harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\gamma_{0}\,=\,\left(\begin{array}{cc}{0}&{\sigma_{3}}\\{\sigma_{3}}&{0}\\\end{array}\right)\,\gamma_{1}\,=\,\left(\begin{array}{cc}{0}&{\sigma_{1}}\\{\sigma_{1}}&{0}\\\end{array}\right)\,\gamma_{2}\,=\,\left(\begin{array}{cc}{0}&{\sigma_{2}}\\{\sigma_{2}}&{0}\\\end{array}\right)\,\gamma_{3}\,=\,\left(\begin{array}{cc}{0}&{iI}\\{-iI}&{0}\\\end{array}\right)
\left.\frac{\partial_{\alpha}\partial^{\beta}\mathcal{V}}{\mathcal{V}}\right|_{0}=\left(\begin{matrix}{2}&{0}&{0}\\{0}&{0}&{0}\\{0}&{0}&{1}\\\end{matrix}\right)\,,\qquad\left.\partial_{\alpha}\partial_{\beta}\mathcal{V}\right|_{0}=0\,.
2{\frac{\partialA^{i}}{\partialz}}=-\Omega_{-}F^{i}=\Omega_{-}K^{-1ij^{*}}\frac{\partial{\calW}^{*}}{\partialA^{*j}},\quad\Omega_{-}\equivi\frac{\langle-iZ_{1}^{*}-Z_{2}^{*}\rangle}{|\langle-iZ_{1}^{*}-Z_{2}^{*}\rangle|}.
L={\frac{1}{2}}\sum_{i=1}^{N}\partial_{\mu}\phi_{i,0}\,\partial_{\mu}\phi_{i,0}+{\frac{\alpha}{2}}\,\biggl(\sum_{i=1}^{N}\phi_{i,0}^{2}-{\frac{1}{g_{0}^{2}}}\biggr)+V_{\mathrm{m}}.
L_{ij}=\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)
\Gamma_{H}~=~\frac14\pi^{2}r_{0}^{2}(a+b)[\frac14+\frac{9}{16}(a+b)^{2}]~\left(e^{\frac{\pi}{2}(a+b)}-1\right)^{-1}\left(e^{\frac32\pi(a+b)}+1\right)^{-1}~\frac{d^{3}k}{(2\pi)^{3}}
\frac{\partial\rho}{\partialt}=\frac{i\hbar}{2m}\nabla\cdot\left(\Psi\nabla\Psi^{\ast}-\Psi^{\ast}\nabla\Psi\right)-\frac{i}{\hbar}\left(\Psi^{\ast}V\star\Psi-\Psi\left(V\star\Psi\right)^{\ast}\right),
V_{x\llb,\,z\llb^{2}/Q_{p}}\toE-{\frac{m_{f}}{2Q_{F}}}(x^{2}+4Q_{p}z)^{\frac{n}{2}}+{\frac{m_{f}J_{\parallel}^{2}}{2E^{2}Q_{F}^{2}}}{\frac{(x^{2}+4Q_{p}z)^{n}}{x^{2}}}.
\left(P^{0}-2\piBR_{1}n\right)^{2}-{\vec{p}}^{~2}=\left(\frac{m}{R_{1}}\right)^{2}+\left(\frac{nR_{1}}{\alpha^{\prime}}\right)^{2}+\frac{2}{\alpha^{\prime}}\left(N+\tilde{N}-2\right)
s_{\mathrm{eff}}[\alpha,\varphi]=\int\!d^{2}x\left({\frac{1}{2}}\,\partial_{\alpha}\varphi\partial_{\alpha}\varphi+{\frac{\alpha-h_{N}}{2}}\,\varphi^{2}-{\frac{\alpha}{2Ng^{2}}}\right)+{\frac{1}{N}}\,\sum_{i=1}^{N-1}s[\alpha-h_{i}],
{\calR}_{M}{}^{N}(z;g)=\left(\begin{array}{cc}{\Omega(z;g)\hat{R}_{\mu}{}^{\nu}(z;g)}&{i\Omega(z;g)^{\frac{1}{2}}(L^{-1}(z;g)\gamma_{\mu}\Sigma^{b}(z;g))^{\beta}}\\{0}&{\Omega(z;g)^{\frac{1}{2}}L^{-1}{}^{\beta}{}_{\alpha}(z;g)U_{a}{}^{b}(z;g)}\\\end{array}\right)\,.
\delta\psi^{i}(x)=\zeta^{i}+i\kappa(\bar{\zeta}^{j}{\gamma}^{\rho}\psi^{j}(x))\partial_{\rho}\psi^{i}(x),\quad\delta{e^{a}}_{\mu}(x)=i\kappa(\bar{\zeta}^{j}{\gamma}^{\rho}\psi^{j}(x))\partial_{[\rho}{e^{a}}_{\mu]}(x),
\omega_{\pm}\equiv{\calQ}_{\pm}(\mp\infty)=\pm2\pi\displaystyle\frac{p-1}{p+1}(S-2S^{\pm})+2\pi\displaystyle\frac{p}{p+1}\left(\displaystyle\frac{\alpha}{\pi}-\delta+2k_{W}^{\pm}-2k_{\pm}\right).
