harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\phi_{0}\rightarrow\frac{\rho^{l}}{\rho_{0}^{l}}\frac{C_{\lambda}\;\Gamma(\nu)\;\Gamma(1-i\lambda)}{\Gamma\left(-i\lambda/2+\nu/2\eta_{+}(\nu)\right)\Gamma\left(-i\lambda/2+\nu/2+\eta_{-}(\nu)\right)}\,.
g_{\alpha\beta}\biggl|_{r\to\infty}=-\nabla_{\alpha}u\nabla_{\beta}u-(\nabla_{\alpha}u\nabla_{\beta}r+\nabla_{\alpha}r\nabla_{\beta}u)+\frac{1}{2}(m_{\alpha}m_{\beta}^{*}+m_{\alpha}^{*}m_{\beta})
\frac{\xi_{0}}{2l_{0}}\approx1+\frac{2l_{0}^{2}}{1-v^{2}}\left[\frac{1}{r^{2}}(1-\frac{r^{\prime\prime}}{r})^{2}+\frac{1}{(R_{0}\pmr)^{2}}\pm\frac{1}{r(R_{0}\pmr)}\right],\nonumber
M_{SM}^{2}=g^{2}v^{2}\left(\begin{array}{ccccc}{2}&{-1}&{0}&{\cdots}&{-1}\\{-1}&{2}&{-1}&{\cdots}&{0}\\{0}&{-1}&{2}&{\cdots}&{0}\\{}&{}&{}&{\cdots}&{}\\{-1}&{0}&{\cdots}&{-1}&{2}\\\end{array}\right).
P_{k,m}^{\nu}=\frac{1}{(k^{2}-m^{2}+i\epsilon)^{\nu}},~~~~~~\overline{{k}}_{j}^{\mu}=~\sum_{n=1}^{L}\omega_{jn}k_{n}^{\mu}+\sum_{m=1}^{E}\eta_{jm}q_{m}^{\mu},
S_{J}=-\intdx^{0}\left(l_{a}J^{a}(\phi)+\pi_{a}{\frac{\partialJ^{a}}{\partial\phi_{i}}}\tau_{i}+\overline{{l}}_{a}\overline{{J}}^{a}(\overline{{\phi}})+{\frac{\partial\overline{{J}}^{a}}{\partial\overline{{\phi}}_{i}}}\overline{{\tau}}_{i}\overline{{\pi}}_{a}\right)
\Psi(r_{1},\!r_{2},\!\psi)\!=\!{\textstyle\frac{2}{g}\sqrt{\frac{n_{1}!\:n_{2}!}{(n_{1}+|l|)!\,(n_{2}+|l|)!}}}\,\left({\textstyle\frac{r_{1}\,r_{2}}{g}}\right)^{\!|l|}\:\mathrm{e}^{-\frac{1}{2g}(r_{1}^{2}+r_{2}^{2})}\,L_{n_{1}}^{|l|}\!\!\left({\textstyle\frac{r_{1}^{2}}{g}}\right)\,L_{n_{2}}^{|l|}\!\!\left({\textstyle\frac{r_{2}^{2}}{g}}\right)\,\mathrm{e}^{\mathrm{i}\,l\,\psi}.
\begin{array}{l}{\lbrackL_{m},L_{n}]=(m-n)L_{m+n}+\fracc8m(m^{2}-1)\delta_{m+n}}\\{\{S_{r},S_{s}\}=2L_{r+s}+\fracc2(r^{2}-\frac14)\delta_{r+s}}\\{\lbrackL_{m},S_{r}]=(\fracm2-r)S_{m+r}}\\\end{array}
f(q,\mu)=\sum_{\rho\neq\sigma\in\Phi_{+},\,R_{\psi}=s_{\rho}s_{\sigma}}g_{|\rho|}\,g_{|\sigma|}\,{\frac{(\sigma^{\vee}\!\!\cdot\rho)}{|\sigma^{\vee}\!\!\cdot\rho|}}\,x_{|\rho|}(\rho\cdotq,(\rho^{\vee}\!\!\cdot\mu)\xi)\,x_{|\sigma|}(\sigma\cdotq,(s_{\rho}(\sigma)^{\vee}\!\!\cdot\mu)\xi)
\mathcal{D}_{h_{1}}^{+}\otimes\mathcal{D}_{h_{2}}\to\sum_{h=h_{1}+h_{2}}^{0}D_{h}+\sum_{h=-h_{1}-h_{2}+2}^{h_{1}-h_{2}}(-)^{h+h_{1}+h_{2}}D_{h}^{+}.
\{Q^{a},Q^{b}\}=2p^{+}\delta^{ab},\quad\{Q^{\dot{a}},Q^{\dot{b}}\}=\delta^{\dot{a}\dot{b}}P^{-},\quad\{Q^{a},Q^{\dot{a}}\}={\frac{1}{\sqrt2}}\gamma_{a\dot{a}}^{I}p^{I},
G_{r}^{(1)}(x)=\sum_{\stackrel{L=0}{L\equivp_{+}(r)[mod2]}}^{\infty}x^{\frac{L}{2}}((K_{\nu})^{L})_{1,r}=\left(\frac{1+p_{+}(r)(\sqrtxK_{\nu}-1)}{1-xK_{\nu}^{2}}\right)_{1,r},
V(r,u)=-(8\pi^{2}\alpha^{\prime})^{-1/2}\int_{0}^{\pi/u_{+}}dl~e^{-r^{2}l/2\pi\alpha^{\prime}}l^{-1/2}\mathrm{tanh}(u)/u\left[\sum_{k=1}^{\infty}C_{k}(ul)^{2k}+\sum_{k=1}^{\infty}\sum_{m=1}^{\infty}C_{k,m}(ul)^{2(k+m)}\right]\quad,
D(P)=\{F_{\psi}\inL^{2}(0,\infty):F_{\psi}\;\mathrm{abs.}\;\mathrm{cont.},\int_{0}^{\infty}\mathrm{d}q\,|{\frac{\mathrm{d}F_{\psi}}{\mathrm{d}q}}|^{2}<\infty,\;F_{\psi}(0)=0\}.
