harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{rcl}{\left({\calV}_{m,1}\otimes{\calV}_{1,2}\right)_{\mathrmf}}&{=}&{{\calV}_{m,2}\,,}\\{\left({\calV}_{m,2}\otimes{\calV}_{1,2}\right)_{\mathrmf}}&{=}&{{\calR}_{m,1}\,,}\\\end{array}
e_{0}^{a}=\left(\begin{array}{c}{gz}\\{0}\\{0}\\{0}\\\end{array}\right)\qquade_{1}^{a}=\left(\begin{array}{c}{0}\\{1}\\{0}\\{0}\\\end{array}\right)\qquade_{2}^{a}=\left(\begin{array}{c}{0}\\{0}\\{1}\\{0}\\\end{array}\right)\qquade_{3}^{a}=\left(\begin{array}{c}{0}\\{0}\\{0}\\{1}\\\end{array}\right)
\phi_{\chi_{j}\otimes\chi_{\bar{j}}}^{\alpha}\;\star\;\phi_{\chi_{k}\otimes\bar{\chi}_{k}}^{\beta}\=\\sumC\left(\chi_{j},\chi_{\bar{j}},\alpha\bigm|\chi_{k},\chi_{\bar{k}},\beta\bigm|\chi_{l},\chi_{\bar{l}},\gamma\right)\,\phi_{\chi_{l}\otimes\chi_{\bar{l}}}^{\gamma}\.
\dot{\varrho}_{m}+3{\frac{\dot{a}}{a}}\left(\varrho_{m}+{\frac{\wp_{m}}{c^{2}}}\right)=-\dot{\varrho}_{\delta\sigma}-\varrho{\frac{\dot{G}_{4}}{G_{4}}}+{\frac{3kc\dot{c}}{4\piG_{4}a^{2}}}.
e^{-2\hat{\phi}}=1+{\frac{2\tilde{m}}{r}}\,\qquadK=1+\sum_{1}^{s}{\frac{2\hat{m}}{r}}\,\qquadA_{n}={\frac{2q_{n}}{r}}\,\qquadn=4,\dots8,\qquadr^{2}\equiv\vec{x}^{2}.
R_{\kappa\lambda\mu\nu}^{4}=g_{\\kappa}^{4\\kappa^{\prime}}g_{\\lambda}^{4\\lambda^{\prime}}g_{\\mu}^{4\\mu^{\prime}}g_{\\nu}^{4\\nu^{\prime}}R_{\kappa^{\prime}\lambda^{\prime}\mu^{\prime}\nu^{\prime}}+K_{\kappa\mu}K_{\lambda\nu}-K_{\kappa\nu}K_{\lambda\mu}.
\hat{H}=\int\left[-\left(\hat{\psi}_{\alpha}^{+}\frac{\nabla^{2}}{2m}\hat{\psi}_{\alpha}\right)+\frac{\lambda}{2}\left(\hat{\psi}_{\beta}^{+}\left(\hat{\psi}_{\alpha}^{+}\hat{\psi}_{\alpha}\right)\hat{\psi}_{\beta}\right)\right]dV,
\frac{1}{(q)_{\infty}}\sum_{r=-\infty}^{\infty}(-1)^{r}q^{r((2k+1)r+2k-2i+1)/2}=\displaystyle\sum_{n_{1}\geq\cdots\geqn_{k-1}\geq0}\frac{q^{n_{1}^{2}+\cdots+n_{k-1}^{2}+n_{i}+\cdots+n_{k-1}}}{(q)_{n_{1}-n_{2}}\cdots(q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}}}.
S=\int{dx^{0}}{dx^{1}}\sqrt{-g}e^{-\overline{{\phi}}}\left[R+\partial_{\alpha}\overline{{\phi}}\partial^{\alpha}\overline{{\phi}}+\frac{1}{8}\mathrm{Tr}\left(\partial_{\alpha}M^{-1}\partial^{\alpha}M\right)\right].
I={\frac{1}{16\piG}}\intd^{4}x\sqrt{-g}\biggl\{R-{\frac{1}{2n^{2}}}\sum_{i=1}^{n}\sum_{j=i+1}^{n}(\partial\sigma_{i}-\partial\sigma_{j})^{2}-{\frac{1}{n}}\sum_{i=1}^{n}e^{-\sigma_{i}}F_{(i)}^{2}\biggr\},
\begin{array}{c}{S\,[x+\deltax\,|\,h_{m(k)}+\deltah_{m(k)}]=S\,[x\,|\,h_{m(k)}]+o(e_{k}^{2})}\\{\deltah_{m(k)}(x)=\epsilon_{m(k-1);\,m}}\\{\deltax^{m}(\tau)=(k-1)\,e_{k}\,{\epsilon^{m}}_{n_{1}...n_{k-2}}\dot{x}^{n_{1}}...\dot{x}^{n_{k-2}}\,({-\frac{\dot{x}^{2}}{m^{2}}})^{1-k/2},}\\\end{array}
C\frac{1}{C^{\dagger}C}C^{\dagger}C\frac{1}{C^{\dagger}C}C^{\dagger}=C\frac{C^{\dagger}C}{C^{\dagger}C}\frac{1}{C^{\dagger}C}C^{\dagger}=C\frac{1}{C^{\dagger}C}C^{\dagger}.
\left\{f(X;P),\;Re\Sigma_{R}(X;P)\right\}=\frac{\partialf(X;P)}{\partialX_{\mu}}\frac{\partialRe\Sigma_{R}(X;P)}{\partialP^{\mu}}-\frac{\partialf(X;P)}{\partialP^{\mu}}\frac{\partialRe\Sigma_{R}(X;P)}{\partialX_{\mu}}.
C_{\nu\lambda}^{\rho}C_{\mu\rho}^{\sigma}e_{\sigma}+[C_{\lambda\nu}^{\rho}C_{\mu\rho}^{\sigma}e_{\sigma}-C_{\mu\lambda}^{\rho}C_{\rho\nu}^{\sigma}e_{\sigma}]=C_{\mu\nu}^{\rho}C_{\rho\lambda}^{\sigma}e_{\sigma}.
