harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\calL}_{G}=-\frac{1}{4}(B_{\mu\nu}-Q_{\mu\nu})^{2}+\frac{1}{2}(A_{\mu}+\partial_{\mu}\theta)^{2}+\frac{1}{2m}\epsilon_{\mu\nu\rho\sigma}(B^{\mu\nu}-Q^{\mu\nu})\partial^{\rho}(A^{\sigma}+\partial^{\sigma}\theta),
\begin{array}{c}{\displaystyle{\sum_{\nu=3/2}^{\infty}\nu^{-(s-k-1)}}\\Re\left\{\displaystyle{\frac{-e^{\displaystyle{-\imath\frac{\pi}{2}(s+1)}}}{\pi}}\,\displaystyle{\int_{-\imathz}^{\infty-\imathz}}t^{-s}\,\displaystyle{D_{TE}^{(k)}(t)}\dt\right\},}\\\end{array}
B_{kn}^{\prime}=-{\calN}_{km}\Bigl({H^{n}}_{m}+e^{\rho}\alpha_{n}\eta^{m}+(B_{1n}+{\frac{1}{2}}a_{1}\cdota_{n}^{\prime})G^{1m}\Bigr)-{\frac{1}{2}}a_{k}^{\prime}\cdota_{n}^{\prime}
\Psi(y)\sim\left[c_{0}-\frac{1}{V_{2}}\sum_{x_{p}}m_{p}G^{4}(x,x_{p})+\sum_{x_{O}}\mu_{O}G^{6}(x,x_{O})-\sum_{i=1}^{n}G^{6}(x,x_{i})\right]^{1/2}~.
i\hbar{\frac{\partial\Phi}{\partialt}}(\{1\},\cdots,\{N\})={\frac{1}{2m}}\sum_{p=1}^{N}\{i\hbar\partial_{i}^{(p)}-{\frac{e}{c}}\sum_{r=1(\nep)}^{N}{\calA}_{i}(p,r)\}^{2}\Phi(\{1\},\cdots,\{N\})\,.
\Psi_{0}(\phi(\vec{x}))\propto\int{\calD}{\tilde{\phi}}e^{-{\frac{1}{2}}\int({\dot{\tilde{\phi}}}^{2}+(\bigtriangledown{\tilde{\phi}})^{2}+\mu^{2}{\tilde{\phi}}^{2})d\vec{x}d\tau}~.
\left(\begin{array}{cc}{k}&{j}\\{0}&{p}\\\end{array}\right)\,\\left(\begin{array}{cc}{0}&{-p}\\{k}&{j}\\\end{array}\right)\,\\left(\begin{array}{cc}{0}&{-p}\\{k}&{j+p}\\\end{array}\right)\,\\0\leqj<k,\\p\neq0\,
\begin{array}{l}{P^{\mu}\left|p_{R},s_{3};\gamma\right\rangle=p_{R}^{\mu}\left|p_{R},s_{3};\gamma\right\rangle\;,}\\{W_{0}\left|p_{R},s_{3};\gamma\right\rangle=0\;,}\\{W_{k}\left|p_{R},s_{3};\gamma\right\rangle=\frac{m}{2}\epsilon_{k\mu\nu0}M^{\mu\nu}\left|p_{R},s_{3};\gamma\right\rangle}\\\end{array}
S=\intd^{D}x\sqrt{|g_{D}|}e^{-\varphi}\left[R_{D}+\left(\nabla\varphi\right)^{2}-\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}+\frac{1}{8}\mathrm{Tr}\left[\nabla_{\mu}{\cal{M}}\nabla^{\mu}{\cal{M}}^{-1}\right]\right]
\sum_{\nu}^{}1/|S_{\nu}|=\sum_{p_{1},\ldots,p_{i}}\sum_{s_{1},\ldots,t_{G}}(\frac{1}{n!})^{2}\sum_{R}^{}d_{R}\chi_{R}(p_{1}\ldotsp_{i}\prod_{i}^{}s_{i}t_{i}s_{i}^{-1}t_{i}^{-1}).
M_{\lambda\sigma}^{\mu\nu}(y)=(C^{-1})^{\mu\nu\alpha\beta}{\tilde{M}}_{\alpha\beta\lambda\sigma}(y)=\frac{M\kappa^{2}}{8}\intd\tau\delta^{4}(y-x(\tau))\left[g^{\mu\nu}{\dot{x}}_{\lambda}{\dot{x}}_{\sigma}+2{\dot{x}}^{\mu}{\dot{x}}^{\nu}{\dot{x}}_{\lambda}{\dot{x}}_{\sigma}\right].
\frac{\partialH}{\partial\rho}+m_{CS}\frac{\partial\phi}{\partial\rho}-\gammaC_{2}\bigg(A_{\varphi}+\frac{1}{\rho^{2}}\int_{0}^{\rho}A_{\varphi}(\rho^{\prime})\rho^{\prime}d\rho^{\prime}\bigg)-2\gammaD_{2}\phi\bigg(\frac{1}{\rho^{2}}\int_{0}^{\rho}A_{\varphi}(\rho^{\prime})\rho^{\prime}d\rho^{\prime}\bigg)=0.
