harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde{H}{}_{n}^{f}=\frac{1}{2}\left[-\frac{d^{2}}{dx^{2}}+W_{+,\nu}^{2}+\nu-1-\frac{1}{2}(W_{+,\nu}^{\prime}+\nu-1)\right]-\frac{1}{4}(W_{+,\nu}^{\prime}+\nu-1)\cdot\sigma_{i}\otimes\sigma_{i}.
\left({\frac{\dot{\bar{a}}_{R}}{\bar{a}_{R}}}\right)^{2}+{\frac{k(c^{2}+R_{\tau}^{2})}{\bar{a}_{R}^{2}}}={\frac{8\pim_{0}e^{2m_{0}R_{0}}G_{5}(1+R_{\tau}^{2}/c^{2})^{3/2}}{3(1-\Omega_{R}^{2})}}\left[\tilde{\varrho}_{1}+\varrho_{2}\Omega_{R}^{4}+{\frac{V_{r}}{c^{2}\sqrt{1+R_{\tau}^{2}/c^{2}}}}\right],
g^{*}h_{\alpha\beta\gamma}^{B}\:=\:\left(h_{\alpha\beta\gamma}^{B}\right)\,\left(\nu_{\alpha\beta\gamma}^{Fg}\right)^{-1}\,\left(\nu_{\alpha\beta\gamma}^{Rg}\right)\,\left(\kappa_{\alpha\beta}^{g}\right)\,\left(\kappa_{\beta\gamma}^{g}\right)\,\left(\kappa_{\gamma\alpha}^{g}\right)
\hat{{\calG}}_{\perp}^{(0)}=\hat{{\calG}}_{\parallel}^{(0)}=\hat{{\calG}}^{(0)}=-\frac{\partial^{2}}{\partial\tau^{2}}-\frac{\partial^{2}}{\partialx^{2}}+\mu^{2}=-\frac{\partial^{2}}{\partial\tau^{2}}+\mu\left[\frac{\partial^{2}}{\partialy^{2}}+1\right]
V_{(p)}(\tilde{\xi})={\frac{1}{4}}(7-p)(8-p){\frac{(\omegaR_{p})^{7-p}}{y^{9-p}}}\Biggl[1+{\frac{(p-3)}{4(8-p)}}{\frac{(\omegaR_{p})^{7-p}}{y^{7-p}}}\Biggr]H_{p}(y)^{-3}\quad.
_0\left<\right|T\left(\Psi^{R}(x_{2})\bar{\Psi}^{R}(x_{1})\right)\left|\right>_{0}=-\frac{1}{(2\pi)^{3}}\int{\mathrmd}^{3}p\frac{-i\hat{p}+m}{2E}e^{ip(x_{2}-x_{1})}
\gamma_{\mu}=\left(\begin{array}{cc}{\sigma_{\mu}}&{0}\\{0}&{-\sigma_{\mu}}\\\end{array}\right)\,,\quad\sigma_{\mu\nu}={\frac{1}{2i}}[\gamma_{\mu},\gamma_{\nu}]=\epsilon_{\mu\nu\rho}\sigma_{\rho}\,;\quad\mu,\nu,\rho=1,2,3\,.
{\bfX}^{MN}=\left(\begin{array}{cc}{\sqrt{\mu_{0}}\,\sigma^{\mu\nu}}&{\delta^{\mu\nu}x_{\nu}^{C.M.}}\\{\delta^{\mu\nu}x_{\nu}^{C.M.}}&{\sqrt{\mu_{0}}\,A_{\mu\nu}}\\\end{array}\right)\,\quadA_{\mu\nu}=\left(\begin{array}{cc}{\vec{0}}&{A}\\{-A}&{\vec{0}}\\\end{array}\right)
K=\sum_{i}K_{i}\chi_{i}\;\;\overrightarrow{\;\;{}_{S}\;\;\;\;}\;\;\tilde{K}=\sum_{i}\Gamma_{i}^{2}\chi_{i}\;\;\;\;\;\mathrm{where}\;\;\;K_{i}=\sum_{b}S_{bi}\Gamma_{b}\Gamma_{b}\;\;.
[(hr^{3}\partial_{r})^{2}+\omega^{2}r^{6}{\mathcalH}-8\frac{hr^{4}r_{a,b}^{4}}{(r^{2}+r_{a,b}^{2})^{2}}(1+\frac{r_{0}^{2}}{r_{a,b}^{2}})]\phi_{a,b}=0,
\bar{L}_{2}=H_{0}{\frac{\delta\bar{L}_{2}}{\deltaH}}-3H_{0}^{2}{\frac{\delta\bar{L}_{2}}{\delta\dot{H}}}={\frac{1}{2}}H_{0}^{2}{\frac{\delta\bar{L}_{2}}{\deltaH_{0}^{2}}}
h_{\mu\nui}^{(1)}=b_{\mu\nui}^{\prime}+V_{\mu\nu}^{k}b_{ki}\,\,\,,\,\,\,h_{\mu\nui}^{(2)}=[V_{\mu}^{k}b_{i\nuk}^{\prime}-(\mu\leftrightarrow\nu)]-\frac{1}{2}\partial_{i}U_{\mu\nu}-{}F_{\mu\nu}^{k}b_{ki}\,\,.
T_{1}^{h}(x)=i\kappa:{\mathcal{L}}_{\scriptscriptstyleHE}^{\scriptscriptstyle(1)}(x):=i\frac{\kappa}{2}\left\{:h^{\alpha\beta}(x)h^{\rho\sigma}(x)_{,\alpha}h_{\rho\sigma}(x)_{,\beta}:\ldots\right\}.
\chi=\prod_{i<j}^{m}(\mu_{i}-\mu_{j})^{\frac{(\alpha_{i},\alpha_{j})}{\kappa}}\prod_{a<b}^{N}(Z_{a}-Z_{b})^{\frac{(\lambda^{a},\lambda^{l})}{\kappa}}\prod_{i,a}^{m,N}(\mu_{i}-Z_{a})^{\frac{(\alpha_{i},\lambda^{a})}{\kappa}}.