\left\{\begin{array}{rcl}{\delta_{1\varepsilon}x^{a}}&{=}&{2\varepsilonp^{a}}\\{\delta_{1\varepsilon}p^{a}}&{=}&{0}\\\end{array}\right\}\sim\left\{\begin{array}{rcl}{\delta_{2\mu}x^{a}}&{=}&{\muj^{a}}\\{\delta_{2\mu}p^{a}}&{=}&{0}\\\end{array}\right\}
\Psi=\left[\begin{array}{c}{\mathrm{L}_{N}^{\left|m\right|}F_{m}}\\{0}\\{c_{2}\mathrm{L}_{N}^{\left|m\right|}F_{m}}\\{d_{2}\mathrm{L}_{N}^{\left|m\right|-1}F_{m+1}}\\\end{array}\right]
\left(\partial_{r}^{2}+{\frac{\partial_{r}}{r}}-{\frac{(j-1/2)^{2}}{r^{2}}}-\kappa^{2}-2e^{2}v^{2}f^{2}\right)h_{4}-i(\kappa+2eN)2e^{2}v^{2}f^{2}({\frac{h_{2}}{\sqrt{2}evf}})=0.
\int_{0}^{\infty}{\frac{r^{2}d\rho}{R_{c}}}=\int_{0}^{1}\psi^{2}y^{p/2-1}(1-y)^{1/2-1}dy<M\int_{0}^{1}y^{p/2-1}(1-y)^{1/2-1}dy=MB(p/2,1/2),
I_{n}(p_{\phi})={\frac{1}{(n-1)!}}{\frac{\Gamma(n\epsilon)}{\epsilon^{n-1}}}{\frac{\Gamma(1-\epsilon)^{n+1}}{\Gamma(2-(n+1)\epsilon)}}\times{\frac{1}{(1-2\epsilon)\cdots(1-n\epsilon)}}\times{\frac{1}{[p_{\phi}^{2}]^{n\epsilon}}}
M=\left(\begin{matrix}{{\frac{{q^{2}}}{{{\left(1+{q^{2}}\right)}^{2}}}}}&{0}&{0}&{0}\\{0}&{{\frac{{q}}{{{2\left(1+{q^{2}}\right)}}}}}&{0}&{0}\\{0}&{0}&{{\frac{{q}}{{{2\left(1+{q^{2}}\right)}}}}}&{0}\\{0}&{0}&{0}&{{\frac{{q^{2}}}{{{\left(1+{q^{2}}\right)}^{2}}}}}\\\end{matrix}\right)
2G^{iji^{\prime}j^{\prime}}=(h^{ii^{\prime}}h^{jj^{\prime}}+h^{ij^{\prime}}h^{ji^{\prime}}-\frac{2}{D}h^{ij}h^{i^{\prime}j^{\prime}})+C~h^{ij}h^{i^{\prime}j^{\prime}},
D^{\mu,\nu}(\varpi)=\prod_{t=1}^{\nu-\mu}{\frac{\sqrt{\Gamma\bigl((\varpi+t-1)h/\pi\bigr)}}{\sqrt{\Gamma((-\varpi-t)h/\pi)}}}\prod_{r=1}^{\mu}\Gamma\bigl((\varpi-r)h/\pi)\bigr)\prod_{s=1}^{\nu}\Gamma\bigl((-\varpi-s)h/\pi)\bigr),
S=\intd^{D}x\left\{\frac{1}{2}\left(\partial_{\mu}\mathbf{\Phi^{a}(x)}\right)^{2}+\frac{m^{2}}{2}\mathbf{\Phi^{a}(x)\Phi^{a}(x)}+\frac{g}{4!}\left({\mathbf{\Phi^{a}(x)\Phi^{a}(x)}}\right)^{2}\right\}
\Pi_{i}\left(r,K\right)=8(T_{i}^{\mu\nu,\,\alpha\beta}T_{_j\,\mu\nu,\,\alpha\beta})^{-1}\bar{P}_{j}\left(r,K\right)\equiv8\left(T_{ij}\right)^{-1}\bar{P}_{j}\left(r,K\right),
B_{0i}(T=0)=\epsilon_{0ij}p^{j}\left(\frac{1}{3\pi}\frac{|\kappa|}{\kappa}\frac{(1+\frac{1}{2}\frac{m_{\sigma}}{M})}{(1+\frac{m_{\sigma}}{M})^{2}}-\frac{v^{2}}{4\pi\kappa}\tilde{p}\right)+{\cal{O}}(\tilde{p}^{\phantom{a}2}).
