harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{l}{y=\sigma^{0}\,\,\,,\,\,\,z=-\sigma^{1}\,\,\,,\,\,\,\tau=\tau\left(\sigma^{0}\right)}\\{\chi=\chi\left(\sigma^{0}\right)\,\,\,,\,\,\,\theta=\theta\left(\sigma^{0}\right)\,\,\,,\,\,\,\omega=\omega\left(\sigma^{0}\right)}\\\end{array}
ds_{t\to0}^{2}\sim\mathrm{e}^{-\frac{2pf_{0}}{q-1}}(-dt^{2}+d\Sigma_{k,+1}^{2}+d\vec{y}_{q-k}^{2})+\mathrm{e}^{2f_{0}}d\vec{x}_{p}^{2}.
\Pi^{1}=R_{vac}\frac{\Psi^{1}}{\Psi^{0}},\quad\Pi^{2}=R_{vac}\frac{\Psi^{2}}{\Psi^{0}},...,\Pi^{i}=R_{vac}\frac{\Psi^{i}}{\Psi^{0}},...,\Pi^{N}=R_{vac}\frac{\Psi^{N}}{\Psi^{0}},
{}~{\displaystyle\sum_{\{k_{i}\},\atopi=1,\cdots,6}}R(\theta_{3},\theta_{5},\theta_{6})_{i_{3},i_{5},i_{6}}^{k_{3},k_{5},k_{6}}R(\theta_{2},\theta_{4},\theta_{6})_{i_{2}i_{4}k_{6}}^{k_{2}k_{4}j_{6}}R(\theta_{1},\theta_{4},\theta_{5})_{i_{1}k_{4}k_{5}}^{k_{1}j_{4}j_{5}}R(\theta_{1},\theta_{2},\theta_{3})_{k_{1}k_{2}k_{3}}^{j_{1}j_{2}j_{3}}
0=F_{u}^{\prime\prime}+\frac{2}{r}F_{u}^{\prime}\left[1+rL_{-}^{\prime}(kr)\right]-\frac{s^{\prime}}{s+\frac{\omega}{m}}\left[F_{u}^{\prime}+F_{u}L_{-}^{\prime}(kr)\right]+W(r,-m)F_{u}-\left[K,F_{u}\right]
100=\left[f(\bar{x},\bar{y})\frac{\partial}{\partial\bar{x}}+g(\bar{x},\bar{y})\frac{\partial}{\partial\bar{y}}\right]A_{0}(\bar{x},\bar{y})=\frac{d}{dt}A_{0}\left(\bar{x}(t),\bar{y}(t)\right)
f_{\mu\nu}(x):=\partial_{\mu}(n^{A}(x){\calA}_{\nu}^{A}(x))-\partial_{\nu}(n^{A}(x){\calA}_{\mu}^{A}(x))-{\frac{1}{g}}{\bfn}(x)\cdot(\partial_{\mu}{\bfn}(x)\times\partial_{\nu}{\bfn}(x)),
G^{ij}=\left(\begin{array}{cccc}{0}&{-\sqrt{\frac{g_{s}^{2}d^{2}+c^{2}w}{c^{2}+g_{s}^{2}d^{2}}}}&{0}&{0}\\{-\sqrt{\frac{g_{s}^{2}d^{2}+c^{2}w}{c^{2}+g_{s}^{2}d^{2}}}}&{0}&{0}&{0}\\{0}&{0}&{\frac{1}{(y^{+})^{2}}\sqrt{\frac{g_{s}^{2}d^{2}+c^{2}w}{c^{2}+g_{s}^{2}d^{2}}}}&{0}\\{0}&{0}&{0}&{\sqrt{\frac{g_{s}^{2}d^{2}+c^{2}w}{c^{2}+g_{s}^{2}d^{2}}}}\\\end{array}\right),
\begin{array}{c}{\dot{\bfx}=\mathrm{i}\left({\bfz}_{1}-{\bf{\bar{z}}}_{1}\right),}\\{\dot{\bfz}_{i-1}=-\mathrm{i}\delta_{1.i-1}{\bfp}+k_{i-1}{\bfz}_{i}-k_{i-2}{\bfz}_{i-2},}\\{\dot{\bfz}_{N}=-\mathrm{i}\delta_{1.N}{\bfp}+\mathrm{i}k_{N}{\bfz}_{N}-k_{N-1}{\bfz}_{N-1},}\\{{\dot{\bfp}}=0,}\\\end{array}
C(k,p)=\left(\begin{array}{ccccccc}{0}&{0}&{ik\delta(\bar{k}+\bar{p})}&{0}&{0}&{0}&{0}\\{0}&{-2i\delta(\bar{k}+\bar{p})}&{0}&{0}&{0}&{0}&{0}\\{i\delta(\bar{k}+\bar{p})}&{0}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{-\delta(0)}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}&{-\delta(0)}\\{0}&{0}&{0}&{\delta(0)}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{\delta(0)}&{0}&{0}\\\end{array}\right)\,,
W[C]=\sum_{n}\int_{s_{1}<s_{2}\ldots<s_{n}}\Gamma_{\mu_{1}\ldots\mu_{n}}^{(n)}(x(s_{1}),\ldotsx(s_{n}))\,\dot{x}_{\mu_{1}}(s_{1})\ldots\dot{x}_{\mu_{n}}(s_{n})\,ds_{1}\ldotsds_{n}
\delta\varphi^{\prime}=-\partial_{\mu}\tilde{\partial}_{+}\chi^{\prime\alpha}\,\sigma_{\alpha\dot{\beta}}^{\mu}\bar{\varepsilon}^{\prime\dot{\beta}},\quad\delta\chi^{\prime\alpha}=\varepsilon^{\prime\alpha}\varphi^{\prime}+\bar{\varepsilon}_{\dot{\beta}}\bar{\sigma}_{\mu}^{\dot{\beta}\alpha}\partial^{\mu}f^{\prime},\quad\deltaf^{\prime}=s\tilde{\partial}_{+}\chi^{\prime\alpha}\varepsilon_{\alpha}
\frac{e^{2}}{(2\pi)^{2}}\int\langleF_{12}(x)F_{12}(0)\ranglee^{-ipx}dx\;\;=\;\;P^{0}(0)-\int_{0}^{\infty}\frac{d\rho(\mu^{2})}{p^{2}+\mu^{2}}\;.
