harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
A=\left[{\frac{-i\mu^{\epsilon}}{(2\pi)^{d}}}\right]^{\ell}\int\prod_{a=1}^{\ell}(d^{d}k_{a}){\frac{S_{0}(q,p)}{\prod_{r=1}^{N}(-q_{r}^{2}+m_{r}^{2}-i\epsilon)}}\,
\hat{\tilde{\mathrm{Q}}}_{\mathrm{R,\mathrm{N}}}|\mathrm{phys};A\rangle\equiv\hbar\vdots\tilde{\rho}(0)\vdots|\mathrm{phys};A\rangle=\hbar[\frac{eb\mathrm{L}}{2\pi}]|\mathrm{phys};A\rangle.
W_{mm^{\prime}}^{j}(u)=q^{(m-j)(m-m^{\prime})}\left[\begin{array}{l}{j+m}\\{m-n}\\\end{array}\right]_{q^{2}}^{1/2}\left[\begin{array}{l}{j-n}\\{m-n}\\\end{array}\right]_{q^{2}}^{1/2}
S=\frac{1}{2}\intd^{4}x\epsilon^{\mu\nu\alpha\beta}\epsilon_{abcd}\bigg[({^+}\tau-{^-}\tau)F_{\mu\nu}^{ab}F_{\alpha\beta}^{cd}+({^+}\tau+{^-}\tau)F_{\mu\nu}^{ab}\tilde{F}_{\alpha\beta}^{cd}\bigg].
R_{1,2}(\zeta_{1}/\zeta_{2})\Phi_{1}^{(i,1-i)}(\zeta_{1})\Phi_{2}^{(1-i,i)}(\zeta_{2})=\Phi_{2}^{(i,1-i)}(\zeta_{2})\Phi_{1}^{(1-i,i)}(\zeta_{1}).
\widetilde{a}_{1}(t)=\widetilde{a}_{1}-\frac{1}{2\lambda}\left(\xi-\frac{1}{6}\right)\left[\left(1-\frac{3\lambdat}{(4\pi)^{2}}\right)^{1/3}-1\right]-\frac{t}{2160(4\pi)^{2}},
{\tilde{\dot{x}}}^{2}=-(1+\mu_{1})^{2}x^{2}+\frac{(\tilde{\dot{x}}x^{\prime})^{2}}{x^{2}}\;\Longrightarrow1+\mu_{1}=\frac{\sqrt{(x^{\prime}\tilde{\dot{x}})^{2}-x^{2}{\tilde{\dot{x}}}^{2}}}{-x^{2}}
\left\langle\intd^{2}x:\Psi^{\dagger}(x)M\Psi(x):\right\rangle=-\frac{1}{2}\left(\sum_{\mathrm{occupied}}\intd^{2}x\psi_{E}^{\dagger}(x)M\psi_{E}(x)-\sum_{\mathrm{unoccupied}}\intd^{2}x\psi_{E}^{\dagger}(x)M\psi_{E}(x)\right).
H_{n}=\intd{\bfx}\intdkdk^{\prime}e^{ikx}e^{ik^{\prime}x}\phi(k)\phi(k^{\prime})(k_{0}^{2}-k_{0}k_{0}^{\prime})\frac{(-k^{2})^{n}-(-k^{2})^{n}}{k^{2}-k^{2}}-\intd{\bfx}{\calL}_{n}
\left(\sum_{\alpha=p}^{\infty}\lambda^{\alpha}\omega^{i_{p},\alpha}\right)\wedge\left(\sum_{\alpha=q}^{\infty}\lambda^{\alpha}\omega^{j_{q},\alpha}\right)=\sum_{\alpha=p+q}^{\infty}\lambda^{\alpha}\sum_{\beta=p}^{\alpha-q}\omega^{i_{p},\beta}\wedge\omega^{j_{q},\alpha-\beta}\;,
S={\frac{1}{4G}}A_{\S}-\int_{\Sigma}\left(8\pia_{1}R+4\pia_{2}R_{\mu\nu}n_{i}^{\mu}n_{i}^{\nu}+8\pia_{3}R_{\mu\nu\lambda\rho}n_{i}^{\mu}n_{i}^{\lambda}n_{j}^{\nu}n_{j}^{\rho}\right)~~~.
\{E_{i\bar{j}},E_{k\bar{l}}\}=u^{2}v^{2}(1,i)[1,\hat{\bar{j}}](1,k)[1,\hat{\bar{l}}]\left({\frac{G_{ik}E_{\bar{j}\bar{l}}}{[\hat{\bar{j}},\hat{\bar{l}}]}}-{\frac{G_{\bar{j}\bar{l}}E_{ik}}{(i,k)}}\right).
F_{\mu\nu}=\partial_{\mu}(\hat{\phi}^{a}W_{\nu}^{a})-\partial_{\nu}(\hat{\phi}^{a}W_{\mu}^{a})-{\frac{1}{g}}\epsilon_{abc}\hat{\phi}^{a}(\partial_{\mu}\hat{\phi}^{b})(\partial_{\nu}\hat{\phi}^{c})
\vartheta_{1}\bigl(\frac{\tau_{12}+i\delta}{2\pit}-\frac{i}{t}|\frac{i}{t}\bigr)=-e^{\frac{\pi}{t}-\frac{\delta}{t}+i\frac{\tau_{12}}{t}}\vartheta_{1}\bigl(\frac{\tau_{12}+i\delta}{2\pit}|\frac{i}{t}\bigr).
Z=\sum_{n_{i}}\frac{1}{\prodn_{i}!}(L^{11+})^{n_{1}}(L^{12+})^{n_{2}}(L^{22+})^{n_{3}}(T^{+})^{n_{4}}{|0\rangle}_{V}\langle0|(b_{1})^{n_{1}}(b_{2})^{n_{2}}(b_{3})^{n_{3}}(b_{4})^{n_{4}}.
