harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
X^{k}=x^{k}+\frac{i\pi_{r}}{m(\omega+m)}\sum_{a=1}^{N}[\psi_{a}^{k},\psi_{a}^{r}]\;,\;\;\theta_{a}^{k}=\psi_{a}^{k}+\frac{\pi_{k}(\omega-m)}{{m\vec{\pi}}^{2}}\pi_{r}\psi_{a}^{r}\;.
\chi_{\Lambda}(\bar{C}_{l}^{\Lambda})=q^{-\frac{l}{2}(\Lambda,\Lambda+2\rho)}\sum_{\nu\in\Pi(\Lambda)}(m_{\nu}^{+}+(-1)^{l}m_{\nu}^{-})q^{\frac{l}{2}(\nu,\nu+2(\Lambda+\rho))}\frac{D_{q}[\Lambda+\nu]}{D_{q}[\Lambda]}\,,~~~~l\in{\bfZ}^{+}
\overline{{V}}=\left[\begin{matrix}{{\frac{1}{\sqrt{2}}}E^{-1}}\\{}&{{\frac{1}{\sqrt{2}}}E^{-1}}\\{}&{}&{I_{24}}\\\end{matrix}\right]\times\left[\begin{matrix}{I_{2}}&{G+B-C}&{-A^{T}}\\{I_{2}}&{-G+B-C}&{-A^{T}}\\{0}&{VLA}&{V}\\\end{matrix}\right]\,
[\hat{y}_{\alpha},\,\hat{y}_{\beta}]=2i\epsilon_{\alpha\beta},\qquad[\hat{\bar{y}}_{\dot{\alpha}},\,\hat{\bar{y}}_{\dot{\beta}}]=2i\epsilon_{\dot{\alpha}\dot{\beta}},\qquad[\hat{y}_{\alpha},\,\hat{\bar{y}}_{\dot{\beta}}]=0\,,
H=\intd^{3}x\{\frac{1}{2}[(\delta_{iI}-\frac{\partial_{i}\partial_{I}}{\partial^{2}})p_{I}]^{2}+\frac{1}{4}F_{ij}^{2}(a)-\frac{1}{2}j_{0}\frac{1}{\partial^{2}}j_{0}-j_{I}a_{I}+j_{3}\partial_{3}^{-1}\partial_{I}a_{I}\}+H_{M}
\Omega_{+q}^{+2i}=0\Rightarrow\psi_{-2\dot{q}}^{1-}=0,\\psi_{+2\dot{q}}^{2+}=0\Rightarrow\Omega_{-q}^{-2i}=0,\\psi_{-2\dot{q}}^{2+}=-\frac{i}{32}\tilde{\gamma}_{\dot{q}q}^{i}\Omega_{-q}^{+2i},
S={\frac{1}{4\pi\alpha^{\prime}}}\int\!d^{2}\sigma\left(\sqrt{-h}h^{\alpha\beta}g_{\mu\nu}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}+\epsilon^{\alpha\beta}B_{\mu\nu}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\right)
\bar{G}_{\dot{\alpha}\dot{\beta}}\equiv-\frac12G_{\mu\nu}(\bar{\sigma}^{\mu})_{\dot{\alpha}\gamma}(\sigma^{\nu})_{\dot{\beta}}^{\gamma}\equiv({E}^{a}+i\,{B}^{a})({\bar{\sigma}}^{a})_{\dot{\alpha}\dot{\beta}}=0\,.
A_{i}^{(g)}(x_{0},{\bfx})=x_{0}^{d/2}\int{\frac{d^{d}k}{(2\pi)^{d}}}e^{-i{\bfk}\cdot{\bfx}}a_{i}({\bfk}){\frac{1}{2}}\left[\Gamma(\nu)\left({\frac{\xi}{2}}\right)^{-\nu}\!\!\!\!+\Gamma(-\nu)\left({\frac{\xi}{2}}\right)^{\nu}+\dots\right].
\begin{array}{cc}{{\calQ}^{a}=\left(\begin{array}{c}{{\calQ}_{+}^{a}}\\{{\calQ}_{-}^{a}}\\\end{array}\right)~~~~}&{~~~~\bar{{\calQ}}_{a}={{\calQ}^{a}}^{\dagger}\Gamma^{0}=(\bar{{\calQ}}_{a}^{-},\bar{{\calQ}}_{a}^{+})}\\\end{array}
\lambda_{\pm}(q_{0})=\lambda(q_{0})\pm\frac{4h(\frac{2}{\pi})^{\frac{1}{2}}(16h)^{\frac{q_{0}}{2}}}{[\frac{1}{2}(q_{0}-1)]!}e^{-4h}\bigg[1+O\bigg(\frac{1}{h}\bigg)\bigg]
\mathcal{L}_{p-brane}^{Polyakov}=\frac{1}{2(d-1)}\,\int\,d^{d}\xi\,\sqrt{-\,\mathrm{det}\,h_{\mu\nu}}\,\left\{\,h^{\rho\sigma}\,\partial_{\rho}X^{\underline{{\mu}}}\,\partial_{\sigma}X^{\underline{{\nu}}}\,g_{\underline{{\mu\nu}}}+\left(d-2\right)\right\}
2\sum_{k=1}^{n_{c}-\tilde{n}_{c}}\frac{\delta_{\ell}-\gamma_{k}}{1-\omega^{2k-2\ell+1}}=(n_{c}-\tilde{n}_{c})\delta_{\ell}-2\sum_{k=1}^{n_{c}-\tilde{n}_{c}}\frac{\gamma_{k}}{1-\omega^{2k-2\ell+1}}.
