harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathrm{~\psi(x)~}\rightarrow\psi^{\prime}(x^{\prime})=\Pi\mathrm{~\psi(x)~},\;\;\;\mathrm{~\overline{{\psi}}(x)~}\rightarrow\overline{{\psi}}^{\prime}(x^{\prime})=\mathrm{~\overline{{\psi}}(x)~}\Pi^{-1},
\bar{J}^{\mu}\Psi_{\mu}=\bar{J}^{\mu}\left\{\frac{\left(\Lambda+3\,M^{2}\right)^{-1}}{\not\!\nabla+M}\left[-M\,\nabla_{\mu}\gamma_{\nu}-\left(\frac{\Lambda}{2}+M^{2}\right)\gamma_{\mu}\gamma_{\nu}+g_{\mu\nu}(\Lambda+3\,M^{2})\right]\right\}J^{\nu}\,.
R_{-\alpha-\beta}=\partial_{\alpha}\partial_{\beta}\bigg\{\frac{2n+4}{(n+4)^{2}}\frac{1}{{x^{-}}^{2}}\Big(n(y^{0})^{2}-\sum_{i=1}^{n}{y^{i}}^{2}\Big)\bigg\}\propto\frac{1}{{x^{-}}^{2}}
\chi=\frac{8+\lambda\dot{\phi}^{2}e^{\phi}-\delta\dot{\sigma}^{2}\xi_{\sigma\sigma}}{4+2(\lambda\dot{\phi}e^{\phi}-\delta\dot{\sigma}\xi_{\sigma})\dot{\omega}+(\dot{\omega}^{2}+ke^{-2\omega})^{2}(3\lambda^{2}e^{2\phi}+\delta^{2}\xi_{\sigma}^{2})}.
g^{*}h_{\alpha\beta\gamma\delta}\:=\:\left(h_{\alpha\beta\gamma\delta}\right)\,\left(\nu_{\beta\gamma\delta}^{g}\right)\,\left(\nu_{\alpha\gamma\delta}^{g}\right)^{-1}\,\left(\nu_{\alpha\beta\delta}^{g}\right)\,\left(\nu_{\alpha\beta\gamma}^{g}\right)^{-1}
\begin{array}{c}{[\,E_{\gamma}\otimesE_{-\gamma}\,,\,E_{\gamma}\otimes1\,]~=~0~,}\\{[\,E_{\gamma}\otimesE_{-\gamma}\,,\,E_{-\gamma}\otimes1\,]~=~H_{\gamma}\otimesE_{-\gamma}~,}\\{[\,E_{\gamma}\otimesE_{-\gamma}\,,\,E_{\delta}\otimes1\,]~=~N_{\gamma,\delta}\,E_{\gamma+\delta}\otimesE_{-\gamma}\quad\mathrm{if~\,~\gamma~\pm~\delta~\neq~0~}~,}\\\end{array}
{\calO}_{1/2}^{[m,0,0]}=[W^{1}(\theta_{2,3,4},\bar{\theta}^{1})]^{m}\quad\mathrm{and}\quad{\calO}_{1/2}^{[0,0,m]}=[\bar{W}_{4}(\theta_{4},\bar{\theta}^{1,2,3})]^{m}\,.
w_{\alpha_{1}\cdots\alpha_{N}}(z_{1},\cdots,z_{N})=\sum_{\varepsilon_{1},\cdots,\varepsilon_{N}}v_{\varepsilon_{1}}\otimes\cdots\otimesv_{\varepsilon_{N}}w_{\alpha_{1}\cdots\alpha_{N}}^{\varepsilon_{1}\cdots\varepsilon_{N}}(z_{1},\cdots,z_{N}),
{\calS}_{\sigma}=\frac{1}{4\pi\alpha^{\prime}}\int_{\Sigma}d^{2}\sigma\sqrt{\gamma}\gamma^{\alpha\beta}g_{MN}(X^{P}(\sigma,\tau))\partial_{\alpha}X^{M}\partial_{\beta}X^{N}~,\qquad\alpha,\beta=\sigma,\tau
S_{1}{0\atopwithdelims[]1}(\infty,y|q)=\sum_{n=0}^{\infty}{\frac{q^{{\frac{1}{2}}m(m-1)}y^{m}}{(q)_{m}}}=(-y)_{\infty}=\prod_{j=0}^{\infty}(1+yq^{j}).
\tau_{1/2}=\frac{E}{P}\sim\frac{N\,T_{0}\,\omega^{2}R^{2}}{\,\kappa^{2}N^{2}\,T_{0}^{2}\,\omega^{12}R^{4}\,}=\left(\kappa^{2}N\,T_{0}\,\omega^{10}R^{2}\right)^{-1}~~.
U_{(12)}\diamond\hat{X}\diamondU_{(12)}^{-1}=\sum_{k=0}^{\infty}\left(\frac{i}{\hbar}\right)^{k}\frac{1}{k!}\underbrace{[A_{(12)},[A_{(12)},\ldots[A_{(12)},}_{\mathrm{k-times}}\hat{X}]\ldots]]
\tilde{\calH}=\omegaq{\frac{d}{dq}}-{\frac{1}{2}}{\frac{d^{2}}{dq^{2}}}-{\frac{g}{q}}{\frac{d}{dq}}=-2\omega\left\{u{\frac{d^{2}}{du^{2}}}+(g+{\frac{1}{2}}-u){\frac{d}{du}}\right\}.
S_{m}=S_{\mathrm{cl}}+(\eta_{A}-\mathrm{~\frac{1}{2}~}m^{2}(P_{+})_{A}^{B}\bar{\phi}_{B})\phi^{A}-(\mathbf{s}_{m}^{a}\phi^{A})\phi_{Aa}^{*}+\bar{\phi}_{A}(\mathrm{~\frac{1}{2}~}\epsilon_{ab}\mathbf{s}_{m}^{b}\mathbf{s}_{m}^{a}+m^{2})\phi^{A}.