\left[i\gamma^{\mu}\partial_{\mu}-m{\frac{1\pmb_{1}\pmb_{2}}{a}}\right]\times\left[i\gamma^{\nu}\partial_{\nu}+{\frac{a}{2m}}\partial^{\nu}\partial_{\nu}+m{\frac{b_{1}^{2}-1}{2a}}\right]\Psi_{1,2}=0\,.
\tilde{L}=\lambda_{a}^{\mu}f_{\mu}^{a}(\psi_{,\mu}^{b})-\lambda_{a}^{\mu}f_{\mu}^{a}(\psi^{a},\lambda_{a}^{\mu})+L\left(\psi^{a},f_{\mu}^{a}(\psi^{a},\lambda_{a}^{\mu})\right).
\frac{1}{\hat{\alpha}_{\mathrm{R}}\hat{g}_{\mathrm{R}}^{2}}=\frac{1}{\alphag^{2}}+\frac{5}{4}\frac{1}{16\pi^{2}}\,t,\quad\Rightarrow\quad\partial_{t}\hat{\alpha}_{\mathrm{R}}=\frac{7}{3}\hat{\alpha}_{\mathrm{R}}\left(1-\frac{15}{28}\hat{\alpha}_{\mathrm{R}}\right)\frac{\hat{g}_{\mathrm{R}}^{2}}{16\pi^{2}}.
F_{n}^{\mathcal{O}|\mu_{1}\ldots\mu_{n}}(\theta_{1},\ldots,\theta_{n})\equiv\langle\mathcal{O}(0)|\mathcal{\,}Z_{\mu_{1}}^{\dagger}(\theta_{1})\ldotsZ_{\mu_{n}}^{\dagger}(\theta_{n})\rangle\,.
\bar{\Pi}_{ij}^{g}=\frac{\partial{\calL}}{\partial(\partial_{0}g_{ij})}\,=\,\frac{1}{\lambda^{2}}\partial^{0}g_{ji}^{-1}\,-\,\frac{n}{4\pi}A_{ji}(g)\,+\,\frac{i\alpha}{4\lambda^{2}}\bar{\psi}_{jk}\gamma^{1}\psi_{km}g_{mi}^{-1}
\delta_{\epsilon}A_{(p)}^{\mu_{1}\ldots\mu_{p-2k}}=\partial_{}^{\left[\mu_{1}\right.}\epsilon_{(p)}^{\left.\mu_{2}\ldots\mu_{p-2k}\right]}+\left(p-2k+1\right)\partial_{\mu}\epsilon_{(p)}^{\mu\mu_{1}\ldots\mu_{p-2k}},\;k=1,\ldots,b_{p},
\bar{F}_{p+1}=\sum_{i_{k},j_{k}}\frac{d}{dx^{-}}\bar{A}_{i_{1}i_{2}\dotsi_{p}}(2x^{-})Q_{j_{1}}^{i_{1}}Q_{j_{2}}^{i_{2}}\cdotsQ_{j_{p}}^{i_{p}}dx^{-}\wedgedx^{j_{1}}\wedgedx^{j_{2}}\wedge\cdots\wedgedx^{j_{p}}~.
L(\rho_{0})=2\sqrt{N}\int_{\rho_{0}}^{\rho_{\mathrm{max}}}d\rho\,\frac{\alpha^{2}+\alphav\left[\alpha^{2}+H(\rho_{0})\right]^{1/2}+H(\rho)}{\left(H(\rho)\left[H(\rho_{0})-H(\rho)\right]\right)^{1/2}}\,.
W_{\mathrm{div}}^{E}[g,\beta]=-{\frac{\eta}{2}}\int_{0}^{\infty}{\frac{ds}{s}}e^{-m^{2}s}{\frac{1}{(4\pis)^{D/2}}}\left(B_{0}+sB_{1}+s^{2}B_{2}\right)~~~.
{\calT}_{semis}\,=\,\left[\begin{array}{cc}{1}&{\qquad0}\\{}&{}\\{0}&{\qquadq^{-1}}\\\end{array}\right]\qquad{\calS}_{semis}\,=\,\frac{(q-q^{-1})}{\sqrt5}\left[\begin{array}{cc}{1}&{\qquad[2]}\\{}&{}\\{{[2]}}&{\qquad-1}\\\end{array}\right]
\begin{array}{l}{:q^{-}p^{+}:\times:q^{-}p^{+}:\,=\,:q^{-}p^{+}q^{-}p^{+}:}\\{\quad\quad+\,<p^{+}q^{-}><q^{-}p^{+}>}\\{\quad\quad+\left(<p^{+}q^{-}>+<q^{-}p^{+}>\right)\,:q^{-}p^{+}:}\\{\quad=\,:q^{-}p^{+}q^{-}p^{+}:\,+\frac14.}\\\end{array}
\left(\kappa\frac{\partial}{\partialz}-\frac{\beta^{2}}{2z}\right)G_{\epsilon}\left(z,\bar{z}\right)=\frac{\beta\beta_{1}\epsilon}{z}\quadG_{\epsilon}\left(z,\bar{z}\right)-\frac{1}{\sqrt{z(1-z)}}H_{\epsilon}(z,\bar{z}),
g(\Lambda)\approx\Lambda^{4}\int\sqrt{-G}d^{3}xdz\approx\Lambda^{4}L^{3}\int_{1/k}^{1/T}\frac{1}{(kz)^{4}}dz\approx\Lambda^{3}L^{3}\Lambda(\frac{k^{4}-T^{4}}{k^{5}})\approx\Lambda^{3}L^{3}\frac{\Lambda}{k}
({\eta}_{\mu\nu})=\left(\begin{array}{ccccc}{1}&{0}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{0}&{-1}\\{0}&{0}&{0}&{-1}&{0}\\\end{array}\right).
{\frac{\delta\sigma[T](x)}{\deltaT^{\pm}(y)}}=\pm{\frac{1}{24\pi}}T_{,x}^{\pm}(x)\left\{\delta_{,x}(x,y)+\partial_{x}\left[(T_{,x}^{\pm}(x))^{-1}\partial_{x}\left((T_{,x}^{\pm}(x))^{-1}\delta_{,x}(x,y)\right)\right]\right\}.
