harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
G_{a}^{m}=\frac{1}{2i}(\Lambda_{1}^{\prime}(\Gamma_{3}\partial_{a}^{m}\Gamma_{3})\Lambda_{1}^{\prime})=\frac{i}{2}(\Lambda_{1}^{\prime\prime}(\Gamma_{3}\partial_{a}^{m}\Gamma_{3})\Lambda_{1}^{\prime\prime})
D_{++}^{p}(q)=\frac{i}{q^{2}-m^{2}+i{\varepsilon}}\frac{m^{2}}{q_{-}^{\;\;2}}=\frac{i}{q^{2}-m^{2}+i{\varepsilon}}\left(-g_{++}+\frac{2n_{+}q_{+}}{q_{-}}-n^{2}\frac{q_{+}^{\;\;2}}{q_{-}^{\;\;2}}\right)-\frac{i}{q_{-}^{\;\;2}},
\lambda\rightarrow\lambda^{\prime}=-\frac{1}{\lambda},\;F_{\mu\nu}\rightarrowF_{\mu\nu}^{\prime}=-\lambda_{2}ML{\tilde{F}}_{\mu\nu}-\lambda_{1}F_{\mu\nu},\;M\rightarrowM,\;G_{\mu\nu}\rightarrowG_{\mu\nu}.
ds^{2}=\alpha^{\prime}\left[\frac{U^{2}}{\sqrt{4\ping_{\mathrm{eff}}}}\left(-(dx^{0})^{2}+(dx^{1})^{2}+(dx^{2})^{2}+(dx^{3})^{2}\right)+\sqrt{4\ping_{\mathrm{eff}}}\left(\frac{dU^{2}}{U^{2}}+d\Omega_{5}^{2}\right)\right].
P(\theta^{2},\bar{\theta}^{2})=1-{\frac{i}{2R}}(\theta^{2}-\bar{\theta}^{2})+{\frac{1}{3!R^{2}}}\theta^{2}\bar{\theta}^{2},\qquad\theta^{2}\equiv\theta^{A}\theta_{A},\quad\bar{\theta}^{2}\equiv\bar{\theta}_{\dot{A}}\bar{\theta}^{\dot{A}},
I_{n}^{\mu_{1}\mu_{2}\ldots\mu_{n}}=\int_{0}^{2\pi}V^{\mu_{1}}(\sigma_{1})d\sigma_{1}\int_{\sigma_{1}}^{\sigma_{1}+2\pi}d\sigma_{2}V^{\mu_{2}}(\sigma_{2})\int_{\sigma_{1}}^{\sigma_{2}}V^{\mu_{3}}(\sigma_{3})\ldots\int_{\sigma_{1}}^{\sigma_{n-1}}d\sigma_{n}V^{\mu_{n}}(\sigma_{n})
\frac{\Bigl(\frac{\partial^{2}Q}{\partialT\partial{\bar{T}}}\Bigr)^{-1}}{12}\frac{\Bigl((\frac{\partialV}{\partial{T_{1}}})^{2}+(\frac{\partialV}{\partial{T_{2}}})^{2}\Bigr)}{V^{2}}\ll1
\tilde{J}_{R}\left(\bar{z}\right)=-ikg^{-1}\left(\bar{\partial}g\right)=ikg_{R}^{-1}\left(\bar{\partial}g_{R}\right)=\tilde{J}^{a}\tau^{a}=\left(\begin{array}{cc}{-\tilde{J}^{2}}&{\tilde{J}^{0}+\tilde{J}^{1}}\\{-\tilde{J}^{0}+\tilde{J}^{1}}&{\tilde{J}^{2}}\\\end{array}\right)
2\kappa^{2}e^{-1}{\calL}=R+{\frac{1}{2}}m^{2}(T^{2}-2T_{ij}T^{ij})-Tr(P_{\mu}P^{\mu})-{\frac{1}{2}}(V_{I}{}^{i}V_{J}{}^{j}F_{\mu\nu}^{IJ})^{2},
P_{(m_{i-\hat{\lambda},\lambda,\mu}n_{i\lambda\mu}\ldots)(q_{i-\hat{\nu},\mu,\nu}n_{i\mu\nu}\ldots)}\cdotP_{(p_{i-\hat{\lambda},\lambda,\mu}q_{i\lambda\mu}\ldots)(p_{i-\hat{\nu},\mu,\nu}m_{i\mu\nu}\ldots)},
U_{\alpha}^{\mu}=\sum_{n=0}^{q}\frac{\epsilon^{\mu\mu_{2}\ldots\mu_{2q+1-2n}\nu_{1}\ldots\nu_{2n}}}{\left(2q-2n\right)!\cdot2^{n}\cdotn!}\,\left[emb^{*}D_{\alpha}^{\left(2q-2n\right)}\right]_{\mu_{2}\ldots\mu_{2q+1-2n}}F_{\nu_{1}\nu_{2}}\cdotsF_{\nu_{2n-1}\nu_{2n}}\quad.
\varphi_{A_{1}...A_{n}a_{1}...a_{m}}(y)=\int_{{\calF}\timesS^{2}}\omega(p,\zeta)\\widetilde{\varphi}(p,\zeta)\Phi(p,\zeta|y)_{A_{1}...A_{n}a_{1}...a_{m}}\\in{\calH}_{E_{o},s}
H_{0}^{2}=-\frac{k}{a_{0}^{2}}+\frac{{\calC}_{\chi}}{a_{0}^{4}}-\frac{1}{3}\frac{{\calQ}^{2}}{a_{0}^{6}}+\frac{1}{144M^{6}}\Big(\rho^{2}+2\sigma\rho\Big).
