harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\psi\left(z,\overline{{z}}\right)\rightarrowe^{i\frac{2\pia}\beta}\psi\left(z,\overline{{z}}\right),\qquad\overline{{\psi}}\left(z,\overline{{z}}\right)\rightarrowe^{-i\frac{2\pia}\beta}\overline{{\psi}}\left(z,\overline{{z}}\right),
\mu(E)=C\cdot\left(\frac{n_{1}}{N}+\alpha^{1}\alpha^{2}\right)\left(\frac{n_{2}}{N}+\alpha^{3}\alpha^{4}\right)\cdot\dots\cdot\left(\frac{n_{g}}{N}+\alpha^{2g-1}\alpha^{2g}\right)
\delta{\calL}_{1}=-\delta\left\{{\frac{\lambda_{--}}{1-\lambda_{++}\lambda_{--}}}A_{+}^{2}+{\frac{\lambda_{++}}{1-\lambda_{++}\lambda_{--}}}A_{-}^{2}+{\frac{1+\lambda_{++}\lambda_{--}}{1-\lambda_{++}\lambda_{--}}}A_{+}A_{-}\right\}
<0|\tilde{F}_{0i}(0)|e_{\pm}^{\lambda},{\vec{p}}>=\mp{\frac{i}{(2\pi)^{3/2}}}\sqrt{\frac{p_{0}}{2}}e_{i}^{\pm}\mathrm{lim}_{p^{2}\rightarrow0}{\frac{1}{1-\Pi(p^{2})}}
\deltaZ^{\underline{M}}E_{\underline{M}}^{\underline{a}}=\deltaz^{M}e_{M}^{A}(e_{A}^{N}\partial_{N}Z^{\underline{M}}E_{\underline{M}}^{\underline{a}})\equiv\deltaz^{M}e_{M}^{A}E_{A}^{\underline{a}}.
S^{\mathrm{add}}\equiv-(p+1)e_{a}e_{b}\intd^{p+1}\xi_{a}\intd^{p+1}\xi_{b}D_{a}^{\mu_{1}\mu_{2}\ldots\mu_{p}}D_{b\mu_{1}\mu_{2}\ldots\mu_{p}}G((X_{}a-X_{b})^{2}).
\{x^{a}\,,\,x^{b}\}^{\ast\ast}=s\frac{\epsilon^{abc}p_{c}}{(-p^{2})^{3/2}}\qquad\{x^{a}\,,\,p_{b}\}^{\ast\ast}=\delta^{a}{}_{b}\qquad\{p_{a}\,,\,p_{b}\}^{\ast\ast}=0
{\omega}_{n,0}[\kappa,\lambda,\sigma,\tau]\lhd\stackrel{P}{\calO}{}\!\!_{n,0}^{(\pm)}\;=\;\sum_{(k,\ell,s,t)\in{\calS}^{3n-3}}\stackrel{P}{K}{}\!\!_{n,0}^{(\pm)}\left(\begin{array}{c}{\!\!\kappa;k\!\!}\\{\!\!\lambda,\sigma,\tau;\ell,s,t\!\!}\\\end{array}\right){\omega}_{n,0}[\kappa+k,\lambda+\ell,\sigma+s,\tau+t]
{\delta}_{\alpha}\equiv\frac\delta{du^{\alpha}}=(\delta_{i}=\partial_{i}-N_{i}^{b}(u)\\partial_{b},\partial_{a}=\frac\partial{dy^{a}})\mathrm{~and~}\delta^{\alpha}\equiv\deltau^{\alpha}=(\delta^{i}=dx^{i},\delta^{a}=dy^{a}+N_{k}^{a}(u)\dx^{k})
R_{\mu\nu}=Tr[\frac{1}{2}J_{\mu}^{p}J_{\nu}^{p}-2p(F_{\mu\lambda}{F^{T}}^{\nu\lambda}-\frac{1}{4}g_{\mu\nu}FF^{T})+pH_{\mu\lambda\sigma}pH_{\nu}^{\lambda\sigma}-\frac{1}{3}g_{\mu\nu}(pH)^{2}]
S_{\mu\nu}(m,n|a,b)=\sum_{p=1}^{\infty}\frac{p}{m}\frac{p}{n}S_{\mu\nu}(p)\oint\frac{dz}{2\pii}z^{p-1}\left(\frac{b^{*}z+a^{*}}{az+b}\right)^{m}\oint\frac{d\bar{z}}{-2\pii}{\bar{z}}^{p-1}\left(\frac{b\bar{z}+a}{a^{*}\bar{z}+b^{*}}\right)^{n}.
S_{0}=\intd^{4}x\bigg(-{\frac{1}{4}}F_{\mu\nu}^{a}F^{a\mu\nu}-{\frac{1}{12}}H_{\mu\nu\lambda}^{a}H^{a\mu\nu\lambda}+{\frac{m}{4}}\epsilon^{\mu\nu\rho\lambda}B_{\mu\nu}^{a}F_{\rho\lambda}^{a}\bigg)\,.
\frac{1}{Z_{k}^{A}}\left\{\frac{1-e^{-(p^{2}+m_{k}^{2})/k^{2}}}{p^{2}+m_{k}^{2}}\left(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)+\frac{1-e^{-(p^{2}/\alpha_{k}+m_{k}^{2})/k^{2}}}{p^{2}/\alpha_{k}+m_{k}^{2}}\frac{p_{\mu}p_{\nu}}{p^{2}}\right\}\delta^{ab}\:.
0=k\cdot\varepsilon_{1,2}=\vec{k}\cdot\vec{\varepsilon}_{1,2}-(k^{0})\varepsilon_{1,2}^{0}=\left[\vec{k}-\frac{k^{0}}{\sqrt{(\vec{n})^{2}+1}}\vec{n}\right]\cdot\vec{\varepsilon}_{1,2}.
