harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\calH}_{Q}=\sum_{j=1}^{r}\left(p_{j}-i{\frac{\partialW}{\partialq_{j}}}\right)\left(p_{j}+i{\frac{\partialW}{\partialq_{j}}}\right)+{\calE}_{0}=\sum_{j=1}^{r}\left(p_{j}+i{\frac{\partialW}{\partialq_{j}}}\right)^{\dagger}\left(p_{j}+i{\frac{\partialW}{\partialq_{j}}}\right)+{\calE}_{0},
=i\int\frac{d^{4}x_{1}d^{4}p^{\prime}}{(2\pi)^{4}}e^{-ip_{\mu}^{\prime}x_{2}^{\mu}}\frac{p_{\nu}^{\prime}\gamma^{\nu}+m}{p_{\tau}^{\prime}p^{\tau\,\prime}-m+i\epsilon}e^{ip_{\eta}^{\prime}x_{1}^{\eta}}N_{\sigma}\gamma^{\sigma}\psi(1)
\hat{T}_{\mu}=\left(\frac{\hbar}{2}\right)^{(D-2)/2}\hat{p}_{\mu}\hat{T},\quad\hat{T}=\hat{\kappa}\frac{\hat{p}^{i}\Sigma^{i}}{\hat{\omega}}-\alpha,\quad\hat{p}_{0}=-\hat{\kappa}\hat{\omega},\quad\alpha=\left(\frac{\hbar}{2}\right)^{-(D-2)/2}\tilde{\alpha},
{\calL}_{n}(z)=-z^{n+1}(\tilde{T}+T_{M^{4}})+{\frac{n+1}{2}}z^{n}\left(-i\alpha_{+}\alpha_{3}\cdot\partial\varphi+\tilde{\xi}_{\alpha_{2}}\tilde{\eta}_{\alpha_{2}}+\tilde{\xi}_{\alpha_{1}+\alpha_{2}}\tilde{\eta}_{\alpha_{1}+\alpha_{2}}\right).
Q(x,y)\;=\;\frac{1}{4}\left(\sum_{k=1}^{K_{xy}}\frac{\partial{\cal{L}}(\lambda^{xy})}{\partial\lambda_{k}^{xy}}\:F_{k}^{xy}\;P(x,y)\:+\:P(x,y)\sum_{k=1}^{K_{yx}}\frac{\partial{\cal{L}}(\lambda^{yx})}{\partial\lambda_{k}^{yx}}\:F_{k}^{yx}\right).
{\calA}=V\,(8\pi^{2}\alpha^{\prime})^{-1/2}\int_{0}^{\infty}{\frac{ds}{2s}}\,s^{-1/2}\left[{\frac{f_{3}^{8}(r)}{f_{1}^{8}(r)}}-{\frac{f_{2}^{8}(r)}{f_{1}^{8}(r)}}\right]
\tilde{G}(x+\hat{\nu}L_{\nu})=G_{0}(x)-\frac{(x+\hat{\nu}L_{\nu})^{2}}{4L_{1}L_{2}}=\tilde{G}(x)-\frac{x_{\nu}L_{\nu}}{2L_{1}L_{2}}-\frac{L_{\nu}^{2}}{4L_{1}L_{2}}.
\left(C^{2}\right)_{\mu\nu}=\frac{\displaystyle1}{\displaystylea^{2}+b^{2}}\left(F_{\mu\nu}^{2}+b^{2}g_{\mu\nu}\right)\,;\;\;\left(B^{2}\right)_{\mu\nu}=\frac{\displaystyle1}{\displaystylea^{2}+b^{2}}\left(F_{\mu\nu}^{2}-a^{2}g_{\mu\nu}\right)\,.
\int\!\!\intd^{d}k_{1}~d^{d}k_{2}~\left(\frac{\partial}{\partialk_{1\mu}}+\frac{\partial}{\partialk_{2\mu}}\right)~\left[k_{1\mu}~P_{k_{1},m_{1}}^{\nu_{1}}P_{k_{1}-k_{2},m_{2}}^{\nu_{2}}P_{k_{2},m_{3}}^{\nu_{3}}\right]\equiv0,
S[A]=\frac{1}{4\pie^{2}}\int_{\mathcal{M}^{3}}\;F_{A}\wedge\astF_{A}=\frac{1}{8\pie^{2}}\int_{\mathcal{M}^{3}}d^{3}x\sqrt{g}\;F_{ij}F^{ij},\;\;\;\;i,j=1,2,3,
\frac{1}{\rho^{2}}Ad(\hat{L})^{2}\hat{a}=\frac{1}{\rho^{2}}\sum_{lm}a_{lm}[\hat{L}_{i},[\hat{L}_{i},\hat{Y}_{lm}]]=\sum_{lm}\frac{l(l+1)}{\rho^{2}}a_{lm}\hat{Y}_{lm}.
\frac{8\pi}{3}\rho_{eff}=(\frac{\dot{\alpha}}{\alpha})^{2}=\frac{(C+E)^{2}g_{S}-|g_{00}|(g_{S}g^{3}+\ell^{2})}{4|g_{00}|g_{rr}g_{S}g^{3}}(\frac{g^{\prime}}{g})^{2}
\begin{array}{cc}{D_{[b_{1}(\beta_{1}}\cdotsD_{b_{l}\beta_{l}}\Psi_{a_{1}\cdotsa_{m}]\alpha_{1}\cdots\alpha_{n})}~~~~}&{~~~~\mathrm{if}~~~\eta=m+\textstyle{\frac{1}{2}}n+\textstyle{\frac{3}{4}}(l-1)\,.}\\\end{array}
\begin{array}{rclcl}{dh^{(p-2[p/2])}}&{=}&{0\,,}&{\quad}&{\mathrm{for}\;n=\left[\frac{p+2}{2}\right]}\\{dh^{(p+2-2n)}+(-1)^{p}h^{(p-2n)}\Pi^{\mu}(C\Gamma_{\mu}\Gamma_{11})_{\alpha\beta}\Pi^{\alpha}\Pi^{\beta}}&{=}&{0\,,}&{\quad}&{\mathrm{for}\;n=\left[\frac{p}{2}\right],\dots,0\,.}\\\end{array}
\gamma=\frac{1}{8\pi}\frac{\alpha(\betav)^{\frac{3}{2}}}{E}\sqrt{1-\frac{E^{2}}{\alpha\betav}}e^{\frac{E^{2}}{6\alpha\betav^{2}}}e^{\frac{\pi^{2}}{12}\frac{\alpha\betav}{E^{2}}}e^{-\pi\alpha\beta^{2}v^{3}\frac{b^{2}}{E^{2}}}.