\phi_{1}(x+y)={\frac{\biggl|\begin{matrix}{\phi_{2}(x)}&{\phi_{2}(y)}\\{\phi_{3}(x)}&{\phi_{3}(y)}\\\end{matrix}\biggr|}{\biggl|\begin{matrix}{\phi_{4}(x)}&{\phi_{4}(y)}\\{\phi_{5}(x)}&{\phi_{5}(y)}\\\end{matrix}\biggr|}}
\int_{0}^{\infty}dt\mathrm{Tr}(\gamma_{5}\deltabe^{-tDD^{\dagger}})e^{-tM_{i}^{2}L_{1}^{2}}=\frac{1}{2\piM_{i}^{2}L_{1}^{2}}\int_{\calT}d^{2}x\deltabeF_{12}(x).
\frac{i}{\sqrt{2p^{+}}}\left[\gamma_{0}(\tau)-{\gamma_{0}^{\dagger}}(\tau)\right]=\frac{i}{2\sqrt{\mup^{+}}}\left(e^{-2i\mu\tau}a_{0}+e^{2i\mu\tau}\bar{a}_{0}\right)-\frac{mR_{1}}{2\mul_{s}^{2}p^{+}}.
\frac{d^{2}x^{a}}{d\hat{s}^{2}}+\{_{mn}^{\;\;a}\}_{hat}\frac{dx^{m}}{d\hat{s}}\frac{dx^{n}}{d\hat{s}}-\frac{\psi_{,n}}{2}(\frac{dx^{n}}{d\hat{s}}\frac{dx^{a}}{d\hat{s}}-\hat{g}^{na})=0,
\lbrack(P+K)_{k},\chi_{\bar{j}}^{\alpha}\rbrack=\lbrack(P+K)_{k},\bar{\chi}^{\bar{j}\alpha}\rbrack=0\qquad\qquad\{\chi_{\bar{l}}^{\alpha},\bar{\chi}_{\beta}^{\bar{j}}\}=\delta_{\bar{l}}^{\bar{j}}\delta_{\beta}^{\alpha}
\left(\begin{array}{r}{\tilde{\hat{F}^{I\prime\}}{}^{\hat{\mu}\hat{\nu}}}\\{\hat{F}^{I\prime\\hat{\mu}\hat{\nu}}}\\\end{array}\right)=\left(\begin{array}{cc}{a}&{b}\\{}&{}\\{c}&{d}\\\end{array}\right)\left(\begin{array}{r}{\tilde{\hat{F}^{I}\}{}^{\hat{\mu}\hat{\nu}}}\\{\hat{F}^{I\\hat{\mu}\hat{\nu}}}\\\end{array}\right)\,,
\Phi(x^{a})=\Phi_{in}(x^{a})-2\intdt^{\prime}\G_{\bfR}(x^{a},Z^{(0)a}(t^{\prime}))\\left[\frac{\kappa_{5}^{2}a(t^{\prime})^{3}}{{\bfk}^{2}}S(t^{\prime})+\left.r\partial_{\perp}(r^{-1}\Phi_{in})\right|_{x^{a}=Z^{(0)a}(t^{\prime})}\right],
S^{ii}(\theta)=\left(\begin{array}{ccc}{(2)_{\theta}}&{(3)_{\theta}(1)_{\theta}}&{-(2)_{\theta}}\\{(3)_{\theta}(1)_{\theta}}&{(2)_{\theta}^{2}}&{(3)_{\theta}(1)_{\theta}}\\{-(2)_{\theta}}&{(3)_{\theta}(1)_{\theta}}&{(2)_{\theta}}\\\end{array}\right),
\Pi_{\mu\nu}^{(\mathrm{tadpole})}=\frac{1}{2}(2e^{2}\delta_{\mu\nu})\frac{1}{\beta}\sum_{n}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{1}{k_{0}^{2}+\omega_{\sigma}^{2}}=e^{2}\delta_{\mu\nu}\frac{\beta}{(2\pi)^{2}}\sum_{n}\int\frac{d^{2}k}{(2\pi)^{2}}\,\frac{1}{n^{2}+(\frac{\beta\omega_{\sigma}}{2\pi})^{2}},
\frac{1}{2}\mathcal{D}_{0}\left(\begin{array}{cccc}{p}&{p}&{q}&{r}\\{0}&{0}&{0}&{0}\\\end{array}\right)\pm\frac{1}{2}\mathcal{D}_{0}\left(\begin{array}{ccc}{p}&{q}&{r}\\{0}&{\frac{1}{2}}&{\frac{1}{2}}\\\end{array}\right)
\psi^{(+)}({\bfk,r})=\psi(r,\theta,\Theta)+\frac{k^{2}}{(2\pi)^{2}}\,\int_{0}^{2\pi}d\theta^{\prime}\,\int_{0}^{2\pi}d\Theta^{\prime}\,\psi(r,\theta-\theta^{\prime},\Theta-\Theta^{\prime})\,\chi(r,\theta^{\prime},\Theta^{\prime})\,+\mathrm{O}(k^{4}\psi)\,,
S_{\scriptscriptstyleYM}=\intd^{4}x\;Tr\left\{-\frac{1}{4}F_{\mu\nu}^{a}F^{a\mu\nu}-\frac{1}{2}\bar{\lambda^{a}}\hat{D}\lambda^{a}+\frac{1}{2}(D^{a})^{2}\right\},
\Psi_{E}^{n\geq\ell+1}\left(r,\theta\right)=\left(\begin{array}{l}{A_{n}^{+}J_{n-\ell-a}\left(kr\right)e^{in\theta}}\\{-i\frac{k}{E-m}A_{n}^{+}J_{n+1-\ell-a}\left(kr\right)e^{i\left(n+1\right)\theta}}\\{A_{n}^{-}J_{n-\ell-a}\left(kr\right)e^{in\theta}}\\{-i\frac{k}{E+m}A_{n}^{-}J_{n+1-\ell-a}\left(kr\right)e^{i\left(n+1\right)\theta}}\\\end{array}\right)\,.
