harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
D_{\mu\nu}^{Weyl}(x)=i\left<0\left|TA_{\mu}(x)A_{\nu}(0)\right|0\right>=\int\frac{d^{4}k}{(2\pi)^{4}}\frac{e^{-ik\cdotx}}{2k_{+}k_{-}-k_{\perp}^{2}+i\epsilon}\left[g_{\mu\nu}-\frac{n_{\mu}^{Weyl}k_{\nu}+n_{\mu}^{Weyl}}{k_{+}+i\epsilon\mathrm{sgn}{k_{-}}}\right],
R(u,\lambda)=\sum_{i=1}^{n}E_{i,i}\otimesE_{i,i}+\sum_{i\neqj}\alpha(u,\lambda_{ij})E_{i,i}\otimesE_{j,j}+\sum_{i\neqj}\beta(u,\lambda_{ij})E_{i,j}\otimesE_{j,i},
\{D_{\alpha\beta},{\bar{D}}_{\dot{\alpha}\dot{\beta}}\}T|={\calA}D_{\alpha\beta}{\bar{D}}_{\dot{\alpha}\dot{\beta}}{\bar{\calA}}|+c.c.=16{\calA}\partial_{(\alpha(\dot{\alpha}}\partial_{\beta)\dot{\beta})}{\bar{\calA}}|+c.c.
\begin{array}{lcr}{P(W_{2})=(y^{2}+x^{3}+z^{6}+\muxyz)+}\\{w(a_{0}z^{n+3}+a_{1}yz^{n}+a_{n-1}z^{{\frac{7}{2}}}x^{{\frac{(n-6)}{2}}}y^{{\frac{1}{2}}})+w^{2}\sum_{i=0}^{n-4}a_{i+1}z^{2(n-i)}x^{i}.}\\\end{array}
m^{2}(l)=\frac{4}{e_{0}^{2}}\left(1-\frac{e_{0}}{2}\right)[l-l_{0}]^{-2}\,,\quadv=-\,\frac{4}{e_{0}^{2}}\left(1+\frac{e_{0}}{2}\right)[l-l_{0}]^{-2}\,,\quadl_{0}=z_{0}+\bar{z}_{0}\,.
\frac{1}{\sqrt{\zeta_{3}+b}}\approx\left[\frac{\upsilon\left[f_{E}(b-h_{E})\overline{{E^{\prime}}}E^{\prime}+f_{U}(b-h_{U})\overline{{U^{\prime}}}U^{\prime}+f_{D}(b-h_{D})\overline{{D^{\prime}}}D^{\prime}\right]}{4\sqrt{2}(b-k)\left[b\left(sM^{4}e^{-2\alpha\phi/M_{p}}+V_{1}\right)-V_{2}\right]}\right]^{1/3}
\pi(d\alpha)=\left(\begin{matrix}{\gamma^{\mu\nu}\partial_{\mu}\alpha_{1\nu}}&{-g^{\mu\nu}\gamma^{5}V_{\nu}(\partial_{\mu}\alpha_{5}+\alpha_{1\mu}-\alpha_{2\mu})}\\{g^{\mu\nu}\gamma^{5}V_{\nu}(\partial_{\mu}\tilde{\alpha}_{5}+\alpha_{1\mu}-\alpha_{2\mu})}&{\gamma^{\mu\nu}\partial_{\mu}\alpha_{2\nu}}\\\end{matrix}\right)
I=\int_{0}^{\pi}d\theta_{k}\int_{0}^{\theta_{k}}d^{k-1}\theta\int_{\theta_{k}}^{\pi}d^{J-k}\thetaf(\theta_{k})=\int_{0}^{\pi}d\theta_{k}\frac{\theta_{k}^{k-1}}{(k-1)!}\frac{(\pi-\theta_{k})^{J-k}}{(J-k)!}f(\theta_{k}).
H=dB-C\vert_{\Sigma_{6}}+2{\frac{T_{3}}{T_{6}}}\Big(\epsilon_{1}\delta^{1}(\Sigma_{2}^{1},\Sigma_{6})+\epsilon_{2}\delta^{1}(\Sigma_{2}^{2},\Sigma_{6})\Big)+2\kappa_{11}^{2}T_{6}\delta^{1}(\Sigma_{7},Q)\vert_{\Sigma_{6}}),
T_{\mu\nu}^{\mathrm{ren}}=T_{\mu\nu}^{\mathrm{reg}}+g_{\mu\nu}\left(\delta\Lambda+\frac{1}{2}\deltam^{2}\phi^{2}(x)+\frac{\delta\lambda}{4!}\phi^{4}(x)\right)+A(g_{\mu\nu}\partial_{\alpha}\partial^{\alpha}-\partial_{\mu}\partial_{\nu})\phi^{2}(x)\;.
x_{m}^{j}=y_{2-m-2j}^{j}-\frac{1}{2\pii}\oint_{c}\frac{\partialH^{\prime}}{\partial\alpha^{j}}\frac{d\zeta}{\zeta^{m+1}}-\frac{1}{2\pii}\oint_{0}\frac{\partialH}{\partial\alpha^{j}}\frac{d\zeta}{\zeta^{m+1}}.
S=\intdt\left(g_{i\bar{\jmath}}{\frac{dz_{i}}{dt}}{\frac{d\bar{z}_{j}}{dt}}+g_{ij}{\frac{dz_{i}}{dt}}{\frac{dz_{j}}{dt}}+g_{\bar{\imath}\bar{\jmath}}{\frac{d\bar{z}_{i}}{dt}}{\frac{d\bar{z}_{j}}{dt}}\right).