M_{r=\infty}=\left(\begin{matrix}{1}&{0}&{0}&{0}\\{1}&{1}&{0}&{0}\\{0}&{1}&{1}&{0}\\{0}&{0}&{1}&{1}\\\end{matrix}\right),\quadM_{c}=\left(\begin{matrix}{1}&{-5}&{-5}&{-5}\\{0}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\\\end{matrix}\right),\quadM_{g}=\left(\begin{matrix}{-4}&{-10}&{-10}&{-5}\\{1}&{1}&{0}&{0}\\{0}&{1}&{1}&{0}\\{0}&{0}&{1}&{1}\\\end{matrix}\right)
S^{E}=\int_{0}^{\beta}d\tau\,{\mathrm{Tr}}\left[\frac{\dot{Y}^{i}\dot{Y}^{i}}{2R}+\frac{1}{\sqrt{\alpha^{\prime}}}\theta^{T}\dot{\theta}+\frac{R}{16\pi^{2}\alpha^{3}}\left[Y^{i},Y^{j}\right]^{2}+\frac{R}{2\pi\alpha^{2}}\theta^{T}\gamma^{j}\left[Y^{j},\theta\right]\right]
{\cal{L}}_{G}=\frac{1}{2\xi_{0}}\left[\partial^{\mu}A_{a}^{\mu}(x)\right]^{2}+{\bar{c}}_{a}(x)\left\{\left[-\delta_{ab}\partial^{\mu}+{\tilde{g}}_{0}f_{abc}A_{c}^{\mu}(x)\right]\partial^{\mu}\right\}c_{b}(x)\,.
F_{r^{\lambda\mu}}=\S_{\alpha}e_{(\lambda,\mu,\alpha)}^{r^{\lambda\mu}}{\Phi}_{r^{\lambda\mu}}^{(\lambda,\mu,\alpha)},\quad\S_{\{\nu_{r}\}=0}^{1}<\chi^{0}\mid\hat{A}\mid\chi>\equiv<\chi^{0}\parallel\hat{A}\parallel\chi>,
[b_{k}^{(\sigma)},b_{k^{\prime}}^{(\sigma^{\prime})\dagger}]=[\bar{b}_{k}^{(\sigma)},\bar{b}_{k^{\prime}}^{(\sigma^{\prime})\dagger}]=\delta_{\sigma\sigma^{\prime}}\delta(\Omega-\Omega^{\prime})\delta(\vec{k}-\vec{k}^{\prime})\,{.}
ds^{2}=-\left(K-{\frac{2M}{r}}+{\frac{Q^{2}}{r^{2}}}\right)dt^{2}+\left(K-{\frac{2M}{r}}+{\frac{Q^{2}}{r^{2}}}\right)^{-1}dr^{2}+r^{2}d^{2}\hat{\Omega}~~.
\omega_{n}\varphi_{\alpha}^{(n)}(x)=-\mathrm{i}(\gamma_{5})_{\alpha\beta}\frac{\partial}{\partialx}\varphi_{\beta}^{(n)}(x)+\frac{Ng^{2}}{4}\int\mathrm{d}y|x-y|\rho_{\alpha\beta}(x,y)\varphi_{\beta}^{(n)}(y)\.
A(p)-1=\frac{\xi\alpha_{0}}{4\pip^{2}}\int_{0}^{\infty}dq^{2}\left[\theta(p^{2}-q^{2})\frac{q^{2}}{p^{2}}+\theta(q^{2}-p^{2})\frac{p^{2}}{q^{2}}\frac{A(p)}{A(q)}\right]
B({\bfX},z_{0};{\bfp})\simeqB({\bfY},x_{0};{\bfp})+(z_{0}-x_{0})\frac{\partialB({\bfY},x_{0};{\bfp})}{\partialx_{0}}+({\bfX}-{\bfY})\cdot\nabla_{\bfY}B({\bfY},x_{0};{\bfp}).
\langle[A_{1}]_{N}({\bfx})\rangle=\frac{i}{\sqrt{2k\omega}}\left\{\frac{1}{2}(1-\frac{m}{\omega})J_{N+1}(kr)e^{i(N+1)\varphi}+\frac{1}{2}(1+\frac{m}{\omega})J_{N-1}(kr)e^{i(N-1)\varphi}\right\}
\varepsilon_{c}(t)=\int_{0}^{\infty}xdx\left[\frac{x}{2}\mathrm{coth}(\frac{\pix}{2t})+\sum_{m=1}^{\infty}x_{m}\mathrm{coth}(\frac{\pix_{m}}{2t})-\int_{0}^{\infty}dm\;\mathrm{coth}(\frac{\pix_{m}}{2t})\right].
C_{n,T}(N)=B_{n,T}(N)-(n!)^{-1}\sum_{k=0}^{n-1}(-1)^{n-k}(\frac{d^{n-k}f}{dx^{n-k}}(x),h(x-T)\frac{d^{k}}{dx^{k}}(x^{n+N}(x-n)^{-N})),
\partial_{t_{k}}n=\hat{\delta}n_{k}+{\frac{1}{2}}(\zeta\delta+\beta\eta)q_{k}-{\frac{1}{2}}(\zeta\delta_{k}+\beta\eta_{k})q+(q\zeta+\beta\Phi+\beta^{\prime})\delta_{k}-(q_{k}\zeta+\beta_{k}\Phi+\beta_{k}^{\prime})\delta
\langleT_{i}^{k}\rangle_{D}^{(1b)}(\xi_{1},\xi)=-A_{d}\delta_{i}^{k}\int_{0}^{\infty}dkk^{d}\int_{0}^{\infty}d\omega\frac{I_{\omega}(k\xi_{1})}{K_{\omega}(k\xi_{1})}F^{(i)}[K_{\omega}(k\xi)]
S_{\mathrm{source}}=\intd^{n}x\,\left(J^{\mu}\starA_{\mu}+J\starF+i(\bar{\eta}\starc-\bar{c}\star\eta)+K^{\mu}\starD_{\mu}c+L\star(-{\frac{1}{2}}c\starc)+H\star({\frac{1}{2}}\bar{c}\starF)\right)
P_{j\,j+1}\overline{{G}}_{\varepsilon}^{(nl)}(\cdots,\zeta_{j+1},\zeta_{j},\cdots)\quad=R_{j\,j+1}(\zeta_{j}/\zeta_{j+1})\overline{{G}}_{\varepsilon}^{(nl)}(\cdots,\zeta_{j},\zeta_{j+1},\cdots),
\mathcal{D}_{P+1}(N,G)=\sum_{g_{P}=0}^{[N/P]}\frac{1}{g_{P}!}\sum_{\begin{array}{l}{{g_{1},g_{2},....,g_{p-1}\geq0}}\\{\sum_{j=1}^{P-1}j.g_{j}=N-Pg_{P}}\\\end{array}}\frac{G!}{g_{1}!g_{2}!....,g_{P-1}!(G-\sum_{j=1}^{P}g_{j})!}
\mathcal{M}_{s_{1}s_{2}\sigma_{1}\sigma_{2}}^{ann}(\mathbf{p},\mathbf{q})=-\frac{ie^{2}}{4m^{2}}\varphi_{s_{1}}^{\dagger}\chi_{\sigma_{2}}^{\dagger}\left\{\overrightarrow{\mathbf{\sigma}}^{\left(+\right)}\cdot\overrightarrow{\mathbf{\sigma}}^{\left(-\right)}\right\}\varphi_{\sigma_{1}}\chi_{s_{2}}.