\delta_{G}z_{1}=\xi\partial_{5}z_{1}+\xi(-G_{5})^{-1}\partial_{5}\tilde{H}^{\mu\nu}(G\tilde{H}G)_{\mu\nu}+2\partial_{\mu}\xi\left({\frac{G^{5\mu}}{G^{55}}}z_{1}+{\frac{G_{5\nu}}{(-G_{5})}}(\tilde{H}G\tilde{H})^{\mu\nu}\right)
<q>=\frac{1}{N_{2}}\frac{1}{2(2\pi)^{3}}\int\phi_{2}^{2}(\vec{q},\vec{n})\frac{[\varepsilon(q,m)+\varepsilon(q,\mu)]}{\varepsilon(q,m)\varepsilon(q,\mu)}\q\d^{3}q\.
S_{1D}=-\intd^{5}\sigma\left(\sqrt{-G}\sqrt{1+z_{1}+\frac{z_{1}^{2}}{2}-z_{2}}+\frac{\epsilon_{\mu\nu\lambda\sigma\tau}C^{\mu}\tilde{{\calH}}^{\nu\lambda}\tilde{{\calH}}^{\sigma\tau}}{8(1+C_{1}^{2})}\right).
H(z_{2}^{\prime},..,z_{\calN}^{\prime},P_{2}^{\prime},..,P_{\calN})\!=\!\!-\!\!\int_{\{z_{0}\}}^{\{z_{B}^{\prime}\}}\!\!\sum_{B}\frac{\partial\beta_{B}}{\partial\mu}(z_{2}^{\prime},..,z_{\calN}^{\prime},z_{A}^{\prime\prime},..,\mu)dz_{B}^{\prime\prime}+f(z_{2}^{\prime},..,z_{\calN}^{\prime}).
B^{\prime\prime}+{\frac{D_{2}}{2}}{B^{\prime}}^{2}+{\frac{D_{1}}{2}}A^{\prime}B^{\prime}={\frac{4\kappa^{2}}{D-2}}\left(-\Lambda-{\frac{2D-D_{1}-4}{D_{2}}}{\frac{F^{2}}{4g^{2}}}\mathrm{e}^{-2B}\right)+k\mathrm{e}^{-B}~,
ds_{P}^{2}=-4{\frac{|aa_{D}^{\prime}-a_{D}a^{\prime}|^{2}}{(a_{D}\bar{a}-\bar{a}_{D}a)^{2}}}|du|^{2}=-{\frac{16}{\pi^{2}}}{\frac{1}{(a_{D}\bar{a}-\bar{a}_{D}a)^{2}}}|du|^{2}=e^{\varphi_{A}}|du|^{2},
\left\{\begin{array}{c}{g_{\rho\sigma}}\\{g_{\mu\nu}\g_{\alpha\beta}}\\\end{array}\right\}=\frac{1}{4}g^{\mu\nu}\delta_{\rho\sigma}^{\alpha\beta}+\frac{1}{4}g^{\alpha\beta}\delta_{\rho\sigma}^{\mu\nu}-\frac{1}{2}\delta_{\rho\sigma}^{\mu(\alpha}g^{\beta)\nu}-\frac{1}{2}\delta_{\rho\sigma}^{\nu(\alpha}g^{\beta)\mu}+\frac{1}{n-2}g_{\rho\sigma}C^{\mu\nu,\alpha\beta}.
\begin{array}{c}{[s^{\alpha}{}_{\beta},s^{\gamma}{}_{\delta}]=\delta^{\alpha}{}_{\delta}s^{\gamma}{}_{\beta}-\delta_{\beta}{}^{\gamma}s^{\alpha}{}_{\delta}\,,}\\{{}}\\{{}[s_{ab},s_{cd}]=-\eta_{ac}s_{bd}+\eta_{ad}s_{bc}+\eta_{bc}s_{ad}-\eta_{bd}s_{ac}\,.}\\\end{array}
J_{ll^{\prime}}=\int_{0}^{\infty}R_{nl^{\prime}}(r;\delta_{1},\delta_{2})R_{nl}(r;\delta_{1},\delta_{2})dr=\frac{2\alpha^{2}}{\left(n+\frac{\delta_{1}+\delta_{2}}{2}\right)^{3}}\,\frac{1}{2l+\delta_{1}+\delta_{2}+1}\,\delta_{ll^{\prime}}.
\bar{V}(\bar{\phi})=\frac{\lambda}{8}\left(\sum_{l=0}^{M}{\bar{\phi_{l}}}^{2}+\frac{2}{\lambda}{m_{0}}^{2}\right)^{2}+\sum_{l=0}^{M}\frac{1}{2}\left({m_{l}}^{2}-{m_{0}}^{2}\right){\bar{\phi_{l}}}^{2},
{\calZ}\=\\int_{\bar{\psi}_{\sigma}\psi_{\sigma}}exp-\{\int_{x}\left[\bar{\psi}\left(\frac{T^{\mu}}{s}~i\partial_{\mu}+M\right)\psi~+~r_{1}\left(\bar{\psi}T^{\mu}\psi\right)^{2}+r_{2}\left(\bar{\psi}\psi\right)^{2}\right]\}
K_{0}(X^{1},\bar{X}^{1},X^{2},\bar{X}^{2})=-X^{1}\bar{X}^{1}+X^{2}\bar{X}^{2}~\sim-X^{1}\bar{X}^{1}+{\frac{1}{2}}(X^{2}+\bar{X}^{2})^{2}\.
\vertC_{a}\rangle=\left[{\frac{2^{d(J_{a})}V(J_{a})}{V(RJ_{a})}}\right]^{\frac{1}{2}}\sum_{{\widetildep}\in2\Lambda(J_{a})}\sum_{{\widetildew}\in{\widetilde\Lambda}(RJ_{a})}\sum_{\bfn}\zeta_{\bfn}V_{L}({\bfn})V_{R}({\bfn})\vert{\widetildew},{\widetildep}\rangle~.
\left[\,\partial_{\mu}\,\partial^{\mu}+{\frac{V_{p}}{(p+1)!}}\,{\frac{\partial^{2}}{\partial\sigma_{\mu_{1}\dots\mu_{p+1}}\,\partial\sigma^{\mu_{1}\dots\mu_{p+1}}}}-(p+1)\,M_{0}^{2}\,\right]\,G\left(\,x-x_{0}\,\sigma\,\right)=\delta^{D}\left(\,x-x_{0}\,\right)\,\delta\left(\,\sigma\,\right)\.