\delta\!f^{\mathrm{D}}(t)=-{\frac{\pi^{2}}{15}}{\frac{(k_{B}T)^{4}}{\hbar^{3}c^{4}}}A\,\delta\dot{q}(t)+{\frac{\zeta(3)}{2\pi}}{\frac{(k_{B}T)^{3}}{\hbar^{2}c^{4}}}A\,\delta\ddot{q}(t).
\begin{array}{ll}{(cc)}&{=\{1,x,\ldots,x^{p-1}\}}\\{}&{}\\{(aa)}&{=\{1,\bar{x},\ldots,\bar{x}^{p-1}\}}\\{}&{}\\{(ca)}&{=\{1,y,\ldots,y^{q-1}\}}\\{}&{}\\{(ac)}&{=\{1,\bar{y},\ldots,\bar{y}^{q-1}\}}\\\end{array}
\begin{array}{ccccc}{\delta_{i}}&{\Longleftrightarrow}&{(\mathbb{E}_{i})_{\mu\nu}}&{=}&{\delta_{0\mu}\delta_{i\nu}-\delta_{0\nu}\delta_{i\mu}-f_{i\mu\nu},}\\{1|\delta_{i}}&{\Longleftrightarrow}&{(1|\mathbb{E}_{i})_{\mu\nu}}&{=}&{\delta_{0\mu}\delta_{i\nu}-\delta_{0\nu}\delta_{i\mu}+f_{i\mu\nu}.}\\\end{array}
s^{M}={\frac{1}{2}}\intd\Omegad\Omega^{\prime}[{\bfp}\times{\bfp}^{\prime}]({\bfr}-{\bfr}^{\prime}){\calN}({\bfr},{\bfp},t){\calN}({\bfr}^{\prime},{\bfp}^{\prime},t),
\left(\sum_{n=0}^{\infty}g_{n}(k+n-2)\frac{\partial}{\partialg_{n+k-2}}+\frac{1}{4N^{2}}\sum_{p=0}^{k-2}p(k-p-2)\frac{\partial^{2}}{\partialg_{p}\partialg_{k-p-2}}\right)Z=0
\begin{array}{c}{L={\tilde{\sigma}}\left[2c\epsilon+c{\bfe^{\prime}}_{1}^{2}+{\bfp}({\bfx^{\prime}}-{\bfe}_{+})+\right.}\\{\left.+{\bfp}_{+}({\bfe^{\prime}}_{+}-{\bfe}_{1})-\sum_{i,j}d_{ij}({\bfe}_{i}{\bfe}_{j}-\eta_{ij})\right],}\\\end{array}
\overline{D}_{\dot{\alpha}}\overline{D}_{\dot{\beta}}X_{\gamma\dot{\gamma}}\bigr|_{\theta=\overline{\theta}=0}=-\varepsilon_{\dot{\alpha}\dot{\beta}}\overline{D}_{\dot{\gamma}}\overline{D}^{\dot{\delta}}X_{\gamma\dot{\delta}}\bigr|_{\theta=\overline{\theta}=0},
S_{Q2f}(M_{n,i})=\sum_{i=0}^{L-1}\frac{1}{N}\mathrm{Tr}\,(2M_{1,i}^{2}+2M_{2,i}^{2})+\frac{2g}{N^{2}}\mathrm{Tr}\,M_{1,i}^{2}\mathrm{Tr}\,M_{2,i}^{2}
{\frac{1}{\omega}}\left(-{\frac{3}{4}}-{\frac{z_{0}^{2}}{l^{2}}}\right)>{\frac{1}{32}}\left(-32\omegaz_{0}^{2}+9l^{2}+3\sqrt{-64\omegaz_{0}^{2}l^{2}+9l^{4}-64l^{2}z_{0}^{2}\omega^{2}}\right)/(\omega^{2}l^{2})\,,
\bar{g}_{\alpha\beta}^{\zeta}(t,t^{\prime})=\frac{\bar{\sigma}(t)}{v(t)}g_{\alpha\beta}^{\zeta}(t,t^{\prime})\frac{\bar{\sigma}(t^{\prime})}{v(t^{\prime})}+\frac{\lambda}{2}\frac{1}{v(t)}I_{2}^{\zeta}(t,t^{\prime})\frac{1}{v(t)}\,,
I_{1}={\frac{1}{2}}(\delta_{\mu\nu}-{\frac{p_{\mu}p_{\nu}}{p^{2}}})\Gamma({\frac{2-\omega}{2}})/(4\pi)^{\omega/2}\Lambda^{3-\omega}\int_{0}^{1}dx[x(1-x)p^{2}]^{(\omega-2)/2}.
(2\pi\alpha^{\prime})d\tilde{\calF}_{(0)}=\tilde{H}_{(0)}-(2\pi\alpha^{\prime})G_{(0)}^{(6)}{\calG}_{(0)}^{(1)}+3(2\pi\alpha^{\prime})^{2}{\calH}_{(0)}^{(3)}d{\calH}_{(0)}^{(3)}\,.
I(Q^{2})=v_{4}\frac{k^{2}}{Q^{2}}\int_{0}^{\frac{1}{2}}dz\left(z^{-2}-\frac{Q^{2}}{k^{2}}z^{-1}-z^{-2}e^{-\frac{Q^{2}}{k^{2}}z}\right).
M_{2}=\left(\begin{array}{crc}{f^{2}+\chi^{2}-f\left(w^{2}e^{2\phi}+v^{2}e^{-2\phi}\right)}&{f\left(\kappawe^{2\phi}-ve^{-2\phi}\right)-u\chi}\\{f\left(\kappawe^{2\phi}-ve^{-2\phi}\right)-u\chi}&{u^{2}-fe^{2\phi}\left(\kappa^{2}+e^{-4\phi}\right)}\\\end{array}\right).
