harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
g_{\mu\nu}A^{\mu}{\frac{dx^{\nu}}{d\tilde{\lambda}}}=\left[2-n^{2}\left({\frac{dt}{d\tilde{\lambda}}}\right)^{2}+a^{2}\sum_{i}\left({\frac{dx^{i}}{d\tilde{\lambda}}}\right)^{2}\right]\left[nn^{\prime}\left({\frac{dt}{d\tilde{\lambda}}}\right)^{2}-aa^{\prime}\sum_{i}\left({\frac{dx^{i}}{d\tilde{\lambda}}}\right)^{2}\right],
\psi_{,TT}^{z}-\psi_{,\sigma\sigma}^{z}+\frac{4\alpha}{1+\alpha}(T+\tilde{\Lambda})^{-1}\psi_{,T}^{z}-\frac{2\alpha(1-\alpha}{(1+\alpha)^{2}}(T+\tilde{\Lambda})^{-2}\psi^{z}=0,
\frac{1}{t^{2}}\sum_{p\leq}(2-\delta_{pq})\left(\frac{d(p+q)+2(p-q)-2pq}{4}-\frac{d(d-2)}{8}\right)\left(\frac{(m^{[p]})^{2}}{V_{d}~p!}\right)\left(\frac{(m^{[q]})^{2}}{V_{d}~q!}\right)
S(\rho)\equiv\frac{A^{2n+2}}{V^{2n+1}}=\frac{4\pi(n+1)}{\lambda}\mathrm{Vol(M)}\,\tilde{\beta^{2}}^{n+1}(\rho^{2}-1)^{n}\left(\frac{(\rho^{2}-1)^{n}}{\int_{1}^{\rho}(s^{2}-1)^{n}ds}\right)^{2n+1}.
F(p^{2},m^{2},x,y,\Lambda_{0})=\int_{0}^{\Lambda_{0}^{2}}d(\vec{k}^{2})\int_{0}^{\Lambda_{0}^{2}}d(\vec{l}^{2})\frac1{\left[\vec{k}^{2}+\vec{l}^{2}+m^{2}-p^{2}D(x,y)-i\epsilon\right]^{2}}\,.
\begin{array}{c}{{\calE}=-\int_{M_{3}}d<\Phi_{A}H^{A}>-\int_{M_{3}}d<\Phi_{B}K^{B}>}\\{=-\int_{S^{2}}<\Phi_{A}H^{A}>-\int_{S^{2}}<\Phi_{B}K^{B}>}\\\end{array}
X_{2}=\frac{1}{\left(\xi^{2}+\eta^{2}\right)}\left\{\xi\eta\left(\partial_{\xi}^{2}+\partial_{\eta}^{2}\right)+\left(-\beta\eta+\gamma\xi\right)\left(\xi^{2}-\eta^{2}\right)-4\alpha\xi\eta\right\}-\partial_{\xi\eta}^{2}\.
N_{R}^{0}=\sum_{n=1}^{\infty}n(a_{n}^{\alpha}{}^{\dagger}a_{n}^{\alpha}+s_{n}^{A}{}^{\dagger}s_{n}^{A}),~~~~~N_{L}^{0}=\sum_{n=1}^{\infty}n(\tilde{a}_{n}^{\alpha}{}^{\dagger}\tilde{a}_{n}^{\alpha}+\tilde{s}_{n}^{A}{}^{\dagger}\tilde{s}_{n}^{A}).
L_{b}={\frac{1}{g^{2}}}\int_{\widetilde{T}^{d}\timesR}d^{d}\sigmadt\tr\left\{-{\frac{1}{4}}F^{2}+{\frac{1}{2}}(D_{\alpha}X^{i})^{2}-{\frac{1}{4}}([X^{i},X^{j}])^{2}\right\}\quad,
\sum_{\rho=0}^{N}\left\{\sum_{\nu=0}^{N}\eta_{\nu\rho}\eta_{\nu\sigma}\ddot{Q}_{\mu\rho}(t)+\left[\sum_{\nu=0}^{N}\omega_{\nu}^{2}\eta_{\nu\rho}\eta_{\nu\sigma}+\sum_{k=1}^{N}\lambda_{k}\left(\eta_{k\rho}\eta_{0\sigma}+\eta_{0\rho}\eta_{k\sigma}\right)\right]Q_{\mu\rho}(t)\right\}=0\;,
\begin{array}{rcl}{\mathcal{D}\phi^{i}}&{=}&{d\phi^{i}+gA^{\mathbf{I}}k_{\mathbf{I}}^{i}(\phi)}\\{\mathcal{D}q^{u}}&{=}&{dq^{u}+gA^{\mathbf{I}}k_{\mathbf{I}}^{u}(q)}\\\end{array}
\Omega_{\tau}^{(4)}(x,q,v,w)=\left(\begin{array}{cccc}{-\frac{1}{2}\xi\vec{v}^{\2}}&{\xi}&{\xiv^{1}}&{\xiv^{2}}\\{}&{}&{}&{}\\{\xi}&{-\frac{1}{2}\xi\vec{v}^{\2}}&{-\xiv^{1}}&{-\xiv^{2}}\\{}&{}&{}&{}\\{-\xiv^{1}}&{\xiv^{1}}&{1+\xi(v^{1})^{2}}&{\xiv^{1}v^{2}}\\{}&{}&{}&{}\\{-\xiv^{2}}&{\xiv^{2}}&{\xiv^{1}v^{2}}&{1+\xi(v^{2})^{2}}\\\end{array}\right)\,.
H_{+}\phi(r,\theta)=\left(\begin{array}{cc}{m}&{ie^{-i\theta}\left(\partial_{r}-\frac{i}{r}\partial_{\theta}-\frac{eB}{2}r\right)}\\{ie^{i\theta}\left(\partial_{r}+\frac{i}{r}\partial_{\theta}+\frac{eB}{2}r\right)}&{-m}\\\end{array}\right)\phi(r,\theta)=E\phi(r,\theta).
