harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
V_{k\sigma,j\sigma^{\prime}}^{(0)}=i\omega_{lk}^{(0)}\sigma\delta_{kj}\delta_{\sigma\sigma^{\prime}},\quadV_{k\sigma,j\sigma^{\prime}}^{(1)}=\sum_{s=\pm}v_{k\sigma,j\sigma^{\prime}}^{s}e^{is\Omegat},
\begin{array}{rll}{S_{eff}[A]}&{=}&{-\intd^{4}x\frac14F_{\mu\nu}^{j}[A(x)]F^{\mu\nu,j}[A(x)]}\\{}&{}&{+g^{2}C_{A}\frac1{(4\pi)^{2}}\frac{11}{6}\intd^{4}xd^{4}yA_{\mu}^{a}(x)H(x-y)(g^{\mu\nu}\partial^{2}-\partial^{\mu}\partial^{\nu})A_{\nu}^{a}(y)}\\\end{array}
[\delta(\epsilon_{1}),\delta(\epsilon_{2})]\zeta^{A}=\frac12\partial_{a}\zeta^{A}\bar{\epsilon}_{2}\gamma^{a}\epsilon_{1}+\frac14\Gamma^{A}\bar{\epsilon}_{2}\epsilon_{1}-\frac14\gamma_{a}\Gamma^{A}\bar{\epsilon}_{2}\gamma^{a}\epsilon_{1}\,.
\biggl\{2K_{0}(Z_{b})+\frac{(\widetilde{q})^{2}}{2}\frac{M_{b}^{2}}{Z_{b}}K_{-1}(Z_{b})+\frac{m^{2}(x+y)(1-x-y)}{M_{b}^{2}}\left(\frac{Z_{b}}{2}\right)K_{1}(Z_{b})\biggr\}
T(0)=\prod_{j=2}^{P}\;\prod_{i=1}^{j-1}R_{ij}^{\iota^{P-j+i}}((\epsilon^{-j}-\epsilon^{-i})\theta^{\prime})\;\;\prod_{j=1}^{P-1}\;\prod_{i=j+1}^{P}R_{ij}^{\iota^{P-j+i}}((\epsilon^{-j}-\epsilon^{-i})\theta^{\prime})
V_{3}=(0(0~0)^{3}\vert\vert{\tilde{e}}^{2}~(-{\tilde{e}}^{2})~0~\vert({\frac{1}{3}}~{\frac{1}{3}}~{\frac{1}{3}}~{\frac{1}{3}}~{\frac{2}{3}})_{r}^{3}\vert0_{r})
j^{1}(x,t)\Bigm|_{x=0}=\frac{1}{2}(\psi_{1}^{\dagger}(0)\psi_{1}(0^{-})-\psi_{2}^{\dagger}(0)\psi_{2}(0^{-})+\psi_{1}^{\dagger}(0^{-})\psi_{1}(0)-\psi_{2}^{\dagger}(0^{-})\psi_{2}(0))=0,
\gamma_{\alpha\dot{\alpha}}^{(-1,1)}(x_{12}^{2},u_{1},u_{2})=2i\partial_{1\;\alpha\dot{\alpha}}\partial_{1}^{--}g^{(1,1)}(x_{12}^{2},u_{1},u_{2})=2i(12)\partial_{1\;\alpha\dot{\alpha}}\;g(x_{12}^{2})\;.
\prod_{p\inP}\bar{T}_{|p|}^{c}(\otimes_{i\inp}f_{i})={\calO}(\hbar^{|I|-|P|}),\quad\prod_{q\inQ}T_{|q|}^{c}(\otimes_{i\inq}f_{i}\otimes\otimes_{j\inq}g_{j})={\calO}(\hbar^{|I^{c}|+m-|Q|}).
\frac{1}{p_{s}^{2}}~{\calI}_{\mu_{1}\cdots\mu_{s},\alpha_{1}\cdots\alpha_{s}}^{(s)}(x)~{\calI}_{\alpha_{1}\cdots\alpha_{s},\nu_{1}\cdots\nu_{s}}^{(s)}(x)=\Im_{\mu_{1}\cdots\mu_{s},\nu_{1}\cdots\nu_{s}}^{(s)},
\begin{array}{l}{\langleN\rangle={\frac{4\piV}{h^{3}}}[{\frac{\sqrt\pi}{2}}({\frac{2m}{\beta}})^{\frac{3}{2}}z+2{\sqrt\pi}({\frac{2m}{\beta}})^{\frac{3}{2}}\delta(p,q)z^{2}+...].}\\\end{array}
\sigma=\int{\frac{d^{2}k}{(2\pi)^{2}}}[\not\!k+\lambda^{2}\gamma^{5}\tilde{\sigma}+\lambda^{2}\tilde{\sigma}]_{\nu\nu}^{-1}=\int{\frac{d^{2}k}{(2\pi)^{2}}}[{\frac{4i\lambda^{2}\sigma}{k^{2}-4\lambda^{4}(\sigma^{2}+\tilde{\sigma}^{2})}}]
M_{A}^{\prime}=\int_{{\calC}_{A}^{(4)}}H_{NS}^{(4)}=\int_{{\calC}_{A}^{(4)}}\widetildeG_{2},\quadM_{A}=\int_{{\calC}_{A}^{(4)}}H_{R}^{(4)}=\int_{{\calC}_{A}^{(4)}}G_{4}\,.
y_{\ast}^{2}={\frac{(4\pi)^{2}\alpha}{2a\alpha_{C}}},~~{\frac{\lambda_{\ast}}{y_{\ast}^{4}}}={\frac{2a}{(4\pi)^{2}}}~{\frac{\alpha_{C}}{\alpha}},~~\xi_{\ast}={\frac{1}{6}}.
