harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
D_{\alpha}^{\mathbfi}=\partial_{\alpha}^{\mathbfi}+i{\bar{\theta}}^{\dot{\alpha}{\mathbfi}}\partial_{\alpha\dot{\alpha}}\,\qquad\quad{\bar{D}}_{\dot{\alpha}{\mathbfi}}=-{\bar{\partial}}_{\dot{\alpha}{\mathbfi}}-i{\theta}_{\mathbfi}^{\alpha}\partial_{\alpha\dot{\alpha}}.
{\mathit{M}}_{n}^{00}=-\frac{\tilde{p}^{2}}{4\alpha^{2}},~~~{\mathit{M}}_{n}^{11}=\frac{\tilde{p}^{2}}{4\alpha^{2}}-\frac{\tilde{p}_{2}^{2}}{2\alpha^{2}},~~~{\mathit{M}}_{n}^{22}=\frac{\tilde{p}^{2}}{4\alpha^{2}}-\frac{\tilde{p}_{1}^{2}}{2\alpha^{2}}\,
\rho=\frac{\dot{a}_{1}}{a_{1}}\frac{\dot{a}_{2}}{a_{2}}+\frac{\dot{a}_{2}}{a_{2}}\frac{\dot{a}_{3}}{a_{3}}+\frac{\dot{a}_{3}}{a_{3}}\frac{\dot{a}_{1}}{a_{1}}\ge0
\vec{\calK}U_{E,\hat{q}}=\left(\begin{array}{r}{\vec{K}^{\prime}\,u_{E,\hat{q}}^{(1)}}\\{E^{-1}\sqrt{V}\,\pi\vec{K}^{\prime}\,u_{E,\hat{q}}^{(1)}}\\\end{array}\right)\,,\quad\vec{K}^{\prime}=\vec{K}+F\,\frac{\vec{\sigma}}{2}\,,
\begin{array}{l}{{\calD}_{0}{\calX}^{0}-{\calX}^{0}{\calD}_{0}=\mu_{0}^{1/2}~,~~~~~~\mu_{0}{\calX}^{0}=q^{2}{\calX}^{0}\mu_{0}~,}\\{\mu_{0}{\calD}_{0}=q^{-2}{\calD}_{0}\mu_{0}~,~~~~~~\mu_{0}^{1/2}\equiv1+(q-1){\calX}^{0}{\calD}_{0}~,}\\\end{array}
\left(\Omega_{\alpha\beta}^{g_{1}g_{2},g_{3}}\right)\,\left(g_{3}^{*}\Omega_{\alpha\beta}^{g_{1},g_{2}}\right)\:=\:\left(\Omega_{\alpha\beta}^{g_{1},g_{2}g_{3}}\right)\,\left(\Omega_{\alpha\beta}^{g_{2},g_{3}}\right)\,\left(\omega_{\alpha}^{g_{1},g_{2},g_{3}}\right)\,\left(\omega_{\beta}^{g_{1},g_{2},g_{3}}\right)^{-1}
ds_{M2}^{2}\approxr^{4}(-dt^{2}+dx_{1}^{2}+dx_{2}^{2})\,k^{-2/3}+k^{1/3}\,\frac{dr^{2}}{r^{2}}+k^{1/3}\,\frac{\Lambda}{6}\,ds_{M^{7}}^{2}.
r_{0}=\frac{\delta}{\varepsilon\varepsilon^{\ast}}\varepsilon=\frac{\sqrt{2}\delta_{0}}{m_{0}}\frac{4\pim_{0}^{2}}{g^{2}N}\sqrt{1+\sqrt{1-\left(\frac{g^{2}N}{4\pim_{0}^{2}\alpha_{0}}\right)^{2}}}\,,
\eta^{A_{1}}\equiv\left(\eta_{1}^{\alpha_{1}},\eta_{2}^{\alpha_{1}},\eta_{3}^{\alpha_{1}}\right),\;\pi^{B_{1}}\equiv\left(\pi_{1}^{\beta_{1}},\pi_{2}^{\beta_{1}},\pi_{3}^{\beta_{1}}\right),\;\lambda^{\gamma_{1}},
\begin{aligned}{P_{m_{1}m_{2}m_{3}m_{4}}{}^{n_{1}n_{2}n_{3}n_{4}}=\,\,}&{{}\delta_{m_{1}m_{2}}^{n_{1}\,n_{2}}\,\delta_{m_{3}m_{4}}^{n_{3}n_{4}}+\frac{4}{d-2}\delta_{m_{1}m_{2}}^{n_{1}[m_{3}}\,\delta_{n_{3}\;n_{4}}^{m_{4}]n_{2}}+\frac{2}{(d-1)(d-2)}\delta_{m_{1}m_{2}}^{m_{3}m_{4}}\,\delta_{n_{1}n_{2}}^{n_{3}n_{4}}\,.}\\\end{aligned}
\dot{z}_{n}^{\prime}=-\sum_{B}\frac{\partial\beta_{B}}{\partial\mu}\frac{\partialz_{B}^{\prime}}{\partialP_{n}^{\prime}}~~~~\mathrm{and}~~~~\dot{P}_{n}^{\prime}=\frac{\partial\beta_{n}}{\partial\mu}+\sum_{B}\frac{\partial\beta_{B}}{\partial\mu}\frac{\partialz_{B}^{\prime}}{\partialz_{n}^{\prime}}
H^{(2)}=\frac{1}{2}\,\intd^{3}x\left[\sum_{i=1}^{3}\left(\pi_{i}-\frac{\theta}{8\pi^{2}}\,{\beta}_{i}\right)^{2}+\sum_{cyclic}\,k_{i}\left(\xi_{i}+\frac{\theta}{8\pi^{2}}\,(\phi_{j}-\phi_{k})\,b_{i}\,\right)^{2}+V(\phi,\chi)\right]\,,
S_{eff}=\intdt[P_{A}\dot{X^{A}}+\dot{C}^{\mu}\bar{C}_{\mu}+\dot{\eta}^{\mu}\bar{\eta}_{\mu}+\Pi_{\mu}\dot{N}^{\mu}+\{Q_{B},~\bar{C}_{\mu}N^{\mu}+\bar{\eta}_{\mu}\chi^{\mu}\}],
\phi^{(i)}\left(x\right)=\sum_{n=1}^{N}\int\widetilde{dp}^{\mu_{n}}\left[M^{\mu_{n}(i)}{}^{(j)}\left(p\right)a_{(j)}^{\mu_{n}}\left(p\right)e^{-ipx}+M^{*\mu_{n}(i)}{}^{(j)}a_{(j)}^{+\mu_{n}}\left(p\right)e^{ipx}\right]\;\,.
