harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\calF}_{N}\star_{N}{\calG}_{N}({\calW})={\calF}_{N}\,{\calG}_{N}({\calW})+\sum_{m=1}^{N}\frac{(N-m)!}{N!m!}\,{\calF}_{N}({\calW})\,\,\vdots\underbrace{(\partial^{^{\!\!\!\!\!\leftarrow}}\,\Omega\,\vec{\bar{\partial}})\cdots(\partial^{^{\!\!\!\!\!\leftarrow}}\,\Omega\,\vec{\bar{\partial}})}_{mfactors}\,\vdots\,{\calG}_{N}({\calW})\,,
S=S_{{\calM}_{6}}+S_{D11}({\hat{F}})-{\frac{1}{6}}\intA^{(3)}\wedgedA^{(3)}\wedgedA^{(3)}-{\frac{1}{2}}\intH^{(3)}\wedge{\hat{F}}^{(4)}\wedge{}^{*}G^{(7)},
A_{1\,1}=\frac{i}{24}\times\frac{1}{3}\times(-3\sqrt{3})\times2\times[3\times2i\sqrt{3}\mathrm{Tr}\tilde{B}_{2}^{9}(-2\mathrm{Tr}(\tilde{B}_{2}^{9})^{2})]=-6\;,
a_{i}^{(1)}(x)={\frac{\theta_{i}}{L_{i}}}+\epsilon_{ij}\partial_{x^{j}}\int\!d\vec{y}G(\vec{x},\vec{y})b^{(1)}(x)+\partial_{x^{i}}\int\!d\vec{y}G_{p}(\vec{x},\vec{y})\partial_{y^{j}}a_{j}^{(1)}(y)\,,
H\;=\;-\frac{1}{2}\sum_{j=1}^{L}\:\left(\,\eta\:\sigma_{j}^{x}\sigma_{j+1}^{x}\;+\;\eta^{-1}\sigma_{j}^{y}\sigma_{j+1}^{y}\right)\;-\;h\,\sum_{j=1}^{L}\;\sigma_{j}^{z}
M_{W1}{}^{1}=-\frac{\kappa}{\pi}B\left[\frac{5}{6}u(\partial^{3}h^{*})+\frac{9}{2}(\partialu)(\partial^{2}h^{*})+\frac{17}{2}(\partial^{2}u)(\partialh^{*})+\frac{17}{3}(\partial^{3}u)h^{*}\right]\,
v_{1}=\sqrt{\frac{1-(\phi_{sph}^{2}-\frac{v^{2}}{2})f_{1}^{-\frac{1}{2}}}{2}},\;\;v_{2}=\sqrt{\frac{1+(\phi_{sph}^{2}-\frac{v^{2}}{2})f_{1}^{-\frac{1}{2}}}{2}},
S[\hat{\phi}]=\intd^{4}x[\frac{1}{2}Z_{3}{\partial}_{\mu}{\hat{\phi}}{\partial}^{\mu}{\hat{\phi}}-\frac{m^{2}+{\delta}m^{2}}{2}{\hat{\phi}}^{2}-\frac{g^{4}}{4!}Z_{1}{\hat{\phi}}^{4}].
\rho(\eta)\simeq\frac{1}{16\pi^{2}\beta(\beta-1)a^{4}}\int_{\calK}\mathrm{d}k\,k^{2}\left(\frac{\eta_{2}}{\eta_{1}}\right)^{2\beta-1}\left|\gamma_{2}\eta_{2}\right|^{2}\left|1+\frac{\gamma_{1}\eta_{1}}{1-2\beta}\right|^{2}\omega(k,\eta),
\left(\begin{matrix}{0}&{0}&{s2}&{s1}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}\\{0}&{s1}&{0}&{0}&{0}&{0}\\{0}&{s2}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{s3}&{0}&{s2}\\{0}&{s3}&{0}&{0}&{0}&{0}\\\end{matrix}\right)
A_{\nu}(q)=\frac{1}{2}\left(\begin{array}{ccc}{0}&{\overline{\psi}(q)\gamma_{\mu}\gamma_{\nu}}&{-(\gamma_{\nu}\gamma_{\mu}\psi(q))^{T}}\\{0}&{0}&{0}\\{0}&{0}&{0}\\\end{array}\right)_{(\mu,\lambda)}\,,
\begin{array}{lll}{x_{1}=c_{q}b_{r},}&{y_{1}=\omegad_{q}a_{r},}&{z_{1}=b_{q}c_{r},}\\{x_{2}=c_{p}a_{q},}&{y_{2}=d_{p}b_{q},}&{z_{2}=b_{p}d_{q},}\\{x_{3}=\omega_{-1}c_{p}b_{r},}&{y_{3}=d_{p}a_{r},}&{z_{3}=b_{p}c_{r},}\\{x_{4}=0,}&{y_{4}=z_{4}.}&{}\\\end{array}
\begin{array}{llc}{\frac{d}{d\tilde{\tau}}\tilde{O}}&{=}&{\{\tilde{O},\tilde{H}\}_{P.B.}+\frac{\partial}{\partial\tilde{\tau}}\tilde{O}_{|m_{1},m_{2},p_{1},p_{2}}}\\{}&{=}&{\frac{\partial}{\partial\tilde{\tau}}\tilde{O}_{|\tilde{u},\tilde{v},\alpha,\beta},}\\\end{array}
H\=\-p_{1}\,\dot{x}\,-\,p_{2}\,\ddot{x}\,-\,p_{3}\stackrel{\ldots}{x}\,-\,L\,=\,-p_{1}\,q_{2}\,-\,p_{2}\,q_{3}\,+\,m\,\sqrt{q_{2}^{2}}\,+\,\alpha\,\frac{\sqrt{g}}{q_{2}^{2}}\,{.}
Tr\epsilon^{r_{1}s_{1}}\cdots\epsilon^{r_{p}s_{p}}u_{r_{1}}^{\A_{1}}u_{s_{1}}^{\B_{1}}\cdotsu_{r_{p}}^{\A_{p}}u_{s_{p}}^{\B_{p}}\Phi_{A_{1}B_{1}\cdotsA_{p}B_{p}}
Z_{u}^{v}\equivq^{{\frac{1}{2}}v^{2}-{\frac{1}{24}}}\prod_{m=1}^{\infty}\left(1+q^{m+v-{\frac{1}{2}}}~e^{-2\piiu}\right)~\left(1+q^{m-v-{\frac{1}{2}}}~e^{2\piiu}\right)~.
z(1-z)\partial_{z}^{2}R(z)+(1-z)\partial_{z}R(z)+\left[\frac{\omega^{2}\ell^{4}}{4r_{H}^{2}}\left(\frac{1}{z}-1\right)+\frac{m^{2}\ell^{2}}{1-z}\right]R(z)=0.
