harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Z[j]=e^{\frac{i}{\hbar}\intdx^{4}{\cal{L}}_{I}(\frac{\hbar}{i}\frac{\delta}{\deltaj})}Z_{0}[j]\textrm{with}Z_{0}[j]=\int_{\Lambda}{\cal{D}}\eta\;e^{\frac{i}{\hbar}\intdx^{4}\left(\frac{1}{2}[(\partial\eta)^{2}-m^{2}\eta^{2}]+j\eta\right)},
H^{2}=\left(\frac{\dot{R}}{R}\right)^{2}=\frac{8\piG_{4}}{3}\,\Lambda_{4}-\frac{k}{R^{2}}+\left(\mu+\frac{8\piG_{4}}{3}\,\hat{\rho}\right)\frac{1}{R^{4}}-\frac{q^{2}}{R^{6}}+\frac{4\piG_{4}}{3\sigma}\,\frac{\hat{\rho}^{2}}{R^{8}}\,.
\left(\frac{x^{-}}{x^{+}}\right)^{2}F(x^{-},x^{+})=\sum_{i}\left|{\frac{L}{\pi}}\langle0|T^{++}(K)|i\rangle\right|_{M_{i}=0}^{2}{\frac{6}{K^{3}\pi^{2}r^{4}}}\,.
T_{jk}^{i}=-\frac14N_{jk}^{i},\quadN_{jk}^{i}=J_{k}^{n}\partial_{n}J_{j}^{i}-J_{j}^{n}\partial_{n}J_{k}^{i}+J_{n}^{i}\partial_{j}J_{k}^{n}-J_{n}^{i}\partial_{k}J_{j}^{n}
\kappa^{2}\delta\rho=-6\left(\dot{\alpha_{0}}\dot{\xi}_{0}^{y}-\dot{\alpha}_{0}^{2}\xi_{0}^{y}-\dot{\alpha}_{0}^{2}\alpha_{1}l^{2}\phi_{0}+\dot{\alpha_{0}}\ddot{\alpha}_{0}e^{\alpha_{0}}lb_{0}-\frac{\dot{\alpha}_{0}\ddot{\alpha}_{0}}{\alpha_{1}}e^{2\alpha_{0}}\dot{\hat{E}}_{0}-\hat{\Psi}_{1}\right),
\sum_{(\tilde{n}_{1},\tilde{n}_{2})\neq(0,0)}^{\infty}\langleN|\mathcal{O}(\tilde{n}_{1}+\tilde{n}_{2})|N\rangle=\sum_{N=1}^{\infty}(N+1)\langleN|\mathcal{O}(N)|N\rangle\,,
W_{\nu}^{M}(z,z^{\prime})=e^{(v+v^{\prime})(\frac{d}{2}+\nu)}\widetilde{W}_{\nu}\left(\eta\left(v,z\right),\eta^{\prime}\left(v^{\prime},z^{\prime}\right)\right)=\frac{1}{2\pi^{\frac{d}2}}\frac{\Gamma(\nu+\frac{d}2)}{\Gamma(\nu+1)}\frac1{{[-(z-z^{\prime})^{2}]}^{\frac{d}2+\nu}}.
\{\tilde{\Theta}_{\alpha}^{\mu}(q,\phi),\tilde{\Theta}_{\beta}^{\nu}(q,\phi)\}=0~~;~~\tilde{\Theta}_{\alpha}^{\mu}(q,\phi)=\Theta_{\alpha}^{\mu}(q)+\Sigma_{n=1}^{\infty}\tilde{\Theta}_{\alpha}^{\mu(n)}(q,\phi)~~;~~\tilde{\Theta}^{\mu(n)}\approxO(\phi^{n}),
\lambdav_{T}-\frac{1}{\beta}\sum_{n\epsilonZ}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{1}{\left[{\bfk}^{2}+v_{T}^{2}+\frac{4\pi^{2}}{\beta}\left(n+\frac{1}{2}\right)^{2}\right]}=0\,\,\,\,.
\begin{array}{ccc}{\tau_{AB}^{1}=2i\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right),}&{\tau_{AB}^{2}=2i\left(\begin{array}{cc}{i}&{0}\\{0}&{i}\\\end{array}\right),}&{\tau_{AB}^{3}=2i\left(\begin{array}{cc}{0}&{-1}\\{-1}&{0}\\\end{array}\right),}\\\end{array}
3\Theta_{\hat{\rho}}^{\parallel}E_{s}\mids\rangle=\Theta_{\hat{\alpha}}^{\parallel}(\gamma^{u}\gamma^{t})_{\hat{\rho}\hat{\alpha}}E_{u}M_{tv}^{\parallel}E_{v}E_{s}\mids\rangle+\Theta_{\hat{\rho}}^{\parallel}E_{s}\mids\rangle-\Theta_{\hat{\rho}}^{\parallel}E_{s}\mids\rangle\frac{1}{\pi^{2}}\int_{-\infty}^{+\infty}e^{-ry^{2}}\frac{1}{2}ry^{2}d^{8}(y\sqrt{r})
