harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\dot{x}^{AB^{\prime}}=o^{A}\bar{o}^{B^{\prime}}-\frac{\rho^{\sigma}}{\rho^{\tau}}\acute{x}^{AB^{\prime}},\,\,\,\,\acute{x}^{AB^{\prime}}=o^{A}\bar{\iota}^{B^{\prime}}+\iota^{A}\bar{o}^{B^{\prime}}.
K_{s}(x)=\sqrt{\frac{\pi}{2x}}e^{-x}\left(1+\sum_{k=1}^{\infty}\frac{1}{(2x)^{k}}\frac{\Gamma\left(s+k+\frac{1}{2}\right)}{k!\Gamma\left(s-k+\frac{1}{2}\right)}\right)\.
{\frac{3}{2}}{\frac{\sigma^{\prime\prime}}{\sigma}}+{\frac{3}{4}}{\frac{\sigma^{\prime}\gamma^{\prime}}{\sigma\gamma}}-{\frac{1}{4}}\left({\frac{\gamma^{\prime}}{\gamma}}\right)^{2}+{\frac{1}{2}}{\frac{\gamma^{\prime\prime}}{\gamma}}=-{\frac{\Lambda_{0}}{4M^{4}}}-{\frac{V_{0}^{i}}{4M^{4}}}\delta(\rho-\rho_{i})
_f<\beta|=\sum_{n=0}^{p}\frac{(\beta^{*})^{m}}{\sqrt{\{n\}!}}<n|\;\rightarrow\sum_{n=0}^{p}\frac{(\beta^{*})^{m}}{z^{n+1}}=\frac1{z-\beta^{*}},\;\;(|z|>|\beta|),
D_{\mu\nu}(k)=-g_{\perp\mu\nu}+{\frac{k_{\perp}^{2}}{(k^{+})^{2}}}\eta_{\mu}\eta_{\nu}+{\frac{1}{k^{+}}}\Bigl(k_{\perp\mu}\eta_{\nu}+k_{\perp\nu}\eta_{\mu}\Bigr)\;,
[{\calX}_{\mu\nu}^{L},{\calX}_{\rho\sigma}^{L}]=\eta_{\nu\rho}{\calX}_{\mu\sigma}^{L}+\eta_{\mu\sigma}{\calX}_{\nu\rho}^{L}-\eta_{\mu\rho}{\calX}_{\nu\sigma}^{L}-\eta_{\nu\sigma}{\calX}_{\mu\rho}^{L}.
\Psi_{E}^{(-)}=\left(\begin{array}{l}{e^{i\frac{\theta}{2}}\sumf_{n}e^{i\left(n+\frac{1}{2}\right)\theta}}\\{e^{-i\frac{\theta}{2}}\sumg_{n}e^{i\left(n+\frac{1}{2}\right)\theta}}\\\end{array}\right)
\left[\left(\frac{H_{0}}{f}\right)^{2}-\left(\frac{f^{\prime}}{f}\right)^{2}\frac{1}{b_{0}\,^{2}}\right]+\frac{2\alpha}{M^{2}}\left[\left(\frac{H_{0}}{f}\right)^{2}-\left(\frac{f^{\prime}}{f}\right)^{2}\frac{1}{b_{0}\,^{2}}\right]^{2}=\frac{\Lambda_{b}}{6M^{3}}~~~.
\left.\left|0(\theta)\right\rangle\!\right\rangle=\prod_{k}(\Gamma_{3_{k}})^{2\delta_{kk}tr\eta^{\mu\nu}}e^{\Gamma_{1_{k}}\left(\tilde{A}_{k}^{\dagger}\cdotA_{k}^{\dagger}\right)}e^{\Gamma_{1_{k}}\left(\tilde{B}_{k}^{\dagger}\cdotB_{k}^{\dagger}\right)}\left.\left|0\right\rangle\!\right\rangle.
\Lambda_{eff}=\frac{1}{2k^{2}}\int_{0}^{\infty}e^{\frac{3}{4}\Phi_{0}}\alpha_{E}^{3/2}\frac{y}{2}dy=\frac{1}{2k^{2}}e^{\frac{3}{4}\Phi_{0}}\alpha_{E}^{3/2}\frac{y^{2}}{4}\Bigg|_{0}^{\infty}\,
L_{SCS}^{\pm}=STr\int_{X}d^{3}x\varepsilon^{ijk}\bigg({{\bfA}^{\pm}}_{i}\partial_{j}{{\bfA}^{\pm}}_{k}+\frac{2}{3}{{\bfA}^{\pm}}_{i}{{\bfA}^{\pm}}_{j}{{\bfA}^{\pm}}_{k}\bigg),
{\cal{L}}_{DJ}^{S}=\frac{1}{2}(f^{\mu}-\partial^{\mu}\phi)(f_{\mu}-\partial_{\mu}\phi)-\epsilon^{\alpha\beta\gamma}f_{\alpha}\partial_{\beta}A_{\gamma}+\frac{m}{2}\epsilon^{\alpha\beta\gamma}A_{\alpha}\partial_{\beta}A_{\gamma}.
S[x^{\mu},g_{\alpha\beta}]=\frac{-1}{4\pi\alpha^{\prime}}\int_{\tau_{1}}^{\tau_{2}}d\tau\int_{0}^{\pi}d\sigma\sqrt{-\mathrm{det}\,g_{\alpha\beta}}\,g^{\alpha\beta}\,\partial_{\alpha}x^{\mu}\partial_{\beta}x^{\nu}\eta_{\mu\nu}\.