{\frac{i}{2}}\left\vert{\frac{Det^{\prime}\left[-(d^{2}/d\tau^{2})+V^{\prime\prime}[x_{c}(\tau)]\right]}{Det\left[-(d^{2}/d\tau^{2})+\omega^{2}\right]}}\right\vert^{-1/2}\,sqrt{{\frac{S_{eo}}{2\pi}}}\d\tau_{c}
S(h,A)=S_{0}+\frac{ik}{2\pi}\int_{D\timesR}dxdtd\tau\mathrm{Tr}\partial_{\tau}[A_{+}*\partial_{x}h*h^{-1}]+\frac{ik}{4\pi}\int_{D\timesR}dxdtd\tau\mathrm{Tr}\partial_{\tau}(A_{+}*A_{x}),
S_{0}=\intdt\Big[-1+\frac{1}{2}\vec{v}\cdot\vec{v}+\frac{15}{16}\frac{(\vec{v}\cdot\vec{v})^{2}}{r^{7}}-\frac{105}{16}\frac{(x^{i}J^{ij}v^{j})(\vec{v}\cdot\vec{v})}{r^{9}}+\cdots\Big]~.
\kappa=\displaystyle\frac{2\Gamma(h/2)}{\pi\Gamma(1-h/2)}\left(\displaystyle\frac{\sqrt{\pi}\Gamma\left(\displaystyle\frac{1}{2-h}\right)}{2\Gamma\left(\displaystyle\frac{h}{4-2h}\right)}\right)^{2-h}\,.
\tilde{R}i\mathrm{tr}\left\{\bar{\psi}(1\pm\gamma_{5})\psi\bar{\psi}(1\pm\gamma_{5})\psi\right\}\tilde{R}^{\dagger}=i\mathrm{tr}\left\{\bar{\psi}(1\pm\gamma_{5})\psi\bar{\psi}(1\pm\gamma_{5})\psi\right\}
inv(j;\omega)=u^{2}\sum_{k=1}^{m}(1,k)^{2}x_{k}^{2}+2v^{2}\sum_{\bar{k}=m+1}^{m+n}[1,\hat{\bar{k}}]^{2}\theta_{\hat{\bar{k}}}\theta_{-\hat{\bar{k}}}\equivu^{2}x^{2}(j)+2v^{2}\theta^{2}(\omega),
\langleC_{s}\left(z,\overline{{z}}\right)\cdots\rangle=\langleC_{s}\left(z\right)\cdots\rangle_{0}+\lambda\intdw\,d\overline{{w}}\,\langleC_{s}\left(z\right)\Phi_{lk}\left(w,\overline{{w}}\right)\cdots\rangle_{0}+O\left(\lambda^{2}\right)\,.
\begin{array}{c}{\delta_{1}^{^{\prime}2}=\left(s_{2}-1/2\right)\left(s_{2}+3/2\right)\quad,\qquad\delta_{2}^{^{\prime}2}=\left(s_{1}-1/2\right)\left(s_{1}+7/2\right)}\\{s_{1},s_{2}=1/2,3/2,...\quad,\qquads_{1}\geqs_{2}}\\\end{array}
\langle\langle\kappa_{\rho},x|ph\rangle\rangle=\frac{2}{|\rho|^{3}}\delta(\rho)(\eta^{\dagger}\eta)^{2}\psi_{s}^{ph}(x)=\frac{2}{z^{\dagger}z}\delta(z^{\dagger}z)(\eta^{\dagger}\eta)^{2}\psi_{s}^{ph}(x)\;,
\begin{aligned}{H^{2}(Y,\mathbbZ)/\mathrm{torsion}}&{{}\congH^{2}(Z/G,\mathbbZ)/\mathrm{torsion}}\\{}&{{}\cong\mathrm{~the~orthogonal~complement~of~the~}}\\{}&{{}\hphantom{\cong}\quad\mathrm{exceptional~divisors~in~}H^{2}(\widetilde{Z/G},\mathbbZ).}\\\end{aligned}
\left[\begin{array}{cc}{0}&{-\left({\frac{2\pin}{m_{l}-m_{r}}}\right)^{2}b_{i}}\\{b_{i}}&{0}\\\end{array}\right],\;\;\;\;\;\left[\begin{array}{cc}{0}&{-\left({\frac{2\pin}{m_{l}-m_{r}}}\right)^{2}b_{i}^{*}}\\{b_{i}^{*}}&{0}\\\end{array}\right]
{\|}\,{(\cal{A}}_{1})^{n}\,{\|}=(ig)^{n}\int{\sfD}(1,\cdots,n)\,\psi_{(1)i_{1}}^{{\alpha_{1}}}(1)\;\psi_{(1)i_{2}}^{{\alpha_{2}}}(2)\;\cdots\psi_{(1)i_{n}}^{{\alpha_{n}}}(n)\;\Pi_{i_{1}}^{\alpha_{1}}(1)\;\Pi_{i_{2}}^{\alpha_{2}}(2)\;\cdots\;\Pi_{i_{n}}^{\alpha_{n}}(n)\;,
W(\phi)=\int_{\phi_{0}}^{\phi}e^{\alpha\eta}\left(\frac{\beta}{2}\eta^{2}-\Lambda\right)d\eta=\left.e^{\alpha\eta}\left[\frac{\beta}{2}\left(\frac{2}{\alpha^{3}}-\frac{2\eta}{\alpha^{2}}+\frac{\eta^{2}}{\alpha}\right)-\frac{\Lambda}{\alpha}\right]\right|_{\phi_{0}}^{\phi}.
\partial^{\mu}\left(-2{\frac{\partial{\calL}}{\partial\sigma}}\partial_{\mu}u-{\frac{\partial{\calL}}{\partial\rho}}\partial_{\mu}u^{*}\right)+{\frac{\partial{\calL}}{\partialu}}=\Lambda\partial^{\mu}({\calF}K_{\mu})+\Phi\partial^{\mu}({\calF}K_{\mu}^{*})
\Big\{g,\partial_{\alpha_{1}\cdots\alpha_{k}}g,\partial_{\alpha_{1}\cdots\alpha_{k-1}}A_{\alpha_{k}}^{a}\Big\}\rightarrow\Big\{g,\partial_{(\alpha_{1}\cdots\alpha_{k-1}}A_{\alpha_{k})}^{a},D_{\alpha_{1}\cdots\alpha_{k-1}}^{(\Theta)}I_{\alpha_{k}}^{a}\Big\}.
