harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\{\tilde{\Theta}_{\alpha}^{\mu}(q,\phi),\tilde{\Theta}_{\beta}^{\nu}(q,\phi)\}=0~~;~~\tilde{\Theta}_{\alpha}^{\mu}(q,\phi)=\Theta_{\alpha}^{\mu}(q)+\Sigma_{n=1}^{\infty}\tilde{\Theta}_{\alpha}^{\mu(n)}(q,\phi)~~;~~\tilde{\Theta}^{\mu(n)}\approxO(\phi^{n})
\tilde{F}(z)=\left(\begin{array}{cc}{\epsilon}&{0}\\{0}&{\zeta}\\\end{array}\right)F(z)^{t}\left(\begin{array}{cc}{\bar{\epsilon}}&{0}\\{0}&{\bar{\zeta}}\\\end{array}\right)=\left(\begin{array}{cc}{i\mathrm{x}_{+}}&{2\tilde{\theta}_{b}}\\{2\tilde{\bar{\theta}}{}^{a}}&{\delta^{a}{}_{b}}\\\end{array}\right)\,.
L^{\psi}V={\binom{L_{0}\L_{1}}{L_{1}\L_{0}}}^{\psi}{\binom{V_{0}}{V_{1}}}={\binom{\psi_{00}L_{0}\\psi_{11}L_{1}}{\psi_{10}L_{1}\\psi_{01}L_{0}}}{\binom{V_{0}}{V_{1}}}.
S_{\mathrm{Dirac}}=\mathrm{\frac{i}{2\pi}}\intd^{2}\!\sigma\,\,{\bar{\Psi}}\gamma^{a}\partial_{a}\Psi=\mathrm{\frac{i}{\pi}}\intd^{2}\!\sigma\,\,{\bar{\Psi}}_{L}{\bar{\partial}}\Psi_{L}-\mathrm{\frac{i}{\pi}}\intd^{2}\!\sigma\,\,{\bar{\Psi}}_{R}{\partial}\Psi_{R}\,
\deltaS=\frac{1}{4\pi}\,\mathrm{Im}\,\,\left\{6\int\!\!d^{8}z_{+}\\sigma({\calF}-\frac{1}{2}{\calA}{\calA}_{D})-\frac{i}{48}\int\!\!d^{12}z\(H-{\bar{H}}){\bar{\calA}}{\calA}_{D}\right\}.
F\left({\calA},G,p,t_{n,l}^{(A)}\right)=\sum_{n=-\infty}^{+\infty}F^{(\mathrm{cov)}}\left(G,p,r_{n,l}^{(A)}\right)\prod_{A=1}^{p}\prod_{l=1}^{\infty}\left[t_{n,l}^{(A)}\right]^{r_{n,l}^{(A)}}\mathrm{e}^{-|r|_{n}{\calA}v(n)}~.
\frac{i(\thetap)^{i}k\thetau}{2\pi}\mpi(\thetau)^{i}=\frac{i(\thetap)^{i}k\thetau-i(\thetau)^{i}(k\thetap)}{2\pi}+\frac{(\thetau)^{i}}{2\pi}i(k\thetap\mp2\pi),
\frac{A^{2n+2}}{V^{2n+1}}=\frac{4\pi(n+1)}{\lambda}\left(\frac{n+1}{2^{n}}\right)^{2n+1}\mathrm{Vol(M)}\,\,\tilde{\beta^{2}}^{n+1}(\rho^{2}-1)^{n}\,\left(\frac{(\rho+1)^{n}}{(\rho-1)\,_{2}F_{1}[1+n,-n,n+2,\frac{1-\rho}{2}]}\right)^{2n+1}.
M=\phi_{1}^{a}(k_{1})\phi_{2}^{b}(k_{2})V_{abm}(k_{1},k_{2})\,\frac{G^{mn}}{s}\,W_{mcd}(k_{3},k_{4})\phi_{3}^{c}(k_{3})\phi_{4}^{d}(k_{4})+(stu)\mathrm{perm.}.
ds^{2}=\frac{4R^{2}}{(2n+2)^{2}}(1-(\frac{a}{R})^{2n+2})(d\tau+A)^{2}+\frac{dR^{2}}{(1-(\frac{a}{R})^{2n+2})}+\frac{R^{2}}{2(n+1)}ds^{2}.
R_{\gamma\alpha\beta}^{\mu}=\partial_{\alpha}\Gamma_{\beta\gamma}^{\mu}-\partial_{\beta}\Gamma_{\alpha\gamma}^{\mu}+\Gamma_{\alpha\lambda}^{\mu}\Gamma_{\beta\gamma}^{\lambda}-\Gamma_{\beta\lambda}^{\mu}\Gamma_{\alpha\gamma}^{\lambda}
M=\left(\begin{array}{cc}{-\delta^{mn}\frac{d}{dt}-i\partial^{m}g^{\astnl}\partial^{\astl}H}&{-i\partial^{\astm}g^{\astnl}\partial^{\astl}H}\\{-i\partial^{m}g^{nl}\partial^{l}H}&{\delta^{mn}\frac{d}{dt}-i\partial^{\astm}g^{nl}\partial^{l}H}\\\end{array}\right).
\dot{\tilde{S}}^{2}+\tilde{V}(\tilde{S})=0,\;\;\;\;\dot{t}=\frac{-E}{M-\frac{J^{2}}{4}\tilde{S}^{2}-(l\tilde{S})^{-2}},\;\;\;\;\dot{f}=\frac{E}{l(M-\frac{J^{2}}{4}\tilde{S}^{2})\tilde{S}^{2}-\frac{1}{l}},
L\left(\frac{q_{0}}{\sqrt{L}}\right)\left[\lambda\left(\frac{q_{0}}{\sqrt{L}}\right)^{2}-m^{2}\right]+\lambda\int_{-L/2}^{L/2}dx\left[3\left(\frac{q_{0}}{\sqrt{L}}\right)\varphi^{2}+\varphi^{3}\right]=0
2r^{2}\left(E_{0}^{-}-\frac{\Ime}{\left|e\right|}E_{1}^{-}\right)\left(E_{0}^{+}+\frac{\Ime}{\left|e\right|}E_{1}^{+}\right)+\left(\frac{\Ree}{|e|}\right)^{2}\left(-\left(E_{1}^{2}\right)^{2}-\delta_{ij}E_{1}^{i}E_{1}^{j}\right)\neq0
C^{(1)}=\frac{e^{2}\xi}2\int\frac{d^{d}k}{(2\pi)^{d}}\frac1{D_{N}\,(k^{2}+m_{gh}^{2})}\left\{(\varphi-2u)k^{2}-\xiu\,e^{2}\varphi^{2})\right\}.