W(\gamma)={\bf\Big<0}|{\calT}\Big(exp\Big[{\frac{ig}{\sqrt{2L}}}\oint_{\gamma}(b_{0}+a_{0}t)dx^{+}\Big]\Big)|{\bf0\Big>}{\bf\Big<0}|{\calT}\Big(exp\Big[ig\oint_{\gamma}\hat{A}(t,x)dx^{+}\Big]\Big)|{\bf0\Big>}=W_{0}\cdot\hat{W},
I=I_{1}(j_{a},m_{a},\bar{m}_{a})\frac{(i\pib^{2})^{2n-2}}{\Gamma[b^{2}(\Sigmaj_{a}+1)]^{2}}\frac{\Upsilon[b]\prod_{a=1}^{3}\Upsilon[(2j_{a}+1)b]}{\Upsilon[(\Sigmaj_{a}+1)b]\prod_{a<b}^{3}\Upsilon[(j_{ab}+1)b]},
\left({\frac{\dot{a}}{a}}\right)^{2}={\frac{(h^{\prime}|g_{00}|g-b^{2})\left[h^{\prime}g_{S}g(C+E)^{2}-(h^{\prime}|g_{00}|g-b^{2})(h^{\prime}g_{S}g^{3}e^{-2\phi}+\ell^{2})\right]}{4h^{\prime}g_{rr}g_{S}g\left[(h^{\prime}|g_{00}|g-b^{2})^{2}g^{2}e^{-2\phi}+b^{2}(C+E)^{2}\right]}}\left({\frac{g^{\prime}}{g}}\right)^{2}.
q_{b}(a,r)=-\frac{1}{2\pia^{3}r^{n}S_{D}\sigma^{D-1}}\sum_{l=0}^{\infty}D_{l}\int_{ma}^{\infty}dz\,z^{3}\frac{\bar{K}_{\nu_{l}}(z)}{\bar{I}_{\nu_{l}}(z)}
\Gamma_{C}\equiv\left(\begin{matrix}{C}&{0}\\{0}&{-C}\\\end{matrix}\right)=\sum_{n=2,4,6}\Gamma^{i_{1}}\cdots\Gamma^{i_{n}}F_{i_{i}\cdotsi_{m}}(X^{a^{\prime}},\theta^{\alpha^{\prime}},h_{ijk},a)\,,
D_{\!x}^{\sigma}\sum_{k=0}^{\infty}a_{k}x^{k}=\!\!\sum_{k\,=\left\lceil\sigma\right\rceil}^{\infty}\!a_{k}\,\frac{\Gamma(1\!+\!k)}{\Gamma(1\!+\!k\!-\!\sigma)}\,x^{k-\sigma}\quad(\sigma>0)
\dot{q}^{\lambda}\bar{\nabla}_{\lambda}(\dot{q}^{\delta}\bar{\nabla}_{\delta}\eta^{\mu})-\bar{R}_{\sigma\rho\lambda}^{\mu}\dot{q}^{\rho}\dot{q}^{\sigma}\eta^{\lambda}-\eta^{\mu}=2H_{\rho\sigma}^{\mu}\dot{q}^{\rho}(\dot{q}^{\delta}\bar{\nabla}_{\delta}\eta^{\sigma}-\eta^{\sigma}),
\{Q_{A},Q_{B}\}=Z_{AB}=\left(C\Gamma^{\widetilde{\mu}_{1}\widetilde{\mu}_{2}}\right)_{AB}\,Z_{[\widetilde{\mu}_{1}\widetilde{\mu}_{2}]}+\left(C\Gamma^{\widetilde{\mu}_{1}\ldots\widetilde{\mu}_{6}}\right)_{AB}\,Z_{[\widetilde{\mu}_{1}\ldots\widetilde{\mu}_{6}]}\,,
r_{\mathrm{CTC}}=(\frac{a^{2}\alpha^{2}b}{1-\frac32a^{2}\alpha^{2}})^{\frac13}\frac{\sqrt{2\sqrt{\overline{{s}}^{2}+4\overline{{q}}^{2}}-\overline{{s}}}-\sqrt{\overline{{s}}}}{2\alpha}
L=\intd^{3}x{\calL}=\sum_{p}y_{p}^{\dagger}(i\partial_{0}-i\gamma^{0}\gamma^{i}p_{i}-m\gamma^{0})y_{p}+\eta_{p}^{\dagger}\gamma^{0}y_{p}+y_{p}^{\dagger}\gamma^{0}\eta_{p}
|\phi_{(D)}>=\prod_{\vec{p},h=\pm1}d_{\vec{p},h}^{+}|0>=\pi_{\alpha}\;e^{-\sum_{\vec{p},h}\frac{\beta_{\vec{p}}}{\alpha_{\vec{p}}}\;b_{\vec{p},h}^{(0)+}d_{\vec{p},h}^{(0)}}\;|\phi_{(0)}>,\;\pi_{\alpha}=\prod_{\vec{\acute{p}}}\alpha_{\vec{\acute{p}}}.
g=\displaystyle\prod_{j=1}^{n}g_{j}^{(1)}=\displaystyle\prod_{j=1}^{n}\Biggl({\frac{a_{0}^{(j)}+e^{-{k_{j}}(r+it)}}{\bar{a}_{0}^{(j)}+e^{{k_{j}}(r+it)}}}\Biggr)
\frac{d}{d\tau}(g_{\mu\nu}{\dot{x}}^{\nu})-\frac{1}{2}\partial_{\mu}g_{\nu\lambda}{\dot{x}}^{\nu}{\dot{x}}^{\lambda}=(\frac{kd_{\tau}}{R}F_{\mu\nu}+\frac{mkR}{\sqrt{\alpha^{\prime}}}\tilde{F}_{\mu\nu}){\dot{x}}^{\nu},
h^{rel}=8a^{2}\biggl[\left(\zeta_{1}^{2}-3\zeta_{2}\right)\partial_{\zeta_{1}}^{2}+(\zeta_{1}\zeta_{2}-9)\partial_{\zeta_{1}\zeta_{2}}^{2}+\left(\zeta_{2}^{2}-3\zeta_{1}\right)\partial_{\zeta_{2}}^{2}+(3\lambda+1)\left(\zeta_{1}\partial_{\zeta_{1}}+\zeta_{2}\partial_{\zeta_{2}}\right)\biggr].
