harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
L^{\dagger}L=\left(\begin{array}{cc}{-\partial_{r}^{2}-{\frac{\partial_{r}+a^{\prime}}{r}}+{\frac{(a+j+1/2)^{2}}{r^{2}}}+f^{2}}&{0}\\{0}&{-\partial_{r}^{2}-{\frac{\partial_{r}}{r}}+{\frac{(j-1/2)^{2}}{r^{2}}}+f^{2}}\\\end{array}\right)
J_{\mathrm{sym}}=\kappa\int_{R_{*}^{2}}d^{2}r\epsilon_{ij}x^{i}\tilde{A}_{j}\epsilon_{kl}\partial_{k}\tilde{A}_{l}=\kappa\int_{R_{*}^{2}}d^{2}x\,\,\partial_{j}\,\left\{\frac{1}{2}x^{j}(\tilde{A}^{k}\tilde{A}_{k})-\tilde{A}_{j}(x^{k}\tilde{A}_{k})\right\}.
\begin{array}{rl}{\phi(x)=}&{-\frac{e\lambda}{L}\Big\{(p_{+}-p_{-})+}\\{}&{+i\sum_{n=1}^{\infty}\sqrt{n}\left((A_{+,n}^{\dagger}+A_{-,n})\,e^{\frac{2i\pi}{L}nx}\,-\,(A_{-,n}^{\dagger}+A_{+,n})\,e^{-\frac{2i\pi}{L}nx}\right)\Big\}\,}\\\end{array}
\omega=\left(\frac{(n-1)\,b\,h_{-}^{\prime}}{2n\,\sigma\,h_{-}}\left(\frac{1}{{\mathcalR}}+\frac{b\,h_{-}^{\prime}}{2\,h_{-}}\right)^{-1}-\omega_{\mathcal{L}}\frac{z^{\prime}}{z}+\frac{f^{\prime}}{2\,f}\right)\frac{2\,f}{n\,f^{\prime}}\,.
F_{8}^{{\mathcal{O}}_{0,2}^{0,1}|++++++--}=\frac{\langle{\mathcal{O}}_{0,2}^{0,1}\rangle(\sigma_{1}^{-})^{3}/(\sigma_{6}^{+})^{1/2}}{2^{6}\,\mathrm{cosh}(\theta_{78}/2)\,\prod_{1\leqi<j\leq6}\mathrm{cosh}(\theta_{ij}/2)}\prod_{i<j}F_{\mathrm{min}}^{\mu_{i}\mu_{j}}(\theta_{ij}),
V_{e}=-\frac{5}{2}\mathcal{H}^{\prime}+\frac{9}{4}\mathcal{H}^{2}+\mathcal{H}\frac{\varphi_{0}^{\prime\prime}}{\varphi_{0}^{\prime}}-\frac{\varphi_{0}^{\prime\prime\prime}}{\varphi_{0}^{\prime}}+2\left(\frac{\varphi_{0}^{\prime\prime}}{\varphi_{0}^{\prime}}\right)^{2}-6K.
D_{\bfp}(\tau,\tau^{\prime})=\frac{1}{W_{\bfp}}(f_{1}^{\omega}(\tau)f_{2}^{\omega}(\tau^{\prime})\theta(\tau^{\prime}-\tau)+f_{2}^{\omega}(\tau)f_{1}^{\omega}(\tau^{\prime})\theta(\tau-\tau^{\prime}))
\delta_{a}\tilde{\Phi}_{\alpha_{0}}=\left(-\right)^{\epsilon_{\alpha_{0}}+1}\bar{\Phi}_{\alpha_{0}}^{(a)},\;\delta_{a}\tilde{\eta}_{b\alpha_{1}}=\left(-\right)^{\epsilon_{\alpha_{1}}}\bar{\eta}_{b\alpha_{1}}^{(a)},
ds_{E}^{2}=\alpha^{\prime}\left[\frac{U^{2}}{R^{2}}\left(f(U)dt^{2}+dx_{i}dx_{i}\right)+\frac{R^{2}f(U)^{-1}}{U^{2}}dU^{2}+R^{2}d\Omega_{5}^{2}\right].
R_{\mathrm{ijkl}}^{\prime}={\frac{\partial\pi^{\alpha}}{\partialp^{\mathrm{i}}}}{\frac{\partial\pi^{\beta}}{\partialp^{\mathrm{j}}}}{\frac{\partial\pi^{\gamma}}{\partialp^{\mathrm{k}}}}{\frac{\partial\pi^{\delta}}{\partialp^{\mathrm{l}}}}{\calL}_{\alpha}^{\mu}{\calL}_{\beta}^{\nu}{\calL}_{\gamma}^{\lambda}{\calL}_{\delta}^{\rho}\;R_{\mu\nu\lambda\rho}.
m^{2}tr\\gamma^{5}G_{R}(x,y)=-m^{2}\int\frac{d^{4}k}{(2\pi)^{4}}e^{ik\cdot(x-y)}\epsilon^{\lambda\mu\rho\sigma}\frac{1}{(m^{2}-k^{2})^{3}}G_{\lambda\mu}(x)G_{\rho\sigma}(x)\,
I_{L}(p)=(-i)^{n+1}\intq.d^{2\omega}\int_{0}^{\infty}d\alpha_{1}\cdotsd\alpha_{n}d\beta\cdotq^{-}.e^{i[\sum_{1}^{n}\alpha_{i}f_{i}(q,p)+\betaq^{+}q^{-}]}.
\kappa^{2}g^{2}\alpha^{\prime}\left(-\frac{1}{2}(\frac{2\cdot2^{8}\sqrt{2}}{3^{5}})^{2}\right)\left(A_{i}A_{j}\right)^{2}=-\kappa^{2}g^{2}\alpha^{\prime}\left(\frac{65536}{59049}\right)\left(2A_{i}A_{i}A_{j}A_{j}\right).
\Pi^{i}=\nablax^{i}-\frac12\Omega^{+2i}x^{-2}-\frac12\Omega^{-2i}x^{+2}-i\pi_{q}^{1+}\gamma_{q\dot{q}}^{i}\theta_{\dot{q}}^{1-}-i\pi_{\dot{q}}^{1-}\tilde{\gamma}_{\dot{q}q}^{i}\theta_{q}^{1+}+(1\leftrightarrow2).