[\:\frac{1}{4}\,(\frac{\ddot{z}^{+}}{\dot{z}^{+}}-\frac{\ddot{z}^{-}}{\dot{z}^{-}})\,l+\frac{1}{2}\,(\,\dot{z}^{+}\,\partial_{+}l-\dot{z}^{-}\,\partial_{-}l\,)\:]\,\sqrt{\frac{\dot{z}^{+}}{\dot{z}^{-}}}=F_{-}\,,
f_{0}=N_{0}/N=1-(1/2\pi^{2}\rho)\int_{0}^{\infty}\mathrm{~}dq\mathrm{~}q^{2}(\frac{-\omega_{{\bf{q}}}+\epsilon_{{\bf{q}}}+\rho_{0}v_{{\bf{q}}}}{2\mathrm{~}\omega_{{\bf{q}}}})
\begin{array}{lcr}{J^{i}=\partialX^{i}+(-)^{i+1}\{X^{i+(-)^{i+1}},X^{9}\},i=1,...,8}&{(a)}\\{J^{9}=\partialX^{9}+\sum_{\nu=1}^{8}{\frac{1}{2}}(-)^{\nu+1}\{X^{\nu},X^{\nu+(-)^{\nu+1}}\},}&{(b)}\\\end{array}
\zeta(\epsilon;b^{2},c^{2},w)=\int_{0}^{1}dx\,\,{\sum_{n_{1},n_{2}=-\infty}^{\infty}}\!\!\!\!\!^{\prime}\left[n_{1}^{2}+wn_{2}^{2}+c^{2}-b^{2}x(1-x)\right]^{-\epsilon},
\Gamma_{\mu}^{\mathrm{BC}}(p,q)=\frac{1}{2}\left[A(p^{2})+A(q^{2})\right]\gamma_{\mu}+\frac{(p+q)_{\mu}}{p^{2}-q^{2}}\left[A(p^{2})-A(q^{2})\right]\frac{\gamma\cdotp+\gamma\cdotq}{2},
K_{\mathrm{{BTZ}}}=i\sum_{n=1}^{\infty}(-1)^{n}\left[\left(d_{\phi,n,+}^{\dagger}d_{\phi,n,-}^{\dagger}-d_{\phi,n,+}d_{\phi,n,-}\right)-\left(d_{\tilde{\phi},n,+}^{\dagger}d_{\tilde{\phi},n,-}^{\dagger}-d_{\tilde{\phi},n,+}d_{\tilde{\phi},n,-}\right)\right]\\.
+\sum_{n}\pi_{n,1}\frac{\partial\overline{{H}}_{0}}{\partialq_{n,0}}+\sum_{n}\sum_{j_{n}=2}^{m_{n}-1}\pi_{n,j_{n}}\left[\frac{\partial\overline{{H}}_{0}}{\partialq_{n,j_{n}-1}}-\mu_{n,j_{n}-1}\right]\.
\left(\dot{y}\left[a\dot{x}+ib\dot{y}^{*}\right]_{<0}-\dot{x}\left[a\dot{y}-ib\dot{x}^{*}\right]\right)_{\leq0}=(\dot{y}\xi_{<0}^{x}-\dot{x}\xi_{<0}^{y})_{\leq0}
i\hbar\frac{\partial\Psi}{\partialt}=H[f,-i\hbar\delta/\deltaf,t]\Psi=-\hbar^{2}\left(\sum_{a,b}\intdx\,\frac{\delta}{\deltaf_{x}^{a}}K_{x}^{ab}[f]\frac{\delta}{\deltaf_{x}^{b}}\right)\Psi+V[f]\Psi,
\begin{array}{cccc}{\beta_{0}=}&{(0,1,0,0)}&{\gamma_{0}=(1,1,1,1)}\\{\beta_{1}=}&{(1,1,0,0)}&{\gamma_{1}=(0,1,1,1)}\\{\beta_{2}=}&{(0,1,1,0)}&{\gamma_{2}=(1,1,0,1)}\\{\beta_{3}=}&{(0,1,0,1)}&{\gamma_{3}=(1,1,1,0)}\\\end{array}
\begin{array}{ccc}{Q^{2}=H,}&{\{Q,\tilde{Q}\}=0,}&{\tilde{Q}^{2}=\frac{1}{2}\,\left(\vec{J}{}^{2}-e^{2}g^{2}+\frac{1}{4}\right),}\\{}&{}&{}\\{\left[Q,H\right]=0,}&{[\tilde{Q},H]=0,}&{[J_{i},H]=0,}\\{}&{}&{}\\{\left[Q,J_{i}\right]=0,}&{\left[\tilde{Q},J_{i}\right]=0,}&{[J_{i},J_{j}]=i\varepsilon_{ijk}J_{k}.}\\\end{array}
A_{\mu}=\left(\begin{matrix}{A_{\mu}^{1}}&{0}&{0}&{0}\\{0}&{A_{\mu}^{2}}&{0}&{0}\\{0}&{0}&{A_{\mu}^{3}}&{0}\\{0}&{0}&{0}&{A_{\mu}^{4}}\\\end{matrix}\right)
\theta(x,t)={\frac{mv}{\hbar}}\int_{-\infty}^{x-vt}(1-{\frac{\rho_{0}^{2}}{\rho^{2}}})-\frac{\lambda}{2}\int_{-\infty}^{x-vt}(\rho^{2}-{\rho_{0}^{2}})-\frac{\lambda}{2}\rho_{0}^{2}x+\theta_{0}
E_{\mu\mathrm{m}}={\frac{1}{2}}\biggl({\frac{\partialu^{\mu}}{\partialX^{\mathrm{m}}}}+{\frac{\partialu^{\mathrm{m}}}{\partialX^{\mu}}}+\sum_{\mathrm{n}=1}^{3}{\frac{\partialu^{\mathrm{n}}}{\partialX^{\mu}}}{\frac{\partialu^{\mathrm{n}}}{\partialX^{\mathrm{m}}}}\biggr)
\Phi^{A}=\Phi^{A}+\frac{\partial}{\partial\Phi_{A}^{*}}f(\Phi,\Phi^{*})\;\qquad\Phi_{A}^{*}=\Phi_{A}^{*}+\frac{\partial}{\partial\Phi^{A}}f(\Phi,\Phi^{*})\.
