harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left(\vec{A}_{(1)}^{\T}\eta\partial\vec{A}_{(2)}\right)\left(\partial\vec{A}_{(2)}^{\T}\eta\vec{A}_{(2)}\right)=-{\textstyle\frac{1}{2}}\left(\vec{A}_{(1)}^{\T}\eta\vec{A}_{(2)}\right)\left(\partial\vec{A}_{(2)}^{\T}\eta\partial\vec{A}_{(2)}\right)\,.
S_{_{\mathrm{4F}}}=\frac{\pi^{2}}{16\rho^{2}g^{2}}\,\varepsilon_{ABCD}\left[\sum_{i=1}^{N-2}{\overline{\mu}}_{i}^{A}\mu_{i}^{B}\right]\left[\sum_{j=1}^{N-2}{\overline{\mu}}_{j}^{C}\mu_{j}^{D}\right]\,.
M_{ij}^{(N)k}=[M_{ij}^{{}k}+{\frac{1}{4\pic}}(x_{i}(\partial_{l}A_{j})F^{lk}-x_{j}(\partial_{l}A_{i})F^{lk})]+{\frac{1}{4\pic}}(-A_{i}F_{j}^{k}+A_{j}F_{i}^{k}),
S[V^{++},\omega]=\frac{1}{2g^{2}}\mathrm{tr}\int\mathrm{d}^{4}x\mathrm{d}^{4}\theta\,W^{2}-\frac{1}{2g^{2}}\mathrm{tr}\intd\zeta^{(-4)}\,\nabla^{++}\omega\nabla^{++}\omega
\begin{array}{l}{{\calF}^{ijk}(X_{1\pm})=D_{1i^{\prime}}^{i}(I(Z_{1})D_{3k^{\prime}}^{k}(I(Z_{1})){\calH}^{i^{\prime}jk^{\prime}}(X_{1\pm})}\\{{}}\\{{\calG}^{ijk}(X_{2\pm})=D_{1i^{\prime}}^{i}(I(Z_{2})){\calH}^{i^{\prime}jk}(I(Z_{2})X_{2\mp})}\\\end{array}
H\,=\,\frac{1}{2\,R}\,\sum_{n=1}^{\infty}\,\sum_{j=1}^{D-2}\,(\alpha_{nj}\alpha_{nj}^{+}\,+\,\alpha_{nj}^{+}\alpha_{nj})\,-\,\frac{1}{2R}\,\sum_{n=1}^{\infty}\,\sum_{j=1}^{D-2}\,(\beta_{nj}\beta_{nj}^{+}\,+\,\beta_{nj}^{+}\beta_{nj})\,{.}
q_{\mu}\Gamma_{abc}^{\mu}(q,p)=\delta_{ab}\left[S_{ca}^{-1}(p+q)-i\epsilon\left[V^{-1}(p_{0}+q_{0})\right]_{ca}^{2}\right]-\delta_{ac}\left[S_{ab}^{-1}(p)-i\epsilon\left[V^{-1}(p_{0})\right]_{ab}^{2}\right]\,.
\Gamma(\tau)\toS\Gamma(\tau)S^{-1}-\frac{dS}{d\tau}S^{-1},\\\\Gamma(\tau)=i\Gamma_{\mu\nu}(\tau)\sigma^{\mu\nu},\\\\sigma^{\mu\nu}\stackrel{\mathrm{def}}{=}\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]
ds_{5}^{2}=\frac{1}{z^{2}}\left[-e^{2T(r,z)}dt^{2}+e^{2R(r,z)}dr^{2}+e^{2S(r,z)}r^{2}d\Omega_{2}^{2}+e^{2Z(r,z)}dz^{2}+2e^{2v(r,z)}drdz\right]
\Gamma^{(2)}(sP,\lambda,m,\mu,T)=s^{2}\Gamma^{(2)}(P,\lambda,{\frac{m}{s}},{\frac{\mu}{s}},{\frac{T}{s}})=s^{2}Z(s)\Gamma^{(2)}(P,\bar{\lambda},{\frac{\bar{m}}{s}},\mu,{\frac{T}{s}}),
A_{4}=\pi^{n/2}\frac{\Gamma(-\frac{1}{2}\alpha(s))\Gamma(-\frac{1}{2}\alpha(t))\Gamma(-\frac{1}{2}\alpha(u))}{\Gamma(-\frac{1}{2}\alpha(s)-\frac{1}{2}\alpha(t))\Gamma(-\frac{1}{2}\alpha(t)-\frac{1}{2}\alpha(u))\Gamma(-\frac{1}{2}\alpha(u)-\frac{1}{2}\alpha(s))}.
\hat{Z}(t)=\left\{\prod_{n=1}^{\infty}\frac{1}{(1-e^{-tn})}\right\}^{D}\left\{\prod_{n,m=1}^{\infty}\frac{1}{1-e^{-t(n+m)}}\right\}^{\frac{D^{2}}{2}}\left\{\prod_{n=1}^{\infty}(1-e^{-2nt})\right\}^{\frac{D}{2}}\,.\nonumber
\frac{\Upsilon_{0,p}[\alpha,\beta]}{\vert\Xi_{0,p}[\alpha,\beta]\vert^{2}{\calP}(\beta)}=\frac{\Upsilon_{0,p}[\alpha,\beta+s]}{\vert\Xi_{0,p}[\alpha,\beta+s]\vert^{2}{\calP}(\beta+s)}\psi_{0,p}[\alpha,\beta,s]
\begin{array}{l}{{[}U_{a},U^{b}{]}=S_{a}^{b}\ing^{0}}\\{{[}U_{a},U_{b}{]}=K_{ab}\ing^{+2}}\\{{[}U^{a},U^{b}{]}=K^{ab}\ing^{-2}}\\{{[}S_{a}^{b},U_{c}{]}=U_{(abc)}\ing^{+1}}\\\end{array}
\langle\varphi^{2}(x)\rangle^{(b)}=\frac{-1}{2^{D-2}\pi^{(D+1)/2}\Gamma\left(\frac{D-1}{2}\right)}\int_{0}^{\infty}dk\,k^{D-2}\int_{0}^{\infty}d\omega\,\frac{\bar{I}_{\omega}(\lambdaa)}{\bar{K}_{\omega}(\lambdaa)}K_{\omega}^{2}(\lambda\xi),\quad\xi>a.
