harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left(\partial_{t}-{\frac{l}{2\pi\alpha^{\prime}}}\partial_{\theta}\right)^{2}X_{4}-\left({\frac{l\,\rho}{2\pi\alpha^{\prime}}}\right)^{2}\left({\frac{1}{\rho^{2}}}\partial_{\theta}^{2}+\partial_{z}^{2}\right)X_{4}+O(l^{3})=0\,,
\left\{\mathrm{Tr}\Big[H^{(1)}H^{(I_{1})}H^{(I_{\pi(2)})}...H^{(I_{\pi(n)})}\Big]-\mathrm{Tr}\Big[H^{(1)}H^{(I_{\pi(2)})}...H^{(I_{\pi(n)})}H^{(I_{1})}\Big]\right\}\;.
{\frac{\delta^{2}\Gamma}{\delta\hat{\mu}(a_{2})\delta\hat{\mu}(a_{1})}}\bigg\vert_{\hat{\mu}=0}=\kappa\partial_{2}^{2}D_{2}\hat{\zeta}(z_{12})+[-{\frac{3}{2}}\partial_{2}\hat{\zeta}(z_{12})-\hat{\zeta}(z_{12})\partial_{2}+{\frac{1}{2}}D_{2}\hat{\zeta}(z_{12})D_{2}]{\frac{\delta\Gamma}{\delta\hat{\mu}(a_{2})}}\bigg\vert_{\hat{\mu}=0}
z_{j}=\mathrm{\frac{~1}{\sqrt{2}~}}\left(\phi_{j}+i\phi_{j+3}\right),\quad\bar{z}^{j}=\mathrm{\frac{~1}{\sqrt{2}~}}\left(\phi_{j}-i\phi_{j+3}\right),\quad\mathrm{~j=2~and~3.}
{\calZ}_{GU}={\displaystyle\int}{\calD}(A^{\mu},\theta)~exp~i{\displaystyle\int}d^{4}x~\left[-\frac{1}{4}F_{\mu\nu}^{2}+\frac{m^{2}}{2}A_{\mu}^{\mu}+\frac{m^{2}}{2}(\partial_{\mu}\theta)^{2}-m^{2}\theta\partial_{\mu}A^{\mu}\right],
K^{(2)}={\frac{l^{3}}{2(d-2)^{2}a^{4}}}\left(R_{\\beta}^{\alpha}R_{\\alpha}^{\beta}-{\frac{3d-4}{4(d-1)^{2}}}R^{2}\right)-{\frac{l^{3}}{2(d-1)(d-2)a^{2}}}\left(R_{\\beta}^{\alpha}R_{\\alpha}^{\beta}-{\frac{1}{2(d-1)}}R^{2}\right)\.
\left\langleM_{fg}\right\rangle=-\frac{1}{16\pi^{2}h}\,\left[\frac{1}{S}\right]_{fg}\left\langle\mathrm{Tr}\,W^{2}\right\rangle\;;~~~~~\left\langle\mathrm{Tr}\,W^{2}\right\rangle=\mathrm{const}\,h\,\sqrt{\mathrm{Pf}(S)}\,\Lambda_{\mathrm{two-fl}}^{2}\;.
E_{\smallCasimir}^{\smallsurface}=+{\frac{1}{48\pi}}\;S\;\hbarc\;K^{3}\;\left[{\frac{(n_{1}-n_{2})^{2}(n_{1}+n_{2})}{n_{1}n_{2}(n_{1}^{2}+n_{2}^{2})}}\right].
\begin{array}{llll}{X_{11}=0,}&{X_{12}=0,}&{X_{13}=2,}&{X_{14}=q.p,}\\{X_{21}=q^{2},}&{X_{22}=0,}&{X_{23}=0,}&{X_{24}=\frac{1}{2}p^{2},}\\{X_{31}=0,}&{X_{32}=0,}&{X_{33}=0,}&{X_{34}=-\frac{1}{4}q^{2},}\\{X_{41}=0,}&{X_{42}=4,}&{X_{43}=2\lambda,}&{X_{44}=0.}\\\end{array}
{\calO}_{\pm}^{a}(\{x\}):=\frac{1}{(N!)^{2}}\sum_{\pi,\pi^{\prime}}\prod_{b=1}^{N}\overline{{\psi}}^{(b)}(x^{(\pi(b))})P_{\pm}\psi^{(b)}(x^{(\pi^{\prime}(b))})\;.
S^{(n+1)}(p)=\frac{ie^{2}}{n+1}\left[-{\calI}(\mu^{2},\kappa^{2})S^{(n)}(p)+\int\frac{d^{2}k}{(2\pi)^{2}}\frac{1}{(k^{2}-\mu^{2}+i\varepsilon)(k^{2}-\kappa^{2}+i\varepsilon)}S^{(n)}(p-k)\right]\;,
\left[-i\partial_{\mu}\gamma^{\mu}+\frac{1}{2\xi}\left(\kappa^{2}+\left(i\frac{u^{\mu}}{c}\partial_{\mu}+\xi\right)^{2}\right)\frac{u_{\mu}\gamma^{\mu}}{c}-\frac{1}{2\xi}\left(\kappa^{2}+\left(i\frac{u^{\mu}}{c}\partial_{\mu}+\xi\right)\left(i\frac{u^{\mu}}{c}\partial_{\mu}-\xi\right)\right)I\right]\psi(x)=0
\psi^{12}\left(p,P\right)=\alphag^{2}iG_{1}\left(p_{1}\right)G_{2}\left(p_{2}\right)\int\frac{d^{4}q}{\left(2\pi\right)^{4}}\frac{\psi^{12}\left(q,P\right)}{\left(p-q\right)^{2}+m_{3}^{2}+\Sigma_{3}\left(\left(p-q\right)^{2}\right)}
[({\mit\Psi})^{\prime},({\mit\Psi}^{\dagger})^{\prime\prime}]=(2\pi)^{-1}\,\delta(p^{\prime}-p^{\prime\prime})\,\int|Z(u_{1}\,,u,{\bfk},p)|^{2}\,e^{i{\bfk}\cdot({\bfr}^{\prime}-{\bfr}^{\prime\prime})}\,d{\bfk}.