\bar{n}=\sum_{n=0}^{\infty}np_{2n}=p_{0}\lambda\frac{\partial}{\partial\lambda}\sum_{n=0}^{\infty}\lambda^{n}q_{2n}(M)\vert_{\lambda=1}=\lambda\frac{\partial}{\partial\lambda}2\,\mathrm{Im}\,W(\lambdaM)\vert_{\lambda=1}=\frac12\mathrm{tr}\,\frac{M}{1\mpM}.
P^{-}=\int_{-\infty}^{\infty}dx^{-}\mathrm{Tr}\biggl[m_{b}^{2}\phi_{+}\phi_{--}{\frac{g^{2}}{2}}{J}^{+}{\frac{1}{\partial_{-}^{2}}}{J}^{+}+{\frac{tg^{2}}{2}}\left[\phi_{+},\phi_{-}\right]^{2}+\sum_{h}{F}_{h}^{\dagger}{\frac{1}{\mathrm{i}\partial_{-}}}{F}_{h}^{\dagger}\biggr]
<\frac{1}{2},m_{s}|{\psi}_{+}^{(\frac{1}{2})}(g)>\equivD_{m_{s}+\frac{1}{2}}^{(\frac{1}{2})}(g)=<g,l|T_{m_{s}+}^{\frac{1}{2}}|g,l+\frac{1}{2}>,
H=\frac{1}{2}\intdx\,dy\left\{q(\vec{\nabla}p-\frac{i}{2}\,\frac{\Psi\vec{\nabla}\bar{\Psi}+\bar{\Psi}\vec{\nabla}\Psi}{q})^{2}+\frac{1}{q}(1+i(\Psi\bar{\partial}\Psi+\bar{\Psi}\partial\bar{\Psi}))\right\},
L_{s}^{m}=\;\frac{q^{2}}{1+q^{2}}\;e^{-1}\;(\dot{x}_{\mu}+q^{1/2}i\chi\psi_{\mu})^{2}+\;\frac{e}{1+q^{2}}\;m^{2}+\frac{i}{2}(\psi_{\mu}\;\dot{\psi}^{\mu}-\psi_{5}\dot{\psi}_{5})-i\chi\psi_{5}m.
g_{\mu\nu}F^{\mu}{\frac{dx^{\nu}}{d\tau}}=g_{\mu\nu}{\frac{Dp^{\mu}}{d\tau}}{\frac{dx^{\nu}}{d\tau}}=m_{0}g_{\mu\nu}{\frac{D}{d\tau}}\left({\frac{dx^{\mu}}{d\tau}}\right){\frac{dx^{\nu}}{d\tau}}+{\frac{dm_{0}}{d\tau}}g_{\mu\nu}{\frac{dx^{\mu}}{d\tau}}{\frac{dx^{\nu}}{d\tau}}=-{\frac{dm_{0}}{d\tau}}.
F_{8}^{{\mathcal{O}}_{1,1}^{1,0}|+++++---}=\frac{-iH^{1,1}((\sigma_{2}^{-})^{2}\sigma_{1}^{+}+\sigma_{5}^{+}+\sigma_{3}^{+}\sigma_{2}^{-})/\sigma_{5}^{+}}{2^{7}\prod_{1\leqi<j\leq5}\mathrm{cosh}(\theta_{ij}/2)\prod_{6\leqi<j\leq8}\mathrm{cosh}(\theta_{ij}/2)}\prod_{i<j}F_{\mathrm{min}}^{\mu_{i}\mu_{j}}(\theta_{ij}),
\nabla\left(\begin{matrix}{V_{AB}}\\{V_{I}}\\{\bar{V}^{AB}}\\{\bar{V}^{I}}\\\end{matrix}\right)=\left(\begin{matrix}{0}&{0}&{{\frac{1}{2}}P_{ABCD}}&{P_{ABJ}}\\{0}&{0}&{{\frac{1}{2}}P_{ICD}}&{P_{IJ}}\\{{\frac{1}{2}}P^{ABCD}}&{P^{ABJ}}&{0}&{0}\\{{\frac{1}{2}}P^{ICD}}&{P^{IJ}}&{0}&{0}\\\end{matrix}\right)\left(\begin{matrix}{V_{CD}}\\{V_{J}}\\{\bar{V}^{CD}}\\{\bar{V}^{J}}\\\end{matrix}\right)
\tilde{Q}=F(0)+\sum_{n=1}^{\infty}\left.\frac{1}{n!}\frac{d^{n}F}{d^{n}t}t^{n}\right|_{t=1}=Q+\sum_{n=1}^{\infty}\frac{(-1)^{n}\epsilon^{n}}{n!}[K,[K,\dots,[K,Q]]]\.