{\calB}^{(\varphi)}=\frac\ell{16\piG}\int_{\calV}\partial_{\varphi}\left[2\,Q_{2}^{-1}\partial_{r}Q_{2}Q^{-1}\partial_{t}Q-\frac12\partial_{t}\Phi\partial_{r}\Phi-2\partial_{t}X\partial_{r}Ye^{-\Phi}\right]\,dr\,d\varphi\,dt\qquad,\qquad
\begin{array}{c}{\left(q^{S_{3}}\otimes{\frac{E}{z}}+{\frac{E}{w}}\otimesq^{-S_{3}}\right)\left|\left({\frac{n}{2}},{\frac{n}{2}}\right)_{p},p\right\rangle=}\\{=\left({\frac{q^{n/2}}{z}}-{\frac{q^{-2p-2+{\frac{n}{2}}}}{w}}\right)\left|{\frac{n}{2}},{\frac{n}{2}}\right\rangle\otimes\left|{\frac{n}{2}},p-{\frac{n}{2}}+1\right\rangle+\ldots}\\\end{array}
I=\frac{1}{16\piG_{N\pm}}\intd^{4}y\sqrt{-\bar{q}_{\pm}}\left[\bar{R}_{\pm}+\frac{1}{6}l_{S\pm}^{2}\bar{R}_{\pm}^{2}+\frac{1}{3}l_{T\pm}^{2}(3\bar{R}_{\pm\nu}^{\mu}\bar{R}_{\pm\mu}^{\nu}-\bar{R}_{\pm}^{2})+a_{\pm}(\bar{R}_{\pm\\rho\sigma}^{\mu\nu}\bar{R}_{\pm\\mu\nu}^{\rho\sigma}-4\bar{R}_{\pm\nu}^{\mu}\bar{R}_{\pm\mu}^{\nu}+\bar{R}_{+}^{2})\right],
Z_{\,\,{\overline{\Psi}}^{\,\,\prime}}\,\,[J_{A}]\,-\,Z_{\,{\overline{\Psi}}}\,[J_{A}]\,=\,-{\frac{i}{\hbar}}\int[d\varphi^{I}]\epsilon^{a}\,{\frac{\partial}{\partial\chi^{a}}}\Big(S_{1}\,+\,\hbarM_{1}\Big)exp{\frac{i}{\hbar}}\Big(S_{_\Sigma}+M_{1}\,+\,J^{A}\,\phi_{A}\Big)
k_{1}^{\mu}T_{\rho\sigma\mu\nu}^{(1)abcd}(p_{1},p_{2};k_{1},k_{2})=C_{1}k_{2\nu}S_{\rho\sigma}^{(1)}(q_{1})+C_{1}\Gamma_{\sigma\nu\rho}^{(1)}(p_{2},k_{2},q_{1})
ds^{2}=e^{2W(z)}\left(e^{2\sqrt{2}b\kappa\phi(t)}dz^{2}-dt^{2}+e^{2\alpha(t)}\left(-2T_{i}dzdx^{i}-2S_{i}dtdx^{i}+\delta_{ij}dx^{i}dx^{j}\right)\right),
{\calB}_{WZWC}^{(\varphi)}+{\calB}_{WZW}^{(\varphi)}=\frac\ell{16\piG}\int_{\calV}\partial_{\varphi}\left[2\,Q_{2}^{-1}\partial_{r}Q_{2}Q^{-1}\partial_{t}Q-Q^{-1}\partial_{r}QQ^{-1}\partial_{t}Q\right]\,dr\,d\varphi\,dt\,.
V=M^{2}-\gamma^{k}M_{,k}+\frac14\bar{R}+{\frac{1}{16}}F^{jk}F_{jk}-\left(eA_{0}+\fraci8F\right)^{2}+{\frac{e}{2}}({\calF}-2\breve{\gamma}^{0}\gamma^{k}\nabla_{k}A_{0})~.
Q(\xi)=\frac{1}{16\pi}\intd\varphi\left\{\frac{\xi^{\bot}}{lr}\left((g_{\varphi\varphi}-r^{2})-2r^{2}(lg^{-1/2}-1)+4r^{2}\left[\phi^{2}+\frac{1}{3}\phi^{4}\right]\right)+2\xi^{\varphi}\pi_{\varphi}^{r}\right\}\,,
\delta\tilde{A}_{\mu}=e^{({\frac{1}{2}}+\beta)A}\bar{\tilde{\epsilon}}_{+i}\tilde{\psi}_{\mu-}^{i}+e^{({\frac{1}{2}}+\gamma)A}\bar{\tilde{\epsilon}}_{+i}\tilde{\Gamma}_{\mu}\tilde{\chi}_{+}^{i}+c.c.\.
\{L_{1}(u),L_{2}(v)\}=[r(u-v),L_{1}(u)+L_{2}(v)],\quadr(u)={\frac{1}{u}}\left(\begin{array}{llll}{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{1}\\\end{array}\right),
\Bigl(J(x)-\hbar\intdy\bigl(C^{\Lambda,\Lambda_{0}}\bigr)^{-1}(x-y)\frac{\delta}{\deltaJ(y)}\Bigr)Z^{\Lambda,\Lambda_{0}}(J)=-\hbar\frac{\delta}{\delta\varrho(x)}\tilde{Z}^{\Lambda,\Lambda_{0}}(\varrho\,;J)|_{\varrho(x)=0}.