J_{\alpha}=\frac{\pi}{2e}\frac{\deltaS}{\delta\bar{\chi}^{\alpha}}=\frac{1}{2}\rho^{\beta}\rho_{\alpha}\bar{\Psi}^{\mu}\partial_{\beta}X_{\mu}-\frac{1}{2}\bar{\Psi}^{\mu}\Psi_{\mu}\rho_{\alpha}\rho^{\beta}\chi_{\beta}=0,
{\hat{g}}_{{\hat{\mu}}{\hat{\nu}}}D_{\hat{\imath}}{\hat{X}}^{\hat{\mu}}D_{\hat{\jmath}}{\hat{X}}^{\hat{\nu}}={\hat{\Pi}}_{{\hat{\mu}}{\hat{\nu}}}\partial_{\hat{\imath}}{\hat{X}}^{\hat{\mu}}\partial_{\hat{\jmath}}{\hat{X}}^{\hat{\nu}}\,,
\int{\calD}\varphi\quade^{+{\frac{1}{2}}(d\varphi,d\varphi)}\quad\prod_{k=1}^{i}e^{2i\sqrt{\frac{\pi}{\xi}}\varphi(y_{k})}\prod_{k=1}^{i}e^{-2i\sqrt{\frac{\pi}{\xi}}\varphi(x_{k})}\propto(\Lambda_{UV})^{2i}K_{i}^{-1}(x,y).
\widetildee^{1}=e^{-(\Phi-\Phi_{0})}e^{1},\qquadd\widetildee^{1}=\widetilde{\Gamma_{bc}^{1}}\tilde{e}^{b}\wedge\tilde{e}^{c}=e^{-(\Phi-\Phi_{0})}(de^{1}-d\Phi\wedgee^{1}),
S_{n}=\sum_{m=1\atopm\nen}^{N}\frac{1}{x_{n}^{2}-x_{m}^{2}}\qquad\mathrm{and}\qquadT_{n}=\sum_{m=1\atopm\nen}^{N}\frac{1}{(x_{n}^{2}-x_{m}^{2})^{2}}\.
\begin{array}{rcl}{A_{n}(a,q)}&{=}&{a^{n}q^{n^{2}}\alpha_{n}(a^{-1},q^{-1}),}\\{B_{n}(a,q)}&{=}&{a^{-n}q^{-n^{2}-n}\beta_{n}(a^{-1},q^{-1}),}\\\end{array}
B(\tau_{2})=\sum_{n,m=-1}^{\infty}a_{n}a_{m}\int_{-1/2}^{+1/2}d\tau_{1}\left|\tau\right|^{-24}e^{\frac{2\pii\tau_{1}}{\left|\tau\right|^{2}}(m-n)}e^{-\frac{2\pi\tau_{2}}{\left|\tau\right|^{2}}(m+n)}.
\begin{array}{rcl}{F_{\sigma}^{(n)}(\zeta_{1},\cdots,\zeta_{2n-2},\zeta_{2n-1},x^{-1}\zeta_{2n-1})^{k\cdotsk_{2n-2}k_{2n-1}k}}&{=}&{\frac{\delta_{k_{2n-2},k}}{\{k\}}F_{\sigma}^{(n-1)}(\zeta_{1},\cdots,\zeta_{2n-2})^{k\cdotsk}}\\\end{array}
r_{p+4}^{3-p}=(2\sqrt{\pi})^{1-p}\Gamma\left({\frac{3-p}{2}}\right)g_{s}N\alpha^{\prime}{}^{\frac{3-p}{2}},\quad\tilde{r}_{p}^{3-p}={\frac{V_{*}}{V}}r_{p+4}^{3-p},
\delta\varphi^{i}(y)=\sum_{k}\intd^{2}x\\left(\Lambda_{k}(x)\phi_{k}^{i(0)}(x,y)+\dot{\Lambda}_{k}(x)\phi_{k}^{i(1)}(x,y)+\ddot{\Lambda}_{k}(x)\phi_{k}^{i(2)}(x,y)\right),
M_{SM}^{2}=g^{2}v^{2}\left(\begin{array}{ccccccc}{1}&{-1}&{0}&{\cdots}&{0}&{0}&{0}\\{-1}&{2}&{-1}&{\cdots}&{0}&{0}&{0}\\{0}&{-1}&{2}&{\cdots}&{0}&{0}&{0}\\{}&{}&{}&{\ddots}&{}&{}&{}\\{0}&{0}&{0}&{\cdots}&{-1}&{2}&{-1}\\{0}&{0}&{0}&{\cdots}&{0}&{-1}&{1}\\\end{array}\right).
{\hat{H}}=\frac{H}{M}=\frac{2\pi}{l}\left(H_{\mathrm{CFT}}+\kappa(h)\frac{l^{2-h}}{\left(2\pi\right)^{1-h}}B\right)\,,\qquadH_{\mathrm{CFT}}=L_{0}+\bar{L}_{0}-\frac{c}{12}{\mathrm{Id}}\:.
S_{NSR}^{ferm}=-\frac{i}{c\alpha^{\prime}}\left(\varphi_{+}^{i}\partial_{-2}\varphi_{+}^{i}-\varphi_{+}^{i}\Omega_{-2}^{ij}\varphi_{+}^{j}\right)-\frac{i}{c\alpha^{\prime}}\left(\varphi_{-}^{i}\partial_{+2}\varphi_{--}^{i}\varphi_{-}^{i}\Omega_{+2}^{ij}\varphi_{-}^{j}\right).
T=\frac{1}{4\pi}\left[\sqrt{\partial_{r}A\partial_{r}\left(\frac{1}{B}\right)}\right]_{r=r_{+}}=\frac{\tilde{d}}{4\pir_{+}}\left[1-\left(\frac{r_{-}}{r_{+}}\right)^{\tilde{d}}\right]^{\frac{\tilde{d}-2}{2\tilde{d}}}.
