harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1}{2}(S^{-}-(S^{-})^{\dagger})=\int_{M}\left(d^{4}x\frac{i}{2}\partial_{\mu}({\tilde{\overline{\Psi}}}_{L}\gamma^{\mu}{\tilde{\Psi}}_{L})-\frac{i}{4}{\tilde{\overline{\Psi}}}_{L}\gamma^{A}{\tilde{\Psi}}_{L}e^{-1}\epsilon_{ABCD}T^{B}\wedgee^{C}\wedgee^{D}\right).
NS:~~~~~~~~~~L_{m}=\frac{1}{2}\sum_{-\infty}^{\infty}:\alpha_{-n}\alpha_{m+n}:+\frac{1}{2}\sum_{-\infty}^{\infty}(r+\frac{1}{2}m):(b_{-r}\cdotb_{m+r}-b_{-r}^{\prime}\cdotb_{m+r}^{\prime}):
\langlez|z^{\prime}\rangle=e^{-(|z|^{2}+|z^{\prime}|^{2})/2}\sum_{J1=0}^{\infty}\frac{(z^{*}z^{\prime})^{J1}}{J1!}=e^{-(|z|^{2}+|z^{\prime}|^{2})/2}e^{z^{*}z^{\prime}}.
\left[a_{s}^{*-}(\mathbf{p}),a_{r}^{+}(\mathbf{p})\right]=\left[a_{s}^{-}(\mathbf{p}),a_{r}^{*+}(\mathbf{p})\right]=\varepsilon_{s}\delta_{sr}\delta(\mathbf{p}-\mathbf{p}^{\prime}).
S_{\mathrm{free}}=\intd^{2}xdud^{2}\theta_{+}^{+}\;\left(iX^{+Y}\partial_{++}X_{Y}^{+}+iP_{++}^{-Y}D^{++}X_{Y}^{+}+{\frac{1}{2}}\Lambda_{+}^{i}D^{++}\Lambda_{+}^{i}\right)\;.
{\bf\Psi}_{n,k_{3}}^{T}(x,y)=\left(c_{1}\Psi_{n-1}(x,y),\;ic_{2}\Psi_{n}(x,y),\;c_{3}\Psi_{n-1}(x,y),\;ic_{4}\Psi_{n}(x,y)\right)\;.
\overline{{g}}_{\mu\nu}=\left[\begin{array}{cc}{1+l(x^{0})^{2}\overline{{\lambda}}(x^{0},x^{1})^{2}}&{l(x^{0})^{2}}\\{l(x^{0})^{2}\overline{{\lambda}}(x^{0},x^{1})^{2}}&{l(x^{0})^{2}}\\\end{array}\right].
J_{{\mu_{0}}{\mu_{1}}}^{ab}\left(x\right)=\intd\taud\sigma\sum_{i}z_{i}^{ab}\,\delta^{d}\left(x_{\mu}-X_{\mu}^{i}\left(\tau,\sigma\right)\right)\,\,\partial_{\tau}X_{[{\mu_{0}}}^{i}\left(\tau,\sigma\right)\,\partial_{\sigma}X_{{\mu_{1}}]}^{i}\left(\tau,\sigma\right)\,\,.
I=\frac{1}{2\kappa^{2}}\intd^{D}x\sqrt{-g}(R-\frac{1}{2}(\nabla\phi)^{2}-\frac{1}{2(q+1)!}e^{-\alpha\phi}F_{q+1}^{2}-\frac{1}{2(d+1)!}e^{\beta\phi}G_{d+1}^{2}),
\lbrack\frac{4}{5!(D+5)}\epsilon_{\mu\nu\lambda}\eta_{\beta\gamma}^{T}\alpha_{-1}^{\mu\nu\lambda\beta\gamma}+\epsilon_{\mu\nu\lambda}(\alpha_{-1}^{\mu}\alpha_{-2}^{\nu\lambda}-\frac{4}{3}\alpha_{-1}^{\mu\nu}\alpha_{-3}^{\lambda})]\left\vert0,k\right\rangle,
\sumq^{\frac{1}{2}p^{2}}=\sum_{n}q^{\frac{3}{2}\left(2n+\frac{m_{1}+m_{2}}{3}\right)^{2}}=\sum_{n}q^{6\left(n+\frac{m_{1}+m_{2}}{6}\right)^{2}}=\theta_{2m_{1}+2m_{2},6}\:(\tau).
\Gamma_{3T(c)}^{\mu^{\prime}\lambda^{\prime}\nu^{\prime}}=t^{\mu^{\prime}\mu}(p_{1})t^{\lambda^{\prime}\lambda}(p_{2})t^{\nu^{\prime}\nu}(p_{3})\Gamma_{3V(c)}^{\mu\lambda\nu}\qquad(c=f\,\,\mathrm{{or}}\,\,d)\,,
A=\left(\begin{array}{ccc}{2\omega_{\eta}}&{-t}&{iv}\\{-t}&{2\omega_{t}}&{-ir}\\{-iv}&{ir}&{2\omega_{b}}\\\end{array}\right)\;\;B=\left(\begin{array}{ccc}{0}&{-it}&{u}\\{-it}&{0}&{s}\\{u}&{s}&{0}\\\end{array}\right)\;,
i\left<A_{\mu}^{3}A_{\nu}^{3}A_{\rho}^{a}A_{\sigma}^{b}\right>_{\mathrm{bare}}=-ig^{2}\delta^{ab}\left[2g_{\mu\nu}g_{\rho\sigma}-\left(1-\frac1\alpha\right)(g_{\mu\rho}g_{\nu\sigma}+g_{\mu\sigma}g_{\nu\rho})\right].
