harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
M_{AB}=\gamma^{\alpha\beta}\partial_{\alpha}\bar{X}^{\mu}\partial_{\beta}\bar{X}^{\nu}{\calG}_{\mu\nuAB}+i\frac{\epsilon^{\alpha\beta}}{\sqrt{\gamma}}\partial_{\alpha}\bar{X}^{\mu}\partial_{\beta}\bar{X}^{\nu}{\calB}_{\mu\nuAB}+\alpha^{(2)}{\calF}_{AB}.
\delta^{2}H=\frac{1}{2}\intdz\bigg[\frac{{\delta}^{2}H}{{\delta}{R^{\prime}}^{2}}(\deltaR^{\prime})^{2}+2\frac{{\delta}^{2}H}{{\deltaR}{\delta{R}^{\prime}}}\deltaR\deltaR^{\prime}+\frac{{\delta}^{2}H}{{\deltaR}^{2}}(\deltaR)^{2}\bigg]
f(a,b,c,d):=(\partial_{z_{1}}-\bar{z_{1}})^{a}(\partial_{\bar{z}_{1}}-z_{1})^{b}(\partial_{z_{2}}-\bar{z_{2}})^{c}(\partial_{\bar{z}_{2}}-z_{2})^{d}e^{-|z|^{2}/2},
\mathrm{Tr}\,K_{n}(s)=\frac{(-s)^{n}}{(4\pis)^{d/2}n}\intdx\,\big<e^{s\Omega_{n}}\big>\,V(x_{1})V(x_{2})...V(x_{n})\Big|_{x_{1}=...=x_{n}=x},\,\,\,n\geq2.
S_{p}^{E}={\frac{1}{2\kappa_{D}^{2}}}\intd^{D}x\sqrt{-g^{E}}\left[{\calR}_{g^{E}}-{\frac{4}{D-2}}(\partial\phi)^{2}-{\frac{1}{2\cdot(p+2)!}}e^{2a_{p}\phi}F_{p+2}^{2}\right],
\left.\frac{\partial\phi}{\partialx}\right|_{x_{n+\frac{1}{2}}}=\mathrm{Scaled}\left[\frac{1}{2}\sum_{b=\{b_{1},b_{2}\}}\phi_{b}(x_{n+1})\right]-\mathrm{Scaled}\left[\frac{1}{2}\sum_{a=\{a_{1},a_{2}\}}\phi_{a}(x_{n})\right].
\left(\begin{matrix}{\phi_{R}(p^{\mu})}\\{\phi_{L}(p^{\mu})}\\\end{matrix}\right)=e^{i\beta_{\mp}}\left(\begin{matrix}{0}&{i\Theta_{[1/2]}}\\{-i\Theta_{[1/2]}}&{0}\\\end{matrix}\right)\left(\begin{matrix}{\phi_{R}^{*}(p^{\mu})}\\{\phi_{L}^{*}(p^{\mu})}\\\end{matrix}\right)=S_{[1/2]}^{c}\left(\begin{matrix}{\phi_{R}(p^{\mu})}\\{\phi_{L}(p^{\mu})}\\\end{matrix}\right)\quad,
\frac{\deltaC(\kappa)}{\delta\sigma(x)}|_{\sigma=0}=+2\kappa^{2}\sqrt{g(x)}\Psi_{\kappa}(x)-2\intd\muC(\mu)\Psi_{\kappa}(x)\Psi_{\mu}(x)\,\frac{\kappa^{2}}{\kappa^{2}-\mu^{2}}\sqrt{g(x)}
\widetildeA_{\mu}^{(C)}(y_{1},x_{2},x_{2}^{\prime})=ig_{\mu0}\frac{1}{4\pi}\frac{1}{|\bf{y}_{1}-\bf{x}_{2}|}\theta(x_{2}^{0}-y_{1}^{0})\theta(y_{1}-x_{2}^{\prime0}).
\ddot{q}_{nn^{\prime}}^{l}+\omega_{ln^{\prime}}^{2}(t)q_{nn^{\prime}}^{l}=2h\sum_{p=1}^{\infty}\dot{q}_{np}^{l}a_{pn^{\prime}}^{l}+\dot{h}\sum_{p=1}^{\infty}q_{np}^{l}a_{pn^{\prime}}^{l}+h^{2}\sum_{p,s=1}^{\infty}q_{np}^{l}a_{ps}^{l}a_{n^{\prime}s}^{l}.
\begin{array}{rl}{\theta_{6}+\theta_{18}-\theta_{-18}-\theta_{-6}}&{=-i~q^{-{\frac{1}{12}}}{\frac{\eta^{3}}{\eta(\tau/2)\eta(2\tau)}}}\\{\theta_{-2}+\theta_{10}-\theta_{22}-\theta_{-14}}&{=q^{-{\frac{1}{12}}}{\frac{\eta^{3}}{\eta(\tau/2)\eta(2\tau)}}.}\\\end{array}
(\partial_{w}^{2}+4f^{\prime}\partial_{w})\psi_{\mu\nu}=-e^{-2f}(\partial_{x}^{2}+\partial_{y}^{2}+\partial_{z}^{2}-\partial_{t}^{2})\psi_{\mu\nu}=-e^{-2f}m^{2}\psi_{\mu\nu},
{\frac{1}{\sqrt{-g}}}\partial_{\mu}[\sqrt{-g}\partial^{\mu}\phi]=-e^{-2\phi}(\partial_{\mu}a)^{2}+{\frac{\alpha^{\prime}}{4g^{2}}}e^{\phi}\left({\cal{R}}_{GB}^{2}-F^{\mu\nu}F_{\mu\nu}\right)
f_{9}^{A}=2\zeta(3)R_{A}e^{-2\phi}+\frac{2\pi^{2}}{3R_{A}}+0+~4\pie^{-\phi}\sum_{m\ne0}\sum_{n\ne0}\left|\frac{m}{n}\right|K_{1}\left(2\piR_{A}e^{-\phi}|mn|\right)e^{2\piimn{\calA}}
{\calL}^{(4)}=\frac{1}{8}h(v^{2})^{2}-\frac{1}{4}h^{-2/3}v^{2}v^{\hat{I}}\partial_{\hat{J}}h\bar{\theta}\Gamma_{~\tilde{I}}^{\tilde{0}~\tilde{J}}\theta+\frac{1}{2}h^{1/3}v^{2}\bar{\theta}(\Gamma^{\tilde{1}}\partial_{1}+\Gamma^{\tilde{2}}\partial_{2}-\Gamma^{\tilde{0}}\partial_{0})\theta.
