harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tau^{1}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\\,\\\tau^{2}=\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right)\\,\\\tau^{3}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\\,
w_{1+\infty}^{(l+1)}=\frac{(-1)^{l+1}}{l+1}:Q_{l+1}^{(+)}[j_{1}+\bar{j}_{1}]+\frac{1}{4}\partial_{P}^{(s-2)}Q_{2}^{2}[\bar{j}_{1}]-\frac{1}{2}{\partial}_{P}^{(s-1)}Q_{2}[\overline{j}_{1}].
{\cal{L}}={\frac{i}{2}}\[\{\bar{\psi}}\gamma^{\mu}(\overrightarrow{\partial}{}_{\mu}+ieA_{\mu})\psi-{\bar{\psi}}\gamma^{\mu}(\overleftarrow{\partial}{}_{\mu}-ieA_{\mu})\psi\]-m{\bar{\psi}}\psi-{\frac{1}{4}}F^{\mu\nu}F_{\mu\nu}\,
\Biggl[-\frac{\partial^{2}}{\partialU^{2}}+\frac{(2l+8-p)(2l+6-p)}{4U^{2}}+\frac{k^{2}g_{\mathrm{YM}}^{2}N}{U^{7-p}}\Biggr]U^{(8-p)/2}\psi=0\.
S_{eff}=\int\!d^{4}\!x~[~\pi_{0}\dot{A}^{0}+\pi_{i}\dot{A}^{i}+\pi_{\rho}\dot{\rho}+B_{2}\dot{N}^{2}+\overline{{\calP}}_{i}\dot{\calC}^{i}+\overline{{\calC}}_{2}\dot{\calP}^{2}~]-H_{total},
{\calL}_{\mathrm{GNJL}}=-{\frac{1}{4}}F_{\mu\nu}^{2}+\bar{\psi}(i\gamma^{\mu}D_{\mu}-m_{0})\psi+{\frac{G_{0}}{2}}[(\bar{\psi}\psi)^{2}+(\bar{\psi}i\gamma_{5}\psi)^{2}],
\left(\begin{matrix}{f_{\alpha}^{I}}\\{\bar{X}^{I}}\\\end{matrix}\right)(\bar{\calN}-\bar{\calN}^{T})_{IJ}\left(\begin{matrix}{f_{\beta}^{J}}&{\bar{X}^{J}}\\\end{matrix}\right)=\left(\begin{matrix}{\langleU_{\alpha},U_{\beta}\rangle}&{\langleU_{\alpha},\bar{V}\rangle}\\{\langle\bar{V},U_{\beta}\rangle}&{0}\\\end{matrix}\right)\.
Q_{2}^{13}=-Q_{2}^{24}=-{\frac{e^{i\theta}}{\sqrt{m}}}\left[{\frac{\partial}{\partialr}}+{\frac{1}{r}}i{\frac{\partial}{\partial\theta}}-{\frac{e^{2}}{c\kappa}}{\frac{\partialG}{\partialr}}\right]
{\calD}_{\mu}^{ab}=\delta^{ab}\partial_{\mu}-i{\calA}_{\mu}[\mathrm{ad}(T_{3})]^{ab}\,\quad[\mathrm{ad}(T_{3})]\equiv\left(\begin{array}{ccc}{0}&{-i}&{0}\\{i}&{0}&{0}\\{0}&{0}&{0}\\\end{array}\right)\.
y^{2}=x^{3}+\sum_{k=-4}^{4}\sum_{l=0}^{8}f_{kl}z^{\primel}z^{4+k}p^{8-l-2k}x+\sum_{k=-6}^{6}\sum_{l=0}^{12}g_{kl}z^{\primel}z^{6+k}p^{12-l-2k}
\left(1+\frac{1}{2}\frac{q_{2}^{2}}{q_{1}^{2}}\right)h_{1}\left(h_{1}^{\prime\prime}+\frac{4-p}{r}h_{1}^{\prime}\right)=\frac{1}{2}\frac{q_{2}^{2}}{q_{1}^{2}}h_{2}\left(h_{2}^{\prime\prime}+\frac{4-p}{r}h_{2}^{\prime}\right)\,,
{\mathcal{K}}_{F}(\hat{X_{F}})|_{X_{F}\to\infty}=\prod_{k=1}^{n_{f}}\left[X_{F}^{(k)}\right]^{2N}\frac{1}{2^{2(Nn_{f}+n_{f}^{2}-n_{f})}N^{2Nn_{f}+n_{f}^{2}}}
\frac{\pi}{2}^{k}\frac{\left[\frac{1}{2}\sum_{i=k+1}^{l}\omega_{i}^{4}-\sum_{i<j}\omega_{1}^{2}\omega_{j}^{2}\right]}{2^{l-k}\prod_{i=k+1}^{l}\omega_{i}}\Gamma(4-l)b^{2l-8}\equivV(b)
v_{1\epsilon\kappa}\left(r\right)=\sqrt{\frac{\epsilon+m_{0}}{2\epsilon}}\,2kr\,j_{l}\left(kr\right)\,,\quadv_{2\epsilon\kappa}\left(r\right)=\sqrt{\frac{\epsilon+m_{0}}{2\epsilon}}\,\frac{\kappa_{0}k}{\epsilon+m_{0}}\,2kr\,j_{\overline{l}}\left(kr\right)
H=-\frac{e}{2}\left({\calP}^{2}-8i\mu\psi^{5}F_{\alpha\beta}{\calP}^{\alpha}\psi^{\beta}+2igF_{\alpha\beta}\psi^{\alpha}\psi^{\beta}-M^{2}\right)-i\left({\calP}_{\alpha}\psi^{\alpha}-M\psi^{5}\right)\chi\,,
M_{ADM}^{2}\geq{\frac{1}{(4G_{4})^{2}}}e^{-\varphi_{\infty}}\left[\vec{Q}_{R}^{2}+\vec{P}_{R}^{2}+2\{\vec{Q}_{R}^{2}\vec{P}_{R}^{2}-(\vec{Q}_{R}^{T}\vec{P}_{R})^{2}\}^{\frac{1}{2}}\right].