M\stackrel{d_{0}}{\longrightarrow}M\oplusM\stackrel{d_{1}}{\longrightarrow}M\oplusM\stackrel{d_{2}}{\longrightarrow}M\stackrel{d_{3}}{\longrightarrow}M\stackrel{d_{0}}{\longrightarrow}M\oplusM\stackrel{d_{1}}{\longrightarrow}\cdots
R^{pc}=3q^{2}[2\epsilon_{\lambda\sigma}^{\gamma\delta}\epsilon_{\omega\nu}^{\sigma\eta}H_{\gamma\delta}^{\nu}H_{\eta}^{\lambda\omega}+\epsilon_{\nu\lambda}^{\rho\eta}\epsilon_{\alpha\beta}^{\lambda\nu}H_{\sigma}^{\alpha\beta}H_{\rho\eta}^{\sigma}]
dl^{2}=2d\phi^{2}+\frac{1}{2}e^{4\phi}d\kappa^{2}+\frac{1}{2f^{2}}\left\{df^{2}+(d\chi+vdu-udv)^{2}\right\}-\frac{1}{f}\left\{e^{-2\phi}dv^{2}+e^{2\phi}(du-\kappadv)^{2}\right\}.
\partial_{p^{+}}D\phi=U\Bigl(-p^{+}\partial_{p^{+}}^{2}-\frac{1}{2}\partial_{p^{+}}+\frac{1}{4p^{+}}(x_{I}^{2}\partial_{J}^{2}-\frac{1}{2}l_{IJ}^{2}+\frac{(d-3)(d-5)}{4})\Bigr)\tilde{\phi}.
(M_{12}^{\beta\gamma},M_{21}^{\beta\gamma})\left(1-T^{\beta\gamma}\left(\begin{matrix}{M_{11}^{\beta\gamma}}&{M_{12}^{\beta\gamma}}\\{M_{21}^{\beta\gamma}}&{M_{22}^{\beta\gamma}}\\\end{matrix}\right)\right)^{-1}\left(\begin{matrix}{T^{\beta\gamma}M_{21}^{\beta\gamma}}\\{T^{\beta\gamma}M_{12}^{\beta\gamma}}\\\end{matrix}\right)+M_{11}^{\beta\gamma}=T^{\beta\gamma}.
\mu_{0}-\mu_{\theta}=\int_{0}^{\infty}drM(r)^{4}\mathcal{L}(r)^{2}\left(\epsilon_{\theta}-\epsilon_{0}\right)=\int_{0}^{\infty}drM(r)^{4}\mathcal{L}(r)^{2}\left\{\frac{K^{\prime}(r)^{2}}{\mathcal{L}(r)^{2}}+\frac{\left[1-K(r)^{2}\right]^{2}}{\mathcal{L}(r)^{4}}+\frac{J(r)^{2}K(r)^{2}}{\mathcal{L}(r)^{2}}\right\}\,.
ds_{t\to\infty}^{2}\sim\mathrm{e}^{-\frac{2pf_{1}}{q-1}}\mathrm{e}^{\frac{2p}{q-1}(\alpha+\frac{ac_{1}}{\chi})t}\left(-2^{\frac{2k}{k-1}}\mathrm{e}^{-kt}dt^{2}+2^{\frac2{k-1}}\mathrm{e}^{-t}d\Sigma_{k,+1}^{2}+d\vec{y}_{q-k}^{2}\right)+\mathrm{e}^{2f_{1}}\mathrm{e}^{-2(\alpha+\frac{ac_{1}}{\chi})t}d\vec{x}_{p}^{2}.
{ds^{(4)}}^{2}=-(\alpha^{2}r^{2}-8M)dt^{2}-8Jdtd\varphi+\frac{dr^{2}}{\alpha^{2}r^{2}-8M+\frac{16J^{2}}{r^{2}}}+r^{2}d\varphi^{2}+dz^{2}.
\begin{array}{rcl}{}&{}&{{^mC_{(i)}}Z_{m}^{n}~,}\\{}&{}&{{^mC_{(i)}}\equiv{^mC_{i_{1}i_{2}\cdotsi_{m}}}\equivC_{1}^{i_{1}}C_{3}^{i_{2}}\cdotsC_{2m-1}^{i_{m}}~,}\\{}&{}&{1\leqi_{1}<i_{2}<\cdots<i_{m}\leqn~,~~i_{k}\geq2k-1~.}\\\end{array}
H_{\mu\nu\rho}=\partial_{[\mu}B_{\nu\rho]}-m^{a}L_{ab}A_{[\mu}^{(2)}A_{\nu}V_{\rho]}^{b}+{\frac{1}{2}}\eta_{IJ}{\calA}_{[\mu}^{I}{\calF}_{\nu\rho]}({\calA}^{J})\,,
S_{\Phi}=\frac{1}{2}\intd^{4}x\,\int_{-\pi}^{\pi}d\phi\sqrt{-G}\left\{G^{AB}\partial_{A}\Phi\partial_{B}\Phi-\left(m^{2}+\frac{2\alphak}{R}(\delta(\phi)-\delta(\phi-\pi))\right)\Phi^{2}\right\}.
ds_{adS*S}^{2}=ds_{adS}^{2}+ds_{S}^{2}=\left({\frac{r}{R}}\right)^{2\omega}dx^{\mu}\eta_{\mu\nu}dx^{\nu}+\left({\frac{R}{r}}\right)^{2}dr^{2}+R^{2}d^{2}\Omega
\left(\left(\overline{{{\calA}}}^{\{a}\circ\Gamma_{\LambdaR_{p-1}}^{\left(p\right)}\right),\Gamma_{\LambdaR_{p-1}}^{\left(0\right)}\right)^{b\}}+\widetilde{V}^{\{a}\left(\overline{{{\calA}}}^{b\}}\circ\Gamma_{\LambdaR_{p-1}}^{\left(p\right)}\right)=-\sum_{q=1}^{p-1}\left(\overline{{{\calA}}}^{\{a}\circ\Gamma_{R_{p-1}}^{\left(p\right)},\Gamma_{R_{p-1}}^{\left(p-q\right)}\right)^{b\}}
\langleW^{(L)}\rangle=\intd\mu_{C}[A]e^{ie(A_{\mu},{j_{\mu}^{W}}^{(L)})}=e^{-\frac{e^{2}}{2}({j_{\mu}^{W}}^{(L)},C_{\mu\nu}{j_{\nu}^{W}}^{(L)})}\;.
