harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S_{B}=\frac{1}{2\pi\alpha^{\prime}}\intd^{2}\sigma[\partial_{+}u\partial_{-}v+\partial_{+}z^{i}\partial_{-}\bar{z}^{i}+if\partial_{+}u(z^{i}\partial_{-}\bar{z}^{i}-\bar{z}^{i}\partial_{-}z^{i})+\partial_{+}x^{A}\partial_{-}x^{A}]~,
\left(\frac{N}{De}\right)_{\mathrm{closed}}\:=\:\:\frac{\frac{c^{2}\beta_{S}}{(m_{0})^{\frac{(D+3)}{2}}\:\left(\frac{D+3}{2}\right)}\:+\:\frac{1}{\left(\frac{D+5}{2}\right)\:(m_{0})^{\frac{(D+5)}{2}}}}{\frac{c^{2}\beta_{S}}{m_{0}^{\frac{(D-1)}{2}}\left(\frac{D-1}{2}\right)}\:+\:\frac{1}{m_{0}^{\frac{(D+1)}{2}}\:\left(\frac{D+1}{2}\right)}}
\left(-\frac{1}{2\mu}\frac{d^{2}}{dx^{2}}-\frac{\mu}{2R^{2}}x^{2}\right)\psi_{n}(x)=e_{n}\psi_{n}(x);\quadE=\sum_{i=1}^{N}e_{n_{i}},\quad\Psi_{f}=\bigwedge_{i=1}^{N}\psi_{n_{i}}
{\calL}_{\mathrm{GF}}=-\frac{1}{2\alpha}\displaystyle\sum_{a=1}^{2}(\partial^{\mu}A_{\mu}^{a}+\alphaM_{W}\chi^{a})^{2}-\frac{1}{2\alpha}(\partial^{\mu}Z_{\mu}+\alphaM_{Z}\chi^{3})^{2}-\frac{1}{2\alpha}(\partial^{\mu}A_{\mu})^{2}.
A_{1}^{(i,0)}(s,t)={\frac{8g_{YM}^{4}}{M_{s}^{4}}}\int_{1/l_{0}}^{\infty}d\tau_{2}\tau_{2}\int_{\eta_{i}>0}d^{4}\eta\delta(1-\sum_{i}\eta_{i})e^{{\frac{2\pi\tau_{2}}{M_{s}^{2}}}(s\eta_{1}\eta_{3}+t\eta_{2}\eta_{4})}e^{-{\frac{2\pi\tau_{2}\mu^{2}}{M_{s}^{2}}}}\.
I=\frac{\Omega}{16\pi}\intd^{2}x\sqrt{-g}\left[\Phi^{3}R+6\Phi(\nabla\Phi)^{2}+\frac{6\Phi}{l^{2}}-\Phi^{3}\mathcal{G}_{ij}\partial_{\alpha}\phi^{i}\partial^{\alpha}\phi^{j}-\frac{1}{2}\Phi^{3}G_{IJ}F_{\alpha\beta}^{I}F^{J\alpha\beta}\right],
\begin{aligned}{\deltaR_{r_{1}r_{2}}{}^{s_{1}s_{2}}=\,\,}&{{}-8D_{[r_{1}}(\bar{\epsilon}\Gamma_{r_{2}]}\psi^{s_{1}s_{2}})}\\{}&{{}+4D_{[r_{1}}(\bar{\epsilon}\Gamma_{r_{2}]}\psi^{s_{1}s_{2}}+2\,\bar{\epsilon}\Gamma^{[s_{1}}\psi^{s_{2}]}{}_{r_{2}]})+\cdots\,,}\\\end{aligned}
G^{(\psi)}=-\frac{1}{\Box-M^{2}}\left(\begin{array}{cc}{\bar{\psi}}&{-\frac{1}{4}\bar{D}^{2}}\\{-\frac{1}{4}D^{2}}&{\psi}\\\end{array}\right)\left(\begin{array}{ll}{\delta_{+}}&{0}\\{0}&{\delta_{-}}\\\end{array}\right)+\ldots
S=\frac{1}{2\kappa^{2}}\int{d^{4}x\sqrt{-g}\left[\muR-\left(\nabla\chi_{1}\right)^{2}+\left(\nabla\psi_{1}\right)^{2}-\left(\mu-1\right)\left(\chi_{1}+\psi_{1}\right)-\frac14\alpha\left(\chi_{1}-\psi_{1}\right)^{2}\right]}.
\frac{\theta(u-\eta)}{-\theta(\eta)}R(u,\etam){\bfe}_{j}\otimes{\bfe}_{k}=\sum_{i,l}W(m,m-{\bfe}_{j},m-{\bfe}_{j}-{\bfe}_{k},m-{\bfe}_{i},u){\bfe}_{i}\otimes{\bfe}_{l},
\frac{1}{2\pii}\oint_{\mathrm{{\scriptsize~edge}}}f(\zeta)\tilde{\chi}d\zeta=-\sum_{r=1}^{g}\left(n_{r}\oint_{a_{r}}f(\zeta)d\zeta+m_{r}\oint_{b_{r}}f(\zeta)d\zeta\right)=-\oint_{c}f(\zeta)d\zeta
\bar{{\calZ}}(z_{2}){\calZ}(z_{1})={\calF}(z_{12})=\left(\begin{array}{cc}{ix_{\bar{2}1}{\cdot\tilde{\sigma}}}&{-2\bar{\theta}_{21b}}\\{-2\theta_{21}^{a}}&{\delta_{~b}^{a}}\\\end{array}\right)
S_{eff}=\intd^{3}x~{\calL}_{M}-\intd^{3}x~d^{3}y~\left(\frac{1}{2}J^{\mu}\frac{1}{\Box}J_{\mu}+\frac{1}{2\theta}\epsilon^{\mu\alpha\beta}J_{\mu}\partial_{\alpha}\frac{1}{\Box}J_{\beta}\right)
T^{-1}(x)\deltaT(x)=\sum_{k,l}\int\frac{dz}{2\pii}\left(\frac{1}{D_{k}}w(-z,q_{kl})X(z)_{kl}E_{kk}-\frac{1}{D_{k}}w(x-z,q_{kl})X(z)_{kl}E_{kl}\right)
\langleT_{i}^{k}\rangle_{D}^{(b)}\left(\xi_{1},\xi_{2},\xi\right)=\langleT_{i}^{k}\rangle_{D}^{(1b)}(\xi_{1},\xi)-A_{d}\delta_{i}^{k}\int_{0}^{\infty}dk\,k^{d}\int_{0}^{\infty}d\omega\frac{K_{\omega}(k\xi_{2})}{K_{\omega}(k\xi_{1})}\frac{F^{(i)}[D_{\omega}(k\xi,k\xi_{1})]}{D_{\omega}(k\xi_{1},k\xi_{2})}.
