harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left[X^{m},X^{n}\right]=-i\frac{\mathfrak{a}}{m}\underbrace{\left(\begin{array}{cc}{-e\mathfrak{a}mK^{\mu\nu}}&{-\eta^{\mu\nu}}\\{\eta^{\mu\nu}}&{-\frac{e}{am}F^{\mu\nu}}\\\end{array}\right)}_{\equiv\tilde{H}^{mn}}
\left.\begin{array}{ccccccc}{}&{}&{{\calC}(3,0,2)}&{}&{{\calC}(3,0,2)}&{\hookleftarrow}&{{\calP}^{>}(0,-2)}\\{\Pi_{2,2}^{-}}&{\hookrightarrow}&{\oplus}&{\stackrel{H=0}{\longrightarrow}}&{\oplus}&{}&{\oplus}\\{}&{}&{{\calC}(-3,0,2)}&{}&{{\calC}(-3,0,2)}&{\hookleftarrow}&{{\calP}^{<}(0,-2),}\\\end{array}\right.
\frac{\partial}{\partialu}\:{\bfC}_{vac}(3)=\:\frac{\partial}{\partialu}\:\Bigl\{\:\frac{r}{8\pi}\:m_{\mu}m_{\nu}\:\frac{1}{\Box}\nabla_{\alpha}\nabla_{\beta}K^{[\mu\alpha][\nu\beta]}(x)+O[R_{..}^{3}]\Bigr\}\biggl|_{{\calI}^{+}}\;.
1=\sum_{k=0}^{\infty}\frac{1}{k\,!}\,\int_{0\leqr_{0},r_{1},\ldots,r_{k}\leq1}\!\!\!\!d^{k}r\,\sum_{n=-\infty}^{\infty}|n;r_{1},\ldots,r_{k}\rangle\langlen;r_{1},\ldots,r_{k}|.
\begin{array}{l}{\xi_{c}\leq\xi\leq\xi_{c}+\frac12F(1-F)\,,\quadd=2\,,}\\{\xi_{c}\leq\xi\leq\xi_{c}+\frac{2F(1-F)}{(d-2)(d-1)d(d+1)}\,,\quadd\geq3\,,}\\\end{array}
\{\hat{\phi}_{1},\hat{\phi}_{2}\}(x)=\int_{D_{+}}\hat{\phi}_{1}^{(d-1)}(y)\hat{\phi}_{2}(x)-\int_{D_{-}}\hat{\phi}_{1}^{(d-1)}(y)\hat{\phi}_{2}(x)=\int_{D}\hat{\phi}_{3}(x)\delta(y-x)=\hat{\phi}_{3}(x),
\lbraceA,B\rbrace{}^{*}=\lbraceA,B\rbrace-\intd^{3}\sigma[\lbraceA,\zeta^{\mu}(\tau,\vec{\sigma})\rbrace\lbrace{\calH}_{\mu}(\tau,\vec{\sigma}),B\rbrace-\lbraceA,{\calH}_{\mu}(\tau,\vec{\sigma})\rbrace\lbrace\zeta^{\mu}(\tau,\vec{\sigma}),B\rbrace],
AII_{0}=-\bar{x}|x|^{2}\int\\mathrm{d}\mu\int{\frac{\mathrm{d}^{2}z~\bar{z}}{|y(z)y(0)y(x)|^{2}}}\,\left[R\,L-{\frac{1}{\bar{x}}}\,L-x\,R+{\frac{x}{\bar{x}}}\right],
[\partial^{2}/\partialr^{2}+(\frac{5}{r}+\frac{f^{\prime}}{f})\partial/\partialr-\frac{12}{r^{2}f(r)}]\phi(r)=16\frac{z^{\frac{5}{2}}}{R^{2}}[\frac{d^{2}}{dz^{2}}-\frac{k}{1-kz}\frac{d}{dz}-\frac{3}{4z^{2}(1-kz)}]\phi(z)
\Biggl\langle{\binom{0}{1}},\Bigl(b_{-n_{1}}\dotsb_{-n_{N}}\bar{b}_{-m_{1}}\dots\bar{b}_{-m_{M}}\vert0\!\!>\Bigr)_{0}{\binom{1}{0}}\Biggr\rangle=\pi(b_{-n_{1}}\dotsb_{-n_{N}}\bar{b}_{-m_{1}}\dots\bar{b}_{-m_{M}}\vert0\!\!>)
N(\zeta,\zeta^{\prime},d_{1}+d_{2})={\frac{\epsilon}{2\pii}}\int_{-i\pi/\epsilon}^{i\pi/\epsilon}d\xiN(\zeta,\xi,d_{1})N(-\xi,\zeta^{\prime},d_{2})\to{\frac{\epsilon}{2\pii}}\int_{-i\infty}^{i\infty}d\xiN(\zeta,\xi,d_{1})N(-\xi,\zeta^{\prime},d_{2})
\Phi_{1}=\left(\begin{array}{cccc}{0}&{0}&{-\phi_{1}}&{0}\\{0}&{0}&{0}&{-\phi_{2}}\\{\phi_{1}}&{0}&{0}&{0}\\{0}&{\phi_{2}}&{0}&{0}\\\end{array}\right),\qquad\Phi_{2}=\Phi_{1}(\phi_{1}\to-\phi_{3},\phi_{2}\to-\phi_{4}),
\delta{\phi^{\prime}}^{I}=-i\overline{{\epsilon}}_{2}^{i}\gamma^{\mu}\epsilon_{1i}\partial_{\mu}{\phi^{\prime}}^{I}+ia\overline{{\epsilon}}_{2}^{i}\epsilon_{1i}\left[3(O^{T}TO)^{IJ}+2\mu(O^{T}QO)^{IJ}\right]{\phi^{\prime}}^{J}
{}^{\ast}\,\Omega_{p}\;=\;{\Omega^{\prime}}_{D-p}=\frac{1}{p!(D-p)!}\epsilon_{\mu_{1}\,\dots\,\mu_{D-p}}{}^{\nu_{1}\,\dots\,\nu_{p}}\;\Omega_{\nu_{1}\,\dots\,\nu_{p}}\;dx^{\mu_{1}}\,\wedge\,\dots\,\wedgedx^{\mu_{D-p}}\;.