S[\phi]=\intdg_{1}dg_{2}dg_{3}\(\phi(g_{1},g_{2},g_{3}))^{2}+\frac{\lambda}{4!}\intdg_{1}\ldotsdg_{6}\\phi(g_{1},g_{2},g_{3})\phi(g_{3},g_{4},g_{5})\phi(g_{2},g_{4},g_{6})\phi(g_{1},g_{5},g_{6}).
F(Y,Z)=\sum_{m,\tilde{m},n,\tilde{n}}\frac{F_{\alpha(m)\dot{\alpha}(\tilde{m}),\beta(n)\dot{\beta}(\tilde{n})}}{m!\tilde{m}!n!\tilde{n}!}y^{\alpha(m)}\bar{y}^{\dot{\alpha}(\tilde{m})}z^{\beta(n)}\bar{z}^{\dot{\beta}(\tilde{n})}
x_{d}(t)={\frac{c}{\mathrm{sn}(ct,k)}},\quady_{d}(t)=-c^{2}{\frac{\mathrm{cn}(ct,k)\,\mathrm{dn}(ct,k)}{\mathrm{sn}^{2}(ct,k)}},\quadz_{d}(t)=-\wp(t),
W=\phi+\theta_{i}^{\alpha}\chi_{\alpha}^{i}+\ldots+\theta_{i_{1}}^{\alpha_{1}}\ldots\theta_{i_{n}}^{\alpha_{n}}\chi_{(\alpha_{1}\ldots\alpha_{n})}^{[i_{1}\ldotsi_{n}]}+\ldots+(\theta)^{2N}\chi
f=({{{f^{a_{1}\ldotsa_{p}}}_{b_{1}\ldotsb_{q}}}^{A_{1}\ldotsA_{r}}}_{B_{1}\ldotsB_{s}})e_{a_{1}}\otimes\ldots\otimese_{a_{p}}\otimes\ldots\otimesE^{B_{1}}\otimes\ldots\otimesE^{B_{s}}
S=\frac{1}{2}\int\left((e^{-\Phi}Q_{B}e^{\Phi})(e^{-\Phi}\eta_{0}e^{\Phi})-\int_{0}^{1}dt(e^{-t\Phi}\partial_{t}e^{t\Phi})\left\{(e^{-t\Phi}Q_{B}e^{t\Phi}),(e^{-t\Phi}\eta_{0}e^{t\Phi})\right\}\right)\.
\frac{\partialF}{\partialv^{1}}=\phi_{1}^{m}+\sum_{A=2}^{m}\phi_{A}^{m}f_{1}^{A},\;\;\;\frac{\partialF}{\partialt}=\phi_{t}^{m}+\sum_{A=2}^{m}\phi_{A}^{m}f_{t}^{A}.
H=-\frac{e}{2}(\pi^{2}-m^{2})+i\sum_{a=1}^{N}(\pi_{\mu}\psi_{a}^{\mu}+sm\psi_{a}^{3})\chi_{a}-i\sum_{a=1}^{N}(\varepsilon^{\mu\nu\lambda}\pi_{\mu}\psi_{a\nu}\psi_{a\lambda}+\frac{i}{2}sm)\kappa_{a}\;.
(\phi_{p}^{\dagger})_{ij}=\frac{ig}{4\sqrt{2}\pi^{2}}\frac{1}{(y^{2}+\rho^{2})}\left\{\left(\frac{N_{c}-2}{2N_{c}}\right)\delta_{ij}\delta_{i,j\leq2}-\frac{1}{N_{c}}\delta_{ij}\delta_{i,j\geq3}\right\}\tilde{\eta}\eta.
I=-\frac{1}{4{\pi}^{2}g}\intd^{3}\xi\bigg\{\sqrt{-det\bigg(g_{\alpha\beta}^{ind}+2\pi{\calF}_{\alpha\beta}\bigg)}+\frac{1}{3!}\epsilon^{\alpha\beta\gamma}A_{\mu\nu\rho}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\partial_{\gamma}X^{\rho}\bigg\}
I=\frac{1}{2\kappa_{10}^{2}}\intd^{10}x\sqrt{g}\frac{1}{4}(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})(\partial_{\mu^{\prime}}A_{\nu^{\prime}}-\partial_{\nu^{\prime}}A_{\mu^{\prime}})g^{\mu\mu^{\prime}}g^{\nu\nu^{\prime}}\.
X_{z_{\pm}}=-iQ_{\pm}^{-1}\displaystyle\displaystyle\frac{\partial}{\partial\overline{{z}}_{\pm}}~,~~~X_{p_{\pm}}=i\displaystyle\displaystyle\frac{\partial}{\partialz_{\pm}}-iQ_{\pm}^{-1}\displaystyle\frac{\partialp_{\pm}}{\partialz_{\pm}}\displaystyle\displaystyle\frac{\partial}{\partial\overline{{z}}_{\pm}}~,
H_{QES}=-P_{4}(y)\frac{d^{2}}{dy^{2}}+\left(\frac{N-1}2P_{4}^{\prime}(y)-P_{2}(y)\right)\frac{d}{dy}+\fracN2\left(P_{2}^{\prime}(y)-\frac{N-1}6P_{4}^{\prime\prime}(y)\right),
J^{(N)}(N\!-\!1;1,\ldots,1)=\frac{2\mathrm{i}^{1-2N}\pi^{(N-1)/2}}{\prodm_{l}}\;\Gamma\left(\frac{N\!+\!1}{2}\right)\int_{0}^{\infty}\ldots\int_{0}^{\infty}\prod\mathrm{d}\alpha_{l}\left(\sum_{i=1}^{N}\frac{\alpha_{i}}{m_{i}}\right)\delta\left(\alpha^{T}\|c\|\alpha\!-\!1\right).
