harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{rcl}{\tilde{S}^{\prime}\[X^{\mu},P_{y},b_{\xi},\gamma]}&{=}&{-{\textstyle\frac{p}{2}}\intd\xi\\sqrt{|\gamma|}\\gamma^{-1}\left[e^{-\frac{2}{3}\phi}g_{\xi\xi}+\gammae^{-\frac{4}{3}\phi}\left(\frac{P_{y}}{p}\right)^{2}\right]}\\{}&{}&{}\\{}&{}&{+\intd\xiP_{y}\left[C^{(1)}{}_{\xi}-\frac{m}{2}\(2\pi\alpha^{\prime})\b_{\xi}\right]\,.}\\\end{array}
\langle\Phi\midT\left(\hat{A}^{0}\left(x_{1}\right)\hat{A}^{0}\left(x_{2}\right)\hat{A}^{0}\left(x_{3}\right)...\right)\mid\Phi\rangle=\left(\frac1i\frac\delta{\deltaJ\left(x_{1}\right)}\frac1i\frac\delta{\deltaJ\left(x_{1}\right)}\frac1i\frac\delta{\deltaJ\left(x_{1}\right)}...Z\left[J\right]\right)_{J=0}.
d\sigma^{m}=-{\textstyle\frac{1}{2}}f_{np}{}^{m}\sigma^{n}\wedge\sigma^{p}\,,\qquadf_{mn}{}^{p}=-2(U^{-1})^{r}{}_{m}(U^{-1})^{s}{}_{n}\,\partial_{[r}U^{p}{}_{s]}\,,
\left\{q_{(u)}^{-A},q_{(u)B}^{-}\right\}=2\delta^{A}{}_{B}\delta_{uv}H_{2(u)}+i\mu\delta_{uv}\left((\gamma^{ij}\Pi){}^{A}{}_{B}J_{(u)}^{ij}+(\gamma^{i^{\prime}j^{\prime}}\Pi^{\prime}){}^{A}{}_{B}J_{(u)}^{i^{\prime}j^{\prime}}-{\tilde{\Pi}}^{A}{}_{B}J_{(u)}^{LR}\right)\,,
\left.+\left(\frac{1}{2}\mathrm{tr}p_{r}^{2}+\frac{1}{12}p_{12}p_{21}+\frac{17}{72}\right)R^{2}+\left(\frac{9}{2}\Lambda+\mathrm{tr}p_{r}p_{l}\right)R+\left(\frac{9}{2}\Lambda^{2}+\frac{1}{2}\mathrm{tr}p_{l}^{2}\right)\right]\;.
\begin{aligned}{j_{3}=}&{{}j_{21}^{\pm}-1-n-mb^{-2}}\\{j_{3}=}&{{}-j_{21}^{\pm}-1-n-mb^{-2}}\\\end{aligned}\qquad\qquad\begin{aligned}{j_{3}=}&{{}j_{21}^{\pm}+n+mb^{-2}}\\{j_{3}=}&{{}-j_{21}^{\pm}+n+mb^{-2}}\\\end{aligned}
\left|\,{p\;q\;\;i\atopp\;p\;(p+2q)/3}\,\right\rangle_{\alpha}=\eta^{b-i}\;{\zeta^{*}}^{i}\;\lambda^{a-q+i}\;{\nu^{*}}^{a+b-p-i}\;\alpha^{p+q-a-b}\;,
\hat{G}^{\hat{\alpha}_{1}\ldots\hat{\alpha}_{4}}=\left(f^{-1}\right)^{\hat{\alpha}_{1}\ldots\hat{\alpha}_{4}}{}_{\hat{\beta}_{1}\ldots\hat{\beta}_{4}}\left(\hat{B}^{\hat{\beta}_{1}\ldots\hat{\beta}_{4}}-{}^{\star}\hat{B}^{\hat{\beta}_{1}\ldots\hat{\beta}_{7}}\hat{A}_{\hat{\beta}_{5}\hat{\beta}_{6}\hat{\beta}_{7}}\right)\,,
Z_{L}^{(n)}(\{q\})=\tilde{Z}^{(n)}(\{q\},L)\prod_{s=1}^{n}\frac{Z^{(1)}(k_{s};l_{1s},l_{2s})}{k_{s}^{(3-2l_{1s})/2}(u_{s}-v_{s}-\mu_{s}\nu_{s})}
ds^{2}=(H_{1}H_{5})^{-\frac{1}{2}}\left[-dt^{2}+dx^{2}+H_{1}\sum_{i=6}^{9}dx_{i}^{2}\right]+(H_{1}H_{5})^{\frac{1}{2}}[dr^{2}+r^{2}d\Omega_{3}^{2}],
\mathcal{A}^{SO(4)}=\left(\begin{array}{cc}{\mathcal{A}_{\mu\nu}\sigma_{\mu\nu}}&{0}\\{0}&{\mathcal{A}_{\mu\nu}\bar{\sigma}_{\mu\nu}}\\\end{array}\right)=\left(\begin{array}{cc}{\mathcal{A}_{+}}&{0}\\{0}&{\mathcal{A}_{-}}\\\end{array}\right)\equiv\mathcal{A}_{a}^{SO(4)}T_{a}\,
E_{2}^{p,q}\:=\:H^{p}\left(S,\underline{{\mathrm{Ext}}}_{{\calO}_{S}}^{q}({\calS}_{1},{\calS}_{2})\right)\Longrightarrow\mathrm{Ext}_{S}^{p+q}\left({\calS}_{1},{\calS}_{2}\right)
\nabla_{j}=\nabla_{j}^{0}+i\left(\begin{array}{cc}{(A_{j})_{i}^{k}}&{\langle(b_{j})_{i}^{\beta}|}\\{|(b_{j})_{\alpha}^{k}\rangle}&{(\hat{D}_{j})_{\alpha}^{\beta}}\\\end{array}\right)\,.
