harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
T_{\mu\nu}=e^{-2\phi}[F_{\mu\rho}F_{\nu}{}^{\rho}-{}^{\star}F_{\mu\rho}{}^{\star}F_{\nu}{}^{\rho}]-2[\nabla_{\mu}\phi\nabla_{\nu}\phi-{\textstyle\frac{1}{2}}g_{\mu\nu}(\nabla\phi)^{2}]\,,
(\hat{V}^{(B=0)})^{-1}\partial_{+}\hat{V}^{(B=0)}={\frac{1-t}{1+t}}\,P_{+},\qquad(\hat{V}^{(B=0)})^{-1}\partial_{-}\hat{V}^{(B=0)}={\frac{1+t}{1-t}}\,P_{-}
L_{\mathrm{eff}}={\frac{\mu}{2}}\dot{r}^{2}+{\frac{\kappa}{4}}\left[{\frac{(\dot{b}+\dot{r})^{2}}{b+r}}+{\frac{(\dot{b}-\dot{r})^{2}}{b-r}}\right]-{\frac{b{\bfJ}^{2}}{2r^{2}(\kappa+\mub)}}-{\frac{bQ^{2}}{2\kappa(b^{2}-r^{2})}}\,\,.
g_{33}(\tau,\tau^{\prime})=\frac{1}{2\omega}\left\{\left[n^{-}+\Theta(\tau-\tau^{\prime})\right]e^{-\omega(\tau-\tau^{\prime})}+\left[n^{+}+\Theta(\tau^{\prime}-\tau)\right]e^{\omega(\tau-\tau^{\prime})}\right\}
\mu\frac{\partial}{\partial\mu}\langleJ_{\mu}(x)J_{\nu}(0)\rangle=\frac{1}{8\pi^{2}}\tilde{b}(\mu)(\Box\delta_{\mu\nu}-\partial_{\mu}\partial_{\nu})\delta^{4}(x)-\frac{\tilde{\beta}(g(\mu))}{2g^{3}}\langleJ_{\mu}(x)J_{\nu}(0)\intd^{4}z\,(F_{\rho\sigma}^{a})^{2}\rangle\,.
\partial_{a}^{3}{\calF}={\frac{\left(a\partial_{a}^{2}{\calF}-\partial_{a}{\calF}\right)^{3}}{4\left[64\pi^{2}b_{1}^{2}\Lambda^{4}+\left({a}\partial_{a}{\calF}-2{\calF}\right)^{2}\right]}},
(\hat{t}\cdot\vec{j})^{2}=A(t)\left[e^{-i\theta}\bar{\psi}_{s}^{\uparrow}\psi_{s}^{\downarrow}+e^{i\theta}\bar{\psi}_{s}^{\downarrow}\psi_{s}^{\uparrow}\right]-2\bar{\psi}_{s}^{\uparrow}\psi_{s}^{\downarrow}
\begin{aligned}{}&{{}\sum_{a}\left(n_{1}^{a}N_{2}^{a}+m_{1}^{a}M_{2}^{a}\right)n_{3}^{a}N_{a}=32,}\\{}&{{}\sum_{a}\left(n_{1}^{a}M_{2}^{a}-m_{1}^{a}N_{2}^{a}\right)M_{3}^{a}N_{a}=\frac{32}{4^{b}},}\\\end{aligned}
\delta^{2}{\calL}=\left(\Lambda_{,m}\epsilon_{(i)}^{m}\partial_{n}\epsilon_{(j)}^{n}+\Lambda\partial_{m}\epsilon_{(i)}^{m}\partial_{n}\epsilon_{(j)}^{n}-\frac{1}{2}\Lambda\partial_{m}\epsilon_{(i)}^{n}\partial_{m}\epsilon_{(j)}^{n}-\frac{1}{2}\Lambda\partial_{m}\epsilon_{(i)}^{n}\partial_{n}\epsilon_{(j)}^{m}\right)\eta^{\mu\nu}\partial_{\mu}\lambda_{(i)}\partial_{\nu}\lambda_{(j)}.
\left[a_{\left[k\right]}^{\mu_{m}}\left(p\right),a_{\left[l\right]}^{+\mu_{n}}\left(q\right)\right]=\pm\delta^{mn}\eta_{\left[k\right]\left[l\right]}\frac{\left|\lambda_{\left[k\right]}^{\mu_{m}}\left(p\right)\right|}{\lambda_{\left[k\right]}^{\mu_{m}}\left(p\right)}
H^{(1)}=\frac{a_{1}}{\rho^{2}}{\bfL}^{2}+\frac{a_{2}}{2\tilde{\mu}^{3}}\left|{\bfr}\right|\left({\bfp}_{r}^{2}\right)^{\frac32}+\frac{a_{3}}{\tilde{\mu}^{4}}\left({\bfp}_{r}^{2}\right)^{2}+\frac{a_{4}}{2\tilde{\mu}\rho^{2}}\frac{\sqrt{{\bfp}_{r}^{2}}{\bfL}^{2}}{\left|{\bfr}\right|}+\frac{a_{5}}{2\tilde{\mu}^{2}\rho^{2}}\frac{{\bfp}_{r}^{2}{\bfL}^{2}}{{\bfr}^{2}},
T_{1}^{00}\geq\pmT_{1}~\frac{i}{2}\mathrm{Tr}\left[\varepsilon_{\mu\nu\rho}DX_{\mu}[X_{\nu},X_{\rho}]\right]=T_{1}~\frac{1}{2}\mathrm{Tr}\left[DX_{\mu}DX_{\mu}-\frac{1}{2}[X_{\mu},X_{\nu}]^{2}\right],
e^{-\frac{i}{2\hbar}\omega_{ij}\hat{L}_{ij}}\,{\alpha_{i}^{\pm}}^{\dagger}\,e^{\frac{i}{2\hbar}\omega_{ij}\hat{L}_{ij}}={\alpha_{j}^{\pm}}^{\dagger}\,R_{ji}\left(\omega_{ij}\right)\\\,
F=\frac{\zeta}{\delta(\delta+\zeta/2)(\delta+\zeta)}\biggl\{f_{3}(dz_{2}\wedged\bar{z}_{2}-dz_{1}\wedged\bar{z}_{1})+f_{+}d\bar{z}_{1}\wedgedz_{2}+f_{-}d\bar{z}_{2}\wedgedz_{1}\biggr\}\,,
\Phi_{1}^{(2)}=\pi^{2}-m^{2},\;\Phi_{2}^{(2)}=\pi_{\mu}\psi_{a}^{\mu}+sm\psi_{a}^{3}\;,\;\;\Phi_{3}^{(2)}=\varepsilon^{\mu\nu\lambda}\pi_{\mu}\psi_{a\nu}\psi_{a\lambda}+\frac{i}{2}sm\;,
V(\phi|g)=g\left\{\frac{1}{4!}\int\mathrm{d}^{D}x\;\phi(x)^{4}+\frac{\zeta^{(1)}}{2}\int\mathrm{d}^{D}x\;\phi(x)(-\bigtriangleup)\phi(x)\right\}+O(g^{2}),
h_{\:,\nu}^{\mu\nu}=0\longrightarrow{h^{\prime}}_{\:,\nu}^{\mu\nu}=h_{\:,\nu}^{\mu\nu}+\underbrace{\Boxf^{\mu}}_{=0}+\underbrace{f_{,\mu\nu}^{\nu}-f_{,\mu\gamma}^{\gamma}}_{=0}=0\,.