\left(\begin{matrix}{0}&{2\partial+i(T_{1}+iT_{2})}\\{2\bar{\partial}+i(T_{1}-iT_{2})}&{0}\\\end{matrix}\right)\left(\begin{matrix}{c}\\{\bar{b}}\\\end{matrix}\right)=(\partial\!\!\!/+iT\!\!\!\!/)\psi=0\.
U(q,q^{\prime})=\int_{-\infty}^{\infty}dq^{\prime\prime}K^{(\frac{1}{2})}\left(q,q^{\prime\prime},t=-\frac{\pi}{2\omega}\right)K^{(a)}\left(q^{\prime\prime},q^{\prime},t=\frac{\pi}{2\omega}\right).
U_{\mathrm{eff}}(\phi)=U_{r}(\phi)-\frac{1}{12}\left[U_{r}^{\prime\prime}(\phi)^{3/2}-m_{r}^{3}-\frac{3}{4}m_{r}\lambda_{r}\phi_{r}^{2}-\frac{3\lambda_{r}^{2}}{32m_{r}}\phi^{4}-\frac{3m_{r}\sigma_{r}}{16}\phi^{4}\right]\;,
<p_{\beta}p_{\gamma}\mathrm{out}\midp_{\alpha}\mathrm{in}>=g_{st}\delta(|p_{\alpha}|-|p_{\beta}|-|p_{\gamma}|)\sqrt{|p_{\alpha}p_{\beta}p_{\gamma}|}\int_{-\infty}^{\infty}f_{3}(\tau)d\tau+O(g_{st}^{3}).
\Gamma_{D}=g_{\mathrm{eff}}~\frac{(\omega^{2}-e^{2})}{\omega}~\rho\left(\frac{\omega+e}{2T_{L}}\right)\rho\left(\frac{\omega-e}{2T_{R}}\right){\frac{d^{4}k}{(2\pi)^{4}}}~.
\tilde{L}_{\omega+}^{**}=\dots-2i\epsilon^{\mu\nu\rho\sigma}\omega_{\mu}^{+ab}\partial_{\nu}G_{\rho\sigmaab}^{+}+{\frac{i}{2}}\epsilon^{\mu\nu\rho\sigma}f_{[cd][ef]}^{[ab]}\omega_{\mu}^{+cd}\omega_{\nu}^{+ef}G_{\rho\sigmaab}^{+}+\dots\\\.
\betaV\cdot{\calN}_{\overline{{\alphaV}}}\equiv\beta{\vec{U}}\cdot{\vec{P}}_{\overline{{\alphaV}}}-\sum_{r}({\betaV})^{r}N_{\overline{{\alphaV}}}^{r}+\sum_{\ell:{\mathrm{right}}}({\betaT})^{\ell}J_{\overline{{\alphaV}}}^{\ell}-\sum_{\ell:{\mathrm{left}}}({\betaT})^{\ell}J_{\overline{{\alphaV}}}^{\ell}~.
\dot{\varrho}+3{\frac{\dot{a}_{0}}{a_{0}}}\left(\varrho+{\frac{\wp}{c^{2}}}\right)=-{\frac{\dot{G}_{4}}{G_{4}}}\varrho+{\frac{3kc\dot{c}}{4\piG_{4}a_{0}^{2}}}-{\frac{3Cc\dot{c}}{4\piG_{4}a_{0}^{4}}}.
\left[-\frac{1}{N^{2}}\,\lambda_{i}\,\frac{\partial^{2}}{\partial\lambda_{i}^{2}}-\frac{1}{N^{2}}\,\sum_{j\neqi}\lambda_{j}\,\frac{1}{\lambda_{j}-\lambda_{i}}\,\left(\frac{\partial}{\partial\lambda_{j}}-\frac{\partial}{\partial\lambda_{i}}\right)+\frac{1}{N}\,(2\eta-1)\,\frac{\partial}{\partial\lambda_{i}}+1\right]Z(\lambda)=0.
ds^{2}=\Omega^{2}[dr^{2}+r^{2}d\omega_{9}^{2}]=\ell^{2}\left(\frac{r\Omega}{\ell}\right)^{2}\left[\left(\frac{r\Omega}{\ell}\right)^{16}\frac{dy^{2}}{64}+d\omega_{9}^{2}\right]
{\calV}=\frac{1}{2}\intd^{3}x\:\mathrm{tr}\:\left\{(a_{I}^{A}K_{A}^{m}\delta_{m}A_{\mu})(a_{I}^{B}K_{B}^{n}\delta_{n}A_{\mu})\right\}=\frac{1}{2}\,g_{mn}\,a_{I}^{A}K_{A}^{m}a_{I}^{B}K_{B}^{n}.
G({\bfx}^{\prime\prime},{\bfx}^{\prime};T\vert{\hat{\bfA}})=\intd{\bfx}_{p}\dots\intd{\bfx}_{1}\langle{\bfx}^{\prime\prime},T\vert{\bfx}_{p},t_{p}\rangle\langle{\bfx}_{p},t_{p}\vert{\bfx}_{p-1},t_{p-1}\rangle\dots\langle{\bfx}_{1},t_{1}\vert{\bfx}^{\prime},0\rangle\,.
