat{\jmath}}_{6}}{\calK}_{{\hat{\jmath}}_{5}{\hat{\jmath}}_{6}}^{(2)}\,,
g=e^{xL_{-1}}e^{t(G_{-1/2}^{+}+G_{-1/2}^{-})}\left(\prod_{n\geq3}e^{\xi_{n-1/2}^{\pm}G_{n-1/2}^{\pm}}e^{u_{n}L_{n}}e^{\phi_{n-1}U_{n-1}}\right)e^{\phi_{1}U_{1}}e^{\xi^{+}G_{3/2}^{+}+\xi^{-}G_{3/2}^{-}}e^{u_{2}L_{2}}.
F=i\frac{dg_{1}\wedged\overline{{g}}_{1}}{(1-g_{1}\overline{{g}}_{1})^{2}}-i\frac{dg_{2}\wedged\overline{{g}}_{2}}{(1+g_{2}\overline{{g}}_{2})^{2}}=-\frac{1}{2}(R_{2}^{1}+R_{4}^{3})~.
\mathrm{T}e^{-i\int_{0}^{T}dt^{\prime}H_{I}(t^{\prime})}=I-i\int_{0}^{T}dt^{\prime}H_{I}(t^{\prime})-{\frac{1}{2}}\int_{0}^{T}dt^{\prime}\int_{0}^{T}dt^{\prime\prime}\\mathrm{T}[H_{I}(t^{\prime})H_{I}(t^{\prime\prime})],
\delta\,S_{CS}^{\prime}=\frac\ell{8\piG}\int_{|r|=\bar{r}}Tr[\partial_{t}g_{1}\,g_{1}^{-1}\,\delta\zeta-\partial_{t}g_{2}\,g_{2}^{-1}\,\delta\tilde{\zeta}\,]|_{\varphi=0}\,dt\qquad,
\hat{Z}_{88}^{1}(\Omega)\stackrel{D_{a_{1}^{-1}a_{2}}}{\longrightarrow}\frac{(det\Im\Omega)^{2}}{4}\biggl\vert\Theta\left[\begin{array}{ll}{1/2}&{1/2}\\{1/2}&{1/2}\\\end{array}\right](0\vert\Omega)\biggr\vert^{8}
{\calK}(\theta,\tau)={\frac{1}{\Gamma(1-\nu)\Gamma(\nu)}}\int_{0}^{1}\Phi\left(\tau,{\frac{t(\theta-\tau)}{2}}\right)\mathrm{e}^{\mathrm{i}(r+\mu)t(\theta-\tau)}(1-t)^{-\nu}t^{\nu-1}dt,\tag{B.33}
E_{6}=\frac{a^{2}}{24\ell^{12}r^{6}}(-8a^{4}X^{4}+5a^{4}X^{6}-2\ell^{4}\pm2a^{2}\ell^{2}\pm8a^{2}X^{4}\ell^{2}\mp9a^{2}\ell^{2}X^{2}+3a^{4}X^{2}+3\ell^{4}X^{2})+O(\frac{1}{r^{8}})
{\calR}^{\frac{1}{2}\frac{1}{2}}=q^{\frac{1}{2}}\left[\begin{array}{cccc}{1}&{}&{}&{}\\{}&{q^{-1}}&{0}&{}\\{}&{1-q^{-2}}&{q^{-1}}&{}\\{}&{}&{}&{1}\\\end{array}\right]
L_{i_{1}}^{(1)}(x):=\intd^{2}y\left[W_{i_{1}j}^{(1)}(x,y)\ddot{\varphi}^{j}(y)+\alpha_{i_{1}}^{(1)}(y)\delta^{2}(x-y)\right]=0,~~~i_{1}=1,\cdots,11,
\ddot{z}+\left({\frac{a^{\prime}}{a}}-2{\frac{n^{\prime}}{n}}+{\frac{d^{\prime}}{d}}\right)\dot{z}^{2}+\left({\frac{nn^{\prime}}{d^{2}}}-{\frac{a^{\prime}}{a}}{\frac{n^{2}}{d^{2}}}\right)=0\,.