\Lambda_{\itpure}^{2(\tilde{n}_{c}-r+1)}=\frac{1}{\Lambda^{n_{c}-\tilde{n}_{c}}}\frac{\prod_{i=1}^{n_{f}}m_{i}}{\prod_{i=1}^{r}m_{i}^{2}}\simm^{2(\tilde{n}_{c}-r+1)}\left(\frac{m}{\Lambda}\right)^{n_{c}-\tilde{n}_{c}}\llm^{2(\tilde{n}_{c}-r+1)}.
\left.\!J^{(N)}\left(n;\nu_{1},\!\ldots\!,\nu_{N}\right)\right|_{\Sigma\nu_{i}=n}\!=2\mathrm{i}^{1-2\Sigma\nu_{i}}\pi^{n/2}\frac{\Gamma\left({\textstyle{\frac{n}{2}}}\right)}{\prod\Gamma\left(\nu_{i}\right)}\;\frac{1}{\prodm_{i}^{\nu_{i}}}\int_{0}^{\infty}\!\ldots\!\int_{0}^{\infty}\prod\alpha_{i}^{\nu_{i}-1}\mathrm{d}\alpha_{i}\;\delta\left(\alpha^{T}\|c\|\alpha\!-\!1\right).
J_{\calC}\left({\frac{\partial}{\partialw^{l}}}\right)=\mathrm{i}{\frac{\partial}{\partialw^{l}}},\qquadJ_{\calC}\left({\frac{\partial}{\partial\bar{w}^{l}}}\right)=-\mathrm{i}{\frac{\partial}{\partial\bar{w}^{l}}}.
e^{iW[\vec{H}]}=\int{\calD}\lambdae^{i\intd^{d}x(\frac{1}{4U}\lambda^{2}(x)+\frac{C^{2}}{2}\lambda(x))-\frac{N}{2}Trln\nabla-i\frac{1}{2}H.\nabla^{-1}.H}
\left(\eta_{MN}-\partial_{a}X_{M}\overline{{g}}^{ab}\partial_{b}X_{N}\right)\dot{X}^{N}=\frac{1}{\overline{{g}}}\left(VP_{M}+\frac{1}{2}\widetilde{\mathcal{F}}\varepsilon^{ab}\partial_{a}X^{N}\partial_{b}X^{P}A_{MNP}\right)
\frac{d^{p}H_{\lambda}(z)}{dz^{p}}=(-1)^{p}p!\frac{i}{2\pi}\oint_{{\Gamma}_{1}}\oint_{{\Gamma}_{2}}dk_{1}\;dk_{2}\;\frac{k_{1}^{\Lambda}F(k_{1})k_{2}^{\Lambda}G(k_{2})}{(z-k_{1}-k_{2})^{p}}
\widetilde{n}^{\pm}={\frac{m+2\pm\sqrt{m^{2}-4}}{2}}\pm{\frac{1}{2}}\sqrt{{\frac{m+2}{m-2}}}{\frac{m^{2}+4\pmm\sqrt{m^{2}-4}}{(1\pm\sqrt{m^{2}-4})(m\pm\sqrt{m^{2}-4})}}\tilde{\epsilon}+O(\tilde{\epsilon}^{2}).
{\calL}=\bar{Y}_{k_{1}l_{1},\ldots,k_{q}l_{q}}^{i_{1}j_{1},\ldots,i_{p}j_{p}}L_{i_{1}j_{1},\ldots,i_{p}j_{p}}^{k_{1}l_{1},\ldots,k_{q}l_{q}}+\ldots
B^{i}=\intd^{2}x\left[x^{i}{\calH}+\kappa\dot{\varphi}\epsilon_{ij}{\frac{\nabla_{j}}{-\nabla^{2}}}\varphi-\dot{\varphi}\epsilon_{ij}{\frac{\nabla_{j}}{-\nabla^{2}}}(\pi_{\chi}+{\frac{\kappa}{2}}\varphi)-(\dot{\pi}_{\chi}+{\frac{\kappa}{2}}\dot{\varphi})\epsilon_{ij}{\frac{\nabla_{j}}{-\nabla^{2}}}\varphi\right]
\int_{X_{\epsilon}}\partial_{z}\phi~\partial_{\bar{z}}\phi~\frac{idz\wedged\bar{z}}{2}=\int(\partial_{z}\phi_{M}\partial_{\bar{z}}\phi_{M}-2\alpha_{1}\phi_{M}\partial_{z}\partial_{\bar{z}}\phi_{B})\frac{idz\wedged\bar{z}}{2}+
\begin{array}{rcl}{-i\bar{\kappa}_{(\alpha)}\gamma^{a}\kappa_{(\beta)}e_{a}}&{=}&{2g\Gamma_{s}(u^{-1})^{T}\mathcal{C}\Gamma_{s}(P^{a})\Gamma_{s}(u^{-1})e_{a}}\\{}&{}&{}\\{}&{=}&{g\mathcal{C}\Gamma_{s}(\hat{M}^{\hat{b}\hat{c}})\Gamma_{\mathrm{Adj}}(u^{-1})^{a}{}_{\hat{b}\hat{c}}e_{a}}\\{}&{}&{}\\{}&{=}&{g\mathcal{C}\Gamma_{s}(\hat{M}^{\hat{b}\hat{c}})k_{(\hat{b}\hat{c})}\,,}\\\end{array}
A_{(l_{1},\dots,l_{p})}(\lambda)=\sum_{\sigma\inS_{p}}\left(\prod_{j=1}^{p}\frac{X_{l_{\sigma(j)}}^{a_{j}+1}}{1-X_{l_{\sigma(1)}}\cdotsX_{l_{\sigma(j)}}}\right)f_{l_{\sigma(1)}}\cdotsf_{l_{\sigma(p)}},
E_{rest}+E_{flux}=\frac{1}{l_{s}}\sqrt{\frac{V}{g_{YM}^{4}l_{s}^{4}}}\left(\frac{|p-q\theta|}{\sqrt{t}}+\frac{t^{3/2}\left[q-\phi(p-q\theta)\right]^{2}}{2(p-q\theta)}\right)
\left[A^{i}(x),\partial_{0}A^{*j}(y)\right]_{x^{0}=y^{0}}=i\delta^{3}(x-y)\delta^{ij},\qquad\left\{\psi_{\alpha}^{i}(x),\bar{\psi}_{\dot{\beta}}^{j}(y)\right\}_{x^{0}=y^{0}}=-\delta^{3}(x-y)\delta^{ij}\sigma_{\alpha\dot{\beta}}^{0}.
