harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Z_{2}:\qquad(A_{\mu},\phi,\bar{\phi},\lambda_{A\alpha},\bar{\lambda}^{A\alpha},G^{\alpha\beta})\rightarrow(A_{\mu},\bar{\phi},\phi,-\bar{\lambda}_{A}^{\!~~\alpha},-\lambda_{\!~~\alpha}^{A},G_{\alpha\beta}).
\sum_{n\ge3}{\frac{1}{n!}}J_{n}\sum_{P_{n}}\prod_{p}C_{r}=J_{int}[\phi]+{\frac{1}{2}}G^{ij}{\frac{\deltaJ_{int}[\phi]}{\delta\phi^{i}\delta\phi^{j}}}+\sum_{n\ge3}{\frac{1}{n!}}\chi_{n}\sum_{P_{n}}^{*}\prod_{p}C_{r}
\begin{array}{rcl}{{\hat{S}}_{t}}&{=}&{{\hat{P}}_{x}\,,}\\{{\hat{P}}_{t}}&{=}&{{\hat{P}}{\hat{S}}_{x}+{\hat{Q}}_{x}\,,}\\{{\hat{Q}}_{t}}&{=}&{{2}{\hat{Q}}{\hat{S}}_{x}\,.}\\\end{array}
K^{\calF}\,A\rightarrow\left(\begin{array}{cc}{0}&{\\0}\\{}&{}\\{0}&{\\4K_{a\mur\nu}^{\calF}\,\eta^{\nu\gamma}+2\beta\,(\eta_{ar}\eta^{\mu\gamma}-\delta_{r}^{\mu}\delta_{a}^{\gamma})}\\\end{array}\right)\left(\begin{array}{c}{\Gamma_{\nu}^{cd}}\\{\tau_{\gamma}^{r}}\\\end{array}\right),
{\calS}[g]=k\left[-\frac{1}{16\pi}\int_{\calM}tr\left(g^{-1}\partial_{\mu}\;g^{-1}\partial^{\mu}g\right)d^{2}x+\frac{1}{24\pi}\int_{\calB}\varepsilon^{\lambda\mu\nu}tr\left(g^{-1}\partial_{\lambda}g\;g^{-1}\partial_{\mu}g\;g^{-1}\partial_{\nu}g\right)d^{3}x\right],
\kappa\,h_{\mu\nu}=-\deltag_{\mu\nu}+\frac12\deltag_{\alpha}^{\alpha}\eta_{\mu\nu}-\frac12\deltag_{\alpha}^{\alpha}\deltag_{\mu\nu}+\deltag_{\mu\alpha}g_{\nu}^{\alpha}+\frac18\left(\deltag_{\alpha}^{\alpha}\right)^{2}\eta_{\mu\nu}-\frac14\deltag^{\alpha\beta}\deltag_{\beta\alpha}\eta_{\mu\nu}+\cdots.
\Lambda_{\pm}(q_{0})=\Lambda(q_{0})\pm\frac{2\kappa\bigg(\frac{2}{\pi}\bigg)^{\frac{1}{2}}}{[\frac{1}{2}(q_{0}-1)]!}\bigg(\frac{1+k}{1-k}\bigg)^{-\frac{\kappa}{k}}\bigg(\frac{8\kappa}{1-k^{2}}\bigg)^{\frac{1}{2}q_{0}}\bigg[1+O\bigg(\frac{1}{\kappa}\bigg)\bigg]
S_{e}[x_{c}(\tau)+\deltax(\tau)]=S_{eo}+\int_{-T/2}^{T/2}\deltax\left[-{\frac{1}{2}}{\frac{d^{2}}{d\tau^{2}}}\deltax+{\frac{1}{2}}V^{\prime\prime}[x_{c}(\tau)]\deltax\right]\d\tau
\pi_{\dot{q}}^{1-}=e^{+2}\psi_{+2\dot{q}}^{1-}+e^{-2}\psi_{-2\dot{q}}^{1-}+e_{p}^{+}h_{+p\dot{q}}^{1-}+e_{\dot{p}}^{-}\chi_{\dot{p}\dot{q}}^{1},
\renewcommand{\arraystretch}{1.5}\begin{array}{rcl}{L_{-2}\Omega}&{=}&{\displaystyle\frac12d_{\alpha\beta}\chi_{-1}^{\alpha}\chi_{-1}^{\beta}\Omega\,,}\\{W_{-3}^{a}\Omega}&{=}&{t_{\alpha\beta}^{a}\chi_{-2}^{\alpha}\chi_{-1}^{\beta}\Omega\,.}\\\end{array}
\frac{1}{({M_{i}}^{2}-{p_{1}}^{2})({M_{i}}^{2}-{p_{2}}^{2})}=\int_{0}^{1}d\,\xi\frac{1}{{[{M_{i}}^{2}-{p_{1}}^{2}-({p_{2}}^{2}-{p_{1}}^{2})\xi]}^{2}}
\omega_{2}~=~-\left[4(\mu-1)(\mu-2)\hat{\Psi}(\mu)~+~\frac{(16\mu^{4}-96\mu^{3}+188\mu^{2}-136\mu+25)}{2(2\mu-3)(\mu-1)}\right]\eta_{1}^{2}~+~\Pi_{2}
W_{cl}=-{\frac{(2\pi\bar{\beta})}{4G}}\int_{x_{+}}^{L}((r^{2})^{\prime}e^{\lambda}g^{\prime}+2ge^{\lambda}(r_{x}^{\prime})^{2}+2Ue^{-\lambda})-{\frac{\pir_{+}^{2}}{G}}.
S=\frac{1}{16\pi}\intd^{4}x\sqrt{-g}\biggl\{-{\calR}+\frac{1}{2}\frac{\partial_{\mu}\lambda\partial^{\mu}\overline{{\lambda}}}{({\mathrm{Im}}\;\lambda)^{2}}-F_{\mu\nu}{}^{\star}\tilde{F}^{\mu\nu}\biggr\}\;.
