harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f^{i}=f_{\,{\hat{\alpha}}{\hat{\beta}}}^{i}{\hat{e}}^{\hat{\alpha}}\wedge{\hat{e}}^{\hat{\beta}}=2{\hat{e}}^{4}\wedge{\hat{e}}^{i}+\varepsilon_{ijk}{\hat{e}}^{j}\wedge{\hat{e}}^{k}
G_{l+\nu}\left(\left.r^{\prime\prime},r^{\prime};E\right|\lambda;a;\sigma\right)=G_{l+\nu}(r^{\prime\prime},r^{\prime};E|\lambda)-\frac{G_{l+\nu}(r^{\prime\prime},a;E|\lambda)\,G_{l+\nu}(a,r^{\prime};E|\lambda)}{G_{l+\nu}(a,a;E|\lambda)-1/\sigma}\;.
\left(f_{\bfk}({\bfx},t),f_{\bfk^{\prime}}({\bfx},t)\right)\equiv-i\intd^{m}x\,n^{2}\left\{f_{\bfk}({\bfx},t)\left[\partial_{t}f_{\bfk^{\prime}}^{*}({\bfx},t)\right]-\left[\partial_{t}f_{\bfk}({\bfx},t)\right]f_{\bfk^{\prime}}^{*}({\bfx},t)\right\}=\delta^{m}({\bfk}-{\bfk^{\prime}}).
{\calL}=\frac{1}{2l_{0}^{2}}[\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)],~~~V(\phi)=-2\frac{\lambda^{2}}{l_{0}^{2}}\phi^{2}+\frac{1}{2}\lambda\frac{\lambda^{2}}{l_{0}^{2}}\frac{\phi^{4}}{\bar{\phi^{2}}}
\tilde{\varphi}_{ij}(t)-2\pi\delta_{ij}:=\int_{-\infty}^{\infty}d\theta\;\varphi_{ij}(\theta)e^{it\theta}-2\pi\delta_{ij}=2\pi\sum_{k=0}^{\infty}(-i)^{k}\eta_{ij}^{(k)}t^{k}\quad,
\begin{array}{l}{\int\partial_{\mu}\phi^{*}\partial^{\mu}(i(L_{1}\cdotsL_{k}+L_{k}\cdotsL_{1}).\phi)=}\\{=-\int\partial_{\mu}(i(L_{1}\cdotsL_{k}+L_{k}\cdotsL_{1}).\phi)^{*}\partial^{\mu}\phi}\\\end{array}
x=1/L,\,\,\,y=1/\gamma_{L},\,\,\,\a=\frac{\Lambda}{M_{7}^{5}},\,\,\,b=\frac{Q^{2}}{M_{7}^{5}},\,\,\,\,d=\frac{2\alpha^{\prime}}{M_{7}^{5}}.
I=-{\frac{1}{4}}\int_{M^{2}}^{}(r^{2}R+2(\nablar)^{2}+2)\sqrt{\gamma}d^{2}x-{\frac{1}{2}}\int_{\partialM^{2}}^{}r^{2}(k-k_{0})dy+\frac12\int\sqrt{\gamma}\varphi_{,\mu}\varphi^{,\mu}.
\Omega_{\mathrm{red}}=d{\bfp}\wedged{\bfx}+s\frac{{\bfx}\timesd{\bfx}\timesd{\bfx}}{|{\bfx}|^{3}},\quad{\bfJ}_{red}={\bfJ}/2={\bfp}\times{\bfx}+s\frac{{\bfx}}{|{\bfx}|}.
{\frac{1}{B^{2}}}\left({\frac{1}{A^{2}}}-{\frac{\ell^{2}}{\tilde{B}^{2}r^{2}}}\right)=0,\qquad{\frac{1}{B^{2}}}\left({\frac{(A^{2})^{\prime}}{A^{4}}}-{\frac{(\tilde{B}^{2}r^{2})^{\prime}\ell^{2}}{\tilde{B}^{4}r^{4}}}\right)=0.
L_{R}=\frac{1}{2}\partial_{\mu}\phi_{1}\partial_{\mu}\phi_{1}+\frac{1}{2}M^{2}\phi_{1}^{2}+\frac{1}{4!}\lambda_{1}\mu^{2\epsilon}(\phi_{1}\phi_{1})^{2}+L_{ct}-\frac{1}{2}\sigma^{2}\phi_{1}\phi_{1}~~,
<\hat{{\cal{J}}}_{\mu_{1}\cdots\mu_{s}}^{(s)}(x)~\hat{{\cal{J}}}_{\nu_{1}\cdots\nu_{s}}^{(s)}(0)>={\prod}_{\mu_{1}\cdots\mu_{s},\nu_{1}\cdots\nu_{s}}^{(s)}\left({\frac{c_{s}(t)}{x^{4}}}\right).
\sum_{j_{12}}(-1)^{2(2(j_{1}+j_{2})(j_{12}+j_{12}^{\prime})+j_{12})}\left\{\begin{array}{ccc}{j_{1}}&{j_{2}}&{j_{12}}\\{j_{1}}&{j_{2}}&{j_{12}^{\prime}}\\\end{array}\right\}^{S}=1~.