\gamma_{\xi}:=\mu\frac\partial{\partial\mu}\xi_{\mathrm{R}}=2\lambda_{\mathrm{R}}\xi_{\mathrm{R}}(\xi_{\mathrm{R}}-1)\left(\xi_{\mathrm{R}}-{\frac{1}{2}}\right)\frac{C_{2}(G)}{(4\pi)^{2}}g_{\mathrm{R}}^{2},
\omega_{\mu\nu\rho}^{L}=-6\,\biggl\{\Omega_{[\mu-}^{(0)}{}^{ab}\partial_{\nu}\Omega_{\rho]-}^{(0)}{}^{ab}-{\frac{1}{3}}\Omega_{[\mu-}^{(0)}{}^{ab}\Omega_{\nu-}^{(0)}{}^{ac}\Omega_{\rho-]}^{(0)}{}^{cb}\biggr\}\.
S_{WZW}=\mathrm{Tr}\intd^{2}z\left[(g^{-1}\partialg)(g^{-1}\bar{\partial}g)-\int_{0}^{1}dt(\hat{g}^{-1}\partial_{t}\hat{g})[(\hat{g}^{-1}\partial\hat{g}),(\hat{g}^{-1}\bar{\partial}\hat{g})]\right],
H^{\prime}=\frac{1}{2m}\sum_{n}\left[\vec{p}_{n}-\frac{e}{c}\vec{A}(x_{n})-\frac{e}{c}\;\vec{a}(x_{n})\right]^{2}+\sum_{n}eA_{0}(x_{n})+\sum_{n<n^{\prime}}V(x_{n}-x_{n^{\prime}})
t_{\\nu}^{\mu}={\frac{1}{32}}\delta_{\nu}^{\mu}\left(R_{\\beta}^{\alpha}R_{\\alpha}^{\beta}-{\frac{1}{3}}R^{2}\right)+{\frac{1}{24}}\left(RR_{\\nu}^{\mu}-{\frac{1}{4}}\delta_{\nu}^{\mu}R^{2}\right)+\tau_{\\nu}^{\mu}+\alpha{\calS}_{\\nu}^{\mu}+{\frac{\beta}{3}}{\calK}_{\\nu}^{\mu}\,
{\calT}_{N}\,=\,\left[\begin{array}{cc}{1}&{\qquad0}\\{}&{}\\{0}&{\qquadq}\\\end{array}\right]\qquad{\calS}_{N}\,=\,\fracq{\sqrt3}\left[\begin{array}{cc}{-1}&{-\frac23(q-q^{-1})}\\{}&{}\\{(q-q^{-1})}&{1}\\\end{array}\right]
A_{N}^{c^{2}}(s,a_{1},a_{2},..,a_{N})=\sum_{n_{1},n_{2}..n_{N}=-\infty}^{\infty}(a_{1}n_{1}^{2}+a_{2}n_{2}^{2}+...+a_{N}n_{N}^{2}+c^{2})^{-s}
:e^{i2\sqrt{\pi}\;\varphi^{(b)}(x)}:_{Q^{W}}\;=\;:e^{i2\sqrt{\pi}\;\varphi^{(b)}(x)}:_{Q^{\mu}}\;\;\Big(\mu\Big)^{\;\sum_{I=2}^{N}(U_{Ib})^{2}}\;.
\begin{array}{rcl}{\displaystyle(\chi_{y}^{\mathrm{LG}})^{(0)}}&{\displaystyle=}&{\displaystyle(-1)^{M-N}y^{r}(\chi_{y^{-1}}^{\mathrm{LG}})^{(0)}\,,}\\{\displaystyle(\chi_{y}^{\mathrm{LG}})^{(\alpha)}}&{\displaystyle=}&{\displaystyle(-1)^{M-N}y^{r}(\chi_{y^{-1}}^{\mathrm{LG}})^{(h-\alpha)}\,,\quad\alpha=1,\ldots,h-1\,.}\\\end{array}
K(\zeta)_{\varepsilon_{1}}^{\varepsilon_{1}^{\prime}}=\sum_{\varepsilon_{2},\varepsilon_{2}^{\prime}=\pm}R(\zeta^{2})_{\varepsilon_{2}^{\prime}-\varepsilon_{1}^{\prime}}^{-\varepsilon_{1}\varepsilon_{2}}K(\epsilonx^{-1}\zeta)_{\varepsilon_{2}}^{\varepsilon_{2}^{\prime}}~~~~(\epsilon=\pm).
F^{(2)}={\frac{1}{2}}dZ^{\underline{N}}dZ^{\underline{M}}F_{\underline{{MN}}}={\frac{1}{2}}dZ^{\underline{N}}dZ^{\underline{M}}(\partial_{\underline{M}}A_{\underline{N}}-(-)^{\underline{{MN}}}\partial_{\underline{N}}A_{\underline{M}}).
{\calL}_{eff}=-\frac{1}{4}(\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+gf^{abc}A_{\mu}^{b}A_{\nu}^{c})^{2}-\frac{1}{2}(\partial_{\mu}A^{a\mu})^{2}
\psi_{\alpha}(x)=\frac{1}{\sqrt{2}}(\psi_{\alpha}^{(1)}(x)+i\psi_{\alpha}^{(2)}(x)),\quad{\calO}_{\mathrm{F1}}(y)\equiv\left(\partial_{y}+2gA_{\mathrm{cl}}(y)\right),\quad{\calO}_{\mathrm{F2}}(y)\equiv\left(-\partial_{y}+2gA_{\mathrm{cl}}(y)\right),
h(r)=\displaystyle\frac{4}{\pi^{2}}\displaystyle\int_{0}^{r+\mu}dr^{\prime}h(r^{\prime})\displaystyle\int_{\epsilon}^{\infty}dk\displaystyle\frac{k^{2}}{|{\calJ}_{+}(k)|^{2}}\phi(k,r)\phi(k,r^{\prime})\nonumber\,+\displaystyle\frac{i}{\pi}\displaystyle\int_{0}^{r+\mu}dr^{\prime}h(r^{\prime})\displaystyle\int_{\gamma}dkk{\calG}_{\xi,l}^{+}(k;r,r^{\prime}).