\left(\gamma_{\alpha}^{Fg_{1},g_{2},g_{3}}\right)\,\left(\gamma_{\alpha}^{Rg_{1},g_{2},g_{3}}\right)^{-1}\,\left(h_{\alpha}^{g_{1}g_{2},g_{3}}\right)\,\left(g_{3}^{*}h_{\alpha}^{g_{1},g_{2}}\right)\:=\:\left(h_{\alpha}^{g_{1},g_{2}g_{3}}\right)\,\left(h_{\alpha}^{g_{2},g_{3}}\right)
L=\left(\begin{matrix}{p_{1}}&{ime^{(q_{1}-q_{2})/2}}&{-ime^{(-q_{1}-q_{2})/2}}&{0}\\{-i\,me^{(q_{1}-q_{2})/2}}&{p_{2}}&{i\,me^{q_{2}}}&{-ime^{(-q_{1}-q_{2})/2}}\\{ime^{(-q_{1}-q_{2})/2}}&{-i\,me^{q_{2}}}&{-p_{2}}&{i\,me^{(q_{1}-q_{2})/2}}\\{0}&{ime^{(-q_{1}-q_{2})/2}}&{-i\,me^{(q_{1}-q_{2})/2}}&{-p_{1}}\\\end{matrix}\right),
\begin{array}{rcl}{Y_{2}A_{1}}&{=}&{L_{2}^{+}SL_{2}^{-}A_{1}}\\{}&{=}&{L_{2}^{+}A_{1}SL_{2}^{-}R_{21}}\\{}&{=}&{A_{1}R_{12}L_{2}^{+}SL_{2}^{-}R_{21}}\\{}&{=}&{A_{1}R_{12}Y_{2}R_{21},}\\\end{array}
U_{k}=\left(\begin{array}{lllllll}{0}&{\cdots}&{0}&{0}&{0}&{\cdots}&{0}\\{\vdots}&{}&{\vdots}&{\vdots}&{\vdots}&{}&{\vdots}\\{0}&{\cdots}&{0}&{0}&{0}&{\cdots}&{0}\\{0}&{\cdots}&{0}&{V}&{0}&{\cdots}&{0}\\{0}&{\cdots}&{0}&{0}&{0}&{\cdots}&{0}\\{\vdots}&{}&{\vdots}&{\vdots}&{\vdots}&{}&{\vdots}\\{0}&{\cdots}&{0}&{0}&{0}&{\cdots}&{0}\\\end{array}\right)_{k,k+1}
H^{1}({\mathbbP}^{1}\timesT^{2},{\calO}_{{\mathbbP}^{1}\timesT^{2}}(3\sigma|_{{\mathbbP}^{1}\timesT^{2}}-3F))|_{b_{-3}^{(1)}}=H^{0}({\calS},{\calO}_{{\calS}}(1))^{*}
\begin{aligned}{\nabla_{m}K_{n}}&{{}=\frac{1}{4}H_{ml_{1}l_{2}}\chi_{n}{^{l_{1}l_{2}}},}\\{\nabla_{m}\omega_{n_{1}n_{2}}}&{{}=\frac{1}{4}H_{ml_{1}l_{2}}{*\chi}_{n_{1}n_{2}}{^{l_{1}l_{2}}},}\\{\nabla_{m}\chi_{n_{1}n_{2}n_{3}}}&{{}=-\frac{3}{2}H_{m[n_{1}n_{2}}K_{n_{3}]}-\frac{1}{4}H_{ml_{1}l_{2}}{*\omega}_{n_{1}n_{2}n_{3}}{^{l_{1}l_{2}}}.}\\\end{aligned}
{\cal{H}}_{\eta_{o}}(u,v)=\left(\frac{v}{u}\right)^{\frac{1}{2}\eta_{o}}\sum_{n=0}^{\infty}\frac{v^{n}}{n!}\frac{({\textstyle{\frac{1}{2}}}\eta_{o})_{n}^{4}}{(\eta_{o})_{2n}(\eta_{o}+1-\mu)_{n}}{}_{2}F_{1}({\textstyle{\frac{1}{2}}}\eta_{o}+n,{\textstyle{\frac{1}{2}}}\eta_{o}+n;\eta_{o}+2n;1-\frac{v}{u}).
\begin{array}{lll}{f_{0}}&{\longleftrightarrow}&{f_{0}^{-1}\;,}\\{f_{i}f_{0}^{-1/2}}&{\longleftrightarrow}&{f_{k-i}f_{0}^{-1/2}\;\;\;i=1,\ldots,k-1\;.}\\\end{array}
{\calT}={\frac{f_{\pi}^{2}}{2}}{\calR}_{0}\cdot{\calR}_{0}-{\frac{1}{2e^{2}}}\biggl[{\calR}_{0}\cdot{\calR}_{\mathrm{i}}\;{\calR}_{0}\cdot{\calR}_{\mathrm{i}}-{\calR}_{0}\cdot{\calR}_{0}\;{\calR}_{\mathrm{i}}\cdot{\calR}_{\mathrm{i}}\biggr]
T^{\mu}\_{\nu}(r)=\left(\begin{array}{cc}{T_{\alpha}^{\alpha}-\Omega(r)^{-1}H(r)}&{0}\\{0}&{\Omega(r)^{-1}H(r)\}\\\end{array}\right)+\Omega^{-1}\left(\begin{array}{cc}{-\beta}&{-\alpha}\\{\alpha}&{\beta\}\\\end{array}\right).
R_{21}^{t_{1}}(\zeta_{2}/\zeta_{1})=\sigma_{1}^{x}R_{12}(x^{-1}\zeta_{1}/\zeta_{2})\sigma_{1}^{x}=-\sigma_{1}^{y}R_{12}(-x^{-1}\zeta_{1}/\zeta_{2})\sigma_{1}^{y}.
\renewcommand{\arraystretch}{1.2}\begin{array}{rcl}{L_{-1}\mu}&{=}&{\chi_{\lambda-1}^{-}\chi_{-\lambda}^{+}\mu\,,}\\{L_{-1}^{2}\mu}&{=}&{(1-\lambda)\chi_{\lambda-2}^{-}\chi_{-\lambda}^{+}\mu-\lambda\chi_{-1-\lambda}^{+}\chi_{\lambda-1}^{-}\mu\,.}\\\end{array}
G_{f}(x,x^{\prime})-G_{g}(x,x^{\prime})=\int\!dldl^{\prime}\!dk\2{\calR}e\left\lbrack\beta_{lk}\beta_{l^{\prime}k}^{*}g_{l}^{*}(x)g_{l^{\prime}}(x^{\prime})-\alpha_{lk}\beta_{l^{\prime}k}^{*}g_{l}(x)g_{l^{\prime}}(x^{\prime})\right\rbrack
g_{\mu\nu}=\left(\begin{matrix}{\epsilon}&{0}&{0}\\{0}&{V\frac{1}{\tau_{2}}}&{V\frac{\tau_{1}}{\tau_{2}}}\\{0}&{V\frac{\tau_{1}}{\tau_{2}}}&{V\frac{\tau_{1}^{2}+\tau_{2}^{2}}{\tau_{2}}}\\\end{matrix}\right),
\pi_{0}(x)={\frac{1}{\sqrt{2\pi}}}\left(p-{\frac{i}{\sqrt{2}}}\sum_{k=1}^{\infty}\sqrt{k}[(a_{(+)k}-a_{(-)k}^{\dagger})e^{-ikx}-(a_{(+)k}^{\dagger}-a_{(-)k})e^{ikx}]\right),
H_{+}\left(A\psi_{n}^{(-)}\right)=AA^{+}\left(A\psi_{n}^{(-)}\right)=A\left(A^{+}A\psi_{n}^{(-)}\right)=AH_{-}\left(\psi_{n}^{(-)}\right)=E_{n}^{-}\left(A\psi_{n}^{(-)}\right)\;.
