harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
G_{\gamma}^{t}\left(\begin{array}{c}{0}\\{1}\\\end{array}\begin{array}{cc}{1}\\{0}\\\end{array}\right)G_{\gamma}-\left(\begin{array}{c}{0}\\{1}\\\end{array}\begin{array}{cc}{1}\\{0}\\\end{array}\right)=2\left(\begin{array}{c}{a_{11}a_{21}}\\{a_{12}a_{21}}\\\end{array}\begin{array}{cc}{a_{12}a_{21}}\\{a_{12}a_{22}}\\\end{array}\right).
\frac{i}{2}\oint_{\gamma_{\infty}}(\phi_{1}+\alpha_{1}\phi_{B})\partial_{\bar{z}}\phi_{1}d\bar{z}-\int(\phi_{1}+\alpha_{1}\phi_{B})\alpha_{1}\partial_{z}\partial_{\bar{z}}\phi_{B}\frac{idz\wedged\bar{z}}{2}.
\Omega(u,v)~=~\left(\begin{array}{cc}{\frac{\partial\beta_{u}(u,v)}{\partialu}}&{\frac{\partial\beta_{u}(u,v)}{\partialv}}\\{\frac{\partial\beta_{v}(u,v)}{\partialu}}&{\frac{\partial\beta_{v}(u,v)}{\partialv}}\\\end{array}\right)~.
\Lambda_{rs}=\oint\frac{d\xi}{2\pi\imath}\oint\frac{d\xi^{\prime}}{2\pi\imath}\xi^{-r-\frac{1}{2}}\xi^{\prime}{}^{-s-\frac{1}{2}}\left(\frac{\partial\lambda(\xi)}{\partial\xi}\right)^{\frac{1}{2}}\frac{1}{\lambda(\xi)-\lambda(\xi^{\prime})}\left(\frac{\partial\lambda(\xi^{\prime})}{\partial\xi^{\prime}}\right)^{\frac{1}{2}}.
I=\int_{\Sigma}d^{3}\!x\sqrt{\gamma}\left(R(\gamma)-{\frac{1}{2}}\left(\left(D\tilde{U}\right)^{2}-e^{-2\tilde{U}}\left(DV\right)^{2}\right)-{\frac{1}{2}}\left(\left(D\tilde{\Phi}\right)^{2}-e^{-2\tilde{\Phi}}\left(D\Psi\right)^{2}\right)\right)\,
S_{\mathrm{gh}}=\int\sum_{\alpha\beta}\left\{\bar{c}^{\alpha\beta}\left(-D_{\mu}^{\alpha\beta}\right)^{2}c^{\beta\alpha}+ig_{\mathrm{YM}}\bar{c}^{\beta\alpha}D_{\mu}^{\alpha\beta}\left[\bar{A}_{\mu},c\right]\right\}.
j=\left(\begin{array}{cccccc}{\theta_{1}}&{}&{}&{}&{}&{}\\{}&{\ddots}&{}&{}&{}&{}\\{}&{}&{\theta_{q+1}}&{}&{}&{b}\\{}&{}&{}&{\ddots}&{}&{}\\{}&{}&{}&{}&{\theta_{p}}&{}\\{}&{}&{c^{t}}&{}&{}&{d}\\\end{array}\right).
\frac{\alpha}{4}g^{2}\epsilon^{mn}\epsilon^{pq}{\overline{{C}}}^{m}{\overline{{C}}}^{n}C^{p}C^{q}\rightarrow-\frac{1}{2\alphag^{2}}\psi^{2}-i\psi\epsilon^{mn}{\overline{{C}}}^{m}C^{n}
\left(P_{\calE}\right)_{\mu\nu}=\frac{1}{2}\left(\bar{n}_{\mu}n_{\nu}+n_{\mu}\bar{n}_{\nu}\right),\qquad\left(P_{\calH}\right)_{\mu\nu}=\frac{1}{2}\left(\bar{m}_{\mu}m_{\nu}+m_{\mu}\bar{m}_{\nu}\right)
\left\langle\frac{1}{N}\mathrm{tr}A\right\rangle_{1}-\frac{c}{3}\left\langle\frac{1}{N}\mathrm{tr}A^{3}\right\rangle_{1}=-\frac{1}{N^{2}}a^{-2/3}\frac{\sqrt{10}}{216\cdot2^{7/3}c^{1/2}s^{2/3}}T^{-1/3}+O(\frac{1}{N^{2}}a^{0}),
\omega_{c-2D}({\bf{q}})=\frac{(\frac{k_{f}|{\bf{q}}|}{m})(1+\frac{2\pi/m}{v_{{\bf{q}}}})}{\sqrt{\frac{4\pi/m}{v_{{\bf{q}}}}+(\frac{2\pi/m}{v_{{\bf{q}}}})^{2}}}
V_{7}(\vec{x})=-{\frac{\alpha}{\sqrt{x^{2}+y^{2}+z^{2}}}}+{\frac{\hbar^{2}}{2M(x^{2}+y^{2})}}\Bigg({\frac{k_{1}^{2}z}{\sqrt{x^{2}+y^{2}+y^{2}}}}+F\bigg({\frac{y}{x}}\bigg)\Bigg)\enspace.
\begin{array}{c}{\left[M^{\alpha\beta},P^{\mu}\right]=i(g^{\alpha\mu}P^{\beta}-(\alpha\longleftrightarrow\beta))}\\{\left[M^{\mu\nu},M^{\rho\sigma}\right]=i\left\{g^{\nu\sigma}M^{\mu\rho}-g^{\mu\rho}M^{\nu\sigma}-\left(\mu\longleftrightarrow\nu\right)\right\}}\\\end{array}
\left(\frac{\left(2+r\right)r}{\left(1+r\right)^{2}}\right)^{2}\frac{d^{2}\hat{a}}{dx^{2}}+\left(2\frac{\left(2+r\right)^{2}}{\left(1+r\right)^{3}}-\frac{4+r}{\left(1+r\right)^{2}}\right)\frac{d\hat{a}}{dx}-\left(\kappa^{2}r^{2}f^{2}-\omega^{2}\right)\hat{a}=0.