\begin{array}{c}{\theta^{(j)}(z)=\sqrt{-1}\omega^{j/2}t^{n(1/2-j/n)^{2}}u^{-1+2j/n}(t^{2n};t^{2n})_{\infty}(t^{2j}u^{2};t^{2n})_{\infty}(t^{2(n-j)}u^{-2};t^{2n})_{\infty},}\\{h(z)=\sqrt{-1}t^{n/4}\displaystyle\frac{(t^{2n};t^{2n})_{\infty}^{3}}{(t^{2};t^{2})_{\infty}^{2}}u^{-1}(u^{2};t^{2})_{\infty}(t^{2}u^{-2};t^{2})_{\infty},}\\\end{array}
\psi_{\gamma}(\lambda)=e^{2\pii\xi(\lambda)}\int_{\gamma}\Phi_{\xi,\Lambda}^{\kappa}(T)\sum_{\sigma\inS_{n}}W_{\sigma,c}(T,X(\lambda))\;dT_{1}\cdotsdT_{n}\;f_{c(\sigma(1))}\cdotsf_{c(\sigma(n))}v_{\Lambda},
\mathrm{d}s_{D}^{2}=r_{0}^{2}\left[\left({\frac{u}{\calR}}\right)^{2}\mathrm{d}x_{d}^{2}+\left({\frac{\calR}{u}}\right)^{2}\mathrm{d}u^{2}+\mathrm{d}\Omega_{\tilde{d}+1}^{2}\right]\quad,\quade^{\phi}=r_{0}^{-{\frac{(D-2)a}{8}}\left({\frac{\beta+1}{\beta}}\right)}\left({\frac{u}{\calR}}\right)^{-{\frac{(D-2)a}{8}}\left({\frac{\beta+1}{\beta}}\right)}\,
\langle\!\langleu,u^{\dagger}|\equiv\langleu,u^{\dagger}|e^{{\mathrm{tr}}\intd^{d}x[u^{\dagger}Qu]}=\langleQ|e^{\sqrt{2}{\mathrm{tr}}\intd^{d}x(u^{\dagger}\hat{\psi}-\hat{\psi}^{\dagger}u)},
L_{11}^{(\pm)}=q^{\mp(F+E)}=q^{\mp\left(\begin{array}{cc}{1}&{0}\\{0}&{0}\\\end{array}\right)},\;\;L_{22}^{(\pm)}=(-1)^{F}q^{\mp(F-E)}=(-1)^{F}q^{\pm\left(\begin{array}{cc}{0}&{0}\\{0}&{1}\\\end{array}\right)},\;\;
[\mathsf{L}_{1},\mathsf{L}_{2}]=i\hbar[\mathsf{L}_{2},\mathsf{\mathcal{C}}_{12}],\qquad[\mathsf{R}_{1},\mathsf{R}_{2}]=i\hbar[\mathsf{R}_{2},\mathsf{\mathcal{C}}_{12}]
{\bfF}(\kappap,\alpha,\mu)=\kappa^{-4}\left({\frac{\bar{g}^{2}(1)}{\bar{g}^{2}(\kappa)}}\right)^{{\bfC}_{\gamma_{\mathrm{f}}}\frac{8\pi^{2}}{2b}}({\bfF}(p,\bar{\alpha}(\kappa),\mu))\left({\frac{\bar{g}^{2}(1)}{\bar{g}^{2}(\kappa)}}\right)^{{\bfC}_{\tilde{\gamma}}\frac{8\pi^{2}}{2b}}.
\Pi_{\mu\nu}^{M}(k)=(g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}})\Pi_{1}^{M}(k^{2})+im\epsilon_{\mu\nu\alpha}k^{\alpha}\Pi_{2}^{M}(k^{2})+\Pi_{GB}^{M}(k^{2})\,\,,
\left.{\calP}_{\delta\sigma_{i}}\right|_{s}=64\pil_{\mathrm{Pl}}^{2}C^{2}(\mu_{i})e^{-2\omega_{i}}\left({\frac{\widetilde{H}}{2\pi}}\right)_{s}^{2}\left({\frac{k}{k_{s}}}\right)^{3-2\mu_{i}}\,,
J_{0}=3\cdot2^{8}{\Big(}R^{MNPQ}R_{RNPS}R_{M}^{~~TUR}R_{~~TUQ}^{S}+{\frac{1}{2}}R^{MNPQ}R_{RSPQ}R_{M}^{~~TUR}R_{~~TUN}^{S}{\Big)}+{\mathcalO}(R_{MN})\,
\tilde{S}=\intd^{d-1}x\sqrt{|g|}\k\\left[R-{\textstyle\frac{1}{4}}k^{2}F_{(2)}^{2}+{\textstyle\frac{(-1)^{(d-2)}}{2\cdot(d-1)!}}F_{(d-1)}^{2}+{\textstyle\frac{(-1)^{(d-3)}}{2\cdot(d-2)!}}k^{-2}F_{(d-2)}^{2}\right]\,,
|V_{B}\rangle=\mathrm{exp}\left(\frac{1}{2}\sum_{r,s=1}^{3}\sum_{m,n=-\infty}^{\infty}\sum_{I=1}^{8}a_{rm}^{I\dagger}\overline{{N}}_{mn}^{rs}a_{sn}^{I\dagger}\right)|0\rangle.
\Bigl<n;t|H|n;t\bigr>=(n+\frac{1}{2})\hbar\omega,~~\bigl<n;t|i\hbar\frac{\partial}{\partialt}|n;t\bigr>=0,~~~(\triangleq)^{2}(\trianglep)^{2}=\frac{\hbar^{2}}{4},
\xi\,(z_{1},\,z_{2})\,=\,\sum_{n=0}^{+\infty}\,\frac{1}{n!}\,\left(\epsilon\,\partial_{z_{1}}\,+\,\eta\,\partial_{z_{2}}\right)^{n}\,\xi(z_{1}^{\prime},\,z_{2}^{\prime})~.