\Lambda^{\mathbfi}{}_{\mathbfj}=-\frac{i}{4}\left(D_{\alpha}^{\mathbfi}\lambda_{\mathbfj}^{\alpha}+{\bar{D}}_{{\dot{\alpha}}{\mathbfj}}{\bar{\lambda}}^{{\dot{\alpha}}{\mathbfi}}-\frac{1}{{\mathcalN}}\delta_{\mathbfj}^{\mathbfi}\left(D_{\alpha}^{\mathbfk}\lambda_{\mathbfk}^{\alpha}+{\bar{D}}_{{\dot{\alpha}}{\mathbfk}}{\bar{\lambda}}^{{\dot{\alpha}}{\mathbfk}}\right)\right)
\left<R\left|\chi_{R_{1}}(U)\chi_{R_{2}}(U)\ldots\chi_{R_{k}}(U)\right|R^{\prime}\right>~=~\int[dU]\chi_{R}^{*}(U)\chi_{R_{1}}(U)\chi_{R_{2}}(U)\ldots\chi_{R_{k}}(U)\chi_{R^{\prime}}(U)
\left.\begin{array}{ll}{\partial_{+}q_{1}={\displaystyle\frac{2}{\lambda}}(1-q_{1}q_{2})e^{\phi},\,\,}&{\partial_{+}q_{2}=-2q_{1}e^{-2\phi}-{\displaystyle\frac{2}{\lambda}}q_{2}^{2}e^{\phi},}\\{\partial_{-}q_{1}=-(q_{2}+\lambdaq_{1}^{2})+q_{1}\partial_{-}\phi,\,\,}&{\partial_{-}q_{2}=\lambda(1-q_{1}q_{2})-q_{2}\partial_{-}\phi,}\\\end{array}\right.
\frac{1}{\Lambda^{2}}\int\frac{d^{2}k}{(2\pi)^{2}}K^{\prime}(k^{2}/\Lambda^{2})\intd^{2}x\frac{\partial^{2}{\calV}}{\partial\phi\partial\phi}(x)=-\gamma_{1}\intd^{2}x\frac{\partial^{2}{\calV}}{\partial\phi\partial\phi}(x)
V_{eff}=\frac{\hbar^{2}}{2}\sum_{\alpha}\Biggl[\dot{u}_{\alpha}^{*}\dot{u}_{\alpha}+\Bigl(m^{2}+{\alpha}^{2}\Bigr)u_{\alpha}^{*}u_{\alpha}\Biggr]+\frac{3\hbar^{4}\lambda}{4}\Biggl[\sum_{\alpha}u_{\alpha}^{*}u_{\alpha}\Biggr]^{2}.
E=\intd^{3}x\Biggl(4\left(J_{i}^{1}(\vec{x})J_{i}^{1}(\vec{x})+J_{i}^{2}(\vec{x})J_{i}^{2}(\vec{x})\right)+2\left(J_{i}^{1}(\vec{x})J_{j}^{2}(\vec{x})-J_{j}^{1}(\vec{x})J_{i}^{2}(\vec{x})\right)^{2}\Biggr)
\times\delta(p_{\theta}-p^{\prime}+\frac{\hbarm}{2})\delta(p_{\theta}-p^{\prime\prime}-\frac{\hbarn}{2})e^{\frac{2i}{\hbar}[-(p_{\theta}-p^{\prime})(\theta-\theta^{\prime\prime})+(p_{\theta}-p^{\prime\prime})(\theta-\theta^{\prime})]}f(\theta^{\prime},p^{\prime})g(\theta^{\prime\prime},p^{\prime\prime}).
\omega_{+}^{-Y^{\prime}}(x,\theta^{+},u)=\tau_{+}^{-Y^{\prime}}(x,u)+\theta_{+}^{+A^{\prime}}l_{A^{\prime}}^{--Y^{\prime}}(x,u)+i(\theta_{+}^{+})^{2}\mu_{-}^{-3Y^{\prime}}(x,u)\;.
S_{\mathrm{eff}}[A_{0}]=\intd^{2}x\left(Z(m,e_{N}{\tilde{A}}_{0}/{{T}})\frac{1}{2T}\vec{\nabla}A_{0}\cdot\vec{\nabla}A_{0}\,-\,N\,V(m,e_{N}A_{0}/{T})\right)\,,
+\intd^{2}z\left(i\bar{c}^{a}D_{\bar{z}}^{ab}c^{b}+A_{\bar{z}}^{a}\pi^{a}-A_{0}^{a}\Phi^{a}-i\bar{b}^{a}b^{a}-\epsilon^{abc}A_{0}^{a}c^{b}\bar{b}^{c}\right)\Biggr]
{\calR}_{n}^{m}\delta_{b}^{a}=ig^{2}f_{b~c}^{~a}f_{~b^{\prime}c^{\prime}}^{c}A_{p}^{b^{\prime}}A_{n}^{c^{\prime}}\eta^{mp}-\frac{1}{2}\delta_{n}^{m}g^{2}f_{ebc}f_{d}^{~ac}A_{p}^{e}A_{q}^{d}\eta^{pq}
{^{[n+1]}{\hat{W}}}(\vec{x},\vec{p},x_{0})=\sum_{k=0}^{n}{\mathcalO}_{(k)}^{n}(\vec{x},\vec{p},x_{0};\vec{\partial}_{x},\vec{\partial}_{p})\{^{[k]}{\hat{W}}}(\vec{x},\vec{p},x_{0})\,,
{\frac{\ddot{a}_{0}}{a_{0}}}+{\frac{1}{2}}{\frac{\dot{I}}{I}}{\frac{\dot{a}_{0}}{a_{0}}}=-{\frac{4\piG_{4}}{3}}\left(\varrho+3{\frac{\wp}{c_{\mathrm{grav}}^{2}}}\right)-{\frac{8\piG_{4}}{3c_{\mathrm{grav}}^{4}\sqrt{I}}}\dot{\Phi}^{2}+{\frac{c_{\mathrm{grav}}^{2}}{3}}\Lambda_{\mathrm{eff}}-{\frac{{\calC}c_{\mathrm{grav}}^{2}}{a_{0}^{4}}},
\begin{array}{l}{U=SO(5,5),\quadK=SO\left(5\right)\otimesSO\left(5\right)}\\{T=SO(4,4),\quadk=SO\left(4\right)_{L}\otimesSO\left(4\right)_{R}}\\{l_{L,R}=1:\quad\left(\sum_{i}r_{i}^{(l_{L,R})}\right)_{L,R}=1_{L,R}}\\{l_{L,R}=2:\quad\left(\sum_{i}r_{i}^{(l_{L,R})}\right)_{L,R}=9_{L,R}}\\{\quad\quad\quad\quad\quad=5_{L,R}^{space}\oplus4_{L,R}^{internal}}\\{l_{L,R}=3:\quadetc.}\\\end{array}
\langleT_{\alpha\beta}(x)T_{\rho\sigma}(0)\rangle={\frac{\pi}{3}}\int_{0}^{\infty}d\nu\,c(\nu,t)\int{\frac{d^{2}p}{(2\pi)^{2}}}\,\mathrm{e}^{ipx}{\frac{(g_{\alpha\beta}p^{2}-p_{\alpha}p_{\beta})(g_{\rho\sigma}p^{2}-p_{\rho}p_{\sigma})}{p^{2}+\nu^{2}}}\quad.