\begin{array}{lcr}{\tau=\pmk\sqrt{-kt^{2}+t_{+}^{2}}}&{\qquad\mathrm{for}\qquad}&{k=\pm1}\\{\tau=\frac{t^{2}}{2|t_{+}|}}&{\qquad\mathrm{for}\qquad}&{k=0\.}\\\end{array}
V_{p^{+},p^{-}}^{r,s}\left(z\right)=Ce^{iX^{-}p^{+}}\,\,e^{-\frac12X_{2}}\,J_{2\sigma}\left(2\sqrt{-p^{+}p^{-}}e^{-\frac12X_{2}(z)}\right)\,e^{iX^{+}p^{-}}
\delta\widetilde{X}^{A}=\epsilon\left(x\right)\left[C\partial\widetilde{X}^{A}-\frac{1}{2}\left(DC\right)D\widetilde{X}^{A}+C\left(\partial\Psi\right)\widetilde{X}^{A}-\frac{1}{2}\left(DC\right)\left(D\Psi\right)\widetilde{X}^{A}\right],
V_{\mathrm{gauge}}=G_{\muI}:(\bar{\partial}X^{I}-\frac{1}{4}k_{\sigma}\bar{S}^{a}\gamma_{a,b}^{I\sigma}\bar{S}^{b})(\partialX^{\mu}-\frac{1}{4}k_{\nu}S^{a}\gamma_{ab}^{\nu\mu}S^{b})e^{ikX}:\,.
{\calG}^{(n)}\=\{\calS}^{(n)}\+\,sum_{0<p\leqn}\\frac{(-1)^{p}}{2^{p}\p!\\left({n\atopp}\right)}\\eta^{p-1}\left[\\eta\{\calS}^{(n)\,[p]}\+\\gamma\{\cal{\not{\!S}}}^{(n)\,[p-1]}\\right]\,
T_{\mu}^{\nu}=\left(\begin{array}{cccc}{-\frac{\alpha\beta}{\kappa}\frac{e^{-\betar}}{r^{2}}}&{0}&{0}&{0}\\{0}&{-\frac{\alpha\beta}{\kappa}\frac{e^{-\betar}}{r^{2}}}&{0}&{0}\\{0}&{0}&{\frac{\alpha\beta^{2}}{2\kappa}\frac{e^{-\betar}}{r}}&{0}\\{0}&{0}&{0}&{\frac{\alpha\beta^{2}}{2\kappa}\frac{e^{-\betar}}{r}}\\\end{array}\right),
\frac{{\partial}^{2}{\eta}_{+}}{{\partial}x_{-}^{2}}-\frac{1}{f}\left(\frac{{\partial}f}{{\partial}x_{-}}\right)\frac{{\partial}{\eta}_{+}}{{\partial}x_{-}}+\frac{1}{4{\hbar}^{2}c^{2}}f^{2}{\eta}_{+}=\frac{m^{2}c^{2}}{{\hbar}^{2}}\frac{f^{2}}{f^{2}-4c^{2}P_{cm}^{2}}{\eta}_{+}.
ds_{adS*S}^{2}=ds_{adS}^{2}+ds_{S}^{2}=\left({\frac{r}{R}}\right)^{2}dx^{\mu}\eta_{\mu\nu}dx^{\nu}+\left({\frac{R}{r}}\right)^{2}dr^{2}+R^{2}d^{2}\Omega
\mu=\left(\begin{array}{rrrrrrrr}{0}&{5}&{-7}&{2}&{-6}&{3}&{-4}&{8}\\{-5}&{0}&{-3}&{6}&{2}&{-7}&{8}&{4}\\{7}&{3}&{0}&{-8}&{-4}&{-5}&{6}&{2}\\{-2}&{-6}&{8}&{0}&{-5}&{4}&{3}&{7}\\{6}&{-2}&{4}&{5}&{0}&{-8}&{-7}&{3}\\{-3}&{7}&{5}&{-4}&{8}&{0}&{-2}&{6}\\{4}&{-8}&{-6}&{-3}&{7}&{2}&{0}&{5}\\{-8}&{-4}&{-2}&{-7}&{-3}&{-6}&{-5}&{0}\\\end{array}\right).