\mathcal{L}=-\frac{1}{4}T^{ab}T_{ab}-\frac{1}{2}\,\frac{(H^{a}+T^{ac}\!A_{c})^{2}}{1-A^{b}A_{b}}+\frac{1}{2}\,\frac{(A_{a}H^{a})^{2}}{1-A^{b}A_{b}}-\frac{1}{4g^{2}}F^{ab}F_{ab}\,,
H=\frac{NT_{0}}{3}\left[\frac{1}{2}\sum_{i=1}^{3}\dot{r}_{i}^{2}+\frac{\alpha^{2}}{2}\Big(r_{1}^{2}r_{2}^{2}+r_{1}^{2}r_{3}^{2}+r_{2}^{2}r_{3}^{2}\Big)\right]~~,
\frac{T_{p+2}}{\kappa}\int_{\mathcal{M}_{p+3}}C_{p+3}=\frac{T_{p+2}}{\kappa}\int_{\mathcal{M}_{p+3}}A_{p+1}\wedge\omega_{2}=\frac{T_{p}}{2\kappa}\int_{\mathcal{M}_{p+1}}\frac{A_{p+1}}{2\pi^{2}\alpha^{\prime}}\,,
\partial_{z}S^{z}\partial_{\bar{z}}S^{\bar{z}}-\partial_{z}S^{\bar{z}}\partial_{\bar{z}}S^{z}=-{\calH}\,,\quad\begin{array}{c}{p=\partial_{\bar{z}}S^{\bar{z}}\partial_{\varphi}S^{z}-\partial_{\bar{z}}S^{z}\partial_{\varphi}S^{\bar{z}}}\\{\bar{p}=\partial_{z}S^{z}\partial_{\varphi}S^{\bar{z}}-\partial_{z}S^{\bar{z}}\partial_{\varphi}S^{z}}\\\end{array}\,.
\begin{array}{rclc}{\left(0,1,0,1,e^{\phi}\right)e_{a}}&{=}&{e_{a}}&{.}\\{\left(V_{1},1,0,1,1\right)e_{a}}&{=}&{-\sqrt{2}V_{1}^{a}\cdotu_{+}+e_{a}}&{.}\\{\left(0,1,V_{2},1,1\right)e_{a}}&{=}&{e_{a}}&{.}\\{\left(0,C_{1},0,1,1\right)e_{a}}&{=}&{\sum_{b=2}^{n-m}\left(C_{1}\right)_{ab}\cdote_{b}}&{.}\\{\left(0,1,0,C_{2},1\right)e_{a}}&{=}&{e_{a}}&{.}\\\end{array}
=3\sum_{k=0}^{\infty}C_{k}(z){\frac{(\sqrt{\sigma})^{2k}}{2z+2k-1}}\left[\zeta_{R}(2z+2k-3,3/2)-\frac14\zeta_{R}(2z+2k-1,3/2)\right]\equiv3f(z,\sigma)~~~.
\prod_{a\inA}\oint_{C_{a}}\frac{dw_{a}}{2\piiw_{a}}\Psi_{b}^{(n)}(w|\zeta_{1}^{-1},\tilde{\zeta})Q_{b}^{(n)}(w|\zeta_{1}^{-1},\tilde{\zeta})\frac{1-x^{2}rz_{1}^{-1}}{1-x^{-2}rz_{1}}.
H_{|2}=\frac1{4l_{p}^{2}}\int_{\Sigma}d^{3}x\sqrt{g}G^{ijkl}\left[K^{-1\bot}\left(x,x\right)_{ijkl}+\left(\triangle_{2}\right)_{j}^{a}K^{\bot}\left(x,x\right)_{iakl}\right]
W=\int\frac{2d^{10}\tilde{p}_{1}}{\tilde{p}_{1}^{2}\tilde{p}_{2}^{2}}\sum_{(\lambda,\lambda^{\prime})}\hat{A}_{(0)}(\{p,\epsilon\}_{(i)},\tilde{p}_{1}\lambda,\lambda^{\prime})\hat{A}_{(n-1)}(\lambda,\lambda^{\prime};\tilde{p}_{1},\{p,\epsilon\}_{(f)})\,,
\sum_{i\neqj}[(\frac{\partial_{i}\theta^{(i)}}{\theta^{(i)}})^{2}\frac{\partial_{j}^{2}\theta^{(j)}}{\theta^{(j)}}-(\frac{\partial_{i}\theta^{(i)}}{\theta^{(i)}})^{2}(\frac{\partial_{j}\theta^{(j)}}{\theta^{(j)}})^{2}]=0.
\begin{array}{lcl}{\tilde{\mathcalL}_{-2}^{(-3,-1)}}&{=}&{{v_{1}\starp}^{-3}+v_{0}\starp^{-2}+v_{-1}\starp^{-1}}\\{}&{=}&{v_{-1}{p}^{-1}+(v_{0}+\thetav_{-1}^{\prime})p^{-2}+(v_{1}+2\thetav_{0}^{\prime}+\theta^{2}u_{-1}^{\prime\prime})p^{-3}}\\\end{array}
S_{eff}\sim\intd^{D}X\sqrt{G}e^{-2\phi}\left[\frac{2}{3}(26-D)+\alpha^{\prime}(R+4\alpha^{\prime}(\partial\phi)^{2}-\frac{1}{12}H_{\lambda\mu\nu}^{2})+\alpha^{2}\frac{1}{4}R_{\lambda\mu\nu\beta}^{2}+...\right]~.
I={\frac{1}{4G_{2}}}\left[\int\sqrt{\gamma}d^{2}x\left(e^{2\phi}R_{2}+2e^{2\phi}(\nabla\phi)^{2}+{\frac{2}{a^{2}}}\right)+2\intdye^{2\varphi}(k-k_{0})\right]~~~,
\left(\frac{\partial^{2}}{\partialt^{2}}-\frac{\partial^{2}}{\partialr^{2}}-\frac{1}{r}\frac{\partial}{\partialr}-4\Omega\theta\frac{\partial}{\partialt}-\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}-4\Omega\thetar\frac{\partial^{2}}{\partialr\partialt}-\frac{\partial^{2}}{\partialz^{2}}\right)\varphi(t,r,\theta,z)=0.
