harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\calL}={\textstyle\frac12}\,i\,\overline{{\psi}}^{a}\gamma^{\mu}\partial_{\mu}\psi^{a}-{\textstyle\frac18}\,g\,B_{\mu}^{ab\,}B^{\mu\,ab}={\textstyle\frac12}\,i\,\overline{{\psi}}^{a}\gamma^{\mu}\partial_{\mu}\psi^{a}+{\textstyle\frac18}\,g\left(\overline{{\psi}}^{a}\psi^{a}\right)^{2}~.
S_{\mathrm{matter}}(\Phi,\bar{\psi},\xi)=\intd^{4}x\;\frac{1}{2}\bar{\psi}\left(\partial^{2}\xi-ig\;\partial_{x+iy}\Phi\;\partial_{0-z}\xi+ig\;\partial_{0-z}\Phi\;\partial_{x+iy}\xi\right)\,,
\begin{array}{ll}{\mathrm{BPS}_{1/2}(3)=[W(3)]^{m+n-2}}&{\mathrm{if~j=0~}}\\{\mathrm{BPS}_{1/4}(3)=[W(3)]^{m+n-2-j}\;[W^{-}(3)]^{j}}&{\mathrm{if~j\neq0~}}\\\end{array}
\left(\begin{matrix}{{}~1}&{~0}&{~0}&{~0}\\{{}~0}&{-1}&{~0}&{~0}\\{{}~0}&{~0}&{~1}&{~0}\\{{}~0}&{~0}&{~0}&{-1}\\{-1}&{-1}&{-1}&{-1}\\\end{matrix}\right)
L=\sum_{i=1}^{\mathrm{N}}\overline{{\psi_{i}}}\left(\triangle_{\mu}\gamma_{\mu}\otimes\mathrm{1}-\frac{1}{2}\gamma_{3}\otimes\gamma_{\mu}\gamma_{3}\Box_{\mu}+g\left(1\otimes\sigma+i\gamma_{3}\otimes\pi\right)\right)\psi_{i}+\frac{\mathrm{Tr}(\sigma^{2}+\pi^{2})}{2},
\eta_{ij}^{(0)}=0\quad\mathrm{and~\quad~}\eta_{ij}^{(2)}=\frac{\pi^{2}B(2-B)}{4^{3}}\left(\begin{array}{lll}{3}&{2}&{1}\\{2}&{4}&{2}\\{1}&{2}&{3}\\\end{array}\right)_{ij}\,\,.
(iL^{\alpha}\mathrm{D}_{\alpha}-\kappa)(iL^{\alpha}\mathrm{D}_{\alpha}+\kappa)\Phi=({\frac{i}{2}}\mathrm{D}_{\alpha}\mathrm{D}^{\alpha}\Phi(\theta)+p_{\alpha\beta}L^{\alpha}L^{\beta})-\kappa^{2})\Phi(\theta)=0.
{\calL}=-{\frac{1}{2}}\mathrm{Tr}\bigl({\bfF}^{\mu\nu}{\bfF}_{\mu\nu}\bigr)\,\mathrm{with}\quad{\bfF}^{\mu\nu}\equiv\partial^{\mu}{\bfA}^{\nu}-\partial^{\nu}{\bfA}^{\mu}+ig\bigl[{\bfA}^{\mu},{\bfA}^{\nu}\bigr]
R_{\alpha}^{\j}\frac{\deltaR_{\beta}^{\i}}{\delta\phi^{j}}-R_{\beta}^{\j}\frac{\deltaR_{\alpha}^{\i}}{\delta\phi^{j}}=D_{\alpha\beta}^{\\\\gamma}R_{\gamma}^{\i}+M_{\alpha\beta}^{\\\ij}\frac{\deltaS_{0}}{\delta\phi^{j}}\,.
\left|\Psi\right\rangle=\sum_{a_{1},\ldots,a_{N}}\Psi^{a_{1}a_{2}\ldotsa_{N}}\left|K_{a_{1}a_{2}}\left(\vartheta_{1}\right)\ldotsK_{a_{N}a_{1}}\left(\vartheta_{N}\right)\right\rangle\,.
\left(\begin{array}{ccc}{1}&{-\bar{t}_{a}R}&{-\bar{t}_{a}R}\\{-\bar{t}_{b}R}&{1}&{-\bar{t}_{b}R}\\{-\bar{t}_{c}R}&{-\bar{t}_{c}R}&{1}\\\end{array}\right)\left(\begin{array}{c}{r_{aa}}\\{r_{ab}}\\{r_{ac}}\\\end{array}\right)=\left(\begin{array}{c}{0}\\{t_{ab}(=t_{ba})}\\{t_{ac}(=t_{ab})}\\\end{array}\right)
\partial_{\xi}\Gamma^{T}(p^{2})=0\quad\mathrm{with}\quad\Gamma_{A_{\mu}A_{\nu}}\equiv\Gamma_{\mu\nu}=(\eta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}})\Gamma^{T}+\frac{p_{\mu}p_{\nu}}{p^{2}}\Gamma^{L}
t_{\mu\nu}(x;\gamma)={\frac{2}{3}}\left(t_{\mu}^{1}t_{\nu}^{2}+t_{\mu}^{2}t_{\nu}^{1}-{\frac{1}{2}}g_{\mu\nu}(t^{1}\cdott^{2})\right)-{\frac{1}{3}}\left(t_{\mu}^{1}t_{\nu}^{1}+t_{\mu}^{2}t_{\nu}^{2}-{\frac{1}{2}}g_{\mu\nu}\right)_{.}
D_{\alpha}\left(\hat{A}_{\beta}^{I}+\hat{A}_{\beta}^{\pm}\right)=\partial_{\alpha}\left(\hat{A}_{\beta}^{I}+\hat{A}_{\beta}^{\pm}\right)-i\,\left[\hat{A}_{\alpha}^{I}+\hat{A}_{\alpha}^{\pm},\,\hat{A}_{\beta}^{I}+\hat{A}_{\beta}^{\pm}\right]-\Gamma_{\alpha\beta}^{\gamma}\,\left(\hat{A}_{\gamma}^{I}+\hat{A}_{\gamma}^{\pm}\right).