{\calS}^{3}=\sum_{n=1}^{N}S_{n}^{3}\,,\qquad{\calS}^{\pm}=\sum_{n=1}^{N}q^{S_{1}^{3}}\ldotsq^{S_{n-1}^{3}}S_{n}^{\pm}q^{-S_{n+1}^{3}}\ldotsq^{-S_{N}^{3}}\,.
r_{H}=\:\left(\frac{16\piGM_{H}}{c^{2}(D-2)A_{D-2}}\right)^{\frac{1}{D-3}}\qquad\qquad,\:\left(A_{D-2}\equiv\:\frac{2\pi^{\frac{(D-1)}{2}}}{\Gamma\left(\frac{D-1}{2}\right)}\right)
\begin{array}{rcl}{\chi_{1,k(r+1)+1}^{(p,kp+1)}(q)}&{=}&{\displaystyle\sum_{n_{1}\geq\cdots\geqn_{k}\geq0}\frac{q^{n_{1}^{2}+\cdots+n_{k-1}^{2}+2n_{k}^{2}+r(n_{1}+\cdots+n_{k-1}+2n_{k})}}{(q)_{n_{1}-n_{2}}\cdots(q)_{n_{k-1}-n_{k}}(q)_{2n_{k}+r}}\overline{{F}}_{p-1;r}^{(2n_{k}+r)}(q).}\\\end{array}
S_{term}[g]=\check{\Gamma}_{\mu\nu}\,\deltag^{\mu\nu}(0)+\int\mathrm{d}^{4}k\,\deltag^{\mu_{1}\nu_{1}}(k)\,\check{\Pi}_{\mu_{1}\nu_{1}\,\mu_{2}\nu_{2}}(k)\,\deltag^{\mu_{2}\nu_{2}}(-k)+\cdots,
\phi(t)=\frac{1}{\Omega}\intd^{3}x\langle\Phi(\vec{x},t)\rangle=\frac{1}{\Omega}\intd^{3}xTr\left[\hat{\rho}(t)\Phi(\vec{x})\right]=\frac{1}{\Omega}\intd^{3}xTr\left[\hat{\rho}(t_{o})U^{-1}(t,t_{o})\Phi(\vec{x},t_{o})U(t,t_{o})\right]
\int_{R_{-}}^{R_{+}}\frac{R^{4}dR}{\sqrt{(R_{+}^{2}-R^{2})(R^{2}-R_{-}^{2})}}=\frac{1}{3}R_{+}^{3}\bigg[-(1-k^{2})K(k)+2(2-k^{2})E(k)\bigg].
\bar{p}_{\phi}=\left(\frac{-{\calD}}{g_{\phi\phi}^{2}}\right)^{1/2}p_{\phi},~~~\bar{p}_{r}=\left(\frac{-{\calD}}{g_{\phi\phi}g_{rr}}\right)^{1/2}p_{r},~~~\bar{p}_{\theta}=\left(\frac{-{\calD}}{g_{\phi\phi}g_{\theta\theta}}\right)^{1/2}p_{\theta},
D_{2}\pi^{\alpha_{1}}=\frac12C_{\;\;\beta_{1}\gamma_{1}}^{\alpha_{1}}\pi^{\beta_{1}}\eta_{2}^{\gamma_{1}}+\frac12C_{\;\;\beta_{1}\gamma_{3}}^{\alpha_{1}}\left(\pi^{\beta_{1}}\eta_{2}^{\gamma_{3}}-\eta_{2}^{\beta_{1}}\pi^{\gamma_{3}}\right),
{\frac{1}{\rho^{2}}}l(l+1)+2\omega^{a}\left(n+{\frac{1}{2}}\right)-m_{\mu}^{a}\equiv\Lambda_{\mu}^{a}(n,l),\mathrm{with~degeneracy~d_l=~2l+1~,~l~\in~{\bf~N},~n~\in~{\bf~N}~}
0={\calM}^{4}-(2\tilde{g}^{2}k_{q}+\frac1{k_{q}k_{T}})(q_{0}^{2}+\bar{q}_{0}^{2}){\calM}^{2}+2\frac1{k_{T}}\tilde{g}^{2}(q_{0}^{2}+\bar{q}_{0}^{2})^{2}.
\hat{H}_{\mathrm{pl}}^{2\!\times\!3}=-\frac{g^{2}}{2}\sum_{i,a=1}^{2}\frac{\partial^{2}}{\partial\tilde{a}_{i}^{a}\partial\tilde{a}_{i}^{a}}+\frac{1}{2\,g^{2}}\big(\tilde{a}_{1}^{1}\tilde{a}_{2}^{2}-\tilde{a}_{1}^{2}\tilde{a}_{2}^{1}\big)^{2}
\mu_{k}(\eta)\,=\,\frac{1}{\sqrt{2k}}\left[C_{+}(k)\frac{a(\eta)}{a_{\mathrm{i}}}-C_{-}(k)\frac{a_{\mathrm{i}}}{\eta_{\mathrm{i}}}a(\eta)\int_{\eta_{\mathrm{i}}}^{\eta}\frac{\mathrm{d}\tau}{a^{2}(\tau)}\right],
\chi_{a,\ell}(z,\tau)=\frac{\displaystyle\sum_{w\in\overline{W}}\mathrm{sgn}(w)\;\Theta_{w(\overline{{a}}+\overline{{\rho}}),g+\ell}(z,\tau)}{\displaystyle\sum_{w\in\overline{W}}\mathrm{sgn}(w)\;\Theta_{w(\overline{{\rho}}),g}(z,\tau)}\:,
\intdt^{\prime}G(x,t;x^{\prime},t^{\prime})=\int\frac{dk^{\prime\prime}}{2\pi}\left(\frac{i}{-k^{2}-4}\right)e^{-ik^{\prime\prime}x^{\prime}}\left(e^{ik^{\prime\prime}x}+K_{0}(k^{\prime\prime})e^{-ik^{\prime\prime}x}\right).
(e^{-2\eta})_{\mathrm{fix}}=\left|\frac{p_{2}\,p_{4}}{q_{1}q_{3}}\right|\,\qquad(e^{-2\sigma})_{\mathrm{fix}}=\left|\frac{p_{2}\,q_{3}}{q_{1}p_{4}}\right|\,\qquad(e^{-2\rho})_{\mathrm{fix}}=\left|\frac{p_{4}\,q_{3}}{q_{1}p_{2}}\right|\.
S_{\delta}=\int\mathrm{d}t\left[{\frac{1}{2}}\left(\mathrm{d}_{t}\phi\right)^{2}+{\frac{1}{2}}\left(m^{2}+{\frac{\lambda}{2m}}\alpha\right)\phi^{2}+\delta{\frac{\lambda}{4}}\left(\phi^{4}-{\frac{\alpha}{m}}\phi^{2}\right)\right].