A_{\mu}^{(1)}(x)=\sum_{n=1}^{\infty}\sum_{m=0}^{n}\intC_{m\,\mu}^{(1)}(x,1,\ldots,m;m+1,\ldots,n)a(1)\cdotsa(m)\mathrm{Tr\;}[a(m+1)\cdotsa(n)]\,.
\left[\begin{array}{c||cccc}{7}&{2}&{2}&{2}&{2}\\\end{array}\right]_{-128}^{1,65}\leftarrow\left[\begin{array}{c||ccccc}{7}&{1}&{1}&{2}&{2}&{2}\\{1}&{1}&{1}&{0}&{0}&{0}\\\end{array}\right]_{-112}^{2,58}\;,
\chi_{j}(\tau,\theta)\rightarrow\tilde{\chi}_{j}=\mathrm{Tr}_{D_{j}}\,e^{2i\pi\tau(\tilde{L}_{0}-c/24)}e^{4i\piz\tilde{J}_{0}^{3}}=e^{-i\pi\tauw^{2}/4}\,e^{i\pizw}\;\chi_{j}(\tau,z-w\tau/2)
\left(F^{(n)},{\tilde{F}}^{(n)}\right)=\sum_{k_{1},k_{2}\ldots,k_{n}\in{\calI}}\overline{{F^{(n)}\left(k_{1},k_{2},\ldots,k_{n}\right)}}{\tilde{F}}^{(n)}\left(k_{1},k_{2},\ldots,k_{n}\right)\;,
B^{T}{\frac{C^{3}}{C^{2}-\lambda^{2}}}B={\frac{1}{\pi}}\left({\frac{\alpha_{3}}{\alpha_{1}\alpha_{2}}}\right)^{2}{\frac{\mathrm{cos}\pi\lambda(1+2\beta)-\mathrm{cos}\pi\lambda}{\lambda\,\mathrm{sin}\pi\lambda}}
d_{a}=x_{ia}^{*}\Lambda^{-1}{}^{ij}{\frac{\partial}{\partialx^{j}}}-(-1)^{\varepsilon(c^{\alphaa})}y^{\alpha}{\frac{\partial}{\partialc^{\alphaa}}}-(-1)^{\varepsilon(B^{\alpha})}\varepsilon_{ab}c^{\alphab}{\frac{\partial}{\partialB^{\alpha}}}+\varepsilon_{ab}\bar{x}_{i}{\frac{\partial}{\partialx_{ib}^{*}}}
\begin{array}{c}{(1-\tilde{q}^{-\mu+1})\int_{0}^{\tau}dt\cdot\omega_{\mu}(t,z,\tau)=\int_{0}^{1}dt\cdot\omega_{\mu}(t,z,\tau)-\tilde{q}^{-1}\int_{0}^{1}dt\cdot\omega_{\mu-2}(t,z,\tau)+}\\{(\tilde{q}^{-1}-1)\int_{0}^{-z}dt\cdot\omega_{\mu}(t,z,\tau)}\\\end{array}
\left(\begin{array}{c}{q_{r}}\\{\bar{q}_{r}^{\dagger}}\\\end{array}\right)=\Biggl\{-{\frac{1}{2}}\left(\begin{array}{cc}{M_{rs}}&{{\widetildeM}_{rs}}\\{-{\widetildeM}_{rs}}&{-M_{rs}}\\\end{array}\right)+{\frac{\sqrt3}{2}}\left(\begin{array}{cc}{0}&{iC_{rs}}\\{-iC_{rs}}&{0}\\\end{array}\right)\Biggr\}\left(\begin{array}{c}{q_{r}}\\{\bar{q}_{r}^{\dagger}}\\\end{array}\right),
\bigl\{T_{2}^{(2)},4\,\eta+\eta^{3\prime\prime}+2\,\lambda\,\eta^{3}\bigr\}_{(\eta)}=-\,8\,\phi^{A}\pi^{A}\,\eta^{1}\delta+2\,\phi^{A}\pi^{A}\,\eta^{4}\delta^{\prime\prime}+2\,\lambda\,\phi^{A}\pi^{A}\,\eta^{4}\delta+\cdots
ds^{2}=-\frac{1}{l^{2}}{\biggl[}\rho^{2}-Ml^{2}\frac{2-\Omega}{2}{\biggr]}dt^{2}-Jdtd\varphi+\frac{l^{2}}{\rho^{2}-Ml^{2}(1-\Omega)}d\rho^{2}+{\biggl[}\rho^{2}+Ml^{2}\frac{\Omega}{2}{\biggr]}d\varphi^{2}\,,
\psi_{n,l}\left(\rho,\theta\right)=\sqrt{\frac{m^{\nu+1}n!}{\pi\left(n+\nu\right)!}}\\mathrm{e}^{il\theta-m\rho^{2}/2}\rho^{\nu}L_{n}^{\nu}\left(m\rho^{2}\right),\qquadE_{n,l}=m\left(2n+1+\nu\right),
\delta_{\epsilon}\hat{\psi}_{\hat{c}}=[\partial_{\hat{c}}-\frac{1}{4}(\hat{\omega}_{\hat{c}}{}^{\hat{a}\hat{b}}-\frac{3}{2}\hat{H}_{\hat{c}}{}^{\hat{a}\hat{b}})\hat{\gamma}_{\hat{a}\hat{b}}]\epsilon\,,
{\frac{1}{\sqrt{2}}}(\hat{\lambda}_{1}\pmi\hat{\lambda}_{2})=e^{\pmi(H_{4}-\kappa+i\phi)}(z),\quad{\frac{1}{\sqrt{2}}}(\hat{\lambda}_{3}\pmi\hat{\lambda}_{4})=e^{\pmi(H_{5}-\kappa+i\phi)}(z).
