harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f_{2}^{i}=\left(\begin{array}{c}{\xi^{i}}\\{\chi^{i}}\\\end{array}\right),\quadA=\left(\begin{array}{ll}{a}&{b}\\{c}&{d}\\\end{array}\right),\quad\tilde{f}_{2i}=\left(\begin{array}{ll}{\tilde{\chi}_{i}}&{\tilde{\xi}_{i}}\\\end{array}\right),\quad\tilde{A}=\left(\begin{array}{ll}{\tilde{d}}&{\tilde{b}}\\{\tilde{c}}&{\tilde{a}}\\\end{array}\right)
(z^{1},z^{2},z^{3})^{\prime}=(\bar{w}^{\bar{1}},w^{2},\bar{w}^{\bar{3}})\,\quad(z^{1},z^{2},z^{3})^{\prime\prime}=(\bar{w}^{\bar{1}},\bar{w}^{\bar{2}},w^{3})\.
\Omega=\frac{1}{2}\frac{\epsilon^{abc}p_{a}\mathrm{d}p_{b}\land\mathrm{d}p_{c}}{(-p^{2})^{3/2}}+\frac{im}{s\sqrt{-p^{2}}}(\eta_{ab}-\frac{p_{a}p_{b}}{p^{2}})\theta^{\alpha}(\sigma^{a})_{\alpha\beta}\mathrm{d}p^{b}\wedge\mathrm{d}\theta^{\beta}-\frac{im}{s\sqrt{-p^{2}}}p_{\alpha\beta}\mathrm{d}\theta^{\alpha}\wedge\mathrm{d}\theta^{\beta}
D_{00}\left(\mathbf{k}\right)=-\frac{1}{\mathbf{k}^{2}},\;\;D_{0l}\left(\mathbf{k}\right)=0,\;\;D_{Al}\left(k^{\mu}\right)=-\frac{1}{k^{\mu}k_{\mu}}\left(\delta_{Al}-\frac{k_{k}k_{l}}{\mathbf{k}^{2}}\right),
{\calH}=\frac{1}{2}\pi_{\varphi}^{2}+\frac{1}{2}\left(\partial_{1}\varphi\right)^{2}+\frac{1}{2}\mu^{2}\varphi^{2}+\frac{1}{2}\left(\frac{\phi}{\mu}\right)^{2}+\left(\frac{\phi}{\mu}\right)(\partial_{1}\varphi)\,
\frac{1}{L}\sum_{n=-\infty}^{\infty}F\left(\frac{2n+1}{L}\right)=\frac{1}{L}\sum_{n=-\infty}^{\infty}F\left(\frac{n}{L}\right)-\frac{1}{L}\sum_{n=-\infty}^{\infty}F\left(\frac{2n}{L}\right)
\left(\frac{1}{G_{4}(\mu)}-\frac{1}{G_{4}^{*}(g)}\right)=\left(\frac{g^{2}(\Lambda)}{g^{2}(\mu)}\right)^{2-cg^{2}}\left(\frac{1}{G_{4}(\Lambda)}-\frac{1}{G_{4}^{*}(g)}\right),
0=g^{\mu\nu}h_{ij}[t_{\mu}\dot{q}^{i}-{(j_{u})_{\mu}}^{\rho}t_{\rho}\dot{q}^{k}{(J_{u})_{k}}^{i}]\,[t_{\nu}\dot{q}^{j}-{(j_{v})_{\nu}}^{\sigma}t_{\sigma}\dot{q}^{l}{(J_{v})_{l}}^{j}]=4t^{2}\dot{q}^{2},
k=\frac{1}{24\pi^{2}}\oint_{S^{3}(\mathrm{space})}d{\mit\Omega}_{\mu}\epsilon_{\mu\nu\rho\sigma}\mathrm{tr}\,\left\{\left(U^{-1}\partial_{\nu}U\right)\left(U^{-1}\partial_{\rho}U\right)\left(U^{-1}\partial_{\sigma}U\right)\right\}\,
{\calL}=\intd^{4}\theta\,\left(\mathrm{tr}\left(\Phi^{\dagger}\Phie^{V}\right)-c\,\mathrm{tr}V\right)+\left(\intd^{2}\theta\;\mathrm{tr}\left(\Phi_{0}\Phi^{T}\Phi)\right)+\mathrm{h.c.}\right).
\begin{array}{lcr}{P(W_{2})=(1+U_{1}+U_{2}+U_{1}^{2}U_{2}^{3})}\\{+U_{3}(a_{0}U_{1}^{2}U_{2}^{3}+a_{1}U_{1}U_{2}+a_{n-1}U_{1}^{{\frac{(5-n)}{2}}}U_{2}^{{\frac{(6-n)}{2}}})+U_{3}^{2}\sum_{i=0}^{n-4}a_{i+1}U_{1}^{3-i}U_{2}^{4-i}.}\\\end{array}
S_{2}=i\frac{\pi}{2}\left[\frac{P}{H}\left(\sum_{i=1}^{3}p_{i}^{-2}-1\right)-3\mathrm{sign}\left(\frac{H}{P}\right)-\sum_{i=1}^{3}\left(12s(q_{i},p_{i})-\frac{q_{i}}{p_{i}}\right)\right]=6\pi\lambda_{CW},
E_{n_{r},l_{D},q}=\pmMc^{2}\left[1+\frac{\alpha^{2}}{\left(\frac12+\frac12\sqrt{\left(2l_{D}+1\right)^{2}-4\left(q^{2}+\alpha^{2}\right)}+n_{r}\right)^{2}}\right]^{-1/2}.
\frac{1}{\pi}\int_{C_{h}}{\left[H_{\nu+m}^{(2)}(zt)\bar{H}_{\nu}^{(1)}(z)-H_{\nu+m}^{(1)}(zt)\bar{H}_{\nu}^{(2)}(z)\right]\frac{F(z)}{\bar{J}_{\nu}(z)}dz},
D=\int_{-Q_{2}}^{Q_{2}}{\rho_{2}(k)dk},\;\;\;D_{m}=\int_{-B_{2}}^{B_{2}}{\sigma_{2}(\Lambda)d\Lambda},\;\;\;E_{2}/N=D^{-1}\int_{-Q_{2}}^{Q_{2}}{k^{2}\rho_{2}(k)dk}.