{\calH}={\frac{1}{2}}(\pi_{\phi}+gA_{1})^{2}+{\frac{1}{2\tilde{\alpha}}}\pi_{1}^{2}+{\frac{1}{2}}\phi^{2}+\pi_{1}A_{0}^{\prime}-gA_{0}\phi^{\prime}-{\frac{1}{2}}ag^{2}(A_{0}^{2}-A_{1}^{2}).
\begin{array}{c}{h_{n}(x;\eta^{\prime})=\Phi(\eta^{\prime},\eta)\h_{n}(x;\eta)=}\\{=\left(\frac{e^{(n+1/2)\pi\beta/L}+e^{i\eta\prime}}{e^{(n+1/2)\pi\beta/L}+e^{i\eta}}\right)h_{n}(x;\eta).}\\\end{array}
\sigma_{\mathrm{abs}}={\frac{4\pi^{3}r_{0}^{2}\;|\alpha^{2}+D|}{\omega}}{\frac{e^{\;4\pii\alpha}-1}{(e^{\;2\pii(\alpha+i\sqrt{D})}-1)(e^{\;2\pii(\alpha-i\sqrt{D})}-1)}}[1+O(\omega^{2}r_{1}^{2})].
{}^{(n-1)}R_{ABCD}={}^{(n)}R_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}^{\pm}\;\;q^{A^{\prime}}{}_{A}\;q^{B^{\prime}}{}_{B}\;q^{C^{\prime}}{}_{C}\;q^{D^{\prime}}{}_{D}\;+K_{AC}^{\pm}\;K_{BD}^{\pm}-K_{AD}^{\pm}\;K_{BC}^{\pm}.
{\cal{H}}=\frac{1}{2}\intd^{2}x\displaystyle\sum_{n=0}^{\infty}(-1)^{n}(\pi_{(n)}^{2}+\varphi_{(n)}^{2})=\frac{1}{4}\intd^{2}x\displaystyle\sum_{n=0}^{\infty}(-1)^{n}[(\pi_{(n)}+\varphi_{(n)}^{\prime})^{2}+(\pi_{(n)}-\varphi_{(n)}^{\prime})^{2}]
\frac{\partial^{2}f_{n}}{\partial\phi_{a}^{2}}=(\lambda_{n}-D)f_{n}+\frac{1}{2}(D-2)\phi_{a}\frac{\partialf_{n}}{\partial\phi_{a}}+2\frac{\partialV_{*}}{\partial\phi_{a}}\frac{\partialf_{n}}{\partial\phi_{a}}.
\left\{{\frac{\partial^{2}}{\partialr^{2}}}+{\frac{5}{r}}{\frac{\partial}{\partialr}}-{\frac{l(l+4)}{r^{2}}}+(\omega^{2}-k_{1}^{2}-k_{2}^{2}-k_{3}^{2})\left(1+{\frac{R_{0}^{4}}{r^{4}}}\right)\right\}\varphi_{B=0}^{l}=0.
[x^{\tilde{i}},x^{\tilde{j}}]=i\theta^{\\!\!\tilde{i}\tilde{j}},\quad[\partial_{\tilde{i}},x^{\tilde{j}}]=i\delta_{\tilde{i}}^{\tilde{j}},\quad[\partial_{\tilde{i}},\partial_{\tilde{j}}]=-i\Phi_{\tilde{i}\tilde{j}}^{\prime}\quad(\tilde{i},\tilde{j}=\tilde{1},\tilde{2})
\begin{array}{rcl}{U_{f}}&{=}&{\displaystyle+[D_{m}(k_{1})D_{m}(k_{2})+D(k_{1})D(k_{2})]D(k_{1}+k_{2}-p)(1+K_{12})}\\{}&{}&{\displaystyle+2D_{m}(k_{1})D(k_{2})D(k_{1}+k_{2}-p)(1-K_{12})}\\{}&{}&{\displaystyle+4[(N-1)D(k_{1})D(k_{2})}\\{}&{}&{\displaystyle+D(k_{1}+p)D(k_{2}+p)K_{12}]D(k_{1}+k_{2}+p),}\\\end{array}
P_{Y_{(2,1)}}^{I_{1}I_{2}I_{3},J_{1}J_{2}J_{3}}=\frac{1}{3}[\delta^{I_{3}J_{3}}(\delta^{I_{1}J_{2}}\delta^{I_{2}J_{1}}+\delta^{I_{1}J_{1}}\delta^{I_{2}J_{2}})-\delta^{I_{3}J_{1}}(\delta^{I_{1}J_{2}}\delta^{I_{2}J_{3}}+\delta^{I_{2}J_{2}}\delta^{I_{1}J_{3}})].
\left(-\partial_{1}^{2}-\partial_{2}^{2}-(1-x^{2})\left(\frac{1}{r_{1}^{2}}+\frac{1}{r_{2}^{2}}\right)\partial_{x}^{2}+r_{1}^{2}\,r_{2}^{2}(1-x^{2})\right)\,\psi=0
\delta\hat{A}_{\alpha}=\hat{A}_{\beta}\partial_{\alpha}\epsilon^{\beta}+\epsilon^{\beta}\partial_{\beta}\hat{A}_{\alpha}+{\frac{1}{2}}M_{\alpha\beta}^{-1}\left(\begin{array}{cl}{\hat{A}_{+}\partial_{-}\epsilon^{+}}\\{\hat{A}_{-}\partial_{+}\epsilon^{-}}\\\end{array}\right)^{\beta}\,.