S={\widehatM}_{P}^{D-3}\int_{\Sigma}d^{D-1}x\sqrt{-{\widehatG}}\left[{\widehatR}-{\widehat\Lambda}\right]+M_{P}^{D-2}\intd^{D}x\sqrt{-G}\left[R-{\frac{4}{D-2}}(\nabla\phi)^{2}-V(\phi)\right]~.
D_{\kappa}{\calG}^{(n)}=D_{\kappa}\left(\sum_{i}\delta_{n}\lambda_{i}{\calS}_{i}\right)=\sum_{i}{\frac{\partial\delta_{n}\lambda_{i}}{\partial\kappa}}{\calS}_{i}=\sigma(S_{\kappa\,div}^{(n)}-D_{\kappa}R^{(n)}).
\delta_{\hat{\xi}}f=\{f,L[\hat{\xi}]\}=\dot{\sqrt{g_{rr}}}=\frac{\hat{\xi}_{r|r}}{f}=f\left(\hat{\xi}_{,r}^{r}+\Gamma_{rr}^{r}\hat{\xi}^{r}\right)=\left(f\hat{\xi}^{r}\right)_{,r}.
\int{\frac{d\rho_{0}d^{4}x_{0}}{\rho_{0}^{5}}}\,\prod_{p=1}^{4}{\left[{\frac{\rho_{0}}{\rho_{0}^{2}+(x_{p}-x_{0})^{2}}}\right]}^{4}=\frac{5\pi^{2}}{108}\prod_{p<q}{\frac{\partial}{\partialx_{pq}^{2}}}\int\prod_{p}d\alpha_{p}{\frac{\delta\left(1-\sum_{q}\alpha_{q}\right)}{(\sum_{p,q}\alpha_{p}\alpha_{q}x_{pq}^{2})^{2}}}\;,
\partial_{\mu}\left(\sqrt{-k}(k^{-1})^{[\mu\nu]}e^{-\phi}-\frac{2T_{p}}{T_{p}^{\prime}}\frac{1}{(n-2)!}\epsilon^{\mu\nu\gamma_{1}\ldots\gamma_{n-2}}(C^{(n-2)})_{\gamma_{1}\ldots\gamma_{n-2}}\right)=0.
\begin{array}{c}{a^{kI}={\frac{1}{\sqrt{2}}}\left({\frac{1}{\sqrt{2\piT}}}P^{kI}+ik\sqrt{2\piT}X^{-kI}\right)}\\{\tilde{a}^{kI}={\frac{1}{\sqrt{2}}}\left({\frac{1}{\sqrt{2\piT}}}P^{kI}-ik\sqrt{2\piT}X^{-kI}\right)}\\\end{array}
E_{1}^{u}=\frac{2~r_{ex}~e^{-2\rho}}{\ell^{2}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,E_{1}^{v}=\frac{1}{r_{ex}}=E_{2}^{v},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,E_{3}^{\rho}=\frac{1}{\ell}.
Z(\beta)=\mathrm{Tr}~e^{-\betaH}=\intd\Phi<\Phi|e^{-\betaH}|\Phi>~\equiv~N\int_{\mathrm{periodic}}[{\calD}\phi]e^{-\int_{0}^{\beta}d\tau\intd^{d}x{\calL}(\phi({\bfx},\tau))}\quad.
g_{\mu\nu}^{\prime}=e^{-2\sigma(x)}g_{\mu\nu}\,,\qquad\phi^{\prime}=e^{\sigma(x)}\phi\,,\qquadW_{\mu}^{\prime}=W_{\mu}\,,\qquad\lambda^{\prime}=e^{\frac{3}{2}\sigma(x)}\lambda\,.
\hat{P^{0}}\,\equiv\,\int_{-R}^{+R}dx^{1}\hat{\Theta}^{00}(x^{0},x^{1})\,=\,\frac{\pi}{R}\sum_{n>0}\omega_{n}\hat{\Lambda}_{n}^{\dagger}\hat{\Lambda}_{n}\,=\,:\hat{H}_{c}^{P}:
({\calD}^{2}-D_{1}\bar{D}_{1}+\frac{1}{2}D_{1}^{2}\bar{P}_{11}-\frac{1}{2}P_{11}\bar{P}_{11}+d\lambda^{2})|\chi\rangle+\lambda(\bar{D}_{2}-D_{1}\bar{P}_{12})|\Phi\rangle=0\,.
\Omega_{MN}^{ij}\left[{\bf\bar{A}}\right]=\left[\begin{array}{ccccc}{0}&{0}&{0}&{0}&{0}\\{0}&{\eta_{_{[AB][CD]}}}&{0}&{0}&{0}\\{0}&{0}&{g_{_{\Lambda_{1}\Lambda_{2}}}}&{0}&{0}\\{0}&{0}&{0}&{0}&{2\delta_{n}^{r}\delta_{\alpha}^{\beta}}\\{0}&{0}&{0}&{-2\delta_{s}^{m}\delta_{\beta}^{\alpha}}&{0}\\\end{array}\right]\otimes-\frac{i}{4}\epsilon^{ijkl}(\partial_{k}\bar{b}
\begin{array}{lll}{g_{rr}=\displaystyle\frac{l^{2}}{r^{2}}-\frac{4l^{2}\chi^{2}}{r^{3}}+O(r^{-4})}&{}&{\displaystyleg_{tt}=-\frac{r^{2}}{l^{2}}+O(1)}\\{g_{tr}=O(r^{-2})}&{}&{g_{\varphi\varphi}=r^{2}+O(1)}\\{g_{\varphir}=O(r^{-2})}&{}&{g_{t\varphi}=O(1)}\\\end{array}
H=-\frac{\hbar^{2}}{2m}\left\{\frac{1}{r^{2}f}[(r\partial_{r})^{2}+(\partial_{\phi}-i\frac{e}{\hbarc}V+\frac{1}{2}i\sigmaW)^{2}]+\sigma\frac{e}{\hbarc}B-\frac{1}{2}K\right\},
4\left(H^{2}+\frac{f(a)}{a^{2}}\right)\left(-3+4\alpha\left(h+3a\frac{dh}{da}\right)\left(H^{2}+\frac{f(a)}{a^{2}}-3\left(H^{2}+\frac{k}{a^{2}}\right)\right)+24\alphaa\frac{dh}{da}\frac{k}{a^{2}}\right)^{2}=\rho^{2}
\langlej|\psi(b_{1})\psi(b_{2})|j\rangle_{\partial}=\frac{jC}{\pi^{2}}\,\mathrm{Im}\,\int\frac{d\vec{k}}{(2\pi)^{3}}\,\langlea_{\vec{k}}a_{-\vec{k}}\psi(b_{1})\psi(b_{2})\rangle_{\partial}
\left(\begin{array}{cc}{m+1}&{0}\\{0}&{-m}\\\end{array}\right)k_{+t}\,+\,k_{+z}\left(\begin{array}{cc}{m+1}&{0}\\{0}&{-m}\\\end{array}\right)+\,\left(2m+1\right)\left(\begin{array}{cc}{0}&{q_{1}}\\{\mpq_{1}^{*}}&{0}\\\end{array}\right)k_{+}\,=\,0\,,\,\,\tag{2.13}
\delta_{12}-\int_{3}S_{13}^{-1}\frac{\deltaW}{\deltaS_{23}^{-1}}=e^{2}\int_{3\cdots7}V_{134}V_{567}D_{47}\left\{\frac{\delta^{2}W}{\deltaS_{23}^{-1}\deltaS_{56}^{-1}}+\frac{\deltaW}{\deltaS_{23}^{-1}}\frac{\deltaW}{\deltaS_{56}^{-1}}\right\}.
