harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\psi_{\alpha\beta(\Gamma)}(\{\theta^{a}\}):=\sum_{i=0}\frac{1}{\mathrm{Trace(I)}}\;(\gamma_{A_{i}})_{\alpha\beta}(\gamma^{\overline{{A}}_{i}})_{\gamma\delta}\psi_{\delta\gamma(\Gamma)}(\{\theta^{a}\}),\,
R_{12}=R_{j_{1}j_{2}}^{i_{1}i_{2}}=\delta_{j_{1}}^{i_{1}}\delta_{j_{2}}^{i_{2}}\left(1+(q-1)\delta^{i_{1}i_{2}}\right)+(q-q^{-1})\delta_{j_{2}}^{i_{1}}\,\delta_{j_{1}}^{i_{2}}\,\theta(i_{1}-i_{2})
V_{1}|_{\phi=0}=\frac{\partialV_{1}}{\partial\phi}|_{\phi=0}=\frac{\partial^{2}V_{1}}{\partial\phi^{2}}|_{\phi=0}=\frac{\partial^{3}V_{1}}{\partial\phi^{3}}|_{\phi=0}=\frac{\partial^{4}V_{1}}{\partial\phi^{4}}|_{\phi=0}=0
S=\intd^{2}xdud^{2}\theta_{+}^{+}\;\left[i{\calL}^{+Y}\partial_{++}X_{Y}^{+}+iP_{++}^{-Y}(D^{++}X_{Y}^{+}+{\calL}_{Y}^{+3})+{\frac{1}{2}}\Lambda_{+}^{i}(\delta^{ij}D^{++}+(V^{++})^{ij})\Lambda_{+}^{j}\right]\;.
\left[\begin{array}{c}{g_{\beta}}\\{\gamma_{\beta}}\\\end{array}\right]=\left[\begin{array}{cc}{1}&{k_{\beta\alpha}^{-1}\phi_{\beta\alpha}}\\{0}&{k_{\beta\alpha}^{-1}}\\\end{array}\right]\left[\begin{array}{c}{g_{\alpha}}\\{\gamma_{\alpha}}\\\end{array}\right],
\mathrm{Area}_{H}=\int_{\rho=\rho_{+}}\,\sqrt{g_{\theta\theta}\,g_{\phi\phi}}\,d\theta\,d\phi\,=\,4\pi\,\rho_{+}^{2}=\,4\pi\,\left(m+\sqrt{m^{2}-|q|^{2}}\right)^{2}
\overline{{\mathrm{\Large~v~}}}_{4}(x)={\frac{4\pi^{2}}{g^{2}\Lambda^{2}}}x-\left({\frac{x^{2}}{2}}-{\frac{{\calE}^{2}}{3}}\right)E_{i}(-x)+{\frac{1}{2}}(1-x){\mathrme}^{-x}-{\frac{1}{2}}\,
\sqrt{-h}e^{-2{\bar{\varphi}}}h^{\bar{\mu}\bar{\mu}^{\prime}}h^{\bar{\nu}\bar{\nu}^{\prime}}(\bar{M}\bar{L})_{\bar{i}\bar{j}}\bar{{\calF}}_{\bar{\mu}^{\prime}\bar{\nu}^{\prime}}^{\bar{j}}=\epsilon^{\bar{\mu}\bar{\nu}\bar{\rho}}\partial_{\bar{\rho}}\psi^{\bar{i}}.
\overline{{s}}_{2m-2}=-(-1)^{m}\frac{1}{(2m-2)!}{k_{\betaj_{1}\ldotsj_{m-2}}}^{\alpha}{C_{i_{1}i_{2}}}^{j_{1}}\ldots{C_{i_{2m-5}i_{2m-4}}}^{j_{m-2}}\pi^{\beta}\pi^{i_{1}}\ldots\pi^{i_{2m-4}}X_{\alpha}\,\,,
e^{\overline{{\phi}}/d}\left(e^{-\overline{{\phi}}/d}\right)^{\prime\prime}=-\frac{1}{d^{2}}\phi^{2}-\frac{1}{4d}\left\{\mathrm{Tr}\left[\left(G^{-1}G^{\prime}\right)_{t}^{2}\right]-\mathrm{Tr}\left[\left(G^{-1}B^{\prime}\right)^{2}\right]\right\},
\left|\frac{1}{u_{s}-v_{s}}+\frac{\mu_{s}\nu_{s}+\hat{o}_{s}(L)}{(u_{s}-v_{s})^{2}}\right|^{2}\left[I_{1}^{(0)}I_{n-1}+I_{1}I_{n-1}^{(0)}+(u_{s}-v_{s})B+(\bar{u}_{s}-\bar{v}_{s})\bar{B}\right]
f^{ab^{\prime}c^{\prime}}f^{ade}f^{c^{\prime}cg}f^{bb^{\prime}g}=-i(T^{a})_{b^{\prime}c^{\prime}}(T^{c})_{c^{\prime}g}(T^{b})_{gb^{\prime}}f^{ade}=-\frac{N}{2}f^{abc}f^{ade}.
e^{H}\int_{D_{\eta}}\frac{1}{|z-z^{\prime}|}e^{v(z^{\prime})}d^{2}z^{\prime}+\int_{D_{\eta}}\frac{1}{|z-z^{\prime}|}|\beta(z^{\prime})|e^{\phi_{B}(z^{\prime})}d^{2}z^{\prime}
\frac{\partial^{2}\Psi}{{\partialz^{\alpha}}{\partial{\bar{z}}^{\alpha}}}+\frac{2\mu}{\hbar^{2}}\left[\frac{E}{4}-\frac{\mu\omega^{2}z{\bar{z}}}{8}\right]\Psi=0\quad\Leftrightarrow\quad{\calH}_{osc}\Psi=E\Psi.