u_{+}(n,k;\rho,\phi)=\frac{1}{\sqrt{2\bar{\omega}_{k}(M+\bar{\omega}_{k})}}\,\left[\begin{array}{c}{ikJ_{n}(k\rho)e^{in\phi}}\\{(M+\bar{\omega}_{k})J_{n+1}(k\rho)e^{i(n+1)\phi}}\\\end{array}\right]
E[\sigma]=\sum_{\ell}N_{\ell}\left[\int_{0}^{\infty}\frac{dk}{2\pi}\sqrt{k^{2}+m^{2}}[\rho_{\ell}(k)]_{N}+\frac{1}{2}\sum_{j}\omega_{j}\right]+\sum_{i=1}^{N}E_{\mathrm{FD}}^{(i)}+E_{\mathrm{CT}}\,,
\int_{-1}^{+1}(1-x^{2})^{L+\frac12}C_{m}^{L+1}(x)C_{n}^{L+1}(x)\,dx=\delta_{m,n}\frac{\pi\,\Gamma(N+L+1)}{n!\,N\left[\Gamma(L+1)\right]^{2}2^{2L+1}}
\sum_{m,n\geq0}d^{m-\delta}z^{n-\delta}\langle\eta_{m}|U_{3}(\psi,w)|\eta_{n}\rangle=\sum_{m,n\geq0}d^{m-\delta}z^{n-\delta}\langleP(\eta_{m},\eta_{n};w^{\ast})|\psi\rangle\,.
\begin{array}{c}{\omega=d{\bfp}\wedged{\bfx}+\sum_{i}d{\bfp}_{i}\wedged{\bfe}_{i},}\\{\;\;}\\{{\calH}=s({\bfp}{\bfe}_{1}-c_{0})+\sum_{i}k_{i-1}\phi_{i-1.i}+{\frac{k_{3}}{2c}}({\bfp}_{3}^{2}-c^{2})+\sum_{i,j}d_{ij}({\bfe}_{i}{\bfe}_{j}-\delta_{ij}),}\\\end{array}
\nabla_{M}\nabla^{M}\phi+2\alpha_{1}\Lambdae^{\alpha_{1}\phi}+\frac{\alpha_{2}T_{3}}{2}\frac{1}{\sqrt{|g|}}\intd^{4}\xi\sqrt{|\gamma|}\delta^{5}(x-X)\gamma^{\alpha\beta}\partial_{\alpha}X^{A}\partial_{\beta}X^{B}g_{AB}e^{\alpha_{2}\phi}=0,
V(x)=\frac{N(N+2)}{12}\left(\frac34\frac{P_{4}^{\prime}{}^{2}}{P_{4}}-P_{4}^{\prime\prime}\right)+\frac{N+1}4\left(2P_{2}^{\prime}-P_{2}\frac{P_{4}^{\prime}}{P_{4}}\right)+\frac14\frac{P_{2}^{2}}{P_{4}}+const,
\deltaC(\theta)=\epsilon(\theta)C^{\prime}(\theta)\,;\;\delta(C,C)=\oint\frac{d\theta}{2\pi}\epsilon(\theta)2C_{\alpha}(\theta)C_{\alpha}^{\prime}(\theta)=-\oint\frac{d\theta}{2\pi}\epsilon^{\prime}(\theta)C_{\alpha}^{2}(\theta)
{\calL}_{0}\left(P_{\alpha}f\right)(\lambda)=\frac{1}{\Gamma\left(\alpha\right)}\int_{\gamma_{0}}\mathrm{e}^{\mathrm{i}\lambda\theta^{\prime}}f\left(\theta^{\prime}\right)d\theta^{\prime}\int_{\gamma_{0}\left(\theta^{\prime}\right)}\mathrm{e}^{\mathrm{i}\lambda\theta^{\prime\prime}}\left(\theta^{\prime\prime}\right)^{\alpha-1}d\theta^{\prime\prime},\tag{B.14}
D_{i}\,t_{\alpha}{}^{i}=0\,,\quadt_{[\alpha\beta]}=0\,,\quadt_{\alpha}{}^{i}=\Sigma_{\alpha}{}^{i}-\frac{1}{2}D_{j}\left(\tau_{\alpha}{}^{ij}-\tau^{ij}{}_{\alpha}+\tau^{j}{}_{\alpha}{}^{i}\right)\,.
\gamma(-\Box)=\frac{1}{12}\biggl(\int_{4m^{2}}^{\infty}d\mu^{2}\,\Bigl(1-\frac{4m^{2}}{\mu^{2}}\Bigr)^{3/2}\frac{1}{\mu^{2}-\Box}-\int_{0}^{\infty}d\mu^{2}\,\frac{1}{\mu^{2}+m^{2}}+\frac{8}{3}\biggr)
\left(\begin{array}{cc}{{\varphi}_{out}^{\prime}}\\{{{\varphi}_{out}^{\prime}}^{*}}\\\end{array}\right)=\left(\begin{array}{cc}{1+\frac{2i}{\pi\epsilon}}&{\frac{2i}{\pi\epsilon}}\\{-\frac{2i}{\pi\epsilon}}&{1-\frac{2i}{\pi\epsilon}}\\\end{array}\right)\left(\begin{array}{cc}{\varphi_{out}}\\{\varphi_{out}^{*}}\\\end{array}\right),
\nabla_{\nu}(\tilde{e})\tilde{e}_{a}^{\mu}=\partial_{\nu}\tilde{e}_{a}^{\mu}+\Gamma_{\sigma\nu}^{\mu}(\tilde{e})\tilde{e}_{a}^{\sigma}-\omega^{b}{}_{a\nu}(\tilde{e})\tilde{e}_{b}^{\mu}=0.