\frac{\beta^{3}}{8\pi^{2}(2\pi\beta)^{3(D-1)/2}}\left(\frac{4}{\beta}\right)^{3D/4}=\frac{1}{8\pi^{2}}\left(\frac{2}{\pi}\right)^{3(D-1)/2}\left[\frac{1}{(4\beta^{3})^{1/4}}\right]^{3(D-2)}.
[a_{ij}(k^{+},{\bfk}^{\perp}),a_{kl}^{\dagger}({\tilde{k}}^{+},{\tilde{\bfk}}^{\perp})]=\delta_{ik}\delta_{jl}\delta(k^{+}-{\tilde{k}}^{+})\delta({\bfk}^{\perp}-{\tilde{\bfk}}^{\perp}).
\begin{array}{rcl}{G^{\pm\pm}}&{=}&{-4x^{\pm}x^{\pm}\left(\phi^{\prime}e^{(\phi-2\psi)}\right)^{\prime}e^{-\phi}\,,}\\{G^{\pmy}}&{=}&{0\,,}\\{G^{\pm\mp}}&{=}&{4\left(-\zeta\phi^{\prime}e^{\phi}\right)^{\prime}e^{-(\phi+2\psi)}\,,}\\{G^{yy}}&{=}&{4\left(-\zeta\psi^{\prime}\right)^{\prime}e^{-2\psi}\,.}\\\end{array}
\omega_{\nu}^{\bar{M}\bar{N}}=(\delta^{i\bar{N}}\delta_{\nu}^{\bar{M}}-\delta^{i\bar{M}}\delta_{\nu}^{\bar{N}})\partial_{i}\phi/\sqrt{|c|\phi^{\prime}},~~~~~\omega_{j}^{\bar{M}\bar{N}}=(\delta^{i\bar{N}}\delta_{j}^{\bar{M}}-\delta^{i\bar{M}}\delta_{j}^{\bar{N}})\partial_{i}\sqrt{\phi^{\prime}}/\sqrt{\phi^{\prime}}.
\left(\begin{array}{c}{\tilde{z}_{a}}\\{\tilde{\mathscr{F}}_{a}}\\\end{array}\right)=\left(\begin{array}{cc}{U}&{V}\\{W}&{Z}\\\end{array}\right)\,\left(\begin{array}{c}{{z}_{a}}\\{{\mathscr{F}}_{a}}\\\end{array}\right)
\partial_{t}U_{k}(\rho)=-v_{d}\,\bigl(NU_{k}^{\prime}(\rho)+2\rhoU_{k}^{\prime\prime}(\rho)\bigr)\int_{0}^{\infty}dw\,w^{d/2-1}\,\frac{\partial_{t}R_{k}(w)}{(Z_{k}w+R_{k}(w))^{2}}+{\calO}(U_{k}^{2}),
{{\mathcal{D}}^{-1}}^{\mu\nu}(k)=\left(k^{2}g^{\mu\nu}-k^{\mu}k^{\nu}\right)-\frac{\kappa}{k^{2}-m_{\mathrm{A}}^{2}}\left(\bar{k}^{2}\bar{g}^{\mu\nu}-\bar{k}^{\mu}\bar{k}^{\nu}\right).
\Pi_{i}^{r}=\partial_{i}X^{\mu}(e_{\mu}^{r}-\frac{1}{4}\bar{\theta}\Gamma^{rst}\theta\hat{\omega}_{\must}+\bar{\theta}\Gamma^{r}T_{\mu}^{\nu\rho\sigma\lambda}\theta\hat{F}_{\nu\rho\sigma\lambda})+\bar{\theta}\Gamma^{r}\partial_{i}\theta+\cdots~,
V_{F}=\frac{\tau_{p}(g_{s})}{4N(2\pil_{s}^{2})^{2}}\left(|\Phi_{1}\Phi_{2}-\epsilon^{-1}\Phi_{2}\Phi_{1}|^{2}+|\Phi_{2}\Phi_{3}-\epsilon^{-1}\Phi_{3}\Phi_{2}|^{2}+|\Phi_{3}\Phi_{1}-\epsilon^{-1}\Phi_{1}\Phi_{3}|^{2}\right),
\begin{array}{ll}{A:\qquadu=u(\sigma_{per}^{+}),}&{\quadv=v(\sigma_{per}^{-})}\\{B:\qquadu=\bar{u}(\sigma_{per}^{-}),}&{\quadv=\bar{v}(\sigma_{per}^{+})}\\{C:\qquadu=c_{1},}&{\quadv=v(\bar{a}(\sigma_{per}^{+}),\bar{b}(\sigma_{per}^{-}))}\\{D:\qquadu=u(a(\sigma_{per}^{-}),b(\sigma_{per}^{+})),}&{\quadv=c_{2}.}\\\end{array}
W^{\mu\nu}(Q^{2},\nu_{B})=\biggl(-\eta^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{q^{2}}\biggr)F_{1}(Q^{2},\nu_{B})+\frac{1}{\nu_{B}}\biggl(p^{\mu}-\frac{\nu_{B}}{q^{2}}q^{\mu}\biggr)\biggl(p^{\mu}-\frac{\nu_{B}}{q^{2}}q^{\nu}\biggr)F_{2}(Q^{2},\nu_{B})\quad.
