harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{s^{2n+1}+t^{2n+1}+u^{2n+1}=stu\left[\sum_{r=1}^{n}\sum_{q=0}^{2n-2r}\right.{\frac{(2n+1)!}{r!(2n+1-r)!}}(-1)^{q}s^{2n-1-r-q}t^{r+q-1}\bigg],}
{\frac{(12)^{2}(34)^{2}}{x_{12}^{4}x_{34}^{4}}}a(s,t)+{\frac{(14)^{2}(23)^{2}}{x_{14}^{4}x_{23}^{4}}}b(s,t)+{\frac{(12)(23)(34)(41)}{x_{12}^{2}x_{23}^{2}x_{34}^{2}x_{41}^{2}}}c(s,t)\;.
R_{\mu\nu}-{\frac{1}{2}}g_{\mu\nu}R=-4\alpha^{\prime}(f_{3})_{\phi_{o},\sigma_{o},b_{o}}(F_{\mu}^{\;\;\sigma}F_{\nu\sigma}-{\frac{1}{4}}g_{\mu\nu}F^{\rho\sigma}F_{\rho\sigma})
\Big\langle\prod_{j=1}^{n}:e^{i\varphi(t_{j})}:_{M}\Big\rangle_{C_{m}}\;=\;e^{\frac{1}{2}\sum_{i=1}^{n}\big(t_{i},C_{M}t_{i}\big)-\frac{1}{2}\sum_{i,j}\big(t_{i},C_{m}t_{j}\big)}\;.
F_{B}=\bigg[2\epsilon_{ij}^{(1)}\epsilon_{il}^{(2)}p^{j}k^{l}-\epsilon_{ij}^{(1)}\epsilon_{ij}^{(2)}k_{l}p_{l}\bigg]{\frac{\Gamma(1+2k_{\parallel}^{2})\Gamma(k\cdotp)}{\Gamma(k\cdotp+2k_{\parallel}^{2}+1)}}\prod_{A=0}^{p}\delta(k^{A}+p^{A}).
\begin{array}{c}{u^{\alpha}=u_{0}^{\alpha}+\xi^{\alpha}s-\frac12\left(\stackrel{s}{\Gamma}_{\beta\gamma}^{\alpha}\right)_{0}\xi^{\beta}\xi^{\gamma}s^{2}-}\\{\frac1{3!}\left(\Gamma^{\alpha}{}_{\beta\gamma\delta}\right)_{0}\xi^{\beta}\xi^{\gamma}\xi^{\delta}s^{3}-\cdots}\\\end{array}
L-H{\frac{\deltaL}{\deltaH}}-I{\frac{\deltaL}{\deltaI}}+[H{\frac{d}{dt}}+H(3H+DI)-\dot{H}]{\frac{\deltaL}{\delta\dot{H}}}+[I{\frac{d}{dt}}+I(3H+DI)-\dot{I}]{\frac{\deltaL}{\delta\dot{I}}}\equiv{\calD}L=0.
\hat{h}_{\mu\nu}^{+}=a^{2}\xi_{\mu\nu}+2a^{2}\left(\int\frac{dz}{a^{4}}\right)\partial_{\mu}\partial_{\nu}\xi-2\frac{a^{2}}{a_{+}^{2}}\left(\int\frac{dz}{a^{2}}\right)\partial_{\mu}\partial_{\nu}\xi-\frac{U_{B}}{2}\frac{a^{2}}{a_{+}^{2}}\xi\eta_{\mu\nu}
T_{\alpha\beta}^{X}=\frac{\Omega^{\delta}}{2}((\partial_{\alpha}-ieA_{\alpha})X(\partial_{\beta}+ieA_{\beta})X^{*}-\frac{1}{2}g_{\alpha\beta}g^{\mu\nu}(\partial_{\mu}-ieA_{\mu})X(\partial_{\nu}+ieA_{\nu})X^{*}),
G_{2}=\left[\frac{\deltaZ}{\delta\widehat{s}}\frac{\delta^{2}\widehat{s}}{\deltas\deltas}+\frac{\delta^{2}Z}{\delta\widehat{s}\delta\widehat{s}}\frac{\delta\widehat{s}}{\deltas}\frac{\delta\widehat{s}}{\deltas}\right]_{s=0}\;.
ds^{2}=N_{f}{\frac{1}{[(V_{1})^{2}+V_{2}^{2}+V_{3}^{2}+V_{4}^{2}]}}[({dV_{1}}^{2})+{(dV_{2}}^{2})+{(dV_{3}}^{2})+{(dV_{4}}^{2})].
g_{5D}=\left(\begin{array}{cc}{g_{\mu\nu}(x,z)+e^{2r(x)}V_{\mu}(x)V_{\nu}(x)}&{e^{2r(x)}V_{\mu}(x)}\\{e^{2r(x)}V_{\nu}(x)}&{e^{2r(x)}}\\\end{array}\right)
f^{\prime\prime}(x)={\frac{{f^{\prime}}^{2}(x)}{2f(x)}}+2f(x)(f_{1}^{2}(x)+f_{1}^{\prime}(x)-\frac{1}{4}f^{2}(x)-f^{\prime}(x)-k_{1}-\frac{1}{2}k_{2})-{\frac{k_{2}^{2}}{2f(x)}},
\phi_{m}=N_{m}e^{{\frac{\nu}{2}}k\rho}\left[J_{\nu-1}\left({\frac{2m}{k}}\right)Y_{\nu}\left({\frac{2m}{k}}e^{{\frac{k}{2}}\rho}\right)-Y_{\nu-1}\left({\frac{2m}{k}}\right)J_{\nu}\left({\frac{2m}{k}}e^{{\frac{k}{2}}\rho}\right)\right]
\begin{array}{ll}{1)}&{h=a=m=0,}\\{2)}&{h=1/2,\quada=1/10,\quadm=0,}\\{3)}&{h=x,\quada=0,\quadm=0,}\\{4)}&{h=1/2+x,\quada=1/10,\quadm=\sqrt{x}.}\\\end{array}
\big<\kappa_{d_{1}-1}\cdots\kappa_{d_{n}-1}\big>=\int_{{\calC}_{n}\left(\overline{{\calM}}_{h}\right)}c_{1}\left(\hat{\calL}_{(1)}\right)^{d_{1}}\wedge\cdots\wedgec_{1}\left(\hat{\calL}_{(n)}\right)^{d_{n}},
U_{I_{14}+\mu}^{(I_{1},I_{2},I_{3},I_{4})}(\eta)=\sum_{\nu}((F^{s})^{I_{1}I_{2}I_{3}I_{4}})_{\mu\nu}S_{I_{34}+\nu}^{(I_{1},I_{2},I_{3},I_{4})}(\eta)\.
