harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{\delta\phi}{2\pi}=4\piG\int_{0}^{\infty}dr\frac{L}{1-{\frac{\gamma}{6}}f^{2}}\left(\varepsilon_{s}+3\varepsilon_{v}+\varepsilon_{sv}+\frac{\lambdav^{4}}{4}(1-f^{2})\left(1+(1-{\frac{\gamma}{3}})f^{2}\right)+v^{2}\frac{N^{\prime}}{N}ff^{\prime}\right)
\left\langle\Theta_{ii}(x)\right\rangle=\frac{1}{3}\sum_{n}\frac{1}{\omega_{n}}\left|\partial_{i}\phi_{n}\right|^{2}+\frac{1}{12}\sum_{n}\omega_{n}\left|\phi_{n}\right|^{2}-\frac{1}{12}\sum_{n}\frac{1}{\omega_{n}}\left|\vec{\nabla}\phi_{n}\right|^{2}-\frac{1}{12}\sum_{n}\frac{1}{\omega_{n}}\left[\phi_{n}\partial_{i}^{2}\phi_{n}^{*}+\left(\partial_{i}^{2}\phi_{n}\right)\phi_{n}^{*}\right]
=-i\frac{3\hbarc}{2}\intd^{3}\vec{r}[\epsilon_{0}(\varphi^{\dagger}\varphi+\chi^{\dagger}\chi)+\varphi^{\dagger}(\vec{\sigma}\cdot\vec{\epsilon})\chi+\chi^{\dagger}(\vec{\sigma}\cdot\vec{\epsilon})\varphi]\,{.}
\tilde{x}_{s}^{\mu}\stackrel{(a,\Lambda)}\longrightarrow\tilde{x}_{s}^{\prime\mu}=\Lambda_{.\nu}^{\mu}\Big[\tilde{x}_{s}^{\nu}+\frac12\bar{S}_{rs}R_{.k}^{r}(\Lambda,p_{s})\frac{\partial}{\partial{p_{s}}_{\nu}}R_{.k}^{s}(\Lambda,p)\Big]+a^{\mu}.
\kappa={\frac{1}{\mathrm{cosh}\delta_{1}\mathrm{cosh}\delta_{2}}}{\frac{\partial_{r}(\Pi-2N)}{4N}}|_{r=r_{+}},\\\\Omega_{i}={\frac{1}{\mathrm{cosh}\delta_{1}\mathrm{cosh}\delta_{2}}}{\frac{l_{i}}{r_{+}^{2}+l_{i}^{2}}},
\begin{array}{cl}{}&{G^{\overbrace{n\cdotsn}^{m}\cdots\hat{i}\cdots\hat{i}\cdots\overbrace{1\cdots1}^{m}\overbrace{i\cdotsi}^{m}}(z^{(n)}|\cdots|z^{(1)},z_{0}\tau^{2n})}\\{=}&{\delta_{i}~G^{i\overbrace{n\cdotsn}^{m}\cdots\hat{i}\cdots\hat{i}\cdots\overbrace{1\cdots1}^{m}\overbrace{i\cdotsi}^{m-1}}(z_{0}|z^{(n)}|\cdots|z^{(1)}),}\\\end{array}
\mathrm{~e}^{-S_{\mathrm{eff}}[A_{0}]}=\sum_{B_{M}}\int[d\vec{a}]\mathrm{~e}^{-\int_{0}^{1/T}d\tau\int_{S^{2}}\left(\frac{1}{2}\dot{\vec{a}}^{2}+\frac{1}{2}(B_{M}^{2}+b^{2})+i\frac{\kappa}{2}\vec{a}\times\dot{\vec{a}}\right)}\cdot\mathrm{~e}^{i\frac{\kappa}{T}\int_{S^{2}}A_{0}(B_{M}+b/2)}
\hat{\phi}(x^{0},x^{1})=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}dk^{1}\frac{1}{\sqrt{k^{1}}}\left[\hat{\Lambda}(k^{1})\,e^{-ik^{1}(x^{0}+x^{1})}\,\,+\,\,\hat{\Lambda}^{\dagger}(k^{1})\,e^{ik^{1}(x^{0}+x^{1})}\right],
\;=\;\frac{e^{2}}{(2\pi)^{2}}\;\mathrm{tr}(I)\,\oint_{{\mathcalC}(\varepsilon)}\,dl\,{\hat{n}}_{\mu}\left[\frac{A(p)p_{\mu}}{A^{2}(p)p^{2}+B^{2}}\right]
L=\intd^{4}x\varepsilon^{\mu\nu\rho\sigma}\bigg(C_{1}{^+}{\calG}_{\mu\nu}^{A}{^+}{\calG}_{\rho\sigma}^{B}+C_{2}{^-}{\calG}_{\mu\nu}^{A}{^-}{\calG}_{\rho\sigma}^{B}\,+C_{3}{^+}{\calF}_{\mu\nu}^{A}{^+}{\calG}_{\rho\sigma}^{B}+C_{4}{^-}{\calF}_{\mu\nu}^{A}{^-}{\calG}_{\rho\sigma}^{B}\bigg)M_{AB}
M_{-\nu}^{(1)}(0,h)=-\bigg(\frac{2+i\sqrt{5}}{3}\bigg)\bigg[\frac{1}{2}(h/2)^{2}.1-\bigg(\frac{3}{2+i\sqrt{5}}\bigg).1.\frac{(h/2)^{-\nu+4}}{(-\nu+4)!}\bigg]
A_{\perp}^{ic}(x)=\sum_{\lambda}\int_{k^{+}>0}{\frac{dk^{+}d^{2}k_{\perp}}{16\pi^{3}k^{+}}}\Bigl[a^{c}(k,\lambda)\epsilon_{\perp}^{i}(\lambda)e^{-ik\cdotx}+a^{c\dagger}(k,\lambda)\epsilon_{\perp}^{i*}(\lambda)e^{ik\cdotx}\Bigr]\;.