\hat{T}_{N}^{M}=\mathrm{diag}[-\hat{\rho},\hat{p},\hat{p},\hat{p},\hat{p}_{5}]+\sum_{i=\mathrm{branes}}\frac{\delta(y_{i})}{b}\mathrm{diag}[-\rho_{i},p_{i},p_{i},p_{i},0].
T_{+}=\left(\begin{array}{cc}{0}&{1}\\{0}&{0}\\\end{array}\right),\quadT_{-}=\left(\begin{array}{cc}{0}&{0}\\{1}&{0}\\\end{array}\right),\quadT_{3}=\frac12\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right).
\left({\frac{d}{d\rho}}\left(F_{,\phi_{i}\dot{\phi}_{j}}+F_{,\dot{\phi}_{k}\dot{\phi}_{j}}{\frac{\partial\dot{\phi}_{k}}{\partial\phi_{i}}}\right)\right)_{\dot{\phi}_{l}}~=~S_{,\rho\phi_{i}\phi_{j}}+S_{,\phi_{i}\phi_{j}\phi_{k}}\dot{\phi}_{k}.
{\frac{1}{T}}={\frac{dS}{dE}}={\frac{d(S_{L}+S_{R})}{dE}}=2\left({\frac{dS_{L}}{dE_{L}}}+{\frac{dS_{R}}{dE_{R}}}\right)=2\left({\frac{1}{T_{L}}}+{\frac{1}{T_{R}}}\right),
M^{2}\simeq\widetilde{M}^{2}+R\left\{-\frac{1}{4}-\frac{\widetilde{M}^{4}}{8B^{2}}+\frac{2B^{2}+\widetilde{M}^{4}}{2B\widetilde{M}^{4}\left[\psi^{\prime}\left(1+\widetilde{M}^{2}/(2B)\right)+\psi^{\prime}\left(\widetilde{M}^{2}/(2B)\right)\right]}\right\}.
\begin{array}{lr}{\bar{\beta}_{00}^{G}=2\alpha^{\prime}D_{0}\partial_{0}\phi(X^{0})=2\alpha^{\prime}\ddot{\phi}\,}&{}\\{\bar{\beta}_{mn}^{G}=\beta_{mn}^{G}}&{(D_{m}\partial_{n}\phi=0\\mathrm{if}\K=\mathrm{const.\and}\\partial_{m}\phi=0)\,}\\{\bar{\beta}^{\phi}=\beta^{\phi}+\frac{\alpha^{\prime}}{a^{2}}\left(\dot{\phi}(X^{0})\right)^{2}\.}&{}\\\end{array}
{\calB}(r)=-\ell(\ell+1)+(kr_{0})^{2}\left(2{\frac{r^{2}}{r_{0}^{2}}}+4{\frac{r}{r_{0}}}-1\right)+(1+2ikr_{0}){\frac{r}{j_{\ell}(kr)}}\;{\frac{dj_{\ell}(kr)}{dr}}
\varepsilon=\frac{1}{R^{4}}\left\{\left[-\frac{\pi^{2}}{90}+\frac{(mR)^{2}}{24}-\frac{(mR)^{3}}{12\pi}+\ldots\right]+\lambda\left[\frac{1}{1152}-\frac{mR}{192\pi}+\ldots\right]\right\}+\ldots
\deltaA(x)=\deltax^{M}\partial_{M}A(x)=a^{\mu}x^{\nu}\eta_{\mu\nu}+{\frac{1}{2}}a_{D}+a_{K}^{\mu}x^{\nu}\eta_{\mu\nu}{\frac{x^{\lambda}x^{\sigma}\eta_{\lambda\sigma}}{r^{2}}}.
Z_{\delta}^{\pm}(q)=\int[dx]\,exp\left(-\int_{-\beta/2}^{\beta/2}dt\left\{{\frac{1}{2}}\left[\dot{x}^{2}+(m^{2}+2\lambda_{\pm})x^{2}\right]\mpiqx+\delta(igx^{3}-\lambda_{\pm}x^{2})\right\}\right).
dX_{\mathrm{cl}}^{0}=\sum_{i=1}^{g}\left(\lambda_{i}\omega_{i}+\bar{\lambda}_{i}\bar{\omega}_{i}\right)~~,~~dX_{\mathrm{cl}}^{1}=\sum_{i=1}^{g}\left(\gamma_{i}\omega_{i}+\bar{\gamma}_{i}\bar{\omega}_{i}\right)
X^{\mu}(\sigma,\tau)=q^{\mu}+2\alpha^{\prime}p^{\mu}\tau+i\sqrt{\alpha^{\prime}}\sum_{n\neq0}\frac{1}{n}[\alpha_{n}^{\mu}\;e^{-in(\tau-\sigma)}+\tilde{\alpha}_{n}^{\mu}\;e^{-in(\tau+\sigma)}]
G_{\alpha\beta}=\left(\begin{array}{ccccc}{-(1-\tilde{v}^{2})}&{0}&{0}&{0}&{-\tilde{v}}\\{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}\\{-\tilde{v}}&{0}&{0}&{0}&{1}\\\end{array}\right),
D_{\mu\mu}=-\frac{g_{\mu\mu}}{32\pi^{2}}\left[\frac{\alpha^{2}}{(3-d)}+32\pi^{2}\alpha\left(\frac{\mu_{R}^{2}}{\lambda_{R}}+\frac{1}{2}\varphi^{2}\right)-16\pi^{2}\frac{\alpha^{2}}{\lambda_{R}}\right]\;\;,
\Pi^{\wedge\vee}={\frac{g^{2}N_{c}}{8\pi^{2}}}\bigg[9\sum_{k=1}^{\infty}{\frac{u^{k}}{k^{2}}}-9\sum_{k=1}^{\infty}f_{k}{\frac{u^{k}}{k}}-\sum_{k=1}^{\infty}h_{k}(1){\frac{u^{k}}{k}}\bigg],
d_{n_{1},n_{2},\cdots,n_{N_{f}+2}}\left\{\begin{array}{ll}{\sim\gamma_{n_{1}}\,(-1)^{N_{1}+\cdots+N_{N_{f}}}\,N_{0}^{4n_{1}-3}\,\prod_{i=1}^{N_{f}}\,{_{n_{1}}}C_{N_{i}}}&{~~~(0\leq{^\forall}N_{i}\leqn_{1})}\\{=0}&{~~~\mathrm{otherwise}}\\\end{array}\right\},
\intd^{8}x\sqrt{-\gamma}\sum_{p=2-2g}^{2g-2}(-)^{p}R^{4}(H_{RR}^{++})^{2g-2+p}(H_{RR}^{--})^{2g-2-p}\hat{\sum_{m,n}}\frac{{T_{2}}^{g}}{(m+nT)^{g+p}(m+n\bar{T})^{g-p}}
{\frac{\partial{\calL}}{\partialt_{k}}}={\calL}{\calA}_{k}-\hat{{\calA}}_{k}{\calL},\,\,\,{\frac{\partial\bar{\calL}}{\partialt_{k}}}=\bar{\calL}{\calA}_{k}-\hat{{\calA}}_{k}\bar{\calL}\,\,\,\;.