\stackrel{\sim}{\gamma}_{_\mu}=\lambda^{-1}\left(\begin{array}{ll}{-\gamma_{_\mu}}&{0}\\{0}&{\gamma_{_\mu}}\\\end{array}\right),\stackrel{\sim}{\gamma_{\mathrm{~}}}_{5}\mathrm{~}=\lambda^{-1}\left(\begin{array}{ll}{h}&{0}\\{0}&{-h}\\\end{array}\right),\gamma=\left(\begin{array}{ll}{0}&{1}\\{1}&{0}\\\end{array}\right).\tag{2.5a}
\Gamma_{(2)}={\frac{1}{2(4\pi)^{2}}}\intd^{12}z{\bf\Psi^{2}\bar{\Psi}^{2}}({\frac{1}{36}}+{\frac{1}{5!}}\sum_{k=1}^{\infty}{\frac{(k+5)(k+4)(k+1)}{(k+3)(k+2)}}X^{k}).
\frac{p_{r}^{2}}{g_{rr}}+\frac{-{g^{\prime}}_{tt}}{-\calD}\left(p_{\phi}+\frac{g_{t\phi}+\Omega_{0}g_{\phi\phi}}{{g^{\prime}}_{tt}}E\right)^{2}=\left(\frac{E^{2}}{-{g^{\prime}}_{tt}}-V\right),
\begin{array}{c}{\Phi_{2}\left(p\right)_{a_{1}...a_{m}b_{1}...b_{n}}=}\\{=\epsilon_{a_{1}b_{1}c_{1}d_{1}}...\epsilon_{a_{n}b_{n}c_{n}d_{n}}p_{a_{n+1}c_{n+1}}...p_{a_{m}c_{m}}\Phi_{2}\left(p\right)^{c_{1}...c_{m}d_{1}...d_{n}}}\\\end{array}
\left[a_{ij}(p^{+}),a_{lk}^{\dagger}(q^{+})\right]=\left\{b_{ij}(p^{+}),b_{lk}^{\dagger}(q^{+})\right\}=\delta(p^{+}-q^{+})\left(\delta_{il}\delta_{jk}-\frac{1}{N}\delta_{ij}\delta_{kl}\right)
M=\frac{\sum_{m=-k+1}^{k}\theta_{\nu}(2mi\lambda,x^{2k})x^{m^{2}/2k+k/8-l/2}(-1)^{(m-l)/2}C_{m}^{l}}{\sum_{m=-k+1}^{k}\theta_{3}((2m-k)i\lambda,x^{2k})x^{(m-k/2)^{2}/2k}C_{m}^{l}}
n_{(r,s,l),(r^{\prime},s^{\prime},l^{\prime})}^{(r^{\prime\prime},s^{\prime\prime},l^{\prime\prime})}=n_{rr^{\prime}}^{(m)\,r^{\prime\prime}}n_{ss^{\prime}}^{(m+2)\,s^{\prime\prime}}n_{l+1,l^{\prime}+1}^{(4)\,l^{\prime\prime}+1},
H_{c}=\intd^{3}x\left[\frac{1}{2}(\pi_{i}^{a})^{2}+\frac{1}{4}F_{ij}^{a}F^{a,ij}+\frac{1}{2}m^{2}\{(A^{a,0})^{2}+(A^{a,i})^{2}\}-A_{0}^{a}\Theta_{2}^{a}\right],
E=\frac{m_{p}(\dot{r}^{2}+r^{2}\dot{\phi}^{2})}{\sqrt{1-H_{0}(\dot{r}^{2}+r^{2}\dot{\phi}^{2})}}+m_{p}H_{0}^{-1}[\sqrt{1-H_{0}(\dot{r}^{2}+r^{2}\dot{\phi}^{2})}-1].
\frac{\partialV_{F}}{\partialz^{a}}(S,\bar{S}_{0},z_{0}^{a},\bar{z}_{0}^{\bar{a}})=0,\qquad\qquadV_{F}(S_{0},\bar{S}_{0},z_{0}^{a},\bar{z}_{0}^{\bar{a}})=0,\qquad\qquadF_{S}(S_{0},z_{0}^{a})\ne0,
\left(\begin{array}{c}{v_{a}}\\{v_{b}}\\{t_{a}}\\{t_{b}}\\\end{array}\right)\toM_{K3}^{\epsilon}\left(\begin{array}{c}{v_{a}}\\{v_{b}}\\{t_{a}}\\{t_{b}}\\\end{array}\right)=\left(\begin{array}{rrrr}{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{0}&{-1}&{-1}&{-1}\\{1}&{1}&{1}&{0}\\\end{array}\right)\left(\begin{array}{c}{v_{a}}\\{v_{b}}\\{t_{a}}\\{t_{b}}\\\end{array}\right)
\begin{array}{ll}{\delta_{S}{\calL}_{0}=}&{2\partial_{z}(\delta_{S}\phi\partial_{\bar{z}}\phi-\bar{\psi}\delta_{S}\bar{\psi}-2\partial_{\bar{z}}\phi\bar{\eta}\bar{\psi}-2\eta\bar{\psi}e^{\phi})+}\\{}&{+2\partial_{\bar{z}}(\delta_{S}\phi\partial_{z}\phi+\psi\delta_{S}\psi-2\partial_{z}\phi\eta\psi-2\bar{\eta}\psie^{\phi}).}\\\end{array}
S=\intd^{4}x\mathrm{Tr}\left[-{\frac{1}{12}}\widetildeH_{\mu\nu\lambda}\widetildeH^{\mu\nu\lambda}-{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}+{\frac{m}{4}}\epsilon^{\mu\nu\rho\lambda}B_{\mu\nu}(F_{\rho\lambda}-\tilde{F}_{\rho\lambda})+{\frac{1}{4}}\epsilon^{\mu\nu\rho\lambda}E_{\mu\nu}\tilde{F}_{\rho\lambda}\right].