W_{J_{1},n_{1}}^{\sigma_{1}}(z)W_{J_{2},n_{2}}^{\sigma_{2}}(0)=\frac{1}{z}{\tilde{f}}_{J_{1}n_{1}\sigma_{1};~J_{2}n_{2}\sigma_{2}}^{J_{3}n_{3}\sigma_{3}}W_{J_{3},n_{3}}^{\sigma_{3}}(0)~+~...~.
\begin{array}{rcrcl}{e^{\pmiH_{1}}}&{=}&{\frac1{\sqrt{k}}(\psi^{1}}&{\mp}&{i\psi^{2})~,}\\{e^{\pmi(H_{2}+\piN_{1})}}&{=}&{{\frac{1}{2\sqrt{k}}}\chi^{3}}&{\mp}&{{\frac{1}{\sqrt{k}}}\psi^{3}~,}\\{e^{\pmi(H_{3}+\piN_{2}+\piN_{1})}}&{=}&{\frac1{2\sqrt{k}}(\chi^{1}}&{\mp}&{i\chi^{2})~,}\\\end{array}
\begin{array}{l}{\displaystyle{\frac{A_{n}}{B_{n}}}\simr_{0}^{2\left(k+\alpha-n\right)}\qquadn-k+\frac{1}{2}-\alpha\leq0}\\{\displaystyle{\frac{B_{n}}{A_{n}}}\simr_{0}^{2\left(n+1-k-\alpha\right)}\qquadn-k+\frac{1}{2}-\alpha>0}\\\end{array}
\left\vert\Phi(P^{+})\right\rangle_{f}=\int_{0}^{P^{+}}dk_{1}dk_{2}\delta(k_{1}+k_{2}-P^{+})\phi_{bf}(k_{1},k_{2}){\frac{1}{N}}\mathrm{tr}[a^{\dagger}(k_{1}),b^{\dagger}(k_{2})]\left\vert0\right\rangle,
S_{\psi}=N\sum_{i=1}^{2}\mathrm{tr}_{N}\psi_{i}^{\dagger}\psi_{i}-N\frac{1}{2}g^{2}\mathrm{tr}_{N}\left\{(\sum_{i=1}^{2}\psi_{i}^{\dagger}\psi_{i})^{2}+\mathrm{e}^{-2\beta}(\sum_{i,j=1}^{2}\psi_{i}^{\dagger}L_{ij}\psi_{j})^{2}\right\}.
\frac{1}{\beta}\sum_{n=-\infty}^{\infty}\int^{\Lambda}\frac{d^{d}{\bfq}}{(2\pi)^{d}}\frac{e^{i\bar{\omega}_{n}\eta}(\rho-\bar{\omega}_{n})}{{\bfq}^{2}+(\rho-\bar{\omega}_{n})^{2}+\Sigma^{2}}=-\frac{ib}{4},
\tilde{R}\,=\,\left(R^{(+)}\right)^{-1}\,=\,\left(\begin{array}{cccc}{Q}&{0}&{0}&{0}\\{0}&{\lambda^{-1}}&{-\sigma}&{0}\\{0}&{0}&{\lambda}&{0}\\{0}&{0}&{0}&{-Q^{-1}}\\\end{array}\right)
{\calF}^{(4)}(\vec{X})=\sum_{n_{1},n_{2}=-\infty}^{+\infty}\frac{1}{\left[(X_{8}-2n_{1}\Omega_{1}-2\kappan_{2}\Omega_{2})^{2}+(X_{7}-2n_{2}\Omega_{2})^{2}+A^{2}\right]^{7}}
g_{nMN}=\sum_{k=0}^{\infty}C_{\xi}^{*k}C_{\eta}^{k}e_{n+k,M}\otimese_{kN}(\frac{q^{n}}{1-q^{2}})^{k}q^{k^{2}}\sqrt{\frac{[n]_{q^{2}}!}{[n+k]_{q^{2}}![k]_{q^{2}}!}}
\mathrm{i}\,\left(\begin{array}{c|c|c|c}{\sigma_{i}}&{0}&{0}&{0}\\\hline{0}&{\sigma_{i}}&{0}&{0}\\\hline{0}&{0}&{\sigma_{i}}&{0}\\\hline{0}&{0}&{0}&{\sigma_{i}}\\\end{array}\right)\quad\mathrm{or}\quad\mathrm{i}\,\left(\begin{array}{c|c|c|c}{\sigma_{i}}&{0}&{0}&{0}\\\hline{0}&{-\sigma_{i}}&{0}&{0}\\\hline{0}&{0}&{\sigma_{i}}&{0}\\\hline{0}&{0}&{0}&{-\sigma_{i}}\\\end{array}\right),
{\calT}_{\mu\nu}={\calT}_{\mu\nu}^{(minimal)}+{\frac{1}{6}}\left(g_{\mu\nu}\nabla^{\rho}\nabla{\rho}|\Phi|^{2}-\nabla_{\mu}\nabla_{\nu}|\Phi|^{2}-{\calG}_{\mu\nu}|\Phi|^{2}\right)
\langle{\calO}_{jm}^{r~+}{\calO}_{j~{-m}}^{-2-r~-}\rangle=(-1)^{j-m}\langle{\calO}_{jj}^{r~+}{\calO}_{j~{-j}}^{-2-r~-}\rangle=(-1)^{j-m}s(j)\frac{1}{(z-z^{\prime})^{2}}.
C_{2}=\frac{1}{2}\tilde{A}_{\mu}^{\nu}\tilde{A}_{\nu}^{\mu}=\frac{P(N+P)(N-1)}{2N},\;\tilde{A}_{\mu}^{\nu}-\frac{\delta_{\mu}^{\nu}}{N}\sum_{\lambda}A_{\lambda}^{\lambda}\;.
\psi_{+}\equiv\frac13d\omega=\hat{e}^{1}\hat{e}^{3}\hat{e}^{5}-\hat{e}^{1}\hat{e}^{4}\hat{e}^{6}-\hat{e}^{2}\hat{e}^{3}\hat{e}^{6}-\hat{e}^{2}\hat{e}^{4}\hat{e}^{5}\,.