1+\frac{h_{2}^{2}}{f_{2}}=1,\\f_{3}=-\tilde{v},\\f_{1}-\frac{h_{1}^{2}}{f_{2}}=1-\tilde{v}^{2},\\f_{2}-\frac{h_{1}^{2}}{f_{1}+f_{3}^{2}}=1,
\Big(t_{8}^{r_{1}\cdotsr_{8}}t_{8}^{s_{1}\cdotss_{8}}\pm\epsilon^{r_{1}\cdotsr_{8}np}\epsilon^{s_{1}\cdotss_{8}}{}_{np}\Big)W_{r_{1}r_{2}s_{1}s_{2}}W_{r_{3}r_{4}s_{3}s_{4}}W_{r_{5}r_{6}s_{5}s_{6}}\,H_{r_{7}r_{8}m}H_{s_{7}s_{8}}{}^{m}\,,
K^{\Omega}(t)=\frac{1}{\left|\Omega\right|}\sum_{x,y\in\Omega}\delta_{x^{y},x^{-1}}\prod_{\xi\in\mathcal{O}_{-}(x,y)}K\left(\frac{\lambda_{\xi}^{2}t}{\left|\xi\right|}\right)\prod_{\xi\in\mathcal{O}_{+}(x,y)}Z\left(\frac{\left|\xi\right|}{2\lambda_{\xi}^{2}it}+\frac{\kappa_{\xi}}{\lambda_{\xi}}\right)
\frac{V_{\perp}}{N}\sime^{-\betam_{Dp}}\sum_{n}e^{-\beta\epsilon_{n}}\intd\vec{p}_{\perp}e^{-\beta\vec{p}_{\perp}^{2}/2m_{Dp}}=e^{-\beta(m_{Dp}+F_{g})}\left(\frac{m_{Dp}}{\beta}\right)^{d_{\perp}/2}
\Gamma_{x}^{n_{x}}(k_{x}^{n_{x}},M)={\frac{\gamma_{x}^{n_{x}}}{M}}\sqrt{[M^{2}-(m_{x}+\mu_{x}^{n_{x}})^{2}][M^{2}-(m_{x}-\mu_{x}^{n_{x}})^{2}]}
\hat{S}_{\hat{A}}=-\frac{1}{4}\intd^{D}x\sqrt{-\hat{G}}e^{-\hat{\phi}}\left(\hat{g}^{\hat{\mu}\hat{\rho}}\hat{g}^{\hat{\nu}\hat{\lambda}}\delta_{IJ}\hat{F}_{\hat{\mu}\hat{\nu}}^{I}\hat{F}_{\hat{\rho}\hat{\lambda}}^{J}\right)
\hat{K}(\Psi)_{ij}=\sum_{k}P_{I}^{L}(\mathcal{I}F^{I})_{ik}\star\Psi_{kj}-\Psi_{ik}\starP_{I}^{L}(\mathcal{I}F^{I})_{kj}=-\sum_{k}P_{I}^{R}(F_{ik}^{I}\Psi_{kj})-P_{I}^{L}(\Psi_{ik}F_{kj}^{I})\,
\dot{x}^{2}(\tau,0)=A^{2}\frac{[f(\tau)-g(\tau)]^{2}}{4\acute{f}(\tau)\acute{g}(\tau)},\;\;\;\;\;\;\dot{x}^{2}(\tau,l)=A^{2}\frac{[f(\tau+l)-g(\tau-l)]^{2}}{4\acute{f}(\tau+l)\acute{g}(\tau-l)}.
\begin{array}{ll}{g_{_E}(\widehat{\mathbf{f}}_{\mu}(x),\widehat{\mathbf{f}}_{\nu}(x))=g(\mathbf{f}_{\mu}(x),\mathbf{f}_{\nu}(x))=g_{\mu\nu}(x)}\\{g_{_E}(\widehat{\mathbf{f}}_{\mu}(x),\mathbf{e}_{k}(x))=0}\\{g_{_E}(\mathbf{e}_{k}(z),\mathbf{e}_{l}(z))=g_{0\,kl}(x)\,.}\\\end{array}
V(\rho)=E\left[1-\frac{1+\frac{E}{2m_{p}}\left(1+\frac{R^{4}}{(\rho^{2}-l^{2})^{2}}\right)}{\left[1+\frac{E}{m_{p}}\left(1+\frac{R^{4}}{(\rho^{2}-l^{2})^{2}}\right)\right]^{2}}\right]+\frac{L^{2}}{2m_{p}\rho^{2}}\frac{1}{\left[1+\frac{E}{m_{p}}\left(1+\frac{R^{4}}{(\rho^{2}-l^{2})^{2}}\right)\right]^{2}}.
M_{\tilde{f}}^{2}=\left(\begin{matrix}{m_{\tilde{f}_{L}}^{2}}&{m_{\tilde{f}_{L/R}}^{2}}\\{\left(m_{\tilde{f}_{L/R}}^{2}\right)^{*}}&{m_{\tilde{f}_{R}}^{2}}\\\end{matrix}\right)\quad.
E^{L}=-{\frac{11}{36}}+{\frac{1}{2}}~(e_{1}+{\tilde{p}}_{L})^{2}+{\frac{1}{2}}~{\tilde{P}}_{SO(10)}^{2}+{\frac{1}{2}}({\frac{e_{1}}{2}}+Q)^{2}+~...~,
\Psi_{\!\mathrm{(gs)}}({\bfr})=\frac{\kappa}{\sqrt{\pi}}\,\frac{\pi^{\epsilon/4}}{[\Gamma(1+\epsilon/2)]^{1/2}}\,r^{\epsilon/2}\,K_{-\epsilon/2}(\kappar)\equivA(\epsilon)\,F_{\epsilon}(\kappar)\;,
S=\frac12(^{+}\tau-^{-}\tau)\int_{M}\varepsilon^{\mu\nu\alpha\beta}\,{\calR}_{\mu\nu}^{ab}\,{\calR}_{\alpha\beta}^{cd}\epsilon_{abcd}-\fraci2(^{+}\tau+^{-}\tau)\int_{M}\varepsilon^{\mu\nu\alpha\beta}\,{\calR}_{\mu\nu}^{ab}\,{^{*}{\calR}}_{\alpha\beta}^{cd}\,\epsilon_{abcd},
C_{5}=\gamma_{1}C_{1}-\overline{{\gamma}}_{2}A+\overline{{B}}=\frac{\gamma_{1}^{2}\mathcal{G}}{\left(g\gamma_{1}^{2}+\mathcal{G}\right)\left(\gamma_{1}^{2}+\overline{{\gamma}}_{2}^{2}\right)}\overline{{B}}\tag{35}