B_{b}^{\mu_{1}\mu_{2}\ldots\mu_{p}}(X)\equiv-e_{b}\sum_{i=0}^{p}\intd\xi_{b}^{0}d\xi_{b}^{1}\ldots\hat{d\xi_{b}^{i}}\ldotsd\xi_{b}^{p}\left[K_{bi}^{\mu_{1}\mu_{2}\ldots\mu_{p}}G((X-X_{b})^{2})\right]_{\xi_{b}^{i}=0}^{\xi_{b}^{i}=l_{b}^{i}}.
\begin{array}{ccc}{\rho^{i}=\left(\begin{array}{cc}{0}&{\bar{\rho}^{a}{}^{t}}\\{\rho_{a}}&{0}\\\end{array}\right)~~~~}&{\mathrm{or~}}&{~~~\rho^{i}=\left(\begin{array}{cc}{\bar{\rho}^{a}{}^{t}}&{0}\\{0}&{-\rho_{a}}\\\end{array}\right)}\\\end{array}
\sum_{\pi(1,2,...n)}\mathrm{Tr}\Big[H^{(1)}H^{(I_{\pi(1)})}.....H^{(I_{\pi(n)})}\Big]\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}}\;.
\Pi^{in}=-\frac{i}{R}\left\{\partial_{x}+\frac{\sigma^{3}}{x}\left[l-l_{0}-\frac{1}{2}\left(1-\sigma^{3}\right)+\vartheta\left(x\right)+\xi\rho_{R}x^{2}\right]\right\}\sigma^{1},\;\rho_{R}=\gammaR^{2}/2\;.
\phi_{0}(x)=2\lambda\psi_{0}(0)\int\frac{d\tau^{\prime}}{(1-H\tau^{\prime})^{2}}{\cal{S}}_{0}(x,0,\tau^{\prime})\,\,\,,\,\,\,\,\phi_{\delta}(x)=2\lambda\psi_{\delta}^{*}(0)\int\frac{d\tau^{\prime}}{(1-H\tau^{\prime})^{2}}{\cal{S}}_{\delta}(x,0,\tau^{\prime})
{\calS}_{1}={\frac{k_{1}}{4\theta_{1}}}\intdt\;\mathrm{{Tr}}\left(i{\bar{Z}_{1}}DZ_{1}-\omega_{1}{\bar{Z}_{1}}Z_{1}\right)+\frac{i}{2}\intdt\;{\bar{\Psi}}_{1}D\Psi_{1}+\frac{k_{1}}{2}\intdt\;{\mathrm{Tr}}A_{1}+hc
{\calL}_{\mathrm{T_{\mathrm{free}}}}={\calL}_{\mathrm{T_{\mathrm{bulk}}}}+i\mu\left(\psi_{2}(0)\psi_{1}^{\phantom{\dagger}}(x)-\psi_{1}(0)\psi_{2}(x)\right)\delta(x)+i\mu^{\dagger}\left(\psi_{2}^{\dagger}(x)\psi_{1}^{\dagger}(0)-\psi_{1}^{\dagger}(x)\psi_{2}^{\dagger}(0)\right)\delta(x).
\tilde{u}_{k}^{*}=\frac{1}{2d\pi^{2}}\left\{\frac{N}{4}[I_{1}^{3}-I_{2}^{3}]+\frac{N}{4}I_{2}^{3}\tilde{g}_{k}^{*}\tilde{\omega}_{k}^{*2}+\tilde{\omega}_{k}^{*4}\tilde{g}_{k}^{*2}[I_{2}^{3}-I_{3}^{3}]\right\},
ds^{2}=-\left(\gamma+\frac{r^{2}}{l^{2}}-\frac{\mu}{r^{d-3}}\right)dt^{2}+\frac{dr^{2}}{\left(\gamma+\frac{r^{2}}{l^{2}}-\frac{\mu}{r^{d-3}}\right)}+r^{2}d\sigma_{\gamma}^{2},
\begin{array}{l}{lim_{T\rightarrow\infty}(T-\tau_{0})^{-1}\int_{\tau_{0}}^{T}d\tauE[((\phi_{\tau},f)-E[(\phi_{\tau},f)])((\phi_{\tau},f^{\prime})-E[(\phi_{\tau},f^{\prime})])]=}\\{\frac{\hbar^{2}}{2i}(f,{\calA}^{-1}f^{\prime})-\frac{\hbar^{2}}{4}lim_{T\rightarrow\infty}(T-\tau_{0})^{-1}({\calA}^{-1}f,\Big(exp(-2i(T-\tau_{0}){\calA})-1\Big){\calA}^{-1}f^{\prime})}\\\end{array}
\partial_{\alpha}\partial^{\alpha}X-X^{\prime\prime}-3{\calH}X^{\prime}+\frac{\partial^{2}V}{\partial\varphi^{2}}a^{2}X-\varphi^{\prime}[4\Psi^{\prime}+\Xi^{\prime}]-2\Xi\,(\varphi^{\prime\prime}+3{\calH}\varphi^{\prime})=0,
[W^{gen},\underline{{m}}\partial_{\underline{{m}}}]=z\int\left\{\hat{\xi}_{A}\frac{m}{e}\frac{\delta}{\delta\varphi_{2}}+\xi_{A}\frac{m}{e}\frac{\delta}{\delta\hat{\varphi}_{2}}\right\}+(\chi\partial_{\xi}+\chi_{A}\partial_{\xi_{A}})z\xi_{A}\frac{m}{e}\int\frac{\delta}{\deltaq_{2}}
q^{-{\frac{1}{2}}L({\frac{L}{n}}-1)}X_{L-1}=\sum_{\{\nu\}}\prod_{a=1}^{n-1}\prod_{j=1}^{\ell}q^{{\frac{1}{2}}({\bfh}|{\bfh})}\biggl[\begin{array}{c}{h_{j}^{(a)}+\sum_{b}\sum_{k}G_{ab}^{-1}C_{jk}h_{k}^{(b)}}\\{h_{j}^{(a)}}\\\end{array}\biggr],
\langlex,\mu|e^{-tH(\alpha)}|x^{\prime},\nu\rangle=\langlex,\mu|e^{-tH(1)}|x^{\prime},\nu\rangle+\int_{t/\alpha}^{t}d\tauD_{\mu}D^{\lambda}\langlex,\lambda|e^{-\tauH(1)}|x^{\prime},\nu\rangle.