(L_{n}-\bar{L}_{-n})|i\rangle\!\rangle=0,\qquad\langle\!\langlei|\tilde{q}^{\frac{1}{2}(L_{0}+\bar{L}_{0}-\frac{c}{12})}|j\rangle\!\rangle=\delta_{ij}\chi_{i}(\tilde{q})\,\,,
\sqrt{-\mathrm{det}\,(\eta_{\mu\nu}+\calF_{\mu\nu})}-\frac{1}{2}\tilde{{\calH}}^{\mu\nu}{\calF}_{\mu\nu}+\frac{1}{8}\epsilon^{\mu\nu\lambda\sigma\tau}C_{\mu}{\calF}_{\nu\lambda}{\calF}_{\sigma\tau}
\mathrm{Tr}e^{iF_{5i}}-2\mathrm{Tr}_{A}e^{iF_{5i}}=m_{i}-2\mathrm{tr}F_{5i}^{2}+2\Big(\mathrm{tr}F_{5i}\Big)^{2}+{\frac{2}{3}}\mathrm{tr}F_{5i}^{4}-{\frac{2}{3}}\mathrm{tr}F_{5i}\mathrm{tr}F_{5i}^{3}+\dots
\int_{0}^{2\pi}\frac{d\theta}{2\pi}e^{-|z|^{2}}\,\prod_{\pm}\left\{e^{xe^{\pmi\theta}|z_{2M+1}^{\pm}|^{2}}\,\prod_{\mu=1}^{M}e^{xe^{\pmi\theta}\left[e^{i\omega_{\mu}}|z_{\mu+}^{\pm}|^{2}+e^{-i\omega_{\mu}}|z_{\mu-}^{\pm}|^{2}\right]}\right\}\\\,
A_{S[K,0;K]}={\frac{2}{r_{*}^{2}}}\intdr^{\prime}\,r^{\prime}\left[{\calO}_{r_{3}}^{[1]}w(r_{1},r_{2},r_{3})\right]_{r_{1}=r,r_{2}=r_{3}=r^{\prime}}{\calO}_{r^{\prime}}^{[2]}{\calA}_{0}^{(1)}(r^{\prime}),
\left(x(s),y(s),z(s)\right)=\left(\frac{1}{5}\left(-2\sqrt{\wp(s)}+\frac{1}{4}\frac{\wp^{\prime}(s)}{\wp(s)}\right),\frac{1}{20}\left(\sqrt{\wp(s)}+\frac{1}{2}\frac{\wp^{\prime}(s)}{\wp(s)}\right),\frac{\sqrt{-g_{3}}}{2\sqrt{12}\wp(s)}\right).
\mathrm{(4a)}+\mathrm{(4b)}=\frac{N}{48\pim}\int_{x}\left[\mathrm{tr}\left\{\partial_{[\mu}\tilde{A}_{\nu]}[\tilde{A}^{\mu},\tilde{A}^{\nu}]\right\}+O(\partial\frac{\partial^{2}}{m^{2}})\right].
d\hat{s}^{2}=\varepsilon^{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},
S_{\mathrm{\scriptsize~eff}}[c,e^{\phi}\widehat{g}_{t}]=S[\widehat{g}_{t},\widetilde{x}_{i}[\phi]]=\int_{0}^{t}dz^{0}\int_{0}^{1}dz^{1}\left(\left({\partial}_{0}x_{cl}\right)^{2}+\left({\partial}_{1}x_{cl}\right)^{2}\right)
\Bigl(i\frac{\partial\lambda_{n}}{\partialt}\Bigr)\vert\lambda_{n}^{a},t\rangle-\Bigl(\hat{\calO}(t)-\lambda_{n}\hat{I}\Bigr)\Bigl[\Bigl(i\frac{\partial}{\partialt}-\hat{H}(t)\Bigr)\vert\lambda_{n}^{a},t\rangle\Bigr]=0,
e_{\alpha}^{\;\;\mu}e_{\beta}^{\;\;\nu}\delta\lambda.\frac{\partial}{\partial\lambda}\beta_{\mu\nu}^{b}={\bar{e}}_{\alpha}^{\;\;i}{\bar{e}}_{\beta}^{\;\;j}\left(\delta\lambda.\frac{\partial}{\partial\lambda}\beta_{ij}^{b}-v_{i}\delta\lambda.\frac{\partial}{\partial\lambda}\beta_{j}^{w}+v_{j}\delta\lambda.\frac{\partial}{\partial\lambda}\beta_{i}^{w}\right).