\left(\begin{array}{c}{\xi}\\{\eta}\\\end{array}\right)=\left(\begin{array}{cc}{\mu\kappa/N_{+}}&{(m_{-}^{2}-\mu^{2})/N_{+}}\\{(m_{+}^{2}-\mu^{2})/N_{-}}&{\mu\kappa/N_{-}}\\\end{array}\right)\left(\begin{array}{c}{\varphi}\\{{(\pi_{\chi}+{\frac{\kappa}{2}}\varphi)/\mu}}\\\end{array}\right)
\frac{1}{2}{\calS}_{RN}=\beta_{RN}\Lambda\left[\frac{r_{s}^{3}-r_{h}^{3}}{12}\right]\,+\,\frac{q^{2}}{4}\beta_{RN}\left(\frac{1}{r_{h}}-\frac{1}{r_{s}}\right)+4\pi\sigmar_{s}^{2}\beta_{RN}\sqrt{f(r_{s})}
\int_{\Lambda_{1}}^{b}d\lambda\frac{1}{\sqrt{2(E_{F}^{(0)}-V)}}=\Analytic\terms\+\int_{\Lambda_{1}}^{b}d\lambda\frac{d}{d\lambda}\left(\frac{1}{\dot{V}}\right)\sqrt{2(E_{F}^{(0)}-V)}
F^{\hat{\nabla}}=\left(\begin{array}{ccc}{0}&{f_{13}\vartheta}&{-f_{12}\vartheta}\\{-f_{13}\vartheta}&{0}&{f_{23}\vartheta^{2}}\\{f_{12}\vartheta}&{-f_{23}\vartheta}&{0}\\\end{array}\right)+\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{0}&{\vartheta}\\{0}&{-\vartheta}&{0}\\\end{array}\right)\,.
E_{0}=\Bigl({\frac{4RT\lambda^{2}}{\eta}}\Bigr)^{{\frac{1}{2}}}\sum_{n\neq0,1}{\frac{1}{|n|^{{\frac{1}{2}}}}}=\Bigl({\frac{4RT\lambda^{2}}{\eta}}\Bigr)^{{\frac{1}{2}}}\Bigl[2\,\zeta(-1/2)-1\Bigr].
V=\begin{array}{l}{\left\langle+,-\right|}\\{\left\langle0,0\right|}\\{\left\langle-,+\right|}\\\end{array}\left(\begin{array}{lll}{-q^{-1}}&{q^{-1/2}}&{\1}\\{-q^{-1/2}}&{\1}&{q^{1/2}}\\{\1}&{-q^{1/2}}&{-q}\\\end{array}\right).
\delta_{\varepsilon}S_{2}^{II}=\intd^{2}\sigma\,\varepsilon^{ij}\,\left(\begin{array}{c}{\bar{\varepsilon}^{1}\gamma^{\mu\rho}\partial_{i}\theta^{1}\bar{\theta}^{1}\gamma_{\mu\nu}\partial_{j}\theta^{1}}\\{+\bar{\varepsilon}^{2}\gamma^{\mu\rho}\partial_{i}\theta^{2}\,\bar{\theta}^{2}\gamma_{\mu\nu}\partial_{j}\theta^{2}}\\\end{array}\right)\lambda_{\rho}\,\lambda^{\nu}.
\Pi_{55}^{AA^{\prime}}(m^{2})=\frac{1}{8}g^{2}\delta^{AA^{\prime}}C_{2}({\calG})\mu^{4-d}\int\frac{d^{d}q}{{(2\pi)^{d}}}\sum_{l=-\infty}^{+\infty}\left(\frac{-4d}{q^{2}+\frac{l^{2}}{{R^{2}}}}+2\frac{2q^{2}+\frac{d(m-2l)^{2}+2l(m-l)}{R^{2}}}{(q^{2}+\frac{l^{2}}{R^{2}})(q^{2}+\frac{(m-l)^{2}}{{R^{2}}})}\right).
J^{(4)}(4;1,1,1,1)={\textstyle{\frac{1}{12}}}\;\mathrm{i}\pi^{2}\;\frac{\Omega^{(4)}}{V^{(4)}}=\frac{2\;\mathrm{i}\pi^{2}}{m_{1}m_{2}m_{3}m_{4}}\;\frac{\Omega^{(4)}}{\sqrt{D^{(4)}}}.
\widehat{\mathcal{L}}=\dot{q}^{i}p_{i}+ik\,\dot{\hat{\chi}}_{+}^{a}\hat{\chi}_{+a}+ik\,\dot{\hat{\chi}}_{-}^{a}\hat{\chi}_{-a}-h^{-}L_{--}h^{+}L_{+}-i\psi^{-}G_{--}i\psi^{+}G_{+}
f_{+}=\left[\frac{\gamma\left(Qb\right)\gamma\left(\frac{Q-2\alpha_{1}}{2}b+\frac{1}{2}\right)}{\gamma\left(\frac{Qb}{2}\right)\gamma\left(\alpha_{1}b+\frac{1}{2}\right)}\right]^{1/2}\frac{\gamma\left(\frac{2\alpha_{2}-Q}{2}b\right)C_{(\alpha_{1}+b/2),(\alpha_{2}),(\alpha_{3})}}{\gamma\left(\frac{2\delta_{1}-Q}{4}b+\frac{1}{4}\right)\gamma\left(\frac{Q-2\delta_{2}}{4}b+\frac{1}{4}\right)},
ds^{2}=\frac{\displaystyle\sum_{i,j=0}^{n-1}\eta_{ij}d\chi^{i}d\chi^{j}}{\left[1+\left(\chi^{0}-\displaystyle\sqrt{\sum_{i=1}^{n-1}\chi^{i}\chi^{i}}\right)^{2}\right]\left[1+\left(\chi^{0}+\displaystyle\sqrt{\sum_{i=1}^{n-1}\chi^{i}\chi^{i}}\right)^{2}\right]}~.