U_{9}=\left(\begin{matrix}{0}&{1}&{0}&{\cdots}&{0}\\{0}&{0}&{1}&{}&{0}\\{\vdots}&{}&{\ddots}&{\ddots}&{\vdots}\\{0}&{0}&{}&{0}&{1}\\{1}&{0}&{\cdots}&{0}&{0}\\\end{matrix}\right).
V_{b}=\frac{e^{2}}{2\pi}\frac{1}{{q^{+}}^{4}}\int_{0}^{q^{+}}\frac{dk^{+}}{q^{-}-\frac{m^{2}}{2k^{+}}-\frac{m^{2}}{2(q^{+}-k^{+})}}\stackrel{q^{+}\rightarrow0}{\longrightarrow}-\frac{e^{2}}{\pi}\frac{1}{6m^{2}}\frac{1}{{q^{+}}^{2}}
\begin{array}{ll}{R_{00}(k,l)}&{=-4\pi{\frac{4}{3}}\alpha_{\mathrm{s}}[4{\frac{p^{2}l^{2}-(pl)^{2}}{(p-l)^{4}}}+{\frac{3}{4}}]-\nonumber}\\{}&{\qquad-\sigma\intd^{3}\vec{\zeta}e^{-i(\vec{k}-\vec{l})\cdot\vec{\zeta}}\vert{\vec{\zeta}}\vert\vert(k_{0}+l_{0})\vert\sqrt{(k_{0}+l_{0})^{2}-(\vec{k}_{\mathrm{T}}+\vec{l}_{\mathrm{T}})^{2}}}\\\end{array}\,.
\hat{\Gamma}_{r}^{\,\alpha,\,\alpha_{0}}(q,p,\Phi_{c})\:=\(2\pi)^{\,4}\left\{\,\delta_{\Phi(p)}\delta_{\Phi(q)}\,L^{\,\alpha,\,\alpha_{0}}(\Phi)\,\vert_{\,\Phi=\tilde{C}_{\alpha}^{\,\alpha_{0}}\!J(\Phi_{c})}\right\}_{r}\quad.
S_{\mathrm{Chern-Simons}}=-\frac{2}{(12)^{4}\,\kappa_{11}^{2}}\int\mathrm{d}^{11}x\,\varepsilon^{\mu_{1}\cdots\mu_{11}}B_{\mu_{1}\mu_{2}\mu_{3}}H_{\mu_{4}\cdots\mu_{7}}H_{\mu_{8}\cdots\mu_{11}}\,.
\dot{x}^{2}(\tau,0)=A^{2}\frac{[f-g]^{2}}{4\acute{f}\acute{g}},\;\;\dot{x}^{2}(\tau,l)=A^{2}\frac{[f_{l}-g_{l}]^{2}}{4\acute{f}_{l}\acute{g}_{l}}.
\vec{v}(\sigma,t+\delta)\=\{\frac{1}{2}}[\vec{v}(\sigma+\delta,t)+\vec{v}(\sigma-\delta,t)]+{\frac{1}{4\delta}}[\vec{f}(\sigma+2\delta,t)-2\vec{f}(\sigma,t)+\vec{f}(\sigma-2\delta,t)]\,
\sigma_{1}=\left(\begin{matrix}{0}&{1}\\{1}&{0}\\\end{matrix}\right)\;\quad\sigma_{2}=\left(\begin{matrix}{\phantom{-}0}&{\mathrm{i}}\\{-\mathrm{i}}&{0}\\\end{matrix}\right)\;\quad\sigma_{3}=\left(\begin{matrix}{1}&{\phantom{-}0}\\{0}&{-1}\\\end{matrix}\right)\.
\intd^{10}\hat{x}\,\sqrt{|\hat{\jmath}|}\,\left\{f_{+}{\textstyle\frac{1}{4\cdot5!}}\left(\hat{G}_{(5)}^{+}\right)^{2}+f_{+}^{-1}{\textstyle\frac{1}{4\cdot5!}}\left(\hat{G}_{(5)}^{-}\right)^{2}\right\}\,,
\int_{-\infty}^{\infty}dq_{-}\,e^{i\omegaq_{-}}\delta\epsilon_{+}(q_{-})=(2\overline{\mu})^{i\omega}\int_{-\infty}^{\infty}dq_{+}\,e^{-i\omegaq_{+}}\sum_{k=1}^{\infty}\frac{\overline{\mu}^{1-k}}{k!}\frac{\Gamma(1+i\omega)}{\Gamma(2-k+i\omega)}\delta\epsilon_{+}(q_{+})^{k}.
\frac{dE}{d\tau}=\frac{d}{d\tau}\int_{V}d^{3}\sigma\:[\frac{\vec{E}_{\perp}^{2}+\vec{B}^{2}}{2}](\tau,\vec{\sigma})=-\int_{S}d\Sigma\:\vec{n}\cdot(\vec{E}_{\perp}\wedge\vec{B})(\tau,\vec{\sigma}).