+\frac12\Biggl[\partial_{\mu}\Biggl(x_{\lambda}\delta_{\nu\rho}-x_{\rho}\delta_{\nu\lambda}\Biggr)+\partial_{\nu}\Biggl(x_{\rho}\delta_{\mu\lambda}-x_{\lambda}\delta_{\mu\rho}\Biggr)\Biggr]{\calD}_{1}\left(x^{2}\right).
t=E{\int}{dr\,\Bigl(1+{\frac{4M}{r}}-{\frac{4q^{2}}{r^{2}}}\Bigr)\left({E^{2}\Bigl(1+{\frac{4M}{r}}-{\frac{4q^{2}}{r^{2}}}\Bigr)-{\frac{L^{2}}{r^{2}}}-m^{2}\sqrt{1+{\frac{4M}{r}}-{\frac{4q^{2}}{r^{2}}}}}\right)^{-1/2}}\.
\begin{array}{rl}{0}&{=\nabla_{\rho}h^{\rhozI_{3}\ldotsI_{s}}=}\\{}&{=\partial_{z}h^{zzI_{3}\ldotsI_{s}}+\partial_{I}h^{zII_{2}\ldotsI_{s}}-{\frac{1}{z}}[d+2s-2]h^{zzI_{3}\ldotsI_{s}}}\\\end{array}
D_{\mu}{\calO}_{i,i+1}^{ab}(x)=\left[\partial_{\mu}{\calO}_{i,i+1}^{ab}(x)+\omega_{\mu;n}^{a}(x,i){\calO}_{i,i+1}^{nb}(x)-{\calO}_{i,i+1}^{an}(x){\omega_{\mu;n}}^{b}(x;i+1)\right]\,.
V_{eff}(\phi)|_{\mathrm{torus}}=-\sum_{I}(-)^{F_{I}}\,\frac{\prod_{i=1}^{d}\,R_{i}}{32\,\pi^{\frac{4-d}{2}}}\,\,\sum_{\vec{n}}e^{2\pii\sum_{i}n_{i}a_{i}^{I}}\,\,\int_{0}^{\infty}dl\,\,l^{\frac{2+d}{2}}f_{s}(l)\,\,e^{-\pi^{2}l\sum_{i}n_{i}^{2}R_{i}^{2}}
\begin{array}{ccc}{\{c_{r}^{a},\overline{{c}}_{s}^{b}\}=2\delta_{rs}\eta^{ab}}&{r,s\neq0}&{}\\{\{d_{r}^{a},\overline{{d}}_{s}^{b}\}=2\delta_{rs}\eta^{ab}}&{}&{}\\\end{array}
\begin{array}{ll}{\lambda_{\theta}=-p_{n}i\sigma^{n}\bar{\psi},}&{\qquad\lambda_{\bar{\theta}}=i\psi\sigma^{n}p_{n},}\\{\lambda_{\rho}=-2p_{n}i\sigma^{n}\lambda_{\bar{\theta}}\approx0,}&{\qquad\lambda_{\bar{\rho}}=2i\lambda_{\theta}\sigma^{n}p_{n}\approx0.}\\\end{array}
\Pi_{\mathrm{L}}=\frac{p^{2}}{\hat{p}^{2}}\,u_{\mu}u_{\nu}\Pi^{\mu\nu},\quad\tilde{\Pi}_{\mathrm{T}}=\frac{\tilde{p}_{\mu}\tilde{p}_{\nu}}{\tilde{p}^{2}}\,\Pi^{\mu\nu},\quad(D-3)\Pi_{\mathrm{T}}=\eta_{\mu\nu}\Pi^{\mu\nu}-\Pi_{\mathrm{L}}-\tilde{\Pi}_{\mathrm{T}}
z_{0}^{2}\varepsilon^{2}<{\frac{1}{32}}\left(-32\omegaz_{0}^{2}+9l^{2}-3\sqrt{-64\omegaz_{0}^{2}l^{2}+9l^{4}-64l^{2}z_{0}^{2}\omega^{2}}\right)/(\omega^{2}l^{2})\,.
\partial_{\mu_{(D-d+1)}}\phi^{a}\partial_{\mu_{(D-d-2)}}\phi^{a_{2}}\cdots\partial_{\mu_{D}}\phi^{a_{d}}\frac\partial{\partial\phi^{a}}\frac\partial{\partial\phi^{a_{1}}}(\frac1{||\phi||^{d-2\}}).
\rho_{m_{0}\rightarrowm_{1}+m_{2}+m_{3}}^{D}\!=\frac{1}{4\pi^{3}}\int_{0}\!\!\int_{0}\!\!\int_{0}\!\!d\mu_{1}\,d\mu_{2}\,d\mu_{3}\frac{\mu_{1}\mu_{2}\mu_{3}}{\lambda_{E}(\mu_{1}^{2},\mu_{2}^{2},\mu_{3}^{2})}\rho_{m_{0}\rightarrowm_{1\mu_{1}}+m_{2\mu_{2}}+m_{3\mu_{3}}}^{D-2}.
\sigma^{;\alpha\beta}=g^{\alpha\beta}-\frac{1}{3}R_{~~\gamma~~\delta}^{\alpha~~\beta~~}\sigma^{;\gamma}\sigma^{;\delta}+\frac{1}{12}R_{~~\gamma~~\delta;\rho}^{\alpha~~\beta~~}\sigma^{;\gamma}\sigma^{;\delta}\sigma^{;\rho}+O(\sigma^{2})~.