\delta\psi_{\mu}={\frac{1}{\sqrt{2\piG_{N}}}}\hat{\nabla}_{\mu}\epsilon={\frac{1}{\sqrt{2\piG_{N}}}}[\nabla_{\mu}-{\frac{\sqrt{G_{N}}}{4}}F_{\nu\rho}\gamma^{\nu}\gamma^{\rho}\gamma_{\mu}]\epsilon.
\gamma(\zeta)=\frac{1}{\zeta}\frac{(q^{2}\zeta;q^{4})_{\infty}(q^{4}\zeta^{-2};q^{4})_{\infty}}{(q^{2}\zeta^{-2};q^{4})_{\infty}(q^{4}\zeta^{2};q^{4})_{\infty}},\qquad(z;p)_{\infty}=\prod_{n=0}^{\infty}(1-zp^{n}).
T_{ij}^{k}=-2Q_{\alpha;(i}^{k}\gamma_{j)l}N^{\alpha\beta}Q_{\beta}^{l}+\gamma_{(il}N^{\alpha\mu}Q_{\mu}^{l}Q_{\alpha}^{m}Q_{\beta;m}^{k}\gamma_{j)n}N^{\beta\nu}Q_{\nu}^{n},
\sum_{s}\sum_{n=1}^{\infty}\bar{N}_{m_{1}n}^{rs}nN_{nm_{2}}^{st}=\frac{1}{m_{1}}\oint\frac{\bar{dy}}{2\pii}\bar{\xi}_{r}(y)^{-m_{1}}\frac{1}{m_{2}}\oint\frac{dz}{2\pii}\xi_{t}(z)^{-m_{2}}F(\bar{y},z),
\begin{array}{l}{D_{[b_{1}(\beta_{1}}\cdotsD_{b_{l}\beta_{l}}\delta\Psi_{a_{1}\cdotsa_{m}]\alpha_{1}\cdots\alpha_{n})}}\\{{}}\\{=2l(-\eta+m+\textstyle{\frac{1}{2}}n+\textstyle{\frac{3}{4}}(l-1))\hat{\bar{\rho}}_{[b_{1}(\beta_{1}}D_{b_{2}\beta_{2}}\cdotsD_{b_{l}\beta_{l}}\Psi_{a_{1}\cdotsa_{m}]\alpha_{1}\cdots\alpha_{n})}\,+\,\mathrm{homogeneous~terms}\,.}\\\end{array}
K_{t}^{\infty}(\bar{r},\bar{r}^{\prime}|-\triangle_{c})=\frac{1}{2\pit}e^{-\frac{r^{2}+r^{2}}{4t}}\int_{0}^{\infty}e^{ik(\varphi-\varphi^{\prime})}I_{k}\left(\frac{rr^{\prime}}{2t}\right)dk
i\int\frac{d^{4}q}{(2\pi)^{4}}\;4e^{2}\,\Gamma_{3}\,\frac{\Sigma(q)}{Z^{2}(q)\,q^{2}+\Sigma^{2}(q)}\,\frac{2p^{2}}{P^{4}}(1\!-\!\xi)\intd^{2}\theta\;\Psi_{+}(-p,\theta)\Psi_{-}(p,\theta).
G^{({\calQ})}(x_{1},x_{2},x_{3},x_{4})=\langle{\calQ}^{i_{1}j_{1}}(x_{1}){\calQ}^{i_{2}j_{2}}(x_{2}){\calQ}^{i_{3}j_{3}}(x_{3}){\calQ}^{i_{4}j_{4}}(x_{4})\rangle\;
\epsilon^{\delta\alpha\beta\gamma}D_{\delta}^{i}(w_{\alpha\ldots\alpha_{m_{1}}}w_{\beta\ldots\beta_{m_{2}}}w_{\gamma\ldots\gamma_{m_{3}}})=0\\rightarrow\\ell=6+{\frac{1}{2}}(J_{1}+2J_{2}+3J_{3})\;.
\left\{b_{\underline{{n}}\sigma},b_{\underline{{n}}\sigma}^{\dagger}\right\}=\delta_{\underline{{n}}^{\prime}\underline{{n}}}\delta_{\sigma^{\prime}\sigma}\,,\;\;\left[a_{\underline{{n}}},a_{\underline{{n}}}^{\dagger}\right]=\delta_{\underline{{n}}^{\prime}\underline{{n}}}\,.