\sum_{r,s>0}rsd\overline{D}_{rs}z^{r+s}\zeta^{-r-s-2}+{\frac{d}{9}}{\frac{1}{\zeta^{2}}}=\sum_{r,s>0}rsd{D_{rs}}^{\ast}z^{-r-s}\zeta^{r+s-2}+{\frac{d}{9}}{\frac{1}{\zeta^{2}}}\,,
\begin{array}{rclcrclc}{\left[K_{a},P_{+}\right]}&{=}&{0}&{\quad.\quad}&{\left[K_{a},P_{-}\right]}&{=}&{P_{a}}&{\quad.}\\{\left[L_{ab},P_{+}\right]}&{=}&{0}&{\quad.\quad}&{\left[L_{ab},P_{-}\right]}&{=}&{0}&{\quad.}\\{\left[K_{a},P_{b}\right]}&{=}&{-\delta_{ab}\cdotP_{+}}&{\quad.\quad}&{\left[L_{ab},P_{c}\right]}&{=}&{\delta_{ac}\cdotP_{b}-\delta_{bc}\cdotP_{a}}&{\quad.}\\\end{array}
\hat{\varphi}(x)=\sum_{\stackrel{\rightharpoonup}{k},\pm}\left\{\frac{1}{\sqrt{2\omega}}e^{-i\omegat+i\stackrel{\rightharpoonup}{k}\cdot\stackrel{\rightharpoonup}{x}}\cdota(\stackrel{\rightharpoonup}{k},+)+\frac{1}{\sqrt{2\omega}}e^{i\omegat+i\stackrel{\rightharpoonup}{k}\cdot\stackrel{\rightharpoonup}{x}}\cdota(\stackrel{\rightharpoonup}{k},-)\right\}
\begin{array}{l}{AF(z)=q^{-j}F(qz)\,,}\\{BF(z)=-\displaystyle{\frac{z}{q-q^{-1}}}(q^{-2j}F(qz)-q^{2j}F(q^{-1}z))\,,}\\{CF(z)=\displaystyle{\frac{1}{z(q-q^{-1})}}(F(qz)-F(q^{-1}z))\,,}\\{DF(z)=q^{j}F(q^{-1}z)\,.}\\\end{array}
W^{\Omega_{z_{1}}^{*},\Omega_{z_{2}}}\left(\begin{array}{cc}{c}&{d}\\{b}&{a}\\\end{array}\right)=(-1)^{n-1+\mu+\kappa}W^{(n-1,1)}\left(\left.\begin{array}{cc}{c}&{d}\\{b}&{a}\\\end{array}\right|z_{1}-z_{2}\right),
\begin{array}{c}{{\displaystyleK_{ab}^{-}~=~\frac{\epsilon}{4}\,(E_{aa}-E_{bb}+E_{\theta(a)\,\theta(a)}-E_{\theta(b)\,\theta(b)})~,}}\\{{\displaystyleM_{ab}^{-}~=~\frac{\epsilon}{4}\,(E_{aa}-E_{bb}-E_{\theta(a)\,\theta(a)}+E_{\theta(b)\,\theta(b)})~,}}\\\end{array}
C^{\Lambda_{0}}=(I\otimes\mathrm{tr})\{[I\otimes\pi_{\Lambda_{0}}(q^{2h_{\rho}})]\Gamma\}=\sum_{s,t}\mathrm{tr}(\pi_{\Lambda_{0}}(q^{2h_{\rho}}a_{s}b_{t}))b_{s}a_{t}
\begin{array}{l}{\hat{B}_{\alpha\beta}=\frac{2\chi_{[\alpha}^{L}\chi_{\beta]}^{R}}{|\vec{\chi}^{R}|^{2}+(\vec{\chi}^{R}\vec{\chi}^{L})}}\\{\nonumber}\\{\hat{B}_{\alpha\mu}=A_{\mu\alpha}^{(2)}+\hat{B}_{\alpha\beta}A_{\mu}^{(1)\beta}}\\\end{array}
J^{\mu\nu}=-\mathrm{i}\sum_{n=1}^{\infty}\left(b_{n-1/2}^{\dagger\mu}b_{n-1/2}^{\nu}-b_{n-1/2}^{\dagger\nu}b_{n-1/2}^{\mu}+{\tilde{b}}_{n-1/2}^{\dagger\mu}{\tilde{b}}_{n-1/2}^{\nu}-{\tilde{b}}_{n-1/2}^{\dagger\nu}{\tilde{b}}_{n-1/2}^{\mu}\right)
S=\int\frac{\mathrm{tr}\,\hat{F}_{\mu\nu}^{2}}{2e^{2}}\,d^{4}x=\int\frac{\mathrm{tr}\,\hat{F}_{\alpha\beta}\,\hat{F}^{\alpha\beta}}{2e^{2}}\,\sqrt{g}\,d^{4}q.
