harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{c}{-\Gamma^{div}=\intd^{4}x\left(\frac{1}{2}\deltaZ_{\varphi}(\partial_{\mu}\varphi)^{2}-\frac{1}{2}\deltam^{2}\varphi^{2}+\right.}\\{\left.+\deltaZ_{\psi}i\bar{\psi}(i\not{\partial}\psi-\deltaM\bar{\psi}\psi-g\deltaZ_{g}\bar{\psi}\psi\varphi+\frac{1}{4}\delta\lambda\varphi^{4}+\frac{1}{3}\delta\sigma\varphi^{3}+\deltac\varphi\right)}\\\end{array}
\begin{array}{c}{\delta_{i}^{+}z^{a}=\epsilon\eta_{i}^{a},}\\{\delta_{i}^{+}\eta_{j}^{a}=\epsilon\left(i\epsilon_{ij}{\bar{U}}_{\bar{b}}g^{\bar{b}a}+\Gamma_{bc}^{a}\eta_{i}^{b}\eta_{j}^{c}\right);}\\{\delta_{i}^{-}z^{a}=0,}\\{\delta_{i}^{-}\eta_{j}^{a}=\epsilon\delta_{ij}{\dot{z}}^{a}}\\\end{array}
\begin{array}{rcl}{{\bf{Q}}_{-{\frac{1}{k}}}^{-}}&{=}&{[{\bf{Q}}_{-{\frac{1}{k}}}^{+}]^{+}}\\{{\bf{Q}}_{1-{\frac{1}{k}}}^{-}}&{=}&{[{\bf{Q}}_{1-{\frac{1}{k}}}^{+}]^{+}}\\\end{array}
\frac{1}{\beta_{0}}\left[\frac{5}{2}u_{2}\Big(3x_{1}-\frac{1}{2}x_{3}\Big)+9u_{1}x_{3}-9x_{4}-2N_{F}\bigg(z_{3}\Big(w_{3}+\frac{1}{2}u_{2}\Big)+(w_{2}-w_{1})z_{4}+z_{5}\bigg)\right]\,=\,u_{3}\;,
(\sigma^{\alpha})^{ab}\bigr((\sigma_{\alpha})_{~~\!c}^{b_{0}}\delta_{a_{0}}^{d}+\delta_{c}^{b_{0}}(\sigma_{\alpha})_{~a_{0}}^{d}\bigr)=(\sigma^{\alpha})^{ab}\bigr((\sigma_{\alpha})_{~c}^{d}\delta_{a_{0}}^{b_{0}}+\delta_{c}^{d}(\sigma_{\alpha})_{~~\!a_{0}}^{b_{0}}\bigr).
(\partial_{\tau}^{2}-\partial_{\sigma}^{2})\hat{X}^{i}(\tau,\sigma)=0,\qquad\hat{X}^{i}(\tau,0)=\tilde{X}_{0}^{i},\quad\hat{X}^{i}(\tau,\pi)=\tilde{X}_{\pi}^{i},
S_{0}[x]:=\frac{1}{2}\int_{0}^{\tau\hbar/m}\mathrm{d}t\left[\dot{x}^{2}(t)-\Phi^{2}(x(t))\right]\quad,\qquad\varphi[x]:=\frac{1}{2}\int_{0}^{\tau\hbar/m}\mathrm{d}t\,\Phi^{\prime}(x(t))\quad.
S_{\mathrm{str}}=(gm_{D})^{2}\frac{N-1}{8N}\int_{\Sigma(C)}^{}d\sigma_{\mu\nu}(\vec{x}(\xi))\int_{\Sigma(C)}^{}d\sigma_{\mu\nu}(\vec{x}(\xi^{\prime}))D_{m_{D}}\left(\vec{x}(\xi)-\vec{x}(\xi^{\prime})\right)
G^{AB}x_{,A}^{\mu}x_{,B}^{\nu}R_{\mu\nu}=\pm2(\mpR_{\mu\nu}l_{\pm}^{\mu}\xi^{\nu})={\frac{2Q^{2}}{\rho^{4}}}l_{\pm}^{\mu}\xi_{\mu}=-{\frac{2Q^{2}}{\rho^{4}}}.
\phi_{i}^{\alpha}\equiv\pi_{i}^{\alpha}+\frac{i}{2}g_{i\bar{j}}\bar{\chi}^{\bar{j}\alpha}=0\qquad\qquad\bar{\phi}_{\bar{j}\alpha}\equiv\bar{\pi}_{\bar{j}\alpha}-\frac{i}{2}g_{i\bar{j}}\chi_{\alpha}^{i}=0
m_{\mu\nu}{}^{i}{}_{k}m_{\rho\sigma}{}^{k}{}_{j}-m_{\rho\sigma}{}^{i}{}_{k}m_{\mu\nu}{}^{k}{}_{j}=-\eta_{\mu[\rho}m_{\sigma]\nu}{}^{i}{}_{j}+\eta_{\nu[\rho}m_{\sigma]\mu}{}^{i}{}_{j}\,.
\sum_{n=0}^{\infty}\left(X_{\mun}{\frac{d{\bar{X}}_{n}^{\mu}}{d\tau}}-{\frac{dX_{\mun}}{d\tau}}\bar{X}_{n}^{\mu}\right)={\frac{1}{2\pii}}\int_{0}^{2\pi}d\sigma\left({\frac{\partialX_{\mu}(\sigma)}{\partial\sigma}}{\frac{\partial{\bar{X}}^{\mu}(\sigma)}{\partial\tau}}-{\frac{\partial^{2}X_{\mu}(\sigma)}{\partial\tau\partial\sigma}}\bar{X}^{\mu}(\sigma)\right),
\langle0|\phi(x^{+},{\bfx})\phi(x^{+},{\bfy})|0\rangle=Z^{-1}\int\frac{dp^{+}d^{2}p_{\perp}}{(2\pi)^{3}}\\frac{\theta(p^{+})}{2p^{+}}\e^{-i{\bfp(x-y)}}\,
\frac{1}{Z^{0}}{\calD}h{\calD}Ae^{-S_{G}[A]-S_{h}[h]}\;\mathrm{det}[\not\!{\partial}-i(e\not{\!\!A}+g^{1/2}\not{\!h})]^{N}\;\;\leadsto\;\;d\mu_{Q}[A]d\mu_{C}[h^{\prime}]\;,
\begin{aligned}{\hat{g}_{\mu\nu}}&{{}=e^{\sqrt{7}\varphi/14}g_{\mu\nu}\,,}\\{\hat{g}_{xx}}&{{}=e^{-\sqrt{7}\varphi/2}\,,}\\{\hat{\mathcal{M}}}&{{}=\Omega(x)\mathcal{M}\,\Omega(x)^{T}\,,}\\\end{aligned}
\Omega_{+2}^{[-2](i)}\tilde{\gamma}_{\dot{B}A}^{i}\lambda_{A}=g_{+3}\theta_{\dot{B}}^{-},\\\Omega_{-2}^{[+2](i)}\gamma_{A\dot{B}}^{i}\lambda_{\dot{B}}=g_{-3}\theta_{A}^{+}.