{\calU}(z,z^{\prime}|s)=\frac{i}{(4\piis)^{2}}e^{is(eW^{\alpha}D_{\alpha}-e\bar{W}_{\dot{\alpha}}\bar{D}^{\dot{\alpha}})}\delta^{4}(\theta-\theta^{\prime})e^{\frac{(x-x^{\prime})^{2}}{4is}}
15\mathrm{Cl}_{2}(\theta_{11})-10\mathrm{Cl}_{2}(2\theta_{11})+\mathrm{Cl}_{2}(5\theta_{11})=Z_{11}=11\sum_{k=1}^{5}\left(\frac{k}{11}\right)\mathrm{Cl}_{2}\left(\frac{2\pik}{11}\right)
\Lambda^{6(N_{c}+1)-2(N_{f}+N_{c}^{\prime})}\Lambda^{\prime6(N_{c}^{\prime}-1)-2N_{c}}=\mu^{2(N_{f}+N_{c}+N_{c}^{\prime})}\tilde{\Lambda}^{\prime4(N_{c}^{\prime}-1)-2(N_{f}+\tilde{N}_{c})}.
{\calL}_{\mathrm{odd/even}}^{1+2}=(-\frac{3}{2}+1)\,\epsilon^{mr_{1}\cdotsr_{9}}t_{8}^{s_{1}\cdotss_{8}}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}}\,H_{r_{7}r_{8}m}H_{s_{7}s_{8}r_{9}}\,,
m^{d+1}\textrm{tr}\left[\sigma^{3}\otimes(\hat{n}(\vec{x})\cdot\vec{\tau})\left(\partial_{i}\hat{n}^{a}\,\partial_{i}\hat{n}^{a}-\epsilon^{ij}\,\epsilon^{abc}\,\partial_{i}\hat{n}^{a}\,\partial_{j}\hat{n}^{b}\,\sigma^{3}\otimes\tau^{c}\right)^{\frac{d}{2}}\right]
\lbrace{\calF}_{i},{\overline{{Q}}}_{a}\rbrace^{\ast}=\frac{i}{2}{(\sigma_{i})}_{a}^{c}{\overline{{Q}}}_{c},\quad\lbrace{\calF}_{i},Q^{a}\rbrace^{\ast}=-\frac{i}{2}{(\sigma_{i})}_{c}^{a}Q^{c}.
exp\bigg(+i\tilde{L}_{D}\bigg)=\int{\calD}G{\calD}B\exp\bigg(+i\int_{X}d^{3}x\varepsilon^{ijk}(aB_{i}^{AB}H_{jkAB}+bA_{i}^{AB}G_{jkAB}+cB_{i}^{AB}G_{jkAB})\bigg).
A_{l}^{(\pm)}\left(\gamma,{\mathbf{k}},r\right)=\omega^{2}J_{l}^{2}(\gammar)+\left(2\xi-\frac{1}{2}\right)\gamma^{2}\left[\left(1\pm\frac{l^{2}}{\gamma^{2}r^{2}}\right)J_{l}^{2}(\gammar)-J_{l}^{^{\prime}2}(\gammar)\right].
\left(\phi_{k,m}\vertV_{\ell}\right)(\tau,z):=\ell^{k-1}\sum_{ad=\ell}\sum_{b=0}^{d-1}d^{-k}\phi_{k,m}\left(\frac{a\tau+b}{d},az\right)\inJ_{k,\ellm}^{\mathrm{nh}}(\mathfrak{g}),
\psi_{CP}=\psi_{C}\circ\psi_{P}:\left(\begin{array}{l}{f_{\pm}^{q}}\\{f_{\pm}^{-q}}\\\end{array}\right)(\theta,\phi)\mapsto\left(\begin{array}{l}{(-1)^{(1\pm1)q}(f_{\mp}^{q})^{*}}\\{(-1)^{(1\mp1)q}(f_{\mp}^{-q})^{*}}\\\end{array}\right)(\pi-\theta,\pi+\phi).
{\calL}_{\mathrm{YM}}=-{\frac{1}{4g_{\mathrm{YM}}^{2}}}G_{\mu\nu}^{a}G^{a\mu\nu}+{\frac{\theta}{32\pi^{2}}}G_{\mu\nu}^{a}{\widetildeG}^{a\mu\nu}=-{\frac{1}{4}}\mathrm{Re}\left(\tau\left[G_{\mu\nu}^{a}G^{a\mu\nu}+iG_{\mu\nu}^{a}{\widetildeG}^{a\mu\nu}\right]\right)~.
\psi=-\intd^{2p+2}x\left(\sum_{k=0}^{a}\tilde{\chi}^{\mu_{1}\ldots\mu_{p-2k-1}}\bar{\eta}_{\mu_{1}\ldots\mu_{p-2k-1}}+\sum_{k=0}^{b}\bar{\chi}^{\mu_{1}\ldots\mu_{p-2k}}\bar{C}_{\mu_{1}\ldots\mu_{p-2k}}\right),
S=\int~d^{4}x\sqrt{-g}~\left[~\frac{\tilde{R}(g,T)}{\kappa}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}\tilde{H}_{\mu\nu\lambda}\tilde{H}^{\mu\nu\lambda}+\frac{1}{\sqrt{\kappa}}T^{\mu\nu\lambda}\tilde{H}_{\mu\nu\lambda}~\right]
\left.(b_{3}c_{1}(\beta)+b_{4}c_{2}(\beta))R_{\mu\nu}n_{i}^{\mu}n_{i}^{\nu}+(b_{5}c_{1}(\beta)+b_{6}c_{2}(\beta))R_{\mu\nu\lambda\rho}n_{i}^{\mu}n_{i}^{\lambda}n_{j}^{\nu}n_{j}^{\rho}\right)~~~
F_{8}^{{\mathcal{O}}_{1,1}^{0,1}|+++-----}=\frac{iH^{1,1}(\sigma_{5}^{-})^{1/2}((\sigma_{2}^{+})^{2}\sigma_{1}^{-}+\sigma_{5}^{-}+\sigma_{3}^{-}\sigma_{2}^{+})/(\sigma_{3}^{+})^{5/2}}{2^{7}\prod_{1\leqi<j\leq3}\mathrm{cosh}(\theta_{ij}/2)\prod_{4\leqi<j\leq8}\mathrm{cosh}(\theta_{ij}/2)}\prod_{i<j}F_{\mathrm{min}}^{\mu_{i}\mu_{j}}(\theta_{ij}),
f_{\mathbf{I}\mathbf{M}}^{\\\mathbf{L}}C_{\mathbf{L}\mathbf{J}}+f_{\mathbf{M}\mathbf{J}}^{\\\mathbf{L}}C_{\mathbf{L}\mathbf{I}}+f_{\mathbf{J}\mathbf{I}}^{\\\mathbf{L}}C_{\mathbf{L}\mathbf{M}}=0
\Gamma_{\mathrm{conf}}=\int_{x}\Big\{\frac{1}{4Z_{\!A}}(Z_{\!A}f_{\mu\nu}+Y\tilde{B}_{\mu\nu})^{2}+\frac{Z_{\!B}}{4}(\partial_{\rho}\tilde{B}_{\mu\nu})^{2}\Big\}.
