harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\omega_{\b\mu}^{a}=e_{\\nu}^{a}\nabla_{\mu}E_{b}^{\\nu}=e_{\\nu}^{a}\partial_{\mu}E_{b}^{\\nu}+e_{\\nu}^{a}\Gamma_{\\mu\lambda}^{\nu}E_{b}^{\\lambda}=\frac{\kappa}{2}\left(\partial_{b}h_{\mu}^{\a}-\partial^{a}h_{b\mu}\right)+O\left(\kappa^{2}\right)
\frac{G^{m}(uq^{-n})}{G^{m}(u)}=\frac{1}{\bar{\kappa}(u)}\frac{(\omega^{-m}q^{2}u^{-2/n};t^{2})_{\infty}(t^{2}\omega^{m}q^{-2}u^{2/n};t^{2})_{\infty}}{(\omega^{m}u^{2/n};t^{2})_{\infty}(t^{2}\omega^{-m}u^{-2/n};t^{2})_{\infty}}.
\omega^{\Lambda}=(\mathrm{d}\varphi_{I}+\vec{W}_{IJ}\cdot\mathrm{d}\vec{x}^{J})\wedge\mathrm{d}x^{\LambdaI}+U_{IJ}\,\varepsilon^{\Lambda\Sigma\Pi}\mathrm{d}x^{\SigmaI}\wedge\mathrm{d}x^{\PiJ}\.
\overline{{{\cal{A}}_{i}^{\gamma}}}\,^{T}({\bfr})={\delta}_{{i}\,{\gamma}}\,{\varphi}_{A}(r)+\frac{r_{i}\,r_{\gamma}}{r^{2}}\,{\varphi}_{B}(r)+{\epsilon}_{i{\gamma}n}\frac{r_{n}}{r}\,{\varphi}_{C}(r)
U_{q}=-\frac{\hbar^{2}}{4}\left(\partial_{\mu}\Gamma^{\mu}+\frac{1}{2}\Gamma_{\mu}\Gamma^{\mu}\right)-\frac{\hbar^{2}}{4}\left(\partial_{\mu}\Theta^{\mu}-\frac{1}{2}\Theta_{\mu}\Theta^{\mu}\right)
\begin{array}{rcl}{sA_{\mu}^{a}}&{=}&{-\,(D_{\mu}c)^{a}\quad,}\\{s\phi^{a}}&{=}&{f^{abc}c^{b}\phi^{c}\quad,}\\{sc^{a}}&{=}&{\frac{1}{2}\,f^{abc}c^{b}c^{c}\quad,}\\{s\bar{c}^{a}}&{=}&{b^{a}\quad,}\\{sb^{a}}&{=}&{0\quad,}\\{s^{2}}&{=}&{0\quad.}\\\end{array}
C_{spin}=\left(\begin{matrix}{0}&{0}&{0}&{0}&{0}&{0}&{0}&{1}\\{0}&{0}&{0}&{0}&{0}&{0}&{-1}&{0}\\{0}&{0}&{0}&{0}&{0}&{-1}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{-1}&{0}&{0}&{0}&{0}&{0}\\{0}&{-1}&{0}&{0}&{0}&{0}&{0}&{0}\\{1}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\\end{matrix}\right)\.
{\mathcalL}_{Sk}=-\frac{f_{\pi}^{2}}{4}Tr(U^{\dagger}\partial_{\mu}UU^{\dagger}\partial^{\mu}U)+\frac{1}{32\alpha^{2}}Tr([U^{\dagger}\partial_{\mu}U,U^{\dagger}\partial_{\nu}U]^{2})
\delta\langleo\midT\left(\overline{{C}}^{a_{1}}(x_{1})\,A^{\mu_{2},a_{2}}(x_{2})\,\partial_{\mu_{3}}A^{\mu_{3},a_{3}}(x_{3})\,\partial_{\mu_{4}}A^{\mu_{4},a_{4}}(x_{4})\right)\mido\rangle=o
{\calI}=(i\pi)T_{2}^{2}\int\frac{d^{2}\tau}{\tau_{2}^{4}}\sum_{n_{1},n_{2},l_{1},l_{2}}Q_{R}{\bar{Q}}_{R}e^{-2\pii{\bar{T}}detA}e^{\frac{-\piT_{2}}{\tau_{2}U_{2}}|n_{1}\tau+n_{2}U\tau-Ul_{1}+l_{2}|^{2}}{\bar{F}}({\bar{\tau}})
C_{S}=\left(\begin{matrix}{\phantom{-}1}&{-2}&{\phantom{-}3}&{-2}\\{-2}&{\phantom{-}1}&{-2}&{\phantom{-}3}\\{\phantom{-}3}&{-2}&{\phantom{-}1}&{-2}\\{-2}&{\phantom{-}3}&{-2}&{\phantom{-}1}\\\end{matrix}\right),
\frac{Z_{3}}{Z_{\lambda}\lambda}\,\langle\,\partialA_{x}^{a}\,\partialA_{y}^{b}\,\rangle\,=\,\widetildeZ_{3}\,\langle\,i(\partialD_{r}c)_{x}^{a}\,\bar{c}_{y}^{b}\,\rangle\;.
S_{r}^{\alpha}={\frac{1}{\sqrt{2}}}\left(\begin{matrix}{f_{I}^{\Lambda}+\mathrm{i}h_{\LambdaI}}&{\bar{f}_{I}^{\Lambda}+\mathrm{i}\bar{h}_{\LambdaI}}\\{f_{I}^{\Lambda}-\mathrm{i}h_{\LambdaI}}&{\bar{f}_{I}^{\Lambda}-\mathrm{i}\bar{h}_{\LambdaI}}\\\end{matrix}\right)
\psi_{_{WKB}}=C{\left(\pi\chi_{0}^{3}a_{0}^{2}(\eta-\frac{1}{2}sin{2\eta})\right)}^{-1/2}exp\left(\frac{i}{\hbar}\pi\chi_{0}^{3}a_{0}^{2}(\eta-\frac{1}{2}sin{2\eta})\right)~.