\mathcal{L}_{\mathrm{bulk}}=i\,\bar{\psi}_{i}\gamma^{\mu}\partial_{\mu}\psi^{i}+\bar{\psi}_{2}\partial_{5}\psi^{1}-\bar{\psi}_{1}\partial_{5}\psi^{2}-\omega(\bar{\psi}_{1}\psi^{1}+\bar{\psi}_{2}\psi^{2})
\begin{array}{cc}{\displaystyle{\{D_{i\alpha},D_{j\beta}\}=-2i\bar{\calE}_{ij}\gamma_{\alpha\beta}^{A}\partial_{A}}~~~}&{~~~\displaystyle{\{\tilde{D}_{\tilde{\imath}}^{\alpha},\tilde{D}_{\tilde{\jmath}}^{\beta}\}=-2i\bar{\tilde{\calE}}_{\tilde{\imath}\tilde{\jmath}}\tilde{\gamma}^{A\alpha\beta}\partial_{A}}}\\\end{array}
{\bfH}_{SUSY}=[Q_{-},Q_{+}]_{+}=\left(\begin{array}{cc}{\mathrm{\bf~A}^{+}\mathrm{\bf~A}^{-}}&{0}\\{0}&{\mathrm{\bf~A}^{-}\mathrm{\bf~A}^{+}}\\\end{array}\right)_{4\mathrm{x}4}=\left(\begin{array}{cc}{{\bfH}_{-}={\bfH}}&{0}\\{0}&{{\bfH}_{+}}\\\end{array}\right),
\begin{array}{}{t]{lll}\left[Au,Av\right]}&{=}&{A\left(\left[Au,v\right]+\left[u,Av\right]\right)-\left[u,v\right],}\\{\left[Du,Dv\right]}&{=}&{D\left(\left[Du,v\right]+\left[u,Dv\right]\right)-\left[u,v\right],}\\{\left[Bu,Bv\right]}&{=}&{B\left(\left[Du,v\right]+\left[u,Dv\right]\right),}\\{\left[B^{*}u,B^{*}v\right]}&{=}&{B^{*}\left(\left[Au,v\right]+\left[u,Av\right]\right)}\\\end{array}
H_{R}(ND)=\sum_{j}\left(N_{F}^{R}(ND)-\frac{1}{24}\right)=\sum_{j}\left(\sum_{r=1/2}^{\infty}r({d_{-r}}^{-j}{d_{r}}_{j}+{d_{-r}}^{j}{d_{r}}_{-j})-\frac{1}{24}\right),
\Theta\left[\begin{array}{lll}{1/2}&{1/2}&{1/2}\\{1/2}&{1/2}&{0}\\\end{array}\right]\stackrel{D_{\alpha_{2}^{-1}\alpha_{3}}}{\longrightarrow}\Theta\left[\begin{array}{lll}{1/2}&{1/2}&{1/2}\\{1/2}&{0}&{1/2}\\\end{array}\right],
<<\prod_{i=1}^{n}\Phi_{8}(z_{i})\Phi_{8}(w_{i})>>_{\bfm}(\Omega_{g})=2^{8-n}A_{g}\left|\frac{\xi_{n}\left[\begin{array}{c}{{\vec{\alpha}}}\\{{\vec{\beta}}}\\\end{array}\right]}{\prod_{j=1}^{n}E^{1/4}(z_{j},w_{j})}\right|^{8}
\left.B_{\tau}=c_{\tau}\,\intd\mu_{I_{\beta}}\,(A)\,\left[1+z(\beta,\rho^{\tau})\,c_{\beta}\,(\rho^{\tau},A)\right]\,\prod_{\rho\in{\calE}_{\tau}}\,\left[1+z(\beta,\rho)\,c_{\beta}\,(A,\rho)\right]\.\right.
R_{itit}=\frac{-aa_{tt}}{(1+\frac{K}{4}\vec{x}\vec{x})^{2}},\;\;\;\;\;\;R_{ijij}=\frac{a^{2}(K+a_{t}^{2})}{(1+\frac{K}{4}\vec{x}\vec{x})^{4}};\;\;\;\;i\neqj
W_{k=0}^{\beta}=\mathrm{}-\frac12\int_{0}^{\infty}\!\frac{{\mathrm{d}}s}{s}\,{\mathrme}^{-sm^{2}}\Big({\mathrmTr}{K}^{\beta}(s)|_{k=0}-{\mathrmTr}K(s)|_{\phi=0}\Big)\Big|_{m^{2}=0}.
H_{W0}=-\intd^{3}x[(\underline{{\psi}}^{+}\overset{\wedge}{H}_{0}\cdot\underline{{\psi}}-\frac{1}{2}(\underline{{\overset{.}{A}}}_{\mu}\cdot\underline{{\overset{.}{A}}}_{\mu}+\partial_{j}\underline{{A}}_{\nu}\cdot\partial_{j}\underline{{A}}_{\nu})],\tag{1.2.8}
\begin{array}{c}{u_{k}=u_{0}+k(u_{0}-u_{-1})}\\{U_{k}(z)=U_{0}(z)+k(u_{0}-u_{-1})}\\{\bar{W}_{k}=U_{0}(\sigma^{+})+U_{0}(\sigma^{-})+(k+1)(u_{0}-u_{-1})-u_{0}}\\\end{array}
{\calL}_{f}=i(\alpha^{\prime}p^{+})\left\{\overline{{\Theta}}\Gamma_{+}\partial_{\tau}\Theta+\overline{{\Theta}}\Gamma_{11}\Gamma_{+}\partial_{\sigma}\Theta+(\alpha^{\prime}p^{+})\overline{{\Theta}}\Gamma_{+}\tilde{\Omega}\Gamma_{-}\Gamma_{+}\Theta\right\}.
