harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Omega_{0}\,=\,\left(\begin{matrix}{{\frac{1}{\sqrt{2}}}}\\{{\bf0}}\\{-{\frac{\mathrm{i}}{\sqrt{2}}}}\\{{\bf0}}\\\end{matrix}\right)\qquad{\hat{\Omega}}\,=\,\left(\begin{matrix}{0}\\{Y^{I}}\\{0}\\{F_{J}}\\\end{matrix}\right)\qquad\Omega_{3}\,=\,\left(\begin{matrix}{{\hat{Y}}_{0}}\\{{\bf0}}\\{{\hat{F}}_{0}}\\{{\bf0}}\\\end{matrix}\right)
S=-\frac{1}{2\pi}\intd^{2}\sigma\left[{\partial}_{\alpha}X^{\mu}{\partial}^{\alpha}X_{\mu}-i\bar{\psi}^{\mu,j}\rho^{\alpha}{\partial}_{\alpha}\psi_{\mu,j}-i\phi^{\mu,k}\rho^{\alpha}{\partial}_{\alpha}\phi_{\mu,k}\right]\;,
\delta_{(I)}^{I_{1}I_{2}\cdotsI_{n}}=\sum_{I_{1}^{\prime}I_{2}^{\prime}\cdotsI_{n}^{\prime}=(I)}\delta_{I_{1}^{\prime}}^{I_{1}}\delta_{I_{2}^{\prime}}^{I_{2}}\cdots\delta_{I_{n}^{\prime}}^{I_{n}}.
\big[\big(C_{1}^{[r]}\big)_{2V,2V}\big]_{\Lambda=0}=\frac{1}{P^{2}}\Big\{\sum_{n=1}^{r}B_{n}^{[r]}(\Lambda=0)\frac{t(P)}{u_{r,2n+1}}B_{n}^{[r]}(\Lambda=0)\Big\}.
S_{E}={\frac{1}{2\kappa_{D}^{2}}}\intd^{D}x\sqrt{-g}\left[{\calR}-{\frac{4}{D-2}}(\partial\phi)^{2}-{\frac{1}{2\cdot(p+3)!}}e^{2a_{p+1}\phi}F_{p+3}^{2}\right]
c^{(0)}=\frac16~,~c^{(1/2)}=-{\frac{N^{(1/2)}}{12}}~,~c^{(1)}={\frac{N^{(1)}}{6}}-1~,~c^{(3/2)}=-{\frac{N^{(3/2)}}{12}}~,~c^{(2)}={\frac{N^{(2)}}{6}}-(d+2)~.
\rho_{Q}(R)\approx{\frac{1}{2\kappa_{5}^{4}|\lambda_{0}|}}\left[\kappa_{5}^{4}\lambda_{0}^{2}-18(\xi_{+}^{2}+\xi_{-}^{2})+{\frac{162}{\kappa_{5}^{4}\lambda_{0}^{2}}}\left(1-9{\frac{\xi_{+}^{2}+\xi_{-}^{2}}{\kappa_{5}^{4}\lambda_{0}^{2}}}\right)(\xi_{+}^{2}-\xi_{-}^{2})^{2}\right]R^{-{\frac{k^{2}}{3}}}.
\langle\Phi_{1}^{\ell}\Phi_{2}^{\ell^{\prime}}\rangle=\frac{\sqrt{(2\ell+1)(2\ell^{\prime}+1)}}{4\pi^{2}}\int_{|t|<1}\frac{dtd\bar{t}}{(1-|t|^{2})^{2}}\int_{-2\pi}^{2\pi}d\psi\\Phi_{1}^{\ell}\bar{\Phi}_{2}^{\ell^{\prime}}\,,
e_{0}=\left(\begin{array}{r}{0}\\{1}\\{1}\\\end{array}\right),\quade_{1}=\frac{1}{\nu}\left(\begin{array}{r}{0}\\{0}\\{1}\\\end{array}\right),\quade_{2}=\frac{1}{\nu}\left(\begin{array}{r}{1}\\{0}\\{0}\\\end{array}\right).
\bar{z}^{2}={\frac{(p^{2}z^{1}-q_{3})}{(p^{0}z^{1}-p^{1})}}\,\quad\mathrm{~and~}\quad\bar{z}^{3}={\frac{(p^{3}z^{1}-q_{2})}{(p^{0}z^{1}-p^{1})}}\.
\Gamma_{+}\,C^{-1}\,\tilde{\Gamma}_{+}=(U_{3}+\sum_{1}^{2}A^{(r)}U_{r}A^{(r)T})\,C^{-1}\,(U_{3}^{-1}+\sum_{1}^{2}A^{(s)}U_{s}^{-1}A^{(s)T})
\langle0|[a_{0},a_{n}]_{4}a_{n}^{\dagger}|0\rangle\hbar^{4}=-\Bigl({\frac{6}{\tilde{g}}}\Bigr)^{4}\Bigl({\frac{7}{3n^{4}}}+{\frac{5\zeta(2)}{6n^{2}}}+{\frac{\zeta(3)}{2n}}\Bigr)\sigma\hbar^{5},
R_{2}^{1}=-2i\frac{dg_{1}\wedged\overline{{g}}_{1}}{(1-g_{1}\overline{{g}}_{1})^{2}},~~~R_{4}^{3}=-2i\frac{dg_{2}\wedged\overline{{g}}_{2}}{(1-g_{2}\overline{{g}}_{2})^{2}},
x_{s}^{({\vec{\eta}}_{+})\mu}(T_{s})=x_{s}^{\mu}(0)+[u^{\mu}(p_{s})+{\frac{1}{N}}\sum_{i=1}^{N}{\frac{\kappa_{ir}}{\sqrt{m_{i}^{2}+{\vec{\kappa}}_{i}^{2}}}}\epsilon_{r}^{\mu}(u(p_{s}))]T_{s}.