\alpha_{(1)}^{(1)}=-\frac{\sqrt{2}}{6}\sigmad(K_{\gamma}(t)-\sqrt{2}bK_{\phi}(t)+f_{\gamma}(t)-\sqrt{2}f_{\phi}(t))+P^{\prime}(0,t)-\frac{\kappa^{2}}{6}(\rho+U)de^{\sqrt{2}b\hat{\phi}}.
\mathrm{\hat{H}|\Psi>=\left[\tilde{R}^{2}\frac{\partial^{2}}{\partial\tilde{R}^{2}}+\tilde{R}\frac{\partial}{\partial\tilde{R}}-6\frac{\partial^{2}}{\partial\tilde{\varphi}^{2}}\right]|\Psi>=0,}
\begin{array}{ll}{\hat{M}_{0}(z)Z_{0}=Z_{0}H(z)\,,}&{H(z)=\left(\begin{array}{cc}{\tilde{\hat{w}}(z)-\frac{1}{2}\hat{\lambda}(z)+i\textstyle{\frac{1}{2}}\hat{\psi}(z)}&{2\hat{\rho}_{b}(z)}\\{0}&{\hat{t}{}_{~b}^{a}(z)+i\textstyle{\frac{2}{{\calN}}}(\hat{\psi}(z)+\delta^{4}{}_{{\calN}}\,\Omega)\delta^{a}{}_{b}}\\\end{array}\right).}\\\end{array}
\delta\mathcal{P}_{a}^{(1)}=-\pi_{a}^{0},\;\delta\mathcal{P}_{a}=\partial_{i}B_{a}^{0i},\;\deltaP_{a}^{(1)i}=p_{a}^{i},\;\deltaP_{a}^{i}=-\partial^{i}\varphi_{a},
\langle0|(E^{V})^{2}|0\rangle=\frac{1}{8}\frac{1}{(2\pi)^{2d}}\int_{V}d^{d}y_{1}\int_{V}d^{d}y_{2}\intd^{d}p\intd^{d}qe^{\imath(\vec{p}+\vec{q})\cdot(\vec{y}_{1}-\vec{y}_{2})}{\left(\frac{\vec{p}\cdot\vec{q}}{\sqrt{pq}}+\sqrt{pq}\right)}^{2}.
\begin{array}{ll}{I:0<\rho<\alpha}&{\Rightarrow\0<\sqrt{\rho_{1}\rho_{2}}<\alpha}\\{II:\alpha<\rho<\alpha+\frac{\pi}{2}}&{\Rightarrow}\\{alpha<\sqrt{\rho_{1}\rho_{2}}<\alpha+\frac{\pi}{2}}\\{III:\alpha+\frac{\pi}{2}<\rho}&{\Rightarrow\alpha+\frac{\pi}{2}<\sqrt{\rho_{1}\rho_{2}}}\\\end{array}
\zeta_{Z.M.}\,\left(s;\frac{\partial^{2}+\xiR}{\mu^{2}}\right)=\frac{\mu^{2s}}{\Gamma\left(s\right)}\frac{\beta}{\sqrt{\pi}}\sum_{n=1}^{\infty}\int_{0}^{\infty}dt\,t^{s-3/2}e^{-n^{2}\beta^{2}/4t}.
\phi^{g}=\frac{1}{2}({\calP}^{2}+m^{2}-iqN_{\theta}\xi_{\mu}\xi_{\nu}{\calF}^{\mu\nu})\approx0,\quad\varrho^{g}=i\epsilon_{\mu\nu\lambda}{\calP}^{\mu}\xi^{\nu}\xi^{\lambda}-2mN_{\theta}\approx0,
{\mathcalJ}^{0}(x)\;=\;\frac{q}{2\pi}~\mathrm{div}\;\underline{{\varphi}}(x)\,,~\underline{{{\mathcalJ}}}(x)\;=\;-\,\frac{q}{2\pi}~\frac{\partial}{\partialt}~\underline{{\varphi}}(x)~.
