harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\kappa_{\perpmax}^{2}=(\sigma^{2}+3m_{G}^{2})x(1-x)-m_{G}^{2}~,~~~\frac{m_{G}^{2}}{(\sigma^{2}+3m_{G}^{2})}\leqx\leq1-\frac{m_{G}^{2}}{\sigma^{2}+3m_{G}^{2}}.
\left[N\frac{\partial}{\partialN}+2\right]F(N,0,1,g,0,0,\ldots)=\sum_{(n_{1},\ldots,n_{k})}\beta_{n_{1}\cdotsn_{k}}(g)\frac{\partialF}{\partialg_{n_{1}\cdotsn_{k}}}(N,0,1,g,0,0,\ldots).
\Omega_{\alpha\beta}^{(1)}=\left(\begin{array}{cccc}{0}&{-m\delta_{ij}}&{e\delta_{ij}\delta(\vec{y}-\vec{q})}&{0}\\{m\delta_{ij}}&{0}&{0}&{0}\\{-e\delta_{ij}\delta(\vec{y}-\vec{q})}&{0}&{2\theta\epsilon_{ij}\delta(\vec{x}-\vec{y})}&{2\theta\epsilon_{ij}\partial^{i}\delta(\vec{x}-\vec{y})}\\{0}&{0}&{-2\theta\epsilon_{ij}\partial^{j}\delta(\vec{x}-\vec{y})}&{0}\\\end{array}\right)
ds_{B}^{2}=a\prod_{i=1}^{4}(z-z_{i})^{-\frac{1}{2}}\prod_{i=1}^{4}(\bar{z}-\bar{z_{i}})^{-\frac{1}{2}}dzd\bar{z}+a^{\prime}\prod_{i=1}^{4}(z^{\prime}-z_{i}^{\prime})^{-\frac{1}{2}}\prod_{i=1}^{4}(\bar{z^{\prime}}-\bar{z_{i}^{\prime}})^{-\frac{1}{2}}dz^{\prime}d\bar{z^{\prime}}.
r^{2}\left\langlef^{\prime}(r)f^{\prime\prime}(r)\right\rangle=\frac{r^{2}}{2}\left\langle{f^{\prime}}^{2}(r)\right\rangle^{\prime}=\frac{1}{4}\left\langlef^{4}(r)\right\rangle^{\prime}-\frac{1}{2}\left\langlef^{2}(r)\right\rangle^{\prime}\left(1+g^{2}(r)\right).
\left(R_{\lambda}^{\mu}-\frac{1}{2}\delta_{\lambda}^{\mu}R\right)D_{\mu}^{l}\frac{\partialD_{l}^{\lambda}}{\partialB_{k}^{a}}=-8\piGU_{a}^{k}(\kappa{\bfB}_{f})=-8\piGT_{\lambda\nu}D^{\nul}\frac{\partialD_{l}^{\lambda}}{\partialB_{k}^{a}},
\psi_{\alpha}(x,t,P)=\,{\frac{\mathrm{Det}\left(\begin{matrix}{h_{\alpha}(P)}&{g_{j}(P)}\\{h_{\alpha}(T_{i})}&{\Theta_{ij}(x,t)}\\\end{matrix}\right)}{\mathrm{Det}\left(\begin{matrix}{\Theta_{ij}(x,t)}\\\end{matrix}\right)}}\,e^{kx+k^{2}t}
\frac{X_{R}^{\mu}\left(\stackrel{i+1}{\tau_{R}+\triangle\tau_{R}}\right)-X_{R}^{\mu}(\stackrel{i}{\tau_{R}})}{\triangle\tau_{R}}=\Pi_{R}^{\mu}\cdot2\pi\alpha^{\prime}=\frac{2p^{\mu}2\pi\alpha^{\prime}}{\triangle\tau_{R}}
\langle\prod_{j=1}^{m}{\calO}_{A_{j}}(z_{j})\rangle=\langle{\calO}_{A_{1}}(z_{1}){\calO}_{A_{2}}(z_{2}){\calO}_{A_{a}}(z_{s1})\rangle\eta^{ab}\langle{\calO}_{A_{b}}(z_{s2})\prod_{j=3}^{m}{\calO}_{A_{j}}(z_{j})\rangle
ds^{2}\approx-d\tilde{U}d\tilde{V}+dX_{\mu}dX^{\mu}+\left[\frac{R^{\prime\prime}}{R}X_{i}^{2}+\frac{(W^{2}R^{3})^{\prime\prime}}{W^{2}R^{3}}X_{4}^{2}\right]d\tilde{U}^{2},
\frac{\alpha^{\prime}}{2}M^{2}=p_{R}^{2}+P_{R}^{2}+2N_{R},\;p_{L}^{2}-p_{R}^{2}=2(N_{R}+1-N_{L})+\frac{1}{2}P_{R}^{2}-\frac{1}{2}P_{L}^{2}=2n^{T}m+b^{T}Cb,
S={\textstyle{\frac{g^{2}}{2}}}\intdt\intd^{3}{x}\intd^{3}{y}\left\{\,-\dot{X}^{ax}(\gamma)\frac{1}{4\pi\midx-y\mid}\dot{X}^{ay}(\gamma)-X^{ax}(\gamma)\,\delta(x-y)\,X^{ay}(\gamma)\,\right\}
{\alpha}=8\pi^{2}\frac{M_{1}-M_{2}}{N_{2}M_{1}-N_{1}M_{2}}\;,\;\;{D}=-\frac{N_{2}}{32\pi^{2}e}\left(1-\frac{N_{1}M_{2}}{N_{2}M_{1}}\right)\left(\frac{-M_{2}}{M_{1}}\right)^{\frac{N_{1}M_{2}}{N_{2}M_{1}-N_{1}M_{2}}}
\begin{array}{lcr}{(|Z_{0}|^{2}+|Z_{1}|^{2})-(|W_{0}|^{2}+|W_{1}|^{2})=R_{1}}\\{Z_{0}W_{0}+Z_{1}W_{1}=\overline{{Z}}_{0}\overline{{W}}_{0}+\overline{{Z}}_{1}\overline{{W}}_{1}=0,}\\\end{array}
ds^{2}=(1-{\frac{qcos\theta}{r}})^{\frac{2}{1+\alpha^{2}}}\lbracedr^{2}+r^{2}d\theta^{2}+r^{2}sin^{2}\thetad\varphi^{2}\rbrace-{\frac{dt^{2}}{(1-{\frac{qcos\theta}{r}})^{\frac{2}{1+\alpha^{2}}}}}\,.