g_{\mu\nu}=\frac{\eta_{\mu\nu}}{r},\quadg_{\mur}=-\frac{\eta_{\mu\nu}x^{\nu}}{r^{2}},\quadg_{rr}=\frac{x^{\mu}x^{\nu}\eta_{\mu\nu}}{r^{3}}+\frac{1}{4r^{2}},
\begin{array}{lcr}{u^{{\pm}{\alpha}}=\epsilon^{{\alpha}{\beta}}u_{\beta}^{\pm};\qquad\overline{{u}}^{+\alpha}=u_{\alpha}^{-};\qquad\epsilon_{\alpha\beta}=-\epsilon_{\beta\alpha}}\\{u^{+\alpha}u_{\alpha}^{-}=1,\qquadu^{+\alpha}u_{\alpha}^{+}=u^{-\alpha}u_{\alpha}^{-}=0.}\\\end{array}
H=\frac{1}{2m}(\frac{4}{u^{2}+v^{2}})[p_{u}^{2}+\frac{1}{u^{2}}p_{\varphi}^{2}+p_{v}^{2}+\frac{1}{v^{2}}p_{\varphi}^{2}]-\frac{4}{u^{2}+v^{2}}e^{2}
\deltaR=\nabla^{2}\left[\left(\overline{{\nabla}}^{2}+\frac{1}{3}\overline{{R}}\right)\overline{{U}}-\frac{3}{4}\nabla_{A\dot{A}}\chi^{A\dot{A}}-3i\left(\nabla_{A}\nabla_{\dot{A}}-\nabla_{\dot{A}}\nabla_{A}\right)\chi^{A\dot{A}}\right]+\ldots
D_{q}^{0}[\lambda_{0}]=\prod_{\alpha\in\Phi_{0}^{+}}\frac{q^{(\lambda_{0}+\rho_{0},\alpha)}-q^{-(\lambda_{0}+\rho_{0},\alpha)}}{q^{(\rho_{0},\alpha)}-q^{-(\rho_{0},\alpha)}}
\dot{\hat{\theta}}^{\pm}=e^{-\frac{\psi_{0}}{2}}e^{\mp\frac{\sqrt{6}}{2}N\;I(a)}(-\frac{9\gammaM}{2a^{3\gamma}}-\frac{9(\alpha+\frac{1}{2})N^{2}}{a^{6}}+\frac{3k}{a^{2}}\pm\frac{6\sqrt{6}N}{a^{3}}\sqrt{\frac{M}{a^{3\gamma}}+\frac{\alphaN^{2}}{a^{6}}-\frac{k}{a^{2}}}),
V(r)\approx\frac{3E\rho_{*}^{4}}{2r^{2}(r^{2}+l^{2})}+\frac{L^{2}}{2m_{p}r^{2}}\left(1-\frac{2\rho_{*}^{4}}{r^{2}(r^{2}+l^{2})}\right).
\gamma_{m,2}(g_{c})~=~-\,\frac{6(\eta_{1}^{\mathrm{o}})^{2}}{(\mu-2)^{2}(2\mu-1)}\left[\mu(\mu-1)\hat{\Theta}-\frac{\mu}{(\mu-1)}+\frac{1}{\mu}+\frac{4\mu}{3}-2\right]
\delta_{v}j^{i_{1}\ldotsi_{s}}=[D-2+s](v\cdotx)j^{i_{1}\ldotsi_{s}}+\sum_{p=1}^{s}(v^{i_{p}}x_{k}-x^{i_{p}}v_{k})j^{i_{1}\ldotsi_{p-1}k\;i_{p+1}\ldotsi_{s}}
\mu^{\prime}=\frac{1}{2}Nx^{2}\phi^{2}+e^{2\gamma\phi}\left[N(K^{2}+H^{2})+\frac{1}{8x^{2}}\left(\left(K^{2}+H^{2}-4\right)^{2}+12K^{2}H^{2}\right)\right]\,
\left[{\frac{dn_{s}(\omega,D)}{d\omega}}\right]_{\mathrm{div}}={\frac{\Gamma\left(1-\fracD2\right)}{(4\pi)^{D/2}}}{\frac{m^{D-4}}{\kappa}}\int_{\Sigma}\left[2\left(m^{2}-\left(\frac16-\xi\right)R\right)-{\frac{\omega^{2}}{\kappa^{2}}}P\right]~~~,
S_{\sigma_{i}\sigma_{j}}^{\sigma_{j}\sigma_{i}}\equivu_{1}=\frac{1}{2}(S^{+}+S^{-}),\qquadS_{\sigma_{i}\sigma_{j}}^{\sigma_{i}\sigma_{j}}\equivu_{2}=\frac{1}{2}(S^{+}-S^{-}),
\overline{{\Psi}}^{\prime}=\overline{{\Psi}}\Gamma\;\;\mathrm{or}\;\;\overline{{\Psi}}^{\prime\prime}=\overline{{\Psi}}S_{F}\qquad\qquad\Psi^{\prime}=\Gamma\Psi\;\;\mathrm{or}\;\;\Psi^{\prime\prime}=S_{F}\Psi
T=\left(\begin{array}{cc}{n+1}&{0}\\{0}&{n}\\\end{array}\right)\quad,\quadq^{H}=\left(\begin{array}{cc}{1+(q-1)x^{1+{\frac{m}{2}}}D_{q}x^{-{\frac{m}{2}}}}&{0}\\{0}&{1+(q-1)x^{1+{\frac{n}{2}}}D_{q}x^{-{\frac{n}{2}}}}\\\end{array}\right)
\begin{array}{c}{U_{+}\left|\alpha_{1},\alpha_{2},...,\alpha_{2N},t\right\rangle=\left|\alpha_{1}^{\prime},\alpha_{2}^{\prime},...,\alpha_{2N}^{\prime},t+a/2\right\rangle}\\{U_{-}\left|\alpha_{1}^{\prime},\alpha_{2}^{\prime},...,\alpha_{2N}^{\prime},t+a/2\right\rangle=\left|\alpha_{1}^{\prime\prime},\alpha_{2}^{\prime\prime},...,\alpha_{2N}^{\prime\prime},t+a\right\rangle}\\\end{array}
