harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\nabla_{z_{\alpha}}^{0}\left(\begin{array}{c}{\langlev|}\\{\hat{V}}\\\end{array}\right)=\frac{1}{\theta_{\alpha}}\left(\begin{array}{c}{\langlev|\widehat{\bar{z}}_{\alpha}}\\{-\lbrack\widehat{\bar{z}}_{\alpha},\hat{V}\rbrack}\\\end{array}\right)\,.
{\calL}=\frac{m^{2}}{2}f_{\mu}f^{\mu}-m^{2}f_{\mu}F^{\mu}+\frac{m^{2}}{2}F^{\mu}A_{\mu}-eJ_{\mu}f^{\mu}-eA^{\mu}g_{\mu}+\frac{\lambda}{2}(\partial_{\mu}A^{\mu})^{2}+{\calL}_{D},
\tilde{\phi}(g_{1},\ldots,g_{4})=\sum_{N_{1}\ldotsN_{4}}\\phi_{\alpha_{1}\ldots\alpha_{4}}^{N_{1}\ldotsN_{4}}\D_{\alpha_{1}\beta_{1}}^{(N_{1})}(g_{1})\ldotsD_{\alpha_{4}\beta_{4}}^{(N_{4})}(g_{4})\w_{\beta_{1}}\ldotsw_{\beta_{4}}.
(4.13)=\frac{2C_{1}C_{2}}{\pi}\int_{0}^{\infty}dw_{1}\;w_{1}^{-\frac{1}{2}}\left(w_{1}+{\sigma}_{1}^{2}\right)^{\frac{\lambda+m}{2}}\int_{0}^{\infty}dw_{2}\;w_{2}^{-\frac{1}{2}}\left(w_{2}+{\sigma}_{2}^{2}\right)^{\frac{\lambda+m}{2}}=
F^{\mu\nu}(x)\,=\,\sum_{\sigma=0,\pm1}\int\frac{d^{3}{\bfp}}{(2\pi)^{3}2E_{p}}\,\left[F_{(+)}^{\mu\nu}({\bfp},\sigma)\,a({\bfp},\sigma)\,e^{-ipx}+F_{(-)}^{\mu\nu}({\bfp},\sigma)\,b^{\dagger}({\bfp},\sigma)\,e^{+ipx}\right]
\left(T(v)y\right)_{\alpha}=\mathrm{Tr}\left(G_{1}(\alpha_{1})...G_{N}(\alpha_{N})\right),\quad\mathrm{with}\quad\left(G_{i}(\alpha)\right)_{\mu\nu}=\sum_{\beta}R_{(}\mu,\alpha;\nu,\beta)g_{i}(\beta).
\tilde{S}_{s,t}f=\Psi_{1}\left(\theta_{\tau_{1}}^{\mathbb{C}}\right)\Psi_{2}\left(\left(\theta_{\tau_{1}}^{\mathbb{C}}\right)^{-1}\theta_{\tau_{2}}^{\mathbb{C}}\right)\cdots\Psi_{n}\left(\left(\theta_{\tau_{n-1}}^{\mathbb{C}}\right)^{-1}\theta_{\tau_{n}}^{\mathbb{C}}\right),
g^{-1}={\frac{4}{\xi_{i}^{n}\xi_{j}^{m}\lambda_{,\bar{z}}^{i}\lambda_{,z}^{j}\{\sigma_{m},\sigma_{n}\}}}[d\bar{z}\otimesd\bar{z}+dz\otimesdz+\xi_{k}^{s}({\frac{\partial\sigma_{s}}{\partialp}}dp-{\frac{\partial\sigma_{s}}{\partialq}}dq)\otimesd\lambda^{k}].
<p_{1},\ldots,p_{l}|{\phi}(x)|0>_{tree}={\int}dp^{\prime}e^{ip^{\prime}x}{\delta}(p^{\prime}-{\sum}_{j}p_{j})(\ldots)=e^{i{\sum}_{j}p_{j}x}(\;a\;function\;of\;\{p_{j}\})
{\hat{\calT}}_{J_{1}}^{R_{1}}{\hat{\calT}}_{J_{2}}^{R_{2}}=\sum_{R,J}\left\langle\begin{array}{l}{R}\\{J}\\\end{array}\right.\left|\begin{array}{ll}{R_{1}}&{R_{2}}\\{J_{1}}&{J_{2}}\\\end{array}\right\rangle{\hat{\calT}}_{J}^{R}.
G_{\mu}\longrightarrowG_{\mu}^{\prime}=-\frac{i}{g}U(\pi)D_{\mu}U^{\dagger}(\pi)\;,\quad\zeta\longrightarrow\zeta^{\prime}=U(\pi)\zeta\;,\quad\zeta^{\dagger}\longrightarrow\zeta^{\dagger}=\zeta^{\dagger}U^{\dagger}(\pi)
T=2\pi\xi_{3}\,n+\int\!\mathrm{d}^{2}x\left\{\left[\frac1{2g}F_{ik}+\frac{g}{4}\left(|q|^{2}-\xi_{3}\right)\varepsilon_{ik}\right]^{2}+\frac12\left|\nabla_{i}\,q+i\varepsilon_{ik}\nabla_{k}\,q\right|^{2}\right\}.
\begin{array}{rclcrcl}{d{\calO}_{\alpha}}&{=}&{\{Q,{\calO}_{\alpha(1)}\},}&{}&{\widetilde{d}\widetilde{\calO}_{\widetilde{\alpha}}}&{=}&{\{\widetilde{Q},\widetilde{\calO}_{\widetilde{\alpha}(1)}\},}\\{d{\calO}_{\alpha(1)}}&{=}&{\{Q,{\calO}_{\alpha(2)}\},}&{}&{\widetilde{d}\widetilde{\calO}_{\widetilde{\alpha}(1)}}&{=}&{0,}\\{d{\calO}_{\alpha(2)}}&{=}&{0,}&{}&{}&{}&{}\\\end{array}
\arrowvertV_{A_{1}}(\theta_{1})V_{A_{2}}(\theta_{2})\cdotsV_{A_{n}}(\theta_{n})\rangle=V_{A_{1}}^{\dagger}(\theta_{1})V_{A_{2}}^{\dagger}(\theta_{2})\cdotsV_{A_{n}}^{\dagger}(\theta_{n})\arrowvert0\rangle.