G_{\\nu}^{\mu}=\frac{\kappa^{2}}{l\Psi_{0}}\left(\stackrel{A}{T_{\\nu}^{\mu}}+(1-\Psi_{0})\stackrel{B}{T_{\\nu}^{\mu}}\right)+\frac{1}{\Psi_{0}}\left(\delta\Psi_{\|\nu}^{|\mu}-\delta_{\nu}^{\mu}\Box\delta\Psi\right)+\frac{l^{2}}{2\Psi_{0}}\left(\frac{d}{l}-\frac{\Psi_{0}}{2}\right){\calS}_{\\nu}^{\mu}-\frac{l}{4}\frac{\Psi_{0}}{1-\Psi_{0}}\Box\chi_{\\nu}^{\mu}\.
\Phi_{(S)+}-\Phi_{(S)-}=\bar{x}_{+}\intdx_{-}e^{\phi_{-}(x_{-})}f_{1}(x_{-})-\bar{x}_{-}\intdx_{+}e^{\phi_{+}(x_{+})}f_{2}(x_{+})+f_{3}(x_{-})-f_{4}(x_{+}),
\sum_{\rho}C_{\Psi_{r}\Psi_{\rho}^{B}}^{B}C_{\Psi_{\sigma}\Psi_{\rho}^{B}}^{B}F_{r\sigma}^{\rho}(1-\eta)=\sum_{m}C_{r\sigmam}A_{\phi_{m}}^{B}F_{r\sigma}^{m}(\eta)
\begin{array}{c}{M_{ABC}=\frac1{12}(M^{ab}\left(\sigma_{ABC}\right)_{ba}+M_{ab}(\widetilde{\sigma}_{ABC})^{ba})}\\{M^{ab}=M^{ABC}(\widetilde{\sigma}_{ABC})^{ab}\,\quadM_{ab}=M^{ABC}\left(\sigma_{ABC}\right)_{ab}}\\\end{array}
\left\{\begin{array}{ccc}{j_{1}}&{j_{2}}&{j_{12}}\\{j_{3}}&{j}&{j_{23}}\\\end{array}\right\}^{S}=\sum_{j_{13}}(-1)^{\Theta^{R}}\left\{\begin{array}{ccc}{j_{1}}&{j_{3}}&{j_{13}}\\{j_{2}}&{j}&{j_{23}}\\\end{array}\right\}^{S}~\left\{\begin{array}{ccc}{j_{2}}&{j_{1}}&{j_{12}}\\{j_{3}}&{j}&{j_{13}}\\\end{array}\right\}^{S}\,.
{\calL}=-{\frac{1}{4}}({\calF}_{\mu\nu}-iegW_{\mu}^{*}W_{\nu})^{2}-{\frac{1}{2}}|{\calD}_{\mu}W_{\nu}-{\calD}_{\nu}W_{\mu}|^{2}+G^{2}\phi^{2}|W_{\mu}|^{2}-{\frac{1}{2}}(\partial_{\mu}\phi)^{2}-V(\phi)
\ddot{\phi}({\bfk}_{n},t)+3H{\dot{\phi}}({\bfk}_{n},t)+\omega_{n}^{2}(t)\phi({\bfk}_{n},t)+\frac{\deltaV_{F}}{\delta\phi(-{\bfk}_{n},t)}+\alpha\Omega({\bfk}_{n},t)=0.
\begin{aligned}{W\inGr^{(k)}(H),\qquadW^{\prime}=W+\lambdaW\inGr^{(k-m)}(H),}\\{W\subsetW^{\prime}\(\mathrm{codimension~}m),\qquad\lambdaW\subsetW^{\prime}\(\mathrm{codimension~}m+1).}\\\end{aligned}
\epsilon_{D}(\lambda,a)=F(D)\sum_{n=1}^{\infty}\int_{0}^{\infty}r^{D-3}\left(r^{2}+\left(\frac{n\pi}{a}\right)^{2}\right)^{\frac{1}{2}}exp\left[-\lambda\left(r^{2}+\left(\frac{n\pi}{a}\right)^{2}\right)^{\frac{1}{2}}\right].
g_{ij}(x^{k})\rightarrowg_{ij}(x^{k}-\varepsilonx^{l}\epsilon_{l}^{~k})-\varepsilon\epsilon_{i}^{~i^{\prime}}g_{i^{\prime}j}(x)-\varepsilon\epsilon_{j}^{~j^{\prime}}g_{ij^{\prime}}(x)
\left(\begin{array}{c}{P_{\vec{k}\,0}\,(t)}\\{X_{\vec{k}\,0}\,(t)}\\\end{array}\right)=\sum_{\vec{k}^{\prime}}\Lambda_{{\vec{k}\,\vec{k}}^{\prime}}\,(t)\left(\begin{array}{c}{\Pi_{\vec{k}^{\prime}\,}}\\{\varphi_{\vec{k}^{\prime}\,}}\\\end{array}\right)
{R^{\mu}}_{\nu}={\frac{1}{2}}\,\partial^{\mu}\phi\,\partial_{\nu}\phi+{\frac{1}{2\,4!}}\,e^{-\,\phi/2}\,\left[4\,F^{\mu\alpha_{2}\alpha_{3}\alpha_{4}}\,F_{\nu\alpha_{2}\alpha_{3}\alpha_{4}}\,-\,{\frac{3}{8}}\,{G^{\mu}}_{\nu}\,F^{2}\right]\,,
\int_{M_{\tau}}\!\!{\calD}^{\mathrm{e}^{\sigma}g_{\mathrm{pl}}}X(z,\bar{z})\mathrm{e}^{-S[\mathrm{e}^{\sigma}g_{\mathrm{pl}},X(z,\bar{z})]}\;=\;({\calA}_{M_{\tau}})^{c^{\mathrm{\scriptscriptstyleT}}}\int_{M_{\tau}}\!\!{\calD}^{g_{\mathrm{pl}}}X(z,\bar{z})\mathrm{e}^{-S[g_{\mathrm{pl}},X(z,\bar{z})]}\,
\begin{array}{lcr}{J_{1}}&{=}&{e^{{\mathrmi}\sigma}J_{m,-m}+e^{-{\mathrmi}\sigma}J_{-m,m}}\\{J_{2}}&{=}&{e^{{\mathrmi}\rho}J_{m,m}+e^{-{\mathrmi}\rho}J_{-m,-m}}\\{J_{3}}&{=}&{e^{{\mathrmi}k_{y}}Q+e^{-{\mathrmi}k_{y}}Q^{-1}}\\\end{array}
{\calL}_{\lambda}=\beta^{2}f(\lambda)+\frac{m^{2}}{2\lambda}A_{\mu}A^{\mu}-\frac14B_{\mu\nu}B^{\mu\nu}+\frac{\chi\,\theta}4\epsilon^{\mu\nu\lambda\rho}B_{\mu\nu}F_{\lambda\rho}\;.