S^{(2)}=\frac{-1}{2\pi\alpha^{\prime}}\intd\taud\sigma\sum_{R=1}^{D-1}[\eta^{ab}(\deltax^{R})_{,a}(\deltax_{R})_{,b}+\frac{m^{2}\alpha^{2}}{l^{2}}\deltax^{R}\deltax_{R}].
\left(j^{k}i,j^{N-k}i\right)\ast\left(j^{p}i^{\prime},j^{N-p}i^{\prime}\right)=\sum_{k}\left(j^{p+k}\left(i\asti^{\prime}\right)_{k},j^{N-\left(p+k\right)}\left(i\asti^{\prime}\right)_{k}\right),
\left[\begin{array}{c}{{}}\\{{\sum\sum}}\\{{}_{j<l}}\\\end{array}\alpha_{j}\alpha_{l}k_{jl}^{2}-\sum\alpha_{i}m_{i}^{2}\right]\Rightarrow-\left[\sum\alpha_{i}^{2}m_{i}^{2}\!+\!2\!\begin{array}{c}{{}}\\{{\sum\sum}}\\{{}_{j<l}}\\\end{array}\alpha_{j}\alpha_{l}m_{j}m_{l}c_{jl}\right],
ds^{2}=1/(\tilde{x}+\tilde{y})^{2}(-{\calF}d\tilde{t}^{2}+{\calF}^{-1}d\tilde{y}^{2}+{\calG}^{-1}d\tilde{x}^{2}+{\calG}d\tilde{z}^{2})\:,
h_{A}^{\phantom{A}B}=\frac{1}{2!}({\calP}_{+}\Gamma^{MN})_{A}^{\phantom{A}B}h_{MN}+\frac{1}{6!}({\calP}_{+}\Gamma^{M_{1}\ldotsM_{6}})_{A}^{\phantom{A}B}h_{M_{1}\ldotsM_{6}}^{+}
(D_{4})_{\alpha}={}-\frac{i}{4}(\gamma^{ad}\theta_{C})_{\alpha}X_{dD}(\theta_{B}\theta_{A})({c_{E}}^{CD}f^{EBA}+{c_{\lambda}}^{CD}f^{\lambdaBA})\nonumber\,\\-\frac{i}{2}(\theta_{C}\theta_{A})\theta_{\alphaB}{d^{BD}}_{E}f^{CAE}.
\left(\mathrm{ad}X_{i}^{-}\right)\left(X_{j}^{-}X_{k}^{-}\right)=\left(\mathrm{ad}X_{i}^{-}\right)(X_{j}^{-})X_{k}^{-}+q_{i}^{a_{ij}/2}X_{j}^{-}\left(\mathrm{ad}X_{i}^{-}\right)(X_{k}^{-})~,
\left.\begin{array}{ll}{\partial_{+}q_{1}=2Y_{0}^{1}e^{\phi}+2Y_{1}^{1}e^{-2\phi},\,\,}&{\partial_{+}q_{2}=2Y_{0}^{2}e^{\phi}+2Y_{1}^{2}e^{-2\phi},}\\{\partial_{-}q_{1}=X_{0}^{1}+X_{1}^{1}\pi_{\phi},\,\,}&{\partial_{-}q_{2}=X_{0}^{2}+X_{1}^{2}\pi_{\phi},}\\\end{array}\right.
V=\left(\begin{array}{ccccc}{1}&{\zeta_{1}}&{\zeta_{1}^{2}}&{\cdots}&{\zeta_{1}^{2j}}\\{1}&{\zeta_{2}}&{\zeta_{2}^{2}}&{\cdots}&{\zeta_{2}^{2j}}\\{\vdots}&{\vdots}&{\vdots}&{\vdots}&{\vdots}\\{1}&{\zeta_{2j+1}}&{\zeta_{2j+1}^{2}}&{\cdots}&{\zeta_{2j+1}^{2j}}\\\end{array}\right)\quad\mathrm{and}\quadC=\mathrm{diag}(c_{-j},\cdots,c_{j}).
S_{G}=\int_{{\calM}_{5}}d^{5}x\left[-\frac{3M_{5}^{3}}{4}\sqrt{-g}e^{-2\phi}\left(G-{\frac{2e^{2\phi}}{3M_{5}^{3}\sqrt{-g}}}{\frac{c}{5!}}\varepsilon^{\alpha\beta\gamma\delta\epsilon}{\calF}_{\alpha\beta\gamma\delta\epsilon}\right)^{2}-\sqrt{-g}{\frac{e^{2\phi}}{5!3M_{5}^{3}}}c^{2}{\calF}^{2}\right].
