harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int\intF^{\lambda_{1}\lambda_{2},\lambda_{3}\lambda_{4},\lambda_{5}\lambda_{6}}=\left(\int_{-\infty}^{\infty}d\lambda_{1}^{\prime}d\lambda_{2}^{\prime}\delta(\lambda_{1}^{\prime}-(\lambda_{1}+\lambda_{1}^{\prime}))\delta(\lambda_{2}^{\prime}-(\lambda_{2}e^{-2i\lambda_{1}^{\prime}Q^{\prime}}+w^{-1}\lambda_{2}^{\prime})),\right.
{\calG}_{\omega}(r,r^{\prime},\theta)=\sum_{n=0}^{\infty}r^{1-D/2}\left[b_{n}J_{n-1+{\frac{D}{2}}}(|\omega|r)+c_{n}J_{-n+1-{\frac{D}{2}}}(|\omega|r)\right]C_{n}^{(-1+D/2)}(z)\quad(r^{\prime}<r<a).
D_{\mu\nu}^{(0)}(p)=\frac{1}{p^{2}+\kappa^{2}}\left[(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}})-\kappa\epsilon_{\mu\nu\lambda}\frac{p_{\lambda}}{p^{2}}\right]+\xi\,\frac{p_{\mu}p_{\nu}}{(p^{2})^{2}}
J_{s}(\alpha)=\frac{\alpha-1}{2(2\alpha-1)}\left(\begin{matrix}{1-\alpha}&{-\alpha}&{0}&{0}\\{\alpha}&{\alpha-1}&{0}&{0}\\{0}&{0}&{1-\alpha}&{\alpha}\\{0}&{0}&{-\alpha}&{\alpha-1}\\\end{matrix}\right)\.
R^{\frac{1}{2}}(\theta)_{\left|\left.a_{k}\dots\frac{1}{2},a_{1},\frac{1}{2}\right|n_{k}\dots,m_{1},n_{1}\right\rangle}^{\left|\left.b_{k}\dots\frac{1}{2},b_{1},\frac{1}{2}\right|n_{k}\dots,m_{1},n_{1}\right\rangle}=R_{a_{1}b_{1}}^{\frac{1}{2}}(\theta)\prod_{i=1}^{k-1}f_{b_{i}b_{i+1}}^{a_{i}a_{i+1}}(w_{m_{i}},\nu_{n_{i+1}},\theta)
V(|\varphi|,|\sigma|)=\frac{\lambda_{\varphi}}{8}(|\varphi|^{2}-\eta^{2})^{2}+f(|\varphi|^{2}-\eta^{2})|\sigma|^{2}+\frac{\lambda_{\sigma}}{4}|\sigma|^{4}+\frac{m_{\sigma}^{2}}{2}|\sigma|^{2}
[\frac{e}{n+\frac{1}{2}}]^{n+\frac{1}{2}}\approx\frac{\sqrt{2\pi}}{n!}\frac{e^{\frac{1}{2}}}{(1+\frac{1}{2n})^{n+\frac{1}{2}}}\approx\frac{\sqrt{2\pi}}{n!}
\gamma_{\Omega^{\prime},3}={\small\left(\begin{array}{cccccccc}{}&{}&{}&{}&{}&{}&{}&{1_{v_{1}}}\\{}&{}&{}&{}&{}&{}&{1_{v_{2}}}&{}\\{}&{}&{}&{}&{}&{\cdots}&{}&{}\\{}&{}&{}&{}&{1_{v_{P}}}&{}&{}&{}\\{}&{}&{}&{-1_{v_{P}}}&{}&{}&{}&{}\\{}&{}&{\cdots}&{}&{}&{}&{}&{}\\{}&{-1_{v_{2}}}&{}&{}&{}&{}&{}&{}\\{-1_{v_{1}}}&{}&{}&{}&{}&{}&{}&{}\\\end{array}\right)}
\theta^{\mu\nu}=1_{mn}\otimes\left(\begin{array}{ccc}{0}&{0}&{0}\\{0}&{0}&{A}\\{0}&{-A}&{0}\\\end{array}\right)\;A=\left(\begin{array}{cc}{0_{k\timesk}}&{0}\\{0}&{\theta1_{N\timesN}}\\\end{array}\right)\.
Z({\calA},G,q)=\mathrm{e}^{F({\calA},G,q)}=\sum_{N=1}^{\infty}Z_{N}({\calA},G)q^{N}=\prod_{n=-\infty}^{+\infty}Z^{(\mathrm{cov)}}(G,\mathrm{e}^{-{\calA}v(n)}q)~.
{\calZ}_{12}(({\tau}_{j}^{i}\otimesid)R_{21}R_{12})_{1}=({Q^{-1}}_{k}^{i}Q_{m}^{l}(({\tau}_{j}^{i}\otimesid)R_{21}R_{12}){Q^{-1}}_{o}^{n}Q_{j}^{p})_{1}({Z^{-1}}_{l}^{k}Z_{p}^{o})_{2}{\calZ}_{12}\,.
[\nabla_{\mu},\nabla_{\nu}]\phi_{\mu_{1}\dots\mu_{k}}=-\sum_{i=1}^{k}R_{\mu_{i}\mu\nu}^{\lambda}\phi_{\mu_{i}\dots\mu_{i-1}\lambda\mu_{i+1}\dots\mu_{k}}+W_{\mu\nu}\phi_{\mu_{1}\dots\mu_{k}},
{\frac{d}{d\tau}}{\frac{1}{N}}\dot{x}^{\mu}+{\frac{1}{N}}\Gamma_{\alpha\beta}^{\mu}\dot{x}^{\alpha}\dot{x}^{\beta}+({\frac{1}{2}}{\calA}R+1)g^{\mu\alpha}{\sqrtg}\epsilon_{\alpha\beta}\dot{x}^{\beta}=0.