\chi_{f_{j}}^{(2k+1)}\left(\psi_{i}^{(-k)},\psi_{j}^{(-k)}\right)={\frac{1}{24(2k+1)\pii}}\oint_{\calC}\psi_{i}^{(-k)}{\calS}_{f_{j}}^{(2k+1)}\psi_{j}^{(-k)}=0,\quad\foralli,j,
\begin{aligned}{\zeta_{AI^{\prime}}\cup\zeta_{BJ^{\prime}}=-2\epsilon_{AB}C_{I^{\prime}J^{\prime}}\\rho}\\{\zeta_{Aa}\cup\zeta_{Bb}=\epsilon_{AB}(\sigma_{1})_{ab}\\rho}\\\end{aligned}\qquad\begin{aligned}{\xi_{0}\cup\sigma}&{{}=\kappa_{0}}\\{\chi_{A}\cup\sigma}&{{}=\kappa_{A}}\\\end{aligned}
\hat{h}_{m}^{I}(y)=c_{2}\sqrt{|y|+y_{0}}\left\{-{\frac{s_{1}Y_{0}(my_{0})-2my_{0}Y_{1}(my_{0})}{s_{1}J_{0}(my_{0})-2my_{0}J_{1}(my_{0})}}J_{0}[m(|y|+y_{0})]+Y_{0}[m(|y|+y_{0})]\right\}\,,
h_{\mu\nu}^{\infty}(x)=-\frac{2x^{\alpha}x^{\beta}}{r^{2}}\int_{r}^{\infty}dr^{\prime}\,r^{\prime}\,R_{\alpha\mu,\beta\nu}^{L}(r^{\prime}\hat{x})+\frac{2x^{\alpha}x^{\beta}}{r^{3}}\int_{r}^{\infty}dr^{\prime}\,{r^{\prime}}^{2}R_{\alpha\mu,\beta\nu}^{L}(r^{\prime}\hat{x})
{}^{*}(dx^{\alpha_{1}}\wedgedx^{\alpha_{2}}\wedge...\wedgedx^{\alpha_{m}})=\frac{\sqrt{|g|}}{(n-m)!}\varepsilon_{\\\\\\\\beta_{m+1}...\beta_{n}}^{\alpha_{1}...\alpha_{m}}dx^{\beta_{m+1}}\wedge...\wedgedx^{\beta_{n}},
\Big[{\frac{M\omega_{D}}{2}}\Big(1-{\sqrt{1-{\frac{(D-1)Q^{2}}{2(D-2)M^{2}}}}}\Big)\Big]^{\frac{1}{D-2}}\lea\le\Big[{\frac{M\omega_{D}}{2}}\Big(1+{\sqrt{1-{\frac{(D-1)Q^{2}}{2(D-2)M^{2}}}}}\Big)\Big]^{\frac{1}{D-2}}.
I_{H}\;=\;I_{H}^{\mathrm{edge}}+I_{H}^{\mathrm{bulk}}\;=\;\sigma_{H}\,\left(\mu_{\ell}-\mu_{r}+\int_{\gamma}\underline{{E}}\left(\underline{{x}},t\right)\,\cdot\,d\underline{{s}}\left(\underline{{x}}\right)\right)~.
\begin{array}{lll}{}&{\phi(u)=\phi_{0}+qu\qquad\mathrm{with}:}&{q^{2}=\frac{k}{6}(25-\frac{3k}{k+2})}\\{k>1:}&{T_{\pm}(u)\sime^{\alpha_{\pm}u}}&{\alpha_{\pm}=\sqrt{k/6}\left(\sqrt{25-\frac{3k}{k+2}}\pm\sqrt{1-\frac{3k}{k+2}}\right)}\\{k=1:}&{T_{1}(u)\sime^{2u}\quad;\quadT_{2}(u)\simue^{2u}}&{.}\\\end{array}
[{\calD}\mu]={\calD}\xi{\calD}\pi_{i}{\calD}A^{\mu}{\calD}\pi_{\rho}{\calD}\rho\prod_{j}\{\delta[\Gamma_{j}(A^{0}+\xi,A^{i},\pi_{i},\rho,\pi_{\rho})]\}det\mid\{\tilde{\Omega}_{i},\Gamma_{j}\}\mid.
