harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f_{{\bf{k}},{\bf{q}}}({\bf{x}})=e^{i{\bf{q.x}}}(\frac{1}{2\mathrm{~}N\mathrm{~}\epsilon_{{\bf{q}}}})\Lambda_{{\bf{k}}}(-{\bf{q}})\omega_{{\bf{k}}}({\bf{q}})+i\mathrm{~}U_{-{\bf{q}}}({\bf{x}})\Lambda_{{\bf{k}}}(-{\bf{q}})
+(-1)^{\epsilon_{\alpha}(\epsilon_{\beta}+\epsilon_{\gamma})}(T_{\beta\gamma,k}^{\delta}R_{\alpha}^{k}-T_{\beta\eta}^{\delta}T_{\gamma\alpha}^{\eta})+(-1)^{\epsilon_{\gamma}(\epsilon_{\alpha}+\epsilon_{\beta})}(T_{\gamma\alpha,k}^{\delta}R_{\beta}^{k}-T_{\gamma\eta}^{\delta}T_{\alpha\beta}^{\eta})
K_{,i\bar{\jmath}}(A,A^{*})(\dot{A}^{i}\dot{A}^{j*}+F^{i}F^{j*})=K_{,i\bar{\jmath}}(A,A^{*})(\dot{A}^{i}\dot{A}^{j*}+J^{i}{}_{\bar{k}}J^{\bar{\jmath}}{}_{l}\dot{A}^{l}\dot{A}^{k*})
F^{2}=-5!\left(\Lambda^{\prime}e^{\Lambda}\right)^{2}e^{-8A-2B}=-5!\frac{64\xi^{2}k^{2}(c_{2}^{2}-1)r_{0}^{8}}{Y^{5/2}r^{10}f_{+}^{5/2}f_{-}^{5/2}}.
I_{C}(k,q)=\frac{1}{|\vec{k}|q^{2}(q+k)^{2}}\left[8q\cdotuk\cdotq-k^{2}k\cdotu-6k\cdotqk\cdotu-8(k\cdotq)^{2}\frac{k\cdotu}{k^{2}}+4k^{2}q\cdotu\right]
R_{\calL}(g^{\otimesn};f^{\otimesm})=\sum_{k=0}^{n}(-1)^{k}\frac{n!}{k!(n-k)!}\bar{T}_{\calL}(g^{\otimesk})T_{\calL}(g^{\otimes(n-k)}\otimesf^{\otimesm}).
\Gamma_{d\timesd}^{A}\equiv\left[\begin{array}{ll}{\Gamma_{d_{b}\timesd_{b}}^{A^{1}}}&{\Gamma_{d_{f}\timesd_{b}}^{A^{2}}}\\{\Gamma_{d_{b}\timesd_{f}}^{A^{3}}}&{\Gamma_{d_{f}\timesd_{f}}^{A^{4}}}\\\end{array}\right],\tag{68}
{\calL}_{B_{+},c}^{U(1)}(2)=-H_{B}^{II3\rho}\{H_{B\rho}^{II3}-(-2\nabla^{\mu}B_{+\mu\rho}^{3}-\partial_{\rho}c^{3})\}-\chi_{B}^{II3\rho}\{-2\nabla^{\mu}\psi_{B\mu\rho}^{3}-\partial_{\rho}\xi^{3}\},
L_{eff}=\intd^{2}{\bfr}(\sum_{m}\partial_{t}\rho(\dot{\bfa}^{m}G^{m})\cdot{\bfr}-\rho\partial_{t}\omega-{\frac{1}{2}}\sum_{m}\rho(\dot{\bfa}^{m}G^{m})\cdot(\dot{\bfa}^{m}G^{m}))
\Sigma^{\prime\prime}|_{0}=(2\pii)^{3}6^{3}~\sum_{n\geq1}\left(-{\frac{q^{3n-1}(1+q^{3n-1})}{(1-q^{3n-1})^{3}}}+{\frac{q^{3n-2}(1+q^{3n-2})}{(1-q^{3n-2})^{3}}}\right)=(2\pii)^{3}6^{3}~\sum_{i\geq1}i^{2}~{\frac{q^{i}(1-q^{i})}{1-q^{3i}}}
\hat{T}{\bff}=0\;,\;\;\hat{T}=\frac{\imathsm}{2\hat{\omega}}\hat{\zeta}\Sigma^{3}\left[\hat{\zeta}\hat{\omega}\Sigma^{3}+\imath\partial_{1}(\imaths\Sigma^{2})+\imath\partial_{2}(-\imaths\Sigma^{1})-sm\right]\;.
H^{\mu\nu\rho(B)}={\frac{1}{3\sqrtg}}e^{2\phi+4\sigma/\sqrt3}\epsilon^{\mu\nu\rho\lambda\sigma}\biggl[(dB)_{\lambda\sigma}+\ell^{a}(dV)_{\lambda\sigma}^{b}L_{ab}+\ell^{2}(dA)_{\lambda\sigma}\biggr]\,.
