harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
R_{Gauss2}=\left(1+z^{2}S_{\chi\chi}\right)\partial_{z}+\left(zS_{\chi}+vz^{2}S_{\chi\chi}\right)\partial_{x}+S_{\chi}\partial_{v}-\left[nz\left(1+\frac{vz}{x}\right)S_{\chi\chi}+\frac{nz}{x}S_{\chi}\right]\partial_{n}\,.
X^{\mu}=x^{\mu}+2\alpha^{\prime}p^{\mu}\tau+i(\alpha^{\prime}/2)^{1/2}\sum_{n\neq0}\frac{1}{n}\left(\alpha_{n}^{\mu}e^{2in(\sigma-\tau)}+{\tilde{\alpha}}_{n}^{\mu}e^{-2in(\sigma+\tau)}\right).
\delta[\frac{1}{4}\vec{R}_{\mu\nu}\vec{R}^{\mu\nu}]=\delta\vec{\Gamma}_{\mu}[\vec{\Gamma}_{\nu}\times\vec{R}^{\mu\nu}]=-\delta\vec{\Gamma}\,\partial_{\mu}[\vec{\Gamma}_{\nu}\times\vec{R}^{\mu\nu}],
{}^{(4)}G_{\mu\nu}=-{\frac{1}{2}}\kappa_{5}^{2}{}^{(5)}\Lambda+KK_{\mu\nu}-K_{\mu}^{~\sigma}K_{\nu\sigma}-{\frac{1}{2}}g_{\mu\nu}\left(K^{2}-K^{\alpha\beta}K_{\alpha\beta}\right)-E_{\mu\nu},
M_{\lambda}:=-i\,\langle\vec{q},r;\vec{k},\lambda)|S^{(1)}|\vec{p},s\rangle=\sum_{l,n}c_{l}^{(+\,e)}{c_{n}^{(-\,e)}}^{\ast}\widetilde{M}_{\lambda}(j_{q},\vec{k},\lambda;j_{p})
\tilde{\psi}_{-,m}^{(n)}(x)=\frac{1}{\tilde{{\calN}}_{m,-}^{(n)}}\left(\begin{array}{c}{i\sqrt{m}(d_{-})^{m-1}\phi_{2}^{(n)}(z,\bar{z})}\\{-(d_{-})^{m}\phi_{2}^{(n)}(z,\bar{z})}\\\end{array}\right)
\hat{A}_{\alpha}^{+}(q)=i\,\Omega_{+}^{-1}(q)\,\partial_{\alpha}\Omega_{+}(q)\qquad\mathrm{or}\qquad\hat{A}_{\alpha}^{-}(q)=i\,\Omega_{-}^{-1}(q)\,\partial_{\alpha}\Omega_{-}(q).
S=\intd^{4}x\sqrt{{g_{4}}}e^{-\Phi_{4}}\Bigl\{{R_{4}}+({\nabla}\Phi_{4})^{2}+\frac{1}{8}Tr\bigl(l\nablam\bigr)^{2}-\frac{1}{12}{H_{4}}_{\mu\nu\lambda}^{2}\Bigr\}
\begin{array}{rccl}{S:}&{{\calC}\otimes{\calC}}&{\to}&{{\calC}}\\{}&{}&{}&{}\\{}&{(s_{1},s_{2})}&{\mapsto}&{\displaystyleS(s_{1},s_{2})=-Z\left(\frac{s_{1}^{*}+s_{2}}{2}\right).}\\\end{array}
\mathrm{E}_{\nu}^{d-1}(x,x^{\prime})=e^{(v+v^{\prime})(-\frac{d}{2}-i\nu)}{W}_{\nu}\left(\eta\left(v,x\right),\eta^{\prime}\left(v^{\prime},x^{\prime}\right)\right)=\frac{\Gamma\left(\frac{d-1}{2}+i\nu\right)\Gamma(-i\nu)}{4\pi^{\frac{d+1}{2}}}\\\left[(x-x^{\prime})^{2}\right]^{-i\nu-\frac{d-1}{2}}
\langle\Psi|\Psi^{\prime}\rangle=\int_{{\calW}_{L}}{\calD}^{\widehat{e}}\widetilde{\varphi}\int_{{\calE}_{L}}{\calD}^{\widehat{e}}\widetilde{x}\;\overline{{\Psi[\widetilde{\varphi},\widetilde{x}]}}\Psi^{\prime}[\widetilde{\varphi},\widetilde{x}]\;\;\;
\tilde{\Pi}_{p}^{(1)}=\frac{4\pi}{\kappa}\frac{m}{p^{2}}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{k^{2}p^{2}-(k\cdotp)^{2}}{k^{2}(k^{2}+m^{2})(k+p)^{2}},
I=\frac{\pi}{8}\intd^{2}x\sqrt{-g}\left[\PhiR+\frac{6}{l_{P}^{2}\Phi^{1/3}}-\Phi{\calG}_{ij}\partial_{\alpha}\phi^{i}\partial^{\alpha}\phi^{j}-\frac{G^{IJ}\tilde{Q}_{I}\tilde{Q}_{J}}{l_{P}^{6}\Phi^{5/3}}\right].
\breve{A}_{2}=\beta\int\sqrt{h}d^{D-1}x\left({\frac{1}{72}}\breve{R}^{2}-{\frac{1}{180}}\breve{R}_{\mu\nu}\breve{R}^{\mu\nu}+{\frac{1}{180}}\breve{R}_{\mu\nu\lambda\rho}\breve{R}^{\mu\nu\lambda\rho}-\frac16\breve{R}V+\frac12V^{2}\right)
{\psi}_{\alpha}(x)=\frac{Z_{\alpha}}{\sqrt{({\gamma}^{0}{\gamma}^{-})_{{\alpha}{\alpha}}}}\mathrm{exp}[-i\sqrt{\pi}{\Lambda}_{{\alpha}r}^{(-)}(x)]{\sigma}_{\alpha}\mathrm{exp}[-i\sqrt{\pi}{\Lambda}_{{\alpha}r}^{(+)}(x)]
\sum_{j=1}^{n}\tau_{c}^{a_{j}}\langle\varphi^{c}(x_{j}\prod_{i\neqj}^{n}\varphi^{a_{i}}(x_{i})\rangle=-K_{ab}\circ\langle\varphi^{a}\varphi^{b}\prod_{j=1}^{n}\varphi^{a_{j}}(x_{j})\rangle\.