\pi(x)={\frac{1}{\sqrt{2\pi}}}\left(p[T]-{\frac{i}{\sqrt{2}}}\sum_{k=1}^{\infty}\sqrt{k}[(a_{(+)k}[T]-a_{(-)k}^{\dagger}[T])e^{-ikx}-(a_{(+)k}^{\dagger}[T]-a_{(-)k}[T])e^{ikx}]\right),
S_{eff}=\int\!d^{4}x~[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^{2}(A_{\mu}+\partial_{\mu}\rho)^{2}-A^{\mu}\partial_{\mu}B+\frac{1}{2}\alpha(B)^{2}+\dot{{\overline{{\calP}}}}\frac{1}{\partial_{i}\partial^{i}}\dot{{\calP}}-{\overline{{\calP}}}{\calP}]
\delta^{(2)}(F_{\mu\nu}^{(0)})_{\overline{{\mathrm{m}}}}+\delta^{(1)}(F_{\mu\nu}^{(1)})_{\overline{{\mathrm{m}}}}+\delta^{(0)}(F_{\mu\nu}^{(2)})_{\overline{{\mathrm{m}}}}=ig[\delta\omega_{\mathrm{m}},(F_{\mu\nu}^{(1)})_{\overline{{\mathrm{m}}}}].
\widetilde{W}_{i}^{T}\simeq-\int_{0}^{\infty}{\frac{dt}{t}}{\frac{n_{i}}{(4\pit)^{D/2}}}\sum_{k=1}^{\infty}\eta_{i}^{k+1}e^{-{\frac{\beta^{2}k^{2}}{4t}}}\int_{\calM}dV(g_{\tau\tau})^{-D/2}e^{-m_{i}^{2}g_{\tau\tau}t}~~~.
G^{+}(x,x^{\prime})=\int\frac{d^{d}\vec{k}}{(2\pi)^{d}}\frac{\pi}{4H_{5}}H_{0}^{(2)}(|\vec{k}|(t-i\epsilon))H_{0}^{(1)}(|\vec{k}|(t+i\epsilon))e^{i\vec{k}\cdot(\vec{x}-\vec{x}^{\prime})}
j_{W}^{(L)}(x)=\left(\begin{array}{l}{\theta(x_{1}+L)\theta(L-x_{1})\;[\delta(x_{2}-L)-\delta(x_{2}+L)]}\\{\;}\\{\theta(x_{2}+L)\theta(L-x_{2})\;[\delta(x_{1}+L)-\delta(x_{1}-L)]}\\\end{array}\right)\;.
\mathbf{J}^{2}=\left(\begin{array}{cccc}{0}&{}&{}&{}\\{}&{\fbox{\ensuremath{\begin{array}{rr}{0}&{1}\\{0}&{0}\\\end{array}}}}&{}&{}\\{}&{}&{\fbox{\ensuremath{\begin{array}{rr}{0}&{1}\\{0}&{0}\\\end{array}}}}&{}\\{}&{}&{}&{\ddots}\\\end{array}\right).
\Phi^{\mu_{1}}\Phi^{\mu_{2}}\Phi^{\mu_{3}}\Phi^{\mu_{4}}\Phi^{m}F_{0\mu_{1}\mu_{2}\mu_{3}\mu_{4}m}=\frac{f}{\lambda^{1/2}}\Phi^{\mu_{1}}\Phi^{\mu_{2}}\Phi^{\mu_{3}}\Phi^{\mu_{4}}\epsilon_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}\Phi^{m}\frac{X^{m}}{|X|}
n^{\mu}{\calW}_{\mu}Z^{\mathrm{\small~c}}=-\intd^{2}x\,j_{b}^{a}\,\Big(\underbrace{-\,j_{c}^{a}+n^{\mu}\partial_{\mu}\frac{\deltaZ^{\mathrm{\small~c}}}{\deltaj_{\bar{c}}^{a}}}_{X^{a}}\Big)=0\quad,
\vartheta\approx-\frac{3}{4\pi}(-\psi_{s})^{-1/12}2^{1/6}\left(\begin{matrix}{\sigma_{+}^{-2/3}\tilde{\epsilon}^{1/3}}\\{\sigma_{-}^{-2/3}\tilde{\epsilon}^{1/3}}\\{\frac{B_{1}}{B_{2}}}\\{\frac{B_{2}}{B_{1}}(\sigma_{-}\sigma_{+})^{-2/3}\tilde{\epsilon}^{2/3}}\\\end{matrix}\right)\.
\left({\frac{a^{\prime}}{a^{3}}}\right)^{2}+k={\frac{M}{a^{2}\phi}}+{\frac{\omega}{6}}{\frac{\phi^{2}}{a^{4}\phi^{2}}}-{\frac{a^{\prime}\phi^{\prime}}{a^{5}\phi}}+{\frac{1}{6a^{4}}}\left(A^{2}+{\frac{B^{2}}{\phi}}\right),
\times\prod_{n>0}\frac{1}{1+e^{-4\pin\frac{l}{t}(1+\frac{1}{v})}-\frac{1+\frac{E^{2}}{\alpha\betav}}{1-\frac{E^{2}}{\alpha\betav}}(e^{-4\pin\frac{l}{t}}+e^{-4\pin\frac{l}{vt}})}.
g^{\mu\nu}=\left(\begin{array}{ccc}{1}&{\dot{z}}&{\dot{\bar{z}}}\\{\dot{z}}&{\dot{z}^{2}}&{\dot{z}\dot{\bar{z}}-2}\\{\dot{\bar{z}}}&{\dot{z}\dot{\bar{z}}-2}&{\dot{\bar{z}}^{2}}\\\end{array}\right),
S_{2}[x]={\frac{1}{4}}\int_{0}^{T}d\tau\dot{x}^{2}+i\int_{0}^{T}d\tau{\calJ}(\tau)\cdot\dot{x}(\tau)+\sum_{i=1}^{M}\alpha_{i}[x(\tau_{a_{i}})-x(\tau_{b_{i}})]^{2}
\sigma_{c}^{\mathrm{np}}=-\frac{2N_{f}}{\pi^{2}}\int_{0}^{\Lambda_{\mathrm{np}}}dp\frac{p^{2}\Sigma(p)}{p^{2}+\Sigma^{2}(p)}=\frac{N_{f}^{2}}{4}\Lambda_{\mathrm{np}}^{2}\frac{d\Sigma}{dp}\bigg|_{p=\Lambda_{\mathrm{np}}}.