+{\binom{-\!1\!-\!1}{m_{2}\!\!+\!\!m_{3}\!\!-\!\!1}}{\binom{-\!1\!+\!1}{m_{1}\!-\!1}}+{\binom{-\!1\!-\!1}{m_{3}\!\!+\!\!m_{1}\!\!-\!\!1}}{\binom{-\!1\!+\!1}{m_{2}\!-\!1}}\Biggr)
S_{\mathrm{BI-YM}}^{(\mathrm{q})}=\sqrt{\,{\frac{N\,V}{4g_{0}^{2}}}\,\mathbf{F}_{cd\,\mu\nu}^{(0)}\,\mathbf{F}^{(0)}{}^{cd\,\mu\nu}\,}\equiv\sqrt{\,{\frac{N\,V}{4g_{0}^{2}}}\mathrm{Tr}\,\mathbf{F}_{\mu\nu}^{(0)}\,\mathbf{F}^{(0)}{}^{\mu\nu}\,}
\begin{array}{c}{\Omega_{+}(z_{+};g)=\Omega(z;g)\Upsilon(z;g)^{\frac{{\calN}}{{\calN}-4}}\,,}\\{{}}\\{\Omega_{-}(z_{-};g)=\Omega(z;g)\Upsilon(z;g)^{-\frac{{\calN}}{{\calN}-4}}\,.}\\\end{array}
\beta(k,p)=\frac{-i}{2\pi}\delta(p_{2}+k_{2})\delta(p_{3}+k_{3})\int\!dx_{1}\frac{e^{ik_{1}x_{1}}}{\sqrt{4\omega_{p}\omega_{k}}}\left\{i\omega_{k}f_{p}^{+\ast}(t^{\prime}(0,x))e^{ip_{1}x_{1}^{\prime}(0,x)}-\partial_{t}\left\{f_{p}^{+\ast}(t^{\prime}(t,x))e^{ip_{1}x_{1}^{\prime}(t,x)}\right\}\left|_{t=0}\right.\right\}
\begin{array}{c}{ds_{IIA}^{2}=e^{2U}ds_{3}^{2}-e^{-2U}ds_{3}^{2}\quad,\quade^{2\phi}=e^{2U}\quad,\quad\vec{F}_{ij}=\sqrt{2}\epsilon_{ijm}\partial_{m}\vec{\chi}\.}\\\end{array}
v_{i}^{(2)}(\vec{R}_{p_{1}})=-\frac{\sigma_{2}}{q_{2}}\sum_{q_{1}}\frac{\varepsilon_{ij}(R_{p_{1}}-R_{q_{2}})^{j}}{\mid\vec{R}_{p_{1}}-\vec{R}_{q_{2}}\mid^{2}}\;\;,\;\;(\;1\leftrightarrow2\;)
{\frac{1}{2}}(\partialh)^{2}+{\frac{1}{2}}(\partialH)^{2}+{\frac{m^{2}}{2}}h^{2}+{\frac{M^{2}}{2}}H^{2}+\lambda_{1}h^{4}+\lambda_{2}H^{4}+\lambda_{3}(h\timesH)^{2}+\lambda_{4}h^{2}H^{2}
S=-{\frac{1}{4}}\,\intd^{4}x\,\left[\sum_{j}{\frac{a_{j}}{g_{5j}^{2}}}{G_{\mu\nu}^{(j)}}^{2}-2\sum_{j}{\frac{a_{j}e^{-2\sigma_{j}}}{g_{5j}^{2}}}\,{G_{\mu5}^{(j)}}^{2}\right]\,
S_{F}(p)\;=\;Z_{2}\,\frac{-i\!p}{p^{2}}\,+\,\int_{m^{2}}^{\infty}d\mu^{2}\,\frac{-i\rho_{1}(\mu^{2})\not\!p\,+\,\rho_{2}(\mu^{2})}{p^{2}\,+\,\mu^{2}}
E={\displaystyle{\frac{1}{2}}}\int_{M}\!d^{2}x\,\left(\,(D_{1}+iD_{2})\phi\,\overline{{(D_{1}+iD_{2})\phi}}+\Omega^{-1}\{F_{12}+{\displaystyle{\frac{\Omega}{2}}}(|\phi|^{2}-1)\}^{2}+F_{12}\right).
{\calA}_{open}=2\,V\,(8\pi^{2}\alpha^{\prime})^{-1/2}\int_{0}^{\infty}{\frac{ds}{2s}}\,s^{-1/2}\left[\mathrm{Tr}_{\mathrm{NS}}\left(r^{(2L_{0}-1)}{\frac{1+\Omega}{2}}\right)-\mathrm{Tr}_{\mathrm{R}}\left(r^{2L_{0}}{\frac{1+\Omega}{2}}\right)\right]~~,
\begin{array}{l}{X^{-}(z)\frac1{P^{+}(w)}=\,:X^{-}(z)\frac1{P^{+}(w)}:\,}\\{\quad\quad\quad\quad\quad\quad-\left(\frac1{P^{+}(w)}\right)^{2}<X^{-}(z)P^{+}(w)>}\\{\quad\quad\quad\quad\rightarrow\frac{-i}{\left(z-w\right)}\left(\frac1{P^{+}(w)}\right)^{2}+\cdots}\\\end{array}
S={\frac{kA_{H}}{4\ell_{P}^{2}}}+{\frac{1}{T_{H}}}\int_{\Sigma}e^{-\phi}\{\varrho_{L}-{\calL}_{E}\}d^{3}r+\int_{\Sigma}{\frac{s}{\sqrt{1-(b/r)}}}d^{3}r.
l({\bfv}),\quad-iv_{0}\frac{\partial}{\partial{\bfv}},\quad-i[\frac{\partial}{\partial{\bfv}}+{\bfv}({\bfv}\frac{\partial}{\partial{\bfv}})+\frac{3}{2}{\bfv}],\quad-iv_{0}({\bfv}\frac{\partial}{\partial{\bfv}}+\frac{3}{2})
q=\frac1{32\pi^{2}}\int\,\epsilon_{\alpha\beta\gamma\delta}\,\mathrm{tr\,}\hat{F}^{\alpha\beta}\,\hat{F}^{\gamma\delta}\,d^{4}q=\frac1{32\pi^{2}}\int\,\epsilon_{abcd}\,\mathrm{tr\,}\hat{F}^{ab}\,\hat{F}^{cd}\,\sqrt{g}d^{4}q.
