harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f_{\mu\nu}(x):=\frac{4}{R_{P}^{4}g^{2}}\frac{\left<\mathrm{Tr}\mathbf{W}[\partialS]\mathrm{Tr}\mathbf{P}_{\mu\nu}(x)\right>-\left<\mathrm{Tr}\mathbf{W}[\partialS]\right>\left<\mathrm{Tr}\mathbf{P}_{\mu\nu}(x)\right>}{\left<\mathrm{Tr}\mathbf{W}[\partialS]\right>}
\begin{array}{ll}{\phi_{\omega}^{E}(\sigma,\tau)=A_{\omega}exp\:(-i\omega(\tau+i\sigma))+B_{\omega}exp\:(-i\omega(\tau-i\sigma))}\\{\phi_{\omega}^{L}(\sigma,\tau)=C_{\omega}exp\:(-i\omega(\tau+\sigma))+D_{\omega}exp\:(-i\omega(\tau-\sigma))}\\\end{array}
\vartheta_{\sigma}^{(n)}(w|\zeta)=\Theta_{x^{2}}\left(-\sigma\prod_{a\inA}w_{a}^{-1}\prod_{j=1}^{2n}\zeta_{j}\right)\prod_{j=1}^{2n}z_{j}^{-n}\prod_{a,b\inA\atopa<b}w_{a}^{-1}\Theta_{x^{2}}(w_{a}/w_{b}).
I{^{\pm}}=\int_{X}\varepsilon^{ijk}\big(2{^{\pm}}A_{i}^{a}\partial_{j}{^{\pm}}A_{k}^{a}+{\frac{2}{3}}\varepsilon_{abc}{^{\pm}}A_{i}^{a}{^{\pm}}A_{j}^{b}{^{\pm}}A_{k}^{c}\big).
Z_{r,p}^{n}(q)=\sum_{\substack{N_{1}\geq\dots\geqN_{p-2}\geq0\,N_{1}+\dots+N_{p-2}=n}}\frac{q^{\sum_{m=r}^{p-2}N_{m}}q^{\sum_{m=1}^{p-2}N_{m}^{2}}}{(q)_{N_{1}-N_{2}}(q)_{N_{2}-N_{3}}\dots(q)_{N_{p-3}-N_{p-2}}(q)_{N_{p-2}}}.
\left.\begin{array}{lc}{\mathrm{tachyon}}&{|\Omega\rangle_{NS}\congce^{-\phi},}\\{\mathrm{massless~NS}}&{\psi_{-1/2}^{\mu}|\Omega\rangle_{NS}\cong\psi^{\mu}ce^{-\phi},}\\{\mathrm{massless~R}}&{|\vec{s}\rangle_{R}\congce^{-\phi/2}\Theta_{\vec{s}},}\\\end{array}\right.
\begin{array}{ll}{\{\theta^{0},a\}=\etac\theta^{1},}&{\{\theta^{1},a\}=-(1+\eta)a\theta^{1},}\\{\{\theta^{0},b\}=\etad\theta^{1},}&{\{\theta^{1},b\}=-(1+\eta)b\theta^{1},}\\{\{\theta^{0},c\}=-(2+\eta)a\theta^{2},}&{\{\theta^{1},c\}=(4+2\eta)a\theta^{0}+(1+\eta)c\theta^{1},}\\{\{\theta^{0},d\}=-(2+\eta)b\theta^{2},}&{\{\theta^{1},d\}=(4+2\eta)b\theta^{0}+(1+\eta)d\theta^{1},}\\\end{array}
\sum_{k}\dot{\bfz}_{k}\,\dot{\bfz}_{k}\cdot\nabla\delta^{D}(x-z_{k})=\frac{\bfj}{\rho}\,(\nabla\cdot{\bfj})=\frac{\bfj}{\rho^{2}}\,({\bfj}\cdot\nabla\rho)=\frac{1}{\rho}({\bfj}\cdot\nabla)\,{\bfj}.
\tilde{\delta}_{\mu\nu}=\delta_{\mu\nu}-u_{\mu}u_{\nu},\qquad\tilde{p}_{\mu}=p_{\mu}-(u\cdotp)u_{\mu},\qquad\overline{{u}}_{\mu}=u_{\mu}-\frac{u\cdotp}{p^{2}}p_{\mu}
e^{-1}{\calL}=-{\frac{1}{2}}R-{\frac{1}{4}}e^{{\frac{2}{3}}\phi}F_{\mu\nu}F^{\mu\nu}-{\frac{1}{4}}e^{-{\frac{4}{3}}\phi}G_{\mu\nu}G^{\mu\nu}+{\frac{1}{6}}(\partial_{\mu}\phi)^{2}-{\frac{e^{-1}}{4\sqrt2}}\epsilon^{\mu\nu\rho\sigma\lambda}F_{\mu\nu}F_{\rho\sigma}B_{\lambda}\.
S^{\upsilon}=\int\left[p_{i}\dot{A}^{i}-\mathcal{H}^{\upsilon}\right]dx\,,\;\mathcal{H}^{\upsilon}=p_{i}\upsilon^{i}-\frac{1}{2}\left(\upsilon^{i}+\partial_{i}\upsilon^{0}\right)^{2}+\frac{1}{4}F_{ik}F^{ik}\,.
p_{1}={\frac{\beta^{2}\left[\left(1-2\beta\right)^{2}\left(-1+4\beta-3\beta^{2}\right)+\left(1-8\beta+17\beta^{2}-12\beta^{3}\right)c+c^{2}-c^{3}\right]}{4\left(1-2\beta\right)^{2}g^{3}}}
+\frac{i\Theta}{3}s_{a}^{(c)}\varepsilon_{\mu\nu\lambda\rho}\intd^{4}xd^{4}y{\calD}_{m}^{(4)}(x-y)j_{\mu}(x)\frac{\partial}{\partialy_{\nu}}\hat{\Sigma}_{\lambda\rho}^{a}(y)-\frac{2i\Theta}{3}s_{a}^{(c)}\hat{L}(\Sigma^{a},C)\Biggr],
W^{(1)}=\frac{B^{\prime}}{4}x_{0}^{4}+\frac{b_{3}}{4}x_{3}^{4}+\frac{b_{4}}{4}x_{4}^{4}+\frac{b_{5}}{4}x_{5}^{4}-\psi_{0}x_{0}x_{3}x_{4}x_{5}\.