\int\prod_{j=1}^{N-1}(d\xi_{j}d\eta_{j})\left(\prod_{j=0}^{N-1}dp_{j}^{\xi}dp_{j}^{\eta}\right)exp\left[\sum_{k}ip_{k}^{\xi}(\xi_{k+1}-\xi_{k})+\sum_{k}ip_{k}^{\eta}(\eta_{k+1}-\eta_{k})\right]\nonumber
\left(D_{ij}\right)=\left(\begin{array}{ccc}{{f^{\prime}}^{2}}&{0}&{0}\\{0}&{A\left(R_{z}+R_{\bar{z}}\right)\left({\bar{R}}_{z}+{\bar{R}}_{\bar{z}}\right)}&{iA\left(R_{z}{\bar{R}}_{z}-R_{\bar{z}}{\bar{R}}_{\bar{z}}\right)}\\{0}&{iA\left(R_{z}{\bar{R}}_{z}-R_{\bar{z}}{\bar{R}}_{\bar{z}}\right)}&{-A\left(R_{z}-R_{\bar{z}}\right)\left({\bar{R}}_{z}-{\bar{R}}_{\bar{z}}\right)}\\\end{array}\right),
{\calV}_{\alpha,\alpha^{\prime}}^{(n)}(u)=\sum_{\beta_{1}\cdots\beta_{n}}w(\beta_{1},\alpha_{1},\beta_{2},\alpha_{1}^{\prime}|u)w(\beta_{2},\alpha_{2},\beta_{3},\alpha_{2}^{\prime}|u)\cdotsw(\beta_{n},\alpha_{n},\beta_{1},\alpha_{n}^{\prime}|u)~,
\alpha_{n}(t):=\delta_{n}(t)+\gamma_{n}(t)\,,~~~\delta_{n}(t):=-\frac{1}{\hbar}\int_{0}^{t}E_{n}(t^{\prime})dt^{\prime}\,,~~~\gamma_{n}(t):=i\int_{0}^{t}\langlen;t^{\prime}|\frac{d}{dt^{\prime}}|n;t^{\prime}\rangle\:dt^{\prime}\;.
e^{\textstyle(J+1)\alpha_{-}\widetilde\Phi(\sigma,\tau)}=c_{J}\sum_{m=-J}^{J}\,{\frac{1}{\lambda_{m}^{(J)}(\varpi)\sqrt{\lfloor\varpi\rfloor}\sqrt{\lfloor\varpi+2m\rfloor}}}\psi_{m}^{(-J-1)}(x_{+})\,{\overline{\psi}_{m}^{(-J-1)}}(x_{-}),
{\hat{\nabla}}_{a}{\hat{\nabla}}_{a}{\hat{A}}=-\sum_{J=0}^{N}\sum_{m=-J}^{J}A_{Jm}[{\hat{S}}_{a},[{\hat{S}}_{a},{\hat{Y}}_{m}^{J}]]=-\sum_{J=0}^{N}\sum_{m=-J}^{J}J(J+1)A_{Jm}{\hat{Y}}_{m}^{J}.
\hat{B}_{x;\hat{k},\hat{l}}+\hat{B}_{x;\hat{n},\hat{k}}+\hat{B}_{x;\hat{l},\hat{n}}+\hat{B}_{x+\hat{n};\hat{l},\hat{k}}+\hat{B}_{x+\hat{l};\hat{k},\hat{n}}+\hat{B}_{x+\hat{k};\hat{n},\hat{l}}=0\.
\bfE_{\mathrm{L}}=\mathrm{\frac{\bf\nabla\mathrm{}}{\bf\nabla\mathrm{^2}}(\rho^{(2)}+\rho^{(3)})\equiv-\frac{1}{4\pi}\int\!d^{3}\!y\frac{1}{\mid\bfy\mathrm{-\bfx\mathrm{\mid}}}\nabla(\rho^{(2)}(y)+\rho^{(3)}(y))\;\;.}
{d^{\dagger}}^{(i)}\psi\;\;=\;\;0\quad\quad,\quad\quad\gamma^{\mu_{1}^{i}}\psi_{{\mu}_{1}^{1}...{\mu}_{p_{1}}^{1}...{\mu}_{1}^{i}...{\mu}_{p_{i}}^{i}...{\mu}_{p_{N}}^{N}}\;\;=\;\;0
ds_{str.}^{2}=-\left(1-\frac{2\eta}{r}\right)^{\frac{m+\sigma}{\eta}}dt^{2}+\left(1-\frac{2\eta}{r}\right)^{\frac{\sigma-m}{\eta}}dr^{2}+\left(1-\frac{2\eta}{r}\right)^{1+\frac{\sigma-m}{\eta}}r^{2}d\Omega^{2}
{R_{\mu\nu\rho}}^{\sigma}=\partial_{\nu}{\Gamma^{\sigma}}_{\mu\rho}-\partial_{\mu}{\Gamma^{\sigma}}_{\nu\rho}+{\Gamma^{\alpha}}_{\mu\rho}{\Gamma^{\sigma}}_{\alpha\nu}-{\Gamma^{\alpha}}_{\nu\rho}{\Gamma^{\sigma}}_{\alpha\mu},
\sigma_{\mathrm{qu}}^{\mathrm{(kink)}}=\frac{1}{2}\int\frac{d^{n-2}k_{\|}}{(2\pi)^{n-2}}\left[(k_{\|}^{2})^{1/2}+(k_{\|}^{2}+\frac{3}{2}m^{2})^{1/2}+\int\frac{dk_{\bot}}{(2\pi)}(k_{\|}^{2}+k_{\bot}^{2}+2m^{2})^{1/2}\;\delta^{\prime}(k_{\bot})\right].
{\mit\Xi}_{[1/2]}\,=\,\left(\begin{array}{cc}{e^{i\phi}}&{0}\\{0}&{e^{-i\phi}}\\\end{array}\right)\quad,\qquad{\mit\Xi}_{[1]}\,=\,\left(\begin{array}{ccc}{e^{i\,2\,\phi}}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{e^{-i\,2\,\phi}}\\\end{array}\right)\quad,
[\tilde{\alpha}_{\omega_{1}},L_{\omega}]=2\sum_{\omega^{\prime}=-N+1}^{N}\sqrt{|\omega_{1}\omega^{\prime}|}\omega_{1}\:C_{-\omega_{1},\omega^{\prime}}^{-\omega}\:\tilde{\alpha}_{\omega^{\prime}}+\sqrt{|\omega_{1}(\omega-\omega_{1})|}\omega_{1}\:\tilde{\alpha}_{\omega_{1}-\omega}
\frac{dT_{-}}{dk_{-}}=\frac{4}{\sqrt{K}}\frac{E(k_{-})}{1-k_{-}^{2}},\;\;\;\;\;\frac{dS_{-}^{\mathrm{cl}}}{dk_{-}}=-\frac{8}{K\alpha^{\prime}}\frac{k_{-}E(k_{-})}{1-k_{-}^{2}}.