\begin{array}{c}{\Omega=d\left(p_{\mu}dx^{\mu}+\theta_{i}^{\mu}g_{\mu\nu}D\theta_{i}^{\nu}\right)=}\\{=dp_{\mu}\wedgedx^{\mu}+\frac12R_{\mu\nu\lambda\rho}\theta_{i}^{\mu}\theta_{i}^{\nu}dx^{\lambda}\wedgedx^{\rho}}\\{+g_{\mu\nu}D\theta_{i}^{\mu}\wedgeD\theta_{i}^{\nu},}\\\end{array}
\beta^{2}=\frac{\gamma}{\pi}\quad,\quad\frac{e^{-2\Theta}}{1+e^{-4\Theta}}=S\quad,\quad\alpha=\alpha^{\prime}+\frac{i\pi}{2}\quad,\quad\alpha_{r}=\alpha_{r}^{\prime}+\frac{i\pi}{2}\,,
Q^{A}=\left(\begin{array}{cc}{u^{A}}&{i\overline{v}_{A^{*}}}\\{iv_{A}}&{\overline{u}^{A^{*}}}\\\end{array}\right)\qquad\overline{Q}^{A}=\left(\begin{array}{cc}{\overline{u}^{A^{*}}}&{-i\overline{v}_{A^{*}}}\\{-iv_{A}}&{u^{A}}\\\end{array}\right)\qquad\begin{array}{l}{u^{A}=q^{A|0}-iq^{A|3}}\\{V^{m}=-q^{A|1}-iq^{A|2}}\\\end{array}
\begin{array}{c}{(\epsilon_{i}-1)a_{i}^{2}=0}\\{a_{i}a_{j}=\kappa_{ij}q^{sgn(j-i)}a_{j}a_{i}}\\{a_{i}a_{j}^{+}=\kappa_{ij}qa_{j}^{+}a_{i},\qquadi\neqj}\\\end{array}
{\ddot{C}}_{0}(N_{v}^{2}+N_{s}^{2})=\intd^{3}x{{\itF}_{n}^{a}({\bfr})}^{(l)}{{\eta_{n}^{a}({\bfr})}_{0}}^{(k)}+\intd^{3}x{{\itF}^{a}({\bfr})}^{(l)}{\eta^{a}({\bfr})}_{0}^{(k)},
\left[\psi_{a,b}^{\left[A,B\right]}(x),\psi_{a^{^{\prime}},b^{^{\prime}}}^{\left[A,B\right]}(y)\right]_{\pm}=0\,\,\,\,\,\,\,\,for\,\,\,\,\,(\,x-y)^{2}<0
V(\phi,\chi)=\frac{1}{2}\lambda^{2}\left(\phi^{2}-\frac{m^{2}}{\lambda^{2}}\right)^{2}+\frac12\alpha^{2}\chi^{2}(\chi^{2}+4\phi^{2})+\alpha\lambda\chi^{2}(\phi^{2}-\frac{m^{2}}{\lambda^{2}}),
\frac{\langleR^{2}\rangle-\langleR^{2}\rangle_{0}}{\langleR^{2}\rangle_{0}}\approx(gN)^{2/3}{\calH}_{4}^{-1/6}f^{2}\frac{\langleT\left[(Tr\tilde{\Phi}^{2})(\tilde{S}_{1})^{2}\right]\rangle_{0}}{\langleTr\tilde{\Phi}^{2}\rangle_{0}},
\delta~\psi_{1}=\xi^{z}\partial_{z}\psi_{1}-(\partial_{z}\xi^{z})\psi_{1}+\xi^{zz}(\partial_{z}^{2}\psi_{1})-(\partial_{z}\xi^{zz})(\partial_{z}\psi_{1})-{\frac{2}{3}}((\partial_{z}^{2}-\tilde{u}_{2})\xi^{zz})\psi_{1}\quad,
c^{\prime\prime}+\frac{1}{r}c^{\prime}-(\frac{1+f^{2}}{2}+\frac{9+12a+4a^{2}}{r^{2}})c=2kf^{2}b-2(k^{\prime}+\frac{2k}{r})(\frac{1+2a}{r}b-b^{\prime}).
L_{+}^{n}|0,\mu_{m}\rangle=S(2\omegaT_{-})^{n}P_{m}=e^{-\omegaT_{-}}e^{-\frac{1}{2\omega}T_{+}}(2\omegaT_{-})^{n}P_{m}=L_{n}^{-2\mu_{m}-1}(2\omegaT_{-})|0,\mu_{m}\rangle.
\sum_{m=0}^{\infty}\frac{1}{m!}\left(\prod_{a=1}^{m+2}\int\!\frac{d^{p+1}k_{a}}{(2\pi)^{p+1}}\right)\,F_{\mu\nu}(k_{1})F^{\mu\nu}(k_{2})\;\left(\prod_{b=3}^{m+2}iM(k_{b})\right)\,J_{m+2}(k_{1},\cdots,k_{m+2})
ds_{BR}^{2}=-{\frac{M_{BR}^{2}}{\rho^{2}}}dt^{2}+{\frac{M_{BR}^{2}}{\rho^{2}}}(d\rho^{2}+\rho^{2}d\Omega)={\frac{M_{BR}^{2}}{|{\vec{y}}|^{2}}}(-dt^{2}+d\vec{y}^{2})\,
\deltaa_{2}=\partial_{\mu}\left(\frac{1}{2}f_{\;\;ab}^{c}A_{c}^{*\mu}\eta^{a}\eta^{b}\right)-\gamma\left(f_{\;\;ab}^{c}A_{c}^{*\mu}A_{\mu}^{b}\eta^{a}+y_{i}^{*}T_{\;\;a}^{i}\eta^{a}\right),
\Lambda^{\mu_{1}\dots\mu_{p-k}\alpha_{1}\dots\alpha_{k}i}\equivb_{k}\epsilon^{ij_{1}\dotsj_{p}}\Pi_{j_{1}}^{\mu_{1}}\dots\Pi_{j_{p-k}}^{\mu_{p-k}}\Pi_{j_{p-k+1}}^{\alpha_{1}}\dots\Pi_{j_{p}}^{\alpha_{k}}\quad.
\frac{\alpha^{2}}T\{\varphi_{n+1}^{\alpha}-2\varphi_{n}^{\alpha}+\varphi_{n-1}^{\alpha}\}-T\alpha^{2}\nabla^{2}\varphi_{n}^{\alpha}=_{c}T\mu_{n}\varphi_{n}^{\alpha},\;\;\;\;\;\varphi_{n}^{\alpha}\varphi_{n}^{\alpha}=_{c}1.
\sum_{\nu=0}^{N}\left[\omega_{\nu}^{2}\eta_{\nu\rho}+\sum_{k=1}^{N}\lambda_{k}\left(\eta_{k\rho}\delta_{0\nu}+\eta_{0\rho}\delta_{k\nu}\right)\right]\eta_{\nu\sigma}=\delta_{\rho\sigma}\Omega_{\sigma}^{2}\;.