(\sum_{n=1}^{15}(-1)^{n}-1)(\frac{g^{2}}{32\pi^{2}})tr\epsilon^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}=-2(\frac{g^{2}}{32\pi^{2}})tr\epsilon^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}
V=q\left({{\calA}_{0}\left(y\right)-{\calA}_{0}\left({y^{\prime}}\right)}\right)=\frac{{q^{2}\sqrt\pi}}{{2g}}\sqrt{\frac{a-1}{{a}}}\left({1-e^{-\frac{g}{{\sqrt\pi}}\sqrt{\frac{a}{{a-1}}}|y-y^{\prime}|}}\right).
\begin{array}{l}{{I_{+}}^{T}={I_{+}}^{*}={(I_{+})}^{+},\quad{(I_{+})}^{2}=1,\quad\midI_{+}\mid=1,}\\{({I_{+}})_{(1,1,\alpha)}^{(2,1,\alpha)}=({I_{+}})_{(2,2,\alpha)}^{(1,2,\alpha)}=0,\quad({I_{+}})_{(1,1,\alpha)}^{(1,1,\alpha)}=({I_{+}})_{(2,2,\alpha)}^{(2,2,\alpha)}=1.}\\\end{array}
W_{\alpha}=\left(\begin{array}{ccc}{0}&{\frac{1}{4}T_{\alpha}}&{\frac{1}{2}W_{3,\alpha}}\\{0}&{0}&{\frac{1}{4}T_{\alpha}}\\{0}&{0}&{0}\\\end{array}\right)\qquad\qquad\mathrm{for}\,\,\alpha=M,G,\mathrm{gh}
\frac{d^{2}\lambda}{d\phi_{i}d\phi_{j}}=\frac{1}{2\pii}\left[-\frac{P(1-\delta_{ij})}{y(x-\phi_{i})(x-\phi_{j})}dx+\frac{P^{3}}{y^{3}(x-\phi_{i})(x-\phi_{j})}dx\right]=
V_{II}(\beta,\phi)=\mu^{D}\sum_{s=1}^{\infty}g^{s}\phi^{2s}h(D,s)\biggl(\frac{1}{2^{\frac{D}{2}-s+2}}\Gamma(s-\frac{D}{2})(\frac{m}{\mu})^{D-2s}+\sum_{n=1}^{\infty}\biggl(\frac{m}{\mu^{2}\betan}\biggr)^{\frac{D}{2}-s}K_{\frac{D}{2}-s}(mn\beta)\biggr)
S_{\mu\nu}=\frac{1}{12}T_{\alpha}^{\alpha}T_{\mu\nu}-\frac{1}{4}T_{\mu\alpha}T_{\nu}^{\alpha}+\frac{1}{24}g_{\mu\nu}\left[3T_{\alpha\beta}T^{\alpha\beta}-(T_{\alpha}^{\alpha})^{2}\right]
\begin{array}{l}{\lbrackq^{-},\alpha_{0}^{+}]=i\,,}\\{\left[\alpha_{n}^{-}\,,\alpha_{m}^{+}\right]=n\,\delta_{n+m,0}\,\,\,,}\\{\left[s_{n}\,,s_{m}\right]=\left(\frack2-1\right)\,n\,\delta_{n+m,0}\,\,\,,}\\{\left[L_{n}^{\prime}\,,L_{m}^{\prime}\right]=(n-m)L_{n+m}^{\prime}+0}\\\end{array}
\left[m+e_{a}^{\mu}\gamma^{a}(p_{\mu}+\frac{1}{2}i\nabla_{\mu})\right]W^{(0)}=-\frac{1}{2}i\kappae_{a}^{\mu}\gamma^{a}R_{\mu\nu}^{bc}\left\{\frac{\partial}{\partialp_{\nu}}W^{(0)},\sigma^{bc}\right\}
{\cal{S}}^{-1}(x,x^{\prime})\equiv\left\{\left(-\frac{\partial^{2}}{\partial\tau^{2}}+\vec{\nabla}^{2}\right)-2\frac{H}{(1-H\tau)}\frac{\partial}{\partial\tau}-\frac{m^{2}}{(1-H\tau)^{2}}\right\}\delta^{(4)}(x-x^{\prime})
2{\frac{\partial{\calW}}{\partialz}}=2{\frac{\partial{\calW}}{\partialA^{i}}}{\frac{\partialA^{i}}{\partialz}}=\Omega_{-}K^{ij^{*}}{\frac{\partial{\calW}}{\partialA^{i}}}\frac{\partial{\calW}^{*}}{\partialA^{*j}}.
S=\int_{E}d^{4}xd^{2}y\left[\frac{1}{2}(\frac{\partial\phi(x,y)}{\partialx^{\mu}})^{2}-\frac{1}{2}\frac{\partial\phi(x,y)}{\partialy^{i}}\frac{\partial\phi(x,y)}{\partialy^{j}}-\frac{1}{2}m_{0}^{2}\phi^{2}(x,y)-\frac{\hat{\lambda}}{4!}\phi^{4}(x,y)\right],
\begin{array}{c}{\varphi(n)=q^{n-1}(S_{0}(n,q^{2})+2\nuS_{0}(n,-q^{2})}\\{=q^{1-n}\{[n]_{q^{2}}^{(a)}+2\nu(-)^{n-1}[n]_{-q^{2}}^{(a)}\}}\\{=[n]_{q}^{(b)}+2\nu(i)^{1-n}[n]_{iq}^{(b)}.}\\\end{array}
\langle\psi^{-}|M^{2}|\hat{\bboxp}j_{3}({\mathsf{s}}_{R}j_{R})^{-}\rangle=\left(M_{R}-i\frac{\Gamma_{R}}{2}\right)^{2}\langle\psi^{-}|\hat{\bboxp}j_{3}({\mathsf{s}}_{R}j_{R})^{-}\rangle\quad\mathrm{for~every~}\psi^{-}\in\Phi_{+}\subset{\mathcalH}\subset\Phi_{+}^{\times}.