F_{2}(q^{2})={\frac{Me^{2}}{4\pi^{2}}}\int_{0}^{1}d\alpha\,\int_{0}^{1}dx\,\,{\frac{m-xM}{\alpha(1-\alpha)\,{\frac{1-x}{x}}\,{\vec{q}}_{\perp}^{2}-M^{2}+{\frac{m^{2}}{x}}+{\frac{\lambda^{2}}{1-x}}}}\.
\begin{array}{l}{L_{f}(x_{f})=J_{\psi}L(x)=J_{\psi}\left[(e^{\mu}\nabla_{\mu}\varphi)^{+}(e^{\mu}\nabla_{\mu}\varphi)-m^{2}\varphi^{+}\varphi\right]=}\\{=J_{\psi}\left[S(B_{f})^{2}(D_{l}\varphi_{f})^{+}(D_{l}\varphi_{f})-m^{2}\varphi_{f}^{+}\varphi_{f}\right].}\\\end{array}
\begin{array}{l}{\Bigl[(r+s)_{23}(x,\mu,\eta)~,~(r+s)_{12}(x,\lambda,\eta)\Bigr]+\Bigl[(r+s)_{23}(x,\mu,\eta)~,~(r+s)_{13}(x,\lambda,\eta)\Bigr]}\\{+\Bigl[(r+s)_{13}(x,\lambda,\eta)~,~(r-s)_{12}(x,\lambda,\mu)\Bigr]+H_{1,23}^{(r+s)}(x,\lambda,\mu,\eta)-H_{2,13}^{(r+s)}(x,\mu,\lambda,\eta)=0,}\\\end{array}
J_{st}=-i\,\left(\underbrace{\vec{q}_{s}\vec{\nabla}_{t}-\vec{q}_{t}\vec{\nabla}_{s}}_{{\bfL}_{st}}+\underbrace{\frac{1}{4}\vec{\Theta}_{\hat{\alpha}}\gamma_{\hat{\alpha}\hat{\beta}}^{st}\vec{\Theta}_{\hat{\beta}}}_{{\bfM}_{st}}\right)\,.
L=i\overline{{\psi}}_{0}{\gamma}^{\mu}\partial_{\mu}{\psi}_{0}+i\overline{{\psi}}_{0}e^{-i{t}^{a}{\theta}_{a}}{\gamma}^{\mu}\left(\partial_{\mu}e^{i{t}^{b}{\theta}_{b}}\right){\psi}_{0}-m\overline{{\psi}}_{0}{\psi}_{0}.
\mu^{A^{\prime}}=\sum_{n=0}^{\infty}\sum_{p,q=1}^{(n+1)(n+2)}\alpha_{n}^{pq}\Bigr[{\widetildem}_{np}(\tau){\overline{\rho}}^{nqA^{\prime}}+r_{np}(\tau)\sigma^{nqA^{\prime}}\Bigr]\;,
\begin{aligned}{}&{{}S_{ij}|s_{1},\dots,s_{i},\dots,s_{j},\dots,s_{N}\rangle=|s_{1},\dots,s_{j},\dots,s_{i},\dots,s_{N}\rangle\,,}\\{}&{{}S_{i}|s_{1},\dots,s_{i},\dots,s_{N}\rangle=|s_{1},\dots,-s_{i},\dots,s_{N}\rangle\,.}\\\end{aligned}
\varepsilon\left(\begin{array}{ccc}{a_{1}}&{\ldots}&{a_{g}}\\{b_{1}}&{\ldots}&{b_{g}}\\\end{array}\right)=\prod_{j=1}^{g}\frac{\vartheta(a_{j},b_{j})\vartheta(a_{j},A_{j-1})}{\vartheta(b_{j},a_{j}^{b_{j}})\vartheta(a_{j},[a_{j},b_{j}])}
V_{I}(\beta,\varphi_{0})=\frac{1}{2}m^{2}\varphi_{0}^{2}+\frac{\lambda}{4!}\varphi_{0}^{4}-\frac{1}{2}\deltam^{2}\varphi_{0}^{2}-\frac{1}{4!}\delta\lambda\varphi_{0}^{4},
R=(d-2)\left[\frac{(d-3)(1-A)}{r^{2}}-\frac{2}{r}\left(\frac{dA}{dr}+A\frac{d\Omega}{dr}\right)\right]-\left[\frac{d^{2}A}{dr^{2}}+2A\frac{d^{2}\Omega}{dr^{2}}+2A\left(\frac{d\Omega}{dr}\right)^{2}+3\frac{dA}{dr}\frac{d\Omega}{dr}\right].
e^{iW_{NL}[\vec{H}]}=\int{\calD}\vec{\Phi}\Pi_{x}\delta^{d}(\vec{\Phi}^{2}(x)-C^{2})e^{i\intd^{d}x(\frac{1}{2}(\partial_{\mu}\vec{\Phi}(x)^{2}+\vec{H}(x)\cdot\vec{\Phi}(x)})
D_{2}(A)\Psi_{_{l,j=l\pm\frac{1}{2},m,q}}(\theta,\phi)=\pm\sqrt{2\lambda\alpha^{2}[(j+\frac{1}{2})^{2}-q^{2}]}\Psi_{_{l,j=l\pm\frac{1}{2},m,q}}(\theta,\phi),
\begin{array}{rcl}{{\calR}(u,c)}&{=}&{\displaystyle\frac{u-icP}{u+ic}}\\{}&{=}&{\displaystyle\frac{1}{u+ic}\left[\begin{array}{cccc}{u-ic}&{0}&{0}&{0}\\{0}&{u}&{-ic}&{0}\\{0}&{-ic}&{u}&{0}\\{0}&{0}&{0}&{u-ic}\\\end{array}\right]~,}\\\end{array}
I\equive^{\frac{t_{0}+Ht_{1}}{x}}\widetilde{I}\equive^{\frac{t_{0}+Ht_{1}}{x}}\sum_{d\geq0}q^{d}\frac{\prod_{m=0}^{3d-1}(K-mx)}{\prod_{m=1}^{d}\prod_{i=1}^{3}(H-\lambda_{i}+mx)},
\mid{\bf\cdot}\,\rangle=\;\midp\,\rangle^{0},\;\;\;\;\;\;P=\;\midp\,\rangle^{0\;0}\langle\,p\mid,\;\;\;\;\;\;\mid\Psi_{P}\,\rangle=\;\midp\,\rangle^{0\;0}\langle\,p\midp\,\rangle.
