harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde{{\calL}}=\bar{\Psi}(x)i\!\not{\!\partial}\Psi(x)+\frac{G_{S}}{2N}(\bar{\Psi}(x)\Psi(x))^{2}-\frac{G_{V}}{2N}(\bar{\Psi}(x)\gamma^{\mu}\Psi(x))^{2}+J\bar{\Psi}(x)\Psi(x),
F_{\mathrm{div}}^{E}[g,\beta,\mu]=-{\frac{\eta}{48\kappa\beta^{2}}}\int_{\Sigma}\left[bf_{1}+a\left(p_{1}{\frac{4\pi^{2}}{\kappa^{2}\beta^{2}}}P+p_{2}R+p_{3}R_{ii}\right)\right]~~~,
q^{+}=q^{i}(x)u_{i}^{+}+\theta^{+\alpha}\psi_{\alpha}(x)+\bar{\theta}_{\dot{\alpha}}^{+}\bar{\kappa}^{\dot{\alpha}}(x)+2i\theta^{+}\sigma^{\mu}\bar{\theta}^{+}\partial_{\mu}f^{i}(x)u_{i}^{-}\,,
W_{ABC}=\frac{1}{2}T_{\underline{{A}}\\\\underline{{B}}\dot{A}\,underline{C}}^{\\\dot{A}},\,\,W_{\dot{A}\dot{B}\dot{C}}=-\frac{1}{2}T_{\\\underline{{\dot{A}}}A\underline{{\dot{B}}}\underline{{\dot{C}}}}^{A}
\Gamma_{2}:=\left\{\,\,\Gamma_{VV}=-D^{-1},\,\,\,\,\Gamma_{G{\bar{G}}}=-{\tilde{D}}^{-1},\,\,\,\,\left\{\Gamma_{F{\bar{F}}}=-S_{F}^{-1}|F=1\ldotsN_{F}\right\}\,\,\right\}\,,
\int_{M}\;\mu\;(2(P^{jk}{}_{,i}D_{\muj}^{l}+P^{mk}{}_{,ij}A_{\mum}P^{jl})-D_{\mui}^{m}P^{kl}{}_{,m}+(\epsilon^{\rho\nu}D_{\rho}X^{i})\epsilon_{\mu\nu}P^{kl}{}_{,ji})C_{l}C_{k}=0.
\begin{array}{ll}{\sigma^{0}=-{\bf1}_{2\times2}={\bar{\sigma}}^{0},}&{\sigma^{1}=\tau_{z}=-{\bar{\sigma}}^{1},}\\{\sigma^{2}=\tau_{x}=-{\bar{\sigma}}^{2},}&{\sigma^{3}=\tau_{y}=-{\bar{\sigma}}^{3},\quad(\mathrm{or}\quad\sigma^{y}=\tau_{y}=-{\bar{\sigma}}^{y}),}\\\end{array}
\tilde{S^{a}}_{R}=\Lambda_{b}^{a}[\bar{\tilde{S}}_{0}^{b}cos\omega_{0}\tau+{\tilde{S^{b}}_{0}}sin\omega_{0}\tau]+\sum_{n\neq0}{\frac{c_{n}}{\sqrt2}}[\phi{_n}{S^{a}}{_n}+{\frac{i}{m}}(\omega_{n}-n)\Lambda_{b}^{a}{\tilde{S^{b}}_{n}}\tilde{\phi_{n}}],
\begin{array}{l}{\displaystyle{D(\tilde{e}Dx^{\mu})=\Omega_{\nu}^{\mu}Dx^{\nu}-\frac{2q}{5}\left(D^{5}x^{\mu}+(D^{3}x)^{2}D^{3}x^{\mu}+3D^{3}x\cdotD^{4}xD^{2}x^{\mu}+\right.}}\\{\displaystyle{\left.+\left[\frac97(D^{4}x)^{2}+\frac{11}7D^{3}x\cdotD^{5}x+\frac25((D^{3}x)^{2})^{2}\right]Dx^{\mu}\right)}\,,\;\;\;\;\;\;\;(Dx)^{2}=0\,,}\\\end{array}
\pi(\alpha)=\sum_{i}a_{i}[D,b_{i}]=\left(\begin{matrix}{\gamma^{\mu}\alpha_{1\mu}}&{V_{\mu}\gamma^{\mu}\gamma^{5}\alpha_{5}}\\{-V_{\mu}\gamma^{\mu}\gamma^{5}\tilde{\alpha}_{5}}&{\gamma^{\mu}\alpha_{2\mu}}\\\end{matrix}\right)
e^{-1}\mathcal{L}=\frac{1}{2}R+g^{2}V-\frac{1}{4}G_{IJ}F_{\mu\nu}^{I}F^{J\mu\nu}-\frac{1}{2}\mathcal{G}_{ij}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}+\frac{e^{-1}}{48}\epsilon^{\mu\nu\rho\sigma\lambda}C_{IJK}F_{\mu\nu}^{I}F_{\rho\sigma}^{J}A_{\lambda}^{K},
{\calL}=\frac{1}{2}D_{+}\rho\,D_{-}\rho+\frac{1}{2}D_{+}\psi\,D_{-}\psi-\frac{1}{2}\mathrm{th}^{2}\rhoD_{+}\omega\,D_{-}\omega+\mathrm{sh}\rho\,e^{\psi}.
\int\sqrt{\gamma}d^{4}\Omegaq_{{\calP}klm}^{(p^{\prime})ij}q_{{\calP}^{\prime}k^{\prime}l^{\prime}m^{\prime}ij}^{(p^{\prime\prime})*}=\delta^{p^{\prime}p^{\prime\prime}}\delta_{{\calP}{\calP}^{\prime}}\delta_{ll^{\prime}}\delta_{mm^{\prime}}
\frac12{\bar{e}}_{\alpha}^{\;\;i}{\bar{e}}_{\beta}^{\;\;j}\frac{\partial\Gamma^{(2)}}{\partialb^{ij}}=-{\bar{e}}_{\alpha}^{\;\;i}{\bar{e}}_{\beta}^{\;\;j}B_{ij}^{b^{\star}(2)}+{e}_{\alpha}^{\;\;\mu}{e}_{\beta}^{\;\;\nu}K_{\mu\nu}^{b}.