X^{i}(\sigma,\tau)=x^{i}+2w^{i}\sigma+\tauG^{ij}(p_{j}-2B_{jk}w^{k})+\frac{i}{2}\sum_{n\not=0}\frac{1}{n}[\alpha_{n}^{i}(E)e^{-2in(\tau-\sigma)}+\tilde{\alpha}_{n}^{i}(E)e^{-2in(\tau+\sigma)}],
(z_{0};z_{1},z_{2})=\left({\frac{1-x_{\mu}^{2}+h^{2}}{2h}}+i{\frac{x_{0}}{h}};{\frac{x_{1}+ix_{2}}{h}},{\frac{x_{3}}{h}}+i{\frac{1+x_{\mu}^{2}-h^{2}}{2h}}\right)
\eta_{\rho\alpha}(x_{1},x_{2})=\int{\frac{d^{d}p_{1}}{(2\pi)^{d/2}}}{\frac{d^{d}p_{2}}{(2\pi)^{d/2}}}e^{i(p_{1}x_{1}-p_{2}x_{2})}\eta_{\rho\alpha}(p_{1},p_{2})
I_{W}(X_{P})=\sum_{\alpha_{1},\alpha_{2},\alpha_{3},\beta}S_{0\beta}^{-2}(ST^{-2}S)_{0\alpha_{1}}(ST^{3}S)_{0\alpha_{2}}(ST^{5}S)_{0\alpha_{3}}S_{\alpha_{1}\beta}S_{\alpha_{2}\beta}S_{\alpha_{3}\beta}
f^{i_{1}\ldotsi_{m_{1}}j_{1}\ldotsj_{m_{2}}\ldotsk_{1}\ldotsk_{m_{N-1}}}\;u_{i_{1}}^{1}\ldotsu_{i_{m_{1}}}^{1}u_{j_{1}}^{2}\ldotsu_{j_{m_{2}}}^{2}\ldotsu_{k_{1}}^{N-1}\ldotsu_{k_{m_{N-1}}}^{N-1}\;.
\deltaW_{5}={\frac{1+\alpha}{48n}}{\frac{1}{(2\pi)^{4}}}\bigg({\frac{(2\pi)^{2}}{2m}}{\frac{\kappa^{4}}{\lambda^{6}}}\bigg)^{1/3}\int_{M^{10}}Q_{2}^{1}\wedgeX_{8},
{\Theta}^{\mu\nu}\equiv\eta^{\mu\nu}-{\frac{\partial^{\mu}\partial^{\nu}}{\Box}}\;,\;\;S^{\mu\nu}\equiv\epsilon^{\mu\alpha\nu}\partial_{\alpha}\;\;\mathrm{and}\;\;\Omega^{\mu\nu}\equiv{\frac{\partial^{\mu}\partial^{\nu}}{\Box}}
G_{l+\nu}^{(n)}(r^{\prime\prime},r^{\prime};E)=\prod_{\alpha=1}^{n}\left[\int_{0}^{\infty}dr_{\alpha}V(r_{\alpha})\right]\,\prod_{\beta=0}^{n}\left[G_{l+\nu}^{(0)}(r_{\beta+1},r_{\beta};E)\right]\;.
Z_{\theta}(\kappa)={\frac{\sqrt{\pi}e^{-1/(8\kappa)}}{4\sqrt{\kappa}}}\left[I_{1/4}\Bigl({\frac{1}{8\kappa}}\Bigr)+I_{-1/4}\Bigl({\frac{1}{8\kappa}}\Bigr)+\theta\Bigl[I_{1/4}\Bigl({\frac{1}{8\kappa}}\Bigr)-I_{-1/4}\Bigl({\frac{1}{8\kappa}}\Bigr)\Bigr]\right].
\frac{1}{4}\tilde{\varphi}^{2}-k^{\mu}\tilde{\varphi}_{\mu\nu}m_{\rho}\tilde{\varphi}^{\rho\nu}=-\frac{1}{2}k^{2}(\tilde{A}^{2}+\tilde{B}^{2})-k^{2}\epsilon_{\mu\nu\rho\sigma}\tilde{A}^{\mu}k^{\nu}m^{\rho}\tilde{B}^{\sigma}\;.
\frac{(-1)^{l}}{2^{n-l}}\int_{0}^{1}{\calD}_{n}\eta\;\eta_{n-1}^{k_{n-1}}\dots\eta_{2}^{k_{2}}\eta_{1}^{k_{1}}V^{(m_{1})}(x_{i_{1}})\dotsV^{(m_{l})}(x_{i_{l}}).
\nabla_{\\\dot{A}}^{B}W_{ABC}=\frac{i}{2}\nabla_{\dot{A}}\nabla^{B}W_{ABC}+\frac{5}{2}iG_{\\\dot{A}}^{B}W_{ABC}=\nabla_{\dot{A}}\nabla_{\underline{{A}}}^{\\\dot{B}}G_{\underline{{C}}\dot{B}}+\frac{5}{2}iG_{\\\dot{A}}^{B}W_{ABC}
i\hbar^{2}\left\{\frac{dA_{\lambda\rho}}{dt},L_{0}\right\}=\hbar\left\{\left[A_{\lambda\rho},H\right],L_{0}\right\}=\hbar\left[\left\{A_{\lambda\rho},L_{0}\right\},H\right]=i\hbar^{2}\left(\lambda+\rho\right)\frac{dA_{\lambda\rho}}{dt}\;.
\sigma_{0}^{\psi_{s}}\Big\vert_{B\to\infty}={\frac{\pi^{4}}{2^{2\sqrt{4+s}-1}\Gamma(\sqrt{4+s})^{2}}}{\frac{(\tilde{\omega}\tilde{R}_{\frac{\pi}{2}})^{2\sqrt{4+s}+4}}{\omega^{5}}}.
H_{c}=\frac{La^{2}}{2}\sum_{\vec{b}}\int_{0}^{L}dz^{-}\int_{0}^{L}dy^{-}\sum_{n}\frac{G_{\bot21}^{\prime}(\vec{b},z^{-})G_{\bot12}^{\prime}(\vec{b},y^{-})}{(\pin-\varphi_{\vec{b}})^{2}}e^{2\piin(z^{--}y^{-})/L}.