\tilde{S}_{d}={\frac{1}{2\kappa_{D}^{2}}}\intd^{D}x\sqrt{-G^{d}}e^{\delta\phi}\left[{\calR}_{G^{d}}+\gamma(\partial\phi)^{2}-{\frac{1}{2\cdot(D-2)!}}F_{D-2}^{2}\right],
\begin{array}{rl}{ds^{2}=}&{(H_{1}H_{5})^{-1/2}(dx_{0}^{2}+dx_{1}^{2})+H_{1}^{1/2}H_{5}^{-1/2}(dx_{2}^{2}+...+dx_{5}^{2})}\\{\nonumber+}&{(H_{1}H_{5})^{1/2}(dx_{6}^{2}+...+dx_{9}^{2})}\\\end{array}
{\calL}=K^{\dagger}K(Q_{i}^{\dagger}Q_{i}+\tilde{Q}_{i}^{\dagger}\tilde{Q}_{i})|_{D}+{\frac{1}{8\pi}}Im\tauW^{\alpha}W^{\alpha}|_{F}+2Re\,m_{ij}Q_{i}\tilde{Q}_{j}|_{F}
S_{-}=\intd^{4}x\sqrt{-g^{0}}\frac{1}{2}V_{0}^{-}\Phi\Bigr|_{y=0}+\intd^{4}x\sqrt{-g^{-\pir_{c}}}\frac{1}{2}V_{\pi}^{-}\Phi\Bigr|_{y=-\pir_{c}}
ds^{2}={\frac{4}{(1-\bar{X}^{2}-\bar{Y}^{2})^{2}}}\left[d\bar{Y}^{2}+d\bar{X}^{2}+{\frac{r_{+}^{2}}{4}}(1+\bar{X}^{2}+\bar{Y}^{2})^{2}d\bar{\phi}^{2}\right],
P^{123}=\int_{\Sigma}\sqrt{-g_{11}}\left(-F^{0123}+\frac{\sqrt{2}}{3(4!)}\eta^{0123m_{5}\cdotsm_{11}}F_{m_{5}m_{6}m_{7}m_{8}}A_{m_{9}m_{10}m_{11}}\right)d^{10}x,
M={\frac{1}{2g_{\mathrm{YM}}^{2}}}\int\!d^{3}x\mathrm{Tr}\left[{\bfE}\!\cdot\!{\bfE}\!+\!{\bfB}\!\cdot\!{\bfB}+{\bfD}\phi^{I}\!\cdot\!{\bfD}\phi^{I}+D_{0}\phi^{I}D_{0}\phi^{I}-\sum_{I<J}\left[\phi^{I},\phi^{J}\right]^{2}\right]\,.
{\calO}_{1}{}^{i}{}_{j}\equiv\left[\begin{matrix}{-{\frac{\partial^{2}{\calW}^{*}}{\partialA_{\mathrm{cl}}^{*i}\partialA_{\mathrm{cl}}^{*j}}}}&{-i\left(-\partial_{1}+i\partial_{2}\right)\delta_{j}^{i}}\\{-i\left(\partial_{1}+i\partial_{2}\right)\delta_{j}^{i}}&{-{\frac{\partial^{2}{\calW}}{\partialA_{\mathrm{cl}}^{i}\partialA_{\mathrm{cl}}^{j}}}}\\\end{matrix}\right]
{\cal{A}}_{ijkl}^{(\eta_{o})}(x_{1},x_{2},x_{3},x_{4})\,\,\leadsto\,\,{\bf{1}}_{ij}\otimes{\bf{1}}_{kl}\,v^{\frac{1}{2}\eta_{o}+\frac{1}{2}}+v^{\frac{1}{2}\eta_{o}+\frac{1}{2}}\,{\cal{T}}_{ijkl}\,O(|x_{12}|,|x_{34}|)\,.
