harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\intdx\,\sqrt{g}\,V[\sigma]=\intdx\,\sqrt{g}\,\frac{N}{2\lambda_{B}}\sigma^{2}+\frac{N}{2}\mathrm{Tr~}\,\int_{1/\Lambda^{2}}^{\infty}\frac{dt}{t}\,e^{-t\sigma^{2}}e^{\not\nabla^{2}t}.
\frac{\pi^{3/2}}{2G\Lambda}=1+\frac{\sqrt\pi(eE)^{1/2}}{2\Lambda}\Bigg(\mathrm{Re}\Bigg[\zeta\big(\frac12,1-i\frac{\mu_{E}^{2}}{2}\big)\Bigg]-\mathrm{Im}\Bigg[\zeta\big(\frac12,1-i\frac{\mu_{E}^{2}}{2}\big)\Bigg]\Bigg)+O\Big(\frac{m^{2}}{\Lambda^{2}}\Big).
\deltaR_{i\chi}=\delta\lgroup{\frac{g_{s}}{4}}F_{iPQ}F_{\chi}^{\;PQ}-{\frac{g_{s}}{4}}CH_{iPQ}F_{\chi}^{\;PQ}+{\frac{g_{s}^{2}}{96}}{\tilde{F}_{iPQRS}}{\tilde{F}_{\chi}^{\;PQRS}}\rgroup,
\mathrm{Im}\,\Pi^{\mathrm{(non\mathrm{-}planar)}}(p)=\frac{\theta(p^{2})\pi^{\frac{n}{2}+1}}{(2\pi)^{n}}\int_{0}^{1}dx\,\frac{1}{|M|^{4-n}}\,\left(\frac{|\bar{\theta}||M|}{2}\right)^{2-\frac{n}{2}}\,J_{\frac{n}{2}-2}(|\bar{\theta}||M|)
S_{eff}={\frac{1}{4\pi}}\mathrm{Im}\,\left(\intd^{4}xd^{2}\thetad^{2}\bar{\theta}\Phi_{D}^{i}\overline{\Phi}_{i}+{\frac{1}{2}}\intd^{4}xd^{2}\theta\tau^{ij}W_{i}W_{j}\right),
{\frac{c_{A}}{24}}\sum_{r}\left[H_{1},A_{i}^{c}(r)\right]\left[H_{1},A_{i}^{c}(-r)\right]\sum_{k}{\frac{1}{\omega_{k}^{3}}}=-{\frac{c_{A}}{24}}{\frac{1}{2\pi^{2}}}log\left({\frac{\Lambda}{\mu}}\right)\intd^{3}x\,E^{2}
\lambda^{2}\frac{1}{2^{15}3^{3}\pi^{6}}\left[\sum_{N=1}^{\infty}[C_{N}^{\frac{3}{2}}(cos{\theta_{4}^{0}})]^{2}\frac{(N+1)(N+2)}{N(N+3)}\right](RK^{2})\frac{1}{\epsilon}\int_{C}d^{4}{\eta}+F.T.
\overline{{\chi}}^{i}\gamma_{\mu_{1}}\gamma_{\mu_{2}}\ldots\gamma_{\mu_{n-1}}\gamma_{\mu_{n}}\psi^{j}=\overline{{\psi}}^{j}\gamma_{\mu_{n}}\gamma_{\mu_{n-1}}\ldots\gamma_{\mu_{2}}\gamma_{\mu_{1}}\chi^{i}
\dot{m}_{+}=m_{+}((-\dot{(h\bar{\Phi})}+\sumY_{j}^{+1}\mu_{j}^{(1}+\sumY_{j}^{+2}\mu_{j}^{(2}+\sumY_{j}^{+3}\mu_{j}^{(3}-\sumY_{j}^{+4})
S\,=\,S_{0}+\phi_{A}^{\ast\,a}\,\delta_{a}\phi^{A}+{\frac{1}{2}}{\bar{\phi}}_{A}\delta_{2}\delta_{1}\phi^{A}\,+\,{\frac{1}{2}}\epsilon^{ab}\phi_{A\,a}^{\ast}\,\pi_{b}^{A}
S_{\mathrm{part},0}^{(\mathrm{N})}=\intdt\,\sum_{\alpha=1}^{N}\left(\xi_{\alpha}^{\underline{{a}}}\left(E_{j,\alpha}^{\underline{{a}}}\dot{x}_{\alpha}^{j}+E_{0,\alpha}^{\underline{{a}}}\right)-{\frac{1}{2}}\xi_{\alpha}^{\underline{{a}}}\xi_{\alpha}^{\underline{{a}}}\right)\,.
F^{-1}=\left(\begin{array}{c|c|c}{f^{-1}-\sum_{i=1}^{2}f^{-1}A_{i}C_{i}^{-1}\tilde{A}_{i}f^{-1}}&{-f^{-1}A_{1}C_{1}^{-1}}&{-f^{-1}A_{2}C_{2}^{-1}}\\\hline{C_{1}^{-1}\tilde{A}_{1}f^{-1}}&{C_{1}^{-1}}&{0}\\\hline{C_{2}^{-1}\tilde{A}_{2}f^{-1}}&{0}&{C_{2}^{-1}}\\\end{array}\right).
F^{AB}\mathrm{\tiny~\wedge~}F_{AB}=R^{ab}\mathrm{\tiny~\wedge~}R_{ab}+\frac{2}{l^{2}}[T^{a}\mathrm{\tiny~\wedge~}T_{a}-R^{ab}\mathrm{\tiny~\wedge~}e_{a}\mathrm{\tiny~\wedge~}e_{b}].
\kappa_{\nu}{}^{ab}=\frac{i}2\Big((\sigma_{\nu})^{\alpha\dot{\beta}}J^{ab}{}_{\alpha\dot{\beta}}+(\sigma^{a})^{\alpha\dot{\beta}}J_{\nu}{}^{b}{}_{\alpha\dot{\beta}}-(\sigma^{b})^{\alpha\dot{\beta}}J_{\nu}{}^{a}{}_{\alpha\dot{\beta}}\Big).