{\calL}=\mathrm{Tr}\left(\frac{1}{2R}(D_{0}X^{a})^{2}+\frac{R}{4}[X^{a},X^{b}]^{2}+\overline{{\Psi}}D_{0}\Psi+iR\overline{{\Psi}}\Gamma_{a}[X^{a},\Psi]\right)
ds^{2}=-\frac{1-\left(\frac{r_{H}}{r}\right)^{7-p}}{1+\frac{\alpha}{r_{H}^{7-p}}}dw^{2}+\frac{r_{H}^{7-p}}{\alpha}dy^{2}+\frac{dr^{2}}{1-\left(\frac{r_{H}}{r}\right)^{7-p}}+r^{2}d\Omega_{8-p}^{2}+dy^{s}dy_{s}.
{\calJ}_{\mu\nu\rho\sigma}^{(4)}=\sum_{\mathrm{symm}}\left[F_{\rho\alpha}^{+}\overleftrightarrow{\partial_{\mu}}\overleftrightarrow{\partial_{\nu}}F_{\alpha\sigma}^{-}-\frac{1}{7}\pi_{\mu\nu}\left(F_{\rho\alpha}^{+}F_{\alpha\sigma}^{-}\right)\right],
u=\frac{1}{2}\rho_{0}+\epsilon_{1}\frac{\sqrt{2q_{1}^{4}+(q_{1}^{2}+q_{2}^{2})\rho_{0}^{2}-\epsilon_{2}2q_{1}^{2}\Lambda}}{2\sqrt{2q_{1}^{2}+q_{2}^{2}}}\,,
f^{\alpha\beta}=0\;\qquad\bar{h}_{A}^{\alpha}=-i\,e_{A}^{\alpha}\qquad\Rightarrow\{\calV}^{\prime}=\left(\begin{matrix}{e_{\alpha}^{A}}&{h_{\alphaA}}\\{0}&{-i\,e_{A}^{\alpha}}\\\end{matrix}\right)\.
{\frac{1}{2}}[\Gamma_{a}\Gamma^{11}e^{-C}\partial_{11}A+\Gamma_{a}\Gamma^{-B}\partial_{\mu}A]\eta+{\frac{\sqrt{2}}{12}}[\Gamma_{a}\Gamma^{5}e^{-6A-C}\partial_{11}D+\Gamma_{a}\Gamma^{\mu}\Gamma^{5}\Gamma^{11}e^{-6A-B}\partial_{\mu}D]\eta=0,
(\bar{3},3):\quadL^{(e)}=\left(\begin{array}{ccc}{\hat{N}_{R}}&{\hat{\tau}_{R}}&{\hat{e}_{R}}\\{\hat{\tau}_{L}}&{\hat{\nu}_{L}}&{\beta_{L}^{e}}\\{e_{L}}&{\nu_{L}^{e}}&{\alpha_{L}^{e}}\\\end{array}\right)
d\left[K^{\left(2q+2\right)}\left(S\right)-C^{\left(2q-1\right)}K^{\left(3\right)}\left(\Gamma_{11}\right)\right]=dK^{\left(2q+2\right)}\left(S\right)+K^{\left(2q\right)}\left(S\Gamma_{11}\right)K^{\left(3\right)}\left(\Gamma_{11}\right)\quad,
\begin{array}{rll}{{\calE}(A,B)=\int_{M_{3}}<(H^{A}\otimesI_{E}-I_{E}\otimesK^{B})}&{\wedge}&{(D^{A}\Phi_{A}\otimesI_{E}-I_{E}\otimesD^{B}\Phi_{B})>}\\{-\int_{M_{3}}<(D^{A}\Phi_{A}\otimesI_{E})}&{\wedge}&{(H^{A}\otimesI_{E})>}\\\end{array}
H_{n,k}^{r_{1},\dots,r_{n}}(z_{1},\dots,z_{n};\mu)=\prod_{i=1}^{n}\left[\frac{(-1)^{r_{i}-1}}{(r_{i}-1)!}\frac{\partial^{r_{i}-1}}{\partial(z_{i}^{2})^{r_{i}-1}}\right]H_{n,k}(z_{1},\dots,z_{n};\mu)\,.
ds^{2}=\frac{1}{2}M_{ij}d{\bfx}_{i}\cdotd{\bfx}_{j}+\frac{g^{4}}{2(4\pi)^{2}}(M^{-1})_{ij}(d\xi_{i}+{\bfW}_{ik}\cdotd{\bfx}_{k})(d\xi_{j}+{\bfW}_{jl}\cdotd{\bfx}_{l})
{\calD}\left(\begin{array}{c}{\delta\Psi_{1}}\\{\vdots}\\{\delta\Psi_{M}}\\{{[\sum_{I}q_{1}^{I}\deltaa_{1}^{I}]+i\sigma_{1}[\sum_{I}q_{1}^{I}\deltaa_{2}^{I}]}}\\{\vdots}\\{{[\sum_{I}q_{M}^{I}\deltaa_{1}^{I}]+i\sigma_{M}[\sum_{I}q_{M}^{I}\deltaa_{2}^{I}]}}\\\end{array}\right)=0\,,
j^{\mu}=\varepsilon^{\nu}(x)T_{\nu}^{\mu}(x)+(\partial\cdot\varepsilon)K^{\mu}+\partial_{\nu}(\partial\cdot\varepsilon)L^{\mu\nu}+(\partial_{\nu}\varepsilon_{\rho}-\partial_{\rho}\varepsilon_{\nu})M^{[\nu\rho]\mu},
\int_{0}^{1}d\lambda\,s(\lambda)=\int_{x_{0}^{+}}^{x_{1}^{+}}d\tilde{x}^{+}s_{+}(\tilde{x}^{+})=\int_{x_{1}^{+}}^{x_{0}^{+}}d\tilde{x}^{+}\left(-s_{+}(\tilde{x}^{+})\right)\leq\frac{1}{4}\left(A_{\mathrm{cl}}(0)-A_{\mathrm{cl}}(1)\right)\,.