\displaystyle{T(H\ggH_{0})=T_{1}\frac{24m^{4}}{\omega^{3}}\bigg[\frac{\omega_{1}}{\omega_{2}\sqrt{4m^{2}-\omega_{2}^{2}}}\mathrm{arctan}\left(\frac{\omega_{2}}{\sqrt{4m^{2}-\omega_{2}^{2}}}\right)+\frac{\omega_{2}}{\omega_{1}\sqrt{4m^{2}-\omega_{1}^{2}}}\mathrm{arctan}\left(\frac{\omega_{1}}{\sqrt{4m^{2}-\omega_{1}^{2}}}\right)-\frac{\omega}{4m^{2}}\bigg]}
M_{1}^{mnp}=m_{1}\delta_{1}^{[m}\delta_{2}^{n}\delta_{3}^{p]}+m_{2}\delta_{1}^{[m}\delta_{5}^{n}\delta_{6}^{p]}+m_{3}\delta_{2}^{[m}\delta_{4}^{n}\delta_{6}^{p]}-m_{4}\delta_{3}^{[m}\delta_{4}^{n}\delta_{5}^{p]}
changed(b_{0}+\tilde{b}_{0})\int_{\pi}^{\infty}ds^{\prime}\,e^{-s^{\prime}(L_{0}+\tilde{L}_{0})}|\mathcal{B}\rangle=\frac{b_{0}+\tilde{b}_{0}}{L_{0}+\tilde{L}_{0}}e^{-\pi(L_{0}+\tilde{L}_{0})}|\mathcal{B}\rangle
\widetilde{f}^{j}(-i{\boldsymbol{\nabla}})A^{j}({\boldsymbol{x}})\equiv\int\!\!{\frac{d^{3}{\boldsymbol{p}}}{(2\pi)^{3}}}{\mathrm{e}}^{i{\boldsymbol{p}}\cdot{\boldsymbol{x}}}\widetilde{f}^{j}({\boldsymbol{p}})\widetilde{A}^{j}({\boldsymbol{p}})=\int\!\!d^{3}y\,f^{j}({\boldsymbol{x}}-{\boldsymbol{y}})A^{j}({\boldsymbol{y}})
V^{AB}=\frac{1}{g^{2}+3}\left(\begin{matrix}{g^{2}-1}&{2+2g\sigma_{y}}&{2-2g\sigma_{y}}\\{2-2g\sigma_{y}}&{g^{2}-1}&{2+2g\sigma_{y}}\\{2+2g\sigma_{y}}&{2-2g\sigma_{y}}&{g^{2}-1}\\\end{matrix}\right)^{AB}
\langleT_{\mu\nu}(x)T_{\rho\sigma}(0)\rangle=\frac{\pi}{3\cdot16}\int_{0}^{\infty}d\mu^{2}\,\tilde{c}(\mu^{2})\int\frac{d^{2}p}{(2\pi)^{2}}e^{ipx}\,\frac{(g_{\mu\nu}p^{2}-p_{\mu}p_{\nu})(g_{\rho\sigma}p^{2}-p_{\rho}p_{\sigma})}{p^{2}+\mu^{2}},\nonumber
T_{\mu}=\frac{(2\imath)^{\frac{D-2}{2}}}{(D-2)!}\epsilon_{\nu\mu\rho_{2}\ldots\rho_{D-1}}\pi^{\mu}\psi^{\rho_{2}}\ldots\psi^{\rho_{D-1}}+\frac{\alpha}{2^{\frac{D-2}{2}}}\pi_{\nu}=0\;.
\tilde{D}^{\mu\nu,\,\alpha\beta}\left(k_{0}=0\right)=\frac1{\vec{k}^{2}}T_{U}^{\mu\nu,\,\alpha\beta}+\frac1{\vec{k}^{2}+\frac{\kappa^{2}\rho}3}T_{S}^{\mu\nu,\,\alpha\beta}+\frac1{\vec{k}^{2}-\kappa^{2}\rho}T_{J}^{\mu\nu,\,\alpha\beta}
S_{B}=\frac{1}{4\pil_{s}^{2}}\intd\tau\int_{0}^{2\pil_{s}^{2}p^{+}}d\sigma\left[\sum_{i=1}^{8}(\partial_{\alpha}x_{i})^{2}+\mu^{2}\sum_{I=3}^{8}x_{I}^{2}+4\mux_{2}\partial_{\sigma}x_{1}\right]
\frac{-2iM_{V}^{2}\gamma}{v}(1-\frac{\epsilon}{2})\eta_{\mu\nu}+\frac{i\gamma\epsilon}{v}(p_{2}\cdotp_{3}\eta_{\mu\nu}-p_{2\nu}p_{3\mu})-\frac{i(2-\epsilon)\gamma}{\alphav}(p_{2\mu}p_{1\nu}+p_{3\nu}p_{1\mu})-\frac{i\gamma\epsilon}{\alphav}p_{2\mu}p_{3\nu}.
{\bar{D}}_{{\dot{\alpha}}{\mathbfi}}\sigma=0\,\qquad{\bar{D}}_{{\dot{\gamma}}{\mathbfi}}\Omega_{\alpha}{}^{\beta}=0\,\qquad(\Lambda^{\mathbfi}{}_{\mathbfj})^{*}=\Lambda^{\mathbfj}{}_{\mathbfi}\,\qquad\Lambda^{\mathbfi}{}_{\mathbfi}=0\.
Q^{\prime}\left(\begin{array}{c}{\phi}\\{a}\\{\psi^{\prime}}\\{\bar{\phi}}\\{\bar{\psi}}\\\end{array}\right)=\left(\begin{array}{c}{0}\\{\psi^{\prime}}\\{-L(\phi)a}\\{\bar{\psi}}\\{-L(\phi)\bar{\phi}}\\\end{array}\right)_{(\phi,a,\psi^{\prime},\bar{\phi},\bar{\psi})}.
A_{i}^{eff}(x)=-{\frac{e}{\muc}}\epsilon_{ij}\partial_{j}\intd\vec{y}G_{p}(\vec{x},\vec{y})J_{0}(y)-{\frac{e}{\muc}}\epsilon_{ij}\partial_{j}\intd\vec{y}G_{np}(\vec{x},\vec{y})J_{0}(y)
\phi_{\mathrm{f}\to\mathrm{inter}}={\frac{1}{u}}\left[\alpha\left({\frac{u}{2}}\right)^{\nu}{\frac{1}{\Gamma(\nu+1)}}+\beta\left({\frac{u}{2}}\right)^{-\nu}{\frac{1}{\Gamma(-\nu+1)}}\right].