Q_{\calM}=\left(\begin{array}{cccccccc}{1}&{1}&{-3}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{-2}&{1}&{0}&{0}\\{0}&{0}&{0}&{1}&{-2}&{1}&{0}\\{0}&{0}&{0}&{0}&{1}&{-2}&{1}\\\end{array}\right)
\Phi^{(b)}(u_{1},u_{2},\ldots,u_{b},g,h)=\sum_{G=0}^{\infty}\sum_{\{k_{j}\},n=0}^{\infty}h^{G}{\calN}_{G}^{(b)}(k_{1},k_{2},\ldots,k_{b};n)\;g^{n}u_{1}^{k_{1}}u_{2}^{k_{2}}u_{3}^{k_{3}}\cdotsu_{b}^{k_{b}}
O(z,\bar{z})=\oint{\frac{\mathrm{d}w}{2\pii}}~\oint{\frac{\mathrm{d}\bar{w}}{2\pii}}{\frac{J^{a}(w)\cdot\bar{J}^{\bar{a}}(\bar{w})\cdot\phi^{a\bar{a}}(z,\bar{z})}{|z-w|^{2}}},
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad=\quad\left(2\sigma\delta_{\alpha\beta}+(2\sigma)^{2}\hat{x}_{\alpha}\hat{x}_{\beta}\right)\frac{e^{-\frac{(x-y)^{2}}{4\sigma(n+1)}}}{(4\pi\sigma(n+1))^{2}}
S=\int_{E}d^{2}x\,\Bigr\{\mathrm{~\frac{1}{4}~}F^{\mu\nu}(A)F_{\mu\nu}(A)+\mathrm{~\frac{1}{2}~}\partial^{\mu}A_{\mu}\partial^{\nu}A_{\nu}+\partial^{\mu}\bar{C}\partial_{\mu}C\Bigr\}\,,
G^{\nu\alpha\lambda\beta}=\frac{1}{k^{2}}\Bigl[g^{\nu\beta}g^{\lambda\alpha}-g^{\nu\alpha}g^{\lambda\beta}+\frac{(1-\alpha)}{k^{2}}\left(g^{\nu\alpha}k^{\lambda}k^{\beta}+g^{\lambda\beta}k^{\nu}k^{\alpha}-g^{\nu\beta}k^{\lambda}k^{\alpha}-g^{\lambda\alpha}k^{\nu}k^{\beta}\right)\Bigr]\.
I_{Bolt}=\frac{-4\pi^{2}(3r_{b}^{6}-(5N^{2}+\ell^{2})r_{b}^{4}-3N^{2}(5N^{2}-2\ell^{2})r_{b}^{2}+3N^{4}(\ell^{2}-5N^{2}))}{3(5r_{b}^{2}+\ell^{2}-5N^{2})}
\mathrm{Re}~W=\int_{{\mathbbR}_{+}\times\widetilde{\mathbb{S}}^{3}}G^{(4)}~\int_{\mathbb{S}^{2}}J=t~\int_{\widetilde{\mathbb{S}}^{3}}C^{(3)}=\theta~t\,,\quadt=\int_{\mathbb{S}^{2}}J\,.
_\pm\langle\{\lambda\},\pmq|\prod_{i=1}^{N}\Psi^{\pm}(z_{i})|\pm(q-N\beta)\rangle=\prod_{i<j}(z_{i}-z_{j})^{\beta}\prod_{i}^{N}(z_{i})^{q-N\beta}J_{\{\lambda\}}^{(\beta)}(z_{1},...z_{N})\(j_{\lambda}^{\beta})^{-1/2}
p_{b\perp}\sim-\varepsilon^{b}\sim\frac{Dap_{b}}{(D-1)(a-r)}\sim\frac{D\Gamma((D+1)/2)(\xi-\xi_{D})}{2^{D}\pi^{(D+1)/2}(a-r)^{D+1}}\left(1-2\delta_{B0}\right).
\Phi({\bfx},\theta)=\chi+\bar{\theta}\lambda+\frac12\bar{\theta}\thetaF,~~D_{\alpha}=\frac{\partial}{\partial\bar{\theta}_{\alpha}}-(\hat{\partial}\theta)_{\alpha},~~\bar{D}_{\alpha}=\frac{\partial}{\partial\theta_{\alpha}}-(\bar{\theta}\hat{\partial})_{\alpha}.
S=\intd^{4}x[\frac{1}{2}{\partial}_{\mu}{\hat{\phi}}*{\partial}^{\mu}{\hat{\phi}}-\frac{m^{2}}{2}{\hat{\phi}}*{\hat{\phi}}-\frac{g^{4}}{4!}({\hat{\phi}}*{\hat{\phi}})*({\hat{\phi}}*{\hat{\phi}})],
L_{eff}^{(0)}=\int\;d^{2}x\;[(\partial_{t}\mid\phi\mid)^{2}+(\partial_{t}\mid\chi\mid)^{2}+\frac{\kappa}{q_{1}}B_{2}\partial_{t}\omega_{1}+\frac{\kappa}{q_{2}}B_{1}\partial_{t}\omega_{2}]\;\;.
\Gamma_{div}^{mat}=\frac{1}{\varepsilon}\int\left(\gamma_{1}\frac{1}{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-\gamma_{2}\frac{\varphi^{4}}{4!}+\gamma_{3}\frac{1}{2}R\varphi^{2}\right)\sqrt{-g}~d^{4}x.