\hat{T}={\frac{1}{\sqrt{3}}}{\frac{\sum_{j=1}^{3}\mathrm{e}^{i{\frac{2\pi}{3}}j}f_{j}}{\sum_{k=1}^{3}f_{k}}},\qquad{\calM}_{j}={\frac{f_{j}}{\sum_{k=1}^{3}f_{k}}}.
{\bar{K}}_{\alpha\beta}(x-y)\,=\,\int{\frac{d^{3}k}{(2\pi)^{3}}}\,e^{ik\cdot(x-y)}\int_{-\infty}^{+\infty}\frac{dk_{3}}{2\pi}\,\frac{\delta_{\alpha\beta}}{k^{2}}\;.
L=\frac{1}{2}\left(\dot{x}^{\mu}+\dot{x}^{\alpha}A_{\alpha}^{\mu}\right)g_{\mu\nu}\left(\dot{x}^{\nu}+A_{\beta}^{\nu}\dot{x}^{\beta}\right)+\frac{1}{2}\dot{x}^{\alpha}g_{\alpha\beta}\dot{x}^{\beta}-U_{q}(x)\,.
f=\left(\begin{array}{cc}{(Z_{1}^{\prime\prime}+Z_{2}^{\prime}+\zeta+\rho^{2})(Z_{\rho}^{\prime\prime}~^{\prime})^{-1}}&{-\sqrt{\zeta+\rho^{2}}(Z_{\rho}^{\prime\prime}~^{\prime})^{-1}\bar{z}_{1}}\\{-\sqrt{\zeta+\rho^{2}}z_{1}(Z_{\rho}^{\prime\prime}~^{\prime})^{-1}}&{(Z_{1}^{\prime}+Z_{2}^{\prime}-\theta_{1}+\rho^{2})(Z_{\rho}^{\prime}~^{\prime})^{-1}}\\\end{array}\right).
{\hat{\pi}}^{i}(\varepsilon)\simeqm{\frac{({\hat{x}}^{i}(\varepsilon)-{\hat{x}}^{i}(0))}{\varepsilon}}+{\frac{e}{2}}({\hat{x}}^{k}(\varepsilon)-{\hat{x}}^{k}(0)){\hat{F}}^{ik}\,.
f(x)*g(x)=\int\frac{d^{n}k}{(2\pi)^{n}}\int\frac{d^{n}q}{(2\pi)^{n}}\tilde{f}(k)\tilde{g}(q)e^{-i(k+q)x}e^{-\frac{i}{2}k_{m}\theta^{mn}q_{n}}\,.
\mathrm{H}=\frac{1}{2}\sum_{n=0}^{\infty}\frac{1}{n+1}h_{n}^{\dagger}h_{n}=\frac{1}{2}\sum_{n=0}^{\infty}\frac{1}{n+1}\sum_{i=0}^{n}a_{n-i}^{\dagger}\sum_{j=0}^{n}a_{j}a_{n-j}.
\begin{aligned}{\int_{\Sigma}\sqrt{W}[-\overline{{\theta}}\Gamma_{-}\Gamma_{r}\mathcal{D}_{r}\theta+\overline{{\theta}}\Gamma_{-}\Gamma_{m}\{X^{m},\theta\}+\Lambda\{\overline{{\theta}}\Gamma_{-},\theta\}]}\\\end{aligned}
\beta_{m}^{(\alpha)}=\frac{2}{3}\frac{\alpha^{2}}{\pi}+\frac{1}{2}\frac{\alpha^{3}}{\pi^{2}}-\frac{3}{2}\frac{\alpha^{3}}{\pi^{2}}+{\calO}(\alpha^{4})=\frac{2}{3}\frac{\alpha^{2}}{\pi}-\frac{\alpha^{3}}{\pi^{2}}+{\calO}(\alpha^{4}),
\begin{matrix}{\hat{\delta}V^{a}=0,}&{\hat{\delta}A=0,}\\{\hat{\delta}\psi_{A}=\psi_{A},}&{\hat{\delta}\psi^{A}=-\psi^{A},}\\{\hat{\delta}F_{ab}^{+}=2F_{ab}^{+},}&{\hat{\delta}F_{ab}^{-}=-2F_{ab}^{-},}\\{\hat{\delta}\zeta_{I}=-\zeta_{I},}&{\hat{\delta}\zeta^{I}=\zeta^{I}.}\\\end{matrix}
G_{i}(x,x;s)=\int\frac{d^{4}p}{(2\pi)^{4}}\left[-\frac{1}{p^{2}-s^{2}}+\left(\frac{1}{3}-\xi_{i}\right)\frac{R}{(p^{2}-s^{2})^{2}}-\frac{2}{3}\frac{{R}^{\mu\nu}p_{\mu}p_{\nu}}{(p^{2}-s^{2})^{3}}\right]+{\calO}(R_{;\mu},R^{2})\,.
\begin{array}{ccc}{x_{\pm}^{\prime\mu}=-\displaystyle{\frac{x_{\mp}^{\mu}}{x_{\mp}^{2}}}~~~~}&{~~~~\theta^{\prime}{}^{at}=\displaystyle{-i\frac{1}{x_{-}^{2}}\,\epsilon^{-1}x_{-}{\cdot\sigma}\bar{\theta}_{b}\,\zeta^{ba}}~~~~}&{~~~~\bar{\theta}_{a}^{\prime}=i\displaystyle{\frac{1}{x_{+}^{2}}\,\bar{\epsilon}^{-1}x_{+}{\cdot\sigma^{t}}\theta^{bt}\,\bar{\zeta}_{ba}}}\\\end{array}
\begin{array}{lcr}{L_{H}=\intd^{2}x\sum_{hypermult}[|D_{\mu}\varphi_{H}|^{2}+|V\varphi_{H}|^{2}+\overline{{\psi}}_{H}\gamma^{\mu}D_{\mu}\psi_{H}+\overline{{\psi}}_{H}V{\psi}_{H}+\overline{{\psi}}_{V}V{\psi}_{H}}\\{+\overline{\varphi}_{H}D\varphi_{H}]}\\\end{array}
\pi_{\mu\nu}=-\tau_{\mu\alpha}\tau_{\nu}^{\phantom{\nu}\alpha}/4+\tau_{\phantom{\alpha}\alpha}^{\alpha}\tau_{\mu\nu}/12+q_{\mu\nu}\tau_{\alpha\beta}\tau^{\alpha\beta}/8-q_{\mu\nu}(\tau_{\phantom{\alpha}\alpha}^{\alpha})^{2}/24,
\frac{1}{2}\langleV_{2}|\Psi,Q\Psi\rangle=\intd^{26}p\;\left\{\phi(-p)\left[\frac{p^{2}-1}{2}\right]\phi(p)+A_{\mu}(-p)\left[\frac{p^{2}}{2}\right]A^{\mu}(p)+\cdots\right\}\,.