\begin{array}{c}{G_{\mu\nu\alpha\beta}=-G_{\mu\nu\beta\alpha}=-G_{\nu\mu\alpha\beta}=G_{\alpha\beta\mu\nu},}\\{G_{\mu\nu\alpha\beta}+G_{\mu\beta\nu\alpha}+G_{\mu\alpha\beta\nu}=0.}\\\end{array}
{\calL}_{m}=-\frac{1}{4}F_{\mu\nu}^{a}F^{\mu\nu,a}-\frac{1}{2}e^{-\gamma\sqrt{8\piG}\Psi}D_{\mu}\Phi^{a}D^{\mu}\Phi^{a}-e^{-2\gamma\sqrt{8\piG}\Psi}V(\Phi^{a})\,
\sum_{n=1}^{\infty}F(2\pin)~\approx~\frac{T~\sqrt{M^{2}\tilde{p}_{2}}}{(~M^{2}+(2\piT)^{2}~)^{3/4}}~e^{-\tilde{p}_{2}\sqrt{~M^{2}+(2\piT)2~}}~~~.\,
S_{loc}~=~\int_{d}\sqrt{g}\left(\Phi_{1}(\phi)R+U(\phi)+\Phi_{2}(\phi)R^{2}+\Phi_{3}(\phi)R_{\mu\nu}R^{\mu\nu}+\Phi_{4}(\phi)R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}\right)
\begin{array}{lll}{L_{KI}^{(-)}L_{IJ}^{(+)}}&{-}&{(-1)^{P_{I}(P_{J}+P_{K})}L_{IJ}^{(+)}L_{KI}^{(-)}=}\\{}&{}&{(q^{-1}-q)(-1)^{P_{I}P_{K}}L_{II}^{(+)}L_{KJ}^{(-)},(I<J<K)}\\\end{array}
R^{-1}(j_{1}\hat{y_{1}}+\frac{j_{2}-\tau_{1}j_{1}}{\tau_{2}}\hat{y_{2}})R=j_{1}^{\prime}\hat{y_{1}}+\frac{j_{2}^{\prime}-\tau_{1}j_{1}^{\prime}}{\tau_{2}}\hat{y_{2}}
V(B)=k\,e^{-B}-\frac{1}{\left(D-2\right)}\left\{\frac{1}{2D_{2}}\left(2\,D-D_{1}-4\right)\,\frac{\kappa^{2}\,F^{2}}{g^{2}}\,e^{-2\,B}-4\,{\kappa}^{2}\,\Lambda\,B\right\}
g^{(2)}(y)=-\frac{2\hbarc\xi^{2}}{\pi^{3/2}}\int_{0}^{1}du\,\frac{y^{3}a^{2}u^{2}}{\sqrt{1-u^{2}}}G_{13}^{30}\left(\frac{1}{2};0,0,1;4a^{2}y^{2}u^{2}\right).
(\Gamma_{\phi}^{\alpha\eta})_{abc}=2e^{3}(\gamma^{\lambda}\gamma^{\phi}\gamma^{\tau})_{\alpha\eta}\int\frac{d^{4}S}{(2\pi)^{4}}(Q+S)_{\tau}(P+S)_{\lambda}[\tau_{3}D(S)]_{ab}[\tau_{3}D(P+S)]_{bc}[\tau_{3}D(Q+S)]_{ca}.
\delta_{\Lambda}^{(1)}S_{0}[h_{{\calO}_{1}},h_{{\calO}_{2}}]=S_{0}[\delta_{\Lambda}^{(1)}h_{{\calO}_{1}},h_{{\calO}_{2}}]+S_{0}[h_{{\calO}_{1}},\delta_{\Lambda}^{(1)}h_{{\calO}_{2}}]
e^{-1}{{\calL}}=-{\frac{1}{2}}R-{\frac{1}{4}}G_{IJ}F_{\mu\nu}{}^{I}F^{\mu\nuJ}-{\frac{1}{2}}g_{ij}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}+{\frac{e^{-1}}{48}}\epsilon^{\mu\nu\rho\sigma\lambda}C_{IJK}F_{\mu\nu}^{I}F_{\rho\sigma}^{J}A_{\lambda}^{K}.
X^{\mu}(\sigma^{1},\sigma^{2}+2\pi)=y^{\mu}(\sigma^{1},\sigma^{2})+2n\beta\sigma^{2}\delta_{0}^{\mu},\quadS(X,g_{ab})=S(y,g_{ab})+{\frac{\pin^{2}x^{2}}{2\tau_{2}}},\quadn\in\mathrm{Z}\quad.
F^{(1)}=\frac{\lambda^{4}\zeta}{(4\pi)^{2}}(D^{\alpha}\Phi)(D_{\alpha}\Phi)(\bar{D}_{\dot{\alpha}}\bar{\Phi})(\bar{D}^{\dot{\alpha}}\bar{\Phi})|{\calL}_{c}^{\prime\prime}(\Phi)|^{-4}+O^{8}(D,\bar{D})
H=\intd^{3}{x}\biggl[{\frac{g^{2}}{2}}\left(P_{ai}-\frac{\theta}{8\pi^{2}}B_{(ai)}\right)^{2}+g^{2}\left(P_{a}-\frac{\theta}{8\pi^{2}}B_{a}\right)^{2}+{\frac{1}{2g^{2}}}B_{ai}^{2}\biggr].
\left[{\frac{dn_{d}(\omega|\mu)}{d\omega}}\right]_{\tiny\mathrm{div}}=r_{d}{\frac{1}{(4\pi)^{2}\kappa}}\int_{\Sigma}\left[2b+a\left({\frac{\omega^{2}}{\kappa^{2}}}{\calP}-{\frac{R}{6}}+{\frac{\calQ}{4}}\right)\right]~~~.