S_{\alpha\beta}^{64}=\Gamma_{\alpha\beta}^{\bar{M}_{1}\bar{M}_{2}\bar{M}_{3}}E_{\bar{M}_{1}\bar{M}_{2}\bar{M}_{3}}+\Gamma_{\alpha\beta}^{\bar{M}_{1}\bar{M}_{2}\cdots\bar{M}_{7}}F_{\bar{M}_{1}\bar{M}_{2}\cdots\bar{M}_{7}}^{+},
\begin{array}{c}{\displaystylec_{-1}=\frac1{(\lambda^{2}-\xi^{2}-\tau^{2})}(\not\!\xi-\lambdaI),}\\{\displaystylec_{-2}=\frac\alpha{(\lambda^{2}-\xi^{2})^{2}}(2\lambda\xi_{\mu}A_{\mu}I-(\lambda^{2}-\xi^{2})\not\!\!A-2\xi_{\mu}A_{\mu}\not\!\xi),}\\\end{array}
H_{D}^{F}=\tilde{\lambda}^{\mu}(\tau)\tilde{H}_{\mu}^{*}(\tau)-\frac12\tilde{\lambda}^{\mu\nu}(\tau)\tilde{H}_{\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]},
\left.\begin{array}{l}{(\gamma_{2})=(D_{2})^{-1}(A)^{-1}[S_{2}](D_{1}),}\\{(\gamma_{3})=(D_{3})^{-1}(B)^{-1}[S_{3}](D_{2}),}\\{(\gamma_{4})=(A)[S_{4}](D_{3}),}\\\end{array}\right.
\sigma_{\mathrm{cl}}=\intdx_{\bot}\left[\frac{1}{2}(\partial_{\bot}h(x_{\bot}))^{2}-\frac{m^{2}}{2}h(x_{\bot})^{2}+\frac{\lambda}{4!}h(x_{\bot})^{4}+\frac{3m^{4}}{2\lambda}\right],
\overline{\zeta}_{3}:=\frac{(d-3)k^{4}}{6(g(k^{2})\pi^{d/2})^{2}}\int\frac{dP_{1}dP_{2}dP_{3}~\delta(k-P_{1})}{P_{1}^{2}P_{2}^{2}P_{3}^{2}(P_{1}-P_{2})^{2}(P_{2}-P_{3})^{2}(P_{3}-P_{1})^{2}}
T_{n}(x_{1},\ldots,x_{n};\hbar)=\sum_{\pi}T_{1}(x_{\pi(1)})\ldotsT_{1}(x_{\pi(n)})\Theta(x_{\pi(1)}^{0}-x_{\pi(2)}^{0})\ldots\Theta(x_{\pi(n-1)}^{0}-x_{\pi(n)}^{0}),
S=S_{0}^{L}\left[A_{\mu}^{a},H_{\mu}^{a},\varphi_{a},B_{a}^{\mu\nu}\right]+\intd^{2}x\left(A_{a}^{*\mu}\partial_{\mu}\eta^{a}+H_{a}^{*\mu}\partial^{\nu}\eta_{\mu\nu}^{a}\right),
Pd\phi_{1}\dotsd\phi_{n}\propto\prod_{i<j}\left|\frac{\epsilon_{i}-\epsilon_{j}}{\epsilon_{i}+\epsilon_{j}}\right|^{2}\left|\frac{\epsilon_{i}+\omega\epsilon_{j}}{\epsilon_{i}-\omega\epsilon_{j}}\right|^{2}d\phi_{1}\dotsd\phi_{n},
\widetilde{M}_{\lambda}(j_{p},j_{q};\vec{k},\lambda)=-i\,\langlej_{q},j_{p}|S^{(1)}|\vec{k},\lambda\rangle=-e\int{d^{4}x}\bar{\psi}_{e}(j_{p},x)\;A_{\mu}^{\lambda}(\vec{k},x)\gamma_{\mu}\;\psi_{p}^{c}(j_{q},x),.
G(x,y;\chi)=\frac{1}{2\pi}\frac{1}{(x-y)^{2}}\left(\begin{array}{cc}{0}&{e^{-\big(\chi,\delta_{n}(x)-\delta_{n}(y)\big)}\;\overline{{(\tilde{x}-\tilde{y})}}}\\{e^{+\big(\chi,\delta_{n}(x)-\delta_{n}(y)\big)}\;(\tilde{x}-\tilde{y})}&{0}\\\end{array}\right)\;.
G_{0}^{(2)}(\vec{r},z,\vec{{r}^{\prime}},z^{\prime})=\frac{1}{(2\pi)^{d-1}}\sum_{n}u_{n}(z)u_{n}^{*}(z^{\prime})\intd^{d-1}p\frac{e^{i\vec{p}.(\vec{r}-\vec{{r}^{\prime}})}}{(\vec{p}^{\,}+k_{n}^{2}+m^{2})}.
\begin{array}{l}{e_{a}\circ_{X}(e_{b}\circ_{X}(e_{c}\circ_{X}e_{d}))=\pme_{7}}\\{\mathrm{~if~exactly~one~of~}a,b,c,d=0\mathrm{~or~}h=7,}\\{e_{a}(e_{b}(e_{c}e_{d}))=\pm1\mathrm{~otherwise.~}}\\\end{array}
\left({x}^{2}+{y}^{2}\right)=4\beta_{0}^{2}(a_{x}^{\dagger}a_{x}^{\prime}+a_{y}^{\dagger}a_{y}^{\prime}+1-ia_{x}^{\prime}a_{y}^{\prime}+ia_{x}^{\dagger}a_{y}^{\dagger}),
{\calL}=\mathrm{Tr}\left[\frac{1}{2r}(D_{t}X_{i})^{2}-\frac{T_{A}^{2}}{4r}[X_{i},X_{j}]^{2}-\theta^{T}D_{t}\theta-\frac{T_{A}^{2}}{4}\theta^{T}\gamma_{i}[\theta,X_{i}].\right],
\iota_{\scriptscriptstyleP\wedgeR}dw=-\iota_{\scriptscriptstyleR}d(\iota_{\scriptscriptstyleP}\omega)-\iota_{\scriptscriptstyle[P,R]_{\scriptscriptstyleSN}}\omega+\iota_{\scriptscriptstyleP}d(\iota_{\scriptscriptstyleR}\omega)
{\frac{1}{4!}}\,e^{\phi/2}\,F^{2}=-\,V^{-2}\,\left(S_{11}\,\left({\partial_{i}\,h_{1}}\right)^{2}+S_{22}\,\left({\partial_{i}\,h_{2}}\right)^{2}+2\,S_{12}\,\left({\partial_{i}\,h_{1}}\right)\left({\partial_{i}\,h_{2}}\right)\right)\,,
{\calA}=\langle\int_{M}(\delta{\tilde{\overline{\Psi}}}_{L}{\frac{\delta{S}^{-}}{\delta{\tilde{\overline{\Psi}}}_{L}}}+\delta{\tilde{\Psi}}_{L}{\frac{\delta{S}^{-}}{\delta{\tilde{\Psi}}_{L}}})\rangle.