{\frac{1}{\sigma_{R}\sqrt{\pi}}}~e^{-{\frac{(E-E_{R})^{2}}{\sigma_{R}^{2}}}}=\intdE^{\prime}~w(E^{\prime})~{\frac{1}{\sigma_{Q}\sqrt{\pi}}}~e^{-{\frac{(E-E^{\prime}-E_{Q})^{2}}{\sigma_{Q}^{2}}}}~~~,
\zeta(s,z)=\frac{z^{1-s}}{s-1}+\frac{z^{-s}}{2}+\frac{s}{12}z^{-s-1}-\frac{s(s+1)}{2!}\int_{0}^{\infty}\frac{\overline{{B}}_{2}(t)}{(z+t)^{s+2}}dt.
G(x)=\psi_{0}(x)I_{4}-\psi_{\mu}(x)\gamma_{\mu}+\frac12\psi_{\mu\nu}(x)\gamma_{[\mu}\gamma_{\nu]}+i\widetilde{\psi}_{\mu}(x)\gamma_{\mu}\gamma_{5}-i\widetilde{\psi}_{0}(x)\gamma_{5}
\hat{\mathrm{H}}_{\mathrm{matter}}=\frac{\hbar}{2}\sum_{\stackrel{p\in\calZ}{p\neq0}}\frac{1}{p}\varepsilon_{\mathrm{R}}(p)\rho_{\mathrm{ex}}(-p)\rho_{\mathrm{ex}}(p)+\frac{e^{2}}{2\pi}\sum_{p>0}\frac{1}{p\varepsilon_{\mathrm{R}}(p)}(j_{p}^{\mathrm{ex}})^{\star}(j_{p}^{\mathrm{ex}}),
{\calL}_{\alpha^{\prime}}=\alpha^{\prime}\lambdae^{-\phi}\Bigl\{c_{1}R_{GB}^{2}+c_{2}\Bigl[R^{\mu\nu}-g^{\mu\nu}R\Bigr]\phi_{,\mu}\phi_{,\nu}+c_{3}\Box\phi\,\phi^{,\mu}\phi_{,\mu}-c_{4}(\phi^{,\mu}\phi_{,\mu})^{2}\Big\},
\begin{array}{c}{\displaystyleOp(\sigma)h(x,t)=\int\sigma(x,t;\xi,\tau)\\hat{h}(\xi,\tau)\e^{i(x\xi+t\tau)}\frac{d\xi}{(2\pi)^{\nu-1}}\\frac{d\tau}{2\pi},}\\{\displaystyleOp^{\prime}(\sigma)h(x,t)=\int\int\tilde{\sigma}(x,t;\xi,s)\\tilde{h}(\xi,s)\e^{ix\xi}\frac{d\xi}{(2\pi)^{\nu-1}}}\\{dfrac{ds}{2\pi},}\\\end{array}
\Sigma_{\mathrm{BFYM}}=\intd^{3}x\,\left\{\varepsilon^{\mu\nu\rho}B_{\mu}^{a}F_{\nu\rho}^{a}+e\,\left(B_{\mu}^{a}+D_{\mu}\eta^{a}\right)^{2}\right\}\,\,,
S(\phi,J)=\sum_{n}a^{D}\left\{\frac{1}{a^{2}}\sum_{\mu}\frac{1}{2}\left(\phi_{n+\mu}-\phi_{n}\right)^{2}+m^{2}\phi_{n}^{2}+\lambda\phi_{n}^{4}+J_{n}\phi_{n}\right\}
H=\frac{1}{2}\left(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}\right)-\frac{1}{2}\left(q_{1}^{2}+q_{2}^{2}+q_{3}^{2}-1\right)^{2}-\frac{\sigma_{2}^{2}}{2}q_{2}^{2}-\frac{\sigma_{3}^{2}}{2}q_{3}^{2}
\gamma_{\Omega^{\prime},3}={\small\left(\begin{array}{cccccccc}{\varepsilon_{v_{0}}}&{}&{}&{}&{}&{}&{}&{}\\{}&{}&{}&{}&{}&{}&{}&{1_{v_{1}}}\\{}&{}&{}&{}&{}&{}&{\cdots}&{}\\{}&{}&{}&{}&{}&{1_{v_{P-1}}}&{}&{}\\{}&{}&{}&{}&{\varepsilon_{v_{P}}}&{}&{}&{}\\{}&{}&{}&{-1_{v_{P-1}}}&{}&{}&{}&{}\\{}&{}&{\cdots}&{}&{}&{}&{}&{}\\{}&{-1_{v_{1}}}&{}&{}&{}&{}&{}&{}\\\end{array}\right)}.
\begin{array}{l}{v(z)=q^{2j}a(q^{j}z)+q^{-2j}d(q^{-j}z)-}\\{\displaystyle{-2(q^{j/2}-q^{-j/2})\left(\left(y_{0}x_{+}(\frac{q^{-j/2-1}}{tk}-q^{j/2+1}tk)+y_{+}x_{0}(q^{-j/2}sk-\frac{q^{j/2}}{sk})\right)z+\right.}}\\{\displaystyle{\left.+\left(y_{0}x_{-}(\frac{q^{j/2}k}{t}-\frac{q^{-j/2}t}{k})+y_{-}x_{0}(\frac{q^{j/2+1}s}{k}-\frac{q^{-j/2-1}k}{s})\right)z^{-1}\right)}.}\\\end{array}
Z_{\pm}=\sum_{h=0}^{\infty}\left(g_{s}^{-1}e^{i\Theta_{\pm}}\right)^{2-2h}\left(\lambda_{1}^{R}\right)^{{\frac{5}{2}}(1-h)}Z_{h},\qquad\Theta_{+}=0,\;\Theta_{-}=\pi/2.