J_{n}=\frac{1}{\Box}\frac{1}{n}{\Big(\frac{\Box^{7}}{{(\Box-M^{2})}^{4}{(\Box-g^{2}|\phi^{2}|)}^{4}}\Big)}^{n}\frac{\lambda^{2n}}{M^{6n}}{|W_{1}^{\prime}|}^{8n}
H^{2}=-\frac{k}{a^{2}}+\frac{\omega_{4}\,M}{a^{4}}-\frac{3\,\omega_{4}^{2}\,Q^{2}}{16\,a^{6}}-\frac{1}{L^{2}}+\frac{4\,\pi}{3\,M_{p}^{2}\,\rho_{0}}\left(\rho_{0}+\rho_{\mathrm{br}}\right)^{2}\,\,,
{\frac{d}{d\tau}}\zeta^{\mu}(\tau,\vec{\sigma})=\lbrace\zeta^{\mu}(\tau,\vec{\sigma}),H_{D}\rbrace=-\lambda^{\mu}(\tau,\vec{\sigma})-{\dot{x}}_{s}^{\mu}(\tau)-{\dot{b}}_{\check{r}}^{\mu}(\tau)\sigma^{\check{r}}\approx0,
\;\;[\;\;{\bfa}^{\mu\nu}(x),{\bfa}^{\lambda\sigma}(y)\;\;]\;\;=\frac{1}{(2\pi)^{3}}\int\frac{d^{3}p}{2p^{o}}[\;\;e^{ip\cdot(x-y)}-e^{-ip\cdot(x-y)}\;\;]\;\;f^{\mu\nu,\lambda\sigma},
W^{\ast}=\frac{1}{12}x_{1}^{12}+\frac{1}{12}x_{2}^{12}+\frac{1}{6}x_{3}^{6}+\frac{1}{6}x_{4}^{6}+\frac{1}{2}x_{5}^{2}-\psix_{1}x_{2}x_{3}x_{4}x_{5}-\frac{1}{6}\phi(x_{1}x_{2})^{6}.
f^{(q)}(u)=\sum_{n=0}^{\infty}f^{\alpha_{1}^{\prime}\ldots\alpha_{2n+q}^{\prime}}u_{(\alpha_{1}^{\prime}}^{+}\ldotsu_{\alpha_{n+q}^{\prime}}^{+}u_{\alpha_{n+q+1}^{\prime}}^{-}\ldotsu_{\alpha_{2n+q}^{\prime})}^{-}\.
P_{N}=\left(\begin{array}{ccccc}{0}&{1}&{}&{}&{}\\{}&{0}&{1}&{}&{}\\{}&{}&{\ddots}&{}&{}\\{}&{}&{}&{\ddots}&{1}\\{1}&{}&{}&{}&{0}\\\end{array}\right),\quadQ_{N}=\left(\begin{array}{ccccc}{1}&{}&{}&{}&{}\\{}&{e^{2\pii/N}}&{}&{}&{}\\{}&{}&{e^{4\pii/N}}&{}&{}\\{}&{}&{}&{\ddots}&{}\\{}&{}&{}&{}&{e^{2\pii(N-1)/N}}\\\end{array}\right)\tag{A.9}
{\calL}_{vec}^{\prime}\,=\,\mathrm{i}\,\Bigl[{\calF}^{-T}\,\bigl(A+B{\bar{\calN}}\bigr)^{T}{\bar{\calN}}^{\prime}(A+B{\bar{\calN}}\bigr){\calF}^{-}\,-\,{\calF}^{+T}\,\bigl(A+B{\calN}\bigr)^{T}{\calN}^{\prime}(A+B{\calN}\bigr){\calF}^{+}\Bigr]
\displaystyle{\frac{1}{\hbar\,\Omega}\,{\calE}(a)=\frac{\left({{n_{1}}}^{3}-{{n_{2}}}^{3}\right)}{6\,\pi}\left(\frac{a\,{\Omega}}{c}\right)^{3}-\frac{1}{12}\,\frac{{\left({{n_{1}}}^{2}-{{n_{2}}}^{2}\right)}^{2}}{{{n_{1}}}^{2}+{{n_{2}}}^{2}}\left(\frac{a\,\Omega}{c}\right)^{2}+{\calO}\left(\frac{a\,\Omega}{c}\right)}.
J^{\mu}({\bfx})=\frac{1}{2}[{\bar{\psi}}({\bfx}+\frac{{\bfl}}{2}),\gamma^{\mu}exp(\imathe\int_{{\bfx}-\frac{{\bfl}}{2}}^{{\bfx}+\frac{{\bfl}}{2}}{\bfA}({\bfx}^{\prime})d{\bfx}^{\prime}){\bar{\psi}}({\bfx}-\frac{{\bfl}}{2})]
\begin{array}{cccc}{R_{1}:}&{\left(x^{2,3,4,5},x^{6,7,8,9}\right)}&{\to}&{\left(x^{2,3,4,5},-x^{6,7,8,9}\right)}\\{R_{2}:}&{\left(x^{2,3,4,5},x^{6,7,8,9}\right)}&{\to}&{\left(-x^{2,3,4,5},x^{6,7,8,9}\right)}\\\end{array}
\Psi^{C}=\frac{1}{16}c\overline{{c}}\left[S^{\alpha}\{{C}_{\alpha}}^{\beta}\overline{{S}}_{\beta}e^{-\frac{3}{2}\phi}e^{-\frac{1}{2}\overline{{\phi}}}-S_{\alpha}\{{C}^{\alpha}}_{\beta}\overline{{S}}^{\beta}e^{-\frac{1}{2}\phi}e^{-\frac{3}{2}\overline{{\phi}}}\right]\,,
2\hat{H}=-g^{-{\frac{1}{2}}}\nabla_{\mu}g^{\frac{1}{2}}g^{\mu\nu}\nabla_{\nu}+R_{\mu\nu\sigma\tau}\hat{\psi}^{\sigma}\hat{\psi}^{*\tau}\hat{\psi}^{*\nu}\hat{\psi}^{\mu}+g^{\mu\nu}{\frac{\partialh}{\partialx^{\mu}}}{\frac{\partialh}{\partialx^{\nu}}}+H_{\mu\nu}[\hat{\psi}^{*\mu},\hat{\psi}^{\nu}],
\langle0|T(q_{4}({\vec{x}})q_{4}({\vec{y}}))|0\rangle=\frac{1}{4g^{2}\beta^{2}}(\frac{\pi^{2}}{3}-2)\delta_{{\vec{m}}^{x},{\vec{m}}^{y}}+{\calO}(\frac{l}{g^{4}\beta^{3}})\;.