S=\frac{1}{8\pi}\int_{\Sigma}d^{2}x\left\{\partial\varphi\bar{\partial}\varphi+e^{-\sqrt{\frac{2}{k}}\varphi}\left((\partial_{t}\tilde{\gamma}^{1})^{2}+(\partial_{x}\tilde{\gamma}^{1})^{2}\right)+2i\partial_{t}\tilde{\gamma}^{1}\right\},
\rho_{m}=\rho_{0}\left[\frac{(-1)^{m}}{m!}\sum_{n_{1},\ldots,n_{m}=0}^{\infty}(\prod_{j=1}^{m}\int_{j}\frac{{\calL}_{I}^{(n_{j})}[\widehat\varphi]}{n_{j}!})\,\langle\eta^{n_{1}}(1)\cdots\eta^{n_{m}}(m)\rangle\,\right]\;,
Res\biggl[E_{N}^{c{2}}(s,a_{1},...,a_{N}),\frac{j}{2}\biggr]=\frac{(-1)^{N-j}\pi^{\frac{j}{2}}}{2^{N}\Gamma(\frac{j}{2})}\sum_{k=0}^{\frac{N-j}{2}}\frac{(-1)^{k}}{k!}c^{2k}\pi^{k}A(2k+j)\,
\begin{array}{c}{\langlee_{a_{k}...a_{1}}\mid=(\mide_{a_{1}...a_{k}}\rangle)^{+}}\\{\langlee_{a_{k}...a_{1}}\mide_{b_{1}...b_{k}}\rangle=\delta_{a_{1}b_{1}}...\delta_{a_{k}b_{k}}}\\\end{array}
\psi_{i}(x)\rightarrowe^{-ig_{i}\frac{1}{\partial^{0}}A^{0}(x)}\psi_{i}(x)\,\\\\\\\,overline\psi_{i}(x)\rightarrowe^{ig_{i}\frac{1}{\partial^{0}}A^{0}(x)}\overline{\psi}_{i}(x)\\\\\(i=1,2)\,
m_{phys.}^{2}=m^{2}+g^{2}\deltam_{0}^{2}+\frac{g^{2}}{32\pi^{2}}\int_{0}^{1}dx\int_{{\calM}_{0}^{2}}^{\infty}d{\calM}^{2}\frac{f_{\lambda_{0}}^{2}}{{\calM}^{2}-m^{2}}=m^{2}+g^{2}\delta\tilde{m}_{0}^{2}\;.
\Phi_{1}^{G}=e+\zeta\pi_{0}^{-1}\;,\;\;\Phi_{2}^{G}=\chi\;,\;\;\Phi_{3\mu}^{G}=b_{\mu}\;,\;\;\Phi_{4}^{G}=x_{0}-\zeta\tau\;,\;\;\Phi_{5}^{G}=\psi^{0}\;,\;\;
\begin{array}{rll}{\mathrm{SU}(n+1):}&{x+y-P_{n+1}(v)=0,}&{2k=0}\\{\mathrm{SO}(2n):}&{x+y-P_{2n}(v^{2})=0,}&{2k=4}\\{\mathrm{SO}(2n+1):}&{x+y-P_{2n}(v^{2})=0,}&{2k=2}\\{\mathrm{Sp}(2n):}&{(x+y)^{2}-P_{2n}(v^{2})=0,}&{2k=-2}\\\end{array}
\begin{array}{c}{\{Q_{a\alpha},\bar{Q}_{\dot{\alpha}}^{b}\}=2\delta_{a}^{~b}\sigma_{\alpha\dot{\alpha}}^{\mu}P_{\mu}\,,}\\{{}}\\{{}[{P}_{\mu},{P}_{\nu}]=[{P}_{\mu},Q_{a\alpha}]=[P_{\mu},\bar{Q}_{\dot{\alpha}}^{a}]=\{Q_{a\alpha},Q_{b\beta}\}=\{\bar{Q}_{\dot{\alpha}}^{a},\bar{Q}_{\dot{\beta}}^{b}\}=0\,,}\\\end{array}
\sum_{p_{1},\ldots,p_{i}\inT_{2}}\sum_{s_{1},t_{1},\ldots,s_{G},t_{G}\inS_{n}}N^{n(2-2G)-i}\left[\frac{1}{n!}\delta(p_{1}\cdotsp_{i}\Omega_{n}^{2-2G}\prod_{j}s_{j}t_{j}s_{j}^{-1}t_{j}^{-1})\right]
\hat{K}=\left(\begin{array}{ccc}{\frac{B}{4}\delta^{\mu\nu,\alpha\beta}}&{0}&{0}\\{0}&{-\frac{B}{16}}&{-\frac{B_{1}}{4}}\\{0}&{-\frac{B_{1}}{4}}&{\frac{B_{1}^{2}}{2B}-A}\\\end{array}\right)
\psi^{\dagger}\gamma_{5}\psi=\left(\begin{array}{cc}{\phi^{\dagger}}&{\chi^{\dagger}}\\\end{array}\right)\left(\begin{array}{c}{-\chi}\\{-\phi}\\\end{array}\right)=-\left(\phi^{\dagger}\chi+\chi^{\dagger}\phi\right)=-\phi^{\dagger}\lambda\left|\vec{v}\right|\phi
L=\frac{1}{2}\left(\partial_{\mu}\varphi\right)^{2}+\frac{1}{2}m^{2}\varphi^{2}+\frac{\lambda}{4!}\varphi^{4}+\frac{1}{2}\xiR\varphi^{2}+\Lambda_{0}+\kappaR+\widetilde{a}_{1}R^{2}.