Z={\frac{1}{N!}}\int_{0}^{\infty}\mathrm{e}^{-{\frac{1}{kT}}\sum_{i=1}^{N}{\frac{p_{i}^{2}}{2m}}}\Omega^{\mathrm{fluid}}dz_{1}d\bar{z}_{1}\cdotsdp_{z_{N}}dp_{\bar{z}_{N}}\.
\begin{array}{rcl}{iT}&{=}&{\displaystyle+\lambdav\int\left[3D_{k}^{\sigma}+D_{k}^{Z}+2D_{k}^{W}\right]}\\{}&{}&{\displaystyle+\frac{n}{2}g^{2}v\int\left[D_{k}^{W}+D_{k}^{Z}\right]}\\{}&{}&{\displaystyle-\frac{1}{2}g^{2}v\int\left[D_{k}^{W}+D_{k}^{Z}\right],}\\\end{array}
i\intdx~e^{ik\cdotx}~\langle0|T(J_{\mu}(x)J_{\nu}(0))|0\rangle=(k_{\mu}k_{\nu}-k^{2}g_{\mu\nu})\left\{\Pi_{0}(k^{2})+\Pi_{2}(k^{2})F^{2}+O(F^{4})\right\}
c(l_{0},k_{1}+1;l_{0},k_{1})d(l_{0},k_{1};l_{0},k_{1}+1)=\frac{(l_{0}-k_{0})a-(l_{0}+k_{0})b}{(k_{1}-l_{0})(k_{1}-l_{0}+1)(k_{1}+l_{0})(k_{1}+l_{0}+1)},
S_{ij}\left(\theta+\theta_{h}+t_{i}\theta_{H}\right)S_{ij}\left(\theta-\theta_{h}-t_{i}\theta_{H}\right)=\prod_{l=1}^{r}\prod_{n=1}^{I_{il}}S_{jl}\left(\theta+(2n-1-I_{il})\theta_{H}\right)\,\quad.
{\frac{1}{\sqrt{g\left(x+\pi(\xi)\right)}}}={\frac{1}{\sqrtg}}\left[1-{\calD}\phi.\xi+{\frac{{\calD}^{2}}{2}}\left(\phi.\xi\right)^{2}-{\frac{\calD}{2}}\phi_{\lambda;\sigma}\xi^{\lambda}\xi^{\sigma}-{\frac{1}{6}}R_{\lambda\sigma}\xi^{\lambda}\xi^{\sigma}+...\right].
\fracd{ds}\left[\begin{array}{c}{\gamma^{0}}\\{\gamma^{1}}\\{\pi^{0}}\\{\pi^{1}}\\\end{array}\right]=\left[\begin{array}{cccc}{2i{\calH}}&{0}&{-2i}&{0}\\{0}&{2i{\calH}}&{0}&{-2i}\\{0}&{-eE}&{0}&{0}\\{-eE}&{0}&{0}&{0}\\\end{array}\right]\left[\begin{array}{c}{\gamma^{0}}\\{\gamma^{1}}\\{\pi^{0}}\\{\pi^{1}}\\\end{array}\right],
\begin{array}{ccccccccccl}{Dp:}&{1}&{2}&{.}&{.}&{p}&{\_}&{\_}&{\_}&{\_}&{\quad\mathrm{probe}\nonumber}\\{F1:}&{\_}&{\_}&{\_}&{\_}&{\_}&{y}&{\_}&{\_}&{\_}&{\quad\mathrm{soliton.}}\\\end{array}
\chi({\bfx},t)=\int\frac{d^{3}{\bfk}}{[(2\pi)^{3}2\omega_{\bfk}]^{1/2}}[c({\bfk})e^{i({\bfk}_{n}\cdotx-\omega_{\bfk}t)}+c^{\dagger}({\bfk})e^{-i({\bfk}_{n}\cdotx-\omega_{\bfk})t)}]
\delta\gamma_{\mu\nu}(x)=-\zeta^{\lambda}\star\partial_{\lambda}\gamma_{\mu\nu}+\partial^{\rho}\zeta_{\mu}\star\gamma_{\rho\nu}+\partial^{\rho}\zeta_{\nu}\star\gamma_{\mu\rho}-\partial^{\rho}\zeta_{\rho}\star\gamma_{\mu\nu}+\frac{1}{\kappa}(\partial_{\nu}\zeta_{\mu}+\partial_{\mu}\zeta_{\nu}-\partial^{\rho}\zeta_{\rho}\eta_{\mu\nu}).
{\calI}=\sum_{\ell=1}^{6}\int\frac{d^{2}\tau}{\tau_{2}^{\;3/2}}{\bar{C}}_{\ell}(\bar{\tau})\,\partial_{\bar{\tau}}(\tau_{2}^{\;1/2}\sum_{p_{L},p_{R}\in\Gamma_{\ell}}e^{\pii\tau\,|p_{L}|^{2}}\,e^{-\pii\bar{\tau\,}p_{R}^{2}})\,
1_{N_{B}}(x_{\eta}^{-1}x_{\zeta})\beta_{i,j}(x_{\eta}^{-1}x_{\zeta})\frac{1}{|K|}\int_{K}\mathrm{d}k\,\phi(kx_{\eta}g_{B}^{-1}x_{\eta}^{-1}x)=1_{N_{B}}(x_{\eta}^{-1}x_{\zeta})\beta_{ij}(x_{\eta}^{-1}x_{\zeta})\frac{1}{|K|}\int_{K}\mathrm{d}k\,\phi(kx)
\psi(2)_{a=c=0}=i\int\frac{d^{4}x_{1}d^{4}p^{\prime}}{(2\pi)^{4}}e^{-ip_{\mu}^{\prime}x_{2}^{\mu}}\;\frac{{p_{\nu}}^{\prime}\gamma^{\nu}}{p_{\tau}^{\prime}p^{\tau\,\prime}-i\epsilon}e^{ip_{\eta}^{\prime}x_{1}^{\eta}}N_{\sigma}\gamma^{\sigma}\psi(1).