{\calS}_{H}=\intd^{4}x\intd\phi~r_{c}~e^{2\sigma(\phi)}~[\eta^{\mu\alpha}\eta^{\nu\beta}\eta^{\lambda\gamma}H_{\mu\nu\lambda}H_{\alpha\beta\gamma}-\frac{3}{r_{c}^{2}}e^{-2\sigma(\phi)}\eta^{\mu\alpha}\eta^{\nu\beta}B_{\mu\nu}\partial_{\phi}^{2}~B_{\alpha\beta}]
{\calJ}_{i}^{\mu}=\frac{-1}{1+4{\bf\psi}^{2}}\left(\left(\delta^{ij}+4\psi^{i}\psi^{j}\right)\varepsilon^{\mu\nu}\partial_{\nu}\psi^{j}+2\varepsilon^{ijk}\psi^{j}\partial^{\mu}\psi^{k}\right).
S=\frac{1}{{g_{o}^{\prime}}^{2}}\intd^{26}x\left[-\frac{1}{2}\partial_{\mu}T(x)\partial^{\mu}T(x)+\frac{1}{2\alpha^{\prime}}T(x)^{2}+\frac{1}{3}\sqrt{\frac{2}{\alpha^{\prime}}}T(x)^{3}\right].
\mathcal{D}_{g}\left(\begin{array}{ccc}{p_{1}}&{\ldots}&{p_{n}}\\{r_{1}}&{\ldots}&{r_{n}}\\\end{array}\right)=\sum_{q}S_{0q}^{2-2g}\prod_{i=1}^{n}\frac{\Lambda_{qp_{i}}(r_{i})}{S_{0q}}
{~}^{*}F^{\mu\nu}=\left(\begin{matrix}{0}&{B_{x}}&{B_{y}}&{B_{z}}\\{-B_{x}}&{0}&{-E_{z}}&{E_{y}}\\{-B_{y}}&{E_{z}}&{0}&{-E_{x}}\\{-B_{z}}&{-E_{y}}&{E_{x}}&{0}\\\end{matrix}\right)
\sum_{n=-\infty}^{\infty}{\frac{(-1)^{n}n^{2}}{[p^{2}+n^{2}]^{2}}}={\tilde{f}}(p^{2})+p^{2}{\frac{d{\tilde{f}}(p^{2})}{dp^{2}}}={\frac{d}{dp^{2}}}[p^{2}{\tilde{f}}(p^{2})]
\begin{array}{rcl}{Y}&{=}&{\frac{1}{2}z^{-2}(x+y)=\frac{1}{2}(z_{1}z_{2})^{-2}(z_{1}^{2n}+z_{2}^{2n})}\\{X}&{=}&{\frac{1}{2}z^{-1}(x-y)=\frac{1}{2}(z_{1}z_{2})^{-1}(z_{1}^{2n}-z_{2}^{2n})}\\{Z}&{=}&{z^{2}=z_{1}^{2}z_{2}^{2}}\\\end{array}
\langleT(z)\Phi_{1}(z_{1})\cdots\Phi_{N}(z_{N})\rangle=\sum_{j=1}^{N}\left(\frac{h_{i}}{(z-z_{j})^{2}}+\frac{1}{z-z_{j}}\frac{\partial}{\partialz_{j}}\right)\langle\Phi_{1}(z_{1})\cdots\Phi_{N}(z_{N})\rangle~.
e^{-\Gamma(F,a)}=e^{iS_{CS}}\sum_{k=0}^{\infty}\Gamma_{k}^{(1)}(F_{\mu\nu})e^{\pi(2k-\Phi(F))a}+\Gamma_{k}^{(2)}(F_{\mu\nu})e^{-\pi(2k-\Phi(F))},
|M_{[\alpha,\psi_{\alpha}]}^{i}|\leqA_{[\alpha,\psi_{\alpha}][\alpha,-\psi_{\alpha}]}^{i}\;\;\;\;\;\mathrm{and}\;\;\;\;\;M_{[\alpha,\psi_{\alpha}]}^{i}=A_{[\alpha,\psi_{\alpha}][\alpha,-\psi_{\alpha}]}^{i}\;\;\;\;\mathrm{mod}\;\;2\;\;\;\;.
G_{(\sigma)}(g_{s}(z),z^{\prime})=(-1)^{2l_{2}-1}(c_{r}z+d_{r})^{-1}\left(G_{(\sigma)}(z,z^{\prime})+\sum_{F_{r}=\mu_{r},\nu_{r}}\tilde{Y}_{\sigma,F_{r}}(z)\Phi_{\sigma,{F_{r}}}^{(0)}(z^{\prime})\right)
A_{\rho}=-\frac{1}{2}\left(\begin{array}{cccc}{0}&{0}&{\left(\overline{\psi}\gamma_{\nu}\gamma_{\rho}\right)^{T}}&{\left(\overline{\psi}\gamma_{5}\gamma_{\nu}\gamma_{\rho}\right)^{T}}\\{0}&{0}&{\gamma_{\rho}\gamma_{\nu}\psi}&{\gamma_{\rho}\gamma_{5}\gamma_{\nu}\psi}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\\end{array}\right)
\sum_{i,j,k,l=1}^{h}\left(U_{l}V_{i}-U_{i}V_{l}\right)A_{ij}A_{lk}\left(\overline{U}_{k}\overline{V}_{j}-\overline{U}_{j}\overline{V}_{k}\right)=\left(\sum_{i,j=1}^{h}U_{i}A_{ij}\overline{U}_{j}\right)^{3}.