I_{\alpha}(z)=\frac{\Gamma(2\alpha)}{\Gamma^{2}(\alpha)}\int_{-i\infty}^{i\infty}\frac{dt}{2\pii}\frac{\Gamma^{2}(t+\alpha)\Gamma(-t)}{\Gamma(t+1-\alpha)(t+\alpha)}(-z)^{t}z^{\alpha}.
C_{2}=\frac{p_{r,n-1}-p_{s,n-1}}{p_{in}-p_{kn}}\sqrt{\frac{(p_{in}-p_{kn}+1)(p_{in}-p_{kn}-1)}{(p_{r,n-1}-p_{s,n-1}+1)(p_{r,n-1}-p_{s,n-1}-1)}}
F_{8}^{{\mathcal{O}}_{1,1}^{1,0}|+-------}=\frac{-iH^{1,1}}{8\prod_{2\leqi<j\leq8}\mathrm{cosh}(\theta_{ij}/2)}\prod_{i<j}F_{\mathrm{min}}^{\mu_{i}\mu_{j}}(\theta_{ij}),
K=\left(\begin{array}{lcccc}{}&{u_{1}}&{u_{2}}&{u_{3}}&{u_{4}}\\{m_{1}\,}&{1}&{0}&{0}&{0}\\{m_{2}\,}&{0}&{1}&{0}&{0}\\{m_{3}\,}&{0}&{0}&{1}&{0}\\{m_{4}\,}&{0}&{0}&{0}&{1}\\{m_{5}\,}&{1}&{1}&{-1}&{0}\\{m_{6}\,}&{1}&{1}&{0}&{-1}\\\end{array}\right).
+\frac{e^{2}\hbar^{2}}{2\pi\mu^{2}m^{2}c^{2}}\frac{1}{\varrho^{2}}\left[\frac{2}{\varrho^{2}}\vec{\varrho}\cdot(\vec{\varrho}\cdot\vec{p})\vec{p}-\vec{p}^{2}\right]+\delta^{2}(\vec{\varrho})\left[\frac{-e^{2}\hbar^{2}}{2m^{2}c^{2}}+\frac{e^{2}\hbar^{2}\vec{p}^{2}}{2\mu^{2}m^{2}c^{2}}\right]~~,
\begin{array}{rcl}{|\psi_{i}>}&{=}&{|q_{i}>\otimes|p_{\lambda}=0>\otimes|-+>=\sum_{n=0}^{\infty}|n>\otimes|p_{\lambda}=0>\otimes|-+><n|q_{i}>\\\,}\\{|\psi_{f}>}&{=}&{|q_{f}>\otimes|p_{\lambda}=0>\otimes|-+>=\sum_{n=0}^{\infty}|n>\otimes|p_{\lambda}=0>\otimes|-+><n|q_{f}>\,}\\\end{array}
{\calL}_{gf}={\frac{1}{2\alpha}}(h_{\mu\rho,}{}^{\mu}-{\frac{1}{2}}h_{,\rho})(h^{\nu\rho}{}_{,\nu}-{\frac{1}{2}}h_{,}{}^{\rho})+{\frac{1}{2\beta}}\sigma_{,\mu}\sigma_{,}{}^{\mu},
{\frac{\psi_{*}}{M_{s}}}=(8\betaN)^{1/4}\,(2\piR\,M_{s})^{3/2}\,g_{s}^{-1/2}=1.48\times10^{4}\,{\frac{(M_{s}Lg_{s})^{1/2}}{(2\piR\,M_{s})^{3}}}\,,
{\calT}={\frac{1}{{\mathrm{dim}}(G)}}\sum_{a=1}^{{\mathrm{\small{dim}}}(G)}{\calT}_{a}={\frac{1}{{\mathrm{dim}}(G)}}\sum_{a=1}^{{\mathrm{\small{dim}}}(G)}{\mathrm{Tr}}\left(g_{a}q^{L_{0}}{\overline{q}}^{{\overline{L}}_{0}}\right)~.
{\calL}_{mod.Sk}={\frac{-F_{0}^{2}}{4}}Tr(L_{\mu}L^{\mu})+{\frac{1}{32e^{2}}}Tr[L_{\mu}\,,\,L_{\nu}]^{2}+{\frac{\gamma}{8e^{2}}}Tr(L_{\mu}L^{\mu})^{2}\,,
\langle\lambda^{\prime}|\langle\mu|\left\{[B_{1}^{-}(\zeta),[B_{1}^{+}(\zeta),\widetildeM^{-}(-z)]]+[A_{1}(\zeta),\widetildeM^{-}(-z)]+M^{+}(-z)\right\}=\langle\lambda^{\prime}|\langle\mu|iS(\lambda,\mu)\,,
L_{2k}(v)\tilde{Z}_{21}^{-1}(u,v)L_{1j}(u)=\tilde{Z}_{21}^{-1}(u.v)L_{1j}(u)\tilde{Z}_{21}(u,v)L_{2k}(v)\tilde{Z}_{21}^{-1}(u,v)\ll{zlzl2u}
\left.\begin{array}{l}{\partial_{\phi}F^{a}=0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\partial_{\pi_{\phi}}G^{a}=0,}\\{F^{b}\partial_{b}G^{a}-G^{b}\partial_{b}F^{a}+\pi_{\phi}\partial_{\phi}G^{a}-2(e^{\phi}-e^{-2\phi})\partial_{\pi_{\phi}}F^{a}=0,}\\\end{array}\right.