\begin{array}{rcl}{Z_{\mu}}&{=}&{\displaystyle\frac{\sqrt{3}}{2}A_{\mu}^{3}-\frac{1}{2}B_{\mu},}\\{A_{\mu}}&{=}&{\displaystyle\frac{1}{2}A_{\mu}^{3}+\frac{\sqrt{3}}{2}B_{\mu}.}\\\end{array}
D_{\mu\nu}=\frac{-i}{p^{2}-\mu^{2}+i0}(g_{\mu\nu}+\frac{\bar{p}_{\mu}\bar{p}_{\nu}}{{\vec{p}}{\,}^{2}}-\frac{n_{\mu}n_{\nu}p_{0}^{2}}{{\vec{p}}{\,}^{2}}+\frac{i\mu}{{\vec{p}}{\,}^{2}}\epsilon_{\mu\nu\lambda}\bar{p}^{\lambda})\,
\partial_{\pi_{\phi}}\widetilde{\phi}=0,\,\,\,\,\,\,\partial_{\phi}\widetilde{\pi_{\phi}}=0,\,\,\,\,\,\,\widetilde{\pi_{\phi}}=\pi_{\phi}(\partial_{\phi}+X_{1})\widetilde{\phi}+X_{0}\widetilde{\phi},
S_{D}=-\intd^{4}\sigma(\sqrt{-\mathrm{det}\,(G_{\mu\nu}+\tilde{\calF}_{\mu\nu}^{\prime})}+\int(C_{4}^{\prime}+C_{2}^{\prime}\wedge\tilde{\calF}^{\prime}-\frac{1}{2}C_{0}\tilde{F}^{\prime}\wedge\tilde{F}^{\prime}).
V(\sigma,\pi)=\frac{4N_{c}\Lambda^{D}}{(4\pi)^{D/2}\Gamma(D/2)}\left[(\frac{1}{g}-\frac{1}{g_{cr}})\frac{\rho^{2}}{2\Lambda^{2}}+\frac{2}{4-D}\frac{\xi_{D}}{D}(\frac{\rho}{\Lambda})^{D}\right]+O(\frac{\rho^{4}}{\Lambda^{4}}),
\begin{array}{ccrc}{}&{\xi_{1}=\left({\begin{array}{cr}{n}\\{1}\\\end{array}}\right)^{\frac12}\xi}\\{}&{\xi_{2}=\left({\begin{array}{cr}{n}\\{2}\\\end{array}}\right)^{\frac12}\xi^{2}}\\{}&{\vdots}\\{}&{\xi_{n}=\left({\begin{array}{cr}{n}\\{n}\\\end{array}}\right)^{\frac12}\xi^{n}}\\\end{array}
\ddot{r}+(1+H^{2}r^{2})H^{2}r\dot{t}^{2}+(1+H^{2}r^{2})r-\frac{H^{2}r\dot{r}^{2}}{1+H^{2}r^{2}}+2(1+H^{2}r^{2})Hr\dot{t}=0,
W^{({D})}({\bfr}_{{D}})=\int\frac{d^{{D}}k_{{{D}}}}{(2\pi)^{{D}}}\;e^{i{\bfk}_{{D}}\cdot{\bfr}_{{D}}}\,\left[\intd^{{D_{0}}}r_{{{D_{0}}}}\;e^{-i{\bfk}_{{D_{0}}}\cdot{\bfr}_{{D_{0}}}}\,W({\bfr}_{{D_{0}}})\right]_{{\bfk}_{{D_{0}}}\rightarrow{\bfk}_{{D}}}\;,
\hat{J}_{a}=-\frac{2s}{m}(\sigma_{a})_{\alpha\beta}\hat{\pi}^{\alpha}\hat{\pi}^{\beta}\qquad\hat{\theta}^{\alpha}=\frac{i}{m}\hat{K}\hat{\pi}^{\alpha}\qquad\hat{B}=-\frac{1}{4}\hat{\calK}(1-\nu\hat{\calK})\;.
{\calL}=\frac{1}{8\pi^{2}g_{YM}^{2}}\left((|\dot{z}_{1}|^{2}+|\dot{z}_{2}|^{2}+|\dot{z}_{3}|^{2})-\frac{4a_{0}^{2}}{L^{2}}|z_{3}|^{2}\right)\,
ds^{2}=\alpha^{\prime}\left[\frac{u^{2}}{R^{2}}\left(dx_{+}dx_{-}+dx_{1}^{2}+dx_{2}^{2}-\frac{b^{2}u^{4}}{R^{4}}dx_{-}^{2}\right)+\frac{R^{2}}{u^{2}}\left(du^{2}+u^{2}d\Omega_{5}^{2}\right)\right],
\psi_{P}:\left(\begin{array}{l}{f_{\pm}^{1/2}}\\{f_{\pm}^{-1/2}}\\\end{array}\right)\mapsto\left(\begin{array}{l}{\pmf_{\mp}^{-1/2}}\\{\mpf_{\mp}^{1/2}}\\\end{array}\right)(\pi-\theta,\pi+\phi),
U=|G(\phi_{i})|^{2}+|p|^{2}\sum_{i}i\left|\frac{\partialG}{\partial\phi_{i}}\right|^{2}+\frac{D}{2e^{2}}+2|\sigma|^{2}\left(\sum_{i}Q_{i}^{2}|\phi_{i}|^{2}+Q_{p}^{2}|p|^{2}\right)\.