{}^{*}(dx^{i_{1}}\cdotsdx^{i_{p}})=\sum_{j_{p+1}<\ldots<j_{D}}g^{i_{1}j_{1}}\ldotsg^{i_{p}j_{p}}\epsilon_{j_{1}\ldotsj_{p}j_{p+1}\ldotsj_{D}}dx^{j_{p+1}}\cdotsdx^{j_{D}},
{\calH}_{c}=\left\{c_{0}+\vec{h}\cdot\vec{c}~,~~~\vec{h}\equiv(i,j,k)~,~~~\vec{c}\equiv(c_{1},c_{2},c_{3})~,~~~~~~~c_{0,1,2,3}\in{\calC}(1,\bbox{\iota})\right\}~,
I_{1}=\intdx\left[\alpha\Omega^{1/2}e^{-\psi_{1}}\alpha^{\prime}\Omega^{\prime}-\frac{3}{4}\alpha^{2}\Omega^{3/2}e^{-\psi_{1}}\psi_{1}^{\prime2}-\frac{\mul^{2}}{2}\Omega^{-1}e^{\psi_{1}}\right],
D={\left(\begin{array}{cccc}{0}&{-\frac{c}{eB}}&{\frac{1}{2}}&{0}\\{\frac{c}{eB}}&{0}&{0}&{\frac{1}{2}}\\{-\frac{1}{2}}&{0}&{0}&{-\frac{eB}{4c}}\\{0}&{-\frac{1}{2}}&{\frac{eB}{4c}}&{0}\\\end{array}\right)}.
I_{ab}=\chi(R_{a},R_{b})=\left(\begin{matrix}{1}&{2}&{6}&{10}&{20}&{30}&{50}&{70}\\{0}&{1}&{2}&{6}&{10}&{20}&{30}&{50}\\{0}&{0}&{1}&{2}&{6}&{10}&{20}&{30}\\{0}&{0}&{0}&{1}&{2}&{6}&{10}&{20}\\{0}&{0}&{0}&{0}&{1}&{2}&{6}&{10}\\{0}&{0}&{0}&{0}&{0}&{1}&{2}&{6}\\{0}&{0}&{0}&{0}&{0}&{0}&{1}&{2}\\{0}&{0}&{0}&{0}&{0}&{0}&{0}&{1}\\\end{matrix}\right)
\bar{Q}=-\frac{i}{\sqrt{m}}\bar{\psi}e^{\mu}(x)\!\left[\p_{\mu}\!-\!A_{\mu}(x)\!-\!\textstyle{\frac{1}{2}}\omega_{\mu}(x)\!+\!\textstyle{\frac{i}{2}}\Gamma_{\\nu\mu}^{\nu}(x)\right].
<n>=\bar{n}_{0}-\beta\frac{\lambda^{2}}{6}\left[\frac{1}{2}((\bar{n^{2}})_{0}-(\bar{n})_{0}^{2})+\frac{3}{2}((\bar{n^{3}})_{0}-\bar{n}_{0}(\bar{n^{2}})_{0})+(\bar{n^{4}})_{0}-\bar{n}_{0}(\bar{n^{3}})_{0})\right]
\overline{{\mathrm{\Large~v~}}}_{D}(x)\equiv{\frac{2(4\pi)^{D/2}}{{\mathrm{tr}}{\bf1}\,\Lambda^{D}}}{\mathrm{\Large~v~}}_{D}(m)={\frac{(4\pi)^{D/2}}{{\mathrm{tr}}{\bf1}\,g^{2}\Lambda^{D-2}}}\left(x-{\frac{\epsilon}{\Lambda^{2}}}\right)+\int_{1}^{\infty}\!\!d\tau\,\tau^{-{\frac{D}{2}}-1}\left\{{\mathrme}^{-\taux}G_{D}(\tau{\calF})-1\right\}\.
\begin{array}{lcr}{|x_{0}|^{2}+|y_{1}|^{2}=1}&{(a)}\\{x_{0}\overline{{y}}_{0}+x_{1}\overline{{y}}_{1}=0}&{(b)}\\{x_{0}{y}_{0}+\overline{{x}}_{1}{y}_{1}=0}&{(c).}\\\end{array}
F_{8}^{{\mathcal{O}}_{0,0}^{0,0}|++------}=\frac{\langle{\mathcal{O}}_{0,0}^{0,0}\rangle(\sigma_{6}^{-}+\sigma_{2}^{-}(\sigma_{2}^{+})^{2}+\sigma_{2}^{+}\sigma_{4}^{-}+(\sigma_{2}^{+})^{3})/(\sigma_{2}^{+})^{3}}{2^{6}\,\mathrm{cosh}(\theta_{12}/2)\,\prod_{3\leqi<j\leq8}\mathrm{cosh}(\theta_{ij}/2)}\prod_{i<j}F_{\mathrm{min}}^{\mu_{i}\mu_{j}}(\theta_{ij}),
\begin{array}{|c|c|c|c|}\hline{}&{P_{1}^{t_{1},\mu_{1}}}&{P_{2}}&{P_{3}}\\\hline{\eta_{1}}&{e^{2\pii\mu_{1}/t_{1}}}&{\infty}&{0}\\\hline{\eta_{2}}&{0^{t_{1}h_{t_{1}}}}&{1}&{\infty^{N_{1}}}\\\hline{\eta_{3}}&{\infty^{t_{1}h_{t_{1}}}}&{0^{N_{1}}}&{1}\\\hline{}\\\end{array}
\frac{F}{V}=-\sum_{\stackrel{n=1}{n~\mathrm{odd}}}^{\infty}\sum_{k=1}^{\infty}\frac{2}{k}\left(\frac{1}{4\pi^{2}\alpha^{\prime}\tau_{2}}\right)^{5}e^{-n^{2}\beta^{2}/4\pi\alpha^{\prime}\tau_{2}}\sum_{M}e^{-\pi\alpha^{\prime}\tau_{2}M^{2}}
(T_{i}\otimesT_{j})_{{\calH}\{\alpha_{\ell},\alpha_{\ell}^{\prime}\}}=\prod_{\ell\not=i,j}^{r}\delta_{\alpha_{\ell},\alpha_{\ell}^{\prime}}\cdot(T^{a})_{\alpha_{i},\alpha_{i}^{\prime}}(T^{a})_{\alpha_{j},\alpha_{j}^{\prime}}
\frac{1}{k!}\epsilon_{\hat{\mu}_{1}\ldots\hat{\mu}_{6}}\gamma^{\hat{\mu}_{1}\ldots\hat{\mu}_{k}}=\frac{1}{G}(-1)^{\frac{k(k+1)}{2}}\bar{\gamma}\gamma_{\hat{\mu}_{k+1}\ldots\hat{\mu}_{6}}.