{\bf\Omega}_{{\bfx},{\bfx^{\prime}}}=m^{2}\delta_{{\bfx},{\bfx^{\prime}}}-{\frac{1}{\sigma^{2}}}\sum_{i=1}^{3}\left(\delta_{{\bfx}+\hat{\sigma}\left(i\right),{\bfx^{\prime}}}-2\delta_{{\bfx},{\bfx^{\prime}}}+\delta_{{\bfx}-\hat{\sigma}\left(i\right),{\bfx^{\prime}}}\right).
S_{a}={\textstyle\frac{1}{16\piG_{N}^{(d)}}}\int{d^{d}x\sqrt{|g|}\left[R+{\textstyle\frac{1}{2}}(\partial\varphi)^{2}+\textstyle{\frac{(-1)^{p+1}}{2(p+2)!}}e^{-2a\varphi}F_{(p+2)}^{2}\right]}\,,
{\calL^{\prime}}_{h}^{M^{\prime}}(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^{\dagger}e^{2V_{h}^{i}}\phi
\Bigl(X_{AB}^{ab}(x,y)\Bigr)=\left(\begin{array}{cc}{0}&{-\,m^{2}\,\delta^{ab}}\\{\delta^{ab}}&{\frac{1}{2}\,g\,f^{abc}\,(D_{i}\pi^{i})^{c}}\\\end{array}\right)\,\delta(x-y)\,.
\bar{\psi}_{\pm}=\psi^{\dagger}\Gamma^{\pm}\Gamma^{\mp}\Gamma^{0}=\pm\frac{1}{\sqrt{2}}\psi^{\dagger}\Gamma^{\pm}\Gamma^{\mp}(\Gamma^{\pm}-\Gamma^{\mp})=\pm\frac{1}{\sqrt{2}}\psi^{\dagger}(\Gamma^{\pm}-\Gamma^{\mp})\Gamma^{\mp}\Gamma^{\pm}=\bar{\psi}\Gamma^{\mp}\Gamma^{\pm},
\begin{array}{llllll}{}&{J_{0}\,\delta_{u}H_{2}}&{=}&{J_{1}\,\delta_{u}H_{1}}&{=}&{{\calR}\,J_{0}\,\delta_{u}H_{1}}\\{=}&{\delta_{\phi}H_{2}}&{}&{}&{=}&{{\calR}\,\delta_{\phi}H_{1}={\calR}\,\phi_{tx}}\\\end{array}
\xi_{0}=\xi_{0}^{\mu}\partial_{\mu}+\xi_{0}^{\alpha}D_{\alpha}-\Lambda^{i}{}_{j}\phi^{j}\partial_{i}-\Big((\alpha+\phi^{j}\Gamma_{j}D\alpha+D\xi)\psi\Big)^{\alpha^{\prime}}\partial_{\alpha^{\prime}},
W_{H}=\Lambda_{\alpha\beta\gamma\;\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\;\alpha^{\prime\prime}\beta^{\prime\prime}\gamma^{\prime\prime}}Tr(M_{\alpha\beta\gamma}V_{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}}V_{\alpha^{\prime\prime}\beta^{\prime\prime}\gamma^{\prime\prime}})\.
P^{+}={\frac{\pi}{L}}\Bigl(\sum_{n=1}^{\infty}\bigl(n\a_{n}^{\dagger}a_{n}+|n-z|d_{n}^{\dagger}d_{n}+|n+z|b_{n}^{\dagger}b_{n}\bigr)+|z|b_{0}^{\dagger}b_{0}-{\frac{1}{2}}(z-{\frac{1}{2}})^{2}-{\frac{3L^{2}}{2\pi^{2}\varepsilon}}\Bigr)\.
\psi_{\alphaA}=\sum_{k\geq1}\bigl(\psi_{+A}^{(s,k)}\otimesD_{(\alpha)}\psi_{(s,k)}+\psi_{-A}^{(s,k)}\otimesD_{(\alpha)}(\rho_{5}\psi_{(s,k)})\bigr)+\bigl(\chi_{+A}^{(s,0)}\otimes\rho_{\alpha}\psi_{(s,0)}+\epsilon_{AB}\chi_{+B}^{(s,0)}\otimes\rho_{\alpha}\rho_{5}\psi_{(s,0)}\bigr)
^{in}\left\langlep_{1}^{\prime},p_{2}^{\prime}....p_{m}^{\prime}\left|S-1\right|p_{1},p_{2}....p_{n}\right\rangle^{in}=\delta(\sum_{i=1}^{m}p_{i}^{\prime}-\sum_{k=1}^{n}p_{k})T(p_{1}^{\prime},...p_{m}^{\prime},p_{1},...p_{n})
\langle2\colonp_{f},\sigma_{f},k_{f},\lambda_{f}\vert1\colonp_{i},\sigma_{i}\rangle=\delta^{(3)}(p_{i}-p_{f})\delta_{\sigma_{i}\sigma_{f}}f(k_{f},\lambda_{f}\colonp_{f}){\frac{e}{(2\pi)^{3/2}}}
\begin{array}{l}{e^{-i\chi_{s}}[-\partial_{r}+s(n-\Phi^{(0)}+\Upsilon)r^{-1}]f_{n}(r)=(E+m)g_{n}(r),}\\{e^{i\chi_{s}}[\partial_{r}+s(n-\Phi^{(0)}+\Upsilon+s)r^{-1}]g_{n}(r)=(E-m)f_{n}(r).}\\\end{array}
S^{\prime}=v\theta\left[\frac{1}{4}\left\{y\left.\frac{\delta^{2}V(y)}{\deltay^{2}}\right|_{y=y_{c}}\right\}^{2}-i\chi\psi\left.\frac{\delta^{2}V(y)}{\deltay^{2}}\right|_{y=y_{c}}\right].