{W}_{\mathrm{ren}}^{\infty}=\mathrm{}-\frac12\frac{\partial}{\partial\epsilon}\left[\frac{\mu^{2\epsilon}}{\Gamma(\epsilon)}\int_{0}^{\infty}\!\frac{{\mathrm{d}}s}{s^{1-\epsilon}}\,{\mathrm{Tr}}{K}(s)\right]_{\epsilon=0},
\begin{array}{lclcl}{M5}&{:}&{\int\Omega_{7}}&{;\qquad}&{\Omega_{7}=dX_{0}\ldotsdX_{5}dH^{-1}\;}\\{D3}&{:}&{\int\Omega_{5}+\tilde{\Omega}_{5}}&{;\qquad}&{\Omega_{5}=dX_{0}\ldotsdX_{3}dH^{-1}\;}\\{BH}&{:}&{\intF}&{;\qquad}&{F=dX^{0}dH^{-1}\.}\\\end{array}
\begin{array}{l}{\,\,\,\,\,\,(-1)^{\varepsilon(f_{1})\varepsilon(f_{3})}d_{S}c_{2}(\xi|f_{1},f_{2},f_{3})=}\\{=2c_{1n}\sigma_{1}\biggl((-1)^{\varepsilon(f_{1})\varepsilon(f_{3})}f_{1}(\xi)[\overleftarrow\partial\!\!_{\alpha}]^{n}\varepsilon^{[\alpha]_{n}}f_{2}(\xi)(\overleftarrow\partial\!\!_{\beta}\lambda_{\beta}\partial_{\beta})^{3}f_{3}(\xi)+\mathrm{cycle}(1,2,3)\biggr).}\\\end{array}
M_{\pm}^{2}~=~{\frac{18}{(16)^{2}}}{\frac{\alpha}{\delta}}\mu^{2}~+~{\frac{81}{2}}(\alpha\gamma)^{2}\mu^{2}\Bigg[1~\pm~\sqrt{1~+~x}~\Bigg],~~~\mathrm{and}~~~x\equiv{\frac{1}{288}}{\frac{\alpha}{\delta}}{\frac{1}{(\alpha\gamma)^{2}}}.\nonumber
S_{g}=\intd^{3}x\left[-\frac{1}{4}F^{a\mu\nu}F_{\mu\nu}^{a}-\frac{1}{4}(G^{a\mu\nu}+f^{abc}F^{b\mu\nu}\rho^{c})(G_{\mu\nu}^{a}+f^{ab^{\prime}c^{\prime}}F_{\mu\nu}^{b^{\prime}}\rho^{c^{\prime}})+\frac{m}{2}\epsilon^{\mu\nu\rho}F_{\mu\nu}^{a}\phi_{\rho}^{a}\right],
\left(V,1,1\right)=\left(\begin{array}{ccc}{1+\frac{V^{2}}{2a}}&{-\frac{V^{2}}{2a}}&{-V^{T}}\\{\frac{V^{2}}{2a}}&{1-\frac{V^{2}}{2a}}&{-V^{T}}\\{-V}&{V}&{{\bf1}_{n-2}}\\\end{array}\right)\quad,
\delta\tilde{T}_{\mu\nu}dy^{\mu}dy^{\nu}=\intd^{3}{\bfk}\left[\tilde{t}_{00}Ydt^{2}+2\tilde{t}_{(L)0}V_{(L)i}dtdy^{i}+(\tilde{t}_{(LL)}T_{(LL)ij}+\tilde{t}_{(Y)}T_{(Y)ij})dy^{i}dy^{j}\right].
s_{m}\pi_{n}^{a}=-\delta_{mn}\lambda^{a}-\frac{1}{2}f_{bc}^{a}\eta_{m}^{c}\pi_{n}^{b}+\frac{1}{12}\varepsilon_{npr}f_{bc}^{a}f_{de}^{c}\eta_{m}^{e}\eta_{p}^{d}\eta_{r}^{b}\,,s_{m}p_{n}^{a}=-\delta_{mn}l^{a}\,,
\left(\begin{array}{c}{\partial_{\tau}X^{M}}\\{\partial_{\tau}P^{M}}\\\end{array}\right)=\left(\begin{array}{cc}{A_{12}}&{A_{22}}\\{-A_{11}}&{-A_{12}}\\\end{array}\right)\left(\begin{array}{c}{X^{M}}\\{P^{M}}\\\end{array}\right).
\frac{1}{\rho}\frac{d}{d\rho}\left(\rho\frac{d\psi_{6}}{d\rho}\right)-\frac{a_{-}^{2}}{\rho^{2}}\psi_{6}-g_{1}\eta^{2}\psi_{6}(\psi_{4}^{2}+\psi_{6}^{2}-1)+g_{2}\eta^{2}\psi_{6}(\psi_{4}^{2}-\psi_{6}^{2}-c)=0.
\lambda(p^{\mu})\,\equiv\left(\begin{matrix}{\left(\zeta_{\lambda}\,\Theta_{[j]}\right)\,\phi_{_L}^{\ast}(p^{\mu})}\\{\phi_{_L}(p^{\mu})}\\\end{matrix}\right)\,\,,\quad\rho(p^{\mu})\,\equiv\left(\begin{matrix}{\phi_{_R}(p^{\mu})}\\{\left(\zeta_{\rho}\,\Theta_{[j]}\right)^{\ast}\,\phi_{_R}^{\ast}(p^{\mu})}\\\end{matrix}\right)\,\,\quad.
\nabla_{\alpha}\eta^{A}-\frac{\sqrt{2}\mathrm{i}}{3}{\calP}_{\;B}^{A}\gamma_{\alpha}\eta^{B}-\delta(y)\delta_{\alpha}^{5}(\omega_{0})_{\;B}^{A}\gamma_{5}\eta^{B}-\delta(y-\pir_{c})\delta_{\alpha}^{5}(\omega_{\pi})_{\;B}^{A}\gamma_{5}\eta^{B}=0\.
\frac{K_{R}}{2}\int_{0}^{s}d\tau\epsilon_{ij}x_{i}\dot{x}_{j}=K_{R}[\frac{1}{2}(v_{1}u_{2}-v_{2}u_{1})+\frac{(v_{i}-u_{i})^{2}}{4(1-C)}(\frac{K_{R}s}{M}-S)].
Z_{F}(k,k;S={\bf1}/(k+{\frac{1}{2}}c_{2}(G)))=J\;Z_{F}(k,k;0)Z_{B}^{\frac{1}{2}}\left(-(k+{\frac{1}{2}}c_{2}(G)),-(k+{\frac{1}{2}}c_{2}(G));0\right)Z_{gh}.