\delta\chi_{a}\;=\;-i\,\sqrt{\,\frac{1}{2}}\,\gamma^{\mu}\,(\partial_{\mu}\,\Phi)\,\varepsilon_{a}\;+\;A(\Gamma_{45})_{ab}\,\varepsilon^{b}\;+\;\frac{1}{2}\,\sqrt{\,\frac{1}{6}}\,\gamma^{\,\mu\nu}\,\left(\,H_{\,\mu\nuab}\;-\;\sqrt{\,2}\,h_{\,\mu\nuab}\,\right)\,\varepsilon^{b}
X=\left(\begin{array}{cccccccc}{0}&{}\\{}&{\ddots}&{}\\{}&{}&{0}\\{}&{}&{}&{0}&{1}\\{}&{}&{}&{}&{0}\\{}&{}&{}&{}&{}&{\ddots}\\{}&{}&{}&{}&{}&{}&{0}&{1}\\{}&{}&{}&{}&{}&{}&{}&{0}\\\end{array}\right)
d{}^{*}d\tau+i\mathrm{e}^{\phi}d\tau\wedge{}^{*}d\tau+\frac{i}{2}G_{3}\wedge{}^{*}G_{3}=2i\kappa^{2}\mathrm{e}^{-\phi}\left[\frac{\delta{\cal{L}}_{b}}{\delta\phi}+i\mathrm{e}^{-\phi}\frac{\delta{\cal{L}}_{b}}{\deltaC_{0}}\right]
H=\frac{1}{L}\left[\sqrt{\Bigl(1+\frac{r^{2}}{L^{2}}\Bigr)\Bigl(P_{\phi}^{2}+\frac{{\tilde{N}}^{2}r^{6}}{L^{6}}\Bigr)}-\frac{{\tilde{N}}r^{4}}{L^{4}}\right]\,,
\left[\begin{array}{c}{\psi^{1}(x)}\\{\psi^{2}(x)}\\{\psi^{3}(x)}\\{\psi^{4}(x)}\\\end{array}\right]=e^{-2\beta}\left[\begin{array}{cccc}{1}&{1}&{0}&{-1}\\{1}&{1}&{1}&{0}\\{0}&{1}&{1}&{1}\\{-1}&{0}&{1}&{1}\\\end{array}\right]\left[\begin{array}{c}{\psi^{1}(x-\hat{1})}\\{\psi^{2}(x-\hat{2})}\\{\psi^{3}(x-\hat{3})}\\{\psi^{4}(x-\hat{4})}\\\end{array}\right],
\begin{array}{l}{\omega_{1\beta}=\displaystyle\frac1{(\Lambda\Pi)^{2}}(\Lambda\Pi_{\sigma})P^{+\sigma\gamma}\Lambda_{n}\Gamma^{n}\tilde{\Gamma}^{m}\Pi_{m\gamma}k_{\beta},}\\{\omega_{2\beta}=\displaystyle\frac1{(\Lambda\Pi)^{2}}(\Lambda\Pi_{\sigma})P^{+\sigma\gamma}\Pi_{n\gamma}\Gamma^{n}\tilde{\Gamma}^{m}\Lambda_{m}s_{\beta}.}\\\end{array}
H_{osc}=2\pi\sum_{n=1}^{\infty}n\left\{-{\frac{1}{2}}(\alpha_{n}^{\dagger}\beta_{n}+\beta_{n}^{\dagger}\alpha_{n}+\tilde{\alpha}_{n}^{\dagger}\tilde{\beta}_{n}+\tilde{\beta}_{n}^{\dagger}\tilde{\alpha}_{n})+a_{n}^{T\dagger}a_{n}^{T}+\tilde{a}_{n}^{\daggerT}\tilde{a}_{n}^{T})\right\}\;.
\begin{array}{lll}{H\left|\Omega(k,k+2)\right\rangle}&{=}&{2\left(Q+Q^{-1}\right)\left|\Omega(k,k+2)\right\rangle+\left|\Omega(k-1,k+2)\right\rangle}\\{}&{}&{}\\{}&{+}&{\left|\Omega(k,k+3)\right\rangle+U_{k+1}\left|\Omega(k,k+2)\right\rangle}\\\end{array}
R=T^{-1}\widetilde{T}^{-1}=\left(\begin{matrix}{0}&{s+\frac{1}{2}}&{s}&{0}\\{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}\\{0}&{s-\frac{1}{2}}&{s}&{0}\\\end{matrix}\right)\.
{\hat{J}}_{(3)}={\frac{i}{\sqrt{2}}}({\bar{\alpha}}_{A}{\bar{\alpha}}^{B}+{\bar{\beta}}_{A}{\bar{\beta}}^{B}){\bar{\pi}}_{B}{\frac{\partial}{\partial{\bar{\pi}}_{A}}}+{\frac{i}{\sqrt{2}}}(\alpha_{A^{\prime}}\alpha^{B^{\prime}}+\beta_{A^{\prime}}\beta^{B^{\prime}})\pi_{B^{\prime}}{\frac{\partial}{\partial\pi_{A^{\prime}}}}.
X_{0}^{+}\tilde{p}^{-\mu}\=\\oint\frac{dz_{1}}{2\piiz_{1}}\oint_{|z_{2}|<|z_{1}|}\frac{dz_{2}}{2\pii}\;X^{+}(z_{1})\;e^{-\varphi^{-}}i\psi^{-\mu}(z_{2}).
