harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1}{2}\int_{\varepsilon^{2}}^{\infty}\frac{dt}{t^{3}}\sum_{k=-\infty}^{\infty}\left[e^{-k^{2}\pit/R^{2}}-e^{-(k+1/2)^{2}\pit/R^{2}}\right]=\frac{1}{R^{4}}\frac{31}{16}\frac{3}{4\pi^{2}}\zeta(5)
\deltaS={\frac{i}{2}}\intd\taud^{D-2}\eta(\Lambda^{qrs}\Gamma_{\underline{b}}E_{s})\left(E_{q}^{\underline{\alpha}}E_{r}^{\underline{\beta}}T_{\underline{\alpha}\underline{\beta}}^{\underline{b}}+2E_{q}^{\underline{A}}E_{r}^{\underline{a}}T_{\underline{a}\underline{A}}^{\underline{b}}\right),
D_{\mu}X^{A}\rightarrowD_{\mu}X^{A}+\frac12\Gamma^{A\dot{A}\ddot{A}}[(Y^{\dot{A}}\stackrel{\leftrightarrow}{\partial_{\mu}}Z^{\ddot{A}})+(Y_{m}^{\dot{A}}\stackrel{\leftrightarrow}{\partial_{\mu}}Z^{m\ddot{A}})].
\langle\rho|U^{\dagger}(\psi_{2},z_{2})U\left(e^{L_{1}}\overline{\phi},1\right)\overline{{U(\psi_{1},z_{1})|\chi\rangle}}=\langle\chi|U^{\dagger}(\psi_{1},z_{1})U(\overline{\phi},-1)\overline{{U(\psi_{2},z_{2})|\chi\rangle}}\,.
{\calA}_{4}={}_{41i}\langleV_{3}|~_{j23}\langleV_{3}|Y_{-2}^{(i)}Y_{-2}^{(j)}\frac{b_{o}^{(i)}}{L^{(i)}}WQ\frac{b_{o}^{(i)}}{L^{(i)}}|V_{2}\rangle_{ij}|\tilde{A}_{4}\rangle_{4}|\tilde{A}_{1}\rangle_{1}|\tilde{A}_{2}\rangle_{2}|\tilde{A}_{3}\rangle_{3}.
f_{TTT}=-{\frac{2i}{\pi}}\frac{j_{T}(T)}{j(T)-j(U)}\left\{\frac{j(U)}{j(T)}\right\}\left\{\frac{j_{T}(T)}{j_{U}(U)}\right\}\left\{\frac{j(U)-j(i)}{j(T)-j(i)}\right\}.
2R=\sum_{\Omega(n,m,m^{\prime})}\left[\frac{x_{m}^{2}}{(x_{m}^{2}-x_{n}^{2})(x_{m}^{2}-x_{m^{\prime}}^{2})}+\frac{x_{m^{\prime}}^{2}}{(x_{m^{\prime}}^{2}-x_{m}^{2})(x_{m^{\prime}}^{2}-x_{n}^{2})}\right]=-R\,
\bigg\langle\sum_{j}(H_{\infty}W(n_{j}))\prod_{i\not=j}W(n_{i})+{\frac{1}{N^{2}}}\sum_{j}\sum_{i\not=j}n_{i}n_{j}W(n_{i}+n_{j})\prod_{k\not=i,j}W(n_{k})\bigg\rangle=0
\frac{1}{6}c_{ijk}\psi^{i}\psi^{j}\psi^{k}+\frac{1}{2}c_{ijk}b^{il}\chi_{l}\psi^{j}\psi^{k}+\frac{1}{2}c_{ijk}b^{il}b^{jm}\chi_{l}\chi_{m}\psi^{k}+\frac{1}{6}c_{ijk}b^{il}b^{jm}b^{kn}\chi_{l}\chi_{m}\chi_{n}.
\begin{array}{c}{\lbracke_{i},f_{j}\rbrack=\delta_{ij}\frac{k_{i}-k_{i}^{-1}}{Q_{i}-Q_{i}^{-1}}}\\{k_{i}e_{j}k_{i}^{-1}=Q_{i}^{a_{ij}}e_{j},~~k_{i}f_{j}k_{i}^{-1}=Q_{i}^{-a_{ij}}f_{j}}\\{k_{i}k_{i}^{-1}=k_{i}^{-1}k_{i}=1,~~k_{i}k_{j}=k_{j}k_{i}}\\\end{array}
\psi_{\zetan}^{{\vec{k}}}=\prod_{\alpha<\beta}\left(\frac{\theta_{1}^{\ast}\left(z_{\alpha}-z_{\beta}\mid\tau\right)}{\theta_{1}\left(z_{\alpha}-z_{\beta}\mid\tau\right)}\right)^{\frac{1}{2\kappa}}\tilde{\psi}_{\zetan}^{{\vec{k}}}
B={\frac{i}{2}}\kappa\\xi\(\bar{\psi}^{1}\gamma_{5}\psi^{1}+\bar{\psi}^{2}\gamma_{5}\psi^{2})-{\frac{1}{2}}\kappa\\theta\(\bar{\psi}^{1}\psi^{1}-\bar{\psi}^{2}\psi^{2})-\kappa\\varphi\\bar{\psi}^{1}\psi^{2}+...\,\,
\begin{array}{l}{\rho_{l}^{\alpha}(x,x_{f})=<e^{\alpha},e_{l}^{f}>=V_{\mu}^{\alpha}(x)D_{l}^{\mu}(\kappa{\bfB}_{f}),}\\{<e_{f}^{\alpha},e_{\beta}^{f}>=<e^{\alpha},e_{\beta}>=\delta_{\beta}^{\alpha}.}\\\end{array}
\{F,G\}=\left(\frac{\partialF}{\partialx_{i}}\frac{\partialG}{\partialp_{j}}-\frac{\partialF}{\partialp_{i}}\frac{\partialG}{\partialx_{j}}\right)\{x_{i},p_{j}\}+\frac{\partialF}{\partialx_{i}}\frac{\partialG}{\partialx_{j}}\{x_{i},x_{j}\}\;,
\bar{\mu}^{\dot{A}}=\left(X^{B\dot{A}}-i\Theta^{B}\bar{\Theta}^{\dot{A}}\right)\lambda_{B}+2\bar{z}^{\dot{A}\dot{B}}\bar{\lambda}_{\dot{B}}-i\bar{\Theta}^{\dot{A}}\bar{\Theta}^{\dot{B}}\bar{\lambda}_{\dot{B}}
Q_{m}=\frac{1}{m}\intV_{\alpha_{+}}(z_{0})\cdotsV_{\alpha_{+}}(z_{m-1})\prod_{i=0}^{m-1}\mathrm{d}z_{i}\left[\frac{\mathrm{e}^{2\pii\alpha_{+}^{2}m}-1}{\mathrm{e}^{2\pii\alpha_{+}^{2}}-1}\right].