S_{\mathrm{linear}}=m_{0}(h_{0i,j}(0)+C_{i,j}(0)){\frac{i}{8}}\theta\gamma^{ij}\theta+m_{0}(B_{ij,k}(0)+C_{0ij,k}^{(3)}(0)){\frac{i}{16}}\theta\gamma^{ijk}\theta+{\calO}(\theta^{4})
\left.\hat{A}_{\alpha}^{\Omega}\right|_{\mathrm{sing}}(q)=\Omega_{-}^{-1}(q)\,\frac{\partialx^{\mu}}{\partialq^{\alpha}}\left.\hat{A}_{\mu}^{I}\right|_{\mathrm{sing}}(q)\,\Omega_{-}(q)+i\,\Omega_{-}^{-1}(q)\frac{\partial\,\Omega_{-}(q)}{\partialq_{\alpha}}\,,
\mathrm{I}_{M}=\int\frac{d^{2\sigma}Q}{(2\pi)^{2\sigma}}\int\frac{d^{2\sigma}K}{(2\pi)^{2\sigma}}\,\frac{K_{\mu}}{(K^{2}-\vec{K}^{2}+i\epsilon)}\,\frac{Q_{\mu}}{(Q^{2}-\vec{Q}^{2}+i\epsilon)}\,,\qquad\epsilon>0;
+\frac{e^{2}}{8}\int_{-\infty}^{\infty}dx^{+}\int_{0}^{L}dx^{-}\int_{0}^{L}dy^{-}j^{+}(x^{-},x^{+})D^{sym}(x^{-},y^{-}|L)j^{+}(y^{-},x^{+}),
\begin{array}{rcl}{{\hat{\psi}}_{\hat{a}}}&{=e^{\varphi/12}\left(\psi_{a}-\frac{1}{6}\Gamma_{a}\Gamma^{i}\lambda_{i}\right)\,,\qquad{\hat{\psi}}_{i}=e^{\varphi/12}\lambda_{i}\,,\qquad\hat{\epsilon}=e^{-\varphi/12}\epsilon\,}\\\end{array}
\psi_{a}^{R}(x)=\sum_{j=1}^{N}\sum_{\sigma^{\prime\prime}\in{\calW}^{\prime\prime}}A_{a}^{R}(\sigma^{\prime\prime}q_{j}^{\prime\prime},id)e^{-\kappa_{j}x_{N}}e^{i(\sigma^{\prime\prime}q_{j}^{\prime\prime}k|x^{\prime\prime})}.
V_{2}^{\mu}\equivp^{\mu}-\left[\frac{(i+\omega)p^{-}}{(1+i+m+\omega)(n\!\cdot\!n^{*})}\right]\;n^{\mu}-\left[\frac{(i+l+\omega)p^{+}}{(1+i+j+l+\omega)(n\!\cdot\!n^{*})}\right]\;n^{*\mu}\,,
ds^{2}\sim\frac{R}{\sqrt{q}}(-\eta_{\mu\nu}dx^{\mu}dx^{\nu})+\frac{\sqrt{q}}{R}dR^{2}+\sqrt{q}Rds_{\mathbf{S}^{3}}^{2}+\frac{\sqrt{q}}{R}\frac{a^{2}}{4}ds_{\mathbf{S}^{2}}^{2}
{\calD}^{AA^{\prime}}\tilde{\gamma}_{A^{\prime}B^{\prime}}^{C}={\calD}^{AA^{\prime}}{\calD}_{B^{\prime}}^{C}\nu_{A^{\prime}}={\calD}_{B^{\prime}}^{C}{\calD}^{AA^{\prime}}\nu_{A^{\prime}}=0,
{\calK}_{i_{1}\dotsi_{3}}^{(3)}={\frac{1}{\sqrt{|\mathrm{det}(\Pi^{(B)})|}}}\Pi_{i_{1}j_{1}}^{(B)}\dots\Pi_{i_{3}j_{3}}^{(B)}\epsilon^{j_{1}\dotsj_{6}}{\calK}_{j_{4}j_{5}j_{6}}^{(3)}\,,
T=\sum_{n}^{occ}\sum_{i=1}^{2}2K\int_{-R/2}^{R/2}\mathrm{d}x\,\frac{1}{{\calN}_{n,i}}\psi_{n,i}^{\dagger}\gamma_{5}\frac{1}{1}{i}\frac{\partial}{\partialx}\psi_{n,i}\,
X+Y+\Psi(X,Y)=\sum_{m=1}^{\infty}{\frac{(-1)^{(m-1)}}{m}}\sum_{\mathrm{\mathrm{~k_i+l_i\geq~1~}\mathrm{~k_i\geq~0,~l_i\geq~0~}}}{\frac{[X^{k_{1}}Y^{l_{1}}\cdotsX^{k_{m}}Y^{l_{m}}]}{k_{1}!l_{1}!\cdotsk_{m}!l_{m}!}},
f(x,p)={\frac{1}{2\pi}}\int\!dy~\Psi^{*}(x-{\frac{\hbar}{2}}y)~e^{-iyp-(ie/\hbar)\int_{x-\hbary/2}^{x+\hbary/2}dzA(z)}~\Psi(x+{\frac{\hbar}{2}}y).
F_{ab}=\left(\begin{matrix}{0}&{\left(\begin{matrix}{\Lambda_{0}}&{0}&{...}&{0}\\{0}&{\Lambda_{1}}&{...}&{0}\\{...}&{...}&{...}&{...}\\{0}&{0}&{...}&{\Lambda_{4}}\\\end{matrix}\right)}\\{\left(\begin{matrix}{-\Lambda_{0}}&{0}&{...}&{0}\\{0}&{-\Lambda_{1}}&{...}&{0}\\{...}&{...}&{...}&{...}\\{0}&{0}&{...}&{-\Lambda_{4}}\\\end{matrix}\right)}&{0}\\\end{matrix}\right),
S_{\sigma}={\frac{1}{4\pi\alpha^{\prime}}}\intd^{2}\sigma[\sqrt{\gamma}\gamma^{\alpha\beta}\partial_{\alpha}X^{M}\partial_{\beta}X^{N}\hat{G}_{MN}+{\epsilon}^{\alpha\beta}\partial_{\alpha}X^{M}\partial_{\beta}X^{N}\hat{B}_{MN}-{\frac{1}{2}}\sqrt{\gamma}\hat{\Phi}{\calR}^{(2)}].
V_{-}(x,a_{0})=\frac{(\omega_{0}-\omega_{1})(3\omega_{0}+\omega_{1})}{4{(\omega_{0}+\omega_{1})}^{2}x^{2}}-\frac{\omega_{1}}{2}+\frac{{(\omega_{0}+\omega_{1})}^{2}x^{2}}{16}~.
