harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\calI}_{\xi}^{(\beta)}(X)\simeq\frac{1}{2^{3/2}\pi}\left({\calM}_{\xi}^{2}(X)\right)^{1/4}\left(T(X)\right)^{3/2}e^{-{\calM}_{\xi}^{2}(X)/T(X)}\;\;\;\;({\calM}_{\xi}^{2}(X)\geq0).
\intd^{2}\zeta_{SS}d^{2}\xi_{SS}\intd^{4}x_{0}\mathrm{Tr}(\phi^{3})=\frac{3}{2}\left(-\frac{g^{2}}{2^{4}\pi^{2}}\right)\left(\mathrm{Tr}(A_{1})-\frac{ig}{4\sqrt{2}\pi^{2}\rho^{2}}\sum_{k=3}^{N_{c}}(\xi_{Mk}\zeta_{Nk}+\xi_{Nk}\zeta_{Mk})\right).
D_{ij}^{ab}=g^{2}\delta^{ab}\frac{\delta_{ij}}{p^{2}}-g^{4}N\frac{\Gamma(2-\omega)\Gamma(\omega)\Gamma(\omega-1)}{(4\pi)^{\omega}\Gamma(2\omega)}\cdot4(2\omega-1)\frac{\delta_{ij}\delta^{ab}}{p^{6-2\omega}}.
\tilde{H}=(2\pi\alpha^{\prime})d\tilde{\calF}-(2\pi\alpha^{\prime})G^{(6)}{\calG}^{(1)}+3(2\pi\alpha^{\prime})^{2}{\calH}^{(3)}d{\calH}^{(3)}+{\textstyle\frac{m}{2}}{\calG}^{(7)}\,.
g_{\tilde{r}\tilde{r}}(G_{\tilde{r}}^{\tilde{r}}+G_{\theta}^{\theta})=16\piG\frac{m}{f}(L_{M}-g^{\tilde{r}\tilde{r}}\frac{\partialL_{M}}{\partialg^{\tilde{r}\tilde{r}}}-g^{\theta\theta}\frac{\partialL_{M}}{\partialg^{\theta\theta}})
\bar{\rho}_{{}_{\mathrm{REG}}}(r)=-\frac{m}{2{\pi}^{2}}\int_{|m|}^{\infty}dq\frac{q}{\sqrt{q^{2}-m^{2}}}[I_{1-F}(qr)K_{1-F}(qr)-I_{F}(qr)K_{F}(qr)],
ds^{2}=\left({\frac{\,d_{ABC}\,\mathrm{Im}Z^{A}\,\mathrm{Im}Z^{B}\,\mathrm{Im}Z^{C}}{\,[d_{ABC}\,\mathrm{Im}Z^{A}\,\mathrm{Im}Z^{B}\,\mathrm{Im}Z^{C}]_{\infty}}}\right)^{-1}\,dt^{2}-\left({\frac{\,d_{ABC}\,\mathrm{Im}Z^{A}\,\mathrm{Im}Z^{B}\,\mathrm{Im}Z^{C}}{\,[d_{ABC}\,\mathrm{Im}Z^{A}\,\mathrm{Im}Z^{B}\,\mathrm{Im}Z^{C}]_{\infty}}}\right)\,d\vec{x}^{2}\.
M^{(-)}(\nu,k)=M_{o}^{(-)}(k)+gM_{1}^{(-)}(k)+g\int_{1}^{\infty}d\alpha(\sum_{\ell=1}^{5}\frac{\Phi_{\ell}^{(2)}(\alpha)}{(\alpha-ik)^{\ell}})[\alpha^{\frac{\nu}{2}}+\alpha^{\frac{-\nu}{2}}]
\begin{array}{l}{\left[J^{0}(w)+J^{1}(w)\right]\timesg(w)\rightarrow\frac{-i}{z-w}\left(\begin{array}{ll}{0}&{0}\\{1}&{0}\\\end{array}\right)g(w)}\\{J^{2}(z)\timesg(w)\rightarrow\frac{i/2}{z-w}\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)g(w)}\\{\left[J^{0}(w)-J^{1}(w)\right]\timesg(w)\rightarrow\fraci{z-w}\left(\begin{array}{ll}{0}&{1}\\{0}&{0}\\\end{array}\right)g(w)}\\\end{array}
\langle\hat{\phi}^{2}\rangle={\frac{b(m)}{16\pi^{2}}}~~,~~\langle\hat{\bar{\psi}}\hat{\psi}\rangle=4m\langle\hat{\phi}^{2}\rangle~~,~~\langle\hat{V}_{\mu}\hat{V}^{\mu}\rangle=3\langle\hat{\phi}^{2}\rangle~~~.