\xi_{\lambda}=a\left(\frac{\alpha}{2\lambdac}\right)^{\nu}r^{\alpha-1}J_{\nu}\left(\frac{2\lambdac}{\alpha}r^{\alpha}\right)+b\left(\frac{\alpha}{2\lambdac}\right)^{-\nu}r^{\alpha-1}J_{-\nu}\left(\frac{2\lambdac}{\alpha}r^{\alpha}\right)
\begin{array}{cc}{(1,2)\mathrm{~D2~charge~}}&{\frac{1}{(2\pi)^{2}}\Sigma_{a=1}^{N}\mathcal{F}_{34}^{a}\mathcal{F}_{56}^{a}}\\{(3,4)\mathrm{~D2~charge~}}&{\frac{1}{(2\pi)^{2}}\Sigma_{a=1}^{N}\mathcal{F}_{56}^{a}\mathcal{F}_{12}^{a}}\\{(5,6)\mathrm{~D2~charge~}}&{\frac{1}{(2\pi)^{2}}\Sigma_{a=1}^{N}\mathcal{F}_{12}^{a}\mathcal{F}_{34}^{a}.}\\\end{array}
S=\intd^{4}x\sqrt{\hat{g}}\Bigl\{\hat{R}+\frac{2\hat{\nabla}_{\mu}\Psi\hat{\nabla}^{\mu}\Psi^{\dagger}}{(\Psi-\Psi^{\dagger})^{2}}+\frac{1}{8}Tr\bigl({\calL}\hat{\nabla}{\calM}\bigr)^{2}+\frac{i}{16}\Psi\hat{\calF}_{\mu\nu}^{-T}{\calM}\hat{\calF}^{-\mu\nu}-\frac{i}{16}\Psi^{\dagger}\hat{\calF}_{\mu\nu}^{+T}{\calM}\hat{\calF}^{+\mu\nu}\Bigr\}
{\Bigl({\frac{d^{2}}{{d\tau}^{2}}}-{\frac{B_{ab}}{\bar{T}}}\Bigl)}^{-1}={\Bigl({\frac{d}{d\tau}}\Bigr)}^{-2}+{\Bigl({\frac{d}{d\tau}}\Bigr)}^{-2}{\frac{B_{ab}}{\bar{T}}}{\Bigl({\frac{d}{d\tau}}\Bigr)}^{-2}+{\Bigl({\frac{d}{d\tau}}\Bigr)}^{-2}{\frac{B_{ab}}{\bar{T}}}{\Bigl({\frac{d}{d\tau}}\Bigr)}^{-2}{\frac{B_{ab}}{\bar{T}}}{\Bigl({\frac{d}{d\tau}}\Bigr)}^{-2}+\cdots
[b_{{\bf{0}}},b_{{\bf{0}}}^{\dagger}]=b_{{\bf{0}}}b_{{\bf{0}}}^{\dagger}-b_{{\bf{0}}}^{\dagger}b_{{\bf{0}}}=exp(-i\mathrm{~}X_{0})\mathrm{~}N_{0}\mathrm{~}exp(i\mathrm{~}X_{0})-N_{0}=1
v^{2}-\frac{2M^{2}r^{2}}{S}u=M^{2}r^{2}\left(1-\frac{2E}{M^{2}}-\frac{r^{2}}{L^{2}}-l^{2}\left(\frac{1}{S^{2}}-\frac{1}{M^{2}L^{2}}\right)\right)\equiv\bar{\theta}
L_{\mu\nu}=\left(\begin{array}{cccc}{0}&{k_{1}-k_{3}}&{-k_{1}+k_{2}}&{k_{3}-k_{2}}\\{-k_{1}+k_{3}}&{0}&{k_{1}}&{-k_{3}}\\{k_{1}-k_{2}}&{-k_{1}}&{0}&{k_{2}}\\{-k_{3}+k_{2}}&{k_{3}}&{-k_{2}}&{0}\\\end{array}\right)
f^{(2)}=\left(\begin{array}{cccc}{0}&{-\delta_{ij}}&{-a_{i}}&{-\pi_{i}}\\{\delta_{ij}}&{0}&{0}&{-a_{i}}\\{a_{i}}&{0}&{0}&{0}\\{\pi_{i}}&{a_{i}}&{0}&{0}\\\end{array}\right)
\partial_{t}\left(\frac{1}{n}ba^{3}\dot{\Phi}\right)-\partial_{y}\left(\frac{1}{b}a^{3}n\Phi^{\prime}\right)+a^{3}n\left[bV^{\prime}+V_{0}^{\prime}\delta(y)+V_{1}^{\prime}\delta(y-1)\right]=0,
|g.s.>=|0,0,-3J,0>=\frac{1}{\sqrt{12}}(2|\uparrow\downarrow\uparrow\downarrow>+2|\downarrow\uparrow\downarrow\uparrow>-|\uparrow\uparrow\downarrow\downarrow>-|\uparrow\downarrow\downarrow\uparrow>-|\downarrow\downarrow\uparrow\uparrow>-|\downarrow\uparrow\uparrow\downarrow>)\quad.
S=\frac{1}{2\pi\alpha^{\prime}}\intd^{2}\sigma\left(\,-\frac{1}{2}\sqrt{-\gamma}\gamma^{mn}\Pi_{m}^{r}\Pi_{n}^{s}\eta_{rs}+\frac{1}{2!}\epsilon^{mn}\Pi_{m}^{A}\Pi_{n}^{B}B_{BA}\,\right)~,
\left[C_{(n,m)\psi_{1,1}}^{\alpha}\right]\left[C_{(1,2)\psi_{1,1}}^{\alpha}\right]=-\frac{s\left(\alpha_{-}^{2}\right)}{s\left(2\alpha_{-}^{2}\right)}\frac{\Gamma^{2}\left(1-\alpha_{-}^{2}\right)}{\Gamma\left(2-2\alpha_{-}^{2}\right)}\left(A+B\right)=A_{n,m}^{\alpha}A_{1,2}^{\alpha}
\begin{array}{rcl}{ds^{2}}&{=}&{ds_{0}^{2}-ds_{n}^{2}}\\{}&{=}&{\displaystyle\frac{(d\chi^{0})^{2}}{[1+(\chi^{0})^{2}]^{2}}-\sum_{\mu,\nu=1}^{n}d\chi^{\mu}d\chi^{\nu}\left(\frac{\delta_{\mu\nu}}{1-\displaystyle\sum_{\alpha=1}^{n}\chi^{\alpha}\chi^{\alpha}}+\frac{\chi^{\mu}\chi^{\nu}}{(1-\displaystyle\sum_{\alpha=1}^{n}\chi^{\alpha}\chi^{\alpha})^{2}}\right)~.}\\\end{array}
[L_{\bar{z}^{l}},\frac{\partial\Phi}{\partial\bar{z}^{l^{\prime}}}]\equiv[A+\nuB,\frac{\partial\Phi_{0}}{\partial\bar{z}^{l^{\prime}}}+\nu(\frac{\partial\Phi_{1}}{\partial\bar{z}^{l^{\prime}}}+\frac{\partial}{\partial\bar{z}^{l^{\prime}}})]\(\mathrm{mod}\\nu^{2}).
\left(\begin{array}{c}{\psi_{1}}\\{\psi_{2}}\\\end{array}\right)\longrightarrow\left(\begin{array}{c}{\widetilde\psi_{1}}\\{\widetilde\psi_{2}}\\\end{array}\right)=\left(\begin{array}{c}{A}\\{C}\\\end{array}\begin{array}{cc}{B}\\{D}\\\end{array}\right)\left(\begin{array}{c}{\psi_{1}}\\{\psi_{2}}\\\end{array}\right).