\Gamma_{\LambdaR_{1}}=\Gamma_{\LambdaR_{1}}^{0}+\hbar\sum_{n=0}\Lambda^{-n}\Gamma_{\LambdaR_{1}}^{\left(1\right)n}+\hbar^{2}\sum_{n=-2}\Lambda^{-n}\Gamma_{\LambdaR_{1}}^{\left(2\right)n}+O\left(\hbar^{3}\right)
\gamma_{i}=\sum_{a=1}^{N}\frac{1}{2}\mathrm{sign}(\pi_{a})\int\frac{\lambda_{a}^{N-i}d\lambda_{a}}{\left|A(\lambda_{a})\right|}\sqrt{\left|\frac{A(\lambda_{a})}{\lambda_{a}^{N+1}-(\alpha-1)\lambda_{a}^{N}+(1-\alpha+\beta)\lambda_{a}^{N-1}}\right|}
f^{2}g^{2}{\frac{\partial^{2}}{\partialr^{*2}}}t+f\{fg^{\prime\prime}+f^{\prime}g^{\prime}\}{\frac{\partial}{\partialr^{*}}}t-\{\sqrt2ff^{\prime}-2f(1-f)\}t-{\frac{\partial^{2}}{\partialt^{2}}}t=0,
\hat{\lambda}_{\hat{\alpha}}=\left(\left(\begin{matrix}{0}\\{\lambda_{\alpha^{\prime}}}\\\end{matrix}\right)\otimes\left(\begin{matrix}{1}\\{0}\\\end{matrix}\right)\right)+\left(\left(\begin{matrix}{\lambda_{\alpha}}\\{0}\\\end{matrix}\right)\otimes\left(\begin{matrix}{0}\\{1}\\\end{matrix}\right)\right)\.
{\calK}\,=\,-\mathrm{log}\left(\mathrm{i}\langle\Omega\,\vert\,\bar{\Omega}\rangle\right)\,=\,-\mathrm{log}\left[\mathrm{i}\left({\bar{X}}^{\Lambda}F_{\Lambda}-{\bar{F}}_{\Sigma}X^{\Sigma}\right)\right]
\deltax(\theta)=\epsilon(\theta)x^{\prime}(\theta)\,;\;\delta(x,x)=\oint\frac{d\theta}{2\pi}\epsilon(\theta)2x_{\alpha}(\theta)x_{\alpha}^{\prime}(\theta)=-\oint\frac{d\theta}{2\pi}\epsilon^{\prime}(\theta)x_{\alpha}^{2}(\theta)
\Psi(x)=\left(f(x)+{\calF}^{\mu}(x)\widehat{b}_{\mu}^{+}+\frac{1}{2!}\varepsilon_{\mu\nu\lambda}{\tilde{\calF}}^{\mu}(x)\widehat{b}^{+\nu}\widehat{b}^{+\lambda}+\frac{1}{3!}\tilde{f}(x)\varepsilon_{\mu\nu\lambda}\widehat{b}^{+\mu}\widehat{b}^{+\nu}\widehat{b}^{+\lambda}\right)|0\rangle.
\left(\begin{array}{ll}{\quad\tilde{d}}&{-\tilde{b}/q}\\{-q\tilde{c}}&{\quad\tilde{a}}\\\end{array}\right)=\left(\begin{array}{ll}{a^{*}}&{c^{*}}\\{b^{*}}&{d^{*}}\\\end{array}\right)\;.
\sum_{\lambda,\kappa=0}^{3}A_{\mu\nu}^{\lambda}(x)C_{\lambda}^{\kappa}\frac{\partial}{\partialy^{\kappa}}=\nabla_{(C\frac{\partial}{\partialy})_{\mu}}(C\frac{\partial}{\partialy})_{\nu}=\sum_{\lambda,\kappa=0}^{3}C_{\mu}^{\lambda}\left(\frac{\partialC_{\nu}^{\kappa}}{\partialy^{\lambda}}+\sum_{\sigma=0}^{3}A_{\lambda\sigma}^{\kappa}(y)C_{\nu}^{\sigma}\right)\frac{\partial}{\partialy^{\kappa}},
\mathrm{d}s^{2}=\frac{r+2m}{r}\,\mathrm{d}r^{2}+\frac{r(r+2m)}{S^{2}}\left[(\Sigma^{1})^{2}+(\Sigma^{2})^{2}\right]+\frac{4rm^{2}}{r+2m}\left[\Sigma^{3}\right]^{2}\,.
R+\frac{\gamma}{4}(\frac{(D\Omega)^{2}}{\Omega^{2}}-2\frac{D\cdotD\Omega}{\Omega})+\frac{d}{d\Omega}(V(\Omega)\Omega)-\frac{1}{4}(1-\frac{\epsilon}{2})\Omega^{-\epsilon/2}F^{2}=0,
\tilde{\Gamma}_{t,s}^{(4)}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4})={\Gamma}_{t}^{(4)}(e^{-s}{\bf{q}}_{1},e^{-s}{\bf{q}}_{2},e^{-s}{\bf{q}}_{3},e^{-s}{\bf{q}}_{4})\;,
\delta{\calE}^{2,0}=\frac{K}{a_{0}^{6}}\left[\left(\frac{a_{0}}{\Lambda}\right)^{3}-\left(\frac{a_{0}}{\Lambda}\right)^{2}+\frac{1}{4}\left(\frac{a_{0}}{\Lambda}\right)\right]e^{-\Lambda/a_{0}}\,{.}
D_{\alpha}=\frac{\partial}{\partial\theta^{\alpha}}-i\sigma_{\alpha\dot{\alpha}}^{\mu}\bar{\theta}^{\dot{\alpha}}\partial_{\mu}\,,\quad\bar{D}_{\dot{\alpha}}=-\frac{\partial}{\partial\bar{\theta}^{\dot{\alpha}}}+i\theta^{\alpha}\sigma_{\alpha\dot{\alpha}}^{\mu}\partial_{\mu}\,\,,
\left|\frac{1}{\eta(\tau)}\right|^{24}\left|\theta_{2}(0,\tau)\right|^{8}=2^{8}\prod_{n=1}^{\infty}\left|\frac{1+e^{2\piin\tau}}{1-e^{2\piin\tau}}\right|^{16}\equiv2^{8}\left|\theta_{4}\left(0,2\tau\right)\right|^{-16}
\,\\\mathrm{or\\as}\\\,\\{\cal{L}}_{BI}=\beta^{2}\,\Biggl[\,\\1-\sqrt{1+\frac{1}{2\beta^{2}}\,({\bfB}^{2}-{\bfE}^{2})-\frac{1}{16\beta^{4}}\,({\bfE}\cdot{\bfB})^{2}}\,\\\,\Biggr]\,.