\eta=\frac{1}{2}\left\langle\sum_{X_{i}\in({\itG}_{1})_{X_{i^{\prime}}^{\prime}}}\sum_{\stackrel{i_{\alpha_{1}}<j_{\alpha_{1}}<i_{\alpha_{2}}<j_{\alpha_{2}}}{\left\{i_{\alpha_{1}},j_{\alpha_{1}},i_{\alpha_{2}},j_{\alpha_{2}}\right\}\inJ_{X_{i}}}}t_{i_{\alpha_{1}}}t_{j_{\alpha_{1}}}{\bf1}_{(w(t_{i_{\alpha_{1}}})=w(t_{j_{\alpha_{1}}}))}\right.\times
\mathcal{D}_{g}\left(\begin{array}{ccc}{p_{1}}&{\ldots}&{p_{N}}\\{r_{1}}&{\ldots}&{r_{N}}\\\end{array}\right)=\sum_{q\in\mathcal{I}}S_{0q}^{2-2g}\prod_{i=1}^{N}\frac{\Lambda_{qp_{i}}(r_{i})}{S_{0q}}
0=\left(\partial_{\beta}T^{(\beta\gamma)\nu}+\partial_{\beta}T^{(\beta\nu)\gamma}\right)+\sqrt{\frac{\left(e^{2}-a\right)}{a}}\frac{1}{2}\left(\epsilon^{\beta\gamma\kappa\lambda}\partial_{\beta}T_{(\kappa\lambda)}{}^{\nu}+\epsilon^{\beta\nu\kappa\lambda}\partial_{\beta}T_{(\kappa\lambda)}{}^{\gamma}\right):=S^{\{\gamma\nu\}},
\mathrm{H}\hat{\psi}(z_{1},...,z_{n})\equiv\sum_{i<k}\frac{1}{z_{i}-z_{k}}\int_{z_{k}}^{z_{i}}\mathrm{d}s\hat{\psi}(z_{1},...,s,...,s,...,z_{n})=\lambda\hat{\psi}(z_{1},...,z_{n}).
c=\frac{1}{4\pi}\sum_{j,m}\left(2j+1\right)\left(\begin{array}{c}{c_{+}\left(j,m;r\right)D_{+1,m}^{j}\left(\theta,\varphi\right)}\\{c_{0}\left(j,m;r\right)D_{0,m}^{j}\left(\theta,\varphi\right)}\\{c_{\tilde{o}}\left(j,m;r\right)D_{\tilde{o},m}^{j}\left(\theta,\varphi\right)}\\{c_{-}\left(j,m;r\right)D_{-1,m}^{j}\left(\theta,\varphi\right)}\\\end{array}\right)
\Psi_{\mathrm{gaugefixing}}=\int_{M}tr(\overline{{c}}\partial_{\mu}A^{\mu}+\frac{1}{2}\overline{{c}}b+c[\overline{{\chi}}_{\mu\nu},\overline{{\chi}}^{\mu\nu}]+\overline{{c}}[\phi,\overline{{\phi}}]).
S_{\mathrm{g.f.}}=\int_{M_{4}}\delta_{B}\{\astb\wedgeD_{0}^{\ast}\Sigma_{f}^{k}+\ast\bar{\lambda}^{k}\wedgeD_{2}\ast\pi_{k}-\ast\beta\wedgeD_{0}^{\ast}\Psi^{k}-\ast\bar{\eta}^{k}\wedgeD_{2}\ast\chi_{k}\}.
A(0,t)=\sum_{n=0}^{\infty}\frac{\rho_{n}(s)}{t-n-\frac{1}{2}}-\sum_{n=0}^{\infty}\left(\frac{r_{n}(t)}{s-n-\frac{1}{2}}+\frac{r_{n}(t)}{n+\frac{1}{2}}\right)\equiv\Psi_{1}(s,t),\\\\\\(s\sim-1,\t\sim0).
O_{M}=\partial_{+}X_{L}^{i_{1}}(\sigma_{+}^{1})\dots\partial_{+}X_{L}^{i_{k}}(\sigma_{+}^{k})\cdot\partial_{-}X_{R}^{j_{1}}(\sigma_{-}^{1})\dots\partial_{-}X_{R}^{j_{q}}(\sigma_{-}^{q})
\begin{aligned}{R(\widehat\omega)_{\mu\nurs}R(\widehat\omega)^{\mu\nu}{}_{tu}\,\epsilon^{rstu}}&{{}=\Big[R_{\mu\nu\rho\sigma}R_{\kappa\lambda}{}^{\rho\sigma}+2\,D_{\mu}A_{\nu}D_{\kappa}A_{\lambda}\Big]\,\frac{1}{\sqrt{g}}\,\epsilon^{\mu\nu\kappa\lambda}}\\{}&{{}\quad-4\Big[D^{\mu}A_{\mu}A^{\nu}A_{\nu}-D_{\mu}A_{\nu}A^{\mu}A^{\nu}\Big]\,-\,8\,D_{\mu}A_{\nu}R^{\mu\nu}\,.}\\\end{aligned}
F_{\mu\nu}^{\;\,m}=\mathbf{f}_{\mu}(A_{\nu}^{m})-\mathbf{f}_{\nu}(A_{\mu}^{m})+t_{\mu\nu}^{\;\;\rho}A_{\rho}^{m}-C_{\mu\nu}^{\;\;\rho}A_{\rho}^{m}.
\sum_{\mu\in\Lambda}\gamma_{\mu}\langle\mu|:e^{B_{1}(z)}:e^{A_{1}(z)}|\chi\rangleV_{0}(\psi,z)=V_{1}(\psi,w-z)\sum_{\mu\in\Lambda}\gamma_{\mu}\langle\mu|:e^{B_{1}(z)}:e^{A_{1}(z)}|\chi\rangle\,,
\Pi_{\vec{k}}\rightarrow\widetilde\Pi_{\vec{k}}=k\Psi_{\vec{k}}\,,~~~~~\Psi_{\vec{k}}\rightarrow\widetilde{\Psi}_{\vec{k}}=-k^{-1}\Pi_{\vec{k}}\,,~~~~~S\rightarrow\widetildeS=S^{-1},
{\frac{\sqrt{2}\vartheta_{2}^{2}\widetilde\vartheta_{3}}{\vartheta_{3}}}+\sqrt{\vartheta_{3}^{2}\widetilde\vartheta_{3}^{2}+\vartheta_{2}^{2}\widetilde\vartheta_{4}^{2}-\vartheta_{4}^{2}\widetilde\vartheta_{2}^{2}}=0.