{\calL}_{TM}=\beta^{2}\sqrt{1+\frac{1}{6m^{2}\beta^{2}}H^{\mu\nu\rho}H_{\mu\nu\rho}-\frac{1}{2\beta^{2}}F^{\mu\nu}F_{\mu\nu}}-\frac{\chi}{2\theta}\varepsilon^{\mu\nu\rho\lambda}B_{\nu\rho}\partial_{\mu}A_{\lambda}\;.
\begin{array}{ccl}{f_{\lambdaAB}}&{=}&{\displaystyle{\int}d^{2}\sigma\sqrt{w(\sigma)}\Phi^{(\lambda)t}(\sigma)\partial_{t}Y_{B}(\sigma)Y_{A}(\sigma),}\\{c_{\lambdaAB}}&{=}&{-2\displaystyle{\int}d^{2}\sigma\epsilon^{rs}\Phi_{r}^{(\lambda)}(\sigma)\partial_{s}Y_{B}(\sigma)Y_{A}(\sigma).}\\\end{array}
\begin{array}{ll}{\Gamma^{i}=\sigma^{i}\otimes\gamma^{(7)}}&{~~~\mathrm{for~}~i=1,2,3}\\{{}}&{{}}\\{\Gamma^{a}=1\otimes\gamma^{a}}&{~~~\mathrm{for~}~a=4,5,6,7,8,9\,.}\\\end{array}
\delta_{(n)}|A\rangle={\calD}_{n}|\Lambda_{n}\rangle\,,\qquad{\calD}_{n}\equiv\sum_{j=0}^{n-1}(-)^{j}\!\!\!\!\sum_{l_{1},\ldots,l_{j+1}=1}^{n}\delta_{nl_{j+1}}\prod_{i=1}^{j}\frac{\varepsilon^{l_{i}l_{i+1}}}{\lambda_{l_{i}n}}{\bfa}_{l_{i+1}l_{i}}^{}D_{l_{1}}\,,
x_{\bar{2}1}^{\mu}=x_{2-}^{\mu}-x_{1+}^{\mu}-2i\theta_{1}^{a}\sigma^{\mu}\bar{\theta}_{2a}=x_{21}^{\mu}+i\theta_{21}^{a}\sigma^{\mu}\bar{\theta}_{21a}=(x_{21}^{\mu})_{-}=-(x_{12}^{\mu})_{+}
r_{*}=\left({\frac{\sqrt{|\Lambda|}M_{N}}{\sqrt{15}\pib}}\right)^{\frac{1}{4}}\left({\frac{M_{Pl}}{M_{N}}}\right)^{\frac{1}{2}}{\frac{1}{M_{N}}}\simeq\left({\frac{\mathrm{TeV}}{b|\Lambda|^{-1/2}}}\right)^{\frac{1}{4}}10^{-10}\mathrm{cm}\,,
\biggl(\sum_{l:l\neqj,j+1}\biggl({\frac{\theta^{\prime}(t_{j}-t_{l})}{\theta(t_{j}-t_{l})}}-{\frac{\theta^{\prime}(t_{j+1}-t_{l})}{\theta(t_{j+1}-t_{l})}}\biggr)\alpha_{c(l)}-\biggl({\frac{\theta^{\prime}(t_{j})}{\theta(t_{j})}}-{\frac{\theta^{\prime}(t_{j+1})}{\theta(t_{j+1})}}\biggr)\Lambda,\alpha_{c(j)})\biggr)=0,
D_{\mu\nu}^{R}(q)=\delta_{\mu0}\delta_{\nu0}{\frac{1}{(\vec{q}^{2}+M^{2})}}+\delta_{\mui}\delta_{\nuj}\left(\delta_{ij}-{\frac{q_{i}q_{j}}{(\vec{q}^{2}+M^{2})}}\right){\frac{1}{q^{2}-M^{2}}}~,
\tilde{\rho}_{B^{2}}(s)=\frac{\beta}{2\pi}\\\\\\\rho_{B^{2}}(s)=\frac{\beta}{2\pi}\sqrt{\frac{b^{2}-s}{s-c^{2}}}\\\\\\\rho_{B}(\eta)=\frac{\beta\eta}{\pi}\sqrt{\frac{b^{2}-\eta^{2}}{\eta^{2}-c^{2}}}
|X(K)|\leq\left[c_{1}{|K|}^{2}+\frac{c_{2}(1+|K|R)}{|\vec{x}-\vec{x}^{\prime}|}+\frac{c_{3}R}{[{|\vec{x}-\vec{x}^{\prime}|}^{2}+R^{2}{(\phi-\phi^{\prime})}^{2}]}\right]e^{-ImK|\vec{x}-\vec{x}^{\prime}|};
e^{\left(\frac{D_{1}}{2}-1\right)A+\left(\frac{D_{2}}{2}+1\right)B}\left\{\partial_{r}^{2}+\left(\left(\frac{D_{1}}{2}-1\right)A^{\prime}+\left(\frac{D_{1}}{2}+1\right)B^{\prime}\right)\partial_{r}\right\}
H=\sum_{n=-\infty}^{+\infty}\omega_{n}\varphi_{n}^{\dagger}\varphi_{n}\+\,int_{0}^{L}dx\left[\frac{1}{2}\left(\frac{\phi}{\mu}\right)\left(\frac{\phi}{\mu}\right)+\left(\frac{\phi}{\mu}\right)(\partial_{1}\varphi)\right]\,
\Psi_{E}^{\ell}\left(r,\theta\right)=\left(\begin{array}{l}{B_{\ell}^{+}J_{a}\left(kr\right)e^{i\ell\theta}}\\{i\frac{k}{E-m}B_{\ell}^{+}J_{a-1}\left(kr\right)e^{i\left(\ell+1\right)\theta}}\\{B_{\ell}^{-}J_{a}\left(kr\right)e^{i\ell\theta}}\\{i\frac{k}{E+m}B_{\ell}^{-}J_{a-1}\left(kr\right)e^{i\left(\ell+1\right)\theta}}\\\end{array}\right)\qquad\mathrm{for}\,a\geq\frac12,
\vert\phi_{1}^{\pm}\rangle=G_{-{\frac{2}{3}}}^{+}\verth;g^{\pm}\rangle{}~,{}~{}~{}~\vert\phi_{2}^{\pm}\rangle=G_{-{\frac{2}{3}}}^{-}G_{0}^{-}\verth;g^{\pm}\rangle{}~.