S(\Phi)=\int_{-\infty}^{0}dx\int_{-\infty}^{\infty}dt\left(\frac{1}{2}\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{a}-\frac{m^{2}}{\beta^{2}}\sum_{i=0}^{r}n_{i}e^{\beta\alpha_{i}\cdot\Phi}\right)
\begin{array}{l}{{\calH}=\frac{N}{2}\left(\frac{P^{m}P_{m}}{\sqrt{\beta}}+\frac{\beta_{ab}}{\sqrt{\beta}}\Pi_{r}^{a}\Pi_{r}^{b}+\sqrt{\beta}\beta^{ab}\partial_{a}X^{m}\partial_{b}X_{m}-\sqrt{\beta}+\frac{1}{2}\sqrt{\beta}\beta^{ac}\beta^{bd}F_{ab}^{r}F_{cd}^{r}\right)}\\{+\Pi_{r}^{a}\partial_{a}A_{0}^{r}+N^{a}\left(\partial_{a}X^{m}P_{m}+\Pi_{r}^{b}F_{ab}^{r}\right)}\\\end{array}
G_{\mu\nu}(p)={\frac{32}{3\pi\theta^{2}}}{\frac{1}{M}}\left(\delta_{\mu\nu}-{\frac{p_{\mu}p_{\nu}}{p^{2}}}\right)+{\frac{16}{\theta}}{\frac{\epsilon_{\mu\rho\nu}p^{\rho}}{p^{2}}},
(-\frac{1}{2}\frac{d^{2}}{{(dx^{1})}^{2}}+\Lambda_{1}^{3}+\frac{1}{2}{(\Lambda_{1}^{3})}^{2}(x^{1})^{2}+\frac{1}{(x^{1})^{2}}(8(m^{1})^{2}-\frac{1}{8}))Y_{1}^{1}(x^{1})=4E_{II}^{1}Y_{1}^{1}(x^{1}),
{\calL}_{bulk}=M^{3}\left\{-\frac{1}{2}K(\chi)(\partial_{y}\chi)^{2}-V(\chi)+\sum_{\vec{n}}\frac{a_{\vec{n}}(\chi)}{M^{|n|}}(\partial\chi)^{n_{1}}...(\partial^{N}\chi)^{n_{N}}\right\},
{\mit\Pi}=\Big({\frac{3}{4\tilde{\hbar}}}\Big)^{3}\Big({\frac{\Theta}{c_{_\mathrm{P}}}}\Big)^{4}\big(1-{\frac{v^{2}}{c_{_\mathrm{P}}^{2}}}\big)^{-3}{\frac{v}{c_{_\mathrm{P}}}}\,
S_{WZW}(g)=\frac1{4\pi}\int_{\Sigma}d^{2}\sigma\Omega_{ab}A_{\alpha}^{a}A^{b\alpha}+\fraci{12\pi}\int_{B}d^{3}\sigma\epsilon_{\alpha\beta\gamma}A^{a\alpha}A^{b\beta}A^{c\gamma}\Omega_{cd}f_{ab}^{d}
2\kappa^{2}e^{-1}{\calL}=R+{\frac{1}{2}}m^{2}(T^{2}-2T_{ij}T^{ij})-Tr(P_{\mu}P^{\mu})-{\frac{1}{2}}(V_{I}{}^{i}V_{J}{}^{j}F_{\mu\nu}^{IJ})^{2}.
\Phi(z,\vec{x},t)\,=\,\sum_{p=1}^{\infty}\,\int{\frac{d\vec{k}}{(2\pi)^{n-1}}}\,{\frac{z^{n/2}\,J_{\nu}(u_{p}z)}{Rw_{p}(\vec{k})\,J_{\nu\,+\,1}(u_{p}R)}}\lbrace{{\bfa}_{p}(\vec{k})\}e^{-iw_{p}(\vec{k})t+i\vec{k}\cdot\vec{x}}\,\,+\,\,c.c.\rbrace\,,
V_{X_{1}X_{2}X_{3}X_{4}}=\intd^{2}z_{1}\intd^{2}z_{2}V(|z_{1}-z_{2}|)\Phi_{X_{1}}^{*}(z_{1})\Phi_{X_{2}}^{*}(z_{2})\Phi_{X_{4}}(z_{2})\Phi_{X_{3}}(z_{1}).
\langleR_{\mu\nu}^{Lab}(x)S_{\rho\sigma}^{Lc}(y)\rangle=[(\partial_{\mu}M_{\nu,\rho}^{ab,cd}\eta_{d\sigma}-\partial_{\nu}M_{\mu,\rho}^{ab,cd}\eta_{d\sigma})-(\rho\leftrightarrow\sigma)].
Y_{i_{1}i_{2}...i_{2n-1}i_{2n}}\equiv\frac{1}{2}Tr\,\left(Y_{i_{1}}Y_{i_{2}}^{\dagger}...Y_{i_{2n-1}}Y_{i_{2n}}^{\dagger}+Y_{i_{1}}^{\dagger}Y_{i_{2}}...Y_{i_{2n-1}}^{\dagger}Y_{i_{2n}}\right)
(F_{L}^{\alpha}i_{L\,\alpha}+F_{R}^{\alpha}i_{R\,\alpha}){\Omega}_{[m]}\sum_{{p+q=s-1}}{\frac{(m-p-1)!(m-q-1)!}{p!q!}}\Big[\Pi_{[p+1,q]}-\Pi_{[p,q+1]}\Big]={\Pi}_{[m,0]}-{\Pi}_{[0,m]}\.
\omega_{0}^{(r)}=\left(\frac{2C_{2}\alpha_{S}(q^{2})}{\pi}\right)^{1/2}\left[1-\left({\frac{\alpha_{S}(q^{2})}{C_{2}}}\right)^{1/3}\pi\left(\frac{3}{8b}(r+\frac{1}{4})\right)^{2/3}\right]
h_{tt}^{\prime\prime}-h_{xx}^{\prime\prime}-\frac{2\Lambda}{\alpha^{\prime}}-4{({\Phi^{0}}^{\prime})}^{2}-8{\Phi^{0}}^{\prime}\varphi^{\prime}+4{\varphi^{\prime\prime}}-8{\Phi^{0}}^{\prime}(h_{tt}^{\prime}-h_{xx}^{\prime})=0.
a^{\dagger}(k,\rho)\vert1\colonp_{i}\rangle=-{\frac{e}{(2\pi)^{3/2}}}{\frac{1}{\sqrt{2k^{+}}}}f^{*}(k,\rho:p_{i})\vert1\colonp_{i}\rangle+\vert2\colonp_{i},k_{i}\rangle\;.