\frac{1}{\mu}\langle\tilde{\psi}^{(0)}|\tilde{x}\tilde{p}^{2}|\tilde{\psi}^{(0)}\rangle=\frac{1}{2}\langle\tilde{\psi}^{(0)}|\tilde{x}^{2}V^{\prime}|\tilde{\psi}^{(0)}\rangle\nonumber
+{\frac{{\frac{(p^{2}\alpha+qa^{2})}{4}}\partial^{2}+q\alpha(\alpha+q)}{({\frac{p+a}{2}})^{2}\partial^{2}+(q+\alpha)^{2}}}\epsilon^{\mu\nu\lambda}B_{\nu\lambda}-{\frac{1}{2g}}B^{\mu}.
S=\intdt\frac{B}{2}\mathrm{Tr}\left\{\epsilon_{ab}(\dot{X}_{a}+i[A_{0},X_{a}])X_{b}+2\thetaA_{0}-\omegaX_{a}^{2}\right\}+\Psi^{\dagger}(i\dot{\Psi}-A_{0}\Phi),
\left(a_{1}^{\dagger}+\sum_{j=2}^{r}\xi_{j}a_{j}^{\dagger}+\sum_{j=2}^{r}\xi_{-j}b_{j}^{\dagger}-(\sum_{j=2}^{r}\xi_{j}\xi_{-j})b_{1}^{\dagger}\right)^{M}|0\rangle.
d_{\mu}(q)=(1-\delta_{q_{3},0})\,\frac{1}{2q^{2}\,q_{3}^{2}}\,q_{\mu}\left(1-\delta_{\mu,3}\right)+\delta_{q_{3},0}\,\frac{q_{\mu}}{(q^{2}+i\epsilon)^{2}}\,(1-\xi)\.
\left\{-\hbar^{2}\frac{\partial^{2}}{\partialx^{2}}+\hbar^{2}\frac{\partial^{2}}{\partialy^{2}}+{(8\pi\chi_{0}^{3})}^{2}\left[x\left(x-a_{0}-\frac{\alpha}{4\pi\chi_{0}^{3}}\right)-y^{2}\right]\right\}\Psi_{1}(x,y)=0~.
Z=(2\pi\beta)^{N/2}{\LARGE\int}d\sigma^{N}{\LARGE|}imI+2\beta\Gamma_{\mu}\sigma_{\mu}{\LARGE|}exp(-\beta\vec{\sigma}^{2})\,Z=(2\pi\beta)^{N/2}{\LARGE\int}d\sigma^{N}(4\beta^{2}\vec{\sigma}^{2}+m^{2})^{M}exp(-\beta\vec{\sigma}^{2})\,M=2^{N/2-1}
[\hat{A}_{1}^{\pm\pm}(n_{1},n_{2}),\hat{A}_{2}^{\pm\pm}(m_{1},m_{2})]=\frac{2i\hbar}{N_{k}}f^{\pm\pm}(n_{1},n_{2})\delta_{n_{1},m_{1}}\delta_{n_{2},m_{2}}\\\,
+2z_{\alpha}(\xi)\frac{\partial}{\partialy_{\alpha}(\xi)}D_{m}^{(4)}(y(\xi)-y(\xi^{\prime}))+z_{\alpha}(\xi)z_{\beta}(\xi^{\prime})\frac{\partial^{2}}{\partialy_{\alpha}(\xi)\partialy_{\beta}(\xi^{\prime})}D_{m}^{(4)}(y(\xi)-y(\xi^{\prime}))\Biggr)\Biggr].
\psi_{2\xi}^{\dagger}(\epsilon)\psi_{1\eta}(-\epsilon)=-e^{\frac{i}{2}[-\theta(\eta)-\theta(\xi)+\frac{1}{2}R_{2}(\epsilon,\xi,\eta)]}e^{-i\pi\intd^{2}y\frac{1}{e}\partial_{i}E_{i}(y)}
S=S_{M}+S_{\partialM}^{(1)}+S_{\partialM}^{(2)}={\frac{\pi^{3}R^{5}}{2\kappa_{10}^{2}}}\int_{0}^{1}dud^{4}x\sqrt{-g}\left({\calR}-2\Lambda\right)+{\frac{\pi^{3}R^{5}}{\kappa_{10}^{2}}}\intd^{4}x\sqrt{-h}K+{\frac{a\pi^{3}}{\kappa_{10}^{2}}}\intd^{4}x\sqrt{-h}\,,
\Box\bar{h}_{\\nu}^{\mu}=-\frac{2\kappa^{2}}{l}\frac{1}{1-\Omega^{2}}\left(\stackrel{A}{T_{\\nu}^{\mu}}+\Omega^{2}\stackrel{B}{T_{\\nu}^{\mu}}\right)+\frac{\kappa^{2}}{3l}\frac{\Omega^{2}}{1-\Omega^{2}}\delta_{\nu}^{\mu}\left(\stackrel{A}{T}+\stackrel{B}{T}\right)\.