=\frac{1}{2}\left\{[\vec{S}_{1}+\vec{S}_{2}+\vec{S}_{3}]^{2}-(\vec{S}_{1}+\vec{S}_{3})^{2}-3/4\right\}+\epsilon(S_{1}^{z}S_{2}^{z}+S_{2}^{z}S_{3}^{z})
{\calA}^{\epsilon}=\frac{M}{2}\epsilong_{\mu\nu}(q)\dot{q}^{\mu}\dot{q}^{\nu}=\frac{M}{2}\epsilong_{\mu\nu}(q^{\prime})\dot{q}^{\prime}{}^{\mu}\dot{q}^{\prime}{}^{\nu}=\frac{M}{2}\epsilong_{\mu\nu}(\bar{q}){\dot{\bar{q}}}^{\mu}{\dot{\bar{q}}}^{\nu},
\left|\left.a_{k}\dots\frac{1}{2},a_{1},\frac{1}{2}\right|n_{k}\dots,m_{1},n_{1}\right\rangle\quad\mathrm{or}\quad\left|\left.\frac{1}{2},a_{k}\dots\frac{1}{2},a_{1},\frac{1}{2}\right|m_{k},n_{k}\dots,m_{1},n_{1}\right\rangle
\begin{array}{rclcrcl}{|e_{R}^{c}\rangle}&{=}&{\displaystyle\bigg|1,\frac{1}{2},\frac{1}{2}\bigg\rangle,}&{\quad}&{-|\nu_{R}^{c}\rangle}&{=}&{\displaystyle\bigg|1,\frac{1}{2},-\frac{1}{2}\bigg\rangle,\nonumber}\\{|e_{L}^{c}\rangle}&{=}&{|2,0,0\rangle,}&{\quad}&{|\nu_{L}^{c}\rangle}&{=}&{|0,0,0\rangle,}\\\end{array}
\frac{1}{\sqrt{-\overline{{g}}}}\partial_{\mu}[\pm\sqrt{-\overline{{g}}}\enspace\overline{{g}}^{\mu\gamma}\overline{{g}}^{\nu\delta}\tilde{F}_{\gamma\delta}]-\frac{\tilde{e}^{2}}{8\omega}\varphi^{2}\overline{{g}}^{\alpha\nu}\tilde{A}_{\alpha}=0
\begin{array}{rcl}{T^{*}}&{=}&{{\calD}T{\calD}^{-1}}\\{}&{=}&{\left(\begin{array}{ccccc}{t_{11}}&{t_{12}}&{-t_{13}}&{t_{14}}&{t_{15}}\\{t_{21}}&{t_{22}}&{-t_{23}}&{t_{24}}&{t_{25}}\\{-t_{31}}&{-t_{32}}&{t_{33}}&{-t_{34}}&{-t_{35}}\\{t_{41}}&{t_{42}}&{-t_{43}}&{t_{44}}&{t_{45}}\\{t_{51}}&{t_{52}}&{-t_{53}}&{t_{54}}&{t_{55}}\\\end{array}\right)~.}\\\end{array}
<\sigma_{\alpha}^{r},\sigma_{\beta}^{r^{\prime}}>=\delta_{rr^{\prime}}\delta_{\alpha\beta},\quad\hat{\sigma}_{r}^{\alpha}=\delta^{\alpha\beta}\hat{\sigma}_{\beta}^{r}={(\hat{\sigma}_{\alpha}^{r})}^{+}=\hat{\sigma}_{\alpha}^{r},\quad\{\hat{\sigma}_{\alpha}^{r},\hat{\sigma}_{\beta}^{r^{\prime}}\}=2\delta_{rr^{\prime}}\delta_{\alpha\beta}I_{2}.
S=\intd^{d}x\sqrt{-g}\,e^{-\phi}\Big\{R+(\partial_{\mu}\phi)^{2}+\frac{1}{8}\partial_{\mu}M^{IJ}\partial^{\mu}M_{JI}-\frac{1}{4}M_{IJ}F_{\mu\nu}^{I}F^{J\mu\nu}-\frac{1}{12}H_{\mu\nu\lambda}^{2}\Big\}\,
\times\Bigg\{\sum_{\sigma\inZ_{N}}\frac{\omega^{\sigmaj_{2}}w(x_{3},x_{13},x_{1}|\sigma)w(x_{4},x_{24},x_{2}|i_{1}+j_{3}+\sigma)}{w(x_{4},x_{14},x_{1}|j_{3}+\sigma)w(x_{3}/\omega,x_{23},x_{2}|j_{1}+\sigma)}\Bigg\}_{0}~~~~~~~~~~~~
\int_{\frac{\lambda^{2}}{4}(Q_{1}-Q_{2})}^{\frac{\lambda^{2}}{4}(Q_{1}+Q_{2})}d\eta\,f(\eta)=\frac{\lambda^{2}}{4}\sum_{j=1}^{Q_{2}}\left[f(\eta_{j})+f(\eta_{j-1})\right]\,+\,O(\lambda^{6}).
S=\int{\frac{d^{2}x}{2\pi}}\left[\partial_{\mu}\eta_{1}\partial_{\mu}\eta_{2}+\Lambda^{\prime}\partial_{\mu}\eta_{1}\partial_{\mu}\eta_{2}+\Lambda^{\prime\prime}\eta_{1}\eta_{2}\partial_{\mu}\eta_{1}\partial_{\mu}\eta_{2}\right]
I_{eff}=\intdtd^{2}{\bfr}(\sum_{m}\partial_{t}\rho(\dot{\bfa}^{m}G^{m})\cdot{\bfr}-\rho\partial_{t}\omega-{\frac{1}{2}}\sum_{m}\rho(\dot{\bfa}^{m}G^{m})\cdot(\dot{\bfa}^{m}G^{m}))
L=\displaystyle\frac{1}{2}g_{jk}\dot{x}^{j}\dot{x}^{k}+\displaystyle\frac{i}{2}g_{jk}\vartheta_{a}^{j}D_{t}\vartheta_{a}^{k}+\displaystyle\frac{1}{4}R_{jkln}\vartheta_{1}^{j}\vartheta_{2}^{l}\vartheta_{1}^{k}\vartheta_{2}^{n}-\displaystyle\frac{1}{2}g^{jk}\frac{\partialW}{\partialx^{j}}\frac{W}{\partialx^{k}}-iW_{j;k}\vartheta_{1}^{j}\vartheta_{2}^{k},
\underline{{\alpha}}(r)=\alpha_{0}\left[1-\frac14\omega_{0}^{2}r^{2}+\frac{1}{64}\left(\omega_{0}^{4}+\frac{9}{r_{0}^{2}}\kappa^{2}\right)r^{4}-\frac{1}{48}\left(\frac{45}{8}\frac{\omega_{0}^{2}\kappa^{2}}{r_{0}^{2}}+12\frac{\kappa^{2}}{r_{0}^{4}}+\frac18\omega_{0}^{6}\right)r^{6}\right].