W=\sqrt{2}\mathrm{Tr}\left(\boldsymbol\Lambda^{1}[\boldsymbol\Phi^{2},\boldsymbol\Phi^{3}]+\boldsymbol\Lambda^{2}[\boldsymbol\Phi^{3},\boldsymbol\Phi^{1}]+\boldsymbol\Lambda^{3}[\boldsymbol\Phi^{1},\boldsymbol\Phi^{2}]\right),
H=Q_{+}Q_{-}+Q_{-}Q_{+}=\frac{1}{2}\bigg\{P_{-},P_{+}\bigg\}+\frac{1}{2}\bigg[P_{-},P_{+}\bigg]\sigma_{3}=\left(\begin{array}{cc}{P_{-}P_{+}}&{0}\\{0}&{P_{+}P_{-}}\\\end{array}\right)
\begin{array}{ccc}{\bigl[{\calW}_{em},{\calW}_{+}\bigr]}&{=}&{-i{\calW}_{+}}\\{\bigl[{\calW}_{em},{\calW}_{-}\bigr]}&{=}&{+i{\calW}_{-}}\\\end{array}\qquad\begin{array}{ccc}{\bigl[{\calW}_{em},{\calW}_{3}\bigr]}&{=}&{0}\\{\bigl[{\calW}_{em},{\calW}_{4}\bigr]}&{=}&{0}\\\end{array}
S=\intd^{4}x\sqrt{\hat{g}_{4}}\Bigl\{\hat{R}_{4}+\frac{2\hat{\nabla}_{\mu}{\calA}\hat{\nabla}^{\mu}{\calA}^{\dagger}}{({\calA}-{\calA}^{\dagger})^{2}}+\frac{1}{8}Tr\bigl(l\hat{\nabla}m\bigr)^{2}-\frac{Q^{2}}{2}\frac{e^{\Phi_{4}}}{b^{6}}\Bigr\}
w^{(n)}=W^{2}g_{n}\left(\frac{v^{3}}{W^{2}}\right)\left(\frac{\Lambda}{v}\right)^{n(3N_{c}-N_{f})}=Sg_{n}\left(\frac{v^{3}}{S}\right)\left(\frac{\Lambda}{v}\right)^{n(3N_{c}-N_{f})}
B_{p_{1}^{M_{1}}p_{2}^{M_{2}}\cdotsp_{l}^{M_{l}}}:=C_{p_{1}^{M_{1}}p_{2}^{M_{2}}\cdotsp_{l}^{M_{l}}}\setminus\biggl(\cup_{m|p_{1}^{M_{1}}p_{2}^{M_{2}}\cdotsp_{l}^{M_{l}},\;\;m\neqp_{1}^{M_{1}}p_{2}^{M_{2}}\cdotsp_{l}^{M_{l}}}(C_{m})\biggr).
\langle\bar{\psi}(x)\psi(x)\,\bar{\psi}(\tilde{x})\psi(\tilde{x})\rangle_{w}\cong-{\frac{\langle1\rangle_{w}}{(4\pi)^{2}}}{\frac{\vert\tilde{x}-x_{o}\vert^{4}}{r_{o}^{4}}}{\frac{2}{\vertx-x_{o}\vert^{4}}}
S_{KM}=\intd^{6}x\left\{-\frac{1}{6}F_{\mu\nu\lambda}F^{\mu\nu\lambda}+\frac{1}{3}\theta_{\mu\nu\lambda,\sigma\rho\delta}^{++}F^{-\mu\nu\lambda}F^{-\sigma\rho\delta}\right\}
{\calL}_{\mathrm{c}}=-{\frac{3}{2}}\left[S_{0}\overline{S}_{0}\,e^{-K/3}\right]_{D}+\left[S_{0}^{3}(ih+\alphaM^{3})\right]_{F}+{\frac{1}{4}}\sum_{a=1}^{2}{\left[(S+\beta^{a}T){\calW}^{a}{\calW}^{a}\right]_{F}},
I=-{\frac{1}{8G}}{\frac{(2\pil)^{2}}{\beta}},\qquad\qquadM={\frac{\partialI}{\partial\beta}}={\frac{1}{8G}}{\frac{(2\pil)^{2}}{\beta^{2}}}={\frac{1}{8G}}{\frac{r_{+}^{2}}{l^{2}}},
\sum_{n=1}^{\calN}P_{ni}\dot{q}_{n}^{i}+\int\pi^{ij}\dot{h}_{ij}\frac{i}{2}dz\wedge\bar{d}z=\sum_{n=2}^{\calN}t_{n}\dot{z^{\prime}}_{n}^{c}+\bar{t}_{n}\dot{{\bar{z}}^{\prime}}_{n}^{c}
\begin{array}{l}{p^{-}=\displaystyle\frac{1}{2}\Big(1-\frac{1}{2b}\tilde{\Gamma}^{[n}\Gamma^{m]}\pi_{n}A_{m}\Big)\equiv\frac{1}{2(n^{\prime}r^{\prime})}\not{\tilde{n}}^{\prime}\notr^{\prime},}\\{p^{+}=\displaystyle\frac{1}{2}\Big(1+\frac{1}{2b}\tilde{\Gamma}^{[n}\Gamma^{m]}\pi_{n}A_{m}\Big)\equiv\frac{1}{2(n^{\prime}r^{\prime})}\not{\tilde{r}}^{\prime}\notn^{\prime},}\\\end{array}
\begin{aligned}{k^{1/2(V_{r}\timesA)}\widetilde{\lambda}_{rA}\mathcal{A}_{s}^{A}-k^{1/2(V_{s}\timesA)}\widetilde{\lambda}_{sA}\mathcal{A}_{r}^{A}+f_{BC}^{A}\mathcal{A}_{r}^{B}\mathcal{A}_{s}^{C}=0}\\\end{aligned}
\frac{d}{d\Lambda}e^{-\frac{1}{\hbar}\left(L^{\Lambda,\Lambda_{0}}(\varphi)+I^{\Lambda,\Lambda_{0}}\right)}=\frac{\hbar}{2}\langle\frac{\delta}{\delta\varphi}\,,\dot{C}^{\Lambda,\Lambda_{0}}\frac{\delta}{\delta\varphi}\rangle\,e^{-\frac{1}{\hbar}\left(L^{\Lambda,\Lambda_{0}}(\varphi)+I^{\Lambda,\Lambda_{0}}\right)}.