\mathrm{Res}_{z\doteqz_{1}=z_{0}}F=\mathrm{Res}_{\bar{z}=\bar{z}_{0}}\sum_{k=1}^{\infty}\frac{(-)^{k}}{k!}\partial_{z}^{k-1}\mathrm{Res}_{z_{1}=z}(z_{1}-z)^{k}F(z,z_{1},z_{0}).
\Psi_{\left(+\right)}\left(x\right)=\left(\begin{array}{c}{\upsilon\left(x\right)}\\{\left({\bf\sigman}\right)\upsilon\left(x\right)}\\\end{array}\right)\,,\;\Psi_{\left(-\right)}\left(x\right)=\left(\begin{array}{c}{u\left(x\right)}\\{-\left({\bf\sigman}\right)u\left(x\right)}\\\end{array}\right)\,,
S\sim\frac{\pi^{2}\ell^{2}}{G\beta}-\frac{3\pi^{2}\ell^{2}}{\beta^{2}}\left(\frac{\partialE_{r}}{\partialr}\right)B+\frac{r^{2}}{\ell^{2}}\left(\frac{\partialE_{r}}{\partialr}\right)B\.
22u\rightarrowu\;,\;\;\;\;\;\;\;v\rightarrowv+\theta(-u)(\tilde{A}y+\textstyle{\frac{1}{4}}\tilde{A}^{2}u+B)\;,\;\;\;\;\;\;\;y\rightarrowy+\textstyle{\frac{1}{2}}\tilde{A}u\theta(-u)\,,
ds^{2}=d\tau^{2}-d\zeta^{2}+(A_{2}^{2}-B_{2}^{2})d\theta^{2}+2\frac{(\alpha_{1}\alpha_{2}-\beta_{1}\beta_{2})}{\sqrt{\alpha_{1}^{2}-\beta_{1}^{2}}}(\zetad\tau-\taud\zeta)d\theta.
\omega=\frac{1}{\gammag_{\phi\phi}}\left(g_{\phi\phi}(\dot{f}+A_{a}\dot{x}^{a})-g_{a\phi}\dot{x}^{a}(n+A_{\phi})\right)^{2}+\frac{(n+A_{\phi})^{2}}{g_{\phi\phi}}.
\intdx\psi^{\dagger}(j,x)\psi(j^{\prime},x)=\delta_{j,j^{\prime}}={\delta_{s,s^{\prime}}\delta_{l,l^{\prime}}}{\delta(p_{3}-p_{3^{\prime}})}{\frac{\delta(p_{\perp}-p_{\perp}^{\prime})}{\sqrt{p_{\perp}p_{\perp}^{\prime}}}}\,,
\left(\begin{matrix}{\psi_{1}^{\prime}}\\{\psi_{2}^{\prime}}\\\end{matrix}\right)=\left(\begin{matrix}{\omega_{1}}&{-\omega_{2}^{*}}\\{\omega_{2}}&{\omega_{1}^{*}}\\\end{matrix}\right)\left(\begin{matrix}{\psi_{1}}\\{\psi_{2}}\\\end{matrix}\right)=\left(\begin{matrix}{\omega_{1}\psi_{1}-\omega_{2}^{*}\psi_{2}}\\{\omega_{2}\psi_{1}+\omega_{1}\psi_{2}}\\\end{matrix}\right)
(\tilde{Z}^{\#}(\theta_{1}^{\prime})...\tilde{Z}^{\#}(\theta_{n}^{\prime})\Omega_{2\pi},\tilde{Z}^{\#}(\theta_{1})...\tilde{Z}^{\#}(\theta_{n})\Omega_{2\pi})=\prod_{i}\delta(\theta_{i}^{\prime}-\theta_{i})F_{n}^{ref}(\theta_{1},...,\theta_{n})
\xi^{t}(\tau,\sigma)=\sum_{n}[(\frac{E}{m}\tilde{A}_{n}^{\parallel}-\frac{\sqrt{E^{2}-m^{2}}}{m}\tilde{C}_{n}^{\parallel})e^{-in(\tau+\sigma)}+(\frac{E}{m}\tilde{B}_{n}^{\parallel}-\frac{\sqrt{E^{2}-m^{2}}}{m}\tilde{D}_{n}^{\parallel})e^{-in(\sigma-\tau)}]+{\calO}(1/r),
\hat{\calQ}_{\alpha}^{2}=-\frac{i}{m}(p\sigma)_{\alpha\beta}\hat{\pi}^{\beta}-m\hat{\theta}_{\alpha}\qquad\hat{{\calK}}=1-2\bar{\theta}\frac{\stackrel{\rightarrow}{\partial}}{\partial\bar{\theta}}\qquad\hat{{\calZ}}=m\;.
I_{s}^{(4)}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4})=I^{(4)}(e^{-s}{\bf{q}}_{1},e^{-s}{\bf{q}}_{2},e^{-s}{\bf{q}}_{3},e^{-s}{\bf{q}}_{4})\;.