Im\,I_{1_{\beta}}\approx\frac{-g^{2}}{8}\int\frac{d^{3}l}{(2\pi)^{2}E_{1}E_{2}}2\delta(E_{1}-E_{2})\Bigg(\frac{2}{e^{\betaE_{1}}-1}+\frac{2}{(e^{\betaE_{1}}-1)^{2}}\Bigg).
2\left(1+\frac{a_{1}}{\epsilon}+...\right)\gamma_{\phi}=\mu\frac{\partial\lambda}{\partial\mu}\frac{\partiala_{1}}{\partial\lambda}\frac{1}{\epsilon}+\mu\frac{\partial\alpha}{\partial\mu}\frac{\partiala_{1}}{\partial\alpha}\frac{1}{\epsilon}+...\;,
\hat{H}_{0}|n_{i}^{\pm}>=\hbarg\omega\left[\sum_{i}(n_{i}^{+}+n_{i}^{-})+d\right]|n_{i}^{\pm}>\\,\\\hat{\phi}|n_{i}^{\pm}>=-\hbar\sum_{i}(n_{i}^{+}-n_{i}^{-})\,|n_{i}^{\pm}>\\.
T^{\mu\nu}=\Sigma_{n}\frac{m_{n}}{\sqrt{g}}\int(1-\alpha\phi)U_{n}^{\mu}U_{n}^{\nu}\delta^{4}(x-x_{n}(\tau_{n}))d\tau_{n}+\frac{1}{\beta}\left[\nabla^{\mu}\phi\nabla^{\nu}\phi-\frac{1}{2}g^{\mu\nu}\left(\nabla_{\alpha}\phi\nabla^{\alpha}\phi-\Lambda^{2}e^{2\phi}\right)\right].
{\calO}_{B}^{\calG}(\widetilde{x})={\frac{\langle\widetilde{e_{\calQ}}|\widehat{\calO}^{\calG}|\widetilde{e_{\calQ}}\rangle_{\widetilde{\calL}}}{[||\widetilde{e_{\calQ}}||^{2}]_{\widetilde{\calL}}}}\equiv{\frac{\langlee_{\calQ}|\widehat{\calO}|e_{\calQ}\rangle_{\calL}^{\calG}}{[||e_{\calQ}||^{2}]_{\calL}^{\calG}}},
\begin{array}{l}{m_{A}^{e}(x)\cdot\Gamma={\hat{A}}^{*\mu}(x_{+})A_{\mu}(x_{-})\cdot\Gamma\,,}\\{n_{A}^{e}(x)\cdot\Gamma={\calB}_{\Gamma}\left[m_{A}^{e}(x)\cdot\Gamma\right]\,,}\\\end{array}
\delta_{{\bfl}\left\{n_{i}\right\}}=\prod_{i=1}^{g}\sum_{j=-\infty}^{\infty}\delta\left(\kappau_{i}+2\pi\left(\Omega-\bar{\Omega}\right)_{ik}^{-1}\sum_{\gamma}\left(\varphi_{k}\left(z_{\gamma}\right)-\bar{\varphi}_{k}\left(z_{\gamma}\right)\right)-\theta_{1i}-2\pi\left(j_{i}p+n_{i}\right)-2\pi\kappal_{i}\right)
{\calS}={\frac{1}{2}}\mathrm{tr}\,[\alpha\alpha^{t}]={\frac{1}{2}}\mathrm{tr}[a^{-1}mbb^{t}m^{t}(a^{-1})^{t}]={\frac{1}{2}}m_{n}m_{k}\mathrm{tr}\,[\alpha_{n}\alpha_{k}^{t}]=M^{t}GM,
S_{c}=\intd^{2}x\left\{-\frac{1}{2}{\partial}_{\mu}{\phi}(x){\partial}^{\mu}{\phi}(x)+\frac{1}{2}{\lambda}_{{\mu}{\nu}}(x)[{\partial}^{\mu}{\phi}(x)-{\epsilon}^{{\mu}{\sigma}}{\partial}_{\sigma}{\phi}(x)][{\partial}^{\nu}{\phi}(x)-{\epsilon}^{{\nu}{\rho}}{\partial}_{\rho}{\phi}(x)]\right\},
\mathbf{A}=\left(\begin{array}{cc}{A_{2}}&{A_{1}}\\{-A_{1}^{\dagger}}&{A_{2}^{\dagger}}\\{I}&{J}\\\end{array}\right),\quad\mathbf{B}=\left(\begin{array}{cc}{\textbf{1}_{k}}&{0}\\{0}&{\textbf{1}_{k}}\\{0}&{0}\\\end{array}\right),\quad\mathbf{B}^{\prime}=\left(\begin{array}{cc}{B_{2}}&{B_{1}}\\{-B_{1}^{\dagger}}&{B_{2}^{\dagger}}\\{K}&{L}\\\end{array}\right)\,.