G(t,t^{\prime})=S_{\sigma}(t,t^{\prime})-\sum_{s=1}^{n}\sum_{p=a,b}\sum_{N_{s}}\int_{C_{p}^{(s)}}S_{\sigma}(t,t_{1})\frac{d\theta_{1}dz_{1}}{2\pii}Y_{p,N_{s}}(t_{1})\chi_{N_{s}}(t^{\prime})
\bigg[\bar{Q}_{1},\bar{Q}_{2},\bar{Q}_{3}\bigg]\to\bigg[\bar{Q}_{(0)}={\frac{\bar{Q}_{1}+\bar{Q}_{2}+\bar{Q}_{3}}{\sqrt6}},\quad\bar{Q}_{(8)}={\frac{\bar{Q}_{1}+\bar{Q}_{2}-2\bar{Q}_{3}}{\sqrt{12}}},\quad\bar{Q}_{(3)}={\frac{\bar{Q}_{1}-\bar{Q}_{2}}{2}}\bigg],
S=\intd^{3}\sigma\Big(-e^{-\phi}\sqrt{-\mathrm{det}\,(G_{\mu\nu}+\calF_{\mu\nu})}+\frac{1}{2}\tilde{H}^{\mu\nu}(F_{\mu\nu}-2\partial_{\mu}A_{\nu})\Big)-\inte^{-\phi}(C_{3}+C_{1}\wedge{\calF}).
{\calO}_{V}(P)=V=V_{{\bar{i}}_{1}{\bar{i}}_{2}\cdots{\bar{i}}_{p}}^{j_{1}j_{2}\cdotsj_{q}}(P)\eta^{\bar{i}_{1}}\eta^{\bar{i}_{2}}\cdots\eta^{\bar{i}_{p}}\psi_{\bar{j}_{1}}\psi_{\bar{j}_{2}}\cdots\psi_{\bar{j}_{q}},
\frac{\partial^{2}a_{n}}{\partialt_{1}^{+2}}=\frac{1}{2}(\frac{1}{a_{n}-1}+\frac{1}{a_{n}})(\frac{\partiala_{n}}{\partialt_{1}^{+}})^{2}-\frac{1}{t_{1}^{+}}\frac{\partiala_{n}}{\partialt_{1}^{+}}-8a_{n}(a_{n}-1)+\frac{2(n+1)^{2}}{t_{1}^{+2}}\frac{a_{n}-1}{a_{n}}.
H=\sum_{n}\frac{1}{2m}\left[\vec{p}_{n}-\alpha\vec{a}_{n}-e\vec{A}^{\mathrm{ex}}(\vec{r}_{n})\right]^{2}+\frac{1}{2}\sum_{n,m(\neqn)}V(|\vec{r}_{n}-\vec{r}_{m}|),
[r^{\mu},r^{\mu}]={\frac{i\hbar}{m^{2}c^{2}}}S^{\mu\nu}={\frac{i\hbarj}{m^{3}c^{3}}}\epsilon^{\mu\nu\lambda}P_{\lambda};~~[r^{\mu},P^{\nu}]=-i\hbarg^{\mu\nu};~~[P^{\mu},P^{\nu}]=0.
\left[\,\,T_{++}-T_{--}\,\,\right]_{\sigma=0,\pi}=0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[\,\,G_{+}^{1}-\eta_{1}G_{-}^{1}\,\,\right]_{\sigma=0,\pi}=0,
\widetilde{\eta}^{i}\equivU^{i}{}_{j}\eta^{j}\,,\qquad\widetilde{\eta}_{i}\equiv\eta_{j}(U^{-1})^{j}{}_{i}\,\qquad\widetilde{\theta}^{i}\equiv\tilde{U}^{i}{}_{j}\theta^{j}\,,\qquad\widetilde{\theta}_{i}\equiv\theta_{j}(\tilde{U}^{-1})^{j}{}_{i}\,
\big\langle0\big|V_{a}(x)V_{-a}(x^{\prime})\big|0\big\rangle=\frac{1}{\left|x-x^{\prime}\right|^{\frac{a^{2}}{4\pi}}}\quad\mathrm{as}\quad\left|x-x^{\prime}\right|\,\rightarrow\,0\:.
g(q,p,t)=\sum_{n=1}^{N}\left[i\lambda\epsilon_{n}t-\mu_{n}{\calF}^{(n)}\cdotq^{(n)}+\mu_{n}\left(\frac{p^{(n)}}{\sqrt{m_{n}}}-\nu_{n}{\calF}^{(n)}\right)^{2}\right],
\deltaI_{C-S}=n\int_{\Omega}\left\langle\nabla(\delta\mathrm{{\bf~A}})\mathrm{\tiny~\wedge~}\mathrm{{\bf~F}}^{n-1}\right\rangle=n\int_{\Omega}d\left\langle\delta\mathrm{{\bf~A}}\mathrm{\tiny~\wedge~}\mathrm{{\bf~F}}^{n-1}\right\rangle.
(3b)=e^{4}i^{3}\int{\calD}q\;\varepsilon_{\alpha\beta\gamma}\frac{q^{\gamma}}{q^{2}}\frac{(2p+q)^{\beta}}{(p+q)^{2}}\int{\calD}k\frac{(2p+k)^{\mu}(2p+2q+k)^{\nu}(2k+2p+q)^{\alpha}}{(p+k)^{2}(p+q+k)^{2}}\;\varepsilon_{\mu\nu\rho}\frac{k^{\rho}}{k^{2}}
\hat{a}_{\pm}={\frac{1}{2}}\left[M^{2}\pm\sqrt{M^{4}-4M^{2}\left(1+\displaystyle{\frac{\bar{g}^{2}}{2\pi}}\right)}\right],\\\\\hat{r}_{\pm}^{2}=1+\frac{{\bar{g}^{2}/2\pi}}{1+{\bar{g}^{2}/2\pi}}\hat{a}_{\pm}.