H=\frac{1}{2}\Pi_{a}^{i}\Pi_{a}^{i}+\frac{1}{4}f_{abc}X_{i}^{b}X_{j}^{c}f_{ade}X_{i}^{d}X_{j}^{e}-\frac{i}{2}f_{abc}X_{a}^{j}\psi_{b}^{\alpha}(\sigma_{j})_{\alpha\beta}\psi_{c}^{\beta}-G_{a}X_{0}^{a}
a_{2}=[a_{0},a_{1}]+b_{0}b_{1}^{t}-b_{1}b_{0}^{t}=2g_{0}g_{1}J^{3}+2\epsilon^{xyz}\bar{a}_{0}^{x}\bar{a}_{1}^{y}\bar{J}^{z}+\frac{1}{2}\vec{\beta}_{0}\cdot\vec{\beta}_{1}(J^{3}+\bar{J}^{3})=0
\Sigma_{2}=-g^{4}N^{2}\frac{\Gamma^{2}(\omega-1)}{2^{7}\pi^{2\omega}(2-\omega)(2\omega-3)}\ointd\tau_{1}\,d\tau_{2}\frac{\left|{\dot{x}}^{(1)}\right|\left|{\dot{x}}^{(2)}\right|-{\dot{x}}^{(1)}\cdot{\dot{x}}^{(2)}}{\bigl[(x^{(1)}-x^{(2)})^{2}\bigr]^{2\omega-3}},
\delta_{B}(f^{(0)}u^{4})+\delta_{F}(f^{(2)}u^{3}\left[\psi^{2}\right])+u^{i}u^{2}\epsilon\dot{N}^{i(0)}\psi+\frac{i}{2}u^{3}\psi\dot{M}^{(0)}\epsilon=0~,
m^{2}\leq{\displaystyle\frac{l^{2}}{27}}\,{\displaystyle\frac{16-24\,e^{2}b-16b\sqrt{1-e^{2}b}\,e^{2}+6b^{2}\,e^{4}+16\,\sqrt{1-e^{2}b}}{e^{6}}}
\mathrm{Im}\int_{r_{o}}^{r_{f}}P_{-}dr=-\pi\int_{0}^{-\omega}{\frac{d{\calH}-\PhidQ}{\kappa(M+{\calH},Q-e)}}~=~-\frac{1}{2}\int_{M,Q}^{M-\omega,Q-e}dS_{BH},
0=\nabla_{(\alpha}\overline{{\xi}}_{\overline{{\beta}})a}=\partial_{(\alpha}\overline{{\xi}}_{\overline{{\beta}})a}-H_{\alpha\overline{{\beta}}}^{\,\,\,\overline{{\gamma}}}\overline{{\xi}}_{\overline{{\gamma}}a}+H_{\alpha\overline{{\beta}}}^{\,\,\,\gamma}\xi_{\gamma}.
{\widehat\gamma}^{1}=\left(\begin{array}{cc}{{\widehatE}_{2}}&{0}\\{0}&{-{\widehatE}_{2}}\\\end{array}\right),{\widehat\gamma}^{2}=\left(\begin{array}{cc}{0}&{\sigma_{x}}\\{-\sigma_{x}}&{0}\\\end{array}\right),
\gamma=\sqrt{\frac{-c_{gh}-c_{M}-r+24(|\rho||\bar{\rho}|-\rho\cdot\bar{\rho})}{48|\bar{\rho}|^{2}}}-\sqrt{\frac{-c_{gh}-c_{M}-r-24(|\rho||\bar{\rho}|+\rho\cdot\bar{\rho})}{48|\bar{\rho}|^{2}}}~,
\left(\begin{matrix}{1}&{0}&{-\sqrt{2}\xi}&{\sqrt{2}\xi}\\{0}&{1}&{-\sqrt{2}\eta}&{\sqrt{2}\eta}\\{\sqrt{2}\xi}&{\sqrt{2}\eta}&{1-(\xi^{2}+\eta^{2})}&{(\xi^{2}+\eta^{2})}\\{\sqrt{2}\xi}&{\sqrt{2}\eta}&{-(\xi^{2}+\eta^{2})}&{1+(\xi^{2}+\eta^{2})}\\\end{matrix}\right).
M^{jk}({\boldsymbol{p}})\equiv\delta^{jk}+i\widetilde{f}^{j*}({\boldsymbol{p}})p^{k}-ip^{j}\widetilde{f}^{k}({\boldsymbol{p}})+{\boldsymbol{p}}^{2}\widetilde{f}^{j*}({\boldsymbol{p}})\widetilde{f}^{k}({\boldsymbol{p}}).
H_{T}=\intdx^{1}\mathrm{Tr}\left\{-\omega_{0}(D_{1}\pi_{\omega_{1}}-[\pi_{\phi},\phi])-(\phi^{2}+\{\pi_{\omega_{1}},\phi\})B+\lambda_{\omega_{0}}\pi_{\omega_{0}}+\lambda_{B}\pi_{B}\right\}.
(s^{2}t^{2})_{\mathrm{fix}}={\frac{|p_{2}|}{|q_{1}|}}\,\qquad(u^{2}t^{2})_{\mathrm{fix}}={\frac{|q_{3}|}{|q_{1}|}}\,\qquad(s^{2}u^{2})_{\mathrm{fix}}={\frac{|p_{4}|}{|q_{1}|}}\,
\left(\frac{1}{2}f^{\prime}\,+\,e^{f}\,\frac{1}{12\sqrt{2}}\bar{g}\left(2e^{-\varphi}+e^{2\varphi}\right)\right)\gamma_{0}\,\varepsilon_{a}\,=\,-\frac{1}{6}\sqrt{\frac{1}{2}}e^{f}e^{\varphi}\frac{1}{2\bar{g}}\gamma^{\nu\rho}R_{\nu\rho}^{\;\;\;ij}\gamma_{ij}\gamma_{0}\,\varepsilon_{a}.
h_{\mu\nu}(z,t)={\frac{16\pi}{2}}{\frac{(z-\betat-\delta/2)^{2}}{(1-\beta^{2})}}\;{\frac{k\ell_{P}^{2}}{\delta^{4}}}\;\left(\eta_{\mu\nu}-4n_{\mu}n_{\nu}\right).
