harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
N=\left(\begin{matrix}{\vec{p}^{~2}+2\kappa^{2}q^{2}\frac{p_{2}^{2}}{\vec{p}^{~2}}}&{-2\kappa^{2}q^{2}\frac{p_{1}p_{2}}{\vec{p}^{~2}}}&{~0~}\\{-2\kappa^{2}q^{2}\frac{p_{1}p_{2}}{\vec{p}^{~2}}}&{\vec{p}^{~2}+2\kappa^{2}q^{2}\frac{p_{1}^{2}}{\vec{p}^{~2}}}&{~0~}\\{{}~0~}&{~0~}&{~0~}\\\end{matrix}\right)
\leq\;\omega^{\sum_{a=1}^{N}2n_{a}}\;\left\langle\prod_{b=1}^{N}\left[\chi_{+}^{(b)}[Q^{M,S}]\right]^{q_{b}}\left[\chi_{-}^{(b)}[Q^{M,S}]\right]^{2n_{b}-q_{b}}\right\rangle_{Q^{M,S}}\;.
\zeta^{\prime}(0)={\frac{Lm_{L}^{3}}{(4\pi)^{\frac{3}{2}}}}(\Gamma(-{\frac{3}{2}})+4\sum_{n=1}^{\infty}\Bigr({\frac{2}{nLm_{L}}}\Bigr)^{\frac{3}{2}}K_{\frac{3}{2}}(nLm_{L})).
\begin{aligned}{\beta}&{{}=a_{1}-\left(a_{2,1}+\frac14a_{2,2}\right)R_{0},}\\{{m_{0}}^{2}}&{{}=\frac{\beta}{3a_{2,1}+a_{2,2}},}\\{{m_{2}}^{2}}&{{}=-\frac{2\beta}{a_{2,2}},}\\\end{aligned}
\lambda_{2}(a)=\lambda_{2}^{(0)}(a)+k^{-2}\lambda_{2}^{(2)}(a)g_{a}^{\mu\nu}\partial_{\mu}\partial_{\nu}+k^{-4}\lambda_{2}^{(4)}(a)g_{a}^{\mu\nu}g_{a}^{\rho\sigma}\partial_{\mu}\partial_{\nu}\partial_{\rho}\partial_{\sigma}+\ldots,
\frac{\Gamma(-\alpha^{\prime}s)\Gamma(-\alpha^{\prime}t)}{\Gamma(1-\alpha^{\prime}s-\alpha^{\prime}t)}=\frac{1}{{\alpha^{\prime}}^{2}st}-\frac{\pi^{2}}{6}-\zeta(3)(s+t)\alpha^{\prime}+{\calO}({\alpha^{\prime}}^{2})\,.
{\bfs}=\left(\begin{matrix}{s}&{0}\\{0}&{s}\\\end{matrix}\right),~~\bar{\bfs}=\left(\begin{array}{cc}{\bar{s}}&{0}\\{0}&{\bar{s}}\\\end{array}\right),~~{\calC}=\left(\begin{array}{cc}{c}&{0}\\{0}&{c^{\prime}}\\\end{array}\right),~~\bar{\calC}=\left(\begin{array}{cc}{{\bar{c}}}&{0}\\{0}&{{\bar{c^{\prime}}}}\\\end{array}\right).
{\calL}=\bar{\psi}^{a}\left(i\gamma^{\mu}\partial_{\mu}-m_{0}\right)\psi^{a}+{\frac{g_{0}}{2N}}\left[\left(\bar{\psi}^{a}\psi^{a}\right)^{2}-\left(\bar{\psi}^{a}\gamma_{5}\psi^{a}\right)^{2}\right].
\gamma^{0}=\left(\begin{array}{cc}{\sigma^{3}}&{0}\\{0}&{\sigma^{3}}\\\end{array}\right)~,~~~\gamma^{j}=\left(\begin{array}{cc}{i\sigma^{j}}&{0}\\{0}&{i\sigma^{j}}\\\end{array}\right)~,~~~\tau=\left(\begin{array}{cc}{0}&{I}\\{-I}&{0}\\\end{array}\right)~,
S_{b}={\frac{-1}{16\piG}}\intd^{4}x\sqrt{-\tilde{g}}\left\{(\varphi_{+}^{2}-\varphi_{-}^{2})\\tilde{\calR}-{\frac{6q}{q+1/2}}\left[(\tilde{\partial}\varphi_{+})^{2}-(\tilde{\partial}\varphi_{-})^{2}\right]\right\}.
\left[h^{\alpha\beta}(x),h^{\mu\nu}(y)\right]=-i{\underbrace{\frac{1}{2}\left(\eta^{\alpha\mu}\eta^{\beta\nu}+\eta^{\alpha\nu}\eta^{\beta\mu}-\eta^{\alpha\beta}\eta^{\mu\nu}\right)}_{\displaystyleb^{\alpha\beta\mu\nu}}}D_{0}(x-y),
K_{\mathrm{m}}^{(2)}=-2{\frac{i\gamma^{0}}{h}}(igh)^{2}\sum_{n^{\prime}=0}^{n}\!\!{}^{\prime}\sum_{n^{\prime\prime}=0}^{n^{\prime}}\!\!{}^{\prime\prime}(-1)^{n+n^{\prime}}(A_{0})_{{\bfm},\overline{{n^{\prime}-1}}}(A_{0})_{{\bfm},\overline{{n^{\prime\prime}-1}}}\psi_{\overline{{\bfm}},n^{\prime\prime}}.
\int_{{\tilde{\calC}_{\alpha}}}^{}R_{\tilde{\calC}}\psi_{\tilde{\calC}}=2\pi\alpha\int_{0}^{\infty}{\frac{u_{\rho}^{\prime}}{u^{\frac{3}{2}}}}\psi_{\tilde{\calC}}(\rho)d\rho=2\pi\alpha\int_{0}^{\infty}{\frac{u_{\rho}^{\prime}}{u^{3/2}}}d\rho\int_{0}^{\rho}{\frac{u^{1/2}-\alpha}{u^{3/2}}}d\rho^{\prime}\equivX(\alpha,a)~~~.