\langle{\calO}\rangle_{A}=\int\prod_{i=1}^{m}[db_{i}^{(0)}]\prod_{j=1}^{{\overline{m}}}[d{\overline{c}}_{j}^{(0)}]\prod_{n\neq0}[db_{n}\,d{\overline{c}}_{n}]\,e^{-\sum_{n\neq0}\lambda_{n}{\overline{c}}_{n}b_{n}}{\calO}(b,{\overline{c}})\,.
\frac{E_{Cas.}(a_{0})-E_{Cas.}(a)}{\hbar\,\Omega}\left(\frac{\Omega}{c}\right)^{-3}\simeq\frac{5}{216\,{\pi}^{2}}\,\left(V_{0}-{V}\right)\simeq2\times10^{-3}\,\left(V_{0}-{V}\right).
\left\langle\Psi_{X}(\xi)\right\rangle_{1}=\frac{1}{N^{2}}a^{-10/3}\hat{s}^{-10/3}5^{3/2}\cdot2^{2/3}c^{1/2}[2w_{(1)D}(y)+u_{(1)}(y)]+O(\frac{1}{N^{2}}a^{-3}),
\Psi^{a_{1}\ldotsa_{K}}\left[A^{g};x_{1},\ldots,x_{K}\right]=g_{a_{1}b_{1}}^{\mathrm{Ad}}(x_{1})\ldotsg_{a_{K}b_{K}}^{\mathrm{Ad}}(x_{K})\Psi^{b_{1}\ldotsb_{K}}\left[A;x_{1},\ldots,x_{K}\right]
S_{\mathrm{dr}}=3\left(\zeta_{\mathrm{d}}-\zeta_{\mathrm{r}}\right)={\frac{\delta\rho_{\mathrm{d}}}{\rho_{\mathrm{d}}}}-{\frac{3}{4}}{\frac{\delta\rho_{\mathrm{r}}}{\rho_{\mathrm{r}}}}\,,
T^{-1}S=\left(\begin{array}{ccc}{D_{\alpha}(Q)D^{\alpha}(B)}&{-(D_{\nu}(B)b)}&{0}\\{cD^{\mu}(B)}&{R^{\mu}{}_{\nu}}&{(D^{\mu}(B)b)}\\{0}&{-D_{\nu}(B)c}&{D_{\alpha}(B)D^{\alpha}(Q)}\\\end{array}\right)(x)\delta(x-y)\,
A_{1}^{++--}=c_{1}\left(e_{2}^{+}\cdote_{4}^{-}\right)^{s_{a}}\left(K_{2}\cdote_{1}^{+}\right)^{s_{a}}\left(K_{4}\cdote_{3}^{-}\right)^{s_{b}}\left(K_{3}\cdote_{4}^{-}\right)^{s_{b}-s_{a}}
V^{\overline{{a}}}{}_{b}=\left[\begin{matrix}{V_{R}}\\{V_{L}}\\\end{matrix}\right]=\left[\begin{matrix}{{\frac{1}{\sqrt{2}}}E^{-1}}\\{}&{{\frac{1}{\sqrt{2}}}E^{-1}}\\{}&{}&{I_{16}}\\\end{matrix}\right]\times\left[\begin{matrix}{I_{4}}&{(G+B+C)}&{-A}\\{I_{4}}&{(-G+B+C)}&{-A}\\{0}&{A^{T}}&{-I_{16}}\\\end{matrix}\right]\,
L\{{{\bfB}^{\prime}},{{\bfE}^{\prime}}\}=\frac{\epsilon_{0}}{2}\intd^{3}x\left(c^{2}{{\bfB}^{\prime}}^{2}-{{\bfE}^{\prime}}^{2}\right)-\rho_{m}\,\psi-{\bfj}_{m}\cdot{\bfW}
{\calL}_{NR}=\psi^{*}\left(iD_{t}+\frac{{\bfD}^{2}}{2m}\right)\psi-\frac{v_{0}}4(\psi^{*}\psi)^{2}+\frac\Theta2\partial_{t}{\bfA}\times{\bfA}-\ThetaA_{0}{\bf\nabla}\times{\bfA}\;,
Z_{L}^{(m,n)}(\tau)=Z_{n/2}^{m/2}(\tau)^{16}+Z_{1+n/2}^{m/2}(\tau)^{16}+Z_{n/2}^{1+m/2}(\tau)^{16}+Z_{1+n/2}^{1+m/2}(\tau)^{16}.
=m_{3/2}\left[\frac{P^{\prime}}{P^{\prime\prime}}\left(P^{\prime}-\frac{\omega^{\prime}(S^{*})}{\omega(S^{*})}\right)+\left(3+(T+T^{*})\frac{h^{\prime}(T^{*})}{h(T^{*})}\right)\left(1+\frac{\sumn_{i}}{3}-\frac{(T+T^{*})}{3}\frac{y_{ijk}^{\prime}(T)}{y_{ijk}(T)}\right)\right]
dl^{2}\sim\biggl(1+\frac{\psi_{i}^{\prime}}{r_{i}^{\prime}}\biggr)^{2+2a}\biggl(1+\frac{\chi_{i}^{\prime}}{r_{i}^{\prime}}\biggr)^{2-2a}(dr_{i}^{\prime2}+r_{i}^{\prime2}d\Omega^{2})\,,
\begin{array}{rcl}{\phi}&{=}&{\displaystyle\phi_{1}+i\phi_{2}=\sqrt{2}\frac{dg/dz}{1-\bar{g}g},}\\{\omega}&{=}&{\displaystyle\omega_{\mu}dx^{\mu}=\frac{i}{2}\Big(\frac{gd\bar{g}-\bar{g}dg}{1-\bar{g}g}\Big),}\\\end{array}
g_{tt}^{-1}=g_{ii}\rightarrow1+{\frac{1}{2r}}\Bigl(e^{\frac{-\eta_{0}-\sigma_{0}-\rho_{0}}{2}}|q_{1}|+e^{\frac{\eta_{0}+\sigma_{0}-\rho_{0}}{2}}|p_{2}|+e^{\frac{-\eta_{0}+\sigma_{0}+\rho_{0}}{2}}|q_{3}|+e^{\frac{\eta_{0}-\sigma_{0}+\rho_{0}}{2}}|p_{4}|\Bigr)+\dots
J^{\mu\nu}=q^{\mu}p^{\nu}-q^{\nu}p^{\mu}-\mathrm{i}\sum_{n=1}^{\infty}\left(a_{n}^{\dagger\mu}a_{n}^{\nu}-a_{n}^{\dagger\nu}a_{n}^{\mu}+{\tilde{a}}_{n}^{\dagger\mu}{\tilde{a}}_{n}^{\nu}-{\tilde{a}}_{n}^{\dagger\nu}{\tilde{a}}_{n}^{\mu}\right)~.