{\dot{y}}_{(\alpha)}=\{H,\{y}_{(\alpha)}\}=\{E+U,\{y}_{(\alpha)}\}={\frac{p_{(\alpha)}}{E}}-{\frac{s\epsilon_{(\alpha)(\beta)(\gamma)}p_{(\gamma)}}{E^{3}}}{\frac{\partialU}{\partial{y}_{(\beta)}}},
ds_{4}^{2}=(1+{\frac{4M}{r}})(dr^{2}+r^{2}d\theta^{2}+r^{2}sin^{2}\varphi^{2})-(1+{\frac{4M}{r}})^{-1}dt^{2}\,;\quade^{2\Phi}={\frac{e^{2\Phi_{0}}}{(1+{\frac{4M}{r}})}}\,.
\tilde{S}=\intd^{2}\xi\left\{\partial_{a}\tilde{X}^{\mu}\partial^{a}\tilde{X}^{\nu}\tilde{G}_{\mu\nu}(\tilde{X})+\epsilon^{ab}\partial_{a}\tilde{X}^{\mu}\partial_{b}\tilde{X}^{\nu}\tilde{B}_{\mu\nu}(\tilde{X})\right\}
{\widehatT_{G}}^{\mu\nu}(q)=-\,{\frac{e^{2}}{q^{2}}}\,\left(\eta^{\mu\nu}q^{2}-q^{\mu}q^{\nu}\right)\,\left(\left.\hat{h}_{G}\right|_{q^{2}=0}+\left(\hat{h}_{G}(q^{2})-\left.\hat{h}_{G}\right|_{q^{2}=0}\right)\right).
\langle\Psi_{1}^{i_{1}}(z_{1})\Psi_{2}^{i_{2}}(z_{2})\rangle=\frac{{D_{1}^{i}}_{i^{\prime}}(I^{-1}(z_{12}))H^{i^{\prime}j}(i(z_{12}))}{(x_{12}^{2}+\theta_{12}^{2}\bar{\theta}_{12}^{2})^{\frac{\eta_{1}+\eta_{2}}{2}}}
\int_{-\infty}^{\infty}dz\int_{0}^{\infty}r\,dr\int_{0}^{\alpha}d\theta\,f_{\lambda^{\prime}n^{\prime}k^{\prime}}^{*}(r,\theta,z)f_{\lambdank}(r,\theta,z)=\delta_{nn^{\prime}}\frac{1}{\lambda}\,\delta(\lambda-\lambda^{\prime})\,\delta(k-k^{\prime})
\begin{array}{rll}{\epsilon_{L}}&{=(\Gamma_{0}\Gamma_{5}\Gamma_{6}\ldots\Gamma_{9})\epsilon_{R}\quad}&{(R5)\quad,}\\{\epsilon_{L}}&{=(\Gamma_{0}\Gamma_{5})\epsilon_{R}\quad}&{(R1)\quad,}\\{\epsilon_{L}}&{=(\Gamma_{0}\Gamma_{5})\epsilon_{L}\quad;\qquad\epsilon_{R}=(\Gamma_{0}\Gamma_{5})\epsilon_{R}\quad}&{(W)\quad,}\\\end{array}
\beta_{i}^{a}(V_{i}^{a}\cdot\overline{{\alpha^{a}V^{a}}}-k_{ij}^{a}\alpha_{j}^{a})+\beta_{i}^{b}(V_{i}^{b}\cdot\overline{{\alpha^{b}V^{b}}}-k_{ij}^{b}\alpha_{j}^{b})=\beta_{i}^{c}(V_{i}^{c}\cdot\overline{{\alpha^{c}V^{c}}}-k_{ij}^{c}\alpha_{j})\pmod{1}
\Omega_{\alpha\beta}^{-1}=\frac{1}{N}\left(\begin{matrix}{N-1}&{-1}&{\ldots}&{-1}\\{-1}&{N-1}&{\ldots}&{-1}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{-1}&{-1}&{\ldots}&{N-1}\\\end{matrix}\right)\\,
\{\varphi_{k}\,,\,\varphi_{h}\}=-{\frac{1}{2}}{s}\left(\frac{k-h}{2N}\right)\quad,\quad\{\bar{\varphi}_{k}\,,\,\bar{\varphi}_{h}\}={\frac{1}{2}}{s}\left(\frac{k-h}{2N}\right)\quad,\quad\{\varphi_{k}\,,\,\bar{\varphi}_{h}\}=0\quad.
F_{\mu\nu}^{a}=-\frac{4\rho^{2}}{(x^{2}+\rho^{2})^{2}}\left\{\bar{\eta}_{\mu\nu}^{a}-2\bar{\eta}_{\mu\rho}^{a}\frac{x_{\rho}x_{\nu}}{x^{2}}+2\bar{\eta}_{\nu\rho}^{a}\frac{x_{\rho}x_{\mu}}{x^{2}}\right\}\.