I_{L}^{p}=\int_{M}\sqrt{g}\delta_{[\alpha_{1}\cdots\alpha_{2p}]}^{[\beta_{1}\cdots\beta_{2p}]}R_{\beta_{1}\beta_{2}}^{\alpha_{1}\alpha_{2}}\cdotsR_{\beta_{2p-1}\beta_{2p}}^{\alpha_{2p-1}\alpha_{2p}}d^{d}x\.
H=\int_{\Sigma}dx^{2}(P^{a}\dot{A}_{a}^{0}+\bar{b}^{a}\dot{\bar{c}}_{a})+\int_{\partial\Sigma}dx^{2}\left[P^{a}(J_{a}-\frac{\alphakl}{2}P_{a})+f_{ab}^{~~c}J_{c}\bar{c}^{b}c^{b}\right]
{\calS}_{0}=\intdx^{0}dx^{3}\sum_{m=0}^{\infty}\left(i\psi_{m}^{(L)*}\partial_{+}\psi_{m}^{(L)}+i\psi_{m}^{(R)*}\partial_{-}\phi_{m}^{(R)}\right)+(\mathrm{massive~modes}),
{\calM}=\frac{1}{\Im\mathrm{m}(\omega_{1}\bar{\omega_{2}})}\left(\begin{array}{cc}{|\omega_{1}|^{2}}&{\Re\mathrm{e}(\omega_{1}\bar{\omega_{2}})}\\{}&{}\\{\Re\mathrm{e}(\omega_{1}\bar{\omega_{2}})}&{|\omega_{2}|^{2}}\\\end{array}\right)\,.
Q_{\alpha}^{(1)}=-\frac{i}{2}\,[A_{\mu},A_{\nu}]^{ij}\Gamma_{\alpha\beta}^{\mu\nu}\,\frac{\partial}{\partial\bar{\psi}_{\beta}^{ij}}+i\Gamma_{\mu\alpha\beta}\psi_{\beta}^{ij}\,\frac{\partial}{\partialA_{\mu}^{ij}}
\sum_{\beta}\left\{\phi_{\beta}[\ddot{q}_{\alpha\beta}(t)+\omega_{\beta}^{2}(t)q_{\alpha\beta}(t)]+2\dot{q}_{\alpha\beta}\dot{\phi}_{\beta}+q_{\alpha\beta}\ddot{\phi}_{\beta}\right\}=0,\quad\omega_{\beta}(t)=j_{l,n}/a(t),
S_{\Omega}=\sum_{n=2}^{\infty}\intd^{3}y_{1}....d^{3}y_{n}F_{\mu_{1}\nu_{1}}(y_{1})....F_{\mu_{n}\nu_{n}}(y_{n})\Omega^{\mu_{1}\nu_{1}...\mu_{n}\nu_{n}}(y_{1},...,y_{n})\;.