\left(P^{\mu_{1}\nu_{1}\,\rho\sigma}\right)^{-1}\,P^{\rho\sigma\,\mu_{2}\nu_{2}}=\frac12\left(\delta_{\mu_{1}}^{\mu_{2}}\delta_{\nu_{1}}^{\nu_{2}}+\delta_{\mu_{1}}^{\nu_{2}}\delta_{\nu_{1}}^{\mu_{2}}\right).
O_{nm}\equiv{\frac{n!m!}{(n+m)!}}\sum_{i=1}^{N}\left(a_{i}^{n}a_{i}^{\daggerm}+a_{i}^{n-1}a_{i}^{\daggerm}a_{i}+\cdots+a_{i}^{\daggerm}a_{i}^{n}\right)
E_{A}=\intd^{2}x\left|\partial_{\pm}|\phi|-i{\frac{e\bar{A}_{\pm}}{\sqrt{1-2g^{2}|\phi|^{2}}}}|\phi|\right|^{2}\pm\intd^{2}x{\frac{e}{\sqrt{1-2g^{2}|\phi|^{2}}}}\epsilon_{ij}\bar{A}_{i}\partial_{j}|\phi|^{2}
S^{tot}=Tr(-\frac{k}{4\pi}\intA\partial_{0}A+\sum_{i=1}^{n}P_{i}dg_{i}g_{i}^{-1})+Tr\intA_{0}(\frac{k}{2\pi}F+\sum_{i=1}^{n}T_{i}\delta(z-z_{i}))\.
S=\frac{1}{16\pi}\mathrm{tr\Im}\left(\tau\intd^{4}xd^{2}\theta\W^{2}\right)+\frac{1}{4}\intd^{4}xd^{4}\theta\sum_{A=1}^{N_{f}}\left(\phi_{A}^{+}e^{-2V}\phi^{A}+\tilde{\phi}^{+A}e^{2V}\tilde{\phi}_{A}\right)
\displaystyle\mathop{\sum_{p\in\Lambda\ast}e^{[-\pi(p+\delta)^{t}C(p+\delta)]+2\piip^{t}\phi]}={V}_{\Lambda}^{-1}{\frac{1}{\sqrtdet{C}}{\sum}_{l\in\Lambda}e^{[-\pi(l+\phi)^{t}C^{-1}(l+\phi)-2\pii{\delta}^{t}(l+\phi)]}},\;\;\;\;\;\;\;\;\;\;\;\;}
\mathrm{Tr}_{IJ}(\frac1{\Gamma}\sum_{g}\hat{g}e^{-2tH_{o}})=\frac{d_{I}\,d_{J}}{\Gamma}\sum_{g}\bar{\chi}^{I}(g)\chi^{J}(g)\mathrm{tr}_{\alpha}(ge^{-2tH_{o}})\;
g_{\mu\nu}\left(x+\pi(\xi)\right)=g_{\mu\nu}+g_{\mu\nu;\lambda}\xi^{\lambda}+{\frac{1}{2}}\left[g_{\mu\nu;\lambda\sigma}+{\frac{1}{3}}\left(R_{~\lambda\mu\sigma}^{\kappa}g_{\kappa\nu}+R_{~\lambda\nu\sigma}^{\kappa}g_{\kappa\mu}\right)\right]\xi^{\lambda}\xi^{\sigma}+...
\tilde{G}_{\mu\nu}^{0A}-\tilde{G}_{\mu\nu}^{0L}=-{\frac{1}{k^{2}}}k_{\mu}k_{\nu}\biggl({\frac{(\lambdak^{2}+\eta^{2})}{(\eta\cdotk)^{2}}}+{\frac{(1-\lambda)}{k^{2}}}\biggr)+{\frac{k_{[\mu}\eta_{\nu]_{+}}}{k^{2}\eta\cdotk}}.
dS^{2}={\frac{1}{I_{0}}}\{{\frac{1}{f_{0}}}[e^{2k_{0}}dz\,d{\bar{z}}+\rho^{2}\,d\phi^{2}]-f_{0}\,dt^{2}\}+I_{0}^{2}(A_{03}\,d\phi+dx^{5})^{2},
\begin{array}{l}{\Bigl(\,\zeta(u-v)-\zeta(u+v)+2\!\zeta(v)\Bigr)\,A_{k}^{\alpha}(q,u)}\\{\mathrm{}-\,\Bigl(\,\Phi(\alpha(q),u-v)\,A_{k}^{\alpha}(q,v)\,-\,\Phi(\alpha(q),u+v)\,A_{k}^{\alpha}(q,-v)\Bigr)~=~0~.}\\\end{array}
R_{\mu\nu}=R_{\mu\lambda\nu}^{\lambda}=\partial_{\rho}\gamma_{\mu\nu}^{\rho}-\partial_{\nu}\gamma_{\mu\rho}^{\rho}-\gamma_{\mu\lambda}^{\rho}\gamma_{\nu\rho}^{\lambda}+\gamma_{\lambda\rho}^{\rho}\gamma_{\mu\nu}^{\lambda}.
I[\overline{{\psi}},\psi,\phi]=\intd^{2D}X\,\left[i\overline{{\psi}}\star\gamma^{\mathrm{MN}}\,\left(a^{\prime}\mathcal{D}_{\mathrm{MN}}+b^{\prime}\overline{{\mathcal{D}}}_{\mathrm{MN}}+c^{\prime}D_{\mathrm{MN}}\right)\psi+g\overline{{\psi}}\star\phi\star\psi+\cdots\right],
u_{i}^{\u}=\frac{1}{\sqrt{1+t\bar{t}}}\left(\begin{array}{cc}{t}&{1}\\{-1}&{\bar{t}}\\\end{array}\right),\u_{u}^{\i}=\frac{1}{\sqrt{1+t\bar{t}}}\left(\begin{array}{cc}{\bar{t}}&{-1}\\{1}&{t}\\\end{array}\right),
{\frac{1}{\vertV\vert}}\int_{V}d^{3}{\mathbf{x}}\\langle\,F_{1}({\mathbf{x}})F_{2}\,\rangle_{b,f}-\langle\,F_{1}\,\rangle_{b,f}\langle\,F_{2}\,\rangle_{b,f}\\\rightarrow\\0\qquad\mathrm{as~}V\nearrow{\mathbbR}^{3}.