{\calA}^{(2k+1)}\left(\psi_{i}^{(-k)},\psi_{j}^{(-k)}\right)={\frac{1}{24(2k+1)\pi}}\int_{\Sigma}\psi_{i}^{(-k)}{\calR}_{J_{H}^{-1}}^{(2k+1)}\psi_{j}^{(-k)},\qquadk=0,1,2,\ldots,
{\hat{G}}(z,z^{\prime};t)={\frac{1}{(4\pit)^{3/2}}}\int_{\rho_{H}}^{\infty}{\frac{{\sqrt{2}}be^{-b^{2}/4t-M^{2}t}}{{\sqrt{\mathrm{Cosh}b-\mathrm{Cosh}\rho_{H}}}}}db,
H=-M+\frac{1}{2M}\left(P-\sum_{n,m=1}^{\infty}G_{mn}p_{n}q_{m}\right)^{2}+\sum_{n=1}^{\infty}\left(\frac{p_{n}^{2}}{2}+\frac{\Omega_{n}^{2}q_{n}^{2}}{2}\right)
\partial_{\mu}\phi^{\prime}\partial^{\mu}\phi^{\prime}=\Bigl(\frac{Q^{2}}{2(GM\rho-Q^{2})}\Bigr)^{2}\Bigl(1-\frac{2GM}{\rho}\Bigr)\rightarrow\frac{G^{2}M^{2}}{\rho^{3}(\rho-2GM)}\propto\frac{1}{x}
L=\frac{1}{2}r_{i}\Gamma^{ij}\dot{r}_{j}-\rho=\frac{1}{2(k-\lambda\alpha)}\left[\lambda\epsilon_{ij}x_{i}\dot{x}_{j}+\alphak\epsilon_{ij}q_{i}\dot{q}_{j}+2kx_{i}\dot{q}_{i}\right]-\frac{1}{2}\left[\frac{x_{i}^{2}}{m}+kq_{i}^{2}\right]
f_{D4}\sim\frac{1}{\sqrt{|x_{1}x_{2}x_{3}|}}e^{-\frac{|x_{1}x_{2}x_{3}|}{yz}}\quad\mathrm{or}\quad\frac{1}{\sqrt{|x_{1}x_{2}x_{3}|}}e^{-\frac{|x_{1}x_{2}x_{3}|}{y\bar{z}}}
\sigma_{x}=\left(\begin{array}{ccc}{0}&{1}\\{1}&{0}\\\end{array}\right);\,\,\,\,\sigma_{y}=\left(\begin{array}{ccc}{0}&{-i}\\{i}&{0}\\\end{array}\right);\,\,\,\,\sigma_{z}=\left(\begin{array}{ccc}{1}&{0}\\{0}&{-1}\\\end{array}\right).
\frac{\partial{\calV}_{\mathrm{eff}}}{\partial\alpha}+C_{i}(\phi,\alpha)\frac{\partial{\calV}_{\mathrm{eff}}}{\partial\phi_{i}}{\}=0~~~,~~~~\frac{\partial{\calV}_{\mathrm{eff}}}{\partial\rho^{ai}}+C_{aij}(\phi,\rho)\frac{\partial{\calV}_{\mathrm{eff}}}{\partial\phi_{j}}{\}=0~~~,
{\calL}(J)=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{J}{2}\phi^{2}+\frac{\lambda\mu^{2\varepsilon}}{4!}\phi^{4}+{\calL}_{\mathrm{CT}}(J)-\mu^{-2\varepsilon}\frac{\zeta}{2}\;J^{2}.