\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2\beta}\left(\partial_{\mu}A^{\mu}\right)^{2}-\frac{1}{2\alpha}\left(n_{\mu}A^{\mu}\right)^{2}=\mathcal{L}_{\mathrm{E}}+\mathcal{L}_{GF}
\intdx^{-}C(x^{+},x^{-},\vec{x})=C(x^{+})\delta(\vec{x}-\vec{x}_{0})-C^{(i)}(x^{+})\partial_{i}\delta(\vec{x}-\vec{x}_{0})+{\frac{1}{2}}C^{(ij)}(x^{+})\partial_{i}\partial_{j}\delta(\vec{x}-\vec{x}_{0})+\cdots
\left[\sum_{i=1}^{N}\frac{\partial^{2}}{\partial{x_{i}}^{2}}+2(a+\frac{1}{2})\sum_{i<j}^{N}\frac{1}{(x_{i}-x_{j})}(\frac{\partial}{\partialx_{i}}-\frac{\partial}{\partialx_{j}})\right]P_{k}(x)=0\,\,\,.
\phi=A(x-y)\Sigma_{lm}\intd\omega(f_{\omegalm}^{b}b_{\omegalm}^{b}+\bar{f}_{\omegalm}^{b}b_{\omegalm}^{b\dagger}+f_{\omegalm}^{a}b_{\omegalm}^{a}+\bar{f}_{\omegalm}^{a}b_{\omegalm}^{a\dagger}),
\Lambda=\left(\begin{array}{cccccc}{0}&{0}&{0}&{\ldots}&{0}&{0}\\{1}&{0}&{0}&{\ldots}&{0}&{0}\\{0}&{1}&{0}&{\ldots}&{0}&{0}\\{0}&{0}&{1}&{\ldots}&{0}&{0}\\{.}&{.}&{.}&{.}&{.}&{.}\\{0}&{0}&{0}&{\ldots}&{1}&{0}\\\end{array}\right)
{\frac{1}{2m_{p+1}}}\,{\frac{\partialS_{0}}{\partial\sigma_{\mu_{1}\dots\mu_{p+1}}}}{\frac{\partialS_{0}}{\partial\sigma^{\mu_{1}\dots\mu_{p+1}}}}-\frac{m_{p+1}}{2}\,p={\frac{\partialS_{0}}{\partial\Omega_{p+1}}}\.
\deltaL=\frac{d}{dt}\left(H\deltat\right)+\frac{dL}{dt}-\left(\dot{p}_{M}-\frac{1}{2}p_{N}\partial_{M}\eta^{NL}p_{L}-\partial_{M}U-p_{N}\circ\varphi^{\mu}{}_{M}\right)\circ\deltax^{N}\,,
\begin{array}{l}{e^{+2}\nabla\psi_{+2\dot{q}}^{2+}+e^{-2}\nabla\psi_{-2\dot{q}}^{2+}+e_{p}^{+}\nabla\chi_{p\dot{q}}^{2}+e_{p}^{-}\nablah_{-p\dot{q}}^{2+}+T^{+2}\psi_{+2\dot{q}}^{2+}+T^{-2}\psi_{-2\dot{q}}^{2+}+T_{p}^{+}\chi_{p\dot{q}}^{2}+T_{p}^{-}h_{-p\dot{q}}^{2+}}\\{=-\frac12e_{p}^{-}\gamma_{p\dot{q}}^{i}\Omega^{+2i}.}\\\end{array}
\begin{array}{||r|c|c|c|c|c|c||}\hline{}&{}&{}&{}&{}&{}&{}\\{\mathrm{fields}:}&{\overline{{c}}}&{b}&{\overline{{\phi}}}&{\eta}&{\overline{{\chi}}_{\mu\nu}}&{b_{\mu\nu}}\\\hline{}&{}&{}&{}&{}&{}&{}\\{\#_{\mathrm{ghost}}:}&{-1}&{0}&{-2}&{-1}&{-1}&{0}\\\hline{}\\\end{array}
\left(\int_{\gamma_{\rho}^{+}}+\sum_{\sigma_{k}=iy_{k},\lambda_{\nu,k}}\int_{C_{\rho}(\sigma_{k})}\right)\frac{\bar{H}_{\nu}^{(1)}(z)}{\bar{J}_{\nu}(z)}f(z)dz-\left(\int_{\gamma_{\rho}^{-}}+\sum_{\sigma_{k}=-iy_{k},-\lambda_{\nu,k}}\int_{C_{\rho}(\sigma_{k})}\right)\frac{\bar{H}_{\nu}^{(2)}(z)}{\bar{J}_{\nu}(z)}f(z)dz,
\begin{aligned}{\bigg(\nabla^{2}\Phi-2[\nabla\Phi]^{2}}&{{}+\frac{1}{12}H_{mnp}H^{mnp}\bigg)\epsilon}\\{}&{{}=-\frac{1}{48}dH_{mnpq}\gamma^{mnpq}\epsilon-\frac{1}{4}e^{2\Phi}\nabla^{m}\left(e^{-2\Phi}H_{mnp}\right)\gamma^{np}\epsilon}\\\end{aligned}
\rho_{r}[\phi_{<f},\phi_{<f}^{\prime},t]=\intd\phi_{<i}\intd\phi_{<i}^{\prime}J_{r}[\phi_{<f},\phi_{<f}^{\prime},t\vert\phi_{<i},\phi_{<i}^{\prime},t_{0}]\rho_{r}[\phi_{<i}\phi_{<i}^{\prime},t_{0}],
ds^{2}=(\rho^{2}-\rho_{+}^{2})\left(-(-\frac{\rho_{+}^{2}}{l^{2}}R^{2}-k)dT^{2}+\frac{dR^{2}}{-\frac{\rho_{+}^{2}}{l^{2}}R^{2}-k}\right)+\frac{d\rho^{2}}{\frac{\rho^{2}-\rho_{+}^{2}}{l^{2}}}+\rho^{2}d\phi^{2}
A_{BPS}=4\pi^{2}[(Q_{1}^{(1)}Q_{1}^{(2)}Q)(1-{\frac{1}{2}}(L_{1}-L_{2})^{2})]^{\frac{1}{2}}=4\pi^{2}[Q_{1}^{(1)}Q_{1}^{(2)}Q-{\frac{1}{4}}J^{2}]^{\frac{1}{2}}.