B_{mn}^{(+)}=-{\frac{i^{k}}{2\pi}}\sqrt{{\frac{n}{m}}}(m+n)^{-k}e^{inT^{+}(0)}\int_{0}^{2\pi}e^{i(m+n)y}{\frac{d^{k+1}\varphi_{t}}{dy^{k+1}}}\,dy\,,
T_{\lambda\mu\nu}^{AVV}=-\int\frac{d^{4}k}{\left(2\pi\right)^{4}}tr\left\{\gamma_{\mu}\left[(\not{k}+\not{k}_{1})-m\right]^{-1}\gamma_{\nu}\left[(\not{k}+\not{k}_{2})-m\right]^{-1}\gamma_{\lambda}\gamma_{5}\left[(\not{k}+\not{k}_{3})-m\right]^{-1}\right\},
\begin{array}{cl}{L_{\mu\nu}=-ig\frac{\tau_{i}}{2}W_{\mu\nu}^{i}+i\frac{g^{\prime}}{2}B_{\mu\nu};~~~R_{\mu\nu}=ig^{\prime}B_{\mu\nu}}\\{D_{\mu}\phi=({\partial}_{\mu}-ig\frac{{\tau}_{i}}{2}W_{\mu}^{i}-i\frac{g^{\prime}}{2}B_{\mu})\phi.}\\\end{array}
L={\frac{1}{2}}g_{ij}(z)\left[\dot{z}^{i}\dot{z}^{j}+i\bar{\psi}^{i}\gamma^{0}D_{t}\psi^{j}\right]+{\frac{1}{6}}R_{ijkl}(\bar{\psi}^{i}\psi^{j})(\bar{\psi}^{k}\psi^{l})
\kappa_{ij}={\frac{g^{2}}{4}}\varepsilon_{k}\varepsilon_{k}^{\prime}\Pi^{\mu\nu}Tr\left[\bar{S}_{i\mu}^{(0)}(\hat{k}_{2}+m)(\hat{k^{\prime}}_{2}+m)S_{j\nu}^{(0)}(\hat{k^{\prime}}_{1}-m)(\hat{k}_{1}-m)\right].
\begin{array}{c}{\int_{0}^{\tau}dt\cdot\omega_{\mu}(t,z,\tau)=-{\frac{\tilde{q}^{\mu-1}}{1-\tilde{q}^{\mu}}}\int_{0}^{1}dt\cdot\omega_{\mu}(t,z,\tau)+{\frac{\tilde{q}^{\mu-2}}{1-\tilde{q}^{\mu-1}}}\int_{0}^{1}dt\cdot\omega_{\mu-2}(t,z,\tau)+}\\{{\frac{\tilde{q}^{\mu-2}(1-\tilde{q})}{(1-\tilde{q}^{\mu-1})(1-\tilde{q}^{\mu})}}\int_{0}^{1}dt\cdot\omega_{-\mu}(t,z,\tau)}\\\end{array}
\left(\begin{matrix}{1}&{w_{1}}&{w_{2}}&{\cdots}&{\cdots}&{w_{n}}\\{0}&{1}&{0}&{\cdots}&{\cdots}&{0}\\{0}&{0}&{1}&{0}&{\cdots}&{0}\\{}&{}&{}&{}&{}&{}\\{\vdots}&{\vdots}&{\vdots}&{\vdots}&{\vdots}&{\vdots}\\{}&{}&{}&{}&{}&{}\\{0}&{0}&{\cdots}&{0}&{1}&{0}\\{0}&{0}&{\cdots}&{\cdots}&{0}&{1}\\\end{matrix}\right)\inCP^{n}.
E(\theta)=\frac{\textrm{sn}\left(\frac{\textbf{K}}{i\pi}2\theta\right)}{\textrm{dn}\left(\frac{\textbf{K}}{i\pi}2\theta\right)}\cdot\frac{\textrm{sn}\left(\frac{\textbf{K}}{i\pi}\theta\right)}{\textrm{cn}\left(\frac{\textbf{K}}{i\pi}\theta\right)}\,\,\,.
V_{II}(\varphi_{0})=\sum_{s=1}^{\infty}\frac{(-1)^{s+1}}{2s}\biggl(\frac{1}{2}\lambda\varphi_{0}^{2}\biggr)^{s}\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{(\omega^{2}+\vec{q}~^{2}+m^{2})^{s}}.
dC^{(n)}=F^{(n+1)}=(-1)^{\frac{n(n+1)}{2}}\hat{\star}F^{(9-n)}=(-1)^{\frac{n(n+1)}{2}}\hat{\star}dC^{(8-n)}=(-1)^{\frac{n(n-1)}{2}}d\hat{\star}C^{(10-n)},
S_{0}=\intd^{D}x\left[\pi^{i}\dot{A}_{i}-{\frac{1}{2}}{\pi^{i}}^{2}-{\frac{1}{4}}F_{ij}^{2}+i\bar{\psi}\gamma^{\mu}D_{\mu}\psi+\lambda^{1}(J^{0}+\partial_{i}\pi^{i})+\lambda^{2}(\partial_{i}A^{i})\right]\,,
O_{\beta\mu_{1}\ldots\mu_{n}}=O_{\beta\mu_{1}\ldots\mu_{n}}^{\mathrm{tw2}}+O_{\beta\mu_{1}\ldots\mu_{n}}^{\mathrm{tw3}}+O_{\beta\mu_{1}\ldots\mu_{n}}^{\mathrm{tw4}}+\ldots\;,
I_{c}=\left[-i\overline{{z}}\frac{dz}{ds}+\epsilon\frac{d\left\langlex^{\mu}\right\rangle_{c}}{ds}\left\langlep_{\mu}\right\rangle_{c}-\overline{{z}}\gamma^{\mu}z\left\langle\pi_{\mu}\right\rangle_{c}\right]ds,
U=|\phi|^{2}\left\{\sigma_{1}\beta_{11}(|\phi|^{2}-c^{2})+\sigma_{2}\beta_{12}(|\chi|^{2}-{c^{\prime}}^{2})\right\}^{2}+|\chi|^{2}\left\{\sigma_{1}\beta_{21}(|\phi|^{2}-c^{2})+\sigma_{2}\beta_{22}(|\chi|^{2}-{c^{\prime}}^{2})\right\}^{2},
{\calY}_{\ell}^{m}(\stackrel{\rightharpoonup}{x_{i}}-\stackrel{\rightharpoonup}{x_{\ell}})=r^{\ell}y_{\ell}^{m}\left(\frac{\stackrel{\rightharpoonup}{x_{i}}-\stackrel{\rightharpoonup}{x_{\ell}}}{r}\right)\\\.