{\calL}_{\mathrm{kin}}=\left(\frac{1}{4}\,\partial_{\mu}\chi\,\partial^{\mu}\chi+\frac{3}{4}\,\partial_{\mu}\beta\,\partial^{\mu}\beta+\frac{\tau}{2}\,e^{\frac{\beta-\chi}{\mp}}\,\partial_{\mu}\varphi\,\partial^{\mu}\varphi\right)\,,
\frac{\partial^{2}}{\partialt^{2}}\phi-\frac{\partial^{2}}{\partialx^{2}}\phi+\frac{\partial}{\partial\phi}V=0,\qquad\frac{\partial^{2}}{\partialt^{2}}\chi-\frac{\partial^{2}}{\partialx^{2}}\chi+\frac{\partial}{\partial\chi}V=0,
\begin{array}{rl}{s_{gauge}=}&{\beta_{1}\int_{0}^{T}dt\left(\frac{1}{S(t)}\frac{dS(t)}{dt}+(r_{1}-r_{b})\right)^{2}+}\\{}&{+\beta_{2}\int_{0}^{T}dt\left(\left(\frac{1}{C(t)}\frac{dC(t)}{dt}+(r_{2}-r_{b})\right)-\alpha(t)\left(\frac{1}{S(t)}\frac{dS(t)}{dt}+(r_{1}-r_{b})\right)\right)^{2}\.}\\\end{array}
\left(1|E_{\left({\frac{dim}{2}}\right)}\right)_{\alpha\beta}=\delta_{0\alpha}\delta_{\beta\left({\frac{dim}{2}}\right)}-\delta_{0\beta}\delta_{\alpha\left({\frac{dim}{2}}\right)}-\epsilon_{\alpha\beta\left({\frac{dim}{2}}\right)}.
\frac{d^{2}}{dt^{2}}\frac{\partialL}{\partial\ddot{h}}+\nabla^{2}\frac{\partialL}{\partial\nabla^{2}h}-\frac{d}{dt}\nabla^{2}\frac{\partialL}{\partial\nabla^{2}\dot{h}}-\frac{d}{dt}\frac{\partialL}{\partial\dot{h}}-\nabla\frac{\partialL}{\partial\nablah}+\frac{\partialL}{\partialh}=0\quad.
\partial_{2}b_{2}-(\frac{4y_{1}(y_{1}^{2}-3y_{2}^{2})}{\zeta(y_{1}^{2}+y_{2}^{2})}+\frac{1}{y_{2}})b_{2}=-\zeta\cdot\frac{8y_{1}^{2}y_{2}}{(y_{1}^{2}+y_{2}^{2})^{2}}-\frac{1}{\zeta}\cdot\frac{32y_{1}^{2}y_{2}}{y_{1}^{2}+y_{2}^{2}}.
\langle\Sigma_{\mu}^{i}(x)\,\Sigma_{\nu}^{j}(0)\rangle={\frac{1}{(|x|\mu)^{h_{ik}(g^{2})}}}\,\,\pi_{\mu\nu}\left({\frac{c_{kl}^{(1)}(g^{2})}{|x|^{4}}}\right){\frac{1}{(|x|\mu)^{h_{jl}(g^{2})}}}\,\left({\frac{1}{4\pi^{2}}}\right)^{2}.
\langle\phi_{n,m}\left(z_{1}\right)\phi_{1,2}\left(z_{2}\right)\phi_{1,2}\left(z_{3}\right)\phi_{n,m}\left(z_{4}\right)\rangle=\left(\prod_{i<j}z_{ij}^{\gamma_{ij}}\right)\eta^{2\alpha_{n,m}\alpha_{1,2}-\gamma_{12}}\left(1-\eta\right)^{2\alpha_{1,2}^{2}-\gamma_{23}}Y\left(\eta\right)
\delta\mathcal{L}\,=\,\delta\mathcal{L}_{\psi}+\delta\mathcal{L}_{\psi^{*}}\,=\,({\bar{\epsilon}}\,,\,{\bar{\epsilon}}^{*})\,A\,\,\left(\begin{array}{c}{\psi}\\{\psi^{*}\}\\\end{array}\right)
H_{\mathrm{f}}=\sum_{n,i}\frac{2\pi}{L}\left(n+\frac{\varphi_{i}}{2\pi}\right)\left(a_{i}^{\dagger}(n)a_{i}(n)-b_{i}^{\dagger}(n)b_{i}(n)\right)+m\sum_{n,i}\left(a_{i}^{\dagger}(n)b_{i}(n)+b_{i}^{\dagger}(n)a_{i}(n)\right)\,
w_{T}(\zeta_{1},\zeta_{2}):=e^{-\left\langle\zeta_{1}\mid\zeta_{2}\right\rangle}K_{T}(\zeta_{1},\zeta_{2})=K_{T}(\zeta_{1},\zeta_{2})e^{-\left\langle\zeta_{1}\mid\zeta_{2}\right\rangle}\in\Lambda_{\overline{{0}}}
\left|\tilde{H}^{\left(s\right),\left(a\right)}\left(\zeta,\zeta^{\prime};t,\lambda_{t}\right)\right|\leqc_{(\varepsilon,\varepsilon^{\prime})}^{(s),(a)}\C_{(\zeta,\zeta^{\prime},t)}^{(H)}\\left|\lambda_{t}-N_{H}\right|^{n_{H}-\frac{d-2}{2}+\varepsilon^{\prime}}\\mathrm{e}^{-\left[\Ree\lambda_{t}-\left(N_{H}+\varepsilon\right)\right]v_{0}}\tag{3.57}
\mathrm{~e}^{ineA_{0}(\vec{x})/T}~\rightarrow~\mathrm{~e}^{ine^{2}/\kappa}~\mathrm{~e}^{ineA_{0}(\vec{x})/T}=\mathrm{~e}^{2\piin\cdotp/q}~\mathrm{~e}^{ineA_{0}(\vec{x})/T}
H^{2}=\frac{1}{2}\frac{(c_{1}+k)^{2}}{r_{0}^{2}}\frac{(1-x^{2})^{5/2}}{x^{\frac{5+3k+c_{1}}{2}}}\left[E^{2}-x^{2k}-\frac{l^{2}}{r_{0}^{2}}\frac{(1-x^{2})^{1/2}}{2x^{1/2}}\right]
{\calR}_{\alpha\nu\mu}^{\\\\\beta}=\partial_{\alpha}\Gamma_{\nu\mu}^{\beta}-\partial_{\nu}\Gamma_{\alpha\mu}^{\beta}+\Gamma_{\alpha\rho}^{\beta}\Gamma_{\nu\mu}^{\rho}-\Gamma_{\nu\rho}^{\beta}\Gamma_{\alpha\mu}^{\rho},
d_{n}^{\calO}(R)=\frac{1}{n!}\frac{1}{(2\pi)^{n}}\intd\theta_{1}\ldotsd\theta_{n}F_{n,n}^{\calO}(\theta_{n},\ldots,\theta_{1}|\theta_{1},\ldots,\theta_{n})e^{-E_{n}R}\,\,\,
\int_{\frac{1}{k}}^{\frac{1}{T}}{\frac{dz^{\prime}}{(kz^{\prime})^{3}}}\psi_{m}(z^{\prime})G_{p}(z,z^{\prime})\sim\psi_{m}(z)\left[{\frac{1}{p^{2}+m_{n}^{2}}}+{\lambda_{2}}(kz)^{2}\right],
{\calC}_{\mu}^{+}=0\,\,\,\,\Leftrightarrow\\,\,\,\,{\calH}_{I}^{+}=0\,\{\calC}_{\mu}^{-}=0\,\,\,\,\Leftrightarrow\,\,\,\,{\calH}_{I}^{-}=0\,
G_{SM}(x-y)=\frac{1}{\calN}\int[d\chi][d\bar{\chi}][d\phi]\,e^{\gamma_{5}\phi(x)}\chi(x)\bar{\chi}(y)e^{\gamma_{5}\phi(y)}J_{F}(\phi)e^{-\intd^{2}x^{\prime}\,{\calL}_{SM}},
|\lambda|>\lambda_{2}\frac{2+N_{2}}{3N_{1}}+|\lambda|\frac{3}{\pi}\left\{-\lambda_{1}\frac{2+N_{1}}{48\pi}+\left[\left(\lambda_{1}\frac{2+N_{1}}{48\pi}\right)^{2}+c_{1}\right]^{\frac{1}{2}}\right\}~~.