\begin{array}{ll}{\mathrm{BPS}_{1/2}(3)=[W(3)]^{m+n}}&{\mathrm{if~j=0~}}\\{\mathrm{BPS}_{1/4}(3)=[W(3)]^{m+n-j}\;[W^{-}(3)]^{j}}&{\mathrm{if~j\neq0~}}\\\end{array}
E^{2}=\frac{1}{p^{2}+\left(\frac{4\pi^{2}\alpha^{\prime}}{L_{1}L_{2}}\right)^{2}q^{2}}\left(\left(\frac{2\pi}{L_{1}}n_{1}\right)^{2}+\left(\frac{2\pi}{L_{2}}n_{2}\right)^{2}\right)
{\mathrm{Tr}}\,\frac{1}{2}\Big(1\pm(-)^{b_{3}}\Big)\times\frac{1}{2}\Big(1-(-)^{b_{2}}\,C\Big[\begin{array}{c}{b_{1}}\\{b_{2}}\\\end{array}\Big]\,\Big)R^{b_{1}}NS^{F+b_{1}}.
\alpha_{(1)}^{\prime}(0,t)=\alpha_{(1)}^{(1)}=\frac{\sqrt{2}}{3}\sigmad(1-6b^{2})f_{\alpha}(t)+g_{\alpha}(t)+k_{\alpha}^{\prime}(0,t)-\frac{\sqrt{2}}{3}\sigmad(3b^{2}-2)K_{\alpha}(t)+P^{\prime}(0,t).
\acute{o}^{A}\bar{o}^{A^{\prime}}+o^{A}\acute{\bar{o}}{}^{A^{\prime}}=o^{A}\dot{\bar{\iota}}{}^{A^{\prime}}+\dot{o}^{A}\bar{\iota}{}^{A^{\prime}}+\iota^{A}\dot{\bar{o}}{}^{A^{\prime}}+\dot{\iota}^{A}\bar{o}{}^{A^{\prime}}.
G(\vec{a},\vec{a}^{\,*},\lambda)=G_{0}(\lambda)+\sum_{k}\left(\tilde{g}_{k}(\lambda)a_{k}^{*}+\tilde{g}_{k}^{*}(\lambda)a_{k}\right)+\sum_{l}\sum_{m}G_{lm}(\lambda)a_{l}^{*}a_{m},
\Gamma_{\alpha}(A)={\frac{1}{T}}(y+2\pi\alpha)\prod_{n=1}^{\infty}{{({\frac{-2\pi}{T}})}^{2}n^{2}}\prod_{n=1}^{\infty}\biggl[1-{\frac{{{({y+2\pi})\alpha}}^{2}}{4n^{2}}}\biggr],
T_{\sigma^{-}\sigma^{-}}\left(\sigma^{-}\right)={\left(\frac{dy^{-}}{d\sigma^{-}}\right)}^{2}T_{y^{-}y^{-}}\left(y^{-}\right)-{\frac{c}{24}}D_{\sigma^{-}}^{S}\left(y^{-}\right)\;,
I_{1}=\frac{1}{4\pi^{2}r^{4}}(1-\eta^{2})\sum_{n=1}^{\infty}\sum_{l=0}^{\infty}(2l+1)\Xi_{n}(1-\eta^{2},\xi,\zeta)Q_{\nu_{l}-1/2}^{(2)}(z_{n}-\eta^{2}\lambda_{n}^{2}).
\begin{array}{ccccc}{EG}&{\leftarrow}&{EG\timesM}&{\rightarrow}&{M}\\{\downarrow}&{}&{\downarrow}&{}&{\downarrow}\\{BG}&{\leftarrow}&{EG\times_{G}M}&{\rightarrow}&{M/G.}\\{}&{\pi}&{}&{\sigma}&{}\\\end{array}
\mathbf{s}_{m}^{a}=\left(\mathbf{s}_{m}^{a}\phi^{A}\right)\frac{\delta_{L}}{\delta\phi^{A}}+V_{m}^{a},\qquad\mathbf{d}_{\alpha}=\left(\mathbf{d}_{\alpha}\phi^{A}\right)\frac{\delta_{L}}{\delta\phi^{A}}+V_{\alpha},\qquad\mathbf{d}_{m}=\left(\mathbf{d}_{m}\phi^{A}\right)\frac{\delta_{L}}{\delta\phi^{A}}+V_{m}.