I_{n}(g=0)=\sum_{\stackrel{i_{1},\ldots,i_{n}=1}{\scriptscriptstylei_{1}<\dots<i_{n}}}^{N}\prod_{k=1}^{n}p_{i_{k}},\qquadJ_{n}(g=0)=\sum_{j=1}^{N}q_{j}\biggl(\sum_{\substack{i_{1}<\dots<i_{n-1}\,i_{1}\neqj\dotsi_{n-1}\neqj}}\prod_{k=1}^{n-1}p_{i_{k}}\biggr).
\mathbf{F_{7}}=\frac{1}{M_{11}^{3}}\sum_{i=1}^{N_{3}}\mathbf{F}_{\mathbf{4},i}(x)\wedge\mathbf{\omega}_{{\mathbf3},i}(y)+*\mathbf{F}_{\mathbf{4},N_{3}+1}(x)\wedge\mathbf{\epsilon_{7}}(y)\.
\mathcal{Q}=Z_{3}\mathcal{Q}^{\mathrm{R}},\quad\mathcal{Q}^{\mathrm{R}}:={\frac{1}{2}}a_{\mu}^{\mathrm{R}}(x)a^{\mu}{}^{\mathrm{R}}(x)+\alpha^{\mathrm{R}}i\bar{C}^{\mathrm{R}}(x)C^{\mathrm{R}}(x).
\begin{array}{rcccl}{S_{\pm\vec{a_{1}}}\psi(\vec{r})}&{=}&{e^{\pm\frac{i}{\hbar}a_{1}(\hat{p}_{1}+\frac{e}{c}By)}\psi(\vec{r})}&{=}&{\psi(\vec{r}\pm\vec{a_{1}})e^{\pm2\pii\alphay/a_{2}},\nonumber}\\{S_{\pm\vec{a_{2}}}\psi(\vec{r})}&{=}&{e^{\pm\frac{i}{\hbar}a_{2}\hat{p}_{2}}\psi(\vec{r})}&{=}&{\psi(\vec{r}\pm\vec{a_{2}}),}\\\end{array}
{D}\left(p,p^{\prime},P\right):=\left(2\pi\right)^{4}\delta^{\left(4\right)}\left(p-p^{\prime}\right)\left({G}_{1}^{\left(2\right)}\right)^{-1}\left(p_{1}\right)\left({G}_{2}^{\left(2\right)}\right)^{-1}\left(-p_{2}\right)
w=\left(\begin{array}{c}{0}\\{0}\\{n^{1}}\\{0}\\{n^{1}-n^{2}}\\{n^{2}}\\\end{array}\right)\;with\;n^{1},n^{2}\;\inZ\;,p=\left(\begin{array}{c}{0}\\{0}\\{m_{1}}\\{-m_{1}}\\{m_{2}}\\{m_{1}-m_{2}}\\\end{array}\right)\;with\;m_{1},m_{2}\;\in\;Z.
\frac{\partial\tilde{\xi}(\Lambda;t)}{\partialt}=\frac{i}{2}\left[\sum_{i=1}^{m}\frac{\partial^{2}}{\partial{\lambda_{i}}^{2}}-\sum_{\alpha=1}^{n}\frac{\partial^{2}}{\partial{\bar{\lambda}}_{\alpha}^{2}}\right]\tilde{\xi}(\Lambda;t),
e^{-\sum_{k}\nu_{k}\Phi_{k}}=e^{-\sum_{j=1}^{n}(\vec{\lambda}\cdot\vec{\lambda}_{j})\mathrm{~ln~}s_{j}(z)-\sum_{j=1}^{n}(\vec{\lambda}\cdot\vec{\lambda}_{j})\mathrm{~ln~}\bar{s}_{j}(\bar{z})}<\lambda|\bar{M}^{-1}(\bar{z})M(z)|\lambda>.
\left[a_{i}^{\dagger}({\bfk}),b_{j}({\bfk^{\prime}})\right]_{+}=\left[b_{i}^{\dagger}({\bfk}),a_{j}({\bfk^{\prime}})\right]_{+}=e^{i\phi(i,j)}\delta_{ij}\delta({\bfk}-{\bfk^{\prime}})
q^{Q}\prod_{k=1}^{Q-1}\frac{qz_{i}-z_{k}}{z_{i}-qz_{k}}=\frac{\left(z_{i}-i\tau\kappa\lambda^{-\frac{1}{2}}q^{\frac{1}{2}}\right)\left(z_{i}+i\kappa\lambda^{\frac{1}{2}}q^{\frac{1}{2}}\right)}{\left(q^{\frac{1}{2}}z_{i}+i\tau\kappa\lambda^{-\frac{1}{2}}\right)\left(q^{\frac{1}{2}}z_{i}-i\kappa\lambda^{\frac{1}{2}}\right)}.
C_{IJK}\tilde{h}^{I}\tilde{h}^{J}\tilde{h}^{K}=a^{3}(y)C_{IJK}h^{I}h^{J}h^{K}=a^{3}(y)\qquad\mathrm{or}\qquada^{2}(y)=h^{I}H_{I}\,.