\gamma_{s}=i\sum_{m\neqs}\int_{\Sigma}\int_{\partial\Sigma={\bfc}}\frac{\langle\hat{\varphi}_{s}|\nabla_{R}H_{1}|\hat{\varphi}_{m}\rangle\times\langle\hat{\varphi}_{m}|\nabla_{R}H_{1}|\hat{\varphi}_{s}\rangle\cdotd\vec{\Sigma}}{\left({\cal\hat{E}}_{s}-{\cal\hat{E}}_{m}\right)^{2}},
E={\frac{8\piA^{2}}{g^{2}}}\int_{r_{c}}^{\infty}r^{2}dr\left(Tr\left(\left[{\frac{d{\bf\Phi}}{dr}}\right]^{2}\right)+{\frac{2}{\A^{2}-C^{2}}}Tr\left(r^{-2}{\frac{d{\bfM_{+}}}{dr}}{\frac{d{\bfM_{-}}}{dr}}\right)\right)
{\overline{{\alphaV}}}\cdot{\overline{{\betaV}}}\equiv\sum_{s=0}^{9}{\overline{{\alphaV}}}^{s}{\overline{{\betaV}}}^{s}-\sum_{s=10}^{19}{\overline{{\alphaV}}}^{s}{\overline{{\betaV}}}^{s}~,
K(\Lambda_{1},\Lambda_{2};t))=\frac{1}{(2\piit)^{(m+n)/2}}\frac{1}{m!}\frac{1}{n!}det(e^{i(\lambda_{1,i}-\lambda_{2,j})^{2}/2t})det(e^{-i({\bar{\lambda}}_{1,\alpha}-{\bar{\lambda}}_{2,\beta})^{2}/2t}),
\Sigma(p)=e^{2}\int\bar{d}^{3}k\;\frac{\gamma_{\mu}(\gamma.p-\gamma.k)\gamma_{\nu}}{(p-k)^{2}}D^{\mu\nu}(k)=-e^{2}\left[\frac{\xi\gamma.p}{16\sqrt{-p^{2}}}+\frac{\sqrt{-p^{2}}}{8\mu}\right]
\begin{array}{rcl}{\{A(x),B(y)\}_{D}}&{=}&{\{A(x),B(y)\}}\\{}&{}&{-\int\{A(x),\phi_{i}(z)\}\,J^{ik}(z,w)\,\{\phi_{k}(w),B(y)\}\,dz\,dw}\\\end{array}
f_{\epsilon}=g_{\epsilon}\frac{\gamma(c_{\epsilon})\gamma(c_{\epsilon}-a_{\epsilon}-b_{\epsilon})}{\gamma(c_{\epsilon}-a_{\epsilon})\gamma(c_{\epsilon}-b_{\epsilon})};\quad\tilde{f}_{\epsilon}=g_{\epsilon}\frac{\gamma(c_{\epsilon})\gamma(a_{\epsilon}+b_{\epsilon}-c_{\epsilon})}{\gamma(a_{\epsilon})\gamma(b_{\epsilon})}.
\hat{\hat{T}}_{\hat{\hat{\mu}}\hat{\hat{\nu}}}{}^{\hat{\hat{\rho}}}=-\left(i_{\hat{\hat{k}}_{(n)}}\hat{\hat{C}}\right)_{\hat{\hat{\mu}}\hat{\hat{\nu}}}Q^{nm}\hat{\hat{k}}_{(m)}{}^{\hat{\hat{\rho}}}\,.
iD_{\mu\nu}(p)=-i\frac{1}{p^{2}-\mu^{2}}\left(g_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)+\mu\frac{1}{p^{2}-\mu^{2}}\frac{1}{p^{2}}\varepsilon_{\mu\nu\rho}p^{\rho}-i\alpha\frac{p_{\mu}p_{\nu}}{p^{4}}\\,
{\hat{G}}(z,z^{\prime};t)=N_{(q,v)}^{2}\int_{-\infty}^{\infty}dq\int_{-\infty}^{\infty}[v\mathrm{Sinh}(\piv)dv]{\sqrt{y}}{\sqrt{y^{\prime}}}K_{iv}(2\pi|q|y)K_{iv}(2\pi|q|y^{\prime}),
J=-\intd^{2}\epsilon_{ij}x^{i}\left[\dot{\varphi}\nabla_{j}\varphi+{\frac{1}{\mu^{2}}}({\dot{\pi}_{\chi}+{\frac{\kappa}{2}}\dot{\varphi}})\nabla_{j}({\pi_{\chi}+{\frac{\kappa}{2}}\varphi})\right]
X^{I}=\left(\begin{array}{cccc}{x_{1}^{I}}&{0}&{\dots}&{0}\\{0}&{x_{2}^{I}}&{\dots}&{0}\\{\dots}&{\dots}&{\dots}&{\dots}\\{0}&{\dots}&{\dots}&{x_{N}^{I}}\\\end{array}\right),\I=1,\dots,26\,
\hat{W}^{\hat{a}\hat{b}\hat{c}}\equiv\hat{\calH}^{\hat{a}\hat{d}\hat{e}}\,\hat{C}_{\hat{d}}^{\hat{b}}\,\hat{C}_{\hat{e}}^{\hat{c}}=\hat{P}^{\hat{a}\hat{b}\hat{d}}\,\hat{C}_{\hat{d}}^{\hat{c}}\,.
{\calL}_{\gamma}=m\gamma(\dot{x}n)+\frac{m}{s}\sqrt{(\gamma^{2}-1)(\dot{x}^{2}+(\dot{x}n)^{2}-\frac{2s}{m}\varepsilon_{abc}\dot{x}^{a}n^{b}\dot{n}^{c}+\frac{s^{2}}{m^{2}}\dot{n}^{2})}+i\gammas\frac{\dot{z}\bar{z}-\dot{\bar{z}}z}{1-z\bar{z}}
\langle\phi(x(\tau_{a}))\phi(x(\tau_{b}))\rangle=\int_{0}^{\infty}d\bar{T}e^{-{\bar{m}}^{2}\bar{T}}{(4\pi\bar{T})}^{-{\frac{D}{2}}}\mathrm{exp}\Biggl[-{\frac{{(x(\tau_{a})-x(\tau_{b}))}^{2}}{4\bar{T}}}\Biggr]\,,
8\pi\;\rho_{H}\;r_{H}^{2}=\left(1-{\frac{Q^{2}}{2M^{2}}}\right){\frac{Q^{2}}{r_{H}^{2}}}={\frac{M-a}{M}}{\frac{Q^{2}}{2M(2M-2a)}}={\frac{a}{2M}}.