\frac{1}{z}\int_{0}^{z}\mathrm{d}s\s^{l}\psi(\frac{s-1}{s})+\int_{0}^{1}\mathrm{d}s\s^{l}\psi(\frac{s-z}{s})+\frac{\psi(1)(-1)^{l}}{2(l+1)}\frac{1-z^{l+1}}{1-z}=\frac{\lambda-1}{2}(-1)^{l}\psi(z),
S_{2}=\frac{1}{2}\intd^{2}\xi\left[\left(\partial_{\tau}\psi_{M}\right)^{2}+\left(\partial_{\sigma}\psi_{M}\right)^{2}+\frac{2}{\tau^{2}}(\psi_{M})^{2}+\frac{c_{m}^{\prime}}{\taua}\left(\psi_{m}\partial_{\sigma}\psi_{0}-\psi_{0}\partial_{\sigma}\psi_{m}\right)\right]\:.
{\mathcal{L}}_{SYM}-\frac{1}{g_{YM}^{2}}{\mathrm{Tr}}\left(\frac{1}{2}\,B^{2}\Phi^{m}\Phi^{m}+\frac{i}{3}\,B\epsilon_{ABC}\Phi^{A}\left[\Phi^{B},\Phi^{C}\right]\right).
x_{i}=(e^{{\frac{2\pi\sqrt{-1}}{n-2}}})^{i}\left(F_{0}(\zeta)+\frac{1}{n-2}\sum_{\kappa=1}^{n-3}(e^{{\frac{2\pi\sqrt{-1}}{n-2}}})^{-i\,n\kappa}\mu_{\kappa}(-p)^{\kappa}F_{\kappa}(\zeta)\right)\,\,\,(i=1,2,...,n-2)
\begin{array}{c}{Q_{1}(p)=a^{\dagger}p^{N+1}{\itP_{-}}+ap^{N}{\itP_{+}}}\\{Q_{2}(p)=i(a^{\dagger}p^{N+1}{\itP_{-}}-ap^{N}{\itP_{+}})}\\{{\itP_{\pm}}=\frac{1}{2}(1\pm(-)^{N}).}\\\end{array}
\Omega^{AB}=\left(\begin{matrix}{\gamma^{\mu}\omega_{1\mu}^{AB}}&{V_{\mu}\gamma^{\mu}\gamma^{5}l^{AB}}\\{-V_{\mu}\gamma^{\mu}\gamma^{5}\tilde{l}^{AB}}&{\gamma^{\mu}\omega_{2\mu}^{AB}}\\\end{matrix}\right)
E({\bfm})={\frac{V_{0}}{4\sqrt{2\pi}}}{\frac{1}{L_{y}}}\sum_{X_{1},X_{2}}e^{-{(X_{1}-X_{2})^{2}/2}}\biggl(1-{\bfm}(X_{1})\cdot{\bfm}(X_{2})\biggr).
(\psi^{-},\phi^{+})=\sum_{j}\int_{m_{0}^{2}}^{\infty}d{\mathsfs}\,\sum_{j_{3}}\int\frac{d^{3}\hat{\bfp}}{2\hat{E}}\langle\psi^{-}|[j,{\mathsfs}],j_{3},\hat{\bfp}^{-}\rangleS_{j}({\mathsfs})\langle^{+}j_{3},\hat{\bfp},[j,{\mathsfs}]|\phi^{+}\rangle\,.
S^{(2)}=-3\intd^{4}x{\frac{-\lambda}{4!}}\left[G(x,x)\right]^{2}-3!\intd^{4}xd^{4}y({\frac{-\lambda}{3!}})^{2}\phi_{cl}(x)\phi_{cl}(y)\left[G(x,y)\right]^{3}\,\,.
\langle0|\bar{\psi}\psi|0\rangle_{J}=-\frac{1}{4\pi^{2}l^{3}}\int_{0}^{1}dy\frac{M(y)}{y+M_{0}^{2}}=\frac{1}{2\pi\alphal^{3}}\left.\left(x\frac{dM(x)}{dx}\right)\right|_{x=1},
M_{ADM}^{2}=e^{-4\Phi_{\infty}^{\prime}}\bigg({\vec{Q}}^{T}{\calM}_{+}{\vec{Q}}+{\vec{P}}^{T}{\calM}_{+}{\vec{P}}+\2\left[({\vec{P}}^{T}{\calM}_{+}{\vec{P}})({\vec{Q}}^{T}{\calM}_{+}{\vec{Q}})-({\vec{P}}^{T}{\calM}_{+}{\vec{Q}})^{2}\right]^{\frac{1}{2}}\bigg).
a_{n-\alpha}=\sum_{k=0}^{n}~\varrho^{2k}~c_{k,n-\alpha}(d)~\bar{a}_{n-k},~~~b_{n-\alpha}=\sum_{k=0}^{n}~\varrho^{2k}~c_{k,n-\alpha}(d-1)~\bar{b}_{n-k},
S=\int_{-\infty}^{\infty}dt\int_{-\infty}^{0}dx\left[\frac{1}{2}((\partial_{t}\Phi)^{2}-(\partial_{x}\Phi)^{2}-m^{2}\Phi^{2})-V(\Phi)\right]-\int_{-\infty}^{\infty}dtU(\Phi(0,t))\qquad.