{\calY}_{\overline{{\betaV}}}^{\overline{{\alphaV}}}=\xi_{\overline{{\betaV}}}^{\overline{{\alphaV}}}Y_{\overline{{\betaV}}}^{\overline{{\alphaV}}}\prod_{\ell=1}^{3}{\overline{X}}_{{\overline{{\betaT}}}^{\,\ell}}^{{\overline{{\alphaT}}}^{\,\ell}}\prod_{\ell=4}^{14}X_{{\overline{{\betaT}}}^{{\,\ell}}}^{{\overline{{\alphaT}}}^{{\,\ell}}}
\left\langle\Phi_{\mu\nu}\Phi^{\mu\nu}\right\rangle=2\left\langleA_{\mu}^{1}A_{\nu}^{2}A^{1\mu}A^{2\nu}\right\rangle-2\left\langleA_{\mu}^{2}A_{\nu}^{1}A^{1\mu}A^{2\nu}\right\rangle.
{\overline{A}}_{\mu}(x,\beta)=-\frac{i}{g}e^{i\theta_{a}T^{a}}\left[{\overline{\sigma}}_{\mu\nu}\frac{\rho^{2}(x-x_{0})_{\mu}}{(x-x_{0})^{2}[(x-x_{0})^{2}+\rho^{2}]}\right]e^{-i\theta_{a}T^{a}}\,,
\mathrm{~\psi(x)~}\rightarrow\psi^{\prime}(x^{\prime})=S_{\pi}\mathrm{~\psi(x)~},\;\;\;\;\;\;\mathrm{~\overline{{\psi}}(x)~}\rightarrow\overline{{\psi}}^{\prime}(x^{\prime})=\mathrm{~\overline{{\psi}}(x)~}S_{\pi}^{-1},
\sum_{[b]}Y_{[b]}\sum_{(a)}(\bar{\psi}_{L})_{r}^{a\tilde{a}}(\tilde{\phi}^{\prime})_{\tilde{a}}^{(a)}v(\psi_{R})_{a}^{[b][1]}+(\bar{\psi}_{L})_{r}^{a\tilde{a}}v^{-1}m_{\tilde{a}[b]}(\phi^{\prime})^{(1)r}(\psi_{R})_{a}^{[b][1]}
\langleT\Phi(t,\vec{x})\Phi(t^{\prime},\vec{x}^{\prime})\rangle={\frac{\ell^{1-d}}{(4\pi)^{\frac{d+1}{2}}}}{\frac{\Gamma({\frac{d}{2}}-\nu)\Gamma({\frac{d}{2}}+\nu)}{\Gamma({\frac{d+1}{2}})}}\_{2}F_{1}\left({\frac{d}{2}}-\nu,{\frac{d}{2}}+\nu;{\frac{d+1}{2}};1-s^{2}(t,\vec{x};t^{\prime},\vec{x}^{\prime})\right)\,,
D_{z}^{\dagger}=\left(\begin{array}{ccc}{(\bar{z}_{2}+\bar{z}_{4})\mathbf{1}_{k\timesk}}&{-(z_{1}+z_{3})\mathbf{1}_{k\timesk}}&{\mathbf{0}_{k\times1}}\\{(\bar{z}_{1}+\bar{z}_{3})\mathbf{1}_{k\timesk}}&{(z_{2}+z_{4})\mathbf{1}_{k\timesk}}&{\mathbf{0}_{k\times1}}\\\end{array}\right)\,.
\overline{{Z}}_{bklm}^{aij}=\frac{1}{2}\left(D_{[k}\right)_{\;\;b}^{a}\delta_{l}^{i}\delta_{m]}^{j},\;\overline{{Z}}_{klmi}^{ab}=-\frac{g}{4}\frac{\delta^{2}W_{cd}}{\delta\varphi_{a}\delta\varphi_{b}}
n\:C\;\alpha_{1}^{1-p}\:(\beta_{1}+\gamma)^{1-q}\left(\prod_{j=2}^{p}\alpha_{1}^{a_{j}-\alpha_{j}}\:\alpha_{j}^{-a_{j}}\right)\left(\prod_{k=2}^{q}(\beta_{1}+\gamma)^{b_{k}-\beta_{k}}\:\beta_{k}^{-b_{k}}\right)\:.
{\cal{L}}^{N}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+(g^{\mu\nu}-\varepsilon^{\mu\nu})\partial_{\mu}\phiA_{\nu}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}A_{\mu}A^{\mu}
\widehat{h}=\displaystyle\frac{\widehat{H}}{{\calM}}=\displaystyle\frac{2\pi}{l}\left(\widehat{L_{0}}+\widehat{\overline{{L}}_{0}}-\displaystyle\frac{c}{12}\widehat{I}+\kappa\displaystyle\frac{l^{2-h}}{\left(2\pi\right)^{1-h}}\widehat{B}\right)\:.
\left.\mathrm{Tr}~K^{(2)}\right|_{{\calC}_{\beta}\times\Sigma}=\left.\mathrm{Tr}~K^{(0)}\right|_{{\calC}_{\beta}}\left.\mathrm{Tr}~K^{(2)}\right|_{\Sigma}+\left.\mathrm{Tr}~K^{(2)}\right|_{{\calC}_{\beta}}\left.\mathrm{Tr}~K^{(0)}\right|_{\Sigma}+\left.\mathrm{Tr}~K^{(1)}\right|_{{\calC}_{\beta}}\left.\mathrm{Tr}~K^{(1)}\right|_{\Sigma}.