\left<tr\frac{\delta}{\delta\gamma_{\mu}(s)}{\cal{F}}_{\mu}(\gamma(0))\right>=\frac{\partial^{2}W(\gamma)}{\partial\gamma_{\mu}^{2}(0)}=-e^{2}\int\delta(\gamma(s)-\gamma(0))W(\gamma_{0s})W(\gamma_{s0})\dot{\gamma_{\mu}}(0)\dot{\gamma_{\mu}}(s)ds
V_{\4T}^{[r,0]}\=\E_{0,T}^{[r]}(p_{2}^{\,2},p_{3}^{\,2},p_{4}^{\,2})\+\\sum_{n=1}^{r}\,\left(\frac{\Lambda^{2}}{p_{1}^{\,2}+u_{2n}\Lambda^{2}}\right)\,E_{n,T}^{[r]}(p_{2}^{\,2},p_{3}^{\,2},p_{4}^{\,2})\,,
M_{r}=\frac{M_{u}M_{v}}{M_{u}+M_{v}}=\frac{2}{3}\,\frac{m^{3}}{\lambda^{2}}\,\frac{(1-\nu_{1}^{2})^{3/2}}{\kappa~}\;,\qquad\kappa=\frac{1}{2}\,(\nu_{1+}^{3}+\nu_{1-}^{3})\;,
g_{i^{\prime}j^{\prime}}(p_{k^{\prime}0}^{\prime\prime})\,[\,g_{i^{\prime}j^{\prime}}^{0}(p_{k^{\prime}0}^{\prime\prime})\,]^{-1}G_{i^{\prime}j^{\prime},ij}^{\,KT}(p_{k^{\prime}0}^{\prime\prime},p_{k0})\,+\cdots
\bar{\xi}_{(m,n)}^{ij}\equivM_{\alpha\beta}^{ij}\xi_{(m)}^{\beta}\xi_{(n)}^{\alpha},\;\;\;\bar{\xi}_{((m,n),l)}^{ij}\equivM_{\alpha\beta}^{ij}\tilde{\xi}_{(m,n)}^{\beta}\xi_{(l)}^{\alpha}.
\hat{\calA}_{1}={\frac{(-1)^{n+1}}{2^{2n-1}n!}}({\frac{g}{\pi}})^{n}\epsilon_{\mu_{1}\nu_{1}...\mu_{n}\nu_{n}}\hat{F}_{\mu_{1}\nu_{1}}...\hat{F}_{\mu_{n}\nu_{n}}.
\mathcal{L}_{F}=\sum_{\sigma=1}^{N}\overline{{\psi}}_{\sigma}v_{\sigma,\mu}\left(\partial_{\mu}-ia_{\mu}\right)\gamma_{\mu}\psi_{\sigma}+\left|\left(\partial_{\mu}-ia_{\mu}\right)b\right|^{2}+V(\left|b\right|^{2}).
D_{\mu\nu}^{(0)}=\frac{1}{p^{2}+\kappa^{2}}\left[(\delta_{\mu\nu}-\frac{p_{\mu}\,p_{\nu}}{p^{2}})-\kappa\epsilon_{\mu\nu\lambda}\,{\frac{p_{\lambda}}{p^{2}}}\right]+\xi\,\frac{p_{\mu}p_{\nu}}{(p^{2})^{2}},
P_{A_{r}}^{{\bfV}_{i}}(x)=\prod_{1\leqj_{1}<\cdots<j_{i}\leqr+1}\Bigl(x-(\tilde{q}_{j_{1}}+\cdots\tilde{q}_{j_{i}})\Bigr)=P_{A_{r}}^{{\bfV}_{r+1-i}}(x)
\hat{H}_{T}=\frac{1}{2m}\vec{p}_{u}^{2}+\frac{m\omega^{2}}{2}\vec{u}^{2}+\frac{1}{2m}\vec{p}_{v}^{2}+\frac{m\omega^{2}}{2}\vec{v}^{2}-e^{2}-\frac{1}{4}g(\vec{v}^{4}-\vec{u}^{4})
a\left(t,z;x\right)\,=\,-\sum_{n=1}^{N}\left(e^{m\theta_{n}z}\right)\int_{t}^{\infty}q_{1}\,\beta_{n_{0}}^{\prime}e^{\theta_{n}\left(m+1\right)t^{\prime}}e^{-i\theta_{n}^{2}\left(2m+1\right)^{2}x}dt^{\prime}\,\,,\tag{3.3a}
\psi^{\dagger}({\bf{x}},t)=exp(\sum_{{\bf{k}},{\bf{q}}\neq0,i}U_{{\bf{k}},{\bf{q}}}^{i}({\bf{x}})b_{i}^{\dagger}({\bf{q}})e^{i\mathrm{~}{\tilde{\omega}}_{i}({\bf{q}})t})exp(-\sum_{{\bf{k}},{\bf{q}}\neq0,i}U_{{\bf{k}},{\bf{q}}}^{*i}({\bf{x}})b_{i}({\bf{q}})e^{-i\mathrm{~}{\tilde{\omega}}_{i}({\bf{q}})t}){\mathcal{R}}_{0}{\mathcal{Z}}_{0}^{*}\sqrt{\rho_{0}}
{{R^{\pm}}_{\mu\nu}}^{ab}=\partial_{\mu}{{\omega^{\pm}}_{\nu}}^{ab}-\partial_{\nu}{{\omega^{\pm}}_{\mu}}^{ab}+{\frac{1}{2}}f_{[cd][ef]}^{[ab]}{\omega^{\pm}}_{\mu}^{cd}{\omega^{\pm}}_{\nu}^{ef},