\textstyle{\int}\,dx\,(\sigma\,C_{2})^{\ast}\,C_{1}\vert_{y=-\infty}^{y=+\infty}=\textstyle{\int}\,d\tilde{x}\,dy\,(p_{y}\,C_{2})^{\ast}\,C_{1}\vert_{q=-q_{0}}^{q=+q_{0}}=0\;.
\frac{\partial^{2}\bar{f}_{\epsilon}}{\partial\bar{t}^{2}}|_{P_{0}}=\frac{\partial^{2}\bar{f}_{\epsilon}}{\partial\bar{t}\partial\bar{x}}|_{P_{0}}=\frac{\partial^{2}\bar{f}_{\epsilon}}{\partial\bar{x}^{2}}|_{P_{0}}=0
h_{D}=\left(\begin{matrix}{ms(r)}&{-i\vec{\sigma}\cdot\vec{\nabla}+mi\vec{\tau}\cdot\hat{x}\,p(r)}\\{-i\vec{\sigma}\cdot\vec{\nabla}-mi\vec{\tau}\cdot\hat{x}\,p(r)}&{-ms(r)}\\\end{matrix}\right)
{\frac{{x_{n+2}}-1}{{x_{n+1}}\,{x_{n+2}}\,{x_{n+3}}-1}}\,\,\,=\,\,\,\,\,\,\,{\frac{{x_{n+1}}-1}{{x_{n}}\,{x_{n+1}}\,{x_{n+2}}-1}}\cdot{{x_{n}}\,{x_{n+1}\,x_{n+2}^{2}}}
\begin{array}{rl}{J=2:}&{G_{zz},\B_{zz},}\\{1:}&{G_{\muz},\G_{yz},\B_{\muz},\B_{yz},}\\{0:}&{G_{\mu\nu},\G_{\muy},\G_{yy},G_{z\bar{z}},}\\{}&{B_{\mu\nu},\B_{\muy},\\Phi,}\\{-1:}&{G_{\mu\bar{z}},\G_{y\bar{z}},\B_{\mu\bar{z}},\B_{y\bar{z}},}\\{-2:}&{G_{\bar{z}\bar{z}},\B_{\bar{z}\bar{z}},}\\\end{array}
{\frac{M_{s}^{-4}\pi^{4}2}{3}}\intd\tau_{2}\tau_{2}d^{4}\eta\delta(1-\sum_{i}\eta_{i})e^{{\frac{2\pi\tau_{2}}{M_{s}^{2}}}(s\eta_{1}\eta_{3}+t\eta_{2}\eta_{4})}e^{-{\frac{2\pi\tau_{2}}{M_{s}^{2}}}\sum_{i}m_{i}^{2}\eta_{i}}\equiv{\frac{\pi^{2}}{6}}B^{\prime}\.
A_{cc}\sim\bigg\{\frac{p_{1}^{+}\cdot(D\cdotp_{1})^{-}p_{2}^{-}\cdot(D\cdotp_{2})^{+}}{s-1}t+(p_{1}^{+}p_{2}^{-})^{2}\bigg\}\frac{\Gamma(s)\Gamma(t)}{\Gamma(s+t)}.
\chi_{r,s}^{(l)}=\sum_{m=0}^{\frac{N}{2}}C_{2m}^{l}(q)\left\{\sum_{j\in{\bfZ},m_{r,s}(j)\equivm}q^{\alpha_{r,s}(j)}-\sum_{j\in{\bfZ},m_{r,-s}(j)\equivm}q^{\alpha_{r,-s}(j)}\right\}
\begin{array}{llll}{\mathrm{for~}\phi=+1}&{~:~}&{(i^{k},1,\pm1,\pm1,\pm1,1)}&{,(k=0,...,3)~,}\\{\mathrm{for~}\phi=-1}&{~:~}&{(i^{k}e^{i\pi/4},1,\pm1,\pm1,\pm1,1)}&{,(k=0,...,3)\;,}\\\end{array}
\psi_{k}^{out}=\frac{1}{\sqrt{\pik_{s}}}2^{n_{s}-1}\Gamma\left(n_{s}\right)a_{s}^{-m}g_{S}^{-l}z_{S}^{2m}\left(\frack{k_{s}}\right)^{-n_{s}}+\frac1{\sqrt{\pik_{s}}}\frac{2^{n_{s}-2}\Gamma\left(n_{s}\right)}{n_{1}(1-n_{1})}a_{s}^{-m}g_{S}^{-l}z_{S}^{n_{1}-1}\left(\frac{k}{k_{s}}\right)^{2-n_{s}}(k_{s}\eta)_{.}^{2n_{1}}
\begin{array}{l}{{\calD}_{-}{\calX}^{--}{\calX}^{-}{\calD}_{-}=\mu_{-}^{-1}~,~~~~~~\mu_{-}{\calX}^{-}=q^{2}{\calX}^{-}\mu_{-}~,}\\{\mu_{-}{\calD}_{-}=q^{-2}{\calD}_{-}\mu_{-}~,~~~~~~\mu_{-}^{-1}\equiv1+(q^{-2}-1){\calX}^{-}{\calD}_{-}~,}\\\end{array}
[T^{a},T^{b}]=f^{abc}T^{c}\qquad[T^{a^{\prime}},T^{b^{\prime}}]=f^{a^{\prime}b^{\prime}c}T^{c}\qquad[T^{a},T^{b^{\prime}}]=f^{ab^{\prime}c^{\prime}}T^{c^{\prime}}
F_{g}=\!\!\!\sum_{\alpha_{j}>1,\,\beta_{j}>1}\!\!\!\left\langle\alpha_{1}\dots\alpha_{s};\beta_{1}\dots\beta_{l}|\alpha\beta\gamma\right\rangle_{g}\frac{M_{\alpha_{1}}\dotsM_{\alpha_{s}}J_{\beta_{1}}\dotsJ_{\beta_{l}}}{M_{1}^{\alpha}J_{1}^{\beta}d^{\gamma}}\,,\qquadg>1,
\zeta_{r}(s)={\frac{1}{3^{s}}}+{\frac{1}{5^{s}}}-{\frac{6}{\pi}}\int_{-\infty}^{\infty}{\frac{(k^{4}+8k^{2}+11)}{(k^{2}+9)(k^{2}+1)(k^{2}+4)^{s+1}}}\dk