S=\intd^{4}x\Phie^{\alpha\phi/M_{p}}\left[-\frac{1}{\kappa}R(\Gamma,g)+\frac{1}{2}g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}\right]+\intd^{4}x\sqrt{-g}e^{\alpha\phi/M_{p}}\left[-\frac{b_{g}}{\kappa}R(\Gamma,g)+\frac{b_{k}}{2}g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}\right]
\begin{array}{rcl}{H^{(-1)}(\lambda)[\psi]}&{=}&{\int_{x_{-}}^{x_{+}}dx\,\psi(\lambda)^{2}\,,}\\{H^{(-2)}(\lambda)[\psi]}&{=}&{\int_{x_{-}}^{x_{+}}dx\,\psi(\lambda)\,\bar{\psi}(\lambda)\,,}\\{H^{(-3)}(\lambda)[\psi]}&{=}&{\int_{x_{-}}^{x_{+}}dx\,\bar{\psi}(\lambda)^{2}\,.\}\\\end{array}
\delta_{\epsilon_{2}}^{(\alpha_{M}^{\prime})^{3}}e_{\mu}{}^{r}\Big|_{\begin{aligned}{}&{{}{}\scriptstyle\psi_{\lambda}\rightarrowD_{\lambda}\epsilon_{1}}\\{}&{{}{}\scriptstyle\psi_{mn}\rightarrow\frac{1}{8}\Gamma^{pq}\epsilon_{1}W_{pqmn}}\\\end{aligned}}-2\bar{\epsilon}_{1}\Gamma^{r}\delta_{\epsilon_{2}}^{(\alpha_{M}^{\prime})^{3}}\psi_{\mu}-(\epsilon_{1}\leftrightarrow\epsilon_{2})\,.
S_{\mathrm{bosonic}}\=\\intd^{4}x\left[1-\left(1+{\frac{1}{2}}F_{mn}F^{mn}+{\frac{1}{8}}(F_{mn}F^{mn})^{2}-{\frac{1}{4}}F_{mn}F^{nk}F_{kl}F^{lm}\right)^{1/2}\right]\.
\mathrm{exp}\biggl(-{\frac{1}{2}}\,u\,\vec{\alpha}^{\ast}\cdot\vec{\alpha}^{\ast}+v\,\vec{\alpha}^{\ast}\,\tilde{\bfR}\,\vec{\gamma}\biggl)=\sum_{n_{1},n_{2}=0}^{\infty}{\frac{\alpha_{1}^{\astn_{1}}}{n_{1}!}}{\frac{\alpha^{\astn_{2}}}{n_{2}!}}{\bfH}_{n_{1},n_{2}}^{\{u{\bfI}_{2}\}}\biggl({\frac{v}{u}}\tilde{\bfR}\vec{\gamma}\biggr)\\.
\gamma_{ab}=\left(\begin{array}{ll}{-N^{2}+N^{A}N^{B}h_{AB}}&{h_{AB}N^{B}}\\{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,h_{AB}N^{B}}&{\,\,\,\,h_{AB}}\\\end{array}\right)\,.
g(m_{i}^{2})\equiv{\frac{1}{16\pi^{2}}}\left[\sum_{s}m_{s}^{D-2}{}~\Gamma\left(1-\fracD2,m_{s}^{2}\delta\right)+2\sum_{d}m_{d}^{D-2}\Gamma\left(1-\fracD2,m_{d}^{2}\delta\right)\right]~~~,
{\calO}_{\alpha_{1}\ldots\alpha_{J_{2}}\;\beta_{1}\ldots\beta_{J_{2}}}^{[a_{1},a_{2}]}=w_{\alpha_{1}\ldots\alpha_{J_{2}}}\;w_{\beta_{1}\ldots\beta_{J_{2}}}^{\prime}\;W^{[a_{1},a_{2}]}\\rightarrow\\ell=4+J_{2}+2(a_{1}+a_{2})\,,
\begin{array}{lclcl}{({\dot{X}}^{(II)}\pm{X^{\prime}}^{(II)})^{2}(\sigma,\tau_{0})}&{=}&{({\dot{X}}^{(I)}\pm{X^{\prime}}^{(I)})^{2}(\sigma,\tau_{0})}&{=}&{0}\\{}&{}&{}&{}&{}\\{({\dot{X}}^{(III)}\pm{X^{\prime}}^{(III)})^{2}(\sigma,\tau_{0})}&{=}&{({\dot{X}}^{(I)}\pm{X^{\prime}}^{(I)})^{2}(\sigma,\tau_{0})}&{=}&{0}\\\end{array}
T_{f}=\left(\frac{1}{h^{2}}\right)^{x/(1-qy)}\left(\frac{\sqrt{m_{Dp}}m^{d_{\perp}-q-1}}{aBV_{p}}\right)^{y/(1-qy)}\sim\left(\frac{1}{V_{p}}\right)^{\frac{x+y/2}{1-qy}}
\frac{dG_{12}}{dt}=-14G_{12}+6wG_{4}G_{12}+3wG_{8}^{2}-6dG_{4}^{3}G_{8}+fG_{4}^{6}+3cg^{2}G_{12}+9eg^{2}G_{4}^{2}G_{8}+hg^{2}G_{4}^{5}.
\left.\begin{array}{llr}{\gamma_{1,3,(0,0,0)}}&{=}&{{\bf1}_{1}}\\{\gamma_{1,\tilde{7}_{3},(0)}}&{=}&{-\alpha{\bf1}_{1}}\\{\gamma_{1,7_{4},(0)}}&{=}&{-\alpha{\bf1}_{2}}\\{\gamma_{1,7_{5},(0)}}&{=}&{\alpha^{2}{\bf1}_{3}}\\\end{array}\right.