{\calE}_{q}^{s}\gamma_{ss^{\prime}}^{i}{\calE}_{r}^{s^{\prime}}+{\calE}_{r}^{s}\gamma_{ss^{\prime}}^{i}{\calE}_{q}^{s^{\prime}}={\frac{2}{D-2}}\delta_{qr}{\calE}_{s}^{t}\gamma_{tt^{\prime}}^{i}{\calE}_{s}^{t^{\prime}}
2{\frac{\partialA^{i}}{\partial\bar{z}}}=-\Omega_{+}F^{i}=\Omega_{+}K^{-1ij^{*}}\frac{\partial{\calW}^{*}}{\partialA^{*j}},\quad\Omega_{+}\equivi\frac{\langle-iZ_{1}^{*}+Z_{2}^{*}\rangle}{|\langle-iZ_{1}^{*}+Z_{2}^{*}\rangle|},
{\calA}_{4}^{1(t)}=\int_{0}^{\infty}dT\oint\frac{dz}{2\pii}\frac{dz}{d\rho}\langle\tilde{{\calO}}_{2}(z_{1})\tilde{{\calO}}_{3}(z_{2}){\calO}_{4}(z_{3}){\calO}_{1}(z_{4})b(z)((\xi(z_{3})-\xi(z_{0}))Y(z_{0})\rangle.
{\calL}=\frac{1}{4}\sum_{i}\int\!\mathrm{d}^{2}\theta\mathrm{d}^{2}\bar{\theta}\,\bar{\Phi}_{i}\Phi_{i}+\left\{\frac{1}{2}\int\!\mathrm{d}^{2}\theta{\calW}(\Phi_{i})+\mathrm{H.c.}\right\}\,,
\begin{array}{l}{\mathrm{B}1)\qquad\qquad\mu_{1}=\mu_{2}=\mu_{3}=m\,,\nonumber}\\{\mathrm{B}2)\qquad\qquad\mu_{1}=\mu_{2}=\sqrt{\sigma}\,,\\\\mu_{3}=m\,,\nonumber}\\{\mathrm{B}3)\qquad\qquad\mu_{1}=\mu_{2}=m,\\\\mu_{3}=\sqrt{\sigma}\,,\nonumber}\\{\mathrm{B}4)\qquad\qquad\mu_{1}=\mu_{2}=\mu_{3}=\sqrt{\sigma}\,,}\\\end{array}
G_{l+\nu}^{(n)}(r^{\prime\prime},r^{\prime};E)=-\frac{2M}{\hbar^{2}}\,\left|{\mathcalV}(r^{\prime\prime}){\mathcalV}(r^{\prime})\right|^{-1/2}\,{\mathcalG}_{l+\nu}^{(n)}(r^{\prime\prime},r^{\prime};E)\;,
{\calL}=\frac{1}{16\pi\alpha^{\prime}}\sqrt{-g}\e^{\mathrm{~-\phi~}}\Biggl\{R+\nabla\phi\cdot\nabla\phi-{\frac{1}{3}}H^{2}-\frac{1}{4}\mathrm{Tr}F^{2}+\cdots\Biggr\}
\begin{array}{ccc}{{\calE}=\int_{M_{3}}<(H^{A}\otimesI_{E}-I_{E}\otimesK^{B})}&{\wedge}&{(D^{A}\Phi_{A}\otimesI_{E}-I_{E}\otimesD^{B}\Phi_{B})>}\\{-\int_{M_{3}}<(D^{A}\Phi_{A}\otimesI_{E})\wedge(H^{A}\otimesI_{E})>}&{+}&{(A\leftrightarrowB,\Phi_{A}\leftrightarrow\Phi_{B}).}\\\end{array}
k_{R}=\frac{1}{Nn}\sum_{\alpha=1}^{n}\sum_{\bfi}\frac{\partial^{2}V}{\partial(\varphi_{\bfi}^{\alpha})^{2}}=2Jd-m^{2}+\lambda\frac{n+2}{Nn}\sum_{\alpha=1}^{n}\sum_{\bfi}(\varphi_{\bfi}^{\alpha})^{2}~.
V_{4(0)}^{b}\sim\frac{i}{2\sqrt{2}}\biggl[(1-i)e^{iH_{3}}e^{iH_{9}}+(1+i)e^{iH_{3}}e^{-iH_{9}}+(1+i)e^{-iH_{3}}e^{iH_{9}}+(1-i)e^{-iH_{3}}e^{-iH_{9}}\biggr]
S={\frac{1}{\sqrt{M^{2}}}}\varepsilon_{\mu\rho\sigma}P^{\mu}N^{\rho\sigma}={\frac{1}{\sqrt{M^{2}}}}\varepsilon_{\mu\rho\sigma}n^{\mu}{}_{i}n^{\rho}{}_{j}X^{\sigma}{}^{\prime}\left(\tilde{D}L^{i}\right)L^{j}\,.
\frac{1}{\mu}\langle\tilde{\psi}^{(0)}|\tilde{x}^{2}\tilde{p}^{3}|\tilde{\psi}^{(0)}\rangle=\frac{i}{\mu}\langle\tilde{\psi}^{(0)}|\tilde{x}\tilde{p}^{2}|\tilde{\psi}^{(0)}\rangle+\frac{2}{3}\langle\tilde{\psi}^{(0)}|\tilde{x}^{3}V^{\prime}\tilde{p}|\tilde{\psi}^{(0)}\rangle-\frac{i}{3}\langle\tilde{\psi}^{(0)}|\tilde{x}^{3}V^{\prime\prime}|\tilde{\psi}^{(0)}\rangle,\nonumber
A^{\prime}=\int_{\calV}d^{4}x\,L^{\prime}(\partial^{2}\phi,\partial\phi,\phi)=\int_{\calV}d^{4}x\,L(\partial\phi,\phi)-\int_{\calV}d^{4}x\\partial_{a}\left[\phi{\frac{\partialL}{\partial(\partial_{a}\phi)}}\right]\equivA-{\calS}
\Theta=\frac{\kappa}{2\pi}\int_{D}d^{2}x~\epsilon^{ij}\left<\partial_{i}g^{-1}\partial_{j}gg^{-1}dg\right>+\frac{\kappa}{2\pi}\oint_{\partial{D}}d\varphi\left<g^{-1}\partial_{\varphi}gg^{-1}dg\right>,
\Lambda/\Lambda_{0}=wA_{N-1}\frac{E(x^{8s}w/a_{N},x^{12s})E(x^{10s}w/a_{N},x^{12s})}{E(x^{4s}w/a_{N},x^{12s})E(x^{2s}w/a_{N},x^{12s})}.