{\calL}_{B}=\frac{1}{2}B_{\mu}^{*}[-i\epsilon_{\mu\nu\lambda}(\partial_{\nu}+ia_{\nu}+iC_{\nu})+M\delta_{\mu\lambda}]B_{\lambda}-\frac{i}{8\pi(\alpha-\frac{1}{2})}\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}\;,
\begin{array}{rcl}{c^{(0)\prime}}&{=}&{-c_{\sigma}^{(1)}\,,}\\{}&{}&{}\\{a_{i\sigma}^{(2)\prime}}&{=}&{-c_{i}^{(1)}\,,}\\{}&{}&{}\\{a_{ij}^{(2)\prime}}&{=}&{-{\tilde{c}}_{ij\sigma}^{(3)}\,,}\\\end{array}
\begin{array}{ccc}{ds^{2}=(\frac{F}{K})^{\frac{2}{3}}\left(dt+\omega_{I}dx^{I}\right)^{2}-(\frac{K}{F})^{\frac{1}{3}}dx_{I}dx_{I}}&{,}&{H_{\mu\nu\rho}=2FK\,A_{[\mu}\partial_{\nu}A_{\rho]}}\\{e^{-\frac{4}{\sqrt{3}}\sigma}=FK}&{,}&{e^{-4\phi}=\frac{K}{F}}\\{A_{\mu}=-K^{-1}(\,1\,,\,\omega_{I}\,)}&{,}&{B_{\mu}=-F(\,1\,,\,\omega_{I})\,.}\\\end{array}
G^{AB}(p)=[(G_{0})^{-1}+i\Pi_{\mathrm{gh}}(p^{2})]_{AB}^{-1}={\frac{1}{-ip^{2}+i\Pi_{\mathrm{gh}}(p^{2})}}\delta^{AB}=(-i){\frac{G_{\mathrm{gh}}(-p^{2})}{-p^{2}}}\delta^{AB},
\widehat{\xi}_{\mu}=\frac{1}{\sqrt{2}}\gamma_{\mu}\otimes\sigma_{1},\quad\widehat{\theta}_{1}=\frac{i}{\sqrt{2}}1\otimes\sigma_{2},\quad\widehat{\theta}_{0}=\frac{1}{\sqrt{2}}1\otimes\sigma_{3},
d\hat{s}^{2}=\epsilon^{2}ds^{2}=-\hat{n}^{2}\hat{c}^{2}d\hat{t}^{2}+\hat{a}^{2}\hat{\gamma}_{ij}d\hat{x}^{i}d\hat{x}^{j}+\hat{b}^{2}d\hat{y}^{2},
\left[\,\left(T_{x}^{g}\right)^{\vee}\otimes\left(h^{*}T_{x}^{g}\right)\,\right]\:\otimes\:\left[\,\left(T_{x}^{h}\right)^{\vee}\otimes\left(g^{*}T_{x}^{h}\right)\,\right]^{\vee}
F^{0}\,\equiv\,dA^{0}\,+\,{\frac{1}{\sqrt{2}}}\,\left[{\bar{\psi}}_{A}\,\wedge\,\psi_{B}\,\varepsilon^{AB}\,+\,{\bar{\psi}}^{A}\,\wedge\,\psi^{B}\,\varepsilon_{AB}\right]\,=\,0
I_{\mathrm{HT/M}}=-\frac{\pi}{2GH_{-}^{2}}\left(1-\eta\sqrt{1-H_{-}^{2}b_{\mathrm{m}}^{2}}\right)-\frac{\pi}{2GH_{+}^{2}}\left(1-\sqrt{1-H_{+}^{2}b_{\mathrm{m}}^{2}}\right).
N_{21}(x,y,a)|_{x=0}=\frac{1}{16ay}-\frac{1}{16ay}\frac{e^{\frac{4\pi\,y}{a}}}{e^{\frac{4\pi\,y}{a}}+1}+\frac{1}{16ay}\frac{1}{e^{\frac{4\pi\,y}{a}}+1}.
\delta{\calL}_{source}=\omega\left[J^{\mu,a}(D_{\mu}c^{a})-i\overline{{\eta}}^{a}(-{\frac{1}{2}}f^{abc}c^{b}c^{c})+iF^{a}\eta^{a}+{\frac{1}{2}}HF^{a}F^{a}\right]
2^{5}\left(v_{4}^{(1)}v_{4}^{(2)}+{\frac{v_{4}^{(2)}}{v_{4}^{(1)}}}+{\frac{v_{4}^{(1)}}{v_{4}^{(2)}}}+{\frac{1}{v_{4}^{(1)}v_{4}^{(2)}}}\right)\int_{\ell\to\infty}\!d\ell,
\frac{\ell^{2}}{2}\intd^{4}x\,H_{\mu\nu\lambda}\,\bar{\psi}\gamma^{\mu}\gamma^{\nu}\gamma^{\lambda}\psi\;=\;\frac\ell2\intd^{4}x\;\partial_{\mu}\varphi\bar{\psi}\gamma^{\mu}\gamma\psi~,
F^{\lambda_{1}\lambda_{2},\lambda_{3}\lambda_{4},\lambda_{5}\lambda_{6}}F_{\lambda_{1}^{\prime}\lambda_{2}^{\prime},\lambda_{3}^{\prime}\lambda_{4}^{\prime},\lambda_{5}^{\prime}\lambda_{6}^{\prime}}=\left(\delta(\lambda_{1}^{\prime}-\lambda_{1})\delta(\lambda_{2}^{\prime}-\lambda_{2}),\right.
{\calL}_{4ferm}^{noninv}=+{\frac{\mathrm{i}}{4}}{\calH}_{\Lambda\vert\mu\nu}^{-}\,{\tau}^{-\Lambda\vert\mu\nu}\,+\,h.c.=+{\frac{\mathrm{i}}{4}}\left({\calN}-\bar{\calN}\right)_{\Lambda\Sigma}{\tau}_{\mu\nu}^{-\Lambda}{\tau}^{-\Sigma\vert\mu\nu}\,+\,h.c.
(2\pi)^{2}\delta(p_{1}+p_{2}+p_{3}+q_{1}-\omega)\delta(p_{1}+p_{2}+p_{3}-q_{1})=\frac{1}{2}(2\pi)^{2}\delta(p_{1}+p_{2}+p_{3}-\frac{\omega}{2})\delta(q_{1}-\frac{\omega}{2}).
V=-{\frac{1}{2\bar{\bfZ}}}\left[\left(\begin{matrix}{{\bf0}}&{{\bfI}}\\{-{\bfI}}&{{\bf0}}\\\end{matrix}\right)\cdot{\bfM}({\calF})+i\left(\begin{matrix}{{\bfI}}&{{\bf0}}\\{{\bf0}}&{{\bfI}}\\\end{matrix}\right)\right]\cdot{\calQ},
A^{22}=-\frac{\psi^{\prime}}{r}\left(1-\frac{b}{r}\right)+\frac{e^{2\psi}\alpha^{2}}{e^{2\psi}\alpha^{2}-\beta^{2}r^{2}}\frac{b^{\prime}r-b}{2r^{3}}+\frac{r^{2}\beta^{2}\left(1-\frac{b}{r}\right)}{e^{2\psi}\alpha^{2}-\beta^{2}r^{2}}\left[\psi^{\prime\prime}-\frac{b^{\prime}r-b}{2r(r-b)}\psi^{\prime}+{\psi^{\prime}}^{2}\right]
\begin{array}{cc}{(3,4,5,6)\mathrm{~D4~charge~}}&{\frac{1}{2\pi}\Sigma_{a=1}^{N}\mathcal{F}_{12}^{a}}\\{(1,2,5,6)\mathrm{~D4~charge~}}&{\frac{1}{2\pi}\Sigma_{a=1}^{N}\mathcal{F}_{34}^{a}}\\{(1,2,3,4)\mathrm{~D4~charge~}}&{\frac{1}{2\pi}\Sigma_{a=1}^{N}\mathcal{F}_{56}^{a}.}\\\end{array}
\left(\begin{array}{cc}{A}&{B}\\{C}&{D}\\\end{array}\right)^{-1}=\left(\begin{array}{cc}{(A-BD^{-1}C)^{-1}}&{-A^{-1}B(D-CA^{-1}B)^{-1}}\\{-D^{-1}C(A-BD^{-1}C)^{-1}}&{(D-CA^{-1}B)^{-1}}\\\end{array}\right).