U_{cs}=\frac{1}{\beta}\left[\frac{1}{2}\tilde{k}^{2}+\left(\betam^{2}+\frac{1}{2}q\tilde{k}\right)|\phi_{1}|^{2}+\left(\betam^{2}-\frac{1}{2}q\tilde{k}\right)|\phi_{2}|^{2}+\frac{1}{8}q^{2}(|\phi_{1}|^{2}-|\phi_{2}|^{2})^{2}\right],
\hat{W}(\gamma)={\calN}^{-1}\Big(exp\Big[g\oint_{\gamma}{\frac{\partial}{\partialJ}}dx^{+}\Big]\Big)\Big[\int{\calD}\hat{A}\,{\calD}\lambda\,exp\,\,i\Big(\intd^{2}x({\calL}+J\hat{A})\Big)\Big]_{\Big|J=0},
E_{c}=\frac{2(m-1)l}{R}\left(k+\frac{r_{+}^{2}}{l^{2}}\right)^{m-1}-\frac{2(m-1)(2m-3)r_{+}}{R}\sum_{i=0}^{m-2}\left(\begin{array}{c}{m-2}\\{i}\\\end{array}\right)\frac{1}{2i+1}\left(\frac{r_{+}}{l}\right)^{2i+1}k^{m-2-i}.
\delta_{\eta}\theta^{I}=\eta^{I}-\frac{i}{4}f^{-1}\partial_{i}h(\bar{\theta}^{J}\Gamma^{-}\eta^{J})\Gamma^{-}\Gamma^{i}\theta^{I}+\frac{i}{4}f^{-1}\partial_{i}h(\bar{\theta}^{J}\Gamma^{i}\eta^{J})\theta^{I}+{\mathcalO}(\theta^{4})~,
\frac{c_{2r-2}^{\nu}}{c_{2r}^{\nu}}=\frac{1}{\displaystyleh^{-2}[s^{2}-(\nu+2r-2)^{2}]-\frac{1}{\displaystyleh^{-2}[s^{2}-(\nu+2r-4)^{2}]-\frac{1}{\displaystyleh^{-2}[s^{2}-(\nu+2r-6)^{2}]\cdot\cdot\cdot}}}
ds^{2}~=~\alpha^{\prime}\left(\frac{u^{2}h(u)}{R^{2}}~e^{\gammaA}dx_{0}^{2}+\frac{u^{2}}{R^{2}}~e^{\gammaC}dx_{i}^{2}+\frac{R^{2}}{u^{2}h(u)}~e^{\gammaB}du^{2}+R^{2}~e^{\gammaD}d\Omega_{5}^{2}\right)~,
{^\ast}F_{\mu\nu}=\left(\begin{array}{cccc}{0}&{B_{x}}&{B_{y}}&{B_{z}}\\{-B_{x}}&{0}&{-E_{z}}&{E_{y}}\\{-B_{y}}&{E_{z}}&{0}&{-E_{x}}\\{-B_{z}}&{-E_{y}}&{E_{x}}&{0}\\\end{array}\right),
\begin{array}{rll}{}&{}&{0<(\psi-\sqrt{\phi^{2}-1})^{1/2}\leqx_{3}\leq(\psi+\sqrt{\phi^{2}-1})^{1/2}\;}\\{}&{\mathrm{~or~}}&{\phantom{0}-(\psi+\sqrt{\phi^{2}-1})^{1/2}\leqx_{3}\leq-(\psi-\sqrt{\phi^{2}-1})^{1/2}<0\;,}\\\end{array}
S_{eff}=\Gamma_{h_{\prime}}^{\alpha\beta}h_{\prime\alpha\beta}\left(0\right)+\frac12\intd^{4}k\Pi_{h_{\prime}}^{\mu\nu,\,\alpha\beta}\left(k\right)\,h_{\prime\mu\nu}\left(k\right)\,h_{\prime\alpha\beta}\left(-k\right)+\cdots.
I_{\nu}(r)\equiv\frac1{2i\pi}\int_{-i\infty}^{i\infty}\frac{\Gamma(2+4\nu)\Gamma(2+2\nu)\Gamma(1-2\nu)\Gamma(-z)}{\Gamma(2+4\nu+z)\Gamma(2+2\nu+z)\Gamma(1-2\nu+z)}[4a\nu^{4}\rho^{-1/\nu}]^{z}\,dz
\begin{array}{c}{\displaystyle{D_{\nu}^{TE}(t)=\nuD_{TE}^{(1)}(t)+D_{TE}^{(0)}(t)+\nu^{-1}D_{TE}^{(-1)}(t)+\nu^{-2}D_{TE}^{(-2)}(t)+\nu^{-3}D_{TE}^{(-3)}(t)},}\\\end{array}
(2)\quad{\mathrm{\bf{i}}}^{2}={\mathrm{\bf{k}}}^{2}={\mathrm{\bf{1}}},\quad{\mathrm{\bf{j}}}^{2}=-{\mathrm{\bf{1}}},\quad{\mathrm{\bf{i}}}{\mathrm{\bf{j}}}={\mathrm{\bf{k}}},\quad{\mathrm{\bf{j}}}{\mathrm{\bf{k}}}={\mathrm{\bf{i}}},\quad{\mathrm{\bf{k}}}{\mathrm{\bf{i}}}=-{\mathrm{\bf{j}}}.