\left\langleT_{22}(\vec{x})\right\rangle=-\frac{1}{32\pi^{2}}F_{0}(\vec{x})+\frac{1}{16\pi^{2}}\left(D_{1}(\vec{x})-D_{2}(\vec{x})\right)-\frac{1}{64\pi^{2}}\left(F_{1}(\vec{x})-F_{2}(\vec{x})\right)
D_{++}^{ab}(x)=-{\frac{i\delta^{ab}}{(2\pi)^{2}}}\intd^{2}k\,e^{ikx}{\frac{\partial}{\partialk_{-}}}P\left({\frac{1}{k_{-}}}\right)=-i\delta^{ab}{\frac{|x^{-}|}{2}}\delta(x^{+})\,,
W=\left(\zeta+a_{0}\frac{x_{1}^{4-n}z^{6-n}}{\zeta}\right)-x^{2}+z^{4}+y^{2}z-2(-c_{0})^{1/2}y+c_{4}z^{3}+c_{3}z^{2}+c_{2}z+c_{1}+\cdots,
Z(\sigma_{(A)};\sigma_{(B)};\sigma_{(C)})=\sigma_{(A)\alpha_{1}\dot{\beta}_{1}}\,\sigma_{(B)\alpha_{2}\dot{\beta}_{2}}\sigma_{(C)\alpha_{3}\dot{\beta}_{3}}G^{\alpha\alpha_{1}\alpha_{2}\alpha_{3}}G^{\beta\beta_{1}\beta_{2}\beta_{3}}
{\cal{G}}={\cal{G}}_{AB}d\varphi^{A}d\varphi^{B}=\frac{1}{2}d\varphi^{2}+\frac{1}{2}dy^{2}+\frac{1}{2}e^{2\varphi}\left(d\sigma_{1}-\chid\sigma_{2}\right)^{2}+\frac{1}{2}e^{\varphi}\left[e^{\sqrt{3}y}d\chi^{2}+e^{-\sqrt{3}y}d\sigma_{2}^{2}\right].
\rho_{n}(x)=\frac{1}{(x^{2})^{\frac{n-2}{2}}}+\sum..\sum_{q\neq0}\{\frac{1}{|x-q|^{n-2}}-\frac{1}{|q|^{n-2}}[1+\frac{(n-2)}{q^{2}}(q\cdotx+\frac{1}{2q^{2}}(n(q\cdotx)^{2}-q^{2}x^{2}))]\}
B=\left(\begin{matrix}{0}&{-\theta_{2}}&{0}&{0}&{0}&{0}\\{\theta_{2}}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{-\theta_{3}}&{0}&{0}\\{0}&{0}&{\theta_{3}}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}\\\end{matrix}\right)~,
\begin{array}{ll}{D=3:}&{W^{+(+)[-]}(x,\theta^{++},\theta^{(++)},\theta^{[-]\{\pm\}},u)}\\{D=4:}&{W^{13}(x,\theta_{2},\theta_{4},\bar{\theta}^{1},\bar{\theta}^{3},u)}\\{D=6:}&{W^{13}(x,\theta^{1},\theta^{3},u)}\\\end{array}
ds^{2}\approxB_{H}[\sigma(r-r_{H})]^{2(1-\alpha)}\bar{g}_{\mu\nu}(x)dx^{\mu}dx^{\nu}+\frac{dr^{2}}{B_{H}(r-r_{H})^{2}}+r_{H}^{2}d\Omega_{d-1}^{2},
\int_{0}^{1}dj_{1}dj_{2}dj_{3}\,\delta(j_{1}+j_{2}+j_{3}-1)(j_{2}e^{2\piinj_{3}}+j_{3}e^{-2\piinj_{2}}+j_{1})|1-e^{2\piinj_{1}}|^{2}=\frac13+\frac{5}{2\pi^{2}n^{2}}
\frac{1}{2}{\calD}_{\mu}\bar{z}_{A}{\calD}_{\mu}z^{A}=\left(\frac{\partial_{\mu}\Phi}{\Phi}\right)^{2}+\frac{1}{2}\Phi^{2}\partial_{\mu}\bar{z}\partial_{\mu}z-\frac{1}{4}U_{\mu}{}^{2}
M_{4}^{Fg}=(stu)^{-1}L_{4}^{Fg},\L_{4}^{Fg}=\frac{\kappa^{2}}{3}\left(\textstyle{\frac{1}{4}}B_{\mu\nu\alpha\beta}^{g}B_{\mu_{1}\nu_{1},\alpha_{1}\beta_{1}}^{F}G^{\mu\mu_{1};\nu\nu_{1}}-C_{\mu\nu\rho;\alpha\beta}^{RF}C_{~~~~~~\alpha_{1}\beta_{1}}^{RF\mu\nu\rho}\right)K^{\alpha\alpha_{1};\beta_{1}\beta},
X_{L}^{9}+X_{R}^{9}\stackrel{T}{\longrightarrow}X_{L}^{9}-X_{R}^{9}\stackrel{\Omega}{\longrightarrow}X_{R}^{9}-X_{L}^{9}\stackrel{T^{-1}}{\longrightarrow}-X_{R}^{9}-X_{L}^{9}
\{d_{m}^{\mu,j},d_{n}^{\nu,j^{\prime}}\}=\eta^{\mu\nu}\delta^{j,j^{\prime}}\delta_{m,-n}~~~~~~~~~and~~~~~\{d_{m}^{\mu,k},d_{n}^{\nu,k^{\prime}}\}=-\eta^{\mu\nu}\delta^{k,k^{\prime}}\delta_{m,-n}.