\begin{aligned}{\Phi^{\prime}(\lambda-\mu,z)\Phi(\mu-\lambda,z)-\Phi(\lambda-\mu,z)\Phi^{\prime}(\mu-\lambda,z)}&{{}=-\wp^{\prime}(\lambda-\mu),}\\{\Phi^{\prime}(\lambda-\nu,z)\Phi(\nu-\mu,z)-\Phi(\lambda-\nu,z)\Phi^{\prime}(\nu-\mu,z)}&{{}=-\Phi(\lambda-\mu)\bigl[\wp(\lambda-\nu)-\wp(\nu-\mu)\bigr].}\\\end{aligned}
dS^{2}={\frac{1}{I}}\{{\frac{1}{f}}[e^{2k_{0}}dz\,d{\bar{z}}+\rho^{2}\,d\phi^{2}]-f(dt-2\sqrt2sA_{03}\,d\phi)^{2}\}+I^{2}(A_{3}\,d\phi+A_{4}\,dt+dx^{5})^{2},
{\calD}D_{n,\mp1/2}^{(j)}D_{.,\pm1/2}^{(1/2)}=-[D_{n,n^{\prime}}^{(j)}(J_{\alpha}^{(j)})_{n^{\prime},\mp1/2}]D_{.,\mu^{\prime}}^{(1/2)}[\frac{1}{2}\sigma_{3},[\frac{1}{2}\sigma_{3},\sigma_{\alpha}]]_{\mu^{\prime},\pm1/2}
I_{k}\left(\mu,d\right)=-\frac{\Gamma\left(\frac{d}{2}+k\right)\mu^{\frac{d-2}{2}}}{4k^{\frac{d+2}{2}}}\frac{d}{d\lambda}\left[\lambda^{-\frac{d}{2}}\,W_{-k,\frac{d-1}{2}}\left(4\muk\lambda\right)\right]_{\lambda=1}.
\tau^{(1)}:=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\;,\;\tau^{(2)}:=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right)\;,\;\tau^{(3)}:=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\;.
D_{L}\equivD_{L}(\Omega_{B})=\frac{V_{0}^{\prime}(1+z)}{2H_{0}(\omega+1)V_{0}^{\frac{1}{2}+\frac{1}{3(\omega+1)}}}\int{\phi(0)}^{\phi(z)}\frac{d\phi}{V^{\frac{3\omega+1}{6(\omega+1)}}},
S_{\mathrm{BI}}=-T_{p}\int\mathrm{Tr}\left(e^{-\phi}\sqrt{-\mathrm{det}(P[E_{ab}+E_{ai}(Q^{-1}-\delta)^{ij}E_{jb}]+(2\pi\alpha^{\prime})F_{ab})\mathrm{det}(Q_{j}^{i})}\right)\,.
i_{e}{}_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}^{N_{1}N_{2}N_{3}N_{4}}{}^{(a)}=\sum_{\Lambda}\M_{e}{}_{\Lambda}^{(a)}\{C}_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}^{N_{1}N_{2}N_{3}N_{4}\,\Lambda},
X_{ij}(x,y)~=~e^{2}\left(\begin{array}{cccc}{-1}&{0}&{0}&{0}\\{0}&{-1}&{0}&{0}\\{e^{2}}&{0}&{1}&{0}\\{0}&{e^{2}}&{0}&{-1}\\\end{array}\right)\delta(x-y)~\equivX_{ij}^{0}\delta(x-y).
S=\left(\frac{H_{1}H_{2}}{H_{0}^{2}}\right)^{\frac13},\qquadT=\left(\frac{H_{0}H_{2}}{H_{1}^{2}}\right)^{\frac13},\qquadU=\left(\frac{H_{0}H_{1}}{H_{2}^{2}}\right)^{\frac13}.
\left(\begin{array}{c}{\hat{\alpha}(k)}\\{\hat{\beta}(k)}\\\end{array}\right)=\left(\begin{array}{c}{{a}(k)}\\{i}\\\end{array}\right)\hat{\xi}(k)+\left(\begin{array}{c}{-i{b}(k)}\\{1}\\\end{array}\right)\hat{\eta}(k)
\left(\begin{array}{c}{M_{1}}\\{M_{2}}\\{M_{3}}\\{M_{4}}\\\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{crrr}{1}&{1}&{-1}&{-1}\\{1}&{-1}&{1}&{-1}\\{1}&{1}&{1}&{1}\\{1}&{-1}&{-1}&{1}\\\end{array}\right)\left(\begin{array}{c}{L_{1}}\\{L_{2}}\\{L_{3}}\\{L_{4}}\\\end{array}\right)
\begin{array}{rl}{P_{\alpha}=}&{(R_{12}-\alpha)(R_{23}R_{12}-\alphaR_{23}+\alpha^{2})}\\{=}&{(R_{23}-\alpha)(R_{12}R_{23}-\alphaR_{12}+\alpha^{2})}\\{=}&{(R_{23}R_{12}-\alphaR_{12}+\alpha^{2})(R_{23}-\alpha)}\\{=}&{(R_{12}R_{23}-\alphaR_{23}+\alpha^{2})(R_{12}-\alpha)}\\\end{array}
Z\;=\;\prod_{j=1}^{M}\Gamma_{\,j},E\;=\;\prod_{j=1}^{M}V_{\,j},V_{\,j}\;=\;\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{0}&{1}\\{0}&{1}&{0}\\\end{array}\right)_{\,j}.
S_{\mathrm{WZ}}=\frac{T_{2}}{2\kappa}\left[\int{d^{3}x\left(\frac{1}{2\pi^{2}\alpha^{\prime}}\frac{Z}{H_{2}H_{6}}-1\right)}+\int_{\mathcal{M}_{3}}2\pi\alpha^{\prime}\mathcal{C}_{\varphi}d\varphi\wedgeF\right]\,.