\nu_{g}=\fracNA=m^{*}\,\,\,(\Omega\,-\,\frac{2\,G\,J}{R^{3}})\frac{R^{2}e^{-\Phi}}A\leq\frac{m^{*}}\pi\,\,\,(\Omega\,-\,\frac{2\,G\,J}{R^{3}})\,.
{\calL}_{M}^{f}=\frac12\bar{\Theta}_{i}{}_{m}^{A}M^{ij}\Theta_{j}{}_{-m}^{A}-i\mum\bar{\Theta}_{i}{}_{m}^{A}\Omega^{i}{}_{-m}^{A}+\frac12\bar{\Omega}^{i}{}_{m}^{A}M_{ij}\Omega^{j}{}_{-m}^{A},
{\calS}\;=\;T_{p}\int_{W_{p}}d^{p}\xi\,\sqrt{|h|}\left[-\textstyle{\frac{1}{2}}h^{ij}\partial_{i}X^{\mu}\partial_{j}X^{\nu}g_{\mu\nu}(X)\;+\;\textstyle{\frac{1}{2}}\left(p-2\right)\right]\;.
\langle0|[iQ_{+},\epsilon^{ab}\bar{C}^{a}\bar{C}^{b}]|0\rangle=2\langle0|\epsilon^{ab}C^{a}\bar{C}^{b}|0\rangle=-\langle0|[iQ_{-},\epsilon^{ab}C^{a}C^{b}]|0\rangle.
{}~\times\frac{w(\omegax_{3}x_{4}z_{1}z_{2}/x_{1}x_{2}z_{3}z_{4}|c+d-a-b)}{w\bigg(\frac{\displaystylex_{4}z_{1}}{\displaystylex_{1}z_{4}}|d-a\bigg)w\bigg(\frac{\displaystylex_{3}z_{2}}{\displaystylex_{2}z_{3}}|c-b\bigg)w\bigg(\frac{\displaystylex_{3}z_{1}}{\displaystylex_{1}z_{3}}|c-a\bigg)w\bigg(\frac{\displaystylex_{4}z_{2}}{\displaystylex_{2}z_{4}}|d-b\bigg)},
A_{\mu},F,F_{A}^{a}=\frac{i}{2}F_{\\\A\dot{A}}^{\dot{A}a},F_{b}^{a}=\frac{1}{2}\left(-\nabla_{b}^{B}F_{B}^{a}+F\overline{{X}}_{\b}^{a}+\overline{{F}}X_{\b}^{a}\right)
{\cal{Q}}^{\dagger}{\cal{Q}}\overline{{\psi}}=\left\{-\frac{d}{dz}-\frac{3}{2}\dot{A}\right\}\left\{\frac{d}{dz}-\frac{3}{2}\dot{A}\right\}\overline{{\psi}}=2m^{2}\overline{{\psi}}\,.
\ddot{\chi}+\dot{\alpha}_{0}\left(1-\frac{\ddot{\alpha}_{0}}{\alpha_{1}^{2}}\right)\dot{\chi}-\frac{1}{3}\left(1+\frac{\ddot{\alpha}_{0}}{\alpha_{1}^{2}}\right)e^{-2\alpha_{0}}\nabla^{2}\chi=0\.
\epsilon^{\alpha}(t)=\intdt^{\prime}\w^{\alpha}(t,t^{\prime})u(t^{\prime}),\quadw^{\alpha}(t,t^{\prime})=\sum_{l=0}^{m_{w}}w_{l}^{i}{\frac{d^{l}}{dt^{l}}}\delta(t-t^{\prime})\.
\left(\begin{array}{l}{\varphi}\\{\varphi^{*}}\\\end{array}\right)\rightarrow\left(\begin{array}{l}{\varphi^{\prime}}\\{\varphi^{*\prime}}\\\end{array}\right)=\left(\begin{array}{ll}{a}&{b}\\{b^{*}}&{a^{*}}\\\end{array}\right)\left(\begin{array}{l}{\varphi}\\{\varphi^{*}}\\\end{array}\right)+\left(\begin{array}{l}{c}\\{c^{*}}\\\end{array}\right)
{\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}}}=-\left[{\calW}^{-1}g^{\rho\sigma}+{\frac{1}{2}}{\frac{dx^{\rho}}{d\tilde{\lambda}}}{\frac{dx^{\sigma}}{d\tilde{\lambda}}}\right]{\frac{dx^{\mu}}{d\tilde{\lambda}}}\partial_{y}\left({\calW}g_{\sigma\mu}\right),
R_{21}=\frac{\Gamma(n+|l|+1-\mu)\Gamma(n-\nu+1+\mu)\Gamma^{2}(1+\nu-\mu)\mu^{2}(1-\mu)^{2}q^{|l|-1}}{\Gamma(n+1-\nu)\Gamma(n+1)\Gamma^{2}(|l|+\nu+1-\mu)}\,f^{2}(\mu)\;.
{\frac{1}{z\,K}}\sqrt{\frac{A_{T^{2}}^{\mathrm{(OM)}}}{A_{T^{2}}^{\mathrm{(bulk)}}}}\sim\left({\frac{l_{p}}{l_{g}}}\right)^{3}\propto{\frac{\alpha^{\prime}}{\alpha_{(eff)}^{\prime}}}\,.
