harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\calL}_{HE}^{(1)\;\prime}(a,B)=\frac{1}{8\pi^{2}}\int_{0}^{\infty}\frac{ds}{s^{3}}e^{-sm^{2}}\biggl[\frac{\sqrt{\pis}}{2a}-\sum_{n=1}^{\infty}e^{-(an)^{2}/s}\biggr]\,\biggl[seB\,{\calM}(seB)+\frac{1}{6}(seB)^{2}\biggr]\;.
\left({\frac{d^{2}}{d\tau^{\prime}{}^{2}}}+{\frac{1}{\tau^{\prime}}}{\frac{d}{d\tau^{\prime}}}+1-{\frac{\nu^{2}}{\tau^{\prime}{}^{2}}}\right)\tau^{\prime}{}^{-{\frac{1}{2}}}\varphi_{n}=0
{\widetilde\xi}_{cl}(q_{\pm})=\frac{i}{cos^{2}(\theta)-sin^{2}(\theta)e^{2i\phi}}(\sqrt{2}sin(\theta)cos(\theta)e^{i\phi}-(cos^{2}(\theta)+sin^{2}(\theta)e^{2i\phi})\xi_{cl}(q_{\pm})).
<p_{2}|T_{\mu\nu}(x)|p_{1}>={\frac{e^{i(p_{2}-p_{1})\cdotx}}{\sqrt{4E_{2}E_{1}}}}\left[2P_{\mu}P_{\nu}F_{1}(q^{2})+(q_{\nu}-\eta_{\mu\nu}q^{2})F_{2}(q^{2})\right]
ds^{2}={\frac{1}{\sqrt{H_{A}H_{B}}}}(-dt^{2}+dz_{I}^{2})+\sqrt{\frac{H_{A}}{H_{B}}}(dz_{b})^{2}+\sqrt{\frac{H_{B}}{H_{A}}}(dz_{a})^{2}+\sqrt{H_{A}H_{B}}(dx_{\perp}^{2})\,,
R=\left(\begin{array}{ccccc}{N-1}&{-1}&{.}&{.}&{-1}\\{-1}&{N-1}&{-1}&{}&{.}\\{.}&{-1}&{.}&{.}&{.}\\{.}&{}&{.}&{.}&{-1}\\{-1}&{.}&{.}&{-1}&{N-1}\\\end{array}\right)\;.
i\psi_{+}\partial_{-}\psi_{+}+i\psi_{-}\partial_{+}\psi_{-}=-\frac{1}{2}\left(\psi_{+},-\psi_{-}\right)\left(\rho^{+}\partial_{+}+\rho^{-}\partial_{-}\right)\left(\begin{array}{c}{\psi_{-}}\\{\psi_{+}}\\\end{array}\right),
\tau_{x}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right),\quad\tau_{y}=\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right),\quad\tau_{z}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)
S=S_{cl}+\int_{-\infty}^{\infty}dt\{-{\frac{1}{2}}\xi^{a}({\frac{d}{dt}}+\lambda_{a}(t))({\frac{d}{dt}}-\lambda_{a}(t))\xi^{a}+\psi^{a}({\frac{d}{dt}}-\lambda_{a}(t))\psi_{a}^{*}\},
\Gamma=W-\intd^{4}x\,d^{4}\theta\,JV-\intd^{4}x\,d^{2}\theta\,\Big(j\,\phi+\tilde{j}\,\tilde{\phi}\Big)-\intd^{4}x\,d^{2}\bar{\theta}\,\Big(j^{*}\phi^{*}+\tilde{j}^{*}\tilde{\phi}^{*}\Big),
\zeta(s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}dt\t^{s-1}e^{-(c-b^{2})t}\left(-{\frac{\partial}{\partial(\alphat)}}\theta_{3}\left(0\left|i{\frac{\alphat}{\pi}}\right.\right)\right)\theta_{3}\left(abt\left|i{\frac{a^{2}t}{\pi}}\right.\right)
P_{\{0,\,n,\varsigma\}}(q,q^{\prime})=\frac{{\tilde{r}}_{n,\varsigma}}{(\Gamma(2n+\varsigma+\gamma+1))^{2}}\;\sum_{k=0}^{n}{\tau}^{2k+\varsigma}\;(\rho\rho^{\prime})^{2(n-k)}c_{2k+\varsigma,\gamma+2n+\varsigma}.
\left(U_{v}^{\mu}\right)=\left(\begin{array}{cccc}{{\bfI}_{2}}&{0}&{0}&{0}\\{0}&{-\frac{1}{\sqrt{2}}}&{\frac{1}{\sqrt{2}}}&{0}\\{0}&{\frac{1}{\sqrt{2}}}&{\frac{1}{\sqrt{2}}}&{0}\\{0}&{0}&{0}&{{\bfI}_{2}}\\\end{array}\right),
<J_{\alpha}(x)~{\calJ}_{\mu\nu\rho}^{(3)}(0)>=(\pi_{\mu\nu}\pi_{\rho\alpha}+\pi_{\mu\rho}\pi_{\nu\alpha}+\pi_{\nu\rho}\pi_{\mu\alpha})\left(\frac{f_{31}(t)}{|x|^{4}}\right)
\left(\begin{array}{cc}{-(1+B_{1}^{2})}&{-B_{1}B_{2}}\\{-B_{1}B_{2}}&{1-B_{2}^{2}}\\\end{array}\right)\left(\begin{array}{cc}{a}&{c}\\{b}&{d}\\\end{array}\right)=\left(\begin{array}{cc}{a}&{c}\\{b}&{d}\\\end{array}\right)\left(\begin{array}{cc}{-\lambda_{-}}&{0}\\{0}&{\lambda_{+}}\\\end{array}\right),
(f*g)(a,\bar{a}):=\intd\mu(v,\bar{v})\intd\mu(u,\bar{u}){\frac{\displaystyle{e^{\Phi(v,\bar{v},a,\bar{a})/\lambda}}}{\displaystyle{e^{\Phi(u,\bar{u},a,\bar{a})/\lambda}}}}f(u,\bar{v})g(v,\bar{u}).