L_{\theta}=\kappa\biggl[G_{\mu\nu}^{3}{\tilde{G}}^{\mu\nu3}+G_{\mu\nu}^{8}{\tilde{G}}^{\mu\nu8}+W_{\mu\nu}^{3}{\tilde{W}}^{\mu\nu3}+Y_{\mu\nu}{\tilde{Y}}^{\mu\nu}\biggr]
J_{\Gamma}(x)=\bar{\chi}(x)\Gamma\chi(x)\equiv\frac{1}{8\pi}\intd\hat{\epsilon}e^{i\omega_{\Gamma}(\hat{\epsilon})}\left\{\left[\chi^{\dagger}(x+\epsilon),\gamma^{0}\Gamma\chi(x-\epsilon)\right],e^{ie\int_{x-\epsilon}^{x+\epsilon}dx_{i}A_{i}^{tr}}\right\}_{|\epsilon|\propto|\eta|}
C_{\pm}\left[\dot{F}\pm2\kappa_{5}^{2}e^{-A-\alpha_{\pm}^{(0)}}(\ddot{A}+\dot{A}^{2})\bar{\tau}_{\pm(LL)}\right]+\eta^{\mu\nu}k_{\mu}k_{\nu}\dot{\Psi}^{(0)}\left[F\mp\kappa_{5}^{2}{\alpha^{\prime}}_{\pm}^{(0)}e^{-A-\alpha_{\pm}^{(0)}}\dot{\Psi}^{(0)}\bar{\tau}_{\pm(LL)}\right]=0,
{\calL}=\bar{\psi}\gamma^{\mu}(i\nabla_{\mu}+A\omega_{\mu}+a_{\mu})\psi+\frac{g^{2}}{4}\bar{\psi}\gamma^{\mu}\psi\bar{\psi}\gamma_{\mu}\psi-\epsilon^{\mu\nu}\rho(\partial_{\mu}a_{\nu}-\frac{B}{2}\epsilon_{ab}e_{\mu}^{a}e_{\nu}^{b}),
\breve{a}_{0,2}=\int\sqrt{h}d^{D-1}x\left({\frac{1}{72}}\bar{R}^{2}-{\frac{1}{180}}\bar{R}_{\mu\nu}\bar{R}^{\mu\nu}+{\frac{1}{180}}\bar{R}_{\mu\nu\lambda\rho}\bar{R}^{\mu\nu\lambda\rho}-\frac16\bar{R}V+\frac12V^{2}\right),
f_{\alpha\beta}^{(1)}(x,y)=\left(\begin{array}{ccc}{f_{\hat{\alpha}\hat{\beta}}}&{O}&{\vec{m}}\\{O}&{f_{\mu\nu}^{GF}}&{\vec{m}_{GF}}\\{-\vec{m}^{T}}&{-\vec{m}_{GF}^{T}}&{0}\\\end{array}\right)\delta(x-y),
I_{TOP}=\mathrm{Tr}\int_{X}\left(\tauR^{+}\wedgeR^{+}+\overline{\tau}R^{-}\wedgeR^{-}\right)=\mathrm{Tr}\int_{X}\left(\tauR^{+}\wedgeR^{+}+\overline{\tau}\overline{{R^{+}}}\wedge\overline{{R^{+}}}\right),
\rho(x_{1},\ldots,x_{p};\{m\})=(-1)^{p(p-1)/2}\frac{\mathrm{Pf}\left[\begin{array}{cccc}{-I^{II}}&{Q^{I}}&{-I^{IR}}&{S^{IR}}\\{-(Q^{I})^{T}}&{0}&{-(Q^{R})^{T}}&{-(P^{R})^{T}}\\{(I^{IR})^{T}}&{Q^{R}}&{-I^{RR}}&{S^{RR}}\\{-(S^{IR})^{T}}&{P^{R}}&{-(S^{RR})^{T}}&{D^{RR}}\\\end{array}\right]}{\mathrm{Pf}\left[\begin{array}{cc}{-I^{II}}&{Q^{I}}\\{-(Q^{I})^{T}}&{0}\\\end{array}\right]}.
\hat{U}(k_{1},k_{2})=\e^{i\pikk_{1}k_{2}}\,e^{ik\left[k_{1}\hat{A}_{2}-k_{2}\hat{A}_{1}\right]}=\e^{ikk_{1}\hat{A}_{2}}\,e^{-ikk_{2}\hat{A}_{1}}\\\.
S_{BF}[e,\omega,\phi]=\intd^{2}x\,\left[-\phi^{0}\left(d\omega-\frac{\Lambda}{2}e^{a}\wedgee^{b}\epsilon_{ab}\right)+\Lambda\delta_{ab}\phi^{a}\left(de^{b}+\omega_{\,\,c}^{b}\wedgee^{c}\right)\right].
\sum_{n>0}\frac{2n^{2}+m^{2}\alpha^{\prime}\lambda}{\sqrt{n^{2}+m^{2}\alpha^{\prime}\lambda}}=-\frac{1}{6}+\frac{(m^{2}\alpha^{\prime})^{2}}{4}\zeta(3)\lambda^{2}-\frac{(m^{2}\alpha^{\prime})^{3}}{4}\zeta(5)\lambda^{3}+{\calO}((m^{2}\alpha^{\prime}\lambda)^{4}).
E_{j}^{(b)}=e^{D\sigma(z_{j})}\int_{z_{1}}^{z_{2}}\langleT_{0}^{0}\rangle_{{\mathrm{ren}}}^{(b)}e^{-(D+1)\sigma(z)}dz=-\frac{J_{D}(B_{1},B_{2})e^{D\sigma(z_{j})}}{2^{D}\pi^{D/2}\Gamma(D/2+1)a^{D}}.