\left[-{\frac{B}{A}}{\frac{d^{2}}{dr^{2}}}+{\frac{B}{A}}\left({\frac{1}{2}}{\frac{A^{\prime}}{A}}-2{\frac{B^{\prime}}{B}}-{\frac{2}{r}}\right){\frac{d}{dr}}+{\frac{l(l+1)B}{r^{2}}}+M_{KG}^{2}B\right]R_{KG}(r)=m_{KG}^{2}R_{KG}(r)\,,\nonumber
H^{(1)}=\left[m^{2}\phi_{0}+{\frac{\lambda}{6}}\phi_{0}^{3}+{\frac{\lambda}{4}}\phi_{0}\int{\frac{d^{\nu}k}{(2\pi)^{\nu}}}{\frac{1}{g(\vec{k})}}\right]\intd^{\nu}x:\eta(\vec{x}):\;,
A_{(j)\mu\nu}=\frac{-\bar{f}_{j}^{2}\eta_{\mu\nu}+f_{j}F_{\mu\nu}+F_{\mu\nu}^{2}-i\bar{f}_{j}\stackrel{*}{F}_{\mu\nu}}{2(f_{j}^{2}-\bar{f}_{j}^{2})},
{\bar{V}}(r)={\frac{1}{4\alphar}}[n_{1}n_{1}^{\prime}\mathrm{Tr}(Y^{2})(1-e^{-\mu_{0}r})+n_{3}n_{3}^{\prime}\mathrm{Tr}(\lambda_{i}\lambda_{j})(1-e^{-\mu_{3}r})]-n_{8}n_{8}^{\prime}\mathrm{Tr}(T_{a}T_{b})\alpha_{s}r\
\left[-\hbar^{2}\left(\int_{0}^{1}ds\sqrt{x^{\,\prime\,2}}\right)^{-1}\int_{0}^{1}{\frac{ds}{\sqrt{x^{\,\prime\,2}}}}{\frac{\delta^{2}}{\deltax^{\mu}(s)\\deltax_{\mu}(s)}}+m^{4}\right]G[C,C_{0};m^{2}]=-\delta[C-C_{0}].
\langle\theta_{\mu\nu}^{\mathrm{reg}}-\theta_{\nu\mu}^{\mathrm{reg}}\rangle_{c}=\frac12\varepsilon_{\mu\nu\alpha\lambda}\partial_{\alpha}\langle\psi_{\varepsilon}^{+}\gamma_{\lambda}\gamma_{5}\psi_{\varepsilon}\rangle_{c}+
\left\langle\frac{1}{2N}\mathrm{Tr}\,H_{S}^{2}\right\rangle-\left\langle\frac{1}{2N}\mathrm{Tr}\,H_{S}\right\rangle^{2}=\left\langle\frac{1}{N}\mathrm{Tr}\,H_{0}^{2}\right\rangle-\left\langle\frac{1}{N}\mathrm{Tr}\,H_{0}\right\rangle^{2}+\frac{3}{8}a^{4}=k^{2}a^{2}+\frac{9}{16}a^{4}
\begin{array}{lcl}{{\calC}_{con}}&{\equiv}&{\{864\psi^{6}+\phi=\pm1\},}\\{{\calC}_{1}}&{\equiv}&{\{\phi=\pm1\},}\\{{\calC}_{\infty}}&{\equiv}&{\{\phi,\psi=\infty\},}\\{{\calC}_{0}}&{\equiv}&{\{\psi=0\}.}\\\end{array}
{\calV}[\Phi]_{Tad}=\frac{T}{2}\sum_{i}\int\frac{d^{3}p}{(2\pi)^{3}}\sum_{n=-\infty}^{\infty}\frac{1}{p^{2}+m_{i}^{2}+(2\pinT)^{2}}c_{i}\intd^{3}x(H_{0}^{2}(\vec{x})-1)
{\calJ}_{+}(e^{\frac{i\pi}{2}}ka)=\displaystyle\frac{2iak}{\pi}e^{-i\pi(|l|+\Phi)/2}\left[I_{l-\xi}(ka)K_{l+\Phi}(ka)+I_{l}(ka)K_{l+\Phi-\xi}(ka)\right],
<\delta^{d}(x-y)\delta^{d}(0)L^{d}F[\sigma]+\intd^{d}z\sigma(z,y){\frac{\deltalnJ}{\delta\sigma(z,x)}}F[\sigma]+\intd^{d}z\sigma(z,y){\frac{\deltaF[\sigma]}{\delta\sigma(z,x)}}+i\intd^{d}z\sigma(z,y){\frac{\deltaS}{\delta\sigma(z,x)}}F[\sigma]>=0.
q^{\mu}\Gamma_{\mu\nu}(q,q^{\prime};p,p^{\prime})=-e\mu^{\frac{\epsilon}{2}}\left\{\Gamma_{\nu}(q^{\prime};p^{\prime}-q^{\prime},p^{\prime})-\Gamma_{\nu}(q^{\prime};p,p+q^{\prime})\right\}\;.
\tilde{x}^{\mu}=\sqrt{\frac{\alpha_{\mathrm{eff}}^{\prime}}{\alpha^{\prime}}}x^{\mu}\,\quad\tilde{r}=\sqrt{\frac{\alpha_{\mathrm{eff}}^{\prime}}{\alpha^{\prime}}}r\,\quad\theta^{01}=1\,\quad\theta^{i,i+1}\,\quadg\,
\delta_{\vec{u}}(t)<W(\gamma)>=-{\frac{3\pii}{k}}\int_{0}^{1}ds\dot{\eta}^{a}(s,t){\eta^{\prime}}^{b}(s,t)\epsilon_{abc}\oint_{\gamma}dy^{c}\delta^{3}(\eta(s,t)-y)<W(\gamma)>.