C_{Jm_{J}ij}^{s_{1}s_{2}\sigma_{1}\sigma_{2}}=C_{Jm_{J}}^{\left(tr\right)\ell_{j}m_{\sigma}}C_{Jm_{J}}^{\left(tr\right)\ell_{i}m_{s}}/\sum_{s_{1}s_{2}m_{J}}\left(C_{Jm_{J}}^{\left(tr\right)\ell_{i}m_{s}}\right)^{2},
F(k)\;=\;e_{N}^{2}\int_{0}^{1}d\alpha\,\int_{0}^{\infty}\frac{dp^{2}}{\pi}\,\frac{1}{\beta}\,\sum_{n=-\infty}^{+\infty}\frac{p^{2}-\alpha(1-\alpha)k^{2}-{\tilde{\omega}}_{n}^{2}+m^{2}}{[p^{2}+\alpha(1-\alpha)k^{2}+{\tilde{\omega}}_{n}^{2}+m^{2}]^{2}}\;,
\begin{array}{l}{<\chi_{\pm}\mid\{{\hat{\gamma}}^{(\lambda,\mu,\alpha)}(p,s),{\hat{\gamma}}_{(\tau,\nu,\beta)}(p^{\prime},s^{\prime})\}\mid\chi_{\pm}>=}\\{=<e^{(\lambda,\mu,\alpha)}(p,s),e_{(\tau,\nu,\beta)}(p^{\prime},s^{\prime})>=\delta_{ss^{\prime}}\delta_{\tau}^{\lambda}\delta_{\mu}^{\nu}\delta_{\beta}^{\alpha}\delta^{(3)}(\vec{p}-\vec{p^{\prime}}).}\\\end{array}
C_{\mu}^{a}\leftrightarrow\eta_{\mu}^{a},\;\bar{C}_{\mu}^{a}\leftrightarrow\bar{\eta}_{\mu}^{a},\;\bar{C}^{\primea}\leftrightarrow\varphi^{(1)a},\;{\calB}_{\mu}^{a}\leftrightarrowb_{\mu}^{a},
\begin{array}{c}{I^{-1}(z_{31})I(Z_{1})I(z_{21})=I(z_{23})}\\{{}}\\{I^{-1}(z_{12})I(Z_{2})I(z_{32})=I(z_{31})}\\{{}}\\{I^{-1}(z_{23})I(Z_{3})I(z_{13})=I(z_{12})}\\\end{array}
\phi_{\pm\nu}^{\mu}(x^{+}=0)=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}\frac{dk^{+}}{\sqrt{2k^{+}}}\left[a_{\nu}^{\mu}(k^{+},\pm)\mathrm{e}^{-\mathrm{i}k^{+}x^{-}}+a_{\nu}^{\dagger\mu}(k^{+},\pm)\mathrm{e}^{\mathrm{i}k^{+}x^{-}}\right].
\Pi_{\mathrm{T}}^{(1)(\beta)}(0)=\frac{C(D)e^{2}}{2^{D-2}\pi^{\frac{D-1}{2}}\Gamma(\frac{D-1}{2})}\int_{m}^{\infty}d\omega_{k}\,\frac{\partial}{\partial\omega_{k}}\left((\omega_{k}^{2}-m^{2})^{\frac{D-1}{2}}\,\frac{n_{F}(\omega_{k})}{\omega_{k}}\right)=0,\qquadD\geq2
{\calM}_{l}(r,\tau)={\calG}_{B}^{l}\left[(\partial_{\tau\tau}^{2}{\calG}_{B})^{2}+(\partial_{rr}^{2}{\calG}_{B})^{2}+2(\partial_{r\tau}^{2}{\calG}_{B})^{2}+2(\partial_{r}{\calG}_{B})^{2}/r^{2}\right]\,.
\Omega_{j,l,m}=\left(\begin{matrix}{{({\frac{l+{\frac{1}{2}}-m}{2l+1}})^{\frac{1}{2}}}Y_{l,{m-{\frac{1}{2}}}}(\theta,\phi)}\\{-{({\frac{l+{\frac{1}{2}}+m}{2l+1}})^{\frac{1}{2}}}Y_{l,{m+{\frac{1}{2}}}}(\theta,\phi)}\\\end{matrix}\right),
\int\mathrm{d}^{2}z\sqrt{\frac{m}{32\pi^{3}\left|z\right|^{3}}}e^{-m\left|z\right|}=2\pi\sqrt{\frac{m}{32\pi^{3}}}\int_{0}^{\infty}\frac{\mathrm{d}r}{\sqrt{r}}e^{-mr}=\frac{1}{2\sqrt{2}}
M_{\alpha_{1}\beta_{1}}^{\beta_{0}\alpha_{0}}=-\left(-\right)^{\epsilon_{\alpha_{1}}\epsilon_{\beta_{1}}}M_{\beta_{1}\alpha_{1}}^{\beta_{0}\alpha_{0}}=-\left(-\right)^{\epsilon_{\alpha_{0}}\epsilon_{\beta_{0}}}M_{\alpha_{1}\beta_{1}}^{\alpha_{0}\beta_{0}}.
x=\frac{R^{2}}{U_{T}}\alpha\sqrt{1-(1+\theta){\alpha}^{4}+\theta{\alpha}^{6}}\int_{1}^{\frac{U\alpha}{U_{T}}}\frac{y^{2}dy}{\sqrt{(y^{2}-1)(y^{2}-{\alpha}^{2})(y^{4}+{\alpha}^{2}y^{2}-\theta{\alpha}^{4})(y^{4}+y^{2}-\theta{\alpha}^{6})}}.
\begin{array}{c}{\displaystyle\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\int\frac{d^{4}k_{2}}{(2\pi)^{4}}D_{m}(k_{2})D_{m}^{2}(k_{1})\overline{{D(k_{1}+p)}}K_{1}.}\\\end{array}
I=Q_{v}B^{\prime\prime}=\intdt\Bigl(q_{i}(\partial_{t}a^{i}-h^{i})+\bar{c}_{i}\partial_{t}c^{i}+\bar{c}_{i}\partial_{j}h^{i}c^{j}-\partial_{t}(\bar{c}_{i}c^{i})\Bigr),
\psi(x)_{ij}=\sqrt{{\frac{1}{4L}}}\sum_{n=\frac{1}{2},\frac{3}{2},\frac{5}{2},\ldots}\left(B_{ij}(n)e^{-in\pix^{-}/L}+B_{ji}^{\dagger}(n)e^{in\pix^{-}/L}\right).