D_{\alpha}=\partial_{\alpha}-\frac{1}{2}i\partial_{\alpha\dot{\beta}}\widetilde{\vartheta}^{\dot{\beta}},\qquad\widetilde{D}_{\dot{\beta}}=-\widetilde{\partial}_{\dot{\beta}}+\frac{1}{2}\vartheta^{\alpha}i\partial_{\alpha\dot{\beta}}
B_{\mu\nu}|_{(k^{2}=s^{2})}=-\frac{1}{s^{2}k_{2}^{2}}\left[\begin{array}{ccc}{s^{2}k_{0}^{2}}&{-isk^{\left(2\right)}k_{0}^{2}}&{0}\\{isk^{\left(2\right)}k_{0}^{2}}&{0}&{-isk_{0}k_{2}^{2}}\\{0}&{isk_{0}k_{2}^{2}}&{-s^{2}k_{2}^{2}}\\\end{array}\right],
\intdt\:\left\{\frac{1}{2}{\calG}_{ab}\dot{z}^{a}\dot{z}^{b}+\frac{i}{2}\psi^{a}D_{t}\psi^{a}+i\eta^{\dagger}{\calD}_{t}\eta+\frac{i}{2}\eta^{\dagger}[{\calF}_{ab}\psi^{a}\psi^{b}]\eta\right\},
\check{\Gamma}_{\alpha\beta}^{\mu\nu}\epsilon_{\mu\nu}^{3,4}=\left(k^{2}+\Pi_{B}\right)\epsilon_{\mu\nu}^{3,4}+2\left(1-\frac{k^{2}}{(k\cdotu)^{2}}\right)\Pi_{6}\epsilon_{\mu\nu}^{5,6}
M_{_{(1)}}=\left(\begin{array}{c}{{\frac{\partial\Omega_{_{(1)}}}{\partialq}}}\\{{\frac{\partial\Omega_{_{(1)}}}{\partial\pi}}}\\\end{array}\right)=\left(\begin{array}{c}{{\frac{\partial\Omega_{_{(1)}}}{\partialq}}}\\{{-I}}\\\end{array}\right)\;\;\;,\;\;\;R_{_{(1)}}=\left({\frac{\partial\Omega_{(1)}}{\partial\alpha}}\right)\;.
H^{(4)}=\intd^{3}x\frac{1}{2}\biggl[\frac{k^{2}}{{\calH}^{2}}\tilde{\Pi}^{2}+2\biggl(\frac{k^{2}}{\calH}-{\calH}-\frac{{\calH}^{\prime}}{{\calH}}\biggr)\tilde{\Pi}\tilde{\Psi}+\biggl(k^{2}-\frac{a^{\prime\prime}}{a}\biggr)\tilde{\Psi}^{2}\biggr].
\left[{\mathcalJ}_{\ell}^{0}\left(t,\underline{{x}}\right)~,~~{\mathcalJ}_{\ell}^{0}\left(t,\underline{{y}}\right)\right]\;=\;\left[\widehat{{\mathcalJ}}_{\ell}^{0}\left(t,\underline{{x}}\right)~,~~\widehat{{\mathcalJ}}_{\ell}^{0}\left(t,\underline{{y}}\right)\right]~.
{f_{mk}}^{i}{\tilde{f}^{jm}}_{\;\;\;\;\;l}-{f_{ml}}^{i}{\tilde{f}^{jm}}_{\;\;\;\;\;k}-{f_{mk}}^{j}{\tilde{f}^{im}}_{\;\;\;\;\;l}+{f_{ml}}^{j}{\tilde{f}^{im}}_{\;\;\;\;\;k}={f_{kl}}^{m}{\tilde{f}^{ij}}_{\;\;\;m}.
V^{\alpha\dot{\alpha}}=\left(\begin{matrix}{V^{1\dot{1}}}&{V^{1\dot{2}}}\\{V^{2\dot{1}}}&{V^{2\dot{2}}}\\\end{matrix}\right)={\frac{1}{\sqrt2}}\,\left(\begin{matrix}{V^{1}+V^{4}}&{V^{2}-V^{3}}\\{V^{2}+V^{3}}&{-V^{1}+V^{4}}\\\end{matrix}\right)\,
\Psi=e^{2\pi\tau_{2}\nu(1-\nu)/2}(1-e^{-2\pi\tau_{2}\nu})\prod_{n=1}^{\infty}{\frac{(1-e^{-2\pi\tau_{2}(n-\nu)})(1-e^{-2\pi\tau_{2}(n+\nu)})}{(1-e^{-n2\pi\tau_{2}})^{2}}}\.
\widehat{\xi}_{\mu}=\frac{1}{\sqrt{2}}\gamma_{*}\gamma_{\mu}\otimes\sigma_{1},\quad\widehat{\theta}_{1}=\frac{1}{\sqrt{2}}\gamma_{*}\otimes\sigma_{1},\quad\widehat{\theta}_{0}=\frac{1}{\sqrt{2}}1\otimes\sigma_{3}.
e_{1}\equiv\frac{M}{12}+\frac{r_{m}}{2l^{2}}\sqrt{Ml^{2}-r_{m}^{2}}\;>\;0\;\geqe_{2}\equiv\frac{M}{12}-\frac{r_{m}}{2l^{2}}\sqrt{Ml^{2}-r_{m}^{2}}\;>\;e_{3}\equiv-\frac{M}{6},
E_{\mu\nu}=-\frac{6{\calU}}{\lambda\kappa_{(4)}^{2}}\left[V_{\mu}V_{\nu}+\frac{1}{3}h_{\mu\nu}^{(4)}+{\calP}_{\mu\nu}+{\calQ}_{\mu}V_{\nu}+{\calQ}_{\nu}V_{\mu}\right].