L_{2k}=-\frac{2y_{1}y_{2}}{\zeta}\cdot[R_{k}-\frac{(y_{1}^{2}-y_{2}^{2})}{2y_{1}y_{2}}\cdot\frac{\partial_{2}P_{k}}{4k}-\frac{\partial_{1}P_{k}}{2k}+\frac{y_{1}^{2}-y_{2}^{2}}{y_{1}y_{2}^{2}}\cdot\frac{P_{k}}{k}],
\begin{array}{cc}{\vartheta^{a}=\left(\begin{array}{c}{\vartheta_{+}^{a}}\\{\vartheta_{-}^{a}}\\\end{array}\right)~~~~}&{~~~~\bar{\vartheta}_{a}={\vartheta^{a}}^{\dagger}\Gamma^{0}=(\bar{\vartheta}_{a}^{-},\bar{\vartheta}_{a}^{+})}\\\end{array}
\bar{\Lambda}(\beta;D)=-\frac{16}{\beta}\;(2\pi\alpha^{\prime})^{-D/2}\int_{F}\frac{d^{2}\tau}{2\pi\tau_{2}^{2}}\;B(\bar{\tau},\tau;D)\left[A_{-+}(\tau;D)-A_{--}(\tau;D)\right]C_{-}^{(-)}(\bar{\tau},\tau;\beta)
G_{5}~m_{1}m_{2}~\int~d\tau~(\dot{x_{0}}\dot{x_{0}}-{\frac{1}{3}}\dot{x_{\mu}}\dot{x^{\mu}})~\int~dm~{\frac{e^{-mr(\tau)}}{r(\tau)}}~|\psi_{m}(0)|^{2}~,
\begin{array}{l}{L_{1}=\mathrm{Tr}e\mathrm{Tr}(e)\mathrm{Tr}(e)\mathrm{Tr}(e),}\\{L_{2}=\mathrm{Tr}(ee)\mathrm{Tr}(e)\mathrm{Tr}(e),}\\{L_{3}=\mathrm{Tr}(e)\mathrm{Tr}(eee),}\\{L_{4}=\mathrm{Tr}(eeee).}\\\end{array}
\varepsilon^{0\mu\nu}\left(\sum_{i=1}^{n}\alpha_{0}.\alpha_{i}\partial_{\lbrack\nu}\mathrm{b}_{\mu]}^{i}-\theta\sum_{i,j=1}^{n}\alpha_{i}.\alpha_{j}\left\{\mathrm{b}_{\mu}^{i},\mathrm{b}_{\nu}^{j}\right\}\right)=0,
I^{(i)}=(\frac{1}{2\pi})^{\frac{1}{2}}\left[{\dot{\phi}}_{c}^{(i)}(\beta){\dot{\phi}}_{c}^{(i)}(-\beta)\right]^{-\frac{1}{2}}\left[\int_{-\beta}^{\beta}\frac{d\tau}{m({\dot{\phi}}_{c}^{(i)})^{2}}\right]^{-\frac{1}{2}}
\epsilon\propto4\frac{v^{2}}{r^{2}}+\frac{2}{3}\left(\frac{w^{\prime}}{r}-\frac{w}{r^{2}}\right)^{2}+4\frac{v^{2}w^{2}}{r^{4}}+\frac{2}{r^{2}}\left(v^{2}-1\right)^{2}.
S=\intd^{3}x\left[\frac{1}{4g^{2}}\left(F_{\mu\nu}^{a}\right)^{2}+\frac12\left(D_{\mu}\Phi^{a}\right)^{2}+\frac{\lambda}{4}\left(\left(\Phi^{a}\right)^{2}-\eta^{2}\right)^{2}\right],
\int_{0}^{\pi}d\phi~e^{ir|p|cos(\phi)}~sin^{n-2}(\phi)~=~\frac{\Gamma(\frac{n-2}{2})\sqrt{\pi}}{(\frac{r|p|}{2}^{\frac{n-2}{2}})}~J_{\frac{n-2}{2}}(r|p|)~~~,
{\calL}^{\prime}=\bar{\psi}^{(i)}\mathrm{i}\partial\!\!\!/\,\psi^{(i)}+\frac{g^{2}}{2}\left(\psi_{R}^{(i)\dagger}\psi_{L}^{(i)\dagger}+\psi_{L}^{(i)}\psi_{R}^{(i)}\right)^{2}\.
\mid\psi(R,\Lambda_{3})\,\rangle\!=\!\langle\,{\bf\cdot}\mid\Psi(x,y)\,\rangle\!=\!\langle\,{\bf\cdot}\mid\Psi_{P}\,\rangle\!=\!\int\!d^{2}\lambda(d\Omega)\sigmau_{R}^{\ast}(\lambda_{1},\lambda_{2})\langle\,0\mid\Psi(x,y)\,\rangle
\sigma\epsilon^{p-3}=\Phi\!\!\left(\!\!\begin{array}{cc}{2}&{1}\\{1}&{3}\\\end{array}\!\!\right)=\Phi\!\!\left(\!\!\begin{array}{cc}{p-3}&{p-3}\\{2}&{1}\\\end{array}\!\!\right)=\sigma^{p-4}\bar{\sigma}^{p-2}
\partial^{\mu}\widetilde{A}_{\mu}(x)=-4\pi\int\delta\xids\,\epsilon_{\mu\nu\rho\sigma}\,\delta^{\mu}(s)\left[\Omega_{\xi}(s,0)L^{\rho\sigma}[\xi|s]\Omega_{\xi}^{-1}(s,0)\right]\dot{\xi}^{\nu}(s)\delta^{4}(x-\xi(s)).
\mathrm{det}M-(\tilde{B}^{A_{1}A_{2}\ldotsA_{N_{f}-N_{c}}}M_{A_{1}}{}^{B_{1}}M_{A_{2}}{}^{B_{2}}\ldotsM_{A_{N_{f}-N_{c}}}{}^{B_{N_{f}-N_{c}}}B_{B_{1}B_{2}\ldotsB_{N_{f}-N_{c}}}),
{\calC}_{0}=i\,\intd{\bf{r}}\,{\textstyle{\calX}^{\alpha}}({\bfr})\,j_{0}^{\alpha}({\bfr})\;,\;\;\;\;\;\mathrm{and}\;\;\;\;\;\;\;{\bar{\calC}}=i\,\intd{\bf{r}}\,\overline{{{\calY}^{\alpha}}}({\bfr})\,j_{0}^{\alpha}({\bfr})\;.