\frac{\kappa_{j}+ik_{j}}{\kappa_{j}-ik_{j}}=-\,\frac{A_{a}(\bar{\sigma}_{N}\sigma^{\prime\prime}q_{j}^{\prime\prime},id)}{A_{a}(\sigma^{\prime\prime}q_{j}^{\prime\prime},id)}\,e^{-ik_{j}L}.
\begin{array}{lll}{D_{u}V}&{=}&{U}\\{D_{u}U}&{=}&{e^{K}C_{uuu}G_{u{\bar{u}}}^{-1}{\bar{U}}}\\{d_{u}{\bar{U}}}&{=}&{G_{u{\bar{u}}}{\bar{V}}}\\{d_{u}{\bar{V}}}&{=}&{0}\\\end{array}
\alpha_{\lambda}=\left(\frac{\omega_{\lambda}^{\mathrm{out}}}{\omega_{\lambda}^{\mathrm{in}}}\right)^{1/2}\frac{\Gamma(1-\imath\omega_{\lambda}^{\mathrm{in}}\eta_{0})\Gamma(-\imath\omega_{\lambda}^{\mathrm{out}}\eta_{0})}{\Gamma(-\imath\omega_{}^{+}\eta_{0})\lambda\Gamma(1-\imath\omega_{\lambda}^{+}\eta_{0}))},
\lambdaH_{c}(\zeta)=\;{\int}_{-1}^{1}\;d{\xi}\;\frac{H_{c}(\xi)}{\xi+\zeta+2\frac{1+\kappa}{1-\kappa}}\;\frac{\sqrt{1-\xi^{2}}}{\sqrt{(\xi+\frac{3+\kappa}{1-\kappa})(\xi+\frac{1+3\kappa}{1-\kappa})}}\;,
a_{\mu}^{\xi}(x):=\langle\Lambda|{\calA}_{\mu}^{\xi}(x)|\Lambda\rangle,\quad{\calA}_{\mu}^{\xi}(x)=\xi^{\dagger}(x){\calA}_{\mu}(x)\xi(x)+{\frac{i}{g}}\xi^{\dagger}(x)\partial_{\mu}\xi(x),
ds_{(b_{0},\bar{b}_{0})}^{2}=l^{2}\left[-d\tau^{2}+\frac{b_{0}}{c/6}\frac{dz^{2}}{z^{2}}+\frac{\bar{b}_{0}}{c/6}\frac{d\bar{z}^{2}}{\bar{z}^{2}}+\left(e^{-2\tau}+\frac{b_{0}\bar{b}_{0}}{(c/6)^{2}z^{2}\bar{z}^{2}}e^{2\tau}\right)dzd\bar{z}\right],
\lambda(p^{\mu})\,:=\left(\begin{array}{c}{\left(\zeta_{\lambda}\,\Theta_{[1/2]}\right)\,\phi_{_L}^{\ast}(p^{\mu})}\\{\phi_{_L}(p^{\mu})}\\\end{array}\right)\,\,,\quad\rho(p^{\mu})\,:=\left(\begin{array}{c}{\phi_{_R}(p^{\mu})}\\{\left(\zeta_{\rho}\,\Theta_{[1/2]}\right)^{\ast}\,\phi_{_R}^{\ast}(p^{\mu})}\\\end{array}\right)\,\,\quad.
\Gamma(p,P)=-e^{2}\int\mathrm{~\,~\frac{d^3q}{(2\pi)^3}~\,~}D(p-q)\gamma_{\mu}S(\mathrm{~\,~\frac{1}{2}~\,~}P+q)\Gamma(q,P)S(-\mathrm{~\,~\frac{1}{2}~\,~}P+q)\gamma_{\mu},
S=\int_{\tau_{1}}^{\tau_{2}}\d\tau\int_{\sigma_{1}(\tau)}^{\sigma_{2}(\tau)}\d\sigma\,sqrt{-g}{\calL}+\int_{\tau_{1}}^{\tau_{2}}\d\tau\L^{(1)}+\int_{\tau_{1}}^{\tau_{2}}\d\tau\L^{(2)}\,
<N\delta_{\beta\rho}\delta^{d}(x-y)F[\sigma]>-<\psi_{\rho}^{a}(y){\frac{\deltaF}{\delta\psi_{\beta}^{a}(x)}}>-i<\psi_{\rho}^{a}(y){\frac{\deltaS}{\delta\psi_{\beta}^{a}(x)}}F[\sigma]>=0
S_{\mathrm{Aux}}=\intd^{3}x(\bar{\pi_{a}}\bar{C_{a}}^{\ast}+\bar{\pi_{a}}^{(1)\mu}\bar{C_{a}}_{\mu}^{(1)\ast}+\cdots\cdots+\bar{\pi_{a}}^{(n)\mu}\bar{C_{a}}_{\mu}^{(n)\ast}+\cdots).
\bar{G}_{2n}=\sum_{l=1}^{n-1}\bar{G}_{2l-2}^{(0)}\bar{G}_{2n-2l}+\bar{G}_{2n-2}-g\bar{G}_{2n-2}\bar{G}_{4}-g\bar{G}_{2n}\bar{G}_{2}+3g\bar{G}_{2n-2}\bar{G}_{2}.
H\;=\;-\sum_{j=1}^{2N-1}\,e_{j}\;=\;-\frac{1}{\sqrt{2}}\,\Bigl(\sum_{j=1}^{N-1}\,\sigma_{j}^{x}\sigma_{j+1}^{x}\;+\;\sum_{j=1}^{N}\,\sigma_{j}^{z}\Bigr)\;-\;\frac{2N-1}{\sqrt{2}}\,.
F_{g}=\sum_{\beta}\sum_{n\geq0}\frac{1}{n!}\langle\omega^{n}\rangle_{g,\beta}=\sum_{\beta}\sum_{n\geq0}\sum_{i_{1},\ldots,i_{n}}\frac{t_{i_{1}}\cdotst_{i_{n}}}{n!}\langle\omega_{i_{1}}\cdots\omega_{i_{n}}\rangle_{g,\beta}\,.