e^{iNw[{\vec{h}}]}=\int{\calD}\lambdae^{iN[(\intd^{d}x(\frac{1}{4u}\lambda^{2}+\frac{1}{2}c^{2}\lambda))+\frac{i}{2}Trln\nabla-\frac{1}{2}h.\nabla^{-1}.h]}
\begin{array}{c}{L_{\mu}=g\partial_{\mu}g^{-1}=-g~(g^{-1}\partial_{\mu}g)~g^{-1}=-g~J_{\mu}{}~g^{-1}}\\{=-\varepsilon_{\mu\nu}~g~\partial^{\nu}\phi~g^{-1}=-\varepsilon_{\mu\nu}\partial^{\nu}(g~\phi~g^{-1})+g~[\partial_{\mu}\phi~,\phi]~g^{-1}.}\\\end{array}
S_{H}=\int\mathrm{d}\tau(p_{m}\dot{x}^{m}+\pi_{\alpha}\dot{\lambda}^{\alpha}+\bar{\pi}_{\dot{\alpha}}{\bar{\lambda}}^{\dot{\alpha}}-\lambdaT_{1}^{int}-\mu\theta^{int}-\bar{\mu}\bar{\theta}^{int}-\nud-\bar{\nu}\bar{d})\,,
\sum_{n=1}^{\infty}\frac{1}{(n^{2}+p^{2})^{k}}=\frac{\sqrt{\pi}\Gamma(k-\frac{1}{2})}{2\Gamma(k)p^{2k-1}}-\frac{1}{2p^{2k}}+\left(\frac{\pi}{p}\right)^{k}\frac{e^{-2\pip}}{\Gamma(k)},
{\calP}_{n}\approx\frac{\left[2\sqrt{T^{2}-4d}+4d-1\right]^{n+\frac12}+(-1)^{n}\left[2\sqrt{T^{2}-4d}-4d+1\right]^{n+\frac12}}{\sqrt{\pin}(1+2T+4d)^{n+\frac12}\left(T^{2}-4d\right)^{\frac14}}.
\begin{array}{c}{\displaystyle{\frac{E(R)}{\hbar\,\Omega}=V\,\frac{{\Omega}^{3}}{16\,\pi^{2}\,c^{3}}+\xi\,S\,\frac{{\Omega}^{2}}{12\,\pi^{2}\,c^{2}}+\dots\}}\\\end{array},
\tilde{q}^{(ab)}(r)=\frac{1}{8\pi^{2}b^{2}r^{2}}\sum_{l=1}^{\infty}(2l+1)\sum_{\lambda=0,1}\int_{0}^{\infty}{z\tilde{\Omega}_{\lambdal}(z,\sigma)F_{\lambdal}^{(q)}(z,zy)dz},
\left[\overline{{F}}^{A},\overline{{p}}_{\mu}\right]=\left[\overline{{a}}_{B}^{A}\,F^{B},a_{\mu}^{\nu}\circp_{\nu}\right]=i\hbara_{\mu}^{\nu}\partial_{\nu}\left(\overline{{a}}_{B}^{A}F^{B}\right)=i\hbar\overline{{\partial}}_{\mu}\overline{{F}}^{A}
\frac{1}{2(\pi-\epsilon)i}\int_{C}ds\,\frac{e^{\rhos}}{s}F(s)=\frac{1}{2(\pi-\epsilon)i}\left(\int_{C_{0}}ds+\int_{C_{1}}ds+\int_{C_{2}}ds\right)\frac{e^{\rhos}}{s}F(s)=0,\nonumber
K_{B_{3}}(x,x^{\prime},s)=\sum_{n=-\infty}^{+\infty}K_{H_{3}}(\rho,~\rho^{\prime},~\psi-\psi^{\prime}+2\pi{\frac{r_{+}}{l}}n,~\theta-\theta^{\prime}+2\pi{\frac{|r_{-}|}{l}}n)~~.
{\calL}_{Weyl}^{nm}=\partial_{+}{A}_{i}\left(\partial_{-}{A}_{i}-\partial_{i}A_{-}\right)+\frac12\left(\partial_{+}{A}_{-}\right)^{2}-\frac14\left(\partial_{i}{A}_{j}-\partial_{j}{A}_{i}\right)^{2}+{A}_{-}{J}^{-}+{A}_{i}{J}^{i}\;,
\omega_{n}(q)\,\sim\,n\,\pi\,+\,\frac{2q}{n\pi}\,-\,\left(1\,+\,\frac{q}{6}\right)\,\frac{(2q)^{2}}{(n\pi)^{3}}+O(n^{-5})\,{,}\quadn\to\infty\,{.}
\tilde{\calE}_{q}[\phi(x)]=\frac{1}{2}\sum_{j}\tilde{\omega}_{j}+\int_{0}^{\infty}\frac{dk}{2\pi}\tilde{\omega}(k)\frac{d}{dk}\tilde{\delta}(k)-\frac{\tilde{m}}{4}+\tilde{\calE}_{ct}[\phi(x)]
[~\varphi(x)~,~\varphi(y)~]_{x^{+}=y^{+}}~=~\frac{\mu^{2}}{N}~\frac{i}{2}\left\{-\frac{1}{2}~\epsilon(x^{-}-y^{-})+\frac{(x^{-}-y^{-})}{2L}\right\}~,
\mathrm{Res}\left[I_{D}(k,q),q_{0}=|\vec{q}+\vec{k}|-k_{0}\right]=-\frac{1}{2|\vec{q}+\vec{k}|}\left.\left(4\frac{k\cdotq}{k^{2}}+3\right)\right|_{q_{0}=|\vec{q}+\vec{k}|-k_{0}}.
{\calV}^{H}(\mathrm{\bfr}_{1},\mathrm{\bfr}_{2})={\frac{i\alpha\hat{\mathrm{\bfk}}\times\mathrm{\bfr}_{12}}{mr_{12}^{2}}}\cdot\boldsymbol{\partial}_{1}+{\frac{\pi}{m}}|\alpha|\delta(\mathrm{\bfr}_{12})+{\frac{\alpha\epsilon\omega_{c}}{2}}.