A_{i}(s)=-{\frac{1}{\Gamma(s)}}\Bigr(r_{+}^{2s}-r_{-}^{2s}\Bigr)\zeta_{H}(2s+i-2;2)\sum_{l=0}^{i}(x_{i,l}+z_{i,l}){\frac{\Gamma(s+l+{\frac{i}{2}})}{\Gamma(l+{\frac{i}{2}})}},\;i=1,3,
[{\calD}\mu]={\calD}B^{\mu}{\calD}\lambda{\calD}\xi\delta(F_{01}+\dot{\lambda})\prod_{\beta=0}^{3}\{\delta(\Gamma_{\beta}[B^{0}+\xi,B^{i},\lambda])\}det\mid\{\tilde{\Omega}_{\alpha},\Gamma_{\beta}\}\mid,
\sigma_{\mu}^{\alpha\beta^{\prime}}=(\tau^{a},i)\,\qquad\bar{\sigma}_{\mu\,\alpha^{\prime}\beta}=(\tau^{a},-i)\,\qquad\sigma_{\mu}^{\alpha\alpha^{\prime}}=\bar{\sigma}_{\mu}^{\alpha^{\prime}\alpha}=\epsilon^{\alpha^{\prime}\beta^{\prime}}\bar{\sigma}_{\mu\,\beta^{\prime}\beta}\epsilon^{\alpha\beta}\,
\begin{aligned}{M_{3(b)}(q)}&{{}=\int{\frac{d^{4}l}{(2\pi)^{4}}}\tau_{1}^{\sigma\rho}(k_{3},k_{3}-l,m_{2})\tau_{1}^{\gamma\delta}(k_{3}-l,k_{4},m_{2})\tau_{2}^{\mu\nu\alpha\beta}(k_{1},k_{2},m_{1})}\\{}&{{}\times\Big[\frac{i{\calP}_{\mu\nu\sigma\rho}}{l^{2}}\Big]\Big[\frac{i{\calP}_{\alpha\beta\gamma\delta}}{(l+q)^{2}}\Big]\Big[\frac{i}{(l-k_{3})^{2}-m_{2}^{2}}\Big]}\\\end{aligned}
S_{2}=\frac{1}{16\pi}\int\sqrt{-g}\,d^{4}x\,(R-(\nabla\sigma)^{2}-\frac{1}{2}(\nabla\varphi)^{2}-\frac{1}{4}e^{-2\sigma-\varphi}(B_{\mu\nu}^{s})^{2}-\frac{1}{4}e^{2\sigma-\varphi}(F_{\mu\nu}^{s})^{2}).
\left(q_{1}q_{2}q_{3}\right)_{1}^{Q}=L_{1}\left(f_{11}^{Q}f_{22}^{Q}f_{33}^{Q}\right)\left(q_{1}q_{2}q_{3}\right)^{Q}=\left({\bff}_{11}^{Q}\bar{{\bff}}_{11}^{Q}q_{1}\right)q_{2}q_{3}=\left(q_{1}q_{2}q_{3}\right)_{1}^{Q}=inv,
H_{p}=\intd^{2}{\bfx}\,\left[-\sum_{n=0}^{\infty}A_{0}^{(n)a}\left(\partial_{i}\Pi_{(0)}^{ia}+f^{abc}\sum_{m=0}^{\infty}A_{i}^{(m)b}\Pi_{(m)}^{ic}+\frac{\kappa}{2}\epsilon^{ij}\partial_{i}A_{j}^{(0)a}\right)+\sum_{n=0}^{\infty}\lambda_{\mu}^{(n)a}\Omega_{(n)}^{\mua}\right].
{\calJ}_{i}^{\mu}=\frac{-1}{1+4{\Phi}^{2}}\biggl(\left(\delta^{ij}+4\Phi^{i}\Phi^{j}\right)\varepsilon^{\mu\nu}\partial_{\nu}\Phi^{j}+2\varepsilon^{ijk}\Phi^{j}\partial^{\mu}\Phi^{k}\biggr).
{\cal{S}}_{(r)}^{[p_{1},...,p_{N}]}\;\;=\;\;-\left(\prod_{i=1}^{N}\frac{1}{p_{i}!}\right)\intd^{D}x\;A^{{\mu}_{1}^{1}...{\mu}_{p_{1}}^{1}...{\mu}_{1}^{i}...{\mu}_{p_{i}}^{i}...{\mu}_{p_{N}}^{N}}{\frac{1}{{\square}^{\,r}}}E_{{\mu}_{1}^{1}...{\mu}_{p_{1}}^{1}...{\mu}_{1}^{i}...{\mu}_{p_{i}}^{i}...{\mu}_{p_{N}}^{N}}
W_{\mathrm{het.}}|_{\beta=\infty}=\int_{\calF}{\frac{d^{2}\tau}{4\tau_{2}^{2}}}(2\pi\tau_{2})^{-5}|\eta(\tau)|^{-16}{\frac{1}{8}}{\frac{1}{{\bar{\eta}}^{4}}}\{({\bar{\Theta}}_{3}^{4}-{\bar{\Theta}}_{4}^{4}-{\bar{\Theta}}_{2}^{4})\left[({\frac{\Theta_{3}}{\eta}})^{8}+({\frac{\Theta_{4}}{\eta}})^{8}+({\frac{\Theta_{2}}{\eta}})^{8}\right]^{2}\}\quad.
\Psi=\psi+\psi_{\alphaA}\lambda_{\alphaA}+\frac{1}{2}\\psi_{\alphaA\betaB}\lambda_{\alphaA}\lambda_{\betaB}+{\dots}+\frac{1}{\Lambda!}\\psi_{\alpha_{1}A_{1},\dots,\alpha_{\Lambda}A_{\Lambda}}\lambda_{\alpha_{1}A_{2}}\dots\lambda_{\alpha_{\Lambda}A_{\Lambda}}\.