\frac{1}{2g^{2}}\mathrm{tr}\intd^{3}xd^{3}y\partial_{i}U^{\dagger}(x)G^{-1}(x-y)\partial_{i}U(y)=\frac{M}{2g^{2}}\mathrm{tr}\intd^{3}x\partial_{i}U^{\dagger}(x)\partial_{i}U(x)+...
H_{\hat{\mu}\hat{\nu}\hat{\rho}}={\frac{1}{6\sqrt{-G}}}G_{\hat{\mu}\hat{\mu}^{\prime}}G_{\hat{\nu}\hat{\nu}^{\prime}}G_{\hat{\rho}\hat{\rho}^{\prime}}\epsilon^{\hat{\mu}^{\prime}\hat{\nu}^{\prime}\hat{\rho}^{\prime}\hat{\lambda}\hat{\sigma}\hat{\eta}}H_{\hat{\lambda}\hat{\sigma}\hat{\eta}},
S=-\intdt\sqrt{(1-4m^{2}r^{2})-{\frac{\dot{r}^{2}}{1-4m^{2}r^{2}}}}=\intdt{\frac{1}{2}}{\frac{\dot{r}^{2}}{(1-4m^{2}r^{2})^{3/2}}}-\sqrt{1-4m^{2}r^{2}}+\ldots\,
d\xi={\frac{e^{(-1)^{s+1}{\frac{\rho}{\alpha}}}}{q^{2}}}(\nablaq_{a}e^{a}+q_{a}\nablae^{a}+(-1)^{s+1}({\frac{1}{\alpha}}-{\frac{\Phi}{\alpha^{2}q^{2}}})d\rho\wedgeq_{a}e^{a})
\left\langle\frac{P_{N-k}(x_{1})\tilde{Q}_{N+l+1}(y_{1})}{x_{1}-\phi_{a}}\right\rangle\rightarrow\oint\frac{dz_{1}}{2\pi\imath}\frac{z_{1}^{k+l}}{p(z_{1},1)-\phi_{a}}
e^{-i\pi/4}\sqrt{2\pi\kappa}(e^{-\pi\omega/{2\kappa}}\phi_{-}(x)-ie^{\pi\omega/{2\kappa}}\phi_{+}(x))\leftrightarrow\frac{-i2\pi}{\Gamma(1+i\omega/\kappa)}\phi_{+s}(x)
K_{Q_{2}}(v)\equivG_{Q_{2}}^{-1}(v)=\frac{1}{v}-\frac{a^{2}v}{2}+\lambda-k-(\lambda-k)\sqrt{\left(1-\frac{a^{2}v}{2(\lambda-k)}\right)^{2}-\frac{a^{2}}{(\lambda-k)^{2}}}
C_{N}(x_{1},x_{2})=\piN\Xi\frac{x_{1}J_{1}(2N\Xix_{1})J_{0}(2N\Xix_{2})-x_{2}J_{0}(2N\Xix_{1})J_{1}(2N\Xix_{2})}{x_{1}^{2}-x_{2}^{2}}\;,
\langle(\varphi^{\dagger}\varphi)_{bJ}^{aI}\rangle_{\mathrm{FT}}^{\mathrm{1-loop,WZ}}\simg^{2}\delta_{J}^{I}\delta_{b}^{a}p^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{\left(k+\frac{p}{2}\right)^{2}\left(k-\frac{p}{2}\right)^{2}}\,.
A_{4B}~\sim~\frac{\Gamma(\frac{1}{2}s-\frac{1}{8}Q^{2})\Gamma(\frac{1}{2}t-\frac{1}{8}Q^{2})}{\Gamma(\frac{1}{2}s+\frac{1}{2}t-\frac{1}{4}Q^{2})}
\begin{array}{ccccccccc}{}&{}&{}&{}&{\frac{1}{l_{p}^{3}}}&{}&{}&{}&{}\\{}&{}&{}&{}&{\mid}&{}&{}&{}&{}\\{\frac{R_{1}}{l_{p}^{3}}}&{-}&{\frac{R_{1}R_{2}}{l_{p}^{6}}}&{-}&{\frac{R_{1}R_{2}R_{3}}{l_{p}^{9}}}&{-}&{\frac{R_{1}R_{2}R_{3}R_{4}}{l_{p}^{9}}}&{-\dots-}&{\frac{1}{R_{d}}}\\\end{array}
\Gamma_{b+c+d}^{(3)}(p_{1},p_{2})=\int\frac{d^{3}k}{(2\pi)^{3}}\,\frac{d^{3}q}{(2\pi)^{3}}\,I_{b+c+d}(k,q,p_{1},p_{2})\,{\calC}_{5}
\frac{k^{2}}{Q^{2}}\int_{0}^{1}{dc}c^{-1}\left(1-e^{-\frac{Q^{2}}{k^{2}}\frac{c}{1+c}}\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+1)!}\left(\frac{Q^{2}}{k^{2}}\right)^{n}\int_{0}^{\frac{1}{2}}dz\frac{z^{n}}{1-z}.
\frac{\partial^{a}}{\partialx_{i_{a}}^{a}}\dots\frac{\partial^{r+1}}{\partialx_{i_{r+1}}^{r+1}}\frac{\partial^{r}}{\partialx_{i_{r}}^{r}}\frac{\partial^{r-2}}{\partialx_{i_{r-1}}^{r-2}}\frac{\partial^{r-1}}{\partialx_{i_{r-2}}^{r-1}}\dots\frac{\partial}{\partialx_{i_{2}}}F_{r_{1},r_{2},\dots,r_{n};\theta_{1},\theta_{2},\dots,\theta_{n}}^{p,g;\iota_{1},\dots,\iota_{n}}
\begin{aligned}{e^{10\lambda}}&{{}=\frac{8}{5}}\\{e^{4\phi}}&{{}=\frac{5}{8}e^{4\lambda}\approx0.75427}\\{m^{2}R^{2}}&{{}=2^{-11/2}5^{5/2}\approx1.2353}\\\end{aligned}
Z=\int{\calD}[f(\xi),g(\xi),X]\,\delta\left(g_{zz}^{f}\right)\delta\left(g_{\bar{z}\bar{z}}^{f}\right)\,\mathrm{det}\left(\frac{g_{zz}^{f}}{\deltaf}\right)\mathrm{det}\left(\frac{g_{\bar{z}\bar{z}}^{f}}{\deltaf}\right)\,e^{-S[g^{f},X]}.