\begin{aligned}{\deltae_{\mu}{}^{r}}&{{}=2\,\bar{\epsilon}\Gamma^{r}\psi_{\mu}\,,}\\{\delta\psi_{\mu}}&{{}=D_{\mu}(\omega)\epsilon+\cdots\,,}\\{\deltaB_{\mu\nu}}&{{}=2\,\bar{\epsilon}\Gamma_{[\mu}\psi_{\nu]}\,.}\\\end{aligned}
\psi_{C}:\left(\begin{array}{l}{f_{\pm}^{q}}\\{(f_{\pm}^{-q})^{*}}\\\end{array}\right)(\theta,\phi)\mapsto\left(\begin{array}{l}{(f_{\pm}^{-q})^{*}}\\{f_{\pm}^{q}}\\\end{array}\right)(\theta,\phi)
I=\intd^{2}x\sqrt{-g}e^{-2\phi}[R-\frac{1}{2}g^{\alpha\beta}\partial_{\alpha}f\partial_{\beta}f+\frac{1}{4}e^{\chif}g^{\alpha\beta}g^{\gamma\delta}F_{\alpha\gamma}F_{\beta\delta}-\frac{1}{2}e^{\chif+4\phi}g^{\alpha\beta}F_{\alpha\theta}F_{\beta\theta}],
\Psi_{0}\left(r,\theta\right)=\left(\begin{array}{l}{e^{-i\frac{\theta}{2}}\sum_{n=-\infty}^{\infty}A_{n}r^{n-k-\alpha}e^{i\left(n+\frac{1}{2}\right)\theta}}\\{e^{i\frac{\theta}{2}}\sum_{n=-\infty}^{\infty}B_{n}r^{k+\alpha-n-1}e^{i\left(n+\frac{1}{2}\right)\theta}}\\\end{array}\right)
P\left({\frac{a\tau+b}{c\tau+d}},{\frac{z}{c\tau+d}}\right)={\epsilon{\binom{a\b}{c\d}}}e^{2\pi\sqrt{-1}(1/2)cz^{2}/(c\tau+d)}P(\tau,z)\,,\quad{\binom{a\b}{c\d}}\inSL(2,{\bfZ})\,.
\begin{array}{c}{\Bigl(a^{+}\left(P^{0}+iP_{1}-P_{2}+P_{3}\right)-a^{-}\left(P^{0}-iP_{1}+P_{2}+P_{3}\right)\Bigr)|\psi\rangle=0,}\\{\Bigl(a^{+}\left(P^{0}-iP_{1}-P_{2}-P_{3}\right)+a^{-}\left(P^{0}+iP_{1}+P_{2}-P_{3}\right)\Bigr)|\psi\rangle=0.}\\\end{array}
\Phi_{k}(x):=\left(\begin{matrix}{y_{k},}&{y_{k+1},}&{\cdots,}&{y_{k+n}}\\{\partialy_{k},}&{\partialy_{k+1},}&{\cdots,}&{\partialy_{k+n}}\\{\vdots}&{}&{}&{\vdots}\\{\partial^{n}y_{k},}&{\partial^{n}y_{k+1},}&{\cdots,}&{\partial^{n}y_{k+n}}\\\end{matrix}\right).
S=\intdt\,dx{\calL}=\intdt\,dx\left[\frac{i\hbar}{2}(\hat{\phi}^{*}\partial_{t}{\phi}-{\phi}\partial_{t}{\phi}^{*})-\frac{\hbar^{2}}{2m}\left|\left(\partial_{x}+i{\frac{\lambda}{2}}\rho^{2}\right)\phi\right|^{2}-V(\rho^{2})\right];
{}~~~~~~~\times\Bigg\{\sum_{\sigma=0}^{N-1}{\frac{w(_{3},x_{13},x_{1}|\sigma+d-h)w(x_{4},x_{24},x_{2}|\sigma+a-g)s(\sigma,b+c-g-h)}{w(x_{4},x_{14},x_{1}|\sigma+e-c)w(x_{3},x_{23},\omegax_{2}|\sigma+f-b)}}\Bigg\}_{0},
P_{2}^{3}=-2r^{2}K(6r^{2}\hat{r}_{2}-2r^{2}\hat{r}_{6}+8r\hat{r}_{2}\hat{r}_{6}+\hat{r}_{2}^{2}\hat{r}_{6}+3\hat{r}_{2}\hat{r}_{6}^{2}),
W={\frac{m^{3}N^{2}}{24g_{\mathrm{YM}}^{2}}}\left[E_{2}(\tau)-{\frac{q}{p}}E_{2}\left({\frac{q}{p}}\tau\right)\right]\to{\frac{m^{3}N^{2}E_{2}(\tau)}{96\pig}}-{\frac{m^{3}N^{3}}{96\pigp^{2}}}\.
\eta_{ij}=\lambda(\sigma_{a})\delta_{ij}:\;\;\frac{\partialx^{\alpha}}{\partial\sigma_{1}}\frac{\partialx^{\alpha}}{\partial\sigma_{1}}=\frac{\partialx^{\alpha}}{\partial\sigma_{2}}\frac{\partialx^{\alpha}}{\partial\sigma_{2}}=\lambda;\;\;\frac{\partialx^{\alpha}}{\partial\sigma_{1}}\frac{\partialx^{\alpha}}{\partial\sigma_{2}}=0.
\begin{array}{l}{q_{0}=\sqrt{\frac{25-D}{6}-\vec{q}^{\;2}}\,}\\{p_{0_{\pm}}=q_{0}\pm\sqrt{q_{0}^{2}-\vec{p}^{\;2}+2\vec{q}\vec{p}-4}=\frac{1}{\sqrt{6}}\left(\sqrt{25-D-6\vec{q}^{\;2}}\pm\sqrt{1-D-6(\vec{q}-\vec{p})^{2}}\right)\.}\\\end{array}
ds^{2}=-(1-\frac{16\piMG_{5}}{3\Omega_{3}r^{2}}+\frac{r^{2}}{l^{2}})dt^{2}+(1-\frac{16\piMG_{5}}{3\Omega_{3}r^{2}}+\frac{r^{2}}{l^{2}})^{-1}dr^{2}+r^{2}d\Omega_{3}^{2},
\frac{e^{-i\frac{\pi}{4}}}{\sqrt{2\pik}}\,\sum_{\ell=-\infty}^{\infty}\,e^{i\ell\varphi}\left(-1\right)=-\frac{e^{-i\frac{\pi}{4}}}{\sqrt{2\pik}}\,2\pi\,\delta[\varphi]=-\sqrt{\frac{2\pi}{ik}}\,\delta[\varphi].