M^{\mu\nu}=\sum_{\sigma=1}^{Q}\left\{x^{\mu\left(\sigma\right)}p^{\nu\left(\sigma\right)}-x^{\nu\left(\sigma\right)}p^{\mu\left(\sigma\right)}-\imath\sum_{n=1}^{\infty}\frac{1}{n}\left(\alpha_{-n}^{\mu\left(\sigma\right)}\alpha_{n}^{\nu\left(\sigma\right)}-\alpha_{-n}^{\nu\left(\sigma\right)}\alpha_{n}^{\mu\left(\sigma\right)}\right)\right\}\tag{41}
\hat{V}_{\mu\nu\rho}^{(2\,x)\,ab}=\hat{V}_{\mu\nu\rho}^{(2\,y)\,ab}=-\frac{1}{4}\,Y_{L}^{(e)}\,\mathrm{Tr}\,\tau^{a}\tau^{b}\,V_{\mu\nu\rho}^{(2)}\,.
\times\prod_{I=2}^{N}\;\Bigg\langle\prod_{b=1}^{N}\;\prod_{j=1}^{n_{b}}:e^{-i2\sqrt{\pi}U_{Ib}\Phi^{(I)}(x_{j}^{(b)})}:_{M^{(I)}}\;\prod_{j=1}^{m_{b}}:e^{+i2\sqrt{\pi}U_{Ib}\Phi^{(I)}(y_{j}^{(b)})}:_{M^{(I)}}\Bigg\rangle_{K^{(I)}}
S=\int_{a}^{b}d\tau\left(\,{\frac{1}{2}}\,g_{\mu\nu}(x)\,\dot{x}^{\mu}\,\dot{x}^{\nu}\,+\,{\frac{i}{2}}\,g_{\mu\nu}(x)\,\psi^{\mu}\,{\frac{D\psi^{\nu}}{D\tau}}\right).
\hat{E}^{A}{}_{M}=\left(\begin{array}{cc}{\begin{matrix}{e^{0}{}_{t}}&{0}\\{0}&{e^{i}{}_{\underline{i}}}\\\end{matrix}}&{0}\\{E^{a}{}_{{\beta}}A^{(1){\beta}}{}_{\underline{i}}}&{E^{a}{}_{\alpha}}\\\end{array}\right)
u^{\mu}(p_{3},0)\mid_{m\rightarrow0}=\left(\begin{matrix}{p_{3}}\\{0}\\{0}\\{{\frac{p_{3}^{2}}{E_{p}}}}\\\end{matrix}\right)\rightarrow\left(\begin{matrix}{\pmE_{p}}\\{0}\\{0}\\{E_{p}}\\\end{matrix}\right)\quad,
ds^{2}=dX^{M}\,{\calG}_{MN}\,dX^{N}=\left({\frac{r}{R}}\right)^{\frac{2}{w}}d{x}^{m}\eta_{mn}dx^{n}+\left({\frac{R}{r}}\right)^{2}[dr^{2}+r^{2}d\Omega^{2}]\,.
\left[a_{ij}(p^{+},n_{\perp}),a_{lk}^{\dagger}(q^{+},m_{\perp})\right]=\left\{b_{ij}(p^{+},n_{\perp}),b_{lk}^{\dagger}(q^{+},m_{\perp})\right\}=\delta(p^{+}-q^{+})\delta_{n_{\perp},m_{\perp}}\delta_{il}\delta_{jk}\,.
E_{Q}-E={\frac{1}{2}}\intdx^{1}\left[{\frac{e^{2}}{\pi+ae^{2}}}\chi^{2}+({\frac{e^{2}}{\pi+ae^{2}}})^{2}\chi(\partial_{1}^{2}-{\frac{e^{2}}{\pi+ae^{2}}})^{-1}\chi\right].
\frac{p_{r}^{2}}{g_{rr}}+\frac{p_{\theta}^{2}}{g_{\theta\theta}}=\frac{p_{\phi}-\left(\frac{g_{\phi\phi}E^{2}}{\calD}+V\right)/\left(\frac{2g_{t\phi}}{\calD}E\right)}{\frac{-{\calD}}{2g_{t\phi}E}},
\mathcal{S}_{quad}=tr\intd^{4}x\left(\frac\alpha4\omega^{2}B^{\mu\nu}B_{\mu\nu}+\frac\beta2\omegaB^{\mu\nu}\varepsilon_{\mu}\xi_{\nu}+\frac\lambda2\varepsilon^{\mu}\varepsilon^{\nu}\xi_{\mu}\xi_{\nu}+\frac\gamma2\varepsilon^{2}\xi^{2}\right)\;,
\xi^{\tau}=4i(\epsilon_{(1)}^{1}\epsilon_{(2)}^{1}+\epsilon_{(1)}^{2}\epsilon_{(2)}^{2})~,~~~\xi^{\sigma}=4i(p^{+})^{-1}(\epsilon_{(1)}^{1}\epsilon_{(2)}^{1}-\epsilon_{(1)}^{2}\epsilon_{(2)}^{2})~.
0\longrightarrow\mathcal{O}_{X}\longrightarrow\mathcal{E}xt_{\mathcal{O}_{X}}^{1}(\mathcal{O}_{X}\toL,\mathcal{O}_{X})\longrightarrow\mathcal{E}xt_{\mathcal{O}_{X}}^{1}(L,\mathcal{O}_{X})\longrightarrow0,
\partial_{\bar{z}}\partial_{z}\zeta_{2}(z,\bar{z})+\partial_{z}\zeta_{2}(z,\bar{z})\partial_{\bar{z}}\zeta_{2}(z,\bar{z})+\frac{1}{2}\left[\partial_{\varphi}^{2}V\left(\varphi\!=\!\varphi_{0}(z,\bar{z})\right)\right]=0\,.