\mathbf{B}=\Big[\varepsilon\Phi(x,y,\xi+\varepsilonz)\mathrm{sin}\left(\frac{\pm\xi}{l(x,y)}+b(x,y)\right);-\varepsilon\Phi(x,y,\xi+\varepsilonz)\mathrm{cos}\left(\frac{\pm\xi}{l(x,y)}+b(x,y)\right);0\Big],
\begin{array}{l}{{\delta}A_{\mu}^{a}=D_{\mu}c^{a}\,,\qquad{\delta}\bar{c}^{a}=B^{a}\,,}\\{\displaystyle{\delta}c^{a}=-\frac12gf^{abc}c^{b}c^{c}\,,\qquad{\delta}B^{a}=0\,.}\\\end{array}
\Big(\begin{array}{cc}{\mathrm{e}^{V}}\\\end{array}\Big)=\frac1{\sqrt{(\bar{z}_{+}\cdotz_{+})(\bar{z}_{-}\cdotz_{-})-(\bar{z}_{+}\cdotz_{-})(\bar{z}_{-}\cdotz_{+})}}\left(\begin{array}{cc}{\bar{z}_{-}\cdotz_{-}}&{-\bar{z}_{-}\cdotz_{+}}\\{-\bar{z}_{+}\cdotz_{-}}&{\bar{z}_{+}\cdotz_{+}}\\\end{array}\right)\,
\begin{array}{ccc}{SU(4)}&{\rightarrow}&{SU(3)\timesU(1)}\\{{\bf4}}&{\rightarrow}&{{\bf3}_{-1}\bigoplus{\bf1}_{3}}\\{{\bf6}}&{\rightarrow}&{{\bf3}_{2}\bigoplus{\bf\bar{3}}_{-2}}\\\end{array}
\sigma_{abs}\sim{\frac{G_{5}(\deltaN_{L})^{2}}{\alpha^{\prime}M\omega}}{\frac{e^{\beta_{L}^{*}\deltaN_{L}+\beta_{R}^{*}\deltaN_{R}}-1}{(e^{\beta_{L}^{*}\deltaN_{L}}-1)(e^{\beta_{R}^{*}\deltaN_{R}}-1)}},
e^{\sumV\left(x_{i}^{(a)}-y_{j}^{(b)}\right)-\frac{1}{2}\sumV\left(x_{i}^{(a)}-x_{j}^{(b)}\right)-\frac{1}{2}\sumV\left(y_{i}^{(a)}-y_{j}^{(b)}\right)}\left(\frac{1}{2\pi}\right)^{N(n+m)}\left[\frac{e^{2}N}{4\pi}e^{2\gamma}\right]^{\frac{N(n-m)^{2}}{2}}\;\;\;.
L_{\xi}=\sum_{i}\xi_{i}D_{i}^{\theta,0}+{\frac{1}{2}}({\calD}_{i}^{0}\xi_{i}){\calD}_{i}+\nu^{2}\sum_{j\nek\nei}\xi_{i}y_{ij}K_{ij}^{\mathrm{tot}}y_{jk}K_{jk}^{\mathrm{tot}}\,,
S_{\mathrm{C}}=\frac{1}{e^{2}}\intd^{4}x\,\sqrt{g}\,\Bigr\{(\nabla^{\mu}T_{\mu\rho}^{-}-iT_{\mu\rho}^{-}V^{\mu})(\nabla_{\nu}T_{+}^{\nu\rho}-iV_{\nu}T_{+}^{\nu\rho})\Bigr\},
\lambda^{(j)}=e^{2\piiv_{j}}\gamma_{1,6^{\prime}}\lambda^{(j)}\gamma_{1,6^{\prime}}^{-1}~~,~~\lambda^{(j)}=\gamma_{\OmegaR,6^{\prime}}\lambda^{(j)^{T}}\gamma_{\OmegaR,6^{\prime}}^{-1}.
V=\left({\it\rho_{1}}+{\frac{{{{\itx_{1}}}^{2}}}{10\,{\it\rho_{1}}}}\right)\,\left({\it\rho_{2}}+{\frac{{{{\itx_{2}}}^{2}}}{10\,{\it\rho_{2}}}}\right)\,\left(\frac{1}{16}+{\frac{x+{\itx_{1}}+{\itx_{2}}}{{{\left({{{\it\rho_{1}}}^{2}}+{{{\it\rho_{2}}}^{2}}+2\,{\it\rho_{1}}\,{\it\rho_{2}}\,\left(1-x\right)\right)}^{{\frac{3}{4}}}}}}\right)
\{Q_{\bar{\alpha}}^{\bar{a}},Q_{\bar{\beta}}^{\bar{b}}\}=\gamma_{\bar{\alpha}\bar{\beta}}^{\mu}(c\tau_{i})^{\bar{a}\bar{b}}P_{\mu}^{i}+\gamma_{\bar{\alpha}\bar{\beta}}^{\mu_{1}\mu_{2}\mu_{3}}c^{\bar{a}\bar{b}}Y_{\mu_{1}\mu_{2}\mu_{3}}+\gamma_{\bar{\alpha}\bar{\beta}}^{\mu_{1}\cdots\mu_{5}}(c\tau_{i})^{\bar{a}\bar{b}}X_{\mu_{1}\cdots\mu_{5}}^{i},
S_{3}=-\frac{m^{2}}{2}\intd^{4}\thetad^{d}xe^{\sigma+\bar{\sigma}}\int\frac{d^{d}k}{{(2\pi)}^{d}}\frac{-A^{2}k^{2}+m^{2}}{k^{2}{(-A^{2}k^{2}+m^{2})}^{2}-81\Lambda\bar{\Lambda}}\frac{\bar{D}_{1}^{2}D_{1}^{2}}{16}\delta_{11}
F=\left(\begin{array}{cc}{f_{1}(x)}&{0}\\{0}&{f_{2}(x)}\\\end{array}\right)~~,~~{\tilde{F}}=\left(\begin{array}{cc}{f_{2}(x)}&{0}\\{0}&{f_{1}(x)}\\\end{array}\right),f_{1}(x),f_{2}(x)\inC^{\infty}(C,{\calM}).
w_{\alpha\beta}=D_{(\alpha}^{i}\Phi_{a}(\gamma^{i}\gamma^{j})_{ab}D_{\beta)}^{j}\Phi_{b}^{\prime}+32i(\Phi_{a}\partial_{\alpha\beta}\Phi_{a}^{\prime}-\partial_{\alpha\beta}\Phi_{a}\Phi_{a}^{\prime})\;.