\leq\;\left\langle\prod_{b=1}^{N}\left[\chi_{+}^{(b)}[Q^{W}]\right]^{q_{b}}\left[\chi_{-}^{(b)}[Q^{W}]\right]^{2n_{b}-q_{b}}\right\rangle_{Q^{W}}\;e^{4\pi\sum_{b=1}^{N}2n_{b}\lambda^{(b)}}\;\;.
\left.\Omega_{s}^{\mathrm{SUSY}}\right|{}_{p_{a}=mn_{a}}\equiv\Omega_{s}^{\mathrm{red}}=-m\mathrm{d}x^{a}\wedge\mathrm{d}n_{a}+\Omega_{{\calL}^{1|2}}\qquad\mathrm{d}n_{a}\equiv\frac{2\xi_{a}\mathrm{d}\bar{z}+2\bar{\xi}_{a}\mathrm{d}z}{(1-z\bar{z})^{2}}\,,
I=\frac{1}{2\kappa^{2}}\intd^{D}x\sqrt{-g}(R-\frac{1}{2}(\nabla\phi)^{2}-\frac{e^{-\alpha\phi}}{2(d_{1}+1)!}F_{d_{1}+1}^{2}-\frac{e^{-\beta\phi}}{2(d_{2}+1)!}F_{d_{2}+1}^{2}),
{\calG}_{\mathrm{rel}}=\left[I_{1}(a)da^{2}+I_{2}(a)(\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2})+\sum_{s,t=4}^{4N-1}\tilde{I}_{3}^{st}(a)\sigma_{s}\sigma_{t}\right],
\begin{array}{l}{2X_{1}=J_{12}-J_{34}\quad,\quad2X_{2}=J_{13}-J_{24}\quad,\quad2X_{3}=J_{14}+J_{23}}\\{D=J_{13}+J_{24}\quad,\quadS=J_{12}-J_{14}+J_{23}+J_{34}}\\\end{array}
\Pi_{Li}^{a}({\bf{r}})=\sum_{{\bf{k}}}\frac{-ik_{i}}{\sqrt{k}}\,[a_{Q}^{a}({\bf{k}})e^{+i{\bf{k}}\cdot{\bf{r}}}-a_{Q}^{a\star}({\bf{k}})e^{-i{\bf{k}}\cdot{\bf{r}}}]\;.
\displaystyle\frac{4}{\pi^{2}}\displaystyle\int_{0^{+}}^{\infty}dk\displaystyle\frac{k^{2}}{|{\calJ}_{+}(k)|^{2}}\phi(k,r)\phi(k,r^{\prime})+\psi_{l}(r)\psi_{l}(r^{\prime})=\delta(r-r^{\prime}).
(\Psi_{q_{1}}\star\Psi_{q_{2}})[X^{(3)}]=\int\mathscr{D}X^{(1)}\mathscr{D}X^{(2)}\,K_{q_{1},q_{2}}(X^{(1)},X^{(2)},X^{(3)})\Psi_{q_{1}}[X^{(1)}]\Psi_{q_{2}}[X^{(2)}].
(b_{-{\frac{5}{2}}}b_{-2}b_{-1}b_{-{\frac{1}{2}}})(b_{-{\frac{7}{2}}}b_{-{\frac{3}{2}}})=b_{-{\frac{7}{2}}}b_{-{\frac{5}{2}}}b_{-2}b_{-{\frac{3}{2}}}b_{-1}b_{-{\frac{1}{2}}}
{V}^{(0)}={\frac{m}{2}}\left[p_{i}(t)p^{i}(t)+\omega^{2}q_{i}(t)q^{i}(t)\right]+eA_{0}(t,\vec{q})+2\theta\intd^{2}x\partial^{i}\Pi_{i}(t,\vec{x})A_{0}(t,\vec{x}).
=\left(-16iF^{ij}F^{jk}F^{ki}-16i\dot{K}^{i}F^{ij}\dot{K}^{j}-16iF^{ij}\dot{K}^{i}\dot{K}^{j}+16iF^{ij}F^{jk}F^{ki}\right)(s_{1})(s_{2})(s_{3})
F_{tm}^{-\,I}=iz\bar{X}^{I}\,\mathrm{e}^{g}\,{\frac{x^{m}}{|\vec{x}|^{2}}}\,,\qquadG_{tm\,I}^{-}=iz\bar{F}_{I}\,\mathrm{e}^{g}\,{\frac{x^{m}}{|\vec{x}|^{2}}}\,.
h_{1}(r)=\frac{i\pi}{4}\int_{r}^{\infty}dr^{\prime}\;H_{1}^{(2)}(\sqrt{4\omega_{0}^{2}-\kappa^{2}}\;r^{\prime})\chi^{\mathrm{ANO}}(r^{\prime})\tilde{f}_{u}(r^{\prime}),
S_{A}^{+}[\gamma](s,t)_{A}^{\quadC}(\sigma_{\hat{a}})_{C\bar{D}}(S_{A}^{+}[\gamma](s,t)^{\dagger})_{\quad\bar{B}}^{\bar{D}}=(\sigma_{\hat{b}})_{A\bar{B}}E_{A}[\gamma](s,t)_{\mathrm{~}\hat{a}}^{\hat{b}},
{[}m_{~\beta}^{\alpha},m_{~\delta}^{\gamma}{]}=\delta_{~\beta}^{\gamma}m_{~\delta}^{\alpha}-\delta_{~\delta}^{\alpha}m_{~\beta}^{\gamma}~,~{[}m_{~j}^{i},m_{~n}^{k}{]}=\delta_{~j}^{k}m_{~n}^{i}-\delta_{~n}^{i}m_{~j}^{k}~,
V_{n,n+1}=\frac{1}{2a^{2}}\sum_{k=1}^{\infty}\frac{\left(a_{n+1}^{\dagger}(k)-a_{n}^{\dagger}(k)\right)\Bigl(a_{n+1}(k)-a_{n}(k)\Bigr)}{k-\frac{1}{2}}{}.