\theta_{3}\Big(0,{\mathrm{e}}^{-\frac{\beta^{2}}{4s}n^{2}}\Big)-1=\frac{\sqrt{4\pis}}{\beta}\left(\sum_{k=-\infty}^{\infty}{\mathrme}^{-\frac{4\pi^{2}s}{\beta^{2}}k^{2}}-\int_{-\infty}^{\infty}{\mathrmd}\kappa\,\,{\mathrme}^{-\frac{4\pi^{2}s}{\beta^{2}}\kappa^{2}}\right).
G_{pq}^{mn}(a_{\mathrm{list}},b_{\mathrm{list}},x)=\frac{1}{2\pii}\int_{L}ds\,\frac{\prod_{j=1}^{m}\Gamma({b_{j}}-s)\prod_{j=1}^{n}\Gamma(1-{a_{j}}+s)}{\prod_{j=m+1}^{q}\Gamma(1-{b_{j}}+s)\prod_{j=n+1}^{p}\Gamma({a_{j}}-s)}{x^{s}}.
\intd^{4}xf^{ilk}A_{\mu}^{l}(x)\langlej^{\mu,k}(x)\rangle=-if^{ilk}\intd^{4}xd^{4}yA_{\mu}^{l}(x)A_{\nu}^{j}(y)\langle0|T[j_{A^{0}}^{\mu,k}(x)j_{A^{0}}^{\nu,j}(y)]|0\rangle=0
\int_{M}a_{6}=\frac{1}{360}\int_{M}Tr\,\left(11F^{\mu\nu;\rho}F_{\mu\nu;\rho}+2F_{\mu\nu}^{\quad;\nu}F_{\quad;\rho}^{\mu\rho}+48F_{\mu\nu}F^{\nu\rho}F_{\,\,\rho}^{\mu}\right)\sqrt{g}d^{4}x.
\hat{H}_{G}=-\frac{\hbar^{2}}{2}\left(\partial_{M}+\Gamma_{M}\right)\eta^{MN}\left(\partial_{M}+\Gamma_{M}\right)=-\frac{\hbar^{2}}{2}\frac{1}{\sqrt{\eta}}\partial_{M}\left(\sqrt{\eta}\,\eta^{MN}\,\partial_{N}\right)\,,
{\calS}_{\sigma}=\frac{1}{4\pi\alpha^{\prime}}\int_{\Sigma}d^{2}\sigma\sqrt{\gamma}\gamma^{\alpha\beta}\partial_{\alpha}X^{M}\partial_{\beta}X_{M}+\frac{1}{4\pi}\int_{\Sigma}d^{2}\sigma\sqrt{\gamma}\Phi(X)R^{(2)}(\sigma,\tau)
\frac{\calL}{N}=\frac{1}{2N}(\partial_{\mu}\phi_{i})^{2}-\frac{1}{2}\mu^{2}\rho-\frac{\lambda_{0}}{4}\rho^{2}-\frac{\eta}{6}\rho^{3}+\frac{\chi}{2}(\rho-\frac{\vec{\phi}\cdot\vec{\phi}}{N}).
P^{\prime}(w^{\prime})=L^{\betak}\;\sum_{\left[n_{i^{\prime}}\right]_{i^{\prime}=0}^{k}}\;\sum_{\left[X_{i}\right]_{i=0}^{n}}\int\prod_{i=0}^{n}dt_{i}\;\prod_{j^{\prime}=0}^{k}\;\delta(\sum_{{i}=m_{j^{\prime}-1}+1}^{m_{j^{\prime}}}\;\;t_{i}-L^{\beta}t_{j^{\prime}}^{\prime})\times
\delta_{\epsilon}\Psi_{M}=\left(\partial_{M}+\frac{1}{4}\omega_{~bM}^{a}\hat{\Gamma}_{a}^{~b}+\frac{1}{144}\Gamma_{M}^{~~NPQR}F_{NPQR}-\frac{1}{18}\Gamma^{PQR}F_{MPQR}\right)\epsilon
\begin{array}{rcl}{F_{\sigma}^{(n)}(\zeta_{1},\cdots,\zeta_{2n-2},\zeta_{2n-1},-x^{-1}\zeta_{2n-1})^{k\cdotsk_{2n-2}k\pm1k}}&{=}&{\delta_{k_{2n-2},k}\frac{e^{-\frac{\pi\sqrt{-1}}{2r}(1\pm2k)}}{\sqrt{-1}\{k\}}F_{\sigma}^{(n-1)}(\zeta_{1},\cdots,\zeta_{2n-2})^{k\cdotsk}}\\\end{array}
H^{B}=\frac{p_{1}^{2}}{2p^{+}}+2\muN_{0}+\sum_{n\neq0}\omega_{n}^{1}N_{n}^{1}+\sum_{n\neq0}\omega_{n}^{2}N_{n}^{2}+\sum_{I=3}^{8}\sum_{n=-\infty}^{\infty}\omega_{n}N_{n}^{I},
L_{R}=\frac{e^{3Ht}}{L^{3}}\sum_{i}\sum_{n}\frac{1}{2}\left[{\dot{\eta}}_{i}({\bfk}_{n},t){\dot{\eta}}_{i}(-{\bfk}_{n},t)-(e^{-2Ht}{\bfk}_{n}^{2}+\mu_{i}^{2})\eta_{i}({\bfk}_{n},t)\eta_{i}(-{\bfk}_{n},t)\right],
\delta{\tilde{e}}_{\mu}^{\\underline{\nu}}=-\bar{\tilde{\epsilon}}_{+i}\tilde{\Gamma}^{\underline{\nu}}\tilde{\psi}_{\mu+}^{i}-2\kappa^{D-2}W\bar{\tilde{\epsilon}}_{+i}\tilde{\Gamma}^{\underline{{\nu\lambda}}}\tilde{\eta}_{-}^{i}\tilde{e}_{\mu\underline{\lambda}}+c.c.\.
e^{-1}{\calL}=-{\frac{1}{2}}R-{\frac{3}{8}}\lambda^{2}F_{\mu\nu}F^{\mu\nu}+{\frac{e^{-1}\lambda^{3}}{8}}\epsilon^{\mu\nu\rho\sigma\lambda}F_{\mu\nu}F_{\rho\sigma}A_{\lambda}\.