\begin{array}{c}{a_{i}^{\mu}a_{j}^{\nu\dagger}-q_{\mu\nu}a_{j}^{\nu\dagger}a_{i}^{\mu}=\delta_{ij}\delta_{\mu\nu},\forall\mu,\nu=1,...,p}\\{a_{i}^{\mu}a_{j}^{\nu}-q_{\mu\nu}a_{j}^{\nu}a_{i}^{\mu}=0,\foralli,j\inS.}\\\end{array}
{\calA}_{5}={\frac{i}{4}}\sum_{a=1}^{3}{\frac{\varpi_{a}(q_{1},q_{2})}{(q-p_{a})^{2}}}S_{1}(p_{a})\langle\prod_{i=1}^{4}V_{i}(k_{i},\epsilon_{i};z_{i},\bar{z}_{i})\rangle\,,
{\frac{\partialL_{P}}{\partial{\bfC}}}=0\quad\longrightarrow\quadD_{[\,\lambda}({\bf\Phi})\,{\bfH}_{\mu\nu\rho\,]}=0\qquad\Rightarrow\qquad{\bfH}_{\mu\nu\rho}=D_{[\,\mu}(\,{\bf\Phi}\,){\bfB}_{\nu\rho\,]}
z(w)=-k\frac{W^{2}}{1+W^{2}}w+2k^{2}\frac{W^{2}}{(1+W^{2})^{2}}w^{2}-k^{3}\frac{W^{2}(7+2W^{2})}{(1+W^{2})^{3}}w^{3}+\mathcal{O}(w^{4}).
\{s_{m}^{a},s_{n}^{b}{}^{\dagger}\}=\delta^{a,b}\delta_{m,n},~~~\{\tilde{s}_{m}^{a},\tilde{s}_{n}^{b}{}^{\dagger}\}=\delta^{a,b}\delta_{m,n},~~~\{s_{0}^{a},s_{0}^{b}{}^{\dagger}\}=\delta^{a,b}
D^{\mu,\nu}(\varpi)=\prod_{t=1}^{\mu-\nu}{\frac{\sqrt{\Gamma\bigl(-(\varpi-t+1)h/\pi\bigr)}}{\sqrt{\Gamma((\varpi-t)h/\pi)}}}\prod_{r=1}^{\mu}\Gamma\bigl((\varpi-r)h/\pi)\bigr)\prod_{s=1}^{\nu}\Gamma\bigl((-\varpi-s)h/\pi)\bigr),
F(v_{i},w_{j})={\frac{\prod_{1\leqi<j\leqp}\vertv_{i}-v_{j}\vert^{2/\beta}\prod_{0\leqi<j\leqq}\vertw_{i}-w_{j}\vert^{2\beta}}{\prod_{i=1}^{p}(1-v_{i}^{2})^{1-1/\beta}\,prod_{j=0}^{q}(w_{j}^{2}-1)^{1-\beta}\prod_{i,j}(v_{i}-w_{j})^{2}}}{\calK}^{2},
(-1)^{\sum_{s=1}^{3}(m_{s}+m_{s}^{\prime})/2}\:\left[\begin{array}{ccc}{j_{1}}&{j_{2}}&{j_{3}}\\{m_{1}}&{m_{2}}&{-m_{3}}\\\end{array}\right]\left[\begin{array}{ccc}{j_{1}}&{j_{2}}&{j_{3}}\\{m_{1}^{\prime}}&{m_{2}^{\prime}}&{-m_{3}^{\prime}}\\\end{array}\right].
{\frac{\deltaf_{a}^{\prime}}{\delta\phi^{A}}}s_{a}\phi^{A}\,+\,{\frac{1}{2}}g_{3}^{\prime}s_{2}\,s_{1}\phi^{A}\,-\,{\frac{1}{2}}\epsilon^{ab}{\frac{\deltaf_{a}^{\prime}}{\delta\phi^{A}}}{\frac{\deltaf_{b}^{\prime}}{\delta{\bar{\phi}}^{A}}}\,\,=\,s_{2}s_{1}\chi\,\,.
\langleT_{00}\rangle_{0}|_{r=0}=\frac{4\xi-1}{2^{D}\pi^{(D+1)/2}\Gamma((D-1)/2)a^{D+1}}\int_{0}^{\infty}dzz^{D}\frac{K_{1}(z)}{I_{1}(z)},\quad\phi_{0}=\pi
J_{orb}(\vec{R})=2\pi\kappa\frac{\sigma_{1}}{q_{1}}R^{i}\varepsilon_{ij}v_{j}^{(2)}(\vec{R})-2\pi\kappa\frac{\sigma_{2}}{q_{2}}R^{i}\varepsilon_{ij}v_{j}^{(1)}(-\vec{R})\;\;,
\begin{array}{l}{n^{\prime}-2(n_{2}+\cdots+n_{j})+\alpha_{i-1j-1}^{(k-1)}\geqa_{1}^{(j)}\geq\cdots\geqa_{n_{j}-n_{j+1}}^{(j)}\geq0,~~~~2\leqj\leqk-1.}\\\end{array}
\phi(x,t)=\Phi_{1}(x-\alpha\t+\int_{-\infty}^{\sigma(x,t)}\frac{{\Phi^{\prime}}_{2}^{2}-p\Phi_{2}^{\prime}}{a+q\Phi_{2}^{\prime}}\d\tilde{\sigma})\+\Phi_{2}(x-\beta\t-\int_{\rho(x,t)}^{\infty}\frac{{\Phi^{\prime}}_{1}^{2}+q\Phi_{1}^{\prime}}{a-p\Phi_{1}^{\prime}}\d\tilde{\rho}).