\lbrackh_{i},e_{j}^{(N^{\prime})}]=N^{\prime}\,A_{ij}e_{j}^{(N^{\prime})}\quad\mathrm{and\quad~}[h_{i},f_{j}^{(N^{\prime})}]=-N^{\prime}\,A_{ij}f_{j}^{(N^{\prime})}\;.
f\star_{\lambda}g=f\g+\sum_{m=1}^{\infty}C_{m}(\lambda)\J^{a_{1}b_{1}}J^{a_{2}b_{2}}\cdotsJ^{a_{m}b_{m}}\\partial_{a_{1}}\cdots\partial_{a_{m}}f\,partial_{b_{1}}\cdots\partial_{b_{m}}g.
\frac{1}{4}Tr(D^{\mu}\PhiD_{\mu}\Phi)=\frac{1}{2}(\partial_{\mu}\phi_{1})^{2}+\frac{1}{2}\left|D_{\mu}^{3}\phi_{2}\right|^{2}+\frac{9}{2}e^{2}f^{2}\left[\left(A_{\mu}^{1}\right)^{2}+\left(A_{\mu}^{2}\right)^{2}\right]+~\cdots\quad,
ds_{(\mu<0)}^{2}=l^{2}\left[-d\tau^{2}+\frac{1+m}{4}\frac{dz^{2}}{z^{2}}+\frac{1+m}{4}\frac{d\bar{z}^{2}}{\bar{z}^{2}}+\left(e^{-2\tau}+\frac{(1+m)^{2}}{16}\frac{1}{z^{2}\bar{z}^{2}}e^{2\tau}\right)dzd\bar{z}\right],
\frac{1}{\sqrt{-g}}\partial_{\mu}[\sqrt{-g}\chi(\tilde{F}_{\alpha\beta}\tilde{F}^{\alpha\beta})g^{\mu\gamma}g^{\nu\delta}\tilde{F}_{\gamma\delta}]-\frac{\tilde{e}^{2}m^{4}}{8}\varphi^{2}\chig^{\alpha\nu}\tilde{A}_{\alpha}=0
S={\frac{kA_{H}}{4\ell_{P}^{2}}}+4\pi{\frac{k}{\hbar}}\int_{H}\;{\frac{\partial{\calL}}{\partialR_{\mu\nu\lambda\rho}}}\;g_{\mu\lambda}^{\perp}g_{\nu\rho}^{\perp}\;\sqrt{{}_{2}g}\;d^{2}x.
K^{\prime}+\bar{K}^{\prime}=\frac{1}{2}\int_{0}^{T}d\tau\Biggl[\frac{m_{1}^{2}}{\mu_{1}(\tau)}+\mu_{1}(\tau)\left(1+\dot{\bfz}^{2}(\tau)\right)+\frac{m_{2}^{2}}{\mu_{2}(\tau)}+\mu_{2}(\tau)\left(1+\dot{\bar{\bfz}}^{2}(\tau)\right)\Biggr],
S=\frac{1}{2\kappa_{5}^{2}}\intdz\,d^{4}x\,\sqrt{g}\left(\frac{1}{2e^{2}}g^{ij}g^{kl}F_{ik}^{a}F_{jl}^{a}+\frac{1}{2}\left({\calM}^{2}\right)_{ab}\,g^{ij}A_{i}{}^{a}A_{j}{}^{b}\right)
\left[\begin{array}{rrrrrrrr}{2}&{-1}&{0}&{0}&{0}&{0}&{0}&{0}\\{-1}&{2}&{-1}&{0}&{0}&{0}&{0}&{0}\\{0}&{-1}&{2}&{-1}&{0}&{0}&{0}&{0}\\{0}&{0}&{-1}&{2}&{-1}&{0}&{0}&{0}\\{0}&{0}&{0}&{-1}&{2}&{-1}&{0}&{0}\\{0}&{0}&{0}&{0}&{-1}&{2}&{-1}&{0}\\{0}&{0}&{0}&{0}&{0}&{-1}&{2}&{-1}\\{0}&{0}&{0}&{0}&{0}&{0}&{-1}&{0}\\\end{array}\right].
{\frac{\dot{G}_{4}}{G_{4}}}=-4{\frac{\dot{c}}{c}}+2{\frac{\dot{G}_{5}}{G_{5}}}+{\frac{16\piG_{5}\sigma_{0}\Omega_{0}^{2}}{3c^{4}(1-\Omega_{0}^{2})}}{\frac{\dot{c}}{c}}-{\frac{4\piG_{5}\sigma_{0}\Omega_{0}^{2}}{3c^{4}(1-\Omega_{0}^{2})}}{\frac{\dot{G}_{5}}{G_{5}}}.
{R^{\alpha}}_{\beta}={\frac{1}{4}}\left(2{f^{\alpha}}_{\betai}{f^{i}}_{\gamma\delta}+{f^{\alpha}}_{\beta\epsilon}{f^{\epsilon}}_{\gamma\delta}-{f^{\alpha}}_{\gamma\epsilon}{f^{\epsilon}}_{\beta\delta}\right)e^{\gamma}\wedgee^{\delta}.
\triangle^{(2)}h_{\mu\nu}=\left(-\nabla^{\alpha}\nabla_{\alpha}\delta_{\mu}^{\rho}\delta_{\nu}^{\sigma}+R_{\mu}^{\rho}\delta_{\nu}^{\sigma}+R_{\nu}^{\sigma}\delta_{\mu}^{\rho}-2R_{\mu}~^{\rho}~_{\nu}~^{\sigma}\right)h_{\rho\sigma}~~~.
I_{Stmod}=-\frac{1}{4}F_{mn[A]}F_{[A]}^{mn}-\frac{1}{2}\mu^{2}[A_{m}+\frac{1}{2\mu}l_{m}][A^{m}+\frac{1}{2\mu}l_{m}]+\frac{1}{4}\epsilon^{mnpq}B_{mn}\partial_{p}l_{q}.