\begin{array}{lll}{Q}&{=}&{\left(\begin{array}{ll}{0}&{\mathrm{e}^{\frac{\theta}{2}-\frac{i\pi}{4}}}\\{\mathrm{e}^{\frac{\theta}{2}+\frac{i\pi}{4}}}&{0}\\\end{array}\right)\;,}\\{\overline{{Q}}}&{=}&{\left(\begin{array}{ll}{0}&{\mathrm{e}^{-\frac{\theta}{2}+\frac{i\pi}{4}}}\\{\mathrm{e}^{-\frac{\theta}{2}-\frac{i\pi}{4}}}&{0}\\\end{array}\right)}\\\end{array}
E=\intd^{3}xT^{00}=\intd^{3}x\left[{\frac{1}{4}}F_{ij}^{a}F^{aij}+{\frac{1}{2}}F_{0i}^{a}F^{a0i}+{\frac{A^{2}}{2}}D_{i}\phi^{a}D^{i}\phi^{a}+{\frac{A^{2}}{2}}D_{0}\phi^{a}D^{0}\phi^{a}\right]
\rho(0)=\sum_{n=0}^{\infty}\sum_{m,m^{\prime}\nen}\frac{1}{\Omega}e^{-n\beta\omega}{C^{\prime}}_{m,n}{C^{\prime}}_{m,^{\prime}n}^{*}|\psi_{+}^{[0]}(m,p-mk)><\psi_{+}^{[0]}(m^{\prime},p-m^{\prime}k\hbar)|
\begin{array}{lcr}{x}&{=}&{\alpha_{1}\displaystyle\frac{\alpha_{3}\pm||\alpha||}{(\alpha_{1}^{2}+\alpha_{2}{}^{2})}\,,}\\{y}&{=}&{\displaystyle-\alpha_{2}\frac{\alpha_{3}\pm||\alpha||}{(\alpha_{1}^{2}+\alpha_{2}^{2})}\,,}\\\end{array}
S_{GF+FP}=-\intd^{4}xi\delta_{B}\left[\bar{C}^{a}\left\{D_{\mu}[a]A^{\mu}+{\frac{\alpha}{2}}B\right\}^{a}-i{\frac{\alpha}{2}}gf^{abi}\bar{C}^{a}\bar{C}^{b}C^{i}-i{\frac{\alpha}{4}}gf^{abc}C^{a}\bar{C}^{b}\bar{C}^{c}\right].
\tilde{\Gamma}_{cad}(p_{1},p_{2},p_{3})=\frac{1}{i^{3}}\,D_{cc^{\prime}}^{-1}(p_{1})\,D_{aa^{\prime}}^{-1}(p_{2})\,D_{dd^{\prime}}^{-1}(p_{3})\,\Gamma_{c^{\prime}a^{\prime}d^{\prime}}(p_{1},p_{2},p_{3})
\frac{\ddot{a}}{a}=\frac{-1}{3M_{Pl}^{2}\sqrt{1+4\alpha\Lambda_{b}/3M^{5}}}\bigg(\frac{\rho_{\mathrm{vis}}}{2}(1+\frac{2\rho_{\mathrm{vis}}}{\Lambda_{2}^{+}})+\frac{3p_{\mathrm{vis}}}{2}(1+\frac{\rho_{\mathrm{vis}}}{\Lambda_{2}^{+}})\bigg)>0
H_{\lambda}\;=\;\frac12\;\sum_{i=1}^{d}\beta\;(P_{i},P_{i})\;+\;\frac12\;\sum_{i_{1}<\cdots<i_{M}}\beta\;\bigl([X_{i_{1}},\cdots,X_{i_{M}}]_{\lambda},[X_{i_{1}},\cdotsX_{i_{M}}]_{\lambda}\bigr),
\Theta^{ij}=\frac{2\pi\alpha^{\prime}e}{g_{1}(1-e^{2})}(\delta_{0}^{i}\delta_{1}^{j}-\delta_{1}^{i}\delta_{0}^{j})+\frac{2\pi\alpha^{\prime}b}{g_{2}(1+b^{2})}(-\delta_{2}^{i}\delta_{3}^{j}+\delta_{3}^{i}\delta_{2}^{j}),
[f_{1},f_{2}]_{*^{(1)}}(\xi)=[f_{1},f_{2}]_{M(\hbar\kappa\lambda)}(\xi)+\hbar^{2}c^{(n)}f_{1}(\xi)[\overleftarrow\partial\!\!_{\alpha}]^{n}\varepsilon^{[\alpha]_{n}}\varepsilon^{[\beta]_{n}}[\partial_{\beta}]^{n}f_{2}(\xi),
{\calH}=\frac{1}{2}\pi^{a}\pi^{a}+V(\phi^{a}\phi^{a})+\frac{1}{2}(\vec{D}\phi)^{a}(\vec{D}\phi)^{a}+\frac{1}{2}(\vec{E}^{a}\vec{E}^{a}+\vec{B}^{a}\vec{B}^{a})\,
E=-\frac{3\xi^{2}}{64a}\left[\sum_{l=1}^{\infty}\nu^{0}-\frac{9}{128}\sum_{l=1}^{\infty}\frac{1}{\nu^{2}}+\frac{423}{16384}\sum_{l=1}^{\infty}\frac{1}{\nu^{4}}+{\calO}\left(\frac{1}{\nu^{6}}\right)\right]{.}
P_{2n}^{(\alpha+1)}(\lambda)=\frac{\tilde{P}_{2n}^{(\alpha+1)}(\lambda)}{\tilde{P}_{2n}^{(\alpha+1)}(0)},\\P_{2n+1}^{(\alpha+1)}(\lambda)=\frac{\tilde{P}_{2n+1}^{(\alpha+1)}(\lambda)}{\tilde{P}_{2n+1}^{(\alpha+1)}{}^{\prime}(0)}.