N_{A}^{\prime}=\int_{{\calC}_{A}^{(2)}}H_{NS}^{(2)}=\int_{{\calC}_{A}^{(2)}}F_{3}^{(2)},\quadN_{A}=\int_{{\calC}_{A}^{(2)}}H_{R}^{(2)}=\int_{{\calC}_{A}^{(2)}}F_{11}^{(2)}\,,
\int\frac{d^{2\sigma}Q}{(2\pi)^{2\sigma}}\frac{Q_{\mu}}{(Q^{2}+\vec{Q}^{2})}\=0\,,\qquad\int\frac{d^{2\sigma}K}{(2\pi)^{2\sigma}}\frac{K_{\mu}}{(K^{2}+\vec{K}^{2})}\=0\,,
P_{Y_{(2r)}}^{I_{1}...I_{r},J_{1}...J_{r}}\delta^{I_{r+1}J_{r+1}}=\sum_{n=0}^{r+1}A_{r,n}P_{Y_{(2r+2-n,n)}}^{I_{1}...I_{r}I_{r+1},J_{1}...J_{r}J_{r+1}}+\mathrm{contraction~terms}
\frac{1}{c_{0}^{2}}\left(\frac{\ddot{a}_{0}}{a_{0}}+\frac{\dot{a}_{0}}{a_{0}}\left(\frac{n-1}{2}\frac{\dot{a}_{0}}{a_{0}}-\frac{\dot{c}_{0}}{c_{0}}\right)\right)=\frac{1}{2{\calR}^{n}}\frac{d}{d{\calR}}\Big(H^{2}{\calR}^{n+1}\Big),
\Gamma^{\mu}\left(D_{\mu}-\partial_{\mu}\Phi\right)\lambda_{\mp}+{\frac{\sqrt{2}}{4}}\Gamma^{\mu}\left(\pm\partial_{\nu}\Phi\Gamma^{\nu}-{\frac{1}{6}}H_{\alpha\beta\gamma}\Gamma^{\alpha\beta\gamma}\right)\psi_{\pm\mu}=0,
{\calG}_{\|}^{[i]}=\{h_{j},\,j\not=i;\;E_{-\vec{e}_{p}+\vec{e}_{q}},\,q\leqp\leqi\mathrm{~and~}q\geqi+1;\;E_{\vec{e}_{p}-\vec{e}_{q}},\,p<q\};
\mathcal{L}=\sqrt{-g}\left[R-\frac{1}{48}G^{MNPQ}G_{MNPQ}\right]+\frac{1}{6\cdot3!4!^{2}}\epsilon^{M_{1}\ldotsM_{11}}G_{M_{1}\ldotsM_{4}}G_{M_{5}\ldotsM_{8}}A_{M_{9}\ldotsM_{11}}.\nonumber
\Pi_{A}^{\Sigma}=\Pi_{00}^{\Sigma}{\frac{p_{b}^{2}}{{\bfp}^{2}}}\;\;\;\Pi_{B}^{\Sigma}=\Pi_{ii}^{\Sigma}-\Pi_{00}^{\Sigma}{\frac{p_{0b}^{2}}{{\bfp}^{2}}}
\varphi\equiv\left[\begin{matrix}{C^{\alpha}}&{\bar{C}_{\alpha}}&{}&{}\\{B}&{\bar{B}}&{Y}&{\bar{Y}}\\\end{matrix}\right]\quad\Rightarrow\quad\star\varphi=\left[\begin{matrix}{-iC^{\alpha}}&{i\bar{C}_{\alpha}}&{}&{}\\{-\bar{B}}&{-B}&{-\bar{Y}}&{-Y}\\\end{matrix}\right],
\lambda~\rightarrow~\frac{a\lambda+b}{c\lambda+d}~~~,~~~F_{\mathrm{NS}}~\rightarrow~d\,F_{\mathrm{NS}}-c\,F_{\mathrm{R}}~~~,~~~F_{\mathrm{R}}~\rightarrow~-b\,F_{\mathrm{NS}}+a\,F_{\mathrm{R}}~~.
\Theta={\frac{1}{16\pi^{2}}}\left[2(2a-c)R_{\mu\nu}R^{\mu\nu}+{\frac{1}{3}}(c-3a)R^{2}+(c-a)R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right]+{\frac{c}{6\pi^{2}}}(F_{\mu\nu})^{2}.
\deltaF[A_{+},\,A_{-}]=-\alpha[A_{+},\,A_{-}]\,(\overline{{{\mathrm{e}}^{-z}}}\,\deltaA_{+}+\overline{{{\mathrm{e}}^{z}}}\,\deltaA_{-})+\frac{1}{8}\,A_{-}\,(A_{-}-2\,A_{+})\,\deltaA_{+}\,.
\Bigl[\frac{1}{R^{2}}e^{R\sigma}\partial_{5}(e^{-R\sigma}\partial_{5})-\Bigl(\frac{1}{4}\pm\frac{1}{2}\Bigr)k^{2}\Bigr]f_{1,2}^{(n)}=e^{2R\sigma}m_{n}^{2}f_{1,2}^{(n)}\;,
t(\tau)=\frac{E}{(1+4H^{2}E^{2})^{\frac{1}{4}}(1+H^{2}r_{m}^{2})}\Pi\left(\frac{H^{2}r_{m}^{2}}{1+H^{2}r_{m}^{2}},\;(1+4H^{2}E^{2})^{\frac{1}{4}}\tau,\;k\right),
(\bar{C}_{\alpha},{\calT}_{\beta}^{\dagger})_{\Omega}=\frac{1}{2}[{\calT}_{\alpha},{\calT}_{\beta}^{\dagger}],\quad({\calT}_{\beta},\bar{C}_{\alpha}^{\dagger})_{\Omega}=\frac{1}{2}[{\calT}_{\beta},{\calT}_{\alpha}^{\dagger}],
{\calL}_{M}=-\frac14F_{\mu\nu}^{a}F^{\mu\nu,a}-\frac12(D_{\mu}\phi^{a})(D^{\mu}\phi^{a})-\frac12(\partial_{\mu}\chi^{a})(\partial^{\mu}\chi^{a})-V(\phi^{a},\chi^{a})\,
{\bfA_{2}A_{1}}f(\bar{z}_{2})=\intd\bar{z}_{0}dz_{0}exp_{q}\{-q^{k}\bar{z}_{0}z_{0}\}\left[\intd\bar{z}_{1}dz_{1}exp_{q}\{-\bar{z}_{1}z_{1}\}A_{2}(\bar{z}_{2},z_{1})A_{1}(\bar{z}_{1},z_{0})\right]f(\bar{z}_{0})\.