\epsilon^{\mu}(x,\tau)=i\sum_{s=\mathrm{polarizations}}\int\frac{d^{4}k}{2\kappa}\kappa\left[\varepsilon_{s}^{\mu}a(k,s)e^{i(k\cdotx+\sigma\kappa\tau)}-\varepsilon_{s}^{\mu*}a^{*}(k,s)e^{-i(k\cdotx+\sigma\kappa\tau)}\right].
\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)^{*}\,\phi_{_R}^{\ast}(p^{\mu})}\\\end{array}\right)\,\,\quad.
\begin{array}{l}{\partial_{-}\partial_{0}=q^{-1}\partial_{0}\partial_{-}~,~~~~~\partial_{0}\partial_{+}=q^{-1}\partial_{+}\partial_{0}~,}\\{\partial_{-}\partial_{+}-\partial_{+}\partial_{-}=(q^{1/2}-q^{-1/2})\partial_{0}\partial_{0}~.}\\\end{array}
\frac32\mathrm{Cl}_{2}(\theta_{7})-\frac32\mathrm{Cl}_{2}(2\theta_{7})+\frac12\mathrm{Cl}_{2}(3\theta_{7})=\frac74\left\{\mathrm{Cl}_{2}\left(\frac{2\pi}{7}\right)+\mathrm{Cl}_{2}\left(\frac{4\pi}{7}\right)-\mathrm{Cl}_{2}\left(\frac{6\pi}{7}\right)\right\}
i\left(\begin{matrix}{\langleU_{\alpha},\bar{U}_{\bar{\beta}}\rangle}&{\langleU_{\alpha},V\rangle}\\{\langle\bar{V},\bar{U}_{\bar{\beta}}\rangle}&{\langle\bar{V},V\rangle}\\\end{matrix}\right)=\left(\begin{matrix}{f_{\alpha}^{I}}\\{\bar{X}^{I}}\\\end{matrix}\right)i({\calN}-{\calN}^{\dagger})_{IJ}\left(\begin{matrix}{\bar{f}_{\bar{\beta}}^{J}}&{X^{J}}\\\end{matrix}\right)
F^{0}(N,g_{j})=\sum_{n=0}^{\infty}a_{j_{1}\cdotsj_{n}}\deltag_{j_{1}}\cdots\deltag_{j_{n}}+\sum_{i}\left(V_{ij}\deltag_{j}\right)^{2-\gamma_{0}^{(i)}}\sum_{n=0}^{\infty}b_{(i)j_{1}\cdotsj_{n}}\deltag_{j_{1}}\cdots\deltag_{j_{n}},
\begin{array}{c}{\deltaX^{a}=\alpha^{\dagger}\Gamma^{a}(i_{*}\xi)^{b}\Gamma_{b}S^{(0)}=}\\{={\frac{1}{2}}\left((i_{*}\xi)^{a}(\alpha^{\dagger}S^{(0)})+\sum_{n}(I_{n}i_{*}\xi)^{a}(\alpha^{\dagger}\hat{I}_{n}S^{(0)})\right)}\\\end{array}
\begin{array}{rcl}{x^{+}(\tau)}&{=}&{e^{-4n\alpha}x^{+}+ve^{\eta}\tau\,,}\\{x^{-}(\tau)}&{=}&{e^{4n\alpha}x^{-}-ve^{-\eta}\tau\,,}\\{y(\tau)}&{=}&{y+4nY_{0}\pm\tau\,.}\\\end{array}
A_{i}^{a}=\frac{1-f_{A}(r)}{gr}\epsilon_{aij}\hat{x}_{j}+\frac{f_{B}(r)}{gr}(\delta_{ia}-\hat{x}_{i}\hat{x}_{a})+\frac{f_{C}(r)}{gr}\hat{x}_{i}\hat{x}_{a}
\gamma_{k}=\left(\begin{matrix}{0}&{-i\sigma_{k}}\\{i\sigma_{k}}&{0}\\\end{matrix}\right)\qquad\gamma_{4}=\left(\begin{matrix}{0}&{I}\\{I}&{0}\\\end{matrix}\right)\qquad\gamma_{5}=\left(\begin{matrix}{I}&{0}\\{0}&{I}\\\end{matrix}\right)
\widetilde{E}_{\\mu}^{\nu}={\frac{1}{d}}\left(\begin{matrix}{d}&{0}&{0}\\{e^{\underline{{2}}}h_{\2}^{\underline{{1}}}-e^{\underline{{1}}}h_{\2}^{\underline{{2}}},}&{h_{\2}^{\underline{{2}}},}&{-h_{\2}^{\underline{{1}}},}\\{e^{\underline{{1}}}h_{\1}^{\underline{{1}}}-e^{\underline{{2}}}h_{\1}^{\underline{{1}}},}&{-h_{\1}^{\underline{{2}}},}&{h_{\1}^{\underline{{1}}}}\\\end{matrix}\right),
\begin{array}{lll}{A_{~\nu}^{\mu}=|\lambda|\delta_{~\nu}^{\mu}~~}&{~{\displaystyle\frac{\partial^{-}\theta^{\prime}{}^{\alpha}}{\partial\theta^{\beta}}=\lambda^{\frac{1}{2}}\delta_{\beta}^{\alpha}}~~}&{~~{\displaystyle\frac{\partial^{+}\bar{\theta}^{\prime}{}^{\dot{\alpha}}}{\partial\bar{\theta}^{\dot{\beta}}}=\bar{\lambda}^{\frac{1}{2}}\delta_{\dot{\beta}}^{\dot{\alpha}}}}\\\end{array}
Z(\kappa_{d-2},\kappa_{d})=\sum_{N_{d}}\mathrm{e}^{-i\kappa_{d}N_{d}}\sum_{N_{d-2}}\mathrm{e}^{i\kappa_{d-2}N_{d-2}}\sum_{T|_{N_{d},N_{d-2}}}\frac{1}{C_{T}},
V|_{j^{n/3}(J-1728)^{m/2}}\stackrel{T\rightarrow\rho}{\rightarrow}\infty,\;\;n\leq2,\,\;V|_{j^{n/3}(J-1728)^{m/2}}\stackrel{T\rightarrow1}{\rightarrow}\infty,\;\;m\leq2.