\int{\calD}\beta{\calD}\gamma\,\delta(\beta(q_{1}))\,\delta(\beta(q_{2}))\,\mathrm{e}^{iS_{\beta\gamma}}=\langle\delta(\beta(q_{1}))\,\delta(\beta(q_{2}))\rangle={\frac{y^{2}(q)}{c^{2}}}\,\times(\mathrm{det}^{\prime}\bar{\partial}_{3/2})^{-1}.
\Gamma_{WZ}(\widetilde{\phi}_{1}\widetilde{\phi}_{2})\=\\Gamma_{WZ}(\widetilde{\phi}_{1})+\Gamma_{WZ}(\widetilde{\phi}_{2})-\frac{i}{4\pi}\int_{\Sigma}Tr(\phi_{1}^{-1}d\phi_{1}\,\phi_{2}d\phi_{2}^{-1})\.
\mu_{\mathbf{R}}^{1}=\zeta(\mathbf{1}_{k\timesk}+\Xi)\,,\quad\mu_{\mathbf{C}}^{1}=\mu_{\mathbf{C}}^{2}=\mu_{\mathbf{C}}^{2\,\,\,\prime}=\mu_{\mathbf{R}}^{3}=\mu_{\mathbf{C}}^{3}=0\,,
{\frac{ik_{-}S_{11}-ik_{3}S_{12}-ikS_{12}}{ik_{-}S_{11}-ik_{3}S_{12}+ikS_{12}}}={\frac{\kappa+\lambda+{\frac{i}{k}}(E\gamma-M{\gamma}^{\prime})}{{\kappa+\lambda-{\frac{i}{k}}(E\gamma-M{\gamma}^{\prime}})}},
{V(S,S^{*},T,T^{*})={{\frac{{\mid\Omega(S)\mid}^{2}}{S_{r}{T_{r}}^{3}}}{\frac{{\midH(T)\mid}^{2}}{{\mid\eta(T)\mid}^{12}}}\{{\midS_{r}{\frac{{\Omega}_{s}}{\Omega}}-1\mid}^{2}+{{\frac{{T_{r}}^{2}}{3}}{\mid{\frac{H^{\prime}}{H}}+{\frac{3}{2\pi}}{\widehat{G}}_{2}}}\mid}^{2}-3\}},
\begin{array}{ccc}{S}&{=}&{\frac12\int\frac{d^{p+1}y}{\Omega_{p}}\sqrt{-\tilde{g}}\{-\partial^{A}x^{0}\partial_{A}x^{0}+a^{i}a^{j}\partial^{A}a_{k}\partial_{A}a_{l}\eta_{ij}^{kl}-2\Lambda\}}\\{}&{}&{}\\{}&{}&{-\frac12\int\frac{d^{p+1}y}{\Omega_{p}}\sqrt{-\tilde{g}}\tilde{R}.}\\\end{array}
Z_{3}(s)=\frac{R^{2s-1}}{61440\,\sqrt{\pi}}\,(1-2s)\,(3-2s)\,(784\,s^{2}-104\,s-235)\,\frac{\Gamma(3/2+s)}{\Gamma(s)}\,\zeta_{\mathrm{R}}(2s+3)\,{.}