T_{\mu\nu}=\frac{\mathrm{e}^{-\phi}}{2\,\,3!}\left[3\,H_{\mu\sigma\rho}H_{\nu}^{\,\,\,\sigma\rho}-\frac{G_{\mu\nu}}{4}H^{2}\right]+\frac{\mathrm{e}^{\phi}}{2\,\,3!}\left[3\,(\tilde{F}_{3})_{\mu\sigma\rho}(\tilde{F}_{3})_{\nu}^{\,\,\,\sigma\rho}-\frac{G_{\mu\nu}}{4}F_{3}^{2}\right]
{\dot{x}}^{\mu\nu}(\sigma)\rightarrow\mathrm{det}\left({\frac{\partialu^{a}}{\partial\sigma^{m}}}\right){\dot{x}}^{\mu\nu}(u)\,\quadN(u)\rightarrow\mathrm{det}\left({\frac{\partialu^{a}}{\partial\sigma^{m}}}\right)N(\sigma)\.
\left[p_{1}^{2}+p_{2}^{2}+\frac{g^{2}}4\left(\tau_{1}+\tau_{2}\right)I_{2}+g\left(p_{1}\tau_{1}^{\frac12}\lambda^{1}+p_{2}\tau_{2}^{\frac12}\lambda^{2}\mp\frac{H_{z}^{3}}2\lambda^{3}\right)\right]\psi_{\pm}=\left(E^{2}-m^{2}\right)\psi_{\pm}
\lambda_{l_{1},l_{2}}^{L}=(L+1)(-1)^{l_{1}+l_{2}}\sum_{k=0}^{L}(-1)^{k}\left(\begin{array}{cc}{L+k+1}\\{k+1}\\\end{array}\right)\left(\begin{array}{cc}{L}\\{k}\\\end{array}\right)T_{k}(l_{1})T_{k}(l_{2}).
\left(\Psi+n^{\mu}\,\nabla_{\mu}\right)\varphi\longrightarrow\Omega^{-{\frac{{\calD}-1}{2}}}\left(\Omega\overline{{\Psi}}\varphi+n^{\mu}\nabla_{\mu}\varphi-{\frac{{\calD}-2}{2}}{\frac{n^{\mu}\partial_{\mu}\Omega}{\Omega}}\varphi\right),
M_{ij}\equiv\left(\begin{array}{ccccc}{2}&{1}&{\cdots}&{\cdots}&{1}\\{1}&{2}&{}&{}&{\vdots}\\{\vdots}&{}&{\ddots}&{}&{\vdots}\\{\vdots}&{}&{}&{\ddots}&{\vdots}\\{1}&{\cdots}&{\cdots}&{\cdots}&{2}\\\end{array}\right).
A_{lm}^{\alpha\beta}=\frac{i}{|P|}\frac{1}{F}\sum_{k}\sum_{f}~C_{k}^{\alpha\beta}(v)\mathrm{Tr}(\gamma_{f}^{\alpha}\lambda_{l}^{(0)})\mathrm{Tr}((\gamma_{f}^{\beta})^{-1}(\lambda_{m}^{(0)})^{2})~,
2iC_{qr}\gamma_{\alpha\beta}^{a}m_{a}^{~b}E_{b}^{\underline{\beta}}v_{\underline{\beta},\gammas}={\frac{1}{4}}\gamma_{\alpha\gamma}^{b}m_{b}^{~a}\hat{\Omega}_{\betar,a}^{~~~~i}(\gamma_{i})_{qs}+{\calD}_{\betar}h_{\alpha\gamma}C_{qs}+(\alphaq)~\leftrightarrow~(\betar).
H_{0}^{\prime}={\frac{e^{2}}{2\pi}}\int_{k>0}{\frac{dk}{4\pi}}\Biggl[\int_{p>0}dp\biggl({\frac{1}{(k-p)^{2}}}-{\frac{1}{(k+p)^{2}}}\biggr)\Biggr]\bigl(b_{k}^{\dagger}b_{k}+d_{k}^{\dagger}d_{k}\bigr)\;.
\delta^{4}(\theta_{3}-\theta_{1}){\frac{{\bar{D}}_{\dot{\alpha}}(k,\theta_{1})D_{\alpha}(k,\theta_{1})}{4}}{\frac{{\bar{D}}^{2}(k,\theta_{1})D^{2}(k,\theta_{1})}{16}}\delta^{4}(\theta_{1}-\theta_{3})={\frac{1}{2}}\sigma_{\alpha{\dot{\alpha}}}^{m}k_{m}\delta^{4}(\theta_{3}-\theta_{1})\,
a_{1}\,\gamma\,-\,a_{2}\alpha\,h^{-1}\,\left(h^{-1}+\mu\mathcal{F}\right)_{S}^{-1}\,h^{-1}\,\left[\mathrm{det}\,\left(h^{-1}+\mu\mathcal{F}\right)\right]^{\alpha}=0
\mathcal{D}_{\beta}=\partial_{\beta}+\frac{1}{4}\partial_{\beta}x^{+}\left[\omega_{\rho\sigma+}\Gamma^{\rho\sigma}-\frac{1}{2\cdot5!}F_{\lambda\nu\rho\sigma\kappa}\Gamma^{\lambda\nu\rho\sigma\kappa}i\sigma^{2}\Gamma_{+}\right]
\int_{g}{}_{\nu\bar{\nu}}\langleO_{2}|\biggl(\Sigma_{i,j=3}^{k}(g^{\dagger}\lambda^{[12]}g)_{j}^{i}(\nu_{i}^{3}{\bar{\nu}}^{4j}-\nu_{i}^{4}{\bar{\nu}}^{3j})\biggr)^{k-2}{|O_{k}\rangle}_{\nu\bar{\nu}}
\left(\begin{matrix}{0}&{1}\\{1}&{0}\\\end{matrix}\right)\,\partial_{0}\left(\begin{matrix}{\psi}\\{\chi}\\\end{matrix}\right)+\left(\begin{matrix}{-1}&{0}\\{0}&{-1}\\\end{matrix}\right)\,\partial_{x}\left(\begin{matrix}{\psi}\\{\chi}\\\end{matrix}\right)=\left(\begin{matrix}{0}\\{0}\\\end{matrix}\right)\,.
\langle0|(X^{1}(\sigma)-\overline{{X^{1}}})^{2}|0\rangle\=\\sum_{m=1}^{\infty}\frac{1}{m^{2}}\langle0|(\alpha_{m}\alpha_{-m}+\tilde{\alpha}_{m}\tilde{\alpha}_{-m})|0\rangle\=2\sum_{m=1}^{\infty}\frac{1}{m}.