\frac{1}{2}\langle\langle\eta_{0}({\bfk}_{n},t)\eta_{0}(-{\bfk}_{n},t^{\prime})+\eta_{0}(-{\bfk}_{n},t^{\prime})\eta_{0}({\bfk}_{n},t)\rangle\rangle_{T\rightarrow\infty}\rightarrow2L^{3}\GammaT\delta(t-t^{\prime}),
\psi_{j}(x)=\int\frac{dp_{j}^{1}}{\sqrt{4\pi}p_{j}^{0}}\left(a_{j}(p)u_{j}(p)e^{-ip_{j}\cdotx}+a_{_{\bar{\jmath}}}^{\dagger}(p)v_{j}(p)e^{ip_{j}\cdotx}\right)\,,\qquad
{\calT}^{a}\equiv\left(\begin{matrix}{(-T^{a})^{*}}&{0}\\{0}&{T^{a}}\\\end{matrix}\right),\qquad\sigma^{1}\equiv\left(\begin{matrix}{0}&{1_{d}}\\{1_{d}}&{0}\\\end{matrix}\right),\qquad\sigma^{3}\equiv\left(\begin{matrix}{1_{d}}&{0}\\{0}&{-1_{d}}\\\end{matrix}\right),
w_{q}^{2}(t)=q^{2}\,\frac{1+Y_{\Lambda}}{1+Z_{\Lambda}}+\frac{V_{\Lambda}^{\prime\prime}}{1+Z_{\Lambda}}+\frac{1}{4}\frac{Z_{\Lambda}^{2}}{(1+Z_{\Lambda})^{2}}\,\dot{\phi}^{2}+\frac{1}{2}\frac{Z_{\Lambda}^{\prime}}{1+Z_{\Lambda}}\,\ddot{\phi}.
|\xi)_{q}=e_{q}^{\xiK_{q}^{+}}|k;0>=e^{\xiT_{K}^{+}}|k;0>=\sum_{n=0}^{\infty}\frac{[2k+n+1]!}{[n]![2k+1]!}\xi^{n}|k;n>=e^{\xiT_{K}^{+}}e^{-\xiK^{+}}|\xi)\,
S={\textstyle\frac{1}{16\piG_{N}^{(11)}}}\int\left\{\Omega_{(11)}\hat{R}-{\textstyle\frac{1}{2}}\hat{G}\wedge{}^{\star}\hat{G}+{\textstyle\frac{1}{6}}d\hat{C}\wedged\hat{C}\wedge\hat{C}\right\}\,.
{\sfC}_{n}=\frac{(-1)^{(n-1)}}{2}\sum_{s=0}^{\left[\frac{n-1}{2}\right]}\frac{(n-s-1)!(n-4s-3)}{(n-2s-1)!(s+1)!}L^{2s}\;\;\;\mathrm{for}\;\;\;n{\geq}2\,,
e^{-\frac{1}{\hbar}\left(L^{\Lambda,\Lambda_{0}}(\varphi)+I^{\Lambda,\Lambda_{0}}\right)}=e^{-\frac{1}{2}\langle\varphi,(\hbarC^{\Lambda,\Lambda_{0}})^{-1}\varphi\rangle}Z^{\Lambda,\Lambda_{0}}((C^{\Lambda,\Lambda_{0}})^{-1}\varphi)
2[\dot{x}^{2}\acute{x}^{AA^{\prime}}-(\dot{x}\acute{x})\dot{x}^{AA^{\prime}}]=v\bar{\pi}^{A}\bar{\pi}_{B}\dot{x}^{BA^{\prime}}+\bar{v}\pi^{A^{\prime}}\pi_{B^{\prime}}\dot{x}^{AB^{\prime}}.
P_{\mathcal{A}}^{N_{1}\ldotsN_{4}\,\Lambda,N_{1}^{\prime}\ldotsN_{4}^{\prime}\,\Lambda^{\prime}}\=:\<\phi_{\mathcal{A}}^{N_{1}\ldotsN_{4}\,\Lambda},\\phi_{\mathcal{A}}^{N_{1}^{\prime}\ldotsN_{4}^{\prime}\,\Lambda^{\prime}}>\=\\frac{1}{4!}\sum_{\sigma}\\\delta^{N_{1}N_{\sigma(1)}^{\prime}}\ldots\delta^{N_{4}N_{\sigma(4)}^{\prime}}\M_{\sigma}^{N_{1}\ldotsN_{4}\,}{}_{\Lambda^{\prime}}^{\Lambda}
\begin{array}{ccl}{\tilde{X}_{rs}^{\mu}}&{\Longrightarrow}&{\tilde{X}^{\mu}(p)\equivA^{\mu}(p)}\\{\tilde{X}_{rs}^{\mu}}&{\Longrightarrow}&{\tilde{X}^{m}(p)\equiv\Phi^{m}(p)}\\{\psi_{rs}}&{\Longrightarrow}&{\psi(p)}\\\end{array}
\left[a_{\left[k\right]}^{\mu_{m}}\left[\varphi\right],a_{\left[l\right]}^{+\mu_{n}}\left[\chi\right]\right]=\pm\delta^{mn}\eta_{\left[k\right]\left[l\right]}\frac{\left|\lambda_{\left[k\right]}^{\mu_{m}}\left(p\right)\right|}{\lambda_{\left[k\right]}^{\mu_{m}}\left(p\right)}\intd^{2}\vec{p}\varphi\left(p\right)\chi^{*}\left(p\right)\;\,,
\left({\frac{\dot{a}}{a}}\right)^{2}={\frac{1}{\tilde{L}^{2}}}\left[(\tilde{E}^{2}-\tilde{\ell}^{2})\left({\frac{\tilde{L}}{a}}\right)^{2(p+1)}-\tilde{\ell}^{2}\left({\frac{\tilde{L}}{a}}\right)^{2p}+2\tilde{E}\left({\frac{\tilde{L}}{a}}\right)^{p+1}-\left({\frac{\tilde{L}}{a}}\right)^{2}\right],
W=\sum_{k}M_{k}q\bar{q}(a\bar{a})^{N-1-k}+\sum_{m}\Big(H_{m}q^{2}a(a\bar{a})^{N-2-m}+\bar{H}_{m}\bar{q}^{2}\bar{a}(a\bar{a})^{N-2-m}\Big)
\int{\calL}=\int\psi(x)^{3}=\int{\frac{d^{4}k_{1}}{(2\pi)^{4}}}{\frac{d^{4}k_{2}}{(2\pi)^{4}}}{\frac{d^{4}k_{3}}{(2\pi)^{4}}}\tilde{\psi}(k_{1})\tilde{\psi}(k_{2})\tilde{\psi}(k_{3})(2\pi)^{4}\delta^{4}(k_{1}+k_{2}+k_{3})
F_{6}^{{\mathcal{O}}_{1,1}^{1,0}|+++++-}=\frac{-H^{1,1}}{4\,\prod_{1\leqi<j\leq5}\mathrm{cosh}(\theta_{ij}/2)}\prod_{i<j}F_{\mathrm{min}}^{\mu_{i}\mu_{j}}(\theta_{ij}).