\frac{d}{dt}{\calF}_{k}(t)=-\frac{1}{t^{2}}\int_{{\calM}_{k}^{\prime}}d_{x}\left(\omegae^{-\frac{1}{t}d_{x}\omega}\right)=\,=-\frac{1}{t^{2}}\int_{{\calM}_{k}^{\prime}}d\left(\omegae^{-\frac{1}{t}d_{x}\omega}\right).
\delta(\varepsilon)\theta_{\alpha\beta\gamma}^{++\,\alpha^{\prime}\beta^{\prime}\gamma^{\prime}}=\left(P_{\theta}^{+}EP_{\theta}^{-}\right)_{\alpha\beta\gamma}^{\,\,\,\,\,\,\,\,\,\,\,\,\alpha^{\prime}\beta^{\prime}\gamma^{\prime}}+\left(P_{\theta}^{+}EP_{\theta}^{-}\right)_{\,\,\,\,\,\,\,\,\quad\alpha\beta\gamma}^{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}}
ds^{2}\,=\,{\frac{\Lambda^{2}}{z^{\prime\,2}}}\,\Big\lbrack\,\,{\frac{\delta^{2}}{(z^{\prime}-\delta)^{2}}}dz^{\prime\,2}\,+\,\,{\frac{(z^{\prime}-\delta)^{2}}{\delta^{2}}}\Big((d\vec{x})^{2}\,-dt^{2}\,\Big)\,\Big\rbrack
\{d(p,s),d^{\dagger}(p^{\prime},s^{\prime})\}=\delta(p^{+}-p^{\prime+)}\delta^{2}(p_{\perp}-p_{\perp}^{\prime})\delta_{s,s^{\prime}}=\delta^{3}(p-p^{\prime})\delta_{s,s^{\prime}}\;,
F_{\tauq}={\calE}_{\tau}^{\underline{\alpha}}{\calE}_{q}^{\underline{\beta}}F_{\underline{{\alpha}}\underline{\beta}}+{\frac{1}{D-2}}\bar{\calE}_{r}\Gamma^{\underline{a}}{\calE}_{r}{\calE}_{q}^{\underline{\beta}}F_{\underline{a}\underline{{\beta}}}.
\left(\begin{matrix}{0}&{g_{2}(2a+2b+1)}&{-2ag_{3}}&{4g_{3}}&{0}&{0}\\{16a+24b+20}&{0}&{g_{2}(b+1/2)}&{4g_{2}(a+b+1)}&{2g_{3}(1-a)}&{0}\\{0}&{8a+24b+12}&{0}&{0}&{g_{2}(2a+2b+5)}&{-4g_{3}(a+1)}\\{0}&{8a+12b+14}&{0}&{0}&{g_{2}(b+1/2)}&{2g_{3}}\\{0}&{0}&{8a+12b+14}&{16(a+3b+3)}&{0}&{g_{2}(2b+3)}\\{0}&{0}&{0}&{0}&{8a+24b+28}&{0}\\\end{matrix}\right).
{\mathcalP}_{n}\left(\partial_{0}\,{^{[0]}W}\right)=-\left[\left(\partial_{0}{\mathcalP}_{n}\right)+\sum_{k=0}^{n}{\mathcalQ}_{(k)}^{n}{\mathcalP}_{k}\right]\,{^{[0]}W}\,,\quad,n\geq1.
F_{\mu\nu}=\left(\begin{array}{cccc}{0}&{E_{1}}&{E_{2}}&{E_{3}}\\{-E_{1}}&{0}&{-B_{3}}&{B_{2}}\\{-E_{2}}&{B_{3}}&{0}&{-B_{1}}\\{-E_{3}}&{-B_{2}}&{B_{1}}&{0}\\\end{array}\right),\quadJ_{\mu}=\left(\rho,-j_{1},-j_{2},-j_{3}\right).
v_{12}^{2}v_{13}^{2}v_{23}^{2}\sim(\dot{X}_{i}^{3})^{2}(\dot{X}_{i}^{8})^{4}+(\dot{X}_{i}^{3})^{6}+(\dot{X}_{i}^{3})^{4}(\dot{X}_{i}^{8})^{2}-(\dot{X}_{i}^{3}\dot{X}_{i}^{8})^{2}(\dot{X}_{i}^{3})^{2}.