\frac{\lambda^{\prime}}{6}\left(\vec{\varphi}^{\,(0)}(X_{0})\right)^{2}\varphi_{i}^{(0)}(X_{0})\to\frac{\lambda^{\prime}}{6}\langle\left(\vec{\varphi}^{\,(0)}(X_{0})\right)^{2}\rangle_{\zeta}\varphi_{i}^{(0)}(X_{0})+\frac{\lambda^{\prime}}{3}\langle\varphi_{j}^{\,(0)}(X_{0})\varphi_{i}^{\,(0)}(X_{0})\rangle_{\zeta}\varphi_{j}^{(0)}(X_{0})
ds_{(\;,1)}^{2}=-\int_{s_{2}}^{s_{3}}\sum_{a,b=1}^{8}\sum_{\mu}{y}_{\mu}^{a}{y}_{\mu}^{b}ds\,\rho_{a}\rho_{b}=-\int_{s_{2}}^{s_{3}}\sum_{\mu}d{t}_{\mu}d{t}_{\mu}.
\begin{array}{rl}{A_{\lambda}^{(nl)}(x|a_{1},\cdots,a_{n}|b_{1},\cdotsb_{l}):=}&{\displaystyle\prod_{j=1}^{n}(x-a_{j}\tau)f_{\lambda}^{(nl)}(x|a_{1},\cdots,a_{n}|b_{1},\cdots,b_{l})}\\{}&{+\tau^{2(l-n+\lambda-1)}\displaystyle\prod_{i=1}^{l}(x-b_{i}\tau^{-1})g_{\lambda}^{(n)}(x|a_{1},\cdots,a_{n}).}\\\end{array}
C=k\frac{2\pi\kappa_{B}}{G_{k}}\frac{\Omega_{d-2}}{\Sigma_{d-2}}r_{+}^{d-2k}\left(\gamma+\frac{r_{+}^{2}}{l^{2}}\right)^{k-1}\left[\frac{r_{+}^{2}+\gammar_{c}^{2}}{r_{+}^{2}-\gammar_{c}^{2}}\right],
H_{4}^{osc}=4r\left\{\frac{\vec{p}^{~2}}{2M}+\frac{(x_{1}p_{2}-x_{2}p_{1})zp_{\alpha}}{Mr(x_{1}^{2}+x_{2}^{2})}\right\}+\frac{1}{2}M\Omega^{2}r
d{*}d\Phi=-\frac{g_{s}e^{-\Phi}}{2}H_{3}\wedge{*}H_{3}+\frac{3g_{s}^{1/2}e^{3\Phi/2}}{4}F_{2}\wedge{*}F_{2}+\frac{g_{s}^{3/2}e^{\Phi/2}}{4}\tilde{F}_{4}\wedge{*}\tilde{F}_{4}
(F_{\mu\nu})_{+1}^{(+)}=+(F_{\mu\nu})_{-1}^{(-)}\,,\,(F_{\mu\nu})_{-1}^{(+)}=+(F_{\mu\nu})_{+1}^{(-)}\,,\,(F_{\mu\nu})_{0}^{(+)}=-(F_{\mu\nu})_{0}^{(-)}\quad.
\Gamma_{(2n)}^{0}=\Gamma_{(2n)}^{0\,T}=\Gamma_{(2n)}^{0+},\quad\Gamma_{(2n)}^{i}=(-1)^{i}\Gamma_{(2n)}^{i\,T}=-\Gamma_{(2n)}^{i\,+},\quadi=1,\dots,2n-1.
|Y_{[f,\psi][0][I]}^{\mathrm{E},t}|\leqN_{[f,\psi][f,\psi][I]}^{\mathrm{E},t}\;\;\;,\;\;\;Y_{[f,\psi][0][I]}^{\mathrm{E},t}=N_{[f,\psi][f,\psi][I]}^{\mathrm{E},t}\;\;\;\;\;\;\mathrm{mod}\;2\;\;\;,
\Psi(r)\;=\;\frac{1}{N^{2}}\,\sum_{-\frac{N}{2}}^{+\frac{N}{2}}\,\left[b_{\pm}(n\mid\varphi)\,u_{\pm}(n\mid\varphi)\,+\,d_{\pm}^{\dagger}(n\mid\varphi)\,v_{\pm}(n\mid\varphi)\right]\,e^{2\piian\cdotr}\;.
d\bar{s}_{\mathrm{string}}^{2}=e^{f+\varphi}\left(-d\xi^{2}+dx^{2}\right)+\frac{\xie^{\varphi}}{d^{2}+c^{2}\xi^{2}e^{2\varphi}}\left(e^{p}dy^{2}+e^{-p}dz^{2}\right)
<\psi_{1}|\psi_{2}>=\int\,d{[A_{\bar{z}},A_{z}]}\:\mathrm{exp}(-\frac{k}{\pi}\int\,d^{2}xA_{\bar{z}}A_{z})\:\overline{{\psi_{1}[A_{\bar{z}}]}}\:\psi_{2}[A_{\bar{z}}].