\ddot{\triangleA(t)}=-4A_{0}\Bigl[2X_{0}Z_{0}\bigl\{\frac{\triangleZ(t)}{Z_{0}}-\frac{\triangleX(t)}{X_{0}}\bigr\}+4Y_{0}^{2}\bigl\{\frac{\triangleX(t)}{X_{0}}-\frac{\triangleY(t)}{Y_{0}}\bigr\}+Y_{0}\bigl\{\frac{\dot{\triangleX(t)}}{X_{0}}-\frac{\dot{\triangleY(t)}}{Y_{0}}\bigr\}\Bigr].
{\frac{\ddot{a}}{a}}-{\frac{\dot{c}_{\mathrm{grav}}}{c_{\mathrm{grav}}}}{\frac{\dot{a}}{a}}=-{\frac{4\piG}{3}}\left(\rho+3{\frac{p}{c_{\mathrm{grav}}^{2}}}\right)+{\frac{c_{\mathrm{grav}}^{2}}{3}}\Lambda,
S_{N}[\Phi]\=\N\,\mathrm{tr}\left({\frac{\Phi^{2}}{2}}(1-b)-g{\frac{\Phi^{4}}{4}}\right)\,+\,{\frac{N^{2}}{2}}\,{\frac{b^{2}}{x}}\,+\,N^{2}\,\rho
\deltab=-2\delta\bar{\theta}\Gamma_{11}\Gamma_{m}d\theta\Pi^{m}+d\Big(-\delta\bar{\theta}\Gamma_{11}\Gamma_{m}\theta\Pi^{m}+{\frac{1}{2}}\delta\bar{\theta}\Gamma_{11}\Gamma_{m}\theta\bar{\theta}\Gamma^{m}d\theta-{\frac{1}{2}}\delta\bar{\theta}\Gamma^{m}\theta\bar{\theta}\Gamma_{11}\Gamma_{m}d\theta\Big).
S=\intd^{4}x\sqrt{|g|}\left[R+2(\partial\phi)^{2}+e^{4\phi}(\partiala)^{2}-e^{-2\phi}\sum_{i=1}^{6}(F^{i})^{2}-a\sum_{i=1}^{6}F^{i}{}^{\star}F^{i}\right]\,,
T_{\mu\nu}^{\,\,\,\,\,\,\,\lambda}=2\Gamma_{[\mu\nu]}^{\,\,\,\,\,\,\,\,\,\,\lambda}=\Gamma_{\mu\nu}^{\,\,\,\,\,\,\,\lambda}-\Gamma_{\nu\mu}^{\,\,\,\,\,\,\,\lambda}.
\begin{array}{ccc}{\itS}\\{\downarrow}\\{{\itS}^{*}}\\\end{array}\begin{array}{ccc}{}&{\longrightarrow}&{}\\{}&{}&{}\\{}&{\longrightarrow}&{}\\\end{array}\begin{array}{ccc}{{\bar{\itS}}}\\{\downarrow}\\{{\bar{\itS}}^{*}}\\\end{array}\begin{array}{ccc}{}&{}&{}\\{}&{}&{}\\{.}\\\end{array}
N_{p_{D},n,r}=\left.\int_{-\infty}^{+\infty}{}^{-}{\calD}_{nn}M^{\ast}(p_{D},p_{-})M(p_{D}^{\prime},p_{-})\frac{dp_{-}}{2\piqE}\right|_{p_{D}=p_{D}^{\prime}}\;.
\int\prod_{a=1}^{N}\prod_{j=1}^{n_{a}}d^{D}z_{j}^{(a)}\bigg|\prod_{b<c}^{N}\;f^{(b,c)}\bigg|\;\leq\;\prod_{a<b}^{N}\Bigg[\int\;\prod_{i=1}^{n_{a}}d^{D}z_{i}^{(a)}\prod_{j=1}^{n_{b}}d^{D}z_{j}^{(b)}\;\bigg|f^{(a,b)}\bigg|^{N-1}\Bigg]^{\frac{1}{N-1}}\;.
S_{c}=\int_{\tau_{1}}^{\tau_{2}}\mathrm{d}\tau\int_{0}^{\pi}\mathrm{d}\sigma\left(\frac{\partial{\cal{L}}}{\partial\dot{x}^{\alpha}}\frac{\mathrm{D}}{\mathrm{d}\tau}r^{\alpha}+\frac{\partial{\cal{L}}}{\partialx^{\alpha}}\frac{\mathrm{D}}{\mathrm{d}\sigma}r^{\alpha}\right).
\begin{array}{l}{R(\theta-\theta^{\prime})[K^{-}(\theta)\otimes1]R(\theta+\theta^{\prime})[K^{-}(\theta^{\prime})\otimes1]=}\\{{[}K^{-}(\theta^{\prime})\otimes1]R(\theta+\theta^{\prime})[K^{-}(\theta)\otimes1]R(\theta-\theta^{\prime})\;\;,}\\\end{array}
L=\frac{1}{2}h^{\mu\nu}\Lambda_{\mu\nu,\rho\sigma}h^{\rho\sigma}-\frac{m^{2}}{2}\left(h^{\mu\nu}h_{\mu\nu}-h^{2}\right)+b^{\mu}\left(\partial^{\nu}h_{\mu\nu}-\frac{1}{2}\partial_{\mu}h+\frac{\alpha}{2}b_{\mu}\right).
\simeq\frac12\int_{0}^{\infty}\frac{\mathrm{d}t}{t}\mathrm{tr}\;\left(-t\right)^{2}\int_{0}^{1}\mathrm{d}u\mathrm{e}^{-\left(1-u\right)p^{2}t}F\mathrm{e}^{-up^{2}t}F.
\sqrt{\,\left\{\,X^{\mu}\,X^{\nu}\,\right\}_{\mathrm{PB}}\left\{\,X_{\mu}\,X_{\nu}\,\right\}_{\mathrm{PB}}\,}\longleftarrow\quad(?)\quad\longrightarrow\sqrt{\,\mathrm{Tr}\mathbf{F}_{\mu\nu}\,\mathbf{F}^{\mu\nu}\,}
\hat{H}=-\left({\frac{\hbar^{2}}{2m}}+\frac{1}{8}k\theta^{2}\right)\partial_{\xi_{1}}^{2}-\left({\frac{\hbar^{2}}{2m}}+\frac{1}{8}k\theta^{2}\right)\partial_{\xi_{2}}^{2}+{\frac{1}{2}}k\thetaL+{\frac{1}{2}}k(\xi_{1}^{2}+\xi_{2}^{2}).