Z=\sum_{N=0}^{\infty}{\frac{1}{N!}}~\bar{J}^{N}\sum_{\{q_{a}\}}\prod_{i=1}^{N}\intd\vec{x}_{i}~\mathrm{exp}\left(-{\frac{2\pi}{g^{2}}}\sum{\frac{q_{a}q_{b}}{\vertx_{a}-x_{b}\vert}}\right)
\langleT_{00}\rangle_{{\mathrm{reg}}}^{{\mathrm{cyl}}}=\frac{4\xi-1}{2^{D}\pi^{(D+1)/2}a^{D+1}\Gamma((D-1)/2)}\sum_{n=-\infty}^{+\infty}\int_{0}^{\infty}dz\,z^{D}
\Phi=\left(\begin{array}{cccc}{-\frac{5}{3}i\Phi_{11}(\frac{5}{3}r)+\frac{1}{3}i}&{}&{}&{}\\{}&{-\frac{5}{3}i\Phi_{22}(\frac{5}{3}r)+\frac{1}{3}i}&{}&{}\\{}&{}&{-\frac{1}{2}iI_{2}+\sum_{j=1}^{3}25f(5r)x_{j}\tau_{j}}&{}\\{}&{}&{}&{-\frac{5}{3}i\Phi_{33}(\frac{5}{3}r)+\frac{1}{3}i}\\\end{array}\right),
\big\{{\widehat\Lambda}^{\alphaA}\,,\,{\widehat\Lambda}_{\alpha}^{~B}\big\}_{~v}^{u}={\frac{4\rho^{2}}{(x^{2}+\rho^{2})^{3}}}\,\Big(\mu^{[Au}\bar{\mu}_{~\,v}^{B]}-{\frac{1}{2}}\mu^{[Ap}\,\bar{\mu}_{~\,p}^{B]}\,\tilde{\delta}_{~v}^{u}\Big)
(i)\quadN^{R}=\left(\begin{matrix}{1}&{0}&{0}&{0}\\{0}&{a}&{a-1}&{0}\\{0}&{a-1}&{a}&{0}\\{0}&{0}&{0}&{1}\\\end{matrix}\right),\quadN^{NS}=\left(\begin{matrix}{a}&{0}&{0}&{a-1}\\{0}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}\\{a-1}&{0}&{0}&{a}\\\end{matrix}\right)
\phi={\frac{3\left(2+\sqrt{7}\right)}{10}}\left(1-{\frac{\left(1+2\sqrt{7}\right)}{3}}\epsilonz+\epsilon^{4/3}\phi_{(4/3)}(z,t)+\epsilon^{5/3}\phi_{(5/3)}(z,t)+{\calO}(\epsilon^{2})\right)\,
\displaystyleZ_{\scriptscriptstyleCPI}^{\scriptscriptstyleBFV}=\int{\calD}\Lambda_{\scriptscriptstyleA}{\calD}\xi^{\scriptscriptstyleA}{\calD}\bar{\Gamma}_{\scriptscriptstyleA}{\calD}\Gamma^{\scriptscriptstyleA}e^{i\intd^{4}x\widetilde{\calL}^{\scriptscriptstyleBFV}}
\Gamma^{(1)}[V^{++}]=\frac{\mathrm{i}}{2}\,\mathrm{Tr}_{(2,2)}\mathrm{ln}\stackrel{\frown}{\Box}-\frac{\mathrm{i}}{2}\,\mathrm{Tr}_{(4,0)}\mathrm{ln}\stackrel{\frown}{\Box}
\partial^{\mu}F_{\mu\nu}-m^{2}\,\Bigl(1+\frac{\partial_{t}^{2}}{\nabla^{2}}+\frac{\partial_{t}^{4}}{\nabla^{4}}\Bigr)\,\frac{\partial^{\mu}}{\nabla^{2}}\,F_{\mu\nu}+\frac{1}{\alpha}\,\partial_{\mu}\partial_{\nu}A^{\mu}=0
R^{(ell)}(u)_{i^{\prime}j^{\prime}}^{ij}=\delta_{i+j,i^{\prime}+j^{\prime}\mathrm{mod}n}\frac{\theta^{(i^{\prime}-j^{\prime})}(u+2\eta)\theta^{(0)}(u)}{\theta^{(i^{\prime}-i)}({2\eta})\theta^{(i-j^{\prime})}(u)}.
\begin{array}{ll}{Z_{12}=}&{\frac{1}{s}(\sum_{l=1}^{g(s)}e^{-4\pii(l-1)/ns}+e^{4\pii/n}\sum_{l=g(s)+1}^{s}e^{-4\pii(l-1)/ns})}\\{}&{\times(\sum_{p=0}^{p_{max}}e^{2\piip/n}\sum_{u(p)>j\geql(p)}e^{-2\pii(j-1)/M}),}\\\end{array}
{\calM}={\frac{1}{\lambda_{2}(x)}}\left(\begin{matrix}{1}&{\lambda_{1}(x)}\\{\lambda_{1}(x)}&{\lambda_{1}^{2}+\lambda_{2}^{2}}\\\end{matrix}\right),\qquad{\calM}^{T}={\calM},\qquad{\cal{MLM}}^{T}={\calL},
\begin{array}{c}{\displaystyle{\frac{e^{\displaystyle{-\frac{i}{2}\,\pi\,\left(1+s\right)}}}{{\nu}^{1+s}\,8\,\pi}}\,\left(\frac{8\,{\nu}^{2}}{1-s}+\frac{4\,\nu}{s}+\frac{1-4\,{\nu}^{2}}{1+s}\right)}\\\end{array}
\frac{T_{p+2}}{\kappa}\int_{\mathcal{M}_{p+3}}C_{p+1}\wedgeB\longrightarrow\frac{T_{p+2}}{\kappa}\int_{\mathcal{M}_{p+3}}\bar{C}_{p+1}b\wedge\omega_{2}=\frac{T_{p}}{2\kappa}\int_{\mathcal{M}_{p+1}}\bar{C}_{p+1}\left(1+\frac{\tilde{b}}{2\pi^{2}\alpha^{\prime}}\right)\,.
S=\pi\left\{\frac{(m^{ijk})^{2}}{t(m^{ij})^{2}}+\frac{(m^{ij})^{2}}{(p^{i})^{2}}xt+\left(p^{i}\right)^{2}yt\right\}-\pixyt+2\piiC_{ijk}m^{ijk}\,.