\frac{dV}{dT}+\frac{2\pi\alpha^{\prime}}{2}\frac{dV}{dT}(T^{\prime})^{2}-2\pi\alpha^{\prime}(VT^{\prime})^{\prime}=\frac{dV}{dT}+\frac{2\pi\alpha^{\prime}}{2}\frac{dV}{dT}(T^{\prime})^{2}-2\pi\alpha^{\prime}(V^{\prime}T^{\prime}+VT^{\prime\prime})=0,
\langle\tilde{\Psi}|\tilde{\Phi}\rangle=\langle\tilde{\mathbf{U}}\Psi|\tilde{\mathbf{U}}\Phi\rangle=\langle\Psi|\tilde{\mathbf{U}}^{\dagger}\tilde{\mathbf{U}}|\Phi\rangle=\langle{\Psi}|{\Phi}\rangle\,,
\delta\phi(x,t)=\int\frac{d^{3}k}{\sqrt{2k}}\left(a_{\bfk}\delta\phi_{\bfk}(t)e^{i{\bfk}\cdot{\bfx}}+a_{\bfk}^{\dagger}\delta\phi_{\bfk}^{*}(t)e^{-i{\bfk}\cdot{\bfx}}\right)
\tilde{\Psi}_{y}^{*}=-\tilde{\Psi}_{xx}^{*}-2u\tilde{\Psi}^{*}+({\theta}_{y}-{\theta}_{x}^{2}+{\theta}_{xx})\tilde{\Psi}^{*}+2{\theta}_{x}\tilde{\Psi}_{x}^{*},
\intd\alpha\d\beta\intdk_{1}\dk_{2}\e^{2ik_{1}\wedgek_{2}}e^{-i(k_{1}+k_{2})\cdoty}e^{-\alpha(k_{1}^{2}+m^{2})}e^{-\beta(k_{2}^{2}+m^{2})}e^{-\frac1{\Lambda^{2}\alpha}-\frac1{\Lambda^{2}\beta}}
\lambda_{{\mathbfi}}^{\alpha}=\frac{i}{24}{\bar{D}}^{4}D^{\alpha{\mathbfj}}L_{{\mathbfi}{\mathbfj}}\qquad\mathrm{and}\qquad\sigma=\frac{i}{144}{\bar{D}}^{4}D^{{\mathbfi}{\mathbfj}}L_{{\mathbfi}{\mathbfj}}\.
v=\ell\int_{0}^{\rho}e^{(\rho_{0}^{2}-{\varrho}^{2})/2\rho_{0}}({\frac{\rho_{0}}{\varrho}})^{{\frac{(D-1)}{2(D-2)}}}d\varrho~=~\elle^{\rho_{0}/2}\Big({\frac{\rho_{0}}{2}}\Big)^{\frac{3D-5}{4(D-2)}}\textstyle\gamma({\frac{D{-}3}{4(D{-}2)}},{\frac{\rho^{2}}{2\rho_{0}}})~,
X^{\lambda}({\calA})=\frac{1}{48\pi^{2}}\epsilon^{\lambda\mu\rho\sigma}\left[\partial_{\mu}{\calA}_{\rho}{\calA}_{\sigma}+{\calA}_{\sigma}\partial_{\mu}{\calA}_{\rho}+\frac{3}{2}{\calA}_{\mu}{\calA}_{\rho}{\calA}_{\sigma}\right]
\tilde{J}_{\mu}^{(\pm)}\equiv\intd^{3}x\,j_{\mu}^{(\pm)}(x)=\sum_{{\bfk}}\frac{k_{\mu}}{\omega_{{\bfk}}}(a_{{\bfk}}^{\dagger}a_{{\bfk}}\pmb_{{\bfk}}^{\dagger}b_{{\bfk}}).
{\widehatT}={\widehatM}_{1}=\left(\begin{array}{cccccc}{1}&{0}&{0}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}&{0}\\{-2}&{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{0}&{0}&{1}\\\end{array}\right)
ds^{2}={H_{5}^{\prime}}^{-1/2}\Big[{\frac{x^{2}}{R_{1}^{2}}}(-dt^{2}+dz^{2})+R_{1}^{2}{\frac{dx^{2}}{x^{2}}}+R_{1}^{2}d\Omega_{3}^{2}\Big]+{H_{5}^{\prime}}^{1/2}(y)(dy^{2}+y^{2}d{\Omega_{3}^{\prime}}^{2}).
d\tilde{s}_{\mathrm{EBTZ}}^{2}=-{\frac{(\tilde{\rho}^{2}-\tilde{\rho}_{0}^{2})^{2}}{\tilde{\rho}^{2}\tilde{\ell}^{2}}}d\tilde{t}^{2}+\tilde{\rho}^{2}\left(d\tilde{\phi}-{\frac{\tilde{J}}{2\tilde{\rho}^{2}}}d\tilde{t}\right)^{2}+{\frac{\tilde{\rho}^{2}\tilde{\ell}^{2}}{(\tilde{\rho}^{2}-\tilde{\rho}_{0}^{2})^{2}}}d\tilde{\rho}^{2}.
\dot{p}\tilde{q}+\dot{\Theta_{1}}\Pi_{1}+\dot{\Theta_{2}}\Pi_{2}-{\calL}=-\frac{1}{2}\tilde{q}\left\{(\vec{\nabla}p-\frac{i}{2}\Theta^{tr}\vec{\nabla}\Theta)^{2}+\frac{1}{\tilde{q}^{2}}+\frac{1}{\tilde{q}}\,\frac{i}{2}\,\Theta^{tr}\vec{\gamma}\times\vec{\nabla}\Theta\right\}.