{\calV}={\calG}^{-1}(n^{2}I+{\calB}-{\calA}^{2}-\dot{{\calA}}){\calG}=\left(\begin{array}{cc}{n^{2}}&{0}\\{0}&{n^{2}-\frac{2Mm^{2}\alpha^{2}}{lr}}\\\end{array}\right),
\lambda_{L}^{(3)}=\bar{\lambda}_{L}\biggr[1+\bar{\lambda}_{L}\frac{2}{\pi}\sum_{i=0}^{L}\frac{k^{2(L-i)}\Lambda^{2i+1}}{2i+1}+\bar{\lambda}_{L}^{2}\biggr(\frac{2}{\pi}\sum_{i=0}^{L}\frac{k^{2(L-i)}\Lambda^{2i+1}}{2i+1}\biggr)^{2}\biggr]
{J_{f}^{\mu}}=\sum_{r=0,2,...}{\tilde{\overline{\Psi}}}_{L_{r}}\sigma^{3}\gamma^{\mu}\tilde{\Psi}_{L_{r}}+\sum_{s=1,3,...}{\tilde{\overline{\Phi}}}_{L_{s}}\gamma^{\mu}\tilde{\Phi}_{L_{s}}.
a_{2}(x)=\frac{1-\alpha^{2}}{r^{4}}\left[(\alpha^{2}-1)\left(\frac1{90}+2\left(\frac16-\xi\right)^{2}\right)+\frac23\left(\frac15-\xi\right)\alpha^{2}\right]+\frac{q^{2}}{6r^{4}}\.
\begin{array}{c}{\displaystyle{X_{(l,m)+}^{2}=\frac{x_{\bar{l}m}^{2}}{x_{\bar{l}1}^{2}x_{\bar{1}m}^{2}}}\,,}\\{{}}\\{X_{(l,m)+}^{\mu}=X_{1(l-1)+}^{\mu}-X_{1(m-1)-}^{\mu}+2i\Theta_{1(l-1)}^{a}\sigma^{\mu}\bar{\Theta}_{1(m-1)a}\,,}\\\end{array}
\Pi_{\mu\nu}^{(\mathrm{rising\sun})}=-(2em)^{2}\,\frac{1}{\beta}\sum_{n}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{\delta_{\mu\nu}\,(k_{0}^{2}+\omega^{2})+k_{\mu}k_{\nu}\,\frac{(k_{0}^{2}+\omega^{2}+\kappa^{2})}{m^{2}}}{((k_{0}+p_{0})^{2}+\omega_{\sigma}^{2})(k_{0}^{2}+\omega_{+}^{2})(k_{0}^{2}+\omega_{-}^{2})}
S_{\mathrm{cin}}=\intd^{4}x\left[\frac{1}{2}(\partial_{\mu}\phi_{+})^{2}+\frac{i\epsilon}{2}\phi_{+}^{2}\right]-\intd^{4}x\left[\frac{1}{2}(\partial_{\mu}\phi_{-})^{2}-\frac{i\epsilon}{2}\phi_{-}^{2}\right].
{\calL}^{(1)}=-\partial_{\mu}\overline{\psi}_{1}\gamma_{\mu\nu}\partial_{\nu}\psi_{1}-\partial_{\mu}\overline{\psi}_{2}\gamma_{\mu\nu}\partial_{\nu}\psi_{2}-m^{2}\overline{\psi}_{1}\psi_{1}+m^{2}\overline{\psi}_{2}\psi_{2}\quad.
{\frac{d^{2}x^{\rho}}{d\tilde{\lambda}^{2}}}+\tilde{\Gamma}_{\mu\nu}^{\rho}{\frac{dx^{\mu}}{d\tilde{\lambda}}}{\frac{dx^{\nu}}{d\tilde{\lambda}}}=-{\frac{1}{2}}{\frac{{\calW}^{\prime}}{\calW}}{\frac{dy}{d\tilde{\lambda}}}{\frac{dx^{\rho}}{d\tilde{\lambda}}}.
\left[C_{a\b}^{\c\d}R_{cd}\,\varphi+2\,\nabla_{c}\nabla_{d}\left(C_{(a\b)}^{\\c\d}\varphi\right)\right]-\left[\overline{C}_{a\b}^{\c\d}\overline{R}_{cd}\bar{\varphi}+2\,\overline{\nabla}_{c}\overline{\nabla}_{d}\left(\overline{C}_{(a\b)}^{\\c\d}\bar{\varphi}\right)\right]\,.
{\calL}_{0}^{\gammafull}=\chi_{x}\upsilon_{t}+\upsilon_{x}\chi_{t}-2{\calH}_{0}^{E}+\lambda\left(u-\frac{\chi_{x}^{\gamma-1}+\upsilon_{x}^{\gamma-1}}{\gamma-1}\right)+\sigma\left(\rho-\chi_{x}\upsilon_{x}\right)
\rho_{m_{0}\rightarrowm_{1}\!+m_{2}\!+m_{3}}^{D}\!=\!\int_{0}^{\infty}\!\!\int_{0}^{\infty}\!\!\int_{0}^{\infty}d\mu_{1}^{2}d\mu_{2}^{2}d\mu_{3}^{2}\frac{\theta(\lambda_{E})}{64\pi^{4}}\rho_{m_{0}\rightarrowm_{1\mu_{1}}\!+m_{2\mu_{2}}\!+m_{3\mu_{3}}}^{D-3}
\left\{x^{\alpha},p_{\beta}\right\}_{D(\Phi_{3n}^{(1)})}=\delta_{\beta}^{\alpha}\,,\,\,\,\,\,\,\,\left\{\psi^{n},\psi^{m}\right\}_{D(\Phi_{3n}^{(1)})}=\frac{i}{2}\eta^{nm}\,,
a_{i}=\oint_{\alpha_{i}}\frac{dx}{2\pii}\left(N-\frac{xF^{\prime}(x)}{F(x)}+\tilde{q}\frac{x^{2N}(NF-xF^{\prime})}{2F(x)^{3}}+\mathcal{O}(\tilde{q}^{2})\right).