L=\frac{1}{2}p_{j}g^{ij}p_{j}+\frac{1}{2}\left(p_{\alpha}-p_{\mu}A_{\alpha}^{\mu}-\frac{i\hbar}{2}\hat{\Gamma}_{\alpha}\right)g^{\alpha\beta}\left(p_{\alpha}-A_{\alpha}^{\mu}p_{\mu}+\frac{i\hbar}{2}\hat{\Gamma}_{\alpha}\right)\,,
ds_{4}^{2}=-(1-{\frac{\mu}{r}})^{{\frac{3}{2}}}dt^{2}+(1-{\frac{\mu}{r}})^{-{\frac{3}{2}}}dr^{2}+(1-{\frac{\mu}{r}})^{{\frac{1}{2}}}r^{2}d\Omega_{2}^{2},
W_{F}=-\frac{\deltaF}{\delta\phi^{A}}(\lambda^{A}+\mathrm{~\frac{1}{2}~}m^{2}(P_{+})_{B}^{A}\phi^{B})-\mathrm{~\frac{1}{2}~}\epsilon_{ab}\pi^{Aa}\frac{\delta^{2}F}{\delta\phi^{A}\delta\phi^{B}}\pi^{Bb}+m^{2}F
i\partial_{\tau}\psi=\mathrm{K}\psi=\left[\frac{1}{2m}(\mathrm{p}-e\mathrm{A}^{(n)})^{2}+\frac{1}{2\lambda}(\pi^{\mu}+y^{T}S^{\mu}\mathrm{p}-e\chi^{\mu})(\pi_{\mu}+y^{T}S_{\mu}\mathrm{p}-e\chi_{\mu})+V\right]\psi
A^{3}=\left(\begin{matrix}{0}&{-\omega_{1}^{3}}&{0}&{0}\\{\omega_{1}^{3}}&{0}&{0}&{0}\\{0}&{0}&{0}&{-\omega_{2}^{3}}\\{0}&{0}&{\omega_{2}^{3}}&{0}\\\end{matrix}\right).
\left(\begin{array}{l}{T_{\alpha{\dot{\alpha}}}}\\{W_{\alpha}}\\{\tau}\\\end{array}\right)\to\left(\begin{array}{rl}{T_{\alpha{\dot{\alpha}}}^{new}=}&{{\widetildeT}_{\alpha{\dot{\alpha}}}-\frac{1}{3}[D_{\alpha},\bar{D}_{\dot{\alpha}}]U}\\{W_{\alpha}^{new}=}&{\frac{1}{2}{\bar{D}}^{2}D_{\alpha}\U}\\{}&{0}\\\end{array}\right)\,
z(1-z)\frac{d^{2}A_{i}}{dz^{2}}+(1-z)\frac{dA_{i}}{dz}+\left[\frac{k_{+}^{2}}{4z}-\frac{k_{-}^{2}}{4}-\frac{L^{2}-2\epsilon_{i}L}{4(1-z)}\right]~A_{i}~=0~.
ds^{2}=-\left(M-\frac{J^{2}}{4r^{2}}\right)dt^{2}+\frac{2}{l}dtd\varphi+\frac{1}{r^{2}}d\varphi^{2}+\left(\frac{r^{2}}{l^{2}}-M+\frac{J^{2}}{4r^{2}}\right)^{-1}dr^{2}
\begin{array}{c}{\Gamma^{M}\Gamma^{N}+\Gamma^{N}\Gamma^{M}=2\eta^{MN},\qquad\eta^{MN}=(+,-,-,\dots),}\\{(\Gamma^{M})^{*}=-\Gamma^{M},\qquad(\Gamma^{0})^{\mathrm{T}}=-\Gamma^{0},\qquad(\Gamma^{i})^{\mathrm{T}}=\Gamma^{i},\quadi=1,\dots,9.}\\\end{array}
\gamma_{l}^{I}=\sum_{n}A_{n}\gamma_{l}^{(n)I}=\gamma_{l}^{I}(A)=(\lambda+2l-1)\frac{\Gamma(\lambda+l-1)}{l!(n+1)!\,\Gamma(\lambda-2)}[1+\frac{1}{2}(-1)^{l}\frac{(l+1)!}{(\lambda-1)_{l-1}}]
\lambda(r)=\left({\frac{\frac{e^{\phi_{0}}}{\sqrt{2}}\left[\lambda_{0}+\frac{\lambda_{0}M+\overline{{\lambda}}_{0}\Upsilon}{r}\right]}{\frac{e^{\phi_{0}}}{\sqrt{2}}\left[1+\frac{M+\Upsilon}{r}\right]}}\right)_{\Upsilon=0}=\{\frac{\lambda_{0}(1+{\frac{M}{r}})}{(1+{\frac{M}{r}})}}=\lambda_{0}\,
\int_{0}^{t}dt^{\prime}\intd^{2}\xi^{\prime}K_{\xi\xi^{\prime}}^{t-t^{\prime}}e^{-\phi(\xi)}2\left((\xi^{\prime}-\xi)^{a}\partial_{a}\phi(\xi)\partial_{a}\chi(\xi)-(\xi^{\prime}-\xi)^{b}\partial_{b}\partial_{a}\chi(\xi)\right)\partial_{a}^{\prime}K_{\xi^{\prime}\xi}^{t^{\prime}}
ds_{deS}^{2}=-{\frac{\ell^{2}}{4}}\left({\frac{d\rho}{\rho}}\right)^{2}+{\frac{1}{\rho}}\,d\vec{x}\cdotd\vec{x}=-{\frac{\ell^{2}}{4}}{\frac{d\rho^{2}}{\rho^{2}}}+{\frac{1}{\rho}}\,\delta_{ab}dx^{a}dx^{b}\,.