\epsilon_{a_{1}\cdotsa_{p}}\frac{\partialX_{I}^{a_{1}}}{\partial\sigma_{I}^{1}}\cdots\frac{\partialX_{I}^{a_{p-2}}}{\partial\sigma_{I}^{p-2}}\left(U^{a_{p-1}}(X_{I})-\frac{\partialX_{I}^{a_{p-1}}}{\partialt}\right)=0\,\,,
X_{\alpha\beta}^{ij}\mid_{e=0}\equivx_{\alpha\beta}^{ij}=\left(\begin{array}{cc}{0}&{-((p.\Lambda)g^{ij}-p^{i}\Lambda^{0j})}\\{g^{ij}}&{{\frac{1}{2}}p_{0}(p.S)\epsilon^{ij}}\\\end{array}\right).
T_{3}=\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{-1}&{0}\\{0}&{0}&{0}\\\end{array}\right)\:,\:Y=\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{2}\\\end{array}\right)
{\bfB}({\bfr})=U_{2}^{-1}({\bfr})\left(\begin{matrix}{{\frac{\hat{\bfy}}{2y^{2}}}}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{-{\frac{\hat{\bfy}}{2y^{2}}}}\\\end{matrix}\right)U_{2}({\bfr})+O(b/y^{3})\,.
Z_{N}(x)\equiv\frac{N}{4}\left[\phi\left(x+\Theta,\frac{\pi\beta^{2}}{2}\right)+\phi\left(x-\Theta,\frac{\pi\beta^{2}}{2}\right)\right]+\sum_{r=1}^{l}\phi(x-\alpha_{r},\pi\beta^{2})+2\pi\beta^{2}\varepsilonl\,.
\frac{\partial_{1}\beta}{\beta}+\frac{\gamma}{8\Omega}+\frac{\beta^{2}}{2}\OmegaV(\Omega)-\frac{\Omega^{\delta}}{4}[(\partial_{1}f)^{2}+\frac{\beta^{2}}{\alpha^{2}}(\partial_{0}f)^{2}]=0
W_{ct}^{bk}={\frac{1}{16\piG}}\int_{\partialM_{r}^{d}}\sqrt{h}\left({\frac{2(d-2)}{l}}+{\frac{l}{d-3}}{\calR}+{\frac{l^{3}}{(d-5)(d-3)^{2}}}({\calR}_{ij}^{2}-{\frac{(d-1)}{4(d-2)}}{\calR}^{2})+...\right)
{\calS}_{WZW}\,=\,-\frac{k}{4\pi}\int_{\Sigma}d^{2}\sigmaA_{\alpha}^{I}A^{J\alpha}\Omega_{IJ}\,-\,\frac{k}{12\pi}\int_{B}d^{3}\sigma\,\epsilon^{\alpha\beta\gamma}A_{\alpha}^{I}A_{\beta}^{J}A_{\gamma}^{L}{f_{IJ}}^{N}\Omega_{NL}\,,
A(u)=\psi(u^{2})\frac{(q^{2}u^{4/n};t^{2},q^{4})_{\infty}(q^{4}u^{-4/n};t^{2},q^{4})_{\infty}}{(t^{2}u^{4/n};t^{2},q^{4})_{\infty}(t^{2}q^{2}u^{-4/n};t^{2},q^{4})_{\infty}},
\begin{array}{c}{G_{\rho\sigma\nu}^{abcd}=\beta_{1}(G_{\rho\sigma\nu}^{(1)}+G_{\rho\sigma\nu}^{(2)})-\beta_{2}(G_{\rho\sigma\nu}^{(1)}+G_{\rho\sigma\nu}^{(3)})}\\{+\beta_{3}(G_{\rho\sigma\nu}^{(3)}-G_{\rho\sigma\nu}^{(2)})=0}\\\end{array}
\int_{C_{4}^{0}}\zeta_{0,n^{\prime}+1}(z_{1},z_{2},w;\tau)dw=\sum_{j=0}^{\infty}\left(\int_{C_{6}^{0}}-\int_{C_{4}^{0}}\right)\zeta_{0,n^{\prime}-j}(z_{1},z_{2},w;\tau)dw
T_{nn}^{(2)}=-\frac{1}{2\pi^{2}}\sum_{m=-N}^{m=+N}\intd^{3}p\frac{\lambda_{m}^{2}}{{[{(\vec{p}-\vec{k})}^{2}+\mu^{2}]}^{2}[p^{2}-(K^{2}-\frac{{(n-m)}^{2}}{R^{2}})-i\epsilon]}.
\begin{array}{c}{F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+[A_{\nu},A_{\mu}]\,,}\\{\tilde{F}_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}+[B_{\nu},A_{\mu}]+[A_{\nu},B_{\mu}].}\\\end{array}
C_{\{d\}m}^{(0)}=n!\prod_{i=1}^{k}d_{i}(n-d_{i}+1),\quad\quadC_{\{d\}m}^{(j)}=-(j-1)\,C_{\{d\}m}^{(0)}\sum_{\sigma_{j}}\,\,\prod_{p\in\sigma_{j}}{\frac{(d_{p}-1)}{(n-d_{p}+1)}},
{\calE}\;\leq\;2^{N}\prod_{a=1}^{N}\sum_{n_{a}=0}^{\infty}\frac{(2\omega\beta^{(a)})^{2n_{a}}}{(2n_{a})!}\left\langle\prod_{b=1}^{N}\bigg[\chi_{+}^{(b)}[Q^{0,S}]\chi_{-}^{(b)}[Q^{0,S}]\bigg]^{n_{b}}\right\rangle_{Q^{0,S}}\;.