{\calH}_{T}=q_{x}\,\pi_{p}+r\,\pi_{q}+\frac{\textstyle{\r_{x}^{\;2}-p_{xx}}}{\textstyle{q_{xx}}}\left(\pi_{r}+\frac{1}{2}q_{x}^{\;2}\right)+\frac{1}{2}p_{x}\,q_{x}
W^{\wedge^{m}(\Omega_{z_{1}}),\Omega_{z_{2}}}(v^{(a_{m},a)}\otimesv^{(b_{m},a_{m})})=\sum_{b}v^{(b_{m},b)}\otimesv^{(b,a)}W^{(m,1)}\left(\left.\begin{array}{cc}{b_{m}}&{a_{m}}\\{b}&{a}\\\end{array}\right|z_{1}-z_{2}\right),
e^{iW[J,J^{\prime}]}=\langleIN|\tilde{T}(e^{-i\intd^{4}x^{\prime}~J^{\prime}(x^{\prime})\Phi(x^{\prime})})T(e^{i\intd^{4}x^{\prime\prime}~J(x^{\prime\prime})\Phi(x^{\prime\prime})})|IN\rangle
\Pi_{\mathrm{gg}}^{33}(p)=\frac{2g^{2}}{L}\sum_{q_{3}}q_{3}^{2}\int\frac{d^{3}q}{(2\pi)^{3}}\sum_{a,b\neq3}D_{\mu\nu}^{ab}(q,q_{3})D^{ab,\mu\nu}(p-q,q_{3})\.
\gamma_{0}=-i\left(\begin{array}{cc}{0}&{-1}\\{1}&{0}\\\end{array}\right),\qquad\gamma_{1}=-i\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right),\qquad\gamma_{5}=\gamma_{0}\gamma_{1}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right),
\left(\begin{array}{c}{\ddot{\phi}_{1n}}\\{\ddot{\phi}_{2n}}\\\end{array}\right)+2{\calA}\left(\begin{array}{c}{\dot{\phi}_{1n}}\\{\dot{\phi}_{2n}}\\\end{array}\right)+{\calB}\left(\begin{array}{c}{{\phi}_{1n}}\\{{\phi}_{2n}}\\\end{array}\right)=\left(\begin{array}{c}{0}\\{0}\\\end{array}\right),
\frac{d^{2}I_{L}}{dtd\Omega}=\frac{8r_{e}mca_{0}^{2}\omega_{0}^{2}}{[1+a_{0}^{2}\theta^{2}]^{3}}\Big[1-\frac{4a_{0}^{2}\theta^{2}(1-\phi^{2})}{[1+a_{0}^{2}\theta^{2}]^{2}}\Big]~.
I=4\intd^{2}xd^{2}\thetaE^{-1}\left(\frac{1}{2\kappa}\left(\nabla_{+}\Psi\nabla_{-}\Psi+\PsiR\right)+2b\nabla_{+}\Phi\nabla_{-}\Phi+Le^{2a\Phi}+\gamma\PhiR\right)
\left[\begin{array}{c}{\,\delta/\delta\xi^{+}\,}\\{\,\delta/\delta\xi^{-}\,}\\\end{array}\right]\Big(S_{\mathrm{rad}}+S_{\mathrm{n-m}}\Big)=\left[\begin{array}{c}{\,\delta/\delta\xi^{+}\,}\\{\,\delta/\delta\xi^{-}\,}\\\end{array}\right]\,S_{\mathrm{rad}}-\frac1{8\piG_{5}}{\bfG}_{\xi}^{T}(\Box){\bfF}^{T}(\Box)\,{\bfR}=0.
\psi_{l}=\prod_{\gamma<\delta}\left(\frac{\theta_{1}^{\ast}\left(z_{\gamma}-z_{\delta}\mid\tau\right)}{\theta_{1}\left(z_{\gamma}-z_{\delta}\mid\tau\right)}\right)^{\frac{1}{2\kappa}}e^{iXv}{\tilde{\psi}}_{l}^{0}
ds_{\mathrm{target}}^{2}=\frac{1}{2}d\varphi^{2}+\frac{1}{2}d\beta^{2}+\frac{1}{2}e^{2\varphi}\left(d\sigma_{1}-\chid\sigma_{2}\right)^{2}+\frac{1}{2}e^{\varphi}\left[e^{\sqrt{3}\beta}d\chi^{2}+e^{-\sqrt{3}\beta}d\sigma_{2}^{2}\right].
U(\prod_{i=1}^{g}\left(g_{a_{i}}^{~}g_{b_{i}}^{~}g_{a_{i}}^{-1}g_{b_{i}}^{-1}\right))=V(\prod_{i=1}^{g}\left(g_{a_{i}}^{~}g_{b_{i}}^{~}g_{a_{i}}^{-1}g_{b_{i}}^{-1}\right))=1~,
\begin{array}{rlcccc}{}&{}&{R_{1}}&{R_{2}}&{}&{SO(2)_{ST}\nonumber}\\{NS:}&{|s_{3}s_{4}\rangle,s_{3}=-s_{4}}&{+}&{i(2s_{4})}&{}&{{\bf0}}\\{R:}&{|s_{1}s_{2}\rangle,s_{1}=-s_{2}}&{+}&{i(2s_{2})}&{}&{\pm{\bf1/2}\nonumber}\\\end{array}
\tilde{m}_{2}={\frac{2\intd\tilde{p}(\Sigma(\tilde{p}))^{2}f(\tilde{p})-{\frac{\beta^{2}}{16\pi}}(\tilde{\mu}^{2}(\varphi_{0})-\tilde{m}_{0}^{2})(\intd\tilde{p}{\frac{\Sigma(\tilde{p})}{f(\tilde{p})}})^{2}}{\intd\tilde{p}(\Sigma(\tilde{p}))^{2}}}\;,
y^{2}=\prod_{a=1}^{\tilde{n}_{c}}(x-\phi_{a})^{2}\prod_{k=1}^{n_{c}-\tilde{n}_{c}}(x-\Lambda\omega^{2k}-\gamma_{k})^{2}+4\Lambda^{2n_{c}-n_{f}}\prod_{i=1}^{n_{f}}(x+m_{i}).