I_{k}\left(\mu,d,\lambda\right):=-\frac{\left(2\mu\right)^{\frac{d}{2}-1}}{\left(2k\lambda\right)^{\frac{d}{2}}}\Gamma\left(\frac{d}{2}+k\right)W_{-k,\frac{d-1}{2}}\left(4\muk\lambda\right).
ds^{2}=dt^{2}+{\frac{1}{4}}t^{2}(\Sigma_{1}^{2}+\Sigma_{2}^{2}+\Sigma_{3}^{2})+16(A_{1}^{2}\alpha_{1}^{2}+A_{2}^{2}\alpha_{2}^{2}+A_{3}^{2}\alpha_{3}^{2}).
\frac{1}{4\pii}\left(\int_{-1}^{-\epsilon}\!\!\!dx\!+\!\int_{\epsilon}^{1}\!\!\!dx\!+\!\int_{C_{\epsilon}}\!\!\!dx\right)e^{-\rhox^{2}}\left(-4xK_{1}(ix)^{2}+x^{3}K_{0}(ix)^{2}-4ix^{2}K_{0}(ix)K_{1}(ix)\right).\nonumber
E_{ik}=(i,k)E_{ik}^{*},\;\;E_{\bar{i}\bar{k}}=[\hat{\bar{i}},\hat{\bar{k}}]E_{\bar{i}\bar{k}}^{*},\;\;E_{i\bar{k}}=u(1,i)v[1,\hat{\bar{k}}]E_{i\bar{k}}^{*},
S_{\mathrm{eff}}\left[A_{\mu}\right]=S_{\mathrm{YM}}\left[A_{\mu},A_{3}=0\right]+S_{\mathrm{gf}}\Bigl[\int_{0}^{L}dz\,A_{\mu}^{3}\Bigr]+M^{2}\intd^{4}x\,\,\Phi_{\mu}^{\dagger}(x)\Phi^{\mu}(x).
\tilde{\calE}_{q}[\phi(x)]=\frac{1}{2}\sum_{j}(\tilde{\omega}_{j}-\tilde{m})+\int_{0}^{\infty}\frac{dk}{2\pi}(\tilde{\omega}(k)-\tilde{m})\frac{d}{dk}\left(\tilde{\delta}(k)-\tilde{\delta}_{1}(k)\right)\,.
\begin{array}{lcr}{U(1)_{R}:X_{j}\longrightarrowX_{j}^{\prime}=X_{j}}\\{\qquadY_{j}\longrightarrowY_{j}^{\prime}=Y_{j}}\\{\qquad\eta_{j}^{+}\longrightarrow\eta\prime_{j}^{+}=e^{i\theta}\eta_{j}^{+}}\\{\qquad\overline{{\eta}}_{j}^{+}\longrightarrow\overline{{\eta}}\prime_{j}^{+}=e^{i\theta}\overline{{\eta}}_{j}^{+}.}\\\end{array}
\begin{array}{c}{F_{\mu\nu\alpha\beta}=-F_{\mu\nu\beta\alpha}=-F_{\nu\mu\alpha\beta}=F_{\alpha\beta\mu\nu},}\\{F_{\mu\nu\alpha\beta}+F_{\mu\beta\nu\alpha}+F_{\mu\alpha\beta\nu}=0,}\\{\partial_{\lambda}F_{\mu\nu\alpha\beta}+\partial_{\mu}F_{\nu\lambda\alpha\beta}+\partial_{\nu}F_{\lambda\mu\alpha\beta}=0.}\\\end{array}
{\calL}_{GF+FP}=-i\delta_{B}[\bar{C}^{A}(\partial_{\mu}{\calA}^{\mu}+{\frac{\alpha}{2}}B)^{A}]=+i\delta_{B}\bar{\delta}_{B}({\frac{1}{2}}{\calA}_{\mu}^{2})+{\frac{\alpha}{2}}i\bar{\delta}_{B}(B^{A}C^{A}),
f(\omega)=\frac{1}{(2-\omega)^{3}}\left[\frac{\Gamma^{2}(3-\omega)\Gamma(2\omega-3)}{\Gamma(5-2\omega)}-{\frac{\Gamma(\omega-1)\Gamma(\omega)(10\omega^{2}-19\omega+10)}{4(2\omega-3)(2\omega-1)}}\right].
M=e^{-\psi/d}\left(\begin{array}{cc}{e^{\psi_{1}/d_{1}}M_{1}^{-1}}&{e^{\psi_{1}/d_{1}}M_{1}^{-1}K}\\{e^{\psi_{1}/d_{1}}K^{T}M_{1}^{-1}}&{e^{-\psi_{2}/d_{2}}M_{2}+e^{\psi_{1}/d_{1}}K^{T}M_{1}^{-1}K}\\\end{array}\right),
\int_{G}dg\,\overline{{\langlel_{1},m_{1}|T_{g}|l_{1},m_{1}^{\prime}\rangle}}\langlel_{2},m_{2}|T_{g}|l_{2},m_{2}^{\prime}\rangle={\frac{\delta^{l_{1}l_{2}}\delta_{m_{1}m_{2}}\delta_{m_{1}^{\prime}m_{2}^{\prime}}}{\mathrm{dim}_{l_{1}}}},
{\hat{\calK}}^{(2)}=2\partial{\hat{\omega}}^{(1)}+l_{p}^{-2}\,D{\hat{X}}^{\hat{\mu}}D{\hat{X}}^{\hat{\nu}}(i_{\hat{k}}{\hat{C}})_{{\hat{\mu}}{\hat{\nu}}}-ml_{p}^{2}\,\partial{\hat{\omega}}^{(0)}{\hat{b}}\,.