\left({\frac{x^{-}}{x^{+}}}\right)^{2}F(x^{-},x^{+})=\left|{\frac{L}{\pi}}\langle0|T^{++}(k)|n\rangle\right|^{2}{\frac{M_{n}^{4}}{8\pi^{2}k^{3}}}K_{4}(M_{n}r)\.
S_{II}\sim\frac{\sigma_{0}}{\pi}\left(\frac{2\pi}{\sigma_{0}}\right)^{2-2\nu}\!\frac{r^{2\nu-2}}{J_{\nu}^{2}(-r\tau_{0})}~\sum_{n\neq0}^{}\frac{J_{\nu}^{2}\left(-\frac{2\pi}{\sigma_{0}}|n|\tau_{0}\right)}{|n|^{2\nu-2}}~|\phi_{n}(\sigma_{0})|^{2}~S_{I}
ds^{2}=({\rho^{2}}+bl^{2})\left(-(bR^{2}-k)dT^{2}+\frac{dR^{2}}{bR^{2}-k}\right)+\frac{d\rho^{2}}{\frac{\rho^{2}}{l^{2}}+b}+\rho^{2}d\phi^{2}
\tilde{R}_{L}^{(n)}(t,t^{\prime};\{q\})\approx\frac{\tilde{S}^{(n)}(t;\{q\};L)+\tilde{\Sigma}^{(n)}(t;\{q\},L)\vartheta^{\prime}}{(z^{\prime}-z_{0})}\equiv\frac{\tilde{E}_{L}^{(n)}(t;\{q\};\vartheta^{\prime})}{(z^{\prime}-z_{0})}\,.
\varepsilon^{\mu\nu}=C_{\perp}\epsilon_{\perp}^{\mu}\epsilon_{\perp}^{\nu}+C_{A}\,\epsilon_{A}^{\mu}\epsilon_{A}^{\nu}+C_{B}\,\epsilon_{B}^{\mu}\epsilon_{B}^{\nu}+C_{C}\,\epsilon_{C}^{\mu}\epsilon_{C}^{\nu}
ds^{2}=-\frac{\bar{r}^{2}}{r_{0}^{2}+\bar{r}^{2}}dt^{2}+e^{2M}(d\bar{r}^{2}+e^{-6K}d\bar{z}^{2})+e^{2K}(r_{0}^{2}+\bar{r}^{2})d\Omega_{3}^{2}
\displaystyle{\int}d^{3}x~\left[\Theta_{\mu}^{~\mu}\left(x\right)\cdot\Gamma\right]_{\mathrm{flat}}\sim\left(\beta_{\kappa}\partial_{\kappa}+\beta_{y}\partial_{y}+\beta_{\lambda}\partial_{\lambda}-\gamma_{A}{\calN}_{A}-\gamma_{\psi}{\calN}_{\psi}-\gamma_{\varphi}{\calN}_{\varphi}\right)\cdot\Gamma_{\mathrm{flat}}\,.
\frac{\tilde{V}_{n,l+1}}{\tilde{V}_{n,l}}=\frac{(1+\tilde{V}_{n-1,l})(1+\tilde{V}_{n-2,l})}{(1+\tilde{V}_{n-1,l+1})(1+\tilde{V}_{n-2,l+1})}.
\begin{array}{c}{S=-\frac{k}{4\pi}\intdtd\overline{{z}}Tr(\overline{{\partial}}WW^{-1}\partial_{t}WW^{-1})-\frac{k}{4\pi}\intdtdzTr(\partialWW^{-1}\partial_{t}WW^{-1})}\\{+\frac{k}{12\pi}\int_{Y}Tr(dWW^{-1})^{3}}\\\end{array}
\{L_{\alpha},L_{\beta}\}=2i(\hat{p}^{\mu}+\Pi_{1}^{\mu})\Gamma_{\alpha\beta}^{\mu}\delta(\sigma-\sigma^{\prime})-2\bar{\theta}^{\gamma}\partial_{1}\theta^{\delta}\Gamma_{\gamma(\delta}^{\mu}(C\Gamma^{\mu})_{\alpha\beta)}\delta(\sigma-\sigma^{\prime}).
\left(\begin{array}{c}{\hat{a}_{\mathbf{k}}}\\{\hat{a}_{-\mathbf{k}}^{\dagger}}\\\end{array}\right)=\left(\begin{array}{cc}{\alpha_{{k}}}&{\beta_{{k}}^{*}}\\{\beta_{{k}}}&{\alpha_{{k}}^{*}}\\\end{array}\right)\left(\begin{array}{c}{a_{\mathbf{k}}}\\{a_{-\mathbf{k}}^{\dagger}}\\\end{array}\right),
C_{\mu\nu}^{-1}=\frac{1}{\nabla^{2}}\left(\begin{array}{cccc}{0}&{0}&{0}&{1}\\{0}&{0}&{-1}&{0}\\{0}&{1}&{0}&{0}\\{-1}&{0}&{0}&{0}\\\end{array}\right)\delta^{2}(x-y).