g={\frac{-2}{f\kappa^{\frac{2}{3}}}}\left(\begin{matrix}{f^{2}+\epsilon^{2}-f\kappa^{2}\psi^{2}}&{-\epsilon}&{-{\frac{1}{2\sqrt2}}(\epsilon\chi+f\kappa^{2}\psi)}\\{-\epsilon}&{1}&{{\frac{1}{2\sqrt2}}\chi}\\{-{\frac{1}{2\sqrt2}}(\epsilon\chi+f\kappa^{2}\psi)}&{{\frac{1}{2\sqrt2}}\chi}&{{\frac{1}{8}}(\chi^{2}-\kappa^{2}f)}\\\end{matrix}\right).
t\,=\,\frac{e^{-2ikL+2a\muL}}{\Omega}\left(\frac{(\mu+\nu)^{2}-\lambda^{2}}{4\mu\nu}\right)\frac{\Gamma^{2}(-\mu-\nu-\lambda)\Gamma^{2}(-\mu-\nu+\lambda)}{\Gamma^{2}(-2\mu)\Gamma^{2}(-2\nu)}
V={\frac{1}{2}}m^{2}\sigma^{2}+{\frac{\lambda}{4}}\sigma^{4}+{\frac{a}{(4\pi)^{2}}}\int_{1/\Lambda^{2}}^{\infty}\frac{ds}{s^{2}}e^{-sM_{F}^{2}}\,\mathrm{coth}(eHs)\,eH\;,
\mathcal{D}=\left(\begin{array}{cc}{A}&{BX}\\{YC}&{-YPX}\\\end{array}\right)=\left(\begin{array}{cc}{1}&{0}\\{0}&{-Y}\\\end{array}\right)\left(\begin{array}{cc}{A}&{B}\\{-C}&{P}\\\end{array}\right)\left(\begin{array}{cc}{1}&{0}\\{0}&{X}\\\end{array}\right)\.
Y_{a}^{o}\hat{t}_{abc}(\phi_{b}^{o}+v^{o}\delta_{bH})c_{c}^{o}\longrightarrowY_{a}^{o}\hat{t}_{abc}(\phi_{b}^{o}-x^{o}\hat{\phi}_{b}^{o}+v^{o}\delta_{bH})c_{c}^{o}-x^{o}Y_{a}^{o}\tilde{I}_{ab}q_{b}^{o}
\partial_{z_{i}^{\prime}}W(z_{1},z_{1}^{\prime})\cdot\cdot\cdotW(z_{n},z_{n}^{\prime})=\frac{-1}{k+g}\sum_{j\neqi}^{n}W(z_{1},z_{1}^{\prime})\cdot\cdot\cdotW(z_{n},z_{n}^{\prime})\frac{\sum_{a}t_{i}^{a}\otimest_{j}^{a}}{z_{j}^{\prime}-z_{i}^{\prime}}
\begin{aligned}{[m]X^{3}+X^{0}}&{{}=U^{1}}\\{X^{3}-X^{0}}&{{}=\Omega^{2}V^{1}}\\{X^{1}}&{{}=\OmegaY^{1}}\\\end{aligned}\qquad\qquad\begin{aligned}{[m]X^{4}+X^{2}}&{{}=U^{2}}\\{X^{4}-X^{2}}&{{}=\Omega^{2}V^{2}}\\{X^{5}}&{{}=\OmegaY^{2}}\\\end{aligned}~,
\frac{d}{d\Omega}I(\omega)\sim\frac{e^{2}}{3\pi^{2}c}\left(\frac{\omega}{\omega_{0}}\right)^{2}\gamma^{-4}\left[K_{2/3}^{2}(\xi)[1+\lambda(1+6\gamma^{2})]+\lambda\frac{24\gamma^{5}\omega_{0}}{\omega}K_{1/3}(\xi)K_{2/3}(\xi)\right]\;,
\begin{array}{rcl}{\nu}&{=}&{\nu_{o}+\frac{\displaystyle\left(E_{q-\mathrm{rot}}(J+2)-E_{q-\mathrm{rot}}(J)\right)}{\displaystylehc}~,}\\{}&{=}&{\nu_{0}+B\left([J+3]_{q}[J+2]_{q}-[J+1]_{q}[J]_{q}\right)~,}\\\end{array}
(*_{\Omega}F^{(0,2)})_{\bar{i}\bar{j}}=-\frac{1}{2}\bar{\Omega}_{\bar{i}\bar{j}\bar{k}\bar{l}}F^{(2,0)\bar{k}\bar{l}},\quad(*_{\Omega}F^{(2,0)})_{ij}=-\frac{1}{2}\Omega_{ijkl}F^{(0,2)kl}.
\sum_{\{p_{v_{q}}\}}^{\{n_{v_{q}}\}}\tilde{\calL}_{\{n_{q}\},\{m_{q}\}}^{\{a_{i_{l}j_{l^{\prime}}},a_{i_{l}l^{\prime}},a_{l^{\prime}j_{l^{\prime}}},a_{ll^{\prime}}\},\{n_{v_{q}}-p_{v_{q}}\}}(z_{q})G_{N\{(p_{v_{q}})\}}^{\{\alpha_{q},\alpha_{v_{q}}\}}(z_{q})=0~~~for~~~\{n_{q}\neq0\}
\zeta(s)=2\left(\frac{\tau_{2}}{2\pi}\right)^{2s}\sum_{Z^{2}}\left[\left(n_{1}-gh_{1}+\frac{1}{2}-\tau_{1}\left(n_{2}-gh_{2}+\frac{1}{2}\right)\right)^{2}+\tau_{2}^{2}\left(n_{2}-gh_{2}+\frac{1}{2}\right)^{2}\right]^{-s},
\widehat{K}_{ij}=\left(\begin{matrix}{Z-{\frac{{C^{\prime}}^{2}}{C}}}&{C^{\prime}/2}&{0}&{q^{\prime}\overline{{\psi}}}\\{C^{\prime}/2}&{0}&{0}&{(q/4)\overline{{\psi}}}\\{0}&{0}&{-{\frac{C}{2}}P^{\mu\nu,\alpha\beta}}&{0}\\{0}&{0}&{0}&{q\hat{1}}\\\end{matrix}\right)
{\bft}_{\mu\nu}=\left(\begin{array}{cccc}{1}&{1}&{1}&{\omega^{k_{1}}}\\{1}&{1}&{1}&{\omega^{k_{2}}}\\{1}&{1}&{1}&{\omega^{k_{a}}}\\{\omega^{-k_{1}}}&{\omega^{-k_{2}}}&{\omega^{-k_{3}}}&{1}\\\end{array}\right).