V(\beta,\phi)=\frac{1}{2}\mu^{2}\phi^{2}+N\mu^{D}\sum_{s=1}^{\infty}p(D,s)a^{\frac{D}{2}-s}(g\phi)^{2s}\sum_{n=-\infty}^{\infty}\frac{1}{(n+\frac{1}{2})^{2s-d}}.
S=\frac{1}{16\piG_{10}}\intd^{10}x\sqrt{-g}\lbraceR-\frac{1}{8}(\partial\Phi)^{2}-\frac{1}{12}e^{-\frac{\Phi}{2}}H^{2}-\frac{1}{4}e^{-\frac{\Phi}{4}}Tr(F^{2})\rbrace
\langle\prod_{j=1}^{k}{\calO}_{A_{j}}(z_{j})\rangle_{d}={}^{\sharp}\{f:\Sigma\stackrel{hol.}{\mapsto}M|\;\mathrm{deg}(f)=d,\;\;f(z_{j})\inPD(A_{j})\quad(j=1,\cdots,k)\}
\begin{array}{ll}{a(\partialc)=q^{2}(\partialc)a+q\mu(\partialc),}&{c(\partialc)=q^{2}(\partialc)c,}\\{b(\partialc)=(\partialc)b+\mu(\partiala)d,}&{d(\partialc)=(\partialc)d,}\\\end{array}
C_{h}^{ag}C_{g}^{br}C_{da}^{e}C_{eb}^{f}(2\delta_{rf}\delta^{hd}-\delta_{r}^{h}\delta_{f}^{d}-\delta_{r}^{d}\delta_{f}^{h}-C_{f}^{ch}C_{rc}^{d})=d(d-1)(d-4)^{2}.
I=-{\frac{\nu}{4}}\Gamma(\nu)\Gamma(-\nu)\int{\frac{d^{d}k}{(2\pi)^{d}}}\left(\sum_{i_{1}<\ldots<i_{p}}a_{i_{1}\ldotsi_{p}}({\bfk})a_{i_{1}\ldotsi_{p}}(-{\bfk})+\sum_{i_{2}<\ldots<i_{p}}{\frac{\beta\bar{\beta}}{|{\bfk}|^{2}}}a_{0i_{2}\ldotsi_{p}}({\bfk})a_{0i_{2}\ldotsi_{p}}(-{\bfk})\right).
\left[{\frac{\partial}{\partialt}}+\intdx\;\left({\frac{\deltaH}{\delta\Pi(x)}}\;{\frac{\delta}{\delta\Phi(x)}}-{\frac{\deltaH}{\delta\Phi(x)}}\;{\frac{\delta}{\delta\Pi(x)}}\right)\right]W[\Phi,\Pi;t]=0\;,
ds_{5}^{2}=-f^{-2/3}\left(1-{\frac{r_{0}^{2}}{r^{2}}}\right)dt^{2}+f^{1/3}\left[\left(1-{\frac{r_{0}^{2}}{r^{2}}}\right)^{-1}dr^{2}+r^{2}d\Omega_{3}^{2}\right]
K_{0}=e^{-\left[\left(P_{\mu}^{\nu}+\xi^{-1}Q_{\mu}^{\nu}\right)k^{2}+\delta_{\mu}^{\nu}X_{0}\right]t}=\left(Pe^{-k^{2}t}+Qe^{-\xi^{-1}k^{2}t}\right)e^{-X_{0}t}
\bar{f}[\bar{t}_{0},\bar{t}_{x},\cdots]\=\\mathrm{extremum\w.r.t.\}t_{0}\mathrm{\of\}\Big(f[t_{0},\cdots]\,+\,t_{0}{\bar{t}_{0}}\,+\,(t_{0})^{2}\bar{t}_{x}\Big)
{\calQ}_{\alpha}^{2}=-\frac{i}{m}p_{\alpha}{}^{\beta}{\calQ}_{\beta}^{1}\qquad{\calK}=-\frac{1}{2m}{\calQ}^{1\,\alpha}{\calQ}_{\alpha}^{1}\qquad{\calZ}=-\frac{p^{2}}{m}=m
D_{i\alpha}D_{j\beta}D_{k\gamma}\tilde{\mathrm{A}}^{\beta\gamma}=D_{j\beta}D_{k\gamma}D_{i\alpha}\tilde{\mathrm{A}}^{\beta\gamma}=-\textstyle{\frac{1}{4}}D_{j\alpha}D_{k\beta}D_{i\gamma}\tilde{\mathrm{A}}^{\beta\gamma}
\left(\begin{array}{c}{\phi}\\{\chi}\\\end{array}\right)=\frac{\sqrt{12}(P(z)Q^{\prime}(z)-Q(z)P^{\prime}(z))}{|q|(|P(z)|^{2}+|Q(z)|^{2})^{3/2}}\left(\begin{array}{c}{P(z)}\\{Q(z)}\\\end{array}\right)\,,
\begin{array}{lcllcl}{P^{i}}&{=}&{p^{i}\;,}&{M^{ij}}&{=}&{x^{i}p^{j}-x^{j}p^{i}\;,}\\{P^{0}}&{=}&{H_{t}\;,}&{M^{i0}}&{=}&{x^{i}H_{t}\;.}\\\end{array}
\nabla_{i}\deltaA^{i}+2i{\frac{e^{3}}{\kappa}}(2|\phi|^{2}-v^{2})(\phi^{*}\delta\phi-\phi\delta\phi^{*})=-{\frac{e}{\kappa}}(\phi^{*}\partial^{0}\delta\phi-\partial^{0}\delta\phi^{*}\phi).
\deltaS_{\psi\psi\lambda\lambda}=\intd^{2}\sigma\left[\frac{1}{2}v^{\rho}\hat{D}^{(+)}{}_{\rho}F_{\mu\nuab}-2iF_{\mu\nucb}\Sigma^{c}{}_{a}\right]\psi^{\mu}\psi^{\nu}\lambda^{a}\lambda^{b}\.