S_{s}=-\intd^{D}x\,\sqrt{-g}\left[\frac12\,g^{MN}\Bigl(R_{MN}+\partial_{M}\,\phi\,\partial_{N}\,\phi\Bigr)+{\frac{1}{2\,n!}}\;e^{\alpha\phi}\,F_{M_{1}...M_{n}}\,F^{M_{1}...M_{n}}\right]\,,
\frac{ds^{2}}{\alpha^{\prime}}=R^{2}\left[u^{2}\hat{h}\left(d\tilde{x}_{0}^{2}+d\tilde{x}_{1}^{2}\right)+\frac{dU^{2}}{U^{2}}+d\Omega_{3}^{2}\right]+\frac{R_{1}}{R_{5}}d\tilde{x_{n}}d\tilde{x_{n}},
\begin{array}{c}{\left[A^{-},A^{+}\right]_{q\left(x\right)}=A^{-}\cdote^{a\left(x\right)}\cdotA^{+}-A^{+}\cdote^{-a\left(x\right)}\cdotA^{-}}\\{q\left(x\right)=e^{a\left(x\right)}}\\\end{array}
H_{c}={\displaystyle\intd^{3}x~{\calH}_{c}}={\displaystyle\intd^{3}x\left(\frac{1}{2}{\pi_{i}}{\pi_{i}}+{\frac{1}{4}}F_{ij}F_{ij}-\frac{m^{2}}{2}(A_{0}^{2}-A_{i}^{2})+A_{0}({\partial_{i}}\pi_{i})\right),}
\intd\xi_{0}^{(3)}d\xi_{0}^{(2)}d\xi_{0}^{(1)}\mathrm{Tr}(\hat{A}_{1}(\xi_{0}^{(1)},\xi)\star\hat{A}_{2}(\xi_{0}^{(2)},\xi)\star\hat{A}_{3}(\xi_{0}^{(3)},\xi))~~~\leftrightarrow~~~\langle\Psi_{1}|\Psi_{2}\star^{W}\Psi_{3}\rangle
{\calA}_{fi}=\intd{\bfx\,}\left[-\frace{w_{{\bfp}}}{\bfA}({\bfx})\,{\bf\cdot\,p}+V({\bfx})\right]\,\mathrm{e}^{-\,i\,({\bfp}^{\prime}-{\bfp})\,{\bf\cdot\,x}}=-\frace{w_{{\bfp}}}\,{\bf\tilde{A}(q})\,{\bf\cdot\,p}+\tilde{V}({\bfq})\;\;\;
\left\langlen\right|H\left|n\right\rangle={\frac{3}{2}}n(n-1)+ng-{\frac{f_{n}^{4}}{4}}-{\frac{2n+1}{4}}f_{n}^{2}+{\frac{n+1}{4}}f_{n+1}^{2}+{\frac{n}{4}}f_{n-1}^{2}-C\;.
\left.|\frac{\partial}{\partialD_{b}}K(X,D)|\leqcN_{p}^{2}\,\sum_{k>0}\,k^{4}\left(\frac{e\beta}{2}\right)^{k}\\,e^{-(|X|-1)\,A_{\beta}}\\,\sum_{Y_{i}:suppY_{i}=X}\,N_{P}(Y_{i})\.\right.
\frac{1}{2}\pi_{a}^{m}\pi_{a}^{m}=\frac{1}{2}\pi_{a}^{\primem}\pi_{a}^{\primem}-F_{a}^{m}\pi_{a}^{\primem}+\frac{1}{2}F_{a}^{m}F_{a}^{m}-\frac{1}{2}[\pi_{a}^{\primem},F_{a}^{m}]
\nu^{2}K_{i}(\infty)X_{i}^{+}X_{I}^{-}V_{\vec{\lambda},\vec{m}}(z)=(x_{i}^{-})^{(0)}X_{I}^{-}V_{\vec{\lambda},\vec{m}}(z)-X_{I}^{-}(x_{i}^{-})^{(0)}V_{\vec{\lambda},\vec{m}}(z)~,
\int_{M}\sum\alpha_{i}\phi^{i}=\sum_{F_{i}\subsetF}\int_{F_{i}}\frac{\sum\alpha_{i}\phi^{i}}{\epsilon_{i}(\phi)}-\int_{\partialM/S^{1}}\sum_{i,k}\tilde{\alpha}_{i}(\mathrm{d}\theta)^{k}\phi^{i-k-1}
\Biggl[-\frac{1}{2}\frac{d^{2}}{dx_{2}^{2}}+\frac{1}{2}\Biggl(k_{1}-qA_{1}(x_{2})\Biggr)^{2}\Biggr]\phi_{\omega,k_{1},k_{3}}(x_{2})=\frac{1}{2}(\omega^{2}-m^{2}-k_{3}^{2})\phi_{\omega,k_{1},k_{3}}(x_{2}).