\frac{1}{\sqrt{-g}}\left(\frac{1}{2}\left(g^{ij}{\frac{\deltaS}{\deltag^{ij}}}\right)^{2}-{\frac{\deltaS}{\deltag^{ij}}}{\frac{\deltaS}{\deltag_{ij}}}-\frac{1}{2}G^{IJ}\frac{\deltaS}{\delta\phi^{I}}\frac{\deltaS}{\delta\phi^{I}}\right)=\sqrt{-g}\,{\cal{L}},
S_{R^{4}}^{A}=\intd^{10}x\sqrt{-g}~2\zeta(3)e^{-2\phi}(t_{8}t_{8}+\frac{1}{8}\epsilon_{10}\epsilon_{10})R^{4}+\frac{2\pi^{2}}{3}(t_{8}t_{8}-\frac{1}{8}\epsilon_{10}\epsilon_{10})R^{4}
\mathcal{B}_{\Sigma}\;\Omega_{\nu}^{-1}\;=\intd^{D}x\;\left(\partial_{\nu}\varphi^{i}\frac{\delta\Sigma}{\delta\varphi^{i}}\;+\;\partial_{\nu}\varphi^{i*}\frac{\delta\Sigma}{\delta\varphi^{i*}}\right)=\mathcal{P}_{\nu}\Sigma\;=0\;\;,
\begin{array}{c}{\{\quad,\quad\}=i{\tilde{g}}^{A\bar{B}}\nabla_{A}\wedge{\bar{\nabla}}_{\bar{B}}+g^{A\bar{B}}\frac{\partial}{\partial\theta^{A}}\wedge\frac{\partial}{\partial{\bar{\theta}}^{\bar{B}}}}\\\end{array}
J^{a}(z,\bar{z})=\frac{i}{2}d_{\alpha\beta}^{a}\mathopen:\Phi^{\alpha}(z,\bar{z})\psi^{\beta}(z)\mathclose:,\qquad\bar{J}^{a}(z,\bar{z})=\frac{i}{2}d_{\alpha\beta}^{a}\mathopen:\Phi^{\alpha}(z,\bar{z})\bar{\psi}^{\beta}(\bar{z})\mathclose:,
\Gamma=\left(\begin{array}{cc}{\sqrt{\frac{\theta_{1}}{2\theta_{1}+\theta_{2}+\rho^{2}}}|0,0\rangle}&{0}\\{-\sqrt{\frac{\zeta+\rho^{2}}{2\theta_{1}+\theta_{2}+\rho^{2}}}|1,0\rangle}&{|0,0\rangle}\\{0}&{0}\\{0}&{0}\\\end{array}\right),
S=\frac{1}{2\pi}\intd^{2}\sigma\left[\frac{2}{\alpha^{\prime}}g_{MN}\partialX^{M}\bar{\partial}X^{N}+\frac{1}{\alpha^{\prime}}g_{MN}\psi^{M}\bar{\partial}\psi^{N}+\frac{1}{\alpha^{\prime}}g_{MN}\bar{\psi}^{M}\partial\bar{\psi}^{N}\right].
\deltaS=-\intd\taud^{n}\eta\Lambda^{AB}F_{AB}=-\intd\taud^{n}\eta\Lambda^{AB}({\calE}_{A}^{\underline{\alpha}}{\calE}_{B}^{\underline{\beta}}F_{\underline{{\alpha}}\underline{\beta}}+2{\calE}_{A}^{\underline{a}}{\calE}_{B}^{\underline{\beta}}F_{\underline{a}\underline{{\beta}}}+{\calE}_{A}^{\underline{a}}{\calE}_{B}^{\underline{b}}F_{\underline{a}\underline{b}}).
p\cdot\partial\,f(x,\,p)\,-\,\frac{\,g^{2}}{2}\,\frac{1}{1-g^{2}V^{(2)}/2p^{2}}\left[\partial^{\mu}V^{(2)}-\frac{\,p^{\mu}}{p^{2}}\,p\cdot\partial\,V^{(2)}\right]\frac{\partialf(x,\,p)}{\partialp^{\mu}}=0\,.
{\calG}_{\mathrm{rel}}=\frac{g^{2}\lambda}{8\pi}\sum_{A}\left[d\rho^{2}+\rho^{2}\frac{\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2}}{4}\right]_{A}=\frac{g^{2}\lambda}{8\pi}\sum_{A}\left[d\rho^{2}+\rho^{2}d\Omega_{3}^{2}\right]_{A}=\frac{g^{2}\lambda}{8\pi}\,{\calG}_{\mathrm{flat}}.
X_{3}=\int_{\sqrt{2(e+d)+e^{2}}}^{1+e}dx{\frac{2}{\left[(1+e)^{2}-x^{2}\right]^{1/2}(x^{2}-2e-e^{2})}}\left[{\frac{1}{x^{2}}}+{\frac{(2+2e+e^{2}-x^{2})}{(x^{2}-2e-e^{2})}}\right].
(D_{1},D_{2},D_{3})\!=\!{\frac{c}{2}}{\frac{\sqrt{1+{\frac{4}{c^{2}}}|\phi|^{4}}}{|\phi|^{2}}}\left(xy^{*}+yx^{*},i(xy^{*}-yx^{*}),|x|^{2}-|y|^{2}\right).
\begin{array}{rcl}{[X_{+}^{-},X_{-}^{-}]}&{=}&{0,~~~~~[k_{+},~k_{-}]=0~,}\\{k_{\pm}X_{\pm}^{-}k_{\pm}^{-1}}&{=}&{q_{\pm}^{-1}X_{\pm}^{-},~~~k_{\pm}X_{\mp}^{-}k_{\pm}^{-1}=-X_{\mp}^{-}~.}\\\end{array}
H_{p}=1+\sum_{i}{\frac{Q_{i}}{[|\vec{x}-\vec{x}_{0\,i}|^{2}+4Q|\vec{z}-\vec{z}_{0}|]^{\frac{8-p}{2}}}},\\\H_{NS}={\frac{Q}{|\vec{z}-\vec{z}_{0}|}}.
{\frac{2p^{-}}{\mu}}=\sum_{n}\left[(1+e^{2U_{0}})^{1/2}\,{\frac{q}{J_{V}}}N_{n}^{r,w}|n|+\sqrt{1+(1+e^{2U_{0}})\,{\frac{q^{2}}{J_{V}^{2}}}\,n^{2}+2\,{\frac{q}{J_{V}}}n}\,\,(N_{n}^{+}+N_{-n}^{-})\right].
D(q^{2})=\frac{Z_{R}(q^{2})}{q^{2}}=\frac{m^{-2}}{c_{1}(q^{2}/m^{2})^{\gamma_{1}}+c_{2}(q^{2}/m^{2})^{\gamma_{2}}+L(q^{2}/m^{2})ln(\lambda_{1}q^{2}/m^{2}+\lambda_{2})}.
\begin{array}{l}{\displaystyle(2\pi)^{-3}\intd^{2}\vec{p}_{\bot}\left(g(x)\star_{x}e^{+i\vec{p}_{\bot}\cdot\vec{x}_{\bot}}\right)\left(h(y)\star_{y}e^{-i\vec{p}_{\bot}\cdot\vec{y}_{\bot}}\right)\int\frac{dp_{3}}{2E_{\vec{p}}}e^{+ip_{3}z_{3}^{\prime}}.}\\\end{array}
\gamma^{1}\;\equiv\;\left(\begin{matrix}{0}&{+1}\\{+1}&{0}\\\end{matrix}\right)\,,\quad\gamma^{2}\;\equiv\;\left(\begin{matrix}{0}&{+i}\\{-i}&{0}\\\end{matrix}\right)\,,\quad\gamma_{\mathrm{S}}\;\equiv\;i\gamma^{1}\gamma^{2}=\left(\begin{matrix}{+1}&{0}\\{0}&{-1}\\\end{matrix}\right)\,,
(\deltap_{i}+p_{j}\frac{\partial(\deltaq^{j})}{\partialq^{i}})\dot{q}^{i}+p_{j}\frac{\partial(\deltaq^{j})}{\partialp_{i}}\dot{p}_{i}=\frac{\partialK}{\partialq^{i}}\dot{q}^{i}+\frac{\partialK}{\partialp_{i}}\dot{p}_{i}.