L\left(\begin{array}{c}{h_{1}(r)}\\{h_{4}(r)}\\\end{array}\right)\equiv\left(\begin{array}{cc}{\partial_{r}+{\frac{a+j+1/2}{r}}}&{f}\\{f}&{\partial_{r}-{\frac{j-1/2}{r}}}\\\end{array}\right)\left(\begin{array}{c}{h_{1}(r)}\\{h_{4}(r)}\\\end{array}\right)=i\omega\left(\begin{array}{c}{h_{2}(r)}\\{h_{3}(r)}\\\end{array}\right)
\nabla_{A}\overline{{R}}=6\alpha\left(\nabla_{A}W^{2}\right)\overline{{\nabla}}^{2}\overline{{W}}^{2}+24i\alphaW^{2}\nabla^{\dot{A}}\nabla_{A\dot{A}}\overline{{W}}^{2}+{\calO}\left(\alpha^{2}\right)
\widehat{A}_{m,n}\left(k\right)=\inte^{-a^{\ast}a}\frac{da^{\ast}da}{2\pii}C^{+}\left(a,k\right)\left(a^{\ast}\right)^{m}\left(\frac{d}{da^{\ast}}\right)^{n}C\left(a^{\ast},k\right)
\left(\delta_{ij}+eA_{j}\tilde{\partial}_{i}\right)^{-1}=\delta_{ij}-eA_{j}\tilde{\partial}_{i}+e^{2}A_{k}\tilde{\partial}_{i}A_{j}\tilde{\partial}_{k}-e^{3}A_{k}\tilde{\partial}_{i}A_{l}\tilde{\partial}_{k}A_{j}\tilde{\partial}_{l}+\cdots
\begin{array}{l}{\Phi(x_{+},\theta)~\rightarrow~\mathrm{sdet}_{c}^{\frac{1}{3}}\Phi(x_{+}^{\prime},\theta^{\prime})}\\{{}}\\{\overline{{\Phi}}(x_{-},\bar{\theta})~\rightarrow~\overline{{\mathrm{sdet}}}_{c}^{\frac{1}{3}}\overline{{\Phi}}(x_{-}^{\prime},\bar{\theta}^{\prime})}\\\end{array}
\sim\int\frac{d^{2}p}{(2\pi)^{2}}D_{00}(p^{0}=0,\vec{p})e^{i\vec{p}\cdot\vec{R}}=\int\frac{d^{2}p}{(2\pi)^{2}}\frac{e^{i\vec{p}\cdot\vec{R}}}{\vec{p}^{2}+m_{el}^{2}}=\frac{1}{2\pi}K_{0}(m_{el}R)\;.
\frac{\partial}{\partialz_{\rho}}\langleJ_{\mu}(x)J_{\nu}(y)J_{\rho}(z)\rangle_{UV}=-\frac{1}{12\pi^{2}}\epsilon_{\mu\nu\rho\sigma}\frac{\partial}{\partialx_{\rho}}\frac{\partial}{\partialy_{\sigma}}\delta(x-z)\delta(y-z)\equiv\frac{16}{9}{\calA}_{\mu\nu}(x,y,z)
W^{\Omega_{z_{1}},\Omega_{z_{2}}}(v^{(d,a)}\otimesv^{(c,d)})=\sum_{b}v^{(c,b)}\otimesv^{(b,a)}W\left(\left.\begin{array}{cc}{c}&{d}\\{b}&{a}\\\end{array}\right|z_{1}-z_{2}\right).
W_{\gamma}({\bfA})=1+\sum_{n=1}^{\infty}\oint_{\gamma}dy_{1}^{a_{1}}\cdots\int_{o}^{y_{n-1}}dy_{n}^{a_{n}}\mathrm{Tr}({\bfA}_{a_{1}}(y_{1})\ldots{\bfA}_{a_{n}}(y_{n})).
G^{(d)}(\{s_{i}\},\!\{m_{s}^{2}\})=i^{L}\left(\frac{\pi}{i}\right)^{\frac{dL}{2}}\!\prod_{j=1}^{N}\frac{i^{-\nu_{j}}}{\Gamma(\nu_{j})}\!\int_{0}^{\infty}\!\!\!\ldots\!\!\int_{0}^{\infty}\frac{d\alpha_{j}\alpha_{j}^{\nu_{j}-1}}{[D(\alpha)]^{\frac{d}{2}}}e^{i[\frac{Q(\{s_{i}\},\alpha)}{D(\alpha)}-\sum_{l=1}^{N}\alpha_{l}(m_{l}^{2}-i\epsilon)]},
\varphi^{+}(x,t)=i\intd^{d-1}\vec{x}^{\prime}(Ht^{\prime})^{-d}G_{F}(x,t;x^{\prime},t^{\prime})(\overrightarrow{\partial_{t^{\prime}}}-\overleftarrow{\partial_{t^{\prime}}})\varphi^{+}(x^{\prime},t^{\prime})
\gamma_{\mu}\;=\;\left(\begin{array}{cc}{0}&{\sigma_{\mu}^{\dagger}}\\{\sigma_{\mu}}&{0}\\\end{array}\right)\;\;,\;\;\gamma_{5}\;=\;\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\;\;,\;\;\sigma_{\mu}\,{\stackrel{d=4}{=}}\,({\vec{\sigma}},1)\;,\;\sigma_{\mu}\,{\stackrel{d=2}{=}}\,(1,i)\;,
\frac{1}{\sqrt{2\theta}}\,(1-\tilde{P}_{1})\left(1+\frac{\mu-\bar{\mu}}{i-\mu}\,\tilde{P}_{2}\right)a\left(1+\frac{\bar{\mu}-\mu}{i-\bar{\mu}}\,\tilde{P}_{2}\right)\tilde{P}_{1}=0.
\frac{a_{o}^{\prime}}{a_{o}}\left(\frac{a_{o}^{\prime}}{a_{o}}+\frac{n_{o}^{\prime}}{n_{o}}\right)-\frac{b_{o}^{2}}{n_{o}^{2}}\left(\frac{\dot{a}_{o}}{a_{o}}\left(\frac{\dot{a}_{o}}{a_{o}}-\frac{\dot{n}_{o}}{n_{o}}\right)+\frac{\ddot{a}_{o}}{a_{o}}\right)-k\frac{b_{o}^{2}}{a_{o}^{2}}=-\frac{1}{6M^{3}}\Lambdab_{o}^{2}+\frac{1}{6M^{3}}T_{55},
{\tilde{\calL}_{E}}=\frac{1}{4\pi^{2}g}\bigg\{-\sqrt{R^{2}(1+{\dot{R}}^{2}+{R^{\prime}}^{2})+{\tilde{B}}^{2}(1+{R^{\prime}}^{2})}+\frac{h}{2}R^{2}\bigg\}.