S_{CP-odd}=i\frac{\thetae^{2}}{16\pi^{2}}\intd^{4}x\,F_{\mu\nu}\tilde{F}_{\mu\nu}=-i\frac{e^{2}\theta}{4\pi^{2}}\intd^{4}x\,A_{0}\,\vec{\nabla}\cdot\vec{B}=-i2q\frac{\theta}{2\pi}\inteA_{0}d\tau
ds^{2}\,=\,-dt^{2}\,\left(1-\frac{2m}{\rho}+\frac{q}{\rho^{2}}\right)+d\rho^{2}\,\left(1-\frac{2m}{\rho}+\frac{q}{\rho^{2}}\right)^{-1}+\rho^{2}\,d\Omega^{2}
\tilde{A}_{i_{1}\ldotsi_{p}}({\bfk})={\frac{1}{2}}\Gamma(\nu)\left({\frac{|{\bfk}|}{2}}\right)^{-\nu}\left(a_{i_{1}\ldotsi_{p}}-i{\frac{\beta}{|{\bfk}|^{2}}}\left\{k_{i_{1}}a_{0i_{2}\ldotsi_{p}}+(-)^{p-1}k_{i_{2}}a_{0i_{3}\ldotsi_{p}i_{1}}+\cdots\right\}\right).
\langle\frac{1}{N}\mathrm{Tr}\xi^{r_{1}}\cdots\frac{1}{N}\mathrm{Tr}\xi^{r_{p}}\rangle=\langle\frac{1}{N}\mathrm{Tr}\xi^{r_{1}}\rangle\cdots\langle\frac{1}{N}\mathrm{Tr}\xi^{r_{p}}\rangle
\hat{U}=\left(\begin{matrix}{d}&{0}&{c}&{0}&{0}\\{0}&{a}&{0}&{-b}&{0}\\{b}&{0}&{a}&{0}&{0}\\{0}&{-c}&{0}&{d}&{0}\\{0}&{0}&{0}&{0}&{{\bf1}_{P}}\\\end{matrix}\right)\,,\quad\hat{V}=(\hat{U}{}^{T})^{-1}=\left(\begin{matrix}{a}&{0}&{-b}&{0}&{0}\\{0}&{d}&{0}&{c}&{0}\\{-c}&{0}&{d}&{0}&{0}\\{0}&{b}&{0}&{a}&{0}\\{0}&{0}&{0}&{0}&{{\bf1}_{P}}\\\end{matrix}\right)\,,
\frac{1}{\sqrt{g}e^{-2\phi}}\frac{\partial\Gamma^{(2)}}{\partial\phi}=8{\tilde{B}}^{\phi(2)}+2B_{\rho\sigma}^{b(1)}B^{b(1)\rho\sigma}+2B_{\rho\sigma}^{g(1)}B^{g(1)\rho\sigma}-2({\tilde{B}}^{\phi(1)})^{2},
\big(F\,,\,G\,\big)^{a}\,\equiv\,{\frac{\delta^{r}F}{\delta\phi^{A}}}{\frac{\delta^{l}G}{\delta\phi_{A}^{\ast\,a}}}\,+\,{\frac{\delta^{r}F}{\delta{\bar{\phi}}^{A}}}{\frac{\delta^{l}G}{\delta\pi_{A}^{a}\,}}\,-\,{\frac{\delta^{r}F}{\delta\phi_{A}^{\ast\,a}}}\,{\frac{\delta^{l}G}{\delta\phi^{A}}}\,-\,{\frac{\delta^{r}F}{\delta\pi_{A}^{a}\,}}{\frac{\delta^{l}G}{\delta{\bar{\phi}}^{A}}}
\stackrel{(2)}{S}=\left(\stackrel{(-1,-1)}{\eta}_{\alpha}^{*(21)}-\stackrel{(-1,-1)}{\eta}_{\alpha}^{*(12)}+\stackrel{(-1,-1)}{\bar{\Phi}}_{i}R_{\;\;\alpha}^{i}\right)\stackrel{(1,1)}{\pi^{\alpha}}+\mathrm{``more"}
S_{BI}=\frac{(2\pi)^{\frac{1-p}{2}}}{g_{s}}\intd^{p+1}\sigmaStr\left(e^{-\phi}\sqrt{det(P[E_{ab}+E_{ai}(M^{-1}-\delta)^{ij}E_{jb}]+F_{ab})det(M_{j}^{i})}\right),
H_{r}=\intd^{2}x~[-\frac{m}{2}\epsilon_{ij}\Pi_{i}A^{j}+\frac{1}{2}(\Pi_{i})^{2}+\frac{m^{2}}{8}(A^{i})^{2}+\frac{1}{2}(\epsilon_{ij}\partial^{i}A^{j})^{2}].
\deltax^{\mu}=0\;,\;\;\deltae=0,\;\delta\psi_{a}^{n}=\sum_{c=1}^{2}t_{ac}\psi_{c}^{n},\;\delta\chi_{a}=\sum_{c=1}^{2}t_{ac}\chi_{c},\;\deltaf_{ab}=\dot{t}_{ab}+\sum_{c=1}^{2}(t_{ac}f_{cb}-t_{bc}f_{ca})\;.