\begin{array}{c}{\left[\begin{array}{c}{\xi_{Y}}\\{\xi_{Z}}\\\end{array}\right]\stackrel{J}{\rightarrow}i\left[\begin{array}{cc}{-{\frac{2+\rho}{1+\rho}}YZ^{*}}&{-{\frac{2+|y|^{-2}+\rho}{1+\rho}}YY^{*}}\\{{\frac{2+|z|^{-2}+\rho}{1+\rho}}ZZ^{*}}&{{\frac{2+\rho}{1+\rho}}ZY^{*}}\\\end{array}\right]\left[\begin{array}{c}{\xi_{Y}^{*}}\\{\xi_{Z}^{*}}\\\end{array}\right]}\\\end{array}
I=iR\int_{0}^{\rho}sds\int\frac{d^{4}p}{(2\pi)^{4}}\frac{1}{(p^{2}-s^{2})^{2}}\rightarrow-R\int_{0}^{\rho}sds\int\frac{d^{4}p}{(2\pi)^{4}}\frac{1}{(p^{2}+s^{2})^{2}}.
S=\intd^{D}xdt\,\,{\calL}\,\Bigl(\phi,\,\,\partial_{\nu_{1}}\phi\,,\,\,\partial_{\nu_{1}}\partial_{\nu_{2}}\phi\,,\,\,\partial_{\nu_{1}}\partial_{\nu_{2}}\partial_{\nu_{3}}\phi\,,\,\,\cdots\Bigr)\,,
m_{A}^{\phantom{A}B}=\frac{1}{2!}({\calP}_{+}\Gamma^{MN})_{A}^{\phantom{A}B}C_{MN}+\frac{1}{6!}({\calP}_{+}\Gamma^{M_{1}\ldotsM_{6}})_{A}^{\phantom{A}B}Z_{M_{1}\ldotsM_{6}}^{+}\;.
+\left(\frac{{\lambda}q+{\lambda}^{-1}q^{-1}}{2}+d_{0}d_{1}\right)\;\left[(S_{k}^{X}S_{k+1}^{X}+S_{k}^{Y}S_{k+1}^{Y})S_{k}^{Z}S_{k+1}^{Z}+\leftrightarrow\right]\,
\sigma_{\mathrm{abs}}={\frac{\kappa_{5}^{2}L_{\mathrm{eff}}}{(32\piT_{\mathrm{eff}})^{2}}}(\omega^{2}+16\pi^{2}T_{L}^{2})(\omega^{2}+16\pi^{2}T_{R}^{2})\\omega{\frac{e^{\frac{\omega}{T_{H}}}-1}{(e^{\frac{\omega}{2T_{L}}}-1)(e^{\frac{\omega}{2T_{R}}}-1)}}
{\mathcal{L}}=\frac{q}{p}\beta^{2}\left(\frac{1}{\beta}A_{\mu}A^{\mu}\right)^{\frac{p}{q}}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\frac{\chi\theta}{2}\varepsilon^{\mu\nu\rho\lambda}\partial_{\mu}B_{\nu\rho}A_{\lambda}\;,
{\tilde{B}}(\mathrm{P},{\Sigma}):=\int_{\Sigma}G(\sigma,t)\epsilon_{\mu\nu\rho\lambda}F^{\rho\lambda}(\sigma,t)T^{\mu}(\sigma,t)S_{\tau}(\sigma,t)S^{\tau}(\sigma,t)d{X_{\sigma,t}^{\nu}}G^{-1}(\sigma,t)d\sigmadt.
{\calB}_{WZW}^{(\varphi,t)}=\frac{\ell}{8\piG}\int_{\calV}Tr[\partial_{\varphi}(Q_{2}^{-1}\partial_{[r}Q_{2}Q^{-1}\partial_{t]}Q)+\partial_{t}(Q_{2}^{-1}\partial_{[\varphi}Q_{2}Q^{-1}\partial_{r]}Q)]dr\,d\varphi\,dt\quad.
F_{\mathrm{D}p}=-c_{p}V_{p}N^{2}\lambda^{-\frac{p-3}{p-5}}T^{\frac{2(7-p)}{5-p}}\left[1+\frac{S_{p}^{1}}{\pi^{2}}\sum_{i}\omega_{i}^{2}+\frac{S_{p}^{2}}{\pi^{4}}\left(\sum_{i}\omega_{i}^{2}\right)^{2}+\frac{S_{p}^{3}}{\pi^{4}}\sum_{i}\omega_{i}^{4}+\ldots\right]
\begin{array}{c}{\partial_{x}\Psi\left(x,0\right)=-l_{x}\Psi\left(x,0\right)=-g^{-1}\partial_{x}g\Psi\left(x,0\right),}\\{\mathrm{so~that~modulo~left~actions~of~constant~loops~}\Psi\left(x,0\right)=g^{-1}\left(x\right).}\\\end{array}
\intd^{4}x\sqrt{-g}\frac{1}{2\kappa^{2}}\sqrt{-g}(R(g)+\overline{{\hat{\psi}}}^{\mu}\hat{\gamma}_{\mu\nu\rho}\hat{D}^{\nu}\hat{\psi}^{\rho}+\frac{3}{2}\overline{{\hat{\psi}_{5}}}\hat{\gamma}_{\mu}\hat{D}^{\nu}\hat{\psi}_{5}).