S=-\frac{1}{2\kappa_{P}^{2}}\intd^{4}x\sqrt{-g}\left[\frac{1}{2}R+\frac{1}{4}(\nabla\varphi)^{2}+\frac{3}{4}(\nabla\beta)^{2}+\frac{q_{5}}{2}e^{(\beta-\varphi)}(\nablaz)^{2}\right]\;,
ds^{2}=-\left(1-\frac{r_{+}^{2}}{r^{2}}\right)d{\hat{t}}^{2}+\left(1-\frac{r_{-}^{2}}{r^{2}}\right)d{\hat{x}}^{2}+{\left(1-\frac{r_{+}^{2}}{r^{2}}\right)}^{-1}{\left(1-\frac{r_{-}^{2}}{r^{2}}\right)}^{-1}\frac{l^{2}dr^{2}}{r^{2}}
S=\int_{M}\varepsilon^{\mu\nu\alpha\beta}\left(-\frac{1}{4^{+}\tau}\,{^+G}_{\mu\nu}^{ab}\,\,{^+G}_{\alpha\beta}^{cd}+\frac{1}{4^{-}\tau}\,{^-G}_{\mu\nu}^{ab}\,\,{^-G}_{\alpha\beta}^{cd}+\cdots\right)\,\epsilon_{abcd}\,,
\Gamma_{div}^{JT}=\frac{1}{4\pi\epsilon}\intd^{d}x\,\sqrt{-g}\,\left\{\frac{24-n}{6}R-\left[8-n\Phi^{\prime}(\phi)^{2}\right]g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+4\Lambda\right\}
Z({\bfp})=\frac{2mM\left[\sqrt{{\bfp}^{2}+m^{2}}+\sqrt{{\bfp}^{2}+M^{2}}\right]}{\sqrt{{\bfp}^{2}+m^{2}}\sqrt{{\bfp}^{2}+M^{2}}\left[\sqrt{{\bfp}^{2}+m^{2}}+\sqrt{{\bfp}^{2}+M^{2}}+E\right]}
\begin{array}{c}{a^{-}a^{+}-qa^{+}a^{-}~=~q^{-N}}\\{a^{-}a^{+}-q^{-1}a^{+}a^{-}~=~q^{N}}\\{q^{N}a^{+}q^{-N}~=~qa^{+}}\\{q^{N}a^{-}q^{-N}~=~q^{-1}a^{-}}\\{q^{N}q^{-N}~=~q^{-N}q^{N}~=1~}\\\end{array}
\int_{0}^{L}\!\!dx_{4}\,\xi_{M}^{+}(x_{4})\,\xi_{M^{\prime}}^{+}(x_{4})=\int_{0}^{L}\!\!dx_{4}\,\xi_{M}^{-}(x_{4})\,\xi_{M^{\prime}}^{-}(x_{4})=\delta_{MM^{\prime}}\,.
\eta_{2}(\beta\sqrt{-\triangle})=\frac{\pi}{4\beta}\,\int_{m}^{\infty}\!{\mathrmd}\tilde{m}\,\Big(1-\frac{2m^{2}}{{\tilde{m}}^{2}}+\frac{m^{4}}{{\tilde{m}}^{4}}\Big)\,\frac{1}{4{\tilde{m}}^{2}-\triangle}
\hbar|\langle\mathrm{tr}\,\phi^{2}\rangle_{\hat{\tau}}-\langle\mathrm{tr}\,\phi^{2}\rangle_{\hat{\tau}_{0}}|\leq{\pi}\mathrm{Im}\,\left(\langle\phi\rangle_{\hat{\tau}}\overline{{\langle\phi_{D}\rangle}}_{\hat{\tau}}\right)\leq2\hbar|\langle\mathrm{tr}\,\phi^{2}\rangle_{\hat{\tau}}-\langle\mathrm{tr}\,\phi^{2}\rangle_{\hat{\tau}_{0}}|.
\mathrm{S}=\intd^{4}xd\tau\left\{\psi^{*}(i\partial_{\tau}+e_{0}a_{5})\psi-\frac{1}{2M}\psi^{*}(-i\partial_{\mu}-e_{0}a_{\mu})(-i\partial^{\mu}-e_{0}a^{\mu})\psi-\frac{\lambda}{4}f_{\alpha\beta}f^{\alpha\beta}\right\}
D_{++}^{ab}(x)={\frac{i\delta^{ab}}{\pi^{2}}}\intd^{2}k\,e^{ikx}{\frac{k_{+}^{2}}{(k^{2}+i\epsilon)^{2}}}={\frac{\delta^{ab}}{\pi}}{\frac{(x^{-})^{2}}{(-x^{2}+i\epsilon)}}\
{\bfJ}^{k}=-i\sum_{\eta}\int\frac{d^{3}{\bfp}}{(2\pi)^{3}}\,\frac{{\bfp}^{k}}{2E_{p}}\left[a_{\eta}b_{-\eta}^{\dagger}+b_{\eta}^{\dagger}a_{-\eta}\right]\quad.
{\mathbfA^{SW}}=\frac{1}{2}\bigl({\mathbfA}_{\mathbfn}+{\mathbfA}_{-\mathbfn}\bigr)=q\frac{({\mathbfn}\cdot{\mathbfr}){\mathbfr}\times{\mathbfn}}{r\bigl(r^{2}-({\mathbfn}\cdot{\mathbfr})^{2}\bigr)},
g^{-1}=\left(\begin{array}{cccc}{0}&{0}&{0}&{1}\\{0}&{0}&{\left(1+e^{2}\right)^{-1}}&{0}\\{0}&{-\left(1+e^{2}\right)^{-1}}&{0}&{0}\\{-1}&{0}&{0}&{0}\\\end{array}\right),
I_{\uparrow}\:\acute{Y}_{R}(y)\:I_{\downarrow}\:\grave{Y}_{L}(x)\:I_{\uparrow}\:+\:\overline{{I_{\uparrow}\:\acute{Y}_{R}(y)\:I_{8}\:\grave{Y}_{L}(x)\:I_{\uparrow}}}\;=\;\kappa(x,y)\:I_{\uparrow}\;,
{\calL}_{B}=-{\frac{a}{4}}C_{\mu\nu}C^{\mu\nu}+{\frac{\alpha}{2}}\epsilon_{\mu\nu\lambda}C^{\mu}C^{\nu\lambda}-{\frac{1}{2g}}B_{\mu}B^{\mu}-{\frac{pe^{2}}{4}}A_{\mu\nu}A^{\mu\nu}+{\frac{qe^{2}}{2}}\epsilon_{\mu\nu\lambda}A^{\mu}A^{\nu\lambda}.