\mu=3\left(l^{-2}+\textstyle{\frac{1}{36}}\kappa^{4}\omega\varrho^{2}\right)y_{0}^{4},\\\\\\\\\q^{2}=2\left(l^{-2}+\textstyle{\frac{1}{72}}\kappa^{4}(1+3\omega)\varrho^{2}\right)y_{0}^{6}.
\langley_{\pm}(x)y_{\pm}(x^{\prime})\rangle={\frac{m}{B^{2}}}\partial_{x}\partial_{x^{\prime}}\langle\varphi_{\pm}(x)\varphi_{\pm}(x^{\prime})\rangle={\frac{m}{B^{2}}}{\frac{1}{(x-x^{\prime})^{2}}}.
S_{br}=\frac{1}{2k^{2}}\intd^{9}x\sqrt{-g^{0}}V_{0}(\Phi)\Bigg|_{y=0}+\frac{1}{2k^{2}}\intd^{9}x\sqrt{-g^{\infty}}V_{\infty}(\Phi)\Bigg|_{y=\infty}\,
<\phi_{m}^{*}(t_{1})\phi_{m}(t_{2})>=\frac{1}{Z}\left[{\calG}_{0}^{(1)}(t_{1},t_{2})+{\calG}_{1}^{(1)}(t_{1},t_{2})+{\calG}_{2}^{(1)}(t_{1},t_{2})\right]\,
\bar{\partial}_{\bar{A}}F_{g}=\bar{C}_{\bar{A}\bar{B}\bar{C}}e^{2K}G^{B\bar{B}}G^{C\bar{C}}\left(D_{B}D_{C}F_{g-1}+\frac{1}{2}\sum_{r}D_{B}F_{r}\cdotD_{C}F_{g-r}\right),
S_{5}=-\intd^{5}x\sqrt{-g}\,\left[i\bar{\Psi}_{M}\gamma^{MNP}D_{N}\Psi_{P}+i\frac{3}{2}\frac{a^{\prime}}{a}\bar{\Psi}_{M}\gamma^{MN}\Psi_{N}+\left(i\bar{J}^{M}\Psi_{M}+h.c.\right)\right]\,.
{\bfE}=-\left({\frac{\partial\phi}{\partialx}}\,{\hat{\bfx}}+{\frac{\partial\phi}{\partialy}}\,{\hat{\bfy}}\right)\qquad;\qquad{\bfB}=\left({\frac{\partial\phi}{\partialy}}\,{\hat{\bfx}}-{\frac{\partial\phi}{\partialx}}\,{\hat{\bfy}}\right),
\begin{array}{lcl}{{\widehatRes}\{{\mathcalL}_{2}\star\tilde{\mathcalL}_{-2}\}}&{=}&{(u_{0}v_{1}+u_{1}v_{0}+u_{2}v_{-1})}\\{}&{+}&{2\theta(u_{1}v_{-1}^{\prime}+2u_{0}v_{0}^{\prime})}\\{}&{+}&{4u_{0}v_{-1}^{\prime\prime}{\theta}^{2},}\\\end{array}
\begin{array}{rl}{\Omega(-1)^{F}:}&{\left[{\frac{f_{3}}{f_{1}}}(q)\right]^{8}\prod_{j=1}^{3}M_{j}}\\{\OmegaR_{i}(-1)^{F}:}&{\left[{\frac{f_{3}}{f_{1}}}(q)\right]^{8}M_{i}\prod_{j\neqi}W_{j}\nonumber}\\\end{array}
g_{00}(\kappa{\bfB}_{f})\approx1+2\kappaB_{k}^{a}(x_{f})\theta_{a}^{k},\quad\theta_{a}^{k}=\left(\frac{\partialD_{0}^{0}(\kappa{\bfB}_{f})}{\partial\kappaB_{k}^{a}}\right)_{0}=\mathrm{const},
\deltaR_{MN}(h)=-{\frac{1}{2}}(\bar{\nabla}_{M}\bar{\nabla}_{N}h_{~~P}^{P}+\bar{\nabla}^{P}\bar{\nabla}_{P}h_{MN}-\bar{\nabla}^{P}\bar{\nabla}_{N}h_{MP}-\bar{\nabla}^{P}\bar{\nabla}_{M}h_{NP}),
V(r)\sim\sum_{\bfk}\left[\sum_{i=1}^{n_{b}}a_{i}({\bfk})\sqrt{r^{2}+b_{i}({\bfk})}-\sum_{i=1}^{n_{f}}c_{i}({\bfk})\sqrt{r^{2}+d_{i}({\bfk})}\right],
\mu_{0}+\mu_{\rho}+2\mu_{\theta}=\alpha\int_{0}^{\infty}dr\left(1-J(r)^{2}\right)M(r)^{4}\mathcal{L}(r)^{2}+M(r)^{4}\mathcal{L}(r)^{2}J(r)J^{\prime}(r)\vert_{0}^{\infty}\,,
(ii)\quadN^{R}=\left(\begin{matrix}{1}&{0}&{0}&{0}\\{0}&{a-1}&{a}&{0}\\{0}&{a}&{a-1}&{0}\\{0}&{0}&{0}&{1}\\\end{matrix}\right),\quadN^{NS}=\left(\begin{matrix}{a}&{0}&{0}&{a-1}\\{0}&{0}&{1}&{0}\\{0}&{1}&{0}&{0}\\{a-1}&{0}&{0}&{a}\\\end{matrix}\right).