A_{i}=\frac{f_{A}+1}{r}\epsilon_{ik}\hat{x}_{k}\left(-\frac{i}{2}\sigma_{3}\right)+\frac{f_{B}}{r}\epsilon_{ik}\hat{x}_{k}\left(-\frac{i}{2}\hat{x}_{i}\sigma_{i}\right)+\frac{f_{C}}{r}\hat{x}_{i}\left(-\frac{i}{2}\epsilon_{kl}\hat{x}_{k}\sigma_{l}\right)
\psi_{1}=\left(1+{\frac{|q|_{1}}{r_{1}}}\right)^{-1},\quad\chi_{2}=\left(1+{\frac{|p|_{2}}{r_{2}}}\right)^{-1},\quad\psi_{3}=\left(1+{\frac{|q|_{3}}{r_{3}}}\right)^{-1},\quad\chi_{4}=\left(1+{\frac{|p|_{4}}{r_{4}}}\right)^{-1},
S_{Bolt}=\frac{4\pi^{2}(15r_{b}^{6}-(65N^{2}-3\ell^{2})r_{b}^{4}+3N^{2}(55N^{2}-6\ell^{2})r_{b}^{2}+9N^{4}(5N^{2}-\ell^{2}))}{3(5r_{b}^{2}+\ell^{2}-5N^{2})}
R_{\beta\\gamma\tau}^{\\alpha}=\delta_{\tau}\Gamma_{\\beta\gamma}^{\alpha}-\delta_{\gamma}\Gamma_{\\beta\tau}^{\alpha}+\Gamma_{\,beta\gamma}^{\varphi}\Gamma_{\\varphi\tau}^{\alpha}-\Gamma_{\,beta\tau}^{\varphi}\Gamma_{\\varphi\gamma}^{\alpha}+\Gamma_{\,beta\varphi}^{\alpha}w_{\\gamma\tau}^{\varphi}.
\hat{G}(0)=\frac{1}{\pi^{2}N}\int_{0}^{m}dy\;\frac{\hat{G}^{3}(0)}{y^{2}+m^{2}}+\frac{1}{\pi^{2}N}\int_{m}^{\infty}dy\;\frac{1}{y^{2}+m^{2}},
\delta_{\mu^{2}}-\delta_{g}a^{2}-\delta_{\lambda}a^{4}=(10a^{2}\lambda+3g)\int\frac{d^{3}k}{(2\pi)^{3}}\frac{i}{k^{2}-m^{2}}+(N-1)(2a^{2}\lambda+g)\int\frac{d^{3}k}{(2\pi)^{3}}\frac{i}{k^{2}},
{\calL}_{eff}=\frac1{12m^{2}}\:H_{\mu\nu\lambda}H^{\mu\nu\lambda}-\frac14\:F_{\mu\nu}F^{\mu\nu}-\frac\chi{2\,\theta}\:\epsilon^{\mu\nu\alpha\beta}\:A_{\mu}\partial_{\nu}B_{\alpha\beta}\;.
E=\sum_{n=0}^{N-1}e_{n}=\int_{E_{0}^{\ast}}^{E_{F}^{\ast}}dEE\rho(E)-\frac{1}{2}E_{F}^{\ast}+\frac{1}{2}E_{0}^{\ast}+\frac{1}{12}\frac{1}{\rho(E_{F}^{\ast})}-\frac{1}{12}\frac{1}{\rho(E_{0}^{\ast})}+\cdots
\sum_{n=-\infty}^{\infty}e^{-{\frac{4\pi^{2}(n+1/2)^{2}}{L^{2}}}t}={\frac{L}{\sqrt{4\pit}}}\Bigl(1+2\sum_{n=1}^{\infty}(-1)^{n}e^{-{\frac{L^{2}n^{2}}{4t}}}\Bigr),
\mathrm{tr}(\gamma^{A}{\calX}_{12}^{\prime}\gamma^{B}{\calX}_{21}^{\prime})=\mathrm{tr}(\gamma^{C}{\calX}_{12}\gamma^{D}{\calX}_{21})R_{C}{}^{A}(z_{1};g)R_{D}{}^{B}(z_{2};g)
S_{p-2}=-\frac{V_{p-2}}{(2\pi)^{p-2}\tilde{g}}\intd\tau\left(\frac{u}{R}\right)^{7-p}\left[\sqrt{\frac{1+(au)^{7-p}}{(au)^{7-p}}}\sqrt{\tilde{f}}-1+\frac{u_{0}^{7-p}}{2u^{7-p}}\right].