\sum_{k=0}^{n-2}(-1)^{n-1+k}q_{n-k}\sigma_{p-m+k}=2\!\!\!\!\sum_{k=0}^{\left[\frac{p-m+n-2}{2}\right]}(-1)^{k}\left(\begin{array}{c}{m+2k+2}\\{m}\\\end{array}\right)\sigma_{p-m+n-2-2k}\,,
G_{\beta}(\vec{x},\tau)={\frac{1}{\beta}}\sum_{n}{\frac{1}{V}}\sum_{\vec{p}}{\frac{1}{2E_{p}}}\left[\frac{e^{i\vec{p}\cdot\vec{x}-i\omega_{n}\tau}}{E_{p}-i\omega_{n}}+\frac{e^{-i\vec{p}\cdot\vec{x}+i\omega_{n}\tau}}{E_{p}-i\omega_{n}}\right]\;.
D_{\mu\nu}^{ab}(p,p_{3})=\frac{\delta^{ab}(1-\delta^{a3})}{p^{2}-p_{3}^{2}-M^{2}+i\epsilon}\left[-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{p_{3}^{2}+M^{2}}\right]
\begin{array}{lll}{\delta\omega^{ab}:}&{\epsilon_{aba_{3}...a_{2n-1}}(R^{a_{3}a_{4}}+l^{-2}e^{a_{3}}e^{a_{4}})\cdot\cdot\cdot(R^{a_{2nd-3}a_{2n-2}}+l^{-2}e^{a_{2nd-3}}e^{a_{2n-2}})T^{a_{2n-1}}}&{=0}\\{\deltae^{a}:}&{\epsilon_{aa_{2}...a_{2n-1}}(R^{a_{2}a_{3}}+l^{-2}e^{a_{2}}e^{a_{3}})\cdot\cdot\cdot(R^{a_{2nd-2}a_{2n-1}}+l^{-2}e^{a_{2nd-2}}e^{a_{2n-1}})}&{=0}\\\end{array}.
\frac{1}{2(\pi-\epsilon)i}\int_{C}ds\,\frac{e^{\rho/s}}{s}F(s)=\frac{1}{2(\pi-\epsilon)i}\left(\int_{C_{0}}ds+\int_{C_{1}}ds+\int_{C_{2}}ds\right)\frac{e^{\rho/s}}{s}F(s)=0,\nonumber
\begin{aligned}{D_{s}\widehat{\Pi}_{r}=\frac{\widehat{\Pi}_{s}^{a}}{\sqrt{W}}\partial_{a}\widehat{\Pi}_{r}=\frac{\epsilon^{ab}}{\sqrt{W}}\partial_{b}\widehat{\Pi}_{s}\partial_{a}\widehat{\Pi}_{r}=\{\widehat{\Pi}_{s},\widehat{\Pi}_{r}\}=\frac{n\epsilon_{rs}}{2}}\\\end{aligned}
\theta=\left(\begin{array}{cccc}{0}&{-ie_{1}}&{-ie_{2}}&{-ie_{3}}\\{ie_{1}}&{0}&{b_{3}}&{-b_{2}}\\{ie_{2}}&{-b_{3}}&{0}&{b_{1}}\\{ie_{3}}&{b_{2}}&{-b_{1}}&{0}\\\end{array}\right),
T_{H}={\frac{\kappa}{2\pi}}=\left.{\frac{(N^{2})^{\prime}}{4\pi}}\right|_{r=r_{+}}={\frac{r_{+}}{2\pia^{2}}}\left(1-{\frac{Gq^{2}a^{2}}{\pir_{+}^{2}}}\right)~~~.
ds_{4}^{2}=-{\frac{1}{\sqrt{-{\frac{1}{6}}H^{0}C_{abc}H^{a}H^{b}H^{c}}}}dt^{2}+\sqrt{-{\frac{1}{6}}H^{0}C_{abc}H^{a}H^{b}H^{c}}d\vec{x}^{2}.
\hat{\calH}=E_{sph}+{\frac{\hat{\pi}_{0}^{2}}{2}}+{\frac{\hat{\pi}_{-1}^{2}}{2}}-{\frac{\Omega^{2}\hat{\phi}_{-1}^{2}}{2}}+{\frac{1}{2}}\sum_{l\ge1}(\hat{\pi}_{l}^{2}+\omega_{l}^{2}\hat{\phi}_{l}^{2})
O_{k}^{I}(x)=\underset{\substack{x_{i}\rightarrowx\,(\mathrm{all}\,i)}}{\mathrm{lim}}[Tr\{:\Phi^{(i_{1}}(x_{1})\Phi^{i_{2}}(x_{2})...\Phi^{i_{k})}(x_{k}):-\mathrm{traces}\}]
0=\frac{\partial{\calL}}{\partialW_{\mu}^{\Lambda}}=2\partial_{\nu}\frac{\partial{\calL}}{\partial{\calF}_{\mu\nu}^{\Lambda}}=2\partial_{\nu}\left(\frac{\partial{\calL}}{\partial{\calF}_{\mu\nu}^{+\Lambda}}+\frac{\partial{\calL}}{\partial{\calF}_{\mu\nu}^{-\Lambda}}\right)
D_{-}^{a}\equiv\partial_{z}+\frac{i}{2}(A_{z}^{1}+\frac{1}{\sqrt{3}}A_{z}^{2}+\cdots+\sqrt{\frac{2}{a(a-1)}}A_{z}^{a-1}+\sqrt{\frac{2(a+1)}{a}}A_{z}^{a}),
G_{n(u)A}{}^{B}=-\frac{\alpha/\sqrt{\alpha^{\prime}}}{1-4\mu\alphaK}(1-2\mu\alphaK(1-i\Omega\Pi))_{A}{}^{C}(P_{(u)nC}^{-1}{}^{B}C_{(u)}^{1/2}C^{1/2}{\bar{N}}^{u})_{n}\,.