D_{\mu}^{ij}[A]\frac{\deltaW}{\deltaA_{\mu}^{j}}+gf^{ijk}(Y_{\mu}^{j}\frac{\deltaW}{\deltaY_{\mu}^{k}}+J_{\mu}^{j}\frac{\deltaW}{\deltaJ_{\mu}^{k}}+\xi^{j}\frac{\deltaW}{\delta\xi^{k}}+\overline{{\xi}}^{j}\frac{\deltaW}{\delta\overline{{\xi}}^{k}})=0,
{\mathrm{log}}\left[1+\sum_{{\bf\vec{k}}}\frac{\vertC({\bf\vec{k}})\vert}{l!}G_{{\bfk}}\prod_{j}x^{k_{j}}\right]=\sum_{{\bf\vec{k}}}\frac{\vertC({\bf\vec{k}})\vert}{l!}G_{{\bfk}}^{(c)}\prod_{j}x^{k_{j}}
\frac{1}{4}{F^{\prime}}^{2}=-\frac{1}{4}F^{2}-\frac{1}{2}{S_{{\nu}{\lambda}}}^{\alpha}A_{\alpha}F^{{\nu}{\lambda}}-{S_{{\nu}{\lambda}}}^{\beta}{S^{{\nu}{\lambda}}}_{\alpha}A_{\beta}A^{\alpha}
\langle\varphi^{2}(x)\rangle_{b}\approx-\frac{\Gamma^{-2}(\nu_{0}+1)}{2^{2\nu_{0}}\pia^{D-1}S_{D}\sigma^{D-1}}\left(\frac{r}{a}\right)^{2\nu_{0}-n}\int_{ma}^{\infty}dz\frac{z^{2\nu_{0}+1}}{\sqrt{z^{2}-m^{2}a^{2}}}\frac{\bar{K}_{\nu_{0}}(z)}{\bar{I}_{\nu_{0}}(z)},\quadr\to0,
T_{\mu\nu}(A_{\alpha})=\frac{1}{4}\overline{{g}}_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}-F_{\mu\alpha}F_{\nu\beta}\overline{{g}}^{\alpha\beta}+\frac{1}{2}m_{A}^{2}(A_{\mu}A_{\nu}-\frac{1}{2}\overline{{g}}_{\mu\nu}A_{\alpha}A^{\alpha})
<F[\sigma]>\equiv<\prod_{i=1}^{m}\sigma_{\alpha_{i}\beta_{i}}(x_{i},y_{i})>={\frac{\intD\psi^{*}D\psie^{iS}\prod_{i=1}^{m}\sigma_{\alpha_{i}\beta_{i}}(x_{i},y_{i})}{\intD\psi^{*}D\psie^{iS}}}
\begin{array}{l}{F=-(u_{0}+v)^{-1}=\alpha+\bar{\beta}}\\{\quad\quad\leftrightarrow\quadv=W=-(\alpha+\bar{\beta})^{-1}-u_{0},}\\{\bar{F}=-(u+v_{0})^{-1}=\bar{\alpha}+\beta}\\{\quad\quad\leftrightarrow\quadu=\bar{W}=-(\bar{\alpha}+\beta)^{-1}-v_{0}.}\\\end{array}
\Sigma_{\mu\nu}\rightarrow\Sigma_{\mu\nu\rho}={\calA}_{\mu\nu\rho}^{F}-8{\calA}_{\mu\nu\rho}^{V},\qquad\Xi_{\mu\nu}\rightarrow\Xi_{\mu\nu\rho}={\calA}_{\mu\nu\rho}^{F}+{\frac{16}{5}}{\calA}_{\mu\nu\rho}^{V},
\frac{3}{8\pi^{2}}\omega_{i}^{(4)}+\frac{1}{8\pi^{2}}\sum\Omega_{ijk}^{(2)}~\omega_{ijk}^{(2)}+\frac{1}{4\pi^{2}}\sum\Omega_{ijklm}^{(4)}=\frac{1}{2},
f_{9}^{A}=2\zeta(3)r_{A}e^{-2\phi}+\frac{2\pi^{2}}{3r_{A}}+0+~4\pie^{-\phi}\sum_{m\ne0}\sum_{n\ne0}\left|\frac{m}{n}\right|K_{1}\left(2\pir_{A}e^{-\phi}|mn|\right)e^{2\piimn{\calA}}
\varphi_{\alpha}(\lambda,\omega)\sim{\frac{1}{(4\pi)^{d/2}}}\sum_{n=0}^{\infty}\left[a_{n}(\omega)~{\frac{\lambda^{d/2-n+\alpha-1}}{\Gamma\left(\fracd2-n+\alpha\right)}}+b_{n}(\omega)~{\frac{\lambda^{(d-1)/2-n+\alpha-1}}{\Gamma\left({\frac{d-1}{2}}-n+\alpha\right)}}\right].
\frac{dT_{+}}{dk_{+}}=\frac{4}{\sqrt{K}}[\frac{E(k_{+})}{k_{+}(1-k_{+}^{2})}-\frac{K(k_{+})}{k_{+}}],\;\;\;\;\;\frac{dS_{+}^{\mathrm{cl}}}{dk_{+}}=-\frac{8}{K\alpha^{\prime}}[\frac{E(k_{+})}{k_{+}^{2}(1-k_{+}^{2})}-\frac{K(k_{+})}{k_{+}^{2}}].
{\calL}_{E}=\bar{\psi}_{a}(i\partial\!\!\!/+m-\frac{e}{\sqrt{N}}A\!\!\!/)\psi_{a}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-i\frac{\kappa}{8\pi}\epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}A_{\rho}.
\frac{1}{2}(1-(-)^{l_{1}+l_{2}})\chi_{l_{1}}^{H_{1}^{+}}\chi_{l_{2}}^{H_{2}^{+}}=\chi_{l_{1}}^{H_{1}^{+}}\chi_{1-l_{1}}^{H_{2}^{+}}\delta_{l_{2},1-l_{1}}\,
Z_{\mathrm{DW}}^{N}(b)=\prod_{\mu=1}^{d}\prod_{1\lei,j\leN}\left[\int_{-\infty}^{\infty}dA_{ij}^{\mu}dA_{ij}^{\mu*}\right]e^{\frac{b}{N}\sum_{\mu\ne\nu}trA_{\mu}A_{\nu}A_{\mu}^{\dagger}A_{\nu}^{\dagger}-\sum_{\mu}trA_{\mu}A_{\mu}^{\dagger}}
W=e\phi(x_{i})=e\phi(q_{i}-\frac{1}{2}\theta_{ij}Q^{j})=e\phi(q)-\frac{1}{2}e\theta_{ij}Q^{j}\partial^{i}\phi(q)+O(\theta^{2})=e\phi(q)-d_{i}E^{i}+O(\theta^{2}),
A_{\mu}^{a}(x)\vert_{\mathcal{F}_{_{o\cdotso}}}=A_{\mu}^{a}(x)-D_{\mu}^{ab}(x)\intd^{4}x^{\prime}H^{bc}(x,x^{\prime};A)[(\partial\cdotD)^{cd}(\partial\cdotA^{d})]_{x^{\prime}},
D_{\mu\nu}^{-1}(k)=Z_{3}[k^{2}g_{\mu\nu}\!-\!k_{\mu}k_{\nu}(1\!-\!\eta^{-1})]+ie^{2}Z_{2}Tr\int\!\frac{d^{n}p}{(2\pi)^{n}}\gamma_{\nu}
\mathrm{d}\mu(z,\bar{z},\theta,\bar{\theta})=-\mathrm{sdet}\left\|\Omega_{A\bar{B}}\right\|\frac{\mathrm{d}z\mathrm{d}\bar{z}}{2\pii}\mathrm{d}\theta\mathrm{d}\bar{\theta}=-2\left(1-\frac{1}{2}\frac{\theta\bar{\theta}}{\zeta}\right)\frac{\mathrm{d}z\mathrm{d}\bar{z}}{2\pii}\frac{\mathrm{d}\theta\mathrm{d}\bar{\theta}}{\zeta}\;.