\Omega_{d}=\frac{1}{(d+1)!\,\mathrm{Area}(S^{d+1})}\intd^{d+1}x\,\epsilon^{\mu_{1}\ldots\mu_{d+1}}\epsilon^{aa_{1}\ldotsa_{d+1}}v_{a}\partial_{\mu_{1}}v_{a_{1}}\cdots\partial_{\mu_{d+1}}v_{a_{d+1}}
{\calB}\left[{\bfA}_{T}^{\prime\prime},{\bfA}_{T}^{\prime},b\right)=\int{\calD}^{2}\!A_{T}^{\iota}\,{\calG}\left[{\bfA}_{T}^{\prime\prime}t^{\prime\prime}|{\bfA}_{T}^{\iota}t_{i}\right)\delta(b-g[\bbox\nabla\times{\bfA}_{T}^{\iota}])\,{\calG}\left[{\bfA}_{T}^{\iota}t_{i}|{\bfA}_{T}^{\prime}t^{\prime}\right),
{\calR}_{\mu\nu}\sigma_{A\dot{A}}^{\mu}\sigma_{B\dot{B}}^{\nu}=\sigma_{\underline{{A}}\underline{{\dot{A}}}}^{\mu}\sigma_{\underline{{B}}\underline{{\dot{B}}}}^{\nu}\left({\calR}_{\mu\nu}-\frac{1}{4}g_{\mu\nu}{\calR}\right)+\frac{1}{2}\varepsilon_{\dot{A}\dot{B}}\varepsilon_{AB}{\calR}
ds^{2}=\left[\left(1-\frac{q^{2}}{r_{1}^{2}}\right)\left(1-\frac{p^{2}}{r_{2}^{2}}\right)\right]^{-\frac{1}{2}}dt^{2}-\left[\left(1-\frac{q^{2}}{r_{1}^{2}}\right)\left(1-\frac{p^{2}}{r_{2}^{2}}\right)\right]^{\frac{1}{2}}d\vec{x}^{2}\,.
(f_{\alpha\beta}^{(1)})^{-1}(x,y)=\left(\begin{array}{cccc}{O}&{I}&{\frac{1}{m^{2}}\vec{\nabla}_{x}}&{\vec{0}}\\{-I}&{O}&{\vec{0}}&{\vec{0}}\\{-\frac{1}{m^{2}}\vec{\nabla}_{x}^{T}}&{\vec{0}^{T}}&{0}&{-\frac{1}{m^{2}}}\\{\vec{0}^{T}}&{\vec{0}^{T}}&{\frac{1}{m^{2}}}&{0}\\\end{array}\right)\delta(x-y).
f_{k}(\eta_{0})=\frac{1}{\sqrt{2W_{k}^{(N)}(\eta_{0})}},\quadf_{k}^{\prime}(\eta_{0})=-\left(iW_{k}^{(N)}(\eta_{0})+\frac{{W_{k}^{(N)}}^{\prime}(\eta_{0})}{2W_{k}^{(N)}(\eta_{0})}\right)f_{k}(\eta_{0})\,.
\frac{\alpha_{n}}{m^{2n+1}}\intd^{3}xF_{\mu\nu}(\partial^{2})^{n}F^{\mu\nu}\;\;,\;\;\;\;\;\;\;\frac{\beta_{n}}{m^{2n}}\intd^{3}x\varepsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}(\partial^{2})^{n}A_{\rho}\;,\;\;\;\;\;\;n\geq1\;,
\frac{\delta}{\deltas_{\mu\nu}(t)}exp\left[-{\frac{1}{4}}\int_{0}^{T}d\tau\dot{x}^{2}(\tau)\right]={\frac{1}{2}}\omega_{\mu\nu}(x(t))\left[-{\frac{1}{4}}\int_{0}^{T}d\tau\dot{x}^{2}(\tau)\right].
\delta_{\alpha{\mathbfi}}{\calR}^{\mu}=16i\left[\frac{1}{192}J_{\alpha{\mathbfi}}^{\mu}-2\sigma_{\alpha\dot{\alpha}}^{\mu}{\bar{D}}_{\mathbfi}^{3\dot{\alpha}}{\bar{\calJ}}|-(a+24)\sigma_{\;\;\alpha}^{\mu\nu\,\,\beta}\partial_{\beta{\mathbfi}}T|\right].
a_{n}^{l}(q^{\prime},q)=\int_{0}^{1}\eta^{n-1}d\eta\left(\frac{1}{2}\frac{\partial^{2}}{\partial\tilde{q}^{2}}-U_{l}(\tilde{q})\right)a_{n-1}^{l}(\tilde{q},q)\Biggl.\Biggr|_{\tilde{q}=q+(q^{\prime}-q)\eta}.