L_{\mathrm{DBI}}\left(\dot{X}^{m}-\bar{\theta}_{1}\Gamma^{m}\dot{\theta}_{1}-\bar{\theta}_{2}\Gamma^{m}\dot{\theta}_{2},\;\dot{A}_{a}-[\bar{\theta}_{1}\Gamma^{m}\dot{\theta}_{1}-\bar{\theta}_{2}\Gamma^{m}\dot{\theta}_{2}]\Pi_{ma}\right)-L_{\mathrm{WZ}}(\bar{\theta}_{1}T_{\mathrm{WZ}}\dot{\theta}_{2}+c.c.)\,
g_{\Gamma}=\frac{1}{\tilde{a}\cdot\tilde{a}}\left(\begin{matrix}{\tilde{a}\cdot\tilde{a}}&{0}&{0}\\{0}&{\tilde{a}_{1}^{1}\tilde{a}_{1}^{1}+\tilde{a}_{2}^{1}\tilde{a}_{2}^{1}}&{\tilde{a}_{2}^{1}\tilde{a}_{2}^{2}}\\{0}&{\tilde{a}_{2}^{1}\tilde{a}_{2}^{2}}&{\tilde{a}_{1}^{1}\tilde{a}_{1}^{1}+\tilde{a}_{2}^{2}\tilde{a}_{2}^{2}}\\\end{matrix}\right)
L{^{\pm}}_{CS}=\int_{\calM}\varepsilon^{ijk}\bigg({^{\pm}}A_{i}^{AB}\partial_{j}{\^{\pm}}A_{kAB}+{\frac{2}{3}}{^{\pm}}A_{iA}^{B}{^{\pm}}A_{jB}^{C}{^{\pm}}A_{kC}^{A}\bigg),
F_{n}^{+}({\frac{z}{\tau}},{\frac{-1}{\tau}})={\frac{i\tau^{\frac{3}{2p}}}{\sqrt{2p}}}\sum_{\mu=1}^{2p}\tilde{q}^{\frac{-n\mu}{2}}\int_{C0}^{\tau}ds\omega_{\mu}(s,z,\tau)
\Big|\!\Big|\!\Big|f^{(a,b)}\Big|\!\Big|\!\Big|\;:=\;\Bigg[\int\;\prod_{i=1}^{n_{a}}d^{D}z_{i}^{(a)}\;\prod_{j=1}^{n_{b}}d^{D}z_{j}^{(b)}\;\bigg|f^{(a,b)}\bigg|^{N-1}\Bigg]^{\frac{1}{N-1}}\;.
\Phi_{p}(y)=\frac{(2\pi)^{2}}{l^{2}\tau_{2}}\sum_{j_{1}j_{2}}<\psi|e^{\frac{2i\pi}{l}(j_{1}(\hat{y_{1}}-y_{1})+\frac{j_{2}-\tau_{1}j_{1}}{\tau_{2}}(\hat{y_{2}}-y_{2})}|\psi>
{\calD}_{C}^{\;\;A}\;\nu_{(\lambda)}^{C}=-{\frac{1}{2}}\biggr({\frac{\partial}{\partial\tau}}\nu_{(\lambda)}^{A}-{\frac{3}{2\tau}}\nu_{(\lambda)}^{A}\biggr)-{_{e}n_{C^{\prime}}^{\;\;\;(C}}\;e^{A)C^{\prime}i}\;{}^{(3)}\nabla_{i}{\nu_{C}}_{(\lambda)}\;.
\begin{array}{c}{R^{\left(2\right)}=K^{\left(2\right)}\left(\Gamma_{11}\right)\quad,}\\{R^{\left(4\right)}=K^{\left(4\right)}\left({\bf1}\right)\quad,}\\{R^{\left(6\right)}=K^{\left(6\right)}\left(\Gamma_{11}\right)\quad,}\\{R^{\left(8\right)}=K^{\left(8\right)}\left({\bf1}\right)\quad,}\\{R^{\left(10\right)}=K^{\left(10\right)}\left(\Gamma_{11}\right)\quad;}\\\end{array}
S_{WZW_{(2n)}}={\frac{i}{4\pi}}\int_{M}\omega^{n-1}\wedge\mathrm{Tr}(h^{-1}\partialh\wedgeh^{-1}\overline{{\partial}}h)+{\frac{i}{12\pi}}\int_{\partialM}\omega^{n-1}\wedge\mathrm{Tr}(h^{-1}dh)^{3}
\mathrm{}[\hat{\Phi}(x^{0},x^{1})\,,\,\hat{\Phi}(x^{0},y^{1})]\,=\,-\frac{i}{2}\left\{\frac{1}{i\pi}{\sum_{n=-\infty}^{+\infty}}^{\prime}\left(\frac{\pi}{R}\right)\frac{e^{\frac{in\pi}{R}(x^{1}-y^{1})}}{\frac{n\pi}{R}}\right\}.
\vec{\calQ}^{\prime}={\frac{1}{\sqrt{2}}}{\calU}^{T}\left(\begin{matrix}{U_{5}({\bfe}_{u}-{\bfe}_{d})}\\{U_{21}\left(\begin{matrix}{{\bfe}_{u}+{\bfe}_{d}}\\{0_{16}}\\\end{matrix}\right)}\\\end{matrix}\right),
{\calL}_{gf}=\frac{2\gamma}{v\alpha}(\partial_{\mu}Z^{\mu})(\partial_{\nu}rZ^{\nu})+\frac{2\gamma}{v\alpha}(\partial_{\mu}W^{\mu+})(\partial_{\nu}rW^{\nu-})+\frac{2\gamma}{v\alpha}(\partial_{\mu}W^{\mu-})(\partial_{\nu}rW^{\nu+}).
{\calL}_{P}=-\frac12G_{\mu\nu}^{\dagger}G^{\mu\nu}-M^{2}\phi_{\mu}^{\dagger}\phi^{\mu}+\frac{\theta}{2}\varepsilon_{\mu\nu\rho}A^{\mu}\partial^{\nu}A^{\rho}+\frac{\lambda}{2}(\partial_{\mu}A^{\mu})^{2}\;,
S_{gauge}=-\,\frac{1}{g_{1}(\mu)^{2}}\intd^{4}x\left\{\frac{1}{2}\partial_{\alpha}\Phi^{i}\partial^{\alpha}\Phi^{i}+\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}\right\}+\frac{\theta_{1}}{32\pi^{2}}\intd^{4}xF_{\alpha\beta}{\tilde{F}}^{\alpha\beta}
\frac{1}{2}\frac{H^{3/2}(\chi)}{2\pi}\frac{\partial}{\partial\chi}\left(\frac{H^{3/2}(\chi)}{2\pi}\frac{\partial}{\partial\chi}\psi_{s}(\chi)\right)-\frac{V^{\prime}(\chi)}{3H(\chi)}\frac{\partial}{\partial\chi}\psi_{s}(\chi)+3H(\chi)\cdot\psi_{s}(\chi)=\lambda_{s}\,\psi_{s}(\chi)\.