\frac{F}{V}=-\sum_{n,p,q}\int_{\calS}d\tau_{1}\frac{d\tau_{2}}{2\tau_{2}}\frac{\left|\eta\left(\tau\right)\right|^{-48}}{(4\pi^{2}\alpha^{\prime}\tau_{2})^{13}}e^{-S_{w.m}(n,p,q)}
f_{n}(y)=N_{n}e^{2\sigma}\left(J_{\nu}\left({\frac{m_{n}}{k}}e^{\sigma}\right)-\left({\frac{j_{\nu}({\frac{m_{n}}{k}})}{y_{\nu}({\frac{m_{n}}{k}})}}\right)Y_{\nu}\left({\frac{m_{n}}{k}}e^{\sigma}\right)\right)
\frac{{\calH}_{+}^{\prime}-2{\calH}_{+}^{2}}{{\calH}_{+}^{\prime}-{\calH}_{+}^{2}}D_{+}=\frac{{\calH}_{-}^{\prime}-2{\calH}_{-}^{2}}{{\calH}_{-}^{\prime}-{\calH}_{-}^{2}}D_{-}\,.
S_{\mathrm{ferm}}=\int\!d^{2}x\,\Bigl(i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi+{\frac{\piF_{e}}{\mu}}\,\bar{\psi}\psi+{\frac{\piF_{m}}{\mu}}\,\bar{\psi}^{C}\psi-{\frac{G}{2}}\,\bar{\psi}\gamma^{\mu}\psi\,\bar{\psi}\gamma_{\mu}\psi\Bigr)
\left({\frac{(2\pi)^{3}}{\delta^{3}\!k\delta^{3}\!x}}\right)^{N}\left({\frac{\delta^{3}\!x}{(2\pi)^{3/2}}}\right)^{2N}=\left(\frac{\delta^{3}\!x}{\delta^{3}\!k}\right)^{N}
\rho_{\alpha\beta}^{(0)}=\left(\begin{array}{ccccc}{0}&{-m\delta_{ij}}&{e\delta_{ij}\delta(\vec{y}-\vec{q})}&{0}&{0}\\{m\delta_{ij}}&{0}&{0}&{0}&{0}\\{-e\delta_{ij}\delta(\vec{y}-\vec{q})}&{0}&{0}&{\theta\delta_{ij}\delta(\vec{x}-\vec{y})}&{0}\\{0}&{0}&{-\theta\delta_{ij}\delta(\vec{x}-\vec{y})}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\\end{array}\right)
H_{0}^{\prime}\equivH_{\left(s\right)0}^{\prime}\equivP_{\left(s\right)0}+H_{\left(s\right)0}\left(t,t_{\left(u\right)\alpha};q_{\left(u\right)a};p_{\left(u\right)a}=\frac{\partialS}{\partialq_{\left(u\right)a}}\right)=0,
K=-\frac{1}{2\thetag^{2}}\sum_{\alpha=1}^{d/2}\mathrm{Tr}[a_{\alpha},\Phi_{0}][a_{\alpha}^{\dagger},\Phi_{0}]=\frac{\phi_{1}^{2}}{\thetag^{2}}\sum_{\alpha=1}^{d/2}\mathrm{Tr}\Bigl(F_{\alpha}^{\dagger}F_{\alpha}+P\Bigr)
\begin{array}{l}{X_{1,1}=\beta^{2}\,\X_{2,4}=\beta^{2}/4\,\}\\{X_{3,3}=\beta^{2}/24-\beta^{4}/12\,\X_{3,5}=-\beta^{2}/2+\beta^{4}/4\,}\\{X_{3,7}=\beta^{4}/16\.}\\\end{array}
\Pi_{\mu\nu}(p)=2N_{c}\lambda\int\Gamma_{\mu}(p,x)\Gamma_{\nu}^{+}(p,x)d^{4}x+2N_{c}\lambda^{2}P\int\Gamma_{\mu}(p,x)\frac{S(x)}{1-\lambdaS(x)}\Gamma_{\nu}^{+}(p,x)d^{4}x
{\calD}_{\alpha}T_{\mu_{1}\ldots\mu_{n}}^{\nu_{1}\ldots\nu_{p}}\equiv\perp_{\mu_{1}}^{\lambda_{1}}\ldots\perp_{\mu_{n}}^{\lambda_{n}}\perp_{\sigma_{1}}^{\nu_{1}}\ldots\perp_{\sigma_{p}}^{\nu_{p}}\perp_{\alpha}^{\beta}\nabla_{\beta}T_{\lambda_{1}\ldots\lambda_{n}}^{\sigma_{1}\ldots\sigma_{p}},
\begin{aligned}{D_{n}\left(ig^{m\bar{n}}F_{m\bar{n}}^{1}\right)}&{{}=-eD_{n}(TT^{*})}\\{D_{n}\left(ig^{m\bar{n}}F_{m\bar{n}}^{2}\right)}&{{}=eD_{n}(T^{*}T)\,}\\\end{aligned}
\begin{array}{rcl}{t_{k\pm1}^{k}(u)}&{=}&{f(u)\bar{t}_{k\pm1}^{k}(u),}\\{\bar{t}_{k\pm1}^{k}(u)}&{=}&{(\sqrt{-1})^{k}(\theta_{1}\theta_{4})\left(\frac{k\mpu}{2r};\frac{\pi\sqrt{-1}}{\epsilonr}\right)v_{+}+(\sqrt{-1})^{-k}(\theta_{2}\theta_{3})\left(\frac{k\mpu}{2r};\frac{\pi\sqrt{-1}}{\epsilonr}\right)v_{-}.}\\\end{array}
\Omega=\int\!dx\,K_{i\bar{j}}\left(\delta\phi_{R}^{i}\delta(\bar{\phi}_{R}^{\bar{j}}+2a^{\bar{j}})+\delta\bar{\phi}_{R}^{\bar{j}}\delta(\phi_{R}^{i}+2a^{i})+\delta\xi_{+}^{i}\delta\xi_{+}^{\bar{j}}-\delta\xi_{-}^{i}\delta\xi_{-}^{\bar{j}}\right).
7\frac{dS_{k}}{du}-\frac{k\beta_{0}}{1-\beta_{0}u}S_{k}=\frac{(1-\delta_{k,1})}{1-\beta_{0}u}\sum_{\ell=1}^{k-1}\beta_{\ell}\left(u\frac{d}{du}+k-\ell\right)S_{k-\ell}(u),\;\;S_{n}(0)=T_{n,0}.