\begin{array}{rl}{}&{{\displaystyleH=\left({\frac{8\piV(\phi)}{3M_{\mathrm{p}}^{2}}}\right)^{1/2}=\left({\frac{2\pi\lambda}{3}}\right)^{1/2}{\frac{\phi^{2}}{M_{\mathrm{p}}}}\,}}\\{}&{{\displaystyle\dot{\phi}=-{\frac{V^{\prime}(\phi)}{3H}}=-\left({\frac{\lambda}{6\pi}}\right)^{1/2}M_{\mathrm{p}}\,\phi}\.}\\\end{array}
\phi_{\omega}(\rho)~\sim{\frac{2^{-({\frac{p-1}{2}})}A}{\Gamma({\frac{p+1}{2}})}}+{\frac{2^{({\frac{p-1}{2}})}\omega^{(1-p)}}{\Gamma({\frac{3-p}{2}})}}~{\frac{B}{r^{p-1}}}
{\calL}~=~-{\frac{1}{2}}{\partial}_{-}{\Omega}^{-}{\vec{\nabla}}^{2}{\partial}_{+}{\Omega}^{+}~-~{\frac{1}{2}}{\partial}_{+}{\Omega}^{+}{\vec{\nabla}}^{2}{\partial}_{-}{\Omega}^{-}-\partial_{+}k^{+}\partial_{-}\Omega^{--}\partial_{-}k^{-}\partial_{+}\Omega^{+}
U(t,t^{\prime})=e^{iH_{0}t/\hbar}e^{-iH(t-t^{\prime})/\hbar}e^{-iH_{0}t^{\prime}/\hbar}=e^{-i\int_{t^{\prime}}^{t}H_{I}^{\prime}(t^{\prime\prime})dt^{\prime\prime}/\hbar},
\zeta(s|A)=\,\mathrm{Tr}\,A^{-s}=\int_{{\calM}}A^{-s}(x,x)d^{N}x=\frac{1}{\Gamma(s)}\int_{0}^{\infty}t^{s-1}\,\mathrm{Tr}\,e^{-tA}\,dt\:,
D_{m~e}^{~~n~f}=\biggl(R-2\Lambda\biggr)\biggl(\delta_{m}^{n}\delta_{e}^{f}-\delta_{m}^{f}\delta_{e}^{n}\biggr)+2R_{~~me}^{fn}+\frac{1}{2}\biggl(R_{m}^{~f}\delta_{e}^{n}-R_{~m}^{n}\delta_{e}^{f}+R_{e}^{~n}\delta_{m}^{f}-R_{~e}^{f}\delta_{m}^{n}\biggr)
f_{D0}^{A}=2\zeta(3)Ve^{-2\phi}+2V\sum_{n^{i}\ne0}\frac{1}{n^{i}g_{ij}n^{j}}+4\piV\tau_{2}\sum_{m\ne0}\sum_{n^{i}\ne0}\frac{|m|}{\sqrt{n^{i}g_{ij}n^{j}}}K_{1}\left(2\pi|m|e^{-\phi}\sqrt{n^{i}g_{ij}n^{j}}\right)e^{2\piimn^{i}{\calA}_{i}}
v=\frac{x_{12}^{2}x_{34}^{2}}{x_{14}^{2}x_{23}^{2}}\,\,\,\,\,,\,\,\,\,\,\frac{v}{u}=\frac{x_{13}^{2}x_{24}^{2}}{x_{14}^{2}x_{23}^{2}}.
g={\frac{1}{f\kappa^{\frac{2}{3}}}}\left(\begin{matrix}{-2f^{2}}&{0}&{0}\\{0}&{2}&{-{\frac{1}{\sqrt{2}}}\chi}\\{0}&{-{\frac{1}{\sqrt{2}}}\chi}&{-{\frac{1}{4}}(\chi^{2}-f\kappa^{2})}\\\end{matrix}\right)
V_{0}(m)={\frac{m^{2}}{2g^{2}}}+{\frac{1}{8\pi^{2}}}\left[{\frac{\Lambda^{4}}{2}}\Bigl(1-{\frac{m^{2}}{\Lambda^{2}}}\Bigr){\mathrm{e}}^{-{\frac{m^{2}}{\Lambda^{2}}}}+{\frac{m^{4}}{2}}E_{1}({m^{2}/\Lambda^{2}})\right],
\frac{\partialU}{\partial\bar{\sigma}_{2}}=\frac{1}{\beta}\left[\left(\beta\tilde{m}^{2}+\frac{1}{2}\alphae\tilde{k}\right)\sigma_{2}-\frac{e^{2}}{4}(|\sigma_{1}|^{2}-|\sigma_{2}|^{2})\sigma_{2}+\frac{\alphaqe}{2}(|\phi_{1}|^{2}-|\phi_{2}|^{2})\sigma_{2}\right]=0
S[\phi]\!\!=\!\!\int_{0}^{\beta}\!\!\!d\tau\int\!\!\!d^{3}x\!\left(\partial_{\nu}\phi_{a}\partial^{\nu}\phi_{a}\!\!+\!m_{o}^{2}\phi^{2}\!\!+\!\frac{\lambda_{o}}{4N}\phi^{4}\!\!-\!\mu^{2}\phi^{2}\!\!+\!2i\mu(\phi_{2}\partial_{\tau}\phi_{1}\!\!-\!\phi_{1}\partial_{\tau}\phi_{2})\right)
\mu_{ab}=\left(\begin{array}{lllllll}{0}&{\alpha}&{0}&{0}&{\dots}&{0}&{0}\\{-\alpha}&{0}&{0}&{0}&{\dots}&{0}&{0}\\{0}&{0}&{0}&{\beta}&{\dots}&{0}&{0}\\{0}&{0}&{-\beta}&{0}&{\dots}&{0}&{0}\\{\cdot}&{\cdot}&{\cdot}&{\cdot}&{\dots}&{\cdot}&{\cdot}\\{0}&{0}&{0}&{0}&{\dots}&{0}&{\lambda}\\{0}&{0}&{0}&{0}&{\dots}&{-\lambda}&{0}\\\end{array}\right)
h_{R\timesH^{D}}\,=\,\frac{2^{[\frac{D+1}{2}]}}{a^{D}}\int_{0}^{\infty}\frac{e^{-\frac{s}{a^{2}}(r^{2}+m^{2}a^{2})}}{(4\pis)^{\frac{1}{2}}}|A(r)|^{2}dr\,,