\sum_{i=1}^{k}(-2\eta_{i}(0))g_{i}=\frac{1}{2\pii}\sum_{j=1}^{g}\left|\begin{array}{cc}{\oint_{a_{j}}\Omega}&{\oint_{a_{j}}d\,\mathrm{log}f_{1}}\\{\oint_{b_{j}}\Omega}&{\oint_{b_{j}}d\,\mathrm{log}f_{1}}\\\end{array}\right|.
\begin{aligned}{a}&{{}=\frac{1}{3}(b_{2}c_{2}-b_{2}^{2}-c_{2}^{2}),}\\{b}&{{}=-\frac{1}{27}(b_{2}+c_{2})(b_{2}-2c_{2})(2b_{2}-c_{2}),}\\{\delta}&{{}=-b_{2}^{2}c_{2}^{2}(b_{2}-c_{2})^{2},}\\\end{aligned}
\langle\dot{y}_{a}\cdot\dot{y}_{b}\rangle=\mathrm{tr}\biggl[\ddot{\calG}_{Bab}+{\frac{1}{2}}{\frac{(\dot{\calG}_{Baa}-\dot{\calG}_{Bab})(\dot{\calG}_{Bab}-\dot{\calG}_{Bbb})}{\bar{T}+{\frac{1}{2}}\Bigl(\bar{\calG}_{ab}+\bar{\calG}_{ba}\Bigr)}}\biggr]\;.
\begin{array}{rcl}{d_{i,\kappa}}&{=}&{\frac{\kappa}{s}+\frac{i-1}{s}+\frac{\gamma}{s\,n}\,\,\,(i=1,2,\dots,n),}\\{e_{i,\kappa}}&{=}&{\frac{\kappa}{s}+\frac{i}{n-s}+\frac{\gamma}{s\,(n-s)}\,\,\,(i=1,2,\dots,n-s),}\\{e_{i,\kappa}}&{=}&{\frac{\kappa}{s}+1+\frac{i-n}{s}+\frac{\delta_{i}}{s}\,\,\,(i=n-s+1,\dots,n-1),}\\\end{array}
G^{(\psi)}(z_{1},z_{2})=\frac{1}{16}\left[\begin{array}{ll}{\bar{D}_{1}^{2}\bar{D}_{2}^{2}G_{++}(z_{1},z_{2})}&{\bar{D}_{1}^{2}D_{2}^{2}G_{+-}(z_{1},z_{2})}\\{D_{1}^{2}\bar{D}_{2}^{2}G_{-+}(z_{1},z_{2})}&{D_{1}^{2}D_{2}^{2}G_{--}(z_{1},z_{2})}\\\end{array}\right]
u\frac{\partialQ^{2}}{\partialu}|_{g}\frac{\partial\tilde{R}}{\partialQ^{2}}|_{g}=\beta\bigl(\frac{\partial\tilde{R}}{\partialg^{2}}|_{u}+\frac{\partialQ^{2}}{\partialg^{2}}|_{u}\frac{\partial\tilde{R}}{\partialQ^{2}}|_{u}\bigr)+\gamma\tilde{R}.
\frac{1}{2\kappa_{11}^{2}}\intd^{11}x(-g)^{1/2}{\mathcalR}-\frac{1}{4\kappa_{11}^{2}}\int\biggl(F_{4}\wedge{\star}F_{4}+\frac{1}{3}A_{3}\wedgeF_{4}\wedgeF_{4}\biggr).
S=\intd^{2}x~\mathrm{D}(\theta){\frac{\partial_{\tau}s}{\partial_{\theta}s}}+{\frac{C\mu}{48\pi}}\int\,d^{2}x~\left[{\frac{\partial_{\theta}^{2}s}{(\partial_{\theta}s)^{2}}}\partial_{\tau}\partial_{\theta}s-{\frac{(\partial_{\theta}^{2}s)^{2}(\partial_{\tau}s)}{(\partial_{\theta}s)^{3}}}\right]~~~,
\bar{\rho}=\rho_{0}+\bar{q}+\bar{q}^{\ast},\quad\rho_{0}=\left|l\right|-1-2\left|q\right|\frac{I_{\left|l\right|-1}\left(2\left|q\right|\right)}{I_{\left|l\right|}\left(2\left|q\right|\right)}\;.
\sigma_{ab}^{(2)\alpha}=0,\qquad\sigma_{a}^{(1)\alpha}=\frac{k_{a}^{\alpha}}{2\pi}d\varphi,\qquad\sigma^{(0)\alpha}=d\theta^{\alpha}-\frac{k_{a}^{\alpha}}{2\pi}x^{a}d\varphi,
\chi_{\\nu}^{\mu}=-{\frac{\kappa^{2}(1-\Psi)}{2\Psi}}\left(T_{\quad\nu}^{A\mu}+T_{\quad\nu}^{B\mu}\right)-{\frac{l}{2\Psi}}\left[\left(\Psi_{\|\nu}^{|\mu}-\delta_{\nu}^{\mu}\Psi_{\|\alpha}^{|\alpha}\right)+{\frac{\omega(\Psi)}{\Psi}}\left(\Psi^{|\mu}\Psi_{|\nu}-{\frac{1}{2}}\delta_{\nu}^{\mu}\Psi^{|\alpha}\Psi_{|\alpha}\right)\right]\.