{\frac{X}{M^{4}}}={\frac{n}{48\pi}}\left({\frac{m}{M}}\right)^{4}\left({\frac{3n^{3}P_{k}^{S}}{128\pi}}\right)^{{\frac{n}{n-4}}}\left({\frac{nL}{4\pi}}\right)^{-{\frac{4(n-1)}{n-4}}}.
{\calL}^{(n)}=\bar{\chi}(i\!\!\not\!\partial+{\not\!\!A}^{(n)})\chi-\frac{1}{4e^{2}}{\left(F_{\mu\nu}^{(n)}\right)}^{2}-\frac{1}{2e^{2}}\left((\Box\phi)^{2}+\epsilon_{\mu\nu}{F}_{\mu\nu}^{(n)}\Box\phi\right)+{\calL}_{c}
I=\frac{1}{2}\frac{\mu}{g^{2}}tr\intd^{3}x[-\mu(v_{m}-U^{-1}\partial_{m}U)(v^{m}-U^{-1}\partial^{m}U)+\epsilon^{mnp}v_{m}(\partial_{n}v_{p}+\frac{2}{3}v_{n}v_{p})],
\begin{array}{lclcccl}{A_{0}}&{=}&{0}&{,}&{A_{1/2}}&{=}&{0\nonumber}\\{A_{1}}&{=}&{4\pi\beta^{2}}&{,}&{A_{3/2}}&{=}&{\frac{\beta^{2}\pi^{3/2}}{2R}\nonumber}\\{A_{2}}&{=}&{0}&{,}&{A_{5/2}}&{=}&{\frac{[(3\pi-128)\beta^{2}+(256-18\pi)\beta^{4}]\pi^{1/2}}{384R^{3}}\nonumber}\\{A_{3}}&{=}&{0}&{,}&{A_{7/2}}&{=}&{\frac{(27\beta^{2}-100\beta^{4}+80\beta^{6})\pi^{3/2}}{24576R^{5}}\nonumber\.}\\\end{array}
\begin{array}{rl}{\mathrm{(A)}}&{\pi^{*}\phi\cdotf(x,t_{1},\dots,t_{n}),}\\{\mathrm{~(B)}}&{\pi^{*}\phi\cdotf(x,t_{1},\dots,t_{n})dt_{1}\wedge\dots\wedgedt_{n},}\\\end{array}
S_{eff}={\frac{1}{16\pi}}\,\intd^{4}x\,\sqrt{-g}\,\,(-R+{\frac{1}{2}}{\frac{\partial_{\mu}\lambda\partial^{\mu}\bar{\lambda}}{(\mathrm{Im}\lambda)^{2}}}-\sum_{n=1}^{N}{F_{\mu\nu}^{(n)}{}^{\star}\tilde{F}^{(n)}{}^{\mu\nu}}),
Z(u)={\frac{\left(\mpi+e^{\eta}\right)\left(-1\pmie^{\eta}+e^{2\eta}\right)\left({e^{\eta}}\pmi\,{e^{u}}\right)\left(-1+{e^{u}}\right)\left(1+{e^{u}}\right){z_{1}}}{2e^{u}\left(\pmi+{e^{\eta}}\right)\,\left(\pmi+e^{3\eta+u}\right)}}
S_{int}=\frac{k}{2}\intd^{4}x[G^{a}V^{a}]_{D}=\frac{k}{2}\intd^{4}x\;Tr\left\{C^{a}D^{a}-A_{\mu}^{a}v^{a\mu}-\bar{\lambda}{}^{a}\chi^{a}\right\}.
\sum_{\nu}x_{i}^{(\nu)}x_{k}^{(\nu)}\left[\frac{e^{-\nu\theta}}{\nu^{2}}+\sum_{m=1}^{\infty}\left(\frac{e^{-(2mh+\nu)\theta}}{(2mh+\nu)^{2}}+\frac{e^{-(2mh-\nu)\theta}}{(2mh-\nu)^{2}}\right)\right]\,,
\int_{0}^{\infty}\frac{d\alpha}{\alpha}\left(1-e^{-i\alpham^{2}}\right)e^{-\varepsilon\alpha-i\eta^{2}/4\alpha}=2K_{0}\left(\sqrt{i\varepsilon\eta^{2}}\right)-2K_{0}\left(\sqrt{-m^{2}\eta^{2}}\right)\mathrm{,}
\begin{array}{rcl}{Y}&{=}&{\alphaS(A)uAD^{-1}u^{-1}}\\{}&{=}&{\alpha<u_{(1)},A>u_{(2)}<u^{-1},A>u^{-1}}\\{}&{=}&{\alpha<u_{(1)}u^{-1},A>u_{(2)}u^{-1},}\\\end{array}
S^{(2)}=-\frac{V}{16\piG}\intdt\intd^{2}x\left(2(\partial_{0}\delta\tilde{\lambda})^{2}+\frac{1}{2}g^{ij}\dot{\xi}_{i}\Box\dot{\xi}_{j}\right)+\frac{V}{16\piG}\frac{1}{2\tau}\intdt\(\partial_{0}\delta\vec{\tau})^{2}.
\left\{\begin{array}{c}{(m-1)\partial_{A}\partial_{B}x^{j}+L^{i}{}_{jk}(x,y)\partial_{A}x^{i}\partial_{B}x^{j}=0}\\{(n-1)\partial_{A}\partial_{B}y^{b}+C^{a}{}_{bc}(x,y)\partial_{A}y^{a}\partial_{B}y^{b}=0}\\\end{array}\right\}
\sum_{x=1}^{N}<g.s.|S_{x}^{3}\cdotS_{x+2}^{3}|g.s>=N~\frac{1}{4}~2~(~P_{3}(\uparrow\uparrow\uparrow)+P_{3}(\uparrow\downarrow\uparrow)-P_{3}(\uparrow\uparrow\downarrow)-P_{3}(\downarrow\uparrow\uparrow)~)\quad.