F_{m}(\phi)=\sum_{n=1}^{\infty}\;f_{mm_{2}\ldotsm_{n}}^{(n)}\phi_{m_{2}}\ldots\phi_{m_{n}},\;\;\;\;\;\;\tilde{F}_{m}(\tilde{\phi})=\sum_{n=1}^{\infty}\;f_{mm_{2}\ldotsm_{n}}^{(n)}\tilde{\phi}_{m_{n}}\ldots\tilde{\phi}_{m_{2}}
y^{\prime}=y+\hat{\xi}\,^{5}(x)\,,~~x^{\mu^{\prime}}=x^{\mu}+\xi^{\mu}(x^{\lambda},y)=x^{\mu}-\frac{\ell}{2}\,e^{2|y|/\ell}\,\partial^{\mu}\,\hat{\xi}\,^{5}(x)+\beta^{\mu}(x^{\lambda})\,,
\alpha_{n}^{\prime}\left(t,z;x\right)\,=\,\sum_{n=1}^{N}\alpha_{n_{0}}^{\prime}e^{-\gamma_{n}\left(\left(m+1\right)z\right)}\left[e^{-\gamma_{n}\left(mt-i\gamma_{n}\left(2m+1\right)^{2}x\right)}\right],\tag{3.2a}
L=\frac{1}{2e}(\dot{x}_{\mu}-iv\epsilon_{\mu\nu\lambda}\xi^{\nu}\xi^{\lambda})^{2}-\frac{e}{2}m^{2}+2i\numv\theta_{1}\theta_{2}-\frac{i}{2}\xi_{\mu}\dot{\xi}{}^{\mu}+\frac{i}{2}\theta_{a}\dot{\theta}_{a},
F_{\pm}=-\frac{V_{p}V(S^{d-1})}{16\piG}h^{(d-2)^{2}/(d-4)}\frac{d-4}{2}\tilde{T}^{(2d-4)/(d-4)}(1+x_{\pm})^{2(d-3)/(d-4)}\left(1+\frac{x_{\pm}}{x_{c}}\right)^{-2\frac{d-2}{d-4}}
Q=\frac{6\cdot16}{4!(2\pi)^{4}}\left(\frac{\zeta\pi}{2}\right)^{2}\left(\frac{\zeta^{\prime}\pi}{2}\right)^{2}\sum\biggl\{\frac{-2\zeta^{2}}{\delta(\delta+\zeta/2)^{2}(\delta+\zeta)}\biggr\}\biggl\{\frac{-2\zeta^{\prime2}}{\delta^{\prime}(\delta^{\prime}+\zeta^{\prime}/2)^{2}(\delta^{\prime}+\zeta^{\prime})}\biggr\}=1\,,
\left(U(\Lambda)\tilde{\psi}\right)\left(\vec{p},s_{3}\right)=\sum_{s_{3}^{^{\prime}}}D_{s_{3}s_{3}^{^{\prime}}}(R(\Lambda,\Lambda^{-1}p)\tilde{\psi}(\Lambda^{-1}\vec{p},s_{3}^{^{\prime}})
H[\varphi]=\int\!d^{d}x\,\left\{\frac{1}{2}\pi^{2}({\bfx})+J\frac{1}{2}[\nabla\!_{d}\varphi({\bfx})]^{2}-\frac{1}{2}\varphi^{2}({\bfx})+\frac{\lambda}{4}\varphi^{4}({\bfx})\right\}.
{\calL}_{++}^{++}(\Lambda)={\frac{1}{2}}\Lambda_{+}^{i}[\delta^{ij}D^{++}+(V^{++})^{ij}]\Lambda_{+}^{j}+\Lambda_{+}^{+Y^{\prime}}[{\frac{1}{2}}(V)_{Y^{\prime}Z^{\prime}}\Lambda_{+}^{+Z^{\prime}}+(V^{++})_{Y^{\prime}}^{-i}\Lambda_{+}^{i}+m\Phi_{+Y^{\prime}}^{+}]\;,
\frac{i}{48}\sqrt{-g_{00(string)}}e^{-\phi}\overline{{\Theta}}\Gamma^{mnp}\ThetaH_{mnp}\rightarrow\frac{i}{48}Z^{-1}\overline{{\Theta}}(\Gamma^{mnp}H_{mnp}-6\Gamma^{iab}w_{iab})\Theta
{M}=\left(\begin{array}{cc}{C}&{\hat{0}}\\{\hat{0}}&{\bar{C}}\\\end{array}\right)\left(\begin{array}{cc}{\sigmaU}&{\bar{\sigma}V}\\{\bar{\sigma}V}&{\sigmaU}\\\end{array}\right)\left(\begin{array}{cc}{D^{T}}&{\hat{0}}\\{\hat{0}}&{\bar{D}^{T}}\\\end{array}\!\right).