\sum_{\pi(1,2,...n)}\mathrm{Tr}\Big[\tau^{(I_{0})}\tau^{(I_{\pi(1)})}.....\tau^{(I_{\pi(n)})}\Big]\frac{1}{\tilde{x}_{0}-\tilde{x}_{\pi(1)}}\frac{1}{\tilde{x}_{\pi(1)}-\tilde{x}_{\pi(2)}}.....\frac{1}{\tilde{x}_{\pi(n)}-\tilde{x}_{0}}\;,
i\gamma\int_{ex}\frac{f_{i}^{2}(\mathbf{x})}{2\tilde{u}}\left\{(\Theta\tilde{u}-3a\tilde{s})\gamma\cdotu+(3a\tilde{u}-\Theta\tilde{s})\gamma\cdots\right\}\psi_{1}(\mathbf{x})d\Omega(\mathbf{x}),
\mathbb{J}^{1}=\left(\begin{matrix}{0}&{1}&{0}&{0}\\{-1}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}\\{0}&{0}&{-1}&{0}\\\end{matrix}\right)\,,\qquad\mathbb{J}^{2}=\left(\begin{matrix}{0}&{0}&{-1}&{0}\\{0}&{0}&{0}&{1}\\{1}&{0}&{0}&{0}\\{0}&{-1}&{0}&{0}\\\end{matrix}\right)\,,\qquad\mathbb{J}^{3}=\left(\begin{matrix}{0}&{0}&{0}&{1}\\{0}&{0}&{1}&{0}\\{0}&{-1}&{0}&{0}\\{-1}&{0}&{0}&{0}\\\end{matrix}\right),
F(r)=8hr^{6}\frac{(r_{1}^{2}-r_{5}^{2})[r_{1}^{2}r_{5}^{2}+r_{1}^{2}r_{K}^{2}+r_{5}^{2}r_{K}^{2}+(r_{1}^{2}+r_{5}^{2}+r_{K}^{2})r^{2}]}{(r_{1}^{2}r_{5}^{2}+r_{1}^{2}r_{K}^{2}+r_{5}^{2}r_{K}^{2}+2(r_{1}^{2}+r_{5}^{2}+r_{K}^{2})r^{2}+3r^{4})^{2}}.
\int\frac{dw}{2\pi}e^{-iw(t^{\prime}-t)}\frac{1}{2\bar{k}}\left(e^{i\bar{k}|x^{\prime}-x|}+K_{a}(w)e^{-i\bar{k}(x^{\prime}+x)}\right),~~~~\bar{k}=\sqrt{w^{2}-m_{a}^{2}}
\tau=\chi+ie^{-\phi}~~~,~~~{\calM}_{ij}=\frac{1}{\mathrm{Im}~\tau}\left(\begin{matrix}{|\tau|^{2}}&{-\mathrm{Re}~\tau}\\{-\mathrm{Re}~\tau}&{1}\\\end{matrix}\right)~~~,~~~F_{3}^{i}=\left(\begin{matrix}{H_{3}}\\{F_{3}}\\\end{matrix}\right)~,
S\left[\phi\right]=\int\mathrm{d^{d}}x\left[{\frac{1}{2}}\phi\left(-\nabla^{2}+{\frac{\nabla^{4}}{\Lambda^{2}}}+m^{2}\right)\phi+{\frac{\lambda}{4}}\phi^{4}+{\frac{1}{2}}\deltam^{2}(\Lambda)\\phi^{2}\right].
(\partial_{++}x^{m}-i\partial_{++}\bar{\theta}\gamma^{m}\theta)+{\frac{1}{2}}{\frac{\bar{\lambda}_{-A}\gamma^{m}\lambda_{-A}}{(\bar{\lambda}_{-B}\gamma^{n}\lambda_{-B})(\bar{\lambda}_{+}\gamma_{n}\lambda_{+})}}\psi_{+}^{I}\partial_{++}\psi_{+}^{I}=\bar{\lambda}_{+}\gamma^{m}\lambda_{+}.
Z^{u}=f_{T}\left(\begin{matrix}{1}&{0.345}&{0}\\{0.345}&{0.132}&{0.639d^{2}}\\{0}&{0.639d^{2}}&{0.333d^{2}}\\\end{matrix}\right);\,\,\,Z^{d}=f_{T}\left(\begin{matrix}{1}&{0.821}&{0}\\{0.821}&{0.887}&{0}\\{0}&{0}&{0.276}\\\end{matrix}\right).
(B\,\,E\,\,\xi\,\,\eta)\left(\begin{array}{cccc}{0}&{0}&{\frac{1}{2}}&{0}\\{0}&{0}&{0}&{\frac{1}{2}}\\{\frac{1}{2}}&{0}&{0}&{0}\\{0}&{\frac{1}{2}}&{0}&{0}\\\end{array}\right)\left(\begin{array}{c}{B}\\{E}\\{\xi}\\{\eta}\\\end{array}\right),
\hat{H}=\frac{1}{2}\frac{1}{\sqrt{\hat{r}}}\hat{p}_{r}\,\hat{r}\,\hat{p}_{r}\,\frac{1}{\sqrt{\hat{r}}}\,+\,\frac{1}{2}\frac{1}{\hat{r}^{2}}\hat{p}_{\theta}^{2}\,+\,\frac{1}{2}\hat{p}_{z}^{2}\,+\,\frac{1}{2}\omega^{2}\hat{r}^{2}\\\,
L_{q}^{new}=eg\,\int\;d^{3}xd^{3}x^{\prime}\,\bar{q}_{\alpha\alpha^{\prime}}\gamma_{\alpha\beta}^{\mu}A_{\mu}\gamma_{\alpha^{\prime}\beta^{\prime}}^{\mu^{\prime}}A_{\mu^{\prime}}^{a}\frac{\lambda^{a}}{2}q_{\beta\beta^{\prime}}.