\xi_{k}(z)=c_{1}\left(\begin{array}{c}{\xi_{k}^{1}(z)^{u}}\\{\xi_{k}^{1}(z)^{d}}\\\end{array}\right)+c_{2}\left(\begin{array}{c}{\xi_{k}^{2}(z)^{u}}\\{\xi_{k}^{2}(z)^{d}}\\\end{array}\right),
\{S^{p},T^{r}\}=\left(\sum_{k_{r}=0}^{p}...\sum_{k_{2}=0}^{k_{3}}\sum_{k_{1}=0}^{k_{2}}\alpha^{k_{1}}\alpha^{k_{2}}...\alpha^{k_{r}}\right)S^{p}T^{r}.
{\bfS}=\int\mathrm{d}^{3}{\bfx}\Big([{\bfA}\times\partial_{0}{\bfA}]+[{\bfB}\times\partial_{0}{\bfB}]+\big[{\bfA}\times[\vec{\partial}\times{\bfB}]\big]-\big[{\bfB}\times[\vec{\partial}\times{\bfA}]\big]\Big)
\begin{aligned}{s^{-}=\omega^{-}\quad,}&{{}\quadx^{+}=z^{+}}\\{s=\phi\,\omega^{-}\quad,}&{{}\quadx=z+\phi\,z^{+}}\\{s^{+}=\omega^{+}+\frac{1}{2}\omega^{-}\,\phi^{2}\quad,}&{{}\quadx^{-}=z^{-}+\phi\,z+\frac{1}{2}z^{+}\,\phi^{2}}\\\end{aligned}
S_{\mathrm{{brane}}}={\frac{\Lambda}{16\piG_{10}}}\intd^{10}x\rho_{\perp}\left\{-V(T)\sqrt{-A}e^{-\Phi}+f(T)\epsilon^{\lambda_{1}\ldots\lambda_{p+2}}C_{[\lambda_{2}\ldots\lambda{p+2}}\partial_{\lambda_{1}]}T\right\},
{\calL}=f^{\prime}(U)\left({\dot{U}}^{2}+U^{2}{\dot{\Omega}}_{2}^{2}\right)+f^{\prime}(U)^{-1}\left({\dot{\sigma}}-{\frac{N(1-\gamma/N)}{8\pi^{2}}}A_{\phi}{\dot{\phi}}\right)^{2}\,
S=-\frac{1}{2\lambda_{s}^{4}}\intd^{6}x\sqrt{-g}e^{-\phi}\biggl[R+g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi+\Lambda-\epsilon\biggl(R_{\mathrm{EGB}}^{2}-g^{\alpha\beta}g^{\rho\sigma}\partial_{\alpha}\phi\partial_{\beta}\phi\partial_{\rho}\phi\partial_{\sigma}\phi\biggl)\biggr],
L^{2}\;\simeq\;\frac{g^{2}l^{5}\mu}{8R^{2}}\,N^{2}\;\leq\;R^{2}\quad\quad\Rightarrow\quad\quadN\;\leq\;N_{\mathrm{max}}\;\simeq\;\frac{1}{g}\,\sqrt{\frac{8}{l^{5}\mu}}\,R^{2}.
\begin{array}{cccccccc}{}&{{\calM}}&{\tilde{\calM}}&{{\calD}}&{\tilde{\calD}}&{{\calQ}}&{\tilde{\calQ}}&{T}\\{U(1)}&{0}&{0}&{1}&{-1}&{1}&{-1}&{0}\\{U(1)^{\prime}}&{1}&{-1}&{1}&{-1}&{0}&{0}&{0,}\\\end{array}
L(x_{i},\dot{x}_{i},\dot{z},q)=\frac{1}{2}\dot{x}_{i}^{2}-g\epsilon_{ij}x_{i}\dot{x}_{j}q+g^{2}\frac{1}{2}q^{2}x_{i}^{2}+\frac{1}{2}(\dot{z}-q)^{2}-V(x_{1}^{2}+x_{2}^{2})
\mathcal{J}_{\mu\nu}=2i\big\{e_{[\mu}{}^{\alpha\dot{\alpha}}e_{\nu]}{}^{\beta}{}_{\dot{\alpha}}\widehatA_{\alpha}^{(1)}\star\widehatA_{\beta}^{(1)}\big\}_{z=0}+\mathcal{L}_{\mu\nu}
\begin{array}{l}{\bar{\Phi}_{i}\bar{\Phi}_{j}=q_{ij}\bar{\Phi}_{j}\bar{\Phi}_{i},}\\{\Phi_{i}\Phi_{j}=p_{ij}^{-1}\Phi_{j}\Phi_{i},}\\{\Phi_{i}\bar{\Phi}_{i}=1+p_{ij}q_{ij}\bar{\Phi}_{i}\Phi_{i}+(p_{ij}q_{ij}-1)\sum_{j=i+1}^{N}\bar{\Phi}_{j}\Phi_{j},}\\{\Phi_{i}\bar{\Phi}_{j}=p_{ij}\bar{\Phi}_{j}\Phi_{i},}\\{\Phi_{j}\bar{\Phi}_{i}=q_{ij}\bar{\Phi}_{i}\Phi_{j},}\\\end{array}
D_{i\alpha}D_{j\beta}\tilde{\mathrm{A}}^{\beta\gamma}=D_{j\alpha}D_{i\beta}\tilde{\mathrm{A}}^{\beta\gamma}=-\textstyle{\frac{1}{4}}\delta_{\alpha}^{~\gamma}D_{i\epsilon}D_{j\eta}\tilde{\mathrm{A}}^{\epsilon\eta}
R_{21}^{V_{z_{2}},V_{z_{1}+nw/2}^{*}}=(R_{12}^{V_{z_{1}},V_{z_{2}}})^{t_{1}},~~~~R_{12}^{V_{z_{1}+nw/2}^{*},V_{z_{2}}}=(R_{21}^{V_{z_{2}},V_{z_{1}+nw}})^{t_{1}},
\varphi(x^{+},x^{-},x^{I})=\int\frac{dp^{+}d^{8}p^{I}}{(2\pi)^{9/2}}e^{i(-x^{-}p^{+}+x^{I}p^{I})}\tilde{\varphi}(x^{+},p^{+},p^{I}),
|\vec{k}_{1},\vec{k}_{2},\cdots,\vec{k}_{n}>=N(\vec{k}_{1},\vec{k}_{2},\cdots,\vec{k}_{n})\,a^{\dagger}(\vec{k}_{1})\,a^{\dagger}(\vec{k}_{2})\cdotsa^{\dagger}(\vec{k}_{n})|0>\,
e^{-2\hat{\phi}}=e^{-2\hat{\phi}_{0}}+\sum_{1}^{s}{\frac{2\tilde{m}_{k}}{r_{k}}}\,\qquadA_{u}=(A_{u})_{0}+\sum_{1}^{s}{\frac{2\hat{m}_{k}}{r_{k}}}\,\qquadA_{i}=(A_{i})_{0}+\sum_{1}^{s}{\frac{2(q_{k})_{i}}{r_{k}}}\,\qquadi=4,\dots9.