G_{F}(x,y)=e^{-{\frac{ie^{2}}{2}}\intd^{2}z_{1}\intd^{2}z_{2}j^{\mu}(z_{1};x,y)G_{\mu\nu}(z_{1},z_{2})j^{\nu}(z_{2};x,y)}~S_{F}(x,y),
Q=\intd^{3}x\left[\sum_{l=1}^{12}\left({\mathbf{\sigma}}\cdot{\bfF}^{\daggerl}\psi^{l}(x)\right)+\sigma^{\mu}\partial_{\mu}\phi^{\dagger}(x)\cdot\psi_{L}(x)+\sigma^{\mu}\partial_{\mu}\phi_{s}^{\dagger}(x)\cdot\psi_{L}(x)+\mathcal{W}~\psi^{1}\right].
\chi_{\Lambda}(C_{m}^{\Lambda_{0}})=\sum_{(\lambda+\lambda_{0}^{\prime},\kappa+\kappa_{0},-s)\inD^{+}}\sum_{s=0}^{\infty}\,\alpha_{\lambda_{0}^{\prime},s}^{m}(\Lambda)(I\otimes\mathrm{tr})\{(I\otimes\pi_{\Lambda_{0}}(q^{2h_{\rho}}))P[\lambda_{0}^{\prime},s]\}
{\cal{L}}_{0}={\mathrm{Tr}}\left\{{\frac{1}{2g}}{\frac{dX^{i}}{dt}}{\frac{dX^{i}}{dt}}+{\frac{g}{4}}\left[X^{i},X^{j}\right]^{2}+\theta^{T}{\frac{d\theta}{dt}}-\theta^{T}\gamma_{i}\left[\theta,X^{i}\right]\right\}\quad.
dz=\frac{1}{2\pi}\sum_{i}N_{i}\frac{d}{dS_{i}}Gdx=-\frac{1}{2\pii}\sum_{i}N_{i}\frac{d}{dS_{i}}\Big[R\Big(x+i\frac{m}{2}\Big)-R\Big(x-i\frac{m}{2}\Big)\Big]\,
\begin{array}{c}{{\tilde{\calH}}=g_{A\bar{B}}V^{A}{\bar{V}}^{\bar{B}}+iV_{,C}^{A}g_{A\bar{B}}{\bar{V}}_{,\bar{D}}^{\bar{B}}\theta^{C}{\bar{\theta}}^{\bar{D}}-}\\{-R_{A{\bar{B}}C{\bar{D}}}V_{,C}^{A}{\bar{V}}_{;\bar{D}}^{\bar{B}}\theta^{A}\theta^{C}{\bar{\theta}}^{\bar{B}}{\bar{\theta}}^{\bar{D}},}\\\end{array}
H_{D}^{F}=\tilde{\lambda}^{\mu}(\tau){\calH}_{\mu}(\tau)-\frac12\tilde{\lambda}^{\mu\nu}(\tau){\calH}_{\mu\nu}(\tau)+\int{d^{3}\sigma\Big[-A_{\tau}(\tau,\vec{\sigma})\Gamma(\tau,\vec{\sigma})+\mu_{\tau}(\tau,\vec{\sigma})\pi^{\tau}(\tau,\vec{\sigma})\Big]},
\mathcal{D}_{\beta}=\partial_{\beta}+\frac{1}{4}\partial_{\beta}x^{+}\left[\left(\omega_{\rho\sigma+}-\frac{1}{2}H_{\rho\sigma+}\sigma^{3}\right)\Gamma^{\rho\sigma}+\frac{1}{2\cdot4!}F_{\lambda\nu\rho\sigma}\Gamma^{\lambda\nu\rho\sigma}\sigma^{1}\Gamma_{+}\right].
{L}_{p\timesp}(\lambda)=\left(\begin{array}{cccccc}{\lambda+S_{0}^{(1)}}&{S_{-}^{(1)}}&{S_{-}^{(12)}}&{S_{-}^{(13)}}&{\ldots}&{S_{-}^{(1,p-1)}}\\{S_{+}^{(1)}}&{}&{}&{}&{}&{}\\{S_{+}^{(12)}}&{}&{}&{}&{}&{}\\{S_{+}^{(13)}}&{}&{}&{\hat{A}}&{}&{}\\{\vdots}&{}&{}&{}&{}&{}\\{S_{+}^{(1,p-1)}}&{}&{}&{}&{}&{}\\\end{array}\right)
\Gamma(p)=-\int\frac{d^{3}q}{(2\pi)^{3}}D(p-q)\,\gamma_{\mu}\,S\left[-(m+\frac{\delta}{2})iv_{\mu}+q_{\mu}\right]\Gamma(q)\,S\left[(m+\frac{\delta}{2})iv_{\mu}+q_{\mu}\right]\gamma_{\mu}.
f^{(III)}=\frac{e^{a[(k_{\|}^{2}+m^{2})^{1/2}-(k_{\|}^{2}+M_{H}(\upsilon)^{2})^{1/2}]}}{2(k_{\|}^{2}+m^{2})^{1/2}}\left(\frac{{f^{(II)}(a)}^{\prime}}{(k_{\|}^{2}{+}M_{H}(\upsilon)^{2})^{1/2}}+f^{(II)}(a)\left[1+\frac{(k_{\|}^{2}+m^{2})^{1/2}}{(k_{\|}^{2}{+}M_{H}(\upsilon)^{2})^{1/2}}\right]\right).