\langle\sigma^{z}\rangle=\langle\sigma^{z}\rangle^{(0)}(-1)=\langle\sigma^{z}\rangle^{(1)}(-1)=-\frac{2(q^{4};q^{4})_{\infty}^{4}}{(q^{2};q^{2})_{\infty}^{8}+(-q^{2};q^{2})_{\infty}^{8}}.
\omega_{\mu}^{\hat{\mu}\hat{\nu}}=\left(\begin{array}{cccc}{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{-{\frac{\delta_{\mu}^{\theta}}{r}}(1-{\frac{r}{2}}\partial_{r}\rho(r))}\\{0}&{0}&{{\frac{\delta_{\mu}^{\theta}}{r}}(1-{\frac{r}{2}}\partial_{r}\rho(r))}&{0}\\\end{array}\right).
H=\frac{\sqrt{\alpha^{\prime}}}{2}g^{\mu\nu}(P_{\mu}-Q_{_{E}}{\calA}_{\mu}-Q_{_{M}}{\tilde{\calA}}_{\mu})(P_{\nu}-Q_{_{E}}{\calA}_{\nu}-Q_{_{M}}{\tilde{\calA}}_{\nu})+\frac{\sqrt{\alpha^{\prime}}}{2k^{2}}({Q_{_{E}}}^{2}+{Q_{_{M}}}^{2}),
A(1,2,3,4):=\tilde{D}_{4,-p_{4}}^{\alpha}\tilde{D}_{\alpha,2,-p_{2}}\tilde{\bar{D}}_{\dot{\alpha},2,-p_{2}}\tilde{\bar{D}}_{1,-p_{1}}^{\dot{\alpha}}\tilde{\delta}_{V}(1,2)\tilde{\delta}_{V}(1,3)\tilde{\delta}_{V}(1,4).
\begin{array}{c}{\frac{1}{2}\left[\begin{array}{cc}{1}&{0}\\{0}&{1}\\\end{array}\right]}\\{\pm\frac{1}{2}\left[\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right]\mathrm{~or~}\pm\frac{1}{2}\left[\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right]\mathrm{~or~}\pm\frac{e_{a}}{2}\left[\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}\right]}\\\end{array}
M(p,p^{\prime})=\intd^{4}xd^{4}x^{\prime}\bar{E}_{p}(x)M(x,x^{\prime})E_{p^{\prime}}(x^{\prime})=\tilde{\Sigma}_{A}(\bar{p})(2\pi)^{4}\hat{\delta}^{(4)}(p-p^{\prime}),
0=\left[e_{\alpha}^{A}h_{A\beta}-h_{A\alpha}e_{\beta}^{A}\right]=e_{\alpha}^{A}\left[h_{A\gamma}(e^{-1})_{B}^{\gamma}-(e^{-1})_{A}^{\gamma}h_{B\gamma}\right]e_{\beta}^{B}=e_{\alpha}^{A}\left[\Pi_{AB}-\Pi_{BA}\right]e_{\beta}^{B}\,
S=\intd^{5}z\Big({\frac12}{\Phi}_{a}D^{2}{\Phi}_{a}+{\frac12}\mu{\Phi}_{a}{\Phi}_{a}-\frac{g}{4}[f{\Phi}_{a}*{\Phi}_{a}*{\Phi}_{b}*{\Phi}_{b}+(1-f){\Phi}_{a}*{\Phi}_{b}*{\Phi}_{a}*{\Phi}_{b}]\Big).
S=-\frac{1}{2}\int\!d\sigmad\tau[\partial_{\mu}\chi_{1}\partial^{\mu}\chi_{1}+2\chi_{2}\partial_{\mu}\chi_{1}\partial^{\mu}\chi_{3}+\frac{1}{1+{\chi_{1}}^{2}}\partial_{\mu}\chi_{2}\partial^{\mu}\chi_{2}+({\chi_{2}}^{2}+\frac{1}{1+{\chi_{1}}^{2}})\partial_{\mu}\chi_{3}\partial^{\mu}\chi_{3}+\partial_{\mu}t\partial^{\mu}t].
v_{o}\,=\,q^{m}P_{m}\,+\,\sum_{j=0}^{m-1}q^{2j(j+1)}\Bigl(P_{j}+\frac{d_{j}^{+}}{[d_{j}^{+}]}N_{j}^{+}+\frac{d_{j}^{-}}{[d_{j}^{-}]}N_{j}^{-}\Bigr)\;,
\begin{array}{rl}{^{(1)}H_{\mu\nu}\=}&{{\displaystyle2\left(\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\nabla^{2}\right)R+2RR_{\mu\nu}-{\frac{1}{2}}g_{\mu\nu}R^{2}\,}}\\{^{(3)}H_{\mu\nu}\=}&{{\displaystyleR_{\mu}^{\\lambda}R_{\lambda\nu}-{\frac{2}{3}}RR_{\mu\nu}-{\frac{1}{2}}g_{\mu\nu}R^{\rho\sigma}R_{\rho\sigma}+{\frac{1}{4}}g_{\mu\nu}R^{2}}\.}\\\end{array}
L_{0}^{osc}=\frac{1}{2}\alpha_{0}^{2}+\sum_{n=1}^{2N}\eta_{\mu\nu}\alpha_{-n}^{\mu}\alpha_{n}^{\nu}+\sum_{n=1}^{2N}\kappa_{n}(b_{-n}c_{n}+c_{-n}b_{n})
\{\chi^{ij}\}=\left(\begin{array}{ccccc}{0}&{0}&{0}&{0}&{0}\\{0}&{\chi^{11}}&{\chi^{12}}&{\chi^{13}}&{0}\\{0}&{\chi^{12}}&{\chi^{22}}&{\chi^{23}}&{0}\\{0}&{\chi^{13}}&{\chi^{23}}&{\chi^{33}}&{0}\\{0}&{0}&{0}&{0}&{0}\\\end{array}\right);~~~~\chi^{11}+\chi^{22}+\chi^{33}=0.