d^{4}x\,du\,Y_{i_{1}j_{1},\ldots,i_{p}j_{p}}^{k_{1}l_{1},\ldots,k_{q}l_{q}}\widetilde{D}_{k_{1}l_{1},\ldots,k_{q}l_{q}}^{i_{1}j_{1},\ldots,i_{p}j_{p}}
\int_{{\bfr}(t^{\prime})={\bfr}^{\prime}}^{{\bfr}(t^{\prime\prime})={\bfr}^{\prime\prime}}\,{\calD}^{(N)}{\bfr}(t)=\prod_{\alpha=1}^{n}\left[\intd^{D}{\bfr}_{\alpha}\right]\;\prod_{\beta=0}^{n}\left[\int_{{\bfr}(t_{\beta})={\bfr}_{\beta}}^{{\bfr}(t_{\beta+1})={\bfr}_{\beta+1}}{\calD}^{(N_{\beta})}{\bfr}(t)\right]\;,
q_{c}=\frac{1}{32a^{3}}\sum_{m=-\infty}^{+\infty}\int_{-\infty}^{+\infty}{dk\sum_{\lambda=0}^{1}\sum_{n=1}^{\infty}\frac{(\gamma_{m,n}^{(\lambda)})^{5-2\lambda}\psi_{\mu}(\gamma_{m,n}^{(\lambda)}/a)}{\sqrt{k^{2}a^{2}+\gamma_{m,n}^{(\lambda)2}}}T_{m}^{AB}(\eta,\gamma_{m,n}^{(\lambda)})f_{\lambdam}^{(q)}(\gamma_{m,n}^{(\lambda)},\gamma_{m,n}^{(\lambda)}x)},
{\calL}^{(1)}={\frac{1}{2}}{\epsilon^{jki}\partial_{j}{A_{k}}^{a}}{\epsilon_{ab}\partial_{0}{A_{i}^{b}}}-{\frac{1}{4}}{F^{a,jk}}{{F^{a}}_{jk}}+{\dot{\lambda}^{(a)}}{A_{3}^{a}}
ds^{2}=\left(\frac{r}{R_{0}}\right)^{2(d_{\perp}-2)/(p+1)}dx^{\mu}\otimesdx^{\nu}\,\eta_{\mu\nu}+\left(\frac{R_{0}}{r}\right)^{2}dy^{I}\otimesdy^{J}\,\delta_{IJ}\.
<{\calT}^{[t]}(A_{\mu})_{ij}{\calT}^{[s]}(A_{\nu}^{\dagger})_{kl}>=e^{\frac{2\pi\imath}{L}(t_{\mu}-s_{\mu})}g_{il}^{[s-t]}g_{kj}^{[t-s]}\delta_{\mu\nu}
{\calL}_{f}/n=-\frac{1}{8\pig}\left(1-\frac{q^{2}}{2n^{2}}+\frac{g^{2}(p-q\chi_{0})^{2}}{2n^{2}}\right)f_{\mu\nu}f^{\mu\nu}+\frac{1}{16\pi}\left(\chi_{0}+\frac{q(p-q\chi_{0})}{n^{2}}\right)\epsilon^{\mu\nu\lambda\sigma}f_{\mu\nu}f_{\lambda\sigma},
\left[K_{\quad\\nu}^{(1)\mu}-\delta_{\nu}^{\mu}K^{(1)}\right]|_{y=0}={\frac{l}{d-2}}\left(R_{\nu}^{\mu}-{\frac{1}{2}}\delta_{\nu}^{\mu}R\right)+\chi_{\nu}^{\mu}={\frac{\kappa^{2}}{2}}T_{\nu}^{\mu}\.
b(g)=B_{1}e^{\chi_{2}{\calX}_{2}}e^{\chi_{3}{\calX}_{3}}+e^{-\chi_{1}{{\calX}_{1}}^{t}}B_{2}e^{\chi_{3}{\calX}_{3}}+e^{-\chi_{1}{{\calX}_{1}}^{t}}e^{-\chi_{2}{{\calX}_{2}}^{t}}B_{3}.
{\calL}=-{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}+(D_{\mu}\phi_{a})^{*}D^{\mu}\phi_{a}-m^{2}\phi_{a}^{*}\phi_{a}-{\frac{ie}{2}}\epsilon^{\mu\nu}F_{\mu\nu}\phi_{a}^{*}(\sigma_{3})_{ab}\phi_{b},
\frac{y^{n}}{n^{r}(n+p)^{k}}=\int_{0}^{y}\Omega^{r-1}\int_{0}^{x}\Omega^{k-1}\Omega_{n}\,x^{p}\frac{dx}{x}=\int_{0}^{y}\Omega^{r-1}\Omega_{p}\Omega^{k-1}\Omega_{n}.
\alpha^{\pm\pm}(n_{1},n_{2})=\sqrt{\frac{1}{f^{\pm\pm}(n_{1},n_{2})}\frac{N_{k}}{4\hbar\tau_{2}}}\,Big[-i\tau\hat{A}_{1}^{\pm\pm}(n_{1},n_{2})+i\hat{A}_{2}^{\pm\pm}(n_{1},n_{2})\Big]\\\,
P_{+}=\int_{-L}^{L}dx^{-}{\calH}(x)=\sum_{n}p_{n+}\left(b_{n}^{a\dagger}b_{n}^{a}-d_{n}^{a}d_{n}^{a\dagger}\right)+2L\left(-{\frac{Nm_{0}}{g_{0}}}\sigma+{\frac{N}{2g_{0}}}\sigma^{2}\right).
\left(L_{3}+\frac{\Sigma_{3}}{2}\right)D^{\pm}\Psi_{ls}=\left(\begin{array}{c}{\left(-\frac{1}{2}\right)+\left(l+1\right)}\\{\left(+\frac{1}{2}\right)+\left(l\right)}\\\end{array}\right)D^{\pm}\Psi_{ls}=\left(l+\frac{1}{2}\right)D^{\pm}\Psi_{ls}
M_{1/2}(u)=\frac{1}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!(2n+1)}\frac{\Gamma(1+(n+1/2)\frac{1}{(1-g)})}{\Gamma(1+(n+1/2)\frac{g}{(1-g)})}u^{2n+1}.
\int_{0}^{\infty}d\rho\int_{0}^{2\pi}d\varphi\,\psi_{n^{\prime},l^{\prime}}^{(j^{\prime})\ast}(\rho,\varphi)\psi_{n,l}^{(j)}(\rho,\varphi)=\delta_{l.l^{\prime}}\delta_{n,n^{\prime}}\,.
{\calO}_{e^{\alpha}}\cdot{\calO}_{e^{\beta}}=C_{\alpha,\beta,\gamma}^{N,k}\eta^{\gamma\delta}{\calO}_{e^{\delta}}=\frac{1}{k}C_{\alpha,\beta,\gamma}^{N,k}{\calO}_{e^{N-2-\gamma}}:=\sum_{d=0}^{\infty}q^{d}\frac{1}{k}C_{\alpha,\beta,\gamma}^{N,k,d}{\calO}_{e^{N-2-\gamma}}.