\langleT_{zz}(z,\bar{z})T_{zz}(0,0)\rangle=\frac{\pi}{3\cdot16}\int_{0}^{\infty}d\mu^{2}\,\tilde{c}(\mu^{2})\int\frac{d^{2}p}{(2\pi)^{2}}\frac{e^{\frac{i}{2}(p\bar{z}+\bar{p}z)}}{p\bar{p}+\mu^{2}}\,\bar{p}^{4},
{\calA}_{m}^{n}=\frac{3}{7}\left(\begin{array}{cccc}{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}\\\end{array}\right)\tau_{1}^{(s=0)}+\mathrm{mean~field~terms}
{\calL}^{\prime}=\frac{\delta{\calL}_{0}}{\delta\phi}\,\delta\phi+\frac{\delta{\calL}_{0}}{\delta\!\left({\partial}_{\mu}\phi\right)}\,\delta\!\left(\partial_{\mu}\phi\right)=\frac{\delta{\calL}_{0}}{\delta\phi}\,\tilde{\phi}+\frac{\delta{\calL}_{0}}{\delta\!\left({\partial}_{\mu}\phi\right)}\,\partial_{\mu}\tilde{\phi}\,,
p_{n}^{\mathcal{O}}(\theta_{2}+i\pi,\underline{{\theta}}^{\prime},\underline{{l}})=g(l_{1},l_{2})p_{n-2}^{\mathcal{O}}(\underline{{\theta}}^{\prime\prime},\underline{{l}}^{\prime\prime})+h(l_{1},l_{2})
\tilde{\Gamma}_{\mu\nu}^{abc,r}(x,y,z)=\int\tilde{G}_{\mu}^{ad}(x-x^{\prime})\tilde{\Gamma}_{\nu}^{dbc,r}(x^{\prime},y,z)dx^{\prime}+(\tilde{\Gamma}^{ir})_{\mu\nu}^{abc}(x,y,z)
\hat{\hat{K}}_{\hat{\hat{a}}\hat{\hat{b}}\hat{\hat{c}}}\;=\;{\textstyle\frac{1}{2}}\left(\hat{\hat{T}\,}_{\hat{\hat{a}}\hat{\hat{c}}\hat{\hat{b}}}\;+\;\hat{\hat{T}\,}_{\hat{\hat{b}}\hat{\hat{c}}\hat{\hat{a}}}\;-\;\hat{\hat{T}\,}_{\hat{\hat{a}}\hat{\hat{b}}\hat{\hat{c}}}\right)\;,
\tilde{\omega}_{n}^{2}=m^{2}\left\{\frac{2\alpha}{\lambda^{2}}(2\alpha+\lambda)-\frac{2\alpha}{\lambda}-\left[\sqrt{\frac{2\alpha}{\lambda^{2}}(2\alpha+\lambda)+\frac14}-(n+\frac12)\right]^{2}\right\}.
\left\{-\delta+\frac{\left|{\bfp}\right|^{2}}{m}+2\mathrm{Re}\Sigma_{-}(p_{3},\left|{\bfp}\right|)\right\}\Phi(p_{3},\left|{\bfp}\right|)=\int\frac{d^{2}{\bfq}}{(2\pi)^{2}}V(\left|{\bfp-q}\right|)\Phi(q_{3},\left|{\bfq}\right|)\chi({\bfp},{\bfq})
\bar{u}_{1}u_{1}-\bar{u}_{2}u_{2}=\sum_{n=1}^{N}\left(\!\begin{array}{cc}{|u_{1,n1}|^{2}-|u_{2,n1}|^{2}}&{\bar{u}_{n1,1}u_{1,n2}-\bar{u}_{n1,2}u_{2,n2}}\\{u_{n1,1}\bar{u}_{1,n2}-u_{n1,2}\bar{u}_{2,n2}}&{|u_{1,n2}|^{2}-|u_{2,n2}|^{2}}\\\end{array}\!\right)\equiv\left(\!\begin{array}{cc}{U_{1}}&{U_{2}}\\{\bar{U}_{2}}&{U_{4}}\\\end{array}\!\right).
\begin{aligned}{}&{{}U_{1}=\mathrm{Pexp}\int_{\Xi_{1}}A^{(1)},}&{}&{{}U_{2}=\mathrm{Pexp}\int_{\Xi_{2}}A^{(2)},}&{}\\{}&{{}V_{1}=\mathrm{Pexp}\int_{\Gamma_{1}}A^{(1)},}&{}&{{}V_{2}=\mathrm{Pexp}\int_{\Gamma_{2}}A^{(2)}}&{}\\\end{aligned}
H_{xxz}=\sum_{n=1}^{N-1}\{\sigma_{n}^{+}\sigma_{n+1}^{-}+\sigma_{n}^{-}\sigma_{n+1}^{+}+\frac{q+q^{-1}}{4}\,\sigma_{n}^{z}\sigma_{n+1}^{z}\+\frac{q-q^{-1}}{4}\,(\sigma_{n}^{z}-\sigma_{n+1}^{z})\},
\Psi(p=0)=\frac{c_{1}}{\sqrt2}\left(\begin{array}{c}{\omega+1}\\{\omega-1}\\{0}\\{0}\\\end{array}\right)+\frac{c_{2}}{\sqrt2}\left(\begin{array}{c}{0}\\{0}\\{\omega-1}\\{\omega+1}\\\end{array}\right)+\frac{c_{3}}{\sqrt8}\left(\begin{array}{c}{\omega-1}\\{\omega+1}\\{\lambda(\omega+1)}\\{\lambda(\omega-1)}\\\end{array}\right),
M_{11}^{2}=\Big(m(2\piR_{11})^{2}\tau_{2}T_{11}\Big)^{2}+{\frac{1}{R_{11}^{2}}}\Big(\ell_{2}^{2}+{\frac{1}{\tau_{2}^{2}}}(\ell_{1}-\ell_{2}\tau_{1})^{2}\Big)+\dots~~,
\tilde{j}_{g\calL}^{\mu}(f)-j_{g\calL}^{\mu}(f)=i\bigl(G^{(1)}(k^{\mu}f,{\calL}g)\bigr)_{g\calL}+i\bigl(G(({\calL},{\calL}_{1})g,k^{\mu}f)\bigr)_{g\calL}.