\sigma^{1}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right),\sigma^{2}=\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right),\sigma^{3}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right).
D_{{\mu}{\nu}}^{-}(k)=\frac{i}{k^{2}+i{\epsilon}}\{-g_{{\mu}{\nu}}+\frac{n_{\mu}k_{\nu}+n_{\nu}k_{\mu}}{k_{-}+i{\epsilon}\mathrm{sgn}(k_{+})}-\frac{n^{2}k_{\mu}k_{\nu}}{(k_{-}+i{\epsilon}\mathrm{sgn}(k_{+}))^{2}}\}.
\intD[C]\int_{0}^{1}dsA[C]\left[{\frac{\hbar^{2}}{2}}{\frac{\delta^{2}A[C]}{\delta\sigma^{\mu\nu}(s)\delta\sigma_{\mu\nu}(s)}}-A[C]\left({\frac{1}{2}}F_{\mu\nu}\left(x(s)\right)F^{\mu\nu}\left(x(s)\right)+m^{4}\right)\right]=0\.
Z_{(a,m)|(a^{\prime},m^{\prime})}(q)=\sum_{(a^{\prime\prime},m^{\prime\prime})}\hat{N}_{(a,m)^{*}\;(a^{\prime},m^{\prime})}{}^{(a^{\prime\prime},m^{\prime\prime})}\hat{\chi}_{(a^{\prime\prime},m^{\prime\prime})}(q)
h_{0}\propto\frac{1}{\sqrt{2n}}\quad,\quad\dot{h}_{0}\propto\sqrt{\frac{n}{2}}\quad,\quad\ddot{h}_{0}\propto\frac{n^{3/2}}{\sqrt{2}}\quad,\quad{\stackrel{...}h}_{0}\propto\frac{n^{5/2}}{\sqrt{2}}\quad.
\begin{array}{rl}{{\calN}=}&{L-u,}\\{{\calA}^{+}=}&{R_{x}+iR_{y},}\\{{\calA}=}&{R_{x}-iR_{y},}\\{{\calA}^{+}{\calA}=}&{\mu^{2}+2H\left(L-1/2\right)^{2}-\lambda\left(L-1/2\right)^{2}\left(\left(L-1/2\right)^{2}-1/4\right)}\\{=\mu^{2}+}&{2H\left({\calN}+u-1/2\right)^{2}-\lambda\left({\calN}+u-1/2\right)^{2}\left(\left({\calN}+u-1/2\right)^{2}-1/4\right)}\\{=}&{\Phi(H,{\calN}).}\\\end{array}
\delta_{gt}|\Phi_{ph}\rangle=(\alpha^{I}\hat{\partial}^{I}+(s-1)\alpha^{z})|\Lambda_{0}\rangle-\frac{\alpha_{J}^{2}}{2s+d-4}\Bigl((\bar{\alpha}^{I}\hat{\partial}^{I}+(s-1)\bar{\alpha}^{z})|\Lambda_{0}\rangle+\hat{\partial}^{+}|\Lambda_{1}\rangle\Bigr)\,.
x_{A}^{\alpha\beta}=x^{\alpha\beta}-i\theta^{\alpha(i}\theta^{\betaj)}(u_{i}^{1}u_{j}^{4}+u_{i}^{2}u_{j}^{3})\;,\quad\theta^{1\alpha}=\theta_{4}^{\alpha}=\theta_{i}^{\alpha}u_{4}^{i}\;,\\theta^{2\alpha}=\theta_{3}^{\alpha}=\theta_{i}^{\alpha}u_{3}^{i}\;.
{\calS}_{YM}=-\,\frac{1}{g_{\mathrm{YM}}^{2}}\intd^{4}\xi\left[\frac{1}{4}F_{\mu\nu}^{A}F_{A}^{\mu\nu}+\frac{1}{2}\,D_{\mu}\bar{\phi}^{A}D^{\mu}\phi_{A}\right]+\frac{\theta_{\mathrm{YM}}}{32\pi^{2}}\intd^{4}\xi\;F_{\mu\nu}^{A}{\tilde{F}}_{A}^{\mu\nu}\;+\;\mathrm{ferm.}
E_{2}^{p,q}\:=\:H^{p}\left(T,{\calE}^{\vee}|_{T}\otimes{\calF}\otimes\Lambda^{q}N_{S/X}|_{T}\right)\:\Longrightarrow\:\mathrm{Ext}_{X}^{p+q}\left(i_{*}{\calE},j_{*}{\calF}\right).
\left\{-\frac{\partial^{2}}{\partialx_{1}^{2}}-\frac{\partial^{2}}{\partialx_{2}^{2}}+{\frac{\omega^{2}}{8}}\left(x_{1}-x_{2}\right)^{2}+{\frac{2\alpha}{\left(x_{1}-x_{2}\right)^{2}}}+\Omega\left(\frac{x_{1}-x_{2}}{\sqrt{2}}\right)\right\}\psi(x_{1},x_{2})=E\psi(x_{1},x_{2}).
\Pi^{\alpha}=\mathcal{P}_{+}d\theta^{\alpha}\,,\qquad\Pi^{\mu}=dx^{\mu}+{\frac{1}{2}}(C\Gamma^{\mu})_{\alpha\beta}\theta^{\alpha}\Pi^{\beta}\,,\qquad\Pi^{\mu_{1}\dots\mu_{5}}=d\varphi^{\mu_{1}\dots\mu_{5}}+{\frac{1}{2}}(C\Gamma^{\mu_{1}\dots\mu_{5}})_{\alpha\beta}\theta^{\alpha}\Pi^{\beta}\,.