\begin{array}{cccc}{\Gamma_{1,\epsilon}}&{\Gamma_{2,\epsilon}}&{\epsilon=1,\;2\nonumber}\\{\Gamma_{1,\epsilon}}&{n_{1}\inZ\+\;\frac{1}{2}}&{n_{2}\;odd\nonumber}\\{\Gamma_{2,\epsilon}}&{n_{1}\inZ}&{n_{2}\;even\nonumber}\\{\Gamma_{3,\epsilon}}&{n_{1}\inZ+\frac{n_{2}+1}{2}}&{n_{2}\;\in\;Z,}\\\end{array}
\Omega_{_{(2)}}^{r}=v_{_{0i}}^{r}{\frac{\partialH}{\partialq_{i}}}+v_{_{0i}}^{r}{\frac{\partial\Omega_{_{(1)}}^{i}}{\partialq_{j}}}{\frac{\partialH}{\partial\pi_{j}}}+u_{_i}^{r}{\frac{\partialH}{\partial\alpha_{i}}\;,}
L_{1}^{\mathrm{(FOG)}}=\int_{\mathcal{M}_{2}}\left[X^{a}\epsilon_{ab}\,\frac{\partial\mathcal{V}}{\partial\,X^{c}}\;e^{c}\wedgee^{b}+Xd\widetilde{\omega}-dX\wedgee^{c}\,\frac{\partial\mathcal{V}}{\partialX^{c}}+\epsilon\mathcal{V}\,\right]\;,
{\mathrm{}^{*}\calL}_{TM_{B\wedgeF}}=\frac1{12m^{2}}\:H_{\mu\nu\lambda}H^{\mu\nu\lambda}-\frac14\:F_{\mu\nu}F^{\mu\nu}-\frac\chi{2\,\theta}\:\epsilon^{\mu\nu\alpha\beta}\:A_{\mu}\partial_{\nu}B_{\alpha\beta}\;,
\left[\langleA_{m_{0}}\rangle\Phi^{2}\vert_{F}~+~\langleA_{g_{0}}\rangle\Phi^{3}\vert_{F}~+~h.c.\right]{}~+~2Re(\langleF_{m_{0}}\rangleA^{2}~+~\langleF_{g_{0}}\rangleA^{3})
\dot{v}_{f}^{i}=\sum_{n>0}W(n+\frac{\epsilon}{\tau})\Biggl[\frac{c_{n}^{i}}{\sqrt{n}}\,\ddot{x}_{n}^{+}+\frac{\tilde{c}_{n}^{i}}{\sqrt{n}}\,\ddot{x}_{n}^{-}+h.c.\Biggr]-\xi^{i}(\sigma,t)
\delta\upsilon^{k}=-\frac{\Lambda}{12}\epsilon^{k}+\rho^{\sigma}\omega_{\sigma}^{k}\,\qquad\delta\theta_{\mu}^{k}=D_{\mu}\epsilon^{k}+2\rho^{\nu}\Sigma_{\nu\mu}^{k}\,\qquad\delta\xi^{\mu}=-\rho^{\mu}\,
S=\intd^{4}xd^{2}\theta\left\{-\frac{1}{8}{\calW}^{a}{\calW}_{a}+d^{2}\bar{\theta}\left[-\frac{1}{2}{\calG}^{2}+\frac{1}{2}m{\calVG}+\frac{1}{16}\bar{\Phi}e^{2h{\calV}}\Phie^{4g{\calG}}\right]\right\}.
|i\rangle=|\Psi_{\mathrm{target}}\rangle\|\Psi_{\mathrm{beam}}\rangle=\int\frac{d^{4}q_{\mathrm{T}}}{(2\pi)^{4}}\frac{d^{4}q_{\mathrm{B}}}{(2\pi)^{4}}\tilde{\psi}_{\mathrm{T}}(q_{\mathrm{T}}-p_{\mathrm{T}})\tilde{\psi}_{\mathrm{B}}(q_{\mathrm{B}}-p_{\mathrm{B}})|q_{\mathrm{T}}\q_{\mathrm{B}}\rangle.
\tau^{1}=\left(\begin{matrix}{0}&{{1}}\\{{1}}&{0}\\\end{matrix}\right),\,\,\,\,\,\,\,\,\,\,\,\tau^{2}=\left(\begin{matrix}{0}&{{-i}}\\{{i}}&{0}\\\end{matrix}\right),\,\,\,\,\,\,\,\,\,\,\,\tau^{3}=\left(\begin{matrix}{1}&{{0}}\\{{0}}&{-1}\\\end{matrix}\right).
\Lambda_{ph}(t)=\Lambda_{0}-\frac{m_{F}^{4}\,\,U^{2}(t)}{2f(t)}+\frac{2\,m_{F}^{4}}{(4\pi)^{2}}\;\int_{0}^{t}U^{2}(t)dt-\frac{2}{(4\pi)^{2}}\,\sum_{i}\,N_{i}\int_{o}^{t}m_{i}^{4}(t)\,dt\,,
\begin{array}{ll}{g^{\mu\nu}\epsilon_{\mu}^{\beta}(k)\epsilon_{\nu}^{\sigma}(k)=g^{\beta\sigma}}&{k^{\mu}\epsilon_{\mu}^{i}(k)=0,\;\;\;i=1,2,3}\\{\epsilon_{\mu}^{0}(k)=k_{\mu}/k}&{\sum_{i=1}^{3}\epsilon_{\mu}^{i}(k)\epsilon_{\nu}^{i}(k)=-M_{\mu\nu}(k)\;.}\\\end{array}
\bar{W}_{\sigma\mu\nu}(g,\bar{\Gamma})=\bar{W}_{\sigmaab}(\bar{\Omega})e_{\mu}^{a}e_{\nu}^{b}=-\left(\bar{\Omega}_{ab\sigma}+\bar{\Omega}_{ba\sigma}\right)e_{\mu}^{a}e_{\nu}^{b}
\displaystyle\mathop{Z_{(\Theta^{4},1)}^{torus}(\tau,\bar{\tau},T,U)={\frac{1}{4}}{\sum_{\stackrel{m_{1},m_{2}\inZ}{n^{1},n^{2}\inZ}}}e^{2\pii\tau(m_{1}\frac{n^{1}}{2}+m_{2}\frac{n^{2}}{2})}e^{{\frac{-\pi\tau_{2}}{T_{2}U_{2}}}|TU{\frac{n^{2}}{2}}+T{\frac{n^{1}}{2}}-Um_{1}+m_{2}|^{2}}}.