P_{12}\cdotsP_{N-1N}\overline{{G}}_{\varepsilon}^{(nl)}(\zeta_{2},\cdots,\zeta_{N},\zeta_{1}q^{-2})=r_{\varepsilon}(\zeta_{1})D_{1}^{(l-n)}\overline{{G}}_{\varepsilon}^{(nl)}(\zeta_{1},\cdots,\zeta_{N})\prod_{j=2}^{N}\frac{\zeta_{j}}{\zeta_{1}}.
\left(\begin{array}{c}{f_{x}}\\{f_{y}}\\{g_{x}}\\{g_{y}}\\\end{array}\right)=\frac{1}{4}\left(\begin{array}{cccc}{1}&{1}&{1}&{1}\\{-i}&{i}&{-i}&{i}\\{-i}&{-i}&{i}&{i}\\{-1}&{1}&{1}&{-1}\\\end{array}\right)\left(\begin{array}{llll}{\kappa_{+}}\\{\kappa_{-}}\\{\lambda_{+}}\\{\lambda_{-}}\\\end{array}\right),
D_{2}^{\mathrm{finite}}=-\frac{\sigma\zeta^{2}}{64\epsilon^{2n+1}}I_{2n+1/2}+\frac{\zeta}{32\epsilon^{n}}\left\{\left[\dot{\alpha}+\alpha\left(\frac{2\dot{A}}{A}+\frac{2\dot{l}}{3l}\right)\right]I_{n+3/2}+\left(\frac{\dot{\epsilon}}{\epsilon}-\frac{2\dot{A}}{A}-\frac{2\dot{l}}{3l}\right)I_{n+1/2}\right\}\;\;,
g_{\mu\nu}^{(d+1)}=\left(\begin{array}{cc}{g_{\alpha\beta}^{(d)}+e^{\psi_{1}}A_{\alpha}A_{\beta}}&{e^{\psi_{1}}A_{\alpha}}\\{e^{\psi_{1}}A_{\beta}}&{e^{\psi_{1}}}\\\end{array}\right),
\phi_{R}\rightarrowB^{\prime}e^{-\gammaax}e^{\gammaaL}\frac{\Gamma(2\gamma)\Gamma(1+2\sigma)}{\Gamma(1+\gamma+\sigma+\lambda)\Gamma(\gamma+\sigma-\lambda)}+B^{\prime}e^{\gammaax}e^{-\gammaaL}\frac{\Gamma(-2\gamma)\Gamma(1+2\sigma)}{\Gamma(1-\gamma+\sigma-\lambda)\Gamma(-\gamma+\sigma-\lambda)}
\mathrm{Im}\,\left(\frac{\nu_{-}^{2}-\mu\,\nu_{+}}{\nu_{+}^{2}+\mu\,\nu_{-}}\right)^{3/2}=0\;,\qquad\mathrm{Im}\,\left(\frac{\nu_{-}^{2}+\mu\,\nu_{+}}{\nu_{+}^{2}-\mu\,\nu_{-}}\right)^{3/2}=0\;~,
\begin{array}{l}{\displaystyle\mu_{\pm}=\frac{n}{2}\left(1\pm\sqrt{1+\frac{4m_{0}^{2}}{n^{2}}}\right)~,}\\{\displaystyle\nu_{\pm}=\frac{n-1}{2}\left(-1\pm\sqrt{1+\frac{4(m_{0}^{2}-\epsilon^{2})}{(n-1)^{2}}}\right)~,}\\{\displaystyle\lambda_{\pm}=\frac{n-1}{2}\left(1\pm\sqrt{1+\frac{4(m_{0}^{2}-\epsilon^{2})}{(n-1)^{2}}}\right)~.}\\\end{array}
{\frac{1}{u}}\rightarrow{\frac{1}{u}}+{\frac{1}{2u}}\;(\partial_{+}\xi^{+}+\partial_{-}\xi^{-})-{\frac{\mathrm{sin}\theta_{r}}{2}}\;(\partial_{+}^{2}\xi^{+}-\partial_{-}^{2}\xi^{-})\.
\big\langle\,T\,{\calO}_{s,n}^{(\mathrm{ren})}(x,t;r)\,{\calO}_{s,n}^{(\mathrm{ren})\dagger}(0,0;r)\,\big\rangle\big|_{\sqrt{x^{2}-t^{2}}=r}=\Big(\frac{t-x}{t+x}\Big)^{\frac{sn}{2}}\,.
+{\frac{2}{3}}\bar{\Theta}\Gamma^{\underline{{a}}}{\calD}_{\beta}\Theta\bar{\Theta}\Gamma^{\underline{{b}}}{\calD}_{a}\Theta)+{\frac{1}{3}}i\bar{\Theta}\Gamma_{\underline{{ab}}}{\calD}_{a}\Theta\bar{\Theta}\Gamma^{\underline{{a}}}{\calD}_{\alpha}\Theta\bar{\Theta}\Gamma^{\underline{{b}}}{\calD}_{\beta}\Theta],
\Sigma_{a_{1},\ldots,a_{p};\;b_{1},\ldots,b_{q};k}^{\;i_{1},\ldots,i_{p};\;j_{1},\ldots,j_{q}}(z)\equiv\sum_{n=1}^{\infty}\frac{(n!)^{2}}{(2n)!}\frac{z^{n}}{n^{k}}
S=\intd^{4}x\sqrt{-g}[\frac{\mathcal{R}}{16\piG}-\frac{1}{4\pi}\frac{1}{2}((\partial_{\mu}\phi)(\partial^{\mu}\phi)+(\frac{g\phi}{2})^{2}A_{\mu}A^{\mu}+V(\phi)+\xi\phi^{2}\mathcal{R})-\frac{1}{4\pi}\frac{1}{4}\midF\mid^{2}].