\phi_{i}(\psi)=\frac{1}{2}\frac{1}{|\psi|^{2}}\bar{z}\sigma_{i}z\,,\quad\phi_{4}(\psi)=-\frac{1}{2}\frac{1}{|\psi|^{2}}(\bar{z}_{1}\theta+z_{2}\bar{\theta})\,,\quad\phi_{5}(\psi)=\frac{1}{2}\frac{1}{|\psi|^{2}}(-\bar{z}_{2}\theta+z_{1}\bar{\theta})\,.
F_{B}=\bigg[2\epsilon_{ij}^{1}\epsilon_{il}^{2}p^{j}k^{l}-\epsilon_{ij}^{1}\epsilon_{ij}^{2}\vec{k}\cdot\vec{p}\bigg]{\frac{\Gamma(1-2k_{0}^{2})\Gamma(\vec{k}\cdot\vec{p}+k_{0}^{2})}{\Gamma(\vec{k}\cdot\vec{p}-k_{0}^{2}+1)}}\,
S_{eff}=\int{d^{4}x\left({\Pi^{i}\stackrel{\bf{\cdot}}{A_{i}}+{\calP}\stackrel{\bf{\cdot}}{C}+{\overline{{\calP}}}\stackrel{\bf{\cdot}}{\overline{C}}+\Pi^{0}\stackrel{\bf{\cdot}}{A_{0}}-H_{0}-\left[{\Omega,\Psi}\right]}\right)},
T_{\alpha\beta}={\frac{1}{4}}\left(\nabla_{\alpha}\tilde{\psi}\gamma_{\beta}\psi+\nabla_{\beta}\tilde{\psi}\gamma_{\alpha}\psi-\tilde{\psi}\gamma_{\alpha}\nabla_{\beta}\psi-\tilde{\psi}\gamma_{\beta}\nabla_{\alpha}\psi\right)~.
W_{{\bfA,B}}({\bfq,p})=2^{N}exp[-2{\bfZZ^{*}}+2{\bfAZ^{*}}+2{\bfBZ^{*}}-{\bfAB^{*}}-\frac{\mid{\bfA}\mid^{2}}{2}-\frac{\mid{\bfB}\mid^{2}}{2}],\nonumber\,
{\calL}(1,\alpha-\frac{1}{2})=\frac{1}{2}B_{\mu}^{*}[-i\epsilon^{\mu\nu\lambda}(\partial_{\nu}+ia_{\nu}+iC_{\nu})+M\delta^{\mu\lambda}]B_{\lambda}-\frac{i}{8\pi(\alpha-1/2)}\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}\;.
{\calV}_{\mathrm{SS}}=-{\textstyle\frac{1}{2}}\,g^{2}{\displaystyle\frac{1}{\Im\mathrm{m}(\tau)}}\left\{(\mathrm{Tr}\,\mathcal{M})^{2}-2\mathrm{Tr}\,(\mathcal{M}^{2})\right\}\,.
\left.\frac{\delta\Gamma}{\deltaB}\right|_{\chi=0=\chi_{A}}=\xiB+\partialA-e\left[(\hat{\varphi}_{1}-\xi_{A}\frac{m}{e})\varphi_{2}-\hat{\varphi}_{2}(\varphi_{1}-\hat{\xi}_{A}\frac{m}{e})\right]
\frac{\kappa}{4\pi\sqrt{-1}}\int_{\widetilde{V}_{M}}\widehat{h}^{\ast}\chi_{SU(2)}-\frac{\kappa}{4\pi\sqrt{-1}}\sum_{j=1}^{N_{0}}\int_{S_{j}^{2}}\left.\widehat{h}\right|_{S_{j}^{2}}^{\ast}\omega_{j}
D^{2}\phi-\frac{{\kappa_{5}}^{2}}{12}(4\lambda-\tau)\Bigl(\frac{d\lambda}{d\phi}+\frac{\partial{\calL}_{m}}{\partial\phi}\Bigr)-\frac{dV}{d\phi}=-\partial_{y}^{2}\phi|_{\mathrm{brane}}.
\rho_{L}(q)=\frac{1}{q^{L-1}}\left[\1+\sum_{1\leqi<j\leqL}(q-1)^{j-i}B_{ij}\+\\sum_{1\leqi<j<k<l\leqL}(q-1)^{j-i+l-k}\B_{i,j,k,l}^{(2)}+\cdots\right]\,
\delta\mu^{\prime\rho}=\partial_{\mu}\left(\varepsilon^{\mu\alpha\beta\gamma}\left(B_{\gamma}^{\;(\sigma)}\partial_{\sigma}A_{\alpha\beta}^{*\;\;(\rho)}-A_{\alpha\beta}^{*\;\;(\sigma)}\partial_{\sigma}B_{\gamma}^{\;(\rho)}\right)\right),
\begin{array}{rcl}{{\calL}_{\mathrm{class}}}&{=}&{\displaystyle-\frac{1}{4}G_{1\mu\nu}G_{1}^{~\mu\nu}-\frac{1}{4}G_{2\mu\nu}G_{2}^{~\mu\nu}}\\{}&{}&{\displaystyle+(D_{\mu}\Phi)^{\dagger}D^{\mu}\Phi+\mu^{2}\Phi^{\dagger}\Phi-\lambda\Phi^{\dagger}\Phi\Phi^{\dagger}\Phi,}\\\end{array}
\begin{array}{lll}{\vec{r}_{1}=(\frac{1}{2},\frac{\sqrt{3}}{2},0)}&{\vec{r}_{2}=(-\frac{1}{2},\frac{\sqrt{3}}{2},0)}&{\vec{r}_{3}=(0,-\frac{1}{\sqrt{3}},\sqrt{\frac{2}{3}})}\\{\vec{r}_{4}=(1,0,0)}&{\vec{r}_{5}=(\frac{1}{2},\frac{1}{2\sqrt{3}},\sqrt{\frac{2}{3}})}&{\vec{r}_{6}=(-\frac{1}{2},\frac{1}{2\sqrt{3}},\sqrt{\frac{2}{3}})}\\{\vec{r}_{i+6}=-\vec{r}_{i}}\\\end{array}
{\calS}({\Gamma})\equiv\int\left\{\partial_{\mu}c\frac{\delta{\Gamma}}{\deltaA_{\mu}}+B\frac{\delta{\Gamma}}{\delta\bar{c}}+\frac{\delta{\Gamma}}{\delta\underline{{Y}}}\frac{\delta{\Gamma}}{\delta\underline{{\varphi}}}+\underline{{q}}\frac{\delta{\Gamma}}{\delta\hat{\underline{{\varphi}}}}\right\}=0
\zeta_{0}^{(p)}(\nu)={\frac{2}{(4\pi)^{D/2}\Gamma(\nu)}}\sum_{n=0}^{\infty}{\frac{1}{2\nu+2n-D}}\sum_{m=n}^{2n}{\frac{\Gamma(m-n+1/2)}{\sqrt{\pi}}}\breve{a}_{2(m-n),m}.