\phi_{j}^{i}=\left(\begin{array}{ccccc}{v}&{}&{}&{}&{}\\{}&{v}&{}&{}&{}\\{}&{}&{v}&{}&{}\\{}&{}&{}&{v}&{}\\{}&{}&{}&{}&{v}\\\end{array}\right)\;\{\bar{\phi}}_{i}^{j}=\left(\begin{array}{ccccc}{{\bar{v}}}&{}&{}&{}&{}\\{}&{{\bar{v}}}&{}&{}&{}\\{}&{}&{{\bar{v}}}&{}&{}\\{}&{}&{}&{0}&{}\\{}&{}&{}&{}&{0}\\\end{array}\right)
{\calZ}=\left(\begin{matrix}{\lambda_{1}e^{i\theta_{1}}\tau}&{0}&{0}&{0}\\{0}&{\lambda_{2}e^{i\theta_{2}}\tau}&{0}&{0}\\{0}&{0}&{\lambda_{3}e^{i\theta_{3}}\tau}&{0}\\{0}&{0}&{0}&{\lambda_{4}e^{i\theta_{4}}\tau}\\\end{matrix}\right)~~~~~~;~~~~~~\tau=\left(\begin{matrix}{0}&{1}\\{-1}&{0}\\\end{matrix}\right)
f\starg=\sum_{s=0}^{\infty}\frac{\kappa^{s}}{s!}\sum_{t=0}^{s}(-1)^{t}\left(\begin{matrix}{s}\\{t}\\\end{matrix}\right)[\partial_{x}^{s-t}\partial_{p}^{t}f][\partial_{x}^{t}\partial_{p}^{s-t}g]\,.
L=2t\int_{\Sigma}d^{2}z(g_{IJ}\partial_{z}\psi^{I}\partial_{\bar{z}}\phi^{J}+i\psi_{-}^{\bar{i}}D_{z}\psi_{-}^{i}g_{{\bar{i}}i}+i\psi_{+}^{\bar{i}}D_{\bar{z}}\psi_{+}^{i}g{{\bar{i}}i}+R_{i{\bar{i}}j{\bar{j}}}\psi_{+}^{i}\psi_{+}^{\bar{i}}\psi_{-}^{j}\psi_{-}^{\bar{j}})
2\hat{H}=-g^{-{\frac{1}{2}}}\nabla_{i}g^{\frac{1}{2}}g^{ij}\nabla_{j}+R_{ijkl}\hat{\psi}^{k}\hat{\psi}^{*}{}^{l}\hat{\psi}^{*}{}^{j}\hat{\psi}^{i}+g^{ij}{\frac{\partialh}{\partialx^{i}}}{\frac{\partialh}{\partialx^{j}}}+H_{ij}\{\hat{\psi}^{*}{}^{j},\hat{\psi}^{*}{}^{i}\},
a_{1}^{d=6}\\rightarrow\a_{1}^{d=4}\,,\qquada_{3}^{d=6}\\rightarrow\\frac{s^{2}}{t^{2}}\,a_{3}^{d=4}\,,\qquadb_{2}^{d=6}\\rightarrow\\frac{s}{t}\,b_{2}^{d=4}\,.
\Phi=\left(\begin{array}{c}{\Theta}\\{\chi}\\\end{array}\right)=\left(\begin{array}{c}{\Theta}\\{\frac{i(m+E)}{\lambda-k_{z}}\sigma^{3}\Theta}\\\end{array}\right),\\Theta=\left(\begin{array}{c}{\Theta_{1}}\\{\Theta_{2}}\\\end{array}\right)
\oint_{C}\frac{w^{-1}dw}{2\pi\sqrt{-1}}\frac{(1-r\zeta^{2})(w^{2}-q^{6})(w-rq^{4})}{(w-q^{2}\zeta^{2})(w-q^{4}\zeta^{2})(w-q^{4}\zeta^{-1})}e^{R(w)+R(q^{6}w^{-1})}|0\rangle^{(0)}=(\zeta\leftrightarrow\zeta^{-1})|0\rangle^{(0)}.
T_{NN}(L,m,d,z)=\frac{g}{2L}\frac{1}{(2\pi)^{d-1}}\sum_{n=-\infty}^{\infty}\intd^{d-1}p\frac{1}{(\vec{p}^{\,2}+(\frac{n\pi}{L})^{2}+m^{2})}+gf_{2}(L,m,d,z).