\begin{array}{l}{x^{-}x^{0}=qx^{0}x^{-}~,~~~~~x^{0}x^{+}=qx^{+}x^{0}~,}\\{x^{+}x^{-}-x^{-}x^{+}=(q^{1/2}-q^{-1/2})x^{0}x^{0}~.}\\\end{array}
j_{M2}^{bs}\cdot\lambda=-\frac{2\kappa_{11}^{2}T_{M2}}{2}\delta^{(6)}(x)\delta^{(2)}(w-e^{-b})\frac{\tilde{f}w\overline{w}}{\rho}dx_{0}\wedge(d\rho+\frac{\tilde{f}}{2V\rho}dx_{9}),
\omega^{-1}=1-\frac{2}{\kappa_{4}^{2}\rho_{o}}\left(\beta+\frac{\gamma}{R^{3}}\right)\Big(1-\frac{\sqrt{3}\alpha}{\sqrt{4\Lambda_{4}-2\Lambda_{5}+3\alpha^{2}+4\kappa_{4}^{2}\rho_{o}}}\Big)^{-1}\,.
A_{P}[h]=\sum_{k=0,k^{\prime}=0}^{\infty}\intd^{d}xh^{m_{1}...m_{k}}(x)P_{\{m_{1}...m_{k}|n_{1}...n_{k^{\prime}}\}}(\partial_{l})h^{n_{1}...n_{k^{\prime}}}(x),
A_{n}=\theta^{2-n}\,\partial_{\bar{z}}\left[\frac{\chi_{n}(z,\bar{z})}{\theta}-\bar{\chi}_{n}(z,\bar{z})\right]\,,\quad\bar{A}_{n}=\theta^{2-n}\,\partial_{z}\left[\frac{\chi_{n}(z,\bar{z})}{\theta}+\bar{\chi}_{n}(z,\bar{z})\right]\,.
{\tilde{H}}_{coll}^{\beta,\pm}=\sum_{n>0}(1-\beta)n\alpha_{n}^{\dagger}\alpha_{n}+N\beta\sum_{n>0}\alpha_{n}^{\dagger}\alpha_{n}\mp\sqrt{\beta}\sum_{n,n^{\prime}>0}(\alpha_{n}^{\dagger}\alpha_{n^{\prime}}^{\dagger}\alpha_{n+n^{\prime}}+\alpha_{n+n^{\prime}}^{\dagger}\alpha_{n}\alpha_{n^{\prime}}).
\partial_{x^{i_{1}}}\ldots\partial_{x^{i_{n-1}}}F_{i_{1}i_{1}^{\prime}\ldotsi_{2k}^{\prime}j_{1}\ldotsj_{l}a_{l+1}\ldotsa_{p}}^{0,(2k+p+1)}Str\left(\partial_{a_{1}}\phi^{j_{1}}\ldots\partial_{a_{l}}\phi^{j_{l}}\phi^{i_{1}}\ldots\phi^{i_{n}}\phi^{i_{2k}^{\prime}}\phi^{i_{2k-1}^{\prime}}\ldots\right).
L^{(2)}=\left(a_{i}-\eta^{(0)\mu}\frac{\partial\Phi_{\mu}^{(0)}}{\partialy^{i}}-\eta^{(1)\mu}\frac{\partial\Phi_{\mu}^{(1)}}{\partialy^{i}}\right){\dot{y}}^{i}-H^{(1)}(y)
b_{l}^{n}=\frac{l}{n+l+2k}(-\partial_{a_{l}})(U_{l}^{n})Str\left(\ldots\phi^{j_{l}}\ldots\right)-\frac{n}{n+l+2k}\left[D_{l}^{n}|_{i_{n}\leftrightarrowa_{l+1}}+\ldots+D_{l}^{n}|_{i_{n}\leftrightarrowa_{p}}\right].
\int_{-\infty}^{+\infty}dx^{1}{\calH}_{c}=H_{c}=\frac{1}{2}\int_{-\infty}^{+\infty}dx^{1}\left[\partial_{1}\phi^{1}\partial_{1}\phi^{1}+\partial_{1}\phi^{2}\partial_{1}\phi^{2}\right],
\begin{array}{c}{\displaystyle{Res\,\zeta_{\nu}(s,x)|_{s=1}=\frac{1}{\pi}\,,}}\\{\displaystyle{Res\,\zeta_{\nu}(s,x)|_{s=0}=0\,,}}\\{\displaystyle{Res\,\zeta_{\nu}(s,x)|_{s=-1}=\frac{1-4\,{\nu}^{2}}{8\,\pi}\,,}}\\\end{array}
p_{1}^{\mu}\langleJ_{\mu}^{a}(p_{1})J_{\nu}^{b}(p_{2})J_{\lambda}^{c}(p_{3})\rangle=-\frac{i}{2^{8}3\pi^{2}}(N_{c}^{2}-1)d^{abc}\epsilon^{\nu\lambda\rho\sigma}p_{2\rho}p_{3\sigma}(2\pi)^{4}\delta^{(4)}(p_{1}+p_{2}+p_{3})
|\Psi({\bfP})\rangle=\sum_{n=1}^{N_{\mathrm{TD}}}\frac{1}{\sqrt{n!}}\int\left[\prod_{i=0}^{n}d^{2}k_{i}\right]\delta^{2}({\bfP}-\sum_{i=1}^{n}{\bfk}_{i})\psi_{n}({\bfk}_{1},{\bfk}_{2},\dots,{\bfk}_{n})\prod_{i=0}^{n}a^{\dagger}({\bfk}_{i})|0\rangle,
\begin{aligned}{S_{lc}^{F}=\frac{i}{4\pil_{s}^{2}p^{+}}\intd\tau\int_{0}^{2\pil_{s}^{2}p^{+}}d\sigma\left[S^{1}\left(\partial_{+}-\frac{\mu}{2}\gamma^{12}-\frac{\mu}{2}\gamma^{34}\right)S^{1}+\right.}\\{\left.+S^{2}\left(\partial_{-}-\frac{\mu}{2}\gamma^{12}-\frac{\mu}{2}\gamma^{34}\right)S^{2}--2\muS^{1}\gamma^{1234}S^{2}\right].}\\\end{aligned}
\mathcal{O}_{+}^{in}(\Omega)=\sqrt{\frac{\mu}{2}}\sum_{Lj}Y_{Lj}^{in*}(\Omega)\mathcal{O}_{+Lj}^{in}\,\Psi^{in}(\Omega)=\sum_{Lj}f_{Lj}Y_{Lj}^{in}(\Omega)\.