\begin{array}{l}{M_{11}^{(c)}(u)\!\!=\!M_{22}^{(c)}(qu^{-1}\!)\!=2x_{0}\displaystyle{\frac{w(su\!-\!s^{-1}u^{-1})\!+\!(k^{2}\!+\!k^{-2})(q^{-1}s^{-1}u\!-\!qsu^{-1})}{(q-q^{-1})^{2}}}\,,}\\{M_{12}^{(c)}(u)=\displaystyle{\frac{x_{+}}{x_{-}}}M_{21}^{(c)}(u)=-x_{+}\displaystyle{\frac{(u^{2}+u^{-2})(q^{-1}u^{2}-qu^{-2})}{(q-q^{-1})^{2}}}\,.}\\\end{array}
\begin{aligned}{L^{12}+L^{21}}&{{}=J^{3}\cdot(L^{11}-L^{22}),}\\{L^{31}+L^{13}}&{{}=J^{2}\cdot(L^{33}-L^{11}),}\\{L^{23}+L^{32}}&{{}=J^{1}\cdot(L^{22}-L^{33}),}\\\end{aligned}
{\calA}_{o}(t,x)=\int_{0}^{1}{d\alphax^{1}}E_{1}^{L}(t,\alphax)=\int_{0}^{1}{d\alphax^{1}\left({1-\frac{{e^{2}}}{\pi}\frac{1}{{\partial_{1}^{2}}}}\right)}_{\alphax}^{-1}\frac{{\partial_{1}^{\alphax}\left({-J^{0}(\alphax)}\right)}}{{\partial_{\alphax}^{2}}},
\frac{1}{g_{YM}^{2}}\rightarrow\frac{1}{g_{YM}^{2}}+\frac{3N-M}{8\pi^{2}}\mathrm{log}\mu,\,\,\,\,\,\theta_{YM}\rightarrow\theta_{YM}-2\alpha\left(N-\frac{M}{3}\right).
\frac{\deltaS}{\delta\theta^{\alpha\beta}}=-\frac{\beta}{2}\intd^{2}x\epsilon^{\mu\nu}{\mathrm{Tr}}\left(\frac{\deltaF_{\mu\nu}}{\delta\theta^{\alpha\beta}}\star\Phi+F_{\mu\nu}\star\frac{\delta\Phi}{\delta\theta^{\alpha\beta}}\right)\,.
\tilde{A}(p)=\frac{3}{2}g^{2}\delta_{ab}\int\frac{d^{4}k}{(2\pi)^{4}}d^{2}\thetad^{2}{\overline{\theta}}\,\frac{m^{2}}{(k^{2}+m^{2})[(p-k)^{2}+m^{2}]}\left\{V^{a}(p,\theta,{\overline{\theta}})V^{b}(-p,\theta,{\overline{\theta}})\right\},
\rho_{m_{0}\rightarrowm_{1}+m_{2}+m_{3}}^{D}\!=\!\!\!\!\sum_{j_{1},j_{2},j_{3}}\!\!\!\frac{(2j_{1}\!+\!1)(2j_{2}\!+\!1)(2j_{3}\!+\!1)}{(4\piR^{2})^{2}}\left(\!\!\begin{array}{ccc}{j_{1}\!}&{\!j_{2}\!}&{\!j_{3}}\\{0\!}&{\!0\!}&{\!0}\\\end{array}\!\!\right)^{2}\!\!\rho_{m_{0}\rightarrowm_{1j_{1}}\!+m_{2j_{2}}\!+m_{3j_{3}}}^{D-2}
\begin{array}{ll}{T_{\pmacde}T_{\pmbcde}}&{=\left(\displaystyle{\frac{1}{4!}}\right)^{2}\epsilon_{acdefghi}\epsilon_{bcdejklm}T_{\pmfghi}T_{\pmjklm}}\\{{}}&{{}}\\{{}}&{=\displaystyle{\frac{1}{4}}\,\delta_{ab}T_{\pmcdef}T_{\pmcdef}-T_{\pmacde}T_{\pmbcde}\,,}\\\end{array}
S_{t}^{(1)}(x)=\frac{1}{4\piL_{1}}\left\{e^{ex_{2}t_{+}}\frac{\vartheta_{1}^{\prime}(0)}{\vartheta_{1}(\bar{t})}\frac{\vartheta_{1}(z+\bar{t})}{\vartheta_{1}(z)}+e^{-ex_{2}t_{-}}\;\frac{\vartheta_{1}^{\prime}(0)}{\vartheta_{1}(t)}\frac{\vartheta_{1}(t-\bar{z})}{\vartheta_{1}(z)}\right\}
N_{i_{\bfk}\,i_{\bfk}^{\prime}}^{n_{\bfk}n_{\bfk}^{\prime}}\equiv\left\{\left(\begin{array}{c}{{n_{\bfk}^{\prime}}}\\{{i_{\bfk}^{\prime}}}\\\end{array}\right)\left(\begin{array}{c}{n_{{\bfk}}}\\{i_{\bfk}}\\\end{array}\right)\frac{1}{i_{\bfk}^{\prime}!\;i_{\bfk}!}\right\}^{1/2}.
{\calS}_{J_{H}^{-1}}^{(2k+1)}=(2k+1){{J_{H}^{-1}}^{\prime}}^{k}\partial_{u}{\frac{1}{{J_{H}^{-1}}^{\prime}}}\partial_{u}{\frac{1}{{J_{H}^{-1}}^{\prime}}}\ldots\partial_{u}{\frac{1}{{J_{H}^{-1}}^{\prime}}}\partial_{u}{{J_{H}^{-1}}^{\prime}}^{k},
{\chi_{1}^{a}=\frac{\partial_{j}A_{j}^{a}}{\nabla^{2}}\,;\quad\chi_{2}^{a}=f^{abc}\frac{\partial_{j}}{\nabla^{2}}\left(\chi_{1}^{b}A_{j}^{c}+\frac12(\partial_{j}\chi_{1}^{b})\chi_{1}^{c}\right)}
\delta_{\epsilon}q^{n}=\epsilon\,\left\{q^{n},\phi\right\}=\epsilon\,\frac{\partialH}{\partialp_{n}}\\,\\\delta_{\epsilon}p_{n}=\epsilon\,\left\{p^{n},\phi\right\}=-\epsilon\,\frac{\partialH}{\partialq^{n}}\\,\\\delta_{\epsilon}\lambda=\dot{\epsilon}\,
{\mathcal{H}}_{i}=-2\left(\gamma_{ik}\frac{\delta\mathcal{S}}{\delta\gamma_{kj}}\right)_{,j}+\frac{\delta\mathcal{S}}{\delta\gamma_{kl}}\gamma_{kl,i}+\frac{\delta\mathcal{S}}{\delta\phi}\phi_{,i}=0~~~.