\sum_{j,j^{\prime};\mu,\mu^{\prime}}z_{j}^{(\mu)}\;\{J_{(\mu)}^{j,-j},J_{(\mu^{\prime})}^{j^{\prime},-j^{\prime}}\}^{*}(y)\;t_{j^{\prime},-j^{\prime}}^{(\mu^{\prime})}=[R(\vec{z};y),y]
\sum_{i=5}^{8}\left.\frac{\partial\xi^{i}}{\partial\omega^{\alpha\beta}}\right|_{\omega=0}\frac{\partial}{\partial\xi^{i}}=\left[{\calL}(n){\calM}_{\alpha\beta}{\calL}(n)^{T}-({\calM}_{\alpha\beta})_{\\nu}^{\mu}n^{\nu}{\calL}(n)\frac{\partial}{\partialn^{\mu}}{\calL}(n)^{T}\right]^{\rho\sigma}y_{\sigma}\frac{\partial}{\partialy^{\rho}}
S_{H}=-\intd^{D}x\sqrt{|g_{D}|}e^{-\varphi}\left[\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}+\frac{1}{4}H_{\mu\nua}H^{\mu\nua}+\frac{1}{4}H_{\muab}H^{\muab}+\frac{1}{12}H_{abc}H^{abc}\right]
\begin{array}{l}{\phi_{k}(\chi)=\displaystyle\frac{1}{\omega_{n-1}}\int_{uu^{\prime}=1}a_{k}(u)P_{\alpha_{k}(m_{0})}(\chi,u)\dot{u}~,}\\{\psi_{k}(\chi)=\displaystyle\frac{1}{2\pi}\int_{uu^{\prime}=1}b_{k}(u)P_{\alpha_{k}(m_{0})}(\chi,u)\dot{u}~.}\\\end{array}
3e^{2}A_{cr}^{2}={\frac{3e^{4}\tilde{\rho}^{2}}{8\pi^{2}}}+{\frac{15e^{6}\tilde{\rho}^{2}}{128\pi^{4}}}\left[{\frac{2}{3}}+{\frac{16\pi^{2}}{5}}C\right].
|\psi_{i}>=\int_{-\infty}^{\infty}dp_{\lambda}\,|n;p_{\lambda};-+>\,\psi_{i}(p_{\lambda})\\,\\|\psi_{f}>=\int_{-\infty}^{\infty}dp_{\lambda}\,|m;p_{\lambda};-+>\,\psi_{f}(p_{\lambda})\,
\begin{array}{rcl}{T_{string}={\frac{L_{2}L_{3}L_{4}L_{5}}{(2\pi)^{5}L_{p}^{6}}}}\\{\lambda_{7}^{4}={\frac{(L_{2}L_{3}L_{4}L_{5})^{3}}{(2\pi)^{2}L_{p}^{12}}}}\\\end{array}
=\sum_{k=o}^{\mu_{mn}}\left(-\frac{2R_{12}}{\sqrt{R_{11}R_{22}}}\right)^{k}\frac{n!m!}{(n-k)!(m-k)!k!}H_{n-k}\left(\frac{\zeta_{1}}{\sqrt{2R_{11}}}\right)H_{m-k}\left(\frac{\zeta_{2}}{\sqrt{2R_{22}}}\right),
\hat{\calL}=\sum_{i=1}^{N_{+}}\bar{\psi}_{i}i\gamma^{\mu}(\nabla_{\mu}+i({\frac{1+\gamma_{5}}{2}})A_{\mu})\psi_{i}+\sum_{i=1}^{N_{-}}\bar{\psi}_{i}i\gamma^{\mu}(\nabla_{\mu}+i({\frac{1-\gamma_{5}}{2}})A_{\mu})\psi_{i}+{\frac{1}{4e^{2}}}F^{\mu\nu}F_{\mu\nu},
f_{\scriptscriptstyle{0,L}}(a)=\frac{1}{2\pi}\int_{0}^{a}dr\int_{0}^{2\pi}d\theta\,e^{i\theta}\,e^{2\phi(r,\theta)}\,\sum_{l=0}^{\infty}b_{\scriptscriptstyle{l,L}}\,r^{l}\,e^{il\theta}\,.
H^{2}=\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi}{3\mp^{2}}\left[\frac{1}{2}\dot{\chi}^{2}+e^{-\alpha\chi}\,\frac{1}{2}\dot{\varphi}^{2}+V(\varphi)+V(\chi)\right]\,,
\sqrt{C_{s}^{n}C_{n}^{2}}C_{n-2}^{2}\Phi^{n}+2\sqrt{C_{s}^{n-2}}C_{n-2}^{2}C_{s-n+2}^{2}\Phi^{n-2}+\sqrt{C_{s}^{n-4}}C_{s-n+4}^{2}C_{s-n+2}^{2}\Phi^{n-4}=0,
T_{\mathrm{grav}}^{\mu\nu}\equiv{\frac{2}{\sqrt{-g}}}{\frac{\deltaL}{\deltag_{\mu\nu}}},\\\\\\\\\\\\\\\T_{\mathrm{ph}}^{\mu\nu}\equiv{\frac{2}{\sqrt{-\hat{g}}}}{\frac{\deltaL}{\delta\hat{g}_{\mu\nu}}},
{P(x,T,T^{*})={\frac{{\midH(T)\mid}^{2}}{{T_{r}}^{3}{\mid\eta(T)\mid}^{6}}}2xA^{2}e^{-2\alphax}{\left({\frac{\alpha-\beta}{1+2x\beta}}\right)}^{2}\left(1+\alpha+\beta+2x\alpha\beta-{1/{4x^{2}}}\right)}
\delta_{\epsilon}{\bfA}=\left[\begin{array}{cc}{\epsilon^{k}\bar{\psi}_{k}-\psi^{k}\bar{\epsilon}_{k}}&{D\epsilon_{j}}\\{-D\bar{\epsilon}^{i}}&{\bar{\epsilon}^{i}\psi_{j}-\bar{\psi}^{i}\epsilon_{j}}\\\end{array}\right],
\displaystyle{\frac{\phi^{(0)}(a,b+1,c|u,v,w+1/4)}{\phi^{(0)}(a,b-1,c|u,v,w+1/4)}}=\displaystyle{\frac{W(a,b+1|A_{0})\,\overline{{W}}(b+1,c|B_{0})}{W(a,b-1|A_{0})\,\overline{{W}}(b-1,c|B_{0})}}.