A(-i\Omega_{m},\stackrel{\rightharpoonup}{p},\stackrel{\rightharpoonup}{l})=T\sum_{n}\frac{1}{(\omega_{n}+i\mu)^{2}+\omega_{l}^{2}}\frac{1}{(\omega_{n}+\Omega_{m}+i\mu)^{2}+\omega_{l+p}^{2}},
\widetilde{J}_{3}^{(1)}=L_{3}+s_{3},\quad\widetilde{J}_{-}^{(1)}=L_{-}+s_{3}n_{-}/(1+n_{3}),\quad\widetilde{J}_{+}^{(1)}=L_{+}+s_{3}n_{+}/(1+n_{3}),
{\calH}^{(0)}=-\frac{2\kappa}{\sqrt{-\gamma}}\frac{\deltaI^{(0)}}{\delta\gamma_{\mu\nu}}\frac{\deltaI^{(0)}}{\delta\gamma_{\lambda\rho}}\left(\gamma_{\mu\rho}\gamma_{\nu\lambda}-\frac{1}{d-1}\gamma_{\mu\nu}\gamma_{\lambda\rho}\right)+\frac{\sqrt{-\gamma}}{\kappa}\Lambda=0,
\begin{array}{lclclcl}{\partial_{a}\,\varepsilon=0}&{;}&{\gamma_{5}\,\varepsilon=\varepsilon}&{;}&{\eta=H^{-1/8}(z,\bar{z},y)\,\theta\,\otimes\,\xi}&{;}&{0=\widehat{\mathcal{D}}\xi}\\{\overline{{\gamma}}_{5}\xi_{-}=-\xi_{-}}&{;}&{\sigma_{3}\,\theta\,=-\theta}&{;}&{0=\partial_{z}\theta}&{;}&{0=\partial_{\bar{z}}\theta}\\\end{array}
G_{\bfk}(x_{1},x_{2};x_{3},x_{4})=\sum_{j=0}^{k_{3}}\sum_{i=0}^{3}(-1)^{2-i_{1}-i_{2}}\left(\begin{matrix}{k_{3}}\\{j}\\\end{matrix}\right)(x_{1}x_{2}x_{3}x_{4})^{\beta}x_{1}^{N_{1}}x_{2}^{N_{2}}x_{3}^{N_{3}}x_{4}^{N_{4}},
s(0)={\frac{2}{(2\pi)^{2}N}}\left[\int_{0}^{\infty}dy{\frac{2s(y)}{(y+0.125)(y^{2}+s^{2}(y))}}+{\frac{1}{12}}\int_{0}^{\infty}dy{\frac{s(0)}{(y+0.125)^{2}(y^{2}+s^{2}(y))}}\right]
{\calZ}~=~\int_{\eta,v_{\mu}}~e^{-S_{\eta}~-~\int_{x}v_{\mu}v_{\mu}~-~2\int{\frac{d\beta}{\beta}}\int_{x}e^{-\intd\tau{\frac{\dot{x}_{\mu}^{2}}{4}}~+~V(x)~+~(-1)^{2s}is~\Omega}}
L_{n}={\frac{1}{2}}\sum_{m=-\infty}^{+\infty}:\alpha_{-m}\cdot\alpha_{n+m}:+{\frac{1}{2}}\sum_{m=-\infty}^{+\infty}:\beta_{-m}\beta_{n+m}:\;-2\sqrt{\beta}ik\beta_{k}+2\beta\delta_{n,0}\;\;\;,
ds^{2}=(Ff_{1}\cdotsf_{7}f_{8})^{1/2}\;(a_{1}(r)\cdotsa_{7}(r))^{1/2}\times\left({\frac{(\sigma_{1})^{2}}{f_{1}^{2}a_{1}^{2}}}+\cdots+{\frac{(\sigma_{7})^{2}}{f_{7}^{2}a_{7}^{2}}}+{\frac{dr^{2}}{f_{8}^{2}}}\right).
E_{x}(x,y,z,t)=q\frac{\Bigl(x-x_{0}-R\frac{at_{0}}{c}\Bigr)\Bigl(1-\frac{a^{2}t_{0}^{2}}{c^{2}}\Bigr)}{\Bigl(R-(x-x_{0})\frac{at_{0}}{c}\Bigr)^{3}}-q\frac{a\Bigl((y-y_{0})^{2}+(z-z_{0})^{2}\Bigr)}{c^{2}\Bigl(R-(x-x_{0})\frac{at_{0}}{c}\Bigr)^{3}}.
A=\int_{0}^{\infty}{\frac{d\tau_{2}}{\tau_{2}^{2}}}\intd^{4}\eta\delta(1-\sum_{i}\eta_{i})e^{{\frac{2\pi\tau_{2}}{M_{s}^{2}}}(s\eta_{1}\eta_{3}+t\eta_{2}\eta_{4})}I^{s/M_{s}^{2}}J^{t/M_{s}^{2}}\,
\bar{J}=\pi^{2}\sum_{k\geq2,\,l\geq0}\frac{(k+l-1)!(k+l)!}{l!(2k+l+1)!}\left(-\frac{p^{2}}{\sigma^{2}}\right)^{k}\left(1-\frac{m_{3}^{2}}{\sigma^{2}}\right)^{l}
\left\{{\frac{Det^{\prime}\left[-(d^{2}/d\tau^{2})+V^{\prime\prime}[x_{c}(\tau)]\right]}{Det\left[-(d^{2}/d\tau^{2})+\omega^{2}\right]}}\right\}^{-1/2}\,sqrt{{\frac{S_{eo}}{2\pi}}}\d\tau_{c}
\sum_{k=0}^{\infty}\frac{q^{2k^{2}+2\mu_{a}k}}{(q)_{2k+\mu_{a}}}=q^{1/48-\mu_{a}/2}\chi_{\mu_{a}+1,1}^{(4)}(q)\equivq^{1/48-\mu_{a}/2}\chi_{\mu_{a}/2}(q).