\qquad\qquad\qquadh_{0}^{(0)}=-k^{2}-m^{2}\,,\qquad\qquadh_{r}^{(0)}=\mu_{r}^{2}f_{r}\big({\frac{k}{\bar{\mu}}}\big)\,,\qquad\qquad\qquad(r=1,2,3)
\left(\begin{array}{cc}{A}&{(m_{i}+\partial_{i})I}\\{(m_{i}-\partial_{i})I}&{-A}\\\end{array}\right)^{2}=(A^{2}+(m_{i}+\partial_{i})(m_{i}-\partial_{i})I)\left(\begin{array}{cc}{I}&{0}\\{0}&{I}\\\end{array}\right)
\tilde{C}_{abc}=\frac{\xi({\frac{b+c-a-1}2})}{\xi(a-1)}\frac{\xi({\frac{a+c-b-1}2})}{\xi(b-1)}\frac{\xi({\frac{a+b-c-1}2})}{\xi(c-1)}\frac{\eta({\frac{a+b+c-1}2})}{\eta(1)}\,
u_{1}={\frac{1}{v_{1}}},\quadu_{2}={\frac{v_{2}}{v_{1}}},\quadv_{1}={\frac{w_{1}}{w_{2}}},\quadv_{2}={\frac{1}{w_{2}}},\quadw_{1}={\frac{1}{u_{2}}},\quadw_{2}={\frac{u_{1}}{u_{2}}},
\begin{array}{c}{A_{q}L_{1}^{n}L_{2}^{n}=L_{2}^{n}L_{1}^{n}D_{q}}\\{L_{1}^{n}B_{q}^{*^{-1}}L_{2}^{n+1}=L_{2}^{n+1}L_{1}^{n}}\\\end{array},
\stackrel{(-1,0)}{\Phi}_{\alpha_{L-2}}^{*(1)}Z_{\;\;\alpha_{L-1}}^{\alpha_{L-2}}=\delta_{1}\left(\stackrel{(-2,0)}{\eta}_{\alpha_{L-1}}^{*(11)}-\stackrel{(-2,0)}{\gamma}_{\alpha_{L}}A_{\alpha_{L-1}}^{\;\;\alpha_{L}}\right).
(\delta_{\beta^{1}}\delta_{\alpha^{1}}-\delta_{\alpha^{1}}\delta_{\beta^{1}})\hat{\psi}=-(\hat{\alpha}\hat{\beta}-\hat{\beta}\hat{\alpha})\hat{\psi}+i((\delta_{\beta^{1}}\hat{\alpha})-(\delta_{\alpha^{1}}\hat{\beta}))\hat{\psi}.
h\propto(C\Gamma_{\mu_{1}\dots\mu_{5}})_{\alpha\beta}\Pi^{\mu_{1}}\dots\Pi^{\mu_{5}}\Pi^{\alpha}\Pi^{\beta}-\frac{15}{2}(C\Gamma_{\mu_{1}\mu_{2}})_{\alpha\beta}\Pi^{\mu_{1}}\Pi^{\mu_{2}}\Pi^{\alpha}\Pi^{\beta}H
\begin{array}{lll}{\{\theta^{0},\theta^{0}\}=-2\theta^{1}\theta^{2},}&{\{\theta^{0},\theta^{1}\}=-2\theta^{0}\theta^{1},}&{\{\theta^{0},\theta^{2}\}=2\theta^{0}\theta^{2},}\\{\{\theta^{1},\theta^{1}\}=0,}&{\{\theta^{1},\theta^{2}\}=0,}&{\{\theta^{2},\theta^{2}\}=0.}\\\end{array}
\Lambda=\left(\begin{array}{ccc}{n}&{{\frac{1}{2}}q^{\prime\prime}+Tq+{\frac{1}{2}}(\zeta\delta+\eta\beta)}&{q\zeta+\beta\Phi+\beta^{\prime}}\\{q}&{n+q^{\prime}}&{\beta}\\{\delta}&{q\eta+\delta\Phi-\delta^{\prime}}&{2n+q^{\prime}}\\\end{array}\right)
\{{\tilde{\phi}_{i+1}},{\tilde{\phi}_{j}^{*}}\}=\left\{\phi_{i+1}-\sum_{k=1}^{i}\frac{\{\phi_{i+1},{\tilde{\phi}_{k}^{*}}\}}{\{{\tilde{\phi}_{k}},{\tilde{\phi}_{k}^{*}}\}}{\tilde{\phi}_{k}}\,\{\tilde{\phi}_{j}^{*}}\right\}.
\begin{array}{lll}{A\alpha_{1}=\alpha_{5},\qquad}&{A\alpha_{2}=\alpha_{4},\qquad}&{A\alpha_{3}=\alpha_{3},}\\{A\alpha_{4}=\alpha_{6},}&{A\alpha_{5}=\alpha_{0},}&{A\alpha_{6}=\alpha_{2},}\\{A\alpha_{0}=\alpha_{1}.}&{}&{}\\\end{array}
T\left(\begin{array}{lcr}{v}\\{u}\\\end{array}\right)=\left(\begin{array}{lcr}{0}&{{\frac{\partial}{\partial\rho}}}\\{-{\frac{\partial}{\partial\rho}}}&{0}\\\end{array}\right)\left(\begin{array}{lcr}{v}\\{u}\\\end{array}\right)=\left(\begin{array}{lcr}{v}\\{u}\\\end{array}\right).
\Omega^{-1}=(\Omega^{AB})=\left(\begin{array}{cc}{\widetilde{\omega}^{ij}}&{-\omega^{il}\partial_{l}\theta_{\gamma}\omega^{\gamma\beta}}\\{\omega^{jl}\partial_{l}\theta_{\gamma}\omega^{\gamma\alpha}}&{\omega^{\alpha\beta}}\\\end{array}\right)
(\sqrt{-g})^{-1}\mathcal{L}=-\frac{M_{\mathrm{P}}^{2}}{2}R-g^{mn}\partial_{m}\phi\partial_{n}\phi^{*}-\mathrm{e}^{\frac{|\phi|^{2}}{M_{P}^{2}}}\left[\left|\mathrm{e}^{\frac{-{|\phi|^{2}}}{M_{\mathrm{P}}^{2}}}{\frac{\partial}{\partial\phi}}\left(\mathrm{e}^{\frac{|\phi|^{2}}{M_{\mathrm{P}}^{2}}}P\right)\right|^{2}-{\frac{3}{M_{\mathrm{P}}^{2}}}|P|^{2}\right],
\frac{d\phi}{dr}=\frac{L}{r^{2}}\,\frac{1+2m\beta(E-V)+\frac{(2\beta-\beta^{\prime})+2m\beta(2\beta+\beta^{\prime})(E-V)}{1+2m(\beta+\beta^{\prime})(E-V)}\left(mr\frac{\partialV}{\partialr}\right)}{\sqrt{2m(E-V)-\frac{L^{2}\left[1+2m\beta(E-V)\right]^{2}}{r^{2}}}}\;.