\sum_{M\in\{l,l-2,l-4,...\}}\frac{G_{M}^{(l)}}{G_{l}^{(l)}}u^{\frac{1}{2}(l-M)}\sum_{n,m=0}^{\infty}\tilde{A}_{nm}^{(M)}\frac{u^{n}(1-v)^{m}}{n!m!}=\sum_{n=0}^{\infty}\frac{u^{n}}{n!}G_{n}(\lambda,l;v).
\chi^{\prime}(|{\bfx-y}|)=\sum_{\bfk}\frac{1}{4k^{2}}\left[(1-\gamma)+\frac{2k^{2}}{m^{2}}-\frac{k^{4}}{m^{2}\omega^{2}}\right]e^{i{\bfk\cdot(x-y)}}\nonumber\,
r_{s}^{d-1}=\frac{4|\tilde{\alpha}|/l^{2}}{1+4|\tilde{\alpha}|/l^{2}}r_{+}^{d-1}\left(1-\frac{l^{2}}{r_{+}^{2}}-\frac{|\tilde{\alpha}|l^{2}}{r_{+}^{4}}\right),
\frac{F}{V}=-\int\frac{d\vec{p}}{(2\pi)^{\smallD-2}}\sum_{k,N=1}^{\infty}\frac{e^{-\frac{\betakN}{\sqrt{2}R}-\frac{\vec{p}^{2}+M^{2}}{2\sqrt{2}N/k\betaR}}}{\sqrt{2}\pi\betaRk}
\{{Q_{11}^{\alpha}},{Q_{11}^{\beta}}\}=({\gamma}^{p})^{\alpha\beta}P_{p}+({\gamma}^{p_{1}p_{2}})^{\alpha\beta}Z_{p_{1}p_{2}}+({\gamma}^{p_{1}\dotsp_{5}})^{\alpha\beta}Z_{p_{1}\dotsp_{5}}\,,
\frac{32\pi}{3}\rho_{i}a_{i}^{3}\frac{{\calH}_{i}}{{\calT}_{i}}\approx8\pi\frac{a^{2}(t_{i})\rho_{(rad.)}(t_{i})}{n^{2}}\approx\frac{3}{16\tau^{2}n^{2}}\varpie^{-\xi_{i}/2}e^{\varpi(1-e^{-\xi_{i}})}<<1\quad,
S_{F}[\overline{{\psi}},\psi,A,h]\;=\;\sum_{b=1}^{N}\intd^{2}x\;\overline{{\psi}}^{(b)}(x)\gamma_{\mu}\Big(\partial_{\mu}-ieA_{\mu}(x)-ig^{1/2}h_{\mu}(x)\Big)\psi^{(b)}(x)\;.
Q_{\lambda}^{(d+1)}(\zeta)=\frac{\Gamma\left(\fracd2\right)}{\sqrt\pi\Gamma\left(\frac{d-1}{2}\right)}\int_{1}^{\infty}(\zeta+it\sqrt{\zeta^{2}-1})^{-\lambda-d+1}(t^{2}-1)^{\frac{d-3}{2}}dt
\{\theta_{i}^{~j},t_{k}^{~l}\}=C_{ik~m}^{~~~jl~s}\theta_{s}^{~m}+C_{ik~i_{1}i_{2}i_{3}}^{~~~jl~j_{1}j_{2}j_{3}}\theta_{j_{1}}^{~i_{1}}\theta_{j_{2}}^{~i_{2}}\theta_{j_{3}}^{~i_{3}}+\ldots~.
{\calK}=\left({\frac{1}{2}}\right)\left({\frac{1}{4}}\right)\sum_{\bfm}q^{{\frac{1}{2}}P_{\bfm}^{2}}\Big[1+(-1)^{m_{1}}+(-1)^{m_{2}}+(-1)^{m_{1}+m_{2}}\big]~,
G_{\mu\nu\phantom{\alpha}\beta}^{\phantom{\mu\nu}\alpha}=\partial_{\mu}B_{\nu\phantom{\alpha}\beta}^{\phantom{\nu}\alpha}-\partial_{\nu}B_{\mu\phantom{\alpha}\beta}^{\phantom{\mu}\alpha}+B_{\mu\phantom{\alpha}\gamma}^{\phantom{\mu}\alpha}B_{\nu\phantom{\gamma}\beta}^{\phantom{\nu}\gamma}-B_{\nu\phantom{\alpha}\gamma}^{\phantom{\nu}\alpha}B_{\mu\phantom{\gamma}\beta}^{\phantom{\mu}\gamma}
\frac{\partial\Omega}{\partialz^{\alpha}}=\frac{1}{n!}\frac{\partialh(x)}{\partialz^{\alpha}}\epsilon_{{\mu}{\nu}\cdots{\rho}}dx^{\mu}dx^{\nu}\cdotsdx^{\rho}+\frac{1}{(n-1)!}h(x)\epsilon_{{\mu}{\nu}\cdots{\rho}}\frac{\partialdx^{\mu}}{\partialz^{\alpha}}dx^{\nu}\cdotsdx^{\rho}.