\mathrm{{Tr}}_{L^{2}}\Bigr(fe^{-tP}\Bigr)\equiv\int_{M}dx\sqrt{\mathrm{{det}}g}\;f(x)\mathrm{{Tr}}U(x,x;t)\sim(4\pit)^{-m/2}\sum_{l=0}^{\infty}t^{l/2}A_{l/2}(f,P,{\calB}).
0=\partial_{+}\left({\frac{2\lambda_{--}}{1-\lambda^{2}}}\partial_{+}\phi+{\frac{1+\lambda^{2}}{1-\lambda^{2}}}\partial_{-}\phi\right)+\partial_{-}\left({\frac{2\lambda_{++}}{1-\lambda^{2}}}\partial_{-}\phi+{\frac{1+\lambda^{2}}{1-\lambda^{2}}}\partial_{+}\phi\right)
\kappa_{I}\epsilon^{ij}\nabla_{j}a_{0}^{I}({\bfr},t)+i\hbar\sum_{p}\frac{q_{p}^{I}}{2m_{p}c}\left[(D_{i}\Psi_{p})^{\dagger}({\bfr},t)\Psi_{p}({\bfr},t)-\Psi_{p}^{\dagger}({\bfr},t)(D_{i}\Psi_{p})({\bfr},t)\right]=0\,,
T^{\mu}\_{\nu}(x)=\left(\begin{array}{cc}{T_{\beta}^{\beta}-\Omega(x)^{-k^{\prime}}H(x)}&{0}\\{0}&{\Omega(x)^{-k^{\prime}}H(x)\}\\\end{array}\right)+\Omega^{k^{\prime}}\left(\begin{array}{cc}{-\eta}&{-\alpha^{\prime}}\\{\alpha^{\prime}}&{\eta\}\\\end{array}\right)
{\mathcalL}^{2\!\times\!2}=\frac{1}{2\,g^{2}}\sum_{i,a=1}^{2}\big(\dot{a}_{i}^{a}-a_{0}\,\epsilon^{ab}a_{i}^{b}\big)\big(\dot{a}_{i}^{a}-a_{0}\,\epsilon^{ac}a_{i}^{c}\big)-{\mathcalV}^{2\!\times\!2}(a_{i}^{a})\;,
0=\delta(y-\pir_{c})\mathrm{i}(\bar{\Psi}_{-})_{\mu}^{A}\gamma^{\mu}\left[2g\alpha_{\pi}+\frac{1}{4\sqrt{2}}\lambda_{0}(1+3\alpha_{\pi}^{2}\epsilon(y)^{2})\right](\delta+\gamma_{5}\sigma_{3})_{A}^{\;B}(\eta_{-})_{B}\,
\Omega^{\alpha\beta}=v_{~\gamma}^{\alpha}(y)v_{~\delta}^{\beta}(y)\left[(\gamma_{a})^{\gamma\delta}e^{a}(x)+R(\gamma_{ab})^{\gamma\delta}\omega^{ab}(x)+\theta^{(\gamma}{\calD}\theta^{\delta)}+(dvv^{-1})^{\gamma\delta}\right],
\alpha_{(0)}^{\prime\prime}+2\alpha_{(0)}^{\prime2}-\gamma_{(0)}^{\prime}\alpha_{(0)}^{\prime}=-\frac{1}{6}\phi_{(0)}^{\prime2}-\frac{1}{6}\left(b^{2}-\frac{2}{3}\right)\sigma^{2}e^{2(\gamma_{(0)}-\sqrt{2}b\phi_{(0)})},
\delta^{4}\left(\sum_{i=1}^{n}p^{i}\right)\Gamma_{n}^{12...n}(p^{1},p^{2},...,p^{n})=(2\pi)^{-4n}\left.\frac{\delta^{n}\Gamma}{\deltab_{1}(-p^{1})...\deltab_{n}(-p^{n})}\right|_{\tilde{\Phi}=0}.
\theta^{\dot{\alpha}}E_{t;\dot{\alpha}}{}^{a}=\theta^{\dot{\alpha}}\Omega_{t;{\dot{\alpha}}}{}^{ab}=0\,,\qquad\theta^{\dot{\alpha}}E_{t;\dot{\alpha}}{}^{\dot{\alpha}}=\theta^{\dot{\alpha}}e_{\dot{\alpha}}{}^{\alpha}(x)\equiv\Theta^{\alpha}\,.
g_{\mu\nu}=\left(\begin{array}{cccc}{-{\sigma}^{2}+A^{2}{\beta^{x}}^{2}+A^{2}{\beta^{z}}^{2}}&{A^{2}\beta^{x}}&{0}&{A^{2}\beta^{z}}\\{A^{2}\beta^{x}}&{A^{2}}&{0}&{0}\\{0}&{0}&{A^{2}l^{2}}&{0}\\{A^{2}\beta^{z}}&{0}&{0}&{A^{2}}\\\end{array}\right)\;\;,
\intd^{4}x\sqrt{g_{{}_{E}\pm}}{\calK}_{{}_{E}\pm}^{n}=\intd^{4}x\left({\frac{z_{\pm}}{z_{0}}}\right)^{(4-n)\beta}{\frac{1}{z_{\pm}^{n}}}\propto\intd^{4}x\;r_{\pm}^{4+(4/3)D_{2}-(n/3)D_{2}},
p_{\perp}\sim\frac{1}{4\pi^{2}a^{4}x^{7}}\int_{0}^{\infty}{z^{2}e_{1}^{2}(z)dz}=\frac{5a^{3}}{16\pi^{2}r^{7}},\quad\varepsilon\sim-4p\sim\frac{a^{3}}{2\pi^{2}r^{7}},\quadr\gga.