\begin{array}{c|ccc}{}&{U(1)_{1}}&{U(1)_{2}}&{U(1)_{3}}\\\hline{A_{1,2}}&{0}&{0}&{-2}\\{{\calQ}}&{2}&{0}&{0}\\{\tilde{\calQ}}&{0}&{2}&{0}\\{q,p}&{0}&{1}&{-1}\\{\tilde{q},\tilde{p}}&{1}&{0}&{-1}\\\end{array}
\eta^{\nu\alpha}\xi_{\alpha}\eta_{\rho}{^{\sigma}}\nabla_{\sigma}\xi^{\rho}=\frac{1}{2}(\eta^{\nu\alpha}\xi_{\alpha})\eta^{\sigma\rho}[\nabla_{\sigma}\xi_{\rho}+\nabla_{\rho}\xi_{\sigma}]=\eta^{\nu\alpha}\xi_{\alpha}\frac{\delta\sqrt{-\gamma}}{\sqrt{-\gamma}},
\partial_{\tau}a_{\varphi}^{0}(x,\tau)=+\frac{1}{\lambda}\\frac12\\frac{e}{4\piR}\epsilon(t-\tau-R/c)\\left[e^{-\bigl|t-\tau-R/c\bigr|/\lambda}+\epsilon(t-\tau+R/c)\e^{-\bigl|t-\tau+R/c\bigr|/\lambda}\right]
L^{\prime}=\sum_{n>0}\left(\frac12|\partial\phi|^{2}-\frac{M^{2}n^{2}}{2}|\phi_{n}|^{2}-2\lambda\phi_{n}\sum_{n>0}|\phi_{n}|^{2}\right)+\frac{\lambda}{3!}\sum_{n,n^{\prime}}{}^{\prime}\phi_{n}\phi_{n^{\prime}}\phi_{n+n^{\prime}}^{*},
G(x,y)=G(x-y)=\frac{1}{(2\pi)^{D}}\intd^{D}ke^{-ik(x-y)}G(k)=\frac{1}{(2\pi)^{D}}\intd^{D}ke^{-ik(x-y)}\frac{1}{k^{2}+M^{2}},
e_{1}=\frac{H^{2}}{6}[-1+3i\sqrt{4H^{2}L^{2}-1}],\;\;e_{2}=\frac{H^{2}}{3},\;\;e_{3}=\frac{H^{2}}{6}[-1-3i\sqrt{4H^{2}L^{2}-1}].
\\Gamma^{3}=\left[\begin{array}{cccc}{0}&{-i}&{0}&{0}\\{-i}&{0}&{0}&{0}\\{0}&{0}&{0}&{-i}\\{0}&{0}&{-i}&{0}\\\end{array}\right]\qquad\Gamma^{4}=\left[\begin{array}{cccc}{0}&{1}&{0}&{0}\\{-1}&{0}&{0}&{0}\\{0}&{0}&{0}&{-1}\\{0}&{0}&{1}&{0}\\\end{array}\right]
{\calL}_{bosonic}^{{\calN}=8}={\frac{1}{g_{_{YM}}^{2}}}\left\{-F^{\underline{m}\underline{n}}F_{\underline{m}\underline{n}}+{\frac{1}{2}}\nabla_{\underline{m}}\overline{{\rho}}_{AB}\nabla^{\underline{m}}\rho^{AB}+{\frac{1}{16}}\left[\overline{{\rho}}_{AB}\rho^{CD}\right]\left[\rho^{AB},\overline{{\rho}}_{CD}\right]\right\}
(\bigtriangleupA)_{k}=\left({\frac{\partial^{2}}{\partialr^{2}}}+{\frac{(d-2)}{r}}{\frac{\partial}{\partialr}}+{\frac{1}{r^{2}}}{}^{(d)}\bigtriangleup-{\frac{(d-1)}{r^{2}}}\right)A_{k}+{\frac{2}{r}}\partial_{k}A_{r}.
S_{\sigma}={\frac{-1}{4\pi\alpha^{\prime}}}\intd^{2}\xi\sqrt{\gamma}\big(\gamma^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}\tilde{g}_{\mu\nu}+\epsilon^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}B_{\mu\nu}-{\frac{1}{2}}\alpha^{\prime}R^{(2)}\Phi\big),
\phi_{h}=\left(\begin{array}{cc}{\frac{y^{2}}{y^{2}+\rho^{2}}A_{1(\mathrm{tl})}+\frac{1}{2}\mathrm{Tr}(A_{1})I_{2}}&{\sqrt{\frac{y^{2}}{y^{2}+\rho^{2}}}A_{2}}\\{\sqrt{\frac{y^{2}}{y^{2}+\rho^{2}}}A_{3}}&{A_{4}}\\\end{array}\right),
\frac{1}{2}\bigg(\bigg|\frac{\alpha_{{k}}}{\beta_{{k}}}\bigg|+\bigg|\frac{\beta_{{k}}}{\alpha_{{k}}}\bigg|\bigg)=\frac{\Omega_{{k}}}{|\Lambda_{{k}}|},\qquad|\alpha_{{k}}|^{2}-|\beta_{{k}}|^{2}=1.
\left(\begin{array}{cc}{M_{11}}&{M_{12}}\\{M_{21}}&{M_{22}}\\\end{array}\right)\left(\begin{array}{c}{W_{22}^{00}}\\{W_{00}^{00}}\\\end{array}\right)=\omega^{2}\left(\begin{array}{c}{W_{22}^{00}}\\{W_{00}^{00}}\\\end{array}\right),
\frac{\deltar_{0}}{r_{0}}=\frac{1}{6kr_{0}}\frac{(1-e^{-2A_{0}})}{m_{r}^{2}M_{Pl}^{2}e^{-2A_{0}}}\left(\rho_{0}-3p_{0}+e^{-2A_{0}}(\rho_{*}-3p_{*})\right)
S_{2}^{\ell}=-{\frac{3}{2\kappa_{10}^{2}{\alpha^{\prime}}^{3}}}\intd^{6}p\,\bar{\rho}_{ij}^{\ell}(p)\,\bar{\rho}^{\ellij}(-p)\left(-1+\sqrt{(\ell+1)^{2}+p^{2}N}\right).