F_{\mu\nu}^{\Lambda}=\tilde{F}_{ab}^{\Lambda}V_{[\mu}^{a}V_{\nu]}^{b}+\mathrm{i}\,f_{i}^{\Lambda}\,\bar{\lambda}^{iA}\gamma_{a}\psi_{[\mu}^{B}V_{\nu]}^{a}\,\epsilon_{AB}+\mathrm{i}\,{\bar{f}}_{{i}^{\star}}^{\Lambda}\,\bar{\lambda}_{A}^{{i}^{\star}}\gamma_{a}\psi_{B[\mu}V_{\nu]}^{a}\,\epsilon^{AB}
f=1+\frac{\tilde{R}^{4}}{r^{4}},\\f_{0}=1+\frac{R^{4}}{r^{4}},\\h=1-\frac{2m}{r^{4}},\\\tilde{h}=1+\frac{l^{2}}{r^{2}}-\frac{2m}{r^{4}}.
\Gamma_{\mu_{1},\cdots,\mu_{N}}(q_{1},\cdots,q_{N})=\int\frac{d^{3}k}{(2\pi)^{3}}\,\tilde{\Gamma}_{\mu_{1},\cdots,\mu_{N}}(k;q_{1},\cdots,q_{N})=\frac{1}{\beta}\sum_{n}\int\frac{d^{2}k}{(2\pi)^{2}}\,\tilde{\Gamma}_{\mu_{1},\cdots,\mu_{N}}(k;q_{1},\cdots,q_{N})
\intdy_{2}d^{2}z_{3}d^{2}z_{4}{\frac{K(1,\bar{2};2,\bar{3};3,\bar{4};4,\bar{1})k_{2}.k_{4}(1-y_{2}^{2})}{(iy_{2}+z_{4})(i+iy_{2})(iy_{2}-\bar{z}_{3})^{2}(z_{3}-\bar{z}_{4})^{2}(z_{4}+i)}}e^{E}.
\begin{array}{lll}{\alpha_{ik}\alpha_{jk}}&{=}&{0,\;\;i\neqj\pmg}\\{\alpha_{ik}\alpha_{i+g,k}}&{=}&{-\alpha_{i+g,k}\alpha_{ik}=\beta_{k},\;\;1\leqi\leqg}\\{\alpha_{ik}\alpha_{jl}}&{=}&{-\alpha_{jl}\alpha_{ik},\;k\neql.}\\\end{array}
\Phi_{(r,r)}(u_{i\alpha},u_{i\beta},\ldots)=\left|\begin{array}{cccc}{u_{i\alpha}}&{u_{i\beta}}&{\cdots}&{u_{i\gamma}}\\{u_{k\alpha}}&{u_{k\beta}}&{\cdots}&{u_{k\gamma}}\\{\cdots}&{\cdots}&{\cdots}&{\cdots}\\{u_{j\alpha}}&{u_{j\beta}}&{\cdots}&{u_{j\gamma}}\\\end{array}\right|=0,
\bar{\Xi}_{\lambda_{2}}^{\lambda_{1}}\otimesl_{\dot{J}}^{\dot{I}}\otimes1\left(\bar{\phi}^{\rho_{1}}\otimess^{\dot{K}}\otimes\phi^{\rho_{2}}\right)=\delta_{\lambda_{2}}^{\rho_{1}}\sum_{\dot{K}_{1}\dot{K}_{2}=\dot{K}}\delta_{\dot{J}}^{\dot{K}_{1}}\bar{\phi}^{\lambda_{1}}\otimess^{\dot{I}\dot{K}_{2}}\otimes\phi^{\rho_{2}}.
B_{\beta\alpha}^{\mu\nu(n)}=\Sigma_{m=0}^{n}\{\tilde{\Theta}_{\beta}^{\nu(n-m)},\tilde{\Theta}_{\alpha}^{\mu(m)}\}_{(q,p)}+\Sigma_{m=0}^{n}\{\tilde{\Theta}_{\beta}^{\nu(n-m)},\tilde{\Theta}_{\alpha}^{\mu(m+2)}\}_{(\phi)}~~;~~n\ge2
\rho=\frac{C}{|x_{\perp}|^{t}},~~~~~~~~~~\nu^{*}={\frac{1}{4}}(2\pi)^{-{\frac{1}{2}}}e^{2i\omega\cdotx}e^{-2|\omega||x_{\perp}|}|\omega|^{-\frac{3}{2}}|x_{\perp}|^{-\frac{3}{2}},
[\bar{C}_{\alpha},{\calT}_{\beta}]=U_{\alpha\beta}^{\gamma}\bar{C}_{\gamma}(-1)^{\varepsilon_{\beta}},\quad[{\calT}_{\beta}^{\dagger},\bar{C}_{\alpha}^{\dagger}]=\bar{C}_{\gamma}^{\dagger}U_{\alpha\beta}^{\dagger\gamma}(-1)^{\varepsilon_{\beta}},
S_{0}={\frac{1}{4\pi\alpha^{\prime}}}\intd^{2}z\left[g_{\mu\nu}\partial_{a}X^{\mu}\partial^{a}X^{\nu}+\epsilon^{ab}B_{\mu\nu}\partial_{a}X^{\mu}\partial_{b}X^{\nu}+{\alpha^{\prime}}\partial_{a}\rho\nabla_{\mu}\Phi\partial^{a}X^{\mu}\right].