\equiv\left(\begin{array}{c|c|c}{M(4,0)}&{+M(3,1)}&{+M[(2)(2)]+}\\\hline{+M(3,1)}&{+M(2,2)}&{+M(1,3)+}\\\hline{+M[(2),(2)]}&{+M(1,3)}&{+M(0,4)}\\\end{array}\right),
{\calM}_{ab}^{og}=\left(\begin{array}{cccc}{0}&{\zeta_{W}^{o}M_{W}^{o2}}&{0}&{0}\\{\zeta_{W}^{o}M_{W}^{o2}}&{0}&{0}&{0}\\{0}&{0}&{\zeta_{Z}^{o}M_{Z}^{o2}}&{0}\\{0}&{0}&{0}&{M_{gA}^{o2}}\\\end{array}\right)
\begin{array}{ll}{1)}&{h=7/16,\quada=7/16,\quadm=0,}\\{2)}&{h=7/16,\quada=3/80,\quadm=0,}\\{3)}&{h=7/16+x,\quada=(3/80,7/16),\quadm=0,}\\{4)}&{h=7/16+x,\quada=3/80,\quadm=\sqrt{x}/2.}\\\end{array}
\int_{-\infty}^{\infty}{\frac{dp}{2\pi}}e^{-\alpha_{1}p^{2}}\,e^{-\alpha_{2}(p+k)^{2}}={\frac{1}{\sqrt{4\pi(\alpha_{1}+\alpha_{2})}}}exp\left[-k^{2}\left({\frac{\alpha_{1}\alpha_{2}}{\alpha_{1}+\alpha_{2}}}\right)\right]
N\sum_{ijk}\int_{x}\Biggl[\tilde{A}_{ij}^{\mu}(x)\tilde{A}_{\mukl}(x)\tilde{A}_{jk}^{\nu}(x)\tilde{A}_{\nuli}(x)W^{ijkl}+\tilde{A}_{ij}^{\mu}(x)\tilde{A}_{\mujk}(x)\tilde{A}_{kl}^{\nu}(x)\tilde{A}_{\nuli}(x)L^{ijkl}\Biggr].
{\calE}=\frac{1}{2}\,\mu^{1-2\nu}\sum_{n=1}^{\infty}\int\frac{d^{2}p}{(2\pi)^{2}}\left\{{\bfp}^{2}+\frac{\pi^{2}}{\ell^{2}}\left[(n+s)^{2}+\frac{g}{3}\right]\right\}^{\nu}\Bigg|_{\nu=1/2}.
\boxed{\begin{aligned}{\hat{V}^{a}(x,\theta)=}&{{}\;\;-\frac{1}{2}\left(\overline{{\theta}}\gamma^{5}\gamma^{\mu}\theta\right)V_{\mu}^{a}(x)-\mathrm{i}\left(\overline{{\theta}}\gamma^{5}\theta\right)\left(\overline{{\theta}}\lambda^{a}(x)\right)-\frac{1}{4}\left(\overline{{\theta}}\gamma^{5}\theta\right)^{2}D^{a}(x)}\\\end{aligned}}\;\;\;,
G_{(3)}^{\mathrm{(TM)}}(\tau)=360\tau^{5}\int_{0}^{\Omega/2}d\Omega^{\prime}\int_{0}^{\Omega^{\prime}}dK\,\frac{K\,\left[K^{2}+\Omega^{\prime}\left(\Omega-\Omega^{\prime}\right)\right]^{2}}{\sqrt{(\Omega^{2}-K^{2})\left[(\Omega-\Omega^{\prime})^{2}-K^{2}\right]}}\,\left(\frac{1}{e^{\Omega^{\prime}}-1}+\frac{1}{e^{\Omega-\Omega^{\prime}}-1}\right).
C_{F}(d-\tilde{\eta}_{o})=C_{F}^{-1}(\tilde{\eta}_{o})=\frac{\Gamma(\tilde{\eta}_{o})\Gamma(\tilde{\eta}_{o}-\mu)\Gamma^{4}(\mu-\frac{1}{2}\tilde{\eta}_{o}+\frac{1}{2})}{\Gamma(d-\tilde{\eta}_{o})\Gamma(\mu-\tilde{\eta}_{o})\Gamma^{4}(\frac{1}{2}\tilde{\eta}_{o}+\frac{1}{2})}\,.
\phi^{4}=\left(\begin{array}{cccc}{k^{1}t\otimes1_{N_{1}\timesN_{1}}}&{0}&{\dots}&{0}\\{0}&{k^{2}t\otimes1_{N_{2}\timesN_{2}}}&{\dots}&{0}\\{\dots}&{\dots}&{\dots}&{\dots}\\{0}&{\dots}&{\dots}&{k^{K}t\otimes1_{N_{K}\timesN_{K}}}\\\end{array}\right)
P(i\rightarrowf)=\sum_{n_{1},\cdots,n_{\ell}=0}^{\infty}\,e^{-im(n_{1}r_{1}+\cdots+n_{\ell}r_{\ell}+D_{G}/2)(t_{f}-t_{i})}\,<\psi_{f}|n_{1},\cdots,n_{\ell}><n_{1},\cdots,n_{\ell}|\psi_{i}>\\\.
S={\frac{1}{4\pi\alpha^{\prime}}}\int_{M}d^{2}\sigma({\sqrt{g}}g^{ab}g_{\mu\nu}+i\epsilon^{ab}B_{\mu\nu})\partial_{a}X^{\mu}\partial_{b}X^{\nu}+i\int_{\partialM}d\lambda(A_{\mu}{\frac{d}{d\lambda}}X^{\mu})\quad,
\epsilon^{\alpha\mu\nu\beta}\gamma_{5}\gamma_{\mu}\partial_{\nu}\Psi_{\beta}^{i}-{\frac{\kappa\epsilon^{ij}}{\sqrt2}}\big[F^{\alpha\beta}+{\scriptstyle{\frac{1}{2}}}\gamma_{5}\epsilon^{\alpha\beta\rho\sigma}F_{\rho\sigma}\big]\Psi_{\beta}^{j}=0\\,
\int_{0}^{\theta}a_{5/2}(f_{\theta^{\prime}},P_{\theta^{\prime}})d\theta^{\prime}={\frac{d}{dD}}a_{5/2}^{D}(P_{\theta})\Big|_{D=5}-{\frac{d}{dD}}a_{5/2}^{D}(P_{0})\Big|_{D=5}.
\Theta^{\rho\sigma}\mathrm{}_{\mu\nu}\Theta_{\rho\sigma,\kappa\lambda}^{-1}=(I)_{\mu\nu,\kappa\lambda}=\left(P^{(2)}+P_{m}^{(1)}+P_{s}^{(0)}+P_{\omega}^{(0)}\right)_{\mu\nu,\kappa\lambda}\.