\mathrm{d}\mathbf{e}^{x_{1},\cdots,x_{k}}=\sum_{y}(\mathbf{e}^{y,x_{1},\cdots,x_{k}}-\mathbf{e}^{x_{1},y,x_{2},\cdots,x_{k}}+\cdots(-1)^{k}\mathbf{e}^{x_{1},\cdots,x_{k},y}),
\mathrm{V}({\rho}_{\mathrm{N}};{\rho}_{\mathrm{N}})=\frac{1}{4}(1+\mathrm{N})^{2}\mathrm{V}({\rho};{\rho})+\frac{1}{4}(1-\mathrm{N})^{2}\mathrm{V}({\sigma};{\sigma})+\frac{1}{2}(1-\mathrm{N}^{2})\mathrm{V}({\rho};{\sigma}).
J(Z_{u})=Z_{u}+\sum_{i}y^{i}Z_{\partial_{i}u}+{\frac{1}{2}}\sum_{i,j}y^{i}y^{j}Z_{\partial_{ij}u}+\cdots=\sum_{n}{\frac{1}{n!}}(\sum_{i}y^{i}\partial_{i})^{n}(Z_{u})\;,
\epsilon^{\alpha\mu\nu\xi}\Omega_{\left(\mu\nu\xi\right)}^{\left\{\beta\sigma\right\}}+\epsilon^{\beta\mu\nu\xi}\Omega_{\left(\mu\nu\xi\right)}^{\left\{\sigma\alpha\right\}}+\epsilon^{\sigma\mu\nu\xi}\Omega_{\left(\mu\nu\xi\right)}^{\left\{\alpha\beta\right\}}=0\,,
\tilde{F}(P_{-},x^{+})=\left|{\frac{L}{\pi}}\langle0|T^{++}(k)|n\rangle\right|^{2}{\frac{1}{2L}}{\frac{\pi^{2}}{L^{2}}}e^{\frac{-iM_{n}^{2}}{2({\frac{k\pi}{L}})x^{+}}}\.
{\vec{\sigma}}^{\prime}=\vec{\sigma}\gamma^{-1}+\vec{V}\gamma^{-2}\left[\frac{\vec{V}\vec{\sigma}}{c^{2}\left(\gamma+\sqrt{\gamma^{2}+\left(\displaystyle\frac{\vec{V}}{c}\right)^{2}}\right)}-\left(\frac{\vec{\sigma}}{c}\right)^{2}\gamma_{0}^{-1}-1\right].
I_{\mathrm{eff}}=-\frac{1}{2\kappa^{2}}\intd^{D}X\sqrt{g}e^{-2\Phi}\left(R+4(\partial_{M}\Phi)^{2}-\frac{1}{12}H_{MNP}^{2}-\frac{2(D-26)}{3\alpha^{\prime}}+\dots\right)
[\hat{\mathrm{G}}_{\pm}^{a}(n),\hat{\mathrm{G}}_{\pm}^{b}(m)]_{-}=ie_{\pm}{\varepsilon}_{abc}\hat{\mathrm{G}}_{\pm}^{c}(n+m)\pme_{\pm}^{2}n\delta_{n,-m}\delta^{ab},\,
T^{+2}=-2i\pi_{q}^{1+}\pi_{q}^{1+}-2i\pi_{\dot{q}}^{2+}\pi_{\dot{q}}^{2+},\quadT^{-2}=-2i\pi_{\dot{q}}^{1-}\pi_{\dot{q}}^{1-}-2i\pi_{q}^{2-}\pi_{q}^{2-}.
\tilde{G}(p,y)=\frac{1}{p^{2}-m^{2}}-\frac{1}{12}\frac{R^{0}}{(p^{2}-m^{2})^{2}}+\frac{2}{3}\frac{{R^{0}}^{\mu\nu}p_{\mu}p_{\nu}}{(p^{2}-m^{2})^{3}}+\mathrm{O}(R_{;\mu},R^{2})\,.
\frac{\ddot{T}}{T|T|^{2}}=\frac{1}{2}X_{11}\bar{X}+\frac{1}{2}|X|^{2}-\frac{1}{2}X{\bar{X}}_{11}-\frac{1}{2}(X_{1})^{2}\frac{\bar{X}}{X}=\frac{A}{2}
\mathrm{diag}(e^{\frac{2\piia_{1}}{N}},e^{\frac{2\piia_{2}}{N}},e^{\frac{2\piia_{3}}{N}},e^{\frac{2\piia_{4}}{N}})\,,\,\,\,\,\,\sum_{\alpha}a_{\alpha}=0\,\,\mathrm{mod}\,\,N\,.