\Theta(x,\vec{k}_{\!\perp}):\qquad\qquad{\frac{m_{1}^{2}+\vec{k}_{\!\bot}^{\,2}}{x}}+{\frac{m_{2}^{2}+\vec{k}_{\!\bot}^{\,2}}{1-x}}\leq(m_{1}+m_{2})^{2}+\Lambda^{2}.
n_{\mu}F^{\mut^{\prime}}=\frac{A\epsilon_{E}^{1/2}G^{\prime}(\xi_{3})^{1/2}\widehat{B}_{E}}{2L\lambda^{3}}\left[1+\frac{G^{\prime\prime}(\xi_{3})}{4G^{\prime}(\xi_{3})}\epsilon_{E}(2\chi+1)\right],
K_{2}(z_{2})W_{2\,1}^{\Omega_{z_{2}},\Omega_{-z_{1}}}K_{1}(z_{1})W_{1\,2}^{\Omega_{z_{1}},\Omega_{z_{2}}}=W_{2\,1}^{\Omega_{-z_{2}},\Omega_{-z_{1}}}K_{1}(z_{1})W_{1\,2}^{\Omega_{z_{1}},\Omega_{-z_{2}}}K_{2}(z_{2}),
\vert\vec{p}_{1},\vec{p}_{2},\cdot\cdot\cdot,\vec{p}_{n}\rangle_{x^{+}}=a^{\dagger}(\vec{p}_{1},x^{+})a^{\dagger}(\vec{p}_{2},x^{+})\cdot\cdot\cdota^{\dagger}(\vec{p}_{n},x^{+})\vert0\rangle
\ddot{r}={\frac{1}{2B^{2}}}\left(-{\frac{(A^{2})^{\prime}}{A^{4}}}+{\frac{(\tilde{B}^{2}r^{2})^{\prime}\ell^{2}}{\tilde{B}^{4}r^{4}}}-(B^{2})^{\prime}\dot{r}^{2}\right)
Z_{EH}={\frac{1}{4}}{\frac{1}{(\eta\bar{\eta})^{4}}}\intdp_{x}dp_{y}\,\,q^{p_{x}\bar{p}_{x}+p_{y}\bar{p}_{y}}\sum_{i}\left|{\frac{\theta_{i}}{\eta}}\right|^{4}+{\frac{1}{4}}\sum_{i,j}\left|{\frac{\theta_{i}}{\theta_{j}}}\right|^{4}\,\,.
H_{QED}={\frac{1}{2}}\,{\frac{1}{Z_{3}(\mu)}}E^{2}+{\frac{1}{2}}\,Z_{3}(\mu)B^{2}+Z_{2}(\mu)\bar{\psi}\;i\;{\vec{\gamma}}\cdot{\vec{\partial}}\psi+Z_{1}(\mu)\,e_{R}\bar{\psi}A\cdot\gamma\psi\cdots
\bar{\Pi}^{\mu\nu,\alpha\beta}\left(k\right)\equiv\Pi^{\mu\nu,\alpha\beta}\left(k\right)-\frac14\left(\eta^{\mu\alpha}\Pi^{\rho}{}_{\rho,}{}^{\nu\beta}+\eta^{\mu\beta}\Pi^{\rho}{}_{\rho,}{}^{\nu\alpha}+\eta^{\nu\alpha}\Pi^{\rho}{}_{\rho,}{}^{\mu\beta}+\eta^{\nu\beta}\Pi^{\rho}{}_{\rho,}{}^{\mu\alpha}\right),
\int_{{\calM}_{\alpha}}R^{\mu\nu\lambda\rho}R_{\mu\nu\lambda\rho}=\alpha\int_{{\calM}_{\alpha=1}}R^{\mu\nu\lambda\rho}R_{\mu\nu\lambda\rho}+8\pi(1-\alpha)\int_{\Sigma}R_{\mu\nu\lambda\rho}n_{i}^{\mu}n_{i}^{\lambda}n_{j}^{\nu}n_{j}^{\rho}+O((1-\alpha)^{2})~~~,
\xi_{k}^{1}(z)^{d}=\frac{k}{\frac{1}{2}+M}z^{\frac{1+M}{2}}(1-z)^{\frac{M}{2}}F(\frac{1}{2}+M-|k|,\frac{1}{2}+M+|k|,\frac{3}{2}+M;z)
ds^{2}=\ell^{2}\frac{dr^{2}}{r^{2}}+\left(\fracr\ell\,dx^{+}+\frac\ellr\,L_{-}(x^{-})\,dx^{-}\right)\left(\fracr\ell\,dx^{-}+\frac\ellr\,L_{+}(x^{+})\,dx^{+}\right)\qquad,
\left[-\gamma_{44}E^{2}+2iE\gamma_{4i}\vec{p}_{i}+\gamma_{ij}\vec{p}_{i}\vec{p}_{j}-m^{2}\right]u_{2}^{\sigma}(\vec{p})=0\quad(\mathrm{or}\,\,v_{2}^{\sigma}(\vec{p}))\quad,
A_{4}A_{1}=i\displaystyle\displaystyle\frac{\partialG(S_{3})}{\partialS_{3}}~,~~~~A_{2}=2i\displaystyle\frac{\partialA_{1}}{\partialS_{3}}~,~~~~A_{3}=-A_{2}~,~~~~A_{5}=2i\displaystyle\frac{\partialA_{2}}{\partialS_{3}}~,
\rho\Bigr(\frac{\delta_{L}}{\deltaB^{\alpha}}S_{m,\mathrm{ext}}^{(0)}+\epsilon^{cd}E_{\alphadc}^{*}\Bigr)=2\xig_{\alpha\beta}B^{\beta}+R_{\alpha}^{j}(A)(g_{ij}Q^{i}-\sigma\bar{Q}_{j})-R_{\alpha,j}^{i}\bar{Q}_{i}Q^{j}+f_{\alpha\gamma}^{\beta}\bar{E}_{\betab}C^{\gammab}.
P^{+}=-{\frac{g^{2}}{4}}\intdx^{-}Tr(\partial_{-}^{2}\sigma)\sigma={\frac{g^{2}}{4}}\intdx^{-}Tr(\partial_{-}\sigma)^{2}={\frac{g^{2}}{8}}\intdx^{-}(\partial_{-}s)^{2}={\frac{g^{2}}{4a}}
S=\intd^{p+1}\xie^{-(\frac{p-1}{2})\phi}(1+e^{2\phi}(C^{(0)})^{2})^{\frac{p-3}{4}}\sqrt{|\mathrm{det}(g+\frac{(2\pi\alpha^{\prime})e^{\phi}}{\sqrt{1+e^{2\phi}(C^{(0)})^{2}}}{\tilde{\calF}})|}\,,
\varepsilon_{\mu\nu\lambda\rho}\partial_{\lambda}\partial_{\rho}\theta_{a}^{\mathrm{sing.}}(x)=2\pi\Sigma_{\mu\nu}^{a}(x)\equiv2\pi\int_{\Sigma_{a}}^{}d\sigma_{\mu\nu}(x_{a}(\xi))\delta(x-x_{a}(\xi)),
Z[A_{\mu}]=\int\!D\bar{\psi}D\psiDB_{\mu}D\thetae^{-\int\!d^{2}x\,[\bar{\psi}i\gamma_{\mu}\partial_{\mu}\psi-j_{\mu}(A_{\mu}+B_{\mu})-\frac{1}{\pi}B_{\mu}\epsilon_{\mu\nu}\partial_{\nu}\theta]}\quad.