\int_{E_{\alpha}}R^{\mu\nu\lambda\rho}R_{\mu\nu\lambda\rho}=\alpha\int_{E}\bar{R}^{\mu\nu\lambda\rho}\bar{R}_{\mu\nu\lambda\rho}+8\pi(1-\alpha)\int_{\Sigma}\bar{R}_{ijij}+O((1-\alpha)^{2})~~~,
\delta_{\epsilon}A=\bar{\epsilon}\Gamma_{11}\Gamma_{m}\thetadX^{m}+{\frac{1}{6}}(\bar{\epsilon}\Gamma_{11}\Gamma_{m}\theta\bar{\theta}\Gamma^{m}d\theta+\bar{\epsilon}\Gamma_{m}\theta\bar{\theta}\Gamma_{11}\Gamma^{m}d\theta).
<\mathrm{in}|N|\mathrm{in}>=\sum_{lmn}<\mathrm{in}|N_{lmn}|\mathrm{in}>=\sum_{l=0}^{\infty}(2l+1)\sum_{n,n^{\prime}=1}^{\infty}|\beta_{n^{\prime}n}^{l}|^{2}
H=\sum_{\vec{x}}a^{3}\left\{\psi_{\gamma\vec{x}}^{\dagger}m\beta_{\gamma\delta}\psi_{\delta\vec{x}}+\frac{i}{2a}\sum_{l=1}^{3}[\psi_{\gamma\vec{x}+\vec{l}}^{\dagger}(\alpha_{l})_{\gamma\delta}\psi_{\delta\vec{x}}-\psi_{\gamma\vec{x}}^{\dagger}(\alpha_{l})_{\gamma\delta}\psi_{\delta\vec{x}+\vec{l}}]\right\}\quad.
(\Box+1)\left(\begin{matrix}{B_{\mu}^{(lm)}}\\{b_{\mu}^{(lm)}}\\\end{matrix}\right)=\left(\begin{matrix}{l(l+1)+2}&{2}\\{2l(l+1)}&{l(l+1)}\\\end{matrix}\right)\left(\begin{matrix}{B_{\mu}^{(lm)}}\\{b_{\mu}^{(lm)}}\\\end{matrix}\right)
<A_{2}|n>=\left(\frac{k\tau_{2}}{2\pi^{2}|\tau|^{2}}\right)^{1/4}\,\frac{(-i)^{n}}{\sqrt{n!}}\,\left(\frac{i\bar{\tau}}{2\tau_{2}}\right)^{n/2}\,e^{\frac{k}{4i\pi\tau}A_{2}^{2}}\,P_{n}\left(\sqrt{\frac{k}{2i\pi\bar{\tau}}}\,A_{2};\frac{i\bar{\tau}}{-i\tau}\right)\\\.
(2)d\mu(z,\bar{z})\equiv\prod_{\alpha=1}^{m}\frac{dz_{\alpha}\wedged\bar{z}_{\alpha}}{-2\pii}=\prod_{\alpha=1}^{m}\frac{dq_{\alpha}dp_{\alpha}}{2\pi\hbar},\;\;\;z=({\bfq}+i{\bfp})/\sqrt{2\hbar},
\int_{C}d^{n}{\eta}Y_{\alpha}^{N}(\eta)D^{2}Y_{\alpha^{\prime}}^{N^{\prime}}(\eta)-\frac{n(n-2)}{4a^{2}}\int_{C}d^{n}{\eta}Y_{\alpha}^{N}(\eta)Y_{\alpha^{\prime}}^{N^{\prime}}(\eta)-[(N,{\alpha})\leftrightarrow(N^{\prime},{\alpha^{\prime}})]
h^{i}\equivf_{+}^{-i}+f_{-}^{+i}=2f_{+}^{+i}={\frac{1}{2}}u^{\underline{{m}}i}u^{\underline{{n}}-}u^{\underline{{k}}+}H_{\underline{{k}}~\underline{{n}}~\underline{{m}}}(X(\xi))
K_{nn_{1}}^{lm}=\int_{0}^{\infty}x^{l+|m|+\delta_{1}+\delta_{2}}\mathrm{e}^{-x}{_1F}_{1}(-n_{1},|m|+\delta_{1}+1;x){_1F}_{1}(-n+l+1,2l+\delta_{1}+\delta_{2}+2;x)\,dx.
Solv(\frac{\mathrm{E}_{5(5)}}{\mathrm{O}(5)\otimes\mathrm{O}(5)})=Solv(\frac{\mathrm{GL}(4,\mathbb{R})}{\mathrm{O}(4)}\otimes\frac{\mathrm{SL}(2,\mathbb{R})}{\mathrm{O}(2)})+({\bf6},{\bf2})^{+}+({\bf1},{\bf1})^{+}.