\{G_{Q}(\eta),G_{Q}(\rho)\}^{*}=G_{Q}([\eta,\rho])+\int_{\partial\Sigma}\omega\mathrm{\tiny~\wedge\,~}\omega\mathrm{\tiny~\wedge\,~}\dots\mathrm{\tiny~\wedge\,~}\omega\mathrm{\tiny~\wedge\,~}\eta_{a}d\rho^{a},
{\frac{1}{\beta}}\sum_{n}{\frac{e^{i\omega_{n}\tau}}{\omega_{n}^{2}+g^{2}(\vec{k})}}={\frac{e^{-g(\vec{k})|\tau|}}{2g(\vec{k})}}+{\frac{1}{2g(\vec{k})}}[e^{g(\vec{k})\tau}+e^{-g(\vec{k})\tau}]{\frac{1}{e^{\betag(\vec{k})}-1}}\;,
h_{\mathrm{red}}=\intd^{2}x_{\bot}\left[-\frac{g^{2}L\eta}{4}\frac{1}{J}\frac{\delta}{\delta\varphi}J\frac{\delta}{\delta\varphi}+\frac{4}{g^{2}L\eta}(\nabla_{\bot}\varphi)^{2}\right].
F_{i}(\zeta,\upsilon+\chi)\equiv\frac{(\upsilon+\chi)f_{i}}{2\sqrt{2}(\zeta+k)^{2}(\zeta+b)^{1/2}}[\zeta^{2}+(3h_{i}-k)\zeta+2b(h_{i}-k)+kh_{i}],\qquadi=E^{\prime},U^{\prime},D^{\prime}.
\bigl(S_{\mu\nu}\bigr)^{-1}=\left(\begin{array}{cc}{0}&{-1}\\{1}&{0}\\\end{array}\right)\,\Bigl(\eta_{\mu\nu}+(\alpha-1)\,\eta_{0\mu}\eta_{0\nu}\Bigr)\,\delta(\vec{x}-\vec{y})\,.
\frac{\partial}{\partialx^{i}}\langleJ^{ai}(x)J^{bj}(y)J^{ck}(z)\rangle|_{anomaly}=-2\alphad^{abc}\epsilon^{ijkl}\frac{\partial}{\partialx^{i}}\delta^{(d)}(x-y)\frac{\partial}{\partialx^{l}}\delta^{(d)}(x-z).
s=\sum_{Q}m_{Q}^{2}S_{Q}(p),\;r=\sum_{Q}m_{Q}^{2}R_{Q}(p),\;\tilde{s}=\sum_{Q}m_{Q}^{4}S_{Q}(p),\;\tilde{r}=\sum_{Q}m_{Q}^{4}R_{Q}(p)
S_{5}=\int\!\!d^{3}x~\left(\alpha_{abcde}\varphi_{a}\varphi_{b}\varphi_{c}\varphi_{d}\varphi_{e}\right),\quadS_{6}=\int\!\!d^{3}x~\left(\alpha_{abcdef}\varphi_{a}\varphi_{b}\varphi_{c}\varphi_{d}\varphi_{e}\varphi_{f}\right),
F_{1}(q^{2})=Z+\int\frac{{\mathrmd}^{2}{\vec{k}}_{\perp}{\mathrmd}x}{16\pi^{3}}\Big[\psi_{+\frac{1}{2}}^{\uparrow\*}(x,{\vec{k}^{\prime}}_{\perp})\psi_{+\frac{1}{2}}^{\uparrow}(x,{\vec{k}}_{\perp})+\psi_{-\frac{1}{2}}^{\uparrow\*}(x,{\vec{k}^{\prime}}_{\perp})\psi_{-\frac{1}{2}}^{\uparrow}(x,{\vec{k}}_{\perp})\Big]\,
X^{a_{1}\ldotsa_{n}}(x_{1},\ldots,x_{n},\gamma)=\oint_{n}^{a_{n}}\ldots\oint_{\gamma}dy_{1}^{a_{1}}\delta(x_{n}-y_{n})\ldots\delta(x_{1}-y_{1})\Theta_{\gamma}(0,y_{1},\ldots,y_{n})
q_{a}^{2}=\frac{\displaystyle\prod_{b=1}^{N}(\bar{\sigma}_{a}^{2}-\lambda_{b})}{\displaystyle\prod_{b=1,b\neqa}^{N}(\bar{\sigma}_{a}^{2}-\bar{\sigma}_{b}^{2})}=\frac{\Lambda(\bar{\sigma}_{a}^{2})}{A^{\prime}(\bar{\sigma}_{a}^{2})}
\tilde{G}_{ab}=\eta_{ab}+2\kappaH_{ab}+2\kappa\Bigl(\partial_{a}X^{i}H_{ib}+H_{aj}\partial_{b}X^{j}\Bigr)+\partial_{a}X^{i}\partial_{b}X^{j}\delta_{ij}+2\kappa\partial_{a}X^{i}\partial_{b}X^{j}H_{ij}~.
S_{\mathrm{eff}}=\int\!d^{3}x\,{\calL}_{A}^{\prime}+iN\,\mathrm{Tr}\,\mathrm{Ln}\!\left[-\,\partial^{2}-m^{2}\right]-iN\sum_{n=1}^{\infty}\frac{1}{n}\,\mathrm{Tr}\!\left[\frac{1}{-\,\partial^{2}-m^{2}}\,\Gamma\right]^{n}.
\left(\begin{array}{cc}{2B}&{B}\\{B}&{-2B}\\\end{array}\right)\left(\begin{array}{c}{\dot{\bar{V}}^{-}}\\{\dot{V}^{+}}\\\end{array}\right)=\lambda\left(\begin{array}{c}{\ddot{\bar{V}}^{-}}\\{\ddot{V}^{+}}\\\end{array}\right),
{\displaystyleQ_{\underline{\alpha}}=\frac{eg}{2F}{\calP}^{\dot{\beta}\beta}F_{\beta\dot{\beta},\underline{\alpha}}+\frac{igm^{2}}{2F}D_{\underline{\alpha}}F,\bar{Q}_{\underline{{\dot{\alpha}}}}=\frac{eg}{2F}{\calP}^{\dot{\beta}\beta}F_{\beta\dot{\beta},\underline{{\dot{\alpha}}}}+\frac{igm^{2}}{2F}\bar{D}_{\underline{{\dot{\alpha}}}}F.}
H(t,M,\kappa_{\tau})=\rho^{\tau[\frac{d-\frac{m}{2}}{2}+1]}\nonumber\,\;\;H(\frac{t}{\rho^{2\tau}},\frac{M}{\rho^{2\tau[\frac{d-\frac{m}{2}}{2}-1]}},\frac{\kappa_{\tau}}{\rho}).