D_{a\alpha}\mathrm{v}^{\beta}{}_{\gamma}(\theta)=\textstyle{\frac{2}{3}}\delta_{\alpha}{}^{\beta}D_{a\delta}\mathrm{v}^{\delta}{}_{\gamma}(\theta)-\textstyle{\frac{1}{3}}\delta^{\beta}{}_{\gamma}D_{a\delta}\mathrm{v}^{\delta}{}_{\alpha}(\theta)\,,
\begin{array}{l}{\overline{{x^{i}}}=C_{ji}x^{j}~,}\\{\overline{{\partial_{i}}}=-q^{-2}C_{ij}\Lambda^{-1}\displaystyle\left(q^{-1/2}[\partial_{+}\partial_{-},x^{j}]+\frac{q}{1+q}[\partial_{0}\partial_{0},x^{j}]\right)~,}\\\end{array}
Z[j_{1},j_{2}]=Tr\left[\rho\bar{T}{e^{i\intd^{4}x(H_{s}+j_{2}\phi_{s}^{*}+j_{2}^{*}\phi_{s})}}T{e^{-i\intd^{4}x(H_{s}-j_{1}\phi_{s}^{\ast}-j_{1}^{\ast}\phi_{s})}}\right]
\gamma(\zeta)=\frac{(p^{2};p^{2})_{\infty}}{(p;p)_{\infty}^{2}}\frac{\Theta_{p^{2}}(q^{2})\Theta_{p^{2}}(p\zeta^{2})}{\Theta_{p^{2}}(q^{2}\zeta^{2})}\frac{g(\zeta^{2})}{\mathrm{snh}(\lambda)},
\left(\begin{matrix}{\vec{F}_{ij}}\\{\vec{H}_{ij}}\\\end{matrix}\right)={\frac{1}{\sqrt2}}\epsilon_{ijm}{\frac{x^{m}}{|\vec{x}|^{3}}}\left(\begin{matrix}{\vec{P}_{\mathrm{gr}}+\vec{P}_{\mathrm{vec}}}\\{\vec{P}_{\mathrm{gr}}-\vec{P}_{\mathrm{vec}}}\\\end{matrix}\right)=\epsilon_{ijm}{\frac{x^{m}}{|\vec{x}|^{3}}}\left(\begin{matrix}{\vec{P}_{F}}\\{\vec{P}_{H}}\\\end{matrix}\right)
\hat{Z}_{L,L^{\prime}}^{(n)}(\{\hat{q}_{N},\overline{{\hat{q}_{N}}}\})=F_{L}(\{\hat{q}_{N}\})\overline{{F_{L^{\prime}}(\{\hat{q}_{N}\})}}Z_{L,L^{\prime}}^{(n)}(\{q_{N},\overline{q}_{N}\})
h={\mathrm{e}}^{a_{0}L_{0}}\;{\mathrm{e}}^{a_{1}L_{1}}\;{\mathrm{e}}^{b_{-1}J_{-1}}\;{\mathrm{e}}^{b_{0}J_{0}}\;{\mathrm{e}}^{b_{1}J_{1}}\;{\mathrm{e}}^{b_{2}J_{2}}\;\tilde{h}\;,
\phi_{(5/3)}(z,t)={\frac{2\phi_{(4/3)}(z,t)\left(2+\sqrt{7}\right)\left({z^{2}}\left(2+\sqrt{7}\right)^{2}-10t\right)z}{27\phi_{(4/3)}^{2}(z,t)-3\cdot5^{\frac{4}{3}}t^{4/3}}}\.
P=-\frac{3}{8}g^{2}+\rhoT\left(1-\frac{2}{3}\frac{\rhoT}{g^{2}}+\frac{16}{9}\left(\frac{\rhoT}{g^{2}}\right)^{2}-\frac{145}{27}\left(\frac{\rhoT}{g^{2}}\right)^{3}+\frac{1472}{81}\left(\frac{\rhoT}{g^{2}}\right)^{4}+\ldots\right)
d{\calT}_{ap;cr}=A_{\mu}^{m}f_{psm}{\calT}_{as;cr}dq^{\mu}+\omega_{\mua}^{b}{\calT}_{bp;cr}dq^{\mu}+\frac{1}{2}R_{ab}{\calT}_{bp;cr}d\tau+f_{psm}F_{ab}^{m}{\calT}_{bs;cr}d\tau
\tilde{D}(j_{3},j_{2},j_{1})=\frac{G(j_{1}+j_{2}+j_{3}+1)G(j_{1}+j_{2}-j_{3})G(j_{1}+j_{3}-j_{2})G(j_{2}+j_{3}-j_{1})}{(\nu^{\prime}(b))^{-j_{1}-j_{2}-j_{3}-1}\;G_{0}^{\prime}\;G(2j_{1})G(2j_{2})G(2j_{3})}.
\langle1|2|3\rangle=C_{123}\prod_{i=1}^{3}\frac{\Gamma(\alpha_{i}/2+1)}{\Gamma(\beta_{i}/2+1)}\\frac{(12)^{\alpha_{1}}(23)^{\alpha_{2}}(31)^{\alpha_{3}}}{({\bf12})^{\beta_{1}}({\bf23})^{\beta_{2}}({\bf31})^{\beta_{3}}}\,,
V_{2b}=\left(\frac{\alpha}{m}\right)^{2b}\,\,e^{4i\alpha\rho\kappa\tau}\,4^{b-i\rho}\,\frac{\Gamma\left(b+i\kappa\right)\Gamma\left(b-i\kappa\right)}{\Gamma\left(i\rho+i\kappa\right)\Gamma\left(i\rho-i\kappa\right)}\,\frac{\Gamma\left(b+i\rho\right)\Gamma\left(b-i\rho\right)}{4\pi\,\Gamma(2b)}\,.
\frac{d^{2}\theta}{dx^{2}}+\left[\frac{2}{x}+\frac{1}{2}\left(\frac{1}{A}\frac{dA}{dx}+\frac{1}{B}\frac{dB}{dx}\right)\right]\frac{d\theta}{dx}-\frac{2e^{2}{\sigma}^{2}{\theta}}{A}=0.
P_{K}(V;{\bfa}_{1},\ldots,{\bfa}_{K},\rho_{1},\ldots,\rho_{K})=\frac{1}{Z(V)}\,\frac{C^{K}}{K!}\prod_{j=1}^{K}\rho_{j}^{\alpha}\\Theta(\{{\bfa}_{j}\},\{\rho_{j}\})\,.