{\calL_{Q}}=-{\frac{1}{4}}\Bigl(1+{\frac{ae^{2}}{\pi}}\Bigr)F^{2}+{\frac{e^{2}}{2\pi}}A^{2}+{\frac{1}{2}}\partial_{\mu}\Sigma\partial^{\mu}\Sigma-{\frac{e}{\sqrt\pi}}\partial_{\mu}\SigmaA^{\mu}-{\frac{e}{\sqrt\pi}}\widetilde{\partial}\cdotA\chi.
\nabla^{2}(\nabla^{2}-s^{2})A_{0}=-\nabla^{2}\rho-s\overrightarrow{\nabla}\times\overrightarrow{j}-sv_{o}\nabla^{2}\varphi+\nabla^{2}\left(\overrightarrow{v}\times\overrightarrow{\nabla}\varphi\right).
S=\intd^{4}x(-{\frac{1}{8}}F_{mn}^{\alpha}F^{\alphamn}-{\frac{1}{4(u_{l}u^{l})}}u^{m}{\calF}_{mn}^{\alpha}{\calF}^{\alphanp}u_{p}-\epsilon^{mnpq}u_{m}\partial_{n}B_{pq})
\hat{F}_{\mu\nu}^{ij}(z)=\int_{T^{4}}d^{4}x\int_{T^{4}}d^{4}x^{\prime}\,\psi^{i}{}^{\dagger}(x;z)\sigma_{\mu}(D_{z}^{\dagger}D_{z})^{-1}(x,x^{\prime};z)\sigma_{\nu}^{\dagger}\psi^{j}(x;z)-[\mu\leftrightarrow\nu].
e(k_{M})\equiv\frac{1}{2}\left(\zeta_{p+1}(k_{M})-i\zeta_{p+2}(k_{M})\right)~,\quad\overline{{e}}(k_{M})\equiv\frac{1}{2}\left(\zeta_{p+1}(k_{M})+i\zeta_{p+2}(k_{M})\right)~.
\frac{\partial}{\partials}\int_{4m^{2}}^{\infty}\frac{d\mu^{2}}{\mu^{2}}\,\Gamma(\mu^{2})\,J_{0}(\mus)=-\frac{2}{s}\int_{2ms}^{\infty}dx\,\Gamma\Bigl(\frac{x^{2}}{s^{2}}\Bigr)\,J_{1}(x)\;.
v=\sum_{i=1}^{5}(x_{i}^{5}+\Pi_{i=1}^{5}x_{i}).{\bfI}_{{\calD}\left(G\right)}+\sum_{i,j,k=1}^{5}\{\calF^{\prime}}_{k_{1}k_{2}k_{3}}(\Phi_{1},\dots,\Phi_{5})\F_{1}^{k_{1}}.F_{2}^{k_{2}}.F_{3}^{k_{3}}.
E_{0}^{(1)}\equiv\Gamma^{(1)}(\rho,A;\bar{\rho})\Big\vert_{{\rho=\bar{\rho}\atopA=0}}=-{\frac{5\bar{\rho}^{4}}{64\pi^{2}}}\left(\lambda^{2}+{\frac{3e^{4}}{10}}\right),
\mathcal{L}_{0}+\mathcal{L}_{\sf{gf}}=-\frac{1}{4}B^{\alpha\beta}\Bigl[g_{\alpha\nu}g_{\beta\lambda}k^{2}\quad+\left(\frac{1}{\alpha}-1\right)\left(g_{\beta\lambda}k_{\alpha}k_{\nu}+g_{\alpha\nu}k_{\beta}k_{\lambda}\right)\Bigr]B^{\nu\lambda}.
\sum_{l=1\atopl\nej}^{r}{\frac{1}{\bar{x}_{j}-\bar{x}_{l}}}+{\frac{k_{1}+k_{2}}{2(\bar{x}_{j}-1)}}+{\frac{k_{2}}{2(\bar{x}_{j}+1)}}=0\quad(j=1,\ldots,r),
\begin{array}{rclcl}{T}&{=}&{\frac{1}{8\piM}}&{\stackrel{\mathrm{\tiny~M\rightarrow~0~}}{\mathrm{\large~\longrightarrow~}}}&{\infty\,,}\\{}&{}&{}&{}&{}\\{S}&{=}&{4\piM^{2}}&{\stackrel{\mathrm{\tiny~M\rightarrow~0~}}{\mathrm{\large~\longrightarrow~}}}&{0\,.}\\\end{array}
H_{C}=\frac{La^{2}}{4}\sum_{\vec{b}}\int_{0}^{L}dz^{-}\int_{0}^{L}dy^{-}\sum_{p,q,n}\,^{\prime}\frac{G_{\botqp}^{\prime}(\vec{b},z^{-})G_{\botpq}^{\prime}(\vec{b},y^{-})}{\left[\pin+(p-q)\varphi_{\vec{b}}\right]^{2}}e^{i2\pin(z^{-}-y^{-})/L}\.