attenzione\langleI\rangle=-\int_{D}d^{2}{\xi}\intd^{2}\theta\,\mu(\theta^{i})\,\left[\left(m^{4}+\kappaR\right)^{1/2}\sqrt{-\gamma}-{\frac{1}{2}}{\dot{x}}^{\mu\nu}\partial_{\,[\mu}S\partial_{\nu]}U\right]
\hat{S}=\intd^{D}x\sqrt{-\hat{G}}e^{-\hat{\phi}}\left(R_{\hat{G}}+\left(\hat{\partial}\hat{\phi}\right)^{2}-\frac{1}{12}\hat{H}_{\hat{\mu}\hat{\nu}\hat{\rho}}\hat{H}^{\hat{\mu}\hat{\nu}\hat{\rho}}\right),
\tilde{h}^{\mu\nu}=-h^{\mu\nu}+\frac12h_{\;\;\lambda}^{\lambda}\,\eta^{\mu\nu}-\frac12h_{\;\;\lambda}^{\lambda}\,h^{\mu\nu}+h^{\mu\lambda}\,h_{\;\;\lambda}^{\nu}+\frac18\left(h_{\;\;\lambda}^{\lambda}\right)^{2}\,\eta^{\mu\nu}-\frac14h_{\alpha\beta}h^{\alpha\beta}\,\eta^{\mu\nu}+\cdots\;\;.
S_{10}=g\int\!\!d^{3}x~f_{abc}\left(\varphi_{a}^{*}\varphi_{b}c_{c}-\partial^{\mu}\varphi_{a}\varphi_{b}A_{c\mu}\right),~S_{11}=\mu_{ab}\int\!\!d^{3}x~\left(A_{a\mu}^{*}\eta_{b}^{\mu}-c_{a}^{*}\rho_{b}\right),
{\bfJ}^{\prime}\frac{\partial}{\partialx_{1}}=\frac{\partial}{\partialp_{m}},\qquad{\bfJ}^{\prime}\frac{\partial}{\partialx_{m}}=\frac{\partial}{\partialp_{1}},\qquad{\bfJ}^{\prime}\frac{\partial}{\partialp_{1}}=-\frac{\partial}{\partialx_{m}},\qquad{\bfJ}^{\prime}\frac{\partial}{\partialp_{m}}=-\frac{\partial}{\partialx_{1}},\qquad
\delta_{\epsilon}A_{(p)}^{\mu_{1}\ldots\mu_{p}}=\partial_{}^{\left[\mu_{1}\right.}\epsilon_{(p)}^{\left.\mu_{2}\ldots\mu_{p}\right]}=Z_{\;\;\nu_{1}\ldots\nu_{p-1}}^{\mu_{1}\ldots\mu_{p}}\epsilon_{(p)}^{\nu_{1}\ldots\nu_{p-1}},
\bar{S}\,[J]=\int\,d^{4}x\,\Bigl[-\frac{1}{4}\,F_{\mu\nu}\,\Bigl(1+\frac{m^{2}}{\Box}\Bigr)\,F^{\mu\nu}-\partial_{\mu}\bar{c}\,\partial^{\mu}c+\frac{1}{2\alpha}\,\Bigl(\partial^{\mu}A_{\mu}\Bigr)^{2}+J_{\mu}A^{\mu}\Bigr]\,,
L_{q}=\frac{1}{2e}\left(\dot{x}_{\mu}-\frac{i}{2}v\varepsilon_{\mu\nu\lambda}\xi_{a}^{\nu}\xi_{a}^{\lambda}\right)^{2}-\frac{1}{2}em^{2}-iqmv\xi_{1}^{\mu}\xi_{2\mu}+\frac{i}{2}\xi_{a}^{\mu}\dot{\xi}_{a\mu},
\begin{array}{c}{\psi_{A}(x)=\psi_{A}^{\left(-\right)}(x)+\psi_{A}^{\left(+\right)}(x),\quad\psi_{B}(x)=\psi_{B}^{\left(-\right)}(x)+\psi_{B}^{\left(+\right)}(x)}\\{\psi(x)=\psi_{A}^{\left(-\right)}(x)+\psi_{B}^{\left(+\right)}(x)}\\\end{array}
\begin{array}{rcccl}{G_{i\mu}}&{\to}&{G_{i\mu}^{\prime}}&{=}&{U_{i}\starG_{i\mu}\starU_{i}^{-1}+ig_{i}^{-1}U_{i}\star\partial_{\mu}U_{i}^{-1},}\\{\Phi}&{\to}&{\Phi^{\prime}}&{=}&{U_{1}\star\Phi\starU_{2}^{-1},}\\\end{array}
<\Psi_{1}|\Psi_{2}>=N\int_{p^{2}=-m^{2}}\frac{d{\bfp}}{p_{0}}\overline{{\Theta}}(p)_{1}^{A(l_{1})B(l_{2})...C(l_{r-1})a(l_{r})}\Theta(p)_{2A(l_{1})B(l_{2})...C(l_{r-1})a(l_{r})}
n_{\mathrm{eff}}^{2}(n,\eta)\,=\,n^{2}-n^{2}\frac{\vertb_{m}\vert}{\ell_{_\mathrm{C}}^{-2m}}\biggl(\frac{n}{2\pi}\biggr)^{2m}\biggl[\ell_{0}-\frac{\ell_{0}-\ell_{\mathrm{b}}}{1+(\eta/\eta_{0})^{2}}\biggr]^{-2m},\quad\frac{a^{\prime\prime}}{a}=\frac{2(\ell_{0}-\ell_{\mathrm{b}})}{\eta_{0}^{2}\ell_{\mathrm{b}}}\frac{1-3(\eta/\eta_{0})^{2}}{[1+(\eta/\eta_{0})^{2}]^{3}},