{\calZ}_{open}=\int_{0}^{\infty}\frac{ds}{s}\left[\mathrm{Tr}_{\mathrm{NS}}\left(\frac{1+{\calP}}{2}\,q^{2L_{0}-1}\right)-\mathrm{Tr}_{\mathrm{R}}\left(\frac{1+{\calP}}{2}\,q^{2L_{0}}\right)\right]~~,
P_{N}^{(i)}(j_{1},\cdots,j_{N})=\frac{G_{N}^{(i)}(z,0|0,\cdots,0)^{j_{1},\cdots,j_{N},j_{N},\cdots,j_{1}}}{\displaystyle\sum_{j_{1},\cdotsj_{N}}G_{N}^{(i)}(z,0|0,\cdots,0)^{j_{1},\cdots,j_{N},j_{N},\cdots,j_{1}}}
\bar{R}_{~\lambda\mu\nu}^{\sigma}(\bar{\Gamma})=\partial_{\mu}\bar{\Gamma}_{~\lambda\nu}^{\sigma}-\partial_{\nu}\bar{\Gamma}_{~\lambda\mu}^{\sigma}+\bar{\Gamma}_{~\alpha\mu}^{\sigma}\bar{\Gamma}_{~\lambda\nu}^{\alpha}-\bar{\Gamma}_{~\alpha\nu}^{\sigma}\bar{\Gamma}_{~\lambda\mu}^{\alpha}
\hat{T}_{\mu\nu}(k)=\frac{1}{\pi}\Big(\delta_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}\Big)\Bigg(1+2\frac{m^{2}}{k^{2}}\mathrm{ln}\Big(\frac{m^{2}}{k^{2}}\Big)+O\Big(\frac{m^{4}}{k^{4}}\mathrm{ln}\Big(\frac{m^{2}}{k^{2}}\Big)\Big)\Bigg)\;.
\frac{a^{\prime\prime}}{a}+\frac{{a^{\prime}}^{2}}{a^{2}}-\frac{a^{\prime}}{a}\frac{b^{\prime}}{b}+\frac{1}{12}\frac{{V^{\prime}}^{2}}{V^{2}}+\frac{\alpha_{0}^{2}}{18}\frac{b^{2}}{V^{2}}=\frac{\sqrt{2}\alpha_{0}}{3}\frac{b}{V}\left(\delta(y)-\delta(y-\pi\rho)\right)
\hat{G}_{\hat{\mu}\hat{\nu}}=\hat{g}_{\hat{\mu}\hat{\nu}}+\hat{B}_{\hat{\mu}\hat{\nu}}-{\textstyle\frac{1}{2}}\left\{\alpha\hat{\Omega}_{\hat{\mu}}{}^{(0)\hat{a}\hat{b}}\hat{\Omega}_{\hat{\nu}}{}^{(0)\hat{a}\hat{b}}+\beta\hat{V}_{\hat{\mu}}^{I}\hat{V}_{I\hat{\nu}}\right\}\,.
\delta_{tot}|\phi\rangle=\Bigl(\xi\partial+\frac{d-2}{2z}\xi^{z}+\frac{1}{2}\partial^{I}\xi^{J}M^{IJ}+M^{IJ}\partial^{+}\xi^{I}\frac{\partial^{J}}{\partial^{+}}-\frac{\partial^{+}\xi^{I}}{2z\partial^{+}}\{M^{zj},M^{jI}\}\Bigr)|\phi\rangle
\Sigma_{A}(\bar{p}_{\parallel})~\simeq~2g^{2}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{\mathrm{e}^{-\hat{q}_{\perp}^{2}}}{q^{2}}\frac{\Sigma_{A}(\bar{p}_{\parallel}-q_{\parallel})}{(\bar{p}_{\parallel}-q_{\parallel})^{2}+\Sigma_{A}^{2}(\bar{p}_{\parallel}-q_{\parallel})},
G_{k_{1},k_{2}}(z_{12})=(z_{12})^{-h_{1}-h_{2}}\left(D_{(h_{1};k_{1})(h_{2};k_{2})}-\int^{z_{12}}\!\frac{\mathrm{d}\zeta}{\zeta^{1-h_{1}-h_{2}}}[G_{k_{1}-1,k_{2}}(\zeta)+G_{k_{1},k_{2}-1}(\zeta)]\right)\,.
\lambda^{2}\left((A^{\dagger}A)_{11}-(A^{\dagger}A)_{22}\right)=\zeta\left((\mathbf{p})^{\dagger}\mathbf{p}+(\mathbf{p}^{\prime})^{\dagger}\mathbf{p}^{\prime}\right)\,,\quad(A^{\dagger}A)_{12}=0\,.