[\bar{D}_{\mu},\frac{1}{\bar{D}^{2}}]=-\frac{1}{\bar{D}^{2}}[\bar{D}_{\mu},\bar{D}^{2}]\frac{1}{\bar{D}^{2}}=-\frac{1}{\bar{D}^{2}}\left(\bar{F}_{\mu\nu}\bar{D}_{\nu}+\bar{D}_{\nu}\bar{F}_{\mu\nu}\right)\frac{1}{\bar{D}^{2}}
A_{i}^{(2)}=T_{3}\frac{f_{2}[(O_{1j}x_{j})^{2}+(O_{2k}x_{k})^{2}]}{(O_{1l}x_{l})^{2}+(O_{2m}x_{m})^{2}}\left(O_{1i}O_{2p}x_{p}-O_{2i}O_{1q}x_{q}\right)
A_{rt}\midsu,n\rangle=A_{rt}N^{\frac{1-n}{2}}\;(\phi)_{su}^{n}\mid0.\rangle=\frac{1}{\sqrt2}N^{\frac{1-n}{2}}\sum_{m}\;(\phi)_{st}^{m}\;(\phi)_{ru}^{n-m-1}\mid0.\rangle
\mathbf{S}_{T}=\mathbf{S}_{Q}+\rho^{\mu}\mathbf{\bar{Q}}_{\mu}-i\xi^{\mu}\mathbf{P}_{\mu}+\bar{\xi}\mathbf{Z}+\xi\mathbf{\bar{Z}}-\rho^{\mu}\frac{\partial}{\partial\xi^{\mu}}+\frac{\partial}{\partial\bar{\xi}}-\rho^{\mu}\rho_{\mu}\frac{\partial}{\partial\xi},
A_{\mu}^{(2)}=\mp{\frac{2i}{g}}\,(H_{3}\,{\frac{v_{\mu}^{1}}{\rho_{1}}}+H_{1}\,{\frac{v_{\mu}^{2}}{\rho_{2}}})+\kappa\,(E_{32}^{\pm}\,R_{\mu}^{1}+E_{12}^{\pm}\,R_{\mu}^{2})\,\delta(R^{1}\cdotR^{2}).
\Biggl[\cdots\biggl[\Bigl[[j_{+}{}^{n},(j_{+}{}^{m})^{\dagger}],(j_{+}{}^{i_{1}})^{\dagger}\Bigr],(j_{+}{}^{i_{2}})^{\dagger}\biggr],\cdots(j_{+}{}^{i_{k+1}})^{\dagger}\Biggr]\left|{\calE}_{N},2N\right\rangle=0,
\begin{array}{c}{\phi_{\omega}(s)=A_{1}{\calJ}_{l+1/2}(z_{1})+B_{1}{\calY}_{l+1/2}(z_{1}),\mathrm{\for\}s<a_{1},}\\{\phi_{\omega}(s)=A_{2}{\calJ}_{l+1/2}(z_{2})+B_{2}{\calY}_{l+1/2}(z_{2}),\mathrm{\for\}a_{1}<s<s_{0},}\\\end{array}
{\bfb}^{12\pm}\sim\sum_{k,l,m}\,j_{l}(kr)\,[\,{\bfc}_{lm}^{\pm}(k)\,Y_{lm}(\theta,\phi)\,e^{-ikt}+{\bfd}_{lm}^{\pm}(k)\,Y_{lm}(\theta,\phi)^{*}e^{ikt}\,]
\left({\bfA}\star{\bfB}\right)({\bfx})=e^{\frac{i}{2}\theta\,\epsilon^{ij}\partial_{i}^{(1)}\partial_{j}^{(2)}}{\bfA}({\bfx}_{1}){\bfB}({\bfx}_{2})\vert_{{\bfx}_{1}={\bfx}_{2}={\bfx}}.
S_{\phi^{2}}=\int_{x}\left(\phi_{\mu}^{\dagger}\a_{\mu\nu}^{-+}\\phi_{\nu}+\phi_{\mu}\a_{\mu\nu}^{+-}\\phi_{\nu}^{\dagger}+\phi_{\mu}\a_{\mu\nu}^{++}\\phi_{\nu}+\phi_{\mu}^{\dagger}\a_{\mu\nu}^{--}\\phi_{\nu}^{\dagger}\right),
T_{\mu\nu}^{\mathrm{tot}}\equivT_{\mu\nu}+{\frac{\widetilde{\kappa}^{4}}{\kappa^{2}}}S_{\mu\nu}-{\frac{1}{\kappa^{2}}}{\calE}_{\mu\nu}-{\frac{\Lambda}{\kappa^{2}}}g_{\mu\nu}\,
\begin{array}{rl}{C_{abcd}^{(1)}}&{\delta_{ab}\delta_{cd}\,,\,\,C_{abcd}^{(2)}=\delta_{ac}\delta_{bd}\,,\,\,C_{abcd}^{(3)}=\delta_{ad}\delta_{bc}\,,}\\{C_{abcd}^{(4)}}&{f_{abn}f_{cdn}\,,\,\,\left[C^{(5)}-C^{(6)}\right]_{abcd}=f_{acn}f_{dbn}-f_{adn}f_{bcn}\,,}\\{C_{abcd}^{(7)}}&{d_{abn}f_{cdn}\,,\,\,C_{abcd}^{(8)}=d_{acn}f_{dbn}\,,\,\,C_{abcd}^{(9)}=d_{adn}f_{bcn}\,.}\\\end{array}
{\frac{\delta\Gamma_{3/2}}{\delta\phi^{i}}}\delta\phi_{1/2}^{i}=-{\frac{\hbar^{2}e_{ij}d_{ijk}\alpha_{k}}{6\pi}}\int\partial^{2}c(K_{\pi_{\gamma}}-B){\frac{\partial^{4}}{\bar{\partial}}}(K_{\pi_{\gamma}}-B).