\phi_{1}=\frac{W+W^{*}}{1+WW^{*}}\,\,\,\,\,\,\,\,\,\phi_{2}=i\frac{W-W^{*}}{1+WW^{*}}\,\,\,\,\,\,\,\,\,\phi_{3}=\frac{1-WW^{*}}{1+WW^{*}}\,\,\,\,\,\,\,\,\,
2\kappa^{2}{\calL}=\sqrt{-g}\left(R+{\frac{1}{2}}m^{2}(T^{2}-2T_{ij}T^{ij})-Tr(P_{\mu}P^{\mu})-{\frac{1}{2}}(V_{I}{}^{i}V_{J}{}^{j}F_{\mu\nu}^{IJ})^{2}\right).
{}_{2}F_{1}(\alpha,\beta;\gamma;z)=\frac{\Gamma(\gamma)}{\Gamma(\alpha)\Gamma(\beta)}\int_{-i\infty}^{i\infty}\frac{dt}{2\pii}\frac{\Gamma(t+\alpha)\Gamma(t+\beta)\Gamma(-t)}{\Gamma(t+\gamma)}(-z)^{t},
H_{Susy}=p_{tA}p_{tA}+\frac{1}{2}\f_{ABC}f_{AB^{\prime}C^{\prime}}q_{sB}q_{tC}q_{sB^{\prime}}q_{tC^{\prime}}+iq_{tC}f_{ABC}\gamma_{\alpha\beta}^{t}\Theta_{\alphaA}\Theta_{\betaB}\,
D[f(\tau)]=D[g(\tau)]=\frac{AK_{1}}{2}\left(\frac{A}{K_{1}\dot{x}_{01}^{2}}-\frac{K_{1}\dot{x}_{01}^{2}}{A}\right)=\frac{AK_{2}}{2}\left(\frac{A}{K_{2}\dot{x}_{02}^{2}}-\frac{K_{2}\dot{x}_{02}^{2}}{A}\right).
ds^{2}=2dUdV-{\frac{1}{E^{2}}}g_{tt}dV^{2}+{\frac{2\mu}{E}}g_{tt}dVd\Phi+(E^{2}g_{\psi\psi}-\mu^{2}g_{tt})d\Phi^{2}+g_{ij}dY^{i}dY^{j}.
E_{1}:\quadI_{grav}^{(s)}+(\frac{6}{\pi^{2}})^{4}(x_{12}^{2}x_{34}^{2})^{-4}[64u^{-4}G_{45}(\frac{1}{u},\frac{v}{u})-32u^{-4}G_{44}(\frac{1}{u},\frac{v}{u})]
\{f,g\}_{1}=\frac{\partialf}{\partialz^{n}}\frac{\partial^{L}g}{\partial{\bar{\eta}}^{n}}+\frac{\partialf}{\partial{\bar{z}}^{n}}\frac{\partial^{L}g}{\partial\eta^{n}}-\frac{\partial^{R}f}{\partial{\bar{\eta}}^{n}}\frac{\partialg}{\partialz^{n}}-\frac{\partial^{R}f}{\partial\eta^{n}}\frac{\partialg}{\partial{\bar{z}}^{n}}
\tau_{0}=\frac12\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\,,\quad\tau_{1}=\frac12\left(\begin{array}{cc}{i}&{0}\\{0}&{-i}\\\end{array}\right)\,,\quad\tau_{2}=\frac12\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right)\,,
H_{NS}(DD)=\sum_{j}\left(N_{F}^{NS}(DD)-\frac{1}{24}\right)=\sum_{j}\left(\sum_{r=1/2}^{\infty}r({b_{-r}}^{-j}{b_{r}}_{j}+{b_{-r}}^{j}{b_{r}}_{-j})-\frac{1}{24}\right)
[E_{m}^{\alpha},E_{n}^{\beta}]=\left\{\begin{array}{ll}{\epsilon(\alpha,\beta)E_{m+n}^{\alpha+\beta}}&{if\\alpha+\beta\\is\a\root\of\{\bfg}}\\{\alpha\cdotH_{m+n}+m\delta_{m+n,0}}&{if\\\alpha+\beta=0}\\{0}&{otherwise}\\\end{array}\right\}
Z=\int\prod_{x}\left\{\mathrm{d}v_{\mu}\mathrm{d}j_{5}^{\mu}\,\mathrm{d}|h|^{2}\,\mathrm{d}\theta\,K[j_{5}^{\mu},|h|]\right\}\,e^{\mathrm{i}\,\int\mathrm{d}^{2}x\,{\calL}[v_{\mu},j_{5}^{\mu},\theta]},
\int_{\gamma_{n-1}}G_{\mu_{1}}G_{\mu_{2}}...G_{\mu_{n-1}}|\Psi\rangle=i\int_{S_{n}}P_{[\mu_{1}}G_{\mu_{2}}G_{\mu_{3}}...G_{\mu_{n}]}|\Psi\rangle=iQ\int_{S_{n}}G_{\mu_{1}}G_{\mu_{2}}...G_{\mu_{n}}|\Psi\rangle,
ds^{2}=-{\frac{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}{r^{2}}}dt^{2}+{\frac{r^{2}dr^{2}}{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}}+r^{2}\left(-{\frac{r_{+}r_{-}}{r^{2}}}dt+d\bar{\phi}\right)^{2}.
V=-{\frac{1}{16}}|z_{1}^{0}-z_{2}^{0}|^{4}(z_{1}^{0}-z_{3}^{0})^{2}(\bar{z}_{1}^{0}-\bar{z}_{3}^{0})(z_{2}^{0}-z_{3}^{0})(\bar{z}_{2}^{0}-\bar{z}_{3}^{0})^{2}.
{\calS}_{\mathrm{toy}}=\intd^{4}x\Biggl\{\frac{1}{2}\left[\hat{\Phi}_{1}^{\dagger}\hat{\Phi}_{1}\right]_{D}+\frac{1}{2}\left[\hat{\Phi}_{2}^{\dagger}\hat{\Phi}_{2}\right]_{D}+\biggl[m\hat{\Phi}_{1}\hat{\Phi}_{2}+g\hat{\Phi}_{1}\hat{\Phi}_{1}\hat{\Phi}_{2}\,\mathrm{+~h.c.}\biggr]_{F}\Biggr\}
(\mathrm{Mass})^{2}={\frac{1}{4\rho_{2}^{(0)}}}\left[{\frac{1}{2}}(\hat{\alpha}_{L})^{2}+N_{L}-1\right]={\frac{1}{4\rho_{2}^{(0)}}}\left[{\frac{1}{2}}(\hat{\alpha}_{R})^{2}+N_{R}-\delta\right]~.