\left[S_{\alpha\beta}^{\mathrm{op}},S_{\gamma\delta}^{\mathrm{op}}\right]=-i\left(\eta_{\alpha\gamma}\,S_{\beta\delta}^{\mathrm{op}}-\eta_{\alpha\delta}\,S_{\beta\gamma}^{\mathrm{op}}+\eta_{\beta\delta}\,S_{\alpha\gamma}^{\mathrm{op}}-\eta_{\beta\gamma}\,S_{\alpha\delta}^{\mathrm{op}}\right)+\frac{1}{m^{2}}[p_{[\alpha}\gamma_{\beta]},p_{[\gamma}\gamma_{\delta]}]\,.
\tilde{S}_{L}=\frac{1}{4\pi\alpha^{\prime}}\int_{\Sigma}[\tilde{g}^{\mu\nu}(\partial_{t}\tilde{x}_{\mu}\partial_{t}\tilde{x}_{\nu}-\partial_{\sigma}\tilde{x}_{\mu}\partial_{\sigma}\tilde{x}_{\nu})+4\pii\alpha^{\prime}\tilde{b}^{\mu\nu}\partial_{t}\tilde{x}_{\mu}\partial_{\sigma}\tilde{x}_{\nu}],
{\calE}_{nl}(t)={\calE}_{nl1}(t)+{\calE}_{nl2}(t)=\frac18\int_{0}^{1}d\rho\left(e^{\frac{1-\rho}{8}{\partial}^{2}}(-q^{2}{\partial}^{2}+1)\psi\right)\overleftrightarrow{\partial}\left(\partiale^{-\frac{1-\rho}{8}{\partial}^{2}}\psi\right).
\frac{1}{2}\alpha^{\prime}M^{2}=N+\bar{N}+\frac{1}{2}\left[\left(\frac{\sqrt{\alpha^{\prime}}}{R}n\right)^{2}+\left(\frac{R}{\sqrt{\alpha^{\prime}}}m\right)^{2}\right]-2\\,\\N-\bar{N}=nm\,
g_{\mathrm{s}}(x)\;=\;{\frac{1}{8\beta\sqrt{x}}}\sqrt{{\frac{x-a^{2}}{x-b^{2}}}}-{\frac{1}{8\pi\beta}}\int{\frac{dy}{x+y}}{\frac{1}{\sqrt{y}}}\sqrt{{\frac{y+a^{2}}{y+b^{2}}}}\;,
\{J_{(\mu)}^{j,m},J_{(\mu^{\prime})}^{j^{\prime},m^{\prime}}\}=\sum_{j^{\prime\prime},\mu^{\prime\prime}}\alpha_{j^{\prime\prime}}^{\mu^{\prime\prime}}J_{(\mu^{\prime\prime})}^{j^{\prime\prime},m+m^{\prime}}
\beta_{\lambda}=\frac{12N}{(trm^{2})^{2}}(trm_{\perp}^{4}(e\beta_{e}-2e^{2}\sum_{i}\frac{m_{i}\beta_{m_{i}}}{trm^{2}})+2e^{2}\sum_{i}m_{i}\beta_{m_{i}}(m_{i}^{2}-\frac{trm^{2}}{N}))
R_{ab}\left(\frac{\lambda}{\lambda^{\prime}}\right)\Pi_{a}(\lambda)\left[\chi_{N}Z_{ab}^{-1}+\bar{\chi}_{N}Z_{ab}\right]\Pi_{b}(\lambda^{\prime})=\Pi_{b}(\lambda^{\prime})\left[\chi_{N}Z_{ab}^{-1}+\bar{\chi}_{N}Z_{ab}\right]\Pi_{a}(\lambda)R_{ab}\left(\frac{\lambda}{\lambda^{\prime}}\right)\,.
\frac{[a^{\prime}]}{a_{0}b_{0}}=-\frac{\kappa^{2}}{3}\rho+\frac{\kappa^{2}}{6n^{2}}\dot{\varphi}^{2},\\\frac{[n^{\prime}]}{n_{0}b_{0}}=\frac{\kappa^{2}}{3}(3p+2\rho)+\frac{\kappa^{2}}{6n^{2}}\dot{\varphi}^{2},
\left\langle\;\mathcal{C}^{(g)}(p)\;\right\rangle=\left.\frac{\mathcal{C}^{(g)}\;\left(\frac{1}{i}\,\frac{\delta}{\deltaJ}\right)\;W_{0}(J)}{W_{0}(0)}\;\right\vert_{J=0}=\;\bar{p}_{3}
r_{i,j}^{l,k}(u)=\delta_{i+j}^{l+k}\left\{(1-\delta_{i}^{l})\frac{\theta^{0}(0)^{\prime}\theta^{i-j}(u)}{\theta^{l-j}(u)\theta^{i-l}(u)}+\delta_{i}^{l}\left(\frac{\theta^{i-j}(u)^{\prime}}{\theta^{i-j}(u)}-\frac{\theta(u)^{\prime}}{\theta(u)}\right)\right\},
H\;=\;-\sum_{j=1}^{2N-1}\,e_{j}\;=\;-\frac{1}{\sqrt{3}}\,\Bigl(\sum_{j=1}^{N-1}\,(\Gamma_{j}\Gamma_{j+1}^{\dagger}+\Gamma_{j}^{\dagger}\Gamma_{j+1})\;+\;\sum_{j=1}^{N}\,(\sigma_{j}+\sigma_{j}^{\dagger})\Bigr)\;-\;\frac{2N-1}{\sqrt{3}}\;.