\frac{y_{\sigma}y_{\rho}}{y^{6}}-\frac{\delta_{\sigma\rho}}{4y^{4}}=\frac{1}{8}\left(\partial_{\sigma}\partial_{\rho}-\frac{1}{4}\,\delta_{\sigma\rho}\,\Box\right)\,\frac{1}{y^{2}}\,,
\mathrm{M.S}:~~~~\mathrm{\bf~Tr}_{{\calH}_{o}}\left[\frac{\Omega}{2}\,(-1)^{F_{s}}e^{-2\pit\,\mathrm{H}_{o}}\right]-\mathrm{\bf~Tr}_{{\calH}_{o}}\left[\frac{\Omega}{2}\,e^{-2\pit\,\mathrm{H}_{o}}\right]
{\calE}_{0}(\beta_{2i^{\prime}j^{\prime}})=4~,~~~{\calE}_{0}(\bar{\beta}_{2i^{\prime}j^{\prime}})=8~,~~~{\calE}_{0}(\beta_{4i^{\prime}j^{\prime}})=2~,~~~{\calE}_{0}(\bar{\beta}_{4i^{\prime}j^{\prime}})=10~.
b\left(\phi_{o}^{4}+\phi^{4}\right){\bfM}^{2}=\frac{1}{6}\left[1+\frac{2{\bfQ}^{2}}{9{\bfM}^{2}}\left(1+\frac{2{\bfQ}^{2}}{a^{2}}\right)^{2}\right]^{2}
\left(\sum_{l=1}^{k}{\frac{{\bfp}_{l}^{2}}{2mM_{l}}}+V_{k,k}-E\right)\Psi_{k}=-V_{k,k-2}\Psi_{k-2}-V_{k,k-1}\Psi_{k-1}-V_{k,k+1}\Psi_{k+1}-V_{k,k+2}\Psi_{k+2}.
\begin{aligned}{[m]\lambda_{1}}&{{}=-m_{1}-m_{2}+m_{3}-m_{4}}\\{\lambda_{2}}&{{}=m_{1}+m_{2}-m_{3}-m_{4}}\\{\lambda_{3}}&{{}=m_{1}+m_{2}+m_{3}-m_{4}}\\{\lambda_{4}}&{{}=-m_{1}-m_{2}-m_{3}-m_{4}}\\{\lambda_{5}}&{{}=-m_{1}+m_{2}+m_{3}+m_{4}}\\{\lambda_{6}}&{{}=m_{1}-m_{2}-m_{3}+m_{4}}\\{\lambda_{7}}&{{}=m_{1}-m_{2}+m_{3}+m_{4}}\\{\lambda_{8}}&{{}=-m_{1}+m_{2}-m_{3}+m_{4}}\\\end{aligned}
{\frac{1}{2}}({\hat{a}}_{(i)(j)}{\bar{\pi}}_{(i)}^{A}\pi_{(j)}^{A^{\prime}}+{\bar{\pi}}_{(i)}^{A}\pi_{(j)}^{A^{\prime}}{\hat{a}}_{(i)(j)})n_{AA^{\prime}}-{\frac{1}{4}}({\hat{a}}_{(i)(i)}{\bar{\pi}}_{(j)}^{A}\pi_{(j)}^{A^{\prime}}+{\bar{\pi}}_{(j)}^{A}\pi_{(j)}^{A^{\prime}}{\hat{a}}_{(i)(i)})n_{AA^{\prime}}.
\begin{array}{rcl}{}&{}&{{^1\Phi}_{h\frac{n-1}{2}}=\displaystyle\sum_{i}a_{i}^{h}~{^1C}_{i}\mid1\rangle~,}\\{}&{}&{\left([2]-{^1E}_{h}\right)a_{i}^{h}+a_{i+1}^{h}+a_{i-1}^{h}=0~,}\\{}&{}&{a_{0}^{h}=a_{n+1}^{h}=0~.}\\\end{array}
({\bfa}_{i}\cdot{\bfa}_{j})=({\bfb}_{i}\cdot{\bfb}_{j})=({\bfc}_{i}\cdot{\bfc}_{j})=\delta_{ij},\quad({\bfa}_{i}\cdot{\bfb}_{j})=-\frac{1}{2},\quad({\bfa}_{i}\cdot{\bfc}_{j})=\frac{1}{2},\quad
\intd^{3}\zeta-\frac{1}{6}\epsilon^{ijk}\Pi_{i}^{A}\Pi_{j}^{B}\Pi_{k}^{C}B_{CBA}=\int\Big[h^{-1}-h^{-2/3}\bar{\theta}(\Gamma^{\tilde{0}}\partial_{0}+\Gamma^{\tilde{1}}\partial_{1}+\Gamma^{\tilde{2}}\partial_{2})\theta
\Gamma^{\nu}D_{\lambda}\Lambda(x,x^{\prime})=D_{\lambda}\Lambda(x,x^{\prime})\Gamma^{\mu^{\prime}}g_{\mu^{\prime}}^{\nu}-(A+C)\Lambda(x,x^{\prime})\Gamma^{\mu^{\prime}}(\delta_{\lambda}^{\nu}n_{\mu^{\prime}}+g_{\lambda\mu^{\prime}}n^{\nu}),
\begin{array}{c}{E_{\dot{5}}^{5}(x)}\\\end{array}\equiv\left(\begin{array}{cc}{\phi_{1}^{-1}(x)}&{0}\\{0}&{\phi_{2}^{-1}(x)}\\\end{array}\right)\equiv\begin{array}{c}{\Phi^{-1}(x)}\\\end{array},
E=T\intdl{\frac{1}{\sqrt{1-v_{\perp}^{2}}}}=T\int{\frac{dl^{2}}{dX^{9}}}=2\piR_{9}T+T\intdX^{9}\sum_{I=1}^{8}\left({\frac{dX^{I}}{dX^{9}}}\right)^{2}
\Gamma_{\mu\nu}^{~~~\lambda}=-\mathrm{~\frac{1}{3}~}\partial_{(\mu}(J^{\alpha})_{\nu)}^{~~\rho}(J_{\alpha})_{\rho}^{\!~~\lambda}-\mathrm{~\frac{1}{6}~}\epsilon^{\alpha\beta\gamma}(J_{\beta})_{(\mu}^{~~\rho}\partial_{|\rho|}(J_{\gamma})_{\nu)}^{~~\sigma}(J_{\alpha})_{\sigma}^{\!~~\lambda}-(J^{\alpha})_{(\mu}^{\!~~~\lambda}\omega_{\nu)\alpha};