(\delta_{i_{1}i_{2}}\delta_{j_{1}j_{2}}-\delta_{i_{1}j_{2}}\delta_{j_{1}i_{2}})\sum_{k,l=1}^{\infty}\big(\frac{2\hbarT}{k^{2}\pi^{2}}\big)\big(\frac{2\hbarT}{l^{2}\pi^{2}}\big)\Big(-[1-(-1)^{k+l}]\frac{kl}{l^{2}-k^{2}}\Big)^{2}.
(k_{1}^{+}{\cdot}k_{3}^{-})(k_{2}^{+}{\cdot}k_{4}^{-})k_{1}{\cdot}k_{4}+(k_{1}^{+}{\cdot}k_{4}^{-})(k_{2}^{+}{\cdot}k_{3}^{-})k_{1}{\cdot}k_{3}=0.
H_{0}^{\prime}=\frac{1}{2}\intd^{3}x\left(B_{a}^{ij}F_{ij}^{a}-A_{i}^{a}A_{a}^{i}+\left(\partial^{i}B_{0i}^{a}\right)\left(\partial_{j}B_{a}^{0j}\right)\right)\equiv\intd^{3}x\,h_{0}^{\prime},
{\calD}_{\alpha\beta}=\left(\begin{matrix}{\partial_{t}^{2}+\beta_{k}-i\alpha_{k}}&{\gamma_{k}\partial_{t}+\textstyle{\frac{1}{2}}\dot{\gamma}_{k}+i\alpha_{k}}\\{-\gamma_{k}\partial_{t}-\textstyle{\frac{1}{2}}\dot{\gamma}_{k}+i\alpha_{k}}&{-\partial_{t}^{2}-\beta_{k}-i\alpha_{k}}\\\end{matrix}\right)\,,
{\frac{\partial\Gamma}{\partial\xi}}=\intd^{n}x\,d^{n}z\,\left.\left({\frac{\delta\Gamma}{\deltaA_{\lambda}(z)}}\star{\frac{\delta^{2}\Gamma}{\deltaH(x)\deltaK^{\lambda}(z)}}-{\frac{\delta\Gamma}{\delta{c}(z)}}\star{\frac{\delta^{2}\Gamma}{\deltaH(x)\deltaL(z)}}+{\frac{\deltaF(z)}{\deltaH(x)}}\star{\frac{\delta\Gamma}{\delta\bar{c}(z)}}\right)\right|_{H=0}
{\calD}X^{\mu}=DX^{\mu}-\frac{l^{2}(k_{\rho}DX^{\rho})-(k\cdotl)(l_{\rho}DX^{\rho})}{k^{2}l^{2}-(k\cdotl)^{2}}\,k^{\mu}\-\frac{k^{2}(l_{\rho}DX^{\rho})-(k\cdotl)(k_{\rho}DX^{\rho})}{k^{2}l^{2}-(k\cdotl)^{2}}\,l^{\mu},
\int_{0}^{\infty}d\omega_{a}^{R}\int_{0}^{\infty}d\omega_{b}^{R}=\int_{0}^{\infty}d\omega_{-}^{R}\int_{\frac{\omega_{-}^{R}}{2}}^{\infty}d\omega_{+}^{R}+\int_{-\infty}^{0}d\omega_{-}^{R}\int_{\frac{-\omega_{-}^{R}}{2}}^{\infty}d\omega_{+}^{R}\equiv\intd{\tilde{\omega}}_{-}^{R}\intd{\tilde{\omega}}_{+}^{R}.
\int\prod_{i}dY_{i}\delta(\sum_{i=1}^{6}w_{i}Y_{i}-t)e^{-\sum_{i}\mu_{i}Y_{i}}\\stackrel{Y_{i}\rightarrowY_{i}+ln\mu_{i}}{\longrightarrow}\int\prod_{i}dY_{i}\delta(\sum_{i=1}^{6}w_{i}Y_{i}-t^{\prime})e^{-\sum_{i}Y_{i}},
\sum_{\pi(1,2,...n)}\mathrm{Tr}\Big[\tau^{(l_{0})}\tau^{(l_{\pi(1)})}.....\tau^{(l_{\pi(n)})}\Big]\frac{1}{\tilde{x}_{0}-\tilde{x}_{\pi(1)}}\frac{1}{\tilde{x}_{\pi(1)}-\tilde{x}_{\pi(2)}}.....\frac{1}{\tilde{x}_{\pi(n)}-\tilde{x}_{0}}\;,
\sum_{k=0}^{r}s\chi_{\left({}_{(k)}^{\{p\}\times(r+1-k)}\right)}(M)+s\chi_{\left({}_{(r+1)}^{\{p\}}\right)}(M)=\sum_{k=0}^{r}(-1)^{k}\Sigma(\lambda,\bar{\lambda})\chi_{\{p\}\times(r+1-k)}(\lambda)\times\chi_{(k)^{T}}(\bar{\lambda})+(-1)^{r+1}\Sigma(\lambda,\bar{\lambda})\chi_{\{p\}}(\lambda)\chi_{(r+1)^{T}}(\bar{\lambda})
\langlep(f),p(g)\rangle=\int_{X_{H}\timesX_{H}}f^{*\alpha\beta}(x){\calW}_{\alpha\beta\alpha^{\prime}\beta^{\prime}}(x,x^{\prime})g^{\alpha^{\prime}\beta^{\prime}}(x^{\prime})d\sigma(x^{\prime})d\sigma(x)\,.
\begin{array}{l}{H\equivP_{+}=\intdx^{-}\big[(1/8)\mathrm{tr}~E^{2}-(m/2)\big(\psi_{-}^{\dagger}(x)\psi_{+}(x)+\psi_{+}^{\dagger}(x)\psi_{-}(x)\big)\big]}\\{\left[A_{-}(x^{-},x^{+}),E(y^{-},x^{+})\right]=i\delta(x^{-}-y^{-})}\\\end{array}