G(z_{1},\bar{z}_{1},z_{2},\bar{z_{2}})\;=\;\frac{1}{2\pi}\,\left(\begin{array}{cc}{\frac{1}{\bar{z}_{1}-\bar{z}_{2}}}&{\frac{1}{z_{1}-\bar{z}_{2}}}\\{}&{}\\{\frac{1}{\bar{z}_{1}-z_{2}}}&{\frac{1}{z_{1}-z_{2}}}\\\end{array}\right)\,.
\intd\xi_{0}^{(3)}d\xi_{0}^{(2)}d\xi_{0}^{(1)}\mathrm{Tr}(\hat{A}_{1}(\xi_{0}^{(1)},\xi)\star\hat{A}_{2}(\xi_{0}^{(2)},\xi)\star\hat{A}_{3}(\xi_{0}^{(3)},\xi))\sim\langle\Psi_{1}|\langle\Psi_{2}|\langle\Psi_{3}|V_{3}\rangle
L_{1}\,=\,L_{0}\,+\,\dot{\lambda}_{1}^{i}\lbrace{\frac{(X_{1}^{i}-X_{0}^{i})}{\epsilon}}\,-\,2\pi\alpha^{\prime}\,M_{ij}^{-1}{\calB}^{jk}P_{0}^{k}\,\rbrace\,+\,\dot{\lambda}_{3}^{i}\lbrace{\frac{(X_{N}^{i}-X_{N-1}^{i})}{\epsilon}}\,-\,2\pi\alpha^{\prime}\,M_{ij}^{-1}{\calB}^{jk}P_{N-1}^{k}\,\rbrace
L_{\mu\nu}=\left(\begin{array}{cccc}{0}&{k_{1}+k_{2}+2k_{3}}&{-2k_{1}-2k_{2}}&{k_{1}+k_{2}-2k_{3}}\\{-k_{1}-k_{2}-2k_{3}}&{0}&{3k_{1}-k_{2}-2k_{3}}&{-2k_{1}+2k_{2}+4k_{3}}\\{2k_{1}+2k_{2}}&{-3k_{1}+k_{2}+2k_{3}}&{0}&{k_{1}-3k_{2}-2k_{3}}\\{-k_{1}-k_{2}+2k_{3}}&{2k_{1}-2k_{2}-4k_{3}}&{-k_{1}+3k_{2}+2k_{3}}&{0}\\\end{array}\right)
f_{-}=\left[\frac{\gamma\left(Qb\right)\gamma\left((\alpha_{1}-Q)b+\frac{1}{2}\right)}{\gamma\left(\frac{Qb}{2}\right)\gamma\left(\frac{Q-2\alpha_{1}}{2}b+\frac{1}{2}\right)}\right]^{1/2}\frac{\gamma\left(\frac{Q-2\alpha_{2}}{2}b\right)C_{(\alpha_{1}-b/2),(\alpha_{2}),(\alpha_{3})}}{\gamma\left(\frac{Q-2\delta_{1}}{4}b+\frac{1}{4}\right)\gamma\left(\frac{2\delta_{2}-Q}{4}b+\frac{1}{4}\right)}.
\begin{aligned}{t=x^{+}+\frac{x^{-}}{R^{2}}\,,\qquad\psi=x^{+}-\frac{x^{-}}{R^{2}}\qquad\mathrm{and}\qquad\theta=\frac{y}{R}\qquad\mathrm{for}\qquadR\rightarrow\infty.}\\\end{aligned}
I(D)=\int_{0}^{\infty}\!\!\!d\hat{T}_{1}\,d\hat{T}_{2}\int_{0}^{1}\!\!\!da\,db\,dc\,dd\,{\biggl[(\hat{T}_{1}+G_{Bab})(\hat{T}_{2}+G_{Bcd})-{\frac{C^{2}}{4}}\biggr]}^{-{\frac{D}{2}}}
J_{k\neq0}^{ab}\Big|_{\bar{\varphi}_{k}^{a}=0}=-\epsilon^{abc}\:a_{1}^{c}+\mathrm{i}\,k\,\frac{2\pi}{L}\,\delta^{ab}\quad\mathrm{and}\quadJ_{0}^{ab}=\delta^{ab}\;.
\int_{M}{\mathrm{Str}}_{\mathrm{\bf~1}}\Big(\frac{1}{2}\mathcal{F}^{2}\Big)=\int_{M}{\mathrm{Str}}_{\mathrm{\bf~1}}\Bigg(Q\Big(\frac{1}{2}{\mathcal{A}}Q{\mathcal{A}}+\frac{1}{3}\mathcal{A}^{3}\Big)\Bigg),
\{N_{\mu}(\sigma),{P_{N}}_{\nu}(\sigma^{\prime})\}=(g_{\mu\nu}-\frac{P_{\mu}P_{\nu}}{P^{2}+N^{2}})\delta~;~\{{P_{N}}_{\mu}(\sigma),{P_{N}}_{\nu}(\sigma^{\prime})\}=(\frac{P_{\mu}X_{\nu}^{\prime}}{P^{2}+N^{2}}(\sigma)+\frac{P_{\nu}X_{\mu}^{\prime}}{P^{2}+N^{2}}(\sigma^{\prime}))\delta^{\prime}
{\bfA}=\left[\begin{matrix}{0}&{1}&{0}&{0}&{0}&{0}\\{0}&{-2}&{0}&{1/4}&{1}&{1}\\{0}&{0}&{-2}&{-1/4}&{-1/2}&{0}\\{0}&{0}&{-e^{2x}}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}\\\end{matrix}\right]
\pi_{k}=\frac{\partialL}{\partialv^{k}}=\frac{\kappa\left[v^{k}-u^{k}u^{0}(1+u^{0}\vec{u}\vec{v})\right]}{\sqrt{(\vec{v})^{2}-(1+u^{0}\vec{u}\vec{v})^{2}}}=-\kappa\omega_{k}
{\bf\Psi}(x)=\left(\begin{array}{c}{\varphi(x)}\\{\Psi(x)}\\\end{array}\right),\;\;\hat{x}^{\mu}=x^{\mu}\;,\;\;\hat{\pi}_{\mu}=-i\partial_{\mu}\;,\;\;\hat{\psi}^{n}=\frac{i}{2}\Gamma^{n},
N_{CS}\equiv\frac{1}{24\pi^{2}}\intd^{3}x\epsilon_{ijk}Tr(U\partial_{i}U^{-1}U\partial_{j}U^{-1}U\partial_{k}U^{-1})=\frac{1}{16\pi^{2}}\intd^{3}x\vec{{\mathcal{A}}}\cdot\vec{{\mathcal{B}}}
\intd^{\nu}x\left[-{\frac{1}{2}}{\frac{\delta^{2}}{\delta\phi(\vec{x})\delta\phi(\vec{x})}}+{\frac{1}{2}}\left(\vec{\nabla}\phi(\vec{x})\right)^{2}+{\frac{1}{2}}m^{2}{\phi^{2}(\vec{x})}+{\frac{\lambda}{4!}}{\phi}^{4}(\vec{x})\right]\Psi[\phi]=E\,\Psi[\phi]\,.