Z=\int{\calD}A_{-}{\calD}\pi^{-}{\calD}\varphi{\calD}\pi_{\varphi}{\calD}\theta^{1}{\calD}\pi^{1}{\calD}\theta^{2}{\calD}\pi^{2}\prod_{i,j=1}^{2}\delta(\widetilde{\top}_{i})\delta(\Gamma_{j})\mathrm{det}\mid\{\widetilde{\top}_{i},\Gamma_{j}\}\mide^{iS},
\Gamma_{\mu_{1}\mu_{2}\mu_{3}}^{a_{1}a_{2}a_{3}}\left(k_{1},k_{2},k_{3}\right)=\frac{g^{3}}{\left(2\pi\right)^{3}}\int\frac{\mathrm{d}^{3}{\bfQ}}{2|{\bfQ}|}N\left(|{\bfQ}|\right)\left[{\calS}_{\mu_{1}\mu_{2}\mu_{3}}^{a_{1}a_{2}a_{3}}\left(k_{1},k_{2},k_{3},Q\right)+Q\rightarrow-Q\right],
h_{00}(x)=\frac{2G_{4}m_{*}}{r}\bigg[1+\frac{2(1-\gamma)}{3(1+\gamma)}\,\frac{\ell^{2}}{r^{2}}\bigg],\,~~~h_{ij}(x)=\frac{2G_{4}m_{*}}{r}\bigg[1+\frac{(1-\gamma)}{3(1+\gamma)}\,\frac{\ell^{2}}{r^{2}}\bigg]\,\delta_{ij}\,.
\begin{array}{ccc}{{\calE}}&{=}&{-\int_{M_{3}}<(D^{A}\Phi_{A}\otimesI_{E})\wedge(H^{A}\otimesI_{E})>}\\{}&{=}&{-\int_{M_{3}}d<\Phi_{A}H^{A}>=-\int_{S^{2}}<\Phi_{A}H^{A}>,}\\\end{array}
L_{\psi}^{\mathrm{cutoff}}=-\frac{1}{2}\int\frac{d\nu}{2\pi}\sum_{i=1}^{2}\frac{\omega^{3}}{\nu^{2}}\frac{v_{i}\bar{\gamma}_{i}^{2}}{\omega^{2}\Sigma((-1)^{i}\bar{R}_{i})}\left(G_{\psi}^{ii}(-i\nu)-G_{\psi}^{ii}(0)\right)\,.
x=\left(\begin{array}{cc}{u_{1}^{\ast}v_{1}+v_{2}^{\ast}u_{2}}&{v_{1}u_{2}^{\ast}-v_{2}^{\ast}u_{1}}\\{v_{2}u_{1}^{\ast}-v_{1}^{\ast}u_{2}}&{v_{2}u_{2}^{\ast}+v_{1}^{\ast}u_{1}}\\\end{array}\right)(\midu_{1}\mid^{2}+\midu_{2}\mid^{2})^{-1}
G_{A}(x,y)=\Sigma\!\!\!\!\!\!\int\frac{d^{4}p}{(2\pi)^{4}}E_{p}(x)\frac{1}{\gamma\cdot\bar{p}+\tilde{\Sigma}_{A}(\bar{p})}\bar{E}_{p}(y),{}~~~\Sigma\!\!\!\!\!\!\intd^{4}p\equiv\sum_{k}\intdp_{0}dp_{2}dp_{3},
S_{AB}^{\prime}=\left(\begin{matrix}{S_{\mu\nu}}&{q_{\mu}}&{0}\\{-q_{\nu}^{T}}&{\tilde{\nabla}}&{0}\\{0}&{0}&{-\tilde{\partial}}\\\end{matrix}\right)\;\qquad\tilde{\partial}\equiv\left(\begin{matrix}{0}&{0}&{0}&{\partial}\\{0}&{0}&{\partial}&{0}\\{0}&{\partial}&{0}&{0}\\{\partial}&{0}&{0}&{0}\\\end{matrix}\right)\.
\theta_{(3)}^{(4,0)}(\tau)=\sum_{(m_{1},m_{2})\in{\bfZ}^{2}}q^{\frac{16}{9}+4m_{1}+3(m_{1}^{2}+m_{2}^{2}-m_{1}m_{2})}=:q^{\frac{7}{9}}{\tilde{\theta}_{(3)}^{2}}(\tau).
S^{\left(\Sigma_{1+1}\right)}=-2\pii\thetaf\left(\overline{{t_{*}}}\right)g\left(t_{*}\right)\int_{\Sigma_{1+1}}\left[i\overline{{\psi}}^{L}\sigma^{a}D_{a}\psi^{L}-\overline{{\psi}}^{L}m\psi^{L}\right],
{\calS}_{\Omega}=s\intd^{3}x_{1}\prod_{j=2}^{n}d^{3}x_{j}F_{\mu_{j}\nu_{j}}(x_{j})\left(\varepsilon_{\mu_{1}\nu_{1}\rho}b^{*\rho}+\sum_{k=1}^{\infty}\frac{{\calM}_{\mu_{1}\nu_{1}}^{k}}{\xi^{k}}\right)\Omega^{\mu_{1}\nu_{1}..\mu_{n}\nu_{n}}\;,
I_{1}(\Omega^{2})-I_{1}(M^{2})=-{\frac{1}{8\pi^{2}}}(\Omega^{2}-M^{2})+{\frac{1}{8\pi^{2}}}\Omega^{2}ln{\frac{\Omega^{2}}{M^{2}}}-{\frac{1}{2}}(\Omega^{2}-M^{2})I_{2}(M^{2})
F_{8}^{{\mathcal{O}}_{1,1}^{0,1}|+++++++-}=\frac{iH^{1,1}(\sigma_{1}^{-})^{7/2}/(\sigma_{7}^{+})^{1/2}}{8\prod_{1\leqi<j\leq7}\mathrm{cosh}(\theta_{ij}/2)}\prod_{i<j}F_{\mathrm{min}}^{\mu_{i}\mu_{j}}(\theta_{ij}).