\left\{\frac{d^{2}}{dr^{2}}+\frac{2M}{\hbar^{2}}\left[E-V(r^{\prime})\right]-\frac{\left(l+\nu\right)^{2}-1/4}{r^{2}}\right\}{\mathcalG}_{l+\nu}(r^{\prime\prime},r^{\prime};E)=\delta(r^{\prime\prime}-r^{\prime})\;,
\beta_{i\bar{\jmath}}=-\frac1{2\pi}R_{i\bar{\jmath}}-\frac{4\zeta(3)}{3(4\pi)^{4}}\partial_{i}\partial_{\bar{\jmath}}\left[R_{k\bar{l}m\bar{n}}R^{p\bar{l}q\bar{n}}R_{p}{}^{k}{}_{q}{}^{m}-R_{k\bar{l}m\bar{n}}R^{m\bar{n}p\bar{q}}R_{p\bar{q}}{}^{k\bar{l}}\right]+\ldots
L_{f}=\frac{i}{2}\overline{{\Psi}}[\gamma^{a}V_{a}^{\mu}(\vec{\partial}_{\mu}+\frac{1}{2}\omega_{\mu}^{cd}\sigma_{cd})-(\overleftarrow{\partial}_{\mu}-\frac{1}{2}\omega_{\mu}^{cd}\sigma_{cd})\gamma^{a}V_{a}^{\mu}]\Psi+U(\overline{{\Psi}}\Psi)
v^{(a)}(y)=\left(\begin{array}{c}{{\frac{1}{4}}\intdz^{-}\epsilon(y^{-}-z^{-})\left[\alpha\partial_{z^{-}}^{2}g_{a}(z)-2\partial_{z^{-}}(g_{a}(z)\partial_{z^{-}}\phi)\right]}\\{g_{a}(y)}\\\end{array}\right)
(1)\quad{\mathrm{\bf{i}}}^{2}={\mathrm{\bf{j}}}^{2}={\mathrm{\bf{1}}},\quad{\mathrm{\bf{k}}}^{2}=-{\mathrm{\bf{1}}},\quad{\mathrm{\bf{i}}}{\mathrm{\bf{j}}}={\mathrm{\bf{k}}},\quad{\mathrm{\bf{j}}}{\mathrm{\bf{k}}}=-{\mathrm{\bf{i}}},\quad{\mathrm{\bf{k}}}{\mathrm{\bf{i}}}=-{\mathrm{\bf{j}}},
{\calF}_{\mu\nu}-{\calH}_{\mu\nu}\equivG_{\mu\nu}^{a}\hat{\Phi}^{a},\qquad{\calH}_{\mu\nu}\equiv-\frac{1}{e}\,\epsilon_{abc}\hat{\Phi}^{a}(D_{\mu}\hat{\Phi})^{b}(D_{\nu}\hat{\Phi})^{c}.
1+\vec{e}_{\tilde{B}}\cdot(\vec{e}_{\tilde{B}}+\vec{e}_{\tilde{F}_{0}})+\vec{h}_{\tilde{B}}\cdot(\vec{h}_{\tilde{B}}+\vec{h}_{\tilde{F}_{0}})+(\vec{e}_{\tilde{B}}+\vec{e}_{\tilde{F}_{0}})\cdot(\vec{h}_{\tilde{B}}+\vec{h}_{\tilde{F}_{0}})\,\mathrm{Pf{\tilde{B}}}>0\,.
(T_{lm}^{(j)})^{\dagger}=(-1)^{m}T_{l,-m}^{(j)},\quad\mathrm{Tr}[T_{lm}^{(j)}T_{l^{\prime}m^{\prime}}^{(j)}]=(-1)^{m}\delta_{ll^{\prime}}\delta_{m+m^{\prime},0}.
Z=\int{\calD}B^{\mu}{\calD}\pi_{\mu}{\calD}\lambda{\calD}\pi_{\lambda}{\calD}\rho{\calD}\pi_{\rho}\prod_{\alpha,\beta=0}^{3}\delta(\tilde{\Omega}_{\alpha})\delta(\Gamma_{\beta})det\mid\{\tilde{\Omega}_{\alpha},\Gamma_{\beta}\}\mide^{iS^{\prime}},
{\calL}_{3}=\frac1{1+4{\bf\psi}^{2}}\left({\textstyle\frac12}\left(\delta^{ij}+4\psi^{i}\psi^{j}\right)\partial_{\mu}\psi^{i}\partial^{\mu}\psi^{j}-\varepsilon^{\mu\nu}\varepsilon^{ijk}\psi^{i}\partial_{\mu}\psi^{j}\partial_{\nu}\psi^{k}\right).
\begin{array}{lll}{{\bfG}(\Lambda)={\displaystyle\int}d^{3}x\Lambda_{AB}{\bfG}^{AB},}&{{\bfL}(\,\epsilon)={\displaystyle\int}d^{3}x\,\epsilon_{A}{\bfL}^{A},}&{{\bfR}(\eta)={\displaystyle\int}d^{3}x\eta_{A}{\bfR}^{A},}\\{{\bfH}(N)={\displaystyle\int}d^{3}xN{\bfH},}&{{\bfH}(\vec{N})={\displaystyle\int}d^{3}xN^{i}{\bfH}_{i},}\\\end{array}
f_{TTT}=\frac{96i}{\pi}\frac{(\Phi_{T}(T))^{2}}{\Phi(T)(\Phi(T)-\Phi(\frac{i}{\sqrt{2}}))}\frac{\Phi^{\frac{1}{2}}(U)}{\Phi_{U}(U)}\frac{{(\Phi(U)-\Phi(\frac{i}{\sqrt{2}}))}^{\frac{1}{2}}}{(\Phi(U)-\Phi(T))}.