S_{BH}={\frac{\pi^{2}(r_{+}^{2}+a^{2})r_{+}}{2G_{5}\Xi}},\,\,T_{BH}={\frac{r_{+}(1+{\frac{a^{2}}{l^{2}}}+2{\frac{r_{+}^{2}}{l^{2}}})}{2\pi(r_{+}^{2}+a^{2})}}.
\alpha_{A}(\varphi_{0})\simeq\alpha_{3}(\varphi_{0})+{\frac{\partial\widehat\sigma}{\partial\varphi_{0}}}\left({\frac{B}{M}}\right)_{A}+{\frac{\partial\widehat\delta}{\delta\varphi_{0}}}\left({\frac{D}{M}}\right)_{A}+a_{3}{\frac{\partial\alpha}{\partial\varphi_{0}}}\left({\frac{E}{M}}\right)_{A}\,
{\calL}=\bar{\psi}_{\alpha}(i\partial\!\!\!/-eA\!\!\!/)\psi_{\alpha}+\frac{1}{2}(\phi^{*}\bar{\psi}_{\alpha}^{c}\psi_{\alpha}-\phi\bar{\psi}_{\alpha}\psi_{\alpha}^{c})-\lambdaN\phi^{*}\phi\,\,\,\,.
\Gamma^{2}=\frac{1}{N^{2}}\,\left[(4!\alpha_{4})^{2}\,+\,\omega_{[0]}\omega_{[4]}\left(\frac{(\alpha_{2})^{2}}{3!\alpha_{0}\alpha_{4}}+2\right)\,+\,8(\alpha_{2})^{2}\,Tr(\hat{F}^{2})\right]
\begin{array}{l}{\delta\theta=\tilde{\Lambda}\tilde{\Pi}_{c}\epsilon^{c},\qquad\delta\phi=-i\phi^{2}(\bar{\psi}\tilde{\Pi}_{c}\epsilon^{c}),}\\{\deltax^{\mu}=4i(\Lambda\Pi_{c})(\bar{\theta}\Gamma^{\mu}\epsilon^{c})+2i(\bar{\theta}\tilde{\Pi}_{c}\epsilon^{c})\Lambda^{\mu},}\\\end{array}
\begin{array}{llllc}{}&{}&{}&{}&{\underline{{{\mathrm{irrep~of~}}~SU(3)}}}\\{\alpha=1}&{\phantom{6}}&{{\bfQ}_{1}^{L}=0}&{\phantom{6}}&{{\bf1}}\\{\alpha=2}&{}&{{\bfQ}_{1}^{L}=\tilde{e}_{2},-\tilde{e}_{1},\tilde{e}_{1}-\tilde{e}_{2}}&{}&{{\bf3}}\\{\alpha=3}&{}&{{\bfQ}_{1}^{L}=-\tilde{e}_{2},\tilde{e}_{1},\tilde{e}_{2}-\tilde{e}_{1}}&{}&{{\bf{\overline{{3}}}}}\\\end{array}
{\ddot{w}}+\frac{1}{2g}\frac{\partialg}{\partialw}{\dot{w}}^{2}+\biggl(\frac{\dot{\lambda}}{\lambda}+\frac{1}{g}\frac{\partialg}{\partial\sigma}\dot{\sigma}\biggr)\dot{w}+\frac{1}{2g}\frac{w+1}{(w-1)^{3}}=0,
\nu=\frac{g^{2}}{16\pi^{2}}\intd^{4}x\,\mathrm{tr}\,F_{\mu\nu}\tilde{F}_{\mu\nu},\quad\mathrm{where}\quad\tilde{F}_{\mu\nu}=\frac12\epsilon_{\mu\nu\rho\sigma}F_{\rho\sigma},
\frac{b_{0}^{j_{1}}}{L_{0}^{j_{1}}}|I\rangle_{i_{1}}\rangle_{j_{1}}\,\,\,\frac{b_{0}^{j_{2}}}{L_{0}^{j_{2}}}|I\rangle_{i_{2}}\rangle_{j_{2}}\,\dots\,\,\,\frac{b_{0}^{j_{N-3}}}{L_{0}^{j_{N-3}}}|I\rangle_{i_{N-3}}\rangle_{j_{N-3}}\,\,\,|Y_{1}\rangle_{1}\,\,\,|Y_{2}\rangle_{2}\,\dots\,\,\,|Y_{N}\rangle_{N}
\omega\left[\frac{E(x^{r-2s}b)}{E(x^{r-2s}/b)}\right]^{N}=-(A_{N-1}b)^{2s/r}\prod_{j=1}^{N-1}\frac{E(x^{r-2s}a_{j}/b)E(x^{r-4s}b/a_{j})}{E(x^{r-2s}b/a_{j})E(x^{r-4s}a_{j}/b)}.
{\mit\Xi}_{[1/2]}\,=\,\left(\begin{array}{cc}{e^{i\phi}}&{0}\\{0}&{e^{-i\phi}}\\\end{array}\right)\quad,\qquad{\mit\Xi}_{[1]}\,=\,\left(\begin{array}{ccc}{e^{i\,2\,\phi}}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{e^{-i\,2\,\phi}}\\\end{array}\right)\quad.