\xi_{2}^{2N}=\frac{\mathrm{A}(\theta_{1},-\theta_{2})}{\mathrm{A}(\theta_{1},\theta_{2})}=\frac{\mathrm{A}(-\theta_{1},-\theta_{2})}{\mathrm{A}(-\theta_{1},\theta_{2})},\quad\xi_{1}^{2N}=\frac{\mathrm{A}(\theta_{2},-\theta_{1})}{\mathrm{A}(\theta_{2},\theta_{1})}=\frac{\mathrm{A}(-\theta_{2},-\theta_{1})}{\mathrm{A}(-\theta_{2},\theta_{1})}
{\frac{1}{N^{4}}}\left\langle:\!\!{\mathrm{Tr}}\:\phi^{2}(x_{1})\!\!:\;:\!\!{\mathrm{Tr}}\:\phi^{2}(x_{2})\!\!:\;:\!\!{\mathrm{Tr}}\:\phi^{2}(x_{3})\!\!:\;:\!\!{\mathrm{Tr}}\:\phi^{2}(x_{4})\!\!:\right\rangle
U(r,z)=-(1+\gamma)^{-1}\,\frac{3}{4\pi}\,\frac{1}{\ellz}\,\bigg(1+\frac{2r^{2}}{3z^{2}}\bigg)\bigg(1+\frac{r^{2}}{z^{2}}\bigg)^{-3/2}\,\bigg[1+{\calO}\big(\ell^{2}/z^{2},\,\ell^{2}/r^{2}\big)\bigg]\,.
\left<W_{R\timesT}\right>=\frac{1}{N_{c}}\sum_{n}^{}\left<\Psi_{ij}(0)\left|n\left>\right<n\right|\Psi_{ji}^{\dagger}(T)\right>=\frac{1}{N_{c}}\sum_{n}^{}\left|\left<\Psi_{ij}(0)|n\right>\right|^{2}\mathrm{e}^{-E_{n}T},
\Sigma(q;\chi)=\frac{1}{i}\int\frac{d^{2}p}{(2\pi)^{2}}\frac{1}{(-p^{2}+\chi)(-(p-q)^{2}+\chi)}=\int\frac{d^{2}p_{E}}{(2\pi)^{2}}\frac{1}{(p_{E}^{2}+\chi)(p_{E}-q_{E})^{2}+\chi)}
{\calH}_{R}=\sum_{n=1}^{\infty}\omega_{n}(a_{n}^{i}{}^{\dagger}a_{n}^{i}+s_{n}^{a}{}^{\dagger}s_{n}^{a}),~~~~~{\calH}_{L}=\sum_{n=1}^{\infty}\omega_{n}(\tilde{a}_{n}^{i}{}^{\dagger}\tilde{a}_{n}^{i}+\tilde{s}_{n}^{a}{}^{\dagger}\tilde{s}_{n}^{a}),
\begin{array}{ll}{D=3:}&{\ell=s+1+a_{2}+\frac{1}{2}a_{3}}\\{D=4:}&{\ell=s+2+2a_{1}+a_{2}}\\{D=5:}&{\ell=s+3+\frac{3}{2}a_{1}}\\{D=6:}&{\ell=s+4+2(a_{1}+a_{2})}\\\end{array}
{\calG}(x-x^{\prime},0)_{\mu\nu,\gamma\delta}\,=\,\langleh_{\mu\nu}(x,0)\,h_{\gamma\delta}(x^{\prime},0)\rangle=\intd^{N}m|\sigma_{m}(0)|^{2}\langle\epsilon_{\mu\nu}^{(m)}(x)\epsilon_{\gamma\delta}^{(m)}(x^{\prime})\rangle\,.
\begin{array}{c}{lim}\\{\varepsilon\rightarrow0}\\\end{array}I^{\prime}(\theta_{0},CC)=\begin{array}{c}{lim}\\{\varepsilon\rightarrow0}\\\end{array}I^{\prime}[\theta,\frac{1}{2}(C+A)\frac{1}{2}(C+A)]+O[\delta\theta]
\begin{aligned}{\partial_{x}M_{ij}=\Psi_{i}M_{m+1\,j},\quadi=1,{\ldots},m,}\\{\left(\partial_{x}-\lambda\right)M_{m+1\,j}+\sum_{i=1}^{m}\Phi_{i}M_{ij}=0,\;\;\;j=1,{\ldots},m+1\,.}\\\end{aligned}
S_{F}({\bar{\psi}},\psi;A)\;=\;\int_{0}^{\beta}d\tau\intdx\,{\bar{\psi}}(\tau,x)e^{-ie({\tilde{\varphi}}-\gamma_{5}{\tilde{\sigma}})}(\not\!\partial+ie\not\!A^{(0)})e^{ie({\tilde{\varphi}}+\gamma_{5}{\tilde{\sigma}})}\psi(\tau,x)
{\tilde{L}}(\lambda|x_{i},\partial_{x_{i}})=\left(\begin{array}{cc}{x_{i}}&{1}\\{-x_{i}y_{i}+\frac{(\lambda+\beta_{i})y_{i-1}}{y_{i-1}-x_{i}}}&{-y_{i}+\frac{(\lambda+\beta_{i})}{y_{i-1}-x_{i}}}\\\end{array}\right)~.
\{T_{\alpha}^{(0)}(a),T_{\beta}^{(0)}(a)\}=T_{\gamma}^{(0)}(a)U_{\alpha\beta}^{\gamma}(a),\quad\{H_{0}(a),T_{\alpha}^{(0)}(a)\}=T_{\gamma}^{(0)}(a)V_{\alpha}^{\gamma}(a)
ds^{2}=-dt^{2}+dy^{s}dy_{s}+\left[1-\left(\frac{r_{H}}{r}\right)^{7-p}\right]dx^{\scriptscriptstyle{10}^{\scriptstyle{2}}}+\frac{dr^{2}}{1-\left(\frac{r_{H}}{r}\right)^{7-p}}+r^{2}d\Omega_{8-p}^{2}.