\Gamma_{(m,\psi_{m})}={\frac{1}{\sqrt{|{\calG}||{\calS}_{m}|}}}\left({\frac{\varepsilon_{1}{P}_{0m}+\varepsilon_{2}{P}_{Jm}}{\sqrt{{S}_{0m}}}}+\psi_{m}{\frac{\delta_{m}}{\sqrt{{S}_{0m}}}}\right)\;\;\;\;,
\begin{array}{c}{F^{-}F^{+}~+~QF^{+}F^{-}~=~Q^{N}}\\{F^{-}F^{+}~+~Q^{-1}F^{+}F^{-}~=~Q^{-N}}\\{Q^{N_{F}}F^{+}Q^{-N_{F}}~=~QF^{+}}\\{Q^{N_{F}}F^{-}Q^{-N_{F}}~=~Q^{-1}F^{-}}\\{Q^{N_{F}}Q^{-N_{F}}~=~Q^{-N_{F}}Q^{N_{F}}~=1~}\\{(F^{+})^{2}=0,(F^{-})^{2}=0}\\\end{array}
ds^{2}={\frac{l^{2}d\rho^{2}}{4\rho^{2}}}+{\frac{1}{\rho}}\left(h_{ij}(x)+h_{ij}^{(1)}(x)\rho+h_{ij}^{(2)}(x)\rho^{2}+...\right)dx^{i}dx^{j}~~,
\begin{array}{rcl}{D_{M}v_{i}^{\mu}}&{=}&{\partial_{M}v_{i}^{\mu}+\Lambda_{M\nu}^{\mu}v_{i}^{\nu}=0}\\{D_{M}e_{\mu}^{i}}&{=}&{\partial_{M}e_{\mu}^{i}-\Lambda_{M\mu}^{\nu}e_{\nu}^{i}=0}\\\end{array}
H_{c}=\intd^{3}x\{\frac{1}{2}\Pi_{i}^{a}\Pi_{i}^{a}-A_{0}^{a}\partial_{i}\Pi_{i}^{a}+gA_{0}^{a}A_{0}^{b}C_{c}^{ab}\Pi_{i}^{c}+\frac{1}{4}F_{ij}^{a}F_{ij}^{a}\}
\sigma(t)=1+\int_{0}^{t}\!dt_{1}\,\psi(t_{1})+\int_{0}^{t}\!dt_{1}\int_{0}^{t_{1}}\!dt_{2}\,\psi(t_{2})\psi(t_{1})+\int_{0}^{t}\!dt_{1}\int_{0}^{t_{1}}\!dt_{2}\int_{0}^{t_{2}}\!dt_{3}\,\psi(t_{3})\psi(t_{2})\psi(t_{1})+\cdots
\begin{array}{c}{\beta_{1}=\frac2N\delta_{ab}\delta_{cd}+d_{abe}d_{cde}}\\{\beta_{2}=\frac2N\delta_{ad}\delta_{bc}+d_{ade}d_{bce}}\\{\beta_{3}=\frac2N\delta_{ac}\delta_{bd}+d_{ace}d_{bde}}\\\end{array}
I_{gh}^{\prime}=\intd^{3}x\mathrm{Tr}(Z_{gh}^{-1}\partial^{\mu}\overline{{c}}^{(r)}\,partial_{\mu}c^{(r)}+\tilde{Z}_{g}^{-1}\partial^{\mu}\overline{{c}}^{(r)}[A_{\mu}^{(r)},c^{(r)}]).
\left[\frac{d^{2}}{dz^{2}}+\frac{4}{\epsilon^{2}}\lambda\,\kappa^{-\epsilon}+\sigma\,\frac{4}{\epsilon^{2}}z^{4/\epsilon-2}-\frac{(\widetilde{l}+\widetilde{\nu})^{2}-1/4}{z^{2}}\right]w_{l}(z)=0\;,
\sum_{j_{12}}(-1)^{[j_{1}+j_{2}+j_{12}+\frac{1}{2}]}\left\{\begin{array}{ccc}{j_{1}}&{j_{2}}&{j_{12}}\\{j_{2}}&{j_{1}}&{j_{12}^{\prime}}\\\end{array}\right\}^{S}=\delta_{j_{12}\,j_{12}^{\prime}}~.
\hat{\nu}\longrightarrow\hat{\nu}-i{\frac{1}{\Box}}\partial_{\alpha}^{\dot{\alpha}}\,\hat{b}_{\dot{\alpha}}^{*,\alpha}+i{\frac{1}{\Box}}D_{\beta}\,\partial_{\alpha}^{\dot{\alpha}}\,\hat{\beta}_{\dot{\alpha}}^{*,(\alpha\beta)}\,,
U^{\Lambda\Sigma}P_{\Lambda}^{i}P_{\Sigma}^{i}=-\frac1{2(z+\bar{z})}\left(P_{1}^{i}P_{1}^{i}+P_{2}^{i}P_{2}^{i}\right)=P_{\Lambda}^{i}P_{\Sigma}^{i}\bar{X}^{\Lambda}X^{\Sigma}\mathrm{e}^{\mathcalK}
\Gamma_{t}^{(4)}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4})-g_{t}\simg_{t}^{2}\gamma^{(4)}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4})+O(g_{t}^{3})\;.
\bar{\alpha}:=\left(\begin{matrix}{\bar{d}_{1}}\\{\bar{d}_{2}}\\\end{matrix}\right),\quad\bar{\beta}:=\left(\begin{matrix}{\bar{d}_{1}}\\{-\bar{d}_{2}}\\\end{matrix}\right),\quad\bar{D}_{0}:=2\left(\begin{matrix}{\bar{d}_{1}}&{0}\\{0}&{\bar{d}_{2}}\\\end{matrix}\right).
r_{1}^{2}=\sqrt{Q_{2}^{2}+\frac{r_{0}^{4}}{4}}-\frac{r_{0}^{2}}{2}\,\r_{2}^{2}=\sqrt{Q^{2}+\frac{r_{0}^{4}}{4}}-\frac{r_{0}^{2}}{2}.
E_{a}=\frac1a\left[\frac{\pi^{4}}{30}\left(\frac{a}{\beta}\right)^{4}-\zeta(3)\left(\frac{a}{\beta}\right)^{3}-\frac{\pi^{2}}{12}\left(\frac{a}{\beta}\right)^{2}+\frac78\left(\frac{a}{\beta}\right)-\frac{11}{32}\right].