\begin{array}{lll}{\left[Q_{k},P_{j}\right]=i\delta_{kj}}&{\left[J_{k},Q_{j}\right]=i\epsilon_{kjm}Q_{m}}&{\left[J_{k},P_{j}\right]=i\epsilon_{kjm}P_{m}\nonumber}\\{\left[J_{k},J_{j}\right]=i\epsilon_{kjm}J_{m}}&{\left[Q_{k},Q_{j}\right]=0}&{\left[P_{k},P_{j}\right]=0.}\\\end{array}
ds^{2}=-2dx^{+}dx^{--}\beta^{2}\sum_{k=1}^{n}a_{k}^{2}\left[(\tilde{x}^{2k-1})^{2}+(\tilde{x}^{2k})^{2}\right]+\sum_{i=1}^{2n}(d\tilde{x}^{i})^{2}+\sum_{i=2n+1}^{9}(dx^{i})^{2},
\kappa^{4}\intER\overline{{R}}\left(\overline{{\nabla}}^{2}R\right)\left(\nabla^{2}\overline{{R}}\right)d^{4}\theta=-\frac{3}{2}\kappa^{4}\int\epsilon\left(\overline{{\nabla}}^{2}+\frac{1}{3}\overline{{R}}\right)\left[R\overline{{R}}\left(\overline{{\nabla}}^{2}R\right)\left(\nabla^{2}\overline{{R}}\right)\right]d^{2}\theta+\mathrm{h.c.}
\begin{array}{ccccccccc}{B_{1}}&{\longleftarrow}&{B_{2}}&{\longleftarrow}&{\cdots}&{\longleftarrow}&{B_{k}}&{\longleftarrow}&{B_{1}}\\{B_{\sigma(1)}}&{\dashrightarrow}&{B_{\sigma(2)}}&{\dashrightarrow}&{\cdots}&{\dashrightarrow}&{B_{\sigma(k)}}&{\dashrightarrow}&{B_{\sigma(1)}}\\\end{array}
\overline{{G}}_{\sigma}^{(n)}(\zeta)^{\varepsilon}=\prod_{a\inA}\oint_{C_{a}}\frac{dw_{a}}{2\pi\sqrt{-1}w_{a}}\Psi_{\sigma}^{(n)}(\{w_{a}\}_{a\inA}|\zeta)Q^{(n)}(\{w_{a}\}_{a\inA}|\zeta)^{\varepsilon},~~~~(\sigma=\pm)
K\cdot\Psi=\int\frac{d\xi}{2\pii}K_{X}\left(\xi\right)\Psi_{X}\left(\xi\right)+\sum_{n=1}^{\infty}\int\prod_{i=1}^{n}\frac{d\xi_{i}}{2\pii}K_{n}\left(\xi_{1},\cdots,\xi_{n}\right)\Psi_{n}\left(\xi_{1},\cdots,\xi_{n}\right),
=\,\frac{1}{2}\int_{0}^{l}dx\left(\dot{y}^{2}\,+\,2a_{3}\dot{y}\stackrel{\ldots}{y}\,-\,a_{2}\dot{y}\dot{y}^{\prime\prime}\,-\,a_{3}\ddot{y}^{2}\,+\,a_{1}y^{2}\right).
\psi_{L}\left(z,\overline{{z}}\right)=e^{-\frac{i\pi}{4}-\frac{\eta+i\theta}{2}}\psi_{L}\left(\eta,\theta\right);\\psi_{R}\left(z,\overline{{z}}\right)=e^{\frac{i\pi}{4}-\frac{\eta-i\theta}{2}}\psi_{R}\left(\eta,\theta\right),
{\calS}(\hat{\Gamma})=\int\left\{\partial_{\mu}c\;\frac{\delta\hat{\Gamma}}{\deltaA_{\mu}}\;+\;B\;\frac{\delta\hat{\Gamma}}{\delta\bar{c}}\;+\;\frac{\delta\hat{\Gamma}}{\delta\underline{{Y}}}\;\frac{\delta\hat{\Gamma}}{\delta\underline{{\varphi}}}\;+\;\underline{{q}}\;\frac{\delta\hat{\Gamma}}{\delta\underline{{\hat{\varphi}}}}\right\}=0
E^{\mu}(\{q,n\})\;:=\;\left\langle\prod_{b=1}^{N}\left[\chi_{+}^{(b)}[Q^{W}]\right]^{q_{b}}\left[\chi_{-}^{(b)}[Q^{W}]\right]^{2n_{b}-q_{b}}\right\rangle_{Q^{\mu}}\;.
L_{E}=-{\dot{\pi}}_{Q}Q+p_{\theta}{\dot{\theta}}+\pi_{\eta}{\dot{\eta}}-\frac{1}{2m}\left[\pi_{\eta}^{2}+\frac{p_{\theta}^{2}}{Q^{2}}\right]-m{\ddot{a}}(Q/2-\eta)-{\dot{a}}(\pi_{Q}-\pi_{\eta}/2)-\lambda(Q-a)-\sigma(\pi_{\eta}-m{\dot{a}}).
e^{i\Gamma_{C}[\vec{\varphi}]}=\langleT_{C}\mathrm{exp}\left[\frac{i}{\delta}\int_{C}\left\{{\calL}^{\prime}(\vec{\phi}(x)+\vec{\varphi}(x))-\frac{\delta\Gamma_{C}[\vec{\varphi}]}{\delta\vec{\varphi}(x)}\cdot\vec{\phi}(x)\right\}\right]\rangle.