S=-M_{0}^{2}\int_{0}^{R}dr\int_{t_{1}}^{t_{2}}dt\left[1+\frac{1}{2}({\acute{\bfu}}^{2}-{\dot{\bfu}}^{2})+\frac{\alpha}{2M_{0}^{2}}{(\Box{\bfu})}^{2}\right]-\beta\int_{t_{1}}^{t_{2}}dt\;\ddot{\bfu}\acute{\bfu},
S:\left(\begin{array}{c}{a_{D}}\\{a}\\\end{array}\right)\longrightarrow\left(\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}\right)\left(\begin{array}{c}{a_{D}}\\{a}\\\end{array}\right)=\left(\begin{array}{c}{a}\\{-a_{D}}\\\end{array}\right).
e^{-1}{\calL}=-{\frac{1}{2}}R-{\frac{1}{4}}G_{IJ}F_{\mu\nu}{}^{I}F^{\mu\nuJ}-{\frac{1}{2}}g_{ij}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}+{\frac{e^{-1}}{48}}\epsilon^{\mu\nu\rho\sigma\lambda}C_{IJK}F_{\mu\nu}^{I}F_{\rho\sigma}^{J}A_{\lambda}^{K}
<O_{n}O_{m}>_{q}=\frac{1}{2^{n+m}}\sum_{k=0}^{n}\sum_{l=0}^{m}\left(\begin{array}{c}{n}\\{k}\\\end{array}\right)\left(\begin{array}{c}{m}\\{l}\\\end{array}\right)<W_{q}^{n-k,k}W_{-q}^{m-l,l}>.
\langle\dot{y}_{a}\cdot\dot{y}_{b}\rangle=\mathrm{tr}\biggl[\ddot{\calG}_{Bab}+{\frac{1}{2}}{\frac{(\dot{\calG}_{Baa}-\dot{\calG}_{Bab})(\dot{\calG}_{Bab}-\dot{\calG}_{Bbb})}{\bar{T}-{\frac{1}{2}}{\calC}_{ab}}}\biggr]\;.
\bigl((J_{1},J_{2})_{-M_{1}\,-M_{2}}^{-N_{2}\,-N_{1}}\bigr)^{*}=\bigl((J_{2},J_{1})_{-N_{2}\,-N_{1}}^{-M_{1}\,-M_{2}}\bigr)^{*}={\overline{{(J_{2},J_{1})}}_{N_{2}\,N_{1}}^{M_{1}\,M_{2}}}
xy^{2}=\left[x\prod_{a=1}^{n_{c}-1}(x-\phi_{a}^{2})(x-2\Lambda^{2}-\beta)+2\Lambda^{2}m_{1}\cdotsm_{n_{f}}\right]^{2}-4\Lambda^{4}\prod_{i=1}^{n_{f}}(x+m_{i}^{2})
u_{i}^{\u}=\left(\begin{array}{cc}{u_{1}^{\+}}&{u_{1}^{\-}}\\{u_{2}^{\+}}&{u_{2}^{\-}}\\\end{array}\right),\u_{u}^{\i}=\left(\begin{array}{cc}{u_{+}^{\1}}&{u_{+}^{\2}}\\{u_{-}^{\1}}&{u_{-}^{\2}}\\\end{array}\right)
\hat{H}_{gh}=\omega\hat{N}_{gh}=\omega(\hat{b}_{1}^{\dagger}\hat{b}_{1}+\hat{b}_{2}^{\dagger}\hat{b}_{2}+\hat{c}_{1}^{\dagger}\hat{c}_{1}-\hat{c}_{2}^{\dagger}\hat{c}_{2})\,,
q^{(\Lambda,\Lambda+2\rho)}\sum_{\nu\in\Pi(\Lambda)}m_{\nu}\,q^{2(\nu,\mu+\rho)}=\sum_{\nu\in\Pi(\Lambda)}m_{\nu}\,q^{(\nu,\nu+2(\mu+\rho))}\frac{D_{q}[\mu+\nu]}{D_{q}[\mu]}\;.
\dot{x}^{\mu}=q^{-1/2}\frac{\partial{\calH}_{c}}{\partialp^{\mu}},\;\dot{p}^{\mu}=-q^{1/2}\frac{\partial{\calH}_{c}}{\partialx^{\mu}},\;\dot{\psi}^{\mu}=-i\frac{\partial{\calH}_{c}}{\partial\psi^{\mu}},\;\dot{\psi}_{5}=i\frac{\partial{\calH}_{c}}{\partial\psi_{5}},
U_{\underline{{m}}}^{++}\sigma_{\underline{{\mu}}\underline{{\nu}}}^{\underline{{m}}}=2V_{\underline{{\mu}}q}^{~+}V_{\underline{{\nu}}q}^{~+},\qquadU_{\underline{{m}}}^{--}\tilde{\sigma}^{\underline{{m}}\underline{{\mu}}\underline{{\nu}}}=2V_{q}^{-\underline{{\mu}}}V_{q}^{-\underline{{\nu}}},\qquad
\hat{H}^{\prime}=\frac{1}{4\Omega_{G}^{2}}\Bigl(\frac{1}{4}\hat{a}_{0}^{\dagger4}+\hat{a}_{0}^{\dagger3}\hat{a}_{0}+\frac{3}{2}\hat{a}_{0}^{\dagger2}\hat{a}_{0}^{2}+\hat{a}_{0}^{\dagger}\hat{a}_{0}^{3}+\frac{1}{4}\hat{a}_{0}^{4}\Bigr).