\begin{array}{l}{\hat{a}\hat{b}=q\hat{b}\hat{a}\,,\;\;\;\hat{b}\hat{d}=q\hat{d}\hat{b}\,,}\\{\hat{a}\hat{c}=q\hat{c}\hat{a}\,,\;\;\;\hat{c}\hat{d}=q\hat{d}\hat{c}\,,}\\{\phantom{a}\!\![\hat{a},\hat{d}]=(q-q^{-1})\hat{b}\hat{c}\,,}\\{\phantom{a}\!\![\hat{b},\hat{c}]=0\,.}\\\end{array}
g(\tau)=\frac{\vartheta_{2}^{4}(\tau)}{\vartheta_{3}^{4}(\tau)+\vartheta_{4}^{4}(\tau)},~~~~~f(\tau)=\frac{\vartheta_{2}^{4}(\tau)\vartheta_{4}^{4}(\tau)}{(\vartheta_{2}^{4}(\tau)-\vartheta_{4}^{4}(\tau))^{2}}~\,,
\hat{Z}(\theta)\left(\begin{array}{cccc}{\mathcal{C}_{+}}&{0}&{0}&{-\mathcal{D}/r}\\{0}&{\mathcal{C}_{+}}&{r\mathcal{D}}&{0}\\{0}&{\mathcal{D}/r}&{\mathcal{C}_{-}}&{0}\\{-r\mathcal{D}}&{0}&{0}&{\mathcal{C}_{-}}\\\end{array}\right)
\frac{dh_{pp^{\prime}}(l)}{dl}=\!\!-\sum_{q}\!\left(\frac{dh_{pq}(l)}{dl}\frac{1}{E_{p}(l)-E_{q}(l)}h_{qp^{\prime}}(l)+h_{pq}(l)\frac{1}{E_{p^{\prime}}(l)-E_{q}(l)}\frac{dh_{qp^{\prime}}(l)}{dl}\right)\,,
\overline{u}(p^{\prime})\Gamma_{\mathrm{a}}^{a,i}u(p)=\frac{g^{3}}{4\pi\Theta}[T^{b}T^{a}T_{b}]\frac{1}{2m}[P^{i}-i\varepsilon^{ij}q_{j}]+{\calO}(\eta^{2})\;,
{\calA}={\frac{{\calF}_{\mathrm{in}}(0)}{{\calF}_{\mathrm{in}}(\infty)}}={\frac{4\sqrt{A_{1}}(x_{+}^{2}-x_{-}^{2})}{\sqrt{1-s}}}{\frac{|C_{1}|^{2}}{|\tilde{\alpha}-i\tilde{\beta}|^{2}}}.
F^{\prime}=\int\!dx~\left(J^{1~j}\Phi^{j}+{\frac{i}{2}}\bar{\chi}^{j}\gamma^{1}\Psi^{j}-{\frac{i}{2}}\bar{\chi}^{j}\gamma^{1}\varepsilon^{jkl}\Phi^{k}\chi^{l}-{\frac{i}{2}}\overline{{\Psi}}^{j}\gamma^{1}\varepsilon^{jkl}{\frac{\Phi^{k}}{(1+4\Phi^{2})}}\Psi^{l}\right).
n_{k}(t)=\int\delta(t_{1}+t_{2}+\ldots+t_{k}+t_{k+1}-t)\,\frac{d^{3}r_{1}}{\ell_{0}^{3}}\ldots\frac{d^{3}r_{k}}{\ell_{0}^{3}}\frac{d^{3}r_{k+1}}{\ell_{0}^{3}}
{\widetildeT}_{\mu\nu}(p)=T_{\mu\nu}(p)-{\widehatM}_{P}^{D-d-2}\biggl[p^{2}H_{\mu\nu}-p_{\mu}p_{\nu}\chi+{\frac{d-1}{d}}{p^{2}}\eta_{\mu\nu}\chi\biggr]~.
{\calQ}=\sum_{j=1}^{r}\psi_{j}^{*}\left(p_{j}+i{\frac{\partialW}{\partialq_{j}}}\right),\qquad{\calQ}^{*}=\sum_{j=1}^{r}\psi_{j}\left(p_{j}-i{\frac{\partialW}{\partialq_{j}}}\right),
M\;\equiv\;T_{[1]}^{(0)}\:T_{[1]}^{(0)}\:\overline{{T_{[0]}^{(-1)}\:T_{[0]}^{(0)}}}\:-\:T_{[0]}^{(-1)}\:T_{[2]}^{(1)}\:\overline{{T_{[0]}^{(-1)}\:T_{[0]}^{(0)}}}\;.
\left\langle0\left|\psi_{n}^{\uparrow}(t_{1})\bar{\psi}_{-n}^{\uparrow}(t_{2})\right|0\right\rangle_{E}=\left\langle0\left|\psi_{n+1}^{\downarrow}(t_{1})\bar{\psi}_{-n-1}^{\downarrow}(t_{2})\right|0\right\rangle_{E}=0.
{\calM}_{CY,J}^{\epsilon}=\left(\begin{matrix}{-1}&{0}&{0}&{0}&{0}&{0}\\{0}&{-1}&{0}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{2}&{1}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{0}&{2}&{1}\\\end{matrix}\right)\.