(A_{i}+\partial_{i}\varphi)(\vec{x})\rightarrowiD_{i}(\vec{x})\equivi\delta_{i}(\vec{x})+\frac{1}{2\pik}\sum_{a}s_{a}\varepsilon_{ij}\frac{(x-x_{a})^{j}}{|\vec{x}-\vec{x}_{a}|^{2}},
\mathcal{M}\left(\gamma^{\mu}p_{\mu}+\gamma^{\perp}p_{\perp}+M\right)S(p,p_{\perp})+\frac{\gamma^{\mu}p_{\mu}+m}{2\pi}\int_{-\infty}^{\infty}dp_{\perp}^{\prime}\,S(p,p_{\perp}^{\prime})=1.
\delta[B_{\mu}\dot{y}^{\mu}+B_{\alpha}\dot{\phi}^{\alpha}-\lambda^{\alpha}\tilde{\chi}_{\alpha}-\tilde{H}]={\frac{d}{dt}}\{[B_{\mu}f^{\mu\nu}{\frac{\partial\tilde{\chi}_{\alpha}}{\partialy^{\nu}}}+B_{\beta}\omega^{\beta\rho}{\frac{\partial\tilde{\chi}_{\alpha}}{\partial\phi^{\rho}}}-\tilde{\chi}_{\alpha}]\epsilon^{\alpha}\}
\Big(R(\textstyle{\frac{6}{\mu}}H)\,,~R(M_{ij})\,,~R(M_{ab})\Big)=\Big(-\textstyle{\frac{1}{2}}\otimes1\,,~-i\textstyle{\frac{1}{2}}\sigma_{ij}\otimes1\,,~1\otimes-i\textstyle{\frac{1}{2}}\rho_{ab}\Big)\,,
\int_{{\calC}_{0}}\left(B_{(2)}\,+\,2\pi\alpha^{\prime}{\calF}\right)=-4\pi^{2}\alpha^{\prime}\,\left(\frac{1}{2}+\frac{1}{4\pi^{2}\alpha^{\prime}}\,\tilde{b}_{1}\right)+4\pi^{2}\alpha^{\prime}=4\pi^{2}\alpha^{\prime}\,\left(\frac{1}{2}-\frac{1}{4\pi^{2}\alpha^{\prime}}\,\tilde{b}_{1}\right)
\tilde{\varphi}_{mn}(\vec{k})=\frac{1}{\sqrt{m!n!}}e^{\frac{k_{c}k_{\bar{c}}}{2}}\left(i\frac{\partial}{\partialk_{c}}\right)^{m}\left(i\frac{\partial}{\partialk_{\bar{c}}}\right)^{n}\tilde{\chi}_{00}(\vec{k}),
\frac{g^{2}}2\intd^{4}xd^{4}x^{\prime}\left\{-\left[\phi(x)\right]D\left(x,x^{\prime}\right)\left\{\phi(x^{\prime})\right\}+i\left[\phi(x)\right]N\left(x,x^{\prime}\right)\left[\phi(x^{\prime})\right]\right\}
(A_{1}A_{2}\cdot\cdot\cdotA_{n})_{\sim}=A_{1}(x)A_{2}(x)\cdot\cdot\cdotA_{n}(x)-\frac{1}{2L}\int_{-L}^{L}dx^{-}A_{1}(x)A_{2}(x)\cdot\cdot\cdotA_{n}(x).
\tilde{\nabla}_{i}=i\sqrt{V}P_{i}+\frac{i}{2}V\sqrt{V}\varepsilon_{ijk}\Sigma_{j}^{*}B_{k}\,,\quad\tilde{\nabla}_{5}=\frac{i}{\sqrt{V}}P_{5}-\frac{i}{2}V\sqrt{V}\vec{\Sigma}^{*}\cdot\vec{B}\,,
ds^{2}~=~\alpha^{\prime}~\left(\frac{u^{2}}{R^{2}}d\vec{x}_{\vert\vert}^{2}~+~\frac{R^{2}}{u^{2}}du^{2}~+~R^{2}~d\Omega_{5}^{2}\right)~=~\alpha^{\prime}~G_{MN}dx^{M}dx^{N}~.
1=\intd^{2}{\bfr}\,|\Psi({\bfr})|^{2}=A_{l}^{2}\,2\pi\kappa^{-2}\,\left\{{\mathcalK}_{i\Theta}(\kappaa)+\left(\frac{\kappa}{\tilde{k}}\right)^{2}\,\left[\frac{K_{i\Theta}(\kappaa)}{J_{l}(\tilde{k}a)}\right]^{2}\,{\mathcalJ_{l}}(\tilde{k}a)\right\}\;,
\frac{1}{2}\intd^{3}x\,d^{3}y\,j_{\mu}^{F}(x)G(x-y)j_{\mu}^{F}(y)\leftrightarrow\frac{1}{2}\intd^{3}x\,d^{3}y\,\mathcal{F}_{\mu}(x)G(x-y)\mathcal{F}^{\mu}(y)\;.
\begin{array}{c|ccccccc}{\,}&{}&{-1}&{0}&{1}&{2}&{3}&{\,}\\\hline{\,}&{}&{}&{}&{\vdots}&{}&{\,}\\{3}&{}&{0}&{0}&{0}&{0}&{0}&{\,}\\{2}&{}&{1}&{1}&{1^{*}}&{1^{\dagger}}&{0}&{\,}\\{1}&{\cdots}&{2}&{2}&{2^{\dagger}}&{1^{*}}&{0}&{\cdots}\\{0}&{}&{2}&{2}&{2}&{1}&{0}&{\,}\\{-1}&{}&{2}&{2}&{2}&{1}&{0}&{\,}\\{\,}&{}&{}&{}&{\vdots}&{}&{\,}\\\end{array}
S=\intd^{4}x\sqrt{\midg\mid}\left({\frac{1}{2}}D_{\mu}\Phi^{\ast}D^{\mu}\Phi-{\frac{\lambda}{4}}(|\Phi|^{2}-v^{2})^{2}-{\frac{1}{12}}{\calR}|\Phi|^{2}-{\frac{1}{4}}{F}_{\mu\nu}{F}^{\mu\nu}+\frac{1}{16\piG}{\calR}\right).