P_{\zeta}(k)=\frac{k^{3}}{2\pi^{2}}|\zeta|^{2}=p2^{2\tilde{\nu}-3}\frac{\Gamma(\tilde{\nu})^{2}}{2\pi^{3}}\left[\frac{{\calH}(1-p)}{M_{\mathrm{{pl}}}p}\right]^{4\tilde{\nu}}\frac{M_{\mathrm{pl}}^{2\tilde{\nu}-3}}{k^{2\tilde{\nu}-3}}\,.
\begin{matrix}{{\calT}_{zz}=T_{zz}^{grav}+{\frac{1}{2}}T_{zz}^{gh},}&{{\calJ}_{z}=J_{z}^{grav}+{\frac{1}{2}}J_{z}^{gh},}\\{{\calG}_{+z}={G}_{+z}^{grav}+{\frac{1}{2}}{G}_{+z}^{gh},}&{{\calG}_{-z}=G_{-z}^{grav}+{\frac{1}{2}}G_{-z}^{gh}.}\\\end{matrix}
\begin{aligned}{\frac{\partial\psi_{ik}(u)}{\partialu_{j}}}&{{}=\beta_{ij}(u)\psi_{jk}(u),\quadi\nej}\\{\sum_{k=1}^{n}\frac{\partial\psi_{ij}(u)}{\partialu_{k}}}&{{}=0,}\\\end{aligned}
\begin{array}{rl}{}&{\hat{\Gamma}(k_{1},k_{2},k_{3},k_{4})}\\{=}&{\displaystyle-g_{\pi}^{2}(2\pi)^{4}\delta^{4}(\sum_{i}k_{i})\sum_{\{\lambda_{j}\}}\sum_{\lambda}\prod_{j=1}^{4}\left[\eta_{\lambda_{j}}(k_{j})\right]\left(A_{s}+A_{t}+A_{u}\right),}\\\end{array}
S_{3}=-\frac{m^{2}}{2}\intd^{8}z\sum_{N=0}^{\infty}\sum_{l=0}^{N}\frac{\sigma^{l}(z)\bar{\sigma}^{N-l}(z)}{l!(N-l)!}G_{+-}(z,z)=\frac{m^{2}}{2}\intd^{8}ze^{\sigma+\bar{\sigma}}G_{+-}(z,z)
f^{(\pm)}(\nu;k;x)=e^{\mpikx}+e^{\mpikx}\sum_{n=0}^{\infty}(\frac{1}{\pi})^{n+1}\int_{1}^{\infty}d\alpha_{o}...\int_{1}^{\infty}d\alpha_{n}\frac{(\prod_{j=0}^{n-1}D(\nu;\alpha_{j})e^{-2\alpha_{j}x})}{[\prod_{j=0}^{n-1}(\alpha_{j}+\alpha_{j+1})](\alpha_{o}\pmik)}
Z_{n}^{\dagger}(\theta_{n})\ldotsZ_{1}^{\dagger}(\theta_{1})\left|0\right\rangle_{\mathrm{out}}=\prod_{1\leqi<j\leqn}S_{ij}(\theta_{i}-\theta_{j})Z_{1}^{\dagger}(\theta_{1})\ldotsZ_{n}^{\dagger}(\theta_{n})\left|0\right\rangle_{\mathrm{in}}\,,
\left(\begin{array}{c}{\psi_{\uparrow}}\\{\psi_{\downarrow}}\\{\chi_{\uparrow}}\\{\chi_{\downarrow}}\\\end{array}\right)=\left(\begin{array}{c}{h_{1}(r)e^{i(j+{\frac{1}{2}}+n)\theta}}\\{h_{2}(r)e^{i(j-{\frac{1}{2}}+n)\theta}}\\{h_{3}(r)e^{i(j+{\frac{1}{2}})\theta}}\\{h_{4}(r)e^{i(j-{\frac{1}{2}})\theta}}\\\end{array}\right)e^{-i\omegat},
\nu_{\alpha,y}^{(1)}(1,x)=\left(\begin{array}{c}{-f_{\hat{\alpha}\hat{\beta}}^{-1}M_{\hat{\beta}1}}\\{1}\\{0}\\\end{array}\right)\delta(x-y),~~~\nu_{\alpha,y}^{(1)}(2,x)=\left(\begin{array}{c}{-f_{\hat{\alpha}\hat{\beta}}^{-1}M_{\hat{\beta}2}}\\{0}\\{1}\\\end{array}\right)\delta(x-y).
\left[N_{i},p_{j}\right]=i\,\delta_{ij}\left({\frac{\kappa}{2}}\left(1-e^{-2{p_{0}/\kappa}}\right)+{\frac{1}{2\kappa}}\vec{p}\,{}^{2}\,\right)-i\,{\frac{1}{\kappa}}p_{i}p_{j},
T(E)={\frac{C+C_{2}^{2}I_{5}+2EC_{2}\left(2-C_{2}I_{3}\right)}{\left(C_{2}I_{3}-1\right)^{2}-I(E)\left[C+C_{2}^{2}I_{5}+2EC_{2}\left(2-C_{2}I_{3}\right)\right]}}
\mathcal{G}_{ij}=\left(\begin{array}{cc}{\frac{1}{T^{2}}}&{\frac{1}{2TU}}\\{\frac{1}{2TU}}&{\frac{1}{U^{2}}}\\\end{array}\right),\qquad\mathcal{G}^{ij}=\frac{4}{3}\left(\begin{array}{cc}{T^{2}}&{-\frac{TU}{2}}\\{-\frac{TU}{2}}&{U^{2}}\\\end{array}\right).