\left(\Gamma_{}^{\left(0\right)0},\Gamma_{\LambdaR_{1}}^{\left(2\right)-1}\right)^{a}+\widetilde{V}^{a}\Gamma_{\LambdaR_{1}}^{\left(2\right)-1}=\alpha_{R_{1}}^{\left(2\right)-2a}-\left(\Gamma_{}^{\left(0\right)1},\Gamma_{\LambdaR_{1}}^{\left(2\right)-2}\right)^{a}
\frac12\epsilon^{\mu\nu\lambda\sigma}\,e_{\lambda}^{a}\,e_{\sigma}^{b}\,\hat{\xi}_{ab}^{+}=\frac12\frac{\epsilon^{\alpha\beta\gamma\delta}}{\sqrt{g}}\frac{\partialx^{\mu}}{\partialq^{\alpha}}\frac{\partialx^{\nu}}{\partialq^{\beta}}e_{\gamma}^{c}\,e_{\delta}^{d}\,\hat{\xi}_{cd}^{+};
<U|T_{\nu}^{\mu}(r)|U>=\left(\begin{array}{cc}{T_{\alpha}^{\alpha}-\Omega^{-1}(r)H(r)}&{0}\\{0}&{\Omega^{-1}H(r)\}\\\end{array}\right)+\Omega^{-1}\left(\begin{array}{cc}{\frac{-\pi}{24a^{2}}+\frac{1}{256\pim^{2}}}&{\frac{-1}{768\pim^{2}}}\\{\frac{1}{768\pim^{2}}}&{\frac{\pi}{24a^{2}}-\frac{1}{256\pim^{2}}\}\\\end{array}\right)
\times\;\prod_{i=1}^{m_{b}^{\prime}}\psi_{2}^{(b)}({y^{\prime}}_{i}^{(b)})\prod_{i=1}^{n_{b}}\overline{{\psi}}_{1}^{(b)}(x_{i}^{(b)}\!+\hat{\tau})\prod_{i=1}^{m_{b}}\psi_{2}^{(b)}(y_{i}^{(b)}\!+\hat{\tau})\prod_{i=1}^{n_{b}^{\prime}}\overline{{\psi}}_{1}^{(b)}({x^{\prime}}_{i}^{(b)})\Big\rangle_{0}\;.
\bar{F}^{+}\equiviv_{\underline{\mu}}^{+}\bar{D}v^{+b\underline{\mu}}=\bar{\Psi}^{+}e^{\Phi+L},~~~\bar{F}^{-}\equiviv_{\underline{\mu}}^{-}\bar{D}v^{-\underline{\mu}}=0,~~~\bar{F}^{0}\equiviv_{\underline{\mu}}^{+}\bar{D}v^{+\underline{\mu}}={\frac{i}{2}}(\bar{D}\Phi-\bar{D}L).
\left[\partial_{t}-\partial_{s}+D-4\right]\tilde{\Gamma}_{t,s}^{(4)}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4})=g_{t}^{2}I_{s}^{(4)}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4})\;.
\begin{array}{c}{D^{B}H^{A}=0,\qquad\qquadD^{B}D^{A}\Phi_{A}=\left[H^{A},\Phi_{B}\right],}\\{D^{A}K^{B}=0,\qquad\qquadD^{A}D^{B}\Phi_{B}=[K^{B},\Phi_{A}].}\\\end{array}
\Phi(x,\theta,\bar{\theta})=\alpha(x)+\theta\psi(x)+\theta^{2}f(x)+i\theta\sigma^{n}\bar{\theta}\partial_{n}\alpha(x)+\frac12\theta^{2}\bar{\theta}\tilde{\sigma}^{n}\partial_{n}\psi+\frac14\theta^{2}\bar{\theta}^{2}\Box\alpha,
\begin{array}{lll}{\langle\alpha|\phi(x)\phi(y)|\alpha\rangle}&{=}&{N_{\alpha}^{2}\left(G_{E}(x,y)+|\gamma|^{2}G_{E}(\bar{x},\bar{y})+\gammaG_{E}(\bar{x},y)+\gamma^{*}G_{E}(x,\bar{y})\right)}\\{}&{\equiv}&{G_{\alpha}(x,y)}\\\end{array}
ds^{2}=-\left(1-\frac{16\pi{\calG}M}{(D-2)\Omega_{D-2}}\frac{1}{r^{D-3}}\right)dt^{2}+\left(1-\frac{16\pi{\calG}M}{(D-2)\Omega_{D-2}}\frac{1}{r^{D-3}}\right)^{-1}dr^{2}+r^{D-2}d\Omega_{D-2}^{2}.
{\bfG^{(D-3)}}=\frac{1}{2}{\bfI}_{q}\oplus({\bfI}_{D-3}+{\bf\Xi}_{D-3})~,~{\bfG_{(D-3)}^{-1}}=2\,{\bfI}_{q}\oplus({\bfI}_{D-3}-\frac{1}{D-2}{\bf\Xi}_{D-3})\,.
\partial_{\alpha}\,\Gamma_{l,s}^{\,\alpha,\,\alpha_{0}}(\vec{p})=\frac{1}{2}\int\frac{d^{\,4}p}{(2\pi)^{\,4}}\(\partial_{\alpha}\,\tilde{C}_{\alpha}^{\,\alpha_{0}}(p))\\hat{\Gamma}_{l-1,s-1}^{\,\alpha,\,\alpha_{0}}(p,-p,\vec{p})\quad,
\Sigma=I+I_{ext}=\intdt(q_{i}(\partial_{t}a^{i}-h^{i})+\bar{c}_{i}\partial_{t}c^{i}+\bar{c}_{i}\partial_{j}h^{i}c^{j}+J_{i}^{a}c^{i}+iJ_{\bar{c}}^{i}q_{i}),
\begin{array}{l}{n_{+}=(\bar{n}_{+}\otimes1)\oplus(\bar{{\calG}}\otimes\lambda{\bfC}[\lambda]),}\\{n_{-}=(\bar{n}_{-}\otimes1)\oplus(\bar{{\calG}}\otimes\lambda^{-1}{\bfC}[\lambda^{-1}]),}\\\end{array}
\delta\eta_{\alpha_{2k+1}}^{*}=\Phi_{\alpha_{2k}}^{*}Z_{\;\;\alpha_{2k+1}}^{\alpha_{2k}}+\Phi_{\alpha_{2k+2}}^{*}A_{\alpha_{2k+1}}^{\;\;\alpha_{2k+2}},\;k=0,\cdots,b,
\begin{aligned}{1.~}&{{}~}&{}&{{}|\beta_{1}^{2}-\beta_{2}^{2}|}&{=}&{{}0\;}\\{2.~}&{{}~0}&{<}&{{}|\beta_{1}^{2}-\beta_{2}^{2}|}&{<}&{{}1\;}\\{3.~}&{{}~}&{}&{{}|\beta_{1}^{2}-\beta_{2}^{2}|}&{=}&{{}1\;}\\{4.~}&{{}~1}&{<}&{{}|\beta_{1}^{2}-\beta_{2}^{2}|}&{<}&{{}2\;}\\{5.~}&{{}~}&{}&{{}|\beta_{1}^{2}-\beta_{2}^{2}|}&{=}&{{}2\.}\\\end{aligned}
\delta_{\epsilon}V^{(1)\mu_{1}\ldots\mu_{p-2k-3}}=\partial^{\left[\mu_{1}\right.}\bar{\epsilon}^{\left.\mu_{2}\ldots\mu_{p-2k-3}\right]}+M\epsilon^{\mu_{1}\ldots\mu_{p-2k-3}}+\partial_{\mu}\bar{\epsilon}^{\mu\mu_{1}\ldots\mu_{p-2k-3}},
\intd^{7}x\sqrt{g_{CY_{3}}}\left[\sum_{i=1,2}\left(\mathrm{tr}F_{lm}^{(i)}F^{(i)lm}-\frac{1}{2}\mathrm{tr}R_{lm}R^{lm}\right)\delta(x^{11}-d^{(i)})\right]\;.