S_{WZW}[x]=\frac{1}{3!}\int_{Y}d^{3}y\epsilon^{\mu\nu\lambda}H_{ijk}\partial_{\mu}x^{i}\partial_{\nu}x^{j}\partial_{\lambda}x^{k}=\int_{W=\partialY}\!\!\!\!\!\!\!\!\!d^{2}\xi\partialx^{i}b_{ij}\bar{\partial}x^{j},
\frac{\partial^{2}\varphi_{E}(\tau)}{\partial\tau^{2}}+3\Bigl(\frac{\partiala(\tau)/\partial\tau}{a(\tau)}\Bigr)\frac{\partial\varphi_{E}(\tau)}{\partial\tau}-\langle0(\tau)\vert\frac{\delta^{2}V(\hat{\phi})}{\delta\hat{\phi}^{2}}\vert0(\tau)\rangle\varphi_{E}(\tau)=0.
f_{ij}^{(1)}(x,y)=m\left(\begin{array}{cccc}{0}&{0}&{0}&{-m}\\{0}&{0}&{1}&{-\partial_{x}^{2}}\\{0}&{-1}&{0}&{\partial_{x}^{1}}\\{m}&{-\partial_{x}^{2}}&{\partial_{x}^{1}}&{0}\\\end{array}\right)\delta^{2}(x-y).
+{\frac{\overline{{\theta}}\theta}{2}}(2i\varepsilon^{jkl}\tau^{j}Z^{k}\chi^{l}-\partial\!\!\!/\chi\cdot\tau-2\varepsilon^{jkl}\tau^{j}{J^{k}}\!\!\!\!\!\!/\\chi^{l}+i\chi\cdot\tau~\overline{{\chi}}\cdot\chi)~,
\epsilon^{\sigma\beta\gamma}\partial_{\beta}(\hat{f}-m\hat{A})_{\gamma}(g_{\sigma}^{\alpha}+\frac{\delta\hat{c}_{\sigma}}{\delta\hat{A}_{\alpha}})=m\epsilon^{\sigma\beta\gamma}\partial_{\beta}\hat{c}_{\gamma}(g_{\sigma}^{\alpha}+\frac{\delta\hat{c}_{\sigma}}{\delta\hat{A}_{\alpha}}),
G_{1,1,{\frac{1}{3}}}^{(10)}\,\left|\,{p\;\,q\;\;i\atopp\;p\;(p+2q)/3}\,\right\rangle=\sqrt{\frac{(p+1)\,(p+q+2)}{p+2}}\;\sum_{i^{\prime}}\;\left|\,{p+1\;\,q\;\;i^{\prime}\atopp\;p\;(p+2q-2)/3}\,\right\rangle\;A_{i^{\prime}\;,i}\;.
y^{\delta}\tilde{\Gamma}^{\alpha}{}_{\beta\delta}\tilde{B}^{\beta}|_{1}=y^{\delta}(\Gamma^{\alpha}{}_{\beta\delta}+\Gamma^{\alpha}{}_{\beta\delta,\sigma}x^{\sigma}/2)[B^{\beta}+(B^{\beta}{}_{,\sigma}+\Gamma^{\beta}{}_{\sigma\tau}B^{\tau})d^{\sigma}/2]
+\frac{1}{6}C^{\alpha}U_{\alpha\beta}^{\gamma}\bar{\calP}_{\gamma}C^{\beta}(-1)^{\varepsilon_{\beta}+\varepsilon_{\gamma}}+\frac{1}{6}\bar{\calP}_{\gamma}U_{\alpha\beta}^{\gamma}C^{\beta}C^{\alpha}(-1)^{\varepsilon_{\beta}+\varepsilon_{\gamma}+(\varepsilon_{\alpha}+\varepsilon_{\beta})(\varepsilon_{\gamma}+1)}.
\begin{array}{rclcrcl}{\widehat{\mathcal{H}}_{\pm\vert\underline{{a_{1}a_{2}a_{3}}}}}&{=}&{\epsilon_{\alpha\beta}\Lambda_{\pm}^{\alpha}\,\mathcal{H}_{\underline{{a_{1}a_{2}a_{3}}}}}&{;}&{\widehat{\mathbf{H}}_{A\vert\underline{{a_{1}a_{2}a_{3}}}}}&{=}&{\epsilon_{\Lambda\Sigma}\,\mathbb{L}_{\phantom{\Lambda}A}^{\Lambda}\,\mathbf{H}_{\underline{{a_{1}a_{2}a_{3}}}}^{\Sigma}\}\\\end{array}
\langle\zeta^{\prime}|\overline{{\psi}}|\zeta^{\prime\prime}\rangle=\overline{{\psi}}^{\prime\prime}\langle\zeta^{\prime}|\zeta^{\prime\prime}\rangle\,,\quad\langle\zeta^{\prime}|\eta^{\mu}|\zeta^{\prime\prime}\rangle=\eta^{\prime\prime}{}^{\mu}\langle\zeta^{\prime}|\zeta^{\prime\prime}\rangle\,.