2\varphi\varphi^{*}=2+\frac{w_{x}}{\lambda^{2}}+\frac{w_{y}}{\lambda^{3}}+\frac{w_{t}}{\lambda^{4}}+\dots+\frac{w_{t_{n}}}{\lambda^{N+1}}+O\biggl(\frac{1}{\lambda^{N+2}}\biggr).
\frac{\partialE}{\partialg_{k}}=\frac{1}{2}\left\{2\sum_{n=0}^{\infty}g_{n}\langleTr\Phi^{n+k-2}\rangle-\frac{1}{N}\sum_{p=0}^{k-2}\langleTr\Phi^{p}Tr\Phi^{k-p-2}\rangle\right\}
\begin{array}{c}{{-\frac{1}{2}(k+1)k(k-1)<O_{k-2}O_{k_{1}}...O_{k_{p-1}}P>_{c}+2t_{2}(2k+4)<O_{k+2}O_{k_{1}}...O_{k_{p-1}}P>_{c}}}\\{+4(k+1)\sum_{i\ink}<O_{i_{1}}...O_{i_{n}}P>_{c}<O_{k}O_{i_{n+1}}...O_{i_{p-1}}P>_{c}}\\{+2\sum_{i=1}^{p-1}(2k+k_{i}+2)<O_{k+k_{i}}O_{k_{1}}...\hat{O}_{k_{i}}...O_{k_{p-1}}P>_{c}=0.}\\\end{array}
\delta(p+q)=-(2\pi)^{\,8}\intd^{\,4}q^{\prime}\\Bigl[\tilde{C}_{\alpha}^{\,\alpha_{0}}(q)\tilde{C}_{\alpha}^{\,\alpha_{0}}(q^{\prime})\\delta_{\Phi_{c}(p)}\delta_{\Phi_{c}(q^{\prime})}\,\Gamma^{\,\alpha,\,\alpha_{0}}(\Phi_{c})
\Gamma^{(2)}[\varphi_{1}]=\frac{\sigma^{2}}{4(2\pi)^{d}}\intd^{d}x\intd^{d}y\,\varphi_{1}^{2}(x)\varphi_{1}^{2}(y)\intd^{d}p\,e^{-ip(y-x)}m_{2}^{d-4}I(p^{2},m_{2}^{2}).
\begin{array}{c}{PR^{\prime}(z)=P_{0}-P_{1}q^{-2}\frac{1-z^{2}q^{2}}{1-z^{2}q^{-2}}+P_{2}q^{-6}\frac{1-z^{2}q^{2}}{1-z^{2}q^{-2}}\frac{1-z^{2}q^{4}}{1-z^{2}q^{-4}}-\ldots}\\\end{array}
\begin{array}{lcr}{V(\phi,\beta)}&{=}&{\mu^{D}\sum_{s=1}^{\infty}a^{\frac{D}{2}-s}h(D,s)(\gamma+g\phi)^{2s}\frac{1}{\Gamma(-\frac{D}{2}+s+1)}}\\{}&{}&{\quad\left(\int_{1}^{\infty}dt\,t^{2s-D}\frac{e^{\frac{t}{2}}}{e^{t}-1}+\sum_{n=0}^{\infty}\frac{B_{n}(\frac{1}{2})}{n!}\frac{1}{2s-D+n}\right).}\\\end{array}
\Sigma_{1}^{\varepsilon}(p)=g^{2}C_{F}{\mu}^{2\varepsilon}\int{\frac{d^{4-2\varepsilon}k}{(2\pi)^{4-2\varepsilon}}}\gamma_{\alpha}S(k)\gamma_{\beta}{\frac{N^{\alpha\beta}(q)}{[q\cdotn]_{ML}(q^{2}+i\epsilon)}},
{\calK}={\frac{1}{2{\mathrm{dim}}(G)}}\sum_{a=1}^{{\mathrm{\small{dim}}}(G)}{\calK}_{a}={\frac{1}{2{\mathrm{dim}}(G)}}\sum_{a=1}^{{\mathrm{\small{dim}}}(G)}{\mathrm{Tr}}\left(\OmegaJ_{a}q^{L_{0}}{\overline{q}}^{{\overline{L}}_{0}}\right)~.
{\calL}_{eff}=-\frac14F^{a\mu\nu}F_{\mu\nu}^{a}+\frac12m^{2}A^{a\mu}A_{\mu}^{a}-\frac1{2\alpha}(\partial^{\mu}A_{\mu}^{a})^{2}-\partial^{\mu}\bar{C}^{a}
I\simeq\frac{2\pi}{m_{H}^{2}}\left[\frac12\left(c^{2}-1+\left(\frac{2}{\mathrm{e}^{\gamma}}\right)^{\frac{8\piT}{g^{2}}}\frac{1-c^{2-\frac{8\piT}{g^{2}}}}{1-\frac{4\piT}{g^{2}}}\right)+\mathrm{e}^{a/\mathrm{e}}-1+\frac{a}{\mathrm{e}}\right].
\tau_{0}=\frac12\left(\begin{array}{cc}{i}&{0}\\{0}&{-i}\\\end{array}\right)\,,\quad\tau_{1}=\frac12\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\,,\quad\tau_{2}=\frac12\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right)\,.
{\calW}_{a}\star_{N}{\calW}_{b}\,(\psi^{\prime},\bar{\psi}^{\prime},N)=\frac{1}{N}\,\bar{\psi^{\prime}}\,(\Lambda_{a}^{(\frac{1}{2})}\,\Lambda_{b}^{(\frac{1}{2})})\,\psi^{\prime}+\frac{N-1}{N}\,(\bar{\psi}^{\prime}\,\Lambda_{a}^{(\frac{1}{2})}\,\psi^{\prime})(\bar{\psi}^{\prime}\,\Lambda_{b}^{(\frac{1}{2})}\,\psi^{\prime})\,.