\sum_{j=-\infty}^{\infty}\left(j^{2}+a^{2}\right)^{-2s}=\int_{-\infty}^{\infty}dj\left(j^{2}+a^{2}\right)^{-2s}+2\int_{-\infty+i\epsilon}^{\infty+i\epsilon}\left(e^{-2\piiz}-1\right)^{-1}\left(z^{2}+a^{2}\right)^{-2s}dz
{\calL}_{m}[h,\Lambda]:={\calL}_{m}[h^{\Lambda}]=-{\frac{1}{4}}(h_{\mu\nu}^{i}+\partial_{\mu}\Lambda_{\nu}^{i}-\partial_{\nu}\Lambda_{\mu}^{i})^{2}+{\frac{1}{12\eta^{2}}}(H_{\mu\nu\lambda}^{i})^{2}+{\frac{1}{4\gamma^{4}}}(\partial^{\lambda}H_{\lambda\mu\nu}^{i})^{2},
B\left(x\right)=\overline{{B}}_{1}\left(T\right)\left(e^{-x\xi_{1}}+e^{-\left(2L-x\right)\xi_{1}}\right)-\overline{{B}}_{2}\left(T\right)\left(e^{-x\xi_{2}}+e^{-\left(2L-x\right)\xi_{2}}\right)+\overline{{B}}
\varphi^{2}ds^{2}=-\frac{4d\zetad\bar{\zeta}}{(1+\epsilon\zeta\bar{\zeta})^{2}}-\frac{dt^{2}}{s(t)}+s(t)\left(dx+2il\frac{\zetad\bar{\zeta}-\bar{\zeta}d\zeta}{(1+\epsilon\zeta\bar{\zeta})}\right)^{2},
J(R)\stackrel{def}{=}-\frac{2\pi\kappa\sigma_{1}\sigma_{2}}{q_{1}q_{2}}+\frac{\partialL_{eff}}{\partial\dot{\Theta}}=-\frac{2\pi\kappa\sigma_{1}\sigma_{2}}{q_{1}q_{2}}+B_{eff}R^{2}+\;O(R^{3})\;\;,
\{f,g\}_{\Lambda}=\omega^{ij}\left(\frac{\partialf}{\partialx^{i}}\frac{\partial_{l}g}{\partial\theta_{j}}-\frac{\partial_{r}f}{\partial\theta_{i}}\frac{\partialg}{\partialx^{j}}\right)+\frac{\partial_{r}f}{\partial\theta^{i}}(\theta^{k}{\partial\omega^{ij}}/{\partialx^{k}})\frac{\partial_{l}g}{\partial\theta_{j}}
{\frac{i}{2\pi}}\intd^{3}x\quad\epsilon^{\mu\nu\rho}\Lambda_{\mu}^{a}{\calF}_{\nu\rho}^{a}(A)-{\frac{i}{24\pi}}\intd^{3}x\quad\epsilon^{\mu\nu\rho}f_{abc}A_{\mu}^{a}A_{\nu}^{b}A_{\rho}^{c},
\left<L(\gamma)\frac{1}{2e^{2}}\inttrF^{2}dx\right>=-\frac{\epsilon}{e^{2}}\left<tr(\nabla_{\mu}F_{\mu\nu}(\gamma(0))\dot{\gamma}_{\nu}(0)U(\gamma))\right>=-\frac{\epsilon}{e^{2}}\frac{\partial^{2}}{\partial\gamma_{\mu}^{2}(0)}W(\gamma)
\big[M_{\mu\nu},Q_{\alpha}\big]=-{\frac{1}{2}}(\sigma_{\mu\nu})_{\alpha}{}^{\beta}Q_{\beta},\qquad\big[M_{\mu\nu},\bar{Q}_{\dot{\alpha}}\big]={\frac{1}{2}}(\tilde{\sigma}_{\mu\nu})^{\dot{\beta}}{}_{\dot{\alpha}}\bar{Q}_{\dot{\beta}}\;,
{\calZ}_{open}^{9-9}=\int_{0}^{\infty}\frac{ds}{s}\left[\mathrm{Tr}_{\mathrm{NS}}\left({{\frac{1+(-1)^{F}}{2}}}{\frac{1+\Omega}{2}}\,q^{2L_{0}-1}\right)-\mathrm{Tr}_{\mathrm{R}}\left({{\frac{1+(-1)^{F}}{2}}}{\frac{1+\Omega}{2}}\,q^{2L_{0}}\right)\right]~~,
\gamma_{\pm}(k)\,=\,\sqrt{(b(k)\pm\mid\lambda\mid)^{2}\,+\,\midC_{2}(ak)\,-\,\tauC_{1}(ak)\mid^{2}}\;,\;b(k)\,=\,\tau_{2}\,\,B(ak)\;,\;\lambda\,=\,\tau_{2}\Lambda\;\;,
\left[\begin{array}{ccccccccc}{0}&{1}&{0}&{0}&{\cdots}&{}&{}&{}&{}\\{0}&{0}&{1}&{0}&{\cdots}&{}&{}&{}&{}\\{}&{}&{}&{\ddots}&{}&{}&{}&{}\\{}&{}&{}&{}&{0}&{0}&{0}&{0}&{\cdots}\\{}&{}&{}&{}&{-1}&{0}&{0}&{0}&{\cdots}\\{}&{}&{}&{}&{0}&{-1}&{0}&{0}&{\cdots}\\{}&{}&{}&{}&{}&{}&{}&{}&{\ddots}\\\end{array}\right]
{\bar{e}}_{\alpha}^{\;\;i}{\bar{e}}_{\beta}^{\;\;j}\left[{B}_{ij}^{\bar{g}^{\prime}(2)}\right]={\bar{e}}_{\alpha}^{\;\;i}{\bar{e}}_{\beta}^{\;\;j}\frac{\partial}{\partial\bar{g}^{ij}}[\Gamma^{(2)}]+{K}_{\alpha\beta}^{g^{\prime}}
G_{I}=-\intdx\;\left({\calL}_{I}\Big(\Phi(x)+{\frac{i\hbar}{2}}{\frac{\delta}{\delta\Pi(x)}}\Big)-{\calL}_{I}\Big(\Phi(x)-{\frac{i\hbar}{2}}{\frac{\delta}{\delta\Pi(x)}}\Big)\right)W[\Phi,\Pi;t]\;.