\begin{array}{l}{{\mathcalD}_{V}f=V^{a}\nabla_{a}f+\frac{n-2}{2n}(\nabla_{a}V^{a})f}\\{\delta_{V}f=V^{a}\nabla_{a}f+\frac{n+2}{2n}(\nabla_{a}V^{a})f.}\\\end{array}
{\calG}(z,z^{\prime})=\triangle^{1/2}(z,z^{\prime})\int_{\delta}^{\infty}ds{\frac{1}{(4\pis)^{2}}}~e^{-{\frac{\sigma^{2}(z-z^{\prime})}{4s}}-m^{2}s}\left(1+a_{1}(z,z^{\prime})s+a_{2}(z,z^{\prime})s^{2}+...\right)~~~.
~-\partial_{y}(K(\chi)\partial_{y}\chi)-\frac{K^{\prime}(\chi)}{2}(\partial_{y}\chi)^{2}+V^{\prime}(\chi)+v_{i}^{\prime}(\chi)\delta(y-y_{i})=\sum_{n>0}\frac{1}{M^{n}}\frac{\deltaS_{h.d.}^{(n)}}{\delta\chi}.
\frac{\mathrm{d}\rho^{2}}{\rho^{2}}\,\mathrm{d}^{2}\bar{\beta}\,\mathrm{d}^{2}\bar{\theta}_{0}\,F(\rho_{\mathrm{inv}}^{2})=16\,\mathrm{d}\rho_{\mathrm{inv}}^{2}\,\delta(\rho_{\mathrm{inv}}^{2})F(\rho_{\mathrm{inv}}^{2})\,.
2\tilde{\phi}=p^{2}+(m-\frac{i}{2}q\epsilon_{\mu\nu\lambda}{\calA}^{\mu}\xi^{\nu}\xi^{\lambda})^{2}-\frac{1}{3}q\epsilon_{\mu\nu\lambda}\xi^{\mu}\xi^{\nu}\xi^{\lambda}\xi_{*}\partial_{\sigma}{\calA}^{\sigma}-iq\epsilon_{\mu\nu\lambda}(p^{\mu}{\calA}^{\nu}+{\calA}^{\nu}p^{\mu})\xi^{\lambda}\xi_{*}\approx0.
S=\intd^{4}x\sqrt{-g}\\left[\frac{1}{16\piGv^{2}}\R\phi^{2}-\frac{1}{4}\left(F_{\mu\nu}^{a}\right)^{2}+\frac{1}{2}\left(D_{\mu}\phi^{a}\right)^{2}-\frac{\lambda}{4}\left(\phi^{2}-v^{2}\right)^{2}\right]
W(x_{B},Q;{\mathcalH})=\left(\begin{array}{cccc}{W^{00}}&{0}&{0}&{0}\\{0}&{W^{11}}&{W^{12}}&{0}\\{0}&{W^{21}}&{W^{22}}&{0}\\{0}&{0}&{0}&{0}\\\end{array}\right)
\begin{array}{l}{\intdy_{1}dy_{2}dy_{3}dy_{4}\langle\phi^{2}(y_{1})h^{n}(y_{1})\partialh(y_{1})\partialh(y_{1})\phi^{2}(y_{2})h^{m}(y_{2})\partialh(y_{2})\partialh(y_{2})}\\{\phi^{2}(y_{3})h^{r}(y_{3})\partialh(y_{3})\partialh(y_{3})h(x_{1})h(x_{2})\rangle}\\\end{array}
H^{(0)}({\alpha},{\bfn})\,\xi_{\bfp}^{(0)}({\alpha},{\bfn})=p_{0}\,\xi_{\bfp}^{(0)}({\alpha},{\bfn}),\quad{\bfP}^{(0)}({\alpha},{\bfn)}\,{\xi}_{\bfp}^{(0)}({\alpha},{\bfn})={\bfp}\,{\xi}_{\bfp}^{(0)}({\alpha},{\bfn}).
\widehat{T_{\k}^{j}}=-\mu{\frac{\partial\Phi^{2}}{\partial\mu}}\mu^{j}\mu_{k}+{\frac{\mu^{2}}{s}}{\frac{\partial\Phi^{2}}{\partials}}s^{j}s_{k}+\mu^{2}c^{2}\left(-\Phi^{2}+s{\frac{\partial\Phi^{2}}{\partials}}+y^{2}{\frac{\partial\Phi^{2}}{\partialy^{2}}}\right)g_{\k}^{j}\.
\times\;\prod_{b=1}^{N}\prod_{1\leqi<j\leqn}\left(y_{i}^{(b)}\!-\!y_{j}^{(b)}\right)^{-2}e^{\frac{1}{2}\sum_{b,b^{\prime},i,j}(1-\delta_{bb^{\prime}}\delta_{ij})\tilde{\tilde{V}}\left(y_{i}^{(b)}-y_{j}^{(b^{\prime})}\right)}\;.