\left[{\frac{dn_{s}(\omega,\epsilon)}{d\omega}}\right]_{\mathrm{div}}={\frac{\calA}{4\pi^{2}\kappa^{3}}}{\frac{\omega^{2}}{\epsilon^{2}}}{}~~~,~~~\left[{\frac{dn_{d}(\omega,\epsilon)}{d\omega}}\right]_{\mathrm{div}}=r_{d}{\frac{\calA}{4\pi^{2}\kappa^{3}}}\left[{\frac{\omega^{2}}{\epsilon^{2}}}+{\frac{\kappa^{2}}{4\epsilon^{2}}}\right]~~~
\int_{0}^{\infty}dr~r^{\frac{n}{2}}~(c^{2}+r^{2})^{m}~J_{\frac{n-2}{2}}(r|p|)~=~(2/|p|)^{m+1}~\frac{c^{\frac{n}{2}+m}}{\Gamma(-m)}~K_{\frac{n}{2}+m}(c|p|)~~~,
\midj,m\\rangle_{j_{1},j_{2}}=\sum_{m_{1},m_{2}}\midj_{1}m_{1}\\rangle\otimes\midj_{2}m_{2}\\rangle\left|\begin{array}{ccc}{j}&{j_{2}}&{j_{1}}\\{m}&{m_{2}}&{m_{1}}\\\end{array}\right|_{q}.
{\calA}=V_{p+1}\,(4\pi^{2}\alpha^{\prime})^{4-p}\int_{0}^{\infty}\!dt\int\frac{d^{9-p}q}{(2\pi)^{9-p}}\,e^{i\vec{q}\cdot(\vec{x}-\vec{y})}\,e^{-\pit\alpha^{\prime}\vec{q}^{2}}(8-8)\prod_{n=1}^{\infty}\frac{(1-e^{-2\pitn})^{8}}{(1-e^{-2\pitn})^{8}}
G_{\mu\nu}(\tau,\tau^{\prime})=\eta_{\mu\nu}\,G(\tau,\tau^{\prime}),\qquad\mathrm{and}\quadG(\tau,\tau^{\prime})=\langle\tau|2\left(\frac{d^{2}}{d\tau^{2}}\right)^{-1}|\tau^{\prime}\rangle.
A(1,2,3,4)=e^{E_{3}^{(4)}}\left(\tilde{D}_{43,-p_{4}}^{\alpha}\tilde{\delta}_{V}(4,3)\right)\left(\tilde{D}_{\alpha,23,-p_{2}}\tilde{\bar{D}}_{\dot{\alpha},23,-p_{2}}\tilde{\delta}_{V}(2,3)\right)\left(\tilde{\bar{D}}_{13,-p_{1}}^{\dot{\alpha}}\tilde{\delta}_{V}(1,3)\right),
\phi_{\mu\nu}(n)=2\bar{\xi}_{n}\xi_{n}(\varepsilon_{\mu}^{n}p_{n,\nu}-\varepsilon_{\nu}^{n}p_{n,\mu})+\frac{4}{T}\bar{\xi}_{n+1}\xi_{n+1}\bar{\xi}_{n}\xi_{n}(\varepsilon_{\mu}^{n+1}\varepsilon_{\nu}^{n}-\varepsilon_{\nu}^{n+1}\varepsilon_{\mu}^{n})\delta(u_{n+1}-u_{n})
i\,(\partial/\partialu)\,[({\mit\Psi})^{\prime},({\mit\Psi}^{\dagger})^{\prime\prime}]=[({\mit\Omega\,\Psi})^{\prime},({\mit\Psi}^{\dagger})^{\prime\prime}]+[({\Psi})^{\prime},(\tilde{\mit\Omega}\,{\mit\Psi}^{\dagger})^{\prime\prime}]=0,
{h_{a}^{(1)}}^{2}=\sigma^{(1)}={\frac{1}{t^{(1)}d_{1}+{\frac{\alpha^{2}}{4}}}},~~~{h_{a}^{(2)}}^{2}=\sigma^{(2)}={\frac{1}{t^{(2)}d_{2}+{\frac{\beta^{2}}{4}}}}
\begin{array}{lll}{S_{a}^{b}\psi^{a}\psi_{b}(z)}&{=}&{-\frac{1}{2\check{g}}(S_{a}^{b}+\bar{S}_{a}^{b})(S_{b}^{a}+\bar{S}_{b}^{a})(z)}\\{\}&{\}&{\}\\{\}&{\}&{+\frac{1}{2\check{g}}S_{a}^{b}S_{b}^{a}(z)-\frac{\check{g}}{2}(\psi^{a}\partial\psi_{a}+\psi_{a}\partial\psi^{a})(z)}\\\end{array}
2\intd^{2}x\,\partial_{+}X^{\mu}\cdot\partial_{-}X^{\nu}\cdot\partial_{[\mu}B_{\nu]}^{\prime}(X)=\intd^{2}x\,\left[\partial_{+}\left(B_{\mu}^{\prime}\partial_{-}X^{\mu}\right)-\partial_{-}\left(B_{\mu}^{\prime}\partial_{+}X^{\mu}\right)\right]\.