\{Q_{\alpha}^{-},Q_{\beta}^{-}\}=4\pi(h_{-})_{\alpha\beta}H_{LC}-\frac{\mu}{3}(h_{-}\gamma^{ij}\gamma^{123})_{\alpha\beta}J^{ij}+\frac{\mu}{6}(h_{-}\gamma^{i^{\prime}j^{\prime}}\gamma^{123})_{\alpha\beta}\widehat{J}^{i^{\prime}j^{\prime}}~,
U=\left(\begin{matrix}{1}&{}&{}&{}&{}\\{}&{q}&{}&{}&{}\\{}&{}&{q^{2}}&{}&{}\\{}&{}&{}&{\ddots}&{}\\{}&{}&{}&{}&{q^{N-1}}\\\end{matrix}\right),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;V=\left(\begin{matrix}{}&{1}&{}&{}&{}\\{}&{}&{1}&{}&{}\\{}&{}&{}&{\ddots}&{}\\{}&{}&{}&{}&{1}\\{1}&{}&{}&{}&{}\\\end{matrix}\right)
\left\langle-p_{0N}\right|h_{\mu\nu}(p_{0})\alpha_{1}^{\mu}\tilde{\alpha}_{1}^{\nu}=\left.\frac{4}{\alpha^{\prime}}h_{\mu\nu}(p_{0})\frac{\partial^{2}}{\partial\bar{\psi}_{1\mu}\partial\psi_{1\nu}}\left\langle\Omega\left(\psi_{n},\bar{\psi}_{n};p_{0}\right)\right|\right|_{\psi,\bar{\psi}=0}~,
\sum_{l_{2}=1}^{*}{\frac{(-1)^{l_{2}}l_{2}^{4}}{(l_{1}^{2}-l_{2}^{2})^{3}}}=(-1)^{l_{1}}\left(-{\frac{1}{32l_{1}^{2}}}+{\frac{5\pi^{2}}{96}}\right).
+i\frac{\pi}{2}\lbrack{-2(i)^{s}\zeta_{H}(s;iz)+z^{-s}+\frac{(2\pi)^{s}}{\Gamma(s)}\sum_{n=1}^{+\infty}\frac{e^{-2n\pi{z}}}{n^{1-s}}}\rbrack-(z)^{-s}(\mathrm{ln}|z|+iarg\:{z})\:.
\begin{aligned}{|X_{2}|^{2}+|X_{4}|^{2}-|X_{3}|^{2}-|X_{0}|^{2}}&{{}=}&{\xi_{2}}\\{|X_{1}|^{2}+|X_{4}|^{2}-2|X_{0}|^{2}}&{{}=}&{\xi_{2}}\\{|X_{2}|^{2}+|X_{5}|^{2}-2|X_{0}|^{2}}&{{}=}&{\xi_{2}}\\{|X_{1}|^{2}+|X_{3}|^{2}+|X_{5}|^{2}-3|X_{0}|^{2}}&{{}=}&{\xi_{2}.}\\\end{aligned}
\pi_{\mu\nu}=-\frac{1}{4}\tau_{\mu\alpha}\tau_{\nu}^{\alpha}+\frac{1}{12}\tau\tau_{\mu\nu}+\frac{1}{8}q_{\mu\nu}\tau_{\alpha\beta}\tau^{\alpha\beta}-\frac{1}{24}q_{\mu\nu}\tau^{2}
(\alpha_{opt}^{2})^{ev}=\sqrt{\frac{2\mid\beta\mid^{2}tanh\mid\beta\mid^{2}+2\mid\beta\mid^{2}cos2\theta_{1}+1}{2\mid\beta\mid^{2}tanh\mid\beta\mid^{2}-2\mid\beta\mid^{2}cos2\theta_{1}+1}}(\alpha_{opt}^{2})^{o},
\int\left[d\Phi^{*}(t)d\Phi(t)\right]_{t_{1}<t\leqt_{2}}\Phi^{n}(t_{2})\prod_{t=t_{1}+1}^{t_{2}}\frac{\left[-S(t)\right]^{\Omega}}{\Omega!}=\alpha^{t_{2}-t_{1}}\Phi^{n}(t_{1})\;.
A_{i}^{(p)}(x_{0},{\bfx})=-2ix_{0}^{d/2}\int{\frac{d^{d}k}{(2\pi)^{d}}}e^{-i{\bfk}\cdot{\bfx}}a_{0}({\bfk}){\frac{k_{i}}{|{\bfk}|^{2}}}\left[{\frac{c}{2}}\Gamma(\nu)\left({\frac{\xi}{2}}\right)^{-\nu}\!\!\!\!\!+{\frac{\bar{c}}{2}}\Gamma(-\nu)\left({\frac{\xi}{2}}\right)^{\nu}+\dots\right],
\Omega(k)=\frac{32}{k^{2}}F\left[\frac{\pi}{2},\frac{8}{k^{2}}-1\right]=\frac{16\pi}{k^{2}}F\left(\frac{1}{2},\frac{1}{2};1,\frac{(8-k^{2})^{2}}{k^{4}}\right).
V_{i}\cdot{\calN}_{\overline{{\alphaV}}}+f_{i}({\overline{{\alphaV}}},{\bfn})+{\frac{\alpha^{*}t^{*}}{2t_{i}}}{\vec{Q}}^{2}({\vec{P}}_{\overline{{\alphaV}}})=\sum_{j}k_{ij}\alpha_{j}+s_{i}-\varphi(V_{i},{\overline{{\alphaV}}})~(\mathrm{mod}~1)~.