\begin{array}{l}{\{Z_{1},Z_{2}\}=i\gamma[Z_{1}rZ_{1},{\calP}]}\\{\{P_{1},P_{2}\}=-i\gamma[P_{1}rP_{1},{\calP}]}\\{\{P_{1},Z_{2}\}=i\gamma[P_{1}Z_{2},r]-{\calP}\quad.}\\\end{array}
\begin{array}{l}{3(3,2)_{(1,-1,0,0,0^{3})}+3(\bar{3},1)_{(-1,0,1,0,0^{3})}+3(\bar{3},1)_{(-1,0,0,1,0^{3})}}\\{+3(1,2)_{(0,1,0,0,\underline{{-1,0,0}})}+3(1,1)_{(0,0,-1,0,\underline{{1,0,0}})}+3(1,1)_{(0,0,0,-1,\underline{{1,0,0}})},}\\\end{array}
S[n,U^{\mu},\varphi,\theta,s,{\tilde{\alpha}}^{r},\beta_{r};{}^{4}g_{\mu\nu}]=-\intd^{4}x\sqrt{{}^{4}g}\Big[\rho(n,s)-nU^{\mu}(\partial_{\mu}\varphi-\theta\partial_{\mu}s+\beta_{r}\partial_{\mu}{\tilde{\alpha}}^{r})\Big]
\left({\frac{1}{2}}\sum_{k=1}^{n-3}{Z}_{n-k}^{F_{1}}{Z}_{k+2}^{F_{1}}(7k+8)(7n+7k-23)-\sum_{k=2}^{n-3}\sum_{j=1}^{k-1}{Z}_{n-k}^{F_{1}}{Z}_{k-j+2}^{F_{1}}{Z}_{j+2}^{F_{1}}\right),\quadn>4.
\frac{\lambda^{2}\sqrt{2}}{m^{3}}E_{\mathrm{NTK3}}=\int_{\bar{\sigma}_{3}^{2}}^{\bar{\sigma}_{2}^{2}}\frac{\lambda_{2}d\lambda_{2}}{\sqrt{1-\lambda_{2}}}+\int_{\bar{\sigma}_{2}^{2}}^{1}\frac{2\lambda_{3}d\lambda_{3}}{\sqrt{1-\lambda_{3}}}=2\left(\sigma_{2}+\sigma_{3}-\frac{\sigma_{2}^{2}+\sigma_{3}^{2}}{3}\right)
{\calG}_{1}^{(1)}(t_{1},t_{2})=-\tauRx^{2(t_{2}-t_{1})}x^{-2}(1+x^{2T})^{\Omega-1}\left[(t_{2}-t_{1})+T(\Omega-1)\frac{x^{2T}}{1+x^{2T}}\right]
\int_{0}^{2\pi}\frac{d\theta}{2\pi}\frac{1}{\left[1-xe^{i\theta}\right]\left[1-xe^{-i\theta}\right]}\,\prod_{\mu=1}^{M}\frac{1}{\left[1-xe^{i(\theta+\omega_{\mu})}\right]\left[1-xe^{i(\theta-\omega_{\mu})}\right]\left[1-xe^{-i(\theta-\omega_{\mu})}\right]\left[1-xe^{-i(\theta+\omega_{\mu})}\right]}\\\.
G^{\prime}=\vec{e}_{R}\cdot\vec{g}=\left(\begin{array}{cccc}{0}&{-g_{3}}&{g_{2}}&{-g_{1}}\\{g_{3}}&{0}&{-g_{1}}&{-g_{2}}\\{-g_{2}}&{g_{1}}&{0}&{-g_{3}}\\{g_{1}}&{g_{2}}&{g_{3}}&{0}\\\end{array}\right)=\vec{e}^{~^{\prime}}\cdot\vec{g}
ds^{2}=-{\frac{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}{r^{2}}}dt^{2}+{\frac{r^{2}dr^{2}}{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}}+r^{2}\left[{\frac{r_{-}}{r_{+}r^{2}}}(r^{2}-r_{+}^{2})dt+d\phi\right]^{2},
\partial_{i,1}\Phi_{j}^{(-n)}=\Phi_{i}\partial_{x}^{-1}\left(\Psi_{i}\Phi_{j}^{(-n)}\right),\quad\partial_{i,1}\Psi_{j}^{(-n)}=-\Psi_{i}\partial_{x}^{-1}\left(\Phi_{i}\Psi_{j}^{(-n)}\right),
\mathrm{ln}S_{B}{0\atopwithdelims[]0}(\infty,y|q)\sim{\frac{k_{b}TM}{2\piv}}\{-{\frac{B}{2}}x^{2}+x\mathrm{ln}y-\int_{0}^{x}dt\mathrm{ln}(1-e^{-t})\}.
\Psi(\tau\to-\infty)=\frac{\Gamma\left(\frac{ip}{\mu_{0}}\right)\Gamma\left(1-\frac{ip}{\mu_{0}}\right)}{\Gamma(-s)\Gamma(1+s)}e^{-ip\tau}+\frac{\Gamma\left(-\frac{ip}{\mu_{0}}\right)\Gamma\left(1-\frac{ip}{\mu_{0}}\right)}{\Gamma\left(-\frac{ip}{\mu_{0}}-s\right)\Gamma\left(1+s-\frac{ip}{\mu_{0}}\right)}e^{ip\tau}\,.
{\calR}=5(\partial\Phi)^{2}-{\frac{9}{2}}\nabla^{2}\Phi+{\frac{(3-p)}{8(p+2)!}}e^{2\Phi}G^{2}+{\frac{1}{8}}\left({\frac{\Lambda\rho_{\perp}}{\sqrt{-g}}}\right)e^{\Phi}V(T)\sqrt{-A}(A^{-1})^{\lambda\sigma}g_{\lambda\sigma}\,,
0=C_{i_{p},\beta\;j_{q},\gamma}^{k_{s},\alpha}C_{l_{t},\rho\;m_{u},\sigma}^{i_{p},\beta}\omega^{j_{q},\gamma}\wedge\omega^{l_{t},\rho}\wedge\omega^{m_{u},\sigma}\quad(\alpha,\beta,\gamma,\rho,\sigma=1,\ldots,N)\;,
c_{2r}^{\pm\nu}(h^{2})M_{\pm\nu}^{(1)}(z,h)=\sum_{l=-\infty}^{+\infty}(-1)^{l}c_{2l}^{\pm\nu}(h^{2})J_{l-r}(he^{-z})J_{\pm\nu+l+r}(he^{z})
\left[Q,K_{2s,t}\right]=\frac{A}{2\pii}\left(c(1)k_{s,t}^{+}(1)-c(-1)k_{s,t}^{+}(-1)\right)+\frac{B}{2\pii}\left(c(1){k}_{s,t}^{-}(1)-c(-1)k_{s,t}^{-}(-1)\right)\.