\mathrm{Re}\,\Pi^{\mathrm{(R)}}(p^{0},\vec{p})=\frac{1}{\pi}\,\mathrm{P}\,\int_{-\infty}^{\infty}d\omega\,\frac{\mathrm{Im}\,\Pi^{\mathrm{(R)}}(\omega,\vec{p})}{\omega-p^{0}}
\sum_{n_{i}=0}^{\infty}\left[\begin{array}{c}{n_{i}+\tilde{n}_{i}+\tilde{V}_{i,r}}\\{n_{i}}\\\end{array}\right]\tilde{x}^{n_{i}}=\frac{1}{(1-\tilde{x})^{\tilde{n}_{i}+1+\tilde{V}_{i},r}}.
\{Q_{m}^{i},P_{n}^{j}\}=\int_{\gamma_{m}}ds\int_{\gamma_{n}}dt\dot{\gamma}_{m}^{a}(s)\dot{\gamma}_{n}^{b}(t)\epsilon_{ab}\delta^{2}(\gamma_{m}(s),\gamma_{n}(t))\delta^{ij}Q_{m}^{i}
J(Z_{u})\bullet_{\hbar}J(Z_{v})=Z_{u}\bullet_{\hbar}^{z}Z_{v}+\sum_{i}y^{i}(Z_{\partial_{i}u}\bullet_{\hbar}^{z}Z_{v}+Z_{u}\bullet_{\hbar}^{z}Z_{\partial_{i}v})+\cdots\;,\quad\forallu,v\inN_{1}.
I^{(i)}=2\beta\sqrt{2K_{1}S_{c}^{(i)}}(\frac{E_{0}^{(i)}}{\pi})^{\frac{1}{2}}\frac{1}{\sqrt{2\pi}}(1-\alpha^{2})^{\frac{1}{4}}\lambda^{\frac{1}{4}}s^{\frac{1}{2}}
S_{b}=-\sum_{i=1}^{2}m_{i}\intdt\sqrt{-\mathrm{~\dot{~}x^\mu_i~}^{2}}-\sum_{i=1}^{2}\kappa_{i}\intdt\frac{\dot{x}_{i}^{\mu}}{\sqrt{-\mathrm{~\dot{~}x^\mu_i~}^{2}}}\left((-1)^{i}t_{\mu\nu}\right)\Big|_{r=(-1)^{i}R_{i}(t)}\frac{d}{dt}\frac{\dot{x}_{i}^{\nu}}{\sqrt{-\mathrm{~\dot{~}x^\mu_i~}^{2}}}\,,
{\hat{q^{\prime}}}_{l}(\zeta)=f(\theta_{+l}^{\prime}(\zeta),\theta_{+k}(\zeta)){\hat{q}}_{k}(\zeta),\quad{\bar{\hat{q}}}_{l}^{\prime}(\zeta)={\bar{\hat{q}}}_{k}(\zeta)f^{*}(\theta_{-l}^{\prime}(\zeta),\theta_{-k}(\zeta)),
\delta\Gamma_{eff}=tr\intd^{d}x\,(D_{\mu}\epsilon^{a})\frac{\delta\Gamma_{eff}}{\deltaA_{\mu}^{a}}=-tr\intd^{d}x\,\epsilon^{a}D_{\mu}\left(\frac{\delta\Gamma_{eff}}{\deltaA_{\mu}^{a}}\right)\;.
\frac{1}{|{\bfx-x^{\prime}}|}=4\pi\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\frac{1}{(2l+1)}\,\frac{r^{\primel}}{r^{l+1}}Y_{lm}^{*}(\theta^{\prime},\phi^{\prime})Y_{lm}(\theta,\phi),
\overline{{{\cal{A}}_{i}^{\gamma}}}\,^{L}({\bfr})={\delta}_{{i}\,{\gamma}}\,{\Phi}(r)+\frac{r_{i}\,r_{\gamma}}{r}\,{\Phi}^{\prime}(r)\;\;\;\mathrm{where}\;\;{\Phi}^{\prime}(r)=\frac{d\Phi}{dr}
\begin{array}{l}{F_{+}(z)=\left(\begin{array}{cc}{-ix_{+}{\cdot\tilde{\sigma}}\,\tau_{+}}&{2\bar{\theta}_{b}\,p_{+}}\\{2\theta^{a}\,\tau_{+}}&{\delta_{~b}^{a}\,p_{+}}\\\end{array}\right)}\\{{}}\\{F_{-}(z)=\left(\begin{array}{cc}{1}&{0}\\{0}&{-\bar{{\calE}}}\\\end{array}\right)F_{+}(-z)^{t}\left(\begin{array}{cc}{1}&{0}\\{0}&{-{\calE}}\\\end{array}\right)}\\\end{array}
ds_{11}^{2}\,=\,\left(1+\frac{k}{r^{\tilde{d}}}\right)^{-\frac{\tilde{d}}{9}}\,dx^{I}\,\,dx^{J}\eta_{IJ}+\,\,\left(1+\frac{k}{r^{\tilde{d}}}\right)^{\frac{d}{9}}\,dy^{\hat{a}}\,\,dy^{\hat{b}}\,\delta_{{\hat{a}}{\hat{b}}}\,.
\sigma_{\mu\nu}=i\left(\begin{array}{cc}{\eta_{a\mu\nu}^{+}\sigma_{a}}&{0}\\{0}&{\eta_{a\mu\nu}^{-}\sigma_{a}}\\\end{array}\right)\;,\;\;\;\gamma_{5}=\left(\begin{array}{cc}{+1}&{0}\\{0}&{-1}\\\end{array}\right)
\Box_{5}\phi(w,t,y)-V^{\prime}[\phi(w,t,y)]=\frac{\Gamma_{D}^{w}}{2k}\delta(w)\frac1N\delta_{t}\phi+\frac{\Gamma_{D}^{y}}{2k_{y}}\delta(y)\frac1N\delta_{t}\phi+\Gamma_{B}\frac1N\delta_{t}\phi.