\mathrm{Re}(\eta\nabla^{2}a)=\frac{1}{\sqrt{2}}\,\frac{\mathrm{Im}(n_{e}+\bar{\tau}_{0}n_{m})}{\mathrm{Im}\,\tau_{0}}\,\frac{\delta(r-r_{0})}{r_{0}^{2}}=-\frac{4\pi}{\sqrt{2}}\,\nu_{m}
{\delta}T(t)=-\frac{1}{12}\frac{d}{dt}ln(1+k[{{\phi}_{0}}^{2}+2{\phi}_{0}{\delta}{\phi}])=-\frac{1}{12(1+k[2{\phi}_{0}{\delta}{\phi}])}(2{\phi}_{0}\frac{d}{dt}{\delta}{\phi})
k=\left(\frac{E}{2mc^{2}}\right)^{1/2},\quadD=\left(\frac{eV_{0}a^{2}}{2mc^{2}}\right)^{1/3},\quadL=\left(\frac{eV_{0}b^{2}}{2mc^{2}}\right)^{1/3},\quadG=\frac{E}{eV_{0}},
H=\sum_{i=1}^{N}\lbrace\sqrt{m_{i}^{2}+{({\vec{p}}_{i}-e_{i}\vec{A}({\vec{x}}_{i},x^{o}))}^{2}}+V({\vec{x}}_{i},x^{o})\rbrace+{\frac{1}{4}}\intd^{3}z[{\vec{\pi}}^{2}(\vec{z},x^{o})+{\vec{B}}^{2}(\vec{z},x^{o})],
l_{p}^{-2}ds^{2}=\hat{H}^{-1/3}\left[-\hat{f}dt^{2}+\sum_{i=1}^{5}(dx^{i})^{2}+\hat{H}\Big(\hat{f}^{-1}d\hat{\rho}^{2}+\hat{\rho}^{2}d\Omega_{4}^{2}\Big)\right]
\psi=\left(\begin{array}{c}{v_{1}}\\{v_{2}}\\{u_{1}}\\{u_{2}}\\\end{array}\right),~~~~~~\hat{\psi}=-i\sigma_{2}\psi^{\ast}=\left(\begin{array}{cc}{-v_{2}^{\ast}}\\{v_{1}^{\ast}}\\{-u_{2}^{\ast}}\\{u_{1}^{\ast}}\\\end{array}\right)
\widetilde{\phi}(t,\sigma^{1},\sigma^{2})\=\\widetilde{\phi}_{1}(t,\sigma^{1},\sigma^{2})\,\widetilde{\phi}_{2}(t,\sigma^{1},\sigma^{2})\cdots\widetilde{\phi}_{D}(t,\sigma^{1},\sigma^{2})\.
H=\frac{z_{A}(z_{A}-1)(z_{A}-z_{3})}{z_{3}(z_{3}-1)}\left\{b_{A}^{2}-(\frac{\mu_{1}}{z_{A}}+\frac{\mu_{2}}{z_{A}-1}+\frac{\mu_{3}-1}{z_{A}-z_{3}})b_{A}+\frac{\kappa}{z_{A}(z_{A}-1)}\right\}.
27\intd^{4}x\,R_{,\mu}R_{,\nu}\Box_{0}^{-2}R^{\mu\nu}=\textstyle{\frac{1}{2}}\intd^{4}x\left[R_{,\mu}R_{,\nu}\Box_{0}^{-1}h^{\mu\nu}-\textstyle{\frac{1}{4}}\,R^{2}h_{\alpha}^{\alpha}+\textstyle{\frac{1}{2}}R^{2}\Box_{0}^{-1}R\right]\;.
J_{1}^{k}|\Psi^{ab}\rangle=J_{1}^{k}J_{-1}^{(a}|\phi^{b)}\rangle=-f_{d}^{k(a}f_{e}^{b)d}|\phi^{e}\rangle~+~{\frac{k}{2}}g^{k(a}|\phi^{b)}\rangle.
h_{\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}}(r_{1},r_{2},j_{1},j_{2})=\sum_{\gamma_{1},\gamma_{2}}c_{\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}}^{\gamma_{1},\gamma_{2}}\varphi_{\gamma_{1},\gamma_{2}}(r_{1},r_{2},j_{1},j_{2}).
\begin{array}{rclrcl}{\vec{F}^{(1)}}&{=}&{\vec{n}F+\vec{p}\{}^{\star}F\,,}&{\rho_{m}}&{=}&{c_{m}\\varphi\,,}\\{}&{}&{}&{}&{}&{}\\{\vec{F}^{(2)}}&{=}&{\vec{m}F+\vec{q}\{}^{\star}F\,,}&{\phi}&{=}&{b\varphi\,,}\\\end{array}
\sum_{k=0}^{1-a_{ij}}(-1)^{k}q_{i}^{-k(1-a_{ij}-k)}\left\vert\left[\begin{array}{c}{1-a_{ij}}\\{k}\\\end{array}\right]_{q_{i}^{-1}}\right.(X_{i}^{+})^{k}X_{j}^{+}(X_{i}^{+})^{1-a_{ij}-k}=0~.
\langle0|T_{\mu}^{\mu}|0\rangle={\frac{\beta(\alpha_{s})}{4\alpha_{s}}}\langle0|({\calF}_{\mu\nu}^{A})^{2}|0\rangle=-{\frac{11N_{c}-2N_{f}}{24}}\langle0|{\frac{\alpha_{s}}{\pi}}({\calF}_{\mu\nu}^{A})^{2}|0\rangle.
H=\intdx\Biggl[\frac{1}{2R}\Pi^{2}(x,t)+\frac{m^{2}R}{2}\Phi^{2}(x,t)+\frac{1}{2R}(\frac{d}{dx}\Phi)^{2}+\frac{\lambdaR}{4}\Phi^{4}(x,t)\Biggr].