\Omega\equiv1-\sqrt{1-\frac{(ml^{2}-r_{+}^{2})^{2}}{(Ml^{2}+2\chi_{\mathrm{m}}^{2})^{2}}\frac{l^{2}J^{2}}{(ml^{2}-r_{+}^{2}-\chi_{\mathrm{m}}^{2})^{2}}}\,.
S_{0}^{\prime}[X]=\int_{0}^{T}dt\intd\thetae(\theta\chi)\left(-\frac{\kappa_{0}}{2}\right)DX\cdotD^{2}X=\int_{0}^{T}dt\left[-\frac{\kappa_{0}}{2}\dot{x}^{2}+i\zeta\cdot\dot{\zeta}+\frac{1}{T}\zeta\cdot\dot{x}\chi\right].
g^{-1}=\left(\begin{array}{ccc}{-2\lambda_{2},}&{-6\lambda_{3},}&{-4\lambda_{4}}\\{-6\lambda_{3},}&{4\lambda_{3}(\lambda_{2}^{2}-4\lambda_{4}),}&{\lambda_{2}\lambda_{3}}\\{-4\lambda_{4},}&{+\lambda_{2}\lambda_{3},}&{-2\lambda_{2}\lambda_{4}+\frac34\lambda_{3}}\\\end{array}\right)
{\frac{E_{Q.}^{(0)}(n)}{E_{Q.}^{(0)}(1)}}={\frac{1+\displaystyle{{\frac{\lambda^{2}}{12}}}}{1+\displaystyle{{\frac{n^{2}\lambda^{2}}{12}}}}}={\frac{1}{n}}\bigl(1-{\frac{\lambda^{2}}{12}}(n^{2}-1)\bigr)+O(\lambda^{4}).
\hat{K}_{ij}=\hat{G}_{ij}\Big|_{\phantom{p}_{\Theta\to1+C^{2}/2\gamma\Phi}},\quad\bigl(\hat{K}^{-1}\bigr)^{ij}={\frac{1}{\sqrtg}}\left(\begin{matrix}{{\frac{1}{f}}g_{\mu\alpha}}&{0}&{0}&{0}\\{0}&{0}&{{\frac{2}{C}}}&{0}\\{0}&{{\frac{2}{C}}}&{-\left({\frac{4}{C^{2}}}+{\frac{2}{\gamma\Phi}}\right)}&{0}\\{0}&{0}&{0}&{{\frac{1}{\gamma\Phi}}P_{\mu\nu,\alpha\beta}}\\\end{matrix}\right),
Z_{\mathrm{gr}}[\varphi,\mu,\bar{\mu}]=e^{K[\varphi,\mu,\bar{\mu}]}\int{\calD}T{\calD}\overline{{T}}\,\,|\chi[\mu,T]|^{2}e^{-{\frac{1}{2}}\intd^{2}w\,\,e^{-\varphi}T\overline{{T}}},
{\calL}_{K}=i\overline{{\psi}}\partial\!\!\!/L\psi+i\overline{{\psi}}^{\prime}\partial\!\!\!/R\psi^{\prime}=i\overline{{\psi}}_{L}\partial\!\!\!/\psi_{L}+i\overline{{\psi}}_{R}^{\prime}\partial\!\!\!/\psi_{R}^{\prime}
\ddot{T}=\left(1+T^{2}\right)^{-1}\left[\left(1-\dot{T}^{2}+T^{2}\right)\left(T^{\prime\prime}-\frac{V_{,T}}{V}\right)+2T^{\prime}\dot{T}\dot{T}^{\prime}-T^{2}T^{\prime\prime}\right]\,
C_{\mathrm{S}}(x^{-},x^{\perp})=m\sqrt{2}\left(\frac{1}{2\pi}\right)^{3}\int\mathrm{d}^{2}p_{\perp}\int_{0}^{\infty}\frac{\mathrm{d}p_{-}}{p_{-}}\mathrm{e}^{\mathrm{i}p_{\perp}x^{\perp}-\mathrm{i}p_{-}x^{-}}
(\kappa)={\frac{1}{\sqrt{1+\vert\nu_{B}\vert^{2}}}}\,\left(\begin{array}{cc}{\kappa_{B}}&{\kappa_{B}\,\nu_{B}}\\{\nu_{B}\,\kappa_{B}}&{\nu_{B}^{c}\,\kappa_{B}\,\nu_{B}}\\\end{array}\right)\,{\frac{1}{\sqrt{1+\vert\nu_{B}\vert^{2}}}}\;,
e^{4U}=4(h_{0}+{\frac{q_{0}}{r}})\left((h^{1}-{\frac{p^{1}}{r}})(h^{2}-{\frac{p^{2}}{r}})(h^{3}+{\frac{p^{3}}{r}})+a(h^{3}+{\frac{p^{3}}{r}})^{3}\right).