{({\eta^{n}\cdot\tau^{n}{\cal{G}}})}_{\mu_{1}...\mu_{s}}\;\;:=\;\;g_{s,n}\,\eta_{(\mu_{1}\mu_{2}}...\eta_{\mu_{2n-1}\mu_{2n}}\,{\cal{G}}_{\mu_{2n+1}...\mu_{s})}^{\{n\}}
a^{\mp}=a^{\mp}(\vec{\sigma}\cdot\vec{L}+{\bf1})=\frac{1}{\sqrt{2}}\left\{\mp\Sigma_{1}\left(\frac{\partial}{\partialr}+\frac{1}{r}\right)\pm\frac{1}{r}(\vec{\sigma}\cdot\vec{L}+{\bf1})\Sigma_{1}\Sigma_{3}-\Sigma_{1}r\right\}
S=\frac{1}{\pi}\intd^{2}x\;[-\partial_{+}\chi\partial_{-}\chi+\partial_{+}\Omega\partial_{-}\Omega+\lambda^{2}{\textstylee}^{\frac{2}{\sqrt{\kappa-2}}(\chi-\Omega)}+\frac{1}{2}\sum_{i=1}^{N}\partial_{+}f_{i}\partial_{-}f_{i}\;]
U=\left(\begin{array}{ccc}{\alphax}&{\alphay}&{\alphaz}\\{\beta/x}&{\beta\omega^{2}/y}&{\beta\omega/z}\\{\gamma\left[\frac{y\omega^{2}}{z}-\frac{z\omega}{y}\right]}&{\gamma\left[\frac{z}{x}-\frac{x\omega^{2}}{z}\right]}&{\gamma\left[\frac{x\omega}{y}-\frac{y}{x}\right]}\\\end{array}\right)
E_{0}\left(\ell,\mu,d\right)=-\frac{C\left(d\right)\,L^{d-1}}{2^{d-1}\pi^{d/2}\Gamma\left(\frac{d}{2}\right)\,\ell^{d}}\sum_{k=1}^{\infty}\frac{\left(-1\right)^{k+1}}{k}I_{k}\left(\mu,d\right),
{\calA}\simeq\frac{G_{o}^{4}}{\alpha_{e}^{\prime}}\frac{1}{\alpha^{\prime}(p+p^{\prime})^{2}}\mathrm{Tr}(\xi\cdot\xi^{\prime})(\zeta\cdot\zeta^{\prime})\alpha^{\prime}k\cdot(p^{\prime}-p)\alpha^{\prime}k^{\prime}\cdot(p^{\prime}-p)+{\calO}(p^{\prime}+p)~,
\left\{\prod_{i=0}^{n}(\gamma_{\mu_{i}})_{21}\frac{1}{\tilde{x}_{0}-\tilde{x}_{\pi(1)}}\frac{1}{\tilde{x}_{\pi(1)}-\tilde{x}_{\pi(2)}}.....\frac{1}{\tilde{x}_{\pi(n)}-\tilde{x}_{0}}\;+\;c.c.\right\}\;,
\overrightarrow{A_{b\bar{b}}}^{N}(a,\bar{a}):=\left(\left(\left(e^{a\bar{b}^{(N+1)}-b^{(N+1)}\bar{a}}*\e^{a\bar{b}^{(N)}-b^{(N)}\bar{a}}\right)*\e^{a\bar{b}^{(N-1)}-b^{(N-1)}\bar{a}}\right)*\\cdots*\e^{a\bar{b}^{(0)}-b^{(0)}\bar{a}}\right).
E=\vec{\dot{\phi}}^{2}+[(\vec{A_{1}^{2}}+\vec{A_{2}^{2}})\vec{\phi}^{2}-(\vec{A_{1}}\cdot\vec{\phi})^{2}-(\vec{A_{2}}\cdot\vec{\phi})^{2}]+\frac{\kappa}{4}(\vec{\phi}^{2}-1)^{2}.
\ddot{R}-{\frac{2R}{1+R^{2}}}\dot{R}^{2}-{\frac{\dot{g}}{g}}\dot{R}-A^{2}g^{2}{\frac{(1-R^{2})(1+R^{2})^{3}}{R^{3}}}+{\frac{R(1-R^{2})}{1+R^{2}}}(h^{2}-l)=0.
\hat{\chi}=\sum_{M=1}^{n}\left(\bar{z}_{+}^{M}\,\,\bar{z}_{-}^{M}\right)\Big(\begin{array}{cc}{\mathrm{e}^{V}}\\\end{array}\Big)\left(\begin{array}{c}{z_{+}^{M}}\\{z_{-}^{M}}\\\end{array}\right)\,
\psi_{n\tof}(z)={\frac{1}{x}}\left[C_{1}E_{1}(x_{+}^{2}-x_{-}^{2})^{(1-\sqrt{1-\mu})/2}x^{\sqrt{1-\mu}}+C_{1}E_{2}(x_{+}^{2}-x_{-}^{2})^{(1+\sqrt{1-\mu})/2}x^{-\sqrt{1-\mu}}\right],
\Gamma\Bigl({\frac{D}{2}}-1\Bigr){{\frac{\dot{x}_{a}\cdot\dot{x}_{b}}{{\Bigl[{(x_{a}-x_{b})}^{2}\Bigr]}^{{\frac{D}{2}}-1}}}\quad}\nonumber\,-{\frac{1-\alpha}{4}}\Gamma\Bigl({{\frac{D}{2}}-2}\Bigr){\frac{\partial}{\partial\tau_{a}}}{\frac{\partial}{\partial\tau_{b}}}{\Bigl[{(x_{a}-x_{b})}^{2}\Bigr]}^{2-{\frac{D}{2}}}.
\Phi^{(2)}={\frac{T^{2}}{8}}\psi_{\mu}(\tau_{2})\psi_{\nu}(\tau_{2})\psi_{\kappa}(\tau_{1})\psi_{\lambda}(\tau_{1})[k_{\mu}\dot{x}_{\nu}(\tau_{2})-k_{\nu}\dot{x}_{\mu}(\tau_{2})][k_{\kappa}\dot{x}_{\lambda}(\tau_{1})-k_{\lambda}\dot{x}_{\kappa}(\tau_{1})]+(1\leftrightarrow2).
A(\hat{p})=\sum_{mn}\frac{1}{2\pis_{1}s_{2}}\int_{0}^{|s_{1}|}dp_{1}\int_{0}^{|s_{2}|}dp_{2}\Phi_{A}(p)e^{2i\pi[(\hat{p_{1}}-p_{1})\frac{m}{s_{1}}+(\hat{p_{2}}-p_{2})\frac{n}{s_{2}}]}.