\begin{matrix}{\bar{C}_{zz}^{\prime\prime}=-b_{zz},}&{C^{+\prime}=c^{z},}\\{\bar{\Gamma}_{+z}^{\prime\prime}=-i\beta_{+z},}&{\Gamma^{+\prime}=-i\gamma^{+},}\\{\bar{\Gamma}_{-z}^{\prime\prime}=\beta_{-z},}&{\Gamma^{-\prime}=-\gamma^{-},}\\\end{matrix}
{\frac{i}{4\pi}}\delta_{ab}\epsilon^{\mu\nu\rho}{\frac{(x-y)_{\rho}}{|x-y|^{3}}},\,\,\,\,\,\,\,\,-igf_{abc}\epsilon_{\mu\nu\rho}\intd^{3}x,\,\,\,\,\,\,\,\,g(T^{a})_{i}^{j}.
\frac{\partial}{\partialu}\:{\bfC}_{vac}=-\frac{\partial}{\partialu}\:\Bigl\{4\pir\:m_{\beta}m_{\nu}\frac{1}{\Box}\Bigl(\:T_{vac}^{\nu\beta}+\:O[R_{..}^{2}]\Bigr)\Bigr\}\biggl|_{{\calI}^{+}}
\begin{array}{rcl}{{\calH}^{(3)\prime}}&{=}&{-{\calK}^{(3)}\,,}\\{}&{}&{}\\{{\tilde{\calF}}^{(6)\prime}}&{=}&{{{\calK}}^{(6)}\,,}\\{}&{}&{}\\{{\calG}^{(1)\prime}}&{=}&{-\left({\tilde{\calK}}^{(1)}+C^{(0)}{\calK}^{(1)}\right)\,,}\\\end{array}
G_{N}(x)=\left|\begin{matrix}{x-\phi_{1}}&{-y_{1}}&{0}&{.}&{.}&{.}&{0}&{0}\\{-1}&{x-\phi_{2}}&{-y_{2}}&{0}&{.}&{.}&{.}&{0}\\{0}&{-1}&{x-\phi_{3}}&{-y_{3}}&{0}&{.}&{.}&{0}\\{.}&{.}&{.}&{.}&{.}&{.}&{.}&{.}\\{.}&{.}&{.}&{.}&{.}&{-1}&{x-\phi_{N-2}}&{-y_{N-1}}\\{0}&{0}&{.}&{.}&{.}&{.}&{-1}&{x-\phi_{N-1}}\\\end{matrix}\right|.
\eta_{+}^{A}=\epsilon_{\pi}(y)e^{-\frac{1}{2}k|y|}\left(\begin{array}{c}{\hat{\eta}_{R}}\\{-\hat{\eta}_{L}}\\\end{array}\right)^{A}\,\qquad\eta_{-}^{A}=\epsilon_{0}(y)e^{-\frac{1}{2}k|y|}\left(\begin{array}{c}{\hat{\eta}_{L}}\\{\hat{\eta}_{R}}\\\end{array}\right)^{A}\,
\begin{aligned}{\frac{d}{dt}\left(Q_{s}+\overline{{Q}}_{s}\right)=\}&{{}\frac{d}{dt}\int_{-\infty}^{0}dx\left(T_{s+1}+\Theta_{s-1}+\overline{{T}}_{s+1}+\overline{{\Theta}}_{s-1}\right)=}\\{=\}&{{}\left(\overline{{T}}_{s+1}+\overline{{\Theta}}_{s-1}-T_{s+1}-\Theta_{s-1}\right)_{x=0}}\\\end{aligned}
S^{(m)}[u(x,t)]=\int_{t_{-}}^{t_{+}}\,\int_{x_{-}}^{x_{+}}\,{\mathcal{P}}^{(m)}[u](x,t)\,\left(u_{t}+\frac{m}{m+3/2}\,(u_{xxx}+12\,u\,u_{x}))\right)\,dx\,dt\,,
A^{(2)}=\left(\begin{array}{ccc}{A^{(2)ij,kl}}&{A^{(2)ij,0}}&{A^{(2)ij,b}}\\{A^{(2)0,kl}}&{A^{(2)00}}&{A^{(2)0b}}\\{A^{(2)a,kl}}&{A^{(2)a0}}&{A^{(2)ab}}\\\end{array}\right)
\vec{a}\,^{\prime}(0)=\left(\begin{array}{c}{cos\,\alpha}\\{sin\,\alpha}\\{0}\\\end{array}\right)\;\;;\;\;\vec{a}\,^{\prime}(2\pi)=\left(\begin{array}{c}{cos\,\alpha}\\{-sin\,\alpha}\\{0}\\\end{array}\right),
\frac{i}{2}\int_{x_{0}}^{x}dx_{j}\,\epsilon^{ml}A_{m}^{(0)}\,\partial_{l}A_{j}^{(0)}=\frac{i}{2}\int_{x_{0}}^{x}dx_{j}\,({\bfA}^{(0)}\times\nablaA_{j}^{(0)})_{3}.\nonumber
{\calA}_{33}=\frac{3g^{2}}{4}\sum_{\pi(p_{1},\ldotsp_{6})}\bar{u}(p_{\pi_{1}})\gamma_{\mu}u(p_{\pi_{4}})\bar{u}(p_{\pi_{2}})\gamma^{\mu}\frac{i}{p_{\pi_{1}}\!\!\!\!\!\!\!/+p_{\pi_{2}}\!\!\!\!\!\!\!/-p_{\pi_{4}}\!\!\!\!\!\!\!/-m}\gamma_{\nu}u(p_{\pi_{5}})\bar{u}(p_{\pi_{3}})\gamma^{\nu}u(p_{\pi_{6}})