M_{\mathrm{BPS}}=e^{-{\frac{\varphi_{\infty}}{2}}}|g_{11\,\infty}^{\frac{1}{2}}\beta_{1}^{({2})}+g_{11\,\infty}^{-{\frac{1}{2}}}\beta_{1}^{(1)}|+e^{\frac{\varphi_{\infty}}{2}}|g_{22\,\infty}^{{\frac{1}{2}}}\alpha_{2}^{(1)}+g_{22\,\infty}^{-{\frac{1}{2}}}\alpha_{2}^{(2)}|.
S_{f}=T\intd^{2}xd^{8}\theta\left\{\theta^{8}\;(i\;\nabla_{-}\Psi^{i}\Psi^{i}+{\frac{1}{8}}\nabla_{\alpha}\Psi^{i}\nabla_{\alpha}\Psi^{i})+P^{\{\alpha\beta\}}\nabla_{\alpha}\Psi^{i}\nabla_{\beta}\Psi^{i}\right\}\;.
\delta_{L}\psi(x)=-{\frac{i}{2}}\epsilon_{ab}\sigma^{ab}\psi,\quad\delta_{L}{e^{a}}_{\mu}(x)=\epsilon{^a}_{b}e{^b}_{\mu}+{\frac{\kappa}{4}}\varepsilon^{abcd}\bar{\psi}\gamma_{5}\gamma_{d}\psi(\partial_{\mu}\epsilon_{bc})
\sum_{\delta}\langle\alpha|\delta\rangle\,\theta_{\delta}(z_{1})\theta_{\delta}(z_{2})\theta_{\delta}(z_{3})\theta_{\delta}(z_{4})=2^{g}\,\theta_{\alpha}(z_{1}^{\prime})\theta_{\alpha}(z_{2}^{\prime})\theta_{\alpha}(z_{3}^{\prime})\theta_{\alpha}(z_{4}^{\prime})\,,
S_{i,\tiny\mathrm{div}}^{C}[g,\beta,D]=\eta\,{\frac{\Gamma\left(1-\fracD2\right)}{(4\pi)^{D/2}}}\,\,{\frac{2\pi^{2}m^{D-4}}{3\kappa\beta}}\,\,\int_{\Sigma}\left[f_{1}m^{2}-\left(p_{1}{\frac{8\pi^{2}}{\kappa^{2}\beta^{2}}}{\calP}+p_{2}R+p_{3}{\calQ})\right)\right]~~~.
{\calL}_{\mathrm{mass}}=\frac{m^{\prime}}{2}(\overline{{N}}_{R}jN_{R}-i\overline{{N}}_{R}jN_{R}i)+\frac{M}{2}\left[\overline{{N}}_{R}(N_{R})^{c}-i\overline{{N}}_{R}(N_{R})^{c}i+\mathrm{H.c.}\right].
A_{ab}^{mn}=\int_{{\bfR}^{2}}\frac{1}{|f|^{2}}(\delta_{ab}z^{*m}z^{n}-\frac{1}{|f|^{2}}c_{a}^{r}c_{b}^{s*}z^{n+r}z^{*m+s})d^{2}x.
\begin{array}{llll}{\hat{c}^{1}|-->=|+->\,\&\hat{c}^{1}|+->=0\,\&\hat{c}^{1}|-+>=|++>\,\&\hat{c}^{1}|++>=0\,}\\{\hat{c}^{2}|-->=|-+>\,\&\hat{c}^{2}|+->=-|++>\,\&\hat{c}^{2}|-+>=0\,\&\hat{c}^{2}|++>=0\,}\\{\hat{b}_{1}|-->=0\,\&\hat{b}_{1}|+->=|-->\,\&\hat{b}_{1}|-+>=0\,\&\hat{b}_{1}|++>=|-+>\,}\\{\hat{b}_{2}|-->=0\,\&\hat{b}_{2}|+->=0\,\&\hat{b}_{2}|-+>=|-->\,\&\hat{b}_{2}|++>=-|+->\,}\\\end{array}
B\equiv\int\left[\prod_{i=1}^{3}\frac{d^{2}k_{i}}{k_{i}^{+}}\right]\delta^{2}({\bfP}-\sum_{i=1}^{3}{\bfk}_{i})\left(E-\sum_{i=1}^{3}\epsilon({\bfk}_{i})\right)^{-1},\quadL\equiv\frac{\Lambda_{n}}{\Lambda_{n+1}}.
X_{ab}^{A\dot{B}}=\{C_{a}^{A},C_{b}^{*\dot{B}}\},\\\\\{C_{a}^{A},C_{b}^{B}\}=0,\\\\X_{ab}^{A\dot{B}}\stackrel{\mathrm{def}}{=}\sigma_{\mu}^{A\dot{B}}X_{ab}^{\mu}
{\hat{\nabla}}^{\alpha}=\nabla^{\alpha}+{\frac{1}{4}}\omega_{a}^{kp}{\varepsilon_{kp}}^{d}{(\gamma^{a}\gamma_{d}\partial)}^{\alpha}+{\frac{1}{2}}{(\gamma^{a}\gamma^{b}\psi_{a})}^{\alpha}\nabla_{b}
\theta_{A}\theta_{\bar{A}}=\theta_{\bar{A}}\theta_{A}=0,\;\theta_{A}\theta_{\bar{B}}=\bar{q}_{AB}\,\theta_{\bar{B}}\theta_{A},\;\theta_{\bar{A}}\theta_{\bar{B}}=q_{AB}\,\theta_{\bar{B}}\theta_{\bar{A}},\;\theta_{\bar{A}}^{2}=0.