\intd^{d}x\left\{f+u^{\dagger}\Gamma(\tau\gamma_{0}\gamma^{i}\partial_{i})u+u^{\dagger}\Xi(\tau\gamma_{0}\gamma^{i}\partial_{i}){v}+{v}^{\dagger}\Pi(\tau\gamma_{0}\gamma^{i}\partial_{i})u+{v}^{\dagger}\Upsilon(\tau\gamma_{0}\gamma^{i}\partial_{i}){v}\right\}.
\theta^{\mu\nu}=\frac{1}{[g_{2}(1-e^{2})+g_{0}b^{2}]E_{c}}\left(\begin{array}{ccc}{0}&{g_{2}e}&{0}\\{-g_{2}e}&{0}&{-g_{0}b}\\{0}&{g_{0}b}&{0}\\\end{array}\right),
{\cal{V}}=-\frac{TV^{2}}{8\pi^{4}\lambda}\frac{m}{N(N+m)}+\frac{1}{2\pi^{2}}(\frac{1}{N}(B^{0})^{2}+\sum_{k}(B^{k})^{2})+\frac{1}{\pi^{2}}\sum_{i}\sum_{j>i}|X^{(ij)}|^{2})+\frac{1}{\pi^{2}}\sum_{i}|X^{i}|^{2}
S[\tilde{B}_{(\tilde{p})},\tilde{A}_{(\tilde{p}+1)}]=\intd^{d}x\\sqrt{|g|}\\left\{{\textstyle\frac{(-1)^{(\tilde{p}+1)}}{2\cdot(\tilde{p}+2)!}}e_{(p)}^{2}\tilde{F}_{(\tilde{p}+2)}^{2}+{\textstyle\frac{(-1)^{\tilde{p}}}{2\cdot(\tilde{p}+1)!}}e_{(p+1)}^{2}\tilde{H}_{(\tilde{p}+1)}^{2}\right\}
DA_{(p)}^{\mu_{1}\ldots\mu_{p-2k}}=\partial^{\left[\mu_{1}\right.}\eta_{(p)}^{\left.\mu_{2}\ldots\mu_{p-2k}\right]}+\left(p-2k+1\right)\partial_{\nu}\eta_{(p)}^{\nu\mu_{1}\ldots\mu_{p-2k}},\;k=1,\ldots,b_{p},
\begin{array}{rcl}{{\hat{c}}^{(6)}{}_{ijklm5}}&{=}&{-5(2\pi\alpha^{\prime})\partial_{[i}v^{(0)}c^{(4)}{}_{jklm]}-5{\textstyle\frac{m}{2}}(2\pi\alpha^{\prime})^{3}v^{(0)}b_{[i}\partial_{j}b_{k}\partial_{l}b_{m]}}\\{}&{}&{}\\{}&{}&{+v^{(5)}{}_{ijklm}\,,}\\\end{array}
\begin{array}{ll}{x_{\mathrm{cm}}=\frac{x_{1}+x_{2}}{2}\;,}&{p_{\mathrm{cm}}=p_{1}+p_{2}\;,}\\{x_{\mathrm{rel}}=x_{1}-x_{2}\;,}&{p_{\mathrm{rel}}=\frac{p_{1}-p_{2}}{2}\;,}\\\end{array}
M_{ij}=\eta_{ij}+F(V)_{ij}+\bar{\theta}\Gamma_{(i}\partial_{j)}\theta+\bar{\theta}\sigma^{3}\Gamma_{[i}\partial_{j]}\theta+\frac{1}{4}\bar{\theta}\Gamma^{a}\partial_{i}\theta\bar{\theta}\Gamma_{a}\partial_{j}\theta+\frac{1}{4}\bar{\theta}\sigma^{3}\Gamma^{a}\partial_{[i}\theta\bar{\theta}\Gamma_{a}\partial_{j]}\theta\,.
Q^{(1)}=\sum_{\sigma}\int\frac{d^{3}p}{(2\pi)^{3}}\,\left[a_{\sigma}^{\dagger}(\vec{p})a_{\sigma}(\vec{p})-b_{\sigma}(\vec{p})b_{\sigma}^{\dagger}(\vec{p})+c_{\sigma}^{\dagger}(\vec{p})c_{\sigma}(\vec{p})-d_{\sigma}(\vec{p})d_{\sigma}^{\dagger}(\vec{p})\right]\,,
\tilde{V}(\varphi)=({\frac{1}{8\pi}}-{\frac{1}{\beta^{2}}})({\tilde{\mu}}^{2}-1)-{\frac{\tilde{m}_{0}^{2}}{8\pi}}ln(\tilde{\mu}^{2})+{\frac{1}{2}}\tilde{m}_{0}^{2}\varphi^{2}\;,
i^{-\frac{1}{6}\left[\left(n^{2}-1\right)\alpha_{+}^{2}+\left(m^{2}+2\right)\alpha_{-}^{2}-2nm\right]}x_{12}^{2-4\alpha_{-}^{2}}\left(2y_{1}\right)^{\frac{1}{2}\alpha_{-}^{2}+2h_{12}+2h_{nm}}\left(2y_{2}\right)^{\frac{1}{2}\alpha_{-}^{2}}
F(x^{+},x^{-},0)=\frac{1}{2\pi}\sum_{K,N_{\perp},\alpha}\frac{1}{2L}\frac{1}{l}\left(\frac{\piK}{L}\right)^{3}e^{-iP_{\alpha}^{-}x^{+}-iP^{+}x^{-}}\frac{|\langleu|\alpha\rangle|^{2}}{lK^{3}|N_{u}|^{2}}\,.