\Lambda_{a}^{\mu}(p,\bar{p})=\lambdae^{3}\int{\frac{d^{2}k_{\perp}}{(4\pi)^{3}}}\int{\frac{dk^{+}}{k^{+}k^{\prime+}k^{\prime\prime+}}}{\frac{N_{a}^{\mu}+yN_{b}^{\mu}}{(p^{-}-k^{-}-k^{\prime-})(p^{-}-k^{-}-k^{\prime\prime-}-q^{-})}}\;,
t_{ij}=\sum_{k}\left[c_{(T)}T_{(T)ij}+c_{(LT)}(D_{i}V_{(T)j}+D_{j}V_{(T)i})+c_{(LL)}(D_{i}V_{(L)j}+D_{j}V_{(L)i})+c_{(Y)}Y\Omega_{ij}\right],
\longrightarrowH^{g+1}(F)\stackrel{pr}{\longrightarrow}H^{g}(F_{rel})\stackrel{\{Q,c_{0}\}}{\longrightarrow}H^{g+2}(F_{rel})\stackrel{i}\longrightarrowH^{g+2}(F)\stackrel{pr}{\longrightarrow}H^{g+1}(F_{rel})\longrightarrow.
\begin{array}{c}{r_{2k+\varsigma,\gamma+2n+\varsigma}=0,\;\;k<0\;or\;k>n;}\\{r_{2k+\varsigma,\gamma+2n+\varsigma}=\frac{(-1)^{k}}{(2k+\varsigma)!}\frac{(2n)!!(2n+2\gamma+2\varsigma+2k-2)!!}{(2n-2k)!!(2n+2\gamma+2\varsigma-2)!!}\;r_{\varsigma,\gamma+2n+\varsigma}\equiv}\\{\equivc_{2k+\varsigma,\gamma+2n+\varsigma}\;r_{\varsigma,\gamma+2n+\varsigma};}\\{k=0,1,2,...,n-1,n,}\\\end{array}
u_{2\epsilon\kappa}^{\prime\prime}-\left[\frac{\kappa\left(\kappa-1\right)}{r^{2}}+\frac{\kappa}{r}\frac{V^{\prime}}{\epsilon-m-V}-V^{2}+2\epsilonV-k^{2}\right]u_{2\epsilon\kappa}+\frac{V^{\prime}}{\epsilon-m-V}\,u_{2\epsilon\kappa}^{\prime}=0\,.
\tilde{{\calL}}^{\prime}=\bar{\Psi}(i\!\not{\!\partial}+J)\Psi-\frac{1}{2G_{S}}\sigma^{2}-\frac{1}{\sqrt{N}}\sigma\bar{\Psi}\Psi+\frac{1}{2G_{V}}\tilde{A}_{\mu}^{2}+\frac{1}{\sqrt{N}}\bar{\Psi}\tilde{\not{\!\!A}}\Psi,
P_{2KN+2K-2}(x)=\frac{\widetilde{\Lambda}^{2KN+2K}}{x^{2}}T_{K}\left(\frac{x^{2}P_{2N}(x)}{\Lambda^{2N+2}}+2\right)-2\frac{\widetilde{\Lambda}^{2KN+2K}}{x^{2}},
\left(\begin{array}{ccc}{j_{1}}&{j_{2}}&{j_{3}}\\{m_{1}}&{m_{2}}&{m_{3}}\\\end{array}\right)_{q}=q^{\frac{1}{3}(m_{2}-m_{1})}\frac{(-)^{j_{1}-j_{2}-m_{3}}}{\sqrt{[2j_{3}+1]_{q}}}<j_{1}m_{1}\j_{2}m_{2}|j_{3}-m_{3}>_{q}
\lambda_{\alpha}={R}\,{\frac{\partialX^{I}}{\partialu^{\alpha}}}{\calW}_{I}={R}\left({\frac{\partialX^{I}}{\partialu^{\alpha}}}\mathrm{d}B_{I}-{\frac{\partialF_{I}}{\partialu^{\alpha}}}\mathrm{d}A^{I}\right)\,.
S=\left(\begin{array}{cccccc}{0}&{1}&{0}&{0}&{0}&{\rightarrow}\\{0}&{0}&{1}&{0}&{0}&{}\\{0}&{0}&{0}&{1}&{0}&{}\\{0}&{0}&{0}&{0}&{1}&{}\\{0}&{0}&{0}&{0}&{0}&{}\\{\downarrow}&{}&{}&{}&{}&{\ddots}\\\end{array}\right)
S_{\mathrm{bdry}}=\intdx^{0}\left(i\overline{{\pi}}\widetilde{D}_{0}\pi-|\varsigma|^{2}|\phi_{1}|^{2}-\phi_{1}\overline{{\pi}}_{a}\beta-\overline{{\phi}}_{1}\overline{{\beta}}\pi-a_{1}\overline{{\pi}}\varsigma\tau_{1}-a_{1}\overline{{\varsigma}}\overline{{\tau}}_{1}\pi\right)
\begin{array}{l}{l_{\mu\nu}=iD_{\mu\nu},}\\{l_{ij}=-\epsilon_{ijk}L_{k}=i\left(x^{i}\partial_{j}-x^{j}\partial_{i}\right),}\\{l_{0j}=-K_{j}=-i\sqrt{1+\left|\vec{x}\right|^{2}}\\partial_{j},}\\\end{array}
\sigma_{\mathrm{ext}}=2^{1-{\frac{\alpha}{2}}}e^{\sqrt\alpha\phi_{0}}\left|\left(e^{\frac{K_{\mathrm{matt}}}{2}}W\right)_{z=0^{+}}\pm\left(e^{\frac{K_{\mathrm{matt}}}{2}}W\right)_{z=0^{-}}\right|\equiv2(\chi_{1}\pm\chi_{2})
u\left(t_{m\,n}^{(l)}\right)=\delta_{m,n}\,q^{2(m-k)}u\left(t_{k\,k}^{(l)}\right),\quadv\left(t_{m\,n}^{(l)}\right)=\delta_{m,n}\,q^{2(k-m)}v\left(t_{k\,k}^{(l)}\right).
\widetildeW_{\beta}^{\mathrm{cl}}[\Phi,\Pi]=\mathrm{exp}\left[-{\frac{1}{2}}\beta\int{\frac{dp}{2\pi}}\;\Bigl(\Pi^{*}(p)\Pi(p)+E^{2}(p)\Phi^{*}(p)\Phi(p)\Bigr)\right]\;.