\begin{array}{rl}{\delta}&{\displaystyle=(-1)^{r(q+1)+1}*d*\sim(-1)^{r(q+1)+1}*s_{2}*=-(-1)^{r(q+1)+1}{\frac{1}{2}}{C_{ij}}^{k}*c^{i}c^{j}\pi_{k}*}\\{}&{\displaystyle=-(-1)^{r(q+1)+1+(q-1)(r-q+1)+q(r-q)}{\frac{1}{2}}{C_{ij}}^{k}*c^{i}**c^{j}**\pi_{k}*={\frac{1}{2}}{C_{ij}}^{k}\pi^{i}\pi^{j}c_{k}=\overline{{s}}_{2}\;.}\\\end{array}
{E_{[a_{1}b_{1}]\dots[a_{k}b_{k}]}}^{c_{1}c_{2}c_{3}}\;\;,\quad{E_{[a_{1}b_{1}]\dots[a_{k}b_{k}]}}^{c_{1}\dotsc_{6}}\;\;,\quad{E_{[a_{1}b_{1}]\dots[a_{k}b_{k}]}}^{c_{0}|c_{1}\dotsc_{8}}
V_{\pm}=\frac{a(a\mp\frac{\mu}{\sqrt{2}})f^{2}+(a(b\mp\frac{\nu}{\sqrt{2}})+b(a\mp\frac{\mu}{\sqrt{2}}))f+b(b\mp\frac{\nu}{\sqrt{2}})}{(f^{\prime})^{2}}\pm\frac{a}{\sqrt{2}}.
H_{c}=\intdx\left[\theta(3{\sigma}^{\prime}{\pi}_{\sigma}+2\sigma{\pi}_{\sigma}^{\prime}+{\pi}_{\phi}{\phi}^{\prime})+{\sigma}^{2}{\pi}_{\sigma}{\pi}_{\phi}-{\frac12}{\sigma}^{3}{\pi}_{\sigma}^{2}+{\frac12}\sigma({\phi}^{\prime})^{2}-{\sigma}^{\prime}{\phi}^{\prime}\right]
\hat{f}(\hat{x},\hat{p})=\frac{1}{(2\pi)^{2}}\int_{-\infty}^{+\infty}dx^{\prime}dp^{\prime}d\taud\sigmae^{i[\tau(\hat{x}-x^{\prime})+\sigma(\hat{p}-p^{\prime})]}f(x^{\prime},p^{\prime}).
S_{boundary}=-\frac{\sqrt{2}}{\kappa_{5}^{2}}\int_{M_{4}^{(1)}}{\sqrt{-g}V^{-1}\alpha^{(1)}}-\frac{\sqrt{2}}{\kappa_{5}^{2}}\int_{M_{4}^{(2)}}{\sqrt{-g}V^{-1}\alpha^{(2)}},
\stackrel{2}{T}{^a}=De^{a}\equivde^{a}+\omega_{\cdotb}^{a}\wedgee^{b},\;\;\;\;\;\stackrel{2}{R}{^{ab}}=d\omega^{ab}+\omega_{\cdotc}^{a}\wedge\omega^{cb}-\frac{1}{4}e^{a}\wedgee^{b}\frac{1}{2}C_{\;\cdotc}^{ab}De^{c},
\frac{d^{p-1}}{dw^{p-1}}\left[\frac{(v-w)^{p}}{(w_{0}-w)^{\beta}}\right]=\frac{(p-1)!}{2\pii}\int_{{\calC}_{w}}dz\frac{(v-z)^{p}}{(z-w)^{p}(w_{0}-z)^{\beta}}.
\Psi\left(x\right)=e^{-i\Gamma}\left(\begin{array}{c}{e^{-i\varphi}\psi_{1}\left(r\right)}\\{i\psi_{2}\left(r\right)}\\\end{array}\right)\,,\,\,\Gamma=k_{0}x^{0}-\left(l-l_{0}\right)\varphi\,,
S(\Lambda,{\calF})={\frac{1}{4}}\Omega_{p-2}\,\Omega_{q}\left({\frac{D-2}{2\Lambda}}\right)^{\frac{D-2}{2}}\left({\frac{p-1}{1-(q-1){\calF}}}\right)^{\frac{p-2}{2}}\left({\frac{q-1}{1+(p-1){\calF}}}\right)^{\frac{q}{2}}\,.
a\,P_{XYZa\verti}\,\frac{d\phi^{i}}{dr}=\frac{b}{2\pi}\frac{e^{U(r)}}{r^{2}}\left(h_{\Sigma}t^{\Sigma}-f^{\Lambda}\bar{{\calN}}_{\Lambda\Sigma}t^{\Sigma}\right)_{[XY}\delta_{Z]}^{b}\epsilon_{ba}\equiv0
\gamma_{ij}dx^{i}dx^{j}=\left(1+\textstyle{\frac{k}{4}}\delta_{mn}x^{m}x^{n}\right)^{-2}\delta_{ij}dx^{i}dx^{j}={\frac{dr^{2}}{1-kr^{2}}}+r^{2}d\Omega_{2}^{2},
\int\intf=\int_{-\infty}^{\infty}d\lambda_{1}^{\prime}d\lambda_{2}^{\prime}e^{\lambda_{1}^{\prime}Q^{\prime}}\tilde{f}(0,\lambda_{2}^{\prime}(1-w^{-1})e^{2i\lambda_{1}^{\prime}Q^{\prime}})+
\mathcal{H}=-\frac{\pi\alpha^{\prime}}{h^{\tau\tau}}\left[G^{\mu\nu}\pi_{\mu}\pi_{\nu}+\frac{1}{(2\pi\alpha^{\prime})^{2}}G_{\mu\nu}Z^{\mu\prime}Z^{\nu\prime}\right]-\frac{h^{\tau\sigma}}{h^{\tau\tau}}\pi_{\mu}Z^{\mu\prime}\,