\delta\mu^{\prime\prime\rho}=\partial_{\mu}\left(\varepsilon^{\mu\alpha\beta\gamma}\left(B_{\gamma}^{\;(\sigma)}\partial^{\rho}A_{\alpha\beta(\sigma)}^{*}-A_{\alpha\beta}^{*\;\;(\sigma)}\partial^{\rho}B_{\gamma(\sigma)}\right)\right).
M_{\mu\nu}^{(p)}=\eta_{\mu\nu}+F_{\mu\nu}+\partial_{\mu}\phi^{i}\partial_{\nu}\phi^{i}-2\bar{\lambda}(\Gamma_{\mu}+\Gamma_{i}\partial_{\mu}\phi^{i})\partial_{\nu}\lambda+\bar{\lambda}\Gamma^{m}\partial_{\mu}\lambda\bar{\lambda}\Gamma_{m}\partial_{\nu}\lambda,
S\left(\begin{array}{cc}{a}&{b}\\{d}&{c}\\\end{array}\right.\left|\begin{array}{cc}{z_{1}}&{z_{2}}\\{z_{2}^{\prime}}&{z_{1}^{\prime}}\\\end{array}\right)=\frac{\overline{{W}}_{ab}(z_{1},z_{2})\overline{{W}}_{bc}(z_{2},z_{1}^{\prime})\overline{{W}}_{cd}(z_{1}^{\prime},z_{2}^{\prime})}{\overline{{W}}_{ad}(z_{1},z_{2}^{\prime})}.
W=\frac{1}{m^{2}}\int{d^{4}x(-g)^{\frac{1}{2}}(aRF^{ij}F_{ij}+bR_{ik}F^{il}{F^{k}}_{l}+cR_{ijkl}F^{ij}F^{kl}+dD_{i}F^{ij}D_{k}{F^{k}}_{j})}
\Omega_{p}=(-1)^{p}\intd^{2p+1}x\,deltaD^{i_{1}...i_{p}}\wedge\deltaA_{i_{1}...i_{p}}=(-1)^{p+1}\intd^{2p+1}x\,deltaG^{\mu_{1}...\mu_{p}0}\wedge\deltaA_{\mu_{1}...\mu_{p}}\.
{\calG}_{\mathrm{ang}}(T=0)=\left\{{\frac{b{\bfJ^{2}}}{r^{2}(\kappa+\mub)}}+\left[{\frac{\mu}{\kappa^{2}}}+{\frac{b}{\kappa(b^{2}-r^{2})}}\right]Q^{2}\right\}dt^{2}\,\,.
f^{\prime\prime}+\left[\frac{d-1}{r}+\left(1+\frac{p}{2}\right)\frac{B^{\prime}}{B}+(1+p)A^{\prime}\right]f^{\prime}-\frac{1}{B}\left[\frac{(d-1)}{r^{2}}+(f^{2}-1)\right]f=0.
v=\left(\begin{matrix}{1}\\{p}\\\end{matrix}\right),\quadw=\left(\begin{matrix}{1}\\{q}\\\end{matrix}\right),\quadv^{\prime}=\left(\begin{matrix}{1}\\{p^{\prime}}\\\end{matrix}\right),\quadw^{\prime}=\left(\begin{matrix}{1}\\{q^{\prime}}\\\end{matrix}\right),
{\frac{i}{2}}\Big(F_{\mu\nu}^{-\,I}-i\bar{L}^{I}T_{\mu\nu}^{-})\gamma^{\mu}\gamma^{\nu}\epsilon={\frac{2}{a}}\Big(iF_{01}^{-\,I}\gamma_{1}+ibF_{01}^{-\,I}+{\bar{L}}^{I}T_{01}^{-}\gamma_{1}+b{\bar{L}}
ds_{(b,\bar{b})}^{2}=l^{2}\left[-d\tau^{2}+\frac{b(z)}{c/6}dz^{2}+\frac{\bar{b}(\bar{z})}{c/6}d\bar{z}^{2}+\left(e^{-2\tau}+\frac{b(z)\bar{b}(\bar{z})}{(c/6)^{2}}e^{2\tau}\right)dzd\bar{z}\right].
a_{k}=e_{k}+\sum_{m,n\geq0;m+n\neq0}\frac{(-1)^{n}(\Lambda^{2N_{c}-N_{f}})^{m+n}}{(m!)^{2}n!2^{2m}}\frac{\partial^{2m+n-1}}{\partiale_{k}}(S_{k}(e)^{m}R_{k}(e)^{n}),
{\calL}=\left(D_{\mu}\phi\right)^{\mathrm{~\ast~}}\!\left(D^{\mu}\phi\right)-\frac{\lambda}{4}\left(\phi^{\mathrm{~\ast~}}\!\phi-\frac{2m^{2}}{\lambda}\right)^{2}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu},
F=V_{p}(8\pi^{2}\alpha^{\prime})^{-p/2}\int_{0}^{\infty}d\tau~\tau^{-4+p/2}e^{-b^{2}/2\pi\alpha^{\prime}{\tau}}\left[e^{-\pi\tau/12}\prod_{n=1}^{\infty}(1-e^{-2\pi\taun})\right]^{-9}\frac{(\theta_{1}(-i\epsilon/\pi\midi\tau))^{4}}{\theta_{1}(-i2\epsilon/\pi\midi\tau)}.