\frac{1}{g^{2}}F(\theta,E)=-\frac{16\pi^{2}}{g^{2}}+\left[\frac{E^{4}}{16\pi^{2}g^{2}v^{4}}\right]^{1/3}\left(3I_{E}(\theta)^{1/3}+4\theta\frac{I_{N}(\theta)}{I_{E}(\theta)^{2/3}}\right)+O(\rho^{6}/g^{2}T^{6})
K_{SPA}(T)=\textrm{Tr}\;e^{-\frac{i}{\hbar}{\cal{H}}T}|_{SPA}=e^{-\frac{i}{\hbar}E[\phi_{cl}]T}\underset{l}{\prod}\intdc_{a_{l}}\int_{c_{a_{l}},0}^{c_{a_{l}},T}{\cal{D}}c_{l}(t)\;e^{\frac{i}{\hbar}\int_{T}dt(\frac{1}{2}\dot{c}_{l}^{2}-\frac{1}{2}\omega_{l}^{2}c_{l}^{2})},
\frac{F}{V}=-\sum_{n,s,k}\int_{\calS}d\tau_{1}\frac{d\tau_{2}}{2k^{2}}\frac{\left|\eta\left(\tau\right)\right|^{-48}}{(4\pi^{2}\alpha^{\prime}\tau_{2})^{13}}e^{-\frac{\beta^{2}n^{2}}{4\pi\alpha^{\prime}\tau_{2}}}\delta^{2}\left(\tau-\frac{s+\tau_{0}n}{k}\right)
\left.\left(S+S_{\mathrm{gf}}+S_{\mathrm{gh}}\right)\right|_{\mathrm{bilinear}}=\intd^{4}x\sqrt{-g}\;\left({\Phi}\hat{H}{\Phi}^{T}+c_{\mu}\hat{H_{\mathrm{gh}}c^{\mu}}\right)\;,
\beta^{2}\equiv{\frac{\beta_{BI}^{2}}{(M_{W})^{4}}}=16{\frac{\beta_{BI}^{2}}{g^{4}v^{4}}}\,\,epsilon\equiv{\frac{1}{2}}({\frac{M_{H}}{M_{W}}})^{2}=2{\frac{\lambda}{g^{2}}}
\psi_{-\chi}^{(+)}(x^{++},v^{+})=\frac1{(2k)^{c}(2\pi)^{\frac32}}\int_{0}^{\infty}\,dp^{--}e^{-\fraci2p^{--}x^{++}}(p^{--})^{c+1}\left(\frac{v_{1}^{+}}{\bar{v}_{1}^{+}}\right)^{-s}a(\vec{p},s)
{\hat{J}}_{(1)}={\frac{1}{\sqrt{2}}}({\bar{\alpha}}_{A}{\bar{\beta}}^{B}+{\bar{\alpha}}^{B}{\bar{\beta}}_{A}){\bar{\pi}}_{B}{\frac{\partial}{\partial{\bar{\pi}}_{A}}}-{\frac{1}{\sqrt{2}}}(\alpha_{A^{\prime}}\beta^{B^{\prime}}+\beta_{A^{\prime}}\alpha^{B^{\prime}})\pi_{B^{\prime}}{\frac{\partial}{\partial\pi_{A^{\prime}}}},
{\tilde{E}}_{2}^{A_{1}A_{2}}=(-1)^{\varepsilon_{A_{1}}(\varepsilon_{A_{2}}+1)}E_{2}^{A_{1}A_{2}}=-(-1)^{(\varepsilon_{A_{1}}+1)(\varepsilon_{A_{2}}+1)}{\tilde{E}}_{2}^{A_{2}A_{1}}.
{\bfH}=\frac{i\omega}{2}\intd^{3}p\,p^{i}\sum_{\lambda}\left[a^{\dagger}(\vec{p},\lambda)\stackrel{\leftrightarrow}{\partial}_{p^{i}}a(\vec{p},\lambda)+b^{\dagger}(\vec{p},\lambda)\stackrel{\leftrightarrow}{\partial}_{p^{i}}b(\vec{p},\lambda)\right]
\delta~(b)=~\frac{1}{4\pi}~\int_{0}^{\infty}\frac{dt}{t}e^{-b^{2}t/2\pi\alpha^{\prime}}~[\prod_{i=1}^{l}\Theta_{3}(0,8i\pi^{2}\alpha^{\prime}t/L_{i}^{2})]~[\prod_{i=p+1}^{c}\Theta_{3}(0,itL_{i}^{2}/2\pi\alpha^{\prime})]\cdot(B\timesJ)~~.
{\calL}_{Kink}=\sqrt{2\kappa\pi}\tau_{p}u\intdx\,|\tilde{T}^{\prime}|\,e^{-{\frac{\kappau^{2}}{2\alpha^{\prime}}}\tilde{T}^{2}}\sqrt{1+{\frac{\partial_{\mu}\tilde{T}\partial^{\mu}\tilde{T}}{(\tilde{T}^{\prime})^{2}}}}\.
{\calL}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2\lambda}(\partial_{\mu}A^{\mu})(\partial_{\nu}A^{\nu})+\frac{i}{2}\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-\frac{i}{2}(\partial_{\mu}\bar{\psi})\gamma^{\mu}\psi-m\bar{\psi}\psi+e\bar{\psi}\gamma^{\mu}A_{\mu}\psi.
\delta\Omega_{\alpha}^{\Lambda}=2\gamma^{\mu}\nabla_{\mu}X^{\Lambda}\epsilon_{\alpha}+{\frac{1}{2}}\gamma^{\nu\lambda}{\calF}_{\nu\lambda}^{+\Lambda}\epsilon_{\alpha\beta}\epsilon^{\beta}+2i\gamma^{\mu}A_{\mu}\epsilon_{\alpha}\.
\vec{V}(z)=\Biggl(z+{\frac{1}{4}}{\frac{({\frac{g^{2}}{2\pi}})^{2}}{1+{\frac{g^{2}}{2\pi}}}}(z-\bar{z}),-{\frac{{\frac{g^{2}}{2\pi}}(2+{\frac{g^{2}}{2\pi}})}{1+{\frac{g^{2}}{2\pi}}}}(z-\bar{z})\Biggr).
\Omega_{C}^{2}(a,b,{\bfk}_{n},{\bfk}_{n^{\prime}})\equiv\Omega_{r}^{ab}({\bfk}_{n})\Omega_{i}^{ba}(-{\bfk}_{n^{\prime}})+\Omega_{i}^{ab}({\bfk}_{n})\Omega_{r}^{ba}(-{\bfk}_{n^{\prime}}).
\hat{T}_{MN}^{\varphi}={\frac{1}{4\epsilon}}(\frac{1}{2}\hat{g}_{MN}\partial_{P}\varphi\partial^{P}\varphi-\partial_{M}\varphi\partial_{N}\varphi)+\frac{V}{\epsilon}e^{-\varphi}\hat{g}_{MN}\:.