\frac{\partialZ_{\Psi}[J,\epsilon]}{\partial\epsilon^{\alpha}(x)}\,=\,{\frac{i}{\hbar}}\,\langle\,-\,\partial_{\mu}j_{\alpha}^{\mu}\,+\,{\frac{\partialW^{\prime}}{\partial\epsilon^{\alpha}}}\,-\,\hbar{\calA}_{\alpha}\,\,-\,G_{\alpha}^{A}[\Phi]\eta_{A}\,\rangle
\Gamma^{(PV)}(A)=\frac{1}{2}\left.\frac{d}{d~s}(\zeta^{PV}(s))\right|_{s=0}=i\frac{\pi}{2}\eta(0)-i\sum_{k=0}^{\infty}(-1)^{k}\frac{m^{(2k+1)}}{2k+1}\eta(2k+1),
\left(\frac{d}{dz}\right)^{2m}\left[z^{\nu}Y_{\nu}(bz)F_{1}(z)\right]=-(2m-1)!!\frac{2^{\nu-m}}{\pib^{\nu-m}}\sum_{k=0}^{m}{\binom{m}{k}}2^{k}\frac{\Gamma(\nu-m+k)}{b^{2k}}\left(\frac{d}{zdz}\right)^{k}F_{1}(z),
\begin{array}{rl}{\delta\tau}&{=\Lambda(\tau,\eta)-{\frac{1}{2}}\eta^{q}D_{q}\Lambda,}\\{\delta\eta^{q}}&{=-{\frac{i}{2}}D^{q}\Lambda,}\\{\deltaD_{q}}&{=-{\frac{1}{2}}\dot{\Lambda}D_{q}+{\frac{i}{4}}[D_{q},D_{r}]\LambdaD^{r}.}\\\end{array}
I_{pk}\left(M_{p}\right):=\int_{0}^{\infty}\,dx\,\left[\left(x^{2}+M_{p}^{2}\right)^{1/2}+\mu\right]^{-k}\left[\left(x^{2}+M_{p}^{2}\right)^{1/2}-\mu\right]^{k}e^{-2k\left(x^{2}+M_{p}^{2}\right)^{1/2},}
T_{+2-2}^{-\dot{q}}=\frac12\tilde{\gamma}_{\dot{q}p}^{i}\left(\Omega_{+2}^{-2i}\psi_{-2p}^{2+}-\Omega_{-2}^{-2i}\psi_{+2p}^{2+}\right),\T_{-\dot{p}\pm2}^{-\dot{q}}=\frac12\tilde{\gamma}_{\dot{q}p}^{i}\Omega_{-\dot{p}}^{-2i}\psi_{\pm2p}^{2+},
(\tilde{Q}_{\alpha}^{+},\tilde{M}^{-a})_{DB}=\theta_{\alphaC}X_{D}^{a}{d^{BD}}_{E}\varphi^{E}+\frac{1}{2}(\gamma^{ad}\theta_{C})_{\alpha}X_{dD}({c_{E}}^{CD}\varphi^{E}+{c_{\lambda}}^{CD}\varphi^{\lambda}).
\sum_{A^{\prime\prime},B^{\prime\prime}\in\{2b-1,2b\}}g^{A^{\prime\prime}B^{\prime\prime}}\bar{F}_{A^{\prime\prime}B^{\prime\prime}}\varphi=2[H_{b}(H_{b}+b-1)+\sum_{a>b}H_{a}]\varphi
S=\frac{1}{4}\int\!\mathrm{d}^{2}x\,\mathrm{d}^{4}\theta\,{K}(\Phi^{i},\bar{\Phi}^{j})+\left\{\frac{1}{2}\int\!\mathrm{d}^{2}x\,\mathrm{d}^{2}\theta\,{W}(\Phi^{i})+\mathrm{H.c.}\right\}\,,
\langleu\rangle_{D}=\frac{1}{\sqrt{\mathrm{Vol}(T^{6})}}\left(\sum_{a=1}^{K}N_{a}\frac{(n_{a}R_{1})^{2}-((m_{a}+bn_{a})R_{2})}{L_{a}}-16\mathrm{Vol}(\mathrm{O}8_{a})\right).
l_{\bar{q}}^{\mu}=\left(k_{g}+k_{\bar{q}}^{\prime}-k_{\bar{q}}\right)^{\mu}={\frac{\vec{k}_{g_{\!\perp}}^{\,2}}{2k_{g}^{+}}}\\eta^{\mu}\quad\mathrm{and}\quadl_{q}^{\mu}=\left(k_{g}+k_{q}-k_{q}^{\prime}\right)^{\mu}={\frac{\vec{k}_{g_{\!\perp}}^{\,2}}{2k_{g}^{+}}}\\eta^{\mu}\.
{\calL}_{{BI}}^{{SUSY}}=\frac{1}{4e^{2}}\left[\int\!d^{2}\theta~W^{2}+\int\!d^{2}\bar{\theta}~\bar{W}^{2}\right]+\sum_{s,t=0}^{\infty}a_{st}^{(\beta)}\int\!d^{4}\theta~W^{2}\bar{W}^{2}~X^{s}Y^{t}
\langlew_{(1)},\sum_{n\in{\mathbbC}}(w_{(3)}^{\prime})_{n}w_{(2)}e^{(-n-1)l_{p}(z)}\rangle_{W_{1}^{\prime}}=\langlew_{(3)}^{\prime},F_{{\calY},p}(w_{(1)}\otimesw_{(2)})\rangle_{W_{3}}.
G_{B\theta}^{\mu\nu}(\tau_{i},\tau_{j};\alpha_{i},\alpha_{j})=g^{\mu\nu}G_{B}(\tau_{i},\tau_{j})-{\frac{i}{T}}\theta^{\mu\nu}\alpha_{ij}(\tau_{i}+\tau_{j})-{\frac{1}{4T}}\alpha_{ij}^{2}(\theta^{2})^{\mu\nu}\,
\gamma^{0}=\left(\begin{array}{cc}{0}&{{-i}}\\{i}&{{0}}\\\end{array}\right),\quad\gamma^{1}=\left(\begin{array}{cc}{0}&{{i}}\\{i}&{{0}}\\\end{array}\right),\quad\gamma^{2}=\left(\begin{array}{cc}{i}&{{0}}\\{0}&{{-i}}\\\end{array}\right).