G_{(3)}^{\mathrm{D}}(\tau)=360\tau^{5}\int_{0}^{\Omega/2}d\Omega^{\prime}\int_{0}^{\Omega^{\prime}}dK\,K\,\sqrt{(\Omega^{2}-K^{2})\left[(\Omega-\Omega^{\prime})^{2}-K^{2}\right]}\left(\frac{1}{e^{\Omega^{\prime}}-1}+\frac{1}{e^{\Omega-\Omega^{\prime}}-1}\right).
\begin{array}{l}{\partial^{i}\left(\Phi^{\frac{d-3}{2}}\partial^{\mu_{1}}J_{i}\partial^{\mu_{2}}J_{i_{2}}\cdots\partial^{\mu_{s}}J_{i_{s}}-\mathrm{traces}_{(i)}\right)=}\\{=-4{\frac{2s^{2}+(3d-11)s+(d^{2}-7d+12)}{2s+d-5}}\left(\Phi^{\frac{d-3}{2}}\partial^{\mu_{1}}\Phi\partial^{\mu_{2}}J_{i_{2}}\cdots\partial^{\mu_{s}}J_{i_{s}}-\mathrm{traces}_{(i)}\right)}\\\end{array}
G_{R}(\alpha)=\frac{\rho(\alpha)}{\rho(Q-\alpha)}=\left(\frac{\pi\mu}{2}\gamma\left(\frac{1+b^{2}}{2}\right)b^{1-b^{2}}\right)^{\frac{Q-2\alpha}{b}}\frac{\gamma\left(\frac{1}{2}-\frac{Qb}{2}+\alphab\right)}{\gamma\left(\frac{1}{2}+\frac{Q}{2b}-\frac{\alpha}{b}\right)}.
\frac{1}{1\mp\frac{\nu}{j+\frac{1}{2}}}=\frac{j+\frac{1}{2}}{j+\frac{1}{2}\mp\\nu}=1\pm\frac{\nu}{j+\frac{1}{2}\mp\nu}=1\pm\frac{\nu}{j+\frac{1}{2}}+\frac{\nu^{2}}{(j+\frac{1}{2})(j+\frac{1}{2}\mp\nu)}
S=\frac{k}{16\pi}\intdzd\bar{z}d\thetad\bar{\theta}\left[-Dg^{s}\bar{D}g^{s\dagger}+\intdtg^{s}\frac{\partialg^{s\dagger}}{\partialt}(Dg^{s}\bar{D}g^{s\dagger}+\bar{D}g^{s}Dg^{s\dagger}\right]
\overline{{L}}_{\alpha}^{a}=(L^{-1})_{a}^{\alpha}\,,\quad\overline{{L}}_{\mu}^{i}=(L^{-1})_{i}^{\mu}\,,\quad\overline{{L}}_{\alpha}^{i}=-\overline{{L}}_{\alpha}^{a}L_{a}^{\mu}\overline{{L}}_{\mu}^{i}
P_{\mu}^{\rho}(\beta)\delta_{\rho\nu}=P_{\mu}^{\mu^{\prime}}(\alpha)P_{\nu}^{\nu^{\prime}}(\alpha)\delta_{\nu^{\prime}\nu^{\prime\prime}}P_{\mu^{\prime}}^{\nu^{\prime\prime}}\left(P_{\rho}^{\sigma}(\alpha)\beta_{\sigma}\right)
B^{-}=\int{\frac{2k^{+}dk^{+}dk^{-}}{(2\pi)^{2}}}{\frac{1}{[p^{+}-k^{+}]_{ML}^{2}}}{\frac{1}{2k^{+}k^{-}m^{2}+i\epsilon}}={\frac{ip^{-}}{\pi(p^{2}-m^{2}+i\epsilon)}}\.
K=c_{M}\hat{M}^{\dagger}Z^{q_{\hat{M}}}\hat{M}+c_{B}\hat{B}^{\dagger}Z^{q_{\hat{B}}}\hat{B}+c_{\bar{B}}\hat{\bar{B}}^{\dagger}Z^{q_{\hat{B}}}\hat{\bar{B}}+\cdots.
f_{\mu}^{a}(\psi_{,\mu}^{b})-f_{\mu}^{a}(\psi^{a},\lambda_{a}^{\mu})-\lambda_{b}^{\nu}\frac{\partialf_{\nu}^{b}}{\partial\lambda_{a}^{\mu}}+\frac{\partialL}{\partial\lambda_{a}^{\mu}}=0.
\zeta(s)=\frac{V_{q}}{(4\pi)^{q/2}}\frac{T^{q/2-s}}{\Gamma(s)}\sum_{\bfp}\sum_{n=1}^{\infty}\frac{e^{-n\beta(\sigma_{\bfp}^{d}-\mu)}}{n^{1+q/2-s}}\;.
u_{\underline{{m}}}^{(0)}\sigma_{\underline{{\mu}}\underline{{\nu}}}^{\underline{{m}}}=v_{\underline{{\mu}}}^{~A}v_{\underline{{\nu}}}^{~A},\qquadu_{\underline{{m}}}^{i}\sigma_{\underline{{\mu}}\underline{{\nu}}}^{\underline{{m}}}=v_{\underline{{\mu}}}^{~A}\Gamma_{AB}^{i}v_{\underline{{\nu}}}^{~B}.
<p_{1},...,p_{n}|S|q_{1},....,q_{m}>=lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}d\tauE\Big[\prod_{j}\prod_{k}(\overline{{f_{p_{j}}}},\partial_{\tau}\phi_{\tau})(f_{q_{k}},\partial_{\tau}\phi_{\tau})\Big]
B_{\beta\alpha}^{(n)}=\Sigma_{m=0}^{n}\{\tilde{\Theta}_{\beta}^{(n-m)},\tilde{\Theta}_{\alpha}^{(m)}\}_{(q,p)}+\Sigma_{m=0}^{n}\{\tilde{\Theta}_{\beta}^{(n-m)},\tilde{\Theta}_{\alpha}^{(m+2)}\}_{(\phi)}~~;~~n\ge2
C=x-x^{2}I_{1}+x^{3}I_{1}^{2}-x^{4}I_{1}^{3}-x^{3}yI_{3}+x^{5}I_{1}^{4}+3I_{1}I_{3}x^{4}y-x^{6}I_{1}^{5}-6x^{5}yI_{1}^{2}I_{3}-{\frac{1}{4}}x^{4}y^{2}I_{5}+...