N^{\calP}~=~~\sum_{\{r_{i}\}}~N_{j_{1}j_{2}}^{~~~~r_{1}}~N_{r_{1}j_{3}}^{~~~~r_{2}}~N_{r_{2}j_{4}}^{~~~~r_{3}}~\dots\dots~N_{r_{p-2}j_{p-1}}^{~~~~~~~~j_{p}}~
\intd^{4}x\sqrt{-g_{4}}\left[f(\phi,\sigma){\calR}^{(4)}+\frac{1}{6}\partial_{\Phi}\partial_{\bar{\Phi}}f\partial_{\mu}\Phi\partial^{\mu}\bar{\Phi}+\frac{1}{6}\partial_{\Sigma}\partial_{\bar{\Sigma}}f\partial_{\mu}\Sigma\partial^{\mu}\bar{\Sigma}\right],
\left(f_{1}\bulletf_{2}\right)(z,\overline{{z}})=\int_{\mathrm{C}^{n}}f_{1}(z,\overline{{v}})f_{2}(v,\overline{{z}})e^{\left(\Phi(z,\overline{{v}})+\Phi(v,\overline{{z}})-\Phi(z,\overline{{z}})-\Phi(v,\overline{{v}})\right)/\hbar}d\mu_{\hbar}(v,\overline{{v}})
S=-\frac{1}{2}\sum_{i,j}\mathscr{M}_{ij}\phi^{i}\phi^{j}-\frac{1}{2}\sum_{a,b}\mathscr{F}_{ab}t^{a}t^{b}-\frac{1}{3}\sum_{i,j,k}\mathscr{G}_{ijk}\phi^{i}\phi^{j}\phi^{k}+\sum\mathscr{G}_{i,a,b}\phi^{i}t^{a}t^{b},
{\calL}\simIm(\tau_{ij})F_{\mu\nu}^{i}F^{\mu\nuj}+Re(\tau_{ij})F_{\mu\nu}^{i}\tilde{F}^{\mu\nuj}+Im\left(\partial_{\mu}u_{i}\partial^{\mu}\frac{\partial{\calF}}{\partialu_{i}}\right)+\mathrm{fermions},
A^{M^{10}}(y)=\int_{0}^{\infty}{\frac{dt}{2t}}{\frac{1}{V_{6}}}\left(8\pi^{2}\alpha^{\prime}t\right)^{6/2}e^{-{\frac{y^{2}}{2\pi\alpha^{\prime}}}t}\cdot2Z_{B}^{o}(t)Z_{F}^{o}(t).
|n_{1},\cdots,n_{\ell}>=N(n_{1},\cdots,n_{\ell})\,\Big[\mathrm{Tr}\left(\alpha^{\dagger}\right)^{r_{1}}\Big]^{n_{1}}\,\cdots\,\Big[\mathrm{Tr}\left(\alpha^{\dagger}\right)^{r_{\ell}}\Big]^{n_{\ell}}|0>\\\.
\left[\varphi({\tau_{2}}),{X_{0}^{\mu}}({\tau_{2}})\right]=\left[\varphi({\tau_{1}}),{X_{0}^{\mu}}({\tau_{1}})\right]+\left[\dot{\varphi}({\tau_{1}}),{\dot{X}_{0}^{\mu}}({\tau_{1}})\right]({\tau_{2}}-{\tau_{1}})+\cdots.
\frac12{\bar{e}}_{\alpha}^{\;\;i}{\bar{e}}_{\beta}^{\;\;j}\frac{\partial\delta\Gamma^{(1)}}{\partialb^{ij}}=-e_{\alpha}^{\;\;\mu}e_{\beta}^{\;\;\nu}\deltaB_{\mu\nu}^{b(2)}+e_{\alpha}^{\;\;\mu}e_{\beta}^{\;\;\nu}X_{\mu\nu}^{b^{\star}}
D^{\mu\nu;\alpha\beta}(h)=\frac{1}{2}\left(\eta_{\alpha\mu}\eta_{\beta\nu}+\eta_{\alpha\nu}\eta_{\beta\mu}-\frac{2}{9}\:\eta_{\alpha\beta}\eta_{\mu\nu}\right)D_{F}\equivG^{\mu\nu;\alpha\beta}D_{F}
P^{b}X^{r}=\sum_{\stackrel{l=0}{n_{1},...,n_{l}=0}}^{\stackrel{\infty}{b}}C_{l}(\theta,r,b,n_{1},...n_{l-1})X^{(r+(n_{1}-1)+(n_{2}-1)...(n_{l}-1))}P^{b-l},
\begin{array}{rcl}{V(\phi,\beta)}&{=}&{\mu^{4}\sum_{s=1}^{\infty}a^{2-s}h(4,s)(\gamma+g\phi)^{2s}\frac{1}{\Gamma(s-1)}}\\{}&{}&{\left(\int_{1}^{\infty}dt\,t^{2s-4}\frac{e^{\frac{t}{2}}}{e^{t}-1}+\frac{B_{0}(\frac{1}{2})}{2s-4}+\frac{B_{2}(\frac{1}{2})}{4s-4}+\sum_{n=3}^{\infty}\frac{B_{n}(\frac{1}{2})}{n!}\frac{1}{2s-4+n}\right).}\\\end{array}
H\equiv\dot{x}_{i}p_{i}+\ddot{x}_{i}\tilde{p}_{i}-{\calL}=-\frac{2\mu}{\kappa^{2}}\,\tilde{p}_{i}^{2}-\frac{2}{\kappa}\,\epsilon_{ij}\,p_{i}\tilde{p}_{j}+\frac{1}{R^{2}}\,x_{i}\tilde{p}_{i}-\frac{\mu}{2R^{2}}\,x_{i}^{2}
\left.2\partial_{\mu}\frac{\partial{g^{\prime}}_{\mu\nu}(x,l)}{\partiall_{i}}\right|_{l=0}+\left(C-\frac{2}{D}\right)\left.\partial_{\nu}\left(\delta^{\alpha\beta}\frac{\partial{g^{\prime}}_{\alpha\beta}(x,l)}{\partiall_{i}}\right)\right|_{l=0}=0
\mathcal{G}_{+,l}(me^{i\pi/2},r)-\mathcal{G}_{-,l}(me^{i\pi/2},r)=\frac{ir}{\pi}\,e^{-i\pi|l|}(e^{2i\delta_{l}^{+}}-e^{2i\delta_{l}^{-}})\,K_{W}^{2}(mr),
\Big\langle\prod_{b=1}^{N}\prod_{i=1}^{n_{b}}\overline{{\psi}}^{(b)}({x}_{i}^{(b)})P_{+}\psi^{(b)}({x}_{i}^{(b)})\prod_{j=1}^{m_{b}}\overline{{\psi}}^{(b)}({y}_{j}^{(b)})P_{-}\psi^{(b)}({y}_{j}^{(b)})e^{ie\sum_{b=1}^{N}(a_{\mu}^{(b)},j_{\mu}^{(b)})}\Big\rangle_{0}^{\theta}\;.