\Gamma_{R}^{\phi_{S}^{0}}(p)=-i\sum_{Q}m_{Q}^{2}/\{[(p^{2}+i\varepsilon)[k+h-is\sqrt{1-R^{2}/S^{2}}]-4[\tilde{k}+\tilde{h}-i\tilde{s}\sqrt{1-R^{2}/S^{2}}]\}
\{f,g\}_{\kappa}=\sum_{s=0}^{\infty}{\frac{\kappa^{2s}}{(2s+1)!}}\sum_{j=0}^{2s+1}(-1)^{j}\left(\begin{array}{c}{2s+1}\\{j}\\\end{array}\right)(\partial_{x}^{j}\partial_{\lambda}^{2s+1-j}f)(\partial_{x}^{2s+1-j}\partial_{\lambda}^{j}g)\,.
u_{-}(n,k;\rho,\varphi)=\frac{1}{\sqrt{2\bar{\omega}_{k}(M+\bar{\omega}_{k})}}\left(\begin{array}{c}{(M+\bar{\omega}_{k})J_{n}(k\rho)e^{in\varphi}}\\{ikJ_{n+1}(k\rho)e^{i(n+1)\varphi}}\\\end{array}\right)
\psi=\left(\begin{array}{c}{\midu\midx_{+}}\\{u}\\\end{array}\right)=\left(\begin{array}{c}{xu}\\{u}\\\end{array}\right),~~~~u=\left(\begin{array}{c}{u_{1}}\\{u_{2}}\\\end{array}\right)=U\frac{1+\sigma_{3}}{2}
\left.Z(\Omega,D)=\int\,d\mu_{I_{\beta}}\,(A)\,\sum_{\tau}\,c_{\tau}\,\left[z(\beta)c_{\beta}(A,D-\Omega)+z(\beta,\rho^{\tau})\,c_{\beta}(A,D+\rho^{\Omega_{\tau}})\right]\times\prod_{\rho\in{\calE}_{\tau}}\,\left[1+z(\beta,\rho)\,c_{\beta}(A,\rho)\right]\.\right.
O=\left[\begin{array}{cccccccc}{1}&{1}&{1}&{1}&{1}&{1}&{1}&{1}\\{1}&{-1}&{1}&{1}&{-1}&{1}&{-1}&{-1}\\{1}&{-1}&{-1}&{1}&{1}&{-1}&{1}&{-1}\\{1}&{-1}&{-1}&{-1}&{1}&{1}&{-1}&{1}\\{1}&{1}&{-1}&{-1}&{-1}&{1}&{1}&{-1}\\{1}&{-1}&{1}&{-1}&{-1}&{-1}&{1}&{1}\\{1}&{1}&{-1}&{1}&{-1}&{-1}&{-1}&{1}\\{1}&{1}&{1}&{-1}&{1}&{-1}&{-1}&{-1}\\\end{array}\right].
{\calL}_{\eta}|\Phi_{ph}\rangle=(\eta\hat{\partial}+\frac{1}{2}\hat{\partial}^{I}\eta^{J}M^{IJ}+\frac{1}{2}\eta^{I}M^{zI})|\Phi_{ph}\rangle-\hat{\partial}^{+}\eta^{I}\alpha^{I}(\bar{\alpha}^{-}|\Phi\rangle)\Bigr|_{\alpha^{+}=0}^{\vphantom{5pt}}\,,
S=-\gamma\intd\tau\intd\sigma\sqrt{{\dot{x}\acute{x}}^{2}-{\dot{x}}^{2}{\acute{x}}^{2}}\,-\,\sum_{i=1}^{2}m_{i}\intd\tau\sqrt{{\left(\frac{dx^{\mu}({\tau}_{i},{\sigma}_{i}(\tau))}{d\tau}\right)}^{2}}
\begin{aligned}{[m]ds^{2}}&{{}=2dx^{+}dx^{-}+\sum_{i,j}A_{ij}x^{i}x^{j}(dx^{-})^{2}+\sum_{i}dx^{i}dx^{i}}\\{F}&{{}=dx^{-}\wedge\Theta,}\\\end{aligned}
\mathrm{\mathrm{}~^*\!~}F_{\mu\nu}(x)=4\pi\{\mathrm{\small~\frac{1}{2}~}\epsilon_{\mu\nu\rho\sigma}[\partial^{\sigma}{\tilde{\lambda}}^{\rho}(x)-\partial^{\rho}{\tilde{\lambda}}^{\sigma}(x)]\},
\zeta_{N}^{c}(2,x^{2})=\frac{\alpha\,R}{8\,x^{3}}+\frac{15\,\alpha}{512\,R\,x^{5}}+\frac{4035}{2^{17}}\,\frac{\alpha}{R^{3}\,x^{7}}+{\calO}(x^{-9})\,{.}
S_{2}=\frac{i}{4}{^+\tau}\int_{M}eF^{\mu\nui}\,F^{\alpha\betaj}g_{ij}\,\,{^+{\calR}}_{\mu\nu}^{ab}\,\,{^+{\calR}}_{\alpha\beta}^{cd}\,\epsilon_{abcd}-\frac{i}{4}{^-\tau}\int_{M}eF^{\mu\nui}\,F^{\alpha\betaj}g_{ij}\,\,{^-{\calR}}_{\mu\nu}^{ab}\,\,{^-{\calR}}_{\alpha\beta}^{cd}\,\epsilon_{abcd}.
\displaystyle{{\calM}=\left(\begin{array}{ccc}{w+\textstyle{\frac{1}{2}}\lambda+i\textstyle{\frac{1}{2}}\psi}&{-ia{\cdot\tilde{\sigma}}}&{2\bar{\varepsilon}_{b}}\\{-ib{\cdot\sigma}}&{\tilde{w}-\textstyle{\frac{1}{2}}\lambda+i\textstyle{\frac{1}{2}}\psi}&{2\rho_{b}}\\{2\bar{\rho}^{a}}&{2\varepsilon^{a}}&{t_{~b}^{a}+i\frac{2}{N}\psi\delta_{~b}^{a}}\\\end{array}\right)}