G(\lambda,\lambda^{\prime},\tau,\bar{\tau})=\left(\frac{2\pii\lambda^{+}\bar{\eta}^{3}}{\bar{\Theta}_{1}(\lambda^{+},\bar{\tau})}\right)\left(\frac{2\pii\lambda^{-}\bar{\eta}^{3}}{\bar{\Theta}_{1}(\lambda^{-},\bar{\tau})}\right)e^{-\frac{\pi(\lambda^{2}+\lambda^{2})}{\tau_{2}}}
\sum_{\alpha\in\Gamma}{\overline{\chi}_{0}}^{\dagger}{\gamma_{\lambda}}^{\dagger}\chi_{\alpha+\lambda}(\chi_{0})e^{\bar{\alpha}\cdot\lambda\phi(1)/3}e^{\bar{\alpha}\cdot\phi(a)/\sqrt3}=\sum_{\alpha\in\Gamma}{\overline{\chi}_{0}}^{\dagger}{\gamma_{\lambda}}^{\dagger}\chi_{-\alpha+\lambda}(\chi_{0})e^{-\bar{\alpha}\cdot\phi(a)/\sqrt3}\,.
\bar{\omega}_{1}\equiv\sum_{n=-\infty}^{\infty}\breve{\omega}_{1}(t,x^{1},x^{2},x^{3},x^{4},x^{5}-n)\simeq-{\frac{2\pim_{0}a}{3\bar{R}^{3}}}x^{2}\,,\qquad\bar{\omega}_{2}\simeq0\,,
S_{CS}[A^{I}]\;=\;\frac{\kappa}{2}\,\langleA_{\mu}^{I}|R_{\mu\nu}|A_{\nu}^{I}\rangle\,=\,\frac{\kappa}{2}\,\langleA_{\mu}|\,{\mathcalI}\,R_{\mu\nu}\,{\mathcalI}\,|A_{\nu}\rangle\;=\;-S_{CS}[A]\;,
W_{LG}=\sum_{i=0,1,2,5,6}e^{-Y_{i}}+e^{-{\frac{t_{1}}{2}}}e^{-{\frac{Y_{1}+Y_{2}}{2}}}+e^{-{\frac{t_{1}}{2}}-t_{2}}e^{{\frac{Y_{1}+Y_{2}}{2}}-2[6Y_{0}-2Y_{5}-3Y_{6}]}+e^{-t_{3}}e^{-[6Y_{0}-2Y_{5}-3Y_{6}]}.
S_{m,\mathrm{ext}}^{(0)}=\tilde{Y}_{m}+\mathrm{\large~\frac{1}{4}~}(1-\tilde{\rho})^{-1}\tilde{\rho}\xi^{-1}g^{\alpha\beta}\frac{\delta\tilde{Y}_{m}}{\deltaB^{\alpha}}\frac{\delta\tilde{Y}_{m}}{\deltaB^{\beta}},
r_{4}=r_{3}\frac{\big|\sqrt{2a^{2}-b}-\sqrt{2}|a|\big|^{\frac{|a|}{\sqrt{2(2a^{2}-b)}}+\frac{1}{2}}}{\big(\sqrt{2a^{2}-b}+\sqrt{2}|a|\big)^{\frac{|a|}{\sqrt{2(2a^{2}-b))}}-\frac{1}{2}}}
\left[A_{\kappa\rho},A^{\lambda\sigma}\right]=\delta_{\kappa}^{\lambda}{M^{\sigma}}_{\rho}+\delta_{\kappa}^{\sigma}{M^{\lambda}}_{\rho}+\delta_{\rho}^{\lambda}{M^{\sigma}}_{\kappa}+\delta_{\rho}^{\sigma}{M^{\lambda}}_{\kappa}\,.
\intd\thetad\psid\phi\quad\left[sin\theta\,D_{mn}^{l}\,D_{m_{1}n_{1}}^{l_{1}}\,D_{m_{2}n_{2}}^{l_{2}}\right]=8\pi^{2}\times\left(\begin{array}{rrr}{l}&{l_{1}}&{l_{2}}\\{m}&{m_{1}}&{m_{2}}\\\end{array}\right)\times\left(\begin{array}{rrr}{l}&{l_{1}}&{l_{2}}\\{n}&{n_{1}}&{n_{2}}\\\end{array}\right).
{\calM}_{10}=\sum_{\sigma}\intd\Omegad^{n-2}p\,\sigma\,\Omega\,[d_{p}^{(\sigma)\dagger}d_{p}^{(\sigma)}+\bar{d}_{\tilde{p}}^{(\sigma)}\bar{d}_{\tilde{p}}^{(\sigma)\dagger}]\,{.}
\displaystyle\widetilde{\calH}_{\scriptscriptstyleK}=\frac{1}{(k+1)!}\Biggl(\lambda_{j}K_{j_{1}j_{2}\cdotsj_{k+1}}^{j}+i\bar{c}_{j}(\partial_{d}K_{j_{1}j_{2}\cdotsj_{k+1}}^{j}c^{d})\Biggr)c^{j_{1}}\cdotsc^{j_{k+1}}
\frac{1}{(2\pi)^{3}}\bar{u}(p^{\prime},s_{3}^{\prime})\left(\gamma_{\mu}F(k^{2})-\frac{i}{2m}(p^{\prime}+p)G(k^{2})+\frac{1}{2m}k_{\mu}H(k^{2})\right)u(p,s_{3})
{\frac{d^{2}t}{d\tilde{\lambda}^{2}}}+{\frac{\dot{n}}{n}}\left({\frac{dt}{d\tilde{\lambda}}}\right)^{2}+{\frac{a\dot{a}}{n^{2}}}\sum_{j}\left({\frac{dx^{j}}{d\tilde{\lambda}}}\right)^{2}=\left[nn^{\prime}\left({\frac{dt}{d\tilde{\lambda}}}\right)^{2}-aa^{\prime}\sum_{j}\left({\frac{dx^{j}}{d\tilde{\lambda}}}\right)^{2}-2{\frac{n^{\prime}}{n}}\right]{\frac{dt}{d\tilde{\lambda}}},
S=\frac{1}{2}\intd^{4}x\int_{-1/\alpha}^{1/\alpha}\left(-e^{-2\sigma(z)}\delta^{IJ}\partial_{I}\Phi\partial_{J}\Phi+\Phi\partial_{z}\left(e^{-4\sigma(z)}\partial_{z}\Phi\right)+M_{5}^{2}e^{-4\sigma(z)}\Phi^{2}\right)
S_{WZ}^{NF1}=\mu_{1}\int_{R^{1+1}}\mathrm{Tr}\left(B^{(2)}+i(2\pi\alpha^{\prime})i_{\Phi}i_{\Phi}C^{(4)}-\frac12(2\pi\alpha^{\prime})^{2}(i_{\Phi}i_{\Phi})^{2}B^{(6)}+\dots\right)\,.