H_{3}\propto\intd\bar{\mu}_{3}\left[\frac{(|\alpha_{1}|+|\alpha_{2}|+|\alpha_{3}|)^{4}}{\alpha_{1}^{2}\alpha_{2}^{2}\alpha_{3}^{2}}\left(P_{\perp}^{2}-P_{\parallel}^{2}\right)s_{1}s_{2}\bar{s}_{3}\right]~.
\left(\begin{array}{cccccccc}{{\calA}_{n},}&{0}\\{{\calB}_{n},}&{{\calA}_{(n-1)}}&{0}&{}&{}\\{0}&{{\calB}_{(n-1)}}&{{\calA}_{(n-2)}}&{0}\\{}&{0}&{{\calB}_{(n-2)}}&{{\calA}_{(n-3)}}&{}\\{}&{}&{}&{}&{}&{0}\\{}&{}&{}&{}&{}&{{\calB}_{2}}&{{\calA}_{1}}&{0}\\{}&{}&{}&{}&{}&{0}&{{\calB}_{1}}&{{\calA}_{0}}\\\end{array}\right)
p_{r}^{2}=\frac{1}{g^{rr}}\left[-g^{tt}({\calE}+qA_{t})^{2}+2g^{t\phi}({\calE}+qA_{t})(m-qA_{\phi})-g^{\phi\phi}(m-qA_{\phi})^{2}-g^{\theta\theta}p_{\theta}^{2}-\mu^{2}\right],
\sum_{j=1}^{n}q^{\rho_{j}-2(n-j)}\Lambda_{j}^{1/2}\mu_{j}^{-1/2}{\calX}^{j}{\calX}^{-j}+\frac{q^{-2n+1}}{1+q}{\calX}^{0}{\calX}^{0}=-\frac{1}{a^{2}}~.
f_{ij}^{(1)}(x,y)=\left(\begin{array}{ccccc}{0}&{0}&{-1}&{0}&{-\frac{m}{2}\partial_{x}^{2}}\\{0}&{0}&{0}&{-1}&{\frac{m}{2}\partial_{x}^{1}}\\{1}&{0}&{0}&{0}&{\partial_{x}^{1}}\\{0}&{1}&{0}&{0}&{\partial_{x}^{2}}\\{-\frac{m}{2}\partial_{x}^{2}}&{\frac{m}{2}\partial_{x}^{1}}&{\partial_{x}^{1}}&{\partial_{x}^{2}}&{0}\\\end{array}\right)\delta^{2}(x-y)
\sum_{n=1}^{\infty}e^{-\betakE_{n}}=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}dx\;e^{-\frac{x^{2}}{4}-\frac{\beta^{2}k^{2}m^{2}}{x^{2}}}\frac{1}{e^{\frac{2\beta^{2}eBk^{2}}{x^{2}}}-1}
Q_{n}[\overline{{\psi}},\psi,A,h]\;:=\;P[\overline{{\psi}},\psi,A,h]\Big(S_{M}[\overline{{\psi}},\psi]\Big)^{n}\;\longrightarrow\;\tilde{Q_{n}}[G,\frac{\delta}{\deltaa},A,h]\;.
{\frac{1}{N}}\sum_{n=n_{1}}^{n_{2}}f({\frac{n}{N}})=\int_{\frac{n_{1}-1/2}{N}}^{\frac{n_{2}+1/2}{N}}f(x)dx-{\frac{1}{24N^{2}}}\mathrm{[}f^{\prime}({\frac{n_{2}+1/2}{N}})-f^{\prime}({\frac{n_{1}-1/2}{N}})\mathrm{]}+O({\frac{1}{N^{4}}}).
\equiv\left(\frac{\alpha}{2\pi}\right)\left[f_{n+1}\gamma_{\nu}-g_{n+1}\frac{(\not{p}+\not{k})\gamma_{\mu}(\not{p}+\not{k})}{(p+k)^{2}}\right]\left(\frac{\mu^{2}}{p^{2}}\right)^{(n+1)x},
\begin{array}{l}{\Omega_{(p)}=(-1)^{p-1}2^{p-1}k_{\alpha_{1}\dots\alpha_{p-1}i_{p}\dotsi_{m-1}b}C_{a_{2p-1}a_{2p}}^{i_{p}}\dotsC_{a_{2m-3}a_{2m-2}}^{i_{m-1}}}\\{\qquad\qquad\qquad\cdot\,\mathcal{W}^{\alpha_{1}}\wedge\dots\wedge\mathcal{W}^{\alpha_{p-1}}\wedge\omega^{a_{2p-1}}\wedge\dots\wedge\omega^{a_{2m-2}}\wedge\omega^{b}\quad,}\\\end{array}
I=\Big(\mathrm{Tr}e^{iF_{1}}+\mathrm{Tr}e^{iF_{2}}-\mathrm{tr}e^{iF_{1}}\,\mathrm{tr}e^{-iF_{2}}-\mathrm{tr}e^{-iF_{1}}\,\mathrm{tr}e^{iF_{2}}\Big)\hat{A}(R)
\begin{array}{ccc}{\psi^{a+}}&{=-\frac1{\sqrt{2}}\left(\chi^{a}+\overline{{\chi}}^{a}\right)}&{}\\{\psi^{a-}}&{=-\fraci{\sqrt{2}}\left(\chi^{a}-\overline{{\chi}}^{a}\right)}&{}\\\end{array}
=-i\left[e^{(a+1)P_{0}/\kappa}\,\frac{\partial}{\partialP_{0}}+\frac{e^{(1+a)P_{0}/\kappa}}{2}\left(\frac{1+a}{\kappa}-\frac{e^{-P_{0}/\kappa}}{f(P_{0})}\right)\right]\Psi_{\xi}(P_{0}),