\left\{Q_{\alphaa},Q_{\betab}\right\}=\gamma_{\alpha\beta}^{\mu}\gamma_{ab}^{ij}P_{\muij}+\gamma_{\alpha\beta}^{\mu_{1}\mu_{2}\mu_{3}}\delta_{ab}T_{\mu_{1}\mu_{2}\mu_{3}}^{+}+\gamma_{\alpha\beta}^{\mu_{1}\mu_{2}\mu_{3}}\gamma_{ab}^{ijkl}Z_{\mu_{1}\mu_{2}\mu_{3}ijkl}^{+}
\Omega_{-2}^{-2i}-\Omega_{+2}^{+2i}=4i\psi_{+2q}^{1+}\gamma_{q\dot{q}}^{i}\psi_{-2\dot{q}}^{1-}-4i\psi_{-2q}^{1+}\gamma_{q\dot{q}}^{i}\psi_{+2\dot{q}}^{1-}+1\leftrightarrow2.
\sum_{J_{t}}(2J_{t}+1)\mathrm{Tr}\left[D^{J_{t}}(U_{T_{1}}^{(t)})\cdotsD^{J_{t}}(U_{T_{n_{t}}}^{(t)})\right]=\delta(U_{T_{1}}^{(t)}\cdotsU_{T_{n_{t}}}^{(t)},1).
\Lambda_{RL}\left(\vec{\theta},\vec{\varphi}\right)\Psi_{RL}=\left(\begin{array}{cc}{e^{i\vec{J}\cdot(\vec{\theta}-i\vec{\varphi})}}&{0}\\{0}&{e^{i\vec{J}\cdot(\vec{\theta}+i\vec{\varphi})}}\\\end{array}\right)\left(\begin{array}{c}{\Phi_{R}}\\{\Phi_{L}}\\\end{array}\right)\,,
\frac{1}{\gamma.D+m}=(\gamma.D-m)\frac{1}{D^{2}+\frac{1}{2}\sigma_{\mu\nu}F_{\mu\nu}-m^{2}}=-(\gamma.D-m)\int_{0}^{\infty}ds\e^{s(D^{2}+\frac{1}{2}\sigma_{\mu\nu}F_{\mu\nu}-m^{2})}
{\calL}=-\frac{1}{4}\,\left(\partial_{[\mu}v_{\nu]}\right)^{2}-\frac{e^{2}}{2\pi}\,v_{\mu}v^{\mu}-\frac{e}{\sqrt{\pi}}\,\epsilon^{\mu\nu}\,{v_{\mu}}(\partial_{\nu}\theta)+\frac{1}{2}\,(\partial_{\mu}\theta)(\partial^{\mu}\theta).
{\calH}_{c}=\intd^{d}x\,\left\{{\frac{1}{2}}\sum_{i=1}^{N}\Bigl[(\partial_{\mu}\Phi_{i})^{2}+r\Phi_{i}^{2}\Bigr]+{\frac{1}{4!}}\Bigl[u(\sum_{i}^{N}\Phi_{i}^{2})^{2}+v\sum_{i}^{N}\Phi_{i}^{4}\Bigr]\right\}.
Z_{\mathrm{\scriptsize~1-loop}}(R,T;t)=e^{-{\frac{1}{2\epsilon^{2}}}\left({\frac{T^{2}}{R^{2}t}}+t\right)}\int{\prod_{n,m\geq0}}^{\prime}d\phi_{nm}e^{-S_{\mathrm{\scriptsize~1-loop}}[\phi]}\;\;\;,
\deltab_{\mu\nu}^{a}=\varphi^{2}(\partial_{\mu}\epsilon^{\rho}\eta_{\rho\nu}^{a}+\partial_{\nu}\epsilon^{\rho}\eta_{\mu\rho}^{a})+\epsilon_{\rho}\partial_{\rho}b_{\mu\nu}^{a}+\partial_{\mu}\epsilon^{\rho}b_{\rho\nu}^{a}+\partial_{\nu}\epsilon_{\mu\rho}^{a}
S=\intd^{3}x\int_{0}^{2\piR}dy\bigg\{-\frac{1}{2}\mathrm{tr}F_{MN}F^{MN}+\sum_{I=1}^{N_{f}}\bar{\psi}_{I}\gamma^{M}D_{M}\psi_{I}\nonumber\,+(D_{M}\phi)^{\dagger}D^{M}\phi-V(\phi)\bigg\},
{\frac{2i(j_{1,c}+j_{2,c}+1)(j_{1,c}-j_{2,c})}{ic}}=2c\left({\frac{j_{1,c}^{2}+j_{2,c}^{2}+j_{1,c}+j_{2,c}}{c^{2}}}-2{\frac{j_{2,c}^{2}+j_{2,c}}{c^{2}}}\right)
S^{ab}S^{cb}=\frac{1}{4}\tau_{ij}^{b}\tau_{kl}^{b}\left(U\tau^{a}U^{\dagger}\right)_{ji}\left(U\tau^{c}U^{\dagger}\right)_{kl}=\frac{1}{2}tr\left(\tau^{a}\tau^{b}\right)=\delta^{ab}
\Box\chi+\frac{\zetaV_{1}^{\prime}+V_{2}^{\prime}}{(\zeta+b)^{2}}=-\frac{\upsilon+\chi}{\sqrt{2}(\zeta+b)^{1/2}(\zeta+k)}[f_{E}(\zeta+h_{E})\overline{{E^{\prime}}}E^{\prime}+f_{U}(\zeta+h_{U})\overline{{U^{\prime}}}U^{\prime}+f_{D}(\zeta+h_{D})\overline{{D^{\prime}}}D^{\prime}],