\biggl[(e^{t^{\prime}}-2229025112064){\frac{d^{24}}{dt^{\prime}\^{24}}}+\sum_{i=1}^{23}\tilde{C}_{i}(t^{\prime}){\frac{d^{i}}{dt^{\prime}\^{i}}}\biggr]\tilde{\Pi}(t^{\prime})=0,
\left[\phi(t,\,z),\dot{\phi}(t,\,z^{\prime})\right]=i\delta(z-z^{\prime})\,,\quad\left\{\psi_{\alpha}(t,\,z),\bar{\psi}_{\beta}(t,\,z^{\prime})\right\}=\left(\gamma^{0}\right)_{\alpha\beta}\delta(z-z^{\prime})
J_{DS}(x)=\left(\begin{array}{cccccc}{0}&{k}&{0}&{0}&{\cdots}&{0}\\{0}&{0}&{k}&{0}&{}&{0}\\{0}&{0}&{0}&{\ddots}&{\ddots}&{\vdots}\\{\vdots}&{\vdots}&{}&{\ddots}&{k}&{0}\\{0}&{0}&{0}&{}&{0}&{k}\\{u_{N}}&{u_{N-1}}&{u_{N-2}}&{\cdots}&{u_{2}}&{0}\\\end{array}\right).
\Gamma_{\alpha\beta}^{\gamma}=\hat{\Gamma}_{\alpha\beta}^{\gamma}-\bar{e}^{\gammaA}C_{\alpha\beta\delta}e_{A}^{\delta}\qquad\mathrm{with}\qquad\hat{\Gamma}_{\alpha\beta}^{\gamma}=e_{A}^{\gamma}\partial_{\beta}e_{\alpha}^{A}\.
\begin{array}{l}{\partial^{\mu}\mathcal{F}_{\mu\nu}=\partial_{\mu}F_{\mu\nu}^{1}=J_{\nu}^{1}}\\{\partial^{\mu}\widetilde{\mathcal{F}}_{\mu\nu}=-\partial_{\mu}F_{\mu\nu}^{2}=J_{\nu}^{2}}\\\end{array}
L^{+}=q^{-1}\left(\begin{array}{ccc}{q^{h_{2}}}&{(q^{-1}-q)\alpha^{+}}&{(q-q^{-1})(-1)^{F}b^{+}}\\{0}&{q^{h_{1}+h_{2}}}&{(q-q^{-1})(-1)^{F}\beta^{+}}\\{0}&{0}&{(-1)^{F}q^{h_{1}+2h_{2}}}\\\end{array}\right)
\langleB\vert\intd^{3}\vec{x}\,\partial_{\mu}\widehatJ^{\mu}(x)\vertA\rangle=f_{\pi}m_{\pi}^{2}\langleB\vert\intd^{3}\vec{x}\,\pi(x)\vertA\rangle\nonumber\,=\langleB\vert\intd^{3}\vec{x}\,f_{\pi}j_{\pi}(x)\vertA\rangle,
{\calL}~=~\partial_{\mu}z_{\alpha}^{*}\partial^{\mu}z^{\alpha}-\frac{2g}{N}(z_{\alpha}^{*}\partial^{\mu}z_{\alpha})(z_{\beta}\partial_{\mu}z_{\beta}^{*})-\lambda(z_{\alpha}^{*}z_{\alpha}-\frac{N}{2g})
n^{\pm\mu}\=\\oint\frac{dz_{2}}{2\pii}\oint_{|z_{1}|<|z_{2}|}\frac{dz_{1}}{2\piiz_{1}}\int_{z_{2}}^{z_{1}}\!\!dw\;e^{-\varphi^{\mp}}i\psi^{\mp\mu}(z_{2})\\partial\xi^{\pm}(w).
\frac{1}{4\pi\ell_{s}^{2}}\epsilon\sum_{I=m+1}^{D-1}g_{Ii}X^{I}e^{\epsilon(\phi_{(0)}-X_{(0)}^{I})}\Theta(X_{(0)}^{I})~\int_{\Sigma}\partial^{\alpha}X^{I}\partial^{\alpha}y_{i}~
\rho=\lambda\left(\sqrt{\frac{6K}{\lambdak_{4}^{2}}\frac1{a^{2}}-\frac{12{\calU}_{0}}{\lambda^{2}k_{4}^{4}}\frac1{a^{4}}+1-\frac{2\Lambda}{\lambdak_{4}^{2}}+\frac{6H_{0}^{2}}{\lambdak_{4}^{2}}}-1\right).
\sum_{n=1}^{\infty}\,f\left(n\right)=-\frac{1}{2}f\left(0\right)+\int_{0}^{\infty}\,dx\,f\left(x\right)+i\int_{0}^{\infty}\,dx\,\frac{f\left(ix\right)-f\left(-ix\right)}{e^{2\pix}-1}.