\frac{1}{\sqrt{-g^{\prime}}}\partial_{\mu}(\sqrt{-g^{\prime}}g^{\prime\mu\nu}\partial_{\nu}\phi)+\frac{1}{\chi}\left[\frac{dV_{1}}{d\phi}-\frac{1}{\chi}\frac{dV_{2}}{d\phi}+\frac{\alpha}{M_{p}}P(\varphi)e^{\alpha\phi/M_{p}}\right]=-\frac{h\gamma}{\sqrt{2}M_{p}}\overline{{\Psi}}^{\prime}\Psi^{\prime}\varphi\frac{e^{\gamma\phi/M_{p}}}{\chi^{3/2}};
(\delta_{12})^{2}(\delta_{34})^{2}=\delta_{\{A_{1}B_{1}\}}^{A_{2}B_{2}}\delta_{\{A_{3}B_{3}\}}^{A_{4}B_{4}}\;,\qquad\delta_{13}\delta_{14}\delta_{23}\delta_{24}=\delta_{\{A_{1}B_{1}\}}^{\{A_{3}\{B_{4}}\delta_{\{A_{2}B_{2}\}}^{A_{4}\}B_{3}\}}\;,
R_{k}^{b}\propto-i\Bigg(\frac{g}{32\pi^{2}}\Bigg)^{k}\intdt\:e^{-2t}\Bigg[(2t)^{k-1}+(k-1)(2t)^{k-2}e^{-2t}f_{2}(\beta)+\cdot\cdot\cdot\Bigg],
4\,\partial_{i}\partial_{i}\phi=-\left({\frac{\partial_{1}E}{E}}\right)^{2}\left(\partial_{i}h_{1}\right)^{2}-\left({\frac{\partial_{2}E}{E}}\right)^{2}\left(\partial_{i}h_{2}\right)^{2}-2{\frac{\partial_{1}E\partial_{2}E-E\partial_{1}\partial_{2}E}{E^{2}}}\left({\partial_{i}h_{1}}\right)\left({\partial_{i}h_{2}}\right),
G_{\mu\nu}=\left(\begin{array}{ccc}{-g_{0}(1-e^{2})}&{0}&{-g_{0}eb}\\{0}&{g_{1}(1-e^{2})+\frac{g_{0}g_{1}}{g_{2}}b^{2}}&{0}\\{-g_{0}eb}&{0}&{g_{2}+g_{0}b^{2}}\\\end{array}\right)
\intd\sigma{\bfg}_{\mu_{1}\mu}(\sigma_{1},\sigma){\bfg}^{\mu\mu_{2}}(\sigma,\sigma_{2})=\delta_{\mu_{1}}^{\mu_{2}}\delta(\sigma_{1}-\sigma_{2})\equiv{\bfI}_{\mu_{1}}^{\mu_{2}}(\sigma_{1},\sigma_{2})
H=\sum_{a=2}^{3}\frac{H_{a}}{\Lambda^{\prime}(\lambda_{a})}=\sum_{a=2}^{3}\left\{-\frac{2A(\lambda_{a})}{\Lambda^{\prime}(\lambda_{a})}\pi_{a}^{2}-\frac{1}{2}\frac{\lambda_{a}^{2}(\lambda_{a}-\bar{\sigma}_{2}^{2})(\lambda_{a}-\bar{\sigma}_{3}^{2})}{\Lambda^{\prime}(\lambda_{a})}\right\}
\begin{array}{c}{\theta_{2}(\xi,\tau,\lambda)=\chi(\vert\xi\vert^{2}+\vert\tau\vert^{2}+\vert\lambda\vert^{2})}\\{\theta_{1}(\xi,\lambda)=\chi(\vert\xi\vert^{2}+\vert\lambda\vert^{2}),}\\\end{array}
{\frac{\delta}{\delta\phi^{a}(x)}}=\intd^{d}z\intd^{d}y{\frac{\delta\sigma(z,y)}{\delta\phi^{a}(x)}}{\frac{\delta}{\delta\sigma(z,y)}}=\intd^{d}z\phi^{*a}(z){\frac{\delta}{\delta\sigma(z,x)}}
\begin{array}{lcr}{q_{n-1}^{n-1}=q_{n}^{n}=-2,\quadq_{n-1}^{n-2}=q_{n}^{n-2}=1}\\{q_{n-2}^{n-1}=q_{n-2}^{n}=q_{n+3}^{n-1}=q_{n+1}^{n}=q_{n+2}^{n}=1}\\{q_{n}^{n-1}=q_{n-1}^{n}=q_{n+1}^{n-1}=q_{n+1}^{n}=q_{n+2}^{n-1}=0.}\\\end{array}
=\;\mathrm{Tr}\Big[\frac{-\not{\!\partial}+m}{-\triangle+m^{2}}\not{\!\!B}\frac{-\not{\!\partial}-m}{-\triangle+m^{2}}\not{\!\!B}\frac{-\not{\!\partial}+m}{-\triangle+m^{2}}\not{\!\!B}\frac{-\not{\!\partial}-m}{-\triangle+m^{2}}\not{\!\!B}\Big]\;.
T_{MN}=-g_{MN}\Lambda_{b}-g_{\mu\nu}\delta_{M}^{\mu}\delta_{N}^{\nu}\Lambda_{1}\delta(y)+4\cdot4!\left(\frac{4}{H^{4}}H_{MNPQ}H_{N}~^{PQR}+\frac12g_{MN}\frac{1}{H^{2}}\right).