f_{D_{n}}=\left(\frac{y^{2}+x_{0}^{2}+x_{1}^{2}}{(y^{2}+x_{0}^{2})^{2}+(y^{2}+x_{0}^{2})P+Q^{2}}\right)^{\frac{n-4}{4}}\,\frac{K_{\frac{n-4}{2}}(S_{1})e^{-iS_{2}}}{\sqrt{y^{2}+x_{0}^{2}}}\,,
\left|R_{N}\right|\le\left|R_{0N}\right|+\left|R_{1N}^{\mathrm{(cut)}}\right|+2\left|R_{1N}^{(1)}\right|_{\mathrm{(I)}}+\left|\widetildeR_{1N}^{(0)}\right|+2\left|\widetildeR_{1N}^{(1)}\right|_{\mathrm{(II)}}+\left|\widetildeR_{1N}^{(1)}\right|_{\mathrm{(III)}}+\left|\widetildeR_{1N}^{(2)}\right|_{\mathrm{(III)}},
\dot{p}_{\mu}\circ\deltaq^{\mu}=\frac{1}{2}\dot{q}^{\mu}(\partial_{\alpha}g_{\mu\nu}\deltaq^{\alpha})\dot{q}^{\nu}-\deltaq^{\mu}\partial_{\mu}U_{q}+\frac{1}{2}[\deltaq^{\mu},\dot{g}_{\mu}]
\begin{aligned}{n_{1}}&{{}=e_{8}-e_{9},}\\{o_{1}}&{{}=3l-e_{1}-\dots-e_{7}-2e_{8}=f-n_{1},}\\{n_{2}}&{{}=l-e_{7}-e_{8}-e_{9},}\\{o_{2}}&{{}=2l-e_{1}-\dots-e_{6}=f-n_{2}.}\\\end{aligned}
\mathcal{D}_{z}=\left(\begin{array}{cc}{\mathbf{a}+\mathbf{b}\mathbf{x}}&{\mathbf{0}}\\{\mathbf{0}}&{\mathbf{a}^{\prime}+\mathbf{b}^{\prime}\mathbf{x}^{\prime}}\\\end{array}\right)\equiv\left(\begin{array}{cc}{D_{z}}&{\mathbf{0}}\\{\mathbf{0}}&{D_{z}^{\prime}}\\\end{array}\right)\,,
\psi(x)=\sum_{r=\pm1}\sum_{\vec{p},p^{02}={\vec{p}}^{2}}\frac{1}{\sqrt{(2\pi)^{3}}}(b^{(0)}{}_{\vec{p},r}u_{\vec{p},r}e^{-ipx}+d^{(0)}{}_{\vec{p},r}v_{\vec{p},r}e^{ipx}).
W_{\alpha}^{2}\bar{W}_{\dot{\alpha}}^{2}\mapstoO_{4}={\frac{1}{4}}[(F^{2})^{2}-(F\tilde{F})^{2}]=F_{\mu\rho}F_{\nu\rho}F_{\mu\sigma}F_{\nu\sigma}-{\frac{1}{4}}(F^{2})^{2}
\Gamma[F,V]=\int_{0}^{\infty}{\frac{\mathrm{d}T}{T}}[4\piT]^{-d/2}\mathrm{e}^{-m^{2}T}{\sum_{n=1}^{\infty}}{\frac{(-T)^{n}}{n!}}\!\!\int\mathrm{d}x_{0}\;\mathrm{tr}\;O_{n}\;.
\left[\bar{\Xi}_{\lambda_{2}}^{\lambda_{1}}\otimesl_{\dot{J}}^{\dot{I}}\otimes1,1\otimesr_{\dot{L}}^{\dot{K}}\otimes\Xi_{\lambda_{4}}^{\lambda_{3}}\right]=\bar{\Xi}_{\lambda_{2}}^{\lambda_{1}}\otimes\left[l_{\dot{J}}^{\dot{I}},r_{\dot{L}}^{\dot{K}}\right]\otimes\Xi_{\lambda_{4}}^{\lambda_{3}},
{\calR}_{\mathrm{i}}^{\mathrm{a}}(A)={\frac{\kappa}{|\vec{x}-\vec{R}_{1}|^{3}}}\bigl(\delta^{\mathrm{ia}}-3\hat{r}_{1}^{\mathrm{i}}\hat{r}_{1}^{\mathrm{a}}\bigr)+O\Biggl({\frac{1}{|\vec{x}-\vec{R}_{1}|^{4}}}\Biggr)
H_{0_{B}}=H_{0}+\intd^{3}x\left(\eta_{1a}^{i}\mathcal{P}_{2i}^{a}-\eta_{1a}\mathcal{P}_{2}^{a}-\eta_{2a}\partial^{i}\mathcal{P}_{2i}^{a}+\eta_{2a}^{i}\partial_{i}\mathcal{P}_{2}^{a}\right)\equiv\intd^{3}xh_{0_{B}},
0=\int_{-{\infty}}^{\infty}dx^{-}\left[{\theta}_{+}^{\;\;+}\right]_{x^{+}=-{\infty}}^{x^{+}={\infty}}-\int_{-{\infty}}^{\infty}dx^{+}\left[B\frac{1}{m^{2}-n_{-}{\partial}_{+}^{\;\;2}}{\partial}_{+}C\right]_{x^{-}=-{\infty}}^{x^{-}={\infty}},
\lambda_{(0)\alpha}^{A}=\frac{1}{2}{\overline{F}}_{\mu\nu}\sigma_{\alpha}^{\mu\nu\,\beta}\frac{1}{\sqrt{\rho_{0}}}\left(\rho_{0}\eta_{\beta}^{A}+(x-x_{0})_{\mu}\sigma_{\beta{\dot{\beta}}}^{\mu}{\overline{\xi}}^{\dot{\beta}A}\right)\,.