\Omega_{\mu\nu,\alpha\beta\gamma}=\frac{1}{3}T_{\alpha\beta\gamma,\mu\nu}+\frac{1}{6}[T_{\mu\beta\gamma,\alpha\nu}-T_{\nu\beta\gamma,\alpha\mu}+(\alpha,\beta,\gamma)]+\frac{1}{3}[T_{\mu\nu\alpha,\beta\gamma}+(\alpha,\beta,\gamma)]
{\bfF}\left(\phi\left(x^{0}+\tau,{\bfx}\right)\right)_{{\bfx},{\bfx^{\prime}}}=\left\{{\frac{1}{2}}g\phi^{2}\left(x^{0},\mathrm{x}\right)+{\frac{1}{\tau^{2}}}\right\}^{-1}\left({\frac{2}{\tau^{2}}}\delta_{{\bfx},{\bfx^{\prime}}}-{\bf\Omega}_{{\bfx},{\bfx^{\prime}}}\right)-{\frac{2}{\tau^{2}}}\delta_{{\bfx},{\bfx^{\prime}}},
\begin{array}{l}{D_{i\alpha}\tilde{\hat{\omega}}_{\beta}{}^{\gamma}(z)=4\delta_{\alpha}^{~\gamma}\bar{\hat{\rho}}_{i\beta}(z)-\delta_{\beta}^{~\gamma}\bar{\hat{\rho}}_{i\alpha}(z)}\\{{}}\\{D_{i\alpha}\hat{\lambda}(z)=-2\bar{\hat{\rho}}_{i\alpha}(z)}\\{{}}\\{D_{i\alpha}\hat{T}_{j}^{~k}(z)=-4\delta_{i}^{~k}\bar{\hat{\rho}}_{j\alpha}(z)-4\bar{\calE}_{ij}\hat{\rho}{}_{\alpha}^{k}(z)}\\\end{array}
\begin{array}{c}{\displaystyle\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\int\frac{d^{4}k_{2}}{(2\pi)^{4}}D_{m}^{2}(k_{1})[D_{m}(k_{2})-D(k_{2})]\overline{{D(k_{1}+p)}}K_{12}.}\\\end{array}
{\calR}_{\alpha\beta}{\eta}^{\alpha}{\eta}^{\beta}=\Gamma_{n+1\alpha,n+1}^{\alpha}-\Gamma_{n+1\,n+1,\alpha}^{\alpha}+\Gamma_{n+1\,\alpha}^{\beta}\Gamma_{\beta\,n+1}^{\alpha}-\Gamma_{n+1\,n+1}^{\alpha}\Gamma_{\alpha\beta}^{\beta}=\kappa^{2}-\dot{h}
S=-2\piRT\intdX^{0}\left[1-{\frac{1}{2}}\sum_{k=-\infty}^{\infty}\left({\frac{dX_{-k}^{I}}{dX^{0}}}{\frac{dX_{k}^{I}}{dX^{0}}}-{\frac{k^{2}}{R^{2}}}X_{-k}^{I}X_{k}^{I}\right)\right]
Z^{t}=(z_{i}^{A})^{t}=\left(\begin{array}{ccccc}{z_{1}^{0}}&{z_{1}^{1}}&{\cdots}&{}&{z_{1}^{d-1}}\\{\vdots}&{\vdots}&{}&{}&{\vdots}\\{z_{r}^{0}}&{z_{r}^{1}}&{\cdots}&{}&{z_{r}^{d-1}}\\\end{array}\right)\quad,\qquadrank\Z=r
U=\frac{r}{l_{\mathrm{s}}^{2}}\,\quadg_{\mathrm{YM}}^{2}=\frac{g_{\mathrm{s}}}{l_{\mathrm{s}}^{3}}\,\quadH-p_{10}=H-\frac{N}{g_{\mathrm{s}}l_{\mathrm{s}}}\.
+\frac{e^{2}}{4\pim^{2}c^{2}}\frac{1}{\varrho^{2}}\vec{\varrho}\cdot(\vec{\varrho}\cdot\vec{p})\vec{p}-\frac{e^{2}\hbar^{2}}{2m^{2}c^{2}}\delta^{2}(\vec{\varrho})~~,
\Omega_{1}^{(2)}=\frac{d}{dt}(\partial^{1}L_{2}^{(0)}+\partial^{2}L_{3}^{(0)}-iz_{1}L_{4}^{(0)}-iz_{2}L_{5}^{(0)}+iz_{1}^{*}L_{6}^{(0)}+iz_{2}^{*}L_{7}^{(0)})+\frac{d^{2}}{dt^{2}}L_{1}^{(0)}\equiv0.
{\calA}\equiv\intdx^{3}\frac{1}{2e^{2}r^{4}}\,[\,\lambdau^{4}-gu^{2}+1\,]\quad\Rightarrow\quad\delta{\calA}=\intdx^{3}\frac{1}{2e^{2}r^{4}}\,[\,4\lambdau^{3}-2gu\,]\,\deltau+\cdots.