I_{N}(p)=\frac{\pi^{(N-1)/2}}{(4\pi)^{(N-1)\ell}}\frac{\Gamma((N-1)/2-\ell(N-1))}{\varpi^{3/2-\ell}\sigma^{N/2+\ell(2-N)}}(1-p^{2}/\sigma^{2})^{\ell(N-1)-(N+1)/2}.
\begin{array}{ll}{\tilde{f}_{0}^{E}=\:[(a/b)_{0}+1]/\sqrt{<\Phi_{0},\Phi_{0}>}}\\{\tilde{f}_{0}^{L}=\:[(c/b)_{0}+(1/b)_{0}]/\sqrt{<\Phi_{0},\Phi_{0}>}}\\\end{array}
m_{i}\frac{d}{d\tau}\left[\frac{\dot{x}^{\mu}(\tau,\sigma_{i}(\tau))+\dot{\sigma}(\tau)\acute{x}^{\mu}(\tau,\sigma_{i}(\tau))}{\sqrt{\dot{x}^{2}(\tau,\sigma_{i}(\tau))(1-{\sigma_{i}}^{2}(\tau))}}\right]=(-1)^{i+1}\gamma\left[\dot{x}^{\mu}(\tau,\sigma_{i}(\tau))+\dot{\sigma}(\tau)\acute{x}^{\mu}(\tau,\sigma_{i}(\tau))\right],
ds^{2}=e^{{\frac{1}{2}}\Phi}\left(e^{{\frac{1}{2}}\xi-5\eta}d\rho^{2}+e^{-{\frac{1}{2}}\xi}dx_{||}^{2}+e^{{\frac{1}{2}}-\eta}d\Omega_{5}^{2}\right),
V_{5(0)}^{b}\sim\frac{i}{2\sqrt{2}}\biggl[(1-i)e^{iH_{5}}e^{iH_{8}}+(1+i)e^{iH_{5}}e^{-iH_{8}}+(1+i)e^{-iH_{5}}e^{iH_{8}}+(1-i)e^{-iH_{5}}e^{-iH_{8}}\biggr]
\partial_{\underline{i}}\epsilon_{I}+{\textstyle\frac{1}{4}}\biggl({\frac{\partial_{\underline{i}}{\overline{{\calH}}_{2}}}{\overline{{\calH}}_{2}}}-{\frac{\partial_{\underline{i}}{\calH}_{2}}{{\calH}_{2}}}\biggr)\epsilon_{I}-{\textstyle\frac{1}{2}}\left(\partial_{\underline{i}}U\right)\epsilon_{I}=0\,.
R_{l}^{\mathrm{WKB}}(r)\propto\frac{1}{\sqrt{\kappa(r)}}\left\{Ae^{-\int_{r_{2}}^{r}dr^{\prime}\kappa(r^{\prime})}+Be^{\int_{r_{2}}^{r}dr^{\prime}\kappa(r^{\prime})}\right\},\;\;\;\kappa(r)\equiv\sqrt{2\mu(V_{\iteff}(r)-E)}
Y^{I}=\left(\begin{array}{cccc}{y_{1}^{I}}&{0}&{\dots}&{0}\\{0}&{y_{2}^{I}}&{\dots}&{0}\\{\dots}&{\dots}&{\dots}&{\dots}\\{0}&{\dots}&{0}&{y_{N}^{I}}\\\end{array}\right),\I=1,\dots,26\,
b_{r,s}^{(l)}\left(e^{2\pii\tau}\right)=\sum_{r^{\prime}=1}^{m-1}\sum_{s^{\prime}=1}^{m+1}\sum_{l^{\prime}=0}^{2}S_{(r,s,l)}^{(r^{\prime},s^{\prime},l^{\prime})}\;b_{r^{\prime},s^{\prime}}^{(l^{\prime})}\left(e^{-2\pii/\tau}\right),
{\left(\rho_{ab}^{(1)}\right)}_{ij}=\left(\begin{array}{ccc}{0}&{-\delta_{ij}}&{0}\\{\delta_{ij}}&{0}&{-\partial_{i}^{x}}\\{0}&{\partial_{j}^{x}}&{0}\\\end{array}\right)\delta_{ab}\delta(\vec{x}-\vec{y}),
R(X)(\varphi)=\sum_{ij}\int_{S^{1}}d\varphi^{\prime}X_{ij}(\varphi^{\prime})\frac{\deltag(D,k)}{\deltaA_{ij}(\varphi^{\prime})}(\varphi)-\frac{1}{2}(\nabla_{(D,k)}g(V_{F_{X}})-V_{F_{X}})
{\calL}_{SQED}=\intd^{4}\theta[Q_{+}^{\dagger}e^{V}Q_{+}|\phi|^{2}+Q_{-}^{\dagger}e^{V}Q_{-}|\Phi|^{2}]+(\intd^{2}\theta\frac{1}{g_{0}^{2}}{\calW}^{2}+\mathrm{h.c.})+\mathrm{gauge\fixing}.
{\delta}B_{\mu\nu}^{p}={\calD}_{\mu}^{pq}{\xi}_{\nu}^{q}-{\calD}_{\nu}^{pq}{\xi}_{\mu}^{q}\equivR_{\mu\nu\alpha}^{pq}{\xi}^{q\alpha},\,\,\,\,\,\,\,\,{\delta}A_{\mu}^{p}=0,
F_{\pm}(\theta,\phi)=\left(\begin{array}{l}{f_{j}}\\{f_{j-1}}\\{\vdots}\\{f_{-j+1}}\\{f_{-j}}\\\end{array}\right)_{\pm}\!\!\!(\theta,\phi)=v\left(\begin{array}{l}{D_{j,j}^{j}}\\{D_{j-1,j}^{j}}\\{\vdots}\\{D_{-j+1,j}^{j}}\\{D_{-j,j}^{j}}\\\end{array}\right)(\theta,\phi,\mp\phi),
h_{\alphaq,\betar}=C_{qr}[h_{a}\gamma_{\alpha\beta}^{a}+{\frac{1}{6}}h_{abc}\gamma_{\alpha\beta}^{abc}]+\gamma_{qr}^{ij}[h_{[ij]a}\gamma_{\alpha\beta}^{a}+h_{[ij]abc}\gamma_{\alpha\beta}^{abc}].
{\dot{\pi}}_{\eta_{i}}({\bfk}_{n},t)=-\frac{e^{3Ht}}{L^{3}}\left[(e^{-2Ht}{\bfk}_{n}^{2}+\mu_{i}^{2})\eta_{i}({\bfk}_{n},t)+\alpha_{i}\phi({\bfk}_{n},t)\right].