U^{r}(\tau,\sigma)=\frac{M}{\sqrt{Mm^{2}+E^{2}}\;\tau^{3}}\sum_{n}\sum_{p}(A_{p\perp}+\tilde{A}_{p\perp})(A_{n-p\perp}+\tilde{A}_{n-p\perp})e^{-in\sigma}+{\calO}(\frac{1}{\tau^{2}}),
e^{-i\psi_{p+1}}=\frac{a_{p}+e^{-i\psi_{p}}}{\bar{a}_{p}+e^{i\psi_{p}}}=e^{-i2\alpha_{p}}\frac{\mida_{p}\mid+e^{-i\chi_{p}}}{\mida_{p}\mid+e^{i\chi_{p}}}
\left[\frac{\partial^{2}}{\partialr^{2}}+(\frac{1}{r}-4i\omega\Omega\thetar)\frac{\partial}{\partialr}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}+(q^{2}-4i\omega\Omega\theta)\right]f(r,\theta)=0.
\sum_{k,l}\tilde{\nu}_{k}K_{kl}\left[\sqrt{2l+1}\;Y_{2l+1,2m}\;+\;Y_{2m,2l+1}\;\frac{2m}{\sqrt{2l+1}}\right]=-\sum_{k}\tilde{\nu}_{k}\;Y_{2m,2k+1}\;\frac{2m}{\sqrt{2k+1}}\,.
\deltae_{\breve{\alpha}}^{--\mu}=-e_{\breve{\alpha}}^{\nu}\partial_{h\nu}^{-}\lambda^{\mu-}+e_{\breve{\alpha}}^{--\nu}\partial_{h\nu}^{+}\lambda^{\mu-}+\lambda_{\breve{\alpha}}^{\breve{\beta}}e_{\breve{\beta}}^{--\mu},
R_{1/2,1/2}^{\mathrm{h}}(u)=\left(\begin{array}{cccc}{1}&{{}}&{{}}&{{}}\\{{}}&{\frac{q^{-1}(1-u)}{1-q^{-2}u}}&{\frac{1-q^{-2}}{1-q^{-2}u}}&{{}}\\{{}}&{\frac{u(1-q^{-2})}{1-q^{-2}u}}&{\frac{q^{-1}(1-u)}{1-q^{-2}u}}&{{}}\\{{}}&{{}}&{{}}&{1}\\\end{array}\right)\,.
V(m^{2})=\frac{Nm^{2}}{4\pi}(-\frac{g}{4\pi}\mathrm{ln}^{2}(\frac{\Lambda^{2}}{m^{2}}+1)+\mathrm{ln}(\frac{\Lambda^{2}}{m^{2}}+1)-\frac{\Lambda^{2}}{m^{2}}\mathrm{ln}(\frac{m^{2}}{\Lambda^{2}}+1)).
{\calL}={\calL}_{D}+{\calL}_{F}=\int\,d^{4}\theta\,K(\Phi_{i},\bar{\Phi}_{i})\,+\,\int\,d^{2}\theta\,W(\Phi_{i})\,\,+\int\,d^{2}\bar{\theta}\,\bar{W}(\bar{\Phi}_{i})
u_{0,l}(r)=\left(\begin{array}{l}{\phi_{0,l,1}(r)}\\{0}\\\end{array}\right),\mathrm{\;}{}l\geq1;\;\;\;u_{0}^{II}(r)=\left(\begin{array}{l}{\phi_{0,1}^{ir}(r)}\\{0}\\\end{array}\right),\;{}l=0.
S^{BH}(G_{\tiny\mathrm{ren}},c_{\tiny\mathrm{ren}}^{i})={\frac{1}{4G_{\tiny\mathrm{ren}}}}{\calA}-\int_{\Sigma}\sqrt{\sigma}d^{2}\theta~(8\pic_{\tiny\mathrm{ren}}^{1}R+4\pic_{\tiny\mathrm{ren}}^{2}{\calQ}+8\pic_{\tiny\mathrm{ren}}^{3}{\calR})~~~.
\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{dR}{dr}\right)-\frac{1}{r^{2}}\left(l+\frac{\delta_{1}+\delta_{2}}{2}\right)\left(l+\frac{\delta_{1}+\delta_{2}}{2}+1\right)R+2\left(E-\frac{\alpha}{r}\right)R=0,
P_{\\nu}^{\mu}=-{\frac{1}{(d-2)^{2}}}\left[R_{\\lambda}^{\mu}R_{\\nu}^{\lambda}-{\frac{d}{2(d-1)}}RR_{\\nu}^{\mu}-{\frac{1}{2}}\delta_{\nu}^{\mu}R_{\\beta}^{\alpha}R_{\\alpha}^{\beta}+{\frac{d+2}{8(d-1)}}\delta_{\\nu}^{\mu}R^{2}\right]\.
S=\int\!{\frac{dk_{+}dk_{-}}{(2\pi)^{2}}}\,\phi_{0}(-k_{+},-k_{-})\left[{\calF}(k_{+},k_{-},z_{B})-{\calF}(k_{+},k_{-},z_{H})\right]\phi_{0}(k_{+},k_{-})\,,
\Psi_{1}=\sum_{N=0}^{\infty}\sum_{l=0}^{N}\sum_{n=-l}^{l}\left\{_1a_{Nln}\phi_{Nln}^{(+)}(t,r,\theta,\phi)+_{1}\!a_{Nln}^{\dagger}\phi_{Nln}^{\ast(+)}(t,r,\theta,\phi)\right\}
{\frac{\Lambda^{2}}{P^{+}}}>\mid{\frac{P_{\perp}^{2}}{P^{+}}}-{\frac{k_{1\perp}^{2}}{k_{1}^{+}}}-{\frac{k_{2\perp}^{2}}{k_{2}^{+}}}-{\frac{k_{3\perp}^{2}}{k_{3}^{+}}}\mid\;.