R({\bfl}^{2})=\frac{mM}E\left\{\frac1{\sqrt{{\bfl}^{2}+m^{2}}\left(\sqrt{{\bfl}^{2}+m^{2}}+t\right)}+\frac1{\sqrt{{\bfl}^{2}+M^{2}}\left(\sqrt{{\bfl}^{2}+M^{2}}+T)\right)}\right\}
S_{\mathrm{massive\D0-brane}}=T\intd\tau\,\Big(e^{-3\hat{\phi}/4}\sqrt{-\dot{X}^{\hat{\mu}}\dot{X}^{\hat{\nu}}\hat{g}_{\hat{\mu}\hat{\nu}}}+\dot{X}^{\hat{\mu}}\hat{A}_{\hat{\mu}}+m\,\hat{V}_{t}\Big)\,,
\frac{\sqrt{-\hat{g}}}{\hat{g}_{str}}\frac{\hat{g}^{00}\hat{g}^{11}F_{01}}{\sqrt{1+(2\pi\alpha^{\prime})^{2}\hat{g}^{00}\hat{g}^{11}F_{01}^{2}}}=B_{23}=\frac{s}{2\piR^{2}},
\begin{array}{ll}{n=0,~1,~2,\ldots}&{\leadsto~~\Phi=e^{ina}\omega^{\alpha_{1}}\cdots\omega^{\alpha_{n}}\psi_{\alpha_{1}\cdots\alpha_{n}}(x),}\\{n=-1,~-2,\ldots}&{\leadsto~~\Phi=e^{ina}\omega^{\dot{\alpha}_{1}}\cdots\omega^{\dot{\alpha}_{n}}\psi_{\dot{\alpha}_{1}\cdots\dot{\alpha}_{n}}(x),}\\\end{array}
\begin{array}{lcl}{{\mathcalL}_{2}^{(0,2)}}&{=}&{{u_{0}\starp}^{2}+u_{1}\starp+u_{2}}\\{}&{=}&{u_{0}{p}^{2}+(u_{1}-2\thetau_{0}^{\prime})p+(u_{2}-\thetau_{1}^{\prime}+\theta^{2}u_{0}^{\prime\prime})}\\\end{array}
\Phi^{\mu\cdots\delta}(x)=\psi_{l}^{\mu}\cdots\psi_{m}^{\delta}R({\bfB}_{f})\Phi_{f}^{l\cdotsm}(x_{f})\equiv{(R_{\psi})}_{l\cdotsm}^{\mu\cdots\delta}\Phi_{f}^{l\cdotsm}(x_{f}),
\frac{E_{\mathrm{cas}}}{A}=\intdk_{x}dk_{y}\left\{\sum_{n}\sqrt{k_{x}^{2}+k_{y}^{2}+\left(\frac{n\pi}{a}\right)^{2}}~-\frac{a}{\pi}\intdk_{z}\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}\right\}
\begin{aligned}{}&{{}U_{2}:}&{}&{{}\qquadu_{2}={\overline{x}}_{2},}&{}&{{}\qquadv_{2}={\overline{y}}_{2}}\\{}&{{}U_{3}:}&{}&{{}\qquadu_{3}={\overline{x}}_{3},}&{}&{{}\qquadv_{3}={\overline{y}}_{3}.}\\\end{aligned}
{\hat{\Pi}}_{{\hat{\mu}}{\hat{\nu}}}{\calD}_{\hat{\imath}}{\hat{X}}^{\hat{\mu}}{\calD}_{\hat{\jmath}}{\hat{X}}^{\hat{\nu}}={\hat{g}}_{{\hat{\mu}}{\hat{\nu}}}D_{\hat{\imath}}{\hat{X}}^{\hat{\mu}}D_{\hat{\jmath}}{\hat{X}}^{\hat{\nu}}\,,
\begin{array}{llc}{1:}&{NS:\;}&{\prod_{r}\left(1+q^{r-1/2}\right)^{8}\nonumber}\\{}&{R:}&{-16\prod_{r}\left(1+q^{r}\right)^{8}}\\{}&{Bos:}&{\prod_{r}\left(1-q^{r}\right)^{-8}\nonumber}\\{R_{j}:}&{NS:\;}&{\prod_{r}\left(1+q^{r-1/2}\right)^{4}\left(1-q^{r-1/2}\right)^{4}\nonumber}\\{}&{R:}&{0}\\{}&{Bos:}&{\prod_{r}\left(1-q^{r}\right)^{-4}\left(1+q^{r}\right)^{-4}\nonumber}\\\end{array}
\frac{\lambda^{2}\sqrt{2}}{m^{3}}E_{\mathrm{TK2}_{\sigma_{3}}}=\int_{\bar{\sigma}_{3}^{2}}^{\bar{\sigma}_{2}^{2}}\frac{\lambda_{2}d\lambda_{2}}{\sqrt{1-\lambda_{2}}}+\int_{\bar{\sigma}_{2}^{2}}^{1}\frac{\lambda_{3}d\lambda_{3}}{\sqrt{1-\lambda_{3}}}=2\sigma_{3}\left(1-\frac{\sigma_{3}^{2}}{3}\right)
{\calL}_{h}^{M}(V_{h},\phi)=\frac{1}{16}\intd^{2}\theta\frac{1}{g_{h}^{2}}W^{a}(V_{h})W^{a}(V_{h})+\mathrm{h.c.}+\intd^{4}\theta\sum_{i}\phi_{i}^{\dagger}e^{2V_{h}^{i}}\phi_{i},
\vec{S}=-{\frac{1}{4}}\overline{{\epsilon}}\vec{\gamma}i\gamma^{5}[(M_{1}-{\frac{1}{2\sqrt{2}}}\gamma^{0}Q_{R}\cdot\Gamma)+(M_{2}+{\frac{1}{2\sqrt{2}}}i\gamma^{5}\gamma^{0}P_{R}\cdot\Gamma)]\epsilon\,
G_{\Psi}^{a}=\sum_{k=1}^{n}(\Psi_{k})_{i}^{\dagger}T_{ij}^{a}(\Psi_{k})_{j}-\sum_{k=n+1}^{N}(\Phi_{k})_{i}^{\dagger}T_{ji}^{a}(\Phi_{k})_{j}~.