{\frac{cdt}{dx}}={\frac{\hat{c}d\hat{t}}{d\hat{x}}},\\\\\{\frac{\hbar}{dEdt}}={\frac{\hat{\hbar}}{d\hat{E}d\hat{t}}},\\\\\{\frac{G_{4}dE}{c^{4}dx}}={\frac{\hat{G}_{4}d\hat{E}}{\hat{c}^{4}d\hat{x}}},\\\\\{\frac{e^{2}}{dEdx}}={\frac{\hat{e}^{2}}{d\hat{E}d\hat{x}}},
\Xi=\left(\begin{array}{c}{\xi_{\alpha}}\\{\bar{\xi}^{\dot{\alpha}}}\\\end{array}\right)\mathrm{~};\mathrm{~}\Lambda=\left(\begin{array}{c}{\lambda_{\alpha}}\\{\bar{\lambda}^{\dot{\alpha}}}\\\end{array}\right)\mathrm{~};\mathrm{~}\Psi=\left(\begin{array}{c}{\psi_{\alpha}}\\{\bar{\psi}^{\dot{\alpha}}}\\\end{array}\right)\mathrm{~},
{\bfS}={\frac{\bfA}{4G_{N}}}=\pi\bigg[{\frac{1}{2}}{\calL}_{ac}{\calL}_{bd}({\vec{v}}^{a\,T}L{\vec{v}}^{b})({\vec{v}}^{c\,T}L{\vec{v}}^{d})\bigg]^{\frac{1}{2}}\.
\pi_{A}^{a\,\prime}\,=\,\phi_{A}^{\ast\,a}\,\,,\,\,\,\,\phi_{A}^{\prime}\,=\,{\bar{\phi}}_{A}\,\,\,\,,\,\,\,\,\phi_{A}^{\ast\,a\,\prime}\,\,=\,\,\,\,\pi_{A}^{\,a}\,\,\,,\,\,\,\,{\bar{\phi}}^{A\,\prime}\,=\,\phi^{A}\,\,,
*(\omega_{i_{1}}\wedge\omega_{i_{2}}\wedge...\wedge\omega_{i_{q}})=\frac{1}{(m-q)!}\delta\left(\begin{array}{cccccc}{1}&{2}&{...}&{...}&{...}&{m}\\{i_{1}}&{...}&{i_{q},}&{j_{1}}&{...}&{j_{m-q}}\\\end{array}\right)\omega_{j_{1}}\wedge\omega_{j_{2}}\wedge...\wedge\omega_{j_{m-q}},
H^{a}=G_{12}^{a}=\frac{1}{2}\varepsilon^{ij}G_{ij}^{a}=-\hat{\phi}^{a}(\frac{\partial\Psi_{1}}{\partialr}+\frac{1}{r}\Psi_{1})=\hat{\phi}^{a}\frac{am}{M}K_{0}(Mr).
S=\frac{1}{16\pi}\intd^{4}\!x\,(-g^{\mathrm{IV}})^{1/2}\left[R^{\mathrm{IV}}-2\Lambda-F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}\sum_{i=1}^{N}(\nabla^{\mathrm{IV}}f_{i})^{2}\right],
g^{\prime}\,=\,e^{f}\left(-\frac{\bar{g}}{6\sqrt{2}}\left(2e^{-\varphi}\,+\,e^{2\varphi}\right)\,+\,\frac{2}{3}\,\frac{\sqrt{2}}{\bar{g}}\,e^{-2g}\,e^{-2\varphi}\right),
\gamma_{\alpha\beta}^{s}\gamma_{\alpha^{\prime}\beta^{\prime}}^{st}+\gamma_{\alpha^{\prime}\beta}^{s}\gamma_{\alpha\beta^{\prime}}^{st}+\gamma_{\alpha\beta^{\prime}}^{s}\gamma_{\alpha^{\prime}\beta}^{st}+\gamma_{\alpha^{\prime}\beta^{\prime}}^{s}\gamma_{\alpha\beta}^{st}=2(\delta_{\alpha\alpha^{\prime}}\gamma_{\beta\beta^{\prime}}^{t}-\delta_{\beta\beta^{\prime}}\gamma_{\alpha\alpha^{\prime}}^{t})
{\calL}_{c}=\bar{\Lambda}^{1/3}\big[\sum_{p_{1}}{(\frac{\Lambda}{\bar{\Lambda}})}^{p_{1}}\rho_{p_{1}}e^{p_{1}\sigma}\partial_{m}\sigma\partial^{m}\sigma+\sum_{p_{2}}{(\frac{\Lambda}{\bar{\Lambda}})}^{p_{2}}\rho_{p_{2}}e^{p_{2}\sigma}\Box\sigma\big]
S_{\mathrm{boundary}}=-M_{0}\intd\tau\,\mathrm{e}^{-\frac12\tilde{\varphi}^{\mathrm{B}}-\frac12\sum_{a}\tilde{\eta}_{a}^{\mathrm{B}}}\sqrt{-G_{00}^{\mathrm{B}}}~+~M_{0}\intd\tau\,\tilde{A}_{0}^{I}~.
\zeta+a_{0}\frac{1}{\zeta}x_{8}^{8-2n}z^{4-n}\to\zeta+a_{0}\frac{1}{\zeta}z\prod_{i=1}^{N_{s}}(x_{8}-m_{i}^{2})\prod_{j=1}^{N_{f}}(z-m_{j}^{2}),
\mathcal{L}=-\frac12\left[\overline{{\Psi}}(x)\left(\Gamma_{\mu}\overrightarrow{D_{\mu}^{(-)}}+m\right)\Psi(x)-\overline{{\Psi}}(x)\left(\Gamma_{\mu}\overleftarrow{D_{\mu}^{(+)}}-m\right)\Psi(x)\right]-\frac14\mathcal{F}_{\mu\nu},
\langleU|S_{\mu}^{\\nu}|U\rangle\to\frac{\alpha^{2}-\beta^{2}}{2r^{2}(24M)^{2}}\left(\begin{matrix}{-1}&{-1}&{0}&{0}\\{1}&{1}&{0}&{0}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\\end{matrix}\right)\,\r\to\infty\.
\begin{array}{c}{\omega_{T}(A)=\omega_{P}^{double}(A)}\\{P=\left(\begin{array}{cc}{T}&{T^{\frac{1}{2}}(1-T)^{\frac{1}{2}}}\\{T^{\frac{1}{2}}(1-T)^{\frac{1}{2}}}&{1-T}\\\end{array}\right)}\\\end{array}
\left(\begin{array}{cc}{\partial_{r}+{\frac{a+j+1/2}{r}}}&{\sqrt{2}evf}\\{\sqrt{2}evf}&{\partial_{r}-{\frac{j-1/2}{r}}}\\\end{array}\right)\left(\begin{array}{c}{h_{1}(r)}\\{h_{4}(r)}\\\end{array}\right)=0.