\frac{x_{3}}{x_{1}}\stackrel{\xi}{\rightarrow}\frac{x_{2}}{x_{3}},~~~\frac{x_{4}}{x_{2}}\stackrel{\xi}{\rightarrow}\frac{x_{1}}{\omegax_{4}},~~~\frac{x_{4}}{x_{1}}\stackrel{\xi}{\rightarrow}\frac{x_{1}}{\omegax_{3}},~~~\frac{x_{3}}{\omegax_{2}}\stackrel{\xi}{\rightarrow}\frac{x_{2}}{\omegax_{4}}
\begin{array}{l}{{\calD}_{-k}{\calX}^{-k}=1+q^{-2}{\calX}^{-k}{\calD}_{-k}~,~~~~~\mathrm{for}~k>0~,}\\{{\calD}_{+k}{\calX}^{+k}=1+q^{2}{\calX}^{+k}{\calD}_{+k}~,}\\{[{\calD}_{i},{\calD}_{j}]=0~,~~~~~[{\calX}^{i},{\calX}^{j}]=0~,}\\{{\calD}_{i}{\calX}^{j}={\calX}^{j}{\calD}_{i}~,~~~~~\mathrm{for}~i\not=j~.}\\\end{array}
{\frac{15}{16}}({\frac{3}{4}})^{n-2}v_{n}^{4}(\vec{v}_{n-1}\cdot\vec{\nabla}_{n})^{2}{\frac{1}{\vert\vec{x}_{n}\vert^{7}}}(\vec{v}_{n-2}\cdot\vec{\nabla}_{n-1})^{2}{\frac{1}{\vert\vec{x}_{n-1}\vert^{5}}}\dots(\vec{v}_{3}\cdot\vec{\nabla}_{4})^{2}{\frac{1}{\vert\vec{x}_{4}\vert^{5}}}(\vec{v}_{12}\cdot\vec{\nabla}_{3})^{2}{\frac{1}{\vert\vec{x}_{3}\vert^{5}}}{\frac{1}{\vert\vec{x}_{12}\vert^{5}}}
\begin{matrix}{I_{1}=\left(\begin{matrix}{0}&{1}&{0}&{0}\\{-1}&{0}&{0}&{0}\\{0}&{0}&{0}&{-1}\\{0}&{0}&{1}&{0}\\\end{matrix}\right),}&{I_{2}=\left(\begin{matrix}{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\\{-1}&{0}&{0}&{0}\\{0}&{-1}&{0}&{0}\\\end{matrix}\right),}&{I_{3}=\left(\begin{matrix}{0}&{0}&{0}&{1}\\{0}&{0}&{-1}&{0}\\{0}&{1}&{0}&{0}\\{-1}&{0}&{0}&{0}\\\end{matrix}\right).}\\\end{matrix}
W_{p,N_{r}}(t)=-5\partial_{z}K(t,t)Y_{p,N_{r}}+E_{p,N_{r}}^{(-2)}\left(G(t,t)\right)-\sum_{N}(-1)^{e(N)e(N_{r})}\chi_{N}(t)\partialq_{N}Y_{p,N_{r}}(t)
f_{UUU}=4\pi^{2}\frac{(T-{\bar{T}})}{(U-{\bar{U}})^{2}}\int\frac{d^{2}\tau}{\tau_{2}^{2}}{\bar{F}}({\bar{\tau}})\sum_{P_{L},P_{R}}P_{L}P_{R}P_{R}P_{R}e^{\pii\tau|P_{L}|^{2}}e^{-\pii{\bar{\tau}}|P_{R}|^{2}}.
\mathcal{H}^{\ast}=\intd^{2}\,x\,\left[\frac{1}{2}\,\left(\pi_{i}^{\perp}\,\pi_{i}^{\perp}+4\,m^{2}A\,_{i}^{\perp}\,A\,_{i}^{\perp}\right)+\frac{1}{2}\left(\,\xi^{\,i\,j}\,\partial_{\,i}\,A\,_{j}^{\perp}\right)^{2}+2\,m\,\left(A_{1}^{\perp}\,\pi\,_{2}^{\perp}-A\,_{2}^{\perp}\,\pi_{1}^{\perp}\right)\right].\tag{19}
\displaystyle{\frac{m_{\pi}^{2}f_{\pi}j_{\pi}(q^{2})}{m_{\pi}^{2}-q^{2}}=\partial^{\mu}J_{\mu}(q)=\frac{q^{2}f_{\pi}j_{\pi}(q^{2})}{m_{\pi}^{2}-q^{2}}+\partial^{\mu}\widehatJ_{\mu}(q).}
\hat{V}^{(B=0)}(x,t)=\mathrm{diag.}\,(\overline{{V}}_{1},\cdots,\overline{{V}}_{4})=\mathrm{diag.}\,(\scriptstyle{{\sqrt{-{\frac{t_{3}}{t_{1}}}}{\frac{t-t_{1}}{t-t_{3}}},\sqrt{-{\frac{t_{4}}{t_{2}}}}{\frac{t-t_{2}}{t-t_{4}}},\sqrt{-{\\frac{t_{1}}{t_{3}}}}{\frac{t-t_{3}}{t-t_{1}}},\sqrt{-{\frac{t_{2}}{t_{4}}}}{\\frac{t-t_{4}}{t-t_{2}}}}})
{\stackrel{\circ}{\omega}}{}^{a}{}_{b\mu}=h^{a}{}_{\rho}\,{\stackrel{\circ}{\nabla}}{}_{\mu}\,h_{b}{}^{\rho}\equivh^{a}{}_{\rho}\left(\partial_{\mu}h_{b}{}^{\rho}+{\stackrel{\circ}{\Gamma}}{}^{\rho}{}_{\nu\mu}h_{b}{}^{\nu}\right)