\frac{-1}{4}\left(\overrightarrow{\partial}_{x}^{2}-\overrightarrow{\partial}_{X}^{2}\right)K_{i\left(P-p\right)}(e^{x+X})\,K_{i\left(P+p\right)}(e^{x-X})=-e^{2x}\,K_{i\left(P-p\right)}^{\prime}(e^{x+X})\,K_{i\left(P+p\right)}^{\prime}(e^{x-X})\;,
\displaystyle{M=\left(\begin{array}{ccc}{\omega+\frac{1}{2}\lambda}&{-i\tilde{\mathrm{a}}}&{2\varepsilon^{j}}\\{-i\mathrm{b}}&{\tilde{\omega}-\frac{1}{2}\lambda}&{2\rho^{j}}\\{2\bar{\rho}_{i}}&{2\bar{\varepsilon}_{i}}&{T_{i}^{~j}}\\\end{array}\right)}
(\kappa_{\tau}\frac{\partial}{\partial\kappa_{\tau}}+\beta_{\tau}\frac{\partial}{\partialu_{\tau}}-\frac{1}{2}N\gamma_{\phi(\tau)}(u_{\tau})+L\gamma_{\phi^{2}(\tau)}(u_{\tau}))\Gamma_{R(\tau)}^{(N,L)}=\delta_{N,0}\delta_{L,2}(\kappa_{\tau}^{-2\tau})^{\frac{\epsilon_{L}}{2}}B_{\tau}(u_{\tau}),
\tilde{\Psi}_{y}=\tilde{\Psi}_{xx}+\left(2(\tilde{\Psi}\tilde{\Psi}^{*})+2(\tilde{\Psi}\tilde{\Psi}^{*})_{x}-(\tilde{\Psi}\tilde{\Psi}^{*})^{2}\right)\tilde{\Psi},
\chi_{1}(x,t)=-\chi_{2}(x,t)=\frac{2}{\sqrt{\pi}}tan^{-1}\left[\frac{a(t)}{cosh(bx)}\right],\quada(t)=\etasin\\omegat,\quadb=\eta\omega,\quad\eta=\sqrt{\frac{4\pim^{2}}{\omega^{2}}-1},
f_{l}(z)=\sqrt{|z|}e^{i\left({\frac{1}{2}}\mathrm{arg}\,z+\pil\right)}\sum_{n=0}^{\infty}b_{n}\left(\sqrt{|z|}e^{i\left({\frac{1}{2}}\mathrm{arg}\,z+\pil\right)}\right)^{-5n}+f_{l}^{n.p.}(z),
\partial_{\alpha}^{a}A_{\beta}^{+}=\partial_{\alpha}^{a}A^{++}=0\\\Rightarrow\\A_{\alpha}^{+}=A_{\alpha}^{+}(x,\theta^{\pma},u)\,\\\A^{++}=A^{++}(x,\theta^{\pma},u)\.
{\calP}_{R}=(q^{I},\;n^{i},\;m_{j})\left(\begin{array}{c}{l_{I}\cdot{\bfe}_{\mu}^{(R)}}\\{{\bar{k}}_{i}\cdot{\bfe}_{\mu}^{(R)}}\\{k^{j}\cdot{\bfe}_{\mu}^{(R)}}\\\end{array}\right){\bfe}_{\mu}^{R}.
\delta\Phi(z,{\bfk}_{\|},\omega)=-e^{i{\kappa}|z|}\,\int{\frac{d\omega_{\mathrm{in}}}{2\pi}}\deltaQ(\omega-\omega_{\mathrm{in}})\partial_{z}\Phi^{0}(0,{\bfk}_{\|},\omega_{\mathrm{in}}),
\mathrm{ln~det}=-\frac{e^{2}}{2\pi}\int\frac{d^{2}k}{(2\pi)^{2}}\,|\hat{B}(k)|^{2}\,\int_{0}^{1}dz\,\frac{z(1-z)}{k^{2}z(1-z)+m^{2}}+\mathrm{ln~det}_{3},
\begin{array}{ccccccc}{}&{H_{B}\psi_{s}(1/t)}&{=}&{D_{1}D_{2}\psi_{s}(1/t)}&{=}&{sD_{1}\psi_{s}(1/t)}&{=}\\{}&{sD_{1}\psi_{1-s}(t)}&{=}&{s(1-s)\psi_{1-s}(t)}&{=}&{s(1-s)\psi_{s}(1/t),}&{}\\\end{array}
x^{\mu}\,\quadu=\frac{r}{\alpha^{\prime}}\,\quad\tilde{\theta}=\alpha^{\prime}\theta\,\quad\tilde{\theta}^{i,i+1}=\alpha^{\prime}\theta^{i,i+1}\,\quadg_{\mathrm{YM}}^{2}=g(\alpha^{\prime})^{3/2}\,
\gamma^{0}=\left(\begin{array}{cc}{\sigma_{3}}&{0}\\{0}&{-\sigma_{3}}\\\end{array}\right),\,\,\gamma^{1}=\left(\begin{array}{cc}{i\sigma_{1}}&{0}\\{0}&{-i\sigma_{1}}\\\end{array}\right),\,\,\gamma^{2}=\left(\begin{array}{cc}{i\sigma_{2}}&{0}\\{0}&{-i\sigma_{2}}\\\end{array}\right).
\stackrel{P}{K}{}\!\!_{0,p}^{(\pm)}\left(\begin{array}{c}{\!\!\alpha\!\!}\\{\!\!\beta,s\!\!}\\\end{array}\right)=\stackrel{P}{\calF}{}\!\!_{0,p}^{(\pm)}\left(\begin{array}{c}{\!\!\alpha\!\!}\\{\!\!\beta\!\!}\\\end{array}\right)\stackrel{\alpha}{\eta}\!\!\lbrack\beta+s\rbrack
{\calS}_{1}\equiv\left(\begin{array}{cccc}{M_{1}}&{0}&{0}&{0}\\{0}&{M_{1}}&{0}&{0}\\{0}&{0}&{\tilde{M}_{1}}&{0}\\{0}&{0}&{0}&{\tilde{M}_{1}}\\\end{array}\right)\quad,\qquad{\calS}_{2}\equiv\left(\begin{array}{cccc}{M_{2}}&{0}&{0}&{0}\\{0}&{\tilde{M}_{2}}&{0}&{0}\\{0}&{0}&{M_{2}}&{0}\\{0}&{0}&{0}&{\tilde{M}_{2}}\\\end{array}\right)\quad.