Z_{R}=z_{R}+{\frac{i}{2}}\sum_{n=1}^{\infty}{\frac{\beta_{n-k/N}}{n-k/N}}~z^{-(n-k/N)}-{\frac{i}{2}}\sum_{n=0}^{\infty}{\frac{\gamma_{n+k/N}^{\dagger}}{n+k/N}}~z^{n+k/N}\;,
A(s,t)=\frac{\Gamma(1-s/8)\Gamma(-t/8)\Gamma(-u/8)}{\Gamma(1+u/8)\Gamma(s/8)\Gamma(1+t/8)}=\frac{s^{2}}{8}\left(\frac{1}{stu}+\frac{\zeta(3)}{256}+\dots\right)\.
A_{\mu}^{(I)}:=\sum_{b=1}^{N}U_{Ib}a_{\mu}^{(b)}\;\;\stackrel{U^{T}=U^{-1}}{\Longleftarrow\!\!=\!\!=\!\!=\!\!=\!\!=\!\!\!\Longrightarrow}\;\;a_{\mu}^{(b)}=\sum_{I=1}^{N}U_{Ib}A_{\mu}^{(I)}\;.
\frac{F}{V}=-\sum_{\stackrel{n=1}{n~\mathrm{odd}}}^{\infty}\sum_{k=1}^{\infty}\sum_{s=0}^{k-1}\frac{1}{k^{2}}\left(\frac{1}{4\pi^{2}\alpha^{\prime}\hat{\tau}_{2}}\right)^{5}2^{9}\left(\theta_{4}\left(0,2\tau_{-}\right)\right)^{-8}\left(\bar{\theta}_{4}\left(0,2\tau_{+}\right)\right)^{-8}e^{-n^{2}\beta^{2}/4\pi\alpha^{\prime}\tau_{2}}
\hat{H}=\frac{1}{2}\hat{p}_{z}^{2}+\hbar\omega\left[a_{+}^{\dagger}a_{+}\,+\,a_{-}^{\dagger}a_{-}\right]\\\,\\\\hat{\phi}=\hat{p}_{z}+\hbarg\left[a_{+}^{\dagger}a_{+}\,-\,a_{-}^{\dagger}a_{-}\right]\\\.
\hat{\calK}^{(3)\\hat{\imath}_{1}\hat{\imath}_{3}\hat{\imath}_{3}}=-\frac{\hat{\epsilon}^{\hat{\imath}_{1}\ldots\hat{\imath}_{6}}}{3!\sqrt{|D_{\hat{\imath}}\hat{X}^{\hat{\mu}}D_{\hat{\jmath}}\hat{X}^{\hat{\nu}}\hat{g}_{\hat{\mu}\hat{\nu}}|}}\hat{\calK}^{(3)}{}_{\hat{\imath}_{4}\hat{\imath}_{5}\hat{\imath}_{6}}\,.
\Theta^{\mu\nu}(x)={\frac{1}{2}}\int_{S}d^{2}S^{\mu\nu}(x(\sigma))\delta^{4}(x-x(\sigma)),\quadd^{2}S^{\mu\nu}:=d\sigma^{1}d\sigma^{2}{\frac{\partial(x^{\mu},x^{\nu})}{\partial(\sigma^{1},\sigma^{2})}}.
\begin{array}{rcl}{\check{R}^{V_{z_{1}},V_{z_{2}}^{*}}}&{=}&{(C\otimesI)^{-1}\check{R}^{V_{z_{1}},V_{z_{2}+(n-1)w}\otimes\cdots\otimesV_{z_{2}+w}}(I\otimesC),}\\{\check{R}^{V_{z_{1}}^{*},V_{z_{2}}}}&{=}&{(I\otimesC)^{-1}\check{R}^{V_{z_{1}+(n-1)w}\otimes\cdots\otimesV_{z_{1}+w},V_{z_{2}}}(C\otimesI).}\\\end{array}
I_{\mathrm{WZW}}[{\bfg},{\bf\alpha}_{w},{\bf\alpha}_{\bar{w}}]=I_{\mathrm{WZW}}[{\bfg}]+2\intd^{2}w\,\,\mathrm{Tr}\left({\bfg}{\bf\alpha}_{w}{\bfg}^{-1}{\bf\alpha}_{\bar{w}}+{\bf\alpha}_{w}{\bfg}^{-1}\partial_{\bar{w}}{\bfg}+{\bf\alpha}_{\bar{w}}{\bfg}\partial_{w}{\bfg}^{-1}\right)
H=\frac{1}{2m}\sum_{k=1}^{N-1}(p_{\rho_{k}}^{2}+\frac{1}{\rho_{k}^{2}}(p_{\phi_{k}}+k\alpha\hbar)^{2}-eB(p_{\phi_{k}}+k\alpha\hbar)+\frac{1}{4}e^{2}B^{2}\rho_{k}^{2}).
{\frac{2K-1}{\pi}}\int_{D_{(1,1)}}{\frac{d\xi^{*}d\xi}{\left({1-|\xi|^{2}}\right)^{2}}}\left\vert{\xi}\right\rangle\left\langle{\xi}\right\vert\equiv\int_{D_{(1,1)}}d\mu(\xi^{*},\xi)\left\vert{\xi}\right\rangle\left\langle{\xi}\right\vert={\bf1}_{K}\quad,
\eta_{\mathrm{\tiny~GL~{}_{q}(2)~}}=\left(\begin{array}{cccc}{-q^{-3}}&{0}&{0}&{0}\\{0}&{0}&{-q^{-1}}&{0}\\{0}&{-q^{-3}}&{0}&{0}\\{0}&{0}&{0}&{-q^{-1}}\\\end{array}\right).