ds_{(3)}^{2}=-\bar{r}^{2}dx^{+}dx^{-}+\gamma_{--}(dx^{-})^{2}+\gamma_{++}(dx^{+})^{2}+\frac{l^{2}\bar{r}^{2}d\bar{r}^{2}}{\bar{r}^{4}-4\gamma_{++}\gamma_{--}}
S_{eff}=\Gamma_{\prime}^{\alpha\beta}h_{_{\prime\,}\alpha\beta}\left(0\right)+\frac12\intd^{4}k\Pi_{\prime}^{\mu\nu,\,\alpha\beta}\left(k\right)\,h_{\prime\,\mu\nu}\left(k\right)\,h_{\prime\,\alpha\beta}\left(-k\right)+\cdots\;\;,
{\cal{L}}_{H}=\frac{1}{2}H^{\mu\nu}\BoxH_{\mu\nu}-\frac{1}{4}H\BoxH-\frac{1}{2}H^{\mu\nu}\partial_{\mu}\partial_{\alpha}H_{\nu}^{\alpha}-\frac{1}{2}H^{\mu\nu}\partial_{\nu}\partial_{\alpha}H_{\mu}^{\alpha}+
\vec{\eta}\,(\vec{x}_{\bot})=\frac{g}{L}\vec{\nabla}_{\bot}\intd^{3}y\,d\left(\vec{x}_{\bot}-\vec{y}_{\bot}\right)\left(\epsilon^{3de}\vec{A}_{\bot}^{d}(\vec{x})\vec{E}_{\bot}^{e}(\vec{x})+\rho_{m}^{3}\left(\vec{x}\right)\right)
\mathcal{L}=-\overline{{\Psi}}(x)\left[\Gamma_{\mu}\left(\partial_{\mu}-ieA_{\mu}-gA_{\mu}^{B}I^{B}\right)+m\right]\Psi(x)-\frac14\mathcal{F}_{\mu\nu}^{2}-\frac14\left(F_{\mu\nu}^{B}\right)^{2}
\lambda(p^{\mu})\,\equiv\left(\begin{array}{c}{\left(\zeta_{\lambda}\,\Theta_{[j]}\right)\,\phi_{_L}^{\ast}(p^{\mu})}\\{\phi_{_L}(p^{\mu})}\\\end{array}\right)\,\,,\quad\rho(p^{\mu})\,\equiv\left(\begin{array}{c}{\phi_{_R}(p^{\mu})}\\{\left(\zeta_{\rho}\,\Theta_{[j]}\right)^{\ast}\,\phi_{_R}^{\ast}(p^{\mu})}\\\end{array}\right)\,\,\quad,
S\sim\tilde{\psi}Q_{o}\tilde{\psi}+g_{s}^{1/2}\tilde{\psi}^{3}+\tilde{\phi}Q_{c}\tilde{\phi}+g_{s}\tilde{\phi}^{3}+\tilde{\phi}B(g_{s}^{1/2}\tilde{\psi})+\cdots.
\mu\nabla_{\boldsymbol{v}}\boldsymbol{v}=\rho_{e}\boldsymbol{E}+\frac1c(\boldsymbol{j}_{e}\times\boldsymbol{B})+\rho_{m}\boldsymbol{B}-\frac1c(\boldsymbol{j}_{m}\times{\boldsymbol{E}}),
(G_{OM}^{-1})_{ab}={\frac{\left((2K^{2}-1)-2K^{2}\sqrt{1-K^{-2}}\right)^{\frac{1}{6}}}{K}}\,\left[\hat{\eta}_{\hat{a}\hat{b}}+\,{\frac{1}{4}}(\hat{{\calH}}^{2})_{\hat{a}\hat{b}}\right]
L(u)=\left(\begin{array}{ll}{y_{0}u^{2}+y_{2}u+y_{1}}&{y_{4}^{+}u+y_{6}^{+}}\\{y_{4}^{-}u+y_{6}^{-}}&{y_{3}}\\\end{array}\right)=\left(\begin{array}{cc}{a}&{b}\\{c}&{d}\\\end{array}\right)(u)
\Phi_{0ij}=Q(\hat{I})_{ij}+Q(\hat{K}_{L}(\hat{I}))_{ij}+\sum_{n=2}^{\infty}\frac{1}{n!}[[Q(\hat{K}_{L}(\hat{I}),\hat{K}_{L}(\hat{I})],\hat{K}_{L}(\hat{I})],\dots],]_{ij}\,
P=\left({\frac{\Gamma(1-a-b)\;\Gamma(a-b)}{\Gamma\left({\frac{1}{2}}-b-c\right)\Gamma\left({\frac{1}{2}}-b+c\right)}}\right)\;\;\mathrm{and}\;\;Q=\left({\frac{\Gamma(1-a-b)\;\Gamma(b-a)}{\Gamma\left({\frac{1}{2}}-a+c\right)\Gamma\left({\frac{1}{2}}-a-c\right)}}\right),
\left(\begin{array}{ccccccccc}{2}&{-1}&{0}&{.}&{.}&{.}&{.}&{0}&{-1}\\{-1}&{2}&{-1}&{0}&{.}&{.}&{.}&{.}&{0}\\{0}&{-1}&{2}&{-1}&{0}&{.}&{.}&{.}&{0}\\{.}&{.}&{.}&{.}&{.}&{.}&{.}&{.}&{.}\\{0}&{.}&{.}&{.}&{.}&{0}&{-1}&{2}&{-1}\\{-1}&{0}&{.}&{.}&{.}&{.}&{0}&{-1}&{2}\\\end{array}\right).