\hat{h}=\frac{\mu}{\mu(q)}\hat{h}^{(0)}+\frac{C_{2}-C_{1}}{\muc^{2}}\frac{\alpha^{2}}{q^{2}}+\frac{\hbar^{2}}{2}\frac{\alpha}{\mu^{2}c^{2}}C_{1}\left(\frac{{\bfq}}{q^{3}}\frac{\partial}{\partial{\bfq}}+4\pi\lambda\delta({\bfq})\right)\,,
0=\left({\int}_{-L}^{L}dx^{-}\left[{\Theta}_{\mu}^{\;\;+}(x)\right]_{x^{+}=-T}^{x^{+}=T}+{\int}_{-T}^{T}dx^{+}\left[{\Theta}_{\mu}^{\;\;-}(x)\right]_{x^{-}=-L}^{x^{-}=L}\right),
L={\frac{\rho}{2f^{2}}}\left[f_{,i}f^{,i}+(\epsilon_{,i}+\psi\chi_{,i})(\epsilon^{,i}+\psi\chi^{,i})\right]+{\frac{\rho}{2f}}\left(\kappa^{2}\psi_{,i}\psi^{,i}+{\frac{1}{\kappa^{2}}}\chi_{,i}\chi^{,i}\right)+{\frac{2}{3}}{\frac{\rho}{\kappa^{2}}}\kappa_{,i}\kappa^{,i}
\tilde{b}_{~[l_{1}\cdotsl_{F+4+k}]}^{~[A_{1}\cdotsA_{F-k}]}=\epsilon_{i_{1}\cdotsi_{2F+8}}x_{i_{1}}^{A_{1}}\cdotsx_{i_{F-k}}^{A_{F-k}}\bar{C}_{l_{1}i_{F+1-k}}\cdots\bar{C}_{l_{k+F+4}i_{2F+4}}W_{i_{2F+5}i_{2F+6}}W_{i_{2F+7}i_{2F+8}}.
p_{2}=\int_{0}^{\infty}\,\frac{d\omega}{2\pi}\int_{0}^{\omega}\,\frac{d\omega^{\prime\prime}}{2\pi}\vert\langleout\,\omega\omega^{\prime\prime}\vertin\rangle\vert^{2}=\frac12\,\int_{0}^{\infty}\,\frac{d\omega}{2\pi}p_{2\omega}.
0=\frac{\partial}{\partial{\bfr}_{C}}U_{\mathrm{eff}}=\sum_{A,B=1}^{r-1}\frac{\partialC^{AB}}{2\partial{\bfr}_{C}}\left(\tilde{q}_{A}\tilde{q}_{B}-\sum_{I=1}^{5}\sum_{A^{\prime},B^{\prime}=1}^{r-1}C_{AA^{\prime}}C_{BB^{\prime}}\tilde{a}_{I}^{A^{\prime}}\tilde{a}_{I}^{B^{\prime}}\right)
\langle0|T_{00}(x)|0\rangle=\frac{1}{2a^{3}S_{D}r^{n}}\sum_{l=0}^{\infty}D_{l}\sum_{k=1}^{\infty}\frac{\lambda_{\nu,k}^{3}T_{\nu}(\lambda_{\nu,k})}{\sqrt{\lambda_{\nu,k}^{2}+m^{2}a^{2}}}f_{\nu}^{(0)}\left[J_{\nu}(\lambda_{\nu,k}r/a)\right],
R(u)=\left(\begin{array}{cccc}{qu\!-\!q^{-1}u^{-1}}&{0}&{0}&{0}\\{0}&{u\!-\!u^{-1}}&{q\!-\!q^{-1}}&{0}\\{0}&{q\!-\!q^{-1}}&{u\!-\!u^{-1}}&{0}\\{0}&{0}&{0}&{qu\!-\!q^{-1}u^{-1}}\\\end{array}\right)
\frac{5}{2}F(c_{2},b,\alpha_{2})-\frac{5}{2}F(-c_{2},b,\alpha_{2})=const[e^{-\frac{5}{2}\alpha_{2}b}(1-c_{2}e^{-\frac{5}{2}{\alpha_{2}}b})^{3}(1+c_{2}e^{-\frac{5}{2}{\alpha_{2}}b})^{-5}]\,
\delta_{\tilde{\lambda}}\delta\hat{A}_{i}-\partial_{i}\delta\hat{\lambda}-i[\delta\hat{\lambda},\hat{A}_{i}]_{\ast}-i[\delta\hat{A}_{i},\hat{\lambda}]_{\ast}=-\frac{1}{2}\delta\theta^{kl}\{\partial_{k}\hat{A}_{i},\partial_{l}\hat{\lambda}\}_{\ast}.