\breve{a}_{0,2}=\int\sqrt{h}d^{D-1}x\left({\frac{1}{72}}\bar{R}^{2}-{\frac{1}{180}}\bar{R}_{\mu\nu}\bar{R}^{\mu\nu}+{\frac{1}{180}}\bar{R}_{\mu\nu\lambda\rho}\bar{R}^{\mu\nu\lambda\rho}-\frac16\bar{R}V+\frac12V^{2}\right)~~,
S^{(N)}\left[{\bfr}_{1},\dots{\bfr}_{N-1}\right]({\bfr}^{\prime\prime},{\bfr}^{\prime};t^{\prime\prime},t^{\prime})\equivS^{(N)}\left[{\bfr}_{1},\dots{\bfr}_{N-1}\right]({\bfr}^{\prime\prime},{\bfr}^{\prime};t^{\prime\prime}-t^{\prime})\;,
{\calE}_{nl}^{(1)}=\frac{\mu\alpha^{4}}{2\hbar^{4}c^{2}}\frac{1}{n^{3}}\left(\frac{C_{1}}{n}-\frac{4(C_{1}-C_{2})}{2l+1}-(2\lambda-1)\frac{2C_{1}}{n^{3}}\delta_{0l}\right)\,.
\int_{-\infty}^{+\infty}dx\,\,\Psi_{k}(x)\Psi_{k\,^{\prime}}(x)=\delta(k-k\,^{\prime}),~~~~~\int_{0}^{+\infty}dk\,\,\Psi_{k}(x)\Psi_{k}(x^{\prime})=\delta(x-x\,^{\prime})
\delta\left(\begin{array}{c}{0}\\{1}\\\end{array}\right)=\frac{uq^{1/2}-u^{-1}q^{-1/2}}{q-q^{-1}}\left(\begin{array}{c}{0}\\{1}\\\end{array}\right)=b\left(\begin{array}{c}{0}\\{1}\\\end{array}\right).
\hat{G}^{i}=\;(T_{i}+\bar{T}_{i})^{-1}+\;\sum_{m}\;c_{im}\frac{\partial_{T_{i}}{\eta\Biggl(\frac{T_{i}}{l_{im}}\Biggr)}}{\eta\Biggl(\frac{T_{i}}{l_{im}}\Biggr)}
I_{\beta}(D,s,m)=\frac{m^{D-2s}}{(2\pi^{1/2})^{D}\Gamma(s)}\left[\Gamma(s-\frac{D}{2})+4\sum_{n=1}^{\infty}\left(\frac{2}{mn\beta}\right)^{D/2-s}K_{D/2-s}(mn\beta)\right]
\begin{array}{lllll}{c_{1}=c~~}&{~c_{2}=\textstyle{\frac{3}{2}}c~~}&{~k_{1}=-12d~~}&{~k_{2}=-6d~~}&{~k_{3}=4d}\\{{}}&{{}}&{{}}&{{}}&{{}}\\{k_{4}=-4c-6d~~}&{~k_{5}=4c+4d~~}&{~k_{6}=-c+d~~}&{~k_{7}=c+d}&{{}}\\\end{array}
I_{1}=-\frac{\Gamma(2-\frac{D}{2})}{(4\pi)^{\frac{D}{2}}}\int_{0}^{1}dx\sum_{n}\frac{(k_{0}+xp_{0})}{[(k_{0}+xp_{0})^{2}+x(1-x)p^{2}+m^{2}]^{2-\frac{D}{2}}}.
\sum_{k=1}^{[n/2]-1}F(n,k)k(n-k-2)\left[\omega_{WP}^{(n)}\right]^{n-4}\cap\left[D_{k}\right]={\frac{1}{2}}\sum_{k=1}^{n-3}F(n,k){\frac{k(n-k-2)}{n-1}}\left(^{\;\;n}_{k+1}\right)\left(^{n-4}_{k-1}\right)V_{k+2}V_{n-k},
\sumq^{\frac{1}{2}p^{2}}=\sum_{n}q^{\frac{3}{2}\left(2n+1+\frac{m_{1}+m_{2}}{3}\right)^{2}}=\sum_{n}q^{6\left(n+\frac{m_{1}+m_{2}+3}{6}\right)^{2}}=\theta_{2m_{1}+2m_{2}+6,6}\:(\tau).
{\calL}=\frac{1}{2}(\partial_{\mu}\sigma)^{2}+\frac{1}{2}(\partial_{\mu}\pi)^{2}\nonumber\,-\frac{1}{2}\mu^{2}(\sigma^{2}+\pi^{2})-\frac{\lambda}{4}(\sigma^{2}+\pi^{2})^{2}+c\sigma\quad.
\chi_{\ell}^{(k)}(\tau,z)=\frac{\theta_{\ell+1,k+2}-\theta_{-\ell-1,k+2}}{\theta_{1,2}-\theta_{-1,2}}(\tau,z):=\sum_{m\in{\bfZ}_{2k}}c_{m}^{\ell}(\tau)\:\theta_{m,k}(\tau,z),
\hat{T}=\frac{\varsigmam}{\hat{\omega}}\hat{\zeta}\Sigma^{2n+1}\left[\hat{\zeta}\hat{\omega}\Sigma^{2n+1}+\imath\partial_{d}\left(\varsigma\Sigma^{2n+1}\Sigma^{d}\right)-\varsigmam\right]\;,\;\;\varsigma=(-1)^{n}s=\pm1\;.