\sum\frac{1}{\omega}\equiv\sum_{-\infty}^{+\infty}\,\!^{\prime}\frac{1}{\omega_{n}}\sim\frac{L}{2\pi}2\int_{0}^{\infty}\frac{1}{\omega(k)}\left[1+\frac{1}{L}\delta^{\prime}(k)\right]dk
\breve{R}^{\Phi_{1}\Phi_{2}}(x)=\sum_{\mu=0}^{-\infty}\prod_{\nu=-1}^{\mu}\frac{1-xq^{2(\Phi_{1}+\Phi_{2}+\nu)}}{x-q^{2(\Phi_{1}+\Phi_{2}+\nu)}}\,{\bfP}_{\mu}^{\Phi_{1}\Phi_{2}}
ds_{10}^{2}=\left(\frac{2\pi^{3}}{15}g_{s}^{\prime}Q\right)^{-1/2}\frac{r^{2}}{l_{s}^{2}}(-d\tau^{2}+dx_{7}^{2}+dx_{8}^{2}+dx_{9}^{2})+\left(\frac{2\pi^{3}}{15}g_{s}^{\prime}Q\right)^{1/2}l_{s}^{2}(\frac{dr^{2}}{r^{2}}+d\Omega_{(5)}^{2})
\omega_{L}(\xi,gg^{\prime})=\sum_{l+m\leL}a_{l,m}^{(L)}\,\bigl(\partial^{l+2m}\omega_{L}(\xi,g)\bigr)\,K_{[l,m]}(\xi)\,\bigl(\partial^{l+2m}\omega_{L}(\xi,g^{\prime})\bigr)
\left[A_{\{(n_{i})\}}^{\{r,r_{i}\}},A_{\{(n_{j})\}}^{\{s,s_{j}\}}\right]=\frac{1}{(2\pii)^{2}}\oint_{0}dz\oint_{z}d\xiU_{\{(n_{i})\}}^{\{r,r_{i}\}}(z)U_{\{(n_{j})\}}^{\{s,s_{j}\}}(\xi)
\bar{\varphi}_{\alpha}(\lambda)\sim{\frac{1}{(4\pi)^{d/2}}}\sum_{n=0}^{\infty}\left[\bar{a}_{n}~{\frac{\lambda^{d/2-n+\alpha-1}}{\Gamma\left(\fracd2-n+\alpha\right)}}+\bar{b}_{n}~{\frac{\lambda^{(d-1)/2-n+\alpha-1}}{\Gamma\left({\frac{d-1}{2}}-n+\alpha\right)}}\right],
\begin{array}{c}{\displaystyle\frac{m+\delta+M_{SC}}{2}\in\mathbbZ\qquad\textrm{for}\qquad\textrm{sineGordon}}\\{\displaystyle\frac{\delta+M_{SC}}{2}\in\mathbbZ\qquad\textrm{for}\qquad\textrm{massiveThirring}.}\\\end{array}
S[\varphi,J]=-\frac{1}{2}\intd^{4}xd^{4}y\varphi(x)G^{-1}(x,y)\varphi(y)-\frac{\lambda}{4!}\intd^{4}x\varphi(x)^{4}+\frac{1}{2}\intd^{4}xJ(x)\varphi(x)^{2},
S_{\mathrm{gh}}=\intd^{3}x\,\left[\partial_{\mu}\bar{c}^{a}D_{\mu}c^{a}+\alphag^{2}\eta\left(\Phi^{3}\bar{c}^{a}c^{a}-\Phi^{a}\bar{c}^{a}c^{3}\right)\right].
\leq\;e^{4\pi\sum_{b=1}^{N}2n_{b}\lambda^{(b)}}\left\langle\prod_{b=1}^{N}\left[\prod_{i_{b}}^{q_{b}}:e^{i2\sqrt{\pi}\varphi^{(b)}(x_{i_{b}}^{(b)})}:_{C^{2}}\prod_{j_{b}}^{2n_{b}-q_{b}}:e^{-i2\sqrt{\pi}\varphi^{(b)}(y_{j_{b}}^{(b)})}:_{C^{2}}\right]\right\rangle_{C^{2}}\;,
\left|\frac{\partial(x_{1}^{\prime},\cdots,x_{D}^{\prime},p_{1}^{\prime},\cdots,p_{D}^{\prime})}{\partial(x_{1},\cdots,x_{D},p_{1},\cdots,p_{D})}\right|=1+\left(\frac{\partial\deltax_{i}}{\partialx_{i}}+\frac{\partial\deltap_{i}}{\partialp_{i}}\right)+\cdots\;.
<\phi|\hat{H}_{0}|\psi>=\int_{(\infty)}d^{3}\vec{x}\,\frac{1}{2}\left[-\hbar^{2}\left(\frac{\delta}{\delta\phi(\vec{x}\,)}\right)^{2}+\left(\vec{\nabla}\phi(\vec{x}\,)\right)^{2}+m^{2}\phi^{2}(\vec{x}\,)\right]\,\Psi[\phi]\.
W(a,<\phi>)=-{\frac{1}{8}}{\frac{2\beta(g)}{g}}(1+{\frac{g}{2\beta(g)}}{\frac{f_{\phi}}{f}})<0|G_{\mu\nu}G^{\mu\nu}|0>+{\frac{c<F(\phi)>}{4}}N_{c}\,cos({\frac{a}{f_{a}N_{c}}}),
(\partial^{\ast}f)_{i_{1}\cdotsi_{p+1}}({\bfm})=\sum_{\epsilon=0,1}\sum_{k=1}^{p+1}(-1)^{\epsilon+k}f_{i_{1}\cdots\widehat{i_{k}}\cdotsi_{p+1}}({\bfm}+\epsilone_{i_{k}}).