Y^{\mu}\leftrightarrow{\frac{1}{2}}\epsilon^{\mu\nu\rho\sigma}\partial_{\nu}B_{\rho\sigma}={\frac{1}{6}}\epsilon^{\mu\nu\rho\sigma}H_{\nu\rho\sigma}=\partial_{\nu}\{}^{*}B^{\mu\nu}\leftrightarrow\partial_{\nu}h^{\mu\nu}.
\begin{array}{lcr}{\varphi_{j}^{+}=X_{j}\eta_{j}^{+}+\gammaY_{j}\overline{{\eta}}_{j}^{+}}\\{\overline{{\varphi}}_{j}^{+}=\overline{{X}}_{j}\overline{{\eta}}_{j}^{+}-\gamma\overline{{Y}}_{j}\overline{{\eta}}_{j}^{+},}\\\end{array}
\int_{x}e^{-ipx}\langle\Psi_{L}^{n}(0)\bar{\psi}_{R}(x)\rangle=\int_{x}e^{-ipx}\left({\frac{1}{a}}\right)\langle\psi_{L}^{i}(0)\bar{\psi}_{R}(x)\rangle\langle\psi_{R}(0)\bar{\psi}_{L}^{i}(0)\rangle+O(g_{2}w(p)),
ds_{11}^{2}\,=\,\left(1+\frac{k}{r^{\tilde{d}}}\right)^{-\frac{\tilde{d}}{9}}\,dx^{\mu}\,\,dx^{\nu}\eta_{\mu\nu}+\,\,\left(1+\frac{k}{r^{\tilde{d}}}\right)^{\frac{d}{9}}\,dX^{I}\,\,dX^{J}\,\delta_{IJ}\,.
\left\{b_{1}\,\cdots\,b_{i_{l_{1}}}\right\}\left\{b_{i_{l_{1}+1}}\,\cdots\,b_{i_{l_{1}+l_{2}}}\right\}\cdot\cdot\cdot\left\{b_{i_{N-{l_{m}}+1}}\,\cdots\,b_{i_{N}}\right\}\,,
c(r)\simeq\frac{3}{\pi^{2}}\sum_{A}M_{A}\int_{0}^{\infty}d\theta\,e^{\theta+x}\,(L_{A}^{-}(\theta)+L_{A}^{+}(\theta))=\frac{3}{\pi^{2}}\sum_{A}M_{A}\int_{x}^{\infty}d\theta\,e^{\theta}\,(L_{A}^{\small{>}}(\theta)+L_{A}^{\small{<}}(\theta)).
\frac{1}{2}\,(z_{0}G\,(z_{0})z_{1}^{2}G^{\prime}\,(z_{1})z_{2}^{3}z_{3}^{3}+11\,\mathrm{moreterms})-2\,(z_{0}G\,(z_{0})z_{1}G\,(z_{1})z_{2}^{3}z_{3}^{3}+5\,\mathrm{more})\,\equiv\,0
\mathrm{d}s^{2}=U_{IJ}(x)\,\mathrm{d}\vec{x}^{I}\cdot\mathrm{d}\vec{x}^{J}+(U^{-1}(x))^{IJ}\Big(\mathrm{d}\varphi_{I}+\vec{W}_{IK}(x)\cdot\mathrm{d}\vec{x}^{K}\Big)\Big(\mathrm{d}\varphi_{J}+\vec{W}_{JL}(x)\cdot\mathrm{d}\vec{x}^{L}\Big)\.
{\calL}_{\sigma}={\frac{c}{2}}\left[\left(\partial_{+}\phi^{a}\right)^{2}-\left(\partial^{-}\phi^{a}\right)^{2}\right]+s\partial_{+}\phi^{a}\partial_{-}\phi^{a}-{\frac{1}{2}}(\partial_{\bot}\phi^{a})^{2}-{\frac{1}{2}}\sigma\phi^{a}\phi^{a}-{\frac{Nm_{0}^{2}}{g_{0}}}\sigma+{\frac{N}{2g_{0}}}\sigma^{2}.
\{F(\bar{\psi},\psi),G(\bar{\psi},\psi)\}=-\frac{i}{2}\intdx~g^{\alpha\beta}\left(\frac{\deltaF}{\delta\bar{\psi}_{\alpha}(x)}\frac{\deltaG}{\delta\psi_{\beta}(x)}-\frac{\deltaG}{\delta\bar{\psi}_{\alpha}(x)}\frac{\deltaF}{\delta\psi_{\beta}(x)}\right)
\left[{\frac{d^{4}}{dr^{4}}}+{\frac{2}{r}}{\frac{d^{3}}{dr^{3}}}-\left(M^{2}+{\frac{l(l+1)}{r^{2}}}\right){\frac{d^{2}}{dr^{2}}}-{\frac{2}{r}}M^{2}{\frac{d}{dr}}\right]c_{l}(Mr)=0,
{\calT}_{\mu\nu}=\frac{1}{1-{\frac{\gamma}{6}}|\Phi|^{2}}\left[{\calT}_{\mu\nu}^{(minimal)}+{\frac{1}{6}}\left(g_{\mu\nu}\nabla^{\rho}\nabla_{\rho}|\Phi|^{2}-\nabla_{\mu}\nabla_{\nu}|\Phi|^{2}\right)\right].
(\bar{\sigma}_{\nu\mu})_{u}^{~v}=\left(\begin{matrix}{0_{[N-2]\times[N-2]}}&{0_{[N-2]\times[2]}}\\{0_{[2]\times[N-2]}}&{(\bar{\sigma}_{\nu\mu})_{\dot{\alpha}}^{~\dot{\beta}}}\\\end{matrix}\right)~~,
\begin{array}{lllllllccl}{}&{r}&{p}&{s}&{t}&{u}&{v}&{x}&{y}&{w}\\{\lambda\}&{0}&{0}&{1}&{1}&{0}&{0}&{4}&{6}&{0}\\{\mu\}&{0}&{0}&{0}&{0}&{1}&{1}&{4}&{6}&{0}\\{\nu\}&{0}&{0}&{0}&{0}&{0}&{0}&{2}&{3}&{1}\\{\sigma\}&{0}&{1}&{1}&{0}&{q}&{0}&{2(q+2)}&{3(q+2)}&{0}\\{\xi\}&{1}&{0}&{1}&{0}&{0}&{q^{\prime}}&{2(q^{\prime}+2)}&{3(q^{\prime}+2)}&{0}\\\end{array}
W^{(2)}(x;B^{\prime},\psi_{0},\psi_{1})=\frac{1}{12}(B^{\prime}x_{0}^{12}+x_{3}^{12})+\frac{1}{3}x_{4}^{3}+\frac{1}{2}x_{5}^{2}-\psi_{0}\,x_{0}x_{3}x_{4}x_{5}-\frac{1}{6}\psi_{1}\,x_{0}^{6}x_{3}^{6}\,
\frac{1}{2}D^{\mu}\partial_{[u}G_{\nu\alpha]}{}^{\nu^{\prime}\alpha^{\prime}}(z,w)-m^{2}G_{\nu\alpha}{}^{\nu^{\prime}\alpha^{\prime}}(z,w)=-\delta(z,w)(g_{\nu}{}^{\nu^{\prime}}g_{\alpha}{}^{\alpha^{\prime}}-g_{\alpha}{}^{\nu^{\prime}}g_{\nu}{}^{\alpha^{\prime}}).