\left.\frac{\delta\Gamma}{\deltaB}\right|_{\chi=0}=\xiB\;+\;\partialA\;-\;e\left[(\hat{\varphi}_{1}-\xi_{A}\frac{m}{e})\varphi_{2}-\hat{\varphi}_{2}(\varphi_{1}-\hat{\xi}_{A}\frac{m}{e})\right]\;,
\langlew^{\prime}|w\rangle\equiv\langleP^{\prime},K^{\prime}|K,P\rangle\equiv{\frac{1}{2}}\langle0|b(P^{\prime})b(K^{\prime})b^{\ast}(K)b^{\ast}(P)|0\rangle={\frac{1}{2}}D_{1}^{\prime}(K,P;K^{\prime}P^{\prime})
G(x^{\prime},t^{\prime};x^{\prime},t^{\prime})=i\int\int\frac{d\omega^{\prime\prime}}{2\pi}\frac{dk^{\prime\prime}}{2\pi}\frac{1}{\omega^{2}-k^{2}-4+i\rho}\left(1+K_{0}(k^{\prime\prime})e^{-2ik^{\prime\prime}x^{\prime}}\right).
\begin{array}{c}{C^{-1}\left(\begin{array}{c}{\gamma_{4}}\\{\gamma_{5}}\\\end{array}\right)C=R_{C}\left(\begin{array}{c}{\gamma_{4}^{T}}\\{\gamma_{5}^{T}}\\\end{array}\right),}\\{\Pi^{-1}\left(\begin{array}{c}{\gamma_{4}}\\{\gamma_{5}}\\\end{array}\right)\Pi=R_{P}\left(\begin{array}{c}{\gamma_{4}}\\{\gamma_{5}}\\\end{array}\right),}\\\end{array}
I_{1}(0)={\frac{(23)^{2}}{x_{23}^{4}}}\left[(12)^{2}(32^{-}){\frac{g(1,2,3)}{x_{12}^{2}}}+(13)^{2}(23^{-}){\frac{g(1,2,3)}{x_{13}^{2}}}+(12)(13)\;2\partial_{2}\cdot\partial_{3}f(1,2,1,3)\right]\.
\omega_{\mathrm{phys}}\,\simeq\,k_{\mathrm{phys}}\sqrt{\epsilon_{1}+\epsilon_{3}}\mathrm{e}^{-k_{\mathrm{phys}}/(2k_{\mathrm{c}})}\quad\quad(k_{\mathrm{phys}}\ggk_{\mathrm{c}}),
M_{11}^{\beta\gamma}=\frac{T^{\beta\gamma}}{1-CI^{\beta\gamma\,-1}T^{\beta\gamma}+T^{\beta\gamma\,2}}=\frac{(CI^{\beta\gamma})^{5}-CI^{\beta\gamma}}{(1-(CI^{\beta\gamma})^{2})(1-3(CI^{\beta\gamma})^{2})}
\bar{\zeta}(s)=\int_{-\infty}^{\infty}\frac{dk}{2\pi}\int_{0}^{\infty}\frac{\lambda\,d\lambda}{(k^{2}+\lambda^{2})^{s}}=\frac{1}{2\sqrt{\pi}}\frac{\Gamma(s-1/2)}{\Gamma(s)}\int_{0}^{\infty}\lambda^{2-2s}d\lambda\,{.}
C_{\bcd}^{a}\equivR_{\bcd}^{a}+{\frac{2}{d-2}}\left(\delta_{\[d}^{a}R_{c]b}+g_{b[c}R_{d]}^{a}\right)+{\frac{2}{(d-1)(d-2)}}\delta_{\[c}^{a}g_{d]b}R\,,
\chi_{({\calV}_{m_{1},n_{1}}\otimes{\calV}_{m_{2},n_{2}})_{\mathrmf}}(\tau)=\sum_{m_{3}=|m_{1}-m_{2}|+1}^{m_{1}+m_{2}-1}\sum_{n_{3}=|n_{1}-n_{2}|+1}^{n_{1}+n_{2}-1}\chi_{m_{3},n_{3}}(\tau)\,,
\nabla=\beta_{g}g\frac{\partial}{\partialg}+\beta_{\xi}\xi\frac{\partial}{\partial\xi}+\beta_{\sigma}\sigma\frac{\partial}{\partial\sigma}+\beta_{\rho}\rho\frac{\partial}{\partial\rho}+\gamma_{Q}\mathbf{N}_{Q}+\gamma_{\bar{Q}}\mathbf{N}_{\bar{Q}}+\gamma_{B}\mathbf{N}_{B}.
{\frac{\phi^{N}}{M^{N-2}}}(H_{L}^{*}H_{L}-H_{R}^{*}H_{R})-m^{2}(H_{L}^{*}H_{L}+H_{R}^{*}H_{R})+\lambda(H_{L}^{*}H_{L}+H_{R}^{*}H_{R})^{2}+\lambda^{\prime}(H_{L}^{*}H_{L}H_{R}^{*}H_{R})
{\calL}_{\mathrm{eff}}=\frac{1}{2}ae^{2}A_{\mu}A^{\mu}-\frac{e^{2}}{2}A_{\mu}\frac{(g^{\mu\nu}+\epsilon^{\mu\nu})\partial_{\nu}\partial_{\rho}(g^{\rho\sigma}-\epsilon^{\rho\sigma})}{\Box}A_{\sigma}~,
\begin{array}{ccccc}{}&{}&{h^{0,0}}&{}&{}\\{}&{h^{1,0}\!}&{}&{\!h^{0,1}\!}&{}\\{h^{2,0}\!}&{\!}&{h^{1,1}}&{\!}&{\!h^{0,2}}\\{}&{h^{2,1}}&{}&{h^{1,2}}&{}\\{}&{}&{h^{2,2}}&{}&{}\\\end{array}=\begin{array}{ccccc}{}&{}&{1}&{}&{}\\{}&{0\!}&{}&{\!0}&{}\\{1}&{\!}&{20}&{\!}&{1}\\{}&{0\!}&{}&{\!0}&{}\\{}&{}&{1}&{}&{}\\\end{array}\.