g_{ab}=\left(\begin{matrix}{g_{\mu\nu}}&{n_{\nu}}\\{n_{\mu}}&{n_{\mu}n^{\mu}+n^{2}}\\\end{matrix}\right),\qquadg^{ab}=\frac1{n^{2}}\left(\begin{matrix}{n^{2}g^{\mu\nu}+n^{\mu}n^{\nu}}&{-n^{\nu}}\\{-n^{\mu}}&{1}\\\end{matrix}\right),
m\,\ddot{x}^{\mu}=eg^{\mu\nu}F_{\nu\rho}\dot{x}^{\rho}-m\Gamma_{\;\:\nu\rho}^{\mu}\dot{x}^{\nu}\dot{x}^{\rho}+\frac{e}{2m}g^{\mu\nu}(\nabla_{\nu}F_{\rho\sigma})S^{\rho\sigma}+\frac12R_{\;\:\nu\rho\sigma}^{\mu}\dot{x}^{\nu}S^{\rho\sigma},
\begin{array}{l}{{\calX}^{0}\vertn,z\rangle_{0}=q^{n}z\vertn,z\rangle_{0}~,}\\{{\calD}_{0}\vertn,z\rangle_{0}=\displaystyle\frac{q^{1-n}}{q-1}z^{-1}\vertn-1,z\rangle_{0}~,}\\{\mu_{0}\vertn,z\rangle_{0}=\vertn-1,z\rangle_{0}~,}\\\end{array}
S_{4}=\frac{1}{2\kappa_{4}^{2}}\intd^{4}x\sqrt{-g}[R-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\sigma\partial_{\nu}\sigma-V(\phi,\sigma)]\quad.
\hat{h}_{\mu\nu}^{\prime\prime}=(5-\hat{d})a^{-1}a^{\prime}\hat{h}_{\mu\nu}^{\prime}+2\left[a^{-1}a^{\prime\prime}+(\hat{d}-4)a^{-2}(a^{\prime})^{2}\right]\hat{h}_{\mu\nu}\,.
S^{(2)}=-i\pi\intd^{2}\xiz_{\nu}(\xi)\varepsilon^{ab}\left(\partial_{a}y_{\mu}(\xi)\right)\Biggl(\left(\partial_{b}z_{\lambda}(\xi)\right)H_{\nu\mu\lambda}[y(\xi)]+\frac{1}{2}z_{\alpha}(\xi)\left(\partial_{b}y_{\lambda}(\xi)\right)\partial_{\alpha}H_{\nu\mu\lambda}[y(\xi)]\Biggr),
{\mathbbE}_{0}^{\mathrm{K}}=E[{\vec{\psi}}_{\mathrm{K}}]+\frac{\hbar}{2}\,\mathrm{Tr}\,(P{\calK})^{\frac{1}{2}}+o(\hbar^{2})=E[{\vec{\psi}}_{\mathrm{K}}]+\frac{\hbar}{2}\,\zeta_{\mathrm{P{\calK}}}(-\textstyle\frac{1}{2})+o(\hbar^{2})\quad.
ds^{2}=H_{p}^{-1/2}[-dt^{2}+\sum_{i=1}^{p-1}(dx^{i})^{2}+H_{5}^{NS}(dx^{p})^{2}]+H_{p}^{1/2}[\sum_{i=p+1}^{6}(dx^{i})^{2}+H_{5}^{NS}(dr^{2}+r^{2}d\Omega_{2}^{2})]
{\frac{1}{3}}-9\,{x^{2}}-36\,{e^{\phi(t)}}\,{x^{2}}+2\,x\,y+27\,{x^{3}}\,y+108\,{e^{\phi(t)}}\,{x^{3}}\,y+2\,{y^{2}}+\,{y^{4}}=0
|B;\pm\rangle_{I}=\int[dPd\chi]e^{\frac{1}{2}\intd\sigma(iP^{i}\partial_{\sigma}P^{j}+\chi^{i}\chi^{j})\omega_{ij}-\intd\sigma(ip_{i}P^{i}-\pi_{i}\chi^{i})}|B;\pm\rangle_{-1,I},
\{F_{X},F_{Y}\}=<\phi,g(A)\left[g(A)^{-1}\int_{S^{1}}(g(A)Xg(A)^{-1})_{ij}(\varphi^{\prime})\frac{\deltag(A)}{\deltaA_{ij}(\varphi^{\prime})},Y\right]g(A)^{-1}>
\deltaP=i\frac{\hbar}{2}\rho_{0}^{2}\left({\phi}^{*}\delta{\phi}-{\phi}\delta{\phi}^{*}\right)\Biggr|_{-\infty}^{+\infty}+i\hbar\int_{-\infty}^{+\infty}d\,x\left(\delta{\phi}\partial_{x}{\phi}^{*}-\delta{\phi}^{*}\partial_{x}{\phi}\right).