g=\left(\begin{array}{cc}{\frac{1}{T^{2}}[1-h(T,U)+Tg(U,T)]}&{\frac{1}{2TU}[1+2h(T,U)+Tg(U,T)+Ug(T,U)]}\\{\frac{1}{2TU}[1+2h(T,U)+Tg(U,T)+Ug(T,U)]}&{\frac{1}{U^{2}}[1-h(T,U)+Ug(T,U)]}\\\end{array}\right)
{\calV}=-{\textstyle\frac{1}{2}}{\displaystyle\frac{|\tau|^{2}}{\Im\mathrm{m}(\tau)}}\left\{[\mathrm{Tr}\,(\mathsf{Q}{\calM})]^{2}-2\mathrm{Tr}\,(\mathsf{Q}{\calM})^{2}\right\}\,,
\begin{array}{lcr}{x}&{=}&{\alpha_{1}\displaystyle\frac{\alpha_{3}\pm||\alpha||}{(\alpha_{1}^{2}+\alpha_{2}{}^{2})}\,}\\{y}&{=}&{\displaystyle-\alpha_{2}\frac{\alpha_{3}\pm||\alpha||}{(\alpha_{1}^{2}+\alpha_{2}^{2})}\.}\\\end{array}
\left[w^{5}{\frac{d}{dw}}w^{-5}{\frac{d}{dw}}+{\frac{Q_{0}\omega^{2}}{64Q_{4}^{3}R^{4}(2N_{0}\omega+k^{2})^{4}}}w^{6}\pm1-{\frac{4m^{2}}{w^{2}}}\right]\varphi(w)=0,
\zeta_{\theta}(s,D)=\mathrm{Tr}\left[\left({\frac{P_{\theta}(D)}{\mu^{2}}}\right)^{-s}\right]={\frac{2\mu^{2s}}{\Gamma(s)}}\int_{0}^{\infty}{\frac{d\xi}{\xi}}\xi^{2s}\\mathrm{Tr}\left[e^{-\xi^{2}P_{\theta}(D)}\right]
\int_{0}^{\infty}\frac{d\kappa^{2}}{\lambda}\rho_{ii}\left(\frac{\kappa^{2}}{\lambda}\right)H(\kappa^{2})\leqC\int_{0}^{\infty}\frac{d\kappa^{2}}{\lambda}\rho_{ii}\left(\frac{\kappa^{2}}{\lambda}\right)G(\kappa^{2}).
dJ_{D}=\eta_{d-1}\lbracee^{-b\Phi}\Lambdae^{\frac{4\phi}{\sqrt{d-2}(d-3)}}-\frac{p}{(d-2)}e^{-a\Phi}{\calH}_{p+1}^{2}e^{-\frac{4p\phi}{\sqrt{d-2}(d-3)}}\rbrace,
{\frac{Dp^{\mu}}{d\tilde{\lambda}}}\equiv{\frac{dp^{\mu}}{d\tilde{\lambda}}}+\Gamma_{\rho\sigma}^{\mu}{\frac{dx^{\rho}}{d\tilde{\lambda}}}p^{\sigma}=-\tilde{m}_{0}{\calW}^{-1}g^{\mu\nu}{\frac{dx^{\rho}}{d\tilde{\lambda}}}\partial_{y}({\calW}g_{\nu\rho}){\frac{dy}{d\tilde{\lambda}}}.
L^{A}\,=\,\frac{1}{2R}\sum_{n\geq0}\left[\omega_{n+\frac{1}{2}}(a_{n}\dot{b_{n}}\,-\,\dot{a_{n}}b_{n})\,\,-\,\,\omega_{n+\frac{1}{2}}^{2}(a_{n}^{2}\,+\,b_{n}^{2})\right].