G_{{\calQ}^{4}}(x_{p})=\frac{3^{4}}{4\pi^{10}}\,g^{8}\,e^{-\frac{8\pi^{2}}{g^{2}}+i\theta}\,\int\frac{d\rho_{0}d^{4}x_{0}}{\rho_{0}^{5}}\:x_{12}^{4}x_{34}^{4}\,\prod_{p=1}^{4}{\left[\frac{\rho_{0}}{{\rho_{0}^{2}+(x_{p}-x_{0})^{2}}}\right]}^{4}.
g(r)=A(r)=C_{1}^{(1)}\sum_{n=0}^{\infty}\frac{(\alpha)_{n}(\beta)_{n}}{n!\left(\frac52\right)_{n}}\,r^{2n}+C_{2}^{(1)}r^{-3}\sum_{n=0}^{\infty}\frac{\left(\alpha-\frac32\right)_{n}\left(\beta-\frac32\right)_{n}}{n!\left(-\frac12\right)_{n}}\,r^{2n},\qquadk=+1.
E_{i}={\frac{1}{\kappa\left(1-{\frac{e^{2}}{\kappa^{2}}}|\Phi|^{2}\right)}}\left(-{\frac{ie}{2}}\right)\epsilon_{ij}\left[\Phi^{\ast}\nabla_{j}\Phi-\Phi\left(\nabla_{j}\Phi\right)^{\ast}\right]\,.
\phi_{I}(x,\tau)=e^{i\mathrm{K}_{0}\tau}\phi_{S}(x)e^{-i\mathrm{K}_{0}\tau}=e^{i\mathrm{K}_{0}\tau}e^{-i\mathrm{K}\tau}\phi_{H}(x,\tau)e^{i\mathrm{K}\tau}e^{-i\mathrm{K}_{0}\tau}
<PO_{k}O_{l}>=\nonumber\,-\frac{1}{\pi}\mathrm{Im}\sum_{n,m}\frac{<n|O_{k}|m><m|O_{l}|n>}{(E_{n}-t_{0})(E_{m}-t_{0})}\frac{(E_{n}-E_{m})^{2}}{p^{2}+(E_{m}-E_{n})^{2}}.
S=\intd^{6}x\left[{\frac16}F_{{\mu}{\nu}{\sigma}}(A)F^{{\mu}{\nu}{\sigma}}(A)+{\frac12}{\lambda}_{{\mu}{\nu}}{\calF}^{{\mu}{\rho}{\sigma}}(A){{\calF}^{\nu}}_{{\rho}{\sigma}}(A)\right],
R(z)\;=\;{\frac{z}{4\beta}}-{\frac{d}{2}}+{\frac{d-2}{2}}\int_{a}^{b}dz^{\prime}{\frac{\rho(z^{\prime})}{z+z^{\prime}}}\;=\;{\frac{z}{4\beta}}-{\frac{d}{2}}-{\frac{d-2}{2}}f(-z)\;.
I=\int_{C}d\tau\;\eta_{ab}\;[p^{a}\partial_{\tau}q^{b}+t^{\mu}(p^{a}e_{\;\mu}^{b}+j^{a}\omega_{\;\mu}^{b})]+\lambda_{1}(p^{2}-m^{2})+\lambda_{2}(W^{2}-m^{2}s^{2})
J^{\mu}=-\sqrt{-\mathrm{det}G}\;G^{\alpha\beta}\;(\Phi_{,\alpha}+A_{\nu}X_{,\alpha}^{\nu})\;X_{,\beta}^{\mu}\equivj\frac{\partialX^{\mu}}{\partial\sigma}-\rho\frac{\partialX^{\mu}}{\partial\tau},
{\frac{d^{2}t}{d\tilde{\lambda}^{2}}}+a_{0}\dot{a}_{0}\sum_{j}\left({\frac{dx^{j}}{d\tilde{\lambda}}}\right)^{2}=-{\frac{F}{(\sigma+\varrho)a_{0}^{2}}}{\frac{dt}{d\tilde{\lambda}}}\sum_{j}\left({\frac{dx^{j}}{d\tilde{\lambda}}}\right)^{2},
\left(\begin{array}{lccc}{C_{o}}&{e^{-iK_{2}}}&{\ldots}&{e^{iK_{2}}}\\{e^{iK_{2}}}&{C_{1}}&{\ldots}&{0}\\{0}&{e^{iK_{2}}}&{\ldots}&{0}\\{\vdots}&{\vdots}&{\vdots}&{\vdots}\\{0}&{0}&{\ldots}&{e^{-iK_{2}}}\\{e^{-iK_{2}}}&{\ldots}&{e^{iK_{2}}}&{C_{q-1}}\\\end{array}\right),
\bar{X}^{\mu}(\sigma)=x^{\mu}+\sum_{n=1}^{\infty}{\frac{1}{\sqrtn}}\bar{X}_{n}^{\mu}e^{-in\sigma},\quadX^{\mu}(\sigma)=p^{\mu}\sigma-\sum_{n=1}^{\infty}{\frac{1}{\sqrtn}}X_{n}^{\mu}e^{in\sigma}.
{\calL}={\frac{1}{16\piG_{6}}}\sqrt{-G}e^{-\Phi}[{\calR}_{G}+G^{\bar{\mu}\bar{\nu}}\partial_{\bar{\mu}}\Phi\partial_{\bar{\nu}}\Phi-{\frac{1}{12}}G^{\bar{\mu}\bar{\mu}^{\prime}}G^{\bar{\nu}\bar{\nu}^{\prime}}G^{\bar{\rho}\bar{\rho}^{\prime}}H_{\bar{\mu}\bar{\nu}\bar{\rho}}H_{\bar{\mu}^{\prime}\bar{\nu}^{\prime}\bar{\rho}^{\prime}}].