\tilde{V}_{mm^{\prime}}=\hat{V}_{mm^{\prime}}+\Phi_{m}^{(r)}(L;\{q\})[\tilde{\Omega}_{L,L^{\prime}}^{(n)}(\{q,\overline{q}\})]_{rs}^{-1}\Phi_{m^{\prime}}^{(s)}(L;\{q\})
\begin{array}{cl}{t^{w}(z,\bar{z})\Phi_{jm\bar{m}}(z^{\prime},\bar{z}^{\prime})}&{\sim(z-z^{\prime})^{-wm}(\bar{z}-\bar{z}^{\prime})^{-w\bar{m}}\Phi_{jm\bar{m}}^{w}(z^{\prime},\bar{z}^{\prime})}\\{}&{=(z-z^{\prime})^{-(m-\bar{m})w}|z-z^{\prime}|^{-2\bar{m}w}\Phi_{jm\bar{m}}^{w}~,}\\\end{array}
G=\left(\begin{array}{llccrr}{R^{2}}&{u}&{v}&{-u}&{-2v-R^{2}}&{-u}\\{u}&{R^{2}}&{u}&{v}&{-u}&{-2v-R^{2}}\\{v}&{u}&{R^{2}}&{u}&{v}&{-u}\\{-u}&{v}&{u}&{R^{2}}&{u}&{v}\\{-2v-R^{2}}&{-u}&{v}&{u}&{R^{2}}&{u}\\{-u}&{-2v-R^{2}}&{-u}&{v}&{u}&{R^{2}}\\\end{array}\right)\;,
{\calC}_{D}={\frac{Q(y_{br})}{2\sqrt{2{\calK}_{1}}}}\,,\qquad{\calC}_{R}={\frac{B(y_{br})}{2\sqrt{2|{\calK}_{2}|}}}\,,\qquad{\calC}_{G}={\frac{\Psi^{(0)}(y_{br})}{2\sqrt{2M_{5}^{3}}}}\;.
\begin{array}{ll}{L=\intd^{4}x\varepsilon^{\mu\nu\rho\sigma}\bigg[\dots+4if_{4c4d}^{ab}(C_{3}{^+}{\calG}_{\rho\sigmaab}-C_{4}{^-}{\calG}_{\rho\sigmaab})A_{\mu}^{4c}A_{\nu}^{4d}}&{}\\{+2if_{4ak}^{i}A_{\nu}^{k}(C_{3}{^+}{\calG}_{~\rho\sigma}^{j}-C_{4}{^-}{\calG}_{~\rho\sigma}^{j})A_{\mu}^{4a}C_{ij}+\dots\bigg],}&{}\\\end{array}
\sum_{m=0}^{\infty}\frac{q^{m(m+\sigma)}}{(q)_{m}}=\frac{1}{(q)_{\infty}}\sum_{j=-\infty}^{\infty}\left\{q^{10j^{2}+(1-4\sigma)j}-q^{10j^{2}+(11-4\sigma)j+3-2\sigma}\right\}\qquad\sigma=0,1,
{\calL}~=~-{\partial}_{-}\left({\frac{1}{2}}{\Omega}^{-}~{\vec{\nabla}}^{2}{\partial}_{+}{\Omega}^{+}+{\partial}_{+}k^{+}~{\Omega}^{-}\right)-{\partial}_{+}\left({\frac{1}{2}}{\Omega}^{+}~{\vec{\nabla}}^{2}{\partial}_{-}{\Omega}^{-}+{\partial}_{-}k^{-}{\Omega}^{+}\right).
Z=\sqrt{\pi}\int{\calD}G\,(detM)^{-1/2}\int{\calD}\bar{A}~exp\bigg(-\int[\alphaG_{\mu\nu}^{a}G_{a}^{\mu\nu}-G_{a}^{\mu\nu}F_{\mu\nu}^{a}(\bar{A})]dx\bigg)\delta(2\bar{A}_{\mu}^{a}+2{M^{-1}}_{\nu\mu}^{ab}\partial_{\rho}G_{b}^{\rho\nu}),
S_{1}=k\intd^{5}x\verte\vert(e_{a}^{[i}e_{b}^{j]}(F_{ij}^{(4)ab}+K_{i}^{a}K_{j}^{b}-K_{j}^{a}K_{i}^{b})+2e_{[5}^{[i}e_{b]}^{j]}(K_{i;j}^{b}-K_{j;i}^{b}))
S^{\mu\nu}(\vec{t})=\frac1{\left(2\pi\right)^{2}t^{2}}\left(e^{2i\theta}v_{\uparrow\downarrow}^{\mu}a_{\uparrow\downarrow}^{\nu}+e^{-2i\theta}v_{\downarrow\uparrow}^{\mu}a_{\downarrow\uparrow}^{\nu}\right),
G_{\sigma}^{(n)}(\zeta_{1},\cdots,\zeta_{2n})=\sum_{\varepsilon_{j}=\pm\atop\varepsilon_{1}+\cdots+\varepsilon_{2n}=0}v_{\varepsilon_{1}}\otimes\cdots\otimesv_{\varepsilon_{2n}}G_{\sigma}^{(n)}(\zeta_{1},\cdots,\zeta_{2n})^{\varepsilon_{1}\cdots\varepsilon_{2n}},~~~~(\sigma=\pm).
\int_{\Sigma}\tilde{f}Bf=\int_{\Sigma}fB\tilde{f}\:\:,\:\:\:\int_{\Sigma}\tilde{f}Df=\int_{\Sigma}fD\tilde{f}\:\:,\:\:\:\int_{\Sigma}\tilde{f}Af=-\int_{\Sigma}fC\tilde{f}
d\Omega_{D}=dC_{4}-C_{2}\wedgedb_{2}-b_{2}\wedge(dC_{2}+C_{0}db_{2}-{\sqrt{1+C_{0}^{2}}}d\tilde{\calF}^{\prime})+({\sqrt{1+C_{0}^{2}}}db_{2}-C_{0}d\tilde{\calF}^{\prime})\wedge\tilde{\calF}^{\prime}.
\begin{array}{l}{\qquad\partial_{0}x^{\mu}=-(p^{\mu}+(pP_{y})P_{y}^{\mu})-i(\bar{\theta}\Gamma^{\mu\nu}\lambda_{\theta})P_{y\nu},}\\{\qquad\partial_{0}p^{\mu}=-\partial_{1}\left[\partial_{1}x^{\mu}-i(\bar{\theta}\Gamma^{\mu\nu}\partial_{1}\theta)P_{y\nu}+i\bar{\theta}\Gamma^{\mu}\lambda_{\theta}\right],}\\{\qquad\partial_{0}\theta^{\alpha}=-\lambda_{\theta}^{\alpha},}\\\end{array}
A_{i}^{\,\,j}(L_{1},L_{2})=\prod_{u=1}^{\mathrm{P}}{\frac{1}{p_{ij}^{(u)}!}}\left(\alpha_{i\,\,;u}^{\,\,j}(L_{1})\right)^{p_{ij}^{(u)}}\prod_{v=1}^{\mathrm{Q}}{\frac{1}{q_{ij}^{(v)}!}}\left(\alpha_{i\,\,;v}^{\,\,j}(L_{2})\right)^{q_{ij}^{(v)}}\prod_{w=1}^{\mathrm{T}}{\frac{1}{t_{ij}^{(w)}!}}\left(\gamma_{i\,\,;w}^{\,\,j}(L_{1},L_{2})\right)^{t_{ij}^{(w)}}\,\,.