I_{5}=\intd^{5}x\sqrt{-\hat{g}}\left\{\hat{R}-{\frac{1}{2}}(\hat{\partial}\phi)^{2}-{\frac{1}{12}}e^{-\sqrt{2/3}\phi}\hat{H}^{2}-{\frac{1}{4}}e^{+\sqrt{2/3}\phi}\hat{F}^{2}\right\}.
S_{\mathrm{{Sugra}}}^{(D)}(\tilde{\hat{g}},\tilde{\hat{B}},\tilde{\hat{\phi}})=S_{\mathrm{Sugra}}^{(D)}(\hat{g},\hat{B},\hat{\phi})+\intd^{D}x\A(\hat{g},\hat{B},\hat{\phi})\partial_{\underline{{x}}}B(\hat{g},\hat{B},\hat{\phi})\,,
<{\calJ}_{\mu_{1}\cdots\mu_{s}}^{(s)}(x)~{\calJ}_{\nu_{1}\cdots\nu_{s}}^{(s)}(0)>={\frac{c_{s}(t)}{|x|^{4+2s}}}{\calI}_{\mu_{1}\cdots\mu_{s},\nu_{1}\cdots\nu_{s}}^{(s)}(x)+\sum_{s^{\prime}=0}^{s-1}{\prod}
\pi\sum_{n=1}^{\infty}\left(\frac{\calE}{n\pi}\right)^{D/2}{\mathrme}^{-{\frac{n\pi}{\calE}}x}={\frac{{\calE}^{D/2}}{\pi^{D/2-1}\Gamma(D/2)}}\int_{0}^{\infty}\!\!dt{\frac{t^{D/2-1}}{{\mathrm{e}}^{t+\pix/{\calE}}-1}}\.
S_{*}=\intd^{4}x\sqrt{|g|}e^{-\varphi}\left[R+\left(\nabla\varphi\right)^{2}-\frac{1}{2}\left(\nabla\beta\right)^{2}-\frac{1}{2}e^{2\varphi}\left(\nabla\sigma\right)^{2}\right]
{\calL}_{\mathrm{mass}}^{\theta}=-{\frac{1}{2}}\left(\begin{array}{ccc}{\theta_{X}}&{\theta_{Y}}&{\theta_{S}}\\\end{array}\right)M_{\theta}^{2}\left(\begin{array}{c}{\theta_{X}}\\{\theta_{Y}}\\{\theta_{S}}\\\end{array}\right),\quadM_{\theta}^{2}={\frac{2\Lambda^{2}}{(64\pi^{2})^{2}}}\left(\begin{array}{ccc}{1}&{1}&{3}\\{1}&{1}&{3}\\{3}&{3}&{9}\\\end{array}\right).
P_{R}[\widetilde{x}_{f},\widetilde{x}_{i}]=\langle\widetilde{x}_{f}|P_{R}|\widetilde{x}_{i}\rangle=\langle\widetilde{x}_{f}|\int_{0}^{\infty}dt\;\eta(t)\mathrm{e}^{-tH_{0}^{x}}|\widetilde{x}_{i}\rangle\;\;\;.
u_{0,l}(r)=\left(\begin{array}{l}{0}\\{\phi_{0,l,-1}(r)}\\\end{array}\right),\;{}l\leq-1;\;\;\;u_{0}^{I}(r)=\left(\begin{array}{l}{0}\\{\phi_{0,-1}^{ir}(r)}\\\end{array}\right),\;{}l=0\;.