J_{2}^{B}=\left(\begin{array}{cc}{\rhoD+D\rho-8\varepsilon^{2}D^{3}}&{3u\,D+2u_{x}}\\{3Du-2u_{x}}&{\begin{array}{c}{8(\rho^{2}D+D\rho^{2})+8\varepsilon^{4}D^{5}}\\{-\varepsilon^{2}[5(\rho\,D^{3}+D^{3}\rho)-3(\rho_{xx}D+D\rho_{xx})]}\\\end{array}}\\\end{array}\right)
\frac{\partial}{\partialX^{\mu_{0}}}\phi_{b}^{\mu_{0}\mu_{1}\ldots\mu_{p}}(X)=-g_{b}\sum_{i=0}^{p}\intd\xi_{b}^{0}d\xi_{b}^{1}\ldots\hat{d\xi_{b}^{i}}\ldotsd\xi_{b}^{p}\left[K_{bi}^{\mu_{1}\mu_{2}\ldots\mu_{p}}G((X-X_{b})^{2})\right]_{\xi_{b}^{i}=0}^{\xi_{b}^{i}=l_{b}^{i}}.
\begin{array}{rrl}{f_{m}(u)}&{:=}&{\displaystyle\sum_{j=0}^{n-1}\omega^{mj}R(z)_{0j}^{j0}}\\{}&{=}&{\frac{1}{\bar{\kappa}(u)}\frac{(\omega^{-m}q^{2}u^{-2/n};t^{2})_{\infty}(t^{2}\omega^{m}q^{-2}u^{2/n};t^{2})_{\infty}}{(\omega^{m}u^{2/n};t^{2})_{\infty}(t^{2}\omega^{-m}u^{-2/n};t^{2})_{\infty}}.}\\\end{array}
\omega^{\nu\sigma}x_{\nu}\partial_{\sigma}h_{\mu|\alpha(n)}^{n}+\omega_{\mu\nu}h^{n\nu|}{}_{\alpha(n)}-{\frac{n}{4}}\omega^{\nu\sigma}\varepsilon_{\nu\sigma\rho}e^{\rho|}{}_{\alpha}{}_{\mu|\alpha(n-1)\beta}^{n},
\langlen_{x,\alpha}n_{0,\beta}\rangle^{(T)}=\phi^{(\alpha,\beta)}\Biggl\{\left(\frac{\gamma}{2\pi}\int_{-\pi}^{\pi}dk\,\vartheta(k)\right)^{2}-\left|\frac{\gamma}{2\pi}\int_{-\pi}^{\pi}dk\,\vartheta(k)\,e^{-ixk}\right|^{2}\Biggr\}
\int_{\calR}d^{n}p^{\prime}\,(\xi_{p}|\hat{e}_{pp^{\prime}}|{\tilde{\pi}}_{p^{\prime}})=\sum_{p^{\prime}\in{\calR}}\,\xi^{\alpha}(p)\,({\calP}\int_{e}A)_{\alpha}^{\beta}\,i[\xi]_{\beta}(p^{\prime})
\frac{1}{2M}\left(-i\hbar\frac{\partial}{\partialx_{j}}-\hbarA_{j}^{a}\hat{T}_{a}\right)^{2}\psi+\frac{\hbar^{2}}{2Mr^{2}}{\hat{T}}^{2}\psi-\frac{\alpha}{r}\psi=\epsilon\psi
\epsilon_{RPA}({\bf{q}},{\tilde{\omega}})=1+\frac{v_{{\bf{q}}}}{V}\sum_{{\bf{k}}}\frac{n_{F}({\bf{k}}+{\bf{q}}/2)-n_{F}({\bf{k}}-{\bf{q}}/2)}{{\tilde{\omega}}-\frac{{\bf{k.q}}}{m}}
Z_{K}(L)=\frac{C^{K}}{K!}\int_{0}^{\frac{L}{\tau}}d\rho_{K}\cdots\int_{0}^{\frac{L}{\tau}-\sum_{j=2}^{K}\rho_{j}}d\rho_{1}\prod_{j=1}^{K}\rho_{j}^{b-2}\,\left(L-\tau\sum_{l=1}^{K}\rho_{l}\right)^{K}\,.
H_{\mathrm{IIT}}=\frac{1}{2m}g^{ij}\left(\Pi_{i}-A_{i}-\lambda^{s}\frac{\partialf_{s}}{\partialq^{i}}\right)\left(\Pi_{j}-A_{j}-\lambda^{t}\frac{\partialf_{t}}{\partialq^{j}}\right)+V(q)+u^{s}\Pi_{s}^{\prime}\;,
I(p,m,\alpha,\varepsilon)=i\int\frac{d^{4}k}{(\alphak_{0}^{2}-|\mathbf{k}|^{2}-m^{2}+i\varepsilon)[\alpha(k+p)_{0}^{2}-|\mathbf{k}+\mathbf{p}|^{2}-m^{2}+i\varepsilon]}
\sigma_{n_{j}n_{j}}=\frac12\mathrm{T}\mathrm{r}\left[\Lambda_{j}{\bfS}\left(\Lambda_{j}{\bfS}-{\bfI}_{2N}\right)\right]+\frac12{\bfy}{\bfR}\left(2{\bfS}\Lambda_{j}-{\bfI}_{2N}\right){\bfS}\Lambda_{j}{\bfS}\Sigma_{x}{\bfR}{\bfy},
\begin{array}{lrcl}{}&{A_{[\mu_{1}...\mu_{p}][\nu_{1}...\nu_{q}]}}&{=}&{A_{\mu_{1}...\mu_{p}\nu_{1}...\nu_{q}}\;\;\;\;\;\;,\;\;\;\;\;\;A_{[\mu_{1}...\mu_{p}\nu_{1}]\nu_{2}...\nu_{q}}\;\;=\;\;0}\\{{\mathrm{and}}\;\;\;\;\;\;\;\;}&{A_{\mu_{1}...\mu_{p}\nu_{1}...\nu_{q}}}&{=}&{A_{\nu_{1}...\nu_{q}\mu_{1}..\mu_{p}}\;\;\;\;\;\;\;\;{\mathrm{if}}\;\;p=q}\\\end{array}
{\calN}_{2}=\sum_{r=n_{c}+2}^{n_{f}}{}_{n_{f}}\!C_{r}\,(r-n_{c}-1)=\sum_{r=0}^{n_{f}}{}_{n_{f}}\!C_{r}\,(r-n_{c}-1)+\sum_{r=0}^{n_{c}+1}{}_{n_{f}}\!C_{r}\,(n_{c}+1-r).