\bar{M}=\left(\begin{matrix}{\bar{G}^{-1}}&{-\bar{G}^{-1}\bar{C}}&{-\bar{G}^{-1}\bar{a}^{T}}\\{-\bar{C}^{T}\bar{G}^{-1}}&{\bar{G}+\bar{C}^{T}\bar{G}^{-1}\bar{C}+\bar{a}^{T}\bar{a}}&{\bar{C}^{T}\bar{G}^{-1}\bar{a}^{T}+\bar{a}^{T}}\\{-\bar{a}\bar{G}^{-1}}&{\bar{a}\bar{G}^{-1}\bar{C}+\bar{a}}&{I+\bar{a}\bar{G}^{-1}\bar{a}^{T}}\\\end{matrix}\right)
\left[\begin{array}{c}{\,h_{\mu\nu}(x,y_{+})\,}\\{\,h_{\mu\nu}(x,y_{-})\,}\\\end{array}\right]=-8\piG_{5}l\,{\bfG}(\Box)\left[\begin{array}{c}{\,T_{\mu\nu}^{+}-\frac{1}{3}\eta_{\mu\nu}T^{+}\,}\\{\,T_{\mu\nu}^{-}-\frac{1}{3}\eta_{\mu\nu}T^{-}\,}\\\end{array}\right]-2l\,{\bfG}(\Box)\left[\begin{array}{c}{\,\xi^{+},_{\mu\nu}\,}\\{\,-\xi^{-},_{\mu\nu}}\\\end{array}\right].
\begin{array}{rcl}{[R_{1},R_{2}]_{P.B.}}&{=}&{\displaystyle\displaystyle\frac{1}{4}((\lambda-8E)L+8\lambdaL^{3}),}\\{[R_{2},L]_{P.B.}}&{=}&{R_{1}~,~~~[L,R_{1}]_{P.B.}=R_{2}~,}\\\end{array}
2[{}^{\ell}Q_{n},{}^{\ell}Q_{m}]=+2\omega\tilde{g}\sum_{t=0}^{n+m-1}\sum_{j,k}\left((\ell_{k}^{+}\ell_{k}^{-})^{t}m_{jk}(\ell_{k}^{+}\ell_{k}^{-})^{n+m-t-1}-(\ell_{k}^{+}\ell_{k}^{-})^{t}m_{jk}(\ell_{k}^{+}\ell_{k}^{-})^{n+m-t-1}\right)=0.
\Psi_{n_{1}n_{2}m}(\mu,\nu,\varphi;\delta_{1},\delta_{2})=\sum_{l=|m|}^{n-1}W_{n_{1}n_{2}m}^{l}(\delta_{1},\delta_{2})\Psi_{nlm}(r,\theta,\varphi;\delta_{1},\delta_{2}).
L_{S}=\int_{V}d^{3}{\bfx}e^{3Ht}\left[\frac{1}{2}\left((\partial_{0}\phi({\bfx},t))^{2}-(e^{-Ht}{\bf\nabla}\phi({\bfx},t))^{2}-m^{2}\phi^{2}({\bfx},t)\right)-V(\phi({\bfx},t))\right],
P(q,q^{\prime})=\sum_{k=0,k^{\prime}=0}^{\infty}\frac{1}{k!k^{\prime}!}q^{n_{1}}...q^{n_{k^{\prime}}}q^{\prime}{}^{m_{1}}...q^{\prime}{}^{m_{k}}P_{\{m_{1}...m_{k}|n_{1}...n_{k^{\prime}}\}}(\partial_{l}).
{\calO}_{k}^{\alpha\beta}~=~\frac{1}{2}\left(\bar{\psi}^{iI}\Gamma_{(k)}{\calT}_{IJ}^{\alpha}\psi^{iJ}\right)\left(\bar{\psi}^{jK}\Gamma_{(k)}{\calT}_{KL}^{\beta}\psi^{jL}\right)~.