{\calL}={\frac{1}{2g\sqrt{\alpha^{\prime}}}}~\mathrm{Tr}~\left({\dot{\Phi}}_{i}^{2}+{\frac{1}{2(2\pi\alpha^{\prime})^{2}}}\left[\Phi_{i},\Phi_{j}\right]^{2}-{\frac{i\lambda}{2\pi\alpha^{\prime}}}\epsilon_{ijk}\left[\Phi_{i},\Phi_{j}\right]{\dot{\Phi}_{k}}\right)~,
\frac{1-\xi_{4}}{\beta}\sum_{N^{\prime}N^{\prime\prime}}\frac{(2\pi)^{3}\alpha^{2}}{N^{\prime}N^{\prime\prime}}Y_{N^{\prime}LM}(\xi)A_{L,N^{\prime}N^{\prime\prime}}^{(c/2\mu)}Y_{N^{\prime\prime}LM}^{*}(\zeta)\frac{1-\zeta_{4}}{\beta}
{\calL}_{\mathrm{int}}=\sum_{i,\tau,l,m}\overline{{\psi}}(x)Q_{i}^{\taulm}\psi(x)\varphi_{\taulm}^{i}(x)\equiv\sum_{i,\tau,l,m}(\psi(x),Q_{i}^{\taulm}\varphi_{\taulm}^{i}(x)\psi(x)),
{\frac{dn^{(\lambda)}(\omega)}{d\omega}}=\int_{\Sigma_{r}}d\Sigma^{\mu}\sum_{k}\tilde{j}_{\mu}(\phi_{\omega,k}^{(\lambda)})=\int_{\Sigma_{r}}\sqrt{h}d^{D-1}x\sum_{k}2\omega|\phi_{\omega,k}^{(\lambda)}|^{2}~~~.
C_{ij}Q_{k}^{i}Q_{l}^{j}=\delta_{kl},\quadC_{ij}(Q^{\prime}{}_{j}^{i}Q_{l}^{j}-Q^{\prime}{}_{k}^{i}Q_{l}^{j})=0,\quadM_{ij}=C_{kl}Q^{\prime}{}_{l}^{k}Q_{j}^{l},
R_{\delta_{0}}^{j}\bigr(F_{\eta_{0}\alpha_{0}}^{\delta_{0}}F_{\beta_{0}\gamma_{0}}^{\eta_{0}}-R_{\alpha_{0}}^{i}F_{\beta_{0}\gamma_{0},i}^{\delta_{0}}+\mathrm{cyclic~perm}(\alpha_{0},\beta_{0},\gamma_{0})\bigr)=0.
\begin{array}{rcl}{d_{i,\kappa}}&{=}&{\frac{\kappa}{2}+\frac{i-1}{2}+\frac{1}{2\,n}\,\,\,(i=1,2,\dots,n),}\\{e_{i,\kappa}}&{=}&{\frac{\kappa}{2}+\frac{i}{n-2}+\frac{1}{2\,(n-2)}\,\,\,(i=1,2,\dots,n-2),}\\{e_{i,\kappa}}&{=}&{\frac{\kappa}{2}+1+\frac{i-n}{2}+\frac{\delta_{i}}{2}\,\,\,(i=n-1),}\\\end{array}
\nabla^{A}\partial_{A}\Phi^{a}=-{\frac{(\partial^{A}\Phi^{b})(\partial_{A}\Phi^{b})}{\eta^{2}}}\Phi^{a}+{\frac{\sqrt{-h}}{\sqrt{-g}}}{\frac{\partial\sigma_{i}(\Phi^{a})}{\partial\Phi^{a}}}\delta(y_{i})\,,
{\calI}-\delta{\calI}=\frac{\Omega_{N/2+1}\Omega_{N/2}\Omega_{N/2-1}}{2^{N+3}\pi^{3N/4+1}}\,\frac{\Gamma(N-1)\Gamma(N/2-1)\Gamma(N/4+1/2)}{\Gamma(N/2)\Gamma(N/2-1/2)}.
\langle\beta,k,s\\mathrm{out}|\alpha\\mathrm{in}\rangle=\frac{-ig(s)}{(2\pi)^{4}}\intd^{5}xe^{-i(k\cdotx+\sigma\kappa\tau)}[\Box+\sigma\partial_{\tau}^{2}]\langle\beta\\mathrm{out}|\varepsilon_{s}\cdota(x,\tau)|\alpha\\mathrm{in}\rangle
\begin{array}{rcl}{D_{\mu,\nu}^{j}(\alpha,\beta,\gamma)}&{=}&{\displaystyle\sum_{k}\frac{(-1)^{k}\sqrt{(j+\mu)!(j-\mu)!(j+\nu)!(j-\nu)!}}{k!(j-\nu-k)!(j+\mu-k)!(k-\mu+\nu)!}\nonumber}\\{}&{}&{\displaystylee^{-i\alpha\mu}(cos(\beta/2))^{2j-\nu+\mu-2k}(sin(\beta/2))^{2k-\mu+\nu}e^{-i\nu\gamma}}\\\end{array}
D(x,x^{\prime})=\int{\frac{d^{4}k}{(2\pi)^{4}}}\int{\frac{d^{2}{\vec{q}}}{(2\pi)^{2}}}{\frac{e^{ik_{\mu}(x^{\mu}-x^{\mu})+i{\vec{q}}\cdot({\vec{x}}-{\vec{x}^{\prime}})}}{k_{4}^{2}+q^{2}}},
\theta=\left(\begin{matrix}{\theta^{\alpha}}\\{\theta^{\alpha^{\prime}}}\\\end{matrix}\right)\,\qquad\kappa=\left(\begin{matrix}{\kappa^{\alpha}}\\{\kappa^{\alpha^{\prime}}}\\\end{matrix}\right)\,\qquad\epsilon=\left(\begin{matrix}{\epsilon^{\alpha}}\\{\epsilon^{\alpha^{\prime}}}\\\end{matrix}\right)