R_{\mu\nu\sigma}^{\lambda}=-\partial_{\sigma}\Gamma_{\mu\nu}^{\lambda}+\partial_{\nu}\Gamma_{\mu\sigma}^{\lambda}+\Gamma_{\rho\nu}^{\lambda}\Gamma_{\mu\sigma}^{\rho}-\Gamma_{\rho\sigma}^{\lambda}\Gamma_{\mu\nu}^{\rho}
\widehat{g}_{{\bfn}{\bfm}}[\tilde{\phi}]\equiv\intd^{d}{\bfx}\;b_{{\bfn}}({\bfx})\,b_{{\bfm}}({\bfx})\left(F^{\prime}[\tilde{\Phi}(t,{\bfx})]\right)^{2}=g_{{\bfn}{\bfm}}[\tilde{\phi}]\;,
\hat{G}^{\xi=0}[a,b;E]=\frac{e^{-\sqrt{2E}|a-b|}}{\sqrt{2E}}+\frac{\frac{\sqrt{2E}}{v^{ren}}+1}{\frac{\sqrt{2E}}{v^{ren}}-1}\frac{e^{-\sqrt{2E}(|a|+|b|)}}{\sqrt{2E}}
\frac{d\lambda}{ds}\frac{d}{d\lambda}\left(\frac{d\lambda}{ds}\frac{dx^{\mu}}{d\lambda}\right)+\sum_{\alpha,\beta=0}^{3}\Gamma_{\alpha\beta}^{\mu}\frac{d\lambda}{ds}\frac{dx^{\alpha}}{d\lambda}\frac{d\lambda}{ds}\frac{dx^{\beta}}{d\lambda}=0
A_{i}^{\mathrm{tr}}(s)=-{\frac{2}{\Gamma(s)}}\Bigr[r_{+}^{2s}+(-1)^{i}r_{-}^{2s}\Bigr]\Bigr[\zeta_{R}(2s+i-2)-\zeta_{R}(2s+i)\Bigr]\sum_{l=0}^{i}x_{i,l}{\frac{\Gamma(s+l+{\frac{i}{2}})}{\Gamma(l+{\frac{i}{2}})}}.
\begin{array}{rcl}{\hat{J}_{\pm}^{\prime}D_{MK}^{J}}&{=}&{\sqrt{[J\mpM][J\pmM-1]}D_{M\pm1,K}^{J}~,}\\{\hat{J}_{3}^{\prime}D_{MK}^{J}}&{=}&{KD_{MK}^{J}~,~~~~\hat{J}_{z}^{\prime}D_{MK}^{J}=MD_{MK}^{J}~.}\\\end{array}
\sum_{n_{1},n_{2}}(\tilde{C}(n_{1},n_{2})\bar{\chi}\partial_{\mu}\chi\tilde{A}_{\mu}(n_{1},n_{2})+c.c.)+\sum_{(n_{1},n_{2})\neq(0,0)}(\tilde{m}\tilde{C}(n_{1},n_{2})\chi\bar{\chi}\tilde{\phi}(n_{1},n_{2})+c.c.)
\frac{1}{4\pi^{2}}A(\vec{x})\wedgeF(\vec{x})=-\frac{1}{8\pi^{2}}\mathrm{tr}\left(J(\vec{x})\wedgedJ(\vec{x})+\frac{2}{3}J(\vec{x})\wedgeJ(\vec{x})\wedgeJ(\vec{x})\right).
\left[\hat{\partial}_{0}\hat{\partial}_{0}+\vec{k}^{2}+|\hat{B}|^{2}+2B_{1}\hat{\partial}_{0}+a\dot{\hat{B}}-a\dot{\hat{A}}^{\dagger}\hat{A}\left(\hat{\partial}_{0}+\hat{B}\right)\right]\theta=0\,.
\hat{H}_{\mathrm{int}}^{(I)}(t)=e^{i(t-t_{0})\hat{H}_{0}}\,\hat{H}_{\mathrm{int}}(t_{0})\,e^{-i(t-t_{0})\hat{H}_{0}}\\,\\\hat{H}_{\mathrm{int}}(t_{0})=\hat{H}-\hat{H}_{0}\.
G_{\psi}^{ij}(\nu)\equiv(-1)^{i+j}\Sigma((-1)^{i}\bar{R}_{i})\Sigma((-1)^{j}\bar{R}_{j})\bar{\gamma}_{i}^{-2}\bar{\gamma}_{j}^{-2}\frac{\partial^{2}}{\partialr\partialr^{\prime}}G(r,r^{\prime},\nu)\Bigg|_{{r^{\prime}=(-1)^{i}\bar{R}_{i}}\atop{r=(-1)^{j}\bar{R}_{j}}}\,.
2\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}+\frac{k}{a^{2}}-2\left(\ddot{\Phi}+2\frac{\dot{a}}{a}\dot{\Phi}-\dot{\Phi}^{2}\right)=-\kappa^{2}e^{\Phi}p_{T},
S={\frac{N}{g}}tr\left({\frac{1}{2}}A(\beta)S^{2}+{\frac{1}{2}}B(\beta)D^{2}+{\frac{1}{4}}S^{4}+{\frac{1}{4}}D^{4}+{\frac{1}{2}}(SDSD+2S^{2}D^{2})\right).
\begin{array}{c}{\phi^{\prime\prime}(u)=0}\\{1+c_{wzw}-26+\frac{6}{k}\,(\phi^{\prime})^{2}=0}\\{-\frac{1}{2k}\,T_{1}^{\prime\prime}-2T_{1}+\frac{1}{k}\,\phi^{\prime}T_{1}^{\prime}=0}\\\end{array}
{\frac{d^{2}t}{d\lambda^{2}}}+{\frac{\dot{n}}{n}}\left({\frac{dt}{d\lambda}}\right)^{2}+2{\frac{n^{\prime}}{n}}{\frac{dt}{d\lambda}}{\frac{dy}{d\lambda}}+{\frac{a\dot{a}}{n^{2}}}\sum_{j}\left({\frac{dx^{j}}{d\lambda}}\right)^{2}=0,
C(\sigma,\phi)=e^{\lambda\phi^{i}\partial_{x^{i}}}C^{0}(\sigma,x^{i})|_{x^{i}=0}=\sum_{n,{i_{n}}}\frac{\lambda^{n}}{n!}\phi^{i_{1}}\ldots\phi^{i_{n}}\partial_{x^{i_{1}}}\ldots\partial_{x^{i_{n}}}C^{0}(\sigma,x^{i})|_{x^{i}=0}
c_{\pm}=\frac{4\pi^{2}n^{2}}{\beta^{2}t^{2}}+\frac{\kappa\rho^{2}\lambda_{1}}{2}\pm\sqrt{\kappa}\,\rho\left[(\lambda_{1}-\lambda_{0})\frac{4\pi^{2}n^{2}}{\beta^{2}t^{2}}+\frac{\kappa\rho^{2}\lambda_{1}^{2}}{4}\right]^{1/2}.