G(\omega,k)=\frac{\Gamma(-\nu)}{\Gamma(\nu)}\frac{\Gamma(h_{+}+\frac{R}{2}(\omega+k))\Gamma(h_{+}+\frac{R}{2}(-\omega+k))}{\Gamma(h_{-}+\frac{R}{2}(\omega+k))\Gamma(h_{-}+\frac{R}{2}(-\omega+k))}\,
q^{(0)}=\frac{1}{8\pi^{2}a^{4}}\sum_{m=-\infty}^{+\infty}\int_{-\infty}^{+\infty}{dk\int_{0}^{\infty}{dz\frac{z^{3}\psi_{\mu}(z/a)}{\sqrt{k^{2}+z^{2}/a^{2}}}f_{m}^{(q)}(zx)}}
\int_{\sigma_{0}^{(1)}}^{\sigma_{0}^{(1)}+\varepsilon}\!d\sigma_{0}^{(1)}b_{\sigma_{0}^{(1)}}=\int_{\sigma_{0}^{(2)}}^{\sigma_{0}^{(2)}-\varepsilon}\!(-d\sigma_{0}^{(2)})(-b_{\sigma_{0}^{(2)}})=-\int_{\sigma_{0}^{(2)}-\varepsilon}^{\sigma_{0}^{(2)}}\!d\sigma_{0}^{(2)}b_{\sigma_{0}^{(2)}}.
\Psi_{(3,3,1)}=\prod_{i<j}(z_{i}^{(1)}-z_{j}^{(1)})^{3}\prod_{i<j}(z_{i}^{(2)}-z_{j}^{(2)})^{3}\prod_{i,j}(z_{i}^{(1)}-z_{j}^{(2)})\;\Psi_{0}.
S=\intdte^{-\phi}\Bigl\{2\Lambda_{7}n-\frac{n}{2}(\frac{P^{2}}{a^{6}}+\frac{Q^{2}}{b^{6}})-\frac{\dot{\phi}^{2}}{n}+\frac{3\dot{a}^{2}}{na^{2}}+\frac{3\dot{b}^{2}}{nb^{2}}\Bigr\}
ds^{2}=\left(\frac{r}{r_{0}}\right)^{-2n}W^{2}(r)\left[\left(\frac{r}{r_{0}}\right)^{2m^{2}}(-dt^{2}+dr^{2})+B^{2}r^{2}d\theta^{2}\right]+\left(\frac{r}{r_{0}}\right)^{2n}\frac{1}{W^{2}(r)}dz^{2},
ds^{2}=-4dx^{+}dx^{-}-\mu^{2}\left(\sum_{I=5}^{8}x_{I}^{2}\right){dx^{+}}^{2}+\sum_{i=1}^{8}{dx_{i}}^{2}+4\mux_{2}dx_{1}dx^{+}+4\mux_{4}dx_{3}dx^{+},
(\partial^{2}-3f\partial_{z})h_{\mu\nu}+4\alpha\kappae^{2A(z)}\Big\{\partial_{z}f\partial_{\lambda}^{2}-f^{2}\partial_{z}^{2}-f\big(2\partial_{z}f-f^{2}\big)\partial_{z}\Big\}h_{\mu\nu}=-\kappaT_{\mu\nu}^{(m)}\,,
\frac{1}{L}{\sum_{n>0}}\{c_{n},c_{n}^{\dagger}\}u_{n}^{+}(x)u_{n}^{+}(y)+\frac{1}{L}\{c_{0},c_{0}\}u_{0}^{+}(x)u_{0}^{+}(y)=\delta(x-y),
ds^{2}=[U^{(-1)}(\rho^{2}/l^{2})]\left(-(bR^{2}-k)dT^{2}+\frac{dR^{2}}{bR^{2}-k}\right)+\frac{d\rho^{2}}{U^{\prime}(U^{(-1)}(\rho^{2}/l^{2}))}+\rho^{2}d\phi^{2}
\begin{aligned}{\deltaA_{\mu}}&{{}=-4\,\bar{\epsilon}\Gamma_{\mu}\chi\,,}\\{\delta\chi}&{{}=\frac{1}{8}\Gamma^{\mu\nu}\epsilonF_{\mu\nu}\,,}\\{\deltaF_{\mu\nu}}&{{}=-8\,\big(D_{[\mu}(\omega)\bar{\epsilon}\big)\Gamma_{\nu]}\chi+8\,\bar{\epsilon}\Gamma_{[\mu}D_{\nu]}(\omega)\chi\,.}\\\end{aligned}
\psi(z,\bar{z})\propto\langlez,\bar{z}\vertn_{1},n_{2},n_{3},\cdots;\bar{n}_{1},\bar{n}_{2},\bar{n}_{3},\cdots\rangle\propto\prod_{j=1}(z_{j})^{n_{j}}\prod_{k=1}(\bar{z}_{k})^{\bar{n}_{k}}.
\begin{array}{lcr}{qJ_{1}J_{2}-q^{-1}J_{2}J_{1}}&{=}&{(q^{2}-q^{-2})J_{3}}\\{qJ_{2}J_{3}-q^{-1}J_{3}J_{2}}&{=}&{(q^{2}-q^{-2})J_{1}}\\{qJ_{3}J_{1}-q^{-1}J_{1}J_{3}}&{=}&{(q^{2}-q^{-2})J_{2}}\\\end{array}
f_{dc}^{a}\tilde{f}_{a}^{rs}=\tilde{f}_{c}^{as}f_{da}^{r}+\tilde{f}_{c}^{ra}f_{da}^{s}-\tilde{f}_{d}^{as}f_{ca}^{r}-\tilde{f}_{d}^{ra}f_{ca}^{s}.
\left(\frac{\dot{R}}{R}\right)^{2}\simeq\frac{1}{R^{2}}\left(\mu+\frac{2}{\ell^{2}}\left(\frac{\hat{\rho}}{\sigma}\right)\right)-\frac{q^{2}}{R^{4}}+\frac{1}{\ell^{2}}\left(\frac{\hat{\rho}}{\sigma}\right)^{2}\frac{1}{R^{6}}\,.
\sum_{k}(\Phi_{0})_{ik}\star(\Phi_{0})_{kj}=\sum_{k,m,l}(e^{-\Psi_{0}})_{ik}\starQ(e^{\Psi_{0}})_{kl}\star(e^{-\Psi_{0}})_{lm}\starQ(e^{\Psi_{0}})_{mj}\.