f_{\omega^{\prime}lm}^{b}=\intd\tilde{\omega}\,d\theta_{0}\,d\eta_{0}\,(\alpha_{\omega\omega^{\prime}lm}^{b}p_{\omega}+\beta_{\omega\omega^{\prime}lm}^{b}\bar{p}_{\omega}+\mathrm{terms~involving~}q_{\omega}),
Q_{0}=\left(\begin{array}{cccc|ccc}{1}&{0}&{0}&{0}&{0}&{\cdots}&{0}\\{0}&{1}&{0}&{0}&{0}&{\cdots}&{0}\\{\vdots}&{}&{}&{}&{}&{}&{\vdots}\\{0}&{0}&{0}&{1}&{0}&{\cdots}&{0}\\\end{array}\right),\qquad\tilde{Q}_{0}=0,
\begin{array}{ccccccccc}{}&{}&{}&{}&{\bigcirc_{0}}&{}&{}&{}&{}\\{}&{}&{}&{}&{\mid}&{}&{}&{}&{}\\{+_{1}}&{-}&{\oplus_{2}}&{-}&{\oplus_{3}}&{-}&{\oplus_{4}}&{-\dots-}&{\oplus_{d-1}}\\\end{array}
\langlex^{L(0)}w_{(1)},{\calY}(x^{-L(0)}w_{(3)}^{\prime},x_{0})x^{-L(0)}w_{(2)}\rangle_{W_{1}^{\prime}}=\langlew_{(1)},{\calY}(w_{(3)}^{\prime},xx_{0})w_{(2)}\rangle_{W_{1}^{\prime}}
\Omega=\left(\begin{array}{cccc}{dd^{\prime}}&{-dc^{\prime}}&{-cc^{\prime}}&{-cd^{\prime}}\\{-db^{\prime}}&{da^{\prime}}&{ca^{\prime}}&{cb^{\prime}}\\{-bb^{\prime}}&{ba^{\prime}}&{aa^{\prime}}&{ab^{\prime}}\\{-bd^{\prime}}&{bc^{\prime}}&{ac^{\prime}}&{ad^{\prime}}\\\end{array}\right)
\partial_{[\,\mu_{1}}\,\deltaB_{\mu_{2}\dots\mu_{2k-1}\,]}={\frac{1}{f}}\deltaX_{\mu_{1}\dots\mu_{2k-1}}={\frac{1}{f\,(2k-1)!}}\,\epsilon_{\mu_{1}\dots\mu_{2k-1}\mu_{2k}}\deltaX^{\mu_{2k}}
\left[\frac{1}{e^{-2\omegaT}+e^{2\omegaX}}\left(\partial_{T}^{2}-\partial_{X}^{2}\right)-\left(\partial_{y}^{2}+\partial_{z}^{2}\right)+i2eB_{x}z\partial_{y}+e^{2}B_{x}^{2}z^{2}-m^{2}\right]\psi=0.
{\calL}_{F}^{(2)}=\frac{\mathrm{i}}{2}(\theta_{i}{\calD}_{0}\theta^{i}+\eta_{i}{\calD}_{0}\eta^{i}-\mathrm{i}\eta_{i}e_{0}^{i}{}_{j}\eta^{j})+e^{2\phi}\eta^{i}C_{ij}^{\prime}({\calD}_{1}\theta^{j}-\mathrm{i}\sqrt{2}e^{\phi}\eta^{j}\partial_{1}x)+h.c.
\phi(x)=\frac{1}{(2\pi)^{3}}\int_{0}^{\infty}\frac{dp^{+}}{2p^{+}}\int_{-\infty}^{\infty}dp^{\bot}\left(a(\vec{p},x^{+})e^{-i\vec{p}\vec{x}}+a^{\dagger}(\vec{p},x^{+})e^{i\vec{p}\vec{x}}\right),
\mathrm{Im}\chi_{\mathrm{vac}}^{\mathrm{EM}}=\mathrm{Im}\chi_{\mathrm{vac}}^{\mathrm{(TE)}}+\mathrm{Im}\chi_{\mathrm{vac}}^{\mathrm{(TM)}}={\frac{\hbarA}{30\pi^{2}}}\omega^{5}.
G(x,t;x^{\prime},t^{\prime})=i\int\int\frac{d\omega}{2\pi}\frac{dk}{2\pi}\frac{e^{-i\omega(t-t^{\prime})}}{\omega^{2}-k^{2}-4+i\rho}\left(e^{ik(x-x^{\prime})}+K_{0}(k)e^{-ik(x+x^{\prime})}\right),
\gamma_{\left[(-1)^{F_{L}}R\right]}=\pm\gamma_{\OmegaR_{q}}\gamma_{\left[(-1)^{F_{L}}R\right]}^{T}\gamma_{\OmegaR_{q}}^{-1}\textrm{}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\textrm{with}/\textrm{withoutvectorstructure}
\left(\begin{matrix}{Z_{AB}}\\{Z_{I}}\\\end{matrix}\right)=\int\left(\begin{matrix}{T_{AB}}\\{T_{I}}\\\end{matrix}\right)=\int\left(\begin{matrix}{T_{AB}^{-}}\\{T_{I}^{-}}\\\end{matrix}\right)=\left(\begin{matrix}{f_{AB}^{\Lambda}}\\{f_{I}^{\Lambda}}\\\end{matrix}\right)e_{\Lambda}-\left(\begin{matrix}{h_{\LambdaAB}}\\{h_{\LambdaI}}\\\end{matrix}\right)g^{\Lambda}
\begin{array}{ll}{D_{\alpha}=\partial_{\alpha}+i\sigma_{\alpha\dot{\alpha}}^{\mu}\bar{\theta}^{\dot{\alpha}}\partial_{\mu},}\\{D_{\dot{\alpha}}=-\bar{\partial}_{\alpha}-i\theta^{\alpha}\sigma_{\alpha\dot{\alpha}}^{\mu}\partial_{\mu}.}\\\end{array}
A_{a}(\vec{\vartheta}^{(1)},\vec{\vartheta}^{(2)})=\sum_{-j\leqm,\;m^{\prime}\leqj}C_{mm^{\prime}}\Psi_{jm}(\vec{\vartheta}^{(1)})\Psi_{jm^{\prime}}(\vec{\vartheta}^{(2)})^{*}.
\langle\Psi|\Psi^{\prime}\rangle=\int_{0}^{\infty}d\alpha\int_{{\calE}_{L}}{\calD}^{\alpha\hat{e}}\widetilde{x}\;\overline{{\Psi[\alpha,\widetilde{x}]}}\Psi^{\prime}[\alpha,\widetilde{x}]\;\;\;.
g_{Dq-GT}^{2}=G_{\mathrm{OS}}^{2}(\alpha_{\mathrm{eff}}^{\prime})^{\frac{q-1}{2}}\,\quad\theta_{Dq-GT}=G_{OS}^{2}(\alpha_{\mathrm{eff}}^{\prime})^{\frac{q+1}{2}}\.