\hbar\frac{\deltaC}{\deltaa_{\mu}^{a}}=\partial^{2}a_{\mu}^{a}+\left[J_{\mu}^{a}(A)+\hbarJ_{\mu}^{a}(A)_{gh}\right]_{A=-\hbarG\frac{\deltaC}{\deltaa}+i\hbarG\frac{\delta}{\deltaa}}1\,.
\begin{array}{lcl}{X^{(II)}(\sigma,\tau)}&{=}&{{\displaystyleX_{0}^{(II)}(\tau)+}}\\{}&{}&{{\displaystyle\frac{\sigma_{0}}{2\pi}~k~\sum_{n\neq0}^{}\left\{\frac{1}{|n|}~F_{\nu}\left(\frac{2\pi}{\sigma_{0}}|n|\tau,\frac{2\pi}{\sigma_{0}}|n|\tau_{0}\right)\phi_{n}(\sigma_{0})~e^{i\frac{2\pi}{\sigma_{0}}n\sigma}\right\}}}\\\end{array}
\tilde{\mathfrakr}^{*}={\mathfrakr}^{*}-1=\tilde{\mathfrakr}_{\mathrm{UV}}+{\mathfrakg\left(r\right)}=\left(\begin{array}{cc}{S\,\tilde{\mathfrakr}_{\mathrm{IR}}\,S^{-1}}&{0}\\{0}&{Ad_{\mathfrakg\left(r\right)}-1}\\\end{array}\right)
\left[\frac{d^{2}}{d\tau^{2}}+q^{2}-1+g\int_{0}^{1}dp\left\{\frac{p^{2}}{p^{2}+L_{R}^{2}}[{\cal{U}}_{p}^{+}(t){\cal{U}}_{p}^{-}(t)-1]\right\}\right]{\cal{U}}_{q}^{\pm}(t)=0
R^{2}-1+{\frac{8\piR^{2}}{3M_{\mathrm{P}}^{2}}}\left(V(f)+{\frac{n^{2}}{8\pi^{4}f^{2}R^{6}}}-{\frac{f^{2}}{2}}\right)+{\frac{1}{R^{2}H_{0}^{2}}}(R^{2}-1)^{2}=0\.
\chi_{\ell}^{(\infty)}(r)=H_{\nu}^{(1)}(kr)\sim\sqrt{\frac{2}{\pik}}\,e^{-i\frac{\pi}{2}(\ell-\frac{\Phi}{2\pi})-i\frac{\pi}{4}}\,\frac{e^{ik\,r}}{\sqrt{r}},
\sigma\sum_{n=0}^{\infty}\intd^{4}z_{1}\dots\intd^{4}z_{n}\left(W_{xz_{1}z_{2}\dotsz_{n}y}\right)_{AB}\frac{e^{-\frac{(x-z_{1})^{2}}{4\sigma}}}{(4\pi\sigma)^{2}}\frac{e^{-\frac{(z_{1}-z_{2})^{2}}{4\sigma}}}{(4\pi\sigma)^{2}}\dots\frac{e^{-\frac{(z_{n}-y)^{2}}{4\sigma}}}{(4\pi\sigma)^{2}}
{\calL}_{\mathrm{B}}=a_{\mathrm{B}}\mathrm{G}_{6}+b_{\mathrm{B}}RR_{\mu\nu}R^{\mu\nu}+c_{\mathrm{B}}R^{3}+d_{\mathrm{B}}R\BoxR+e_{\mathrm{B}}R_{\mu\nu}\BoxR^{\mu\nu},
N_{\alpha_{1}...\alpha_{n}\beta_{1}...\beta_{n}}^{(L)}:=\Big\langle\psi_{\alpha_{1}}^{(1)}(-L,0)...\psi_{\alpha_{n}}^{(n)}(-L,0)U({\calC}^{(L)})^{n}\overline{{\psi}}_{\beta_{1}}^{(1)}(L,0)...\overline{{\psi}}_{\beta_{n}}^{(n)}(L,0)\Big\rangle\;.
H_{o}=\pi\sum_{\mu=0}^{p}p_{\mu}p^{\mu}+\pi\sum_{i=6}^{9}p_{i}p^{i}+\pi\sum_{m=0,\atopm\nel.c.}^{9}\left(\sum_{n>0}\alpha_{-n}^{m}\alpha_{n}^{m}+\sum_{r>0}r\psi_{-r}^{m}\psi_{r}^{m}\right)+\piC_{o}\,
{\calL}_{3}=-\,\frac{1}{4}\,F_{\mu\nu}F^{\mu\nu}+\frac{m^{2}}{4}\,F_{\mu\nu}\,\Bigl(1+\frac{\partial_{t}^{2}}{\nabla^{2}}\Bigr)\,\frac{1}{\nabla^{2}}\,F^{\mu\nu}-\,\frac{1}{2\alpha}\,\bigl(\partial_{\mu}A^{\mu}\bigr)^{2}\,.
ds^{2}=H_{8}^{-1/2}H_{4}^{-1/2}\eta_{ij}dx^{i}dx^{j}+H_{8}^{1/2}H_{4}^{1/2}dz^{2}+H_{8}^{-1/2}H_{4}^{1/2}(dr^{2}+r^{2}d\Omega_{3}^{2}),
{\calL}_{1}=\kappa\left\{\phantom{|}_{abc}^{\mu\nu\alpha}\right\}f_{\mu}{}^{a}f_{\nu}{}^{b}f_{\alpha}{}^{c},\qquad\delta_{1}T_{\alpha\beta\gamma,a}=\kappa\varepsilon_{abc[\alpha\beta}f_{\gamma]}{}^{b}\eta^{c}
{\calT}_{\mu\nu}^{\alpha\beta}\sim\epsilon_{~\mu}^{\alpha~\rho\sigma}\epsilon_{~\nu}^{\beta~\lambda\gamma}\big(F_{\rho\sigma}F_{\lambda\gamma}+\Lambda\tilde{F}_{\rho\sigma}F_{\lambda\gamma}+\kappa\tilde{F}_{\rho\sigma}\tilde{F}_{\lambda\gamma}\big)\\,.