{\calD}_{\mu\nu}(x-y)=i\left<0\left|T\bar{A}_{\mu}(x)\bar{A}_{\nu}(y)\right|0\right>+\frac{g_{\mu-}g_{\nu-}}{(1+\alpha)^{2}}\frac{1}{\partial_{-}^{2}}*\delta(x-y)
\left\{Q_{\alpha}^{a},Q_{\beta}^{b}\right\}=\delta^{ab}\gamma_{\alpha\beta}^{\mu}\,\,P_{\mu}+\sum_{p=0,1,\cdots}\gamma_{\alpha\beta}^{\mu_{1}\cdots\mu_{p}}\,\,\,Z_{\mu_{1}\cdots\mu_{p}}^{ab}.
=(-1)^{l_{1}+1/2}c_{0}\sqrt{\frac{1-(-1)^{l_{1}+l_{0}}}{2(l_{1}-l_{0}-1)(l_{1}+l_{0})}+\frac{1+(-1)^{l_{1}+l_{0}}}{2(l_{1}+l_{0}+1)(l_{1}-l_{0})}},
[\hat{\nabla}_{\hat{r}},\hat{\nabla}_{\hat{i}}]=-[H^{\frac{1}{6}}\partial_{i}({\frac{\partial_{k}H}{6H^{\frac{3}{2}}}})\gamma^{k}-{\frac{(\partial_{k}H)^{2}}{36H^{\frac{7}{3}}}}\gamma_{i}+{\frac{(\partial_{k}H)(\partial_{i}H)}{36H^{\frac{7}{3}}}}\gamma^{k}]\gamma_{r}(1+i\gamma^{0}\gamma^{1}\gamma^{2})
c_{\mathrm{extended}}=c_{s\ell(2)_{k_{1}}}+c_{s\ell(2)_{k_{2}}}+c_{\mathrm{scalar}}=\frac{3k_{1}}{k_{1}+2}+\frac{3k_{2}}{k_{2}+2}+1\stackrel{\eqref{kk1}}{=}0+1\,.
\mu^{Aui}=w_{\dot{\alpha}}^{~ui}\,\bar{\zeta}^{\dot{\alpha}A}~~,~~{M^{\prime}}_{~~ij}^{\betaA}=-{\bar{\zeta}}_{\dot{\alpha}}^{~A}\,(\bar{\sigma}^{\mu})^{\dot{\alpha}\beta}\,{a^{\prime}}_{\muij}~~,
\langle{\calO}_{{\bar{i}},2}{\calO}_{{\bar{j}},2}\prod_{l=1}^{N}{\calO}_{l,0}\rangle_{g}=\langle{\calO}_{{\bar{j}},2}{\calO}_{{\bar{i}},2}\prod_{l=1}^{N}{\calO}_{l,0}\rangle_{g}\,
\Psi_{n+2}=\mu\Psi_{n},\;\;\;\tilde{\Psi}_{n+2+\frac{1}{3}}=\mu\tilde{\Psi}_{n+\frac{1}{3}},\;\;\;\hat{\Psi}_{n+2+\frac{2}{3}}=\mu\hat{\Psi}_{n+\frac{2}{3}},
U_{\phi^{4}}={\frac{N}{g}}tr\left({\frac{1}{2}}A^{2}+{\frac{1}{2}}B^{2}+{\frac{1}{4(1-c)^{2}}}A^{4}+{\frac{1}{4(1+c)^{2}}}B^{4}+{\frac{1}{2(1-c)(1+c)}}(ABAB+2A^{2}B^{2})\right).
h_{xy}^{\prime\prime}+{\frac{1-3u^{2}}{uf}}h_{xy}^{\prime}+{\frac{1}{(2\piT)^{2}f^{2}u}}\biggl(f{\frac{\partial^{2}h_{xy}}{\partialz^{2}}}-{\frac{\partial^{2}h_{xy}}{\partialt^{2}}}\biggr)
\frac{\mathrm{d}}{\mathrm{d}z}:=\left.\frac{\partial}{\partialz}\right|_{\bar{z},\varphi=\mathrm{const}}+\left.\Phi\frac{\partial}{\partial\varphi}\right|_{\bar{z},z=\mathrm{const}}\,,\quad\frac{\mathrm{d}}{\mathrm{d}\bar{z}}:=\left.\frac{\partial}{\partial\bar{z}}\right|_{z,\varphi=\mathrm{const}}+\bar{\Phi}\left.\frac{\partial}{\partial\varphi}\right|_{z,\bar{z}=\mathrm{const}}\,.
\dot{x}_{f}^{i}=\sum_{n>0}W(n+\frac{\epsilon}{\tau})\Biggl[\frac{c_{n}^{i}}{\sqrt{n}}\,\dot{x}_{n}^{+}+\frac{\tilde{c}_{n}^{i}}{\sqrt{n}}\,\dot{x}_{n}^{-}+h.c.\Biggr]-\eta^{i}(\sigma,t)
\sigma_{\mathrm{qu}}=\frac{1}{2}\int\frac{d^{n-2}k_{\|}}{(2\pi)^{n-2}}\sum_{i}\left[\sqrt{{\vec{k}}_{\|}^{2}+(\epsilon_{i})^{2}}-\sqrt{{\vec{k}}_{\|}^{2}+(\epsilon_{i}^{(0)})^{2}}\right].
E_{n_{1}}=\frac{\pi}{2\lambda_{L}^{2}e^{2}}\frac{V}{L_{1}^{2}}\left(\frac{n_{1}}{N_{f}}\!+\!\frac{\Phi}{\Phi_{0}}\!+\!\frac{Q}{2}\right)^{\!2}\!\!+\!2\piN_{f}^{2}\lambda_{L}^{2}e^{2}\frac{Q^{2}}{V}.