\varepsilon=-{r_{0}}\sqrt{1-{\frac{2M}{{r_{0}}}}+{\frac{a^{2}+Q^{2}}{{r_{0}}^{2}}}}\,\,\Bigg[1-{\frac{a^{2}}{6{r_{0}}^{2}}}\bigg(1+{\frac{2M}{{r_{0}}}}-{\frac{Q^{2}}{{r_{0}^{2}}}}\bigg)+O\bigg({\frac{a^{4}}{{r_{0}}^{4}}}\bigg)\Bigg]\\,
\overrightarrow{{\bfX}^{(n)}}(x)=\overrightarrow{{\bfX}^{(0)}}(x)+{\bfY}(x)\int_{x_{0}}^{x}{\bfY}^{(-1)}(t)\overrightarrow{{\bfF}}(t,\overrightarrow{{\bfX}^{(n-1)}}(t))dt
\eta^{(i)}(\tau)=y^{(i)}(\tau)+\dot{\phi}_{c}^{(i)}(\tau)\int_{-\beta}^{\tau}\frac{{\ddot{\phi}}_{c}^{(i)}(\tau^{\prime)}}{{\dot{\phi}}_{c}^{(i)}(\tau^{\prime})}y^{(i)}(\tau^{\prime})d\tau^{\prime}
\bar{\varphi}_{\alpha}(\lambda,\omega)\sim{\frac{1}{(4\pi)^{d/2}}}\sum_{n=0}^{\infty}\left[\bar{a}_{n}(\omega)~{\frac{\lambda^{d/2-n+\alpha-1}}{\Gamma\left(\fracd2-n+\alpha\right)}}+\bar{b}_{n}(\omega)~{\frac{\lambda^{(d-1)/2-n+\alpha-1}}{\Gamma\left({\frac{d-1}{2}}-n+\alpha\right)}}\right].
\varphi=e^{\zeta}\Biggl(1-\frac{w}{2\lambda}+O\biggl(\frac{1}{\lambda^{2}}\biggr)\Biggr),\quad\varphi^{*}=e^{-\zeta}\Biggl(1+O\biggl(\frac{1}{\lambda}\biggr)\Biggr).
B_{ij}^{n}(q)=(-1)^{n(1-\delta_{ij})}n^{-1}\frac{q_{ij}^{n}-q_{ij}^{-n}}{q_{j}-q_{j}^{-1}}\frac{q-q^{-1}}{q_{i}-q_{i}^{-1}}\,,~~~~q_{ij}=q^{(\alpha_{i},\alpha_{j})}\,,~~~q_{i}\equivq_{\alpha_{i}}
{\calE}_{\mathrm{RCS}}=\intd^{2}x\left[|D_{0}\phi\pm\frac{i}{\kappa}(|\phi|^{2}-v^{2})\phi|^{2}+|D_{\pm}\phi|^{2}+V(|\phi|)-\frac{1}{\kappa^{2}}|\phi|^{2}(|\phi|^{2}-v^{2})^{2}\mpv^{2}B\right]
[G_{R}Q]^{2}\{sI\}=\int_{0}^{s}ds^{\prime}\int_{o}^{s^{\prime}}ds^{\prime\prime}(s-s^{\prime})[Q^{\mu}{}_{\alpha}(s^{\prime})](s^{\prime}-s^{\prime\prime})[Q^{\alpha}{}_{\nu}(s^{\prime\prime})]s^{\prime\prime}.
f(r,z,{\bar{z}})*h(r,z,{\bar{z}})=e^{{\frac{i}{2}}[\Theta^{rz}(\partial_{r}^{1}\partial^{2}-\partial_{r}^{2}\partial^{1})+\Theta^{r{\bar{z}}}(\partial_{r}^{1}{\bar{\partial}}^{2}-\partial_{r}^{2}{\bar{\partial}}^{1})]}f(r_{1},z_{1},{\bar{z}}_{1})h(r_{2},z_{2},{\bar{z}}_{2})|_{1=2}.
\left(\begin{matrix}{1}&{0}&{0}\\{0}&{1}&{0}\\{\xi}&{\eta}&{1}\\\end{matrix}\right)\left(\begin{matrix}{A_{1}}\\{A_{2}}\\{A_{u}}\\\end{matrix}\right)=\left(\begin{matrix}{A_{1}}\\{A_{2}}\\{A_{u}+\xiA_{1}+\etaA_{2}}\\\end{matrix}\right).
\frac{df}{d\gamma}\;=\;\frac{1}{3\gamma}E\;+\;\frac{1}{3\gamma}\sum_{n=1}^{\infty}\Big[2\lambda_{n}|c_{n}|^{2}\;+\;\kappa_{n}c_{n}\;+\;\bar{\kappa}_{n}c_{-n}\Big]\;-\;\frac{1}{96}\;.
F(t=1)=A=e^{-\Phi_{0}}Q_{0}(e^{\Phi_{0}})=Q_{0}(\Phi_{0})+\sum_{n=2}^{\infty}\frac{1}{n!}\overbrace{[[Q_{0}(\Phi_{0}),\Phi_{0}],\dots,]\Phi_{0}]}^{n-1}\.
L_{(1)}^{\kappa\lambda\mu\nu}=\,\delta^{\kappa\lambda}\delta^{\mu\nu},\quadL_{(2)}^{\kappa\lambda\mu\nu}=\,\delta^{\kappa\mu}\delta^{\nu\lambda},\quadL_{(3)}^{\kappa\lambda\mu\nu}=\,\delta^{\kappa\nu}\delta^{\lambda\mu},
t_{\mu\nu}=\frac{1}{8\pi{\calG}}\left[R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R_{\;\;\;\alpha}^{\alpha}-R_{\;\;\;\;\mu\nu}^{(1)}+\frac{1}{2}\eta_{\mu\nu}R_{\;\;\;\;\;\;\;\alpha}^{(1)\,\alpha}\right]\,.