K_{ij}^{0}(\,p_{ij0}^{\prime},\,p_{ij0},\,P_{ij0})\,=\beta_{i}\beta_{j}\,\Lambda_{ij}^{++}\,\beta_{i}\beta_{j}\,K_{ij}^{T}(\,s_{ij},\,s_{ij},\,P_{0}-h_{k}\,)\,\Lambda_{ij}^{++}
\langle\phi_{1}(z_{1},\bar{z}_{1})\phi_{2}(z_{2},\bar{z}_{2})\phi_{3}(z_{3},\bar{z}_{3})\rangle=C_{12}^{3}z_{12}^{\gamma_{12}}z_{13}^{\gamma_{13}}z_{23}^{\gamma_{23}}\bar{z}_{12}^{\bar{\gamma}_{12}}\bar{z}_{13}^{\bar{\gamma}_{13}}\bar{z}_{23}^{\bar{\gamma}_{23}}
{\calH}=\sum_{\nu}{\frac{(E+p)^{4\nu}-M^{4\nu}}{(E+p)^{4\nu}+M^{4\nu}}EC_{\nu}}+\beta\sum_{\nu}{\frac{2EM^{2\nu}(E+p)^{2\nu}}{(E+p)^{4\nu}+M^{4\nu}}B_{\nu}},
+\frac{\omega\left(\epsilon,t\right)}{\left(\alpha\left(t\right)+3\right)\left(\alpha\left(t\right)+4\right)}\sum_{n=0}^{\infty}\frac{\left(2\right)_{n}\left(\frac52\right)_{n}\left(2\right)_{n}}{\left(\frac{\alpha\left(t\right)+5}2\right)_{n}\left(\frac{\alpha\left(t\right)+6}2\right)_{n}n!}\left(-1\right)^{n},
\phi_{3}(\vec{q}_{1},\vec{q}_{2},\vec{q}_{3},\vec{n})=\frac{g\phi_{2}(q^{\prime},z^{\prime})}{(s_{123}-m^{2})(1-x_{2})}+\frac{g\phi_{2}(q^{\prime\prime},z^{\prime\prime})}{(s_{123}-m^{2})(1-x_{3})}\.
\hat{G}_{xx}^{(10)}\to\hat{G}_{xx}^{(10)\,\prime}={\frac{1}{\hat{G}_{xx}^{(10)}}},\\\\\hat{G}_{\mu\nu}^{(10)}\to\hat{G}_{\mu\nu}^{(10)}\\(\mu,\nu\neqx),\\\\e^{2\hat{\Phi}}\toe^{2\hat{\Phi}^{\prime}}={\frac{e^{2\hat{\Phi}}}{\hat{G}_{xx}^{(10)\,\prime}}}.
|{\Psi_{\alpha}}_{2}(p)\rangle_{tree}=\Big\{[U_{2}^{(2)}(\alpha)+U_{2}^{(11)}(\alpha)]|\Psi_{0}(p)\rangle+U_{1}(\alpha)|\Psi_{1}(p)\rangle+U_{0}(\alpha)|\Psi_{2}(p)\rangle\Big\}_{tree}=|\Psi_{2}(p_{\alpha})\rangle\;.
\begin{array}{rclcrcl}{|\nu_{L}\rangle}&{=}&{\displaystyle\bigg|-1,\frac{1}{2},\frac{1}{2}\bigg\rangle,}&{\quad}&{|e_{L}\rangle}&{=}&{\displaystyle\bigg|-1,\frac{1}{2},-\frac{1}{2}\bigg\rangle,}\\{|e_{R}\rangle}&{=}&{|-2,0,0\rangle,}&{\quad}&{|\nu_{R}\rangle}&{=}&{|0,0,0\rangle,}\\\end{array}
\{X(x),Y(y)\}_{PB}=\intd^{3}z\,\biggl(\frac{\deltaX(x)}{\deltaA_{\mu}^{C}(z)}\frac{\deltaY(y)}{\delta\Pi^{\muC}(z)}-\frac{\deltaY(y)}{\deltaA_{\mu}^{C}(z)}\frac{\deltaX(x)}{\delta\pi^{\muC}(z)}\biggr)
u_{\pm}=\frac{1}{4\sqrt{2}}\Big((F_{\mu\nu}F_{\mu\nu}+F_{\mu\nu}^{*}F_{\mu\nu})^{\frac{1}{2}}\pm(F_{\mu\nu}F_{\mu\nu}-F_{\mu\nu}^{*}F_{\mu\nu})^{\frac{1}{2}}\Big)
D\left(\begin{array}{l}{f_{\pm}^{q}}\\{f_{\pm}^{-q}}\\\end{array}\right)=d\left(\begin{array}{l}{f_{\pm}^{q}}\\{f_{\pm}^{-q}}\\\end{array}\right)-iqA_{\pm}\sigma_{3}\left(\begin{array}{l}{f_{\pm}^{q}}\\{f_{\pm}^{-q}}\\\end{array}\right).
{\calL}_{\mathrm{eff}}=\intd^{4}\vartheta\,\,{\calK}_{\mathrm{eff}}[\Sigma,\bar{\Sigma}]+\intd^{2}\vartheta\,\,{\calW}_{\mathrm{eff}}[\Sigma]+\intd^{2}\bar{\vartheta}\,\,\bar{{\calW}}_{\mathrm{eff}}[\bar{\Sigma}]