dv_{A}^{~\underline{{\mu}}}\equiv-{\frac{1}{4}}\Omega^{\underline{{b}}\underline{{a}}}(\sigma_{\underline{{b}}\underline{{a}}})_{A}^{~B}v_{B}^{~\underline{{\mu}}}={\frac{1}{2}}f^{i}\Gamma_{AB}^{i}v_{B}^{~\underline{{\mu}}}+{\frac{1}{4}}A^{ij}\Gamma_{AB}^{ij}v_{B}^{~\underline{{\mu}}}
R_{\mu\nu}(W)=\partial_{\rho}W_{\mu\nu}^{\rho}-\frac{1}{2}\left(\partial_{\nu}W_{\mu\rho}^{\rho}+\partial_{\mu}W_{\nu\rho}^{\rho}\right)-W_{\sigma\nu}^{\rho}W_{\mu\rho}^{\sigma}+W_{\sigma\rho}^{\rho}W_{\mu\nu}^{\sigma}
{\calG}_{\mathrm{np}}^{ab}(z_{i},z_{j})=\alpha^{\prime}G^{ab}{\calG}(z_{i},z_{j})+\displaystyle\frac{(\theta\cdotG\cdot\theta)^{ab}}{2\pi\alpha^{\prime}t}(x_{i}-x_{j})^{2}-\frac{2i}{t}\theta^{ab}(x_{i}-x_{j})(y_{i}+y_{j}).
\delta(P^{\alpha}-k^{\alpha})\,delta(y_{2}^{\prime}\cdotk-{\frac{4}{3}}\nu_{2}c^{2}-{\frac{2}{3}}\nu_{3}c^{2})\,delta(y_{3}^{\prime}\cdotk-{\frac{4}{3}}\nu_{3}c^{2}-{\frac{2}{3}}\nu_{2}c^{2})\\psi
E=-E_{n}\,,\;n\geq1\;\;\psi_{1}^{(-)}=\sqrt{\frac{E_{n}+m}{2E_{n}}}{e^{-ipx_{2}-\frac{1}{2}{\xi}^{2}}}\left(\begin{array}{c}{\frac{\sqrt{E_{n}^{2}-m^{2}}}{E_{n}+m}N_{n}H_{n}(\xi)}\\{N_{n-1}H_{n-1}(\xi)}\\{0}\\{0}\\\end{array}\right)
Q=\left(\begin{array}{ccccccccr}{0}&{0}&{0}&{0}&{0}&{-1}\\{1}&{0}&{0}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}&{0}&{-1}\\{0}&{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}&{-1}\\{0}&{0}&{0}&{0}&{1}&{0}\\\end{array}\right).
\begin{array}{cl}{}&{A_{\lambda}^{(m)}(x|z^{(n)}|\cdots|z^{(1)})|_{z_{m}^{(k)}=a\tau^{2n-2k}~~(k=1,\cdots,n)}=(x-a\tau^{n-1})}\\{\times}&{\displaystyle\sum_{\rho=0}^{n-1}(-1)^{\rho}\sigma_{\rho}(a\tau,a\tau^{3},\cdots,a\tau^{2n-3})A_{\lambda-\rho}^{(m-1)}(x|z^{(n)}|\cdots|z^{(1)}).}\\\end{array}
Z=\int{\calD}g(\sigma,\tau)\int{\calD}\gamma{\calD}Xe^{-{\calS}_{\sigma}[\gamma,X]}\delta(\gamma_{++}^{g})\delta(\gamma_{--}^{g})\mathrm{det}\left(\frac{\delta\gamma_{++}^{g}}{\deltag}\right)\mathrm{det}\left(\frac{\delta\gamma_{--}^{g}}{\deltag}\right)
R_{\mu\nu}=Tr\{2J_{(\mu}^{z}\bar{J}_{\nu)}^{z}+i(z-\bar{z})({\calF}_{\mu\lambda}{{{\calF}^{+}}_{\nu.}}^{\lambda}+\bar{\calF}_{\mu\lambda}{{{\calF}^{T}}_{\nu.}}^{\lambda})\},
\sigma=-\mathrm{sign}\left(\sum_{i=1}^{n}\frac{q_{i}}{p_{i}}\right)+\sum_{i=1}^{n}\mathrm{sign}\left(\frac{p_{i}}{q_{i}}\right)+\sum_{i=1}^{n}\sum_{j=1}^{t_{i}-1}\mathrm{sign}(a_{j}^{(i)})
0=\intd[A,\Phi]\,\delta[\mathrm{div}\,{\bfA}]e^{iS\left[A,\Phi\right]}e^{i\intd^{4}x\,\left(J^{\mu}A_{\mu}+k^{\star}\Phi+k\Phi^{\star}\right)}\intd^{4}x\,\alpha(x)\left(\partial_{\mu}j^{\mu}+ik^{\star}\Phi-k\Phi^{\star}\right).\nonumber
\hat{D}^{\Lambda,\Lambda_{0}}(q\,;\varphi):=\hat{L}_{(1)}^{\Lambda,\Lambda_{0}}(q\,;\varphi)+\hat{I}_{(1)}^{\Lambda,\Lambda_{0}}(q)-(2\pi)^{4}\frac{\delta}{\delta\hat{\varphi}(q)}L^{\Lambda,\Lambda_{0}}(\varphi).
\sum_{j,j^{\prime};\mu\mu^{\prime}}z_{j}^{(\mu)}\;\{J_{(\mu)}^{j,-j},J_{(\mu^{\prime})}^{j^{\prime},-j^{\prime}}\}^{*}(y)\;t_{j^{\prime},-j^{\prime}}^{(\mu^{\prime})}=-ad_{y}\left(\frac{1}{1+Lad_{w}}F(\vec{z})\right)
\nabla_{a}\nabla^{a}h=Tr\left[\Phi\nabla_{a}\left(J^{(0)a}-J^{(1)a}\right)\right]+Tr\left[\Phi\left(J_{a}^{(0)}J^{(0)a}+J_{a}^{(1)}J^{(1)a}-2J_{a}^{(0)}J^{(1)a}\right)\right]
Q(z)=\sum_{n}\left[\frac{1-\mu_{n}}{4(z-z_{n})^{2}}+\frac{\beta_{n}}{2(z-z_{n})}\right]+\sum_{B}\left[-\frac{3}{4(z-z_{B})^{2}}+\frac{\beta_{B}}{2(z-z_{B})}\right],
{\calT}^{(\mathrm{grav})}(p_{1},p_{2};q)^{\mathrm{odd}}=i{\calE}_{i}^{\mu}(p_{1},p_{1}-q){\calE}_{j}^{\nu}(p_{2},p_{2}+q)D_{\mu\nu}^{ij}(q)^{\mathrm{odd}}\equiv0
\left[a^{2}\dot{\phi}_{0}^{2}\partial_{y}\frac{1}{a^{2}\dot{\phi}_{0}^{2}}\partial_{y}a^{2}-\frac{2\kappa}{3}a^{2}\dot{\phi}_{0}^{2}+m^{2}\left(1+2\sum_{\sigma}\lambda_{\sigma}\delta(y-y_{\sigma})\right)\right]Y^{(1)}=0,
|\Psi\rangle=\int\frac{d^{26}k}{(2\pi)^{26}}\,\Bigl\{T(k)|0,k\rangle+iA_{\mu}(k)\alpha_{-1}^{\mu}|0,k\rangle+B(k)b_{-1}c_{0}|0,k\rangle+\ldots\,\Bigr\}.