L=-{\frac{1}{4}}\,Tr\,{\bfF}_{\mu\nu}(\,{\bfA}\,){\bfF}^{\mu\nu}(\,{\bfA}\,)-{\frac{1}{2}}\,Tr\,\left[\,m\,\left(\,{\bfA}_{\mu}-{\bf\Phi}_{\mu}\,\right)-D_{\mu}({\bf\Phi}){\bfC}\,\right]^{2}
S=-\frac{1}{4\pi\alpha^{\prime}}\int\mathrm{d}^{2}\!\sigma\,\left(G_{IJ}\,\partial_{\alpha}\theta^{I}\partial_{\alpha}\theta^{J}+\epsilon^{\alpha\beta}B_{IJ}\,\partial_{\alpha}\theta^{I}\partial_{\beta}\theta^{J}\right)\,,
\beta=\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{\omega_{2d}}&{0}\\{0}&{0}&{\omega_{2d}^{-1}}\\\end{array}\right)~~~~~~~\gamma=\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{0}&{i}\\{0}&{i}&{0}\\\end{array}\right)
S=-m_{f}\intd\tau{\frac{(x^{2}+4Q_{p}z)^{\frac{n}{2}}}{Q_{F}}}\left[\sqrt{1-{\frac{Q_{F}v_{\parallel}^{2}}{(x^{2}+4Q_{p}z)^{\frac{n}{2}}}}-{\frac{Q_{p}Q_{F}v_{\perp}^{2}}{z(x^{2}+4Q_{p}z)^{\frac{n}{2}}}}}-1\right],
\mathrm{solv}\bigl({\frac{\mathrm{O}(5,Q)}{\mathrm{O}(5)\times\mathrm{O}(Q)}}\bigr)=\mathrm{solv}\bigl({\frac{\mathrm{O}(1,Q)}{\mathrm{O}(Q)}}\bigr)+\mathrm{solv}\bigl({\frac{\mathrm{O}(4,Q)}{O(4)\times\mathrm{O}(Q)}}\bigr)
N_{B}=i(2\pi)^{p+1}g^{N}\delta(\sum_{i=1}^{N}{\bfp}_{i})(4\pi^{2}\alpha^{\prime}l)^{-(p+1)/2}\mathrm{det}({\bf1}+{\bfB})\prod_{i=1}^{N}\intds_{i}{\sqrt{g}}\quad.
{\calL}_{GRAV}=\sqrt{g}R+\frac1{2\mu}{\calE}^{\mu\lambda\nu}\Gamma^{\rho}{}_{\lambda\sigma}\left(\partial_{\mu}\Gamma^{\sigma}{}_{\nu\rho}+\frac23\Gamma^{\sigma}{}_{\mu\xi}\Gamma^{\xi}{}_{\nu\rho}\right)\;\,.
H={\frac{1}{2\pi\alpha^{\prime}P^{+}}}\intd\sigma\left({\frac{1}{2}}(\partial_{\tau}X^{i})^{2}+{\frac{1}{2}}(\partial_{\sigma}X^{i})^{2}+{\frac{C_{1}}{2\tau^{2}}}(X^{i})^{2}\right).
\left({\frac{\kappa_{5}^{2}\rho_{0}}{3\sqrt{|\xi_{+}^{2}-\xi_{-}^{2}|}}}\right)^{\frac{3}{3m-k^{2}}}\lesssimR\lesssim\left({\frac{\rho_{0}}{|\lambda_{0}|}}\right)^{\frac{3}{3m-k^{2}}}.
H_{F}\=\\left(\vec{q}_{1}+i\,\vec{q}_{2}\right)\;\cdot\;\left(\vec{\lambda}\wedge\vec{\lambda}\right)-\left(\vec{q}_{1}-i\,\vec{q}_{2}\right)\cdot\left(\frac{\partial}{\partial\vec{\lambda}}\wedge\frac{\partial}{\partial\vec{\lambda}}\right)\.
(\partial_{\bar{z}}A_{0}^{\bar{z}})(\partial_{z}A_{0}^{z})-(\partial_{z}A_{0}^{\bar{z}})(\partial_{\bar{z}}A_{0}^{z})=\partial_{z}\varphi_{0}\partial_{\bar{z}}\varphi_{0}-V(\varphi_{0})={\calL}_{0}\,.
\int\left[\mathrm{d}\phi\right]e^{-\int\mathrm{d^{d}}x\left[{\frac{1}{2}}\left(\partial_{\mu}\phi\right)^{2}+{\frac{1}{2}}m^{2}\phi^{2}+(\phiD\phi/C_{N})\right]}\left(\int\mathrm{d^{d}}x\phi^{4}\right)^{n}\left(\int\mathrm{d^{d}}x\\phiD\phi\right)^{m}
g(x_{\mu};q)=\left[\left(-\frac{(1-q)^{2}}{4}\rho_{+}^{2};q^{2}\right)_{\infty}\right]^{1/4}\left[\left(-\frac{(1-q)^{2}}{4}\rho_{-}^{2};q^{2}\right)_{\infty}\right]^{1/4}\cdot1.
L_{1}^{\left(0\right)}\,=\,-\frac{a}{2}A_{\mu}^{\left(0\right)}\,A^{\mu\left(0\right)}\,+\,\frac{b}{2}\left(F_{\mu\nu}^{\left(0\right)}\right)^{2}\,+\,\theta\,\epsilon^{\mu\nu\rho}\,\,A_{\mu}^{\left(0\right)}\,\partial_{\nu}\,A_{\rho}^{\left(0\right)}.