{\frac{d}{dt}}\left(\begin{matrix}{p}\\{q}\\\end{matrix}\right)={\frac{1}{\hbar}}\left(\begin{matrix}{0}&{-1}\\{1}&{0}\\\end{matrix}\right)\left(\begin{matrix}{{\frac{\deltaH_{1}}{\deltap}}}\\{{\frac{\deltaH_{1}}{\deltaq}}}\\\end{matrix}\right)
D_{c}^{(n)}(\eta,{\eta}^{\prime})=\frac{\Gamma(\frac{n}{2}-1)}{4\pi^{\frac{n}{2}}}\frac{1}{|{\eta}-{\eta}^{\prime}|^{n-2}}-\frac{\Gamma(\frac{n}{2}-1)}{4\pi^{\frac{n}{2}}}\frac{1}{|\frac{a_{{\eta}^{\prime}}}{a_{B}}{\eta}-\frac{a_{B}}{a_{{\eta}^{\prime}}}{\eta}^{\prime}|^{n-2}}
\sum_{n}^{\infty}{\frac{a^{n}}{n!}}\;H_{n}(x)H_{n}(y)={\frac{1}{\sqrt{1-4a^{2}}}}\;\mathrm{exp}\left[{\frac{4axy-4t^{2}(x^{2}+y^{2})}{1-4a^{2}}}\right]\;
\begin{array}{ll}{[M_{\mu\nu},M^{\rho\sigma}]=-2\delta_{[\mu}^{[\rho}M_{\nu]}{}^{\sigma]}\;\qquad}&{\[P_{\mu},M_{\nu\rho}\]=\eta_{\mu[\nu}P_{\rho]}\nonumber}\\{\[K_{\mu},M_{\nu\rho}\]=\eta_{\mu[\nu}K_{\rho]}\;\qquad}&{\[P_{\mu},K_{\nu}\]=2(\eta_{\mu\nu}D+2M_{\mu\nu})\,\nonumber}\\{\[D,P_{\mu}\]=P_{\mu}\;\qquad}&{\[D,K_{\mu}\]=-K_{\mu}\.}\\\end{array}
G_{\calQ}^{(2)}(x,x^{\prime},y,z,g_{\calQ})={\frac{1}{(x-x^{\prime})^{2}}}\int_{\calM}\omega_{y}^{(4)}\,\omega_{z}^{(1)}(x-x^{\prime})^{2}\,\gamma_{(2)}(x,x^{\prime},\rho,x_{0}),
Q^{-}\sim\sqrt{g^{2}N}\times\frac{K}{\sqrt{N}}\times\sum_{n,l=1}^{\infty}\left(\frac{1}{n}+\frac{1}{l}\right)\times\left(\Gamma^{\dagger}(l+n)\Gamma(l)\Gamma(n)+\Gamma^{\dagger}(n)\Gamma^{\dagger}(l)\Gamma(n+l)\right),
\zeta(s,x|L_{b})={\frac{a^{2s-N}}{2^{N}\pi^{\rho_{N}}\Gamma\left(\fracN2\right)\Gamma(s)}}\sum_{n=1}^{\rho_{N}}{\frac{(2n-1)!!}{2^{n}}}c_{2n}^{N}(a^{2}b)^{n+\frac12-s}\Gamma\left(s-n-\frac12\right).
G_{bb}^{-2}(x,y)=\left[-\frac{{N^{2}}}{g^{2}}\frac{{4}}{f^{4}({\hat{\phi}})}\nabla_{x}^{2}+\frac{{1}}{2}\frac{{\hbar}}{f({\hat{\phi}})}G_{aa}^{-1}(z,z)\right]\delta(x-y).
\Sigma(p,\beta)={\frac{C}{f(p,\beta)(2f(p,\beta)-m_{2})}}\biggl[ch(2g(p,\beta))+{\frac{\int{\frac{dpch^{2}(2g(p,\beta))}{f^{2}(p,\beta)(2f(p,\beta)-m_{2})}}}{1+{\frac{\nu^{4}(\varphi,\beta)}{16(2\pi)^{D}}}\int{\frac{dpch(2g(p,\beta))}{f^{2}(p,\beta)(2f(p,\beta)-m_{2})}}}}\biggr]
\begin{array}{l}{\tilde{Z}(\tilde{z}^{\prime})=G(g)^{-1}\tilde{Z}(\tilde{z})\tilde{K}(\tilde{z};g)~~~}\\{{}}\\{\bar{\tilde{Z}}(\tilde{z}^{\prime})=\bar{\tilde{K}}(\tilde{z};g)\bar{\tilde{Z}}(\tilde{z})G(g)}\\\end{array}
\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}}-F_{,\phi_{i}\phi_{j}}+F_{,\dot{\phi}_{i}\dot{\phi}_{l}}{\frac{\partial\dot{\phi}_{l}}{\partial\phi_{k}}}{\frac{\partial\dot{\phi}_{k}}{\partial\phi_{j}}}~=~0.
z_{1}\left(\tau,\sigma\right)=\sum_{n\inZ}\frac{i}{\sqrt{2\omega_{n\left(q\right)}}}e^{-in\left(q\right)\sigma}\left(e^{-i\omega_{n\left(q\right)}\tau}b_{n}-e^{i\omega_{n\left(q\right)}\tau}d_{-n}^{\dagger}\right),
\left\langle0\right|T\left[\bar{\psi}^{\uparrow}(t,\theta)\psi^{\downarrow}(0)\right]\left|0\right\rangle=\left\langle0\right|T\left[\bar{\psi}^{\downarrow}(t,-\theta)\psi^{\uparrow}(0)\right]\left|0\right\rangle.