\partial_{\bar{a}}C_{i_{1}i_{2}i_{3}i_{4}}=fact({\bar{C}}_{\bar{a}})-\sum_{i=1}^{n}{\bar{C}}_{{\bar{a}}i}^{l}C_{\sigma_{1}(l)i_{1}..{\hat{i}}..i_{4}}+{\bar{C}}_{{\bar{a}}0}^{l}C_{\sigma_{2}(l)i_{1}i_{2}i_{3}i_{4}}
\begin{array}{l}{\beta_{mn}^{G}=\left[(D-1)\frac{\alpha^{\prime}\epsilon}{K^{2}}+(D-1)\left(\frac{\alpha^{\prime}\epsilon}{K^{2}}\right)^{2}\right]G_{mn}\,}\\{\beta^{\phi}=\frac{1}{6}(D-25)+\frac{D(D-1)}{8}\left(\frac{\alpha^{\prime}\epsilon}{K^{2}}\right)^{2}\.}\\\end{array}
S^{\mathrm{total}}=-\intd^{p+1}\xi(-\sigma\cdot\sigma)^{\frac{1}{2}}+g\intd^{p+1}\xi\sigma^{\mu_{0}\mu_{1}\ldots\mu_{p}}\psi_{\mu_{0}\mu_{1}\ldots\mu_{p}}+\intd^{D}X{\calL}.
{\calL^{\prime}}=m\epsilon^{\mu\nu\lambda}\partial_{\mu}A_{\nu}B_{\lambda}+\frac{m^{2}}2\left(B_{\mu}+\partial_{\mu}\omega\right)\left(B^{\mu}+\partial^{\mu}\omega\right)+\frac{m^{2}}2\left(A_{\mu}+\partial_{\mu}f\right)\left(A^{\mu}+\partial^{\mu}f\right),
\begin{array}{ll}{(a)~\mathrm{For~each~(three~point)~vertex}}&{:-i(2\pi)^{2}c^{abc},}\\{(b)~\mathrm{For~each~propagator}}&{:\frac{i}{(2\pi)^{2}(p_{a}^{2}-m_{a}^{2})},}\\{(c)~\mathrm{Loop~integration}}&{:\intd^{2}l.}\\\end{array}
{\eta(\frac{\tau}2)}^{-L}=\left[q^{\frac{L}{24}}\prod_{n=1}{(1-q^{n})}^{L}\right]^{-1}\left[q^{-\frac{L}{48}}\prod_{n=1}{(1-q^{n-1/2})}^{L}\right]^{-1}
j_{\mu\alpha}^{can}=\sigma_{\mu}{}^{\nu}{}_{\alpha}{}^{\beta}D_{\beta}\left(\frac{1}{6}\sigma_{\nu}^{\gamma{\dot{\gamma}}}D_{\gamma}\phi{\bar{D}}_{\dot{\gamma}}{\bar{\phi}}K_{\phi{\bar{\phi}}}+\frac{4i}{3}\partial_{\nu}{\bar{\phi}}K_{\bar{\phi}}\right)\displaystyle{\biggl\vert}\,
\bar{S}=\intd^{4}x\,\Bigl[{\frac{1}{2}}\,\bar{A}_{\mu}\,\bigl(\Box-m^{2}\bigr)\,\bar{A}^{\mu}-\,{\frac{1}{2}}\,\partial_{\mu}\,\bar{A}^{\mu}\,\Bigl(1-{\frac{1}{\alpha}}-{\frac{m^{2}}{\Box}}\Bigr)\,\partial_{\nu}\,\bar{A}^{\nu}\Bigr]+S_{ghost}
P^{\mu}=\sum_{\sigma}\intd^{3}\vec{p}\,p^{\mu}\Bigl(a_{j}^{\dagger}(p,\sigma)a_{j}(p,\sigma)+a_{j-\frac{1}{2}}^{\dagger}(p,\sigma)a_{j-\frac{1}{2}}(p,\sigma)\Bigr)\:.
K=\sqrt{1+\frac{1}{24}\tilde{C}^{2}}\,\quad\tilde{C}^{2}=\tilde{g}^{\mu_{1}\nu_{1}}\tilde{g}^{\mu_{2}\nu_{2}}\tilde{g}^{\mu_{3}\nu_{3}}\tilde{C}_{\mu_{1}\mu_{2}\mu_{3}}\tilde{C}_{\nu_{1}\nu_{2}\nu_{3}}\.
I_{nn^{\prime\prime}}(\hat{q}_{1},\hat{q}_{2})\equiv\int_{-\infty}^{\infty}d\eta\mathrm{e}^{i\eta\mathrm{sgn}(eH)\hat{q}_{1}}D_{n}(\eta-\hat{q}_{2})D_{n^{\prime\prime}}(\eta+\hat{q}_{2}),
G^{(1)}~=~\frac{i}{16{(2\pi)}^{3}}\int_{\Gamma}\frac{d^{2}\tau}{\tau_{2}^{2}}\bar{\eta}^{-2}\mathrm{Tr}_{R}F(-1)^{F}q^{L_{0}-\frac{c}{24}}\bar{q}^{\bar{L}_{0}-\frac{\bar{c}}{24}}.