I=\intdx[x\dot{\Omega}\dot{\rho}-\frac{\mu}{8}e^{2\rho}\Omega^{1-\lambda-\gamma/4}-\frac{1}{4}\Omega^{\delta}x\dot{f}^{2}+\frac{1}{4}e^{-2\rho}\Omega^{(\gamma-2\epsilon)/4+1}\dot{A}^{2}
I_{2}^{\zeta}(t,t^{\prime};k)_{\alpha\beta}=\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}g_{\alpha\beta}^{\zeta}(t,t^{\prime};k^{\prime})g_{\alpha\beta}^{\zeta}(t,t^{\prime};k^{\prime}+k)\,.
\Gamma_{A^{\mu}A^{\nu}}(p,-p)\equiv\Gamma_{\mu\nu}(p,-p)=(\eta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}})\Gamma^{T}(p^{2})+\frac{p_{\mu}p_{\nu}}{p^{2}}\Gamma^{L}(p^{2})
I(k^{2})=-{\frac{\lambda^{2}}{96\pi^{2}\mu^{2}}}\int_{0}^{1}dx\int_{0}^{1-x}dy{\frac{f^{2}(x)f^{2}(y)}{[{\frac{k^{2}}{\mu^{2}}}xy(1-x-y)-xy-(x+y)(1-x-y)]}}.
ds^{2}={\frac{(r^{2}-r_{+}^{2})(r^{2}+\tilde{r}_{-}^{2})}{r^{2}}}d\tau^{2}+{\frac{r^{2}dr^{2}}{(r^{2}-r_{+}^{2})(r^{2}+\tilde{r}_{-}^{2})}}+r^{2}\left[-{\frac{\tilde{r}_{-}}{r_{+}r^{2}}}(r^{2}-r_{+}^{2})d\tau+d\phi\right]^{2}.
{\calA}_{D_{2}}(k_{1},\epsilon_{1};k_{2},\epsilon_{2})=g_{c}^{2}e^{-\lambda}\int_{0}^{1}dy\langlec(z)V_{\epsilon_{1}}(z,\bar{z})c(w)\tilde{c}(\bar{w})V_{\epsilon_{2}}(w,\bar{w})\rangle|_{z=iy,w=i}~,
\mu_{\rho}=\frac{1}{2}\int\frac{d^{2}p}{(2\pi)^{2}}\left[\left(\frac{g^{2}}{2\pi^{2}}p^{2}D(p)+2z\right)^{-1}-\left(\frac{1}{\pin}p^{2}+2z\right)^{-1}\right]
\lambda_{(0)\alpha}^{u}=\frac{1}{2}{\overline{F}}_{\mu\nu}\sigma_{\alpha}^{\mu\nu\,\beta}\frac{1}{\sqrt{\rho_{0}}}\left(\rho_{0}\eta_{\beta}^{u}+(x-x_{0})_{\kappa}\sigma_{\beta\dot{\beta}}^{\kappa}\bar{\xi}^{\dot{\beta}u}\right)\,.
\begin{array}{rcl}{\mathrm{integral}}&{=}&{\displaystyle-2m^{2}(4\pi)^{-2}\int_{0}^{1}dx\int\frac{d^{4}k_{1}}{(2\pi)^{4}}D_{m}^{2}(k_{1})\overline{{D(k_{1}+p)}}K_{0}\left(\sqrt{xm^{2}\tilde{k_{1}}^{2}}\right).}\\\end{array}
R_{ab}^{ef}(\vartheta-\vartheta^{\prime})\left[t_{ec}(\vartheta)\right]_{\alpha\gamma}\left[t_{fd}(\vartheta^{\prime})\right]_{\gamma\beta}=\left[t_{ae}(\vartheta^{\prime})\right]_{\alpha\gamma}\left[t_{bf}(\vartheta)\right]_{\gamma\beta}R_{ef}^{cd}(\vartheta-\vartheta^{\prime})
\delta_{\epsilon}\Phi^{\alpha_{0}}=Z_{\;\;\alpha_{1}}^{\alpha_{0}}\epsilon^{\alpha_{1}},\;\delta_{\epsilon}\Phi^{\alpha_{2}}=A_{\alpha_{1}}^{\;\;\alpha_{2}}\epsilon^{\alpha_{1}}+Z_{\;\;\alpha_{3}}^{\alpha_{2}}\epsilon^{\alpha_{3}},
e=\frac{i}{2}\left\{d\tau+\sum_{a=1}^{N}\frac{GS_{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}{rr}{0}&{dz}\\{d\bar{z}}&{0}\\\end{array}\right)\,,
G_{\pm}^{(2)}(\rho,u_{\pm})|_{\rho=0}=\mp\frac{1}{(2\pi)^{d-1}L}\sum_{n=1}^{\infty}cos({\frac{n\pi{u_{\pm}}}{L}})\int{d}^{d-1}p\frac{1}{(\vec{p}^{2}+m^{2}+(\frac{n\pi}{L})^{2})},
L=\frac{1}{2}h^{\mu\nu}\Lambda_{\mu\nu,\rho\sigma}h^{\rho\sigma}-\frac{m^{2}}{2}\left(h^{\mu\nu}h_{\mu\nu}-\frac{1}{2}h^{2}\right)+b^{\mu}\left(\partial^{\nu}h_{\mu\nu}-\frac{1}{2}\partial_{\mu}h+\frac{\alpha}{2}b_{\mu}\right).