\left(r_{c}^{2}-r^{2}\right)\partial_{r}^{2}R(r)+\left(\frac{2}{r}-\frac{4r}{r_{c}^{2}}\right)\partial_{r}R(r)+\left(\frac{\omega^{2}r_{c}^{2}}{(r_{c}^{2}-r^{2})}-\frac{\nu(\nu+1)}{r^{2}}-m^{2}\right)R(r)=0,
\left(\begin{array}{c}{L}\\{M-gxg^{-1}{\bf1}_{B}}\\\end{array}\right)=\left(\begin{array}{cc}{\tilde{g}}&{0}\\{0}&{\rho\cdotg}\\\end{array}\right)\left(\begin{array}{c}{L}\\{M-x{\bf1}_{B}}\\\end{array}\right)g^{-1}\cdot\rho^{-1}
ds_{str}^{2}=\frac{(1-\frac{\mu}{r^{2}})}{(1-\frac{\kappa\mu}{r^{2}})^{2}}d\tau^{2}+\frac{dr^{2}}{(1-\frac{\mu}{r^{2}})}+r^{2}\mathrm{cosh}^{2}t(d\Omega_{2}^{2})-r^{2}dt^{2}
ds_{\mathrm{5D}}^{2}={\frac{1}{\xi^{2}}}e^{{\frac{4k}{\sqrt{k^{2}+6}}}Q(x)}\,r^{{\frac{2}{3}}(k^{2}-3)}\,dr^{2}+e^{-{\frac{2k}{\sqrt{k^{2}+6}}}Q(x)}\,r^{2}\,h_{\mu\nu}(x)\,dx^{\mu}dx^{\nu}
\sum_{\stackrel{i,j=1}{i\neqj}}^{N}\frac{1}{\xi_{i}-\xi_{j}}\,\left(\xi_{i}\frac{\partial}{\partial\xi_{i}}-\xi_{j}\frac{\partial}{\partial\xi_{j}}\right)\,=\sum_{k=1}^{N}(N-k+1)(N-k)\tau_{k-1}\frac{\partial}{\partial\tau_{k}}\,.
F(\tau,\tau^{\prime})=\sum_{n}\int\frac{d\omega}{2\pi}\,\frac{e^{-i\omega(\tau-\tau^{\prime})}}{\omega^{2}+\omega_{n}^{2}}=\sum_{n}\frac{e^{-\omega_{n}|\tau-\tau^{\prime}|}}{2\,\omega_{n}}\qquad(\omega_{n}>0).
\vec{q}^{\vec{m}}=(q^{1})^{m_{1}}(q^{2})^{m_{2}}(\bar{q}^{1})^{m_{1}^{\prime}}(\bar{q}^{2})^{m_{2}^{\prime}},\;\;\;m_{1}+m_{2}=m_{1}^{\prime}+m_{2}^{\prime}
H_{\infty}\equivT(\lambda)a_{1}+\sum_{n>0}n\bigg[{\frac{1}{\lambda}}\big(a_{n+1}^{\dagger}-(1-\delta_{n,1})a_{n-1}^{\dagger}\big)+a_{n}^{\dagger}+\sum_{p=1}^{n-1}a_{p}^{\dagger}a_{n-p}^{\dagger}\bigg]a_{n}+\mathrm{~n<0~terms}
i\,[Q_{(1)}^{ab},T_{\mu\nu}]=-\,\frac12f\,\left(\epsilon_{\mu\lambda}\partial^{\lambda}J_{\nu}^{ab}+\epsilon_{\nu\lambda}\partial^{\lambda}J_{\mu}^{ab}\right)=-f\,\epsilon_{\mu\lambda}\partial_{\nu}^{ab}\;.
\overline{{F}}_{\sigma}^{(n)}(\zeta)^{kk_{1}\cdotsk_{2n-1}k+2l}=\prod_{a\inA_{-}^{\prime}}\oint_{C_{a}}\frac{dw_{a}}{2\pi\sqrt{-1}w_{a}}\Psi_{\sigma}^{(n)}(w|\zeta)Q^{(n)}(w|\zeta)^{kk_{1}\cdotsk_{2n-1}k+2l},
\mu_{k}^{\mathrm{out}}=\frac{\alpha_{k}}{\sqrt{2\omega_{k}^{\mathrm{out}}}}\mathrm{e}^{-i\omega_{k}^{\mathrm{out}}\eta}\,+\,\frac{\beta_{k}}{\sqrt{2\omega_{k}^{\mathrm{out}}}}\mathrm{e}^{i\omega_{k}^{\mathrm{out}}\eta}\,.
{\cal{V}}_{{\bf{k}},{\bf{k}}^{\prime}}^{\Lambda,\Lambda_{0}}\{\varphi\}=\frac{\delta^{2}{\cal{G}}_{\mathrm{ir}}^{\Lambda,\Lambda_{0}}\{\varphi\}}{\delta\varphi_{\bf{k}}\delta\varphi_{{\bf{k}}^{\prime}}}\;.