\frac{1}{V}\frac{\langle0_{m}|H|0_{m}\rangle}{\langle0_{m}|0_{m}\rangle}=\frac{B^{2}}{2}-\frac{eB}{2\pi}\sum_{n=0}^{\infty}\sqrt{2neB+m^{2}}+\frac{|m|}{V}\frac{\langle0_{m}|{\bar{\psi}}(\vec{x})\psi(\vec{x})|0_{m}\rangle}{\langle0_{m}|0_{m}\rangle}
M_{\mu}(k)=1-2l_{1s}+\frac{(-1)^{2l_{2s}}l_{1s}\sqrtk}{[1+(-1)^{2l_{2s}}\sqrtk]}\quad\mathrm{and}\quadM_{\nu}(k)=1-2l_{1s}+\frac{3l_{1s}[1+(-1)^{2l_{2s}}k])}{[1+(-1)^{2l_{2s}}\sqrtk]}.
M_{ADM}^{2}={\frac{1}{2}}{\vec{\tilde{\alpha}}^{T}}\mu_{+}{\vec{\tilde{\alpha}}}+{{\frac{1}{2}}}e^{-4\Phi_{\infty}^{\prime}}{\vec{\beta}}^{T}\mu_{+}{\vec{\beta}}+e^{-2\Phi_{\infty}^{\prime}}\left[({\vec{\beta}}^{T}\mu_{+}{\vec{\beta}})({\vec{\alpha}}^{T}\mu_{+}{\vec{\alpha}})-({\vec{\beta}}^{T}\mu_{+}{\vec{\alpha}})^{2}\right]^{\frac{1}{2}},
{\tilde{I}}_{1}(B)=-\frac{eB}{\pi^{\frac{3}{2}}\beta}\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}\int_{0}^{\infty}dxe^{-\frac{x^{2}}{4}-\frac{\beta^{2}k^{2}m^{2}}{x^{2}}}\left[\frac{1}{e^{\frac{2\beta^{2}eBk^{2}}{x^{2}}}-1}-\frac{x^{2}}{2eB\beta^{2}k^{2}}\right].
\left(\begin{array}{c}{a_{D}}\\{a}\\\end{array}\right)\rightarrow\left(\begin{array}{cc}{0}&{-1}\\{1}&{0}\\\end{array}\right)\left(\begin{array}{c}{a_{D}}\\{a}\\\end{array}\right)=\left(\begin{array}{c}{-a}\\{a_{D}}\\\end{array}\right),
L=-\frac{1}{4}G_{a}^{\mu\nu}G_{a\mu\nu}+\frac{1}{2}\left(D_{\mu}S\right)_{a}^{*}\left(D^{\mu}S\right)_{a}+\frac{1}{2}\left(D_{\mu}\phi^{\dagger}\right)\left(D^{\mu}\phi\right)-V(S,\phi)
G_{Q_{\lambda}}^{-1}(v)\equivK_{Q_{\lambda}}(v)=\frac{1}{v}-\frac{a^{2}}{4}v-2k-(\lambda-k)\sqrt{\left(1-\frac{a^{2}v}{2(\lambda-k)}\right)^{2}-\frac{a^{2}}{(\lambda-k)^{2}}}
{\calF}_{2}=-\frac{\mathrm{ln}Z_{2}}{\betaV}=\frac{1}{(4\pi)^{2}}g^{2}M_{r}T^{2}\left(\frac{M_{r}}{T}+2\mathrm{ln}(1-e^{-M_{r}/T})\right)^{2}+{\calO}(g^{4})\;.
\left.{\frac{\partialS}{\partialK_{A}^{\prime}}}\right|_{\Phi^{\prime}}={\frac{\partial\Phi^{\primeA}}{\partialg}}-{\frac{\partialK_{B}}{\partialg}}{(M^{-1})^{B}}_{C}F^{CA}=\left.{\frac{\partial\Phi^{\primeA}}{\partialg}}\right|_{\Phi,K}.
{\bfF}(\Box)=-\frac{1}{J_{2}^{+}\,Y_{2}^{-}-J_{2}^{-}\,Y_{2}^{+}}\left[\begin{array}{cc}{\,\sqrt{\Box}z_{+}u_{+}(z_{-})}&{-\displaystyle{\frac2\pi}}\\{-\displaystyle{\frac2\pi}}&{\sqrt{\Box}z_{-}u_{-}(z_{+})\,}\\\end{array}\right].
{\calT}~=~\left(\begin{matrix}{1}&{0}&{-1}&{-2}&{-7}&{}\\{0}&{1}&{2}&{3}&{10}\\\end{matrix}\right)\equiv\left(\begin{matrix}{1}&{0}&{-1}&{-3}&{-2}&{-7}&{}\\{0}&{1}&{2}&{5}&{3}&{10}\\\end{matrix}\right)
A_{0\ldotsp}^{\prime}=\tilde{F}_{r0\ldotsp}-\tilde{b}^{\prime}\,C_{0\ldotsp}=(3-p)\frac{q_{1}}{r^{4-p}}\frac{1}{H}\left[1+\frac{r}{3-p}\frac{q_{2}^{2}}{q_{1}^{2}}h_{3}\left(\frac{h_{2}}{h_{1}}\right)^{\prime}\right]\,,
S(x-y)=\sum_{k=-\infty}^{\infty}S^{(k)}(x-y)=-i\frac{1}{Z^{(0)}[0,0,0]}\sum_{k=-\infty}^{\infty}e^{ik\theta}\left.\frac{\delta^{2}Z^{(k)}[\eta,\overline{{\eta}},J]}{\delta\overline{{\eta}}(x)\delta\eta(y)}\right|_{J,\eta,\overline{{\eta}}=0}\;.
R_{12}^{\Phi_{1}\Phi_{2}}(x)R_{13}^{\Phi_{1}\Phi_{3}}(xy)R_{23}^{\Phi_{2}\Phi_{3}}(y)=R_{23}^{\Phi_{2}\Phi_{3}}(y)R_{13}^{\Phi_{1}\Phi_{3}}(xy)R_{12}^{\Phi_{1}\Phi_{2}}(x).