L_{B_{N}}=L_{D_{N}}+ig_{1}\sum_{j=1}^{N}x(q^{j})(\sigma_{+}^{(j)}-\sigma_{-}^{(j)}),\quadM_{B_{N}}=M_{D_{N}}+ig_{1}\sum_{j=1}^{N}y(q^{j})(\sigma_{+}^{(j)}+\sigma_{-}^{(j)}).
\eta=-\frac{b_{0}\frac{g^{2}}{(4\pi)^{2}}+b_{1}\,\frac{g^{2}}{(4\pi)^{2}}w_{2}}{1+d_{1}\,\frac{g^{2}}{(4\pi)^{2}}+d_{2}\,\frac{g^{2}}{(4\pi)^{2}}w_{2}},
Q_{BRST}=\ointdz\left\{c(-{\frac{1}{2}}\partialX\partialX+{\frac{1}{2}}\psi\partial\psi-{\frac{1}{2}}(\partial\phi)^{2}-2\partial^{2}\phi)+c\partialcb+{\frac{1}{2}}e^{\phi}\eta\psi\partialX-{\frac{1}{4}}\eta\partial\etab\right\}.
E_{\\nu}^{\underline{{\mu}}}=\left(\begin{matrix}{\begin{matrix}{1,}\\\end{matrix}}&{\begin{matrix}{0,}&{\ldots}&{0}\\\end{matrix}}\\{\begin{matrix}{e^{\underline{{1}}}}\\{\vdots}\\{e^{\underline{{d}}}}\\\end{matrix}}&{\begin{matrix}{h_{\i}^{\underline{{a}}}}\\\end{matrix}}\\\end{matrix}\right)
\{g(z,\bar{z}),J_{0}(z_{1})\}=\frac{1}{2}\,(t_{0}\,g(z,\bar{z}))\,\delta(z-z_{1}),~~~~\{g(z,\bar{z}),\bar{J}_{0}(\bar{z}_{1})\}=\frac{1}{2}\,(g(z,\bar{z})\,t_{0})\,\delta(\bar{z}-\bar{z}_{1}),
0=\left[\frac{1}{P}\frac{dP(A)}{dA}-4a\frac{d\bar{\phi}}{dA}\right]\left[\frac{d^{2}\bar{\phi}}{dA^{2}}+2a\left(\frac{d\bar{\phi}}{dA}\right)^{2}-\frac{\bar{f}_{0}^{2}}{4P^{2}}\right].
e=\frac{i}{2}\left\{d\tau+\sum_{a=1}^{N}\frac{S_{a}}{2i}\left({\tt\frac{1}{z-z_{a}}}dz-{\tt\frac{1}{\bar{z}-\bar{z}_{a}}}d\bar{z}\right)\right\}\left(\begin{array}{rr}{1}&{0}\\{0}&{-1}\\\end{array}\right)+\frac{i}{2}\left(\begin{array}{cc}{0}&{dz}\\{d\bar{z}}&{0}\\\end{array}\right)\,,
|0;0\rangle=\frac{1}{2}\left(|q_{+};q_{+}\rangle+|q_{+};q_{-}\rangle+|q_{-};q_{+}\rangle+|q_{-};q_{-}\rangle\right)\equiv\sum_{q_{\pm}^{a},q_{\pm}^{b}}\psi(q_{\pm}^{a},q_{\pm}^{b})|q_{\pm}^{a};q_{\pm}^{b}\rangle.
L^{Kr}=\sum_{i}p_{i}E_{ii}-\frac{\nu}{2\pii}\sum_{i\neqj}\frac{\sigma(z+\frac{\alpha_{ij}(D)}{\pik}{\mathrm{I}m\tau})}{\sigma(z)\sigma(\frac{\alpha_{ij}(D)}{\pik}{\mathrm{I}m\tau})}E_{ij}
m=(\omega_{2},\omega_{1})\Omega\left(\begin{matrix}{\upsilon_{2}}\\{\upsilon_{1}}\\\end{matrix}\right)=(\omega_{2},\omega_{1})\left(\begin{matrix}{0}&{-1}\\{1}&{0}\\\end{matrix}\right)\left(\begin{matrix}{\upsilon_{2}}\\{\upsilon_{1}}\\\end{matrix}\right)=\omega_{1}\upsilon_{2}-\omega_{2}\upsilon_{1}\.
Z_{het}(\tau,\bar{\tau})=\tau_{2}^{-4}\frac{\bar{\Theta}(\bar{\tau})}{\bar{\eta}^{24}(\bar{\tau})}\eta^{-12}(\tau)[\theta_{3}^{4}(0|\tau)-\theta_{3}^{4}(0|\tau)-\theta_{2}^{4}(0|\tau)]\;,
\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\piG_{N}}{3}\left[\frac{1}{2}\dot{\phi}^{2}+V_{eff}(\phi)\right]-\frac{K}{a^{2}}+\frac{\Lambda_{4}}{3}+\frac{\mu}{a^{4}}+O(\epsilon^{2}),
\sum_{l=0}^{N-1}\frac{w(v_{3},-l)\gamma(b,l)}{w(v_{4},a-l)}=\varphi_{2}(v_{3},v_{4})\frac{w(v_{3}^{\prime},a-b)}{w(v_{4}^{\prime},-b)w(v_{4}/v_{3},a)},
\left(\mathcal{R}^{\left(e\right)}\mathcal{T}^{\left(e\right)}\right)\mathcal{V}^{\left(e\right)}=\mathcal{V}^{\left(e\right)},\quad\mathrm{versus}\quad\mathcal{R}^{\left(e\right)}\left(\mathcal{T}^{\left(e\right)}\mathcal{V}^{\left(e\right)}\right)=0.