S=\frac{1}{4}\int\!\mathrm{d}^{4}x\,\mathrm{d}^{4}\theta\,\Phi\bar{\Phi}+\frac{1}{2}\int\!\mathrm{d}^{4}x\,\mathrm{d}^{2}\theta\,{\calW}(\Phi)+\frac{1}{2}\int\!\mathrm{d}^{4}x\,\mathrm{d}^{2}\bar{\theta}\,\bar{\calW}(\bar{\Phi})\,.
\frac{1}{i\partial_{-}-e\bar{B}_{-}}-\frac{1}{i\partial_{-}}=\left(\frac{1}{i\partial_{-}-e\bar{B}_{-}}e\bar{B}_{-}\right)*\frac{1}{i\partial}=\frac{1}{i\partial}\ast\left(e\bar{B}_{-}\frac{1}{i\partial_{-}-e\bar{B}_{-}}\right)\,
\hat{{\calJ}}_{a}=-i\epsilon_{abc}p^{b}\frac{\partial}{\partialp_{c}}+\hat{J_{a}}\qquad\hat{\calP}_{a}=p_{a}\qquad\hat{\calQ}_{\alpha}^{1}=i(p\sigma)_{\alpha\beta}\hat{\theta}^{\beta}-\hat{\pi}_{\alpha}\;.
\left.\frac{\partialU_{4}}{\partialv}\right|_{v=v_{s}}=-\frac{1}{v_{s}^{3}}+v_{s}+\frac{3uv_{s}^{3}}{16\pi^{2}}[\mathrm{ln}\frac{v_{s}^{2}u}{M^{2}}-\frac{1}{2}]=0.
0\rightarrow{\calO}_{dP_{9}}(6\sigma|_{dP_{9}}-5F)\stackrel{f_{C}}{\rightarrow}{\calO}_{dP_{9}}(9\sigma|_{dP_{9}}-F)\rightarrow{\calO}_{dP_{9}}(9\sigma|_{dP_{9}}-F)|_{C}\rightarrow0.
G^{tttt}(\omega,q)=-4{\frac{\delta^{2}S_{\epsilon}}{\delta(H_{tt}^{0})^{2}}}={\frac{3N^{2}\pi^{2}T^{4}q^{2}}{2(3\omega^{2}-q^{2})}}-{\frac{3\pi^{2}}{8}}N^{2}T^{4}=3P\frac{5q^{2}-3\omega^{2}}{3\omega^{2}-q^{2}}\,.
\stackrel{(0)}{S}=S_{0}\left[A_{\alpha\beta}^{\;\;\;(\sigma)}\right]+\intd^{D}x\left(A_{\;\;\;\;(\sigma)}^{*\alpha\beta}\partial_{\left[\alpha\right.}\eta_{\left.\beta\right]}^{\;\;\;(\sigma)}+\eta_{\;\;\;(\sigma)}^{*\alpha}\partial_{\alpha}C^{(\sigma)}\right),
\!\!\!\!\!\!\!\!H=\hat{p}^{2}+\left(W^{\prime}(\hat{q})\right)^{2}-\hbarW^{\prime\prime}(\hat{q})+E_{g}=\left(\hat{p}+iW^{\prime}(\hat{q})\right)\left(\hat{p}-iW^{\prime}(\hat{q})\right)+E_{g}
\hat{X}_{\mu}\hat{W}\equiv-\frac{\partial}{\partialp_{\nu}}\left(\int_{0}^{1}(1-\frac{1}{2}s)e^{-\frac{i(1-s)}{2}\bigtriangleup}F_{\mu\nu}\hat{W}ds+\hat{W}\int_{0}^{1}\frac{1-s}{2}e^{\frac{1}{2}is\bigtriangleup}F_{\mu\nu}ds\right)
H^{\prime}=\frac12\intd^{3}x\left(B_{a}^{ij}F_{ij}^{a}-A_{i}^{a}A_{a}^{i}+\left(\left(D^{i}\right)_{\;\;b}^{a}B_{0i}^{b}\right)\left(D_{j}\right)_{a}^{\;\;c}B_{c}^{0j}\right)\equiv\intd^{3}x\,h^{\prime},
g_{\mu\nu}^{(1)}=-{\frac{l^{2}}{2}}\left({\frac{1}{a^{2}}}-1\right)\left(R_{\mu\nu}-{\frac{1}{6}}h_{\mu\nu}R\right)-{\frac{l}{2}}\left({\frac{1}{a^{4}}}-1\right)\chi_{\mu\nu}\,
p_{N{\mathrm{(int)}}}^{(1)}(\xi_{1},\xi_{2})=\frac{A_{d}}{2\xi_{1}^{2}}\int_{0}^{\infty}dkk^{d-2}\int_{0}^{\infty}d\omega\frac{K_{\omega}^{\prime}(k\xi_{2})(1+\omega^{2}/k^{2}\xi_{1}^{2})}{K_{\omega}^{\prime}(k\xi_{1})N_{\omega}^{\prime}(k\xi_{1},k\xi_{2})}.