T=\left(\begin{array}{rrrr}{e^{J+J_{1}}}&{e^{-i\varepsilon-J_{1}}}&{e^{-J+J_{1}}}&{e^{i\varepsilon-J_{1}}}\\{e^{i\varepsilon-J_{1}}}&{e^{J+J_{1}}}&{e^{-i\varepsilon-J_{1}}}&{e^{-J+J_{1}}}\\{e^{-J+J_{1}}}&{e^{i\varepsilon-J_{1}}}&{e^{J+J_{1}}}&{e^{-i\varepsilon-J_{1}}}\\{e^{-i\varepsilon-J_{1}}}&{e^{-J+J_{1}}}&{e^{i\varepsilon-J_{1}}}&{e^{J+J_{1}}}\\\end{array}\right)
L_{eff}=-R=\frac{1}{2}A_{ij}^{\mu\nu}\;\partial_{\mu}{\deltay}^{i}\,\partial_{\nu}{\deltay}^{j}+B_{ij}^{\mu}\;{\deltay}^{i}\,\partial_{\mu}\deltay^{j}+\frac{1}{2}C_{ij}\;\deltay^{i}\deltay^{j}\,,
\deltaB\vert_{\Sigma_{2}^{i}}=-2\epsilon_{i}{\frac{T_{3}}{T_{6}}}\chi_{2}^{1}(N_{i})-2\kappa_{11}^{2}T_{6}\chi_{2}^{1}(N^{\prime})\vert_{\Sigma_{2}^{i}},\i=1,2,
\langle\hat{\Phi}^{\dagger}(p)\hat{\Phi}(p)\rangle_{t}=\langle\hat{\Phi}^{\dagger}(p)\hat{\Phi}(p)\rangle_{0}\;\Big(1+(\mu^{2}+m^{2})t^{2}+{\calO}\big(\omega_{-}^{4}(p)t^{4}\big)\Big)\;.
{\calE}_{\pm}^{\mathrm{tot}}=L\lambdaB^{2}+{\calE}_{\pm}(L,m\lambda,eB\lambda^{2})=\frac{\pi^{2}}{e^{2}}\frac{\Phi^{2}}{\lambdaL}+{\calE}_{\pm}(L,m\lambda,\pi\Phi\lambda/L)
B=-\frac{\verte\vertH}{(2\pi)^{2}}\frac{2\sqrt{\pi}\verte\vertH}{\Gamma(1+\delta)}\int_{0^{+}}^{\infty}d\eta\;\eta^{-3/2+1+\delta}\mathrm{e}^{-(4+\beta^{2}m^{2})\eta}\mathrm{csch}^{2}(\beta^{2}\verte\vertH\eta)
{}~{\displaystyle\sum_{\{k_{i}\},\atopi=1,\cdots,6}}R(u_{3},u_{5},u_{6})_{i_{3},i_{5},i_{6}}^{k_{3},k_{5},k_{6}}\overline{{R}}(u_{2},u_{4},u_{6})_{i_{2}i_{4}k_{6}}^{k_{2}k_{4}j_{6}}R(u_{1},u_{4},u_{5})_{i_{1}k_{4}k_{5}}^{k_{1}j_{4}j_{5}}\overline{{R}}(u_{1},u_{2},u_{3})_{k_{1}k_{2}k_{3}}^{j_{1}j_{2}j_{3}}
A=\left(\begin{array}{cc}{a}&{b}\\{c}&{d}\\\end{array}\right),\,R=\left(\begin{array}{cccc}{q}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{0}&{\lambda}&{1}&{0}\\{0}&{0}&{0}&{q}\\\end{array}\right).
{\calD}_{Vv}^{\mathrm{sym}}\equiv{\calD}_{Vv}^{r_{\eta}\,(l)}-r^{(1)}\bigl(8{\calD}_{g^{2}}^{(l+1)}+8l({\calN}_{V}^{(l+1)}-8(l+1)r^{(1)}\delta{\calN}_{V}^{(l+2)})\bigr).
F=\left(\begin{array}{cccc}{0}&{-B_{z}}&{B_{y}}&{E_{x}}\\{B_{z}}&{0}&{-B_{x}}&{E_{y}}\\{-B_{y}}&{B_{x}}&{0}&{E_{z}}\\{-E_{x}}&{-E_{y}}&{-E_{z}}&{0}\\\end{array}\right).
\int\!{\calD}_{\mu}\bar{\phi}{\calD}_{\mu}\phi\,\mathrm{d}^{4}x=-\int\!\bar{\phi}{\calD}^{2}\phi\mathrm{d}^{4}x+\int\!\mathrm{d}\Omega_{\mu}\partial^{\mu}\left(\bar{\phi}{\calD}_{\mu}\phi\right)\mathrm{d}^{4}x=\int\!\mathrm{d}\Omega_{\mu}\,\partial^{\mu}\left(\bar{\phi}{\calD}_{\mu}\phi\right)\mathrm{d}^{4}x\,.
S=\frac{1}{16\piG}\int_{M}\sqrt{-^{(3)}g}(^{(3)}R+\frac{2}{l^{2}})d^{3}x+\frac{1}{8\piG}\int_{\partialM}\sqrt{-^{(3)}g}(2D_{\mu}n^{\mu}+\frac{1}{l})d^{2}x.
\begin{array}{llll}{\gamma_{1}(e)=+1}&{\gamma_{1}(h_{1})=+1}&{\gamma_{1}(h_{2})=+1}&{\gamma_{1}(h_{3})=+1,}\\{\gamma_{2}(e)=+1}&{\gamma_{2}(h_{1})=+1}&{\gamma_{2}(h_{2})=-1}&{\gamma_{2}(h_{3})=-1,}\\{\gamma_{3}(e)=+1}&{\gamma_{3}(h_{1})=-1}&{\gamma_{3}(h_{2})=+1}&{\gamma_{3}(h_{3})=-1,}\\{\gamma_{4}(e)=+1}&{\gamma_{4}(h_{1})=-1}&{\gamma_{4}(h_{2})=-1}&{\gamma_{4}(h_{3})=+1.}\\\end{array}
e^{-F(\vec{x}_{1},\ldots,\vec{x}_{m},\vec{y}_{1},\ldots,\vec{y}_{n})/T}=\langleP(\vec{x}_{1})\ldotsP(\vec{x}_{m})P^{\dagger}(\vec{y}_{1})\ldotsP^{\dagger}(\vec{y}_{n})\rangle
diag\big({\frac{q}{3}}+{\frac{r}{2}}-{\frac{s}{3}},{\frac{q}{3}}-{\frac{r}{2}}-{\frac{s}{3}},-{\frac{2q}{3}}-{\frac{s}{3}},{\frac{t}{2}}+{\frac{s}{2}},-{\frac{t}{2}}+{\frac{s}{2}}\big)
S_{\psi}^{\mu\nu}=\frac14L_{.\alpha}^{\mu}(p_{s},\stackrel{o}p_{s})L_{.\beta}^{\nu}(p_{s},\stackrel{o}p_{s})\int{d^{3}\sigma\stackrel{o}{\bar{\psi}}(\tau,\vec{\sigma})\big[\gamma^{\bar{o}},\sigma^{\alpha\beta}\big]_{+}\stackrel{o}{\psi}(\tau,\vec{\sigma})},
{\calL}=-\frac{1}{k^{2}}R(e,\psi)+\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}\bar{\psi}_{\mu}\gamma_{5}\gamma_{\nu}\nabla_{\rho}\psi_{\sigma}+\frac{1}{3}(A_{\mu}^{2}-S^{2}-P^{2})\,,
Z_{4}=\frac{(mL)^{4}}{4!}I_{1}^{4}-(mL)^{3}I_{1}^{2}I_{2}+\frac{(ML)^{2}}{2}\left[\frac{2}{3}I_{1}I_{3}+\left(\frac{I_{2}}{2}\right)^{2}\right]-\frac{(mL)}{4}I_{4}\,\,\,.