\frac{kn+2}{2kn+2}\,r_{n,k}=\frac{1}{2\pii}\int_{\Gamma}\,\frac{1}{z^{(k-1)n+3}}\,f_{n-1}(z^{-1})\,\left(z^{n}\,l_{n}(z^{-1})\right)^{k-1+\frac{1}{n}}\,dz
\hat{\calI}_{\mu\nu}^{\alpha\beta}={\itg_{1}}\bigg(\hat{\theta}_{\rho}\big(-i\tilde{F}_{\lambda\gamma,\sigma}+F_{\lambda\gamma}\hat{P}_{\sigma}-{\sqrt2}i\hat{\theta}_{5}F_{\lambda\gamma,\sigma}\big)+\hat{\theta}_{\lambda}\big(-i\tilde{F}_{\rho\sigma,\gamma}+F_{\rho\sigma}\hat{P}_{\gamma}-{\sqrt2}i\hat{\theta}_{5}F_{\rho\sigma,\gamma}\big)\bigg)\,,
E_{s}^{a}\;=\;\Pi^{a}\:E_{s}\:+\:s\:\frac{\varepsilon}{2\mu_{k}}\left(\Pi^{a}\:E_{s}\:{\cal{B}}_{0}\:E_{\bar{s}}+E_{\bar{s}}\:{\cal{B}}_{0}\:\Pi^{a}\:E_{s}\right)\:+\:{\cal{O}}(\varepsilon^{2})
{\calA}^{T}=\left(\begin{matrix}{\omega-{\frac{e^{+-}}{\sqrt{3}}}}&{(\omega^{-}+e^{-})}&{\sqrt{2}e^{--}}\\{(\omega^{+}-e^{+})}&{{\frac{2e^{+-}}{\sqrt{3}}}}&{(-\omega^{-}+e^{-})}\\{\sqrt{2}e^{++}}&{(-\omega^{+}-e^{+})}&{-\omega-{\frac{e^{+-}}{\sqrt{3}}}}\\\end{matrix}\right)
\begin{array}{c}{M_{0}=J_{23}p_{2}p_{3}x^{1}+J_{31}p_{3}p_{1}x^{2}+J_{12}p_{1}p_{2}x^{3}=0\}\\{M_{1}=p_{0}(x^{2}p_{3}-x^{3}p_{2})-J_{23}p_{2}p_{3}x^{0}=\nu\}\\{M_{2}=p_{0}(x^{3}p_{1}-x^{1}p_{3})-J_{31}p_{3}p_{1}x^{0}=0\}\\{M_{3}=p_{0}(x^{1}p_{2}-x^{2}p_{1})-J_{12}p_{1}p_{2}x^{0}=0.}\\\end{array}
\omega(A,B)_{i_{1}\cdotsi_{m+n-2}}=\sum_{(i_{1}^{\prime},\cdots,i_{m+n-2}^{\prime})}(-1)^{e(i_{1}^{\prime},\cdots,i_{m+n-2}^{\prime})}A_{ri_{1}^{\prime}\cdotsi_{m-1}^{\prime}}\omega^{rs}B_{si_{m}^{\prime}\cdotsi_{m+n-2}^{\prime}},
Q_{N}(V,\beta)={\frac{1}{N!}}({\frac{2}{(2\pi)^{3}}}){}^{N}\int_{V}\prod_{i=1}^{N}d^{3}\eta_{i}d^{3}\kappa_{i}e^{-\beta\sqrt{m^{2}+{\vec{\kappa}}_{i}^{2}}}.
S_{\mathrm{gh}}=\int\sum_{\alpha\beta}\left\{\bar{c}^{\alpha\beta}\left(-D_{\mu}^{\alpha\beta}\right)^{2}c^{\beta\alpha}+ig_{\mathrm{YM}}\bar{c}^{\beta\alpha}D_{\mu}^{\alpha\beta}\left[\bar{A}_{\mu},c\right]\right\}
\frac{da^{\mu}}{d\tau}=\tilde{c}_{1}F^{\mu\nu}a_{\nu}+\tilde{c}_{2}\langle{u^{\mu}u_{\nu}}\rangleF^{\nu\lambda}a_{\lambda}+\tilde{c}_{3}\frac{\pi^{\mu}}{m}\frac{\pi_{\nu}}{m}F^{\nu\lambda}a_{\lambda}
g_{tt}=g_{ii}^{-1}=\left[{\frac{\mathrm{Im}Z^{s}\Bigl((\mathrm{Im}Z^{1})^{2}-(\mathrm{Im}Z^{a})^{2}\Bigr)}{\left[\mathrm{Im}Z_{s}\Bigl((\mathrm{Im}Z^{1})^{2}-(\mathrm{Im}Z^{a})^{2}\Bigr)\right]_{\infty}}}\right]^{-1}.
\begin{array}{lcl}{K[a]}&{=}&{1+a_{1}p^{-1}+(a_{2}+\thetaa_{1}^{\prime})p^{-2}+(a_{3}+\theta^{2}a_{1}+2\thetaa_{2}^{\prime})p^{-3}+...}\\{}&{=}&{1+\sum_{m=1}^{\infty}\sum_{s=0}^{m-1}{\theta^{s}}c_{m-1}^{s}a_{m-s}^{(s)}p^{-m}}\\\end{array}
e_{\mu}^{a}\,e_{\nu}^{b}\,\hat{\xi}_{ab}=\Omega^{-1}\,\hat{\eta}_{\mu\nu}\Omega\qquad\mathrm{or}\qquad\Omega\,\hat{\xi}_{ab}=e_{a}^{\mu}\,e_{b}^{\nu}\,\hat{\eta}_{\mu\nu}\Omega.
n^{A}=\langle\Lambda|\xi(w)^{\dagger}\sigma^{A}\xi(w)|\Lambda\rangle=\left(\begin{matrix}{\bar{\phi}_{1}}&{0}\\\end{matrix}\right)\left(\begin{matrix}{1}&{\bar{w}}\\{0}&{1}\\\end{matrix}\right)\sigma^{A}\left(\begin{matrix}{1}&{0}\\{w}&{1}\\\end{matrix}\right)\left(\begin{matrix}{\phi_{1}}\\{0}\\\end{matrix}\right).