\Gamma^{9}=i\left(\begin{array}{cc}{0}&{I_{16}}\\{I_{16}}&{0}\\\end{array}\right)\,\quad\Gamma^{\mu}=\left(\begin{array}{cc}{\gamma^{\mu}}&{0}\\{0}&{-\gamma^{\mu}}\\\end{array}\right)\quad\mu=0,\cdots,8.
T(w)_{cylinder}=\Bigg(\frac{2\pi}{R}\Bigg)^{2}\Big(\sum_{n\in\mathbb{Z}}z^{-n}L_{n}-\frac{c}{24}\Big)=\Bigg(\frac{2\pi}{R}\Bigg)^{2}\sum_{n\in\mathbb{Z}}\Big(z^{-n}L_{n}-\frac{c}{24}\delta_{n,0}\Big).
u^{2}a^{2}y^{2}K_{1}^{2}(2ayu)=u^{2}a^{2}y^{2}\frac{\sqrt{\pi}}{2}G_{13}^{30}\left(\frac{1}{2};-1,0,1;4a^{2}y^{2}u^{2}\right)=\frac{\sqrt{\pi}}{8}G_{13}^{30}\left(\frac{3}{2};0,1,2;4a^{2}y^{2}u^{2}\right)
\begin{array}{lll}{}&{}&{\partial_{u}\partial_{v}\phi-\displaystyle{\frac{\rho}{2}}V(\phi)\,,}\\{}&{}&{\partial_{u}\partial_{v}M+\displaystyle{\frac{\partial_{u}M\partial_{v}M}{N-M}}=0\,,}\\{}&{}&{\partial_{u}\phi\,\partial_{u}M=0\,,~~~~~~\partial_{v}\phi\,\partial_{v}M=0\,.}\\\end{array}
\hat{U}_{m}(F)=\mathrm{exp}\bigr\{\frac{\deltaF}{\delta\phi^{A}}(\frac{\delta}{\delta\bar{\phi}_{A}}-\mathrm{~\frac{1}{2}~}m^{2}(P_{-})_{B}^{A}\frac{\delta}{\delta\eta_{B}})-(\hbar/i)\mathrm{~\frac{1}{2}~}\epsilon_{ab}\frac{\delta}{\delta\phi_{Aa}^{*}}\frac{\delta^{2}F}{\delta\phi^{A}\delta\phi^{B}}\frac{\delta}{\delta\phi_{Bb}^{*}}+(i/\hbar)m^{2}F\bigr\}.
R(x,\alpha)=-\frac{(1+\alpha)^{2}m^{2}[(1+\alpha)^{2}m^{2}+\{(1+\alpha)^{2}-(1+x)\}(\alpha-x)M^{2}}{\alphaM^{2}[(\alpha-x)(1+x)M^{2}-(1+\alpha)^{2}m^{2}][x(\alpha-x)M^{2}+(1+\alpha)m^{2}]}.
\sum_{ij}{\calA}_{j}[a_{i}^{r},a_{j}^{\dagger}]{\calA}_{j}^{\prime}=\sum_{i}{\calA}_{i}ra_{i}^{r-1}{\calA}_{i}^{\prime}+\lambda\sum_{il}{\calA}_{i}\sum_{s=0}^{r-1}a_{i}^{r-1-s}a_{l}^{s}K_{il}{\calA}_{i}^{\prime}-\lambda\sum_{ij}{\calA}_{j}\sum_{s=0}^{r-1}a_{i}^{r-1-s}a_{j}^{s}K_{ij}{\calA}_{j}^{\prime}
\mathcal{K}_{ij}\left(p,q\right)=\underset{\sigma_{1}\sigma_{2}s_{1}s_{2}m_{J}}{-i\sum}C_{Jm_{J}ij}^{s_{1}s_{2}\sigma_{1}\sigma_{2}}\intd\hat{\mathbf{p}}\,d\hat{\mathbf{q}}\,\mathcal{M}
N(h)=\left[(h^{0})^{3}-\frac{3}{2}h^{0}\delta_{ij}h^{i}h^{j}-\frac{1}{\sqrt{2}}(h^{1})^{3}+\frac{3}{\sqrt{2}}h^{1}[(h^{2})^{2}+\ldots+(h^{n})^{2}]\right].
{\calL}_{1}\,=\,\frac{1}{2}\left(\dot{{\bfx}}_{1}^{2}\,-\,{\bfx}_{1}^{2}\right),\,{\calL}_{2}\,=\,\frac{\epsilon}{2}\left(\dot{{\bfx}}_{2}^{2}\,-\,{\bfx}_{2}^{2}\right)\,-\,\frac{{\bfx}_{2}^{2}}{2}.
\zeta_{N}^{V}(s)=\frac{(-1)^{\frac{N}{2}-s}V_{N}}{(4\pi)^{N/2}}\frac{\Gamma\left(s-\frac{N}{2}\right)}{\Gamma(s)}\left[(N-1)+(1-a)^{-N/2}\right]P^{\frac{N}{2}-s}\,
x_{+}={\sqrt{\frac{m\Omega}{2\omega_{+}}}}x_{1}-{\sqrt{\frac{1}{2w_{+}m\Omega}}}p_{2};~~~x_{-}={\sqrt{\frac{m\Omega}{2\omega_{-}}}}x_{1}+{\sqrt{\frac{1}{2w_{-}m\Omega}}}p_{2},
\tilde{\Gamma}^{(1)}=\int_{-\infty}^{\infty}d\tau\left[-\frac{15}{16}\frac{v^{4}}{r^{7}}+\frac{315}{128}\frac{v^{6}}{r^{11}}-\frac{3465}{128}\frac{v^{4}(v\cdotr)^{2}}{r^{13}}+O(v^{8})\right].