(r^{2}-r_{H}^{2})\partial_{r}^{2}R_{(\gamma)}(r)+\frac{1}{r}\left[2r^{2}+\gamma(r^{2}-r_{H}^{2})\right]\partial_{r}R_{(\gamma)}(r)+\frac{\omega^{2}\ell^{4}}{(r^{2}-r_{H}^{2})}R_{(\gamma)}(r)=0.
\mathrm{I=\left\{\int_{0}^{1}dx\int_{0}^{x}dG\int_{G}^{1-x+G}dy\int_{0}^{x-G}dv\right\}\int_{0}^{\infty}dAA^{4}\int\frac{d^{4}q}{(2\pi)^{4}}~{\itq}_{i}~e^{-E}~,}
{\widetildeH}_{GU}=H_{c}+{\displaystyle\int}dx\left[\frac{\displaystyle\left(\pi_{1}+(\alpha-1)\partial_{1}A_{1}\right)}{\displaystyle{\alpha-1}}\;Q_{1}+\frac{\displaystyle(\partial_{1}Q_{1})^{2}}{\displaystyle2e^{2}(\alpha-1)}+\frac{\displaystyleQ_{1}^{2}}{\displaystyle2(\alpha-1)^{2}}\right],
\stackrel{\frown}{\Box}{\calF}^{ij}=({\calD}^{m}{\calD}_{m}+\frac{1}{2}\{\bar{W},W\}){\calF}^{ij}+\frac{\mathrm{i}}{3}{\calD}^{\alpha(i}W{\calD}_{\alpha|k|}{\calF}^{j)k}+\frac{\mathrm{i}}{3}\bar{\calD}_{\dot{\alpha}}^{(i}\bar{W}\bar{\calD}_{|k|}^{\dot{\alpha}}{\calF}^{j)k}\;.
D_{++}^{(CPV)ab}(x)=D_{++}^{(CPV)}(x)\delta^{ab}=-{\frac{i\delta^{ab}}{(2\pi)^{2}}}\intd^{2}k\,e^{ikx}{\frac{\partial}{\partialk_{-}}}CPV\left({\frac{1}{k_{-}}}\right)=-{\frac{i\delta^{ab}}{2}}|x^{-}|\delta(x^{+})\,
g_{n,{\vec{c}}}=\theta\left[\begin{array}{c}{{\vec{0}}}\\{\frac{{\vec{c}}}{\ell}}\\\end{array}\right]\left({\vec{z}}-\frac{\tau}{\omega\kappa}\left(n+\frac{\beta}{2\pi}\right){\hat{e}}+\frac{\alpha}{2\pi\omega\kappa}{\hat{e}},\Omega\right)
{\bf\Phi}_{1}^{(n_{\rho},m)}={\bf\Phi}_{1}^{(n_{\rho},m)}(\rho,k)=\left(\begin{array}{c}{\phi_{1}^{(n_{\rho},m)}(\rho,k)}\\{\phi_{2}^{(n_{\rho},m)}(\rho,k)}\\\end{array}\right)
\begin{array}{c}{{\bfg}=ig_{a\bar{b}}\eta^{a}\sigma_{0}{\bar{\eta}}^{b},}\\{D\eta_{i}^{a}=d\eta_{i}^{a}+\Gamma_{bc}^{a}\eta_{i}^{a}dz^{a},\quadi=1,2}\\\end{array}
k=\mathrm{ind}(A)=-\mathrm{sign}(\mathrm{Pfaff}(\theta))\left(\frac{i}{2\pi}\right)^{d/2}\frac{(d/2-1)!}{(d-1)!}\int_{S^{d-1}}\mathrm{tr}(a^{-1}da)^{d-1}
\tilde{g}_{tt}=M-\frac{J^{2}}{4r^{2}},\;\;\;\tilde{g}_{rr}=(\frac{r^{2}}{l^{2}}-M+\frac{J^{2}}{4r^{2}})^{-1},\;\;\;\tilde{g}_{\phi\phi}=\frac{1}{r^{2}},\;\;\;\tilde{g}_{t\phi}=\frac{1}{l},
E^{\mathrm{(surf)}}(a)=\frac{4\xi-1}{2\pia}\sum_{l=0}^{\infty}D_{l}\int_{ma}^{\infty}\frac{zdz}{\sqrt{z^{2}-m^{2}a^{2}}}\left[1-\left(\beta+\frac{n}{2}\right)\frac{\left(\tilde{I}_{\nu}(z)\tilde{K}_{\nu}(z)\right)^{\prime}}{z\tilde{I}_{\nu}^{\prime}(z)\tilde{K}_{\nu}^{\prime}(z)}\right]\;
Z=\sum_{M,N=0}^{\infty}\frac1{M!N!}\sum_{e_{i},g_{j}}\int\!\prod_{i=1}^{M}d^{2}x_{i}\prod_{j=1}^{N}d^{2}y_{j}\,e^{-S(\vec{x}_{i},e_{i};\vec{y}_{j},g_{j})}
T_{\mu\nu}=\:\frac{i}{4}\bar{\psi}\left(\gamma_{\mu}\stackrel{\leftrightarrow}{\partial}_{\nu}+\gamma_{\nu}\stackrel{\leftrightarrow}{\partial}_{\mu}\right)P_{\pm}\psi:\=\frac{1}{2}\left(T_{\mu\nu}^{V}\pmT_{\mu\nu}^{A}\right),
S_{R}=-\intd^{2}x\,\sqrt{\widetilde{g}}\,\left[\frac{1}{2}\widetilde{g}^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi+C\widetilde{R}\Phi+{V}-\frac{V^{\prime}}{\epsilonC}+\frac{1}{4}\left(1-\frac{f^{\prime}(\Phi)}{4\epsilonC}\right)\widetilde{g}^{\mu\alpha}\widetilde{g}^{\nu\beta}F_{\mu\nu}F_{\alpha\beta}\right].