\begin{array}{c}{\left\{Q_{\alpha},Q_{\beta}\right\}=\left(S_{A}\right)_{\alpha\beta}}\\{\left(S_{A}\right)_{\alpha\beta}=\left(C\gamma^{M_{1}M_{2}}\right)_{\alpha\beta}\,\,\,Z_{M_{1}M_{2}}+\left(C\gamma^{M_{1}\cdotsM_{6}}\right)_{\alpha\beta}\,\,\,Z_{M_{1}\cdotsM_{6}}^{+}}\\\end{array}
\begin{array}{c}{(\ell^{\prime},h^{\prime})(h\alpha,h\dot{\alpha}+\ell(h\alpha))=(h^{\prime}h\alpha,h^{\prime}h\dot{\alpha}+(h^{\prime}\ellh^{\prime-1})(h^{\prime}h\alpha)+\ell^{\prime}(h^{\prime}h\alpha))}\\{=(h^{\prime}h\alpha,h^{\prime}h\dot{\alpha}+(\ell^{\prime}+Adh^{\prime}\ell)(h^{\prime}h\alpha))=}\\{(\ell^{\prime}+Adh^{\prime}\ell,h^{\prime}h)(\alpha,\dot{\alpha})}\\\end{array}
T_{\pm\pm}=\partial_{\pm}X^{\mu}\;\partial_{\pm}X_{\mu}+\frac{i}{2}\psi_{\pm}^{\mu,j}\;\partial_{\pm}\psi_{\pm\mu,j}-\frac{i}{2}\phi_{\pm}^{\muk}\;\partial_{\pm}\phi_{\pm\mu,k}=0
g=\left(\begin{array}{ccccc}{-R_{0}^{2}}&{0}&{0}&{\dots}&{0}\\{0}&{(g_{ij})}&{\dots}&{0}&{0}\\{0}&{\dots}&{1}&{\dots}&{0}\\{0}&{\dots}&{\dots}&{\ddots}&{0}\\{0}&{0}&{\dots}&{\dots}&{1}\\\end{array}\right)
\deltaq_{\mu\nu}dy^{\mu}dy^{\nu}=\intd^{3}{\bfk}\left[\sigma_{00}Ydt^{2}+2\sigma_{(L)0}V_{(L)i}dtdy^{i}+(\sigma_{(LL)}T_{(LL)ij}+\sigma_{(Y)}T_{(Y)ij})dy^{i}dy^{j}\right].
K(z,\bar{z})=z\bar{z}+\sum{\frac{1}{L_{R}^{n-2}}}K_{I_{1}..I_{p}\bar{I}_{p+1}..\bar{I}_{n}}z^{I_{1}}..z^{I_{p}}\bar{z}^{I_{p+1}}..\bar{z}^{n}
\left[D_{a}^{b},D_{c}^{d}\right]=\delta_{c}^{b}D_{a}^{d}-\delta_{a}^{d}D_{c}^{b}+\delta_{\bar{b}}^{d}D_{c}^{\bar{a}}-\delta_{c}^{\bar{a}}D_{\bar{b}}^{d}
\!\!\prod_{a}\beta_{a}^{-\mathrm{i}M_{a}^{z\bar{z}}}f({\bfX})=\prod_{a}\beta_{a}^{-\lambda_{a}}f({\bfX}(\beta))\,,\quad\!\!e^{-q(\zeta^{2}\bar{\zeta}+b\zeta)}f(\zeta)=(1+q\zeta)^{-b}f\Bigl(\frac{\zeta}{1+q\zeta}\Bigr)\,,
\begin{array}{lcl}{X_{n}^{(II)}(\tau_{0})}&{=}&{{\displaystylei\sum_{m=-\infty}^{\infty}\frac{1-e^{im\sigma_{0}}}{m\sigma_{0}-2\pin}X_{m}^{(I)}(\tau_{0})}}\\{}&{}&{}\\{{\dot{X}}_{n}^{(II)}(\tau_{0})}&{=}&{{\displaystylei\sum_{m=-\infty}^{\infty}\frac{1-e^{im\sigma_{0}}}{m\sigma_{0}-2\pin}{\dot{X}}_{m}^{(I)}(\tau_{0})}}\\\end{array}
S=\int_{0}^{1}\left[-\frac{1}{2e}\left(\dot{x}^{\mu}-i\psi^{\mu}\chi+i\varepsilon^{\mu\nu\rho\zeta}b_{\nu}\psi_{\rho}\psi_{\zeta}+{\frac{\alpha}{2}}b^{\mu}\right)^{2}-i\psi_{\mu}\dot{\psi}^{\mu}\right]d\tau\;,
\gamma(-\Box)=\frac{1}{12}\Biggl[\frac{8}{3}+\int_{4m^{2}}^{\infty}d\mu^{2}\,\frac{\Bigl(1-\frac{\textstyle4m^{2}}{\textstyle\mu^{2}}\Bigr)^{3/2}}{\mu^{2}-\Box}-\int_{0}^{\infty}d\mu^{2}\,\frac{1}{\mu^{2}+m^{2}}\Biggr]
\frac{3}{16}{\sqrtg}e^{2\phi}H_{\mu\nu\rho}^{i}{\calM}_{ij}^{-1}H^{j\mu\nu\rho}+\frac{1}{8}\epsilon^{\mu\nu\rho\lambda\sigma}{\calB}_{\mu}^{i}\biggl(\partial_{\nu}H_{\rho\lambda\sigma}^{j}-2\partial_{\nu}C_{\rho}\partial_{\lambda}{\calA}_{\sigma}^{j}\biggr){\calL}_{ij}\,,
\frac{p_{\alpha}(p+k_{1})_{\beta}(p+k_{1}+k_{2})_{\gamma}(p+k)_{\delta}}{p^{2}(p+k_{1})^{2}(p+k_{1}+k_{2})^{2}(p+k)^{2}}=\frac{p_{\alpha}p_{\beta}p_{\gamma}p_{\delta}}{p^{6}(p+k_{1})^{2}}+\ldots\;.