G(x^{\prime},t^{\prime};x^{\prime},t^{\prime})=\frac{1}{2}\int\frac{dk_{1}}{2\pi}\frac{1}{\sqrt{k_{1}^{2}+4}}\left(\frac{ik_{1}+2\sigma_{1}}{ik_{1}-2\sigma_{1}}\right)e^{-2ik_{1}x^{\prime}}.
P_{t}^{\beta}(x_{1},y_{1})=\int_{x_{1}-a}^{x_{1}+a}\!\!\!dx_{1}dx_{1}^{\prime}\!\!\!\int_{i=2,..,N}\!\!dx_{i}dx_{i}^{\prime}dy_{i}\:\langlex_{1},x_{i}|\rho|x_{1}^{\prime},x_{i}^{\prime}\rangle\langlex_{1}^{\prime},x_{i}^{\prime}|U_{t}^{\dagger}|y_{1},y_{i}\rangle\langley_{1},y_{i}|U_{t}|x_{1},x_{i}\rangle
g_{\mathrm{YM}}^{2}=G_{\mathrm{D4}}^{4/5}\ell_{\mathrm{D4}}^{3}\,\quad\Theta^{56}=\tilde{\theta}^{56}=G_{\mathrm{D4}}^{-4/5}\ell_{\mathrm{D4}}^{2}\.
G^{-1}dG=L^{A}T_{A}=L_{a}P^{a}+L_{a^{\prime}}P^{a^{\prime}}+\frac{1}{2}L^{ab}J_{ab}+\frac{1}{2}L^{a^{\prime}b^{\prime}}J_{a^{\prime}b^{\prime}}+L^{\alpha\alpha^{\prime}I}Q_{\alpha\alpha^{\prime}I}
{ds_{\mathrm{6D}}^{p}}^{2}=\left(1+{\frac{R^{2}}{r^{2}}}\right)^{-1}\left(-dudv+{\frac{p(u)}{r^{2}}}du^{2}\right)+\left(1+{\frac{R^{2}}{r^{2}}}\right)\left(dr^{2}+r^{2}d\Omega_{3}^{2}\right).
\alpha_{1}^{\lambda}=\alpha_{1}^{\lambda}\left(\left[V^{*\alpha(\lambda)}\right],\left[A_{\;\;\;(\lambda)}^{*\alpha}\right]\right),\;\alpha_{1}^{\prime\lambda}=\alpha_{1}^{\prime\lambda}\left(\left[V^{*\alpha(\lambda)}\right],\left[A_{\;\;\;(\lambda)}^{*\alpha}\right]\right).
\intdzd\overline{{z}}\left(\frac{\delta\mathcal{S}}{\deltab}\frac{\delta\mathcal{S}}{\deltab^{*}}\;+\frac{\delta\mathcal{S}}{\deltac}\frac{\delta\mathcal{S}}{\deltac^{*}}\;\right)=\;\frac12\mathcal{B}_{\mathcal{S}}
\begin{array}{lcl}{\tilde{T}_{0}^{bc}}&{=}&{-T_{0}^{bc}}\\{\tilde{T}_{1}^{bc}}&{=}&{-T_{1}^{bc}+\lambda\sum_{f}(-1)^{p(f)}T_{0}^{bf}T_{0}^{fc}}\\{\tilde{T}_{2}^{bc}}&{=}&{-T_{2}^{bc}+\lambda\sum_{f}(-1)^{p(f)}(T_{0}^{bf}T_{1}^{fc}+T_{1}^{bf}T_{0}^{fc})}\\{}&{}&{-\lambda^{2}\sum_{f,g}(-1)^{p(f)+p(g)}T_{0}^{bf}T_{0}^{fg}T_{0}^{gc}}\\\end{array}\;.
\partial_{\sigma}\left(\frac{\chi^{2}e^{2\gamma}}{\sqrt{-G}V}z^{\prime}\right)={\frac{1}{2\sqrt{-G}}}\frac{\partial}{\partialz}\left(\frac{\chi^{2}e^{2\gamma}}{V}\right)\left({\rho^{\prime}}^{2}+z^{\prime\;2}\right),
\hat{G}_{\mu\nu}^{(n)}=-\frac{1}{2(n+2)}\epsilon^{\mu\nu\lambda\rho\sigma}\hat{G}_{\lambda\rho}^{(n)}\hat{G}_{\sigma}^{(n)}=-\frac{1}{2(n+2)}\epsilon^{\mu\nu\lambda\rho\sigma}\hat{G}_{\lambda}^{(n)}\hat{G}_{\rho}^{(n)}\hat{G}_{\sigma}^{(n)}.
L_{0}^{m}+L_{0}^{gh}=\frac{1}{2}p^{i}p_{i}+\sum_{n=1}^{\infty}n(N_{bn}+N_{cn}+\sum_{i=d+1}^{25}N_{in})+\sum_{r=1/2}^{\infty}\sum_{a=0}^{d}rN_{ar}\,,
{\frac{1}{3!}}[f_{i}^{1}f_{j}^{2}f_{k}^{3}+q_{ij}^{-1}f_{j}^{1}f_{i}^{2}f_{k}^{3}+q_{jk}^{-1}f_{i}^{1}f_{k}^{2}f_{j}^{3}+q_{ik}^{-1}q_{jk}^{-1}f_{k}^{1}f_{i}^{2}f_{j}^{3}+q_{ij}^{-1}q_{ik}^{-1}f_{j}^{1}f_{k}^{2}f_{i}^{3}+q_{ij}^{-1}q_{ik}^{-1}q_{jk}^{-1}f_{k}^{1}f_{j}^{2}f_{i}^{3}].