Y_{Q(z)}^{\prime}(v,x)=\biggl(1-\frac{y_{1}}{z}\biggr)^{L(0)}Y_{Q(z)}^{\prime}\biggl(\biggl(1-\frac{y_{1}}{z}\biggr)^{-L(0)}v,\frac{x}{1-y_{1}/z}\biggr)\biggr(1-\frac{y_{1}}{z}\biggr)^{-L(0)}.
I_{n}(p_{\phi})=\int{\frac{d^{4-2\epsilon}r_{1}}{r_{1}^{2}(r_{1}-r_{2})^{2}}}\int{\frac{d^{4-2\epsilon}r_{2}}{r_{2}^{2}(r_{2}-r_{3})^{2}}}\cdots\int{\frac{d^{4-2\epsilon}r_{n}}{r_{n}^{2}(r_{n}-p_{\phi})^{2}}}
K_{ij}(x_{0},x;y_{0},y;t)=\frac{1}{\beta(2\pi)^{3}}\sum_{r=-\infty}^{\infty}\intd^{3}k\,e^{i(k(x-y)+2\pir(x_{0}-y_{0})/\beta)}e^{-tk^{2}}e^{-t(2\pir/\beta-a_{0}(y))^{2}}\left(e^{-(A+B)}\right)_{ij}
S=\left(\begin{array}{cccccc}{0}&{0}&{0}&{C}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{C}\\{0}&{0}&{0}&{0}&{C}&{0}\\{C}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{C}&{0}&{0}&{0}\\{0}&{C}&{0}&{0}&{0}&{0}\\\end{array}\right),
G=\frac{1}{\sqrt{2}}\cdot\left(\begin{array}{cccccccc}{1}&{0}&{0}&{0}&{0}&{0}&{0}&{1}\\{0}&{1}&{0}&{0}&{0}&{0}&{-1}&{0}\\{0}&{0}&{1}&{0}&{0}&{-1}&{0}&{0}\\{0}&{0}&{0}&{1}&{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}&{-1}&{0}&{0}&{0}\\{0}&{0}&{-1}&{0}&{0}&{-1}&{0}&{0}\\{0}&{-1}&{0}&{0}&{0}&{0}&{-1}&{0}\\{1}&{0}&{0}&{0}&{0}&{0}&{0}&{-1}\\\end{array}\right)
E_{\vec{m}^{\prime}}\;E_{\vec{m}^{\prime\prime}}\cdotsE_{\vec{m}^{(M)}}\;-\;\omega^{\gamma(\vec{m}^{\prime},\vec{m}^{\prime\prime},\cdots,\vec{m}^{(M)})}\;\cdot\;E_{\vec{m}^{\prime}+\cdots+\vec{m}^{(M)}}
L_{_{\mathrm{GF+FP}}}^{1}=\mathrm{Tr}\left(\frac{1}{4}B^{\mu\nu}[\lambdaF_{\mu\nu}-\frac{1}{2}S_{\mu\nu\alpha\beta}F^{\alpha\beta}]-\chi_{\mu\nu}\left[X^{[\mu},\psi^{\nu]}\right]+\frac{1}{8}aB^{\mu\nu}B_{\mu\nu})\right)
\left(\begin{matrix}{V+1}&{0}&{-\partial_{z}}&{-\partial_{\rho}-{\frac{(m_{z}+{\frac{1}{2}})}{\rho}}+\it{A}}\\{0}&{V+1}&{-\partial_{\rho}+{\frac{(m_{z}-{\frac{1}{2}})}{\rho}}-\it{A}}&{\partial_{z}}\\{\partial_{z}}&{\partial_{\rho}+{\frac{(m_{z}+{\frac{1}{2}})}{\rho}}-\it{A}}&{V-1}&{0}\\{\partial_{\rho}-{\frac{(m_{z}-{\frac{1}{2}})}{\rho}}+\it{A}}&{-\partial_{z}}&{0}&{V-1}\\\end{matrix}\right).
K=-K^{\prime}=\left(\begin{matrix}{{\frac{1}{4}}}&{0}&{0}&{0}\\{0}&{{\frac{{q^{2}}}{{{\left(1+{q^{2}}\right)}^{2}}}}}&{0}&{0}\\{0}&{0}&{{\frac{{q^{2}}}{{{\left(1+{q^{2}}\right)}^{2}}}}}&{0}\\{0}&{0}&{0}&{{\frac{1}{4}}}\\\end{matrix}\right)
\begin{array}{rcl}{G_{i\mu\nu}}&{=}&{\partial_{\mu}G_{i\nu}-\partial_{\nu}G_{i\mu}-ig_{i}[G_{i\mu},G_{i\nu}],}\\{D_{\mu}\Phi}&{=}&{\partial_{\mu}\Phi-ig_{1}G_{1\mu}\Phi+ig_{2}\PhiG_{2\mu},}\\\end{array}
F_{B}={\frac{1}{3}}H_{ijl}^{1}(k)H_{ijl}^{2}(p){\frac{\Gamma(1-2k_{0}^{2})\Gamma(\vec{k}\cdot\vec{p}+k_{0}^{2})}{\Gamma(\vec{k}\cdot\vec{p}-k_{0}^{2}+1)}}\.