\tilde{\Gamma}_{IJ}^{i}=\Gamma_{IJ}^{i}-(T^{i}+\bar{T^{i}})G_{IJ}-{\frac{2}{3}}(T^{i}+\bar{T^{i}})(T^{k}+\bar{T^{k}})K_{Tkj}\Gamma_{IJ}^{j}
A_{ijk}\mapsto(i\gammaT+\delta)^{3}\zeta_{W}^{-1}\frac{\sum_{mnp}(Ay)_{mnp}C_{mi}^{\dagger}C_{nj}^{\dagger}C_{pk}^{\dagger}}{\sum_{m^{\prime}n^{\prime}p^{\prime}}y_{m^{\prime}n^{\prime}p^{\prime}}C_{m^{\prime}i}^{\dagger}C_{n^{\prime}j}^{\dagger}C_{p^{\prime}k}^{\dagger}}.
A_{\Sigma}=\delta\left(\sum_{r=1}^{N}p_{r}^{+}\right)\delta\left(\sum_{r=1}^{N}p_{r}^{-}\right)\int\prod_{I=2}^{N-2}d^{2}(\rho_{I}-\rho_{1})\;W^{\scriptscriptstyle\mathrm{T}}W^{\scriptscriptstyle\mathrm{B}}W^{\scriptscriptstyle\mathrm{L}}
\frac{K}{2}\int_{R}\sqrt{-g}g^{rr}\phi(-k,r)\partial_{r}\phi(k,r)\frac{d^{4}k}{(2\pi)^{4}}-\frac{K}{2}\int_{L}\sqrt{-g}g^{rr}\phi(-k,r)\partial_{r}\phi(k,r)\frac{d^{4}k}{(2\pi)^{4}}\,
\widetilde{Z}(R(S),\langle\cdot\,,\cdot\rangle_{H},\langle\cdot\,,\cdot\rangle)=\widetilde{Z}(R(S),\langle\cdot\,,\cdot\rangle){\cdot}\Psi(R(S),\langle\cdot\,,\cdot\rangle_{H},\langle\cdot\,,\cdot\rangle)
S_{D,p}=\int_{{\calM}^{p+1}}^{}{\calL}_{p+1}=\int_{{\calM}_{p+1}}^{}\left(-{\frac{(-1)^{p}}{p!}}(E^{a}e^{a_{1}}...e^{a_{p}}-{\frac{p}{(p+1)}}e^{a}e^{a_{1}}...e^{a_{p}})\varepsilon_{aa_{1}...a_{p}}+{\calL}_{p+1}^{WZ}\right)
i\partial_{\mu}^{x}[T_{1/1}^{\mu}(x),T_{1}(y)]+i\partial_{\mu}^{y}[T_{1}(x),T_{1/1}^{\mu}(y)]\stackrel{def}{=}i\partial_{\mu}^{x}D_{2/1}^{\mu}(x,y)+i\partial_{\mu}^{y}D_{2/2}^{\mu}(x,y).
\begin{array}{c}{e^{-B}Ae^{B}=\sum_{n=0}^{\infty}\displaystyle\frac1{n!}[A,B]_{(n)},}\\{{}[A,B]_{(0)}\equivA,\qquad[A,B]_{(n+1)}=[[A,B]_{(n)},B].}\\\end{array}
\Gamma_{R(\tau)}^{(N,L)}(\rhok_{i(\tau)},\rhop_{i(\tau)},u_{\tau}^{*},\kappa_{\tau})=\rho^{\tau[N+(d-\frac{m}{2})-\frac{N(d-\frac{m}{2})}{2}-2L]-\frac{N\gamma_{\phi(\tau)}^{*}}{2}+L\gamma_{\phi^{2}(\tau)}^{*}}\Gamma_{R(\tau)}^{(N,L)}(k_{i(\tau)},p_{i(\tau)},u_{\tau}^{*},\kappa_{\tau}).
y^{2}-y(B_{N_{c}}(v^{2})+\Lambda_{N=2}^{2N_{c}+2-N_{f}}\frac{2\prod_{j=1}^{f}im_{j}}{v^{2}})+\Lambda_{N=2}^{4N_{c}+4-2N_{f}}v^{-4}\prod_{j=1}^{f}(v^{2}-m_{j}^{2})=0.
\int_{0}^{\infty}d\nu\int_{m}^{\infty}dk\[k^{2}-m]^{1/2-s}\frac{\partial}{\partialk}\frac{t^{j}}{\nu^{n}}=\frac{(mR)^{2-n}}2\frac{\Gamma(\frac32-s)\Gamma(1+\frac{j-n}2)\Gamma(s+\frac{n-3}2)}{\Gamma(\fracj2)}.
\zeta_{nG-n+1,G}^{n,0}=\sum_{R}\left({\frac{n!}{d_{R}^{2}}}\right)^{G}(n!)^{G-1}\left({\frac{d_{R}}{\sqrt{n!}}}\right)^{2}=\sum_{R}({\frac{n!}{d_{R}}})^{2G-2}.