\biggl[\bar{z},u_{1}+\sum_{j=1}^{n-1}\bar{z}^{m_{j}}u_{j+1}\biggr]=0\qquad\Rightarrow\qquadu_{1}=-\sum_{j=1}^{n-1}\bar{z}^{m_{j}}u_{j+1}+f(t,\bar{z}),
\sum_{b=1}^{N}\Big(a_{\mu}^{(b)},j_{\mu}^{(b)}\Big)\;=\;\sum_{I=1}^{N}\sum_{b=1}^{N}\Big(U_{Ib}A_{\mu}^{(I)},j_{\mu}^{(b)}\Big)\;:=\;\sum_{I=1}^{N}\Big(A_{\mu}^{(I)},J_{\mu}^{(I)}\Big)\;,
\stackrel{x}{\partial}_{\mu}H(x,x^{\prime})=2\pi\int\frac{d^{n-1}k}{(2\pi)^{n-1}}e^{ik(x-x^{\prime})}\frac{(1+f)}{2|\omega|}\lbrackik_{\mu}+F_{\mu}-\Omega_{\mu}\rbrack
D_{\lambda}\hat{\xi}_{\mu\nu}(x)=D_{\lambda}\;\frac{\partialq^{\alpha}}{\partialx^{\mu}}\frac{\partialq^{\beta}}{\partialx^{\nu}}\,e_{\alpha}^{a}\,e_{\beta}^{b}\,\hat{\xi}_{ab}=\partial_{\lambda}e_{\mu}^{a}\,e_{\nu}^{b}\,\hat{\xi}_{ab}+e_{\mu}^{a}\,e_{\nu}^{b}\,\left[\Omega^{-1}\,\partial_{\lambda}\Omega,\,\hat{\xi}_{ab}\right]=0,
\Omega_{k=0}(\Lambda)=\frac{\sqrt{\pi}}{2}\frac{{\calR}}{2}\frac{2}{5}\left\{2{\sqrt\pi}\Lambda^{5}\left(Erf(\Lambdax)-1\right)+{x^{-5}}\left(1-e^{-\Lambda^{2}x^{2}}(1+\Lambda^{2}x^{2}-2\Lambda^{4}x^{4})\right)\right\}
S_{\mathrm{YM}}\left[A_{\mu}^{i}\right]=\frac14\intd^{4}x\left(f_{\mu\nu}+C_{\mu\nu}+\left(F_{\mu\nu}^{\mathrm{sing.}}\right)^{3}\right)^{2}+\frac14\intd^{4}x\left(S_{\mu\nu}^{a}+\left(F_{\mu\nu}^{\mathrm{sing.}}\right)^{a}\right)^{2},
\left[\frac{1}{2}\alpha_{-1}\cdot\alpha_{-1}+\frac{1}{2}\alpha_{-1}^{25}\alpha_{-1}^{25}+\frac{5}{2}k\cdot\alpha_{-2}+\frac{3}{2}\left(k\cdot\alpha_{-1}\right)^{2}\right]\left|k,l=0,i,j\right\rangle
G_{NS}(\alpha)=\frac{\nu(\alpha)}{\nu(Q-\alpha)}=\left(\frac{\pi\mu}{2}\gamma\left(\frac{1+b^{2}}{2}\right)b^{1-b^{2}}\right)^{\frac{Q-2\alpha}{b}}\frac{b^{2}\gamma(\alphab-\frac{Qb}{2})}{\gamma\left(-\frac{\alpha}{b}+\frac{Q}{2b}\right)},
\partial_{i}\left(\sqrt{-{\mathrm{det}\hat{g}}}\,\hat{g}^{ij}\partial_{i}X^{\nu}g_{\mu\nu}\right)-\frac{1}{2}\sqrt{-{\mathrm{det}\hat{g}}}\,\hat{g}^{ij}\partial_{i}X^{\nu}\partial_{j}X^{\rho}\partial_{\mu}g_{\nu\rho}=0\,,
\left[\frac{sh\frac{\gamma}{2}(\lambda_{j}+2is)}{sh\frac{\gamma}{2}(\lambda_{j}-2is)}\right]^{L}=-\prod_{k=1}^{M}\frac{sh\frac{\gamma}{2}(\lambda_{j}-\lambda_{k}+2i)}{sh\frac{\gamma}{2}(\lambda_{j}-\lambda_{k}-2i)}
\sum_{c}\sum_{Q^{\prime\prime}}\Gamma_{S}^{Q^{\prime}bQ^{\prime\prime}c}(p)\left[\delta_{Q^{\prime\prime}Q}\delta^{ca}-N_{Q^{\prime\prime}}^{ca}(p)g_{Q^{\prime\prime}Q}{(-1)}^{a+1}\right]=\frac{i}{2}g_{Q^{\prime}Q}\delta^{ba}{(-1)}^{a+1},\\Q^{\prime},Q=U,D,\\b,a=1,2,
\left[\overline{{\gamma}}_{a_{1}},\tilde{H}^{*}\right]=-\frac1\lambda\left(Z_{\;a_{1}}^{b}\left[C_{b}^{0},f^{a}\left(C\right)\right]+\left[Z_{\;a_{1}}^{a},\tilde{H}^{*}\right]\right)\gamma_{a}.
\bigl[\bigl(\omega^{ij}\bigr)^{ab}\bigr]=\left(\begin{array}{cccc}{0}&{-\,1}&{0}&{0}\\{-\,1}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}\\{0}&{0}&{-1}&{0}\\\end{array}\right)\,\delta^{ij}\,.
\beta_{r}(q_{13}^{0},|\vec{q}_{13}|,q_{24}^{0},|\vec{q}_{24}|)=\frac{-(q_{13}^{0}-q_{24}^{0})^{2}+\vec{q}_{13}^{2}+\vec{q}_{24}^{2}+m_{r}^{2}-i\varepsilon}{2|\vec{q}_{13}||\vec{q}_{24}|}\,.