\begin{array}{l}{\bar{\beta}^{T}=-\frac{1}{2}\alpha^{\prime}D^{2}T-2T+\alpha^{\prime}\partial^{\mu}\phi\partial_{\mu}T~,}\\{\bar{\beta}^{\phi}=\frac{1}{6}(D-26)-\frac{1}{2}\alpha^{\prime}D^{2}\phi+\frac{1}{16}\alpha^{2}R_{\mu\nu\alpha\lambda}^{2}+\alpha^{\prime}\partial^{\mu}\phi\partial_{\mu}\phi~.}\\\end{array}
\Phi_{\sigma,N_{s}}^{(0)}=\sum_{N_{r}=\mu_{r},\nu_{r}}\tilde{M}_{N_{s},N_{r}}(\{\sigma_{q}\})\chi_{N_{r}}^{(0)}\quad\mathrm{{where}}\quad\tilde{M}_{N_{s},N_{r}}(\{\sigma_{q}\})=\sum_{p=a,b}\int_{C_{p}^{(r)}}\Phi_{\sigma,N_{s}}^{(0)}(z)\frac{d\thetadz}{2\pii}Y_{p,N_{r}}^{(0)}(t).
S(t_{j}^{i})=\overline{{t}}{}_{j}^{i}\,,\\\S(\overline{{t}}{}_{j}^{i})=\Omega_{m}^{i}t_{n}^{m}\,\overline{{\Omega}}{}_{j}^{n}\,,\\\S(E_{i})=-E_{j}\,\overline{{t}}{}_{i}^{j}
\{\{...\{p_{0}\xi_{0},\underbrace{S\},S\},...\},S\}}_{n}=\left\{\begin{array}{lr}{Q(2i)^{n}\Psi_{n}(p,x,\vec{\xi})\xi_{5}\xi_{0}}&{n-odd}\\{}&{}\\{Q(2i)^{n}\Psi_{n}(p,x,\vec{\xi})\xi_{0}}&{n-even,n\neq0,}\\\end{array}\right.
G_{\sigma}^{(n)}(\zeta_{1},\cdots,\zeta_{2n})=(\sqrt{-1})^{-n}\sum_{k_{0},k_{1},\cdotsk_{2n}}t_{k_{1}}^{k_{0}}(u_{1}-u_{0})\otimes\cdots\otimest_{k_{2n}}^{k_{2n-1}}(u_{2n}-u_{0})F_{\sigma}^{(n)}(\zeta_{1},\cdots,\zeta_{2n})^{k_{0}k_{1}\cdotsk_{2n}}.
\phi_{m}(\rho)=e^{\frac{c}{4}(n+3)\rho}\left[C_{1}J_{\frac{1}{2}(n+3)}(\frac{2m}{c}e^{\frac{c}{2}\rho})+C_{2}Y_{\frac{1}{2}(n+3)}(\frac{2m}{c}e^{\frac{c}{2}\rho})\right]~,
\begin{array}{cl}{}&{F^{(i)}(z_{1},z_{2},\cdots,z_{N})^{a_{0},a_{1},a_{2},\cdots,a_{N-1},a_{N}}}\\{=}&{{}_{B}\langlek-\omega_{i},k|\phi^{(a_{0},a_{1})}(z_{1})\phi^{(a_{1},a_{2})}(z_{2})\cdots\phi^{(a_{N-1},a_{N})}(z_{N})|k-\omega_{i},k\rangle_{B},}\\\end{array}
L_{\mathrm{BRS}}=\mathrm{}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{m^{2}}{2}\left(A_{\mu}-\frac{1}{m}\partial_{\mu}\theta\right)^{2}+b\left(\partial^{\mu}A_{\mu}+\frac{\alpha}{2}b\right)+i\bar{c}\Boxc
{\bff}(\mathrm{~\left|~{\bf~p}~\right|~},p_{3};M)=\intdq_{3}\intd\mathrm{~\left|~{\bf~q}~\right|~}\,K(\mathrm{~\left|~{\bf~p}~\right|~},p_{3};\mathrm{~\left|~{\bf~q}~\right|~},q_{3};M){\bff}(\mathrm{~\left|~{\bf~q}~\right|~},q_{3};M),
H^{1}({\mathbbP}^{1}\timesT^{2},{\calO}_{{\mathbbP}^{1}\timesT^{2}}(3\sigma|_{{\mathbbP}^{1}\timesT^{2}}-3F))|_{b_{-3}^{(1)}}\stackrel{m_{1}}{\rightarrow}H^{1}({\mathbbP}^{1}\timesT^{2},{\calO}_{{\mathbbP}^{1}\timesT^{2}}(6\sigma|_{{\mathbbP}^{1}\timesT^{2}}-2F))|_{c_{-1}^{(1)}}.
{\widehatT}={\widehatM}_{1}=\left(\begin{array}{cccccc}{1}&{0}&{0}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}&{0}\\{-2}&{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{0}&{0}&{1}\\\end{array}\right)\;.
r_{12}=\sum_{i,j=1\atopi\nej}^{N}\frac{1}{x_{j}-x_{i}}e_{ij}\otimese_{ji}+\frac{1}{2}\sum_{i,j=1\atopi\nej}^{N}\frac{1}{x_{j}-x_{i}}e_{ii}\otimes(e_{ij}-e_{ji}),
\frac{1}{m}E_{\mathrm{tot}}=\frac{1}{g^{2}}{\calE}_{\mathrm{cl}}(w,b_{1},b_{2},c)+\left(1-Q_{\mathrm{vac}}\right)\epsilon_{1}(w,b_{1},b_{2},c)+{\calE}_{\mathrm{vac}}(w,b_{1},b_{2},c)
F_{l}(x)=x^{2}\left(1+{\frac{l(l+1)}{x^{2}}}\right)-{\frac{1}{4}}\left({\frac{d}{dx}}e_{l}s_{l}\right)^{2}-x^{2}\left[2\left(1+{\frac{l(l+1)}{x^{2}}}\right)e_{l}s_{l}-{\frac{1}{2}}{\frac{d^{2}}{dx^{2}}}e_{l}s_{l}\right]^{2}.