U_{\underline{{m}}}^{--}\sigma_{\underline{{\mu}}\underline{{\nu}}}^{\underline{{m}}}=2V_{\underline{{\mu}}\dot{q}}^{~-}V_{\underline{{\nu}}\dot{q}}^{~-},\qquadU_{\underline{{m}}}^{++}\tilde{\sigma}^{\underline{{m}}\underline{{\mu}}\underline{{\nu}}}=2V_{\dot{q}}^{+\underline{{\mu}}}V_{\dot{q}}^{+\underline{{\nu}}}.\qquad
\tilde{\hat{g}}_{t\varphi}=\frac{1}{2\Omega}\left\{\hat{g}_{tt}\left[\hat{A}^{I}{}_{t}\hat{A}^{I}{}_{\varphi}-2\hat{B}_{t\varphi}\right]-\hat{g}_{t\varphi}\hat{A}^{I}{}_{t}\hat{A}^{I}{}_{t}\right\}\,,
\mathbf{Q}_{m}^{a}(\zeta)\hat{W}_{m}(\zeta)=-\mathrm{~\frac{1}{2}~}m^{2}(\sigma^{A})_{~b}^{a}\mathbf{Q}_{m}^{b}(\zeta)\mathbf{Q}_{A}(\zeta),\qquad\mathbf{Q}_{A}(\zeta)\hat{W}_{m}(\zeta)=\hat{W}_{m}(\zeta)\mathbf{Q}_{A}(\zeta),
\intd^{4}x\sqrt{-g}\frac{1}{2\kappa^{2}}(R(g)+\overline{{\hat{\psi}}}^{\mu}\hat{\gamma}_{\mu\nu\rho}\hat{D}^{\nu}\hat{\psi}^{\rho}+\overline{{\hat{\psi}^{\mu}}}\hat{\gamma}_{\mu\nu}\gamma^{5}\hat{D}^{\mu}\hat{\psi}_{5}+\overline{{\hat{\psi}^{5}}}\gamma_{5}\hat{\gamma}_{\mu\nu}\hat{D}^{\mu}\hat{\psi}^{\nu})
{\calI}=\frac{1}{4\pi}\int\left(\frac{1+|z|^{2}}{1+|R|^{2}}\left|\frac{\mathrm{d}R}{\mathrm{d}z}\right|\right)^{4}\frac{2i~\mathrm{d}z\mathrm{d}{\bar{z}}}{(1+|z|^{2})^{2}}.
{\calL}=\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi-\frac{1}{2}m^{2}\varphi^{2}-\frac{\lambda}{4!}\varphi^{4}+\frac{1}{2}\deltaZ~\partial_{\mu}\varphi\partial^{\mu}\varphi-\frac{1}{2}\deltam^{2}\varphi^{2}-\frac{1}{4!}\delta\lambda~\varphi^{4}\,
{\hat{\calM}}^{ab}:=i{\pi}_{(i)}^{({A^{\prime}}}{\frac{\partial}{\partial{\pi}_{{B^{\prime}})}^{(i)}}}\epsilon^{AB}+i{\bar{\pi}}_{(i)}^{({A}}{\frac{\partial}{\partial{\bar{\pi}}_{{B)}}^{(i)}}}\epsilon^{{A^{\prime}}{B^{\prime}}},
K(t,t^{\prime})=K_{s}^{(1)}(t,t^{\prime})+\sum_{r\neqs}\int_{C_{r}}K_{s}^{(1)}(t,t_{1})\frac{d\theta_{1}dz_{1}}{2\pii}K(t_{1},t^{\prime})-\varphi_{s}(t)\int_{C_{b}^{(s)}}f_{s}(t_{1})\frac{d\theta_{1}dz_{1}}{2\pii}K(t_{1},t^{\prime})
Z=\int\prod_{n=1}^{\mathcal{N}}D[P_{n}]D[\pi^{ij}]D[h_{ij}]D[N^{i}]D[N]\delta(\chi)\prod_{i=1}^{2}\delta(\chi^{i})|\mathrm{Det}\{\chi^{\mu},H_{\nu}\}|e^{iS}.