\Psi=\left(\begin{array}{c}{L_{1}}\\{S_{1}}\\\end{array}\right)\otimes\left(\begin{array}{c}{L_{2}}\\{S_{2}}\\\end{array}\right)=\left(\begin{array}{c}{L_{1}L_{2}}\\{L_{1}S_{2}}\\{S_{1}L_{2}}\\{S_{1}S_{2}}\\\end{array}\right)=\left(\begin{array}{c}{a}\\{b}\\{c}\\{d}\\\end{array}\right)
\mu_{11}[l_{1}=l_{2}]={\frac{\pi^{(D-3)/2}}{2^{D+2}}}{\frac{B((D-1)/2,-D/2+3)}{\Gamma((D-1)/2)}}\left(\sum_{l=1}^{\infty}l^{D-2}\right){\frac{\hbarL^{D-1}}{a^{D}}},
\tilde{\calL}_{class}=-\varphi_{1}\Box\varphi_{2}+m^{2}\varphi_{1}\varphi_{2}+{\frac{\lambda}{4}}\varphi_{1}^{2}\varphi_{2}^{2}+J_{1}\varphi_{1}+J_{2}\varphi_{2}+K\varphi_{1}^{2}.
\begin{array}{lcr}{\partial_{\sigma}X^{\mu}+(-)^{\mu+1}\{X^{\mu+(-)^{\mu+1}},X^{9}\}=0,\mu=1,...,8}&{(a)}\\{\partial_{\sigma}X^{9}+\sum_{\mu=1}^{8}{\frac{1}{2}}(-)^{\mu+1}\{X^{\mu},X^{\mu+(-)^{\mu+1}}\}=0,}&{(b)}\\\end{array}
\bar{F}_{k}(\lambda^{-1})\equiv\bar{L}_{k}(\lambda^{-1})\bar{L}_{k-1}(\lambda^{-1})\equiv\left(\begin{array}{cc}{\bar{F}_{11}^{(k,k-1)}(\lambda^{-1})}&{\bar{F}_{12}^{(k,k-1)}(\lambda^{-1})}\\{\bar{F}_{21}^{(k,k-1)}(\lambda^{-1})}&{\bar{F}_{22}^{(k,k-1)}(\lambda^{-1})}\\\end{array}\right)
|B\rangle_{bulk}=\frac{N}{2(2\piR)}\sum_{\ell}\frac{T_{0}}{2}e^{i\frac{N}{R}\ell\hat{y}}\Big[|0,+\rangle_{NS}-(-1)^{k\ell}|0,-\rangle_{NS}+|0,+\rangle_{R}+(-1)^{k\ell}|0,-\rangle_{R}\Big].
F_{B}^{bosonic}={\frac{\Gamma(2-2k_{0}^{2})\Gamma(\vec{k}\cdot\vec{p}+k_{0}^{2}-1)}{\Gamma(\vec{k}\cdot\vec{p}-k_{0}^{2}+2)}}\bigg[\vec{k}\cdot\vec{p}H_{0ij}^{1}(k)H_{0ij}^{2}(p)-{\frac{1-k_{0}^{2}}{3}}H_{ijl}^{1}(k)H_{ijl}^{2}(p)\bigg]\,
{\calL}_{\mathrm{extra}}=A_{0}^{0\pi}{\bar{\Psi}}T^{2}\gamma_{1}\Psi+A_{0}^{\pi0}{\bar{\Psi}}T^{1}\gamma_{2}\Psi-A_{1}^{0\pi}{\bar{\Psi}}T^{2}\gamma_{0}\Psi-A_{2}^{\pi0}{\bar{\Psi}}T^{1}\gamma_{0}\Psi
C_{V}=\sum_{n,w}\int_{\calF}[d\tau]e^{-y}\left[4t^{2}(-{\frac{n^{2}}{x^{2}}}+{\frac{w^{2}x^{2}}{4}})^{2}-2t(3{\frac{n^{2}}{x^{2}}}+{\frac{w^{2}x^{2}}{4}})\right].
J_{0}^{(r)}(k^{2})=I_{0}^{(r)}(k^{2})+\Lambda^{2}\,\sum_{s=1}^{r}\,\frac{I_{s}^{(r)}(k^{2})-I_{s}^{(r)}(-u_{r,2s}\Lambda^{2})}{k^{2}+u_{r,2s}\Lambda^{2}}\,,
\begin{array}{lll}{{\widetildeT}_{\alpha{\dot{\alpha}}}}&{=}&{-\frac{1}{8}W_{\alpha}{\bar{W}}_{\dot{\alpha}}+\frac{1}{12}{\nabla}_{\alpha}\Phi{\bar{\nabla}}_{\dot{\alpha}}\bar{\Phi}-\frac{i}{6}\bar{\Phi}\stackrel{\leftrightarrow}{{\nabla}}_{\alpha{\dot{\alpha}}}\Phi\,}\\{{\widetildeJ}}&{=}&{\frac{1}{4}(q-1){\bar{\nabla}}^{2}\left(\bar{\Phi}\Phi\right)\.}\\\end{array}
\bar{\beta}_{\omega\omega^{\prime}lm}^{b}=-\frac{2\bar{N}\bar{C}_{T}\omegae^{i\omega^{\prime}t_{0}}e^{-im\varphi_{0}}}{\sqrt{2\pi}|G^{\prime}(\xi_{4})||1-e^{-2\pi\omega^{\prime}/\kappa}|^{1/2}}\intdz\,e^{i\omegar}e^{i\omega^{\prime}z}e^{-m|z|}f_{l\omega^{\prime}}(p)\tilde{n}_{lm}(p),
\sigma_{1}=\left\|\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right\|\quad\sigma_{2}=\left\|\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right\|\quad\sigma_{3}=\left\|\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right\|
L=\frac{64k^{2}}{a^{8}\xi}\\\\\\\\M=\frac{D-1}{4\xi}\\\\\\\\N=\frac{4}{3a^{2}\xi}\left(1+\frac{D-1}{2}-\frac{k^{2}}{a^{2}}\right)-\frac{4}{3a^{4}}