\int_{0}^{\infty}ds\,s^{1/2}e^{-as}=\int_{0}^{\infty}ds\,s\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}dp\,e^{-(p^{2}+a)s}=\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}dp\,(p^{2}+a)^{-2}
\oint_{\infty}\frac{d\zeta}{2\pii}k\zeta^{k-1}\oint_{\infty}\frac{dw}{2\pii}\frac{w^{n-1}}{w\bar{w}(\zeta)-1}=\oint_{\infty}\frac{d\zeta}{2\pii}\frac{k\zeta^{k-1}}{\left(\bar{w}(\zeta)\right)^{n}}
\alpha_{0}(\sigma,\bar{\sigma})=\frac{1}{2}\left(\mathrm{artanh}\frac{\sigma}{4}+\bar{\sigma}\right)\,\,\mathrm{and}\,\,\,\bar{\alpha}_{0}(\sigma,\bar{\sigma})=\frac{1}{2}\left(\mathrm{artanh}\frac{\sigma}{4}-\bar{\sigma}\right)
\begin{array}{c}{{\bf1}^{2}={\bf1},\quad{\bfi}^{2}=\epsilon_{1}{\bf1},\quad{\bfj}^{2}=\epsilon_{2}{\bf1},\quad{\bfk}^{2}=-\epsilon_{1}\epsilon_{2}{\bf1},}\\{{\bfi}{\bfj}=-{\bfj}{\bfi}={\bfk},\quad{\bfj}{\bfk}=-{\bfk}{\bfj}=-\epsilon_{2}{\bfi},\quad{\bfk}{\bfi}=-{\bfi}{\bfk}=-\epsilon_{1}{\bfj},}\\\end{array}
(d_{1}(\sum_{j=0}^{k}f_{j}s^{j}t^{k-j}),d_{2}(\sum_{j=0}^{k}f_{j}s^{j}t^{k-j}),\ldots,d_{N}(\sum_{j=0}^{k}f_{j}s^{j}t^{k-j}))\,\simeq(d_{1},d_{2},\ldots,d_{N})
*G_{\it3}=\frac{\sqrt{-g}}{3!}\epsilon_{mnpqst}g^{qq^{\prime}}g^{ss^{\prime}}g^{tt^{\prime}}G_{q^{\prime}s^{\prime}t^{\prime}}dy^{m}\wedgedy^{n}\wedgedy^{p}\wedgegZ(r)\chi_{\it4}\
\frac{dX^{1}}{ds}=\frac{c_{1}}{e^{\lambdaT}}\,\\\\frac{dX^{2}}{ds}=c_{2}\,\\,\frac{dX_{3}}{ds}=c_{3}\,\\\\\frac{dT}{ds}=\sqrt{\epsilon+\frac{c_{1}^{2}}{e^{\lambdaT}}+c_{2}^{2}+c_{3}^{2}}
g(u)=\left(\begin{array}{c|c}{1}&{\displaystyle\frac{\vec{\sigma}^{\mathrm{T}}}{c}\gamma_{0}^{-2}}\\\hline{\displaystyle\frac{\vec{\sigma}}{c}\gamma_{0}^{-2}}&{\displaystyle-I+\frac{\vec{\sigma}\otimes\vec{\sigma}^{\mathrm{T}}}{c^{2}}\gamma_{0}^{-4}}\\\end{array}\right),
L_{\eta}=\int\lambda\left\{\etaA_{a}^{\alpha}\sigma_{3}^{\alpha\beta}\kappa_{ab}\dot{A}_{b}^{\beta}+\etaA_{a}^{\alpha}\epsilon^{\alpha\beta}\kappa_{ab}\dot{A}_{b}^{\beta}-\lambdaA_{a}^{\alpha}A_{a}^{\alpha}\right\}
I=\intd^{4}x[-\frac{1}{4}a_{mn}a^{mn}-\frac{1}{12}H_{mnp}H^{mnp}-\frac{1}{4}\mu\epsilon^{mnpq}B_{mn}a_{pq}+\frac{1}{2}\epsilon^{mnpq}a_{mn}\partial_{p}a_{q}].
\theta=\left({\frac{mv}{\hbar}}\mp{\frac{\lambda}{2}}\sqrt{\xi}\rho_{0}^{2}\right)\int_{-\infty}^{x-vt}\!\!\!\!\!\!d\,y\left(1-{\frac{\rho_{0}^{2}}{\rho^{2}}}\right)-{\frac{\lambda}{2}}\int_{-\infty}^{x-vt}\!\!\!\!\!\!d\,y(\rho^{2}-\rho_{0}^{2})-\frac{\lambda}{2}\rho_{0}^{2}\left(1\mp\sqrt{\xi}\right)x+\theta_{0}(t),
\begin{array}{lcr}{|X_{0}|^{2}+|X_{1}|^{2}-(|Y_{0}|^{2}+|Y_{1}|^{2})=R_{1}}&{(a)}\\{X_{0}\overline{{Y}}_{0}+X_{1}\overline{{Y}}_{1}=0}&{(b)}\\{\overline{{X}}_{0}{Y}_{0}+\overline{{X}}_{1}{Y}_{1}=0.}&{(c)}\\\end{array}
h_{a}^{\prime\prime}-\frac{h_{a}^{\prime}}{r}+\left[\Omega_{a}^{2}-\kappa^{2}(F^{\mathrm{ANO}}(r))^{2}\right]h_{a}=F^{\mathrm{ANO}}(r)\chi^{\mathrm{ANO}}(r)f_{u}(r),
G(x,y;B^{(b)})=\frac{1}{2\pi}\frac{1}{(x\!-\!y)^{2}}\!\left(\begin{array}{cc}{0}&{e^{-[\chi^{(b)}(x)-\chi^{(b)}(y)]}\;\overline{{(\tilde{x}\!-\!\tilde{y})}}}\\{e^{+[\chi^{(b)}(x)-\chi^{(b)}(y)]}\;(\tilde{x}\!-\!\tilde{y})}&{0}\\\end{array}\right)\;,
{\calF}_{0}=-\frac{\mathrm{ln}Z_{0}}{\betaV}=\frac{1}{\betaV}\sum_{n}\sum_{\bfp}\mathrm{ln}\left(\frac{\betaM^{2}}{\rho}\right)+2T\int\frac{d^{2}p}{(2\pi)^{2}}\left[\beta\omega+ln(1-e^{-\beta\omega})\right]\;,
lA_{1}+nB_{1}=lU_{l}^{n}Str\left(\ldots\phi^{i_{2k-1}^{\prime}}\ldots\underbrace{\partial_{a_{l}}\phi^{j_{l}}\phi^{i_{2k}^{\prime}}}\ldots\right)+nU_{l}^{n}Str\left(\ldots\phi^{i_{2k-1}^{\prime}}\ldots\underbrace{\phi^{i_{n}}\phi^{i_{2k}^{\prime}}}\ldots\partial_{a_{l}}\phi^{j_{l}}\ldots\right)
<g_{i}(z_{1})g^{j\dagger}(z_{2})g^{k\dagger}(z_{3}^{*})g_{l}(z_{4}^{*})>=[(z_{1}-z_{4}^{*})(z_{2}-z_{3}^{*})]^{-1+1/N}\sum_{p=0}^{1}U_{p}[\delta_{i}^{j}\delta_{l}^{k}{\calF}_{1}^{(p)}(x)+\delta_{l}^{j}\delta_{i}^{k}{\calF}_{2}^{(p)}(x)]
B=\left(\begin{matrix}{0}&{0}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{-\theta_{2}}&{0}&{0}&{0}&{0}\\{0}&{\theta_{2}}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{-\theta_{3}}&{0}&{0}\\{0}&{0}&{0}&{\theta_{3}}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}&{-\theta_{4}}\\{0}&{0}&{0}&{0}&{0}&{\theta_{4}}&{0}\\\end{matrix}\right)~.