S_{BH}\,=\,\frac{Area_{H}}{4G_{N}}\,\stackrel{\smallG_{N}=1}{=}\,\pi\sqrt{P_{(4)}(p,q)}\,=\,2\pi\sqrt{(p^{3})^{3}q_{0}+p^{1}q_{1}p^{3}q_{3}-\left(p^{1}q_{1}+p^{3}q_{3}\right)^{2}}
\rho_{n+1}(x^{\prime},y^{\prime})=\sum_{(x,y)\inf^{-1}(x^{\prime},y^{\prime})}\frac{\rho_{n}(x,y)}{\lambda|\partialT/\partialx|}=\sum_{x\inT^{-1}(x^{\prime})}\frac{\rho_{n}(x,(y^{\prime}-\tau^{1/2}x)/\lambda)}{\lambda|\partialT/\partialx|}
\left(\begin{array}{c}{u^{B}}\\{u^{y}}\\{u^{R}}\\\end{array}\right)_{L}\;\;\;\left(\begin{array}{c}{d^{B}}\\{d^{y}}\\{d^{R}}\\\end{array}\right)_{L}\;\;\;\left(\begin{array}{c}{u^{B}}\\{u^{y}}\\{u^{R}}\\\end{array}\right)_{R}\;\;\;\left(\begin{array}{c}{d^{B}}\\{d^{y}}\\{d^{R}}\\\end{array}\right)_{R}\;\;\;\left(\begin{array}{c}{e_{R}}\\{e_{L}}\\{\nu_{L}}\\\end{array}\right)
\left(e^{2\lambdar_{c}}-\frac{M_{0}\pi}{\lambda^{2}}\right)\left(e^{2\lambdar_{c}}-\frac{M_{2}\pi}{\lambda^{2}}\right)=\left(e^{2\lambdar_{c}}-\frac{M_{1}\pi}{\lambda^{2}}\right)\left(e^{2\lambdar_{c}}-\frac{\tilde{M}_{1}\pi}{\lambda^{2}}\right)
\begin{array}{c}{F_{\mu\nu}^{k}=\partial_{\mu}A_{\nu}^{k}-\partial_{\nu}A_{\mu}^{k}+C_{ij}^{k}\,A_{\nu}^{i}\,A_{\mu}^{j}\,,}\\{\tilde{F}_{\mu\nu}^{k}=\partial_{\mu}B_{\nu}^{k}-\partial_{\nu}B_{\mu}^{k}+C_{ij}^{k}\,B_{\nu}^{i}\,A_{\mu}^{j}+C_{ij}^{k}\,A_{\nu}^{i}\,B_{\mu}^{j}\,,}\\\end{array}
I(\nu)=\frac{2W(1)}{(\nu^{2}-\frac{1}{4})}-\int_{1}^{\infty}d\alpha\{\alpha^{\frac{3}{2}}(\frac{d}{d\alpha}[W(\alpha)\alpha^{\frac{3}{4}}])\}[{\frac{\alpha^{\frac{\nu}{2}-\frac{5}{4}}}{\frac{\nu}{2}+\frac{1}{4}}}-{\frac{\alpha^{\frac{-\nu}{2}-\frac{5}{4}}}{\frac{\nu}{2}-\frac{1}{4}}}].
\Gamma_{a,s}(l_{1s})=\tilde{\Gamma}_{s}^{-1}\Gamma_{a,s}^{(o)}(l_{1s})\tilde{\Gamma}_{s},\qquad\Gamma_{b,s}(l_{2s})=\tilde{\Gamma}_{s}^{-1}\Gamma_{b,s}^{(o)}(l_{2s})\tilde{\Gamma}_{s}
S(U,V)=\frac{1}{2}\mathrm{Tr}\;U^{2}+\frac{1}{2}\mathrm{Tr}\;V^{2}-c\mathrm{Tr}\;UV-\frac{g}{3}(\mathrm{Tr}\;U^{3}+\mathrm{Tr}\;V^{3})\,
\widehat{T}^{\check{A}\check{B}}(\tau,\vec{\sigma}_{o})=\left(\widehat{\rho}(\tau,\vec{\sigma}_{o})+\widehat{p}(\tau,\vec{\sigma}_{o})\right)\,\widehat{U}^{\check{A}}(\tau,\vec{\sigma}_{o})\,\widehat{U}^{\check{B}}(\tau,\vec{\sigma}_{o})-\widehat{p}(\tau,\vec{\sigma}_{o})\,g^{\check{A}\check{B}}(\tau,\vec{\Sigma}(\tau,\vec{\sigma}_{o})).
X^{M}(\sigma,\tau)=\hat{x}_{0}^{M}-i\alpha^{\prime}\hat{p}_{0}^{M}\tau+i\sqrt{\frac{\alpha^{\prime}}{2}}\sum_{n\neq0}\left(\frac{\alpha_{n}^{M}}{n}e^{-n(\tau+i\sigma)}+\frac{\tilde{\alpha}_{n}^{M}}{n}e^{-n(\tau-i\sigma)}\right)~,
g_{aa}=-\frac{1}{4}\frac{\Lambda^{\prime}(\lambda_{a})}{A(\lambda_{a})}=-\frac{1}{4}\frac{\displaystyle\prod_{b\neqa,b=1}^{N}(\lambda_{a}-\lambda_{b})}{\displaystyle\prod_{b=1}^{N}(\lambda_{a}-r_{b})},\g_{ab}=0,\foralla\neqb
\frac{d({\calH}^{2}e^{4\alpha})}{d\alpha}=\frac{d}{d\alpha}\left[e^{4\alpha}\frac{\rho^{2}}{36}\right]-\frac{e^{4\alpha}}{9}\frac{r_{B}\rho}{\calH}-\frac{2}{3}e^{4\alpha}q_{B}-\frac{d}{d\alpha}\left(e^{4\alpha}\frac{k}{e^{2\alpha}}\right).
F_{B(F)}\left(\beta,V,\frac{1}{m}\right)=-\frac{V}{2^{\frac{d}{2}+1}\pi^{\frac{d}{2}}}\int_{0}^{\infty}ds\;s^{-1-\frac{d}{2}}\left[\theta_{3(4)}\left(0\left|\frac{i\beta^{2}}{2\pis}\right.\right)-1\right]e^{-\frac{m^{2}s}{2}}\;.
Q_{D}=\left(\begin{matrix}{0}&{0}&{0}&{0}&{0}&{1}&{1}&{-1}&{-1}&{1}&{-1}&{-2}&{0}&{2}\\{0}&{0}&{0}&{0}&{0}&{0}&{0}&{-1}&{1}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{-1}&{1}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{-1}&{-1}&{2}&{1}&{-1}&{1}&{1}&{0}&{-2}\\\end{matrix}\right).