{\frac{ds_{11}^{2}}{\ell_{p}^{2}}}=e^{-2\phi/3}{\frac{ds_{IIA}^{2}}{\alpha^{\prime}}}+e^{4\phi/3}({\frac{dx^{11}}{R}}-{\frac{C_{1}}{\sqrt{\alpha^{\prime}}}})^{2}\,
\frac{2}{k}\int_{\Sigma_{\tau}}d\bar{\eta}d\eta~~{\calE}(-\eta)q_{i}-\frac{1}{k}\sum_{i\neqj}\int_{\Sigma_{\tau}}d\bar{\eta}d\eta~~(\frac{u_{j}}{u_{i}}w_{ij}v_{ij}+\frac{u_{i}}{u_{j}}w_{ji}v_{ji})+2h_{i}=\frac{2}{n}\sum_{j}\frac{(Xu)_{j}}{u_{j}}
\int\sqrt{\gamma}d^{3}xq_{{\calP}lm}^{(p)ij}q_{{\calP}^{\prime}l^{\prime}m^{\prime}ij}^{(p^{\prime})*}=\delta^{pp^{\prime}}\delta_{{\calP}{\calP}^{\prime}}\delta_{ll^{\prime}}\delta_{mm^{\prime}}
\begin{array}{ll}{D=3:}&{W^{+(+)[+]}(x,\theta^{++},\theta^{(++)},\theta^{[+]\{\pm\}},u)}\\{D=4:}&{W^{12}(x,\theta_{3},\theta_{4},\bar{\theta}^{1},\bar{\theta}^{2},u)}\\{D=5:}&{W^{1}(x,\theta^{1},u)}\\{D=6:}&{W^{12}(x,\theta^{1},\theta^{2},u)}\\\end{array}
f_{0}\left(w\right)=f_{2}\left(w\right)=\left(\begin{array}{cc}{q^{-1}}&{0}\\{0}&{q}\\\end{array}\right),\f_{1}\left(w\right)=\left(\begin{array}{cc}{q^{-2}}&{0}\\{0}&{q^{2}}\\\end{array}\right),\f_{k}\left(I\right)=1,\tag{2.8}
{\calW}_{{\calC}}^{U(N)}=1-\frac{1}{2}~Ng^{2}{\calA}_{{\calC}}+\frac{1}{8}(\frac{2}{3}N^{2}+\frac{1}{3})~g^{4}{\calA}_{{\calC}}^{2}-\frac{1}{48}(\frac{5}{15}N^{3}+\frac{10}{15}N)~g^{6}{\calA}_{{\calC}}^{3}+{\calO}(g^{8}),
\widehat{F_{z\bar{z}}}=F_{z\bar{z}}+{\frac{1}{4}}\theta^{ij}\left[2(F_{zi}F_{\bar{z}j}+F_{\bar{z}j}F_{zi}-(A_{i}(\nabla_{j}F_{z\bar{z}}+\partial_{j}F_{z\bar{z}})+(\nabla_{j}F_{z\bar{z}}+\partial_{j}F_{z\bar{z}})A_{i}))\right]
\frac{1}{\calN}\int[d\chi][d\bar{\chi}]\,\chi_{\pm}(x)\chi_{\pm}^{\dagger}(y)e^{\intd^{2}x^{\prime}\,\bar{\chi}{i\gamma_{\mu}\partial_{\mu}}\chi}\int[d\phi]\,e^{\pm\phi(x)\mp\phi(y)}e^{-\intd^{2}x^{\prime}\,\frac{1}{2}\phi\hat{O}\phi},
(\gamma_{a}{\calC})_{\alpha\betaij}x^{a}(z,\bar{z})=\left(\begin{matrix}{x^{ij}(z,\bar{z})\epsilon_{\alpha\beta}}&{0}\\{0}&{x^{*ij}(z,\bar{z})\epsilon_{\dot{\alpha}\dot{\beta}}}\\\end{matrix}\right)\equivX(z,\bar{z})\,
D_{1}\pi^{\alpha_{1}}=\frac12C_{\;\;\beta_{1}\gamma_{1}}^{\alpha_{1}}\pi^{\beta_{1}}\eta_{1}^{\gamma_{1}}+\frac12C_{\;\;\beta_{1}\gamma_{3}}^{\alpha_{1}}\left(\pi^{\beta_{1}}\eta_{1}^{\gamma_{3}}-\eta_{1}^{\beta_{1}}\pi^{\gamma_{3}}\right),
{\left(\gamma^{a}\right)_{\alpha}}^{\beta}=\left[\left(\begin{array}{cc}{i}&{0}\\{0}&{-i}\\\end{array}\right),\,\left(\begin{array}{cc}{0}&{i}\\{i}&{0}\\\end{array}\right),\,\left(\begin{array}{cc}{0}&{i}\\{-i}&{0}\\\end{array}\right)\right]\,.
\zeta_{\mathrm{T}}(s)\simeq\zeta(s)+\frac{2}{(2\pi)^{3/2}}\sum_{n=0,1/2,\dots}a_{n}\left(\frac{\hbar}{2\piT}\right)^{2s-3+2n}\frac{\Gamma(s-3/2+n)}{\Gamma(s)}\,\zeta_{\mathrm{R}}(2s+2n-3),
{D_{w}=\tau_{2}{\calD}=i\left(\tau_{2}{\frac{\partial}{\partial\tau}}-i{\frac{w}{2}}\right),\qquad\bar{D}_{\hat{w}}=\tau_{2}\bar{\calD}=-i\left(\tau_{2}{\frac{\partial}{\partial\bar{\tau}}}+i{\frac{\hat{w}}{2}}\right)}
q^{3}\sigma_{q}^{-1}\gamma_{1}+q^{-5}\sigma_{q}\gamma_{1}=q^{-1}\gamma_{1},\mathrm{\qquad~}q^{-3}\gamma_{1}+q^{5}\sigma_{q}^{-2}\gamma_{1}=q\sigma_{q}^{-1}\gamma_{1}\,\,,
S_{\mathrm{anomalous}}=-{\frac{1}{T_{H}}}\int_{\Sigma}e^{-\phi}{\calL}_{E}\;d^{3}r=-{\frac{k}{\hbar}}\int_{\Omega}{\calL}_{E}\;\sqrt{g_{E}}d^{4}x=-k\{\alpha\chi+\betap\}.
\begin{array}{lcr}{F^{++}=if^{+}\overline{{f}}^{+}=iu_{(\alpha}^{+}u_{\beta)}^{+}f^{(\alpha}\overline{{f}}^{\beta)}}\\{F^{0}={\frac{i}{2}}(f^{+}\overline{{f}}^{-}+f^{-}\overline{{f}}^{+})={\frac{i}{2}}u_{(\alpha}^{+}u_{\beta)}^{-}f^{(\alpha}\overline{{f}}^{\beta)}}\\{F^{--}=if^{-}\overline{{f}}^{-}=iu_{(\alpha}^{-}u_{\beta)}^{-}f^{(\alpha}\overline{{f}}^{\beta)}.}\\\end{array}
\lbrackK]_{q\bar{q}}=\left(\begin{array}{ll}{q\bar{q}+q^{-1}\bar{q}^{-1}}&{-1}\\{-(1+\bar{q}^{2}+\bar{q}^{-2})}&{q\bar{q}^{3}+q^{-1}\bar{q}^{-3}}\\\end{array}\right)
\left.\sum_{[v]:m_{v}=k}\,\prod_{p\subsetX}\,\tilde{I}_{\beta}\,(v_{p})=\sum_{Y_{i}:suppY_{i}=X}\\sum_{[v]:m_{v}=k,v_{p}\neq0}\,\prod_{p\inY_{i}}\,\tilde{I}_{\beta}\,(v_{p})\.\right.