W(p,q)\equivW_{4}(p,q;0)=\left(\begin{array}{cccc}{1+\frac{p^{2}+q^{2}}{2}}&{p}&{q}&{-\frac{p^{2}+q^{2}}{2}}\\{p}&{1}&{0}&{-p}\\{q}&{0}&{1}&{-q}\\{\frac{p^{2}+q^{2}}{2}}&{p}&{q}&{1-\frac{p^{2}+q^{2}}{2}}\\\end{array}\right)
(\bar{\nabla}_{\mu}+{\frac{2-\epsilon}{2}}\partial_{\mu}\bar{\Phi})({\calF}^{\mu\nu}-\bar{F}_{\alpha}^{~~\nu}h^{\alpha\mu}-\bar{F}_{~~\beta}^{\mu}h^{\beta\nu})+\bar{F}^{\mu\nu}(\delta\Gamma_{\sigma\mu}^{\sigma}(h)+{\frac{2-\epsilon}{2}}(\partial_{\mu}\phi))=0,
\begin{array}{l}{{\calD}_{-k}{\calX}^{-k}=1+q^{-2}{\calX}^{-k}{\calD}_{-k}~,~~~~~\mathrm{for}~k>0~,}\\{{\calD}_{0}{\calX}^{0}=1+q{\calX}^{0}{\calD}_{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}
D_{\mu\nu}^{R}(q)J^{\mu}J^{\nu}=-{\frac{1}{q^{2}-M^{2}}}\left(J_{\nu}^{2}+{\frac{(J_{i}q_{i})^{2}-(J_{0}q_{0})^{2}}{(\vec{q}^{2}+M^{2})}}\right)
\langle\bar{\psi}\psi\rangle\Big|_{m^{2}d^{2}={\calO}(0.1)}\;\approx\;\frac{1}{4\pi^{2}}\;m\;\rho\;\;c_{0}\Bigl(m^{2}\,d^{2}\Bigr)\Big|_{m^{2}d^{2}={\calO}(0.1)}\;\approx\;\;0.058\;\frac{1}{4\pi^{2}}\;\rho\,/\,d\;,
\hat{h}_{\mu\nu}=a^{2}\chi_{\mu\nu}+2a_{0}^{2}a^{2}\left(\int\frac{dz}{a^{4}}\right)\partial_{\mu}\partial_{\nu}\xi_{z}-2a^{2}\left(\int\frac{dz}{a^{2}}\right)\partial_{\mu}\partial_{\nu}\xi_{z}-\frac{U_{B}}{2}a^{2}\xi_{z}\eta_{\mu\nu}
g_{h}(R)=(2h-3)!\2^{-2h}\sum_{n=0}^{h}\left({\frac{1}{R}}\right)^{2n}{\frac{(2^{2(h-n)}-2)(2^{2n}-2)\vertB_{2(h-n)}B_{2n}\vert}{[2(h-n)]![2n]!}}
Q^{A}=\left(\begin{array}{cc}{u^{A}}&{i\overline{v}_{A^{*}}}\\{iv_{A}}&{\overline{u}^{A^{*}}}\\\end{array}\right)\qquad\overline{Q}^{A}=\left(\begin{array}{cc}{\overline{u}^{A^{*}}}&{-i\overline{v}_{A^{*}}}\\{-iv_{A}}&{u^{A}}\\\end{array}\right)\qquad\begin{array}{l}{u^{A}=q^{A|0}-iq^{A|3}}\\{v^{A}=-q^{A|1}-iq^{A|2}\,.}\\\end{array}
\pi_{i_{1}\ldotsi_{p-2k-2}}=-\frac1{\triangle}\left(\partial^{i}\tilde{\gamma}_{ii_{1}\cdotsi_{p-2k-2}}^{(2)}+\frac1{p-2k-2}\partial_{\left[i_{1}\right.}^{}\tilde{\gamma}_{\left.i_{2}\cdotsi_{p-2k-2}\right]}^{(2)}\right),
\left(\Sigma^{\mu\nu}\right)^{\rho\tau\alpha\beta}=\frac12\left[\eta^{\mu\rho}\eta^{\nu\alpha}\eta^{\tau\beta}+\eta^{\mu\tau}\eta^{\nu\beta}\eta^{\rho\alpha}+\left(\alpha\leftrightarrow\beta\right)-\left(\mu\leftrightarrow\nu\right)\right]\;\,,
R_{\kappa\lambda\mu\nu}\,=\,R_{\kappa\mu}g_{\lambda\nu}-R_{\lambda\mu}g_{\kappa\nu}+R_{\lambda\nu}g_{\kappa\mu}-R_{\kappa\nu}g_{\lambda\mu}-\frac{R}{2}\left(g_{\kappa\mu}g_{\lambda\nu}-g_{\kappa\nu}g_{\lambda\mu}\right).
\zeta_{D}(s)=\frac{1}{(4\pi)^{\frac{D-1}{2}}}\frac{\Gamma(s-\frac{D-1}{2})}{\Gamma(s)}\frac{1}{L}E_{1}^{m^{2}}(s-\frac{D-1}{2};\frac{\pi^{2}}{L^{2}}),
{\epsilon_{\mu_{1}\,\dots\;\mu_{p}}}{}^{\mu_{p+1}\,\dots\;\mu_{p+q}}\;=\;g_{\mu_{1}\nu_{1}}\,g_{\mu_{2}\nu_{2}}\,\dots\,g_{\mu_{p}\nu_{p}}\,\epsilon^{\nu_{1}\,\dots\,\nu_{p}\mu_{p+1}\dots\mu_{p+q}}
S_{A}=\frac{c}{4!}\int_{{\calM}_{5}}d^{5}x\left[-\varepsilon^{\alpha\beta\gamma\delta\epsilon}(\partial_{\epsilon}G){\calA}_{\alpha\beta\gamma\delta}+\sum_{i=1}^{2}2\alpha_{i}\varepsilon^{\mu\nu\rho\sigma}{\calA}_{\mu\nu\rho\sigma}\delta(x^{5}-x_{i}^{5})\right]\,.
\oint{d}\hat{s}(\xi^{\prime\prime}){\hat{K}_{D}}(\xi,\xi^{\prime\prime}){\hat{G}_{K}}(\xi^{\prime\prime},\xi^{\prime})={\hat{\delta}_{B}}(\xi-\xi^{\prime})-{\frac{1}{\oint{d}\hat{s}(\xi^{\prime\prime\prime})}}.
{\frac{1}{8\omega+3}}\left(-3-{\frac{5z_{0}^{2}}{l^{2}}}\right)+{\frac{z_{0}^{2}}{\omegal^{2}}}-{\frac{9}{32\omega^{2}}}>{\frac{3}{32\omega^{2}}}\sqrt{-64\omegaz_{0}^{2}l^{2}+9l^{4}-64l^{2}z_{0}^{2}\omega^{2}}\,.