\,+\sqrt{\left(\tilde{n}_{1}\right)^{2}+\left(\frac{V_{s}}{g_{\mathrm{YM}}^{2}}\right)^{2}\left(\tilde{n}_{4}\right)^{2}+\left(\frac{V_{s}}{g_{\mathrm{YM}}^{2}}\right)^{4}\left(\tilde{n}_{1;6}\right)^{2}}
|\phi_{s}\rangle\equiv\zeta^{i_{1}}\ldots\zeta^{i_{s}}\phi^{i_{1}\ldotsi_{s}}\,,\qquad|\phi_{s-1}\rangle\equiv\zeta^{i_{1}}\ldots\zeta^{i_{s-1}}\phi^{i_{1}\ldotsi_{s-1}}\,.
{\calF}_{2}\left(a,\epsilon_{1},\epsilon_{2}\right)=\frac{20a^{2}+7\epsilon_{1}^{2}+16\epsilon_{1}\epsilon_{2}+7\epsilon_{2}^{2}}{\left(4a^{2}-\epsilon^{2}\right)^{2}\left(4a^{2}-\left(2\epsilon_{1}+\epsilon_{2}\right)^{2}\right)\left(4a^{2}-\left(\epsilon_{1}+2\epsilon_{2}\right)^{2}\right)}.
C\Big[i\pi\epsilon(p_{0})e^{-{\frac{m^{2}}{2\mu^{2}}}}\Bigg(\delta(p^{2})+\theta(p^{2}){\frac{m}{2\mu^{2}}}{\frac{1}{\sqrt{p^{2}}}}I_{1}\Big({\frac{m\sqrt{p^{2}}}{\mu^{2}}}\Big)\Bigg)e^{-p^{2}/2\mu^{2}}\Big].
\Phi(s,t)\left[{\frac{(12)^{2}(34)^{2}}{x_{12}^{4}x_{34}^{4}}}+{\frac{(14)^{2}(23)^{2}}{x_{14}^{4}x_{23}^{4}}}s+{\frac{(12)(23)(34)(41)}{x_{12}^{2}x_{23}^{2}x_{34}^{2}x_{41}^{2}}}(t-s-1)\right]\;.
(\theta^{-1})_{\mu\nuk\lambda}=X\overset{(2)}{P}+Y\overset{(1)}{P}_{\!\!m}+Z\overset{(0)}{P}_{\!\!s}+W\overset{(0)}{P}_{\!\!w}+T\overset{(0)}{P}_{\!\!sw}+R\overset{(2)}{P}_{\!\!ws}+MS_{1}+NS_{2}\.
\begin{array}{lll}{S_{m}(x)}&{\approx-ix^{-{\frac{1}{2}}}\left[1-\left(m+{\frac{1}{2}}\right)(ix)^{-1}+\cdots\right]~,}&{\quad|x|>\!\!>1\nonumber}\\{S_{m}(x)}&{\approx\left(m-{\frac{1}{2}}\right)^{-1}x^{\frac{1}{2}}~~,}&{\quad|x|\approx0\mathrm{~\mathrm{{and}}~}m>0\nonumber}\\{S_{0}(x)}&{\approx\pi^{\frac{1}{2}}e^{-i\pi/4}~~,}&{\quad|x|\approx0~~.}\\\end{array}
\hbar\Gamma(b_{i}\tob_{f}+\gamma)=\xi\left({\frac{\hbar\omega}{m_{P}}}\right)^{2}\left({\frac{A_{H}}{\ell_{P}^{2}}}\right)\left(\hbar\omega\right)\left(2\;\;{\frac{2J_{f}+1}{2J_{i}+1}}\right){\frac{N_{f}}{N_{i}}}.
\begin{array}{l}{0=lim_{T\rightarrow\infty}(T-\tau_{0})^{-1}\int_{\tau_{0}}^{T}d\tau\partial_{\tau}E[\Phi(\phi_{\tau}(\phi))]=}\\{=lim_{T\rightarrow\infty}(T-\tau_{0})^{-1}\int_{\tau_{0}}^{T}d\tau{\calG}_{\phi}E[\Phi(\phi_{\tau}(\phi))]={\calG}_{\phi}<\Phi(\phi_{\tau}(\phi))>}\\\end{array}