\begin{array}{c}{S(X,y,z,T)=-k_{y}y\pmi\int\sqrt{m^{2}-\lambda^{2}+\left(k_{y}+\omega_{o}z\right)^{2}}dz\pm}\\{\pm\int\sqrt{k-\lambda^{2}e^{2\omegaX}}dX\pm\int\sqrt{k+\lambda^{2}e^{-2\omegaT}}dT,}\\\end{array}
R_{MK}-\frac{1}{2}g_{MK}R=\Lambdag_{MK}+\sigma\delta(z)\left(g_{(4)}^{\mu\kappa}+T_{(4)}^{\mathrm{matter}\;\mu\kappa}\right)\sqrt{\left|\frac{g_{(4)}}{g_{(5)}}\right|}g_{\muM}g_{\kappaK}.
\bigl\lbrackA,\lbrackB,C\bigr\rbrack\rbrack_{\mu_{1}...\mu_{d-2}}+(cycl.perm.)=\partial_{\lbrack\mu_{1}}\epsilon_{\mu_{2}...\mu_{d-2}\rbrack\nu\lambda\sigma}\xi^{\nu}(A)\xi^{\lambda}(B)\xi^{\sigma}(C)
\partial_{-}\deltaA_{+}+4i{\frac{e^{3}}{\kappa^{2}}}(2|\phi|^{2}-v^{2})\phi^{*}\delta\phi-2i{\frac{e^{2}}{\kappa}}|\phi|^{2}{\calQ}=2{\frac{e}{\kappa}}\phi^{*}\partial_{0}\delta\phi,
\Bigl(\,{\frac{w^{\prime\prime}(s)}{2\,w(s)}}\,-\,{\frac{w^{\prime\prime}(t)}{2\,w(t)}}\,\Bigr)\,w(s+t)~=~w(s)\,w^{\prime}(t)\,-\,w^{\prime}(s)\,w(t)~.
(\mathrm{i}{\bfP}^{J}\frac{\partial}{\partial{\bfP}^{I}}-\mathrm{i}{\bfP}^{I}\frac{\partial}{\partial{\bfP}^{J}}+\sum_{a}M_{a}^{IJ})|h_{3}\rangle=0\,,\qquad(\sum_{a}\beta_{a}\frac{\partial}{\partial\beta_{a}}+{\bfP}^{I}\frac{\partial}{\partial{\bfP}^{I}})|h_{3}\rangle=0\,.
D_{\mu\nu}^{(\mathrm{static})}=P_{\mu\nu}\,{\frac{1}{\vec{p}^{2}}}+Q_{\mu\nu}\,{\frac{1}{\vec{p}^{2}+M_{+}^{2}}}-{\frac{(\kappa+\Pi_{\mathrm{odd}})}{\vec{p}^{2}(\vec{p}^{2}+M_{+}^{2})}}\,\epsilon_{\mu\nu\lambda}p_{\lambda}.
|{\calZ}_{1,2}|^{2}={\textstyle\frac{1}{2}}\displaystyle{\sum_{n=1}^{N}}|\Gamma^{(n)}|^{2}\pm{\textstyle\frac{1}{2}}\left[\left(\displaystyle{\sum_{n=1}^{N}}|\Gamma^{(n)}|^{2}\right)^{2}-\left|\displaystyle{\sum_{n=1}^{N}}\left(\Gamma^{(n)}\right)^{2}\right|^{2}\right]^{1/2}\,,
U(\omega)={\frac{1}{\sqrt{A}}}e^{-{\frac{4}{A}}}{\frac{1}{2\sqrt{\pi}}}\int_{0}^{+\infty}[e^{-{\frac{(\Omega-\xi)^{2}}{4A}}}-e^{-{\frac{(\Omega+\xi)^{2}}{4A}}}]({\frac{\phi(\xi)}{\phi(\Omega)}})d\xi,
H_{0}^{J,\mu}=\int_{x}[{\frac{1}{2}}\Pi_{x}^{2}+{\frac{1}{2}}(\partial_{x}\phi_{x})^{2}+{\frac{1}{2}}\mu^{2}\phi_{x}^{2}-J_{x}\phi_{x}-{\frac{1}{2}}f_{xx}+{\frac{1}{2}}\int_{y}J_{x}h_{xy}^{-1}J_{y}],
\{g_{_E}(x,z)\}=\left(\begin{array}{cc}{g_{\mu\nu}(x)+A_{\mu}^{\,\,i}(x)\,A_{\nu}^{\,\,j}(x)\,g_{0\,ij}(x)}&{A_{\mu}^{\,\,i}(x)\,g_{0\,il}(x)}\\{g_{0\,ki}(x)\,A_{\nu}^{i}(x)}&{g_{0\,kl}(x)}\\\end{array}\right)
N_{i_{e}f_{e,\mu}g_{e,\mu}}=\frac{1}{2}[N_{i_{0}fg}N_{\bar{0}\bar{f}\bar{g}}+e^{-\frac{1}{2}\pii(m+\bar{m})}\breve{N}_{i_{0}fg}\breve{N}_{\bar{0}\bar{f}\bar{g}}]\;\;\;,
\begin{array}{l}{d\phi_{a}=i\hbar\Box\phi_{a}d\tau+ig\hbar(\phi_{1}^{2}+\phi_{2}^{2})\phi_{a}d\tau-ie\epsilon_{ab}A_{\mu}\partial_{\mu}\phi_{b}d\tau-\frac{i}{\hbar}e^{2}A_{\mu}A^{\mu}d\tau+\sqrt{2}\hbardB_{a}}\\{\equivi\hbar\Box_{A}\phi_{a}d\tau+ig\hbar(\phi_{1}^{2}+\phi_{2}^{2})\phi_{a}d\tau+\sqrt{2}\hbardB_{a}}\\\end{array}
N^{\calP}~=~\sum_{m_{1}=-j_{1}}^{j_{1}}\cdots\sum_{m_{p}=-j_{p}}^{j_{p}}\left[~{\delta}_{(\sum_{n=1}^{p}m_{n}),0}~-~\frac12~{\delta}_{(\sum_{n=1}^{p}m_{n}),1}~-~\frac12~{\delta}_{(\sum_{n=1}^{p}m_{n}),-1}~\right].