\left(\begin{array}{c}{Q_{j+1}}\\{P_{j+1}}\\\end{array}\right)=\left(\begin{array}{cc}{a}&{b}\\{c}&{d}\\\end{array}\right)\left(\begin{array}{c}{Q_{j}}\\{P_{j}}\\\end{array}\right)=T\left(\begin{array}{c}{Q_{j}}\\{P_{j}}\\\end{array}\right)
\frac{\theta(-r^{0})}{r^{+}-i\epsilon}+\frac{\theta(r^{0})}{r^{+}+i\epsilon}=PV\frac{1}{r^{+}}-i\pi\delta(r^{+})\frac{(r^{+}+r^{-})}{|r^{+}+r^{-}|}\,\,,
J_{q}(L,r)=\sumq^{X_{r}(L)}\prod_{i=1}^{\nu-2}\left[\begin{array}{c}{n_{i}+m_{i}}\\{n_{i}}\\\end{array}\right]_{q}=\sumq^{X_{r}(L)}\prod_{i=1}^{\nu-2}\left[\begin{array}{c}{n_{i}+\tilde{n}_{i}-V_{i,r}}\\{n_{i}}\\\end{array}\right]_{q};\;\;L+r\equiveven\;\;
\begin{array}{l}{(M_{1})_{n,m}=\frac{2}{\pi}\left(\frac{2n}{2m-1}\right)^{1/2}\;\frac{(-)^{n+m}}{2n-(2m-1)}\,,}\\{(M_{2})_{n,m}=\frac{2}{\pi}\left(\frac{2n}{2m-1}\right)^{1/2}\;\frac{(-)^{n+m}}{2n+2m-1}\,.}\\\end{array}
L=-2c\sqrt{\frac{\dot{A}^{+}(x^{+}(\tau))\dot{A}^{-}(x^{-}(\tau))}{[A^{+}(x^{+}(\tau))-\epsilonA^{-}(x^{-}(\tau))]^{2}}},~~~~~~~(c:=\frac{m_{0}}{m}).
{\calL}=-\frac{k}{8}\:\varepsilon^{\mu\nu\lambda}a_{\mu}f_{\nu\lambda}+i\Psi^{\ast}(\partial_{t}+ia_{0})\Psi-\frac{1}{2}|{\bfD}\Psi|^{2}+\frac{g}{2}|\Psi|^{4}.
\frac{1}{[(k+p)^{2}-m^{2}]}=\sum_{j=0}^{N}\frac{\left(-1\right)^{j}\left(p^{2}+2p\cdotk\right)^{j}}{\left(k^{2}-m^{2}\right)^{j+1}}+\frac{\left(-1\right)^{N+1}\left(p^{2}+2p\cdotk\right)^{N+1}}{\left(k^{2}-m^{2}\right)^{N+1}[\left(k+p\right)^{2}-m^{2}]}\cdot
\partial_{+}\tilde{j}^{+}(x)=-i[\tilde{j}^{+}(x),P^{-}]=im(:\psi_{L}^{\dagger}\psi_{R}(x)-\psi_{R}^{\dagger}\psi_{L}(x):)_{\sim}+{\frac{e}{2\pi}}\partial_{-}\tilde{A^{-}}.
u_{-}(n,k;\rho,\phi)=\frac{1}{\sqrt{2\bar{\omega}_{k}(M+\bar{\omega}_{k})}}\,\left[\begin{array}{c}{(M+\bar{\omega}_{k})J_{n}(k\rho)e^{in\phi}}\\{ikJ_{n+1}(k\rho)e^{i(n+1)\phi}}\\\end{array}\right],
\begin{array}{ll}{b_{\frac{j}{2}}|h\rangle=0}&{\mathrm{~if~}\{{\sigma_{h}}\}_{j}\mathrm{~is~a~corner}}\\{b_{-\frac{j}{2}}|h\rangle=0}&{\mathrm{~if~}\{{\sigma_{h}}\}_{j}\mathrm{~is~a~straight~line}}\\\end{array}
A_{\mu}(x_{i})=\frac{\partialY^{\alpha}}{\partialx_{i}^{\mu}}(x_{i})a_{j\alpha}(Y_{ji}(x_{i})),\quadG_{\mu\nu}(x_{i})=\frac{\partialY^{\alpha}}{\partialx_{i}^{\mu}}(x_{i})\frac{\partialY^{\beta}}{\partialx_{i}^{\nu}}(x_{i})g_{j\alpha\beta}(Y_{ji}(x_{i}))
\Sigma(p_{f})={\frac{e^{2}}{\beta}}\sum_{n}\int{\frac{d^{2}k}{(2\pi)^{2}}}{\frac{{\calM}(k_{f})}{k_{f}^{2}+{\calM}^{2}(k_{f})}}\left[D_{L}+D_{T}+{\frac{\epsilon(q_{b})}{q_{b}^{2}}}\right]
Z[\eta,\overline{{\eta}}]:=\int[d\psi][d\overline{{\psi}}][dA]e^{-S[\psi,\overline{{\psi}},A]+\sum_{a}\Big\{(\overline{{\eta}}^{(a)},\psi^{(a)})+(\overline{{\psi}}^{(a)},\eta^{(a)})\Big\}}\;.