W^{(1,m)}\left(\left.\begin{array}{cc}{c}&{d}\\{b}&{a}\\\end{array}\right|z\right)=\sum_{\sigma\in\mathfrak{S}_{m}}\mathrm{sgn}\,\sigma\prod_{j=1}^{m}W\left(\left.\begin{array}{cc}{d_{j}}&{d_{j-1}}\\{a_{j}^{\sigma}}&{a_{j-1}^{\sigma}}\\\end{array}\right|z+\frac{m+1}{2}-j\right).
{\calR}_{\lambda=1}=(-1)^{(\zeta-H)\otimes(\zeta-H)}Q^{2(\zeta\otimesH+H\otimes\zeta)}\left(1\!\mathrm{l}\otimes1\!\mathrm{l}+\sigmaQ^{\zeta}X_{+}\otimesgQ^{-\zeta}X_{-}\right)\,.
H_{S}=\sum_{n}\frac{1}{2}\left[e^{-3Ht}L^{3}\pi_{\phi}({\bfk}_{n},t)\pi_{\phi}(-{\bfk}_{n},t)+\frac{e^{3Ht}}{L^{3}}(e^{-2Ht}{\bfk}_{n}^{2}+m^{2})\phi({\bfk}_{n},t)\phi(-{\bfk}_{n},t)\right]+\frac{e^{3Ht}}{L^{3}}V_{F},
C_{i_{1}i_{2}...i_{n}}=\sqrt{\frac{[r_{1}]_{q^{2}}!}{\left([r_{1}]_{q}!\right)^{3}}\cdot\frac{[r_{2}]_{q^{2}}!}{\left([r_{2}]_{q}!\right)^{3}}\cdot...\cdot\frac{[r_{m}]_{q^{2}}!}{\left([r_{m}]_{q}!\right)^{3}}}\mathrm{\quad~with~\quad}m\len
\left(\frac{\dot{a_{0}}}{a_{0}}\right)^{2}+\frac{\ddot{a_{0}}}{a_{0}}=\frac{1}{12M^{3}}\left[\Lambda-\frac{(\rho+b_{0}v_{b})(\rho+3p-2b_{0}v_{b})}{48M^{3}}\right].
E_{n_{x},n_{y}}^{0}=m\omega^{2}\hbar^{2}\left(\frac{1}{m^{2}\omega^{2}\hbar^{2}}+\frac{\theta^{2}}{4\hbar^{4}}\right)^{\frac{1}{2}}\left(n_{x}+n_{y}+1\right)-\frac{m\omega^{2}\theta}{2}\left(n_{x}-n_{y}\right).
I=(\beta-1)\frac{\Gamma^{q}(\beta)}{2\pii}\prod_{0\leqi<j}^{q}(w_{i}-w_{j})^{2\beta-1}\prod_{i=1}^{p}\prod_{j=0}^{q}(w_{j}-v_{i})^{-1}\int_{{\calC}_{w}}dz\frac{\prod_{i=1}^{p}(v_{i}-z)}{\prod_{i=1}^{q}(z-w_{i})^{\beta}(w_{0}-z)^{\beta}}
\frac{1}{4}~\mathrm{Tr}\left(\frac{1}{Y}[A_{\mu},A_{\nu}]^{2}\right)\rightarrow\frac{i}{2}~\mathrm{Tr}\left(\epsilon_{m}\left(\Gamma_{0}\Gamma_{\mu}\right)_{mn}\psi_{n}[A_{\nu},\{[A^{\mu},A^{\nu}],Y^{-1}\}]\right),
\psi_{L,R}^{(n)}(\hat{x})\,=\,P_{L,R}\psi^{(n)}(\hat{x})\;\;,\;\;\;\;\;\overline{{\psi}}_{L,R}^{(n)}(\hat{x})\,=\,\overline{{\psi}}^{(n)}(\hat{x})P_{R,L}\;,
{\calD}^{(j)}[\Lambda]=\left(\begin{matrix}{D^{(j)}[\Lambda]}&{0}\\{0}&{\overline{D}^{(j)}[\Lambda]}\\\end{matrix}\right),\quadD^{(j)}[\Lambda]=\overline{D}^{(j)}[\Lambda^{-1}]^{\dagger}\quad,\quad{\calD}^{(j)}[\Lambda]^{\dagger}=\beta{\calD}^{(j)}[\Lambda^{-1}]\beta\,\,,
\begin{array}{ll}{S=}&{-\frac{1}{2}\int_{\Sigma}d^{2}\sigma\left(G_{mn}\partial_{a}X^{m}\partial^{a}X^{n}+B_{mn}\epsilon^{ab}\partial_{a}X^{m}\partial_{b}X^{n}\right)}\\{}&{-\int_{\partial\Sigma}dsA_{m}\partial_{s}X^{m}+\frac{1}{4\pi}\int_{\Sigma}d^{2}\sigma\,\sqrt{\gamma}R^{(2)}\Phi-\frac{1}{2\pi}\int_{\partial\Sigma}dsk\Phi\,,}\\\end{array}
\begin{array}{ll}{F_{12}+F_{34}+F_{56}=0\,,~}&{~~F_{13}+F_{42}+F_{57}=0\,}\\{{}}&{{}}\\{F_{14}+F_{23}+F_{76}=0\,,~}&{~~F_{15}+F_{62}+F_{73}=0\,,}\\{{}}&{{}}\\{F_{16}+F_{25}+F_{47}=0\,,~}&{~~F_{17}+F_{35}+F_{64}=0\,,}\\{{}}&{{}}\\{F_{27}+F_{63}+F_{54}=0\,,~}&{~~F_{a8}=0\,.}\\\end{array}
E_{i}^{q_{i}^{-1}}=I,\qquad\prod_{l=1}^{m}E_{l}\prod_{k=1}^{n}P_{k}\prod_{j=1}^{h}\left(H_{2j-1}H_{2j}{H_{2j-1}^{-1}}{H_{2j}^{-1}}\right)=I,
{\calL}_{\mathrm{Y-M}}^{\prime}~=~G_{\alpha\beta}^{I}G_{\gamma\delta}^{I}\epsilon^{\alpha\beta\mu\nu}\epsilon^{\gamma\delta\rho\tau}\nabla_{\mu}\phi^{a}\nabla_{\nu}\phi^{b}\nabla_{\rho}\phi^{c}\nabla_{\tau}\phi^{d}\eta_{ac}\eta_{bd}.