+\frac{2}{AB-1}(AX_{n}^{0}\tilde{X}_{n}^{1}-BX_{n}^{1}\tilde{X}_{n}^{0})e^{-nl_{0}}e^{-nl_{1}}]exp\sum_{n\geq1}[X_{-n}^{0}\tilde{X}_{-n}^{1}-X_{-n}^{1}\tilde{X}_{-n}^{0}]|0>
{\bfg}_{\mu\nu}=\eta_{\mu\nu}-\kappa\gamma_{\mu\nu}+\kappa^{2}{\gamma_{\mu}}^{\alpha}\star{\gamma_{\alpha}}_{\nu}-\kappa^{3}{\gamma_{\mu}}^{\alpha}\star\gamma_{\alpha_{\beta}}\star{\gamma_{\nu}}^{\beta}+O(\kappa^{4}).
(\tilde{A}^{2})_{\mu\nu}=\tilde{g}^{\mu_{1}\nu_{1}}\tilde{g}^{\mu_{2}\nu_{2}}\tilde{A}_{\mu_{1}\mu_{2}\mu}\tilde{A}_{\nu_{1}\nu_{2}\nu}\,\quad\tilde{A}^{2}=\tilde{g}^{\mu\nu}(\tilde{A}^{2})_{\mu\nu}\,\quadK=\sqrt{1+{\frac{1}{24}}\tilde{A}^{2}}\.
L_{_{\mathrm{GF+FP}}}^{1}=-\frac{i}{4}\mathrm{Tr}\left(\delta_{1}(\chi^{\mu\nu}[\lambdaF_{\mu\nu}-\frac{1}{2}S_{\mu\nu\alpha\beta}F^{\alpha\beta}+\frac{1}{2}aB_{\mu\nu}])\right).
W_{\mathrm{tree}}=\sqrt{2}\sum_{i<j\atopk<l}\tilde{A}_{ij}\left(\Phi^{i}{}_{k}\delta^{j}{}_{l}+\delta^{i}{}_{k}\Phi^{j}{}_{l}+m\delta^{i}{}_{k}\delta^{j}{}_{l}\right)A^{kl}.
{\calV}=e^{K/M_{Pl}^{2}}\{(\frac{\partialW}{\partial\tilde{q}^{i}}+\frac{W}{M_{Pl}^{2}}\frac{\partialK}{\partial\tilde{q}^{i}})K^{-1~ij}(\frac{\partialW^{\dagger}}{\partial\tilde{q}^{j\dagger}}+\frac{W^{\dagger}}{M_{Pl}^{2}}\frac{\partialK}{\partial\tilde{q}^{j\dagger}})~-~3\frac{|W|^{2}}{M_{Pl}^{2}}\}+\frac{g^{2}}{2}(\frac{\partialK}{\partial\tilde{q}}t^{a}\tilde{q})^{2},
S=\frac{1}{4\pi}\intd\Omega\left[-{\calL}_{i}\Phi{\calL}_{i}\Phi+\mu^{2}\Phi^{2}+\sum_{n}\frac{\lambda^{(n)}}{n!}\Phi^{n}\right],\quad\mathrm{where}\quad{\calL}_{i}=-i\epsilon_{ijk}x_{j}\partial_{k}.
\widehat{\phi}={\frac{2\left(1+\sqrt{7}\right)}{5}}\left(1-\sqrt{7}\epsilon^{}z+\epsilon^{4/3}\widehat{\phi}_{(4/3)}(z,t)+\epsilon^{5/3}\widehat{\phi}_{(5/3)}(z,t)+{\calO}(\epsilon^{2})\right)\,
\prod_{1\leqk<j\leq2n}\left[Q^{2}(x_{j}-x_{k},\tau_{j}-\tau_{k})\right]^{\epsilon_{j}\epsilon_{k}\lambda^{2}/4\pi}=\prod_{1\leqk<j\leq2n}\left[Q^{2}(\sum_{l=k+1}^{j}x_{l}^{\prime},\sum_{l=k+1}^{j}\tau_{l}^{\prime})\right]^{\epsilon_{j}\epsilon_{k}\lambda^{2}/4\pi}
K_{N}(z^{2},w^{2})=c_{2N}\frac{w\varphi_{2N}^{(f(\tilde{f}),\tilde{\alpha}+\frac{1}{2})}(z)\varphi_{2N-1}^{(f(\tilde{f}),\tilde{\alpha}+\frac{1}{2})}(w)-z\varphi_{2N-1}^{(f(\tilde{f}),\tilde{\alpha}+\frac{1}{2})}(z)\varphi_{2N}^{(f(\tilde{f}),\tilde{\alpha}+\frac{1}{2})}(w)}{z^{2}-w^{2}}\.
C_{l}=-{\frac{J_{l+2}(s)N_{l+2}^{\prime}(s)+N_{l+2}(s)J_{l+2}^{\prime}(s)-{\frac{\alphaL}{qR}}J_{l+2}(s)N_{l+2}(s)}{2J_{l+2}(s)J_{l+2}^{\prime}(s)-{\frac{\alphaL}{qR}}J_{l+2}^{2}(s)}}\.
\prod_{n=0}^{\infty}\left(1-q^{n+1}\right)=q^{-4+\frac{5}{24}}\prod_{n=0}^{\infty}\left(1-q^{15n+15}\right)^{-6}\prod_{r=0}^{6}\left[\theta_{2r+1,30}(q)-\theta_{29-2r,30}(q)\right].