A_{Li}^{a}({\bf{r}})=\sum_{{\bf{k}}}\frac{k_{i}}{2k^{\frac{3}{2}}}\,[a_{R}^{a}({\bf{k}})e^{+i{\bf{k}}\cdot{\bf{r}}}+a_{R}^{a\star}({\bf{k}})e^{-i{\bf{k}}\cdot{\bf{r}}}]\;,
{\calS}_{o}\,=\,\fracq{\sqrt3}\left[\begin{array}{cccc}{0}&{-\frac13(q-q^{-1})}&{\frac13(q-q^{-1})}&{0}\\{}&{}&{}&{}\\{\frac23(q-q^{-1})}&{-1}&{0}&{-\frac23(q-q^{-1})}\\{}&{}&{}&{}\\{-\frac73(q-q^{-1})}&{-1}&{0}&{-\frac23(q-q^{-1})}\\{}&{}&{}&{}\\{-1}&{\frac23(q-q^{-1})}&{\frac13(q-q^{-1})}&{1}\\\end{array}\right]
ds^{2}=2\frac{(c^{3}+L^{3})^{\frac13}}L\left\{dx^{i}d\bar{x}^{i}-\frac{c^{3}}{L(c^{3}+L^{3})}\bar{x}^{i}dx^{i}x^{j}d\bar{x}^{j}\right\},
\Lambda_{1}(\lambda,\lambda_{1})=z(\lambda_{1}-\lambda)x_{1}^{N}(\lambda)+\varepsilon^{2}\frac{z(\lambda-\lambda_{1})}{\omega(\lambda-\lambda_{1})}x_{2}^{N}(\lambda)+\frac{x_{2}(\lambda-\lambda_{1})}{x_{3}(\lambda-\lambda_{1})}x_{3}^{N}(\lambda)
A^{(1)}=\left(\begin{array}{ccc}{A^{(1)ij,kl}}&{A^{(1)ij,0}}&{A^{(1)ij,b}}\\{A^{(1)0,kl}}&{A^{(1)00}}&{A^{(1)0b}}\\{A^{(1)a,kl}}&{A^{(1)a0}}&{A^{(1)ab}}\\\end{array}\right)
\{\varepsilon^{\mu\nu}\}=\left(\begin{array}{cccc}{0}&{0}&{0}&{0}\\{0}&{0}&{c}&{-b}\\{0}&{-c}&{0}&{a}\\{0}&{b}&{-a}&{0}\\\end{array}\right),~~~~~~~\varepsilon^{\mu}=-i\left(\begin{array}{c}{0}\\{a}\\{b}\\{c}\\\end{array}\right)
S_{T2}^{\mathrm{ghost}}=\intd^{8}z\,[c^{\prime\daggera}(e^{{\calV}_{B}})^{ab}c^{b}+c^{\daggera}(e^{{\calV}_{B}})^{ab}c^{\primeb}-2\xib^{\daggera}(e^{{\calV}_{B}})^{ab}b^{b}].
\langle0p_{1}\ldotsp_{n}\rangle\sim\sum_{a,b,c,d=3}^{n}\theta_{a}^{+}\theta_{b}^{+}\theta_{c}^{+}\theta_{d}^{+}f_{abcd}^{p_{1}\vertp_{2}\vert\ldotsp_{a}-1\ldotsp_{b}-1\ldotsp_{c}-1\ldotsp_{d}-1\ldots\vertp_{n}}
\frac{1}{r^{2}}\frac{d^{2}f}{d\theta^{2}}(\theta)=f(\theta)\left\{-1+2f(\theta)^{2}+(2+\lambda_{1})\left[f(\theta-\frac{2\pi}{3})^{2}+f(\theta+\frac{2\pi}{3})^{2}\right]\right\},
Solv_{r+1}\,\equiv\,Solv\left(E_{r+1(r+1)}/{\calH}_{r+1}\right)=Solv\left(SL(2,R)/O(2)\right)\,\oplus\,Solv\left(\frac{GL(r)}{SO(r)}\right)\,\oplus\,{\bar{\calW}}_{r+1}
\frac{E_{V}}{A}=\frac{\mu}{2}\int_{-\infty}^{\infty}\frac{dk_{y}dk_{z}}{(2\pi)^{2}}\sum_{n=1}^{\infty}((\frac{n\pi}{a\mu})^{2}+(\frac{k_{y}}{\mu})^{2}+(\frac{k_{z}}{\mu})^{2})^{-\frac{s}{2}}\rfloor_{s=-1}\,.
S_{\Sigma}=\frac{1}{8\pi}\intd^{2}\sigma\,\biggl\{\Bigl(\sqrt{g}g^{ab}G_{\mu\nu}(X)+i\epsilon^{ab}B_{\mu\nu}(X)\Bigr)\partial_{a}X^{\mu}\partial_{b}X^{\nu}+2\sqrt{g}R\Phi(X)\biggr\},
\delta\varphi^{i}(y)=\sum_{k}\intd^{2}x\left(\Lambda_{k}(x)\phi_{k}^{i(0)}+\dot{\Lambda}_{k}(x)\phi_{k}^{i(1)}(x,y)+\ddot{\Lambda}_{k}(x)\phi_{k}^{i(2)}(x,y)\right),
\frac{1}{2}\int\frac{d^{4}k}{(2\pi)^{2}}\Bigg(\frac{1}{k^{2}(k+p)^{2}}-\sum_{i}c_{i}\frac{1}{\Big(k^{2}+M_{i}^{2}\Big)\Big((k+p)^{2}+M_{i}^{2}\Big)}\Bigg).
S=-\frac{1}{4\pi\alpha^{\prime}}\int\!d^{2}\xi[\sqrt{-h}h^{\alpha\beta}G_{AB}{\partial_{\alpha}}X^{A}{\partial_{\beta}}X^{B}+\epsilon^{\alpha\beta}B_{AB}{\partial_{\alpha}}{X^{A}}{\partial_{\beta}}{X^{B}}],
T_{0}(z^{\prime})T(z,\theta)=\frac{\frac{c(\theta)}{2}\Phi_{0}(z,\theta)}{(z^{\prime}-z)^{4}}+\frac{2+\theta}{(z^{\prime}-z)^{2}}T(z,\theta)+\frac{\partial_{z}T(z,\theta)}{z^{\prime}-z}+\cdots\:\:\:,
Y_{\lambda\mu\rho}=\big(T^{\alpha}{}_{\rho}\partial_{\alpha}T_{\lambda\mu}+T_{\lambda\alpha}\partial_{\mu}T^{\alpha}{}_{\rho}+T_{\alpha\mu}\partial_{\lambda}T^{\alpha}{}_{\rho}-\partial_{\rho}(T^{\alpha}{}_{\lambda}T_{\alpha\mu})\big).