\deltaI_{CS}=\frac{1}{8\pi^{2}}\int_{T^{2}}d^{2}x\,\epsilon^{ij}\mathrm{Tr}[G_{m}\partial_{i}G_{m}^{-1}({\calU}_{s}^{-1}\partial_{j}{\calU}_{s}-{\calU}_{s}\partial_{j}{\calU}_{s}^{-1})].
{\frac{1}{N^{2}}}\left<\Psi\right|{T_{1}}{T_{2}}\left|\Psi\right>={\frac{1}{N^{2}}}\left<\Omega\right|{T_{1}}{T_{2}}\left|\Omega\right>-{\frac{1}{N^{2}}}\int{\left[T_{2},T\right]}\int{\left[T_{1},T\right]}.
s^{b}({\bfr})=\left\{\left[-\left(1-L^{2}\right){\delta}_{ba}+\left(2{\calL}_{(1)}^{ba}+{\calL}_{(2)}^{ba}\right)\right]{\nabla}^{2}+\left(2{\delta}_{ba}-{\calL}_{(1)}^{ba}\right)\frac{{\nabla}^{4}}{\kappa}-\frac{{\nabla}^{6}}{{\kappa}^{2}}\right\}j_{0}^{b}({\bfr}).
S_{gf}=\int_{M_{n}}\sqrt{g}(g^{ij}u_{a}D_{i}e_{j}^{a}+g^{ij}v_{ab}D_{i}\omega_{j}^{ab}-g^{ij}\bar{c}_{a}\tilde{D}_{i}D_{j}c^{a}-g^{ij}\bar{f}_{ab}\tilde{D}_{i}D_{j}f^{ab}+\bar{c}_{a}e_{id}D^{i}f^{ab}).
\underbrace{(\rho_{i-\hat{\lambda},\lambda,\mu}\otimes\rho_{i\lambda\mu}^{\ast})\otimes\cdots}_{\lambda\in\{1,\ldots,\mu-1\}}\otimes\underbrace{(\rho_{i-\hat{\nu},\mu,\nu}^{\ast}\otimes\rho_{i\mu\nu})\otimes\cdots}_{\nu\in\{\mu+1,\ldots,d\}}
\frac{f(-s\triangle)-1}{s\triangle}=\int_{0}^{1}\!\!{\mathrm{d}}\alpha_{1}\alpha_{1}(1-\alpha_{1})\!\int_{0}^{1}\!\!{\mathrm{d}}\alpha_{2}\,{\mathrme}^{\alpha_{1}(1-\alpha_{1})\alpha_{2}s\triangle},
\rho(z_{1};\sigma_{1}|z_{2};\sigma_{2})=\langle\Psi_{\sigma_{1}}^{\dagger}(z_{1})\Psi_{\sigma_{2}}^{\dagger}(z_{2})\Psi_{\sigma_{2}}(z_{2})\Psi_{\sigma_{1}}(z_{1})\rangle\,.
I_{E}=2\pi\beta\,e^{(\Gamma-\Lambda)/2}\,[\,(\Gamma^{\prime}-\phi^{\prime})\,r^{2}+\frac{\alpha^{\prime}e^{\phi}}{g^{2}}\,(1-e^{-\Lambda})\,(\Gamma^{\prime}+\phi^{\prime})-\frac{\alpha^{\prime}e^{\phi}}{g^{2}}\,re^{-\Lambda}\Gamma^{\prime}\phi^{\prime}\,]\,|_{r_{h}}^{\infty}
\frac{3}{\sqrt{g}}g_{\mu\mu^{\prime}}g_{\nu\nu^{\prime}}\partial_{\alpha}[\sqrt{g}\partial_{[\alpha^{\prime}}a_{\mu^{\prime\prime}\nu^{\prime\prime}]}g^{\alpha\alpha^{\prime}}g^{\mu^{\prime\prime}\mu^{\prime}}g^{\nu^{\prime\prime}\nu^{\prime}}]=-\frac{2i\kappa}{3}{\dot{F}}_{\mu\nu\rho\sigma\tau}\partial^{[\rho}a^{\sigma\tau]},
{\Bigr({\calP}_{+}\Bigr)_{\;\nu}^{\mu}}_{;j}=i\gamma_{5}{\hat{k}}_{j}\left(n^{\mu}n_{\nu}-{\frac{1}{2}}\delta_{\nu}^{\mu}\right)+i\gamma_{5}{\hat{n}}\Bigr(K_{j}^{\mu}n_{\nu}+n^{\mu}K_{\nuj}\Bigr)\;.
\begin{array}{c}{a_{0}+a_{1}^{\prime}-a_{1}^{\prime\prime}+a_{1}^{\prime\prime\prime}=a_{1}+a_{0}^{\prime}-a_{2}^{\prime\prime}+a_{2}^{\prime\prime\prime}=0,}\\{a_{2}-a_{2}^{\prime}+a_{0}^{\prime\prime}+a_{3}^{\prime\prime\prime}=a_{3}-a_{3}^{\prime}+a_{3}^{\prime\prime}+a_{0}^{\prime\prime\prime}=0,}\\\end{array}