\deltaS_{T}=M\,\big((\chi^{A}-\tilde{\chi}^{A})T_{AC}K_{\,\,B}^{C}(\chi^{B}-\tilde{\chi}^{B})+{\frac{1}{2}}(\chi^{A}-\tilde{\chi}^{A})\deltaT_{AB}\,(\chi^{B}-\tilde{\chi}^{B})\big)
S=\intd^{4}x\,\mathrm{tr}\,\sum_{n=-\infty}^{\infty}\Bigl[-\,\frac{1}{4}F_{(n)}^{\mu\nu}F_{(-n)\mu\nu}-\frac{1}{2}\Bigl(\frac{2n\pi}{R}A_{(n)}^{\mu}+i\,D^{\mu}\phi_{(n)}\Bigr)\Bigl(\frac{2n\pi}{R}A_{(-n)\mu}-i\,D_{\mu}\phi_{(-n)}\Bigr)\Bigr]
\langle\psi_{p}^{(0)}|i\tilde{x}^{2}\tilde{p}^{4}+\tilde{x}\tilde{p}^{3}|\psi_{p}^{(0)}\rangle=0,\qquad\langle\psi_{p}^{(0)}|2i\tilde{x}\tilde{p}^{3}+\tilde{p}^{2}|\psi_{p}^{(0)}\rangle=0.\nonumber
G\=\G_{0}e^{\frac{i}{2}\int\frac{d^{4}k}{(2\pi)^{4}}(1-\xi\frac{k_{0}^{2}}{k^{2}+i\epsilon})\frac{\left|g_{1}(e^{ik.x_{1}}-e^{ik.x_{1}^{\prime}})-g_{2}(e^{ik.x_{2}}-e^{ik.x_{2}^{\prime}})\right|^{2}}{(k^{2}+i\epsilon)k_{0}^{2}}}\.
H\psi=-\frac12\left[\bigg(\frac{\partial}{\partialx_{1}}-ib_{2}x_{3}\bigg)^{2}+\bigg(\frac{\partial}{\partialx_{2}}-ib_{3}x_{1}\bigg)^{2}+\bigg(\frac{\partial}{\partialx_{3}}-ib_{1}x_{2}\bigg)^{2}\right]\psi(x_{1},x_{2},x_{3}).
E\,=\,\frac{\rhoF}{2}\int_{0}^{l}dx\,\left(\dot{y}^{2}\,+\,r^{2}\dot{\psi}^{2}\right)\,+\,\frac{EI}{2}\int_{0}^{l}dx\,\psi^{2}\,+\,\frac{kFG}{2}\int_{0}^{l}dx\,\left(y^{\prime}\,-\,\psi\right)^{2}.
A_{\mu}\toA_{\mu}\,,\,\,\,\lambda_{\alpha}\toe^{-i\alpha}\lambda_{\alpha}\,,\,\,\,\psi_{\alpha}^{f}\toe^{i\alpha/3}\psi_{\alpha}^{f}\,,\,\,\,\phi^{f}\toe^{-2i\alpha/3}\phi^{f}\,.
\begin{aligned}{\delta\hat{\psi}_{\hat{\mu}}}&{{}=(D_{\hat{\mu}}+\frac{1}{4}ie^{\hat{\phi}}\partial_{\hat{\mu}}\hat{\chi})\,\hat{\epsilon}\,,}\\{\delta\hat{\lambda}}&{{}=(\not\!\partial\hat{\phi}+ie^{\hat{\phi}}\!\not\!\partial\hat{\chi})\,\hat{\epsilon}^{*}\,,}\\\end{aligned}
F=\frac{2\pi\sqrt{2}}{g}\left(\tilde{\mathcal{G}}_{S}^{\mu\nu}\partial_{\mu}T\partial_{\nu}T+V(T)+f(T)\tilde{\mathcal{G}}_{S}^{\mu\nu}\partial_{\nu}\theta_{R}^{T}\Gamma^{0}\partial_{\nu}\theta_{L}\right)
Det_{2}\left(\begin{matrix}{\langlei\parallelK\paralleli\rangle}&{\langlei\parallelX_{i}^{-}K\paralleli\rangle}\\{\langlei\parallelKX_{i}^{+}\paralleli\rangle}&{\langlei\parallelX_{i}^{-}KX_{i}^{+}\paralleli\rangle}\\\end{matrix}\right)=[i+1][i-1]
\begin{array}{ccccccccccccc}{0\!\!\!}&{\!\stackrel{}{\longrightarrow}\!\!\!}&{\!\Omega^{n}\!\!\!}&{\!\stackrel{d_{n-1}^{*}}{\longrightarrow}\!\!\!}&{\!\cdots\!}&{\!\stackrel{d_{k+2}^{*}}{\longrightarrow}\!\!\!}&{\!\Omega^{k+2}\!\!\!}&{\!\stackrel{d_{k+1}^{*}}{\longrightarrow}\!\!\!}&{\!\Omega_{k+1}^{\prime\prime}\!\!\!}&{\!\stackrel{d_{k}d_{k}^{*}}{\longrightarrow}\!\!\!}&{\;\;\;0,}\\\end{array}
D_{++}^{(P)ab}(x)=-{\frac{i\delta^{ab}}{(2\pi)^{2}}}\intd^{2}k\,e^{ikx}{\frac{\partial}{\partialk_{-}}}P\left({\frac{1}{k_{-}}}\right)=-{\frac{i\delta^{ab}}{2}}|x^{-}|\delta(x^{+})\.
dC_{4}-C_{2}\wedgedb_{2}=\frac{i}{6}\bar{L}\tau_{3}\tau_{1}\hat{L}^{3}L+\frac{1}{30}(\varepsilon^{a_{1}...a_{5}}L^{a_{1}}\wedge\cdots\wedgeL^{a_{5}}+\varepsilon^{a_{1}^{\prime}...a_{5}^{\prime}}L^{a_{1}^{\prime}}\wedge\cdots\wedgeL^{a_{5}^{\prime}}).