M_{4}=\left(\begin{array}{cccc}{c_{11}}&{c_{13}}&{0}&{d_{13}}\\{c_{13}}&{c_{33}}&{-d_{13}}&{0}\\{0}&{-d_{13}}&{c_{11}}&{c_{13}}\\{d_{13}}&{0}&{c_{13}}&{c_{33}}\\\end{array}\right)
\mathrm{and~\thinspace~\thinspace~}U(R(\Lambda,p))\left|p_{R},s_{3}\right\rangle=\sum_{s_{3}^{^{\prime}}}\left|p_{R},s_{3}^{^{\prime}}\right\rangleD_{s_{3}^{^{\prime}}\,s_{3}}(R(\Lambda,p))\,\,\,\,\,\,\,\mathrm{we~obtain:}
{\delta_{\rho}}{\mathcal{F}_{D}^{0}}={\frac{Q^{2}}{8\pi}}\int{d^{2}}\xi\sqrt{\hat{g}}\hat{R}\rho+{\frac{Q^{2}}{8\pi}}\oint{d}\hat{s}(\xi)\int{d^{2}}\xi^{\prime}\rho(\xi){\partial_{\hat{n}}}{\hat{G}_{D}}(\xi,\xi^{\prime})\sqrt{\hat{g}(\xi^{\prime})}\hat{R}(\xi^{\prime}).
\partial_{\mu}\Omega(\Sigma,x)=-\int_{\Sigma}d^{D-1}\tilde{\sigma}_{\mu}\delta^{D}(x-\bar{x}(\sigma))-\int_{\Sigma}d^{D-1}\tilde{\sigma}_{\nu}(\delta_{\mu\nu}\partial^{2}-\partial_{\mu}\partial_{\nu})D(x-\bar{x}(\sigma))\.
\hat{\mathcal{A}}_{\pm}=\mathcal{A}_{\pm}^{(1)}\otimes\left(\begin{matrix}{1}&{0}\\{0}&{0}\\\end{matrix}\right)+\mathcal{A}_{\pm}^{(2)}\otimes\left(\begin{matrix}{0}&{0}\\{0}&{1}\\\end{matrix}\right)+\mathcal{B}_{\pm}^{*}\otimes\left(\begin{matrix}{0}&{1}\\{0}&{0}\\\end{matrix}\right)+\mathcal{B}_{\pm}\otimes\left(\begin{matrix}{0}&{0}\\{1}&{0}\\\end{matrix}\right)
T_{~\mu}^{\mu}=-\frac{T_{3}V(T)}{\sqrt{-X}}\left[(4+3\partial_{\rho}T\partial^{\rho}T)+\left((1+\frac12\partial_{\rho}T\partial^{\rho}T)\delta_{~\sigma}^{\mu}-\partial^{\mu}T\partial_{\sigma}T\right)F_{\mu\nu}F^{\sigma\nu}\right].
\sqrt{\langle\rho^{2}\rangle}\sim\sqrt{N_{4}g_{s}l_{s}}\sqrt{l_{s}^{2}/r_{\perp0}}=(g_{s}N_{4})^{1/3}l_{s}\sqrt{\frac{(g_{s}N_{4})^{1/3}l_{s}}{r_{\perp0}}}.
\varsigma_{R}\varsigma_{R^{\prime}}{\frac{\vec{\alpha}^{(R)}\vec{\alpha}^{(R^{\prime})}}{2}}-{\frac{d_{R}d_{R^{\prime}}}{D-2}}+\sum_{L=0}^{D-1}\Upsilon_{RL}\Upsilon_{R^{\prime}L}=0,\;R\not=R^{\prime},
\langle{\vec{u}}(\vec{x}).{\vec{u}}(0)\rangle=\sum_{i=j}^{3}iS_{ij}^{F}({\vec{x}}-{\vec{x^{\prime}}})=\int{\frac{\mathrm{d}^{3}{\vec{k}}}{(2\pi)^{3}}}{{\frac{e^{i{\vec{k}}.({\vec{x}}-{\vec{x^{\prime}})}}}{\omega_{k}}}}
+m\sum_{n=2}^{N}b_{n}\varepsilon^{\mu\nu\sigma}h_{\mu|}^{n}{}^{\alpha(n)}e_{\nu|\alpha(2)}h_{\sigma|\alpha(n-2)}^{n-2}+m\sum_{n=0}^{N-2}\lambda_{\alpha(n+2)}^{n+2}e^{\mu|\alpha(2)}h_{\mu|}^{n}{}^{\alpha(n)},
\mathcalL_{\mathrm{mass}}=-\left(A_{1}^{\dagger}\A_{2}^{\dagger}\right)\left(\begin{array}{cc}{p_{5}^{2}+(\omega/R)^{2}}&{i\,2\omega\,p_{5}/R}\\{-i\,2\omega\,p_{5}/R}&{p_{5}^{2}+(\omega/R)^{2}}\\\end{array}\right)\left(\begin{array}{c}{A^{1}}\\{A^{2}}\\\end{array}\right)\.
\psi_{Q}=\left(\begin{array}{c}{\frac{\psi_{(0)}}{\sqrt{N_{f}}}I_{N_{f}}+\psi_{(1)}}\\{\psi_{(2)}}\\\end{array}\right)\;,\qquad\quad\psi_{\bar{Q}}=\left(\begin{array}{cc}{\frac{\psi_{(\bar{0})}}{\sqrt{N_{f}}}I_{N_{f}}+\psi_{(\bar{1})}}&{\psi_{(\bar{2})}}\\\end{array}\right)
0=-k_{1\beta}\nabla_{\beta\mu}-\frac{1}{3}k_{1\beta}k_{1\alpha}k_{1\nu}\Box_{\alpha\beta\mu\nu}+\frac{1}{3}k_{1}^{2}k_{1\nu}\triangle_{\nu\mu}+\frac{2}{3}k_{1\mu}k_{1\alpha}k_{1\beta}\triangle_{\alpha\beta}.