N_{\mathrm{max}}\simeq\mathrm{Int}\left\lbrack0\cdot15\left(\frac{L_{DS}}{L_{S}}\right)^{2}\right\rbrack=\mathrm{Int}\left\lbrack0\cdot15\left(\frac{c^{3}}{\alpha^{\prime}\hbarH^{2}}\right)\right\rbrack
S=\intd^{3}x\left\{-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{\kappa}{2}\epsilon^{\mu\nu\lambda}A_{\mu}\partial_{\nu}A_{\lambda}\right\}+\sum_{i}e_{i}\int_{\Gamma_{i}}dx^{\mu}A_{\mu}
(-)^{a}\sum_{j=0}^{a}(-1)^{j}\left(\begin{array}{c}{j}\\{a}\\\end{array}\right)\mathrm{exp}\left[2\pii\frac{Q}{P}(h_{e_{1}+Pe,n_{1}+Pn-\frac{P}{a}j}-h_{e_{1},n_{1}})\right]|e_{1}+Pe,n_{1}+Pn-\frac{P}{a}j>
D\bigl(\frac{n-1}{2}+\varepsilon,\frac{1}{2},\ldots,\frac{1}{2}\bigr)\underset{\varepsilon\to0}{\longrightarrow}D\bigl(\frac{n-1}{2},\frac{1}{2},\ldots,\frac{1}{2}\bigr)+D\bigl(\frac{n+1}{2},\frac{1}{2},\ldots,\frac{1}{2},\nu-\frac{1}{2}\bigr).
L_{mass}=A\sigma^{\frac{1}{1-2\delta}}(f_{\mu}f^{\mu})^{\frac{\delta}{2\delta-1}}-\frac{D\delta}{2\delta-1}\epsilon^{\mu\nu\rho}f_{\mu}\partial_{\nu}f_{\rho}+\frac{1}{2}(\partial_{\mu}\phi)^{2}.
T_{\theta}(\tau)={\frac{1}{2}}\left[{\frac{\Lambda_{R}^{m}(\tau)}{\eta_{R}(\tau)}}+{\frac{\Lambda_{-R}^{m}(\tau)}{\eta_{-R}(\tau)}}\right]+{\frac{1}{2}}\mathrm{Tr}S\left[{\frac{\eta_{R}(\tau)}{\eta_{R}({\frac{\tau}{2}})}}+(\tau\mapsto\tau+1)\right]\,,
{{((B_{1}^{s})^{I_{1}I_{2}I_{3}I_{4}})}^{\lambda}}_{\mu}=\delta_{\mu}^{\lambda}(-1)^{I_{1}+I_{2}-I_{34}-\mu}q^{(I_{34}+\mu)(I_{34}+\mu+1)-I_{1}(I_{1}+1)-I_{2}(I_{2}+1)}\,
G(z_{out},z_{in})=\frac1{16\sqrt2}e^{\gamma^{+}\frac\partial{\partial\mu}}e^{\gamma^{-}\frac\partial{\partial\nu}}\langlez_{out}|{({\hat{p}}^{2})}^{-1}\left(\mu{\hat{p_{\theta}^{\prime}}}^{2}{\hat{p_{\bar{\theta}}^{\prime}}}^{2}+\nu{\hat{p_{\bar{\theta}}^{\prime}}}^{2}{\hat{p_{\theta}^{\prime}}}^{2}\right)|z_{in}\rangle|_{\mu=\nu=0},
\tilde{F}(s)=\frac{1}{48}\int_{0}^{\infty}\!\!\!d\mu^{2}\,\tilde{c}(\mu^{2})\mus\left(({\mu}^{3}{s}^{3}\!+\!24\mus)K_{0}(\mus)\!+\!(8\mu^{2}s^{2}\!+\!48)K_{1}(\mus)\right).
g^{mn}E_{m}^{\underline{\beta}}\left[E_{n}^{\underline{a}}\Gamma_{\underline{a}}(1-\bar{\Gamma})\right]_{\underline{\beta}\underline{\alpha}}=0,\quadg_{mn}(\xi)=e_{m}^{a}e_{na}(\xi)=E_{m}^{\underline{a}}E_{n}^{\underline{b}}\eta_{\underline{{ab}}}.
\begin{array}{l}{\phi(\tau,x)=exp(i\hbar(\tau-\tau_{0})\Box_{A})\phi+ig\hbar\int_{\tau_{0}}^{\tau}exp(i\hbar\Box_{A}(\tau-s))(\phi_{1}^{2}+\phi_{2}^{2})\phi(s)ds+}\\{+\sqrt{2}\hbar\int_{\tau_{0}}^{\tau}exp\Big(i\hbar\Box_{A}(\tau-s)\Big)dB(s)}\\\end{array}
\left(2\varphi_{\bar{z}}(z,\bar{z})\right)^{l}e^{-k\varphi(z,\bar{z})}=\left(2\widetilde\varphi_{\bar{w}}(w,\bar{w})+2\bar{w}_{\bar{z}\bar{z}}/(\bar{w}_{\bar{z}})^{2}\right)^{l}e^{-k\widetilde\varphi(w,\bar{w})}w_{z}^{-k}\bar{w}_{\bar{z}}^{-k+l},
\delta(t)=\frac{\delta(0)}{1-\frac{\delta(0)}{E_{d}^{(0)}}\int_{0}^{t}\left(\partial^{0}\tilde{\phi}_{d\,(\rightarrow)}^{(1)}\,\partial^{\xi}\tilde{\phi}_{d\,(\rightarrow)}^{(1)}\right)\mid_{\xi=1+\delta}dt^{\prime}}.
\begin{aligned}{|\Phi_{1}|^{2}+|\Phi_{2}|^{2}+|\Phi_{3}|^{2}}&{{}=r_{1}\,,}\\{|\Phi_{1}|^{2}+|\Phi_{4}|^{2}}&{{}=r_{2}\,,}\\{|\Phi_{2}|^{2}+|\Phi_{5}|^{2}}&{{}=r_{3}\,,\,\,\,\,\,r_{1}\geqr_{2},r_{1}\ger_{3}\,,}\\\end{aligned}
S_{GB}^{volume}=2\left(1-(e^{U}R^{\prime})^{2}\right)|_{r=r_{h}}-2(e^{2U})^{\prime}|_{r=r_{h}})^{-1}\left[(e^{2U})^{\prime}\left(1-(e^{U}R^{\prime})^{2}\right)\right]_{r=r_{0}}\.
n_{1}^{2}+n_{2}^{2}+n_{3}^{2}={\frac{1}{3}}k_{1}^{2}+{\frac{2}{3}}(k_{2}^{2}+k_{3}^{2}+k_{2}k_{3}),\qquad\quadn_{1}+n_{2}+n_{3}=k_{1}