I_{1}=\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{d^{4}l}{(2\pi)^{4}}\,\frac{d^{4}q}{(2\pi)^{4}}\,\frac{2(q^{2}+q_{\mu}k_{\mu})}{k^{2}\Big(1+k^{2n}/\Lambda^{2n}\Big)\,l^{2}\Big(1+l^{2n}/\Lambda^{2n}\Big)\,q^{4}(q+k)^{4}(q+l)^{2}}
\tilde{F}(P_{-},x^{+})={\frac{1}{2L}}\langle{\calO}(P_{-},x^{+}){\calO}(-P_{-},0)\rangle\=\sum_{i}{\frac{1}{2L}}\langle0|{\calO}(P_{-})|i\ranglee^{-iP_{+}^{i}x^{+}}\langlei|{\calO}(-P_{-},0)|0\rangle\.
\begin{array}{l}{f_{lk}^{(+)}f_{cd}^{(+)}-f_{ld}^{(+)}f_{ck}^{(+)}=\|f^{(+)}\|\varepsilon_{lcm}\varepsilon_{kdn}f_{nm}^{(-)},}\\{f_{kl}^{(-)}f_{dc}^{(-)}-f_{dl}^{(-)}f_{kc}^{(-)}=\|f^{(-)}\|\varepsilon_{lcm}\varepsilon_{kdn}f_{mn}^{(+)},}\\\end{array}
\begin{array}{rclrclrcl}{\deltax}&{=}&{i\epsilon\psi_{+},}&{\delta\psi_{+}}&{=}&{-\epsilon\dot{x},}&{\delta\psi_{-}}&{=}&{-\epsilonU(x),}\\{\tilde{\delta}x}&{=}&{i\epsilon\psi_{-},}&{\tilde{\delta}\psi_{+}}&{=}&{\epsilonU(x),}&{\tilde{\delta}\psi_{-}}&{=}&{-\epsilon\dot{x},}\\\end{array}
Z_{k}(\lambda_{1}^{2},\lambda_{2}^{2},q^{2};\lambda^{2})=\int_{0}^{1}dzz^{k}ln\left(\frac{q^{2}z(1-z)+(\lambda_{1}^{2}-\lambda_{2}^{2})z-\lambda_{1}^{2}}{(-\lambda^{2})}\right).
\langle\mathrm{exp}\left(\sum_{a}\alpha_{Aa}I_{A}(\Sigma_{a})+\lambda_{A}{\calO}_{A}\right)\rangle_{A}=\sum_{\rho}\mathrm{e}^{a_{\rho}\chi+b_{\rho}\sigma}\mathrm{exp}\left(\frac{1}{2}\eta_{A\rho}\sum_{a,b}\alpha_{Aa}\alpha_{Ab}\#(\Sigma_{a}\cap\Sigma_{b})+\lambda_{A}\langle{\calO}_{A}\rangle_{\rho}\right),
\m^{2}(\beta)=m_{0}^{2}+\lambda_{0}\beta^{2-D}\sum_{k=0}^{\infty}f(D,k)\left(m(\beta)\beta\right)^{2k}\int_{m(\beta)\beta}^{\infty}d\tau\frac{\tau^{D-3-2k}}{e^{\tau}-1},
\tilde{H}_{v}-\tilde{E}_{v}=2\int_{\calB}d\sigma_{\nu}\zeta_{\mu}\nabla_{\rho}\nabla_{\sigma}\left({\frac{\partial{\tilde{L}}_{v}}{\partialR_{\mu\sigma\rho\nu}}}+{\frac{\partial{\tilde{L}}_{v}}{\partialR_{\nu\sigma\rho\mu}}}\right)~~~.
\Gamma(x_{1},...,x_{m})=\pi^{-\frac{D}{2}}\int_{z_{0}>0}\frac{d^{D+1}}{z_{0}^{D+1}}\prod_{i=1}^{m}(\frac{z_{0}}{z_{0}^{2}+(\vec{z}-\vec{x_{i}})^{2}})^{\alpha_{i}}
{\hat{\vartheta}}=\left(\begin{array}{cc}{\displaystyle\frac{\partial^{2}}{\partial\tau^{2}}}&{\displaystyle0}\\{0}&{\displaystyle\frac{\partial^{2}}{\partial\tau^{2}}}\\\end{array}\right),\;\;\;\hat{h}=\bigg(\begin{array}{cc}{\displaystyle\hat{h}_{u}}&{\displaystyle0}\\{0}&{\displaystyle\hat{h}_{v}}\\\end{array}\bigg),
\Omega_{\;B}^{A}\;=\;\left(\begin{array}{cc}{\gamma^{\mu}\,\omega_{1\mu}\;_{\;B}^{A}}&{k\,\gamma^{5}\,e^{-\sigma}\;{\binom{\omega_{1}\;_{\;B}^{A}}{\omega_{2}\;_{\;B}^{A}}}}\\{k^{*}\,\gamma^{5}\,e^{-\sigma}\,(\tilde{\omega}_{1}\;_{\;B}^{A},\;\tilde{\omega}_{2}\;_{\;B}^{A})}&{\gamma^{\mu}\,\omega_{2\mu}\;_{\;B}^{A}}\\\end{array}\right)
\frac{H_{1}^{\prime}(\rho_{b})}{H_{1}(\rho_{b})}+\frac{2(d-1)(d-2)\rho_{b}^{2d-6}-2(d^{2}-d-2)\omegaM\rho_{b}^{d-3}+(d-1)^{2}\omega^{2}M^{2}}{\rho_{b}(\rho_{b}^{d-3}-\omegaM)[2(d-2)\rho_{b}^{d-3}-(d-1)\omegaM]}=0.
\left[\iint_{\hat{R}}\rho\cdot1dxdy\right]^{2}\leq\iint_{\hat{R}}\rho^{2}dxdy\iint_{\hat{R}}dxdy\cdot1=ab\iint_{\hat{R}}\rho^{2}dxdy\mathrm{~\quad\quad.}\tag{2.15}
\zeta(s,D)\sim2{\frac{\mu^{2s}}{\Gamma(s)}}\left\{\sum_{n=0}^{\infty}{\frac{\Lambda^{n-D+2s}}{n-D+2s}}a_{n/2}^{D}(P_{\theta})+\int_{\Lambda}^{\infty}{\frac{d\xi}{\xi}}\xi^{2s}\mathrm{Tr}\left[e^{-\xi^{2}P_{\theta}(D)}\right]\right\},
X_{0}^{A{A^{\prime}}}:=i{\sqrt2}(\sigma^{A}\beta^{A^{\prime}}-\varsigma^{A}\alpha^{A^{\prime}})=-i{\sqrt2}({\bar{\sigma}}^{A^{\prime}}{\bar{\beta}}^{A}-{\bar{\varsigma}}^{A^{\prime}}{\bar{\alpha}}^{A}),
{\mathrmRe}\partial_{p^{2}}\Gamma_{+-}^{T}(p^{2})\big|_{p^{2}=\kappa_{W}^{2}}=1\quad{\mathrmRe}\partial_{p^{2}}\Gamma_{ZZ}^{T}(p^{2})\big|_{p^{2}=\kappa_{Z}^{2}}=1\quad{\mathrmRe}\partial_{p^{2}}\Gamma_{AA}^{T}(p^{2})\big|_{p^{2}=\kappa_{A}^{2}}=1
\left[\frac{E(x^{7}b,x^{60})E(x^{13}b,x^{60})E(x^{37}/b,x^{60})E(x^{43}/b,x^{60})}{E(x^{7}/b,x^{60})E(x^{13}/b,x^{60})E(x^{37}b,x^{60})E(x^{43}b,x^{60})}\right]^{N}=b^{2N}.