{\frac{(abcd)_{1234}}{(abce)_{1234}}}={\frac{(abc)_{123}(abed)_{1234}}{(abe)_{123}(abce)_{1234}}}+{\frac{(ab)_{12}(aed)_{123}}{(ae)_{12}(abe)_{123}}}+{\frac{a_{1}(ed)_{12}}{e_{1}(ae)_{12}}}+{\frac{d_{1}}{e_{1}}}\,
=\frac{\hbar}{8\pi^{2}}\alpha^{4}\phi_{c}^{4}\sum_{N=1}^{m}\frac{K_{2}\left(2N\alphaa\phi_{c}\right)}{\left(N\alphaa\phi_{c}\right)^{2}}+\frac{\hbar}{8\pi^{2}}\alpha^{4}\phi_{c}^{4}\sum_{N=m+1}^{\infty}\frac{K_{2}\left(2N\alphaa\phi_{c}\right)}{\left(N\alphaa\phi_{c}\right)^{2}}.
\sum_{b=1}^{M}{\frac{v_{b}^{2}}{2}}e^{\alpha\phi+2sB+2\sum_{i=1}^{N}\Lambda_{ai}r_{i}F_{i}+2\chi_{b}}\lbrack\partial_{\alpha}\chi_{b}\partial_{\beta}\chi_{b}-{\frac{\delta_{\alpha\beta}}{2}}(\partial\chi_{b})^{2}\rbrack.
\left(\xi_{0}a_{0}^{\dagger}+a_{1}^{\dagger}+\sum_{j=2}^{r}\xi_{j}a_{j}^{\dagger}+\sum_{j=2}^{r}\xi_{-j}b_{j}^{\dagger}-({\frac{\xi_{0}^{2}}{2}}+\sum_{j=2}^{r}\xi_{j}\xi_{-j})b_{1}^{\dagger}\right)^{M}|0\rangle.
S_{1}=-(d-2)L\left.\frac{d}{ds}\zeta_{A}(s/2)\right|_{s=0},\,\\\zeta_{A}(s/2)=\sum_{n=-\infty}^{\infty}\int_{-\infty}^{+\infty}\frac{dk}{2\pi}\,(k^{2}+y_{+}^{2})^{-s/2}(k^{2}+y_{-}^{2})^{-s/2},
\begin{array}{rlll}{SU(r+1)}&{\;\;\;q=r+1-N_{f}/2}&{}&{}\\{SO(2r+1)}&{\;\;\;q=2r-1-N_{f}}&{\;\;a=0}&{\;\;b=2}\\{Sp(2r)}&{\;\;\;q=2r+2-N_{f}}&{\;\;a=2}&{\;\;b=0}\\{SO(2r)}&{\;\;\;q=2r-2-N_{f}}&{\;\;a=0}&{\;\;b=4}\\\end{array}
g_{q}(n)=\frac{||g_{nMN}||^{2}}{[n]_{q^{2}}!}=\sum_{k=0}^{\infty}(\frac{q^{n}}{1-q^{2}})^{2k}q^{2k^{2}}\frac{1}{[n+k]_{q^{2}}![k]_{q^{2}}!}
\hat{R}^{\epsilon}=\left[\begin{array}{cccc}{\lambda}&{0}&{0}&{-q}\\{0}&{q}&{0}&{0}\\{0}&{0}&{q}&{0}\\{-q^{-1}}&{0}&{0}&{0}\\\end{array}\right]\quad,\qquad(\hat{R}^{\epsilon})^{-1}=\left[\begin{array}{cccc}{0}&{0}&{0}&{-q}\\{0}&{q^{-1}}&{0}&{0}\\{0}&{0}&{q^{-1}}&{0}\\{-q^{-1}}&{0}&{0}&{-\lambda}\\\end{array}\right]\;,
Z_{(r_{1},s_{1},l_{1})|(r_{2},s_{2},l_{2})}=\sum_{r=1}^{m-1}\sum_{s=1}^{m+1}\sum_{l=0}^{2}n_{(r_{1},s_{1},l_{1}),(r_{2},s_{2},l_{2})}^{(r,s,l)}b_{rs}^{(l)}(q),
S=2\pi\,\frac{\sqrt{(y^{2}+x_{0}^{2}+x_{1}^{2})(y^{2}+x_{0}^{2}+x_{2}^{2})(y^{2}+x_{0}^{2}+x_{3}^{2})}}{y^{2}+x_{0}^{2}}-2\pii\frac{x_{0}x_{1}x_{2}x_{3}}{y(y^{2}+x_{0}^{2})}\,.
=\;\Big\langleJ_{\mu_{1}}^{(1)}(x_{1})J_{\mu_{2}}^{(1)}(x_{2}).....J_{\mu_{n}}^{(1)}(x_{n})\Big\rangle\;\;\Big\langleJ_{\nu_{1}}^{(I_{1})}(y_{1})J_{\nu_{2}}^{(I_{2})}(y_{2}).....J_{\nu_{k}}^{(I_{k})}(y_{k})\Big\rangle_{0}^{\theta}\;.
l(s)=\int_{0}^{1}dyG(\Phi_{s}(y),\bar{\Phi}_{s}(y))\frac{d\Phi_{s}}{dy}e^{-i\delta}\left(\left.\frac{d\bar{W}}{d\bar{\Phi}}\right|_{\Phi=\Phi_{s}(y)}\right)^{-1}\,.
\bar{Q}_{~\mu\nu}^{\sigma}(\bar{\Gamma})=\bar{Q}_{~\mu\nu}^{a}(e,\bar{\Omega})e_{a}^{~\sigma}=-\frac{1}{2}\left(\partial_{\mu}e_{~\nu}^{a}-\partial_{\nu}e_{~\mu}^{a}+\bar{\Omega}_{~b\mu}^{a}e_{~\nu}^{b}-\bar{\Omega}_{~b\nu}^{a}e_{~\mu}^{b}\right)e_{a}^{~\sigma}
p_{(a)i}=\frac{\epsilon}{2\pi\alpha^{\prime}}\left[g_{ij}\dot{x}_{(a)}^{j}+2\pi\alpha^{\prime}B_{ij}\frac{1}{\epsilon}\left(x_{(a+1)}^{j}-x_{(a)}^{j}\right)-2\pi\alpha^{\prime}\frac{1}{\epsilon}A_{i}\delta_{a,0}\right].