\frac{3S^{\prime\prime}}{S}-12\alpha\frac{S^{\prime\prime}{S^{\prime}}^{2}}{S^{3}}+12\alpha\frac{kS^{\prime\prime}}{S^{3}}+12\alpha\frac{S^{\prime\prime}\dot{S}^{2}}{N^{2}S^{3}}+..=-\rho\delta(w)
S=\intd\tau\Bigl(-m_{1}\sqrt{\dot{x}^{+}\dot{x}^{-}}-m_{2}\sqrt{\dot{x}^{+}\dot{x}_{a}^{-}}-e_{1}e_{2}\Biggl({\frac{\dot{x}^{-}}{x_{a}^{-}-x^{-}}}+{\frac{\dot{x}^{+}}{x_{a}^{+}-x^{+}}}\Biggr)\\Bigr)
S_{*}=\intd^{4}x\sqrt{|\tilde{g}|}\left[\tilde{R}-\frac{1}{2}\left(\tilde{\nabla}\varphi\right)^{2}-\frac{1}{2}\left(\tilde{\nabla}\beta\right)^{2}-\frac{1}{2}e^{2\varphi}\left(\tilde{\nabla}\sigma\right)^{2}\right]
P_{-}\lambda_{\bfg}^{i}\eta^{i}=\lambda^{i}\eta^{i}P_{-}\\\\;\\\\P_{-}(\eta^{i}-\lambda_{\bfg}^{i}\lambda_{\bfg}^{j}\eta^{j})=(\eta^{i}-\lambda_{\bfg}^{i}\lambda_{\bfg}^{j}\eta^{j})P_{+}
S=\int\left[\lambda_{A}\bar{\lambda}_{\dot{A}}(dx^{A\dot{A}}-i\theta^{A}d\bar{\theta}^{\dot{A}}+id\theta^{A}\bar{\theta}^{\dot{A}})+\lambda_{A}\lambda_{B}(dy^{AB}-i\theta^{A}d\theta^{A})+c.c.\right],
r_{v}=-\frac{1}{8\pi}\frac{\rho}{v}\left(\frac{\kappa}{4\pi}\right)^{2/3}\int_{CY_{3}}\omega\wedge\Big(\mathrm{tr}(F_{(1)}\wedgeF_{(1)})-\frac{1}{2}\mathrm{tr}(R\wedgeR)\Big)\;,
\bigg|\frac{\pi_{a}}{\pi_{1}}\bigg|^{2}=1+\frac{(M^{2}/\Lambda^{2})(\big|\pi_{1}\big|^{2}-\big|\pi_{a}\big|^{2})}{G_{S}W_{\mathrm{eff};\lambda}^{\ast}\left(1-\alpha\right)+\betaX_{B}+(M^{2}/\Lambda^{2})\big|\pi_{a}\big|^{2}}.
\begin{array}{c}{\displaystyle{z_{0}^{-1}\delta\left(\frac{z_{1}-z_{2}}{z_{0}}\right)Y(u,z_{1})Y(v,z_{2})-[v_{g},v_{h}]z_{0}^{-1}\delta\left(\frac{z_{2}-z_{1}}{-z_{0}}\right)Y(v,z_{2})Y(u,z_{1})}}\\{\displaystyle{=z_{2}^{-1}\delta\left(\frac{z_{1}-z_{0}}{z_{2}}\right)Y(Y(u,z_{0})v,z_{2})}}\\\end{array}
\frac{1}{[(k+k_{i})^{2}-m^{2}]}=\sum_{j=0}^{N}\frac{\left(-1\right)^{j}\left(k_{i}^{2}+2k_{i}\cdotk\right)^{j}}{\left(k^{2}-m^{2}\right)^{j+1}}+\frac{\left(-1\right)^{N+1}\left(k_{i}^{2}+2k_{i}\cdotk\right)^{N+1}}{\left(k^{2}-m^{2}\right)^{N+1}\left[\left(k+k_{i}\right)^{2}-m^{2}\right]},
\mathrm{Tr}\,K_{n}(s)=-\frac{s}{(4\pis)^{d/2}}\intdx\,\underbrace{V\frac{1}{\Box}V\frac{1}{\Box}...V\frac{1}{\Box}}_{n-1}\,V(x)+O\left(\frac{1}{s^{d/2}}\right).
\alpha_{ln}=\left(\frac{\omega_{ln}^{\mathrm{out}}}{\omega_{ln}^{\mathrm{in}}}\right)^{1/2}\frac{\Gamma(1-\imath\omega_{ln}^{\mathrm{in}}t_{0})\Gamma(-\imath\omega_{ln}^{\mathrm{out}}t_{0})}{\Gamma(-\imath\omega_{ln}^{+}t_{0})\Gamma(1-\imath\omega_{ln}^{+}t_{0}))},
\Psi(x,z,n)\sim\prod_{\alpha<\beta}^{M}(z_{\alpha}-z_{\beta})^{\frac{\kappa^{2}}{\lambda}}\prod_{i<j}^{N}(x_{i}-x_{j})^{\lambda}\prod_{i,\alpha=1}^{N,M}(x_{i}-z_{\alpha})^{\kappa}L_{n}^{-2\mu_{\kappa}-1}(2\omega{\calT}_{-}).
{\calL}=\sqrt{(1-B_{3}^{2})+(1-b_{1}^{2})\phi_{\tau}^{2}-2b_{1}B_{2}\phi_{\tau}\phi_{x}+(1-B_{2}^{2})\phi_{x}^{2}+2b_{1}B_{3}\phi_{\tau}+2B_{2}B_{3}\phi_{x}}.
S_{B}[A]\;=\;i\,\frac{1}{\eta}\,S_{CS}[A]\;+\;R[\widetilde{F}]+\frac{i}{\theta}\intd^{3}xd^{3}yj_{\mu}^{T}(x){\mathcalK}_{\mu\nu}(x-y)j_{\nu}^{T}(y)\;,
\begin{array}{c}{\partial_{+}(G\,\,\partial_{-}u)+\partial_{-}(G\,\,\partial_{+}u)=\frac{\partialG}{\partialv}(\partial_{+}u\partial_{-}v+\partial_{+}v\partial_{-}u)}\\{\partial_{+}(G\,\,\partial_{-}v)+\partial_{-}(G\,\,\partial_{+}v)=\frac{\partialG}{\partialu}(\partial_{+}u\partial_{-}v+\partial_{+}v\partial_{-}u)}\\{\partial_{+}u\partial_{+}v=0=\partial_{-}u\partial_{-}v\,\,\,.}\\\end{array}
G^{(2,2,2,2)}(1\vert2\vert3\vert4)={\frac{(12)^{2}(34)^{2}}{\hat{x}_{12}^{4}\hat{x}_{34}^{4}}}a(\hat{s},\hat{t})+{\frac{(14)^{2}(23)^{2}}{\hat{x}_{14}^{4}\hat{x}_{23}^{4}}}b(\hat{s},\hat{t})+{\frac{(12)(23)(34)(41)}{\hat{x}_{12}^{2}\hat{x}_{23}^{2}\hat{x}_{34}^{2}\hat{x}_{41}^{2}}}c(\hat{s},\hat{t})\;.