J_{C_{(n)}}=-\frac{T_{p}}{16\sqrt{2}\,n!}\,V_{p+1}\,C_{\mu_{1}\ldots\mu_{n}}\,\mathrm{Tr}\left(\Gamma^{\mu_{n}\ldots\mu_{1}}\Gamma^{0}\cdots\Gamma^{p}\,;\mathrm{e}^{-\frac{1}{2}{\hat{F}}_{\alpha\beta}\Gamma^{\alpha}\Gamma^{\beta}};\right)~~.\nonumber
R_{\kappa,i}(x;q)=\sum_{n\geq0}\frac{x^{\kappan}q^{\kappan+\kappan^{2}+n-in}(1-x^{i}q^{2ni+i})(-1)^{n}q^{\frac{n(n-1)}{2}}}{(q)_{n}(xq^{n+1})_{\infty}}.
L^{P}\,\equiv\,\int_{-R}^{+R}dx^{1}{\calL}\,=\,\frac{1}{2R}\sum_{n>0}\left[\omega_{n}(a_{n}\dot{b_{n}}\,-\,\dot{a_{n}}b_{n})\,\,-\,\,\omega_{n}^{2}(a_{n}^{2}\,+\,b_{n}^{2})\right].
H^{(D)}=\sum_{{\vec{p},h}}\;H_{\vec{p},h}^{(D)}=\sum_{{\vec{p},h}}\;(H_{\vec{p},h}^{(0)}+H_{\vec{p},h}^{(1D)})=\sum_{{\vec{p},h}}\;|p^{0}|(b_{\vec{p},h}^{+}b_{\vec{p},h}-d_{\vec{p},h}^{+}d_{\vec{p},h})
\frac{dn_{i}}{d\rho_{i}}=\frac{1}{16\piG_{11}T_{2}}\intF_{\rho_{i}\phi_{i}\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}\alpha_{5}}^{(7)}d\phi_{i}d^{5}\alpha=8N^{2}\frac{q_{i}}{L_{7}^{6}}\rho_{i}\,.
S=\alpha_{IJ}\mathcal{A}_{I}\mathcal{A}_{J}+\beta_{IJK}\mathcal{A}_{I}\mathcal{A}_{J}\mathcal{A}_{K}+\gamma_{IJKL}\mathcal{A}_{I}\mathcal{A}_{J}\mathcal{A}_{K}\mathcal{A}_{L}.
g=Z_{2}=Z_{0}^{-2}\Biggl(\frac{a_{0}+Z_{0}}{\bar{a}_{0}+Z_{0}^{-1}}\Biggr)\Biggl(\frac{\mu_{+}+Z_{0}}{\bar{\mu}_{+}+Z_{0}^{-1}}\Biggr)\Biggl(\frac{\mu_{-}+Z_{0}}{\bar{\mu}_{-}+Z_{0}^{-1}}\Biggr)
S[\varphi]=\int\,{\mathrm{d}}x\,{\mathrm{d}}y\,\left\{\theta(x)\,\left(\frac{1}{2}(\nabla\varphi)^{2}+\frac{m^{2}}{2}\varphi^{2}\right)+\frac{c}{2}\delta(x)\varphi^{2}\right\},
\left[\prodz_{ij}^{2\rho}\prodz_{\hat{i}\hat{j}}^{2\rho^{\prime}}\prodz_{i\hat{j}}^{2\sigma}\right]_{(\hat{2}2\hat{1}1)}=z_{12}^{2\rho}z_{\hat{1}\hat{2}}^{2\rho^{\prime}}(z_{1\hat{1}}z_{1\hat{2}}z_{\hat{1}2}z_{2\hat{2}})^{2\sigma}.
\hat{R}_{12}(\sigma^{1}K^{\prime}\sigma^{1})_{1}\hat{R}_{12}^{-1}(\sigma^{1}K^{\prime}\sigma^{1})_{1}=(\sigma^{1}K^{\prime}\sigma^{1})_{1}\hat{R}_{12}^{-1}(\sigma^{1}K^{\prime}\sigma^{1})_{1}\hat{R}_{12}\;.
V_{m}^{a}=\epsilon^{ab}\phi_{Ab}^{*}\frac{\delta}{\delta\bar{\phi}_{A}}-\eta_{A}\frac{\delta}{\delta\phi_{Aa}^{*}}+m^{2}(P_{+})_{Ab}^{Ba}\bar{\phi}_{B}\frac{\delta}{\delta\phi_{Ab}^{*}}-m^{2}\epsilon^{ab}(P_{-})_{Ab}^{Bc}\phi_{Bc}^{*}\frac{\delta}{\delta\eta_{A}}
\displaystyle{E_{Cas.}(a)-E_{Cas.}(a_{0})=\hbar\Omega\left(\frac{\Omega}{c}\right)^{3}\left\{\frac{{\left({n_{2}}-1\right)}^{2}\,\left(2+{n_{2}}\right)}{16\,{\pi}^{2}}\,\left(V-{V_{0}}\right)-\frac{3\,{\left({n_{1}^{2}}-1\right)}^{2}}{64\,{\pi}^{2}}\,\frac{{V_{0}}}{V}\,\left(V-{V_{0}}\right)+{\calO}{\left({n_{1}^{2}}-1\right)}^{3}\right\}}.