\Phi=(\varphi^{I},c^{\alphab},B^{\alpha}),\quad\bar{\Phi}=(\bar{\varphi}_{I},\bar{c}_{\alphab},\bar{B}_{\alpha}),\quad\Phi_{a}^{*}=(\varphi_{Ia}^{*},c_{\alphab|a}^{*},B_{\alphaa}^{*}).
\stackrel{\circ}{T}{^a}=\stackrel{\circ}{D}e^{a}\equivde^{a}+\stackrel{\circ}{\omega}{_{\,\cdotb}^{\,a}}\wedgee^{b},\;\;\;\;\;\stackrel{\circ}{R}{^{ab}}=d\stackrel{\circ}{\omega}{^{ab}}+\stackrel{\circ}{\omega}{^a_{\cdotc}}\wedge\stackrel{\circ}{\omega}{^{cb}},
\Phi({\bfr}_{1}^{(1)},\ldots,{\bfr}_{n_{1}}^{(1)},\ldots,{\bfr}_{1}^{(M)},\ldots,{\bfr}_{n_{M}}^{(M)},t)=\left\langle0\left|\prod_{p=1}^{M}\frac{1}{\sqrt{n_{p}!}}\Psi_{p}({\bfr}_{1}^{(p)},t)\cdots\Psi_{p}({\bfr}_{n_{p}}^{(p)},t)\right|\Phi\right\rangle
{\calE}[\phi]=\intdx\left[\frac{1}{2}\left(\frac{d\phi_{1}}{dx}\right)^{2}+\frac{1}{2}\left(\frac{d\phi_{2}}{dx}\right)^{2}+\frac{1}{8}\left(4\phi_{1}^{2}+2\sigma\phi_{2}^{2}-1\right)^{2}+2\sigma^{2}\phi_{1}^{2}\phi_{2}^{2}\right]\qquad,
T=\left(\begin{array}{cccc}{0}&{-g_{z}}&{g_{y}}&{f_{x}}\\{g_{z}}&{0}&{-g_{x}}&{f_{y}}\\{-g_{y}}&{g_{x}}&{0}&{f_{z}}\\{-f_{x}}&{-f_{y}}&{-f_{z}}&{0}\\\end{array}\right).
{X}^{\underline{{m}}}=\tilde{X}^{\underline{{m}}}(\tau),\quad{\Theta}^{1\underline{{\mu}}}=\tilde{\Theta}^{1\underline{{\mu}}}(\tau),\quad{\Theta}_{\underline{{\mu}}}^{2}=\tilde{\Theta}_{\underline{{\mu}}}^{2}(\tau)\quad:\quad{\calM}^{1}\rightarrow{\underline{{{\calM}}}}^{(10|32)},
V_{\alpha}(x,\theta)=\xi_{\alpha}(x)-\frac{i}{2}(\gamma_{5}\gamma_{\mu}\theta)_{\alpha}\frac{A^{\mu}(x)}{g}+\frac{1}{2}\theta_{\alpha}\phi(x)-\frac{1}{2}N(x)(\gamma_{5}\theta)_{\alpha}-\frac{1}{2}\bar{\theta}\theta\sqrt{2}\Psi_{\alpha}(x),
\begin{array}{lcr}{\xi_{a}^{++}=i\zeta_{a}^{+}\overline{{\zeta}}_{a}^{+}=iu_{(\alpha}^{+}u_{\beta)}^{+}\zeta_{a}^{(\alpha}{\overline{{\zeta}}_{a}^{\beta)}}}\\{\zeta_{a}^{\pm}=u_{\alpha}^{\pm}\zeta_{a}^{\alpha};\qquad\overline{\zeta}_{a}^{\pm}=u_{\alpha}^{\pm}\overline{{\zeta}}_{a}^{\alpha},}\\\end{array}
\psi(x,{\bfp};s_{1},s_{2})={\frac{1}{2P^{+}}}\int{\frac{dp^{-}}{2\pi}}\bar{u}(xP^{+},{\bfp};s_{1})\gamma^{+}\Phi(p)\gamma^{+}v((1-x)P^{+},-{\bfp};s_{2}),
{\calN}=\left(\frac{[2j_{1}]_{q}![2j_{2}]_{q}![2J+1]_{q}}{[j_{1}+j_{2}-J]_{q}![j_{1}-j_{2}+J]_{q}![-j_{1}+j_{2}+J]_{q}]![j_{1}+j_{2}+J+1]_{q}!}\right)^{\frac{1}{2}}.
[\hat{p}_{\alpha}+\beta_{\alpha\beta}\hat{\epsilon}^{\beta},\hat{p}_{\alpha^{\prime}}+\beta_{\alpha^{\prime}\beta^{\prime}}\hat{\epsilon}^{\beta^{\prime}}]=i(\beta_{\alpha\alpha^{\prime}}+\beta_{\beta\alpha^{\prime}}\partial_{\alpha}\epsilon^{\beta}+\beta_{\alpha\beta}\partial_{\alpha^{\prime}}\epsilon^{\beta}).