\nabla^{2}\varphi_{x}-\int_{-\infty}^{\infty}{\frac{d\alpha}{2\sqrt{\pi}}}e^{-{\frac{\alpha^{2}}{4}}}V^{(1)}({\frac{\alpha}{2}}\sqrt{J_{1}(g)-I_{1}(Q^{2})}+\varphi_{x})=0\;,
c_{2r}^{\nu}(h^{2})=c_{-2r}^{-\nu}(h^{2})\;\;\;and\;\;\;\frac{c_{2r}^{\nu}(h^{2})}{c_{0}^{\nu}(h^{2})}=\frac{c_{-2r}^{-\nu}(h^{2})}{c_{0}^{-\nu}(h^{2})}
\intd\mu_{B}\=\\int{\frac{d^{3}X}{(2\pi)^{3/2}}}\S_{\mathrm{cl}}^{3/2}\int_{0}^{2\pi}{\frac{d\theta}{(2\pi)^{1/2}}}\{\frac{S_{\mathrm{cl}}^{1/2}}{M_{W}}}\.
I_{\psi,{\widetilde\psi}}\equiv{\frac{1}{2}}\int_{M}\epsilon^{\mu\nu\rho\sigma}\left[{\widetilde\psi}_{\;\;\;\mu}^{A^{\prime}}\;e_{AA^{\prime}\nu}\;D_{\rho}\;\psi_{\;\;\sigma}^{A}-\psi_{\;\;\mu}^{A}\;e_{AA^{\prime}\nu}\;D_{\rho}\;{\widetilde\psi}_{\;\;\;\sigma}^{A^{\prime}}\right]d^{4}x\;,
y_{1}~=~-\,\frac{(2\mu-3)\Gamma(2\mu)}{16\Gamma(\mu)\Gamma(2-\mu)\pi^{2\mu}}{}~~,~~~z_{1}~=~-\,\frac{\Gamma(2\mu-1)}{4\Gamma(1-\mu)\Gamma(\mu-1)\pi^{2\mu}}
{\calI}(M^{2})=-\mathrm{Tr}\,\gamma_{5}{\frac{M^{2}}{-(\gamma\cdotD)^{2}+M^{2}}}=-\intd^{3}x\langlex|\mathrm{tr}\,\gamma_{5}{\frac{M}{\gamma\cdotD+M}}|x\rangle\,.
4f(k,z)={\frac{\alpha}{\pi}}\int_{-1}^{1}dz^{\prime}\int_{0}^{1}d\gamma\;A(\theta,\theta^{\prime},\gamma)\left\{\gamma^{3/2}f(\gammak,z^{\prime})+\gamma^{-5/2}f({\frac{k}{\gamma}},z^{\prime})\right\}
W=\frac{1}{2}\intd^{D}x\hat{A}_{R}[g^{BC}(-\partial_{B}+\hat{\Gamma}_{B})(\partial_{C}+\hat{\Gamma}_{C})]_{RS}\hat{A}_{S}\equiv\frac{1}{2}\int\hat{A}_{R}(-\triangle_{EM})_{RS}\hat{A}_{S}
\frac{1}{m}\oint\frac{\bar{dz}}{2\pii}\bar{\xi}_{t}(z)^{-m}\;\;(I)=\frac{1}{m}\oint\frac{\bar{dz}}{2\pii}\bar{\xi}_{t}(z)^{-m}\frac{1}{(\bar{h}_{r}(0)-\bar{z})}=-\bar{N}_{0m}^{rt}
\Gamma\left(q,P\right)=-i\frac{g^{2}}{\left(2\pi\right)^{4}}\intd^{4}k\,D^{\mu\nu}\left(q-k\right)\gamma_{\mu}G_{1}\left(k_{+}\right)\Gamma\left(k,P\right)G_{2}\left(k_{-}\right)\gamma_{\nu}\,.
J_{\mu}^{a,\mathrm{reg}}(x)\cdotZ_{E}[A]=-\frac\delta{\deltaA_{\mu}^{a}(x)}\Gamma_{E}[A]_{c}=j_{\mu}^{a,\mathrm{reg}}(x)\cdotZ_{E}[A]-\int_{0}^{1}\mathrm{d}t\,t\int\mathrm{d}
\{\bar{\psi},\chi\}_{_{\sim}}=\left\{\left(\begin{array}{c}{\psi_{rB}^{\ast}}\\{0}\\\end{array}\right),\left(\begin{array}{c}{0}\\{\chi_{l}^{B}}\\\end{array}\right)\right\}+\left\{\left(\begin{array}{c}{0}\\{\psi_{l}^{\ast\dot{A}}}\\\end{array}\right),\left(\begin{array}{c}{\chi_{r\dot{A}}}\\{0\}\\\end{array}\right)\right\}
S_{qf}(\mathrm{ex})\simN_{eff}~H_{max}^{3}V\sime^{-\phi_{exit}}M_{s}^{3}~V\sim\left(l_{s}^{2}/l_{P}^{2}\right)_{exit}~Vl_{s}^{-3}\simS_{HB}(\mathrm{ex})~,
S={\frac{1}{4\pi}}\int_{\calM}d^{2}\!\sigma\,\left\{{\frac{1}{\alpha^{\prime}}}\partialX^{\mu}\bar{\partial}X_{\mu}+\psi^{\mu}\bar{\partial}\psi_{\mu}+\tilde{\psi}^{\mu}\partial\tilde{\psi}_{\mu}\right\}
={\frac{\varepsilon^{m_{1}\cdotsm_{6}}}{\sqrt{-g}}}\left[{\frac{1}{6!}}F^{\underline{{a}}}{}_{m_{6}\cdotsm_{1}}-{\frac{1}{(3!)^{2}}}\left(F^{\underline{{a}}}{}_{m_{6}m_{5}m_{4}}\hat{H}_{m_{3}m_{2}m_{1}}-E_{n}^{\underline{{a}}}F^{n}{}_{m_{6}m_{5}m_{4}}\hat{H}_{m_{3}m_{2}m_{1}}\right)\right],
\left[\left({\frac{H_{1}^{\prime}}{H_{1}}}+{\frac{H_{2}^{\prime}}{H_{2}}}\right)+{\frac{2}{r}}\right](h_{1}^{\prime}+h_{2}^{\prime})=-4\left({\frac{H_{1}^{\prime}}{H_{1}}}-{\frac{H_{2}^{\prime}}{H_{2}}}\right)\varphi^{\prime},
\frac{d\zeta_{0}}{d\Lambda}=-\frac{1}{2}\frac{\lambda_{\varphi}\,\frac{\partial}{\partial\zeta}\left.\left(\frac{\partialU_{\Lambda}}{\partial\Lambda}\right)\right|_{(\rho_{0},\zeta_{0})}+\lambda_{\varphi\chi}\,\frac{\partial}{\partial\rho}\left.\left(\frac{\partialU_{\Lambda}}{\partial\Lambda}\right)\right|_{(\rho_{0},\zeta_{0})}}{\lambda_{\varphi}\lambda_{\chi}-\lambda_{\varphi\chi}^{2}}\;.