\begin{array}{rl}{{\calH}=\frac{1}{2}\pi_{1}^{2}+}&{\frac{1}{8\pi}\left(\partial_{1}\phi_{+}+\partial_{1}\phi_{--}\frac{2\pi}{e\lambda}\partial_{1}\pi_{1}+\frac{2\pi}{e\lambda}\partial_{1}\pi_{1}\right)^{2}}\\{+}&{\frac{1}{8\pi}\left(\partial_{1}\phi_{+}-\partial_{1}\phi_{--}2e\lambdaA^{1}\right)^{2}\,}\\\end{array}
I=\int\!\!d^{2\omega}k_{1}\:d^{2\omega}k_{2}\frac{k_{1}^{\mu}\,k_{1}^{\nu}\,k_{2}^{\rho}}{(k_{1}-q)^{2}(k_{1}-k_{2})^{4}k_{2}^{2}\,[[n(k_{1}+k_{2}+s)]]\,[[n(k_{1}+p)]]\,[[nk_{1}]]^{2}},
\prod_{i,j}\frac{\partial^{2}\tilde{A}}{\partialp_{i}\partial\tilde{p}_{j}}\,dp_{i}\,d\tilde{p}_{j}=\prod_{i,j}\frac{\partial^{2}\tilde{A}}{\partialp_{k}^{\prime}\partial\tilde{p}_{l}^{\prime}}dp_{i}^{\prime}\,d\tilde{p}_{j}^{\prime}
D_{\alpha\beta}F_{{\mathbfi}{\mathbfj}}=i(D_{\alpha\beta}D_{{\mathbfi}{\mathbfj}}\Phi-D_{\alpha\beta}{\bar{D}}_{{\mathbfi}{\mathbfj}}{\bar{\Phi}})=8\partial_{(\alpha\dot{\alpha}}D_{\beta)({\mathbfi}}{\bar{D}}_{{\mathbfj})}^{\dot{\alpha}}{\bar{\Phi}}=0
{\cal{R}}_{GB}^{2}={\frac{48M^{2}(r^{2}-A^{2}{cos}^{2}\theta)[(r^{2}+A^{2}{cos}^{2}\theta)^{2}-16r^{2}A^{2}{cos}^{2}\theta]}{(r^{2}+A^{2}{cos}^{2}\theta)^{6}}}
S_{cs}[\Sigma]=c_{1}\Lambda^{2}\int_{\sigma}\sqrt{g}-c_{2}\frac{\Lambda^{2}}{m^{2}}\int_{\sigma}\sqrt{g}(\partial_{a}t_{\mu\nu})^{2}+c_{3}me^{2}\oint_{C}d\tau\left(\frac{dx_{\mu}}{d\tau}\frac{dx_{\mu}}{d\tau}\right)^{\frac{1}{2}}
\sum_{l^{\prime},m^{\prime}}g_{\mu\nu}V_{\tau^{\prime\prime}l^{\prime\prime}m^{\prime\prime},\tau^{\prime}l^{\prime}m^{\prime}}^{\mu}W_{\tau^{\prime}l^{\prime}m^{\prime},\taulm}^{\nu}=b_{\tau\tau^{\prime}}v_{\tau\tau^{\prime}}w_{\tau^{\prime}\tau}\delta_{\tau^{\prime\prime}\tau}\delta_{l^{\prime\prime}l}\delta_{m^{\prime\prime}m},
H=\intd^{3}{x}\,\biggl[\,\frac{1}{2}\,\left(P_{ai}-\frac{\theta}{8\pi^{2}}\,B_{ai}^{(+)}\right)^{2}+\left(\sum_{n=0}^{\infty}(1/g)^{n}\,a_{i}^{(n)}(S,P-\frac{\theta}{8\pi^{2}}\,B^{(+)})\right)^{2}+\frac{1}{2}\,V(S)\,\biggr]\,,
\left({\frac{\dot{a}}{a}}\right)^{2}={\frac{1}{\rho^{2}}}\left({\frac{dt}{d\eta}}\right)^{2}\left({\frac{d\rho}{dt}}\right)^{2},\\\\\\\\\{\frac{\ddot{a}}{a}}=\left[1+{\frac{1}{2}}a{\frac{\partial}{\partiala}}\right]\left({\frac{\dot{a}}{a}}\right)^{2}.
{\calS}_{M}[{\vec{n}}]=s\sum_{\vec{r}}{\calS}_{WZ}[{\vec{n}}({\vec{r}})]-{\int}_{0}^{T}dx_{0}Js^{2}\sum_{<{\vec{r}},{\vec{r}}^{\prime}>}\left\{{\vec{n}}_{\perp}({\vec{r}},x_{0}).{\vec{n}}_{\perp}({\vec{r}}\;^{\prime},x_{0})+\lambda{\vec{n}}_{3}({\vec{r}},x_{0}).{\vec{n}}_{3}({\vec{r}}\;^{\prime},x_{0})\right\}
{\calE}[\phi]=\intdx\left[\frac{1}{2}\left(\frac{d\phi_{1}}{dx}\right)^{2}+\frac{1}{2}\left(\frac{d\phi_{2}}{dx}\right)^{2}+\left(4\phi_{1}^{2}+2\sigma\phi_{2}^{2}-1\right)^{2}+16\sigma^{2}\phi_{1}^{2}\phi_{2}^{2}\right]
P_{0}^{i}\,=\,{\frac{\Pi_{0}^{i}}{\epsilon}}\,\,\,,\,\,\,P_{1}^{i}\,=\,\Pi_{1}^{i}\,\,\,\,,...\,\,\,,P_{n}^{i}\,=\,\Pi_{n}^{i}\,\,\,\,,\,...\,\,,P_{N-1}^{i}\,=\,\Pi_{N-1}^{i}\,\,,\,\,P_{N}^{i}\,=\,{\frac{\Pi_{N}^{i}}{\epsilon}}\,\,.
S=\intd^{4}x\sqrt{g}\left({\frac{m_{s}^{8}v_{1}v_{2}v_{3}}{(2\pi)^{7}\lambda^{2}}}R+{\frac{1}{4}}{\frac{m_{s}^{4}v_{2}v_{3}}{(2\pi)^{5}\lambda}}F^{2}+{\frac{1}{4}}{\frac{m_{s}^{4}v_{1}v_{3}}{(2\pi)^{5}\lambda}}\tilde{F}^{2}+\dots\right)
\begin{array}{rcl}{\delta\bar{\chi}}&{=}&{-\bar{\epsilon}+\eta^{\mu}\partial_{\mu}\bar{\chi}\,,}\\{}&{}&{}\\{\deltaV_{\mu}}&{=}&{-\frac{1}{2}\bar{\epsilon}\Gamma_{\mu}\chi-\frac{1}{12}\bar{\epsilon}\Gamma^{a}\chi\bar{\chi}\Gamma_{a}\partial_{\mu}\chi+\eta^{\rho}\partial_{\rho}V_{\mu}+(\partial_{\mu}\eta^{\rho})V_{\rho}\,,}\\\end{array}
\vertw\vert=\mathrm{det}{w^{a}}_{\mu}=\mathrm{det}({e^{a}}_{\mu}+{t^{a}}_{\mu}),\quad{t^{a}}_{\mu}={\frac{\kappa^{4}}{2i}}\sum_{j=1}^{10}(\bar{\psi}^{j}\gamma^{a}\partial_{\mu}{\psi}^{j}-\partial_{\mu}{\bar{\psi}^{j}}\gamma^{a}{\psi}^{j}),