V={x^{7}}\,{y^{7}}-{x^{5}}\,{y^{5}}\,\left({x^{3}}+{y^{3}}\right)+{x^{3}}\,{y^{3}}\,\left({x^{6}}+{y^{6}}\right)+x\,y\,\left({x^{9}}+{y^{9}}\right)\,
\sum_{{\bf{q}}_{1}\neq{\bf{0}},{\bf{q}}}{\bar{n}}_{{\bf{k}}-{\bf{q}}/2+{\bf{q}}_{1}}e^{i\mathrm{~}\theta({\bf{k}}+{\bf{q}}_{1}/2,-{\bf{q}}+{\bf{q}}_{1})}e^{-i\mathrm{~}\theta({\bf{k}}-{\bf{q}}/2+{\bf{q}}_{1}/2,{\bf{q}}_{1})}=0
J_{3}=\sum_{r=1}^{m}\;\left[2\,\left(\partial\theta_{r}\right)^{2}\,\left(\partial\theta_{r+1}-\partial\theta_{r-1}\right)+\left(\partial\theta_{r}\right)\,\left(\partial^{2}\theta_{r+1}-\partial^{2}\theta_{r-1}\right)\right],
e^{2k}=e^{({\frac{\alpha^{2}+1}{2\alpha^{2}}})2k_{g}}e^{-{(\gammaq_{1}q_{2}-{\frac{\tau_{o}^{2}}{2\alpha^{2}}})}2k_{e}}e^{-({\frac{\tau_{o}}{2\alpha^{2}}})2k_{s}}\equive^{2K_{2}k_{g}}e^{2K_{1}k_{e}}e^{2K_{3}k_{s}},
\deltam_{\lambda}^{2}=\deltam_{0}^{2}+\frac{1}{32\pi^{2}}\int_{0}^{1}dx\int_{{\calM}_{0}^{2}}^{\infty}d{\calM}^{2}\frac{f_{\lambda}^{2}-f_{\lambda_{0}}^{2}}{{\calM}^{2}-m^{2}}\;,
\frac{\partialz_{A}}{\partialz_{3}}=\frac{\partialH}{\partialb_{A}}=\frac{z_{A}(z_{A}-1)(z_{A}-z_{3})}{z_{3}(z_{3}-1)}\left[2b_{A}-\frac{\mu_{1}}{z_{A}}-\frac{\mu_{2}}{z_{A}-1}-\frac{\mu_{3}-1}{z_{A}-z_{3}}\right]
\begin{array}{c}{q_{0}^{J_{0}}J_{\pm}q_{0}^{-J_{0}}=q_{0}^{\pm1}J_{\pm}}\\{\left[J_{+},J_{-}\right]=\displaystyle\frac{q_{0}^{2J_{0}}-q_{0}^{-2J_{0}}}{q_{0}-q_{0}^{-1}}}\\\end{array}
\mathcal{H}^{\ast}=\intd^{2}x\left[\frac{1}{2}\left(\xi_{5}^{\ast2}+\xi_{6}^{\ast2}\right)+\frac{1}{2}\left(\partial_{1}\xi_{3}^{\ast}-\partial_{2}\xi_{2}^{\ast}\right)^{2}+\frac{1}{2}m^{2}\left(\xi_{2}^{\ast2}+\xi_{3}^{\ast2}\right)+m\left(\xi_{2}^{\ast}\xi_{6}^{\ast}-\xi_{3}^{\ast}\xi_{5}^{\ast}\right)\right].
\begin{array}{c}{\Omega=i\partial\bar{\partial}\left(K(z,\bar{z})-ig_{A\bar{B}}\theta^{A}{\bar{\theta}}^{\bar{B}}\right)=}\\{=i(g_{A\bar{B}}-iR_{A\bar{B}C\bar{D}}\theta^{C}\bar{\theta}^{\bar{D}})dz^{A}\wedged{\bar{z}}^{\bar{B}}}\\{+g_{A\bar{B}}D\theta^{A}\wedgeD{\bar{\theta}}^{\bar{B}},}\\\end{array}
iA^{+}=\left[\begin{array}{ll}{<\psi_{1}^{(0)}|d\psi_{1}^{(0)}>}&{<\psi_{2}^{(0)}|d\psi_{1}^{(0)}>}\\{<\psi_{1}^{(0)}|d\psi_{2}^{(0)}>}&{<\psi_{2}^{(0)}|d\psi_{2}^{(0)}>}\\\end{array}\right]=iA_{a}dx^{a},
V=-\frac{\Gamma(-1/2)}{\sqrt{8\pi^{2}\alpha^{\prime}}}[(\frac{r^{2}}{2\pi\alpha^{\prime}}-\pi\epsilon)^{1/2}+(\frac{r^{2}}{2\pi\alpha^{\prime}}+\pi\epsilon)^{1/2}-2(\frac{r^{2}}{2\pi\alpha^{\prime}})^{1/2}]
\begin{array}{c}{TA_{1}(t_{1})A_{2}(t_{2})....A(t_{n})=A_{i_{1}}(t_{i_{1}})A_{i_{2}}(t_{i_{2}})....A_{i_{n}}(t_{i_{n}})}\\{for\quadt_{i_{1}}\geqt_{i_{2}}\geq....\geqt_{i_{n}}}\\\end{array}
\eta^{\mu\nu\rho\sigma}=\frac{\epsilon^{\mu\nu\rho\sigma}}{\sqrt{-g}}\quad,\quadf=\frac{e^{\phi}}{8g^{2}}\quad,\quad\tilde{R}_{\,\,\,\,\kappa\lambda}^{\mu\nu}=\eta^{\mu\nu\rho\sigma}R_{\rho\sigma\kappa\lambda}\,\,.
R_{\\\alpha\beta}^{\mu\nu}R_{\\\lambda\rho}^{\alpha\beta}R_{\\\mu\nu}^{\lambda\rho}=\kappa^{3}\partial_{\rho}\partial_{\alpha}h_{\beta\nu}\partial_{\beta}\partial_{\lambda}h_{\mu\alpha}\partial_{\mu}\partial_{\nu}h_{\lambda\rho}+\ldots
\tilde{A}_{\mu}(x)=-\frac{2\sqrt{6}}{\bar{N}}\int\delta\xids\omega(\xi(s_{+}))T_{\mu\nu}[\xi|s]\omega^{-1}(\xi(s_{+}))\dot{\xi}^{\nu}(s)\dot{\xi}(s)^{-2}\delta(x-\xi(s)).