W=\frac{\mu^{3}(-c-b)e^{-\frac{c}{b}-1}}{{j(S)}^{\frac{1}{2\pib}}\eta^{2}(S)\eta^{6}(T)}=\frac{\mu^{3}{\alpha}_{o}}{j^{\frac{1}{2\pib}}\eta^{2}(S)\eta^{6}(T)},
Y_{8}={\frac{1}{24}}\Big({\frac{1}{8}}\mathrm{tr}R^{4}+{\frac{1}{32}}(\mathrm{tr}R^{2})^{2}+(\mathrm{tr}F_{1}^{4}-\mathrm{tr}F_{2}^{4})-{\frac{1}{8}}\mathrm{tr}R^{2}(\mathrm{tr}F_{1}^{2}-\mathrm{tr}F_{2}^{2})\Big),
\omega_{\;b\mu}^{a}(x)\longrightarrow\omega_{\;b\mu}^{\primea}(x)=\Lambda_{c}^{a}(x)\Lambda_{b}^{d}(x)\omega_{\;d\mu}^{c}(x)+\Lambda_{c}^{a}(x)\partial_{\mu}\Lambda_{b}^{c}(x).
\delta_{\epsilon}A_{\;\;\;(\sigma)}^{\alpha\beta}=\varepsilon^{\alpha\beta\gamma\delta}\partial_{\gamma}\epsilon_{\delta(\sigma)}\equiv\stackrel{(0)}{R}_{\;\;\;(\sigma)}^{\prime\alpha\beta\;\;\;\delta(\rho)}\epsilon_{\delta(\rho)},\;\delta_{\epsilon}B_{\alpha}^{\;(\sigma)}=0,
\frac{1}{c_{0}^{2}}\left(\frac{\ddot{a}_{0}}{a_{0}}+\frac{\dot{a}_{0}}{a_{0}}\left(\frac{n-1}{2}\frac{\dot{a}_{0}}{a_{0}}-\frac{\dot{c}_{0}}{c_{0}}\right)\right)=\frac{1}{2R^{n}}\frac{d}{dR}H^{2}R^{n+1},
\left(\begin{matrix}{a_{D}}\\{a}\\\end{matrix}\right)\Bigl(\overline{z}\Bigr)=\left(\begin{matrix}{-1}&{2}\\{0}&{1}\\\end{matrix}\right)\left(\begin{matrix}{\overline{a}_{D}}\\{\overline{a}}\\\end{matrix}\right)\Bigl(z\Bigr),
SW_{j}^{\alpha}W_{\alpha}^{j}|_{\theta^{2}}=-\lambda_{j}^{\alpha}\lambda_{\alpha}^{j}Z+i\sqrt{2}\delta^{\alpha}\lambda_{\alpha}^{j}D_{j}-\frac{1}{\sqrt{2}}(\sigma^{\mu}\cdot\bar{\sigma}^{\nu})_{\alpha}{}^{\beta}F_{\mu\nu}{}^{j}\lambda_{j}^{\alpha}\delta_{\beta}
\begin{array}{cl}{}&{F_{2N}^{(i)}(z,z^{\prime}|y_{1},\cdots,y_{N},z_{N},\cdots,z_{1})^{j_{1}^{\prime},\cdots,j_{N}^{\prime},j_{N},\cdots,j_{1}}}\\{=}&{{}_{i}\langleB|A_{SW}^{(i)}(z,z^{\prime})\phi^{k_{1}}(y_{1}-z^{\prime})\cdots\phi^{k_{N}}(y_{N}-z^{\prime})\phi^{j_{N}}(z_{N}-z^{\prime})\cdots\phi^{j_{1}}(z_{1}-z^{\prime})A_{SW}^{(i)}(z,z^{\prime})|B\rangle_{i},}\\\end{array}
W[L]=\langle\mathrm{Tr}TP\mathrm{exp}\biggl(\oint_{L}A_{\mu}dx^{\mu}\biggr)\rangle=\langle\mathrm{tr}TP\mathrm{exp}\biggl(\sum_{i=1}^{N}\oint_{C_{i}}A_{\mu}dx^{\mu}\biggr)\rangle.