{\calD}_{\mu}z^{i}=\left(\begin{array}{c}{\partial_{\mu}a\nonumber}\\{\partial_{\mu}b-m^{a}L_{ab}(V_{\mu}^{b}+v^{b}A_{\mu}^{(2)})\nonumber}\\{\partial_{\mu}v^{a}+m^{a}(A_{\mu}+aA_{\mu}^{(2)})}\\\end{array}\right)\,,
\begin{aligned}{H}&{{}\mapstol_{1}+l_{2}-e_{1}\,,}\\{E_{1}}&{{}\mapstol_{2}-e_{1}\,,}\\{E_{2}}&{{}\mapstol_{1}-e_{1}\,,}\\{E_{a+1}}&{{}\mapstoe_{a}\,,\,\,\,\,a=2,\cdots,k-1\,.}\\\end{aligned}
\frac{\left[(E^{\prime}+E^{2})^{\prime\prime}-E(E^{\prime}+E^{2})^{\prime}\right]^{\prime}}{(E^{\prime}+E^{2})^{\prime\prime}-E(E^{\prime}+E^{2})^{\prime}}=2E=2\frac{h^{\prime\prime}}{h^{\prime}},
g_{I}=\left(H_{1}^{-\frac{1}{2}}H_{5}^{-\frac{1}{2}}H_{W}\right)\left(H_{1}^{\frac{1}{2}}H_{5}^{-\frac{1}{2}}\right)^{4}=H_{1}^{\frac{3}{2}}H_{5}^{-\frac{5}{2}}H_{W}
\begin{array}{ll}{n\left(\frac{1-k}{1+k}\right)=2G\left(U-T-J^{2}-\nuS^{2}\right)}\\{B^{2}=1-8G\left(U+\frac{(J^{2}+\nuS^{2})}{2}\right)}\\{m^{2}=4G(J^{2}+\nuS^{2}).}\\\end{array}
v_{0}=u-w_{1}a^{-\gamma}-(\mathrm{~\frac{1}{12}~}u^{\prime\prime}+u^{2})a^{-2\gamma}+\mathrm{~\frac{2}{5}~}(\mathrm{~\frac{1}{6}~}w_{1}^{\prime\prime}+uw_{1})a^{-3\gamma}+\mathrm{O}(a^{-4\gamma})
\begin{array}{ccll}{\{g_{t,i},g_{t,k}\}}&{=}&{0,}&{\mathrm{~if~i-k~even}}\\{\{g_{t,i},g_{\tilde{t},k}\}}&{=}&{1,}&{\mathrm{~if~i-k~odd,~i<k,\;~|i-k|<2p~},}\\{\{g_{t,i},m_{t}^{(k)}\}}&{=}&{0}&{\mathrm{~(~i-k~odd),}}\\{\{g_{t,i},m_{\tilde{t}}^{(k)}\}}&{=}&{2}&{\mathrm{~(~i-k~even),}}\\{\{m_{t}^{(1)},m_{\tilde{t}}^{(2)}\}}&{=}&{0}\\\end{array}
\left(\begin{matrix}{-\psi_{2}}\\{\psi_{1}}\\\end{matrix}\right)=\left(\begin{matrix}{\,\frac{\partial\psi_{1}}{\partialt}-\frac{\partial\psi_{1}}{\partialx}}\\{\frac{\partial\psi_{2}}{\partialt}+\frac{\partial\psi_{2}}{\partialx}}\\\end{matrix}\right)\.
V=\int_{0}^{\frac{4\pi\,C}{n-2}}dt\int_{C}^{r}s^{n-1}\,ds\,\,\,\mathrm{Vol(M)}=\mathrm{Vol(M)}\,\frac{4\pi\,C}{n(n-2)}\left(r^{n}-C^{n}\right)
\Psi\rightarrow\left(\begin{array}{c}{\psi_{+}}\\{\overline{{\psi}}_{+}}\\\end{array}\right),~\Psi^{\dagger}\rightarrow\left(\psi_{-}~\overline{{\psi}}_{-}\right);~~\overline{{\Psi}}\equiv\Psi^{\dagger}\gamma^{0};~~\gamma_{0}=\left[\begin{array}{cc}{0}&{~1}\\{1}&{~0}\\\end{array}\right]\,~~\gamma_{1}=\left[\begin{array}{cc}{0}&{~1}\\{-1}&{~0}\\\end{array}\right]\nonumber
ds^{2}\,:=\,B_{\mathrm{H}}[\sigma(r-r_{\mathrm{H}})]^{2(1-\alpha)}\,\bar{g}_{\mu\nu}dx^{\mu}dx^{\nu}+\frac{dr^{2}}{B_{\mathrm{H}}(r-r_{\mathrm{H}})^{2}}+r_{\mathrm{H}}^{2}d\Omega_{n-1}^{2}.
\left(h_{\alpha}^{g_{1},g_{2}g_{3}}\right)\,\left(h_{\alpha}^{g_{2},g_{3}}\right)\:=\:\left(g_{3}^{*}h_{\alpha}^{g_{1},g_{2}}\right)\,\left(h_{\alpha}^{g_{1}g_{2},g_{3}}\right)
E^{-1}\mathcal{L}=\frac{1}{2}R-\frac{1}{2}g_{XY}\partial_{M}q^{X}\partial^{M}q^{Y}-\frac{1}{2}g_{xy}\mathcal{\partial}_{M}\phi^{x}\mathcal{\partial}^{M}\phi^{y}-\mathcal{V}(\phi,q),
\nabla_{a}{\calM}^{a\mu\nu}={\frac{1}{2}}\left[2{\calQ}^{ab}e^{\mu}{}_{a}e^{\nu}{}_{b}-\sqrt{-\gamma}\;L^{ab}{}_{i}K_{ab\,j}n^{\mu\,i}n^{\nu\,j}-(\mu\leftrightarrow\nu))\right]\,.