\hat{K}_{ij}=\left(\begin{matrix}{fg^{\mu\alpha}}&{0}&{0}&{0}\\{0}&{Z-{\frac{{C^{\prime}}^{2}}{C}}}&{C^{\prime}/2}&{0}\\{0}&{C^{\prime}/2}&{0}&{0}\\{0}&{0}&{0}&{-{\frac{C}{2}}P^{\mu\nu,\alpha\beta}}\\\end{matrix}\right)
{\calL}_{\mathrm{mass}}^{W}=-{\frac{1}{2}}\left(\begin{array}{ccc}{\psi_{X}^{\dot{\alpha}}}&{\psi_{Y}^{\dot{\alpha}}}&{\psi_{S}^{\dot{\alpha}}}\\\end{array}\right)M_{\psi}^{W}\left(\begin{array}{c}{\psi_{X\dot{\alpha}}}\\{\psi_{Y\dot{\alpha}}}\\{\psi_{S\dot{\alpha}}}\\\end{array}\right)+\mathrm{h.c.},
=\sum_{m_{1},m_{2},m_{3}}q^{\frac{2}{3}(m_{1}-m_{3})}\left(\begin{array}{ccc}{j_{1}}&{j_{2}}&{j_{3}}\\{m_{1}}&{m_{2}}&{m_{3}}\\\end{array}\right)_{q}|j_{1}m_{1}>|j_{2}m_{2}>|j_{3}m_{3}>=
{\mathbbP}H^{0}(dP_{9},{\calO}_{dP_{9}}(3\sigma|_{dP_{9}}+4F))\simeq{\mathbbP}H^{0}({\calS},{\calO}_{{\calS}}(4)\oplus{\calO}_{{\calS}}(2){\calO}_{{\calS}}(1))\simeq{\mathbbP}^{9}.
\left.\mathrm{Tr}K_{\delta}^{(0)}(s)\right|_{{\calC}_{\beta}}={\frac{\beta}{2\pi}}\left.\mathrm{Tr}K^{(0)}(s)\right|_{R^{2}}+{\frac{\beta}{24\pi}}\left(\left({\frac{2\pi}{\beta}}\right)^{2}-1\right)-{\frac{\delta}{4\pi\beta}}(2\pi-\delta)~~~.
H(\tilde{z},t)=\frac{F_{\pi}}{2\pi}\\left(1.6+3.7\,\,\sqrt{4{{\calJ}_{z}}^{2}+25(\,\,{\calI}^{2}-{{\calJ}_{z}}^{2}\,\,)}+20{{\calJ}_{z}}^{2}+50(\,\,{\calI}^{2}-{{\calJ}_{z}}^{2}\,\,)\right)
{\calX}{\bar{H}}_{\mathrm{\scriptsize~I}}=\Xi^{\dagger}{\calA}^{\dagger}H_{\mathrm{\scriptsize~I}}+\left(1-{\calX}\right)H_{0}-\left(1-\Xi^{\dagger}{\calA}^{\dagger}\right)H_{0}+{\calX}{\bar{H}}\left(1-\Xi^{\dagger}{\calA}^{\dagger}\right).
K(i,j)=\int_{0}^{1}dw_{i}w_{i}^{-2+\hat{\beta}^{2}}(1-w_{i})^{\gamma_{i}}\int_{0}^{w_{i}}dw_{j}w_{j}^{-2+\hat{\beta}^{2}}(1-w_{j})^{\gamma_{j}}(w_{i}-w_{j})^{-\hat{\beta}^{2}},
F^{\mu\nu}=\left(\begin{array}{cccc}{0}&{-l_{x}}&{-l_{y}}&{-l_{z}}\\{l_{x}}&{0}&{-w_{z}}&{w_{y}}\\{l_{y}}&{w_{z}}&{0}&{-w_{x}}\\{l_{z}}&{-w_{y}}&{w_{x}}&{0}\\\end{array}\right).
G^{(A_{\mu})}(p,n)=\left[\frac{-i}{p^{2}-\mathcal{M}^{n}(\mathcal{M}^{n})^{T}}\left(g_{\mu\nu}-\frac{(1-\xi)p_{\nu}}{p^{2}-\xi\mathcal{M}^{n}(\mathcal{M}^{n})^{T}}\right)\right]^{AB},
\left.\pi^{\frac{1}{2}}\frac{\Gamma(s-{\frac{1}{2}})}{\Gamma(s)}\zeta_{\varepsilon_{n}}(s-1/2)\right|_{s=0}=\pi^{\frac{1}{2}}\Gamma\left(-{\frac{1}{2}}\right)A_{-1}=-2\piA_{-1}.
V\simG_{N}{\frac{m_{1}m_{2}}{r}}+\int_{0}^{\infty}{\frac{dm}{k}}G_{N}{\frac{m_{1}m_{2}e^{-mr}}{r}}e^{3ky_{0}}\simG_{N}{\frac{m_{1}m_{2}}{r}}\left(1+{\frac{e^{3ky_{0}}}{kr}}\right)\;.