D_{\mu\nu}(p)=D_{e}(p)\left[g_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right]+D_{\perp}(p)\left[g_{\mu\nu}^{\perp}-\frac{p_{\mu}^{\perp}p_{\nu}^{\perp}}{p_{\perp}^{2}}\right]+D_{o}(p)\;i\epsilon_{\mu\nu\rho}p^{\rho}+\xi\,\frac{p_{\mu}p_{\nu}}{(p^{2})^{2}},
(-m_{\pi}^{2}+\partial_{\bot}^{2})\,\omega_{\pi}=\frac{\lambda}{2L}\int_{-L}^{L}dx^{-}(\varphi_{\pi}^{3}+\varphi_{\pi}\varphi_{\sigma}^{2}+2v\varphi_{\pi}\varphi_{\sigma}),
S_{\mathrm{BFV}}=\intd\tau\left(p\dot{x}+P_{n}\dot{\psi}^{n}+\pi_{a}\dot{\lambda}_{a}+\overline{{\theta}}_{a}\dot{\theta}_{a}-T_{a}\lambda_{a}-\overline{{B}}_{a}B_{a}-\frac{i}{2}\overline{{B}}_{2}c_{1}\lambda_{1}\right)\,.
M_{ij}=e^{\mathscr{G}/2}\,\mathscr{G}_{i\bar{m}}^{-1/2}\,[\mathscr{G}_{mn}+\mathscr{G}_{m}\mathscr{G}_{n}-\mathscr{G}_{mn\bar{k}}\mathscr{G}_{\bar{k}l}^{-1}\mathscr{G}^{l}]\,\mathscr{G}_{\bar{n}j}^{-1/2};
\widetilde{\bigtriangleup}_{\mu\nu}^{A}(p)=\bigtriangleup_{\mu\nu}^{A}+\bigtriangleup_{\mu\mu^{\prime}}^{A}V^{\lambda\sigma,\mu^{\prime}}\bigtriangleup_{\lambda\sigma,\lambda^{\prime}\sigma^{\prime}}^{B}V^{\lambda^{\prime}\sigma^{\prime},\nu^{\prime}}\bigtriangleup_{\nu^{\prime}\nu}^{A}+\cdots.
S_{h}=\int\!\!d^{6}x\sqrt{-{\calG}_{6}}e^{-2\phi_{6}^{(h)}}\left\{R+4\big(\nabla\phi_{6}^{(h)}\big)^{2}-{\frac{1}{12}}(H_{6}^{(h)})^{2}-{\frac{1}{4}}{\calF}_{6}^{2}\right\}
f_{2}(L,m,d,z)=\frac{1}{2}h(d)\int_{0}^{\infty}dp\frac{{p}^{d-2}}{\sqrt{p^{2}+m^{2}}}\frac{cosh((L-2z)\sqrt{p^{2}+m^{2}})}{sinh(L\sqrt{p^{2}+m^{2}})}.
T\equiv\beta_{H}^{-1}=\left.\frac{\tilde{A}^{\prime}}{4\pi\sqrt{\tilde{A}\tilde{B}}}\right|_{r_{+}}\nonumber\,=\frac{\tilde{d}}{4\pir_{+}}\left[1-\left(\frac{r_{-}}{r_{+}}\right)^{\tilde{d}}\right]^{(\tilde{d}-2)/2\tilde{d}},
\left(\sigma_{M_{5}\timesS^{1}}^{\delta\phi}\right)_{l=0,K_{5}=0}=\left({\frac{\pi^{3}R^{8}}{64}}\right)2\piR(\omega^{2}+16\pi^{2}T_{R}^{2})(\omega^{2}+16\pi^{2}T_{L}^{2})\omega{\frac{e^{\omega/T_{H}}-1}{[e^{\omega/2T_{R}}-1][e^{\omega/2T_{L}}-1]}}.
\begin{array}{lcl}{F_{-1\mu}+F_{n\mu}}&{=}&{(X_{-1\lambda}+X_{n\lambda}){X_{\mu}}^{\lambda}+(X_{n,-1}X_{-1,\mu}-X_{-1,n}X_{n,\mu})-\frac{1}{2}n(X_{-1,\mu}+X_{n\mu})}\\{}&{}&{}\\{}&{=}&{P_{\lambda}({X_{\mu}}^{\lambda}+(D+1-\frac{n}{2})\delta_{\mu}^{\lambda})}\\\end{array}
\partial_{\chi}\Gamma_{Y_{i}\varphi_{i}}(\kappa_{i}^{2})=-\bigl(\partial_{p^{2}}\partial_{\chi}\Gamma_{Y_{i}\varphi_{i}}\bigr)\Big|_{p^{2}=\kappa_{i}^{2}}\Gamma_{\varphi_{i}\varphi_{i}}(\kappa_{i}^{2})
S_{1^{j}}(z)=\sum_{{\scriptstylek_{1}+2k_{2}+\cdots+mk_{m}=j}\atop{\scriptstylek_{1}\geq0,\cdots,k_{m}\geq0}}(-1)^{k_{2}+k_{4}+\cdots+k_{2\left[\fracm2\right]}}\frac1{k_{1}!\cdotsk_{m}!}z_{1}^{k_{1}}\cdotsz_{m}^{k_{m}}
\begin{array}{lll}{g(x=0,\Psi,\phi_{+})}&{=g_{-}(\rho=\frac{\pi}{4},\Psi,\phi_{+}-\Psi)}&{=e^{iqH(\phi_{+}-\Psi)}}\\{g(x=1,\Psi,\phi_{+})}&{=g_{+}(\rho=\frac{\pi}{4},\Psi,\phi_{+})}&{=e^{iqH\phi_{+}}}\\\end{array}
t={{\calZ}^{F_{1}}}^{3}-{\calZ}^{F_{1}}{{\calZ}^{F_{1}}}^{\prime\prime}-{\frac{1}{2}}\left({{\calZ}^{F_{1}}}^{\prime}\right)^{2}+{\frac{1}{10}}{{\calZ}^{F_{1}}}^{(4)}.
a_{0}^{I\dagger}=\frac{1}{\sqrt{2m}}(p_{0}^{I}+\mathrm{i}mx_{0}^{I})\,,\qquad{a}_{0}^{I}=\frac{1}{\sqrt{2m}}(p_{0}^{I}-\mathrm{i}mx_{0}^{I})~,