\Phi=\frac{4\pi\nu}{e}\int_{0}^{\infty}\frac{(2\nur^{2\nu-1})}{(1+r^{2\nu})^{2}}dr=\frac{4\pi\nu}{e}\int_{1}^{\infty}\frac{dw}{w^{2}}=\frac{4\pi\nu}{e}~~,
\xi^{i}=-{\frac{\rhodx_{0}^{i}}{D^{3/2}}}-{\frac{(x-x_{0})^{i}[2\rho^{2}+(x-x_{0})^{2}]d\rho}{\rho^{2}D^{3/2}}},\quad\quad\omega_{(0,1)}^{u}=q^{t}I_{u}\xi=-{\frac{(x-x_{0})^{t}I^{u}dx_{0}}{D^{2}}}.
{\bf70}\,\stackrel{Usp(4)\,\times\,SU(4)\,\times\,U(1)}{\longrightarrow}\,\left({\bf1},{\bf1},{\bf1}+{\bar{\bf1}}\right)\,\oplus\,\left({\bf5},{\bf6},{\bf1}\right)\,\oplus\,\left({\bf1},{\bf6},{\bf1}\right)\,\oplus\,\left({\bf4},{\bf4},{\bf1}\right)\,\oplus\,\left({\bf4},{\bf4},{\bf1}\right)
\widetildeV_{1}(\psi,z)=:\prod_{a=1}^{M}{\frac{i}{(m_{a}-1)!}}{\frac{d^{m_{a}}}{dz^{m_{a}}}}C_{+}^{i_{a}}(z)\prod_{b=1}^{N}{\frac{i}{(n_{b}-1)!}}{\frac{d^{n_{b}}}{dz^{n_{b}}}}C_{-}^{j_{b}}(z)e^{i{\frac{\bar{\lambda}}{\alpha}}\cdotC_{+}(z)}e^{i{\frac{\lambda}{\alpha}}\cdotC_{-}(z)}:\gamma_{\lambda}\,,
eT_{\mu\nu}={\tilde{\overline{\Psi}}_{L_{0}}}{\gamma_{\mu}}i\left(D_{\nu}-\frac{1}{2}\Gamma_{\alpha\nu}^{\alpha}\right){\tilde{\Psi}}_{L_{0}}-\frac{i}{2}g_{\mu\nu}\partial_{\alpha}({\tilde{\overline{\Psi}}}_{L_{0}}\gamma^{\alpha}{\tilde{\Psi}_{L_{0}}}).\nonumber\,
\sum_{\ell=1}^{\infty}p^{\ell}\ell^{k-1}\sum_{\substack{ad=\ell\,a>0}}\sum_{b=0}^{d-1}d^{-k}\sum_{n,\boldsymbol{\gamma}}D(n,\boldsymbol{\gamma})\mathbf{e}\left[bn/d\right]q^{na/d}(\boldsymbol{\zeta}^{\boldsymbol{\gamma}})^{a}\,.
u_{1\epsilon\kappa}^{\prime\prime}-\left[\frac{\kappa\left(\kappa+1\right)}{r^{2}}-\frac{\kappa}{r}\frac{V^{\prime}}{\epsilon+m-V}-V^{2}+2\epsilonV-k^{2}\right]u_{1\epsilon\kappa}+\frac{V^{\prime}}{\epsilon+m-V}\,u_{1\epsilon\kappa}^{\prime}=0\,.
V_{i}\cdotV_{j}=\left(\begin{array}{ccc}{-1}&{-1/2}&{-1/2}\\{-1/2}&{-1/3}&{-1/3}\\{-1/2}&{-1/3}&{0}\\\end{array}\right)~,\quad\quadk_{ij}=\left(\begin{array}{ccc}{0}&{0}&{0}\\{1/2}&{0}&{0}\\{1/2}&{2/3}&{1/2}\\\end{array}\right)~.
Z_{m}(\kappa)={\frac{me^{-1/(4\kappa)}}{2^{(1+(-1)^{m})/2}\sqrt{2\pi}(2\kappa)^{1/2-1/m}}}\int_{-\infty}^{+\infty}\!dx\,e^{x^{m}-\kappax^{2m}}\,,
\varepsilon_{0}^{l}=-\frac{2}{(2\pi)^{2}}\left[\frac{\mu^{s+1}}{\Gamma(s/2)}\int_{0}^{\infty}dtt^{s/2-1}e^{-m^{2}t}\int_{-\infty}^{\infty}dp_{x}e^{-p_{x}^{2}t}\int_{-\infty}^{\infty}dp_{y}e^{-p_{y}^{2}t}\right]_{s=-1},
\int_{G}D\mu(\omega)f(-i\omega^{-1}(x)\partial_{\mu}\omega(x))=\int_{G}D\mu(\omega)f(-i\omega^{-1}(x)\partial_{\mu}\omega(x)-i\omega_{0}^{-1}(x)\partial_{\mu}\omega_{0}(x)),\forall\omega_{0}\inG
\sum_{n=\pm1,\pm2,\cdot\cdot\cdot}\frac{\pi}{L}\intdp^{-}\frac{F(p^{+}p^{-})}{2p^{+}p^{-}-m^{2}+i0}=C\sum_{n=\pm1,\pm2,\cdot\cdot\cdot}\frac{\pi}{L}\delta(p_{n}^{+})=0,
c\frac{d^{2}x_{k}^{\mu}}{ds_{k}^{2}}=-\frac{1}{c}\eta^{\mu\mu}\sum_{{j=1}\atop{j\neqk}}^{n}\sum_{\nu=0}^{3}F_{\mu\nu}(x_{k};x_{j})\frac{dx_{k}^{\nu}}{ds_{k}},
Z[J(x)]=\int{\mathcalD}[\phi]\,e^{i\intd^{4}x\,[\frac{1}{2}\partial^{\mu}\phi(x)\partial_{\mu}\phi(x)-\frac{m^{2}}{2}\phi^{2}(x)-\frac{\lambda}{24}\phi^{4}(x)+\frac{1}{2}J(x)\phi(x)+\frac{1}{2}\phi(x)J(x)]}~.