{\bfP}_{\alpha_{ab}}={diag(1,\alpha_{ab},}\alpha_{ab}^{2},...,\alpha_{ab}^{d+1});\quad{\bfQ}=\left(\begin{array}{ccccccc}{0}&{0}&{0}&{.}&{.}&{.}&{1}\\{1}&{0}&{0}&{.}&{.}&{.}&{0}\\{0}&{1}&{0}&{.}&{.}&{.}&{0}\\{.}&{.}&{.}&{.}&{.}&{.}&{.}\\{.}&{.}&{.}&{.}&{.}&{.}&{.}\\{0}&{0}&{0}&{.}&{1}&{0}&{0}\\{0}&{0}&{0}&{.}&{.}&{1}&{0}\\\end{array}\right)
\langle{\calC}^{11}(x_{1}){\calC}_{11}^{\dagger}(x_{3}){\calK}_{{\bf1}}(x_{2})\rangle_{_{(0)}}={\frac{N^{2}-1}{3(2\pi)^{6}(x_{13})^{2}(x_{12})^{2}(x_{23})^{2}}}\;.
\sum_{\epsilon_{l}\leq0}u_{\epsilon_{l}l}\left(r\right)u_{\epsilon_{l}l}\left(r^{\prime}\right)+\int_{0+}^{\infty}\frac{\mathrm{d}k}{2\pi}\,\left[u_{kl}\left(r\right)u_{kl}\left(r^{\prime}\right)-v_{kl}\left(r\right)v_{kl}\left(r^{\prime}\right)\right]=0\,,
I_{d,d}[\Phi]=2^{d/2}\int_{{\calF}}\frac{d^{2}\rho}{\rho_{2}^{2}}Z(g/2,b/2,\rho)\left[\Phi\left(\frac{\bar{\rho}}{2}\right)+\Phi\left(-\frac{1}{2\bar{\rho}}\right)+\Phi\left(\frac{\bar{\rho}+1}{2}\right)\right]\.
\int_{C_{s}}F_{1/2,s}(t)dtK_{s}^{(1)}(t,t^{\prime})=-\int_{C_{b}^{(s)}}F_{1/2,s}(t)dt2\pii\eta_{s}^{(1)}(t^{\prime})+\int_{C_{b}^{(s)}}F_{1/2,s}(t)dt\varphi_{s}(t)f_{s}(t^{\prime})
\frac{\partial}{\partialt}~\underline{{\varphi}}(x)\;=\;\fracih~\left[H-\sum_{\ell=1}^{L}\mu_{\ell}N_{\ell},\underline{{\varphi}}(x)\right]\;+\;\fracih\sum_{\ell=1}^{L}\mu_{\ell}\left[N_{\ell},\underline{{\varphi}}(x)\right]~,
\left({\frac{1}{2}}\sigma_{\mu\nu}{\calF}_{\mu\nu}T_{3}\right)^{2}=\left({\frac{\sqrt{B^{2}+{\bboxE}^{2}}}{2}}\right)^{2}{\bfI}\equiv\left({\frac{F_{+}}{2}}\right)^{2}{\bfI}\.
Z_{0}=\frac{1}{(2\pi)^{N}}\int_{M}\mathrm{e}^{H(x)}\sqrt{det\omega_{ij}}d^{2N}x=\frac{1}{\pi^{N}}\int_{{\calM}}\mathrm{e}^{H-F}d^{2N}xd^{2N}\theta,
\hat{\mathrm{H}}_{\mathrm{F}}^{s}=\intdx{\calH}_{\mathrm{F}}^{s}=\frac{1}{2}\hbar\intdx(\psi_{\mathrm{R}}^{\daggers}d\psi_{\mathrm{R}}^{s}-\psi_{\mathrm{R}}^{s}d^{\star}\psi_{\mathrm{R}}^{\daggers}).
\begin{array}{c}{PR^{\prime}\left(\sqrt{\frac{z}{w}}\right)\left(q^{S_{3}}\otimes{\frac{E}{z}}+{\frac{E}{w}}\otimesq^{-S_{3}}\right)=}\\{=\left(q^{S_{3}}\otimes{\frac{E}{w}}+{\frac{E}{z}}\otimesq^{-S_{3}}\right)PR^{\prime}\left(\sqrt{\frac{z}{w}}\right)}\\\end{array}
A(n_{1},k_{1};n_{2},k_{2};\dotsn_{M},k_{M};n_{I},k_{I})\frac{1}{n_{I}!}\frac{1}{p^{n_{I}+k_{I}^{2}/2}-p}A(n_{I},k_{I};n_{M+1},k_{M+1}\dotsn_{N},k_{N}),
L_{0}\{\varphi,\partial_{\mu}\,\varphi\}={{\frac{1}{2}}}\intd^{4}x\left[\left({\frac{\partial\varphi(x)}{\partialx_{k}}}\right)^{2}-\left({\frac{\partial\varphi(x)}{\partialt}}\right)^{2}~-~m^{2}\left(\varphi(x)\right)\right]\quad.
J_{1lin}=\left[\frac{a(t_{o})}{a(t)}\right]^{3}\frac{T_{o}m(T_{o})}{\pi^{2}}\int_{1}^{\infty}dz\left\{a^{3}(t)m(T_{o})\sqrt{z^{2}-1}\frac{\mid\varphi_{k}^{H}(t)\mid^{2}}{2}-\frac{a(t)}{2a(t_{o})}\right\}
{\calE}=\frac{1}{-g^{tt}}\left\{-g^{t\phi}p_{\phi}\pm\left[(g^{t\phi}p_{\phi})^{2}+(-g^{tt})(\mu^{2}+g^{\phi\phi}p_{\phi}^{2}+g^{rr}p_{r}^{2}+g^{\theta\theta}p_{\theta}^{2})\right]^{1/2}\right\}\equiv{\calE}_{\pm}.