(H-\epsilon_{a})J_{a}\big|_{m_{c}=0}\propto\left|\begin{array}{ccc}{0}&{(\epsilon_{b}-\epsilon_{a})m_{b}}&{v_{ab}m_{b}}\\{\epsilon_{c}-\epsilon_{a}}&{v_{bc}}&{v_{ac}}\\{0}&{\epsilon_{b}-\epsilon_{a}}&{v_{ab}}\\\end{array}\right|\,.
ds^{2}=\alpha^{\prime}\left[\frac{U^{2}}{\sqrt{4\ping_{s}}}g_{s}\,e^{2V_{0}}\left(-(dx^{0})^{2}+\cdots+(dx^{3})^{2}\right)+\sqrt{4\ping_{s}}\left(\frac{dU^{2}}{U^{2}}+d\Omega_{5}^{2}\right)\right].
i\hbar\partial_{t}\psi=\frac{\hbar^{2}}{2m}\biggl\{\Bigl(\frac{1}{i}\partial_{x_{1}}\pm\frac{\varkappa^{2}}{x_{1}-x_{2}}\Bigr)^{2}+\Bigl(\frac{1}{i}\partial_{x_{2}}\pm\frac{\varkappa^{2}}{x_{2}-x_{1}}\Bigr)^{2}\biggr\}\psi.
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_{2}\tau_{1}-\ell_{1})^{2}\Big)+\dots~~,
{\calL}=\psi^{\dagger}i\partial_{t}\psi+\psi^{\dagger}\frac{\nabla^{2}}{2M}\psi-\frac{1}{2}C_{0}(\psi^{\dagger}\psi)^{2}\,-\frac{1}{2}C_{2}(\psi^{\dagger}\nabla^{2}\psi)(\psi^{\dagger}\psi)+h.c.+\ldots.
K_{0-0}=\mathrm{Tr}_{1/\theta}\Bigl(\frac{1}{2}(\partial_{t}\deltaz_{i})^{2}+\frac{1}{2}(\partial_{t}\deltaa^{I})^{2}+\frac{1}{4}([\tilde{\nabla}_{i}^{0},\deltaz_{j}]-[\tilde{\nabla}_{j}^{0},\deltaz_{i}])^{2}+\frac{1}{2}[\tilde{\nabla}_{i}^{0},\deltaa^{I}]^{2}\Bigr)
m\epsilon_{\mu\alpha\beta}\partial^{\alpha}\hat{A}^{\beta}=\epsilon_{\mu\alpha\beta}\partial^{\alpha}(\hat{f}^{\beta}-m\hat{a}^{\beta}+\hat{h}^{\beta})-\epsilon_{\nu\alpha\beta}(m\partial^{\alpha}\hat{A}^{\beta}-\partial^{\alpha}\hat{f}^{\beta})\frac{\delta\hat{a}^{\nu}}{\delta\hat{A}^{\mu}}+O(\theta^{2}).
\left(\sum_{a}\frac{L_{1}^{a}L_{2}^{a}L_{3}^{a}}{R_{A}R_{B}v}-16\sqrt{2}(u+\frac{1}{4^{b}}\frac{1}{u})\right)^{2}+\left(\sum_{a}\frac{L_{1}^{a}L_{2}^{a}}{R_{A}R_{B}v}\frac{(n_{3}^{a}R_{1})^{2}-(M_{3}^{a}R_{2})^{2}}{L_{3}^{a}}-16\sqrt{2}(u-\frac{1}{4^{b}}\frac{1}{u})\right)^{2}\nonumber
{\calF}_{\mathrm{het}}=-ST^{2}-\alphaT^{3}-c+\frac{1}{8\pi^{3}}\sum_{j\geq0,k\geq1}n_{k,j}Li_{3}(e^{-2\pi(jS+(k-j)T)})-\frac{\beta}{4\pi^{3}}Li_{3}(e^{-2\pi(S-T)}),
\{P_{(z,w)}(x^{-}),O_{(z^{\prime},w^{\prime})}(x^{-^{\prime}})\}_{PB}=-g_{st}^{2}(zw^{\prime}-z^{\prime}w)O_{(z+z^{\prime},w+w^{\prime})}\delta(x^{-}-x^{-^{\prime}}).
\begin{array}{c}{\left\{L_{1}^{n},L_{2}^{n}\right\}=\left(a-\alpha\right)L_{1}^{n}L_{2}^{n}-L_{1}^{n}L_{2}^{n}\left(a+\alpha\right),}\\{\left\{L_{1}^{n},L_{2}^{n+1}\right\}=L_{1}^{n}\left(\alpha-s\right)L_{2}^{n+1},}\\{\left\{L_{1}^{n},L_{2}^{m}\right\}=0,\left|n-m\right|\geq2.}\\\end{array}
\Gamma_{n+1}^{\tilde{e}^{0}(k^{0})...\tilde{e}^{n}(q^{n})}(\tilde{k}^{0},...,\tilde{k}^{n})=const.\times\sum_{P}\left(\frac{d^{i_{0}}}{d^{i_{1}}}+\frac{d^{i_{1}}}{d^{i_{0}}}-1\right)...\left(\frac{d^{i_{n-1}}}{d^{i_{n}}}+\frac{d^{i_{n}}}{d^{i_{n-1}}}-1\right)+o(\omega^{2})
9B_{\alpha\beta\gamma\delta}=R_{\alpha\rho\gamma\sigma}\;R_{\beta}~\!\!^{\rho}~\!\!_{\delta}~\!\!^{\sigma}+\tilde{R}_{\alpha\rho\gamma\sigma}\;\tilde{R}_{\beta}~\!\!^{\rho}~\!\!_{\delta}~\!\!^{\sigma}\;,
\frac{\partial\mathrm{~\cal~P~}}{\partialt_{2l-1}}=\{\mathrm{~\cal~B~}_{2l-1},\mathrm{~\cal~P~}\},~~~~~(~\mathrm{~\cal~B~}_{2l-1}=(\mathrm{~\cal~L~}^{2l-1})_{+}~)
\left[\begin{array}{rrrrrr}{2}&{-1}&{0}&{0}&{0}&{0}\\{-1}&{2}&{-1}&{0}&{0}&{0}\\{0}&{-1}&{2}&{-1}&{0}&{0}\\{0}&{0}&{-1}&{2}&{-1}&{0}\\{0}&{0}&{0}&{-1}&{0}&{1}\\{0}&{0}&{0}&{0}&{-1}&{2}\\\end{array}\right].