P\circ\Bigl(\bigotimes_{\sigma\in\Lambda^{(4)}}Z_{\sigma}^{(V),(\epsilon_{\sigma})}\Bigr)=\Phi\,\Bigl[P\circ\Bigl(\bigotimes_{\sigma\in\Lambda^{(4)}}Z_{\sigma}^{(\tilde{V}),(\epsilon_{\sigma})}\Bigr)\Bigr]\,\Phi^{-1}.
\chi_{S}^{(2k+1)}\left(\psi_{i}^{(-k)},\psi_{j}^{(-k)}\right)={\frac{1}{24(2k+1)\pii}}\oint_{\calC}\psi_{i}^{(-k)}{\calS}_{J_{\Omega}^{-1}}^{(2k+1)}\psi_{j}^{(-k)},\qquadk=0,1,2,\ldots,
T_{n}=\intd^{4}x_{1}\cdots\intd^{4}x_{n}\f_{n\;a_{1}\cdotsa_{n}}^{\mu_{1}\cdots\mu_{n}}(x_{1},\cdots,x_{n})A_{\mu_{1}}^{a_{1}}(x_{1})\cdotsA_{\mu_{n}}^{a_{n}}(x_{n}).
S=\intdt\left[\mathrm{e}^{-\varphi}\left(3\dot{\alpha}^{2}-\dot{\varphi}^{2}+6k\mathrm{e}^{-2\alpha}+6\dot{\beta}^{2}-2\Lambda\right)+\frac12\mathrm{e}^{\varphi+6\alpha}\dot{\sigma}^{2}-Q^{2}\mathrm{e}^{c\beta+3\alpha}\right]\,
{\tilde{\Omega}}_{u,v}=\left(1+i\epsilon\Big(\frac{1}{2}\sum_{0<k<p}(\alpha_{p-k}\alpha_{k}-\bar{\alpha}_{p-k}\bar{\alpha}_{k})+\alpha_{p}\alpha_{u,v}-\bar{\alpha}_{p}\bar{\alpha}_{u,v}\Big)\right)\Omega_{u,v}.
\Pi_{\mu\nu}(k,m)=-\frac{g^{2}}{3\pi}\Lambdag_{\mu\nu}+\left(g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}\right)\Pi^{(1)}(k^{2},m)+i\epsilon_{\mu\nu\lambda}k^{\lambda}\Pi^{(2)}(k^{2},m)
R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=\frac{1}{l^{4}}\big[40+\frac{48M^{2}z^{4}}{\Sigma^{6}}(r^{2}-a^{2}cos^{2}{\theta})(r^{4}-14a^{2}r^{2}cos{\theta}+a^{4}cos^{4}{\theta})\big]
\begin{array}{c}{x_{\bar{2}1}^{\mu}=x_{2-}^{\mu}-x_{1+}^{\mu}-2i\theta_{1}^{a}\sigma^{\mu}\bar{\theta}_{2a}=x_{21}^{\mu}+i\theta_{21}^{a}\sigma^{\mu}\bar{\theta}_{21a}=(x_{21})_{-}^{\mu}\,,}\\{{}}\\{x_{\bar{1}2}^{\mu}=x_{1-}^{\mu}-x_{2+}^{\mu}-2i\theta_{2}^{a}\sigma^{\mu}\bar{\theta}_{1a}=x_{12}^{\mu}+i\theta_{12}^{a}\sigma^{\mu}\bar{\theta}_{12a}=(x_{12})_{-}^{\mu}\,.}\\\end{array}
\left[-{\frac{1}{2}}{\frac{d^{2}}{dq^{2}}}+{\frac{m^{2}}{2}}\{\psi({\frac{1}{2}}+q-M(q))+\psi({\frac{1}{2}}-q+M(q))\}\right]\varphi(q)=\tilde{\epsilon}\varphi(q),
H=\int\mathrm{d}^{2}r\\left\{{\frac{1}{2m}}\left[\left({\bfD}\Phi\right)^{\dagger}\cdot{\bfD}\Phi+\left({\bfD}\Psi\right)^{\dagger}{\bfD}\Psi\right]-{\frac{e}{2mc}}:B\\rho_{F}:-{\frac{e^{2}}{2mc\kappa}}:\rho_{B}^{2}:-3{\frac{e^{2}}{2mc\kappa}}\rho_{b}\rho_{F}\right\}
G_{r}^{(2)}(x)=\sum_{n_{0},n_{1},\ldots,n_{\nu-2}=0}^{\infty}x^{\frac{r-1}{2}+\sum_{i=0}^{\nu-2}n_{i}(\nu-1-i)}\times\prod_{i=1}^{\nu-2}\left[\begin{array}{c}{n_{i}+\tilde{n}_{i}+\tilde{V}_{i,r}}\\{n_{i}}\\\end{array}\right].
(\sigma_{0})_{\alpha\beta}=\left(\begin{array}{cc}{1}&{0}\\{0}&{1}\\\end{array}\right)\,\quad(\sigma_{1})_{\alpha\beta}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\,\quad(\sigma_{2})_{\alpha\beta}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)
A_{jk}(x+y)={\frac{\biggl|\begin{matrix}{c_{2}A_{mk}(x)}&{c_{2}A_{mk}(y)}\\{C_{mk}(x)}&{C_{mk}(y)}\\\end{matrix}\biggr|}{\biggl|\begin{matrix}{G_{mk}(x)}&{G_{mk}(y)}\\{1}&{1}\\\end{matrix}\biggr|}},
\hat{A}\left(\left\{l_{2n}\right\},x_{0},\left\{r_{2n}\right\}\right)=\int\,\hat{A}\left(\left\{l_{2n}\right\},x_{0},\left\{z_{2n}\right\}\right)\,\hat{B}\left(\left\{z_{2n}\right\},x_{0},\left\{r_{2n}\right\}\right)\prod_{k}dz_{2k}.