S_{\mathrm{min}}=\int_{0}^{s_{1}}d\tau_{1}\int_{0}^{s_{2}}d\tau_{2}\delta(z_{10}-z_{20})\vert{\bfz}_{1}-{\bfz}_{2}\vert\int_{0}^{1}d\lambda\Big\{{\dot{z}_{10}^{2}}{\dot{z}_{20}^{2}}-(\lambda\dot{\vec{z}}_{1\mathrm{T}}\dot{z}_{20}+(1-\lambda)\dot{\vec{z}}_{2\mathrm{T}}\dot{z}_{10})^{2}\Big\}^{\frac{1}{2}}\,
\times[D_{\nu_{n}}^{F},\ldots,[D_{\nu_{2}}^{F},[D_{\nu_{1}}^{F},Q_{\pm}]]\ldots]\left(D^{B\nu_{1}}D^{B\nu_{2}}\ldotsD^{B\nu_{n}}\right)_{(0,1)00,(0,n+1)lm},
\begin{array}{|c||c|c|}\hline{\strut\mathrm{Theory}}&{\quadO9\quad}&{\quadO5\quad}\\\hline{\strutSo\mathrm{~tensor~/~BZDP}}&{\quad-\quad}&{\quad+\quad}\\{So\mathrm{~hyper~/~GP}}&{\quad-\quad}&{\quad-\quad}\\{Sp\mathrm{~tensor~}}&{\quad+\quad}&{\quad-\quad}\\{Sp\mathrm{~hyper~}}&{\quad+\quad}&{\quad+\quad}\\\hline{}\\\end{array}
V(e^{t}\sigma;\lambda,e^{t}\Lambda;e^{t}M)=V(e^{t}\sigmaZ_{S}^{1/2}(t);\lambdaZ_{Q}^{-1}(t)Z_{S}^{-1/2}(t),e^{t}\LambdaZ_{Q}(t)^{1/2};M),
\begin{array}{ccccc}{\mathrm{D-string}}&{\mathrm{D3-brane}}&{\mathrm{D5-brane}}&{\mathrm{F-string}}&{\mathrm{NS5-brane}}\\{\Gamma^{01}\eta^{\pm}=\pmi\eta^{\mp},}&{\Gamma^{0123}\eta^{\pm}=\mpi\eta^{\pm},}&{\Gamma^{012345}\eta^{\pm}=\pmi\eta^{\mp},}&{\Gamma^{01}\eta^{\pm}=\eta^{\mp},}&{\Gamma^{012345}\eta^{\pm}=\eta^{\mp}.}\\\end{array}
\sigma_{\vec{\alpha}}(z)=\vartheta\left[\begin{array}{c}{\alpha_{2}/n+1/2}\\{\alpha_{1}/n+1/2}\\\end{array}\right](z,\tau),~~~~~~\theta^{(j)}(z)=\vartheta\left[\begin{array}{c}{1/2-j/n}\\{1/2}\\\end{array}\right](z,n\tau),
Y_{u}(z,\bar{w})=\prod_{k=1}^{N}\phi_{u}(z_{k};\bar{w}_{k},\bar{w}_{k+1})=\prod_{k=1}^{N}(z_{k}-\bar{w}_{k})^{-s-iu}\,(z_{k}-\bar{w}_{k+1})^{-s+iu}\,.
z_{j}\longrightarrowz_{j}^{^{\prime}}=\frac{az_{j}+b}{cz_{j}+d}\;\;,\;\;\overline{{z}}_{j}\longrightarrow\overline{{z^{^{\prime}}}}_{j}=\frac{\overline{{a}}\overline{{z}}_{j}+\overline{{b}}}{\overline{{c}}\overline{{z}}_{j}+\overline{{d}}}
C_{+\ldots+}^{k}(x_{1},\ldots,x_{k})=M_{(k)}{\frac{m^{k}}{\mu^{k^{2}}}}\Bigl({\frac{2}{e^{\gamma}}}\Bigr)^{k(k-1)}e^{-2{\sum\atop{i>j}}K_{0}^{x_{i}x_{j}}}=C_{-\ldots-}^{-k}(x_{1},\ldots,x_{n}).
ig\left[\delta^{ij}\delta^{kl}T_{JI}^{a}T_{LK}^{a}\Gamma_{(n)\,\alpha}^{~~~~~\beta}\Gamma_{~~~\gamma}^{(n)~\delta}~-~\delta^{il}\delta^{kj}T_{LI}^{a}T_{JK}^{a}\Gamma_{(n)\,\alpha}^{~~~~~\delta}\Gamma_{~~~\gamma}^{(n)~\beta}\right]~.
\renewcommand{\arraystretch}{1.5}\begin{array}{c}{ds_{1}^{2}=e^{-\gamma}ds_{0}^{2}}\\{ds_{2}^{2}=e^{+\gamma}ds_{0}^{2}}\\\end{array}~~~~~\mathrm{where}~~ds_{0}^{2}=-dt^{2}+e^{\sigma/3}\delta_{\mu\nu}dx^{\mu}dx^{\nu}
\begin{array}{rclrclrclrcl}{A}&{=}&{T_{a}A^{a},\quad\hat{\psi}}&{=}&{T_{a}\hat{\psi}^{a},\quad\psi}&{=}&{\Sigma_{\alpha}\psi^{\alpha},\quad\hat{A}}&{=}&{\Sigma_{\alpha}\hat{A}^{\alpha},}&{}&{}&{}\\{\hat{a}}&{=}&{T_{a}\hat{a}^{a},\quad\alpha}&{=}&{T_{a}\alpha^{a},\quad\hat{\alpha}}&{=}&{\Sigma_{\alpha}\hat{\alpha}^{\alpha},\quada}&{=}&{\Sigma_{\alpha}a^{\alpha}.}&{}&{}&{}\\\end{array}
({\frac{\delta\Gamma}{\deltaA}},{\frac{\delta\Gamma}{\deltaK}})+({\frac{\delta\Gamma}{\deltac}},{\frac{\delta\Gamma}{\deltaL}})+({\frac{\delta\Gamma}{\delta\bar{c}}},b)+{\frac{\delta\Gamma}{\deltaV}}.{\frac{\delta\Gamma}{\deltaM}}=0\.
\int_{-1/K}^{1/K}dy\,u_{0}^{2}(y)w(y)\sim\left.(1-K|y|)^{\frac{4(D-2p-3)+(D-2)^{2}a^{2}}{(D-2)^{2}a^{2}}}\right|_{-1/K}^{1/K}
\delta_{\mathrm{T}}=\left(\frac{13}6-\frac\lambda2\right)\frac{(g\mu^{-\epsilon})^{2}}{(4\pi)^{2}}\frac{C_{2}(G)}{\epsilon},\quad\delta_{\mathrm{L}}=-\lambda\xi(1-\xi)\frac{(g\mu^{-\epsilon})^{2}}{(4\pi)^{2}}\frac{C_{2}(G)}{\epsilon}.