\sum_{m}Y_{l,m}^{q}(\theta,\phi)(Y_{l,m}^{q}(\theta^{\prime},\phi^{\prime}))^{*}=\sqrt{(2l+1)/4\pi}Y_{l,-q}^{q}(\gamma,0)e^{iq(\phi-\phi^{\prime})}e^{-iq(R+R^{\prime}-\pi)}\,
\intd^{4}x{\calL}_{s}=\intd^{4}x\frac{\partial}{\partial\theta}\frac{\partial}{\partial\lambda}\left[\bar{X}^{i\prime}(\bar{x})\tilde{\Lambda}_{i}^{j}A_{j}(\bar{x})+\left\{(\epsilon^{\mu}a\lambda+\deltab\theta)\partial_{\mu}\bar{X}^{i}(\bar{x})+\delta\cdotxb\bar{X}_{,\lambda}^{i}-\epsilon\cdotxa\bar{X}_{,\theta}^{i}\right\}A_{i}(\bar{x})\right]
{\frac{\partial^{2}G}{\partialq_{i}\partialq_{j}}}={\frac{\partial^{2}G}{\partialp_{i}\partialp_{j}}}\,,\quad{\frac{\partial^{2}G}{\partialq_{i}\partialp_{j}}}=-\,{\frac{\partial^{2}G}{\partialq_{j}\partialp_{i}}}.
\frac{\lambda^{3}}{3!\,\beta}\cdot\frac{96}{64}\int_{0}^{\beta}d\tau_{1}\cdots\int_{0}^{\beta}d\tau_{3}\left[\langle\eta^{2}(\tau_{1})\eta(\tau_{2})\eta(\tau_{3})\rangle_{0,c}+\mathrm{cyclic~perm.}\right]=\frac{3\lambda^{3}}{280}\,\beta^{4}.
S_{\mathrm{{QH}}}=\int_{\calM}d^{3}x~\left(\psi^{\dagger}\,i\left(\partial_{0}-iA_{0}-iea_{0}\right)\psi+\frac{1}{2m_{e}}\psi^{\dagger}\left(\nabla-iA-iea\right)^{2}\psi\right)+\frac{1}{m}S_{\mathrm{CS}}^{[U(1)]}[A]
{\bfG}_{\xi}(\Box)=-\frac2{l^{2}\Box}\,\frac{a^{2}}{1-a^{2}}\left[\begin{array}{cc}{\,\,1\,\,}&{\,-1\,\,}\\{\,\,1\,\,}&{\,-1\,\,}\\\end{array}\right]+O(l^{2}\Box),
ds^{2}=-(\frac{r^{2}}{l^{2}}-M+\frac{J^{2}}{4l^{2}})dt^{2}+\frac{dr^{2}}{(\frac{r^{2}}{l^{2}}-M+\frac{J^{2}}{4r^{2}})}+r^{2}(d\phi-\frac{J^{2}}{2r^{2}}dt)^{2}
D_{mn}^{j}(u)=\sqrt{\frac{(j+m)!(j-m)!}{(j+n)!(j-n)!}}\sum_{\mu}\left(\begin{array}{l}{j+n}\\{\mu}\\\end{array}\right)\left(\begin{array}{l}{j-n}\\{j-\mu-m}\\\end{array}\right)\cdot
\begin{array}{rl}{\langle\psi^{\prime},c^{i}\psi\rangle=}&{\displaystyle\int_{G}\psi^{\prime}\wedge*c^{i}\psi=\displaystyle(-1)^{q(r-q)}\int_{G}\psi^{\prime}\wedge*c^{i}**\psi=(-1)^{q}\int_{G}\psi^{\prime}\wedge\pi^{i}*\psi}\\{=}&{\displaystyle\int_{G}\pi^{i}\psi^{\prime}\wedge*\psi=\langle\pi^{i}\psi^{\prime},\psi\rangle\quad,}\\\end{array}
U=\frac1{2e^{2}}\sum_{l=1}^{s}D_{l}^{2}+\sum_{i=1}^{N}|F_{i}|^{2}+2\sum_{k,l=1}^{s}\sum_{i=1}^{N}\bar{\sigma}_{k}\sigma_{l}Q_{i}^{(k)}Q_{i}^{(l)}|\phi_{i}|^{2},
\tilde{\Gamma}_{a}\ast\tilde{\Gamma}_{b}\equiv({\frac{\delta\tilde{\Gamma}_{a}}{\deltaA}},{\frac{\delta\tilde{\Gamma}_{b}}{\deltaK}})+({\frac{\delta\tilde{\Gamma}_{a}}{\deltac}},{\frac{\delta\tilde{\Gamma}_{b}}{\deltaL}})+{\frac{\delta\tilde{\Gamma}_{a}}{\deltaV}}\cdot{\frac{\delta\tilde{\Gamma}_{b}}{\deltaM}}\,
\left(\begin{matrix}{\tilde{\mathcal{F}}^{+}}\\{\tilde{G}_{+}}\\\end{matrix}\right)=\mathcal{S}\left(\begin{matrix}{\mathcal{F}^{+}}\\{G_{+}}\\\end{matrix}\right)=\left(\begin{matrix}{A}&{B}\\{C}&{D}\\\end{matrix}\right)\left(\begin{matrix}{\mathcal{F}^{+}}\\{G_{+}}\\\end{matrix}\right)\,.
\prod_{n=1\atopn\equiv1-j(\mathrm{mod}2)}^{\infty}(1+q^{n})=\displaystyle\sum_{r=0}^{\infty}\frac{q^{r^{2}+jr}}{(q)_{r}(-q)_{r}}=\displaystyle\sum_{r=0}^{\infty}\frac{q^{r^{2}+jr}}{(q^{2};q^{2})_{r}},{}~~~~j=0,1.