E_{0}^{slab}\=\\sum_{n}\frac12\omega_{n}^{slab}\=\\frac{\hbarcL^{2}}{4\pi^{2}}\int_{0}^{\infty}dk_{1}dk_{2}\sum_{n}^{\infty}\sqrt{k_{1}^{2}+k_{2}^{2}+(2\pin/d)^{2}}\.
\frac{\partial}{\partial\tau}\Sigma_{\mu\nu}(\tau,\tau^{\prime})-2e_{0}{\tilde{g}}\int\delta(\tau-\lambda)F_{\mu}^{\alpha}\Sigma_{\alpha\nu}(\lambda,\tau^{\prime})\;d\lambda=\eta_{\mu\nu}\frac{\partial}{\partial\tau}\delta(\tau-\tau^{\prime}),
S=-\frac{T}{2}\intd^{3}\sigma\left[\sqrt{-\gamma}\left(\gamma^{\alpha\beta}\Pi_{\alpha}^{a}\Pi_{\beta}^{b}\eta_{ab}-1\right)+\epsilon^{\alpha\beta\gamma}\Pi_{\alpha}^{A}\Pi_{\beta}^{B}\Pi_{\gamma}^{C}B_{ABC}\right]
S[X]=\intd\tau\int_{0}^{\pi}d\sigma{\frac{1}{2\pi}}\left\{\eta^{\alpha\beta}\partial_{\alpha}X^{I}(\tau,\sigma)\partial_{\beta}X^{I}(\tau,\sigma)+B^{IJ}\varepsilon^{\alpha\beta}\partial_{\alpha}X^{I}(\tau,\sigma)\partial_{\beta}X^{J}(\tau,\sigma)\right\}\,
\left.+\left(b_{1}\left(\kappa_{s}+\kappa_{t}\right)+b_{2}\kappa_{s}+\frac{\Lambda}{3}b_{3}\right)\dot{\phi}^{2}+\frac{\Lambda}{3}b_{4}\ddot{\phi}+c_{1}\dot{\phi}^{4}+c_{2}\dot{\phi}^{2}\ddot{\phi}+c_{3}\ddot{\phi}^{2}\right]+(\mathrm{s.t.})\;,
\begin{array}{l}{AB=qBA,\;\;\;BD=qDB\,,}\\{DC=qCD,\;\;\;CA=qAC\,,}\\{\!\!\phantom{a}[B,C]=\displaystyle{\frac{A^{2}-D^{2}}{q-q^{-1}}}\,,}\\{\!\!\phantom{a}[A,D]=0\,.}\\\end{array}
\sum_{k=1}^{2m}\frac{1}{1-\omega^{2k+1}}=\sum_{k=1}^{m}\frac{1}{1-\omega^{2k+1}}+\frac{1}{1-\omega^{2m+1}}+\sum_{k=m+2}^{2m+1}\frac{1}{1-\omega^{2k+1}}.
\Omega_{\gamma}=tr\{dP\,i\,u^{-1}\,du-\frac{1}{2\gamma}(e^{i\gammaP}u^{-1}\,du\,e^{-i\gammaP}\,u^{-1}\,du+L_{+}^{-1}dL_{+}L_{-}^{-1}dL_{-})\}.
\gamma_{\xi}^{(\beta)}(X)\simeq\frac{\pi^{2}}{45}b_{\xi}(X){\calT}_{\xi}^{4}(X)-\frac{1}{12}c_{\xi}(X){\calM}_{\xi}^{2}(X){\calT}_{\xi}^{2}(X)+\frac{1}{6\pi}{\calM}_{\xi}^{3}(X)\,{\calT}^{(\xi)}(X).
\begin{array}{c}{\displaystyleZ_{N}(+\infty)=N\pi+\pi\displaystyle\frac{p-1}{p+1}S+2\pi\,\mathrm{sign}(p-1)\,M_{W\downarrow}+2\omega}\\{Z_{N}(-\infty)=-N\pi-\pi\displaystyle\frac{p-1}{p+1}S-2\pi\,\mathrm{sign}(p-1)\,M_{W\uparrow}+2\omega}\\\end{array}
\theta_{\mu\nu\lambda}^{++\,\rho\sigma\delta}=6\left(\frac{\partial_{\mu}a\partial^{\rho}a}{\left(\partial_{\chi}a\partial^{\chi}a\right)}\delta_{\nu}^{\sigma}\delta_{\lambda}^{\delta}\right)^{++}=6\left(N_{\mu\nu\lambda}^{\quad\rho\sigma\delta}\right)^{++}
\zeta^{a}=e^{-\mathrm{i}\beta\sigma_{3}f(y)}\hat{\zeta}^{a}\,\quad\Psi^{A}=e^{-\mathrm{i}\beta\sigma_{3}f(y)}\hat{\Psi}^{A}\,\quad\xi=e^{\mathrm{i}2\betaf(y)}\hat{\xi}
{\bfe}\Bigl[\frac{(\beta^{\{p\}}-1)(1-\alpha)}{8}-\frac{p}{8n}\left(\alpha^{2}-2\alpha\right)-\frac{(\tilde{\beta}^{\{p\}}-1)(1-\tilde{\alpha})}{8}+\frac{p}{8n}\left(\tilde{\alpha}^{2}-2\tilde{\alpha}\right)\Bigr].
\left(\begin{array}{cc}{-\varrhom^{2}\qquad\qquadm^{2}+2E\left(\vec{J}\cdot\vec{p}\right)}&{+2\left(\vec{J}\cdot\vec{p}\right)^{2}}\\{m^{2}-2E\left(\vec{J}\cdot\vec{p}\right)+2\left(\vec{J}\cdot\vec{p}\right)^{2}}&{-\varrhom^{2}}\\\end{array}\right)\left(\begin{array}{c}{\Phi_{R}\left(\vec{p}\right)}\\{\Phi_{L}\left(\vec{p}\right)}\\\end{array}\right)=0.