\begin{array}{lcl}{\frac{\partial}{\partialt_{4}}{u_{2}}}&{=}&{\frac{4}{3}(u_{2}v_{3}+2\theta^{2}v_{3}^{\prime\prime})^{\prime}}\\{\frac{\partial}{\partialt_{4}}{v_{3}}}&{=}&{\frac{4}{3}\{v_{3}v_{3}^{\prime}-\theta^{2}u_{2}u_{2}^{\prime\prime\prime}-\frac{1}{3}(u_{2}^{2}u_{2}^{\prime}+2\theta^{4}u_{2}^{(5)})\}}\\\end{array}
\left.\partial_{0}\phi(x_{0})\right|_{x_{0}=\epsilon}=\left\{{\frac{\partial_{\epsilon}{\calO}(\epsilon)}{\phi_{h}(0)}}\left(1-{\frac{{\calO}(\epsilon)}{\phi_{h}(0)}}+...\right)-{\frac{\lambda}{\epsilon}}\right\}\phi_{\epsilon}.
\left(D_{\mu}\mp\frac{1}{4}H_{\muNP}^{5}\Gamma^{NP}\right)\epsilon_{r}=\left(D_{\mu}\mp\frac{1}{2\ell}e_{\mu\alpha}(1\otimes1\otimes\gamma^{\alpha})\pm\frac{i}{2\ell}A_{\mu}^{I}K_{I}^{m}(1\otimes\sigma_{m}\otimes1)\right)\epsilon_{r}=0\.
{\itL}=\sqrt{-g}\biggl[\phi\biggl(R-\omega\frac{\phi_{;\rho}\phi^{;\rho}}{\phi^{2}}-\frac{\Psi_{;\rho}\Psi^{;\rho}}{\phi^{2}}\biggr)-\frac{1}{2}\xi_{;\rho}\xi^{;\rho}\biggr]+L_{\mathrm{r}},
I=\int_{M}d^{5}x\sqrt{-g}\left[\frac{1}{2\kappa^{2}}R_{5}+a\kappa^{2}R_{5}^{4}-\Lambda\right]+\int_{\Sigma}d^{4}y\sqrt{-q}(bR_{4}^{2}-\lambda)+\int_{\bar{\Sigma}}d^{4}\bar{y}\sqrt{-\bar{q}}(\bar{b}\bar{R}_{4}^{2}-\bar{\lambda}),
{}~~~~~~~~~~~~~\times\Bigg\{\sum_{\sigma\inZ_{N}}\frac{w(x_{8}x_{13},x_{1}u,x_{3}x_{58}|\sigma+k_{3}+k_{2})w(x_{2}x_{67},x_{2}v,x_{6}x_{24}|\sigma)\omega^{\sigmak_{1}}}{w(x_{5}x_{13},\omegax_{1}u,\omegax_{1}x_{58}|\sigma+k_{2})w(x_{4}x_{67},x_{2}v,x_{7}x_{24}|\sigma+k_{4}-k_{1})}\Bigg\}_{0}
\deltaS=\int_{0}^{1}d\tau\\sum_{n}\Biggl(-\dot{p}_{n}^{\mu}+{\frac{\partialL}{\partialx_{n}^{\mu}}}\Biggr)\deltax_{n}^{\mu}\+\,sum_{n}p_{n}^{\mu}\deltax_{n}^{\mu}\Bigl|_{0}^{1}=0
{\calW}_{F}^{(2;cr)}=-{\frac{C_{F}C_{A}}{16{\pi}^{4}}}{\frac{\Gamma^{2}(D/2-1)}{(D-4)^{4}}}\left[{\calE}+8{\calB}\left(1-{\frac{\Gamma^{2}(3-D/2)}{\Gamma(5-D)}}\right)\right],
0<Q^{2}<\frac{16}{3\omega_{4}^{2}}\,\biggl[-\frac{\ell^{2}}{3}\left(\omega_{4}M+\frac{2\ell^{2}}{9}\right)+\frac{2\ell}{3\sqrt{3}}\left(\omega_{4}M+\frac{\ell^{2}}{3}\right)^{3/2}\biggr]\,,
I_{E}[\phi,g_{\mu\nu},\beta]=\frac12\int_{{\calM}_{\beta}}\left(\phi^{,\mu}\phi_{,\mu}+m^{2}\phi^{2}+\xiR\phi^{2}\right)\sqrt{-g}~d^{4}x+2\pi\xi\left(1-{\frac{\beta}{\beta_{H}}}\right)\int_{\Sigma}\phi^{2}\sqrt{\gamma}d^{2}x~~~.
C_{1}b_{n}^{i}C_{1}^{-1}=\oint_{\xi_{0i}=0}d\xi_{0i}\(\xi_{0i})^{n+1}\left(\frac{d}{d\xi_{0i}}V_{i}^{-1}V_{0i}\xi_{0i}\right)^{2}b^{i}\left(V_{i}^{-1}V_{0i}\xi_{0i}\right)\.
e^{AA^{\prime}j}\;{}^{(3)}\nabla_{j}\Bigr({_{e}n_{\;\;A^{\prime}}^{B}}\;\rho_{B}^{nq}\Bigr)=\Bigr(n+{\frac{3}{2}}\Bigr){_{e}n^{AA^{\prime}}}\;\Bigr({_{e}n_{\;\;A^{\prime}}^{B}}\;\rho_{B}^{nq}\Bigr)\;,