T_{+p+2}^{-\dot{q}}=-\frac{\chi}{2}\tilde{\gamma}_{\dot{q}p}^{i}\Omega_{+2}^{-2i},\T_{+p-2}^{-\dot{q}}=-\frac{\chi}{2}\tilde{\gamma}_{\dot{q}p}^{i}\Omega_{-2}^{-2i}+\frac12\tilde{\gamma}_{\dot{q}q}^{i}\Omega_{+p}^{-2i}\psi_{-2q}^{2+},
T_{2\psi3\lambda}=\intd^{2}\sigma\left\{\left(e^{\rho}{}_{a}\hat{D}_{\rho}^{(+)}F_{\mu\nubc}+6F_{\mu\nuka}f^{k}{}_{bc}\right)\psi^{\mu}\psi^{\mu}\lambda^{a}\lambda^{b}\lambda^{c}\right\}\.
T(x^{9})_{diag}=T_{v}\left(\begin{array}{cc}{-1_{N-k}}&{0}\\{0}&{1_{k}}\\\end{array}\right),x^{9}<0,\T(x^{9})_{diag}=T_{v}\left(\begin{array}{cc}{1_{N-k}}&{0}\\{0}&{-1_{k}}\\\end{array}\right),x^{9}>0
\zeta^{D}(\alpha,R)\equiv\frac{1}{2\piR}\sum_{n=-\infty}^{\infty}\int\frac{d^{D-1}p}{(2\pi)^{D-1}}\frac{-i}{p^{2}+\left(\frac{n+\alpha}{R}\right)^{2}+m^{2}}.
\langlet|\Psi\rangle={\frac{1}{\sqrt{2\piB}}}\left({\frac{B}{\pi}}\right)^{1/4}e^{-Bt^{2}/2}e^{-{\frac{1}{4B}}\left({\frac{\partial}{\partialt}}\right)^{2}}e^{-Bt^{2}}\Psi(-i\sqrt{2B}t).
\begin{array}{ccl}{e_{1}(\tau)=}&{\frac{1}{3}(\theta_{2}^{4}(0,\tau)+\theta_{3}(0,\tau))=}&{\frac{2}{3}+16q+16q^{2}+\cdots,}\\{e_{2}(\tau)=}&{-\frac{1}{3}(\theta_{1}^{4}(0,\tau)+\theta_{3}^{4}(0,\tau))=}&{-\frac{1}{3}-8q^{1/2}-8q-32q^{3/2}-8q^{2}+\cdots,}\\{e_{3}(\tau)=}&{\frac{1}{3}(\theta_{1}^{4}(0,\tau)-\theta_{2}^{4}(0,\tau))=}&{-\frac{1}{3}+8q^{1/2}-8q+32q^{3/2}-8q^{2}+\cdots,}\\\end{array}
\left\{Q_{\alpha},\,Q_{\beta}\right\}=\Gamma_{\alpha\beta}^{\mu}P_{\mu}+2\Gamma_{\,\alpha\beta}^{\mu\nu}Z_{\mu\nu}+\frac{1}{5!}\Gamma_{\,\alpha\beta}^{\mu\nu\rho\sigma\tau}Z_{\mu\nu\rho\sigma\tau}^{5}\,.
\hat{J}^{\mu\nu}=\hat{T}_{+}\hat{J}^{\mu\nu}+\hat{T}_{-}\hat{J}^{\mu\nu}=\hat{T}_{+}\hat{J}^{\mu\nu}\hat{T}_{+}+\hat{T}_{-}\hat{J}^{\mu\nu}\hat{T}_{-},
\mathcal{P}\intd\theta\left(\varphi_{ij}\left(\theta+\theta_{h}+t_{i}\theta_{H}\right)+\varphi_{ij}\left(\theta-\theta_{h}-t_{i}\theta_{H}\right)\right)e^{ik\theta}=\sum_{l=1}^{r}I_{il}\,\,\widetilde{\varphi}_{lj}(k)\,\,.
\{X_{i}^{1},X_{j}^{2}\}=-r_{+}X_{i}^{1}X_{j}^{2}-X_{i}^{1}X_{j}^{2}r_{+}+X_{i}^{1}r_{+}X_{j}^{2}+X_{j}^{2}r_{+}X_{i}^{1},
S_{\mathrm{GF}}=\int\,d^{4}x\{B_{\mu}D_{\nu}B^{\nu\mu}+\varepsilon_{ab}B^{a}D_{\mu}C^{\mub}-\frac{1}{2}B_{\mu}B^{\mu}-\frac{1}{2}\varepsilon_{ab}B^{a}B^{b}\},
(-g)^{-1/2}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\varphi)+\sigma,_{\mu}\varphi^{,\mu}+V^{\prime}(\varphi)+\tilde{e}^{2}\varphig^{\alpha\beta}\tilde{A}_{\alpha}\tilde{A}_{\beta}=0,
\frac{\alpha^{\prime}}{\alpha_{\mathrm{eff}}^{\prime}}G^{ij}=-(\delta_{0}^{i}\delta_{0}^{j}-\delta_{1}^{i}\delta_{1}^{j})+\frac{\theta_{1}}{\theta_{0}b}(\delta_{2}^{i}\delta_{2}^{j}+\delta_{3}^{i}\delta_{3}^{j}),
{\tilde{J}}_{0}\,=\,\frac{1}{2\pii}\,\epsilon_{kl}\partial_{k}({\tilde{\phi}}^{*}\partial_{l}{\tilde{\phi}})\;\;\;{\tilde{J}}_{k}\,=\,\frac{1}{2\pii}\,\epsilon_{kl}[\partial_{l}({\tilde{\phi}}^{*}\partial_{0}{\tilde{\phi}})-\partial_{0}({\tilde{\phi}}^{*}\partial_{l}{\tilde{\phi}})]
\begin{array}{rclrcl}{F_{0i}(A)}&{=}&{-2k^{-2}\partial_{\underline{{i}}}k\,,}&{F_{0i}(B)}&{=}&{0\,,}\\{}&{}&{}&{}&{}&{}\\{F_{ij}(A)}&{=}&{-4k^{-1}A_{[\underline{{i}}}\partial_{\underline{{j}}]}k+F_{\underline{{i}}\underline{{j}}}(A)\,,}&{F_{ij}(B)}&{=}&{k^{2}F_{ij}(A)\,.}\\\end{array}