S_{gf}+S_{gh}=\frac{1}{e^{2}}\mathrm{tr}\intd^{4}x\Big(\alphaB^{2}-2\partial_{\mu}B\,A^{\mu}-i\alphaB\{c\,\bar{c}\}-2i\partial_{\mu}{\bar{c}}{\calD}^{\mu}c+\alphac^{2}{\bar{c}}^{2}\Big),
\langle+\infty|-\infty\rangle_{\lambda+\delta\lambda}=\langle+\infty|-\infty\rangle_{\lambda}-\delta\lambda\frac{1}{\tilde{N}}{\int}d\mu[B(t)X^{\omega}(t)+ig\bar{c}(t)Y^{\omega}(t)c(t)]\exp\{i{\int}({\calL}+{\calL}_{g})dt\}
\frac{\partialc^{\mu}(x,x_{0},\tau)}{\partialx^{\rho}}\vert_{\tau\rightarrow\tau_{i}}+\frac{\partialc^{\mu}}{\partial{\tau}}\vert_{\tau\rightarrow\tau_{i}}\frac{\partial\tau_{i}(x)}{\partialx_{\rho}}=\frac{\partialc^{\mu}(x,x_{0},\tau_{i}(x))}{\partialx^{\rho}}
\left(A_{\mu}^{a},{\lambda}_{i\;\alpha}^{a},{C}^{a}\right)\sim\left(J^{a}({\bar{z}})\;{\partialX}_{\mu}(z),J^{a}(\bar{z})\,V_{i\,\alpha}(z),J^{a}(\bar{z})\;{\partialX}^{\pm}(z)\right)
b_{1}=-k^{2}D^{2}\sqrt{\frac{g_{2}^{2}}{X(g_{1}+g_{2})p_{3}}},\qquadb_{2}=\frac{g_{2}D^{2}}{\sqrt{Xg_{1}p_{1}}},\qquadb_{3}=\frac{g_{1}D^{2}k^{\prime2}}{\sqrt{Xg_{2}p_{2}}},
\mathrm{tr}\left[e^{-tm^{2}}f_{n}(t,A)\right]=\frac{t^{n}}{n!}\sum_{i_{1},i_{2},\ldots,i_{n}}^{3}c_{i_{1}i_{2}\ldotsi_{n}}(t)\mathrm{str}\left(A_{i_{1}},A_{i_{2}},\ldots,A_{i_{n}}\right),
\frac{1}{4}[\hat{G}_{\mu},[\hat{G}_{\mu},\hat{Y}_{r_{1}r_{2}}]]=-\frac{1}{8}[\hat{G}_{\mu\nu},[\hat{G}_{\mu\nu},\hat{Y}_{r_{1}r_{2}}(\hat{G}_{\mu},\hat{G}_{\mu\nu})]]-2r_{2}(r_{2}+1)\hat{Y}_{r_{1}r_{2}}.
F^{k_{s}}=dA^{k_{s}}+\frac{1}{2}c_{i_{p}\;j_{q}}^{k_{s}}\;A^{i_{p}}\wedgeA^{j_{q}}\quad,\quaddF^{k_{s}}=c_{i_{p}\;j_{q}}^{k_{s}}\;F^{i_{p}}\wedgeA^{j_{q}}\quad,
B=\left(\begin{array}{ccccccc}{-\beta_{-n}}&{b_{-n}}&{0}&{}&{}&{}&{}\\{b_{-n}}&{-\beta_{-n+1}}&{b_{-n+1}}&{}&{}&{}&{}\\{0}&{b_{-n+1}}&{-\beta_{-n+2}}&{}&{}&{}&{}\\{}&{}&{}&{\ddots}&{}&{}&{}\\{}&{}&{}&{}&{-\beta_{n-2}}&{b_{n-1}}&{0}\\{}&{}&{}&{}&{b_{n-1}}&{-\beta_{n-1}}&{b_{n}}\\{}&{}&{}&{}&{0}&{b_{n}}&{-\beta_{n}}\\\end{array}\right),
\biggl(\sum_{i}(N_{i}^{(1)}+N_{i}^{(2)})^{\frac{3}{2}}\biggr)^{\frac{2}{3}}\leq\biggl(\sum_{i}(N_{i}^{(1)})^{\frac{3}{2}}\biggr)^{\frac{2}{3}}+\biggl(\sum_{i}(N_{i}^{(2)})^{\frac{3}{2}}\biggr)^{\frac{2}{3}}~,
D_{i\alpha}\tilde{\mathrm{A}}^{\beta\gamma}(\theta)=\textstyle{\frac{1}{3}}\left(\delta_{\alpha}^{~\beta}D_{i\delta}\tilde{\mathrm{A}}^{\delta\gamma}(\theta)-\delta_{\alpha}^{~\gamma}D_{i\delta}\tilde{\mathrm{A}}^{\delta\beta}(\theta)\right)
\begin{array}{l}{\displaystylea(r)=2\pi^{2}\,r\frac{d}{dr}G_{\rhom_{_W}}(r)+\frac{1}{\nu}f^{a}(r),}\\{\displaystyleh(r)=-2\pi^{2}\,G_{\rhom_{_H}}(r)+\frac{1}{\nu}f^{h}(r),}\\{\displaystylef^{a}(r)=-\pi^{2}\,r\frac{d}{dr}G_{\rho\mu_{_W}}(r),}\\{\displaystylef^{h}(r)=\pi^{2}\,G_{\rho\mu_{_H}}(r),}\\\end{array}
I_{\nu_{1}\nu_{2}}^{(d)}(q^{2},m_{1}^{2},m_{2}^{2})=\int\frac{d^{d}k_{1}}{[i\pi^{d/2}]}P_{k_{1},m_{1}}^{\nu_{1}}P_{k_{1}-q,m_{2}}^{\nu_{2}}.