S=\int_{0}^{L}dy\intd^{4}x\sqrt{-G}\left(-\frac{1}{2}G^{MN}\partial_{M}\Phi\partial_{N}\Phi+\frac{1}{\sqrt{-G}}\partial_{M}\left(\sqrt{-G}G^{MN}\Phi\partial_{N}\Phi\right)-\frac{1}{2}V(\Phi)\right)\.
M\in\mathcal{G}_{iso}\quad\Leftrightarrow\quadM_{\Lambda_{1}}^{\phantom{\Lambda}\Sigma_{1}}\,M_{\Lambda_{2}}^{\phantom{\Lambda}\Sigma_{2}}\,M_{\Lambda_{3}}^{\phantom{\Lambda}\Sigma_{3}}\,d_{\Lambda_{1}\Lambda_{2}\Lambda_{3}}\,=\,d_{\Sigma_{1}\Sigma_{2}\Sigma_{3}}
\begin{array}{l}{\Phi(H,{\calN})=F_{m,n}(H,{\calN}+u)=}\\{=\prod_{k=1}^{m}\left(H/2+{\calN}+u-\frac{2k-1}{2m}\right)\prod_{\ell=1}^{n}\left(H/2-{\calN}-u+\frac{2\ell-1}{2n}\right).}\\\end{array}
V_{(\ell,m)\;(a_{1},m_{1})}{}^{(a_{2},m_{2})}=\sum_{(a^{\prime},m^{\prime})}\frac{S_{(\ell,m)}{}^{(a^{\prime}-1,m^{\prime})}\;\Psi_{(a_{1},m_{1})}{}^{(a^{\prime},m^{\prime})}\;{\Psi_{(a_{2},m_{2})}{}^{(a^{\prime},m^{\prime})}}^{*}}{S_{(0,0)}{}^{(a^{\prime}-1,m^{\prime})}}\in{\mathbbN}
D_{a}=\frac{\partial}{\partial\theta^{a}}-\frac{i}{2}\overline{{\theta}}_{a}\frac{\partial}{\partialt},\quad\overline{{D}}^{a}=\frac{\partial}{\partial\overline{{\theta}}_{a}}-\frac{i}{2}\theta^{a}\frac{\partial}{\partialt},
m_{I}\;f_{L_{1}\ldotsL_{r}}^{K_{1}\ldotsK_{q}}(u)=(\delta_{K_{1}I}-\delta_{K_{1}N}-\delta_{L_{1}I}+\delta_{L_{1}N}+\ldots)f_{L_{1}\ldotsL_{r}}^{K_{1}\ldotsK_{q}}(u)
f(z,\bar{z};\zeta,\bar{\zeta})={\frac{\theta_{1}({\frac{z-\zeta}{2}}|\tau)\theta_{1}({\frac{z+\zeta}{2}}|\tau)}{\theta_{1}({\frac{z-\bar{\zeta}}{2}}|\tau)\theta_{1}({\frac{z+\bar{\zeta}}{2}}|\tau)}}
{\frac{\kappa}{4\pi}}\{\xi_{\alphaL}(x),\xi_{\betaR}(z)\}\;=\;\Bigl(L^{T\;-1}(x)H(z)R^{-1}(z)\Bigr)_{\alpha\beta}\;+\;\Bigl(\tilde{h}(x)R^{-1}(z)\Bigr)_{\alpha\beta}\quad,
|B,\eta>_{RR}^{NSNS}=exp(\sum_{\mu=2}^{9}(-\sum_{n=1}^{\infty}\frac{1}{n}\alpha_{-n}^{\mu}\tilde{\alpha}_{-n}^{\mu}-\sum_{r>0}i\eta\psi_{-r}^{\mu}\tilde{\psi}_{-r}^{\mu}))|B,\eta>_{RR}^{(0)NSNS}
q_{I}^{adj}(x,y+L)=U_{g}q_{I}^{adj}(x,y)U_{g}^{\dagger},\qquad\phi_{qI}^{adj}(x,y+L)=\mathrm{e}^{i\beta}U_{g}\phi_{qI}^{adj}(x,y)U_{g}^{\dagger}.
\mathrm{d}s_{K}^{2}=\sum_{m,n\neqk}^{\infty}{\frac{\partial^{2}K(z_{(k)},{\bar{z}}_{(k)})}{\partialz_{(k)}^{m}\partial{\bar{z}}_{(k)}^{n}}}\,\mathrm{d}z_{(k)}^{m}\mathrm{d}{\bar{z}}_{(k)}^{n}.
S_{2}:=\sum_{n\ge0}\frac{2n+1}{(3n+1)^{2}(3n+2)^{2}}=\frac{2}{3^{3/2}}\mathrm{Cl}_{2}(2\pi/3)=\frac{4}{3^{5/2}}\mathrm{Cl}_{2}(\pi/3)
\langleV[\phi]\rangle=\sum_{n}\frac{1}{(2n)!}\frac{d^{2n}V[\phi_{c}]}{d\phi_{c}^{2n}}\langle(\phi-\phi_{c})^{2n}\rangle=\sum_{n}\frac{1}{2^{n}n!}G^{n}\frac{d^{2n}V[\phi_{c}]}{d\phi_{c}^{2n}}
W_{N}(g,x)=\sum_{n=0}^{N}c_{n}g^{n}{\frac{1}{x^{3n-1}}}\sum_{k=0}^{N-n}{\frac{\Gamma(3n/2+k-1/2)}{k!\,\Gamma(3n/2-1/2)}}\left(1-{\frac{1}{x^{2}}}\right)^{k},
{\langle}\mathrm{{\Omega}}|T\left({\Gamma}_{i}(x){\Gamma}_{j}(y)\right)|{\mathrm{\Omega}}{\rangle}=\frac{1}{(2{\pi})^{4}}\intd^{4}qD_{ij}^{g}(q)e^{-iq\cdot(x-y)}
\sum_{n=-\infty}^{n=\infty}\mathrm{e}^{i\pin-\left(n\beta\right)^{2}/2t}=\frac{\sqrt{2\pit}}{\beta}\sum_{n=-\infty}^{n=\infty}\mathrm{e}^{-2\pi^{2}\,t\left(n+1/2\right)^{2}/\beta^{2}},