\tilde{V}_{\nu}(r)=-\omega^{2}(f-1)+h\left[{\frac{3}{4r^{2}}}\left(1+{\frac{3r_{0}^{2}}{r^{2}}}\right)+{\frac{8R^{4}}{r^{2}(r^{2}+R^{2})^{2}}}\right],
S=\frac{i}{2}\int_{0}^{2\pi}d\tauF_{ab}X^{a}(\tau)\dot{X}^{b}(\tau)=\pi\alpha^{\prime}\sum_{n=1}^{\infty}nF_{ab}X_{-n}^{a}X_{n}^{b}\,F_{ab}=\theta_{ab}\,
\begin{array}{l}{z_{1}^{(3)}=z_{1}^{(4)}\tau^{2}=\cdots=z_{1}^{(n)}\tau^{2(n-3)}=z_{0}\tau^{2(n-2)},}\\{z_{i}^{(2)}=z_{i}^{(3)}\tau^{2}=\cdots=z_{i}^{(n)}\tau^{2(n-2)},~~2\leqi\leqm,}\\{z_{1}^{(1)}=z_{1}^{(2)}\tau^{2},}\\\end{array}
\gamma_{q\dot{q}}^{i}h_{+p\dot{q}}^{1-}+\gamma_{p\dot{q}}^{i}h_{+q\dot{q}}^{1-}=0,\\gamma_{q\dot{q}}^{i}h_{-p\dot{q}}^{2+}+\gamma_{p\dot{q}}^{i}h_{-q\dot{q}}^{2+}=0,\-\gamma_{q\dot{q}}^{i}\chi_{\dot{q}p}^{1}+\gamma_{p\dot{q}}^{i}\chi_{\dot{q}q}^{2}=0.
\int_{\infty}^{\sigma}{\frac{d\sigma^{\prime}}{\sqrt{1+4{\sigma^{\prime}}^{3}}}}=-{\frac{4^{-1/3}}{3}}B\left({\frac{1}{3}},\,{\frac{1}{6}}\right)+\sigma\,{_{2}F_{1}}\left({\frac{1}{2}},\,{\frac{1}{3}};\,{\frac{4}{3}};\,-4{\sigma}^{3}\right)=ig_{3}^{\frac{1}{6}}(\eta_{\infty}-\eta).
V_{\mathrm{eff}}=\frac{g^{2}}{\cal{D}}j(l_{e})^{\mu}j(l_{\bar{e}})_{\mu}+\frac{g^{2}}{\cal{D}}\frac{T^{*}-\omega}{|x-x^{\prime}|}j(l_{e})^{+}j(l_{\bar{e}})^{+}.
{\calL}=\frac14F^{a\mu\nu}F_{\mu\nu}^{a}-\frac12F^{a\mu\nu}(\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+gf^{abc}A_{\mu}^{b}A_{\nu}^{c})+\frac12
e_{a\mu}=\left(\begin{array}{cc}{N(\tau)}&{0}\\{0}&{a(\tau)E_{\hat{a}i}}\\\end{array}\right)~,~e^{a\mu}=\left(\begin{array}{cc}{N(\tau)^{-1}}&{0}\\{0}&{a(\tau)^{-1}E^{\hat{a}i}}\\\end{array}\right)~,
H(t)=\sum_{\alpha}H_{\alpha}(t)=\sum_{\alpha}\left[\frac{1}{2}\frac{1}{R^{3}(t)}\pi_{\alpha}^{2}+\frac{1}{2}R^{3}(t)\left(m^{2}+\xi{\bfR}+\frac{{\bfk}^{2}}{R^{2}(t)}\right)\phi_{\alpha}^{2}\right].
A(x_{i},p_{i})\starB(x_{i},p_{i})=e^{\kappa\sum_{i=1}^{n}(\partial_{x_{i}}\partial_{\tilde{p}_{i}}-\partial_{p_{i}}\partial_{\tilde{x}_{i}})}\left.A(x_{i},p_{i})B(\tilde{x}_{i},\tilde{p}_{i})\right|_{\tilde{x}_{i}=x_{i},\tilde{p}_{i}=p_{i}}
\bigl<\eta;t|\hat{\gamma}|\psi\bigr>=\Theta^{\ast}(\eta)\intdq\bigl<\eta;t|q\bigr>\bigl<q|\hat{\gamma}|\psi\bigr>=\Bigl(\frac{\hbar}{i}\frac{\partial}{\partial\eta}-\frac{\hbar}{i\Theta^{\ast}}\frac{\partial\Theta^{\ast}}{\partial\eta}-\frac{\eta}{2\tilde{\varphi}(t)}\Bigr)\bigl<\eta;t|\psi\bigr>.
I=-\nu{\frac{\Gamma(-\nu)}{\Gamma(\nu)}}\int{\frac{d^{d}k}{(2\pi)^{d}}}\tilde{A}_{i}({\bfk})\left[(\delta_{ij}-{\frac{k_{i}k_{j}}{|{\bfk}|^{2}}})+{\frac{\bar{c}}{c}}{\frac{k_{i}k_{j}}{|{\bfk}|^{2}}}\right]\left({\frac{|{\bfk}|}{2}}\right)^{2\nu}\tilde{A}_{j}(-{\bfk}).
U(z_{1})U(z_{2})={\frac{1}{c}}\left({\frac{80GU-32GG^{\prime}}{z_{1}-z_{2}}}\right)\quad,\quad\bar{U}(z_{1})\bar{U}(z_{2})=-{\frac{1}{c}}\left({\frac{80\bar{G}\bar{U}+32\bar{G}\bar{G}^{\prime}}{z_{1}-z_{2}}}\right)\;.
{\frac{d}{dx}}S_{N}^{\mathrm{(ODM)}}=-{\frac{d\beta}{dx}}{\frac{\beta^{N}}{(1-\beta)^{\sigma+1}}}\sum_{n=0}^{N}c_{n}{\frac{\Gamma(N+(\alpha-1)n-\sigma+1)}{(N-n)!\,\Gamma(\alphan-\sigma)}}\rho^{n}.
{\calM}_{\pi}^{2}(X)=M_{\pi}^{2}(X)+\frac{\lambda}{6}\,\vec{\varphi}^{\,2}(X),\;\;\;\;\;\;{\calM}_{\sigma}^{2}(X)=M_{\sigma}^{2}(X)+\frac{\lambda}{2}\,\vec{\varphi}^{\,2}(X).
S=\frac{1}{2\kappa^{2}}\int{d^{4}x\sqrt{-g}\left[\left(\left[1+\frac{1}{2}\pi\right]g^{\mu\nu}-\pi^{\mu\nu}\right)R_{\mu\nu}+\frac{1}{4}m^{2}\left(\pi_{\mu\nu}\pi^{\mu\nu}-\pi^{2}\right)\right]}.