f_{\alpha\beta}^{(0)}(x,y)=\left(\begin{array}{ccccc}{O}&{-I}&{\vec{0}}&{\vec{0}}&{\vec{0}}\\{I}&{O}&{\vec{0}}&{\vec{0}}&{\vec{0}}\\{\vec{0}^{T}}&{\vec{0}^{T}}&{0}&{-1}&{-c_{2}}\\{\vec{0}^{T}}&{\vec{0}^{T}}&{1}&{0}&{0}\\{\vec{0}^{T}}&{\vec{0}^{T}}&{c_{2}}&{0}&{0}\\\end{array}\right)\delta(x-y)
|V\rangle=(\Pi_{Z})_{-1+\eta}|l\rangle\Psi^{(l-\frac{1}{2})}(\boldsymbolp,\boldsymbol\theta)+(d_{--})_{-1+\eta}|l\rangle\Phi^{(l-1)}(\boldsymbolp,\boldsymbol\theta)+\theta_{-1+\eta}^{++}|l\rangle\Xi^{(l-1)}(\boldsymbolp,\boldsymbol\theta).
A={\frac{1}{2}}\kappa\\xi\(\bar{\psi}^{1}\psi^{1}+\bar{\psi}^{2}\psi^{2})+{\frac{i}{2}}\kappa\\theta\(\bar{\psi}^{1}\gamma_{5}\psi^{1}-\bar{\psi}^{2}\gamma_{5}\psi^{2})+i\kappa\\varphi\\bar{\psi}^{1}\gamma_{5}\psi^{2}+...\,\,
H_{HP}^{(1)}=\frac{g^{2}\sqrt{N}}{4\pi}\int_{0}^{\infty}\frac{dR}{\sqrt{R}}\int_{0}^{1}dz\sum_{nml}\left(K_{nml}(z)-\tilde{K}_{nml}(z)\right)\left(b_{R}^{(n)\dagger}b_{Rz}^{(m)}b_{R(1-z)}^{(l)}+\mathrm{h.c.}\right),
=\frac12\int\mathrm{d}^{D}p\int\frac{\mathrm{d}^{D}q}{\left(2\pi\right)^{D}}\int_{0}^{1}\mathrm{d}u\frac{\tilde{F}\left(p\right)\tilde{F}\left(-p\right)}{\left[\left(1-u\right)\left(p+q\right)^{2}+uq^{2}\right]^{2}}.
\hat{I}[a_{cl}+\deltaa]\approx\hat{I}_{cl}+\frac{1}{2}\int_{0}^{T}dt\deltaa(t)\left[-2\Lambdaa_{cl}-2\ddot{a}_{cl}-2\dot{a}_{cl}\frac{d}{dt}-2a_{cl}\frac{d^{2}}{dt^{2}}\right]\deltaa(t)
\sqrt{2}\left[a\left(\frac{1+s^{2}}{2}\right)+a^{\dagger}\left(\frac{1-s^{2}}{2}\right)\right]\psi_{ss}(s^{2},x_{0}+is^{2}p_{0})=[x_{0}+is^{2}p_{0}]\psi_{ss}(s^{2},x_{0}+is^{2}p_{0}).
\bar{D}_{\dot{\gamma}}\bar{D}_{\dot{\alpha}}\Phi^{i*}=J^{\bar{\imath}}\!_{j\dot{\alpha}}\!^{\beta}{}_{,\bar{k}}\bar{D}_{\dot{\gamma}}\bar{\Phi}^{k}D_{\beta}\Phi^{j}+2iJ^{\bar{\imath}}\!_{j\dot{\alpha}}\!^{\beta}\delta_{\beta\dot{\gamma}}\partial_{t}\Phi^{j}\.
(Z,Z)\equiv\frac{1}{2}tr(\bar{Z}Z)=z^{1}z^{3}-q^{2}z^{2}z^{4}\qquad(\,\bar{Z}=\left(\begin{array}{cc}{z^{3}}&{-q^{2}z^{4}}\\{-q^{2}z^{2}}&{z^{1}}\\\end{array}\right)\,)\;.
\frac{d^{2}w}{dt^{2}}=\biggl(\frac{1}{2w}+\frac{1}{w-1}\biggr)\biggl(\frac{dw}{dt}\biggr)^{2}-\frac{1}{t}\frac{dw}{dt}-\frac{1}{2}\frac{w(w+1)}{w-1}.
E_{\mathrm{gap}}\approx\frac{(2n+1)\pi}{3^{(1/4)}8\sqrt{\eta}}\left(1-\frac{\eta}{2\sqrt{3}}\right)\,\qquad\Gamma_{\mathrm{gap}}\approx\frac{64\sqrt{3}\eta}{\pi^{3}(2n+1)^{3}}\,\qquadn=0,1,2...
\psi_{L}=(1-\gamma_{5})/2\left(\begin{array}{c}{\psi_{1}}\\{\psi_{2}}\\{\psi_{3}}\\{\psi_{4}}\\{.}\\\end{array}\right),\\\psi_{R}=(1+\gamma_{5})/2\left(\begin{array}{c}{\psi_{4}}\\{\psi_{5}}\\{\psi_{6}}\\{.}\\{.}\\\end{array}\right)
c_{n}=\frac{(-1)^{n(n+1)/2}}{x^{n(n-1)/2}(x^{2};x^{2})_{\infty}^{n(n-1)/2}}\frac{(x^{2};x^{4},x^{4})_{\infty}^{2n}}{(x^{8};x^{4},x^{4})_{\infty}^{2n}}.
[W_{\alpha\beta}(x)-\frac{1}{2}g_{\alpha\beta}W_{\gamma}^{\gamma}(x)]^{;\alpha}=\frac{1}{4}(\BoxW(x))_{;\beta}-\frac{1}{2}m^{2}W_{;\beta}(x)+2v_{1}(x)_{;\beta}+\frac{1}{2}R_{\alpha\beta}W^{;\alpha}(x)-\frac{1}{2}\xiRW_{;\beta}(x)