{\mathcalL}_{GF}+{\mathcalL}_{FP}=-\frac{1}{2\alpha}(D_{\mu}^{mn}A^{\mun})^{2}+i{\overline{{C}}}^{m}D_{\mu}^{mp}D^{\mupn}C^{n}-ig^{2}\epsilon^{mq}\epsilon^{pn}{\overline{{C}}}^{m}C^{n}A^{\mup}A_{\mu}^{q}+\frac{\alpha}{4}g^{2}\epsilon^{mn}\epsilon^{pq}{\overline{{C}}}^{m}{\overline{{C}}}^{n}C^{p}C^{q}
\left.\frac{m_{2}}{m_{1}}\right|_{g_{2}^{(1)}}=\sqrt{3},\qquad\left.\frac{m_{2}}{m_{1}}\right|_{d_{4}^{(3)}}=\sqrt{\frac{\sqrt{3}+1}{\sqrt{3}-1}}=\frac{\sqrt{3}+1}{\sqrt{2}}
\begin{array}{rcl}{\hat{\phi}_{(I)}(t,\vec{x}\,)}&{=}&{e^{i(t-t_{0})\hat{H}_{0}}\,\hat{\phi}(t_{0},\vec{x}\,)\,e^{-i(t-t_{0})\hat{H}_{0}}\,}\\{}&{}&{}\\{\hat{\pi}_{(I)}(t,\vec{x}\,)}&{=}&{e^{i(t-t_{0})\hat{H}_{0}}\,\hat{\pi}(t_{0},\vec{x}\,)\,e^{-i(t-t_{0})\hat{H}_{0}}\,}\\\end{array}
0=\frac{d}{d\lambda}\vert_{\lambda=0}\frac{\delta(S+\lambdaF)}{\delta\varphi(y)}(f+\lambdah)=\frac{\deltaF}{\delta\varphi(y)}(f)+\intdz\,\frac{\delta^{2}S}{\delta\varphi(y)\varphi(z)}(f)h(z)
\langleB_{i}(\mathbf{r},t)B_{j}(\mathbf{r},t)\rangle_{0}=\left({\frac{\pi}{a}}\right)^{4}{\frac{2}{3\pi}}\left[\left(-\delta^{\Vert}+\delta^{\bot}\right)_{ij}\;{\frac{1}{120}}-\delta_{ij}F(\xi)\right]\,.
\begin{array}{c}{G({\bfr},{\bfr}_{0};m^{2})=<0|\varphi(y,x)\,{\cal{D}}\,\varphi(y_{0},-x_{0})|0>_{E}+}\\{+<0|\varphi(y,x)\varphi(y_{0},x_{0})\,{\cal{D}}|0>_{E}}\\\end{array}
\int_{\infty}d\Omega_{(1)}{}^{*}H_{\Omega_{(1)}}^{a}+\int_{\infty}d\Omega_{(2)}{}^{*}H_{\Omega_{(2)}}^{a}=\int_{\infty}d\Omega_{(3)}{}^{*}H_{\Omega_{(3)}}^{a},
\xi=\left(\begin{matrix}{1}&{w_{1}}&{w_{2}}&{\cdots}&{\cdots}&{w_{n}}\\{0}&{1}&{w_{n+1}}&{\cdots}&{\cdots}&{w_{2n-1}}\\{0}&{0}&{1}&{w_{2n}}&{\cdots}&{w_{3n-3}}\\{}&{}&{}&{}&{}&{}\\{\vdots}&{\vdots}&{\vdots}&{\vdots}&{\vdots}&{\vdots}\\{}&{}&{}&{}&{}&{}\\{0}&{0}&{\cdots}&{0}&{1}&{w_{n(n+1)/2}}\\{0}&{0}&{\cdots}&{\cdots}&{0}&{1}\\\end{matrix}\right)\inF_{n}.
\alpha_{\omega\omega^{\prime}}=\frac{1}{2\pi}\sqrt{\frac{\omega^{\prime}}{\omega}}I_{\omega\omega^{\prime}}^{+}\quad\mathrm{and}\quad\beta_{\omega\omega^{\prime}}=\frac{1}{2\pi}\sqrt{\frac{\omega^{\prime}}{\omega}}I_{\omega\omega^{\prime}}^{-}
ds_{E}^{2}=H_{p}^{-{\frac{D-p-3}{D-2}}}\left[-dt^{2}+dx_{1}^{2}+\cdots+dx_{p}^{2}\right]+H_{p}^{\frac{p+1}{D-2}}\left[dy_{1}^{2}+\cdots+dy_{D-p-1}^{2}\right],
E_{0}=\frac{m}{3p^{+}}\left(2a^{i_{N}\dagger}a^{i_{N}}+a^{i_{N}^{\prime}\dagger}a^{i_{N}^{\prime}}+i\psi_{0}\gamma^{4}\Omega\psi_{0}-\frac{i}{2}\psi_{0}^{\prime}\gamma^{4}\Omega\psi_{0}^{\prime}+e_{0}\right)~,
\langle0|T(\Phi(x,t)\Phi(x^{\prime},t^{\prime}))|0\rangle=\int\frac{d\omega}{2\pi}\frac{e^{-i\omega(t-t^{\prime})}}{2k(\omega)}(e^{ik(\omega)|x-x^{\prime}|}+R(k)e^{-ik(\omega)(x+x^{\prime})})+\dots
curl\,{\bfH}=\frac{4\pi}{c}q{\bfV}\delta({\bfR}(t))+\frac{1}{c}\frac{\partial{\bfE}}{\partialt}-\frac{1}{c}{\bfV}\,div\,{\bfE}+\frac{1}{c}curl({\bfV}\times{\bfE}).
\sum_{i=j}^{3}iS_{ij}^{F}({\vec{x}}-{\vec{x^{\prime}}})=\int{\frac{\mathrm{d}^{3}{\vec{k}}}{(2\pi)^{3}}}{{\frac{e^{i{\vec{k}}.({\vec{x}}-{\vec{x^{\prime}})}}}{\omega_{k}}}}.
T_{\phi\phi}=\frac{1}{(k-2)}g_{ab}{\large:}J_{\phi}^{a}(z)J_{\phi}^{b}(z){\large:}=\frac{1}{k-2}\Sigma{\large:}J_{n-m}^{a}J_{m}^{a}{\large:}z^{-n-2}=\SigmaL_{n}z^{-n-2},
\tilde{S}(\tilde{\Psi})=S(\Psi_{0}+\tilde{\Psi})=S_{0}-\frac{1}{2}\int\tilde{\Psi}\star\tilde{Q}\tilde{\Psi}-\frac{g}{3}\int\tilde{\Psi}\star\tilde{\Psi}\star\tilde{\Psi}\,.
A_{x_{\perp}}^{21}=\int_{R^{2}}d^{2}z\,\psi_{-\frac{1}{2}}^{\dagger}(z+\omega)\frac{\partial}{\partialx_{\perp}}\psi_{\frac{1}{2}}(z-\omega)=\frac{\bar{x}_{\perp}}{2\pi|x_{\perp}|}e^{2ix\cdot\omega}(I+J).