\gammaB_{a}^{0i}=2\partial_{j}\eta_{a}^{ij},\;\gammaB_{a}^{ij}=2\eta_{a}^{(1)ij},\;\gammaH_{i}^{a}=-C_{i}^{(1)a},\;\gammaH_{0}^{a}=\partial^{i}C_{i}^{a},
\omega=-\frac{1}{2}d[S\,\mathrm{tr}\,(M_{3}R^{T}(\varphi,\vartheta)dR(\varphi,\vartheta))]+\frac{1}{2}d[S\,\mathrm{tr}\,(M_{3}F^{T}(\lambda,\mu,\nu)dF(\lambda,\mu,\nu))].
\delta_{\epsilon}A_{\mu}^{a}=\partial_{\mu}\epsilon^{a},\;\delta_{\epsilon}H_{\mu}^{a}=\partial^{\nu}\epsilon_{\mu\nu}^{a},\;\delta_{\epsilon}\varphi_{a}=0,\;\delta_{\epsilon}B_{a}^{\mu\nu}=\partial_{\rho}\epsilon_{a}^{\mu\nu\rho},
\frac{R}{R_{0}}=\left(\frac{V}{V_{0}}\right)^{-\frac{1}{3(1+\omega)}},\;\;\;\frac{\dot{R}}{R_{0}H_{0}}=\left(\frac{V}{V_{0}}\right)^{-\frac{3\omega+5}{6(1+\omega)}}\frac{V^{\prime}}{V_{0}^{\prime}},
\frac{1}{V}<H^{\theta}>=\frac{1}{8\pi}[\frac{4}{3}\Lambda^{3}+\frac{1}{3}(1+\frac{8\theta^{3}g^{3}}{\eta^{3}})m^{3}]-\frac{1}{4}\frac{g^{2}}{\pi^{2}}m^{2}
\begin{array}{l}{\displaystyle{H^{B}=\intd\sigma\left(\Pi_{i}\dot{x}_{i}-\mathcal{L}_{B}\right)=}}\\{\displaystyle{\qquad\frac{1}{4\pil_{s}^{2}}\int_{0}^{2\pil_{s}^{2}p^{+}}d\sigma\left[\sum_{i=1}^{8}({\dot{x}_{i}}^{2}+{{x}_{i}^{\prime}}^{2})+\mu^{2}\left(\sum_{I=3}^{8}x_{I}^{2}+4x_{2}^{2}\right)+4\mux_{2}\partial_{\sigma}x_{1}\right]}}\\\end{array}
I_{3}\le4\pi\int_{\verta\vert^{4}}^{\infty}\frac{1+\vert\beta\vert^{2}\vertz\vert^{2}}{(1+\vert\beta\vert^{2}\vertz\vert^{2}+\vert\lambda\vert^{2}\vertz\vert^{4}/16)^{2}}d\vertz\vert^{2}\le\frac{2^{10}\pi\vert\beta\vert^{2}}{\vert\lambda\vert^{4}\verta\vert^{8}}.
|\Phi\rangle=\sum_{r,s}\,b_{\overline{\jmath}_{1}...\overline{\jmath}_{s}}^{i_{1}...i_{r}}\,\lambda_{i_{1}}\dots\lambda_{i_{r}}\,\psi^{\overline{\jmath}_{1}}\dots\psi^{\overline{\jmath}_{s}}|0\rangle~,
\begin{array}{ccccc}{p_{j}=-p_{i},}&{}&{\alpha_{n}^{(j)}=-\alpha_{-n}^{(i)}=-\alpha_{n}^{(i)\dagger},}&{}&{\bar{\alpha}_{n}^{(j)}=-\bar{\alpha}_{-n}^{(i)}=-\bar{\alpha}_{n}^{(i)\dagger}.}\\\end{array}
\gamma=8\piG\eta_{1}^{2}\\,\\alpha=e^{2}/\lambda_{1}\\,\\q=\frac{\eta_{2}}{\eta_{1}}\\,\\{\bar{\Lambda}}=\frac\Lambda{\lambda_{1}\eta_{1}^{2}}\\,\\,beta_{i}^{2}=\frac{\lambda_{i}}{\lambda_{1}}\\,\\i=1,2,3\.
\kappa_{11}^{2}\Big(\delta_{2}S_{\mathrm{CJS}}+\deltaS_{T}\Big)=\frac{1}{4}e\,T_{mn}{}^{\nu}\bar{\epsilon}\Gamma_{\nu}{}^{\kappa\sigmamn}{\calT}_{\kappa}\cdot\hat{H}\psi_{\sigma}+(\epsilon\leftrightarrow\psi_{\sigma})\,.
<\eta>=\delta(p_{1}-p_{2}){\frac{-ig^{2}}{4\sqrt{N}}}\sigma^{0}(p_{1})\sigma^{0}(p_{2})\int{\frac{d^{d}k}{(2\pi)^{d}}}\Big[\sigma^{0}(k)-{\frac{ig^{2}}{4}}\int{\frac{d^{d}l}{(2\pi)^{d}}}\Big[(\sigma^{0}(l))^{2}\sigma^{0}(k)+\sigma^{0}(k)\sigma^{0}(l)\sigma^{0}(k+l-p_{1})\Big]\Big]
\int_{{\calM}_{\alpha}}R^{\mu\nu}R_{\mu\nu}=\alpha\int_{{\calM}_{\alpha=1}}R^{\mu\nu}R_{\mu\nu}+4\pi(1-\alpha)\int_{\Sigma}R_{\mu\nu}n_{i}^{\mu}n_{i}^{\nu}+O((1-\alpha)^{2})~~~,
\widehat{D}=\left(\begin{array}{cccc}{0}&{1+t\nabla_{x}}&{1}&{1}\\{-1-t\nabla_{-x}}&{0}&{-1}&{1}\\{-1}&{1}&{0}&{1+t\nabla_{y}}\\{-1}&{-1}&{-1-t\nabla_{-y}}&{0}\\\end{array}\right).