F_{A_{/1}}=0\Longleftrightarrowq^{a_{i/1}\left[i\right]}=\phi^{a_{i/1}}\left(q^{b_{i/1}\left[i-1\right]},q^{b_{i/2}\left[i-1\right]},q^{b_{i/2}\left[i\right]}\right)\,.
i\partial_{\tau}\psi=\mathrm{K}\psi=\left[\frac{\mathrm{p}^{2}}{2m}+\frac{1}{2\lambda}(\pi^{\mu}+y^{T}S^{\mu}\mathrm{p})(\pi_{\mu}+y^{T}S_{\mu}\mathrm{p})+V\right]\psi,
[\delta_{1},\delta_{2}]\phi^{I}=2i\overline{{\epsilon}}_{1}^{i}\gamma^{\mu}\epsilon_{2i}\partial_{\mu}\phi^{I}+a\frac{3}{2}\overline{{\epsilon}}_{2}^{i}\left(\sigma^{I}{\sigma^{J}}^{\dagger}M-M\sigma^{J}{\sigma^{I}}^{\dagger}\right)_{ij}\epsilon_{1j}\phi^{J}.
\Omega_{a\nu}(az,bz)=\frac{\bar{K}_{\nu}^{(b)}(bz)/\bar{K}_{\nu}^{(a)}(az)}{\bar{K}_{\nu}^{(a)}(az)\bar{I}_{\nu}^{(b)}(bz)-\bar{K}_{\nu}^{(b)}(bz)\bar{I}_{\nu}^{(a)}(az)},
\left\{\varphi,\psi\right\}\left(g,l_{t}\right)=\left\langle\mathrm{D}^{\prime}\psi,X_{\varphi}\right\rangle-\left\langle\mathrm{D}^{\prime}\varphi,X_{\psi}\right\rangle+\left\langlel_{t},\left[X_{\varphi},X_{\psi}\right]\right\rangle.
V\propto\sum_{i,j}tr\left([\phi_{i},\phi_{j}][\phi_{i},\phi_{j}]^{\dagger}\right)+m^{2}\sum_{i}|\phi_{i}|^{2}+m\sum_{ijk}\epsilon^{ijk}\left([\phi_{i},\phi_{j}]\phi_{k}^{\dagger}+[\phi_{i},\phi_{j}]^{\dagger}\phi_{k}\right)
\frac{\partial^{2}\phi}{\partialt^{2}}-\frac{\partial^{2}\phi}{\partialz^{2}}=-\frac{\partialV}{\partial\phi^{*}},\qquad\frac{\partial^{2}\chi}{\partialt^{2}}-\frac{\partial^{2}\chi}{\partialz^{2}}=-\frac{\partialV}{\partial\chi^{*}}.
\left\langle\begin{array}{cc}{r}\\{t}\\\end{array}\right\rangle_{n-1}\left\langle\begin{array}{cc}{r}\\{q}\\\end{array}\right\rangle_{n-1}=\left\langle\begin{array}{cc}{r}\\{q}\\\end{array}\right\rangle_{n-1}\left\langle\begin{array}{cc}{r}\\{t}\\\end{array}\right\rangle_{n-1}
{\calH}=\frac{1}{2}g^{tt}P_{t}^{2}+\frac{1}{2}g^{rr}P_{r}^{2}+\frac{1}{2}g^{\phi\phi}P_{\phi}^{2}+g^{t\phi}P_{t}P_{\phi}+\frac{1}{2}g_{\phi\phi},
\left(\begin{array}{c|c}{\frac{1}{\bar{D}^{2}}\bar{D}_{\alpha}\bar{D}_{\sigma}}&{-i\frac{1}{bar{D}^{2}}\bar{D}_{\alpha}}\\{i\bar{D}_{\sigma}}&{1}\\\end{array}\right)=\left(\begin{array}{c}{\frac{1}{\bar{D}^{2}}\bar{D}_{\alpha}}\\{i}\\\end{array}\right)\left(\bar{D}_{\sigma}\,\-i\right)
\Omega^{\alpha}\equiv\frac{1}{(2m-2)!}{k_{\betaj_{1}\dotsj_{m-2}}}^{\alpha}{C_{i_{1}i_{2}}}^{j_{1}}\ldots{C_{i_{2m-5}i_{2m-4}}}^{j_{m-2}}c^{\beta}c^{i_{1}}\ldotsc^{i_{2m-4}}\quad,
\begin{array}{c}{L=-\left(A_{x}+\frac\lambda2\left(B_{x}+B_{t}\right)+\frac1{2\lambda}\left(B_{x}-B_{t}\right)\right),}\\{T=-\left(A_{t}+\frac\lambda2\left(B_{x}+B_{t}\right)-\frac1{2\lambda}\left(B_{x}-B_{t}\right)\right).}\\\end{array}
{\frac{\varrho_{m}}{\varrho_{\delta\sigma}}}={\frac{2b_{0}c_{0}^{6}B}{\tilde{G}_{5}^{2}\sigma_{0}\delta\sigma}}a^{-3w+6{\sfn}-2{\sfq}-3}+{\frac{9{\sfn}kc_{0}^{8}b_{0}}{16\pi^{2}\tilde{G}_{5}^{2}\sigma_{0}\delta\sigma(3w+2{\sfn}+1)}}a^{8{\sfn}-2{\sfq}+1}+{\frac{6{\sfn}-2{\sfq}}{3w-6{\sfn}+2{\sfq}+3}}.
{\bfJ}^{k}={\frac{1}{2}}\epsilon^{ijk}J^{ij}=\epsilon^{ijk}\intd^{3}{\bfx}\left[F^{0i}(\partial_{\mu}F^{\muj})+F_{\mu}^{\quadj}(\partial^{0}F^{\mui}+\partial^{\mu}F^{i0}+\partial^{i}F^{0\mu})\right]\quad.
{\calL}^{(k)}=\frac{f_{1}^{\mu\nu}}{2}(\partial_{\mu}\rho_{j}\partial_{\nu}\rho_{j})+\frac{f_{2}^{\mu\nu}}{\rho^{2}}(\rho_{j}\partial_{\mu}\rho_{j})(\rho_{i}\partial_{\nu}\rho_{i})