\Gamma_{PC}^{odd}[A]=\frac{1}{2}{\frac{d}{ds}}\zeta_{{D\!\!\!\!/}^{~2}}(0)+\sum_{k=1}^{\infty}(im)^{2k}\frac{\zeta_{{D\!\!\!\!/}^{~2}}(k)}{2k}+\Gamma_{Loc}^{odd}[A],
\left[\Gamma_{TR\Phi}+\frac{b}{\sqrt{(1+b^{2})W_{1}-b^{2}}}\Gamma_{T}\Gamma_{\natural}\right]\epsilon_{0}=\frac{\sqrt{(1+b^{2})W_{1}}}{\sqrt{(1+b^{2})W_{1}-b^{2}}}\epsilon_{0}.
{}~~~~~~~~~~~~~\times\Bigg\{\sum_{\sigma\inZ_{N}}\frac{w(x_{4}x_{13},x_{1}x_{34},x_{3}x_{14}|\sigma-a+b+n)w(x_{23},x_{34},x_{24}|\sigma)s(\sigma,d)}{w(x_{13},\omegax_{34},\omegax_{14}|\sigma+n)w(x_{4}x_{23},x_{2}x_{34},x_{3}x_{24}|\sigma+c-d)s(\sigma,a)}\Bigg\}_{0}
S_{1D}=-\intd^{5}\sigma\left(\sqrt{-G}\sqrt{1+z_{1}+\frac{z_{1}^{2}}{2}-z_{2}}+\frac{\epsilon_{\mu\nu\lambda\sigma\tau}C^{\mu}\tilde{\calH}^{\nu\lambda}\tilde{\calH}^{\sigma\tau}}{8(1+C_{1}^{2})}\right),
S={\frac{1}{2\kappa_{5}^{2}}}\intd^{4}x\sqrt{-g}{\frac{\bar{K}_{+}-\bar{K}_{-}}{2\bar{K}_{+}\bar{K}_{-}}}\left[{\calR}_{g}-{\frac{4}{3}}\bar{K}_{+}\bar{K}_{-}\right].
G_{i}^{(2)a}=\left(\delta_{\;\;d}^{a}\delta_{\;\;i}^{j}-\left(D_{i}\right)_{\;\;b}^{a}\bar{M}_{\;\;c}^{b}\left(D^{j}\right)_{\;\;d}^{c}\right)\gamma_{j}^{(2)d}\approx0.
\left(\begin{array}{c}{{\mathcal{F}^{\prime}}_{\mu\nu}}\\{\widetilde{\mathcal{F}}_{\mu\nu}^{\prime}}\\\end{array}\right)=\left(\begin{array}{cc}{a}&{c}\\{b}&{d}\\\end{array}\right)\left(\begin{array}{c}{\mathcal{F}_{\mu\nu}}\\{\widetilde{\mathcal{F}}_{\mu\nu}}\\\end{array}\right),
det\left|\begin{array}{ccc}{Q_{1}(\lambda-3i/2)}&{Q_{1}(\lambda\pmi/2)}&{Q_{1}(\lambda+3i/2)}\\{P_{1}(\lambda-3i/2)}&{P_{1}(\lambda\pmi/2)}&{P_{1}(\lambda+3i/2)}\\{R_{1}(\lambda-3i/2)}&{R_{1}(\lambda\pmi/2)}&{R_{1}(\lambda+3i/2)}\\\end{array}\right|=T^{\pm}(\lambda).
\begin{array}{c}{F(r)=\frac{r-n}{35(r+n)^{4}\ell^{2}}\{35r^{5}+175nr^{4}+(300n^{2}+5\ell^{2})r^{3}+(25n\ell^{2}+100n^{3})r^{2}}\\{+(47n^{2}\ell^{2}-295n^{4})r-315n^{5}+35\ell^{2}n^{3}\}}\\\end{array}
\mu_{u}^{A}=\left(\mu_{1}^{A},\dots,\mu_{N-2}^{A},0,0\right),\qquad(x^{\alpha})^{v}=\left(0,\dots,0,x^{\mu}\sigma_{\mu}^{\alpha\beta^{\prime}}\right)\quad\mathrm{with}\quadN-2+\beta^{\prime}=v.
\begin{array}{lc}{R(\alpha_{i})=\alpha_{\sigma(i)},}&{R(\alpha_{i}\alpha_{j})=\alpha_{\sigma(i)}\alpha_{\sigma(j)},}\\{R(\alpha_{i}\alpha_{j}\alpha_{k})=\alpha_{\sigma(i)}\alpha_{\sigma(j)}\alpha_{\sigma(k)},}&{\cdots;}\\\end{array}
U_{i_{1}\,\,i_{2}}^{\,\,j_{1}\,\,j_{2}}=\sum_{\rho\subset\lambda\otimes\mu}U(\lambda\mu\rho)\left(\begin{array}{ccc}{\lambda}&{\mu}&{\rho}\\{i_{1}}&{i_{2}}&{m}\\\end{array}\right)^{\ast}\left(\begin{array}{ccc}{\lambda}&{\mu}&{\rho}\\{j_{1}}&{j_{2}}&{m}\\\end{array}\right)\,\,.
m_{F,0}^{2}-m_{B,0}^{2}\simeq\frac{4N(g_{1}^{2}\phi_{1}^{2})\cdots(g_{N}^{2}\phi_{N}^{2})\theta^{2}}{\sum_{k=1}^{N}(g_{1}^{2}\phi_{1}^{2})\cdots\widehat{(g_{k}^{2}\phi_{k}^{2})}\cdots(g_{N}^{2}\phi_{N}^{2})}
\Gamma\lbrackA\rbrack=-2{\displaystyle\int_{0}^{\infty}}{\frac{dT}{T}}e^{-m^{2}T}{\displaystyle\int}{\calD}\,X\mathrm{tr}{\calP}\mathrm{e}^{-\int_{0}^{T}d\tau\intd\theta\,\Bigl[-{\frac{1}{4}}XD^{3}X-\mathrm{ie}DX_{\mu}A_{\mu}(X)\Bigr]},