k_{1}^{\mu}T_{\rho\sigma\mu\nu}^{(2)abcd}(p_{1},p_{2};k_{1},k_{2})=C_{2}k_{2\nu}S_{\rho\sigma}^{(2)}(q_{2})+C_{2}\Gamma_{\rho\nu\sigma}^{(2)}(p_{1},k_{2},q_{2})
S={\frac{kA_{H}}{4\ell_{P}^{2}}}+{\frac{1}{T_{H}}}\int_{\Sigma}e^{-\Psi}\{\varrho_{L}-{\calL}_{E}\}\sqrt{g_{3}}d^{3}x+\int_{\Sigma}s\sqrt{g_{3}}d^{3}x.
\rho(x)=\frac{2}{x^{2}}+\sum_{q\neq0}\frac{16(q\cdotx)^{3}-8q^{2}x^{2}(q\cdotx)+2q^{2}x^{4}-8x^{2}(q\cdotx)^{2}}{q^{6}(x-q)^{2}}
\frac{\phi^{\prime}}{\left[1-(\phi^{\prime})^{2}+(y^{\prime})^{2}\right]^{\frac{1}{2}}}=-\frac{e}{r^{p-1}},\;\;\;\frac{-y^{\prime}}{\left[1-(\phi^{\prime})^{2}+(y^{\prime})^{2}\right]^{\frac{1}{2}}}=\frac{c}{r^{p-1}}
2i(U^{\mathrm{T}}W)_{IJ}+(U^{\mathrm{T}}V)_{I}{}^{K}\,F_{KJ}+F_{IK}\,(Z^{\mathrm{T}}W)^{K}{}_{J}-\textstyle{\frac{i}{2}}F_{IK}\,(Z^{\mathrm{T}}V)^{KL}\,F_{LJ}
{\calS}(\Gamma)\equiv\left\{\partial_{\mu}c\frac{\delta\Gamma}{\deltaA_{\mu}}+B\frac{\delta\Gamma}{\delta\bar{c}}+\frac{\delta\Gamma}{\delta\underline{{Y}}}\frac{\delta\Gamma}{\delta\underline{{\varphi}}}+\underline{{q}}\frac{\delta\Gamma}{\delta\underline{{\hat{\varphi}}}}\right\}+\chi\partial_{\xi}\Gamma+\chi_{A}\partial_{\xi_{A}}\Gamma\;=\;0
J[{\hat{\xi}}]={\frac{1}{8\piG}}\int_{r=r_{+}}\!\!d^{n-2}x\,\Bigl\{n^{a}\nabla_{a}{\hat{\xi}}^{t}\sqrt{\sigma}+{\hat{\xi}}^{a}\pi_{a}{}^{r}+n_{a}{\hat{\xi}}^{a}K\sqrt{\sigma}\Bigr\}
\hat{H}{_W}=\frac{1}{2}(\hat{p}_{x}^{2}+\hat{x}^{2})=\frac{1}{2}[\hat{a}^{-},\hat{a}^{+}]_{+}=\frac{1}{2}(\hat{a}^{-}\hat{a}^{+}+\hat{a}^{+}\hat{a}^{-}),
\begin{array}{rcl}{3\partial_{[m}B_{np]}}&{=}&{6(A_{[m}^{(1)}F_{np]}^{(2)}+A_{[m}^{(2)}F_{np]}^{(1)})}\\{}&{=}&{\epsilon_{mnp}(A_{l}^{(1)}\partial_{l}\chi^{(2)}+A_{l}^{(2)}\partial_{l}\chi^{(1)})\.}\\\end{array}
\frac{(-1)^{N}}{\vertG\vert}\sum_{{\boldsymbol\alpha},{\boldsymbol\beta}\inG}\prod_{\stackrel{\scriptstylei}{\omega_{i}\alpha_{i}\in{\bfZ}\\mathrm{and}\\omega_{i}\beta_{i}\in{\bfZ}}}\left(1-\frac{1}{\omega_{i}}\right)\,,
F(x^{+},x^{-},0)_{F}=\frac{i}{4(2\pi)^{3}}\frac{m^{5}}{x}\left(\frac{x^{+}}{x^{-}}\right)^{2}\left[K_{7/2}(mx)K_{3/2}(mx)-K_{5/2}^{2}(mx)\right]\,.
x_{\pi}=\frac{1}{(1-r^{2})^{2}}\left\{(\lambda_{0}^{2}-r^{2})(1+r^{2})-\left[(1-r^{2})^{2}+4r^{2}(\lambda_{0}^{2}-r^{2})^{2}\right]^{1/2}\right\}
\left\{j_{M}^{0}\left(t,\sigma\right),j_{N}^{0}\left(t,\sigma^{\prime}\right)\right\}_{PB}\;=\;\pmC_{MN}^{\;K}\cdotj_{K}^{0}\left(t,\sigma\right)\cdot\delta\left(\sigma-\sigma^{\prime}\right)\quad,
\begin{matrix}{0}&{0}&{0}&{0}&{0}&{0}&{{\frac{1}{q}}}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\{0}&{{\frac{1}{q}}}&{0}&{0}&{0}&{0}&{0}&{q}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{-{q^{-3}}}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{-{\frac{1}{q}}}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}&{-{\frac{1}{q}}}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\\end{matrix}