\begin{array}{rcl}{{\calR}(u)}&{=}&{\displaystyle\frac{1}{4}\sum_{i=0}^{3}w_{i}\sigma^{i}\otimes\sigma^{i}}\\{}&{=}&{\rho\left[\begin{array}{cccc}{a}&{}&{}&{}\\{}&{b}&{c}&{}\\{}&{b}&{c}&{}\\{}&{}&{}&{a}\\\end{array}\right]~,}\\\end{array}
S={\frac{1}{2\kappa^{2}}}\intd^{10}x{\sqrtg}[{\frac{1}{2}}g^{mn}\partial_{m}\phi\partial_{n}\phi]={\frac{V}{4\kappa^{2}}}\intd^{4}x\int^{\rho}drr^{5}[(\partial_{r}\phi)^{2}+h(r)\eta^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi],
{\calS}_{E}=\frac{\pi}{4}\tau\nu\sum_{n=0}n^{2}X_{n0}^{2}+\frac{\pi}{2}\tau\sum_{n=0}\sum_{k=1}\left(\rho\left(\frac{2\pik}{\tau}\right)^{2}+n^{2}\nu\right)|X_{nk}|^{2},
W_{n}:={\frac{1}{(2\pi)^{2n}}}{\frac{1}{n!}}\intd^{n}zd^{n}\bar{z}\prod_{a=1}^{n}\bigg({\frac{1}{z_{a}-1}}{\frac{1}{\bar{z}}}-{\frac{1}{\bar{z}-1}}{\frac{1}{z_{a}}}\bigg)
\begin{array}{rcl}{ds^{2}}&{=}&{\left(1+\frac{P}{2\rho^{2}}\right)^{-1/4}\left[-(dt)^{2}+\delta_{ij}dx^{i}dx^{j}\right]+\left(1+\frac{P}{2\rho^{2}}\right)^{3/4}\left(d\rho^{2}+\rho^{2}d\Omega_{3}^{2}\right)}\\{e^{2\phi}\,\,\,}&{=}&{\,\,\,\left(1+\frac{P}{2\rho^{2}}\right),\qquadH^{(1)}\,\,\,=\,\,\,P\epsilon_{3}.}\\\end{array}
S~=~-\frac{1}{2}\inte^{-{\bf\Phi}}\Bigl\{\epsilon_{abc}{\bfe}^{a}{\bfR}^{bc}+d{\bf\Phi}~{^{*}d}{\bf\Phi}-2{\bfH}~{^{*}{\bfH}}+{\frac{\alpha^{\prime}}{8}}Tr{\bfF}~{^{*}{\bfF}}+\frac{\Lambda(\Phi)}{3}\epsilon_{abc}{\bfe}^{a}{\bfe}^{b}{\bfe}^{c}~\Bigr\}
\frac{1}{(g_{\mathrm{YM}}^{\mathrm{f}})^{2}(\mu)}=\frac{1}{(g_{\mathrm{YM}}^{\mathrm{f}})^{2}}\left(1-\frac{(g_{\mathrm{YM}}^{\mathrm{f}})^{2}N}{4\pi\mu}\right)\,.
A=\int...\intdk_{1}...dk_{r}A_{I}(q_{j},k_{i})\prod_{n=1}^{r}\frac{W_{n}(k_{n}^{2},q)}{k_{n}^{2}+m_{n}^{2}}A_{II}(q_{j}^{\prime},k_{i})\delta^{4}(q-k_{1}-...-k_{r}),
G^{(m)}(u)=\omega^{-m}\frac{\varphi(u)}{\varphi(1)}\frac{(q^{2};q^{2})_{\infty}^{2}}{(t^{2};t^{2})_{\infty}^{2}}\frac{(t^{2}\omega^{m}u^{2/n};t^{2})_{\infty}(t^{2}\omega^{-m}u^{-2/n};t^{2})_{\infty}}{(q^{2}\omega^{m}u^{2/n};q^{2})_{\infty}(q^{2}\omega^{-m}u^{-2/n};q^{2})_{\infty}}.
[{\calD}\mu]={\calD}B^{\mu}{\calD}\lambda{\calD}\pi_{\lambda}{\calD}\rho{\calD}\pi_{\rho}{\calD}\xi\prod_{\beta=0}^{3}\{\delta(\Gamma_{\beta}[B^{0}+\xi,B^{i},\lambda,\rho])\}det\mid\{\tilde{\Omega}_{\alpha},\Gamma_{\beta}\}\mid,
\alpha_{1}(\zeta_{1}/\zeta_{2})\sum_{m\equivn~(2)}\langle\varphi_{m}^{(l\pm1,l)}\varphi_{n}^{(l,l\mp1)}\zeta_{2}^{-m}\zeta_{1}^{-n}\rangle=(\zeta_{2}\leftrightarrow\zeta_{1}).
\int_{0}^{1}\,d\rho\frac{1}{[(\rhox-y)^{2}]^{\frac{N}{2}-1}}=\int_{0}^{\infty}d\rho\,\frac{1}{[(\rhox-y)^{2}]^{\frac{N}{2}-1}}-\int_{0}^{1}d\rho\,\frac{\rho^{N-4}}{[(x-\rhoy)^{2}]^{\frac{N}{2}-1}}
n^{2}b_{1}\biggl(\frac{\ell_{_\mathrm{C}}}{\lambda}\biggr)^{2}=n^{2}\quad\Rightarrow\quad\vert\eta_{\mathrm{j}}\vert=\vert\eta_{1}^{\prime}\vert=\biggl(\frac{n\ell_{_\mathrm{C}}}{2\pi\ell_{0}}\biggl)^{-1}b_{1}^{-1/2}.
\tilde{A}_{\mu}(x)\longrightarrow[1+i\tilde{g}\tilde{\Lambda}(x)]\tilde{A}_{\mu}(x)[1-i\tilde{g}\tilde{\Lambda}(x)]-2i\tilde{g}\partial_{\mu}\int\delta\xids\tilde{\Lambda}_{\nu}[\xi|s]\dot{\xi}^{\nu}(s)\delta(\xi(s)-x),
2B_{1}\equiv\hat{B}+\hat{B}^{\dagger}=-3\dot{a}\frac{\alpha_{1}}{\alpha}+{\frac{3a}{2\alpha}}M_{P}^{-2}\left(\mathbf{m}\alpha_{2}+\alpha_{2}^{\dagger}\mathbf{m}^{\dagger}\right)=a\frac{\dot{\alpha}}{\alpha}+3\dot{a}\,.