\cdots~\to~\underbrace{\pi_{1}(G)}_{=0}~\to~\underbrace{\pi_{1}(Y)}_{=0}~\to~\pi_{1}(X)~\to~\underbrace{\pi_{0}(G)}_{=G}~\to~\underbrace{\pi_{0}(Y)}_{=0}~\to~\underbrace{\pi_{0}(X)}_{=0}
G_{{D}}^{(+)}({\bfr},{\bfr}^{\prime};k)={\mathcalG}_{{D}}^{(+)}({\bfR};k)\stackrel{\scriptscriptstyle(r\rightarrow\infty)}{\sim}-\frac{1}{4\pi}\left(\frac{k}{2\pi}\right)^{({D}-3)/2}\,e^{i\gamma_{{D}}}\,\frac{e^{ikr}}{r^{({D}-1)/2}}\,e^{-i{\bfk}^{\prime}\cdot{\bfr}^{\prime}}\;,
Z=\int{\calD}Gexp\big(-\int\alphaG_{\mu\nu}^{a}G_{a}^{\mu\nu}dx\big)\int{\calD}Aexp\bigg[\int(2\partial_{\mu}G_{a}^{\mu\nu}A_{\nu}^{a}-M_{ab}^{\mu\nu}A_{\mu}^{a}A_{\nu}^{b})dx\bigg],
\begin{array}{rl}{{\calH}=\frac{1}{2}\pi_{1}^{2}}&{+\frac{\pi}{2e^{2}}\left(\partial_{1}\pi_{1}\right)^{2}+\frac{e^{2}}{2\pi}\left(A^{1}-\frac{1}{2e\lambda}\partial_{1}(\phi_{+}-\phi_{-})\right)^{2}}\\{}&{+\frac{\pi}{2e^{2}}\phi^{2}-\frac{\pi}{e^{2}}\phi(\partial_{1}\pi_{1})\.}\\\end{array}
V_{eff}(z)=\frac{15}{4\big(|z|+\ell\big)^{2}}-\frac{3}{\ell}\,\delta(z)\bigg[1-\frac{4\gamma\,\ell}{3(1-\gamma)}\,sgn(z)\,\bigg(\partial_{z}+\frac{3}{2\ell}\,sgn(z)\bigg)\bigg]
\gamma_{x}=\left[\begin{array}{cc}{0}&{0}\\{1}&{0}\\\end{array}\right]\otimes1,\;\;\gamma_{y}=\left[\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right]\otimes\left[\begin{array}{cc}{0}&{0}\\{1}&{0}\\\end{array}\right],\;\;\gamma_{\bar{x}}=\gamma_{x}^{T},\;\;\gamma_{\bar{y}}=\gamma_{y}^{T}
\partial_{-}h_{m+}=-\partial_{i}h_{im}-e^{A}\left(\frac{D_{1}-2}{2}A^{\prime}h_{rm}+h_{m\underline{l};}^{\quad\underline{l}}-\kappa^{2}\left(\phi^{\dagger}D_{m}\Phi+(D_{m}\Phi)^{\dagger}\phi+V_{\underline{l}}F_{n}^{\quad\underline{l}}\right)\right)~.
\left\{k^{ij}\left(\vec{x},t\right),\pi^{0\mu}\left(\vec{y},t\right)\right\}^{*}=\frac18\left(\eta^{\mui}\partial^{j}+\eta^{\muj}\partial^{i}+3\eta^{ij}\partial^{\mu}\right)\delta^{2}\left(\vec{x}-\vec{y}\right)\;\,,
\begin{array}{l}{q_{0}=\sqrt{\frac{26-D}{6}-\vec{q}^{\;2}}~,}\\{p_{0_{\pm}}=q_{0}\pm\sqrt{q_{0}^{2}-\vec{p}^{\;2}+2\vec{q}\vec{p}-4}=\frac{1}{\sqrt{6}}\left(\sqrt{26-D-6\vec{q}^{\;2}}\pm\sqrt{2-D-6(\vec{q}-\vec{p})^{2}}\right)~.}\\\end{array}
\epsilon^{\mu\nu}{\partial}_{\mu}B_{\nu}^{ab}=-\epsilon^{\mu\nu}B_{\mu}^{ac}B_{\nu}^{cb}-{\textstyle\frac12}~\epsilon^{\mu\nu}(A_{\mu}^{ac}B_{\nu}^{cb}+B_{\mu}^{ac}A_{\nu}^{cb})~.
S=\intd^{6}x\sqrt{g}\left[{\frac{1}{2}}(\partial\phi)^{2}+{\frac{1}{2}}m^{2}\phi^{2}+{\frac{g_{4}}{4!}}\phi^{4}+\cdots\right]+\intd^{4}x\sum_{n=0}^{\infty}{\frac{\lambda_{2n}}{(2n)!}}\phi^{2n}
g=\frac{4\pi^{2}R}{e^{2}}\,\left(C_{AB}d\vec{r}_{A}\cdotd\vec{r}_{B}+(C^{-1})_{AB}(d\psi_{A}+\vec{\omega}_{AC}\cdotd\vec{r}_{C})(d\psi_{B}+\vec{\omega}_{BD}\cdotd\vec{r}_{D})\right).
\begin{array}{rl}{x^{\mu}(\tau,\sigma)=\sqrt{2\alpha^{\prime}}\left[\right.}&{q^{\mu}+\alpha_{0}^{\mu}(\tau-\sigma)+\bar{\alpha}_{0}^{\mu}(\tau+\sigma)}\\{}&{}\\{}&{\left.+\frac{1}{2}i{\sum_{n}}^{\prime}\frac{1}{n}\left(\alpha_{n}^{\mu}e^{-2in(\tau-\sigma)}+\bar{\alpha}_{n}^{\mu}e^{-2in(\tau+\sigma)}\right)\right]\,}\\\end{array}
0=\Omega_{ij}^{u}k_{\Sigma}^{i}k_{\Gamma}^{j}+{\frac{1}{2}}{f^{\Lambda}}_{\Sigma\Gamma}{\calP}_{\Lambda}^{u}-{\frac{1}{2}}\varepsilon^{uvz}{\calP}_{\Sigma}^{v}{\calP}_{\Gamma}^{z}
\left(\Gamma_{j}\right)_{ab}=\left(\begin{array}{cc}{0}&{{\mathbbE}_{j}}\\{-{\mathbbE}_{j}}&{0}\\\end{array}\right),\quad\quad\Lambda_{j}=\left(\begin{array}{cc}{1|{\mathbbE}_{j}}&{0}\\{0}&{1|{\mathbbE}_{j}}\\\end{array}\right).
k^{2}+m^{2}+\overline{A}(\overline{x})+\frac{i}{2}(\overline{\partial}_{\mu}\overline{A})(T^{\mu}-v_{g}^{\mu}/\omega)-\overline{{\partial^{\mu}\gamma_{\mu}}}-F^{2}-2ik^{\mu}F_{\mu}-i\gamma^{\mu}k_{\mu}-\gamma^{\mu}F_{\mu}=0.