\Big(\mathrm{tr}e^{iF_{9i}}-\mathrm{tr}e^{-iF_{9i}}\Big)\Big(\mathrm{tr}e^{iF_{\bar{9}i}}-\mathrm{tr}e^{-iF_{\bar{9}i}}\Big)=-4\mathrm{tr}\,\mathrm{sin}F_{9i}\mathrm{tr}\,\mathrm{sin}F_{\bar{9}i}.
ds^{2}=-\left(1-\frac{r_{0}^{2}}{r^{2}}\right)dudv+\frac{p(u)}{r^{2}}du^{2}+\left(1-\frac{r_{0}^{2}}{r^{2}}\right)^{-2}dr^{2}+r^{2}d\Omega_{3}.
S^{+}\otimesS^{+}\,\Psi^{\mu}\,\Psi^{\nu_{1}}\Psi^{\nu_{2}}\,\Psi^{\nu_{4}}\Psi^{\nu_{4}}\to\Gamma_{\nu}\Gamma_{\rho}\Gamma_{\lambda}\,\timest_{8}^{\nu\rho\lambda\mu\nu_{1}\nu_{2}\nu_{3}\nu_{4}}\,.
ds^{2}={\frac{1}{4p^{2}}}\left(p{\frac{d}{dp}}\right)^{2}K\,dp^{2}+\left(p{\frac{d}{dp}}\right)K\,d\hat{z_{i}}d\hat{\bar{z_{i}}}+\left(p^{2}{\frac{d^{2}}{dp^{2}}}\right)K\,|\hat{z_{i}}d\hat{\bar{z_{i}}}|^{2},
R\left(\frac{sL_{1}^{2}}{4};\mu_{s}^{2}L_{1}^{2},\mu_{u}^{2}L_{1}^{2},\mu_{t}^{2}L_{1}^{2};m_{0}L_{1},w\right)\equiv16\pi^{2}\frac{\sigma^{(\infty)}(s)-\sigma^{(0)}(s)}{g\sigma^{(0)}(s)}.
\zeta(s,z:\omega,a,b:q)=\sum_{k_{1},k_{2},\cdotsk_{n}=0}^{\infty}\frac{q^{s(a_{1}k_{1}+\cdots+a_{n}k_{n}+b)}}{[k_{1}\omega_{1}+k_{2}\omega_{2}+\cdots+k_{n}\omega_{n}+z]_{q}\,^{s}}.
Q_{B}=\sum_{n}(c_{n}L_{-n}^{m}+\tilde{c}_{n}\tilde{L}_{-n}^{m})+\sum_{m,n}{\frac{(m-n)}{2}}:(c_{m}c_{n}b_{-m-n}+\tilde{c}_{m}\tilde{c}_{n}\tilde{b}_{-m-n}):+a\,(c_{0}+\tilde{c}_{0}),
S_{int,\psi}~=~g^{2}\intY\Psi\ast\Psi\ast\frac{b_{0}}{L}X(\Psi\ast\Psi)-g^{2}\intY\Psi\ast\Psi\ast\frac{b_{0}}{L}W\frac{b_{0}}{L}YQ(\Psi\ast\Psi).
F_{\mathrm{L}}^{\mu\nu}:=\left(\begin{matrix}{F_{\mathrm{L}}^{00}}&{F_{\mathrm{L}}^{0N}}\\{F_{\mathrm{L}}^{M0}}&{F_{\mathrm{L}}^{MN}}\\\end{matrix}\right)=\left(\begin{matrix}{0}&{q\psi_{\mathrm{L}}^{N}}\\{-q^{-1}\psi_{\mathrm{L}}^{M}}&{\varepsilon^{MN}{}_{C}\,\psi_{\mathrm{L}}^{C}}\\\end{matrix}\right),
s^{\prime}\;\left(\frac{1}{2\pi}\right)^{2M}\prod_{b=1}^{N}\prod_{i,j=1}^{n_{b}+n_{b}^{\prime}}\Big(w_{i}^{(b)}-z_{j}^{(b)}\Big)^{-2}\prod_{1\leqi<j\leqn_{b}+n_{b}^{\prime}}\Big(w_{i}^{(b)}-w_{j}^{(b)}\Big)^{2}\Big(z_{i}^{(b)}-z_{j}^{(b)}\Big)^{2}\;,
\intd\sigma^{p+1}Tr(aR_{ijkl}\lbrack\tilde{\Phi}^{i}\tilde{\Phi}^{j}\rbrack\lbrack\tilde{\Phi}^{k}\tilde{\Phi}^{l}\rbrack+bR_{ijkl}\tilde{\Phi}^{i}\tilde{\Phi}^{k}\tilde{\Phi}^{j}\tilde{\Phi}^{l})
\begin{array}{ll}{A^{n}=\sum_{k=0}^{n}\left(\begin{array}{c}{n}\\{k}\\\end{array}\right)_{+}A_{1}^{k}A_{2}^{n-k},\}&{\mathrm{where}\\\left(\begin{array}{c}{n}\\{k}\\\end{array}\right)_{+}=\frac{(n)_{+}!}{(k)_{+}!(n-k)_{+}!},}\\{}&{\mathrm{with}\\(n)_{+}=\frac{q^{2n}-1}{q^{2}-1},}\\\end{array}
\mathrm{~\frac{1}{2}~}(S_{m},S_{m})^{a}+V_{m}^{a}S_{m}=0,\qquad\mathrm{~\frac{1}{2}~}\{S_{m},S_{m}\}_{\alpha}+V_{\alpha}S_{m}=0,\qquad\mathrm{~\frac{1}{2}~}\{S_{m},S_{m}\}+V_{m}S_{m}=0,
\Psi_{\lambda}^{i_{1}}\Psi_{\mu}^{i_{2}}=\sum_{\rho\subset\lambda\otimes\mu}\sum_{i_{3}=1}^{d(\rho)}\left(\begin{array}{ccc}{\lambda}&{\mu}&{\rho}\\{i_{1}}&{i_{2}}&{i_{3}}\\\end{array}\right)\Psi_{\rho}^{i_{3}}\,\,.