\vec{r}^{\;(3)}=c^{(3)}\left(\begin{array}{c}{1}\\{.}\\{.}\\{.}\\{1}\\{-(N-2)}\\{0}\\\end{array}\right)\;,........\;,\;\vec{r}^{\;(N)}=c^{(N)}\left(\begin{array}{c}{\;1}\\{-1}\\{0}\\{.}\\{.}\\{0}\\{0}\\\end{array}\right)\;,
\tilde{\nabla}_{\delta\dot{\delta}}Z={\calD}_{\delta\dot{\delta}}\left(\begin{array}{c}{\omega^{\alpha}}\\{\pi_{\dot{\alpha}}}\\\end{array}\right)+i\left(\begin{array}{cc}{0}&{\delta_{\delta}^{\alpha}\delta_{\dot{\delta}}^{\dot{\beta}}}\\{{\calP}_{\delta\dot{\delta},\beta\dot{\alpha}}}&{0}\\\end{array}\right)\left(\begin{array}{c}{\omega^{\beta}}\\{\pi_{\dot{\beta}}}\\\end{array}\right)=0,
\langle{\calO}_{1}(x_{1})...{\calO}_{1}(x_{r}){\calO}_{2}(x_{r+1})...{\calO}_{2}(x_{r+r^{\prime}})\rangle=\langle{\calO}\rangle_{\Omega}^{r}\langle{\calO}\rangle_{\Omega^{\prime}}^{r^{\prime}}\langle1\rangle,
\left(1-\frac{wN}{6}e^{2\phi}\right)\left(\dot{\phi}^{\prime}-\dot{\rho}\phi^{\prime}-\rho^{\prime}\dot{\phi}\right)-\dot{\phi}\phi^{\prime}=\frac{N}{12}e^{2\phi}\left[\dot{Z}Z^{\prime}+2\dot{Z}^{\prime}-2\left(\dot{\rho}Z^{\prime}+\rho^{\prime}\dot{Z}\right)\right].
H_{m}^{M}=H_{m}^{L}+H_{m}^{R}=\frac{1}{2}m_{L}\intd^{3}{x}(\bar{\psi}_{L}\psi_{L}^{c}+h.c)+\frac{1}{2}m_{R}\intd^{3}{x}(\bar{\psi}_{R}\psi_{R}^{c}+h.c).
y^{\delta}\tilde{\Gamma}^{\alpha}{}_{\beta\delta}\tilde{B}^{\beta}|_{1+3}=y^{\delta}dx^{\sigma}[\Gamma^{\alpha}{}_{\beta\delta}B^{\beta}{}_{,\sigma}+\Gamma^{\alpha}{}_{\beta\delta}\Gamma^{\beta}{}_{\sigma\tau}B^{\tau}+\Gamma^{\alpha}{}_{\beta\delta,\sigma}B^{\beta}].
{\calL}^{+4}(x_{h}^{+},u^{+},u^{-})={\calL}_{normal}^{+4}(x_{h}^{+},u^{-})+\sum_{n,m}u^{+a_{1}}\dotsu^{+a_{n}}u^{-b_{1}}\dotsu^{-b_{m}}f_{(a_{1}\dotsa_{n}b_{1}\dotsb_{m})}^{4-n+m}(x_{h}^{+})\,
H^{(1)}=H+\lambda_{a}\Phi_{a}^{(1)},\,\,\,\mathrm{where}\,\,\,\,H=\left.\left(\frac{\partial_{r}L}{\partial\dot{q}}\dot{q}-L\right)\right|_{\frac{\partial_{r}L}{\partial\dot{q}}=P},\,\,\,\,\,\,q=(x,e,\chi,\psi^{n})\,\,\,,
\breve{\Phi}(\omega)\simeq{\frac{2\omega^{D-2}}{(4\pi)^{(D-1)/2}}}\sum_{n=0}^{\infty}\left[{\frac{\breve{c}_{n}}{\Gamma\left({\frac{D-1}{2}}-n\right)}}\omega^{-2n}+{\frac{\breve{d}_{n}}{\Gamma\left({\frac{D-2}{2}}-n\right)}}\omega^{-(2n+1)}\right]~~~,
{\calG}_{k}(U_{*},U_{*})=-i\frac{g_{YM}^{4}}{\sqrt{\lambda}k}U_{*}^{\frac{1}{2}(p-9)}\frac{H_{\alpha}^{(1)}\left(i(\alpha-1)\sqrt{\lambda}kU_{*}^{\frac{1}{2}(p-5)}\right)}{H_{\alpha-1}^{(1)}\left(i(\alpha-1)\sqrt{\lambda}kU_{*}^{\frac{1}{2}(p-5)}\right)}~,
\langle\Phi^{*}(x)\Phi(y)\rangle\sim\langle\mathrm{e}^{-\mathrm{i}\theta(x)}\mathrm{e}^{\mathrm{i}\theta(y)}\rangle\sim\mathrm{e}^{-2\piG(x-y)/N}\sim\frac{1}{|x-y|^{1/N}}\.
\left[\hat{x}^{\mu},\hat{\pi}_{\nu}\right]_{-}=\imath\left\{x^{\mu},\pi_{\nu}\right\}_{D(\Phi_{3}^{(1)})}=\imath\delta_{\nu}^{\mu}\;,\;\;\left[\hat{\psi}^{\mu},\hat{\psi}^{\nu}\right]_{+}=\imath\left\{\psi^{\mu},\psi^{\nu}\right\}_{D(\Phi_{3}^{(1)})}=-\frac{1}{2}\eta^{\mu\nu}\;.