\langle\deltaS\rangle_{f}=|\phi_{lm}|^{2}\frac{\lambda_{j}}{4!}\left[\frac{4(4j+1)}{2j(2j+1)+\mu_{j}^{2}}+(-1)^{l}\frac{2(2j+1)(4j+1)}{2j(2j+1)+\mu_{j}^{2}}\left\{\begin{array}{ccc}{j}&{j}&{l}\\{j}&{j}&{2j}\\\end{array}\right\}\right].
\begin{array}{l}{Z=\intD_{\hat{g}}X\e^{-S}\\,}\\{S=\frac{1}{4\pi\alpha^{\prime}}\int_{M}d^{2}z\sqrt{\hat{g}}\left(\hat{g}^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}G_{\mu\nu}+\alpha^{\prime}\hat{R}\phi+\alpha^{\prime}T\right)\.}\\\end{array}
\quad+{\frac{t_{2}(t_{1}^{2}-1)}{(t_{1}-t_{2})^{2}}}(\delta(\epsilon,t_{1})-\delta(\epsilon,t_{2}))+{\frac{t_{1}(1-t_{2}^{2})}{t_{1}-t_{2}}}~{\frac{\partial}{\partialt_{2}}}\delta(\epsilon,t_{2})\Big)g.
\begin{matrix}{\mathrm{ev}_{i}:}&{\overline{{\mathcal{M}}}_{g,n}(Y,\beta)}&{\longrightarrow}&{Y}\\{}&{(\Sigma_{g},p_{1},\ldots,p_{n},\varphi)}&{\longmapsto}&{\varphi(p_{i})\,,}\\\end{matrix}
A_{2}^{a}\equiv\alpha_{R_{1}}^{\left(2\right)-1a}-\left(\Gamma_{\LambdaR_{1}}^{\left(0\right)1},\Gamma_{\LambdaR_{1}}^{\left(2\right)-1}\right)^{a}+\left(\Gamma_{\LambdaR_{1}}^{\left(0\right)2},\Gamma_{\LambdaR_{1}}^{\left(2\right)-2}\right)^{a}
\int\frac{d^{d}{\bfk}}{({\bfk}^{2}+a_{i}^{2})}=\pi^{\frac{d}{2}}\frac{\Gamma(s-\frac{d}{2})}{\Gamma(s)}\frac{1}{(a_{i}^{2})^{s-\frac{d}{2}}}
S={\frac{1}{2\kappa^{2}}}\intd^{10}x{\sqrtg}[{\frac{1}{2}}g^{MN}\partial_{M}\phi\partial_{N}\phi]={\frac{V}{4\kappa^{2}}}\intd^{4}x\int^{\rho}drr^{5}[(\partial_{r}\phi)^{2}+h(r)\eta^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi],
\mathrm{}^{*}{\calL}(\mathrm{}^{*}A,\mathrm{}^{*}B)=\frac1{12m^{2}}H_{\alpha\beta\lambda}H^{\alpha\beta\lambda}-\frac18F_{\alpha\beta}F^{\alpha\beta}+\frac12\epsilon^{\alpha\beta\lambda\rho}\partial^{\sigma}F_{\sigma\rho}H_{\alpha\beta\lambda}
S_{\mathrm{topol}}={\frac{8\pi^{2}}{g_{\mathrm{str}}^{2}}}={\frac{\pi}{4}}\alpha^{\prime}M_{\mathrm{P}}^{2}={\frac{\pi}{4}}M_{\mathrm{P}}^{2}l_{\mathrm{str}}^{2}=\pi\left({\frac{M_{\mathrm{P}}}{M_{\mathrm{str}}}}\right)^{2}\.
I=\int_{-{\frac{\pi}{2}}}^{\frac{\pi}{2}}d\mu\,f^{-\tilde{\gamma}}\left[(K^{\prime})^{2}+(\tilde{\gamma}^{2}-9/4)K^{2}\right]>{-{\frac{9}{4}}}\int_{-{\frac{\pi}{2}}}^{\frac{\pi}{2}}d\mu\,f^{-\tilde{\gamma}}K^{2}\,
dt^{2}\tof(r)dt^{2}\,\qquaddx_{1}^{2}+\cdots+dx_{d-1}^{2}\tof^{-1}(r)dr^{2}+r^{2}d\Omega_{d-2}^{2}\,\qquadf(r)=1-{\frac{\mu^{d-3}}{r^{d-3}}}\,
\hat{U}^{-1}(\theta_{0})|A_{\bar{z}}(z,\bar{z})>=e^{-\frac{ik}{4\pi}\int_{T_{2}}dz\wedged\bar{z}\left[\partial_{\bar{z}}\chi\partial_{z}\theta_{0}-\partial_{\bar{z}}\theta_{0}\overline{{\partial_{\bar{z}}\chi}}\,\right]}\|A_{\bar{z}}(z,\bar{z})+\partial_{\bar{z}}\theta_{0}(z,\bar{z})>\\\,
dS^{2}={H_{6}}^{-1/2}[dudv-\mu^{2}\sum_{i=1}^{4}{z^{i}}^{2}du^{2}+\sum_{i=1}^{4}dz^{i}dz^{i}+d\tilde{x}^{2}]+{H_{6}}^{1/2}(dr^{2}+r^{2}d{\Omega_{2}}^{2}),
\Omega_{p}^{red}=\int_{V}\lambda_{p}\left(\delta\Pi_{A_{2}...A_{p}}^{(1)}\wedge\deltaQ_{(1)}^{A_{2}...A_{p}}+(-1)^{p+1}\delta\Pi_{A_{2}...A_{p}}^{(2)}\wedge\deltaQ_{(2)}^{A_{2}...A_{p}}\right)\.
\frac{dF}{ds}=\{F,\\tilde{H}_{T}\}_{p,q,t,\pi_{t}}\approx\tilde{\mu}\{F,\\pi_{t}+H_{0}\}_{q,p,t,\pi_{t}}^{*}+\tilde{\mu}^{A}\left\{F,\\psi_{A}\right\}_{q,p,t,\pi_{t}},
\mathrm{G_{\mu\nu}^{ab}({\itq})=\frac{-i\delta^{ab}}{(2\pi)^{4}(q^{2}+i\varepsilon)}\left[g_{\mu\nu}-\left(\frac{n^{2}q_{\nu}-q\cdotn(q_{\mu}n_{\nu}+q_{\mu})}{-\vec{q}^{2}}\right)\right]~,~~~~~\varepsilon>0~,}