F^{\mu}=e^{-\tau}\partial_{\nu}\left(e_{a}^{~\mu}e^{a\nu}e^{\tau}\right)+a_{1}\bar{\Omega}_{~~\alpha}^{\mu\alpha}+a_{2}\bar{\Omega}_{~~a}^{a\mu}+a_{3}\bar{\Omega}_{~a}^{a~\mu}
e^{f(0)L_{-1}}\;f^{\prime}(0)^{L_{0}}\prod_{n=1}^{\infty}e^{S_{n}L_{n}}V(\psi,z)\prod_{n=\infty}^{1}e^{-S_{n}L_{n}}\;f^{\prime}(0)^{-L_{0}}\;e^{-f(0)L_{-1}}=f^{\prime}(z)^{h}\;V(\psi,f(z)).
[D_{\nu}{\hat{f}}_{m}^{\alpha}](\theta)=[D_{r+1}P_{1-{\nu}_{1}}{\hat{f}}_{m}^{\alpha}](\theta)=[DP_{1-{\nu}_{1}}D_{r}{\hat{f}}_{m}^{\alpha}](\theta)+\sum_{p=0}^{r-1}[D_{p}f](0)\frac{\theta^{p-\nu}}{\Gamma(p-\nu+1)}\tag{B.39}
\left(\begin{array}{cc}{(l-1)(l-2)-4}&{2}\\{4}&{(l-1)(l-2)-2}\\\end{array}\right)\left(\begin{array}{c}{s_{1}}\\{s_{2}}\\\end{array}\right)\equiv{\calA}\left(\begin{array}{c}{s_{1}}\\{s_{2}}\\\end{array}\right)=\left(\begin{array}{c}{0}\\{0}\\\end{array}\right),
\frac{1}{2}\hat{\tilde{F}}_{I}(A^{-1})_{IJ}\hat{\tilde{F}}_{J}=\frac{1}{2(p^{\prime}+1)!}\sqrt{-\hat{g}}\hat{g}^{\mu_{1}\nu_{1}}\cdots\hat{g}^{\mu_{p^{\prime}}\nu_{p^{\prime}}}\hat{g}^{zz}\hat{\tilde{F}}_{\mu_{1}\cdots\mu_{p^{\prime}}z}\hat{\tilde{F}}_{\nu_{1}\cdots\nu_{p^{\prime}}z}
S=\;\left(\begin{array}{lll}{-\mathrm{cos}\;2\vartheta}&{\quad}&{\mathrm{sin}\;2\vartheta}\\{\;\;\mathrm{sin}\;2\vartheta}&{\quad}&{\mathrm{cos}\;2\vartheta}\\\end{array}\right)\\\\mathrm{and}\\\S^{\prime}=\;\left(\begin{array}{lll}{\;\mathrm{cos}\;2\vartheta}&{\quad}&{-\mathrm{sin}\;2\vartheta}\\{\;\mathrm{sin}\;2\vartheta}&{\quad}&{\;\;\mathrm{cos}\;2\vartheta}\\\end{array}\right)\.
{\calW}_{\gamma}(L,T)=1+{\frac{1}{N}}\sum_{n=2}^{\infty}(ig)^{n}\oint_{1}^{\mu_{1}}\cdots\oint_{\gamma}dx_{n}^{\mu_{n}}\theta(x_{1}>\cdots>x_{n})\mathrm{Tr}[G_{\mu_{1}\cdots\mu_{n}}(x_{1},\cdots,x_{n})]\,
\tilde{j}^{1}=-\gamma\frac{i}{2}(\bar{\phi}D_{0}\phi-\phi\overline{{D_{0}\phi}})-\frac{1}{2}B^{2}-\frac{1}{2}|D_{1}\phi|^{2}+\frac{1}{2}|D_{2}\phi|^{2}+\frac{\lambda}{8}(1-|\phi|^{2})^{2}-\gammax_{2}E_{2}
\left(\begin{array}{cc}{m_{11}}&{m_{12}}\\{m_{21}}&{m_{22}}\\\end{array}\right)\to\left(\begin{array}{cc}{a}&{b}\\{c}&{d}\\\end{array}\right)\left(\begin{array}{cc}{m_{11}}&{m_{12}}\\{m_{21}}&{m_{22}}\\\end{array}\right),
Z=\frac{1}{2}\frac{(\mathrm{Im\tau})^{-d}}{|\eta(q)|^{2d}}\sum_{b_{0}=0}^{2K-1}\sum_{b_{1},\dots,b_{r}=0,1}\sum_{\lambda,\mu}^{\mathrm{ev},\beta}(-1)^{b_{0}}M_{\lambda\lambda^{\prime}}\chi_{\mu}^{\lambda}(q)\chi_{\mu+b_{0}\beta_{0}+\sum_{j=1}^{r}b_{j}\beta_{j}}^{\lambda^{\prime}}(\bar{q})\;,
\left[i\sum_{r>0}{\frac{1}{r}}\beta_{r}^{i}z^{-r}\,,\,i\sum_{s<0}{\frac{1}{s}}\beta_{s}^{j}w^{-s}\right]=-g^{ij}\mathrm{ln}\left({\frac{\sqrtz-\sqrtw}{\sqrtz+\sqrtw}}\right),
C_{1,-3}\;=\;\left\{(J,J_{1},\varepsilon):\pi/4\leq\varepsilon\leq\pi/2\;,\;J=0\;\mathrm{and}\;\varepsilon\geq\frac{\pi}{4}+\mathrm{arcsin}\left(\frac{e^{2J_{1}}}{\sqrt{2}}\right)\right\}.
\Lambda_{ij,\,lm}(\hat{k})=\delta_{il}\delta_{jm}-2\hat{k}_{j}\hat{k}_{m}\delta_{il}+\frac{1}{D-2}\left(-\delta_{ij}\delta_{lm}+\hat{k}_{l}\hat{k}_{m}\delta_{ij}+\hat{k}_{i}\hat{k}_{j}\delta_{lm}\right)+\frac{D-3}{D-2}\hat{k}_{i}\hat{k}_{j}\hat{k}_{l}\hat{k}_{m}\,.