\Lambda(a_{1})={\frac{1}{2}}\int_{ST_{2}}d\tau_{2}\partial_{1}{\hat{\zeta}_{1,2}}{\hat{\mu}}(a_{2})-{\frac{1}{4}}\int_{ST_{2}}d\tau_{23}[({\hat{\zeta}_{1,2}}\partial_{2}-D_{1}{\hat{\zeta}_{1,2}}D_{2})\partial_{2}{\hat{\zeta}_{2,3}}]{\hat{\mu}}(a_{2}){\hat{\mu}}(a_{3}),
\phi(g_{1},\ldots,g_{4})=\sum_{N_{1}\ldotsN_{4}}\sum_{\Lambda}\left(\phi_{\alpha_{1}\ldots\alpha_{4}}^{N_{1}\ldotsN_{4}}\C_{\alpha_{1}\ldots\alpha_{4}}^{N_{1}\ldotsN_{4}\,\Lambda}\right)\left({C}_{\beta_{1}\ldots\beta_{4}}^{N_{1}\ldotsN_{4}\,\Lambda}D_{\beta_{1}\gamma_{1}}^{(N_{1})}(g_{1})\ldotsD_{\beta_{4}\gamma_{4}}^{(N_{4})}(g_{4})\w_{\gamma_{1}}\ldotsw_{\gamma_{4}}\\right),
\tilde{f}(q,p,\phi)\equivf(\tilde{q},\tilde{p})=f(q,p)+\Sigma_{n=1}^{\infty}\tilde{f}(q,p,\phi)^{(n)}~~;~~\tilde{f}^{(1)}=-\phi_{c}^{\beta}\omega_{\beta\gamma}^{cb}X_{bd}^{\gamma\delta}\{\theta_{\delta}^{a},f\}_{(q,p)}
{\calB}_{WZWC}^{(\varphi)}=\frac{\ell}{16\piG}\int_{\calV}Tr[\partial_{\varphi}(Q_{2}^{-1}\partial_{r}Q_{2}\,Q_{2}^{-1}\partial_{t}Q_{2}-Q_{1}^{-1}\partial_{r}Q_{1}\,Q_{1}^{-1}\partial_{t}Q_{1})]\,dr\,d\varphi\,dt
{\calL}_{\mathrm{BI}}=-\sqrt{-\mathrm{det}\,\big(\eta_{\mu\nu}+F_{\mu\nu}+\bar{\chi}\Gamma_{\mu}\partial_{\nu}\chi+{\textstyle{\frac{1}{4}}}\bar{\chi}\Gamma^{a}\partial_{\mu}\chi\bar{\chi}\Gamma_{a}\partial_{\nu}\chi\big)}\,\,.
L_{D}^{\ast}=\int_{\calM}d^{3}x\varepsilon^{ijk}\bigg\{-{\frac{3}{4a}}(a\partial_{i}B_{jAB}+bG_{ijAB})[{\bfR}^{-1}]_{kn}^{ABCD}\varepsilon^{lmn}(a\partial_{l}B_{mCD}+bG_{lmCD})+c\alpha_{i}^{AB}G_{jkAB}\bigg\},
\Gamma(\sigma,\pi)=-i\sum_{j=1}^{N}\mathrm{~Tr~Ln}\left[i\gamma^{\mu}D_{\mu}-(\sigma_{j}+i\gamma_{5}\pi_{j})\right]-\frac{1}{2G}\intd^{4}x(\sigma_{j}^{2}+\pi_{j}^{2}).
\langleT_{\\mu}^{\mu}\rangle=\frac{1}{48\pi}\Biggl[\left(\partial_{\mu}\partial_{\nu}h^{\mu\nu}-\partial_{\mu}\partial^{\mu}h_{\\nu}^{\nu}\right)\mp\frac{1}{2}\varepsilon_{\mu\lambda}\partial^{\lambda}\partial_{\nu}h^{\mu\nu}\Biggr]\,.
\delta_{f}L=\frac{d}{dt}\left[\frac{\kappa}{2\pi}\int_{D}d^{2}x\epsilon^{ij}A_{i}^{a}D_{j}(f^{k}A_{k})^{a}\right]+\frac{\kappa}{2\pi}\oint_{\partial{D}}d\varphif^{i}A_{i}^{a}{\dot{A}}_{\varphi}^{a}.
\begin{array}{l}{\Phi_{2}=(\phi_{2}^{+})^{-1}\{D_{2}\}_{p,q},}\\{\bar{\Phi}_{2}=\phi_{2}^{+},}\\{\Phi_{1}=(\phi_{1}^{+})^{-1}\{D_{1}\}_{p,q}p^{D_{2}},}\\{\bar{\Phi}_{1}=\phi_{1}^{+}q^{D_{2}}.}\\\end{array}
\frac{1}{8\piG_{d}}\int_{M}d^{d}x\sqrt{\hat{g}}e^{-a\Phi}\bar{F}^{2}=\frac{1}{4\piG_{d}}\int_{\partialM}d^{d-1}x\sqrt{b}e^{-a\Phi}\bar{F}^{\mu\nu}n_{\mu}\bar{A}_{\nu},
{\calA}=2\pi\gammai\sum_{r=1}^{N}\left(2R^{2}\frac{\bar{z}_{r}}{1+|z_{r}|^{2}}+\tilde{b}_{r}\right)dz_{r}-2\pi\gammaid\left(\frac{1}{2}{\calB}+R^{2}\sum_{r=1}^{N}\mathrm{log}\left(1+|z_{r}|^{2}\right)\right),
\frac{{\calH}_{+}^{2}}{{\calH}_{+}^{\prime}-{\calH}_{+}^{2}}D_{+}=\frac{{\calH}_{-}^{2}}{{\calH}_{-}^{\prime}-{\calH}_{-}^{2}}D_{-}+\frac{S_{-}{\calH}_{-}-S_{+}{\calH}_{+}}{a^{2}}\,.