\left(\begin{array}{c}{D^{+}}\\{D^{-}}\\\end{array}\right)\left|j_{3},s=1\right\rangle=\left(\begin{array}{c}{\left|j_{3},s=1;\,l+1,-\frac{1}{2}\right\rangle}\\{\left|j_{3},s=1;\,l,+\frac{1}{2}\right\rangle}\\\end{array}\right),
T_{0}^{t}h_{3k,3k+1}(q,J)T_{0}=\frac{J}{4}\xi^{2}[\sigma_{k}^{x}\sigma_{k+1}^{x}+\sigma_{k}^{y}\sigma_{k+1}^{y}+\frac{q+q^{-1}}{2}\sigma_{k}^{z}\sigma_{k+1}^{z}
\partial_{\mu}J^{\mu}=-g{\frac{\bfa}{(2\pi)^{3}}}\epsilon^{\mu\nu\rho\sigma\tau\omega}F_{\mu\nu}^{5}\left[\,e^{2}\,F_{\rho\sigma}F_{\tau\omega}+g^{2}F_{\rho\sigma}^{5}F_{\tau\omega}^{5}\right]
\langle\intd^{4}x\,{\calO}_{\chi}\rangle=-\left.\frac{\partial}{\partial\chi_{0}}\langlee^{-\chi_{0}\int{\calO}_{\chi}}\rangle\right|_{\chi_{0}=0}=\left.-\frac{\partialS_{\mathrm{sugra}}(\chi_{0})}{\partial\chi_{0}}\right|_{\chi_{0}=0}
u={\calG}_{3}(\tau)={\frac{2\wp\left({\frac{1+\tau}{2}}\right)-\wp\left({\frac{\tau}{2}}\right)-\wp\left({\frac{1}{2}}\right)}{\wp\left({\frac{\tau}{2}}\right)-\wp\left({\frac{1}{2}}\right)}}=1-2\left[\frac{\Theta_{2}(0|\tau)}{\Theta_{3}(0|\tau)}\right]^{4},
S_{eff}=\int{dx\left({\Pi^{1}\stackrel{\bf{\cdot}}{A_{1}}+{\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)}.
\int_{0}^{\infty}drr^{\beta+2}G_{n\ell}\left(r,u\right)G_{n\ell^{\prime}}\left(r,v\right)=\sum_{\eta\eta^{\prime}}C_{\eta\eta^{\prime}}\left(n,\beta,\ell,\ell^{\prime}\right)u^{\eta}u^{\eta^{\prime}}.
\hat{T}{\bf\Psi}=0\;,\;\;\hat{T}=\frac{ism\Sigma^{3}}{2\hat{\omega}}\hat{\zeta}\left[\hat{\zeta}\hat{\omega}\Sigma^{3}+i\partial_{1}(i\Sigma^{2})+i\partial_{2}(-i\Sigma^{1})-sm\right]\;.
S_{E}^{vol}=\sum_{a}\frac{V_{a}}{4G_{d}}+\frac{\beta}{16\piG_{d}}\sum_{a}\int_{M_{a}^{d-2}}F\wedge\Psi-\frac{\beta}{8\piG_{d}}\int_{\partial\Sigma_{\infty}}J_{D}^{i}d\sigma_{i}.
\begin{array}{l}{2(a_{2}b_{2}+a_{4}b_{0})+a_{3}b_{1}-a_{4}a_{1}=0,}\\{2(2a_{1}b_{2}+a_{3}b_{0})-a_{3}a_{1}-4a_{4}a_{0}=0}\\{a_{0}(a_{3}-2b_{2})=0,\quada_{3}b_{2}+2a_{4}b_{1}=0}\\{a_{4}a_{0}=0,\quada_{4}a1=0,\quada_{3}^{2}-4a_{4}a_{2}=0}\\\end{array}
\times\Bigg\{\sum_{\sigma\inZ_{N}}\frac{w(\omegau_{2}^{\prime}u_{3},\sigma+j_{2}+j_{3})w(qu_{2}^{\prime\prime},\sigma)s(\sigma,j_{1})}{w(q^{-1}u_{2}^{\prime},\sigma+j_{2})w(u_{2}^{\prime\prime}u_{3},\sigma+i_{3})}\Bigg\}_{0}
\nabla_{c}\nabla_{d}C_{(a\b)}^{\\c\d}=e^{-2\sigma}\left\{\overline{\nabla}_{c}\overline{\nabla}_{d}\overline{C}_{(a\b)}^{\\c\d}+(d-3)\overline{C}_{(a\b)}^{\\c\d}\sigma_{;cd}+2(d-4)\sigma_{d}\overline{\nabla}_{c}\overline{C}_{(a\b)}^{\\c\d}+(d-3)(d-5)\overline{C}_{(a\b)}^{\\c\d}\\sigma_{c}\sigma_{d}\right\}\,.
U_{N}(\vec{r}^{N})=\mathrm{exp}[i\{\sum_{L=1}^{N-1}\alpha_{L}(\vec{r}^{N})\hat{J}_{L}([\vec{r}^{N}])+\sum_{j=1}^{k-1}\sum_{k=2}^{N}\sum_{i=1}^{2}\alpha_{jk}^{(i)}(\vec{r}^{N})\hat{J}_{jk}^{(i)}([\vec{r}^{N}])\}]
\delta\Gamma^{\sigma}\left(p^{\prime},p\right)=\int\frac{d^{4}k}{(2\pi)^{4}}\Biggl\{\frac{g^{2}}{m^{2}}\Lambda_{s^{\prime}r^{\prime}}^{\alpha\beta}(p-k)\left[iG_{\alpha\beta\mu\nu}(p-k)\right]\Lambda_{rs}^{\mu\nu}(p-k)\left[iS_{F}(k)_{lr}\right]\left(-\gamma_{l^{\prime}l}^{\sigma}\right)\left[iS_{F}(k^{\prime})_{r^{\prime}l^{\prime}}\right]\Biggr\}.
S\;=\;\intd^{3}x\;\left[-\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi-\frac{1}{2}{\bar{\psi}}\not\!\partial\psi-\frac{g^{2}}{2}(\varphi^{2}-(\frac{\kappa}{g})^{2})^{2}-g\varphi{\bar{\psi}}\psi\right]\;.
\left[e^{i\gamma\varphi}\right]_{n}(\theta_{1,}\theta_{2,}\dots,\theta_{n})=\left[e^{i\gamma\varphi}\right]_{1}(\theta_{1})\,\left[e^{i\gamma\varphi}\right]_{n-1}(\theta_{2,}\dots,\theta_{n})+O(e^{-\theta_{1}})\,.