{\calL}_{St}=3D-{\frac{1}{4}}A_{\mu\nu}A^{\mu\nu}+{\frac{\mu}{=4}}\epsilon_{\mu\nu\lambda}A^{\mu\nu}A^{\lambda}+{\frac{m^{2}}{2}}(A_{\mu}=-\partial_{\mu}\theta)(A^{\mu}-\partial^{\mu}\theta),
\Gamma^{(int)}=\frac{i}{2^{n}n!^{2}}\varepsilon_{a_{1}a_{2}\ldotsa_{d-1}a_{d}}S_{v}^{a_{1}a_{2}}S_{v}^{a_{3}a_{4}}\ldots=\frac{i}{n!^{2}}\varepsilon_{a_{1}a_{2}\ldotsa_{d-1}a_{d}}\theta^{a_{1}}p_{\theta}^{a_{2}}\theta^{a_{3}}p_{\theta}^{a_{4}}\ldots.
\pi:=R_{k-1;k}^{V_{z_{1}},V_{z_{2}}}R_{k-2,k-1;k}^{V_{z_{1}}\otimesV_{z_{2}},V_{z_{3}}}\cdotsR_{1,\cdots,k-1;k}^{V_{z_{1}}\otimes\cdots\otimesV_{z_{k-1}},V_{z_{k}}}.
\begin{array}{l}{\,\,\,\,\,[f_{1},f_{2}]_{*^{(2)}}(\xi)=[f_{1},f_{2}]_{*_{0}}(\xi)+}\\{+\hbar^{2}c_{n}\biggl(f_{1}(\xi)(1-{\frac{1}{2}}\overleftarrow\partial\!\!_{\alpha}\xi^{\alpha})\varepsilon^{[\beta]_{n}}[\partial_{\beta}]^{n}f_{2}(\xi)-f_{1}(\xi)[\overleftarrow\partial\!\!_{\alpha}]^{n}\varepsilon^{[\alpha]_{n}}(1-{\frac{1}{2}}\xi^{\beta}\partial_{\beta})f_{2}(\xi)\biggr).}\\\end{array}
P_{+}=\int_{-{\infty}}^{\infty}\{i{\Psi}_{+}^{\ast}{\partial}_{+}{\Psi}_{+}-({\partial}_{-}A_{+})^{2}\}dx^{-}+\int_{-{\infty}}^{\infty}T_{++}(x^{+},x^{-}={\pm}{\infty})dx^{+}.
N_{I,\bar{I}}=\frac{1}{2^{r}}\prod_{i=1}^{r}\left(\delta_{l_{i},\bar{l}_{i}}\delta_{m_{i},\bar{m}_{i}}+\delta_{k_{i}-l_{i}\bar{l}_{i}}\delta_{m_{i}+k_{i}+2,\bar{m}_{i}}\right)~.
\frac{t}{d}=\oint\frac{dz}{2\pi\imath}U_{eff}^{\prime}\left(z+\frac{R(t)}{z}\right)=\left(\frac{m_{0}^{2}}{d}-\Gamma_{2}\right)R(t)-\sum_{n>2}\frac{(2n)!}{(n!)^{2}}\Gamma_{2n}R(t)^{n}
T_{\uparrow\downarrowX}=\frac{e^{4}}{m\theta^{2}}\int_{0}^{\Lambda}\frac{d^{2}k}{(2\pi)^{2}}\frac{(k-p)_{-}}{({\vec{k}}-{\vec{p}})^{2}}\,\frac{(k-p^{\prime})_{+}}{({\vec{k}}-\vec{p}^{\phantom{x}\prime})^{2}}+\frac{e^{4}}{\theta^{2}}\int_{\Lambda}^{\infty}\frac{d^{2}k}{(2\pi)^{2}}\frac{1}{w_{k}\vec{k}^{2}},
V^{\prime}+\frac{2\pi\alpha^{\prime}}{2}V^{\prime}(T^{\prime})^{2}-2\pi\alpha^{\prime}V^{\prime}(T^{\prime})^{2}-\frac{2\pi\alpha^{\prime}}{2}V((T^{\prime})^{2})^{\prime}=V^{\prime}-\frac{2\pi\alpha^{\prime}}{2}(V(T^{\prime})^{2})^{\prime}=0,
F(s+{\frac{3}{2}},1,s+2,{\frac{3}{4}})={\frac{\Gamma(s+2)}{\Gamma(1)\\Gamma(s+1)}}\\int_{0}^{1}(1-t)^{s}\\left(1-{\frac{3t}{4}}\right)^{-s-{\frac{3}{2}}}\dt
\Omega_{\rho}(z_{1},z_{2})=\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}A(n_{1}\!+\!\rho_{1},n_{2}\!+\!\rho_{2})z_{1}^{n_{1}+\rho_{1}}z_{2}^{n_{2}+\rho_{2}},
\left\langle\left[X^{a}\left(x\right),j^{0b}\left(y\right)\right]\right\rangle=\frac{i}{2\pi}\left(\partial_{1}^{x}\delta^{ac}+f^{aec}A_{1}^{e}\left(x\right)\right)\delta\left(x-y\right)\cdot{\mathrmtr}t^{b}t^{c}\,
\vec{y}(\omega)=-{\frac{e\vec{E}^{e}(\omega)/\omega^{2}}{m_{0}+{\frac{2}{3}}e^{2}{\frac{1}{(2\pi)^{3}}}\int{\frac{d\vec{k}}{k^{2}-\omega^{2}}}\,|f(\vec{k})|^{2}}}~~.
J_{0}(0)=-\frac{2\pi}{|\tau|}\left\{-\frac{B_{2}(\tilde{t}_{2})}{2}-\frac{B_{2}(1-\tilde{t}_{2})}{2}\right\}=\frac{2\pi}{|\tau|}B_{2}(\tilde{t}_{2})=\frac{2\pi}{|\tau|}\left(\tilde{t}_{2}^{2}-\tilde{t}_{2}+\frac{1}{6}\right).