ie\left(2\pi\right)^{4}\delta(p^{\prime}-p-q)\Gamma_{abc}^{\mu}(q,p)=\intdxdydze^{i\,(p^{\prime}z-py-qx)}{\frac{\delta^{3}\Gamma}{\delta\bar{\psi}_{c}(z)\delta\psi_{b}(y)\deltaA_{a}^{\mu}(x)}}\,.
\begin{array}{lll}{A(1,2,3,4)}&{=}&{-g_{s}\alpha^{\prime}(\lambda^{1}\lambda^{2}\lambda^{3}\lambda^{4}+\lambda^{4}\lambda^{3}\lambda^{2}\lambda^{1})\,\int_{0}^{1}dx\,x^{-1-\alpha^{\prime}s}(1-x)^{-1-\alpha^{\prime}t}\frac{1}{[F(1-x).F(x)]^{1/2}}}\\{}&{}&{\times\left[\overline{{u}}^{(1)}\gamma_{\mu}u^{(2)}\overline{{u}}^{(4)}\gamma^{\mu}u^{(3)}\right]\sume^{-S_{cl}(x)}}\\\end{array}
\left[A_{\{(n_{i})\}}^{\{r,r_{i}\}},V_{\{(n_{j})\}}^{\{\lambda,\lambda_{j}\}}\right]=\frac{1}{(2\pii)^{2}}\oint_{0}dz\oint_{z}\frac{d\xi}{\xi}U_{\{(n_{i})\}}^{\{r,r_{i}\}}(z)U_{\{(n_{j})\}}^{\{\lambda,\lambda_{j}\}}(\xi)
\frac{N}{4g^{2}}F_{\mu\nua}F_{\mu\nua}+\frac{N}{g^{2}}\left(iB_{a}(\partial_{\mu}A_{\mua})+\frac{\alpha}{2}B_{a}B_{a}+\psi_{a}(\partial_{\mu}(\mathrm{D}_{\mu}\phi)_{a})\right)
S_{f}=-\frac{1}{2g^{2}}Tr\left(\sum_{i=0}^{3}\overline{{\psi_{i}}}\Gamma^{\mu}[A_{\mu},\psi_{i}]+2\sum_{i=1}^{3}\overline{{(\psi_{i})^{c}}}\bar{\Gamma}^{(i)}[B_{i}^{\dagger},\psi_{0}]+\sum_{i,j,k=1}^{3}|\epsilon_{ijk}|\overline{{(\psi_{i})^{c}}}\Gamma^{(j)}[B_{j},\psi_{k}]+h.c.\right)
G=ig_{\mu\bar{\nu}}\,dz^{\mu}\wedged\bar{z}^{\bar{\nu}}={\frac{i}{\sqrt{F_{2}F_{6}}}}h_{\mu\bar{\nu}}\,dz^{\mu}\wedged\bar{z}^{\bar{\nu}}\equiv{\frac{K}{\sqrt{F_{2}F_{6}}}}
\delta\psi_{\mu}=\partial_{\mu}\epsilon-{\frac{1}{4}}\omega_{\mu}^{ab}\gamma_{a}\gamma_{b}\epsilon+{\frac{i}{288}}\left(\gamma_{[\mu}\gamma^{\alpha}\gamma^{\beta}\gamma^{\gamma}\gamma^{\delta]}-8\delta_{\mu}^{\alpha}\gamma^{[\beta}\gamma^{\gamma}\gamma^{\delta]}\right)F_{\alpha\beta\gamma\delta}\epsilon
\mathcal{L}_{fermion}^{kin}=\intd^{4}x\left[2\left({\bar{\psi}}_{\mu}^{A}\,\gamma_{\nu}\nabla_{\rho}\psi_{A\vert\mu}+\mathrm{h.c}\right)-\mathrm{i}\,\sqrt{-g}\,\frac{1}{24}\left({\bar{\chi}}^{ABC}\gamma^{\mu}\,\nabla_{\mu}\chi_{ABC}-\mathrm{h.c.}\right)\right]
Z_{cs}=N\int{\calD}a_{\mu}{\calD}c{\calD}\bar{c}\mathrm{exp}\left(i\intd^{3}x(\frac{1}{2}\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}+a_{\mu}j_{\mu}+i\bar{c}\partial^{2}c+\frac{1}{2\rho}(\partial_{\mu}a_{\mu})^{2})\right)\;.
M_{N}=\left(\begin{matrix}{\phi_{1}}&{y_{1}}&{0}&{\dots}&{z}\\{1}&{\phi_{2}}&{y_{2}}&{\dots}&{0}\\{0}&{1}&{\phi_{3}}&{\dots}&{0}\\{.}&{.}&{.}&{\dots}&{y_{N-1}}\\{y_{0}z^{-1}}&{.}&{.}&{1}&{\phi_{N}}\\\end{matrix}\right)
A\left(\bar{x},x_{2n},p_{2n}\right)=\sum_{n_{1},n_{2},\cdots\geq0}\phi^{\left(n_{1},n_{2},\cdots\right)}\left(x_{0}\right)\,\,A_{\left(n_{1},n_{2},\cdots\right)}\left(x_{2n},p_{2n}\right),