\pi^{I}=\frac{\dot{x}^{I}}{2\pil_{s}^{2}},\quad\pi^{1}=\frac{\dot{x}^{1}+2\mux^{2}}{2\pil_{s}^{2}},\quad\pi^{2}=\frac{\dot{x}^{2}}{2\pil_{s}^{2}}.
\frac12\int\eta(t,\rho+a(t))\left[-\frac{\partial^{2}}{\partialt^{2}}-\hat{\Omega}\right]\eta(t,\rho+a(t))dtd\rho+\intm_{\eta^{2}}(t)\left(\dot{a}(t)\right)^{2}dt+S_{0}^{(int)}[\eta,a]
\left[\partial_{t}+2D-6+\sum_{i=1}^{6}{\bf{q}}_{i}\cdot\nabla_{{\bf{q}}_{i}}\right]\Gamma_{t}^{(6)}({\bf{q}}_{1},\ldots,{\bf{q}}_{6})=g_{t}^{3}I^{(6)}({\bf{q}}_{1},\ldots,{\bf{q}}_{6})\;,
\hat{H}\Psi=\sum_{a=0}^{\infty}\frac{1}{a!}\prod_{n=1}^{a}\intdx_{n}dy_{n}u^{\dagger}(x_{n})\gamma^{5}u(y_{n})e^{ie\int_{y_{n}}^{x_{n}}\calA}\tilde{f}^{a}(x_{1},y_{1},\ldots,x_{a},y_{a}).
\stackrel{\lbrack1]}{a^{\prime}}_{1}=\varepsilon^{\alpha\lambda\mu}\left(V_{\alpha(\lambda)}^{*}\left(c_{1}^{\prime}C_{(\mu)}+c_{2}^{\prime}\eta_{(\mu)}\right)+A_{\alpha(\lambda)}^{*}\left(c_{3}^{\prime}C_{(\mu)}+c_{4}^{\prime}\eta_{(\mu)}\right)\right),
S=-\frac{1}{2\pi}\intd^{2}\sigma~e~\left[h^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X_{\mu}-i\bar{\Psi}^{\mu}\rho^{\alpha}\nabla_{\alpha}\Psi_{\mu}+2\bar{\chi}_{\alpha}\rho^{\beta}\rho^{\alpha}\Psi^{\mu}\partial_{\beta}X_{\mu}+\frac{1}{2}\bar{\Psi}^{\mu}\Psi_{\mu}\bar{\chi}_{\beta}\rho^{\beta}\rho^{\alpha}\chi_{\alpha}.\right].
{\calH}_{n}^{(2)}={}-\frac12\frac{d^{2}}{dr^{2}}+\frac{c^{2}}{32}r^{6}+\frac{c_{0}c}8r^{4}+\frac18\left(c_{0}^{2}-2c(2n-m)\right)r^{2}+\frac{m^{2}-\frac14}{2r^{2}}-\frac12(n-m)c_{0}.
i{\frac{2^{1/4}\pi\sqrt3}{g\sqrt{NL}}}\left(B_{ij}\right)=\left(\begin{array}{cccc}{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{-{\frac{9}{2}}}\\{0}&{-3\sqrt{\frac{3}{2}}}&{0}&{0}\\{0}&{{\frac{3}{2}}\sqrt{3}}&{0}&{0}\\\end{array}\right).
i\Pi^{\mu\nu}(q)\simi\frac{g^{2}}{16\pi^{2}}\left\{\frac{10}{3}\left(g^{\mu\nu}q^{2}-q^{\mu}q^{\nu}\right)\mathrm{ln}(q^{2}\tilde{q}^{2})+32\frac{\tilde{q}^{\mu}\tilde{q}^{\nu}}{\tilde{q}^{4}}-\frac{4}{3}\frac{q^{2}}{\tilde{q}^{2}}\tilde{q}^{\mu}\tilde{q}^{\nu}\right\}\,,
\mu\frac{d}{d\mu}\tilde{y}=\tilde{y}(\frac12\gamma_{1}+\frac12\gamma_{2}+\frac12\gamma_{3})+\tilde{y}(D/2-2)=\frac{\tilde{y}}{2}(\gamma_{1}+\gamma_{2}+\gamma_{3}+D-4).
\left<0,t\right|\hat{N}(t)\left|0,t\right>=\sum_{\alpha}\left<\alpha,0,t\right|\hat{N}_{\alpha}(t)\left|\alpha,0,t\right>=\frac{l^{3}}{(2\pi)^{3}}\intd{\bfk}|u_{2}(k,t,t_{0})|^{2}.
\begin{array}{ll}{<L_{1}^{\pm},M_{2}>=R_{12}^{\pm}\;,\quad}&{\qquad<\tilde{L}_{1}^{\pm},\tilde{M}_{2}>=R_{12}^{\pm}\;,}\\{<L_{1}^{\pm},\tilde{M}_{2}>=A_{12}\;,\quad}&{\qquad<\tilde{L}_{1}^{\pm},M_{2}>=\tilde{A}_{12}\;.}\\\end{array}
\frac12(\partial_{a}v_{b}^{~j}-\partial_{b}v_{a}^{~j})=\varepsilon^{jkl}v_{a}^{~k}v_{b}^{~l}=\varepsilon_{jab}+{\frac{\delta^{ja}\varphi^{b}-\delta^{jb}\varphi^{a}}{\sqrt{1-{\varphi}^{2}}}}~.
\times\;\prod_{i=1}^{n_{b}^{\prime}}\overline{{\psi}}^{(b)}({x^{\prime}}_{i}^{(b)})P_{+}\psi^{(b)}({x^{\prime}}_{i}^{(b)})\prod_{i=1}^{m_{b}^{\prime}}\overline{{\psi}}^{(b)}({y^{\prime}}_{i}^{(b)})P_{-}\psi^{(b)}({y^{\prime}}_{i}^{(b)})\Big\rangle_{0}