\tilde{F}^{\mu\nu(1)}=-\frac12\,(\thetaF)F^{\mu\nu}-\frac14\,\theta^{\mu\nu}F^{2}+\theta_{\alpha\beta}F^{\mu\alpha}F^{\nu\beta}+(\theta^{\mu\beta}F^{\alpha\nu}-\theta^{\nu\beta}F^{\alpha\mu})F_{\alpha\beta}
H_{\mathrm{eff}}=\intd^{2}x_{\bot}\left(-\frac14\widetilde{A}_{a}\nabla_{\bot}^{2}\widetilde{A}_{a}+\frac{m^{2}}4\widetilde{A}_{a}\widetilde{A}_{a}+\frac3{32L}\lambda\left(\widetilde{A}_{a}\widetilde{A}_{a}\right)^{2}\right)\;,
D(x,y)=-\frac{\Gamma(\frac{N}{2}-1)}{4\pi^{N/2}}\,\frac{1}{N-4}\int_{0}^{1}d\rho\,\frac{1}{[(\rhox-y)^{2}]^{\frac{N}{2}-1}}\,[\rho^{N-4}-1].
e^{2\varphi}={\frac{8Q_{F}^{2}}{\pi^{2}Q_{H}^{2}}}\equiv\lambda^{6};\\Q_{H}\equiv{\frac{\sqrt{2}}{4\pi^{2}}}\int_{S^{3}}e^{-4\varphi/3}\starG,\Q_{F}\equiv{\frac{\sqrt{2}}{16\pi}}\int_{S^{3}}e^{2\varphi/3}\starF.
S\approx{\frac{p-2}{2}}\Omega_{p-1}\intd\tau[\sum_{k}m_{k}(\epsilon)v_{k}^{2}+(p-2)\Omega_{p-1}\sum_{k<\ell}q_{k}q_{\ell}{\frac{|\vec{v}_{k}-\vec{v}_{\ell}|^{2}}{|\vec{x}_{k}-\vec{x}_{\ell}|^{p-2}}}],
\mathrm{Tr}e^{-t\breve{H}^{2}(\lambda)}\simeq{\frac{1}{(4\pit)^{(D-1)/2}}}\sum_{n=0}^{\infty}\left[\breve{a}_{n}(\lambda)t^{n}+\breve{b}_{n}(\lambda)t^{n+\frac12}\right]~~~,
\delta_{\tilde{T}_{R}}x_{\alpha\dot{\alpha}}=\epsilon_{\tilde{T}_{R}}v_{\alpha}\bar{v}_{\dot{\alpha}},\quad\delta_{\tilde{T}_{R}}z_{\alpha\beta}=\epsilon_{\tilde{T}_{R}}v_{\alpha}v_{\beta},\quad\delta_{\tilde{T}_{R}}\bar{z}_{\dot{\alpha}\dot{\beta}}=\epsilon_{\tilde{T}_{R}}\bar{v}_{\dot{\alpha}}\bar{v}_{\dot{\beta}},
{\calZ}_{\delta}(x,z;p,\epsilon)\equiv\langle\alpha|\delta\rangle\,\theta_{\delta}(0)^{-1}\langle\psi(x_{1})\!\cdot\!\psi(x_{2})\prod_{i=1}^{4}p_{i}\!\cdot\!\psi(z_{i})\,\epsilon_{i}\!\cdot\!\psi(z_{i})\rangle_{\delta}\,.
E=\frac{1}{2m}(\dot{\vec{A_{1}^{2}}}+\dot{\vec{A_{2}^{2}}})+\vec{\dot{\phi}}^{2}+\frac{1}{2m}[\vec{A_{1}^{2}}\vec{A_{2}^{2}}-(\vec{A_{1}}\cdot\vec{A_{2}})^{2}]+[(\vec{A_{1}^{2}}+\vec{A_{2}^{2}})\vec{\phi^{2}}-(\vec{A_{1}}\cdot\vec{\phi})^{2}-(\vec{A_{2}}\cdot\vec{\phi})^{2}]+\frac{\kappa}{4}(\vec{\phi}^{2}-1)^{2}.
\displaystyle\mathop{\sum_{p\in\Lambda\ast}e^{[-\pi(p+\delta)^{t}C(p+\delta)]+2\piip^{t}\phi]}={V}_{\Lambda}^{-1}{\frac{1}{\sqrtdet{C}}{\sum}_{l\in\Lambda}e^{[-\pi(l+\phi)^{t}C^{-1}(l+\phi)-2\pii{\delta}^{t}(l+\phi)]}}\;\;\;\;\;\;\;\;\;\;\;\;}
g(z)=\frac{1}{2i}\left[\frac{\bar{H}_{\nu}^{(1)}(\lambdaz)}{\bar{H}_{\nu}^{(1)}(z)}+\frac{\bar{H}_{\nu}^{(2)}(\lambdaz)}{\bar{H}_{\nu}^{(2)}(z)}\right]\frac{h(z)}{C_{\nu}^{AB}(\lambda,z)},\quadf(z)=\frac{h(z)}{\bar{H}_{\nu}^{(1)}(z)\bar{H}_{\nu}^{(2)}(z)},
\frac{1}{4\pig^{2}}L_{\mathrm{rel}}^{\prime}=\frac{\tau_{2}(z)}{2}\left(|\dot{z}|^{2}+\dot{\chi}^{2}\right)+\frac{1}{2\tau_{2}(z)}\left(\dot{t}+\tau_{1}(z)\dot{\chi}\right)^{2},
{\tilde{Z}}_{2}(-\frac{1}{\tau})\left({\tilde{Z}}_{+}(-\frac{1}{\tau})-{\tilde{Z}}_{-}(-\frac{1}{\tau})\right)=\left(\sqrt{\frac{\tau}{i}}\right)^{-16}{\tilde{Z}}_{2}(\tau)\left({\tilde{Z}}_{+}(\tau)-{\tilde{Z}}_{-}(\tau)\right).