p_{a}=\frac{1}{\sqrt{k}}\times\frac{1}{l_{s}}\left(\frac{n_{a}}{r_{a}}+w^{a}r_{a}\right)~,~~~\bar{p}_{a}=\frac{1}{\sqrt{k}}\times\frac{1}{l_{s}}\left(\frac{n_{a}}{r_{a}}-w^{a}r_{a}\right)~,
\frac{x_{23}}{x_{13}}\rightarrow\frac{x_{67}}{x_{13}},{}~~\frac{x_{13}}{x_{1}}\rightarrow\frac{x_{13}}{x_{1}},~~\frac{x_{13}}{x_{14}}\rightarrow\frac{x_{5}x_{13}}{x_{1}x_{58}},~~\frac{x_{24}}{x_{23}}\rightarrow\frac{x_{1}x_{24}}{x_{5}x_{67}},
\rho_{l}=A(r)\frac{2r}{r+3}\gamma_{s}^{3}B^{\frac{2r}{r+3}}\int_{x_{1}}^{x_{2}}dxx^{-3(r+1)/(r+3)}I_{0}(\frac{R_{H}x}{B\gamma_{s}})\mu/R_{H}^{2}
\langle\psi^{-}|\phi^{bg}\rangle=\int_{m_{0}^{2}}^{-\infty_{II}}d{\mathsfs}\,\langle\psi^{-}|{\mathsfs}^{-}\rangleS_{j}({\mathsfs})\langle^{+}{\mathsfs}|\phi^{+}\rangle=\int_{m_{0}^{2}}^{\infty}d{\mathsfs}\,\langle\psi^{-}|{\mathsfs}^{-}\rangleb_{j}({\mathsfs})\langle^{+}{\mathsfs}|\phi^{+}\rangle\,,
\left(\begin{array}{cc}{u_{1}^{+}}&{u_{1}^{-}}\\{u_{2}^{+}}&{u_{2}^{-}}\\\end{array}\right)={\frac{1}{\sqrt{1+y\bar{y}}}}\left(\begin{array}{cc}{1}&{-\bar{y}}\\{y}&{1}\\\end{array}\right)\;.
{\frac{-1}{(8\pi^{2}\alpha^{\prime})^{2}}}\,\int_{0}^{\infty}{\frac{dt}{t^{3}}}\,{\frac{(1-z)^{4}}{z}}\,\sum_{n=0}^{\infty}z^{2n}=-\Big({\frac{\theta}{8\pi^{2}\alpha^{\prime}}}\Big)^{2}\,\int_{0}^{\infty}{\frac{du}{u^{3}}}\,e^{u}\,(1-e^{-u})^{4}\,\sum_{n=0}^{\infty}e^{-2nu}\,.
Z^{2}\Big[{\frac{\mu^{2}}{\mu^{2}+A^{2}}}+{\frac{A^{2}(\mu^{2}+A^{2})^{k-1}\gamma^{k}}{(k+1)^{2}B^{2k}}}\Big(1-k({\frac{\mu|B|Z}{\sqrt{\gamma}\sqrt{\mu^{2}+A^{2}}}})^{{\frac{1}{n}}}\Big)^{2(n+1)}\Big]=1.
S_{0}(\beta)={\frac{\displaystyle\strut\Gamma({\frac{1}{2}}+{\frac{\beta}{2\pii}})\displaystyle\strut\Gamma(-{\frac{\beta}{2\pii}})}{\displaystyle\strut\Gamma({\frac{1}{2}}-{\frac{\beta}{2\pii}})\displaystyle\strut\Gamma({\frac{\beta}{2\pii}})}}.
{\calF}(a)+{\frac{1}{4}}v^{t}\left[G_{\gamma}^{t}\left(\begin{array}{c}{0}\\{1}\\\end{array}\begin{array}{cc}{1}\\{0}\\\end{array}\right)G_{\gamma}-\left(\begin{array}{c}{0}\\{1}\\\end{array}\begin{array}{cc}{1}\\{0}\\\end{array}\right)\right]v,
\frac{dx_{\mu}}{ds\,}\frac{d^{2}x_{\mu}}{ds^{2}\,}\,=\,0;\qquad\frac{dx_{\mu}}{ds\,}\frac{d^{3}x_{\mu}}{ds^{3}\,}\,=\,k_{1}^{2};\qquad\frac{d^{2}x_{\mu}}{ds^{2}\,}\frac{d^{3}x_{\mu}}{ds^{3}\,}\,=\,-\,k_{1}\,\frac{dk_{1}}{ds\,}\,{.}
S=\frac{1}{2}\intd^{4}x\,\left\{g_{ij}\partial_{m}f^{i}\partial^{m}f^{j}+\bar{g}^{ij}\partial_{m}\bar{f}_{i}\partial^{m}\bar{f}_{j}+2h_{~j}^{i}\partial_{m}f^{j}\partial^{m}\bar{f}_{i})-V(f)\right\}
{\calI}_{s}^{-}=\left\{\begin{array}{cl}{1}&{\qquadq_{1}=0}\\{8|q_{1}|}&{\qquad0<|q_{1}|<|\tilde{a}_{1}|\quad\mathrm{and}\quadq_{1}a_{1}<0}\\{0}&{\qquad\mathrm{otherwise}}\\\end{array}\right\}.
4\overline{{\chi}}_{(A}\gamma_{a}\psi_{B)}\overline{{\psi}}_{C}\gamma^{A}\psi^{C}-6\overline{{\chi}}_{C}\psi^{C}\overline{{\psi}}_{(A}\psi_{B)}+\overline{{\chi}}_{C}\gamma_{7}\gamma_{ab}\psi^{C}\overline{{\psi}}_{(A}\gamma^{7}\gamma^{ab}\psi_{B)}=0.