\sum_{n,k=0}^{\infty}e^{-\epsilonkz-\epsilon^{2}nt}k^{1/3}n^{{-7}/3}e^{-k^{2}/n}=f(\epsilonz,\epsilon^{2}t)+2^{-4/3}\Gamma(4/3)\Gamma(-4/3)\epsilon^{4/3}\Phi(z,t/4)+{\calO}(\epsilon^{5/3})\,
S_{\mathrm{IIB}}=\frac{1}{2\kappa^{2}}\intd^{10}x\sqrt{-g}\left[R+\frac{1}{4}\,\mathrm{Tr}\left(\partial{\cal{M}}\,\partial{\cal{M}}^{-1}\right)-\frac{1}{12}\,{\bfF}^{T}{\cal{M}}{\bfF}\right]
L=\frac{1}{2}\,\dot{q}^{i}\dot{q}^{i}+i\,\bar{\psi}^{i}\dot{\psi}^{i}+\frac{1}{2}\,d^{i}d^{i}-d^{i}\,\frac{\partialv}{\partialq^{i}}+\psi^{i}\bar{\psi}^{j}\,\frac{\partial^{2}v}{\partialq^{i}\partialq^{j}}\,,
G_{T}W_{\mathrm{eff};a}^{\ast}\frac{\pi_{\lambda}}{\pi_{a}}\left(1-\alpha\right)=G_{S}W_{\mathrm{eff};\lambda}^{\ast}\left(1-\alpha\right)+\betaX_{B}+M^{2}\big|\frac{\pi_{a}}{\Lambda}\big|^{2},
P=\frac{2^{6}}{3^{2}\cdot5\cdot7\cdot11}\frac{2\kappa^{2}N^{2}T_{0}^{2}}{\Omega_{8}}(\omega_{1}^{12}R_{1}^{4}+\omega_{2}^{12}R_{2}^{4}+\omega_{3}^{12}R_{3}^{4})
\tilde{A}=\left(\begin{array}{cccccccc}{\;9}&{-9}&{\;3}&{-3}&{\;3}&{-3}&{\;3}&{-3}\\{\;3}&{-3}&{\;9}&{-9}&{\;3}&{-3}&{\;3}&{-3}\\{\;3}&{-3}&{\;3}&{-3}&{\;9}&{-9}&{\;3}&{-3}\\{\;3}&{-3}&{\;3}&{-3}&{\;3}&{-3}&{\;9}&{-9}\\\end{array}\right)\;,
\deltaS=\frac{i}{48}\intd^{12}z\(H-{\bar{H}})(F^{{\mathbfi}{\mathbfj}}F_{{\mathbfi}{\mathbfj}}-96m^{2}\Phi{\bar{\Phi}})+\frac{3}{10}(H-{\bar{H}})_{{\mathbfi}{\mathbfj}{\mathbfk}{\mathbfl}}F^{{\mathbfi}{\mathbfj}}F^{{\mathbfk}{\mathbfl}}\.
{\frac{1}{a_{1}-\kappa_{2}}}{\frac{\alpha+a_{1}}{a_{1}-\kappa_{1}}}ln\biggl[{\frac{\kappa_{1}-a_{1}\kappa_{2}}{a_{1}-a_{1}\kappa_{1}}}\biggr]=-{\frac{(a_{1}+\alpha)(1-i\xi_{1})^{2}(1+\xi_{1}^{2})}{2\xi_{1}^{2}}}
\begin{array}{ll}{{[L_{n},L_{m}]}=(n-m)L_{n+m},}&{n,m=0,\pm1,}\\{{[L_{n},F_{r}]}=\left(\frac{1}{2}n-r\right)F_{n+r},}&{r=\frac{1}{2},-\frac{1}{2},}\\{{\{F_{r},F_{s}\}}=2L_{r+s},}&{r,s=\frac{1}{2},-\frac{1}{2}.}\\\end{array}
\epsilon=e^{-{\frac{g\etat}{2\sqrt{2}}}\epsilon_{ij}}H^{-1/8}\left[\sqrt{f^{1/2}+1}-\sqrt{f^{1/2}-1}\gamma_{\overline{{r}}}\right]e^{-{\frac{1}{2}}\gamma_{\overline{{\thetar}}}}e^{-{\frac{1}{2}}\gamma_{\overline{{\phi\theta}}}}(1-\eta\gamma_{\overline{{0}}}\epsilon^{ij})\epsilon_{0},
\begin{array}{c}{[{\calM}_{1},{\calM}_{2}]={\calM}_{3}+i\textstyle{\frac{1}{2}}\psi_{3}\,1\,,}\\{{}}\\{\psi_{3}=2i(\varepsilon_{2}^{a}\rho_{1a}-\varepsilon_{1}^{a}\rho_{2a}+\bar{\rho}_{2}^{a}\bar{\varepsilon}_{1a}-\bar{\rho}_{1}^{a}\bar{\varepsilon}_{2a})\,,}\\\end{array}