\vec{B}=\frac{\Phi_{0}}{2\pi\lambda^{2}}\,\vec{b}\\,\\\frac{\vec{E}}{c}=\frac{\Phi_{0}}{2\pi\lambda^{2}}\,\vec{e}\\,\\F_{\mu\nu}=\frac{\Phi_{0}}{2\pi\lambda^{2}}\,f_{\mu\nu}\\,\\A^{0}=\frac{\Phi_{0}}{2\pi\lambda}\,\varphi\\,\\\vec{A}=\frac{\Phi_{0}}{2\pi\lambda}\,\vec{a},
\left\langle0\left|\psi_{n}^{\uparrow}(t_{1})\bar{\psi}_{-n-1}^{\downarrow}(t_{2})\right|0\right\rangle_{E}=\left[\begin{array}{c}{\theta(t_{1}-t_{2})\theta(-n-\frac12)}\\{-\theta(t_{2}-t_{1})\theta(n+\frac12)}\\\end{array}\right]\frac{t_{1}^{n}}{t_{2}^{n+1}},
{\frac{\delta}{\delta\psi_{\alpha}^{a}(x)}}=\intdz\intdy{\frac{\delta\sigma_{\beta\rho}(y,z)}{\delta\psi_{\alpha}^{a}(x)}}{\frac{\delta}{\delta\sigma_{\beta\rho}(y,z)}}=-\intdy\bar{\psi}_{\beta}^{a}(y){\frac{\delta}{\delta\sigma_{\beta\alpha}(y,x)}}
\begin{array}{c}{D_{i}X_{i}^{n}=[n]_{q}\X_{i}^{n-1}}\\{D_{1}=\frac{1}{X_{1}}[X_{1}\partial_{1}]_{q}\q^{-X_{2}\partial_{2}}}\\{D_{2}=\frac{1}{X_{2}}[X_{2}\partial_{2}]_{q}\q^{X_{1}\partial_{1}}}\\\end{array}
F_{12}=\left(\begin{array}{cccc}{F^{1}\otimes1_{N_{1}\timesN_{1}}}&{0}&{\dots}&{0}\\{0}&{F^{2}\otimes1_{N_{2}\timesN_{2}}}&{\dots}&{0}\\{\dots}&{\dots}&{\dots}&{\dots}\\{0}&{\dots}&{\dots}&{F^{K}\otimes1_{N_{K}\timesN_{K}}}\\\end{array}\right)
D_{++}^{ab}(x)=-{\frac{i\delta^{ab}}{(2\pi)^{2}}}\intd^{2}k\,e^{ikx}{\frac{\partial}{\partialk_{-}}}\textrm{P}\left({\frac{1}{k_{-}}}\right)=-i\delta^{ab}{\frac{|x^{-}|}{2}}\delta(x^{+})\,,
S_{|T_{l=a}|}^{WZ}(\eta)=\frac{\kappa}{4\pi\sqrt{-1}}\int_{V_{M}}\widehat{h}^{\ast}\chi_{SU(2)}-\frac{\kappa}{4\pi\sqrt{-1}}\sum_{j=1}^{N_{0}}\int_{U_{\rho^{2}(j)}}\left.\widehat{h}\right|_{U_{\rho^{2}(j)}}^{\ast}\omega_{j},
(\psi^{out},S\phi^{in})=(\psi^{-},\phi^{+})=\sum_{jj_{3}}\int_{(m_{1}+m_{2})^{2}}^{\infty}d{\mathsf{s}}\intd\mu(\hat{\bboxp})\langle\psi^{-}|\hat{\bboxp}j_{3}{\mathsf{s}}j^{-}\rangleS_{j}({\mathsf{s}})\langle^{+}\hat{\bboxp}j_{3}{\mathsf{s}}j|\phi^{+}\rangle
\Psi_{ls=+1}=\frac{1}{\sqrt{4\piE}}\left[\left(\begin{array}{c}{\sqrt{E+m}J_{l}\left(p^{\bot}\rho\right)e^{i(l\varphi)}}\\{0}\\{0}\\{0}\\\end{array}\right)+\left(\begin{array}{c}{0}\\{0}\\{0}\\{i\sqrt{E-m}J_{l+1}\left(p^{\bot}\rho\right)e^{i(l+1)\varphi}}\\\end{array}\right)\right],
\left(\begin{array}{c}{-p_{0}au_{3}i+p_{x}jau_{3}i}\\{p_{y}au_{3}i+p_{z}jau_{3}i}\\{p_{0}au_{1}i-p_{x}jdu_{1}i+p_{y}au_{2}i-p_{z}jdu_{1}i}\\\end{array}\right)=m\;\left(\begin{array}{c}{u_{1}}\\{u_{2}}\\{u_{3}}\\\end{array}\right)
\delta\psi_{M}=(D_{M}+i\frac{e^{\phi}}{4}\,\partial_{M}C)\,\varepsilon+\frac{i}{480}\Gamma^{M_{1}...M_{5}}F_{M_{1}...M_{5}}\Gamma_{M}\varepsilon+i\frac{e^{\frac{1}{2}\phi}}{96}\left(\Gamma_{M}^{PQR}G_{PQR}-9\Gamma^{PQ}G_{MPQ}\right)\varepsilon^{*}
Y_{IJK}^{(\mathcal{R}_{1}\mathcal{R}_{2}\mathcal{R}_{3})}=2\sqrt{2\pi\alpha_{G}}\int_{Z}\Omega\wedgeu_{I}^{x}(\mathcal{R}_{1})\wedgeu_{J}^{y}(\mathcal{R}_{2})\wedgeu_{K}^{z}(\mathcal{R}_{3})f_{xyz}^{(\mathcal{R}_{1}\mathcal{R}_{2}\mathcal{R}_{3})}
\underset{M_{-}}{\overset{M_{+}}{\Pi}}\intdc_{a_{n}}\int_{c_{a_{n}},t^{\prime}}^{c_{a_{n}},t^{\prime\prime}}B_{n}(T){{\mathcal{D}}}c_{n}(t)\;e^{\frac{i}{\hbar}\int_{T}dt(\frac{1}{2}\dot{c}_{n}^{2}-\frac{1}{2}\omega_{n}^{2}c_{n}^{2})}
\begin{array}{lcr}{\{X^{\mu},X^{\nu}\}}&{=}&{\sum_{\stackrel{m_{1},n_{1}}{m_{2},n_{2}}}X_{m_{1}n_{1}}^{\mu}(\sigma)X_{m_{2}n_{2}}^{\nu}(\sigma)\{L_{m_{1}n_{1}},L_{m_{2}n_{2}}\}}\\{}&{=}&{-\sum_{\stackrel{m_{1},n_{1}}{m_{2},n_{2}}}X_{m_{1}n_{1}}^{\nu}(\sigma)X_{m_{2}n_{2}}^{\mu}(\sigma)\{L_{m_{1}n_{1}},L_{m_{2}n_{2}}\}.}\\\end{array}
V(r,u)=-2(8\pi^{2}\alpha^{\prime})^{-1/2}\int_{0}^{\infty}dle^{-r^{2}l/2\pi\alpha^{\prime}}{l}^{1/2}{\frac{\mathrm{tanh}(u)e^{-u^{2}l/\pi}}{\mathrm{Sin}(ul)}}[22+2\mathrm{Cos}(2ul)]\quad.
T[\{k_{i},\epsilon_{\mu_{i}}^{(i)},a_{i}\}\rightarrow\{p_{j},\epsilon_{\nu_{j}}^{(j)},a_{j}\}]=\left.\int\prod_{i}e^{-ik_{i}x_{i}}\epsilon_{\mu_{i}}^{(i)}\frac{\delta}{\deltaa_{\mu_{i}}^{a_{i}}}\prod_{j}e^{ip_{j}y_{j}}\epsilon_{\nu_{j}}^{(j)}\frac{\delta}{\deltaa_{\nu_{j}}^{a_{j}}}{\calF}[a]\right|_{a=0}\,.