\int_{\Lambda}\left[\prod_{i=1}^{n}d^{2}x_{i}d^{2}y_{i}\right]\prod_{j=1}^{2n}\Bigg|\frac{1}{w_{j}-z_{\pi(j)}}\Bigg|^{\alpha}\;\leq\;\bigg[\Xi\Big(\alpha\Big)\bigg]^{2n}\;,
\vartheta_{\sigma}^{(n)}(w|\zeta)=\prod_{j=1}^{2n}z_{j}^{-n}\prod_{a\inA}\frac{\Theta_{x^{4}}(w_{a}^{2})}{1-xrw_{a}}\prod_{a,b\inA\atopa<b}\frac{w_{b}\Theta_{x^{2}}(w_{a}/w_{b})\Theta_{x^{2}}(w_{a}w_{b})}{x^{-1}-xw_{a}w_{b}}.
V_{i}(k_{i},\epsilon_{i};z_{i},\bar{z}_{i})=(\epsilon_{i}\cdot\partialX(z_{i})+ik_{i}\cdot\psi(z_{i})\epsilon_{i}\cdot\psi(z_{i}))\,\mathrm{e}^{ik_{i}\cdotX(z_{i},\bar{z}_{i})}.
H^{c}=\frac{1}{2}\left[p_{a}^{\prime\prime}p_{a}^{\prime\prime}+\partial_{a}^{\prime\prime}W^{\prime\prime}(x^{\prime\prime})\partial_{a}^{\prime\prime}W^{\prime\prime}(x^{\prime\prime})+\hbar\partial_{a}^{\prime\prime}\partial_{b}^{\prime\prime}W^{\prime\prime}(x^{\prime\prime})\chi_{ab}^{(j)}\right]
{g}_{MN}=\left(\begin{matrix}{e^{\varphi}g_{\mu\nu}+\sum_{i,j}\xi_{ij}A_{\mu}^{i}A_{\nu}^{j}}&{\sum_{i}A_{\mu}^{i}\xi_{ij}}\\{\sum_{j}A_{\nu}^{j}\xi_{ij}}&{\xi_{ij}}\\\end{matrix}\right)
\left(\begin{matrix}{N_{1}-N_{2}}&{N-N_{1}+{\frac{1}{2}}N_{2}}&{N-N_{1}+{\frac{1}{2}}N_{2}}\\{N-N_{1}+{\frac{1}{2}}N_{2}}&{N_{1}}&{N_{1}-N}\\{N-N_{1}+{\frac{1}{2}}N_{2}}&{N_{1}-N}&{N_{1}-2N}\\\end{matrix}\right)\;.
\mathcal{G}_{\mu\nu}^{\mathrm{CS}}=-\frac1{4\pi}\varepsilon_{\mu\nu\rho}\partial^{\rho}\frac1{\left|x-y\right|}\;=\frac1{4\pi}\varepsilon_{\mu\nu\rho}\frac{(x-y)^{\rho}}{\left|x-y\right|^{3}}\;,
\partial_{z_{i}}W(z_{1},z_{1}^{\prime})\cdot\cdot\cdotW(z_{n},z_{n}^{\prime})=\frac{-1}{k+g}\sum_{j\neqi}^{n}\frac{\sum_{a}t_{i}^{a}\otimest_{j}^{a}}{z_{i}-z_{j}}W(z_{1},z_{1}^{\prime})\cdot\cdot\cdotW(z_{n},z_{n}^{\prime})
(P_{q})_{a}{}^{c}(P_{q})_{c}{}^{b}=\int\mathrm{d}^{6}\bar{z}^{\prime\prime}\,{\mathrm{\small{~\frac12~}}}{\calD}_{z}^{2}\,\delta^{8}(z-z^{\prime\prime})\,{\mathrm{\small{~\frac12~}}}{\calD}_{z^{\prime\prime}}^{2}\,\delta^{8}(z^{\prime\prime}-z^{\prime})={\mathrm{\small{~\frac12~}}}{\calD}_{z}^{2}\,\delta^{8}(z-z^{\prime})=(P_{q})_{a}{}^{b},
\mathrm{Vol}=i\int_{T^{6}}\Omega_{0}\wedge\bar{\Omega}_{0}=i\left[\mathrm{det}\,\tau-\mathrm{det}\,\bar{\tau}+\mathrm{tr}(\tau\,\mathrm{Cof}\,\bar{\tau})-\mathrm{tr}(\bar{\tau}\,\mathrm{Cof}\,\tau)\right]\,,
{\calM}=\mathrm{diag}(-{\textstyle{\frac{r}{r-m}}},\overbrace{1,...,1}^{6},-{\textstyle{\frac{r-m}{r}}},\overbrace{1,...,1}^{22},-{\textstyle{\frac{r}{r-m}}},-{\textstyle{\frac{r-m}{r}}}),
H_{gonimetric}^{2d}=-k\sum_{\vec{r},\vec{\alpha}}\sigma_{\vec{r}}\sigma_{\vec{r}+\vec{\alpha}}+\frac{k}{2}\sum_{\vec{r},\vec{\alpha},\vec{\beta}}\sigma_{\vec{r}}\sigma_{\vec{r}+\vec{\alpha}+\vec{\beta}}-\frac{1-k}{2}\sum_{\vec{r},\vec{\alpha},\vec{\beta}}\sigma_{\vec{r}}\sigma_{\vec{r}+\vec{\alpha}}\sigma_{\vec{r}+\vec{\alpha}+\vec{\beta}}\sigma_{\vec{r}+\vec{\beta}}
\Phi=\left(\begin{array}{c}{\Theta}\\{\chi}\\\end{array}\right)=\left(\begin{array}{c}{\Theta}\\{\frac{iE\sigma^{3}}{k_{z}-\lambda}\Theta}\\\end{array}\right),\Theta=\left(\begin{array}{c}{\Theta_{1}}\\{\Theta_{2}}\\\end{array}\right)