ds_{P}^{2}={\frac{\left|{\partial_{a}^{3}{\calF}}\right|^{2}}{\left(\mathrm{Im}\,\tau\right)^{2}}}|da|^{2}={\frac{\left|\partial_{u}\tau\right|^{2}}{\left(\mathrm{Im}\,\tau\right)^{2}}}|du|^{2}=e^{\varphi}|du|^{2},
A_{d}=\frac{2^{2-d}\pi^{\frac{1}{2}d+\frac{1}{2}}\Gamma(2-\frac{1}{2}d)\Gamma(\frac{1}{2}d-1)}{(2\pi)^{d}\Gamma(\frac{1}{2}d-\frac{1}{2})}\,\,\,\,\mathrm{and}\,\,\,\,v^{2}=\frac{4M_{\ast}^{2}}{p^{2}+\omega_{n}^{2}}\,
B_{p}=\left(v_{\alpha}^{+}{\left(\frac{p^{--}}{2k}\right)}^{\frac12}{\left(\frac{v_{1}^{+}}{\bar{v}_{1}^{+}}\right)}^{-\frac12}\,,\,v_{\alpha}^{-}(v^{+}){\left(\frac{p^{--}}{2k}\right)}^{-\frac12}{\left(\frac{v_{1}^{+}}{\bar{v}_{1}^{+}}\right)}^{\frac12}\right),
24\mathrm{Cl}_{2}(\theta_{5,3})-12\mathrm{Cl}_{2}(2\theta_{5,3})-8\mathrm{Cl}_{2}(3\theta_{5,3})+6\mathrm{Cl}_{2}(4\theta_{5,3})=Z_{15}=15\sum_{k=1}^{7}\left(\frac{k}{15}\right)\mathrm{Cl}_{2}\left(\frac{2\pik}{15}\right)
a^{2}~t^{2}~+~t~(2~a)^{n}~(\deltay-\deltaa_{1})~(\deltay-\deltaa_{2})~\ldots~(\deltay-\deltaa_{n})~+~~\Lambda_{SU}^{2b_{0}^{SU}}~\Lambda_{SO}^{b_{0}^{SO}}~a^{2}~=~0~.
7.12\Gamma_{\hat{h}}(\phi)\=\\frac{1}{96\pi}\int_{\Sigma}d^{2}\xi\sqrt{|\hat{h}(\xi)|}\\left(\hat{h}^{\alpha\beta}(\xi)\,\partial_{\alpha}\phi(\xi)\,\partial_{\beta}\phi(\xi)\;+\;4r_{\hat{h}}(\xi)\,\phi(\xi)\right)\.
{\calZ}={\frac{1}{\prod_{i}m_{i}}}\sum_{\alpha,\beta}\mathrm{Tr}(q^{H_{\overline{{\alphaV}}}^{L}}~{\overline{q}}^{H_{\overline{{\alphaV}}}^{R}}g^{-1}({\overline{{\betaV}}}))\,={\frac{1}{\prod_{i}m_{i}}}\sum_{\alpha,\beta}C_{\overline{{\betaV}}}^{\overline{{\alphaV}}}{\calZ}_{\overline{{\betaV}}}^{\overline{{\alphaV}}}~,
\Pi^{\mu\nu}=\left(\begin{array}{ccc}{-\left(\mathit{\Pi}_{\mathit{0}}+\mathit{\Pi}_{\mathit{0}}\,^{\prime}\,k^{2}\right)}&{0}&{-i\mathit{\Pi}_{\mathit{1}}k}\\{0}&{0}&{0}\\{i\mathit{\Pi}_{\mathit{1}}k}&{0}&{\mathit{\Pi}_{\,\mathit{2}}k^{2}}\\\end{array}\right)\tag{3}
J\,=\,{|\epsilon^{1}(0)|}^{2}\,\frac{\theta}{|\theta|}\intd^{2}k\,a^{+}(\vec{k})a^{-}(\vec{k})\,+\,i\epsilon^{jl}{|\epsilon^{1}(0)|}^{2}\intd^{2}k\,a^{+}(\vec{k})\,k^{l}\frac{\partial}{\partialk^{j}}\,a^{-}(\vec{k}).
{\Gamma^{\prime}}_{A^{\mu}\varphi_{1}\varphi_{2}}^{(n)}(p,p_{1},p_{2})=\Gamma_{A^{\mu}\varphi_{1}\varphi_{2}}^{(n)}(p,p_{1},p_{2})-ie(p_{1}^{\mu}-p_{2}^{\mu})\deltaz_{e}^{(n)}(\xi)
V_{I}(\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},
\Lambda_{\nu}^{\mu}(x,y)=\delta_{\nu}^{\mu}\,\delta^{3}(x-y)-J^{\mu\alpha}\intd^{\,3}\!\rho\;d^{\,3}\!\sigma\,\delta_{\alpha(x)}\varphi^{i}(\rho)\,J_{ij}(\rho,\sigma)\,\delta_{\nu(y)}\varphi^{j}(\sigma)
\Gamma^{(n)}(x_{1},x_{2},..,x_{n})=\frac{1}{(2\pi)^{n}}\intd^{4}k_{1}d^{4}k_{2}..d^{4}k_{n}(2\pi)^{4}\delta(k_{1}+k_{2}+..k_{n})\e^{i(k_{1}x_{1}+..k_{n}x_{n})}\tilde{\Gamma}^{(n)}(x_{1},x_{2},..,x_{n}).