P^{0}=E=\frac{1}{2e^{2}{\phi_{0}}^{2}}\intd^{2}x\left[\frac{1}{2}F_{ij}^{2}+\vertD_{i}\Phi\vert^{2}+\frac{e^{2}}{4}(\vert\Phi\vert^{2}-{\Phi_{0}}^{2})^{2}\right]
T^{\mu\nu}={\frac{2}{\sqrt{-g}}}{\frac{\partial({\calL}\sqrt{-g})}{\partialg_{\mu\nu}}}=F^{\mu\rhoa}F_{\rho}^{\nua}+D^{\mu}\phi^{a}D^{\nu}\phi^{a}+g^{\mu\nu}{\calL}
\mu^{2-2\omega}\int\frac{d^{2\omega}k}{\left(2\pi\right)^{2\omega}}\sum_{n=-\infty}^{\infty}\left(k^{2}+\frac{1}{L^{2}}\left(2\pin+\chi\right)^{2}\right)^{1/2}=-\frac{\pi^{2}}{45L^{3}}\left[1-\frac{15}{8}\left(\left(\frac{\chi}{\pi}-1\right)^{2}-1\right)^{2}\right]
0\rightarrow{\calL}_{b}\stackrel{d_{0}}{\rightarrow}{\calL}_{b_{1}}\oplus{\calL}_{b_{1}^{\prime}}\stackrel{d_{1}}{\rightarrow}{\calL}_{b_{2}}\oplus{\calL}_{b_{2}^{\prime}}\stackrel{d_{2}}{\rightarrow}\cdots
g_{-}^{-1}(u)E_{jj}^{(n)}g_{-}(u)=g_{-}^{-1}(u)\left(\frac{\partial}{\partialu_{j}^{(n)}}g_{-}(u)\right)+\left(\frac{\partial}{\partialu_{j}^{(n)}}g_{+}(u)\right)g_{+}^{-1}(u)\,.
xy^{2}=\left[x^{n_{c}}(x-\phi_{n_{c}}^{2})\right]^{2}-4\Lambda^{4}x^{2n_{c}}=x^{2n_{c}}(x-\phi_{n_{c}}^{2}-2\Lambda^{2})(x-\phi_{n_{c}}^{2}+2\Lambda^{2}).
\left[\frac{1}{c(1-\vec{n}\cdot\vec{\beta})}\frac{d}{dt^{\prime}}\left(\frac{1}{c(1-\vec{n}\cdot\vec{\beta})}\frac{d}{dt^{\prime}}\frac{\vec{n}c(t-t^{\prime})}{(1-\vec{n}\cdot\vec{\beta})R}\right)\right]_{\mathrm{ret}}\;,
H_{\pm}=\left(\begin{array}{cc}{m}&{ie^{\mpi\theta}\left(\partial_{r}\mp\frac{i}{r}\partial_{\theta}\mp\frac{eB}{2}r\right)}\\{ie^{\pmi\theta}\left(\partial_{r}\pm\frac{i}{r}\partial_{\theta}\pm\frac{eB}{2}r\right)}&{-m}\\\end{array}\right).
\int_{\sigma}^{\infty}{\frac{d\sigma^{\prime}}{\sqrt{1+4{\sigma^{\prime}}^{3}}}}={\frac{1}{3}}\int_{0}^{\infty}{\frac{X^{-2/3}dX}{\sqrt{1+4X}}}-{\frac{1}{3}}\int_{0}^{\sigma^{3}}{\frac{X^{-2/3}dX}{\sqrt{1+4X}}}=ig_{3}^{\frac{1}{6}}(\eta-\eta_{\infty}),
Z_{E}\left({\frac{a\tau+b}{c\tau+d}},{\frac{z}{c\tau+d}}\right)={\epsilon{\binom{a\b}{c\d}}}^{r-D}e^{2\pi\sqrt{-1}(r/2)cz^{2}/(c\tau+d)}Z_{E}(\tau,z)\,,\quad{\binom{a\b}{c\d}}\inSL(2,{\bfZ})\,,
{\bfP}_{A}=-i\partial_{A}\quad,\qquad{\bfM}_{AB}=-i\left(x_{A}\partial_{B}-x_{B}\partial_{A}+\left(\sigma_{AB}\right)_{a}\!^{b}\left(\lambda_{b}\partial^{a}+\overline{{\lambda}}_{b}\overline{{\partial}}^{a}\right)\right)
ds^{2}=-4dx^{-}dx^{+}-\mu^{2}\left(4x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+\sum_{I=5}^{8}x_{I}^{2}\right)d{x^{+}}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+dx_{4}^{2}+\sum_{I=5}^{8}dx_{I}^{2}
V_{eff-tot}=\left[e^{2}\left(a-\overline{{a}}\right)^{i}\left(a-\overline{{a}}\right)_{i}-\frac{1}{2\alpha^{\prime}}+\left(e\left(\phi-\overline{{\phi}}\right)_{a}-\frac{\delta^{a}}{2\pi\alpha^{\prime}}\right)^{2}\right]T\overline{{T}}+2*\frac{e^{2}}{2}\left(T\overline{{T}}\right)^{2}
\partial_{x}\Gamma_{i}^{*}(t,\lambda)=\Phi_{i}(t)\psi_{BA}^{*}(t,\lambda)\;\;;\;\;-\partial_{x}\psi_{BA}^{*}(t,\lambda)-\sum_{i=1}^{m}\Psi_{i}\Gamma_{i}^{*}(t,\lambda)=\lambda\psi_{BA}^{*}(t,\lambda)\,.