\epsilon^{+}(z)=\frac{z}{2\pi}+\frac{i}{\pi}\sum_{n>0}\frac{e^{-in(z-i\varepsilon)}}{n},~~~~\epsilon^{-}(z)=\frac{z}{2\pi}+\frac{i}{\pi}\sum_{n<0}\frac{e^{-in(z+i\varepsilon)}}{n}.
\mathrm{Tr}\,\left[\left(Pe^{-ig\int_{0}^{\beta}dt^{\prime}\,A_{0}(t^{\prime},\vec{x})}\right)B(0,\vec{x})\right]=\mathrm{Tr}\,\left[\left(Pe^{-ig\int_{x_{0}}^{x_{0}+\beta}dt^{\prime}\,A_{0}(t^{\prime},\vec{x})}\right)B(x_{0},\vec{x})\right]
\widetilde{\Psi}_{R}^{c}(x;k)=\left(\begin{array}{c}{\frac{\Psi_{R}^{1}(x)}{\frac{\pi}{L}(k+\frac{1}{2})-gq}}\\{\frac{\Psi_{R}^{2}(x)}{\frac{\pi}{L}(k+\frac{1}{2})+gq}}\\\end{array}\right)^{c},
{\calL}_{J}=\frac{N}{G}\Biggl[(\partial_{\mu}z)^{\dagger}(\partial^{\mu}z)-J_{\mu}J^{\mu}+\thetaG\,\epsilon^{\mu\nu\rho}J_{\mu}\partial_{\nu}J_{\rho}+\frac{1}{4}\theta^{2}G^{2}J_{\mu}(\partial^{2}g^{\mu\nu}-\partial^{\mu}\partial^{\nu})J_{\nu}\Biggr],
\int_{G/H}dx\sqrtgD_{A\zeta,q}^{(n)*}(g_{x}^{-1})D_{A\xi,p}^{(n^{\prime})}(g_{x}^{-1})=Vd_{n}^{-1}d_{D}\delta_{\zeta\xi}\delta_{pq}\delta_{nn^{\prime}}.
\Psi_{k,g}^{[I]_{a}[J]_{b}}=\sum_{g_{1}\inS_{I}}\sum_{g_{2}\inS_{J}}\Psi_{k,g_{1}gg_{2}^{-1}}^{[I][J]}(-1)^{-\hat{Q}_{g_{1}}(k)}e_{a}(g_{1})e_{b}(g_{2}^{-1})\\.
\frac{1}{2}\epsilon^{abcd}\gamma_{5}\gamma_{b}e_{c}^{\mu}e_{d}^{\nu}\left(e_{\mu}^{e}p_{e}+\frac{1}{2}i\nabla_{\mu}\right)W_{\nu\rho}=\hat{X}_{\mu\rho}^{(3/2)~\nu\lambda}W_{\nu\lambda}
E=-\frac{1}{4\pi\alpha^{\prime}}\int_{0}^{2\pi}d\sigma[\partial_{0}t+if(z^{i}\partial_{-}\bar{z}^{i}-\bar{z}^{i}\partial_{-}z^{i}+2i\bar{\psi}_{R}^{i}\psi_{R}^{i})]=-\frac{1}{2}\frac{l}{\alpha^{\prime}}+2fJ_{R}~.
c_{(\pm)k}[T]=\pm{\frac{1}{2\pi\sqrt{k}}}\int_{0}^{2\pi}e^{\pmikx}T_{,x}^{\pm}\left[\sum_{n=1}^{\infty}\sqrt{n}(a_{(\pm)n}e^{-inT^{\pm}(x)}-a_{(\pm)n}^{\dagger}e^{inT^{\pm}(x)})\right]\,dx,
S=\frac{1}{4\pi\alpha\acute{}}\int_{\Sigma}d^{2}\sigma[g^{\alpha\beta}\eta_{\mu\nu}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}+\epsilon^{\alpha\beta}B_{\mu\nu}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}]+\frac{1}{2\pi\alpha\acute{}}\oint_{\partial\Sigma}d\tauA_{i}\partial_{\tau}X^{i},
{\calL}=-\frac12\varphi\partial^{2}\varphi+\frac{i}{2}\overline{\psi}\not\!\partial\psi+\frac12F^{2}+\sigma\varphi_{j}F_{j}+\frac12\lambda\varphi^{2}-\frac12\sigma\overline{\psi}\psi-\overline{\xi}\psi_{j}\varphi_{j}-\frac{N}{2g}\lambda,
{\calV}_{G}(\varphi)={\frac{1}{2}}I_{0}(\mu)-{\frac{\mu^{2}}{4}}I_{1}(\mu)+\int_{-\infty}^{\infty}{\frac{d\alpha}{2\sqrt{\pi}}}e^{-{\frac{\alpha^{2}}{4}}}V({\frac{\alpha}{2}}\sqrt{I_{1}(\mu)}+\varphi)
G^{\mu\nu\mu^{\prime}\nu^{\prime}}=g^{\mu\mu^{\prime}}g^{\nu\nu^{\prime}}+g^{\mu\nu^{\prime}}g^{\nu\mu^{\prime}}-{\frac{2}{D}}g^{\mu\nu}g^{\mu^{\prime}\nu^{\prime}}+Cg^{\mu\nu}g^{\mu^{\prime}\nu^{\prime}}\:.
\tilde{\sigma}_{i}=\left(\begin{array}{cc}{\sigma_{i}}&{0}\\{0}&{\sigma_{i}}\\\end{array}\right)\quad\quad\tilde{C}=\left(\begin{array}{cc}{C}&{0}\\{0}&{-C}\\\end{array}\right)\quad\quad\tilde{\varepsilon}=\left(\begin{array}{cc}{0}&{I_{2\times2}}\\{I_{2\times2}}&{0}\\\end{array}\right)\,.