L^{\prime}=L+b_{\alpha}\chi^{\alpha}+\frac{1}{2}b_{\alpha}b_{\beta}\gamma^{\alpha\beta}-\bar{c}_{\alpha}s(q^{r})\frac{\partial\chi^{\alpha}}{\partialq^{r}}-\bar{c}_{\alpha}s(\dot{q}^{r})\frac{\partial\chi^{\alpha}}{\partial\dot{q}^{r}}.
F\approx-64\frac{\gamma\zeta(5)}{\pih^{2}}\sqrt{\frac{h}{2M}}\left(\frac{M}{\beta}\right)^{5}-\frac{3}{8\pik}\gammaL^{3}\int_{0}^{\infty}d\omega\frac{\omega^{4}}{e^{\beta\omega}-1}.
\hat{\Phi}(x^{0},x^{1})=\frac{1}{\sqrt{2\pi}}\int_{0^{+}}^{\infty}dk^{1}\frac{1}{\sqrt{k^{1}}}\left[\hat{\Lambda}(k^{1})\,e^{-ik^{1}(x^{0}+x^{1})}\,\,+\,\,\hat{\Lambda}^{\dagger}(k^{1})\,e^{ik^{1}(x^{0}+x^{1})}\right],
\left[\psi^{a}\psi^{b},\psi^{c}\psi^{d}\right]=\eta^{bc}\psi^{a}\psi^{d}-\eta^{bd}\psi^{a}\psi^{c}+\eta^{ad}\psi^{b}\psi^{c}-\eta^{ac}\psi^{b}\psi^{d},
\frac{1}{2}\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{-1}&{0}\\{0}&{0}&{0}\\\end{array}\right),\quad\frac{1}{2\sqrt3}\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{-2}\\\end{array}\right).
\chi_{1}(x)=\left(\begin{matrix}{e^{i\nu(x)}}\\{ie^{i\zeta(x)}e^{i\theta}}\\\end{matrix}\right)e^{ikx}\mathrm{~~~~and~~~~}\chi_{2}(x)=\left(\begin{matrix}{e^{-i\nu(x)^{\ast}}}\\{ie^{-i\zeta(x)^{\ast}}e^{-i\theta}}\\\end{matrix}\right)e^{-ikx}
\langle\!\langleY_{-2}|\mathcal{O}_{1},\mathcal{O}_{2},\mathcal{O}_{3}\rangle\!\rangle=e^{i\pi\sumh_{j}}(-1)^{\frac{N_{odd}\left(N_{odd}-1\right)}{2}}\langle\!\langleY_{-2}|\mathcal{O}_{3},\mathcal{O}_{2},\mathcal{O}_{1}\rangle\!\rangle.
\sum_{t}\phi_{t}\frac{\partial^{2}\phi_{t}}{\partiala_{Dm}\partiala_{Dl}}=-\sum_{i,j,k,t}\phi_{t}\frac{\partial\phi_{t}}{\partiala_{Dk}}\frac{\partial\phi_{j}}{\partiala_{Dl}}\frac{\partial\phi_{i}}{\partiala_{Dm}}\frac{\delta^{2}a_{Dk}}{\delta\phi_{j}\delta\phi_{i}}.
\begin{array}{rcccl}{\partial_{p^{2}}\Gamma_{ZZ}^{T}(p^{2}=M_{Z}^{2})}&{=}&{1}&{\Rightarrow}&{z_{ZZ}}\\{\partial_{p^{2}}\Gamma_{AA}^{T}(p^{2}=0)}&{=}&{1}&{\Rightarrow}&{z_{AA}}\\{\Gamma_{AZ}^{T}(p^{2}=0)}&{=}&{0}&{\Rightarrow}&{\theta_{V}}\\{\Gamma_{AZ}^{T}(p^{2}=M_{Z}^{2})}&{=}&{0}&{\Rightarrow}&{z_{AZ}}\\{\Gamma_{ZZ}^{T}(p^{2}=M_{Z}^{2})}&{=}&{0}&{}&{}\\\end{array}
\Lambda_{\mu}(p,q;\xi)=\frac{-1}{16\pi^{2}}\gamma_{\alpha}\gamma_{\mu}\gamma_{\beta}\frac{\partial^{2}}{\partialp_{\alpha}\partialq_{\beta}}\intd^{3}x\,d^{3}y\,e^{i(p\cdotx-q\cdoty)}e^{-e^{2}\xi\left|x-y\right|/8\pi}\frac{1}{|x|^{3}|y|^{3}}.
W[Y]=\int\frac{d^{4}x}{32\pi^{2}}\int_{0}^{\infty}\frac{dT}{T^{3}}\rho(T,\Lambda^{2})e^{-TK}\mathrm{tr}\left(e^{-TM\tau_{3}}[1-f_{1}(T,Y)+f_{2}(T,Y)-\bar{f}_{3}(T,Y)+\ldots]\right),
\phi(\ell,m,n;z)\equiv\sum_{k=0}^{m}\left(\begin{matrix}{m}\\{k}\\\end{matrix}\right)\left(\begin{matrix}{\ell-k+n}\\{n}\\\end{matrix}\right)\,\frac{(z-1)^{k}}{(z+1)^{\ell-k+n+1}}\;,
\ddot{\vec{\phi}}=-g^{2}[(\vec{A_{1}^{2}}+\vec{A_{2}^{2}})\vec{\phi}-(\vec{A_{1}}\cdot\vec{\phi}\vec{A_{1}}+\vec{A_{2}}\cdot\vec{\phi}\vec{A_{2}})]-\frac{1}{2}\frac{\partialV}{\partial\vec{\phi}}.
\varphi_{3}(x^{+},x^{-})={\frac{a_{0}(x^{+})}{\sqrt{4\pi}}}+{\frac{1}{\sqrt{4\pi}}}\sum_{n=1}^{\infty}w_{n}\Bigl(a_{n}(x^{+})\mathrm{e}^{-ik_{n}x^{-}}+a_{n}^{\dagger}(x^{+})\mathrm{e}^{+ik_{n}x^{-}}\Bigr)\.