U_{e}=\frac{1}{\beta}\left[\frac{e^{2}}{8}\left(|\sigma_{1}|^{2}-|\sigma_{2}|^{2}\right)^{2}+\left(\beta\tilde{m}^{2}-\frac{1}{2}\alphae\tilde{k}\right)|\sigma_{1}|^{2}+\left(\beta\tilde{m}^{2}+\frac{1}{2}\alphae\tilde{k}\right)|\sigma_{2}|^{2}\right],
\int_{-\pi/2+3\epsilon/2}^{\pi/2-\epsilon/2}d\theta_{2}\int_{-\pi/2+\epsilon/2}^{\theta_{2}-\epsilon}d\theta_{1}~f(\theta_{1},\theta_{2})=\int_{-\pi/2+\epsilon/2}^{\pi/2-3\epsilon/2}d\theta_{2}\int_{\theta_{2}+\epsilon}^{\pi/2-\epsilon/2}d\theta_{1}~f(\theta_{2},\theta_{1})
\tilde{\mathrm{G}}_{d}=\mathrm{G}_{d}-{\frac{2^{n}d}{2(d-1)}}f_{d}^{0}\Box^{n-1}R+\sum_{j=1}^{J}f_{d}^{j}{\calD}_{j}=\mathrm{G}_{d}+\nabla_{\alpha}J_{d}^{\alpha},
\left(\begin{matrix}{\Gamma_{\phi^{-}}^{11}}&{\Gamma_{\phi^{-}}^{12}}\\{\Gamma_{\phi^{-}}^{21}}&{\Gamma_{\phi^{-}}^{22}}\\\end{matrix}\right)=M_{C}^{-1}\left(\begin{matrix}{\Gamma_{R}^{\phi^{-}}(p)}&{0}\\{0}&{{\Gamma_{R}^{\phi^{-}}}^{*}(p)}\\\end{matrix}\right)M_{C}^{-1}
S_{\mathrm{\small{CS}}}={\frac{1}{2\pi\alpha^{\prime}}}\int_{\mathrm{D3}}C_{R}\wedge\left[\mathrm{Tr}\left(e^{2\pi\alpha^{\prime}{\calF}_{1}}\right)-\mathrm{Tr}\left(e^{2\pi\alpha^{\prime}{\calF}_{2}}\right)\right]~.
f_{D1}^{B}=4\piV\sum_{l\ne0}\sum_{p\wedgeq=1}\sum_{n^{ij}}{\frac{e^{-2\pil|p+q\tau|\sqrt{n^{ij}~g_{ik}g_{jl}~n^{kl}}+2\piiln^{ij}(q{\calB}_{ij}-pB_{ij})}}{\sqrt{n^{ij}~g_{ik}g_{jl}~n^{kl}}}}
\begin{array}{rl}{}&{(g(x)h(y))\starD(z,m^{2})}\\{=}&{\displaystyle(2\pi)^{-3}\intd^{2}\vec{p}_{\bot}\left(g(x)\star_{x}e^{+i\vec{p}_{\bot}\cdot\vec{x}_{\bot}}\right)\left(h(y)\star_{y}e^{-i\vec{p}_{\bot}\cdot\vec{y}_{\bot}}\right)\int\frac{dp_{3}}{2E_{\vec{p}}}e^{-i(E_{\vec{p}}z_{0}-p_{3}z_{3})},}\\\end{array}
\frac{1}{2}\int\int\mathrm{d}w\mathrm{d}x\left(\left(\partial_{\mu}\tilde{y}\right)\left(\partial_{\mu}\tilde{y}\right)+\left(\partial_{\mu}\tilde{z}\right)\left(\partial_{\mu}\tilde{z}\right)\right)=\frac{1}{2}\int\mathrm{d}t\left(\tilde{y}\left(n.\partial\tilde{y}\right)+\tilde{z}\left(n.\partial\tilde{z}\right)\right)
\begin{aligned}{[t]\chi_{mn}^{\pm}}&{{}\mapsto-\chi_{mn}^{\mp}}\\{(\psi_{m},\tilde{\psi}_{n})}&{{}\mapsto(-\tilde{\psi}_{m},-\psi_{n})}\\{(\eta,\tilde{\eta})}&{{}\mapsto(-\tilde{\eta},-\eta)}\\{V_{m}}&{{}\mapsto-V_{m}}\\{\mu}&{{}\mapsto-\mu~.}\\\end{aligned}
{\calF}(\sigma)\equiv\int_{0}^{\infty}dy\,y^{-\sigma-1}\,F(y)={\frac{8}{\pi}}\int_{0}^{1}{\frac{du}{\sqrt{1-u^{2}}}}\,u^{2}\,\int_{0}^{\infty}dy\,y^{-\sigma+1}\,K_{1}^{2}(2uy).
\omega=\mathrm{Im}(a^{*}\,a_{D})\,\frac{n_{E}^{(1)}n_{M}^{(2)}-n_{M}^{(1)}n_{E}^{(2)}}{\left|\,n_{E}^{(1)}\,a+n_{M}^{(1)}\,a_{D}\right|\left|\,n_{E}^{(2)}\,a+n_{M}^{(2)}\,a_{D}\right|}+{\calO}(\omega^{2})\,.
\displaystyle{M=\left(\begin{array}{ccc}{w+\textstyle{\frac{1}{2}}\lambda}&{ia{\cdot\gamma}}&{\sqrt{2}\varepsilon^{b}}\\{ib{\cdot\gamma}}&{w-{\textstyle\frac{1}{2}}\lambda}&{\sqrt{2}\rho^{b}}\\{-\sqrt{2}\bar{\rho}_{a}}&{-\sqrt{2}\bar{\varepsilon}_{a}}&{t_{a}{}^{b}}\\\end{array}\right)}\,,
g_{cr}\ge\frac{1}{1-\frac{L\pi}{2\Lambda^{2}}+\frac{M^{2}}{3\Lambda^{2}}(\frac{\pi}{2L}-1/\Lambda^{2})-\frac{M}{\Lambda^{2}}\mathrm{ln}2}\approx\frac{1}{1-\frac{\pi|e||f_{1}|^{1/2}}{2\Lambda^{2}}-\frac{|e|}{\Lambda^{2}}(\frac{f_{2}^{2}}{|f_{1}|})^{1/2}\mathrm{ln}2}.