\eta_{2}={\frac{C^{2(D-3)}}{(D-4)!}}\left(1-{\frac{C^{\prime^{2}}}{B^{2}}}+{\frac{A^{\prime}C^{\prime}C}{AB^{2}}}-{\frac{A^{\prime\prime}C^{2}}{AB^{2}}}+{\frac{A^{\prime}B^{\prime}C^{2}}{AB^{3}}}+{\frac{CC^{\prime\prime}}{B^{2}}}-{\frac{B^{\prime}C^{\prime}C}{B^{3}}}\right)
\tilde{H}=\frac{N}{2}\mathbb{I}_{N}+\sum_{j,k=1}^{N}\left[D_{j,k}c_{j}^{\dagger}c_{k}^{}+\frac{1}{2}\left(E_{j,k}^{}c_{j}^{\dagger}c_{k}^{\dagger}+E_{k,j}^{\ast}c_{j}^{}c_{k}^{}\right)\right],
\phi_{n}(x)={\frac{1}{\sqrtL}}e^{-i\epsilon_{n}x-ie\int_{0}^{x}A(z)dz},\quad\epsilon_{n}={\frac{2\pi}{L}}\left(n+{\frac{1}{2}}-{\frac{eL}{2\pi}}c\right)\equiv{\frac{2\pin}{L}}+{\frac{\pi}{L}}-ec.
M^{2}\left(\begin{matrix}{\phi^{\prime}}\\{{\overline{{\phi}}}^{*\prime}}\\\end{matrix}\right)=\left(\begin{matrix}{m^{2}}&{m^{2}x}\\{m^{2}x}&{m^{2}}\\\end{matrix}\right)\left(\begin{matrix}{\phi^{\prime}}\\{{\overline{{\phi}}}^{*\prime}}\\\end{matrix}\right)\.
\rho={\frac{na_{H}^{n}}{16\piG_{n+2}a^{n+1}}}\left({\frac{a_{H}}{L}}+k{\frac{L}{a_{H}}}+{\frac{nw_{n+1}^{2}Q^{2}L}{8(n-1)a_{H}^{2n-1}}}\right),
V(\phi_{1},\phi_{2})=\frac{1}{2}m_{1}^{2}\phi_{1}\phi_{1}+\frac{1}{2}m_{2}^{2}\phi_{2}\phi_{2}+\frac{1}{4!}\lambda_{1}\mu^{2\epsilon}(\phi_{1}\phi_{1})^{2}+\frac{1}{4!}\lambda_{2}\mu^{2\epsilon}(\phi_{2}\phi_{2})^{2}+\frac{1}{4}\lambda\mu^{2\epsilon}(\phi_{1}\phi_{1})(\phi_{2}\phi_{2})
\intdwd\alphad\beta\\bar{f}\left[V\frac{(N-1)(\alpha-\beta)}{\alpha(c_{\alpha}\alpha-\beta)}+\frac{(\alpha-\beta)^{2}}{\beta(c_{\alpha}\alpha-\beta)}\frac{\partialV}{\partial\alpha}\right]|k\rangle_{1i}\dots|\Psi\rangle_{N}=0\.
{\calF}(q,x)=q^{m+n}\Bigl[\Bigl(\frac{d}{dq}\Bigr)^{m+1}{\calH}_{q}X_{2}(x)\Bigr]\Bigl[\Bigl(\frac{d}{dq}\Bigr)^{n+1}{\calH}_{q}X_{3}(x)\Bigr]\quad.
\langle\sigma^{z}\rangle^{(i)}(r)=\frac{(P_{+}^{(i)}(1;r))^{2}-(P_{-}^{(i)}(1;r))^{2}}{(P_{+}^{(i)}(1;r))^{2}+(P_{-}^{(i)}(1;r))^{2}}.
\frac{1}{2}\sum_{n\neq0}\left\{\frac{\omega_{n}}{2}\chi_{n}^{2}+\frac{1}{2\omega_{n}}[(\partial_{\,z}+F^{\prime})\chi_{n}]^{2}-\frac{\omega_{n}}{2}\chi_{n}^{2}-\frac{1}{2\omega_{n}}[(\partial_{\,z}+F^{\prime})\chi_{n}]^{2}\right\}\equiv0\,,
S=\intd^{4}x\mathrm{Tr}\bigg(-{\frac{1}{12}}H_{\mu\nu\lambda}H^{\mu\nu\lambda}-{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}+{\frac{m}{4}}\epsilon^{\mu\nu\rho\lambda}B_{\mu\nu}F_{\rho\lambda}\bigg).
{\bfi}_{\Phi}C_{(2)}=\Phi^{i}C_{ij}(\Phi)dX^{j}\;\quad{\bfi}_{\Phi}{\bfi}_{(2)}(\Phi)=\Phi^{j}\Phi^{i}C_{ij}(\Phi)={\frac{1}{2}}[\Phi^{i},\Phi^{j}]C_{ij}(\Phi)\,
\mathcal{L}_{0}=-\sum_{n}\left(\frac{n+\omega}{R}\right)^{2}\bar{A}^{(n)}A^{(n)}-\sum_{m,n}\bar{A}^{(m)}\left[\delta(0)\\frac{W^{2}}{\piR}+i\,\frac{W}{\pi}\\frac{m-n}{R^{2}}\right]A^{(n)}\,.
\bar{f}_{\pm}=\left.e^{-\alpha_{\pm}^{(0)}}\left[F-2\dot{A}\phi_{\pmw}-{\alpha_{\pm}^{\prime}}^{(0)}(e^{-2A}\varphi+\phi_{\pmw}\dot{\Psi}^{(0)})\right]\right|_{w=w_{\pm}}.