L_{2}=\frac{1}{2}(\partial_{t}\mathrm{x})^{2}+\frac{1}{2}\mu^{2}\mathrm{x}^{2}-\frac{1}{4}\lambda\mathrm{x}^{4}-\frac{\mu^{4}}{4\lambda}=\frac{1}{2}(\partial_{t}\mathrm{x})^{2}-\frac{1}{4}\lambda(\mathrm{x}^{2}-\frac{\mu^{2}}{\lambda})^{2}
\begin{array}{c}{[A_{ab},A_{cd}]=i(\delta_{ac}A_{bd}-\delta_{ad}A_{bc}-\delta_{bc}A_{ad}+\delta_{bd}A_{ac})\,,}\\{{}}\\{{}[A_{ab},Q^{c}]=i(\delta_{a}{}^{c}\delta_{bd}-\delta_{b}{}^{c}\delta_{ad})Q^{d}\,,}\\{{}}\\{{}[A_{ab},S^{c}]=i(\delta_{a}{}^{c}\delta_{bd}-\delta_{b}{}^{c}\delta_{ad})S^{d}\,,}\\{{}}\\{{}[A_{a}{}^{b},P_{\mu}]=[A_{a}{}^{b},K_{\mu}]=[A_{a}{}^{b},M_{\mu\nu}]=0\,.}\\\end{array}
S_{l.c.}=\frac{1}{2}\intd^{d}x\sum_{s^{\prime}=0}^{s}(\phi_{s^{\prime}}^{i_{1}\ldotsi_{s^{\prime}}}\Box\phi_{s^{\prime}}^{i_{1}\ldotsi_{s^{\prime}}}-\frac{1}{z^{2}}A_{s^{\prime}}|\phi_{s^{\prime}}|^{2})\,.
S_{\mathrm{eff}}^{(k)}=kS_{0}+\\int_{0}^{\beta}d\tau\\frac{1}{2}\left[\\tilde{g}_{pq}\d_{\tau}Y^{p}d_{\tau}Y^{q}+\\tilde{g}_{pq}\i\bar{\alpha}^{p}\gamma^{0}D_{\tau}\alpha^{q}+\\frac{1}{12}\tilde{R}_{pqrs}(\bar{\alpha}^{p}\alpha^{r})(\bar{\alpha}^{q}\alpha^{s})\right]
L_{ij}=\left(\begin{array}{cccc}{0}&{L_{12}}&{L_{13}}&{L_{14}}\\{-L_{12}}&{0}&{L_{23}}&{L_{24}}\\{-L_{13}}&{-L_{23}}&{0}&{L_{34}}\\{-L_{14}}&{-L_{24}}&{-L_{34}}&{0}\\\end{array}\right)
\frac{d\rho}{dt}=-\frac{i}{\hslash}\left[H,\rho\right]-\gamma(x-x^{\prime})\left(\frac{\partial\rho}{\partialx}-\frac{\partial\rho}{\partialx^{\prime}}\right)-\frac{2m\gammak_{B}T}{\hslash^{2}}
\begin{array}{rl}{c_{00}+c_{04}\stackrel{S}{\longrightarrow}~{\frac{1}{2\sqrt{6}}}\sqrt{\frac{i}{\tau}}\big(2c_{00}+2c_{04}+4c_{02}+4c_{20}+4c_{22}\big)}&{={\frac{1}{2\sqrt{6}}}\sqrt{\frac{i}{\tau}}\big({\frac{\eta_{1/6}^{5}}{\eta^{2}~\eta_{1/12}^{2}~\eta_{1/3}^{2}}}\big)}\\{}&{\stackrel{S}{\longrightarrow}{\frac{\eta_{6}^{5}}{\eta^{2}~\eta_{12}^{2}~\eta_{3}^{2}}},}\\\end{array}
\langlel_{1}^{\vphantom\prime}l_{2}^{\vphantom\prime}|V|l_{1}^{\prime}l_{2}^{\prime}\rangle=\frac{\kappa}{B}(l_{1}+l_{2}+2)\delta_{l_{1}^{\vphantom\prime}l_{1}^{\prime}}\delta_{l_{2}^{\vphantom\prime}l_{2}^{\prime}}\,
S_{l.c.}=\frac{1}{2}\intd^{d}x\Bigl(\phi^{I}\Box\phi^{I}-\frac{(d-2)(d-4)}{4z^{2}}|\phi_{1}|^{2}-\frac{(d-4)(d-6)}{4z^{2}}|\phi_{0}|^{2}\Bigr)\,.
\left(\Gamma_{45}\right)_{a}^{\,b}\;\varepsilon_{b}\,=\,i\varepsilon_{a}\,,\,\left(\Gamma_{3}\right)_{a}^{\,b}\;\varepsilon_{b}\,=\,\pm\varepsilon_{a}\,\rightarrow\,\bar{g}\left(\frac{a}{\sqrt{2}}\,\pm\,b\right)\,=\,-1,
\Omega_{\xi}(s,0)\{{\calD}_{\mu}(s){\tilde{\Lambda}}_{\nu}[\xi|s]\}\Omega_{\xi}^{-1}(s,0)=\frac{\delta}{\delta\xi^{\mu}(s)}\{\Omega_{\xi}(s,0){\tilde{\Lambda}}_{\nu}[\xi|s]\Omega_{\xi}^{-1}(s,0)\},
G^{E}(2\pinT,k)=\pi\,C_{\cal{O}}{\frac{(2\piTn)^{2}+k^{2}}{4}}\left[\psi\left(1+{\frac{n}{2}}-{\frac{ik}{4\piT}}\right)+\psi\left(1+{\frac{n}{2}}+{\frac{ik}{4\piT}}\right)\right]\,.
z_{1}=\frac{1}{\sqrt{1+\vert\psi\vert^{2}}}~,~z_{\alpha+1}=\frac{\psi_{\alpha}}{\sqrt{1+\vert\psi\vert^{2}}},\quad~;~Z_{1}^{T}=(z_{1},z_{2},\cdots,z_{N}),
E_{z}^{a}=\frac{N(z,t)}{f^{\prime}(z,t)}\left(\begin{array}{c}{f}\\{1}\\{f^{2}}\\\end{array}\right)\;,\;\;E_{u}^{a}=\frac{n(u,t)}{\dot{f}(u,t)}\left(\begin{array}{c}{f}\\{1}\\{f^{2}}\\\end{array}\right)