\mathrm{H}_{31}^{F}=-\frac{1}{4}{\calF}_{31}\left(9\alpha_{31}+6\beta_{31}\right)\intd^{3}{\vec{x}}\;i\pi({\vec{x}})\tau_{a}\partial_{a}{\xi}(\vec{x})\left(\frac{\ell_{P}}{{\calL}}\right)^{2\left(1+\Upsilon\right)}
\tilde{A}_{2}(p)=2\,\gamma\,g^{2}\delta_{b}^{a}\delta_{J}^{I}\int\frac{d^{4}k}{(2\pi)^{4}}d^{2}\thetad^{2}{\overline{\theta}}\,\frac{[(p-k)^{2}+p^{2}]}{k^{4}[(p-k)^{2}+m^{2}]}\left\{\Phi_{aI}^{\dagger}(p,\theta,{\overline{\theta}})\Phi^{bJ}(-p,\theta,{\overline{\theta}})\right\}\,.
A_{k}^{i}=\frac{1}{\sqrt{k}}\alpha_{k}^{i},\qquadA_{k}^{i\dagger}=\frac{1}{\sqrt{k}}\alpha_{-k}^{i},\qquadB_{k}^{i}=\frac{1}{\sqrt{k}}\beta_{k}^{i},\qquadB_{k}^{i\dagger}=\frac{1}{\sqrt{k}}\beta_{-k}^{i},
\rho_{1}(s)={\frac{i}{\pi}}{\frac{\Gamma(s)}{\Gamma^{2}(s)+\left({\frac{\theta}{16}}\right)^{2}s}},\quad\rho_{2}(s)={\frac{i}{\pi}}{\frac{1}{\Gamma^{2}(s)+\left({\frac{\theta}{16}}\right)^{2}s}}
\alpha^{\prime}\left(\frac{1}{E}\right)^{\mu\nu}\approx\frac{\theta^{\mu\nu}}{2\pi}~,\quad\alpha^{\prime}\left(\frac{1}{E^{T}}\right)^{\mu\nu}\approx-\frac{\theta^{\mu\nu}}{2\pi}~,\quad\alpha^{\prime}\left(g^{-1}Ng^{-1}\right)^{\mu\nu}\approx\frac{1}{4\pi^{2}\alpha^{\prime}}\left(\thetaG\theta\right)^{\mu\nu}~.
\left(e-\frac{1}{2}\right)\,[D,e]^{4}=\left[\begin{matrix}{t-\frac{1}{2}}&{q}\\{q^{*}}&{\frac{1}{2}-t}\\\end{matrix}\right]\,\left[\begin{matrix}{dt}&{dq}\\{dq^{*}}&{-dt}\\\end{matrix}\right]^{4}
=\left\vert\begin{array}{cccc}{(x_{1}-x_{r+1})^{2}}&{(x_{1}-x_{r+1}).(x_{2}-x_{r+1})}&{\ldots}&{(x_{1}-x_{r+1}).(x_{r}-x_{r+1})}\\{(x_{2}-x_{r+1}).(x_{1}-x_{r+1})}&{(x_{2}-x_{r+1})^{2}}&{\ldots}&{(x_{2}-x_{r+1}).(x_{r}-x_{r+1})}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{(x_{r}-x_{r+1}).(x_{1}-x_{r+1})}&{(x_{r}-x_{r+1}).(x_{2}-x_{r+1})}&{\ldots}&{(x_{r}-x_{r+1})^{2}}\\\end{array}\right\vert
{\calT}_{\mu}{}^{\sigma_{1}\cdots\sigma_{n}}=\frac{n-1}{2(d-2)n!}\left(\Gamma_{\mu}{}^{\sigma_{1}\cdots\sigma_{n}}-\frac{n}{n-1}(d\!-\!n\!-\!1)\,\delta_{\mu}{}^{[\sigma_{1}}\Gamma^{\sigma_{2}\cdots\sigma_{n}]}\right)\,.
\begin{array}{rcl}{\left[\mathfrak{h},\mathfrak{h}\right]}&{\subset}&{\mathfrak{h}\,,}\\{}&{}&{}\\{\left[\mathfrak{k},\mathfrak{h}\right]}&{\subset}&{\mathfrak{k}\,,}\\{}&{}&{}\\{\left[\mathfrak{k},\mathfrak{k}\right]}&{\subset}&{\mathfrak{h}\,.}\\\end{array}
P^{\mu_{1}\nu_{1}\,\mu_{2}\nu_{2}}=c_{A}{\calT}_{A}^{\mu_{1}\nu_{1}\,\mu_{2}\nu_{2}}+c_{B}{\calT}_{A}^{\mu_{1}\nu_{1}\,\mu_{2}\nu_{2}}+c_{C}{\calT}_{A}^{\mu_{1}\nu_{1}\,\mu_{2}\nu_{2}}+\sum_{i=4}^{14}c_{i}T_{i}^{\mu_{1}\nu_{1}\,\mu_{2}\nu_{2}}.
|\lambda_{j}>=\left(\begin{array}{c}{H_{13}(H_{22}-\lambda_{j})-H_{23}H_{12}}\\{H_{23}(H_{11}-\lambda_{j})-H_{13}H_{21}}\\{|H_{12}|^{2}-(H_{11}-\lambda_{j})(H_{22}-\lambda_{j})}\\\end{array}\right),
X^{a}{}_{b}=\left(\begin{array}{cccc}{-k_{1}^{\,2}A+k^{2}m^{2}}&{-k_{0}k_{1}A}&{k_{0}k_{2}\,k^{2}}&{k_{0}k_{3}\,k^{2}}\\{k_{0}k_{1}A}&{k_{0}^{\,2}A+k^{2}m^{2}}&{-k_{1}k_{2}\,k^{2}}&{-k_{1}k_{3}\,k^{2}}\\{-k_{0}k_{2}\,k^{2}}&{-k_{1}k_{2}\,k^{2}}&{k_{3}^{\,2}B-k^{2}l^{2}}&{-k_{2}k_{3}B}\\{-k_{0}k_{3}\,k^{2}}&{-k_{1}k_{3}\,k^{2}}&{-k_{2}k_{3}B}&{k_{2}^{\,2}B-k^{2}l^{2}}\\\end{array}\right)
\langle\psi^{\dagger}({\bf{0}},0)\psi({\bf{x}},t)\rangle=exp\mathrm{~}[\mathrm{~}(\frac{1}{4N_{0}})\sum_{{\bf{q}}}f_{{\bf{q}}}({\bf{x}},t)\mathrm{~}]\langle\psi^{\dagger}({\bf{0}},0)\psi({\bf{x}},t)\rangle_{free}
{\frac{dV_{T}(\varphi)}{d\varphi}}=\int_{-\infty}^{+\infty}{\frac{d\alpha}{2\sqrt{\pi}}}e^{-{\frac{\alpha^{2}}{4}}}U^{(1)}({\frac{\alpha}{2}}\sqrt{J_{1}(g)-I_{1}(M^{2})}+\varphi)\;.