R(\sigma)_{p}={\frac{3\cdot2^{19}\,b}{\chi\,q_{1}^{\,3}}}\;\frac{\delta_{pq}+\hat{x}_{p}\hat{x}_{q}}{|\vec{x}|}\;\sum_{A\neq1}q_{A}\,\,\frac{(x_{A})^{q}}{|\vec{x}_{A}|^{3}}+{\calO}(b^{2})\,,
\hat{\calK}^{(3)}=3\left[\partial\hat{\omega}^{(2)}+{\textstyle\frac{1}{3(2\pi\alpha^{\prime})}}D\hat{X}^{\hat{\mu}}D\hat{X}^{\hat{\nu}}D\hat{X}^{\hat{\rho}}\hat{C}_{\hat{\mu}\hat{\nu}\hat{\rho}}-{\textstyle\frac{m}{2}}(2\pi\alpha^{\prime})\hat{b}\partial\hat{b}\right]\,,
W=2\sqrt{2}me^{{\frac{5}{\sqrt{15}}}\tilde{\varphi}}-e^{{\frac{2}{\sqrt{15}}}\tilde{\varphi}}(\sqrt{2}\mue^{{\frac{3}{\sqrt{10}}}\tilde{f}}+{\frac{R_{4}}{2\sqrt{2}\mu}}e^{-{\frac{2}{\sqrt{10}}}\tilde{f}}).
C_{\ell}\simeq\frac{2}{\pi}\sum_{\omega=0}^{\omega_{c}}\int\frac{dk}{k}k^{3}\left[\left(\Theta_{\omega}+\Psi_{\omega}\right)\jmath_{\ell}(kd)+v_{\gamma\omega}\jmath_{\ell}^{\prime}(kd)\right]^{2},
\left(B_{r}^{(1)}\right)^{\vee}=A_{2r-1}^{(2)},\quad\left(C_{r}^{(1)}\right)^{\vee}=D_{r+1}^{(2)},\quad\left(F_{4}^{(1)}\right)^{\vee}=E_{6}^{(2)},\quad\left(G_{2}^{(1)}\right)^{\vee}=D_{4}^{(3)}.
(L_{-2}+3/2L_{-1}^{2})\mid0,p\rangle=\left(L_{-2}^{(\alpha)}+\frac{3}{2}L_{-1}^{(\alpha)^{2}}\right)\mid,0,p\rangle+\left(L_{-2}^{(b)}+\frac{3}{2}L_{-1}^{(b)^{2}}\right)\mid0,p\rangle+\left(L_{-2}^{(b^{\prime})}+\frac{3}{2}L_{-1}^{(b^{\prime})^{2}}\right)\mid0,p\rangle
E(\theta)\;\longrightarrow\;\frac{1-\frac{h^{2}}{2\textrm{M}}-\textrm{sn}\left[\frac{2i\textbf{K}}{\pi}\left(i\frac{\pi}{2}-\theta\right)\right]}{1-\frac{h^{2}}{2\textrm{M}}+\textrm{sn}\left[\frac{2i\textbf{K}}{\pi}\left(i\frac{\pi}{2}-\theta\right)\right]}\,E(\theta)\,.
\{f,z\}={\frac{f^{\prime\prime\prime}}{f^{\prime}}}-{\frac{3}{2}}\left({\frac{f^{\prime\prime}}{f^{\prime}}}\right)^{2}=-2(f^{\prime})^{\frac{1}{2}}((f^{\prime})^{-{\frac{1}{2}}})^{\prime\prime},
S_{\mathrm{b}}(B)=\frac{1}{2}\intd^{2}x\,\partial_{\mu}B^{i}\left[\frac{\delta_{ij}+B^{i}B^{j}}{1+B^{2}}\delta_{\mu\nu}+i\epsilon^{\mu\nu}\frac{\epsilon^{ijk}B^{k}}{1+B^{2}}\right]\partial_{\nu}B^{j}.
P=\frac{1}{2\alpha^{\prime}\dot{t}}\;\frac{\dot{R}^{2}-R^{2}}{(1+\frac{K}{4}R^{2})^{2}}=\frac{1}{2\sqrt{b}\alpha^{2}}[\frac{\dot{r}^{2}}{1-Kr^{2}}-r^{2}]=\frac{b\alpha^{2}-2r^{2}}{2\sqrt{b}\alpha^{2}}.
V_{GEP}(\phi_{0})={\frac{\lambda}{4!}}\phi_{0}^{4}+{\frac{m^{2}}{2}}\phi_{0}^{2}+{\frac{1}{2}}\int{\frac{d^{\nu}k}{(2\pi)^{\nu}}}g(\vec{k})-{\frac{\lambda}{32}}\left[\int{\frac{d^{\nu}k}{(2\pi)^{\nu}}}{\frac{1}{g(\vec{k})}}\right]^{2}.