R_{bc}^{\bar{b}^{\prime}c^{\prime}}(\beta_{2})R_{ac^{\prime}}^{\bar{a}^{\prime}d}(\beta_{1})S_{\bar{b}^{\prime}\bar{a}^{\prime}}^{\bar{b}\bar{a}}(\beta_{1}-\beta_{2})=S_{ab}^{a^{\prime}b^{\prime}}(\beta_{1}-\beta_{2})R_{a^{\prime}c}^{\bar{a}c^{\prime}}(\beta_{1})R_{b^{\prime}c^{\prime}}^{\bar{b}d}(\beta_{2}).
\delta_{H}^{2}\simeq\frac{512\pi}{75}\,\frac{A}{q^{3}{M_{P}}^{6}}\,\frac{\phi^{2+q}}{\phi_{0}^{q}}\left[1+\xi(3q-\frac{3\kappa^{2}\phi^{2}}{2}+\frac{\kappa^{2}\phi^{2}}{q})\right]^{2}
2{\calD}_{\bar{z}}A^{i}=\Omega_{+}K^{-1ij^{*}}\frac{\partial{\calW}^{*}}{\partialA^{*j}},\quad{\calD}_{\bar{z}}=\frac{1}{2}({\calD}_{1}+i{\calD}_{2}),\quad{\calD}_{\mu}A^{i}=\left({\frac{\partial}{\partialx^{\mu}}}+i{\frac{e_{i}}{2}}v_{\mu}\right)A^{i}.
\alpha=\omega\,\displaystyle\frac{p+1}{p}+\chi_{\infty}\left(\left\lfloor\displaystyle\frac{1}{2}+\displaystyle\frac{S}{p+1}+\displaystyle\frac{\omega}{\pi}\right\rfloor-\left\lfloor\displaystyle\frac{1}{2}+\displaystyle\frac{S}{p+1}-\displaystyle\frac{\omega}{\pi}\right\rfloor\right)
\Phi_{Gn}^{\mathrm{in}}=\frac{e^{-i\omega_{n}^{\mathrm{in}}t}}{\sqrt{2\pi\omega_{n}^{\mathrm{in}}}}e^{ik_{n}^{\mathrm{in}}z}e^{im\phi}J_{|m|}(k_{n}^{\mathrm{in}}r)\,,
\begin{array}{ll}{\Pi_{J_{0}}^{e}\equiv\Omega_{\beta_{0}}^{e}}&{\Pi_{\beta_{l}}^{r^{k}}\equiv\Omega_{\beta_{l}}^{r^{k}}\oplus\Omega_{\beta_{-l}}^{r^{-k}}}\\{\Pi_{J_{1}}^{e}\equiv\Omega_{\beta_{0}}^{e}}&{\Pi_{\gamma_{0}}^{s}\equiv\Omega^{s}}\\{\Pi_{\alpha_{k}}^{e}\equiv\Omega_{\beta_{k}}^{e}\oplus\Omega_{\beta_{-k}}^{e}}&{\Pi_{\gamma_{1}}^{s}\equiv\Omega^{s}.}\\\end{array}
V=\frac{1}{\muc^{2}}C_{1}\frac{\alpha}{q}\hat{h}^{(0)}+\frac{C_{2}-C_{1}}{\muc^{2}}\frac{\alpha^{2}}{q^{2}}+\frac{\hbar^{2}}{2}\frac{\alpha}{\mu^{2}c^{2}}C_{1}\left(\frac{{\bfq}}{q^{3}}\frac{\partial}{\partial{\bfq}}+4\pi\lambda\delta({\bfq})\right)\,,
{\calM}(\nu(p_{1})\to\nu(p_{2})\gamma(q))=-{\frac{G_{F}}{\sqrt{2}e}}Z\epsilon^{\alpha}\bar{u}(p_{2})\gamma^{\beta}(1-\gamma_{5})u(p_{1})\left(g_{V}\Pi_{\alpha\beta}(-q)-g_{A}\Pi_{\alpha\beta}^{5}(-q)\right),
\left[a_{m}^{+(i)},\left.a_{n}^{+(j)}\right.^{\dagger}\right]=\delta_{mn}\,\delta^{ij}\qquad\mathrm{and}\qquad\left[a_{m}^{-(i)},\left.a_{n}^{-(j)}\right.^{\dagger}\right]=\delta_{mn}\,\delta^{ij}
\chi_{1,kr}^{(p,kp+1)}(q)=\sum_{n_{1}\geq\cdots\geqn_{k}\geq0}\frac{q^{n_{1}^{2}+\cdots+n_{k-1}^{2}+2n_{k}^{2}+(r-1)(n_{1}+\cdots+n_{k-1}+2n_{k})}}{(q)_{n_{1}-n_{2}}\cdots(q)_{n_{k-1}-n_{k}}(q)_{2n_{k}+r-1}}\overline{{F}}_{p-1;r}^{(2n_{k}+r-1)}(q).