\left.\mathrm{Trln}{\calG}^{-1}\right\vert^{div}=\frac{iV_{4}}{(4\pir^{2})^{2}\epsilon}\left[-\frac{7574}{45}+\frac{67552}{315}(\Lambdar^{2})-\frac{13504}{135}(\Lambdar^{2})^{2}+\frac{1664}{81}(\Lambdar^{2})^{3}-\frac{128}{81}(\Lambdar^{2})^{4}\right].
{{\calG}^{-1}}_{\mu\nu}^{\alpha\beta}\eta_{\alpha\beta}^{\mathrm{Tr}}=\left(k^{2}-\Lambda_{l}+\frac{1}{2r^{2}}\frac{N(N-1)(N-2)}{N+d-2}-\frac{N+d}{N+d-2}\Lambda\right)\eta_{\mu\nu}^{\mathrm{Tr}}.
\frac{1}{(1-v^{2})^{n}}=\left(\frac{{\calE}}{m}\right)^{2n}(1+2n\frac{gM}{m^{2}}\frac{{\calE}}{m}BS_{B})=\left(\frac{{\calE}}{m}\right)^{2n}(1+2n\frac{M}{m}\left(\frac{{\calE}}{m}\right)^{2}S_{B}\omega).
{\frac{\delta}{\deltaA_{a}^{i}(x)}}U_{\alpha}[A]=\int_{0}^{1}dt\,\dot{\alpha}^{a}(t)\,\delta^{3}\left(x,\alpha(t)\right)\left[U_{\alpha}(0,t)T_{(n)}^{i}U_{\alpha}(t,1)\right]
\left[\widehat{{\mathcalJ}}_{\ell/r}^{0}\left(t,\underline{{x}}\right),~\widehat{{\mathcalJ}}_{\ell/r}^{0}\left(t,\underline{{y}}\right)\right]\;=\;\pm~i~\frac{q^{2}}{4\pi^{2}}~\left(\underline{{B}}\left(\underline{{x}},t\right)\,\cdot\,\underline{{\nabla}}\right)\;\delta\left(\underline{{x}}-\underline{{y}}\right)~,
\begin{matrix}{G_{\infty}}&{}&{\rightarrow}&{L_{\infty}}\\{}&{d}&{[\cdot,\cdot]}&{[\cdot,\cdot,\cdot]}&{\cdots}\\{\downarrow}&{\cdot}\\{\mathrm{comm.}~A_{\infty}}&{\{\cdot,\cdot,\cdot\}}\\{}&{\vdots}\\\end{matrix}.
a_{n}=\frac{\left(E+g-\frac{3}{2}\right)\left(E+g-\frac{n}{2}\right)}{n!~(2n+1)!!}\prod_{i=1}^{n}\left(E^{2}-E+2g-g^{2}+\frac{1}{4}-\left(i+1\right)^{2}\right),
H=-M^{11}-\left(\begin{array}{cc}{M^{12}}&{M^{21}}\\\end{array}\right)\left(\begin{array}{cc}{1-HM^{11}}&{-HM^{12}}\\{-HM^{21}}&{1-HM^{22}}\\\end{array}\right)^{-1}\left(\begin{array}{c}{HM^{21}}\\{HM^{12}}\\\end{array}\right),
h=\frac{1}{Q^{3}}\left(\begin{array}{c}{\begin{array}{cc}{-1-{y^{\prime}}^{2}}&{y^{\prime}{\phi}^{\prime}}\\{y^{\prime}{\phi}^{\prime}}&{1-{{\phi}^{\prime}}^{2}}\\\end{array}}\\\end{array}\right),\;\;{\hat{H}}^{\dagger}={\hat{H}},
\psi^{2}\hat{T}(G_{n}^{\lambda})=\sum_{r=0}^{\lambda-1}\sum_{i<j}(\frac{\Theta_{r}^{ij}(G_{n}^{\lambda})}{\hat{x}}+O(\hat{x}^{-2})):\hat{J}_{ij}^{(r)}\hat{J}_{ij}^{(-r)}:
G_{n}(x_{1},\dots,x_{n})=\left<\Omega\right|T\phi_{H}(x_{1})\dots\phi_{H}(x_{n})\left|\Omega\right>=\frac{\left<0\right|T\phi_{D}(x_{1})\dots\phi_{D}(x_{n})\;{\cal{S}}\left|0\right>}{\left<0\right|{\cal{S}}\left|0\right>}.
\delta\varphi^{rs}=\left[\eta_{i}^{\mu}\eta_{j}^{\nu}\delta^{ir}\delta^{js}-(\eta_{o}^{\mu}\eta_{j}^{\nu}-\eta_{o}^{\nu}\eta_{j}^{\mu})\frac{\delta^{ir}\delta^{js}p^{j}}{p^{o}+\sqrt{p^{2}}}\right]\delta\omega_{\mu\nu}.