I=\frac{1}{2}\mu\intd^{3}x[-\epsilon^{mnp}A_{m}^{i}F_{np(a+v)}^{i}-\mua_{m}^{i}a^{mi}+\epsilon^{mnp}v_{m}^{i}(\partial_{n}v_{p}^{i}+\frac{1}{3}\epsilon^{ijk}v_{n}^{j}v_{p}^{k})],
R_{H}^{\mu\nu}(x)=\partial^{\mu}\partial_{\mu}a_{H}^{\mu\nu}(x)-\partial^{\mu}\partial_{\lambda}a_{H}^{\lambda\nu}(x)-\partial^{\nu}\partial_{\lambda}a_{H}^{\lambda\mu}(x)+\partial^{\mu}\partial^{\nu}a_{H\lambda}^{\lambda}(x).
\langle\,\rho_{a}\,\rangle\;\;\propto\;\;\langle\;\prod_{I}|\eta(t^{I})|^{2(b-b_{a})/b_{a}}\;\rangle\;=\;|\eta(1)|^{6(b-b_{a})/b_{a}}\;\approx\;e^{-\pi(b-b_{a})/2b_{a}}.
M^{12}=R^{5,1}\timesCP^{3}\sim\frac{Poincar\acute{e}\,group}{\left[SO\left(2\right)\timesSO\left(3\right)\right]R}\simR^{5,1}\times\frac{SO\left(5\right)}{SO\left(2\right)\timesSO\left(3\right)}
\Gamma_{\mu}(\partial,\stackrel{\leftarrow}{\partial}^{\dagger})=\Lambda_{\mu}+\sum_{l=1}^{N-1}\sum_{k=0}^{l}\stackrel{\leftarrow}{\partial}^{\dagger\;\;\mu_{1}}...\stackrel{\leftarrow}{\partial}^{\dagger\;\;\mu_{k}}\Lambda_{\mu_{1}...\mu_{k}\mu\mu_{k+1}...\mu_{l}}\partial^{\mu_{k+1}}...\partial^{\mu_{l}}
\frac{\mathrm{d}w}{\mathrm{d}z}=\kappa\left(e^{i\phi_{1}}-z\right)^{-\alpha_{1}}\left(e^{i\phi_{2}}-z\right)^{-\alpha_{2}}\ldots\left(e^{i\phi_{n}}-z\right)^{-\alpha_{n}}
\gamma_{\alpha\beta}^{0}=\left(\begin{array}{cc}{1}&{0}\\{0}&{1}\\\end{array}\right)\,,\quad\gamma_{\alpha\beta}^{1}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\,,\quad\gamma_{\alpha\beta}^{2}=\left(\begin{array}{cc}{-1}&{0}\\{0}&{1}\\\end{array}\right)\,.
\mathrm{Re}\,\Pi_{00}^{\mathrm{dispersion}}(k_{0},s)=\mathrm{Re}\,\Pi_{00}^{\mathrm{dispersion}}(k_{0},k)={\frac{1}{\pi}}\int_{-\infty}^{\infty}d\omega\,{\frac{\mathrm{Im}\,\Pi_{00}(\omega,k)}{\omega-k_{0}}}={\frac{2}{\pi}}\int_{0}^{\infty}d\omega{\frac{\omega\mathrm{Im}\,\Pi_{00}(\omega,k)}{\omega^{2}-k_{0}^{2}}}
\bar{f}_{+}=-\frac{\kappa_{5}^{2}e^{-2A_{+}-\alpha_{+}^{(0)}}}{\int_{w_{-}}^{w_{+}}dw^{\prime}e^{-3A(w^{\prime})}}e^{-2A_{+}-\alpha_{+}^{(0)}}\bar{\tau}_{+(LL)}
\begin{array}{rcl}{u}&{=}&{\displaystyle\frac{1}{\sqrt{2}}\sqrt{r^{2}-n^{2}}\frac{d\bar{\zeta}}{1+\frac{\kappa}{4}\zeta\bar{\zeta}}\,}\\{v}&{=}&{\displaystyle\frac{1}{\sqrt{2}}\left[\frac{dr}{\sqrt{V(r)}}+\mathrm{i}\sqrt{V(r)}\left(d\tau+{\frac{i}{2}}n{\frac{\zetad\bar{\zeta}-\bar{\zeta}d\zeta}{1+{\frac{\kappa}{4}}\zeta\bar{\zeta}}}\right)\right]\,.}\\\end{array}
C_{l}=\frac{1}{8}\left(\!\!\begin{array}{ccccccccc}{2}&{-2}&{4}&{-2}&{0}&{0}&{-2}&{2}&{-2}\\{1}&{5}&{-3}&{-1}&{-1}&{-1}&{5}&{-3}&{1}\\{-1}&{-1}&{1}&{1}&{-1}&{-1}&{-1}&{3}&{3}\\{1}&{-3}&{-1}&{-3}&{3}&{3}&{5}&{1}&{-3}\\{-2}&{2}&{0}&{6}&{0}&{0}&{-6}&{-2}&{2}\\\end{array}\!\right)
(\:{\tilde{\Gamma}}_{{\bar{j}}{\bar{i}}}^{\bar{l}}\:G_{k{\bar{l}}}+\partial_{\bar{i}}G_{k{\bar{j}}}\:)\:|\sigma_{1}\rangle+d\:G_{k{\bar{j}}}\:|{\calO}_{{\bar{i}},2}\rangle=(\:{\tilde{\Gamma}}_{{\bar{i}}{\bar{j}}}^{\bar{l}}\:G_{k{\bar{l}}}+\partial_{\bar{j}}G_{k{\bar{i}}}\:)\:|\sigma_{1}\rangle+d\:G_{k{\bar{i}}}\:|{\calO}_{{\bar{j}},2}\rangle\,