2\pi\alpha^{\prime}S_{b}=\intd\tau\int_{0}^{2\pi\alpha^{\prime}p^{+}}d\sigma\left(\,{\frac{1}{2}}G_{MN}\,\partial_{a}X^{M}\partial^{a}X^{N}+B_{MN}\partial_{\tau}X^{M}\partial_{\sigma}X^{N}\right).
v=v_{\hat{j},\,\hat{m}}\equiv\left(\begin{matrix}{\alpha_{1}\;\Psi_{\hat{j},\;\hat{m}}\,(\vec{\vartheta})}\\{\alpha_{2}\Psi_{\hat{j},\,\hat{m}+2}(\vec{\vartheta})}\\{\alpha_{3}\Psi_{\hat{j},\,\hat{m}+1}(\vec{\vartheta})}\\{\alpha_{4}\Psi_{\hat{j},\,\hat{m}+1}(\vec{\vartheta})}\\\end{matrix}\right)
\langle\sigma^{z}\rangle_{\infty}=-\frac{F_{-}(-q^{-1},1)}{F_{+}(-q^{-1},1)}=-\frac{(q^{2};q^{2})_{\infty}^{4}(-p;p)_{\infty}^{4}}{(-q^{2};q^{2})_{\infty}^{4}(p;p)_{\infty}^{4}}.
g_{\Lambda\Pi}^{(4+n)}\equiv\left[\begin{matrix}{\mathrm{e}^{-{\frac{1}{\alpha}}\varphi}g_{\mu\nu}+\mathrm{e}^{\frac{2\varphi}{n\alpha}}\rho_{ij}A_{\mu}^{i}A_{\nu}^{j}}&{\mathrm{e}^{\frac{2\varphi}{n\alpha}}\rho_{ij}A_{\lambda}^{j}}\\{\mathrm{e}^{\frac{2\varphi}{n\alpha}}\rho_{ij}A_{\pi}^{i}}&{\mathrm{e}^{\frac{2\varphi}{n\alpha}}\rho_{ij}}\\\end{matrix}\right],
S=\frac{1}{2\alpha_{0}}\intd^{2}\sigma\,\left[\rho^{-1}\partial^{2}X^{\mu}\partial^{2}X_{\mu}+\lambda^{ab}\left(\partial_{a}X^{\mu}\partial_{b}X_{\mu}-\rho\delta_{ab}\right)\right]+\mu_{0}\intd^{2}\sigma\,\rho,
|N_{\frac{1}{4}}-N_{\frac{1}{4}}^{(0)}|\leq\frac{(26)^{2}}{4}\{Max_{|k|=\frac{1}{4}}|M_{o}^{(-)}(k)[M^{(-)}(\nu,k)]^{\prime}-[M_{o}^{(-)}(k)]^{\prime}M^{(-)}(\nu,k)|\}\leq(\frac{169\lambda}{T_{o}})<1,
S_{s}(m_{s,i})={\frac{b(m_{s,i})}{48\pi}}{\calA}~~,~~S_{d}(m_{d,i})={\frac{2b(m_{d,i})}{48\pi}}{\calA}~~,~~S_{v}(m_{v,i})={\frac{3b(m_{v,i})}{48\pi}}{\calA}~~.
S=\frac{1}{16\pi}\int\left\{R-2\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}e^{4\phi}{\partial_{\mu}}\kappa\partial^{\mu}\kappa-e^{-2\phi}F_{\mu\nu}F^{\mu\nu}-\kappaF_{\mu\nu}{\tilde{F}}^{\mu\nu}\right\}\sqrt{-g}d^{4}x,
e^{2\phi}\bigl(\partial_{\underline{i}}a\bigr)-e^{2U}\bigl(\partial_{\underline{i}}b\bigr)={\frac{i}{\overline{{\calH}}_{2}}}\partial_{\underline{i}}{\overline{{\calH}}_{2}}-{\frac{i}{{\calH}_{2}}}\partial_{\underline{i}}{\calH}_{2}\,,
\frac{\epsilon_{\alpha\beta\gamma\delta}\,\mathrm{tr\,}\hat{F}^{\alpha\beta}\,\hat{F}^{\gamma\delta}}{32\pi^{2}}\,=\partial_{\alpha}K^{\alpha}=\partial_{\alpha}\frac{\epsilon^{\alpha\beta\gamma\delta}}{16\pi^{2}}{\,\mathrm{tr\,}}\left(\hat{A}_{\beta}\,\hat{F}_{\gamma\delta}+\frac{2i}3\hat{A}_{\beta}\hat{A}_{\gamma}\hat{A}_{\delta}\right).
\Psi_{nqm}(\xi,\eta,\varphi;R;\delta_{1},\delta_{2})=\sum_{n_{1}=0}^{n-|m|-1}U_{nqm}^{n_{1}}(R;\delta_{1},\delta_{2})\Psi_{n_{1}n_{2}m}(\mu,\nu,\varphi;\delta_{1},\delta_{2}),
S=\frac{1}{\gamma^{2}}\int_{-\infty}^{+\infty}d\tau\int_{0}^{2\pi}d\sigma\Bigl(\frac{1}{2}\partial_{\alpha}\varphi\cdot\partial^{\alpha}\varphi-\mu^{2}\sum_{a=1,2}\mathrm{e}^{\alpha^{a}\cdot\varphi}\Bigr)~,
(3n_{a}^{2}+3n_{d}^{2}+3n_{e}^{2})\neq0,\;\Q^{l}=n_{c}^{1}(Q_{a}-3Q_{d}-3Q_{e})-\frac{3{\tilde{\epsilon}}\beta^{2}(n_{a}^{2}+n_{d}^{2}+n_{e}^{2})}{2\beta^{1}}Q_{c}
I=\intd^{3}x[-\mu\epsilon^{mnp}A_{m}\partial_{n}a_{p}-\frac{1}{2}\mu^{2}a_{m}a^{m}-\mu\epsilon^{mnp}A_{m}\partial_{n}v_{p}+\frac{1}{2}\mu\epsilon^{mnp}v_{m}\partial_{n}v_{p}]