{\frac{\pi^{2}}{16a}}\int_{\sqrt{2(e+d)+e^{2}}}^{\lambda}{dx{\frac{1}{(\lambda^{2}-x^{2})^{1/2}\left[-(\lambda^{2}-x^{2})^{1/2}+1\right]}}\left[{\frac{1}{x^{2}}}+{\frac{1}{\left[-(\lambda^{2}-x^{2})^{1/2}+1\right]}}\right]}.
ds_{11}^{2}=\frac{-f(r)dt^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+dx_{4}^{2}+dx_{5}^{2}}{H^{1/3}(r)}+H^{2/3}(r)\left(\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{4}^{2}\right)\,,
\sum_{n=1}^{\infty}\left|\int_{0}^{\infty}ds\,e^{-sn^{2}-\frac{\pi^{2}m^{2}}{s}}\right|\leq\sum_{n=1}^{\infty}\left|\int_{0}^{\infty}ds\,e^{-sn^{2}}\right|=\sum_{n=1}^{\infty}\frac{1}{n^{2}}=\zeta(2)=\frac{\pi^{2}}{6}~.
\begin{array}{rcllcccc}{\mathrm{\underline{{sector}}}}&{{}}&{\mathrm{\underline{{state}}}}&{{}}&{{}}&{\mathrm{\underline{{~R_1~}}}}&{\mathrm{\underline{{~R_2~}}}}&{\mathrm{\underline{{~R_3~}}}}\\{\mathrm{NS}}&{{}}&{|2s_{3},2s_{4}\rangle,}&{s_{3}=s_{4}}&{{}}&{+}&{+}&{-2is_{3}}\\{\mathrm{R}}&{{}}&{|+,2s_{1},2s_{2}\rangle,}&{s_{1}=-s_{2}}&{{}}&{+}&{-}&{+2is_{1},}\\\end{array}
\hat{V}^{\dagger}(\gamma_{2})\,\hat{V}^{\dagger}(\gamma_{1})\,\hat{\phi}(\sigma)\,\hat{V}(\gamma_{1})\hat{V}(\gamma_{2})=\hat{V}^{\dagger}(\gamma_{1}\gamma_{2})\,\hat{\phi}(\sigma)\,\hat{V}(\gamma_{1}\gamma_{2}),
\beta_{\xi}\xi=\tilde{\beta}_{\xi}f_{\xi}\;,\;\gamma_{s}=\tilde{\gamma}_{s}f_{s}\;,\;\hat{\gamma}_{s}=\tilde{\hat{\gamma}}_{s}\hat{f}_{s}\;,\;\bar{\gamma}_{s}=\tilde{\bar{\gamma}}_{s}\bar{f}_{s}\;,\;\gamma_{B}=\tilde{\gamma}_{B}f_{B}
\mathrm{bpz}^{-1}\circ\mathrm{hc}\left(\frac{i}{\sqrt{2}}B_{\mu}\alpha_{-2}^{\mu}c_{1}|0\rangle\right)=\mathrm{bpz}^{-1}\left(\frac{-i}{\sqrt{2}}\langle0|B_{\mu}^{*}\alpha_{2}^{\mu}c_{-1}\right)=-\frac{i}{\sqrt{2}}B_{\mu}^{*}(-\alpha_{-2}^{\mu})c_{1}|0\rangle,
\textrm{Tr}((\triangleV)^{\nu}\triangleI)^{(d)}\sim-\frac{m^{d+1}}{2\pid(m^{2}-z^{2})^{d}}\intd^{2}x\,\frac{q(\vec{x})}{|\vec{v}(\vec{x})|}\left[\left(\partial_{i}\hat{n}^{a}\,\partial_{i}\hat{n}^{a}+|\vec{v}(\vec{x})|\right)^{d/2}-\left(\partial_{i}\hat{n}^{a}\,\partial_{i}\hat{n}^{a}-|\vec{v}(\vec{x})|\right)^{d/2}\right]
\deltag_{\mu\nu}(x,\tau,\sigma,f)=(P\xi)_{\mu\nu}(x)+f^{\star}\left[\left(2\delta\sigma+\frac{\bar{g}^{\alpha\beta}}{D}\frac{\partial\bar{g}_{\alpha\beta}}{\partial\tau_{i}}\delta\tau_{i}+\frac{\bar{g}^{\alpha\beta}}{D}(\bar{F}\bar{\xi})_{\alpha\beta}\right)\bar{g}_{\mu\nu}(\sigma,\tau)\right](x)+k_{\mu\nu}^{i}(x)\delta\tau_{i}.
Q_{\mu}^{\mathrm{NS}}\equiv{\frac{Q^{\mathrm{NS}}}{v_{1}}}\int_{{\calM}_{1}}dX^{\mu}\,\quad\mathrm{with}\quadQ^{\mathrm{NS}}={\frac{1}{\mathrm{Vol}\,S^{7}}}\int_{S^{7}}e^{-2\Phi}\,{}^{*}H^{(3)}
0=({\calM}^{2})^{2}-\frac1{4k_{q}}\tilde{\gamma}\tilde{g}^{2}(q_{0}^{2}+\bar{q}_{0}^{2}){\calM}^{2}+\frac1{4k_{T}k_{q}}(\tilde{\gamma}\tilde{g}^{2}-\frac1{k_{T}})q_{0}^{2}\bar{q}_{0}
\varphi[x_{\mathrm{qc}}]={\textstyle\frac{1}{2}}\left[\mathrm{sgn}(\dot{x}^{\prime})a(x^{\prime})-\mathrm{sgn}(\dot{x}^{\prime\prime})a(x^{\prime\prime})\right]+n_{R}a(x_{R})-n_{L}a(x_{L}).
[(\epsilon_{1}^{+}\epsilon_{5}^{+}p_{2}^{-}p_{6}^{-})(\epsilon_{2}^{+}\epsilon_{4}^{+}p_{3}^{-}p_{5}^{-})(\epsilon_{3}^{+}\epsilon_{6}^{+}p_{1}^{-}p_{4}^{-})]~=~0\,