\overline{{V}}_{i}(x,t)={\frac{t_{d+i}}{t_{i}}}\,{\frac{t-t_{i}}{t-t_{d+i}}}\,E_{i}^{-1}=\sqrt{-{\frac{t_{d+i}}{t_{i}}}}\,{\frac{t-t_{i}}{t-t_{d+i}}}
\left.\overline{{R}}\right|,\left.\nabla_{A}\overline{{R}}\right|,\left.\nabla^{2}\overline{{R}}\right|,\left.G_{A\dot{A}}\right|,\left.\nabla_{\underline{{A}}}G_{\underline{{B}}\dot{A}}\right|,\left.\nabla_{\underline{{\dot{A}}}}\nabla_{\underline{{A}}}G_{\underline{{B}}\underline{{\dot{B}}}}\right|,\left.W_{ABC}\right|,\left.\nabla_{\underline{{D}}}W_{\underline{{ABC}}}\right|
\begin{array}{lcll}{\mathrm{floor}}&{\mathrm{SU(3/1)}}&{\mathrm{field}}&{\mathrm{helicity}}\\{\mid\mathrm{gnd}>}&{(0~0~-\frac{7}{2})}&{e_{\mu}^{a}}&{+2}\\{\mid\mathrm{1st}>}&{(0~1~-\frac{7}{2})}&{\overline{3}\Psi_{\mu}}&{+\frac{3}{2}}\\{\mid\mathrm{2nd}>}&{(1~0~-\frac{5}{2})}&{3A_{\mu}}&{+1}\\{\mid\mathrm{3rd}>}&{(0~0~-\frac{3}{2})}&{\lambda}&{+\frac{1}{2},}\\\end{array}
W_{i_{2}j}^{(2)}(x,y)=\left(\begin{array}{cc}{\widetilde{W}_{ij}^{(0)}}\\\hline{}\\{\begin{array}{ccccc}{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{\kappa}&{-{\sqrt\kappa}}&{0}\\{0}&{-\kappa}&{0}&{0}&{-{\sqrt\kappa}}\\{0}&{\kappa\partial_{x}^{2}}&{-\kappa\partial_{x}^{1}}&{{\sqrt\kappa}\partial_{x}^{1}}&{{\sqrt\kappa}\partial_{x}^{2}}\\\end{array}}\\\end{array}\right)\delta^{3}(x-y),
\delta\omega^{A}=\frac{\partial\Theta^{Ab}}{\partiale^{a}}\pi_{b}\,deltae^{a}+\frac{\partialU^{A\beta}}{\partiale^{a}}\lambda_{\beta}\,deltae^{a}+\Theta^{Aa}\delta\pi_{a}+U^{A\alpha}\delta\lambda_{\alpha},
\deltag_{MN}dx^{M}dx^{N}=\sum_{k}\left[h_{ab}Ydx^{a}dx^{b}+2h_{(L)a}V_{(L)i}dx^{a}dx^{i}+(h_{(LL)}T_{(LL)ij}+h_{(Y)}T_{(Y)ij})dx^{i}dx^{j}\right],
\frac{d}{dt}\tilde{W}(t|\bar{x}(t),y)=\left[\,\frac{\partial}{\partialt}+\frac{(x-y)^{\mu}}{t}\frac{\partial}{\partial\bar{x}^{\mu}}\right]\tilde{W}(t|\bar{x},y)=e^{t\overset{-}{\Box}}f(t|\,\bar{x}(t),y),
H_{p}={\frac{Q}{|\vec{z}-\vec{z}_{0}|^{7-p}}},\\\\H_{W}=1+{\frac{Q_{W}}{(x^{2}+{\frac{4Q}{(p-5)^{2}}}|\vec{z}-\vec{z}_{0}|^{p-5})^{\frac{p^{2}-8p+19}{2(p-5)}}}}.
\langleT_{i}^{k}\rangle_{N}^{(b)}\left(\xi_{1},\xi_{2},\xi\right)=\langleT_{i}^{k}\rangle_{N}^{(1b)}(\xi_{1},\xi)-A_{d}\delta_{i}^{k}\int_{0}^{\infty}dk\,k^{d}\int_{0}^{\infty}d\omega\frac{K_{\omega}^{\prime}(k\xi_{2})}{K_{\omega}^{\prime}(k\xi_{1})}\frac{F^{(i)}[N_{\omega}(k\xi,k\xi_{1})]}{N_{\omega}^{\prime}(k\xi_{1},k\xi_{2})},
B_{(m,\psi_{m})[\alpha,\psi_{\alpha}]}={\frac{\sqrt{|\calG|}}{\sqrt{|{\calS}_{m}|}|{\calS}_{\alpha}|}}{\frac{\sum_{J}\psi_{m}(J)\psi_{\alpha}(J){S}_{m\alpha}^{J}}{\sqrt{{S}_{0m}}}}\;\;\;\;,
{{}_{2}F_{1}}^{*}(\alpha,\beta;\gamma;z)=\sum_{n=1}^{\infty}\frac{(\alpha)_{n}(\beta)_{n}}{(\gamma)_{n}}\left[\sum_{k=0}^{n-1}\left(\frac{1}{\alpha+k}+\frac{1}{\beta+k}-\frac{2}{\gamma+k}\right)\right]\frac{z^{n}}{n!},
\epsilon^{r_{1}\cdotsr_{9}m}\epsilon^{s_{1}\cdotss_{9}}{}_{m}\,W_{r_{1}r_{2}s_{1}s_{2}}W_{r_{3}r_{4}s_{3}s_{4}}W_{r_{5}r_{6}s_{5}s_{6}}\,\Big(H_{r_{7}r_{8}r_{9}}H_{s_{7}s_{8}s_{9}}-3\,H_{r_{7}r_{8}s_{9}}H_{s_{7}s_{8}r_{9}}\Big)=0\,.
S_{c}={\frac{1}{2\pi}}\int\sqrt{-g}d^{2}x\,\Bigl[\mathrm{\mathrm{e}}^{-2\phi}\Bigl\{R+4(\nabla\phi)^{2}+4\lambda^{2}\Bigr\}-{\frac{1}{2}}\sum_{i=1}^{N}(\nablaf_{i})^{2}\Bigr]
\nabla\Pi=\sqrt{-\gamma}\;\Pi{}^{\prime}=-{\frac{1}{\sqrt{-\gamma}}}\left[\left({\frac{L_{2}}{\kappa_{1}}}\right)^{\prime}\;\eta_{2}+{\frac{L_{2}\kappa_{2}}{\kappa_{1}}}\;\eta_{1}-{\frac{L_{2}\kappa_{3}}{\kappa_{1}}}\;\eta_{3}\right]\,.
{\calP}^{a\mu}=\alpha_{2}\sqrt{-\gamma}\left[\left(K_{cd}{}^{i}K^{cd}{}_{i}\gamma^{ab}-2K^{a}{}_{c\,i}K^{bc\,i}\right)e^{\mu}{}_{b}+2(\widetilde\nabla_{b}K^{ab\,i})n^{\mu}{}_{i}\right]\,.
\epsilon=-g_{\mu\nu}{\frac{dx^{\mu}}{d\lambda}}{\frac{dx^{\nu}}{d\lambda}}={\calW}\left[\left({\frac{dt}{d\lambda}}\right)^{2}-{\frac{d{\bfx}}{d\lambda}}\cdot{\frac{d{\bfx}}{d\lambda}}\right]-\left({\frac{dy}{d\lambda}}\right)^{2},