A_{1}(l_{0})={\frac{g_{s}^{2}}{M_{s}^{10}}}l_{0}\intd^{4}\eta\delta(1-\sum_{i}\eta_{i})E_{2}(-{\frac{2\pi}{l_{0}M_{s}^{2}}}(s\eta_{1}\eta_{3}+t\eta_{2}\eta_{4}))+\dots
q^{-{\frac{1}{16}}}\prod(1-q^{n})(1-q^{n-1/2})\longrightarrowe^{-2\pii/16}~q^{-{\frac{1}{16}}}\prod(1-q^{n})(1+q^{n-1/2})=e^{-2\pii/16}~q^{-{\frac{1}{12}}}{\frac{\eta^{3}}{\eta(\tau/2)\eta(2\tau)}}
\partial_{z}W(z)\vert_{z=z_{0}}=-e^{W(z_{0})-\sqrt{2}b\kappa\Xi(z_{0})}\frac{\sqrt{2}}{6}\lambda_{0},\quad\partial_{z}\Xi(z)\vert_{z=z_{0}}=-e^{W(z_{0})-\sqrt{2}b\kappa\Xi(z_{0})}b\kappa^{-1}\lambda_{0}.
S_{Pl}=S_{cm}-\frac{1}{2}\sum_{p=1}^{D-1}\left\{\,\intd^{p+1}\sigma\sqrt{\gamma_{p}}\left[\,\gamma_{p}^{mn}\,\partial_{m}\,Y_{p}^{\mu}\,\partial_{n}\,Y_{\mu\,p}-p\,m_{p+1}\,\right]\,\right\}
U(\varphi)=\left({\frac{M_{b}}{M_{p}}}\right)^{2}\left[{\frac{k_{1}M_{b}^{2}}{2}}\;\varphi^{2}+{\frac{k_{2}}{3!}}\;\varphi^{3}+{\frac{k_{4}}{4!\,M_{b}^{2}}}\;\varphi^{4}+\cdots\right],
V=\left(\begin{array}{lll}{(-q)^{-n}\Theta_{n}}&{\quadA_{n1}}&{\quad\Theta_{n}}\\{-A_{n1}^{t}}&{\quad1}&{-(-q)^{n}A_{n1}^{t}}\\{\quad\Theta_{n}}&{(-q)^{n}A_{n1}}&{\\(-q)^{n}\Theta_{n}}\\\end{array}\right)
\rho_{s}^{(N_{f})}(z,m)=\frac{z}{2}\frac{B^{(N_{f})}(z,m)\sum_{f=0}^{N_{f}}\widetilde{D}_{f}^{(N_{f})}(z,m)-D^{(N_{f})}(z,m)\sum_{f=0}^{N_{f}}\widetilde{B}_{f}^{(N_{f})}(z,m)}{[C^{(N_{f})}(m)]^{2}\prod_{f=1}^{N_{f}}(z^{2}+m_{f}^{2})}\:,
l_{p}^{-2}ds^{2}=\hat{H}^{-1/3}\left[-dt^{2}+\sum_{i=1}^{5}(dx^{i})^{2}+\hat{H}\Big(d\hat{\rho}^{2}+\hat{\rho}^{2}d\Omega_{4}^{2}\Big)\right]
\frac{\partialf}{\partialt}=F(f,\overline{f},f~^{\prime},{\overline{f}}~^{\prime},{\overline{D}}f,D{\overline{f}}...),\quad\frac{\partial{\overline{f}}}{\partialt}={\overline{F}}(f,\overline{f},f~^{\prime},{\overline{f}}~^{\prime},{\overline{D}}f,D{\overline{f}}...),
|\pi_{i=1\simN_{c}-1}|=\Lambda_{T}^{2},\\|\pi_{u}|=\Lambda_{T}^{N_{c}-1},\\|\pi_{i=N_{c}\simN_{f}}|=\xi\Lambda_{T}^{2},\\|\pi_{\lambda}|\sim\Lambda^{3}\xi^{\frac{\rho+N_{f}-N_{c}+1}{N_{f}-N_{c}}},
H_{yukawa}=\sum_{\alpha,\gamma}h_{\alpha,\,\gamma}Q_{L}^{\gamma}(\bar{U}_{I}H_{U\,I}^{\alpha}+{\bar{U}_{II}}{H}_{U\,II}^{\alpha})\,+(\{\bar{U},H_{U}\}\to\{\bar{U}^{\prime},H_{U}^{\prime}\}),
K=i\int_{0}^{\infty}dT\;e^{-iT(\gamma^{\mu}P_{\mu}^{\prime}-m)}\delta^{4}(P^{\prime\prime}-P^{\prime})={\frac{\delta^{4}(P^{\prime\prime}-P^{\prime})}{\gamma^{\mu}P_{\mu}^{\prime}-m-i0_{+}}}\,.
\frac{\partial\Theta}{\partial\lambda_{k}^{1}}=\frac{H_{n-k}(r)\;\Im\;[e^{i(k-n)\theta}+\sum_{l=0}^{n-1}\lambda_{l}^{\star}H_{n-l}(r)e^{i(k-l)\theta}]}{\mid1+\sum_{s=0}^{n-1}\lambda_{s}H_{n-s}(r)e^{-i(n-s)\theta}\mid^{2}}\;\;.
E=\intd^{4}x\left\{\frac12\left(F_{12}+s(|T|^{2}-\zeta)\right)^{2}+|D_{1}T-isD_{2}T|^{2}+s\zetaF_{12}+V(T)-\frac12(|T|^{2}-\zeta)^{2}\right\}
\langle\prod_{j=1}^{m}{\calO}_{e^{\alpha_{j}}}(z_{j})\rangle=\langle{\calO}_{e^{\alpha_{1}}}(z_{1}){\calO}_{e^{\alpha_{2}}}(z_{2}){\calO}_{e^{a}}(z_{s1})\rangle\eta^{ab}\langle{\calO}_{e^{b}}(z_{s2})\prod_{j=3}^{m}{\calO}_{e^{\alpha_{j}}}(z_{j})\rangle.