L={\frac{1}{2}}(\partial_{\mu}\Phi)(\partial^{\mu}\Phi)+{\frac{1}{2}}(\partial_{\mu}\Psi)(\partial^{\mu}\Psi)+{\frac{1}{2}}m^{2}\Phi^{2}+{\frac{1}{2}}\omega^{2}\Psi^{2}+{\frac{\lambda_{\Phi}}{24}}\Phi^{4}+{\frac{\lambda_{\Psi}}{24}}\Psi^{4}+{\frac{\lambda_{\Phi\Psi}}{4}}\Phi^{2}\Psi^{2}~,
Q^{+}=\mathrm{i}2^{1/4}\sum_{|n^{\perp}|\leT}\int_{0}^{\infty}dk\sqrt{k}\left[b_{ij}^{\dagger}(k,n^{\perp})a_{ij}(k,n^{\perp})-a_{ij}^{\dagger}(k,n^{\perp})b_{ij}(k,n^{\perp})\right]\,.
\bar{\rho}_{\alpha}^{\prime}i\partial^{\alpha\dot{\alpha}}\rho_{\dot{\alpha}}^{\prime\prime}+\rho_{\dot{\alpha}}^{\prime}i\partial^{\alpha\dot{\alpha}}\bar{\rho}_{\alpha}^{\prime\prime}+\bar{\rho}_{\alpha}^{\prime\prime\prime}i\partial^{\alpha\dot{\alpha}}\rho_{\dot{\alpha}}^{\prime\prime\prime}\,
\delta^{NL}\psi^{i}(x)={\frac{1}{\kappa^{2}}}\zeta^{i}+i\kappa^{2}(\bar{\zeta}^{j}{\gamma}^{\rho}\psi^{j}(x))\partial_{\rho}\psi^{i}(x),\quad\delta^{NL}{e^{a}}_{\mu}(x)=i\kappa^{2}(\bar{\zeta}^{j}{\gamma}^{\rho}\psi^{j}(x))\partial_{[\rho}{e^{a}}_{\mu]}(x),
M_{\pm}^{2}=\frac{(\kappa+\Pi_{\mathrm{odd}})^{2}+M_{1}^{2}+M_{2}^{2}\pm((M_{1}^{2}-M_{2}^{2})^{2}+2(M_{1}^{2}+M_{2}^{2})(\kappa+\Pi_{\mathrm{odd}})^{2}+(\kappa+\Pi_{\mathrm{odd}})^{4})^{1/2}}{2}.
g=\int_{S_{\infty}^{2}}d\vec{\sigma}\cdot\vec{B}=\frac{1}{2ev^{3}}\int_{S_{\infty}^{2}}d^{2}\sigma^{i}\,\varepsilon_{ijk}\varepsilon^{abc}\Phi_{a}\partial^{j}\Phi_{b}\partial^{k}\Phi_{c}=\frac{4\pin}{e}\,,
G(q,q_{0};t,0)=\left(\fracM{2\pii\hbara_{2}}\right)^{\frac12}exp\left[\frac{iM}{2\hbara_{2}}\left(a_{1}q_{0}^{2}+\dot{a}_{2}e_{1}^{\etat}q^{2}-2q_{0}q\right)\right]{\it\mathrm{.}}
\sum_{i,j,k,l=0}^{\infty}\frac{(a_{+}a_{-}+b_{+}b_{-})^{i}}{i!}\frac{(a_{+}b_{-})^{j}(b_{+}a_{-})^{k}}{j!k!}\frac{(i+j+k+l)!}{(i+j+k)!l!}Q^{2i+j+k+2l}.
{\calL}_{m}^{(1)}=\vec{E}(-\partial_{0}\vec{A}-\vec{\nabla}A^{0})-\frac{1}{2}[\vec{E}^{2}+(\vec{\nabla}\times\vec{A})^{2}-m^{2}(A_{0}^{2}-\vec{A}^{2})]-A^{0}\rho+\vec{A}\vec{j}
2{\calF}-a{\frac{\partial{\calF}}{\partiala}}=8\piib_{1}\Lambda^{2}\left\{2\left[\frac{\Theta_{2}\left(0|\partial_{a}^{2}{\calF}\right)}{\Theta_{3}\left(0|\partial_{a}^{2}{\calF}\right)}\right]^{4}-1\right\},
ds_{11}^{2}=dx^{+}dx^{-}+l_{p}^{9}\sum_{n=1}^{N}\sum_{n_{0}=-\infty}^{\infty}\frac{p_{-n}\delta(x^{-}-x_{n}^{-}+n_{0}R)}{|\vec{r}-\vec{r}_{n}|^{7}}dx^{-}dx^{-}+dx_{1}^{2}+\\cdots\+dx_{9}^{2},
W_{12}\,\psi\,=\,(h_{1}+h_{2}+\Lambda_{12}^{++}\,V_{12}\,\Lambda_{12}^{++}+\Lambda_{2}^{+}\,V_{2}^{+}\,\Lambda_{2}^{+}+\Lambda_{1}^{+}\,V_{1}^{+}\,\Lambda_{1}^{+})\,\psi
\lambda_{\pm}(x_{0},r;z;q)=\frac{1}{2}\left\{\sum_{n=0}^{\infty}\frac{(\frac{\rho_{+}}{\rho_{-}};q)_{n}}{(q;q)_{n}}(i\rho_{-}(1-q)z)^{n}\pm\sum_{n=0}^{\infty}\frac{(\frac{\rho_{-}}{\rho_{+}};q)_{n}}{(q;q)_{n}}(i\rho_{+}(1-q)z)^{n}\right\}.