m_{b}=\frac{1}{2}\left[\sum_{i=0}^{k}\left({k\atopi}\right)\frac{(-1)^{i}N^{2i}r_{b}^{2k-2i-1}}{(2k-2i-1)}+\frac{(2k+1)}{\ell^{2}}\sum_{i=0}^{k+1}\left({k+1\atopi}\right)\frac{(-1)^{i}N^{2i}r_{b}^{2k-2i+1}}{(2k-2i+1)}\right]
e^{+2\sqrt{\pi}\sqrt{\frac{\pi}{\pi+gN}}U_{1a}\big(\delta(x)-\delta(y),C_{m_{d}}\alpha^{(1)}\big)}\prod_{I=2}^{N}e^{+2\sqrt{\pi}U_{Ia}\big(\delta(x)-\delta(y),C_{0}\alpha^{(I)}\big)}\;\rho(x-y)+\;O(m^{3})\;.
G_{\mu\nu}=\left(\begin{array}{ccc}{-\frac{2\pi\alpha^{\prime}}{\theta_{0}}}&{0}&{-\frac{2\pi\alpha^{\prime}}{\theta_{1}}}\\{0}&{\frac{2\pi\alpha^{\prime}}{\theta_{0}}+\biggl(\frac{2\pi\alpha^{\prime}}{\theta_{1}}\biggr)^{2}}&{0}\\{-\frac{2\pi\alpha^{\prime}}{\theta_{1}}}&{0}&{1}\\\end{array}\right),
\left.\mathrm{Res}\right|_{s\rightarrow-\frac12}\zeta_{\mathrmint}(s)+\left.\mathrm{Res}\right|_{s\rightarrow-\frac12}\zeta_{\mathrmext}(s)=\frac1{R^{2}}\left[-\frac{z^{3}}{3\pi}-\frac1{15}\frac{z}{\pi}\right]
\left[\widehat{x}_{\mu},\widehat{x}_{\nu}\right]=\theta_{\mu\nu}(\widehat{x})=\theta_{\mu\nu}^{(0)}+\theta_{\mu\nu}^{(1)}{}^{\rho}\widehat{x}_{\rho}+\theta_{\mu\nu}^{(2)}{}^{\rho\tau}\widehat{x}_{\rho}\widehat{x}_{\tau}+\ldots
\mathrm{l.h.s.}:\begin{array}{l}{R_{23}R_{13}K_{2}R_{12}K_{1}R_{21}R_{31}R_{32}Y_{3}R_{32}^{-1}R_{31}^{-1}}\\{+\eta_{3}R_{12}R_{13}K_{1}R_{31}{\calP}_{32}+\eta_{3}R_{12}{\calP}_{13}R_{23}R_{21}K_{2}\quad,}\\\end{array}
\beta_{10}^{2}=\frac{4}{35}\frac{(r-5L)(5r^{7}+25r^{6}L+97r^{5}L^{2}+485r^{4}L^{3}+2495r^{3}L^{4}+12475r^{2}L^{5}+62235L^{6}+7L^{7})}{(r^{2}-L^{2})^{4}}.
\sum_{t=1}^{N}\sum_{n=1}^{\infty}V_{mn}^{rt}V_{n0}^{ts}=V_{m0}^{rs},\\\\sum_{t=1}^{N}\sum_{m=1}^{\infty}V_{nm}^{rt}V_{ml}^{ts}=\delta_{n,l}\delta_{r,s}.
\lambda_{\pm}(x_{0},r;z;q)=\frac{1}{2}\left\{\sum_{n=0}^{\infty}\frac{(\frac{\rho_{+}}{\rho_{-}};q)_{n}}{(q;q)_{n}}(i\rho_{-}(1-q)z)^{n}\pm(\rho_{+}\leftrightarrow\rho_{-})\right\}.
\epsilon\simeq\frac{(2\pi)^{2}g_{s}^{2}F^{2}}{2(M_{s}^{2}A_{T})^{2}}\left(\frac{M_{P}}{\Psi}\right)^{2},\qquad\eta\simeq-\frac{2\pig_{s}F}{(M_{s}^{2}A_{T})}\left(\frac{M_{P}}{\Psi}\right)^{2}.
\begin{array}{l}{\varphi_{n}(t)=C\,e^{\frac{-i}{q}\int_{0}^{t}dt^{\prime}[\lambda_{n}-m(q)+V^{\prime\prime}(x)]}}\\{\lambda_{n}={\frac{2n\piq}{T}}+m(q)-{\frac{1}{T}}\int_{0}^{T}V^{\prime\prime}(x)\qquad\;;n\in{Z\!\!\!Z}.}\\\end{array}
\left(\frac{2}{\kappa}+\frac{1}{\xi}\right)\left[\left(1+\frac{\kappa}{4\xi}\right)\ddot{\xi}-\left(\frac{3}{2\xi}+\frac{2}{\kappa}+\frac{\kappa}{2\xi^{2}}\right){\dot{\xi}}^{2}\right]=0\,.
\begin{array}{rcl}{\Phi(q,r)}&{=}&{1+p_{q}q+p_{r}r+p_{qq}q^{2}+p_{rr}r^{2}+p_{qr}(qr+rq)+p_{qqq}q^{3}+p_{rrr}r^{3}}\\{}&{}&{\qquad+p_{qqr}(qqr+qrq+rqq)+p_{qrr}(qrr+rqr+rrq)+\ldots}\\\end{array}