\frac{f^{\prime}\left(\frac{x_{1}}{\sqrt{\gamma}}+i\sqrt{\gamma}x_{2}\right)}{f\left(\frac{x_{1}}{\sqrt{\gamma}}+i\sqrt{\gamma}x_{2}\right)}=\frac{\left(1-\frac{2\gamma_{1}\hbar}{\gamma\theta}\right)x_{1}-i\gammax_{2}+z\left(\frac{\sqrt{\gamma}}{2}+\frac{\gamma_{1}\hbar}{\theta\sqrt{\gamma}}\right)}{\frac{\theta\sqrt{\gamma}}{2}-\frac{\gamma_{1}\hbar}{\sqrt{\gamma}}};
{\hat{\calG}}^{a}({\bfr})=\partial_{j}\Pi_{j}^{a}({\bfr})+g{\epsilon}^{abc}A_{j}^{b}({\bfr})\Pi_{j}^{c}({\bfr})+j_{0}^{a}({\bfr})\;\;\mathrm{and}\;\;j_{0}^{a}({\bf{r}})=g\,\,\psi^{\dagger}({\bf{r}})\,{\textstyle\frac{\tau^{a}}{2}}\,\psi({\bf{r}})
T_{h}=\frac{1}{2}\left(\!\!\!\begin{array}{cccccccccccccc}{-1}&{4}&{-2}&{1}&{-2}&{-2}&{0}&{2}&{0}&{0}&{2}&{-1}&{0}&{-1}\\{0}&{0}&{-1}&{1}&{-1}&{3}&{-1}&{0}&{-1}&{-1}&{1}&{0}&{-1}&{1}\\{1}&{0}&{0}&{-3}&{0}&{0}&{-6}&{6}&{2}&{0}&{0}&{-3}&{0}&{3}\\\end{array}\!\right)^{\!\!\sim}
Y_{\alpha}^{N}(\eta)=a^{-\frac{n}{2}}e^{\pmim_{n-1}\theta_{1}}\prod_{k=0}^{n-2}(sin{\theta_{n-k}})^{m_{k+1}}C_{m_{k}-m_{k+1}}^{m_{k+1}+\frac{1}{2}(n-k)}(cos{\theta_{n-k}})
D_{n}=\intd^{d}x_{1}...\intd^{d}x_{n}\intd^{d}y_{1}...\intd^{d}y_{n}{\frac{\delta^{n}S}{\delta\sigma_{\alpha_{1}\beta_{1}}(x_{1},y_{1})...\delta\sigma_{\alpha_{n}\beta_{n}}(x_{n},y_{n})}}|_{\sigma_{0}}\eta_{\alpha_{1}\beta_{1}}(x_{1},y_{1})...\eta_{\alpha_{n}\beta_{n}}(x_{n},y_{n})
(-)^{i}\phi_{i}^{*}\phi_{j}^{*}f^{ji}{}_{a_{1}}=(-)^{a}y_{a}y_{b}g^{ba}{}_{a_{1}}+(-)^{i}\phi_{i}^{*}y_{a}g^{ai}{}_{a_{1}}+(-)^{i}\phi_{i}^{*}\phi_{j}^{*}g^{ji}{}_{a_{1}}\.
{\calL}^{kin}\cong2A_{\sigma}^{(s)}\partial_{+}\partial_{-}A^{(s)\sigma}+\frac{1}{2}A_{\sigma}^{(m)}\BoxA^{(m)\sigma}+i\bar{\psi}_{-}^{(s)}\gamma_{-}\partial_{+}\psi_{-}^{(s)}\,+\frac{i}{4}\bar{\psi}_{-}^{(m)}\frac{\gamma_{-}}{\partial_{-}}\Box\psi_{-}^{(m)}
H^{++\mu-}=x_{h}^{-{\breve{\alpha}}}\partial_{h{\breve{\alpha}}}^{-}H^{++{\breve{\beta}}+}=2g(\sigma_{3})_{\nu}^{\mu}(x_{h}^{\nu-}\rho^{++}+x_{h}^{\nu+}(x_{h}^{1-}x_{h}^{2+}+x_{h}^{1+}x_{h}^{2-}))
\delta\psi_{A\mu}=\mathcal{D}_{\mu}\,\epsilon_{A}-\frac{1}{3}\,\mathcal{T}_{AB}^{\rho\sigma}\left(g_{\mu\rho}\,\gamma_{\sigma}-\frac{1}{8}\,\epsilon_{\mu\rho\sigma\lambda\nu}\,\gamma^{\lambda\nu}\right)\,\epsilon^{B}
D_{\mu\nu}^{LC}(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}^{LC}k_{\nu}+n_{\mu}^{LC}}{\left[k_{-}\right]_{CPV}}\right],
N=6:\quadA={\frac{1}{3}}\left(\begin{matrix}{-1}&{1}&{2}\\{2}&{1}&{0}\\{3}&{{\frac{3}{2}}}&{0}\\\end{matrix}\right);\quadB={\frac{1}{3}}\left(\begin{matrix}{1}&{2}&{0}\\{1}&{-1}&{2}\\{{\frac{3}{2}}}&{3}&{0}\\\end{matrix}\right)
\widetilde\Sigma_{\lambda}(p^{2})\=\C_{2}^{\mathrm{adj}}\,g^{2}\,(3+\xi)\int{{\frac{dk^{2}}{16\pi^{2}}}\,{\frac{k^{2}}{{\calM}^{2}}}\,{\frac{A_{\lambda}({\calM}^{2})}{A_{\lambda}(m^{2})}}\,{\frac{\widetilde\Sigma_{\lambda}(k^{2})}{k^{2}+\widetilde\Sigma_{\lambda}^{2}(k^{2})}}}\,
{\calD}_{r}=\left[\frac{1}{{\kappa}^{2}}\left({\nabla}_{r}^{2}-\frac{{\ell}({\ell}+1)}{r^{2}}\right)^{3}-\frac{2}{\kappa}\left({\nabla}_{r}^{2}-\frac{{\ell}({\ell}+1)}{r^{2}}\right)^{2}+\left[1-{\ell}({\ell}+1)\right]\left({\nabla}_{r}^{2}-\frac{{\ell}({\ell}+1)}{r^{2}}\right)+{\kappa}{\ell}\left({\ell}+1\right)\right].