\!\!\!{\calE}_{n}^{(1)}\propto\!\!\!\int_{\infty}^{\infty}\Pi_{j=1}^{n}dz_{j}(\chi_{n}^{(0)})^{*}(\sum_{i<j}^{n}\delta(z_{i}-z_{j})\partial_{z_{i}})\chi_{n}^{(0)}\propto\!\!\!\int_{\infty}^{\infty}\Pi_{j=1}^{n}dz_{j}(\chi_{n}^{(0)})^{*}\chi_{n}^{(0)}(\sum_{i\neqk}^{n}\mathrm{sign}(z_{i}-z_{k}))=0,
\{(Q_{\alpha}^{+\,I},Q_{\beta}^{+\,J}\}=\delta^{IJ}(C\Gamma^{\mu}{\calP}^{+})_{\alpha\beta}P_{\mu}+\tilde{S}_{IJ}(C\Gamma_{\mu}{\calP}^{+})_{\alpha\beta}Z^{\mu}
\sum_{i}\gamma_{i}^{\left(D\right)}\left(a,b,c\right)\,\alpha_{il}\left(a,b,c\right)\alpha_{il^{\prime}}\left(a,c,b\right)=\left(-1\right)^{\left(m-2\right)\left(l-1\right)}\gamma_{l}^{\left(D\right)}\left(b,a,c\right)\delta_{ll^{\prime}}.
Z(\epsilon,t)=-{\frac{1}{2}}\sum_{s=2,3,4}(-)^{s}\;{\frac{\theta_{s}\left({\frac{\epsilont}{2}}\left|\right.{\frac{it}{2}}\right)}{\theta_{1}\left({\frac{\epsilont}{2}}\left|\right.{\frac{it}{2}}\right)}}\;{\frac{\theta_{s}^{3}\left(0\left.\right|{\frac{it}{2}}\right)}{\eta^{9}\left({\frac{it}{2}}\right)}}\.
{\calL}_{0}=s\Psi_{0}=\sum_{i}\bar{\theta}_{i}C^{a}(x_{i})+\sum_{i}\bar{\gamma}_{i}\left(\phi^{a}(x_{i})+{\frac{1}{2}}f_{\phantom{a}bc}^{a}C^{b}(x_{i})C^{c}(x_{i})\right).
S=S^{(0)}\,+\,S^{(1)}=-\frac{1}{2\lambda_{s}^{4}}\int\,d^{6}x\,\sqrt{-g}\,e^{-\phi}\,\biggl[R\,+\,g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi\,+\,\Lambda-\epsilon\,R_{\mu\nu\alpha\beta}\,R^{\mu\nu\alpha\beta}\biggr],
\lambda={\frac{iq_{1}A_{q}^{1/2}-q_{2}}{iq_{2}A_{q}^{1/2}+q_{1}}}={\frac{q_{1}q_{2}(A_{q}-1)+i(q_{1}^{2}+q_{2}^{2})A_{q}^{1/2}}{q_{1}^{2}+q_{2}^{2}A_{q}}}\.
2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4}=-u_{1}+u_{2}\rightarrowB_{2}^{1}=\left[B_{\bar{1}}^{0},B_{2}^{0}\right]=D_{10}^{\overline{{11}}}+D_{12}^{\bar{9}}-D_{13}^{\bar{7}}
B\left(x\right)=\frac{\xi_{1}^{2}\kappa+\eta}{\xi_{1}}\left(C_{1}e^{-x\xi_{1}}-C_{2}e^{x\xi_{1}}\right)+\frac{\xi_{2}^{2}\kappa+\eta}{\xi_{2}}\left(C_{3}e^{-x\xi_{2}}-C_{4}e^{x\xi_{2}}\right)+C_{5}
a_{n>}^{i}=A_{n}a_{n<}^{i}-B_{n}^{*}\tilde{a}_{n<}^{i^{\dagger}},\qquad\tilde{a}_{n>}^{i}=A_{n}\tilde{a}_{n<}^{i}-B_{n}^{*}a_{n<}^{i^{\dagger}}.
\langleR_{\alpha_{1}}^{\epsilon_{1}}(z_{1})R_{\alpha_{2}}^{\epsilon_{2}}(z_{2})V_{\alpha_{3}}(Z_{3})\rangle=\left(C^{\epsilon_{1},\epsilon_{2}}+\theta_{3}\bar{\theta}_{3}\frac{|z_{1}-z_{3}||z_{2}-z_{3}|}{|z_{1}-z_{2}|}\tilde{C}^{\epsilon_{1},\epsilon_{2}}\right)\prod_{i<j}|z_{i}-z_{j}|^{\delta_{ij}}
\mathcal{A}_{4}^{cl}=(2\pi)^{2}16g_{c}^{2}(2\alpha^{\prime})^{(d-4)/2}B\left(-\frac{1}{2}\alpha\left(s\right),-\frac{1}{2}\alpha\left(t\right),\frac{1}{2}\alpha\left(s\right)+\frac{1}{2}\alpha\left(t\right)+1\right).
\check{Q}_{ib}(\tau)\check{\vec{A}}_{a\perp}(\tau,{\vec{\eta}}_{i}(\tau))=Q_{ia}(\tau)[P\,e^{\Omega_{sa}^{(\hat{\gamma})}(\eta^{(A)}(\tau,-{\vec{\eta}}_{i}(\tau))){\hat{T}}^{a}}]_{ab}\check{\vec{A}}_{b\perp}(\tau,{\vec{\eta}}_{i}(\tau)),
\begin{array}{l}{R(\theta-\theta^{\prime})[1\otimesK^{+}(\theta)]R(\theta+\theta^{\prime})[1\otimesK^{+}(\theta^{\prime})]=}\\{{[}1\otimesK^{+}(\theta^{\prime})]R(\theta+\theta^{\prime})[1\otimesK^{+}(\theta)]R(\theta-\theta^{\prime})\;\;,}\\\end{array}
K(X^{1},X^{2},X^{3})=g^{2}e^{x^{A}K^{AB}y^{B}},\\\\\K^{AB}=g^{-1}\left(\begin{matrix}{0}&{1}&{-1}\\{-1}&{0}&{1}\\{1}&{-1}&{0}\\\end{matrix}\right)^{AB},