\langle{\calO}_{1}(x_{1})\cdots{\calO}_{n}(x_{n})\rangle=\frac{\langle{\calO}_{1}(x_{1})\cdots{\calO}_{n}(x_{n})e^{-\lambda\intd^{2}z\Phi_{(1,3)}}\rangle_{{\calM}_{m}}}{\langlee^{-\lambda\intd^{2}z\Phi_{(1,3)}}\rangle_{{\calM}_{m}}}
g_{\mu\nu}\left(x\right)=\eta_{mn}\,e_{\mu}^{m}\left(x\right)e_{\nu}^{n}\left(x\right)\,\,,\quadg^{\mu\nu}\left(x\right)=\eta^{mn}\,e_{m}^{\mu}\left(x\right)e_{n}^{\nu}\left(x\right)\,\,,
\langlep_{f},+T|\prod_{a=1}^{n}\phi_{s_{a}}(x_{a},t_{a})|p_{i},-T\rangle=\intd\theta_{i}d\theta_{f}\psi_{p_{f}}^{*}(\theta_{f})\psi_{p_{i}}(\theta_{i})\langle\theta_{f},+T|\prod_{a=1}^{n}\phi_{s_{a}}(x_{a},t_{a})|\theta_{i},-T\rangle
L_{eff}=-R=\frac{1}{2}{\calA}_{ij}{\delta\dot{y}}^{i}{\delta\dot{y}}^{j}+{\calB}_{ij}{\delta\dot{y}}^{i}\deltay^{j}+\frac{1}{2}{\cal}C_{ij}\deltay^{i}\deltay^{j}\,,
X\equiv\left(\begin{array}{c}{\Theta_{1}}\\{H_{1}}\\{\Theta_{2}}\\{H_{2}}\\\end{array}\right)=e^{i\lambdat}\left(\begin{array}{c}{a_{1}}\\{b_{1}}\\{a_{2}}\\{b_{2}}\\\end{array}\right)=e^{i\lambdat}A
\tilde{F}_{\sigma}^{(n)}(\zeta)^{kk_{1}\cdotsk_{2n}}=\tilde{c}_{n}\prod_{1\leqslantj<k\leqslant2n}\zeta_{j}^{\frac{r-1}{r}}g(z_{j}/z_{k})\times\overline{{\tilde{F}}}_{\sigma}^{(n)}(\zeta)^{kk_{1}\cdotsk_{2n}},
{\calF}^{\prime\prime}D^{\alpha\beta}{\calA}|=D^{\alpha\beta}{\calF}^{\prime}|-{\calF}^{\prime\prime\prime}D^{\alpha{\mathbfi}}{\calA}D_{\mathbfi}^{\beta}{\calA}|\stackrel{|x|\to\infty}{\longrightarrow}D^{\alpha\beta}{\calF}^{\prime}|.
\gamma(\beta\sqrt{-\triangle})=\frac{2\pi^{2}}{\beta\sqrt{-\triangle}}+\frac{8\pi}{\beta\sqrt{-\triangle}}\left(\sum_{k=1}^{\infty}{\mathrmarctan}\Big(\frac{\beta\sqrt{\triangle}}{4\pik}\Big)-\int_{0}^{\infty}\!{\mathrmd}\kappa\,{\mathrmarctan}\Big(\frac{\beta\sqrt{\triangle}}{4\pi\kappa}\Big)\right).
dz\otimesdz\longrightarrowdz\otimesdz+\mu_{\bar{z}}^{z}(dz\otimesd\bar{z}+symm)+(\mu_{\bar{z}}^{z})^{2}d\bar{z}\otimesd\bar{z}+\underbrace{\mu_{\bar{z}}^{zz}d\bar{z}+more}_{new\:terms}.
CV_{N}^{rs}=CV_{2N}^{rs}+CV_{2N}^{r,s+N}=CV_{3N}^{rs}+CV_{3N}^{r,s+N}+CV_{3N}^{r,s+2N}=\cdots=\sum_{l=0}^{k-1}CV_{kN}^{r,s+lN}.
\begin{array}{rcl}{v_{+}}&{=}&{1{}-{}\frac{\alpha^{2}}{2}\left[\left(E_{2}^{\prime}{}-{}B_{3}^{\prime}\right)^{2}{}+{}\left(E_{3}^{\prime}{}+{}B_{2}^{\prime}\right)^{2}\right]}\\{v_{-}}&{=}&{1{}-{}\frac{\alpha^{2}}{2}\left[\left(E_{2}^{\prime}{}+{}B_{3}^{\prime}\right)^{2}{}+{}\left(E_{3}^{\prime}{}-{}B_{2}^{\prime}\right)^{2}\right]}\\\end{array}\quad,
u(p,\xi)=f_{\lambda}^{\rho\chi}(\xiB_{p}),\,\,\,\,\,\,\,f_{\lambda}^{\rho\chi}(\xi)=\left|\xi_{2}\right|^{2c-2}e^{-i\lambda\Phi_{1}}J_{l_{0}-\lambda}(\frac\rho\kappa\left|z\right|)e^{il_{0}\Phi_{2}}
G_{AB}=\left(\begin{matrix}{\gamma_{\mu\nu}+y^{k}y^{l}A_{\mu}^{kh}A_{\nu}^{lh}}&{y^{k}A_{\mu}^{jk}}\\{y^{k}A_{\nu}^{ik}}&{\delta^{ij}}\\\end{matrix}\right),
[g_{IJ}]=\frac{R_{2}}{\mathrm{Im}\,\tau}\left(\begin{matrix}{1}&{\mathrm{Re}\,\tau}\\{\mathrm{Re}\,\tau}&{|\tau|^{2}}\\\end{matrix}\right)~~~~~~,~~~~~~[\beta_{IJ}]=\left(\begin{matrix}{0}&{R_{1}}\\{-R_{1}}&{0}\\\end{matrix}\right)
f(\hat{Z}_{-})=\Bigl(1-{\frac{Z_{-}^{2}}{2N}}\Bigr)\rho+{\calO}(1/N^{2})\,,\quadf(\hat{Z}_{1})=\Bigl(1-{\frac{Z_{1}^{2}}{2N}}\Bigr)\rho+{\calO}(1/N^{2})\,.
\bar{f}_{3}(T,Y)=f_{3}(T,Y)+\frac{T^{3}}{8}\left[\left(\frac{1}{3}+2c_{1}^{(3)}-c_{2}^{(4)}\right)(\partialY)^{2}+\left(\frac{1}{3}-2c_{1}^{(3)}+c_{2}^{(4)}\right)(\tau_{3}\partialY)^{2}\right].