V(\Phi)=-\frac{1}{4}m_{0}^{2}\Phi^{2}+\frac{1}{4}g_{0}\Phi^{4}+\frac{3}{8}\frac{m_{0}^{4}}{g_{0}}=\frac{1}{4!}g_{0}\left(\Phi^{2}-v_{0}^{2}\right)^{2}\,,
P(l)=\frac{2\pi\lambda}{l!^{2}(l+1)!^{2}}\left(\frac{\omegar_{0}}{2}\right)^{2l+2}\left|\frac{\Gamma^{2}\left(\frac{l+2}{2}-\frac{i\lambda}{2}\right)}{\Gamma\left(1-i\lambda\right)}\right|^{2}\,.
\frac{\delta}{\delta\xi^{\mu}(s)}\{\Omega_{\xi}(s,0)\Lambda_{\nu}[\xi|s]\Omega_{\xi}^{-1}(s,0)\}=\Omega_{\xi}(s,0)\{{\calD}_{\mu}(s)\Lambda_{\nu}[\xi|s]\}\Omega_{\xi}^{-1}(s,0).
U(t)=I-\frac{i}{\hbar}\int_{0}^{t}dt^{\prime}H(t^{\prime})+\left(-\frac{i}{\hbar}\right)^{2}\int_{0}^{t}dt^{\prime}\int_{0}^{t^{\prime}}dt^{\prime\prime}H(t^{\prime})H(t^{\prime\prime})+\cdots
\sum_{\sigma_{i},\tau_{i}}S(\{\sigma_{i}\}_{i=1}^{k},\{\tau_{i}\}_{i=1}^{k})F_{\sigma_{2}\tau_{2}}^{\sigma_{1}\tau_{1}}F_{\sigma_{3}\tau_{3}}^{\sigma_{2}\tau_{2}}\dotsF_{(\zeta\sigma_{1})(\xi\tau_{1})}^{\sigma_{k}\tau_{k}}
\begin{array}{c}{K^{(0)}=-2|k_{1}+k_{2}|^{2}-3m^{2}+2\frac{(|k_{1}|^{2}+m^{2})(|p_{2}|^{2}+m^{2})+(|p_{1}|^{2}+m^{2})(|k_{2}|^{2}+m^{2})}{|k_{1}-p_{1}|^{2}+m^{2}}}\\\end{array}
\pi_{\mu\nu}=-{\frac{1}{4}}T_{\mu}^{\lambda}T_{\lambda\nu}+{\frac{1}{4(d-1)}}TT_{\mu\nu}+{\frac{1}{8}}g_{\mu\nu}\left(T^{\alpha\beta}T_{\alpha\beta}-{\frac{1}{d-1}}T^{2}\right)\.
A=4\beta_{\phi}^{a}G^{bcd}\frac{\delta\Gamma}{\deltaG^{abcd}}+\beta_{G}^{ab}\Big(-\frac{\delta\Gamma}{\deltaG^{ab}}+6\overline{\phi}^{c}\frac{\delta\Gamma}{\deltaG^{abc}}-13\overline{\phi}^{c}\overline{\phi}^{d}\frac{\delta\Gamma}{\deltaG^{abcd}}\Big)
\left(\begin{array}{ll}{-\frac{d^{2}}{d\rho^{2}}+\frac{m^{2}-\frac{1}{4}}{\rho^{2}}-2{\tilde{E}}}&{\;\;\;\;\;\;\frac{-2F}{\rho}}\\{\frac{-2F}{\rho}}&{-\frac{d^{2}}{d\rho^{2}}+\frac{(m+1)^{2}-\frac{1}{4}}{\rho^{2}}-2{\tilde{E}}}\\\end{array}\right)\left(\begin{array}{l}{\phi_{1}^{(n_{\rho},m)}}\\{\phi_{2}^{(n_{\rho},m)}}\\\end{array}\right)=0,
U_{\epsilon}(t,t_{0})=\sum_{n=0}^{\infty}{\frac{(-i)^{n}}{n!}}\int_{t_{0}}^{t}dt_{1}...\int_{t_{0}}^{t}dt_{n}\,e^{-\epsilon(|t_{1}|+...+|t_{n}|)}\,\,\,T[H_{I}(t_{1})...H_{I}(t_{n})]
\tilde{k}_{m}\:\tilde{k}_{m^{\prime}}\;=\;\delta(m-m^{\prime})\:\sum_{\beta_{1},\beta_{2}=0}^{\infty}(-i\pi)^{2\beta_{1}+2\beta_{2}}\:b_{m}^{<}\:(k_{m}\:b_{m})^{2\beta_{1}}\:p_{m}\:(b_{m}\:k_{m})^{2\beta_{2}}\:b_{m}^{>}\;.
\begin{array}{rcl}{C_{ij}}&{=}&{{Y^{k}}_{i}\bulletC_{kl}{Y^{l}}_{j}}\\{}&{=}&{C_{kl}{(Y_{1}\bulletY_{2})^{kl}}_{ij}}\\{}&{=}&{C_{kl}{(R_{12}^{-1}Y_{1}R_{12}Y_{2})^{kl}}_{ij},}\\\end{array}
\sum_{i=1}^{n}\frac{1}{m_{i}}\nabla_{\vec{r^{\prime}}_{i}}^{2}=\frac{1}{M}\nabla_{\vec{R}}^{2}+\sum_{i=1}^{n}\frac{1}{m_{i}}\nabla_{\vec{r}_{i}}^{2}-\frac{1}{M}(\sum_{i=1}^{n}\nabla_{\vec{r}_{i}})\cdot(\sum_{j=1}^{n}\nabla_{\vec{r}_{j}})
\begin{array}{ccc}{\tilde{P}\timesV}&{\mapsto}&{\tilde{P}}\\{\downarrow\tilde{\pi}_{2}}&{}&{\downarrow\tilde{\pi}_{3}}\\{{\calV}=\tilde{P}\times_{SO(V)}V}&{\stackrel{\mapsto}{\pi_{1}}}&{P=\tilde{P}/SO(V)}\\{\downarrow\pi_{2}}&{}&{\downarrow\pi_{3}}\\{E={\calV}/G}&{\stackrel{\mapsto}{\pi_{4}}}&{M=P/G=\tilde{P}/(SO(V)\timesG).}\\\end{array}