\begin{aligned}{\mathrm{solution~3:}\quad}&{{}\eta=7a_{1}+5a_{2}+5a_{3},}\\{\mathrm{solution~4:}\quad}&{{}\eta=8a_{1}+5a_{2}+5a_{3},}\\{\mathrm{solution~5:}\quad}&{{}\eta=11a_{1}+5a_{2}+5a_{3},}\\\end{aligned}
(W^{12})^{p}=\phi^{\stackrel{\underbrace{\mathrm{\scriptsize~1\ldots~1}}}{p}\stackrel{\underbrace{\mathrm{\scriptsize~2\ldots~2}}}{p}}+\ldots+\theta^{1\alpha}\theta^{1\beta}\theta^{2\gamma}\theta^{2\delta}A_{[\alpha\beta][\gamma\delta]}^{\stackrel{\underbrace{\mathrm{\scriptsize~1\ldots~1}}}{p-2}\stackrel{\underbrace{\mathrm{\scriptsize~2\ldots~2}}}{p-2}}+\mathrm{\small~d.t.}
\begin{array}{lcr}{\varphi_{j}^{+}=X_{j}\eta^{+}+Y_{j}\overline{{\eta}}^{+}}\\{\overline{{\varphi}}_{j}^{+}=-\overline{{Y}}_{j}\eta^{+}+\overline{{X_{j}}}\overline{{\eta}}^{+},}\\\end{array}
\Pi_{r}^{\alpha}=\prod_{A}(\Phi_{r}^{A})^{n_{\alpha}^{A}}=e^{\sum_{I}q_{I}^{\alpha}g^{I}/2}\Pi^{\alpha},\;\;\;\;\Phi_{r}^{A}=e^{\sum_{I}q_{I}^{A}g^{I}/2}\Phi^{A},
\eta^{\prime\prime}+{\frac{1}{2}}\big({\frac{1}{\xi-a_{1}}}+{\frac{1}{\xi-a_{2}}}+{\frac{1}{\xi-a_{3}}}\big)\eta^{\prime}-{\frac{m(m+1)\xi+\varepsilon}{4(\xi-a_{1})(\xi-a_{2})(\xi-a_{3})}}\eta=0
\alpha(\rho)={\frac{1}{2}}\beta^{\frac{1}{3}}\biggl[{\frac{1}{3\pi}}\Gamma({\frac{2}{3}})\Gamma({\frac{1}{6}})^{2}-{\frac{3}{5}}y^{\frac{5}{3}}{}_{2}F_{1}({\frac{5}{6}},{\frac{5}{3}};{\frac{8}{3}};y)\biggr]
\tilde{H}=\frac{1}{2}\intd^{3}x\left(B_{a}^{ij}F_{ij}^{a}-A_{i}^{a}A_{a}^{i}+\left(\left(D^{i}\right)_{\;\;b}^{a}B_{0i}^{b}\right)\left(D_{j}\right)_{a}^{\;\;c}B_{c}^{0j}\right)\equiv\intd^{3}x\,\tilde{h},
(G_{o}(k)^{ab})=\left(\begin{array}{cc}{G_{0}(k)}&{0}\\{0}&{\widetilde{G}_{0}(k)}\\\end{array}\right)=\left(\begin{array}{cc}{\frac{1}{k^{2}-m^{2}+i\epsilon}}&{0}\\{0}&{\frac{-1}{k^{2}-m^{2}-i\epsilon}}\\\end{array}\right).
\zeta^{(T)}\left(z,\beta,\sigma\right)=\sum_{n=0}^{\infty}\sum_{l=1}^{\infty}D_{n}^{T}(\Lambda_{n,l}^{T}(\sigma))^{-z}+\sum_{n=1}^{\infty}\bar{D}_{n}^{T}(\Lambda_{n,0}^{T}(\sigma))^{-z}~~~,
j_{\mu}^{matter}=\bar{\varphi}_{2}\partial_{\mu}\bar{\varphi}_{1}-(\bar{\varphi}_{1}+\mathrm{~\frac~me~})\partial_{\mu}\bar{\varphi}_{2}+eA_{\mu}(\bar{\varphi}_{2}^{2}+(\bar{\varphi}_{1}+\mathrm{~\frac~me~})^{2})+O(\hbar)
S=\frac{1}{4\pi\alpha^{\prime}}\int_{\Sigma}d^{2}\sigma\sqrt{-h}h^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}-\int_{\partial\Sigma}ds\left(A_{\mu}\frac{\partial}{\partials}X^{\mu}+\phi^{i}\frac{\partial}{\partial\sigma}X^{i}\right)
f(\lambda)=h\biggl\{\frac{1}{N\lambda^{N-1}}\frac{\partial}{\partial\lambda}+\sum_{k=1}^{\infty}\frac{k}{N}\lambda^{k-N}t_{k}+\frac{w}{2N\lambda^{N+1}}+O\biggl(\frac{1}{\lambda^{N+2}}\biggr)\biggr\}\cdot1.
p_{2,1}=\{{\frac{1}{2}\frac{1}{\midG\mid}\sum_{allg\inG}\sum_{h\inG\atop{I=even}}\prod_{\tilde{\theta}_{i}^{g}=\tilde{\theta}_{i}^{h}=0}(1-{\frac{1}{q_{i}}})}\}-1.
\sum_{n}\left\{\frac{(-1)^{n}}{2(n+1)!}(Ad_{\Phi})^{n}\partial^{2}\Phi^{a}+\frac{(-1)^{n+1}}{2(n+2)!}\sum_{k=0}^{n-1}(Ad_{\Phi})^{k}Ad_{\partial_{\mu}\Phi}(Ad_{\Phi})^{n-k}\partial^{\nu}\Phi^{a}\right\}=0,
{\calL}=\frac{1}{2}(\partial_{\mu}\phi_{i})^{2}-\frac{1}{2}\mu^{2}\vec{\phi}\cdot\vec{\phi}-\frac{\lambda_{0}}{4N}(\vec{\phi}\cdot\vec{\phi})^{2}-\frac{\eta}{6N^{2}}(\vec{\phi}\cdot\vec{\phi})^{3}
Z(\beta\,;R(S_{\nabla}),\langle\cdot\,,\cdot\rangle_{H},\langle\cdot\,,\cdot\rangle)=\pi^{\zeta/2}\,e^{-\frac{i\pi}{4}((\frac{2\theta}{\pi}\mp1)\zeta\,\pm\,\eta)}\,|\beta|^{-\zeta/2}\,\tau(M,\chi,\langle\cdot\,,\cdot\rangle_{H})^{1/2}