S=-{\frac{1}{4\pi\alpha^{\prime}}}\intd\xi^{0}d\xi^{1}\sqrt{-h}\left\{h^{ij}\partial_{i}X^{\mu}\partial_{j}X_{\mu}-i\Thetae_{I}^{j}(\Gamma^{0})_{AB}\overline{\Psi}^{A}\gamma_{j}{\nabla^{I}}\Psi^{B}\right\}.
\sum_{\lambda}{\calE}^{\lambda}(z,\xi)\,{\calE}^{\lambda}(z,\xi)\left(Hz\cdot\xi\right)^{\sigma}=-\left(\theta-\frac{\bar{\xi}\bar{\xi}}{(Hz\cdot\xi)^{2}}\right)\left(Hz\cdot\xi\right)^{\sigma}=-\frac{\sigma}{\sigma-1}\left[\theta-\frac{H^{2}D_{2}D_{1}}{2\sigma^{2}}\right]\left(Hz\cdot\xi\right)^{\sigma}\,.
<q|p>=\frac{e^{i\varphi(q_{0},p)}}{(2\pi\hbar)^{n/2}}\frac{\Omega[P(q_{0}\rightarrowq)]}{g^{1/4}(q)h^{1/4}(p)}\,e^{\frac{i}{\hbar}(q-q_{0})\cdotp}\\\,
E_{TM}(a)=\frac{\hbarc}{a}\,\left\{-\frac{\,{n_{0}}^{3}\,a^{4}\,{\Omega}^{4}}{24\,\pi\,c^{4}\,\left(1+s\right)}+\frac{n_{0}a^{2}\Omega^{2}}{4\pic^{2}(1+s)}+\mathcal{O}(s+1)^{0}\right\}.
{\calW}_{ML}^{(2;cr)}=-{\frac{C_{F}C_{A}}{16{\pi}^{4}}}{\frac{\Gamma^{2}(D/2-1)}{(D-4)^{4}}}\left\{2{\calE}{\frac{D-2}{D-3}}+8{\calB}\left[1-2{\frac{\Gamma^{2}(3-D/2)}{\Gamma(5-D)}}\right]\right\}
\begin{array}{lll}{{\widetildeT}_{\alpha{\dot{\alpha}}}}&{=}&{-\frac{4}{3}D_{\alpha}G{\bar{D}}_{\dot{\alpha}}G({\mathcalF}^{\prime\prime}-2G{\mathcalF}^{\prime\prime\prime})+\frac{4}{3}G{\mathcalF}^{\prime\prime}\[D_{\alpha},{\bar{D}}_{\dot{\alpha}}]G\,}\\{{\widetildeJ}}&{=}&{4{\bar{D}}^{2}\left(G{\mathcalF}^{\prime}-{\mathcalF}\right)\.}\\\end{array}
\int_{m}^{\infty}dk\[k^{2}-m]^{1/2-s}\frac{\partial}{\partialk}t^{j}=-m^{1-2s}\frac{\Gamma(\frac32-s)\Gamma(s+\frac{j-1}{2})}{\Gamma(\fracj2)}\frac{\left(\frac{\nu}{mR}\right)^{j}}{\left(1+\left(\frac{\nu}{mR}\right)^{2}\right)^{s+\frac{j-1}{2}}}.
\langle0p_{1}p_{2}p_{3}p_{4}\rangle={\frac{\xi_{123}^{2}\xi_{124}^{2}}{(12)^{2}}}{\calF}^{p_{1}-2\vertp_{2}-2\vertp_{3}-2\vertp_{4}-2}(x,u)+O(\theta^{5}\bar{\theta})\;.
\begin{array}{ll}{\alpha^{+}=e^{-\phi}(-y^{2}dx+dy)\}&{\\bar{\alpha}^{+}=e^{\bar{\phi}}d\bar{x}}\\{\alpha^{-}=e^{\phi}dx\}&{\\bar{\alpha}^{-}=e^{-\bar{\phi}}(-\bar{y}^{2}d\bar{x}+d\bar{y})}\\{\alpha^{3}=-2ydx+d\phi\}&{\\bar{\alpha}^{3}=2\bar{y}d\bar{x}-d\bar{\phi}.}\\\end{array}
\bar{f}_{\epsilon}(0,\bar{x})=\epsilon\bar{x}\\\\,\\\\,frac{\partial\bar{f}_{\epsilon}}{\partial\bar{t}}|_{P_{0}}=1\\\\,\\\\,frac{\partial\bar{f}_{\epsilon}}{\partial\bar{x}}|_{P_{0}}=\epsilon\bar{\alpha}\frac{\partial\bar{f}_{\epsilon}}{\partial\bar{t}}|_{P_{0}}=\epsilon
L=\frac{1}{2}(p_{M}-\frac{i\hbar}{2}\Gamma_{M})\eta^{MN}(p_{M}+\frac{i\hbar}{2}\Gamma_{M})=\frac{1}{2}p_{M}\eta^{MN}p_{N}+\frac{\hbar^{2}}{4}\left(\partial_{M}\Gamma^{M}+\frac{1}{2}\Gamma_{M}\Gamma^{M}\right)\,,
\begin{array}{c}{\frac{\tau_{+}^{2}}{\ell^{2}}+2m-1=0\Rightarrow\tau_{+}=\ell\sqrt{1-2m}}\\{\RightarrowS^{d=2}=\frac{\left(1-2m\right)\beta_{H}V_{1}}{8\piG}}\\\end{array}
\deltaL^{+4}=(\epsilon_{b}^{\beta}\theta^{+b}\partial_{\beta}^{-}-\epsilon^{+b}\partial_{b}^{-})L^{+4}+{\frac{1}{2}}(\theta^{+b}\epsilon_{b}^{\beta})\partial^{+\alpha}\mathrm{Tr}(A_{\alpha}^{+}A_{\beta}^{+})\,
\left\langle\alpha_{1},\alpha_{2},...,\alpha_{2N}\right|U_{+}\left|\beta_{1},\beta_{2},...,\beta_{2N}\right\rangle=\prod_{i=1}^{N}R_{\alpha_{2i-1},\alpha_{2i}}^{\beta_{2i-1},\beta_{2i}}(\lambda-\lambda_{i})