S=-\frac1{g_{YM}^{2}}\intd^{10}x\\left[\e^{-a|T|^{2}}|D_{\mu}T|^{2}+\frac14e^{-b|T|^{2}}(1+c|T|^{2})F_{\mu\nu}F^{\mu\nu}+V(T)\\right]
\Phi_{0}=\left(\begin{array}{c}{q^{6}+iq^{7}}\\{q^{8}+iq^{9}}\\\end{array}\right),\qquad\Phi_{1}=\left(\begin{array}{c}{\bar{q}^{6}+i\bar{q}^{7}}\\{\bar{q}^{8}+i\bar{q}^{9}}\\\end{array}\right).
({\bfe}_{\underline{a}},\;sk_{\underline{a}},\;sk_{{\underline{a}}-1},s)\to(\mathrm{i}{\bfe}_{\underline{a}},\;\mathrm{i}sk_{\underline{a}},\;\mathrm{i}sk_{{\underline{a}}-1},(-\mathrm{i})^{\delta_{1{\underline{a}}}}s).
ds^{2}=\left(1-\left(\frac{r_{H}}{r}\right)^{7-p}\right)^{\frac{1}{8}}\left[-dt^{2}+dy^{s}dy_{s}+\frac{dr^{2}}{1-\left(\frac{r_{H}}{r}\right)^{7-p}}+r^{2}d\Omega_{8-p}^{2}\right],
\left(\sigma_{AdS_{3}\timesS^{3}}^{\delta\phi}\right)_{m=0,l^{\prime}=0}^{\omega\to0}={\frac{1}{3}}\left[{\frac{\tilde{\calA}_{H}^{6D}}{4}}\left({\frac{r_{0}}{R}}\right)^{4}\right],~~\tilde{\calA}_{H}^{6D}={\calA}_{H}^{6D}.
\left(1-u^{2}\right)\left(E+gu\right)\frac{\mathrm{d}^{2}G}{\mathrm{d}u^{2}}-\left(\left(4u+1\right)\left(E+gu\right)+g\left(1-u^{2}\right)\right)\frac{\mathrm{d}G}{\mathrm{d}u}+
\mathrm{exp}\left[\frac{2i\pi}{k}\taue(\tilde{n}-1/2)+\frac{2i\pi}{k}e(\widetilde{n_{0}}z_{1}+(1-\widetilde{n_{0}})z_{2}-w)+\frac{2i\pi}{k}(\tilde{n}-1/2)e_{0}(z_{1}-z_{2})\right]dw
\int\frac{d^{3}k}{(2\pi)^{3}}\left\{D_{\mu\nu}^{\mathrm{T}}(k-\ell)\mathrm{tr}\left[\gamma_{\mu}S(k)\Gamma_{\nu}^{\mathrm{BC}}(k,\ell)S(\ell)\right]+\frac{1}{(k-\ell)^{2}}F(k,\ell)\right\}=0,
\tilde{\eta}=<(e^{\frac{\alphaG}{2}})_{,A}>\phi^{A}+<(e^{\frac{\alphaG}{2}})_{,\bar{A}}>\chi^{\bar{A}},\qquad\qquad\bar{\tilde{\eta}}=<(e^{\frac{\alphaG}{2}})_{,A}>\chi^{A}+<(e^{\frac{\alphaG}{2}})_{,\bar{A}}>\phi^{\bar{A}}.
K_{\theta}(2,1)\psi_{\theta}(1)=(e^{i\theta_{2}/2}e^{-i\theta_{1}^{\prime}/2})e^{i\theta_{1}/2}=e^{i(\theta_{2}-\theta_{1}^{\prime}+\theta_{1})}=e^{i\theta_{2}/2}=\psi_{\theta}(2).
b_{n}J_{n-1+{\frac{D}{2}}}^{\prime}(|\omega|r^{\prime})+c_{n}J_{-n+1-{\frac{D}{2}}}^{\prime}(|\omega|r^{\prime})-a_{n}J_{n-1+{\frac{D}{2}}}^{\prime}(|\omega|r^{\prime})={\frac{(2n+D-2)\Gamma\left({\frac{D-2}{2}}\right)}{4(\pir^{\prime})^{\frac{D}{2}}|\omega|}}
\omega^{red}=d{\bfp}\wedged{\bfX}+\frac{c^{2}}{c_{0}}{dp\wedgedq},\quad{\calH}^{red}=\frac{sc^{2}}{c_{0}}\left(p^{2}/2+\frac{c_{0}^{2}}{2c^{2}q^{2}}-\frac{{\bfp}^{2}-c_{0}^{2}}{2c^{2}}\right).
S_{\mathrm{cf}}=\int\!d^{2}x\,[\psi_{+}^{\dagger}i\partial_{-}\psi_{+}+\psi_{-}^{\dagger}i\partial_{+}\psi_{-}+\psi_{-}^{\dagger}B_{+}\psi_{-}+{b}_{+}\partial_{-}c_{+}]
\frac{\partial{\calV}}{\partial\varphi}={\textstyle\frac{2}{n}}\frac{1}{\mathrm{Tr}({\calM}^{2})}\,{\calM}_{mn}\frac{\partial{\calV}}{\partial{\calM}_{mn}}\,,\,\,\,\,\Rightarrow{\calM}^{mn}{\calM}_{pq}\frac{\partial{\calV}}{\partial{\calM}_{pq}}-\mathrm{Tr}({\calM}^{2})\frac{\partial{\calV}}{\partial{\calM}_{mn}}=0\,.
{\calJ}_{(\mu_{1}\dots\mu_{n};\mu_{n+1})}^{(n)}+\frac{\partial{\calJ}_{(\mu_{1}\dots\mu_{n}}^{(n)}}{\partial\psi^{\sigma}}\Gamma_{\mu_{n+1})\lambda}^{\sigma}\psi^{\lambda}=\frac{i}{2}\psi^{\rho}\psi^{\sigma}R_{\rho\sigma\nu(\mu_{n+1}}{{\calJ}^{(n+1)\nu}}_{\mu_{1}\dots\mu_{n})}