\begin{array}{rl}{\hat{U}^{-1}(k_{1},k_{2})}&{|A_{\bar{z}}(z,\bar{z})>=}\\{}&{}\\{=}&{e^{i\pikk_{1}k_{2}}\,e^{-\frac{1}{2}ik\left[(k_{1}\tau-k_{2})\overline{{A_{\bar{z}}^{++}(0,0)}}+(k_{1}\bar{\tau}-k_{2})A_{\bar{z}}^{++}(0,0)\right]}\|A_{\bar{z}}(z,\bar{z})-\frac{i\pi}{\tau_{2}}(\tauk_{1}-k_{2})>\\\,}\\\end{array}
\left(\frac{x^{-}}{x^{+}}\right)^{2}F(x^{-},x^{+})=\left.\left|{\frac{L}{\pi}}\langle0|T^{++}(k)|n\rangle\right|^{2}{\frac{M_{n}^{4}}{8\pi^{2}k^{3}}}K_{4}(M_{n}r)\right|_{M_{n}=0}=\left|{\frac{L}{\pi}}\langle0|T^{++}(k)|n\rangle\right|_{M_{n}=0}^{2}{\frac{6}{k^{3}\pi^{2}r^{4}}}.
\Omega_{i}^{2}=\frac{r_{0}^{d-4}}{h^{d-2}}\left(\frac{l_{i}}{r_{0}}\right)^{2}\left[1+\frac{4}{d-2}\sum_{j}\left(\frac{l_{j}}{r_{0}}\right)^{2}-2\left(\frac{l_{i}}{r_{0}}\right)^{2}+\ldots\right]
\begin{array}{rl}{v_{01}}&{=v_{0}+{{\frac{p^{-}}{\sqrt{2}}}}}\\{u_{01}(\sigma_{per}^{+},\sigma_{per}^{-})}&{=\frac{1}{v_{0}+\frac{p^{-}}{\sqrt{2}}}\left\{1-{\frac{[1-(v_{0}+{{\frac{p^{-}}{\sqrt{2}}}})(u_{0}+p^{+}\sigma_{per}^{+})]\times[1-(v_{0}+{\frac{p^{-}}{\sqrt{2}}})(u_{0}+p^{+}\sigma_{per}^{-})]}{[1-(v_{0}+{\frac{p^{-}}{\sqrt{2}}})(u_{0}-{\frac{p^{+}}{\sqrt{2}}})]}}\right\}}\\\end{array}
{\calP}_{R}^{2}={\tilde{v}}^{T}{\tilde{\phi}}{\tilde{\phi}}^{T}{\tilde{v}}={\tilde{v}}^{T}{\phi}_{c}{\tilde{\phi}}_{c}^{\dagger}{\tilde{v}}=|{\tilde{v}}^{T}{\phi}_{c}|^{2}.
D_{\mu}\langle\widetilde{j}_{5\mu}^{a,\mathrm{reg}}\rangle=2\mathrm{i}m\langle\widetilde{\psi}_{\varepsilon}^{+}T^{a}\gamma_{5}\widetilde{\psi}_{\varepsilon}\rangle+\widetilde{{\calA}}_{5}^{a}
\Big\{-\frac{\hbar^{2}}{2}\left[\partial_{r}^{2}+\frac{1}{r}\partial_{r}+\left(\frac{1}{r^{2}}+g^{2}\right)\partial_{\theta}^{2}\right]\,+\,\frac{1}{2}\omega^{2}r^{2}\Big\}\psi(r,\theta)=E\psi(r,\theta)\\\.
\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}.
\psi(\vec{k},t)=\frac{1}{\sqrt{V}}\sum_{\vec{p}}\sqrt{\frac{mc^{2}}{|E|}}\{\sum_{r=1,2(E>0)}b_{\vec{p}}^{(r)}u^{(r)}(\vec{p})e^{\frac{i}{\hbar}(\vec{p}\cdot\vec{x}-Et)}+\sum_{r=3,4(E<0)}b_{\vec{p}}^{(r)}u^{(r)}(\vec{p})e^{\frac{i}{\hbar}(\vec{p}\cdot\vec{x}-Et)}\}
0=\frac{\delta\Gamma}{\deltaK_{\mu}}\frac{\delta\Gamma}{\deltaQ_{\mu}}-\frac{\delta\Gamma}{\deltaK}\frac{\delta\Gamma}{\deltac}+\left[\frac{1}{\alpha}D_{\mu}(A)Q_{\mu}-\frac{\chi}{2\alpha}\overline{{c}}\right]\frac{\delta\Gamma}{\delta\overline{{c}}}+\chi\frac{\delta\Gamma}{\delta\alpha}\,;
\frac{1}{(aq)_{\infty}}\sum_{n=0}^{\infty}a^{kn}q^{kn^{2}}\alpha_{n}=\sum_{n_{1}\geq\cdots\geqn_{k}\geq0}\frac{a^{n_{1}+\cdotsn_{k}}q^{n_{1}^{2}+\cdots+n_{k}^{2}}\beta_{n_{k}}}{(q)_{n_{1}-n_{2}}\cdots(q)_{n_{k-1}-n_{k}}}.
\left\langle\sigma_{n_{1}}\ldots\sigma_{n_{s}}\cdot\widetilde{P}\ldots\widetilde{P}\right\rangle=\int_{{\bfR}{\calM}}\iota^{\ast}c_{1}({\calL}_{(1)})^{n_{1}}\wedge\ldots\wedge\iota^{\ast}c_{1}({\calL}_{(s)})^{n_{s}}
{\calE}(B+\deltaB(\vec{x}))-{\calE}(B)=\frac{1}{2}\intd^{2}x\,\left\{1+\alpha\,(e^{2}/\sqrt{eB})\left[1-\beta\,(eB)^{-1}|\vec{\partial}\deltaB/\deltaB|^{2}\right]\right\}(\deltaB)^{2}
2\pin=-\frac{i}{240\pi^{2}}\int_{S^{5}}dtd^{4}x\epsilon^{ijklm}tr\left[U^{-1}\partial_{i}UU^{-1}\partial_{j}UU^{-1}\partial_{k}UU^{-1}\partial_{l}UU^{-1}\partial_{m}U\right]\\.