V=\left(U^{\Lambda\Sigma}+\mathrm{e}^{\mathcalK}\bar{X}^{\Lambda}X^{\Sigma}\right)P_{\Lambda}^{x}P_{\Sigma}^{x}=-\frac12\left(\Im{\mathcalN}_{\Lambda\Sigma}\right)^{-1}P_{\Lambda}^{x}P_{\Sigma}^{x}
Z_{12}=\left(\begin{array}{cccc}{\bar{q}\overline{{R}}}&{\cdot}&{\cdot}&{\cdot}\\{\cdot}&{\bar{q}\overline{{R}}}&{\bar{q}\overline{{R}}-\beta}&{\cdot}\\{\cdot}&{\cdot}&{\beta}&{\cdot}\\{\cdot}&{\cdot}&{\cdot}&{-\beta}\\\end{array}\right)\,.
i\left<\bar{A}_{\mu}^{a}\bar{A}_{\nu}^{b}\right>_{\mathrm{counter}}=-i\delta_{T}(p^{2}g_{\mu\nu}-p_{\mu}p_{\nu})\delta^{ab}-\fraci{\alpha_{\mathrm{R}}}\delta_{L}p_{\mu}p_{\nu}\delta^{ab},
F_{\mu}(y-z,x-z)=\int\Big(e^{-iq\cdot(y-z)}-e^{-iq\cdot(x-z)}\Big)\frac{(x-y)_{\mu}}{q\cdot(x-y)}\frac{d^{4}q}{(2\pi)^{4}}+F_{\mu}^{T}\;,
\frac{M_{2}}{\lambda}\equiv\frac{M_{0}+M_{b}}{\lambda}=\frac{M_{0}}{\lambda}+e^{-\lambdar_{b}(0)}\left(e^{\lambdar_{b}(0)}+\frac{\sqrt{M_{0}\pi}}{\lambda}\right)\left(-\frac{\partialS_{b0}}{\partialr_{b}(0)}\right)
(\partial_{A}\omegaK^{AB}\partial_{B}\omega^{\prime})^{n}=\sum_{l+2i=n}d_{[l,i]}\,\bigl[(\partial\omega)^{l}K_{[l]}(\partial\omega^{\prime})^{l}\bigr]\left[(\partial\partial\omega)K_{[0,1]}(\partial\partial\omega^{\prime})\right]^{i}(\omega\omega^{\prime})^{i}\;.
R^{(2)}(j)=\intDx(t)\delta[x(s)-x(0)]\delta[x(s_{1})-x(s_{2})]exp\left(-{\frac{1}{4}}\int_{0}^{s}\,dt\,\dot{x}^{2}+i\sum_{n=1}^{M}\hat{k}(t_{n})\cdotx(t_{n})+i\int_{0}^{s}dt\,j\cdot\dot{x}(t_{n})\right)
{{\calG}^{-1}}_{\mu\nu}^{\alpha\beta}\eta_{\alpha\beta}^{\mathrm{LLT}}=\left(k^{2}-\Lambda_{l}+\frac{1}{2r^{2}}(N-1)-\frac{N+d}{N+d-2}\Lambda\right)\eta_{\mu\nu}^{\mathrm{LLT}}.
\partial_{\mu}\langle{\mathcalJ}_{\ell}^{\mu}(x)\rangle_{A}\;=\;2\,q\,{\mathcalA}(x)~,~~\partial_{\mu}\langle{\mathcalJ}_{r}^{\mu}(x)\rangle_{A}\;=\;-2\,q\,{\mathcalA}(x)~.
\left(\begin{array}{cccc|cc}{\times}&{\times}&{}&{}&{}&{}\\{\times}&{\times}&{}&{}&{}&{}\\{\times}&{\times}&{\times}&{\times}&{\times}&{\times}\\{\times}&{\times}&{\times}&{\times}&{\times}&{\times}\\\hline{\times}&{\times}&{}&{}&{\times}&{\times}\\{\times}&{\times}&{}&{}&{\times}&{\times}\\\end{array}\right)
\frac{\lambda^{2}\sqrt{2}}{m^{3}}E_{\mathrm{TK2}_{\sigma_{2}}}=\int_{\bar{\sigma}_{2}^{2}}^{1}\frac{\lambda_{3}d\lambda_{3}}{\sqrt{1-\lambda_{3}}}=2\sigma_{2}\left(1-\frac{\sigma_{2}^{2}}{3}\right)
{\frac{1}{\prod_{i}m_{i}}}\sum_{\beta}C_{\overline{{\betaV}}}^{\overline{{\alphaV}}}g^{-1}({\overline{{\betaV}}})=e^{2\pii\alphas}\xi({\overline{{\alphaV}}},{\bfn})\eta({\overline{{\alphaV}}},{\calN}_{\overline{{\alphaV}}},{\bfn})~,
\mathbf{I}^{\dagger}(D_{\theta^{x}})=-\mathbf{I}(D_{\theta^{x}})\mathcal{R}_{x}\#,\qquad\mathbf{I}(D_{\theta^{x}})=-\mathbf{I}^{\dagger}(D_{\theta^{x}})\mathcal{R}_{x}^{-1}\#.
u\in\mathrm{USp}(2N):\qquadu_{i}^{I}u_{J}^{i}=\delta_{J}^{I}\;,\\u_{i}^{I}\Omega^{ij}u_{j}^{J}=\Omega^{IJ}\;,\\u_{i}^{I}=(u_{I}^{i})^{*}\;.
\gamma\eta_{1}^{a}=0,\;\gamma\eta_{2}^{a}=-\frac12f_{\;\;bc}^{a}\eta_{2}^{b}\eta_{2}^{c},\;\gamma\eta_{1}^{i}=0,\;\gamma\eta_{2}^{i}=\partial^{i}\eta,\;\gamma\eta=0,
T_{1R}=\left(\begin{array}{ccc}{0}&{i}&{0}\\{-i}&{0}&{0}\\{0}&{0}&{0}\\\end{array}\right)\;\T_{2R}=\left(\begin{array}{ccc}{0}&{0}&{0}\\{0}&{0}&{i}\\{0}&{-i}&{0}\\\end{array}\right)\;\T_{3R}=\left(\begin{array}{ccc}{0}&{0}&{-i}\\{0}&{0}&{0}\\{i}&{0}&{0}\\\end{array}\right)