\partial_{\mu}S_{0}^{\mu}=\frac{1}{3}\sum_{i}\left(\gamma_{i}^{\ast}+\gamma_{i}\right)\Phi_{i}^{\alpha}\frac{\partialW}{\partial\Phi_{i}^{\alpha}}\Bigg|+\frac{1}{48\pi^{2}}\left[3T(G)-\sum_{i}T(R_{i})(1-(\gamma_{i}^{\ast}+\gamma_{i}))\right]F_{\muv}^{a}\widetilde{F}^{\muva}\,,
\zeta(\nu|\beta)\simeq{\frac{2\varrho^{-2\nu}\kappa^{D-1-2\nu}}{(4\pi)^{(D-1)/2}\Gamma(\nu)}}\sum_{n=0}^{\infty}\zeta_{R}\left(2\nu+2n+1-D\right)\Gamma\left(\nu+n-{\frac{D-1}{2}}\right)\breve{c}_{n,\nu}\kappa^{-2n}~~~,
F=-\frac{2}{3\pi}\mu^{4}r_{h}^{3}\int\frac{d{\hat{E}}}{e^{{\hat{\beta}}{\hat{E}}}-1}\int_{1+{\hat{\epsilon}}}^{{\hat{L}}}dx\,x^{6}(x-1)^{-4}(x+1)^{-4}\left[{\hat{E}}^{2}-\frac{1}{x^{2}}(x-1)^{2}(x+1)^{2}\right]^{\frac{3}{2}}.
I_{sp}=\int_{M}d^{4}x\\left[\overline{{\psi}}_{+}\gamma^{\mu}\left(\partial_{\mu}+A_{\mu}\right)\psi_{+}+\overline{{\psi}}_{+}\gamma^{\mu}\sigma_{\mu}\psi_{+}+\left(\mathrm{terms~with~~}+\rightarrow-\right)\right].
P_{12}\cdotsP_{N-1N}G_{\varepsilon}^{(nl)}(\zeta_{2},\cdots,\zeta_{N},\zeta_{1}q^{-2})=r_{\varepsilon}^{(l-n)}(\zeta_{1})D_{1}^{(l-n)}G_{\varepsilon}^{(nl)}(\zeta_{1},\cdots,\zeta_{N}).\,
\lambda^{a,\alpha\beta}\lambda_{a~\delta}^{~\gamma}-\lambda^{a,\gamma\alpha}\lambda_{a~\delta}^{~\beta}=\frac{2}{\sigma_{0}}\left(\Omega^{\alpha\beta}\delta_{~\delta}^{\gamma}-2\Omega^{\beta\gamma}\delta_{~\delta}^{\alpha}+\Omega^{\gamma\alpha}\delta_{~\delta}^{\beta}\right),
\frac{1}{16\gamma^{3}R_{11}^{3}M_{11}^{3}}\intd^{3}x^{\prime}\,\frac{(\partial_{0}^{\prime}X^{i}\partial_{0}^{\prime}X^{i})^{2}}{X^{3}}e^{-(X/\gamma-iX^{11})/R_{11}}\.
\oint\left(\Pi\mathrm{d}\psi+\overline{{\Pi}}\mathrm{d}\overline{{\psi}}\right)=\int_{0}^{T_{E}}\mathrm{d}t\left(\Pi(t)\dot{\psi}(t)+\overline{{\Pi}}(t)\dot{\overline{{\psi}}}(t)\right),
\delta_{B}(f^{(4)}u^{2}\left[\psi^{4}\right])+\delta_{F}(f^{(6)}u\left[\psi^{6}\right])+u^{i}\epsilon\dot{N}^{i(4)}\left[\psi^{4}\right]\psi+\frac{i}{2}u\psi\dot{M}^{(4)}\left[\psi^{4}\right]\epsilon=0~,
t^{\alpha\beta}=\delta^{\mathrm{mn}}e_{\mathrm{m}}^{\mathrm{i}}{\frac{\partial\pi^{\alpha}}{\partialp^{\mathrm{i}}}}e_{\mathrm{n}}^{\mathrm{j}}{\frac{\partial\pi^{\beta}}{\partialp^{\mathrm{j}}}}.
\sum_{n}\left[\psi_{+,n}({\bf{x})\overline{{\psi}}_{+,n}({y})+\sigma_{3}\psi_{+,n}^{c\ast}({x})\overline{{\psi_{+,n}^{c}}}^{\ast}({y})\sigma_{3}}\right]=\delta({\bf{x}-{y})\;.}
\Phi_{\alpha}(\vec{q},t)=\\mathrm{exp}\biggl\{-{\frac{|\alpha|^{2}}{2}}+{\frac{1}{2}}{\frac{\mu_{p}^{\ast}}{\mu_{p}}}\,\vec{\alpha}\cdot\vec{\alpha}+{\frac{i}{\hbar\mu_{p}}}\vec{q}\\tilde{\bfR}\vec{\alpha}\biggr\}\Phi_{0}(\vec{q},t)\\.
C=e^{\pi\sqrt{-1}[(n-1)(n+2)+2(n-1)^{2}\tau]/2n}\displaystyle\frac{\prod_{i=0}^{n-1}(\sum_{j=0}^{n-1}\omega^{ij}\theta^{(j)}(1/2))}{h(n/2-(n-1)\tau/2)\prod_{k=1}^{n-1}h(k\tau/n)^{n-k}},