E(z)=\sum_{m=-\infty}^{\infty}c_{m}^{(0)}z^{m}=\sum_{m=0}^{\infty}a_{m}^{(0)}z^{m}+\sum_{m=1}^{\infty}\frac{b_{m}^{(0)}}{z^{m}}\mathrm{~,}\tag{116}
dS^{2}={\frac{1}{I}}\{{\frac{1}{f}}[e^{2k_{0}}dz\,d{\bar{z}}+\rho^{2}\,d\phi^{2}]-f(dt+qA_{0}\,d\phi)^{2}\}+I^{2}(A_{3}\,d\phi+A_{4}\,dt+dx^{5})^{2},
S_{i\;j}^{k\;\ell}\left(\begin{array}{cc}{a}&{d}\\{b}&{c}\\\end{array}\right):{\calV}_{a,b}(\theta_{1})\otimes{\calV}_{b,c}(\theta_{2})\rightarrow{\calV}_{a,d}(\theta_{2})\otimes{\calV}_{d,c}(\theta_{1})
\frac{i}{\hbar}[\tilde{Y}_{\alpha},\tilde{Y}_{\beta}]=\omega_{\alpha\beta}(\tilde{Y}),\quadK_{\mu}^{\prime}(\tilde{Y}(q^{\prime},\partial_{q^{\prime}},j))=K_{\mu}(\tilde{X}(q,\partial_{q},j))=\kappa_{\mu}(j)\,.
\phi(g)=\sum_{\Lambda}\\phi_{\alpha\beta}^{\Lambda}\D_{\alpha\beta}^{(\Lambda)}(g)=\int_{SO(3)}dh\\phi(gh)=\sum_{\Lambda}\\int_{SO(3)}dh\\phi_{\alpha\beta}^{\Lambda}\D_{\alpha\gamma}^{(\Lambda)}(g)\D_{\gamma\beta}^{(\Lambda)}(h).
\chi_{\mu}(C_{m}^{\Lambda})=\sum_{\nu\inM(\Lambda)}q^{-m\beta_{\nu}(\mu)}\frac{D_{q}[\mu+\nu]}{D_{q}[\mu]}\,\sum_{s=0}^{\infty}m_{\nu-s\delta}\,q^{-2s\{g+m(\Lambda+\mu+\rho,\delta)\}}\;.
{\bfH}_{\mu\nu\rho}(\,{\bfB}\,)\equivD_{[\,\mu}(\,{\bfV}\,)\,{\bfB}_{\nu\rho\,]}=\partial_{[\,\mu}\,{\bfB}_{\nu\rho\,]}-i\,\left[\,{\bfV}_{[\,\mu},{\bfB}_{\nu\rho\,]}\,\right]
H^{-1}(z)=\left[1-\frac{kd_{L}^{2}(z)}{a_{0}^{2}(1+z)^{2}}\right]^{-1/2}\frac{d}{dz}\left[\frac{d_{L}(z)}{1+z}\right]\to\frac{d}{dz}\left[\frac{d_{L}(z)}{1+z}\right]
\frac{x_{5}}{\omegax_{2}}=q^{-1}u_{1},~~\frac{u_{2}^{\prime}}{u_{2}^{\prime\prime}}=qu_{2},{}~~\frac{x_{2}x_{13}}{\omegax_{6}x_{58}}=qu_{2}^{\prime},~~\frac{x_{2}x_{67}}{x_{6}x_{24}}=qu_{2}^{\prime\prime},~~\frac{x_{6}}{x_{2}}=q^{-1}u_{3}
\{Q_{\alpha}^{I},Q_{\beta}^{J}\}=\delta^{IJ}({\calC}\gamma^{\mu})_{\alpha\beta}P_{\mu}+\sum_{p=0,1,...}({\calC}\gamma^{\mu_{1}\cdots\mu_{p}})_{\alpha\beta}Z_{\mu_{1}\cdots\mu_{p}}^{IJ},
\begin{array}{lll}{\{I,{p^{\prime}}^{2}\}=2\left(\frac{D}{L^{2}}-\frac{{p^{\prime}}^{2}}{S}\right),}&{\{I,{r}^{2}\}=-\left(\frac{D}{M^{2}}-\frac{r^{2}}{S}\right),}&{\{I,D\}=\frac{r^{2}}{L^{2}}-\frac{{p^{\prime}}^{2}}{M^{2}}}\\{\{{p^{\prime}}^{2},r^{2}\}=4DI,}&{\{D,{p^{\prime}}^{2}\}=-2{p^{\prime}}^{2}I,}&{\{D,r^{2}\}=2r^{2}I}\\\end{array}
i{t}_{a}{A_{\mu}^{a}}^{\prime}(x)=e^{i{t}^{a}{\omega}_{a}}e^{-i{t}^{b}{\theta}_{b}}\partial_{\mu}\left(e^{i{t}^{c}{\theta}_{c}}e^{-i{t}^{l}{\omega}_{l}}\right),
h=\left(\begin{array}{cc}{1}&{0}\\{X}&{1}\\\end{array}\right)\left(\begin{array}{cc}{e^{-\frac{1}{2}\tilde{\Phi}}}&{0}\\{0}&{e^{\frac{1}{2}\tilde{\Phi}}}\\\end{array}\right)\left(\begin{array}{cc}{1}&{Y}\\{0}&{1}\\\end{array}\right).
\bar{g}(s)=\frac{gs^{\frac{y}{2}}}{1-\frac{\piA(y)b(y)g}{y}(s^{\frac{y}{2}}-1)}=\frac{g_{IR}^{*}|\tilde{g}|s^{y/2}}{1+|\tilde{g}|s^{y/2}},\\\\\\\tilde{g}=\frac{g}{g-g_{IR}^{*}}\\in\\(-\infty,0).
L_{2n-1}=\kappa\sum_{p=0}^{n-1}\frac{1}{D-2p}\left(\begin{array}{c}{n-1}\\{p}\\\end{array}\right)l^{2p-D}\epsilon_{a_{1}\cdotsa_{2n-1}}R^{a_{1}a_{2}}\wedge\cdots\wedgeR^{a_{2p-1}a_{2p}}\wedgee^{a_{2p+1}}\wedge\cdots\wedgee^{a_{2n-1}}.