S=\intd^{5}x\sqrt{-\hat{g}}\left[{\frac{1}{2\kappa_{5}^{2}}}{\calR}-{\frac{4}{3}}\partial_{M}\phi\partial^{M}\phi-e^{-2\alpha\phi}\Lambda\right]+\intd^{4}x\sqrt{-g}\left[{\calL}_{mat}-\sigmae^{-\alpha\phi_{0}}\right],
\left(\begin{array}{cc}{\displaystyle{\frac{\mathrm{e}^{i\varphi/2}}{\sqrt{2}}}}&{\displaystyle{\frac{\mathrm{e}^{-i\varphi/2}}{\sqrt{2}}}}\\{}&{}\\{\displaystyle{\frac{-i\mathrm{e}^{i\varphi/2}}{\sqrt{2}}}}&{\displaystyle{\frac{i\mathrm{e}^{-i\varphi/2}}{\sqrt{2}}}}\\\end{array}\right)
R_{12}^{\,\prime}~=~g_{1}^{}\,g_{2}^{}\biggl(R_{12}^{}\,+\,g_{1}^{-1}\{g_{1}^{}\!,L_{2}^{}\}\,+\,{\textstyle\frac{1}{2}}\left[\,g_{1}^{-1}g_{2}^{-1}\,\{g_{1}^{}\!,g_{2}^{}\}\,,L_{2}^{}\,\right]\biggr)\,g_{1}^{-1}g_{2}^{-1}~.
W^{(ab)}(r,r^{\prime})=\frac{\bar{I}_{\nu}^{(a)}(az)}{\bar{I}_{\nu}^{(b)}(bz)}\frac{\bar{K}_{\nu}^{(b)}(bz)}{\bar{K}_{\nu}^{(a)}(az)}\left[\frac{G_{\nu}^{(a)}(az,zr)G_{\nu}^{(b)}(bz,zr^{\prime})}{\bar{K}_{\nu}^{(a)}(az)\bar{I}_{\nu}^{(b)}(bz)-\bar{K}_{\nu}^{(b)}(bz)\bar{I}_{\nu}^{(a)}(az)}-I_{\nu}(zr^{\prime})K_{\nu}(zr)\right].
{\frac{1}{4}}\intd^{3}xd^{3}y<\lambda_{iH}^{a}(x)G^{-2}(x-y)\lambda_{iH}^{a}(y)>_{U}=V{\frac{N^{2}-1}{2}}\int_{M}^{\Lambda}{\frac{d^{3}k}{(2\pi)^{3}}}G^{-1}(k)
F_{1}({\bf{k}})=1-\sum_{i,{\bf{q}}}\frac{1-n_{F}({\bf{k}}+{\bf{q}})}{({\tilde{\omega}}_{i}(-{\bf{q}})+\frac{{\bf{k.q}}}{m}+\epsilon_{{\bf{q}}})^{2}}g_{i}^{2}(-{\bf{q}})
x_{\ast}^{2}=-{\frac{\alpha}{\alpha_{c}}}\left[{\frac{1}{g_{4R}(\mu)}}-{\frac{1}{g_{4R}^{\ast}}}\right]\left({\frac{-R}{4\mu^{2}}}\right)^{{\frac{\alpha}{\alpha_{c}}}}~-{\frac{R}{6\mu^{2}}}\left({\frac{-R}{4\mu^{2}}}\right)^{{\frac{\alpha}{2\alpha_{c}}}}
[J_{m}^{+},\left(J_{+1}^{-}\right)^{n}]=-n\left(2J_{m+1}^{3}-km\delta_{m+1}\right)\left(J_{+1}^{-}\right)^{n-1}-n(n-1)J_{m+2}^{-}\left(J_{+1}^{-}\right)^{n-2}
v^{\mu}\rightarrow{v^{\prime}}^{\mu}=\big(W_{i}(v)\big)^{\mu}\equivv^{\mu}-2\frac{w_{i\nu}v^{\nu}}{w_{i\rho}w_{i}^{\rho}}w_{i}^{\mu}\;\;\;\;(\mathrm{no~sum~over~i~)}
\frac{1}{(k\!\cdot\!n)^{2}\:(p-k)\!\cdot\!n}\rightarrow\frac{1}{(p\!\cdot\!n)^{2}}\left[\frac{p\!\cdot\!n}{(k\!\cdot\!n)^{2}}+\frac{1}{k\!\cdot\!n}+\frac{1}{(p-k)\!\cdot\!n}\right]\:,\:\:\:\:\:\:\:p\!\cdot\!n\neq0
{\breve{\omega}}^{A}={\omega}^{A}+iT^{A{A^{\prime}}}{\pi}_{A^{\prime}},\qquad\qquad\qquad{{\bar{\breve{\omega}}}}^{A^{\prime}}={{\bar{\omega}}}^{A^{\prime}}-iT^{A{A^{\prime}}}{\bar{\pi}}_{A}.
V(\alpha^{*},\alpha)=\frac{g}{3!}\left(V_{abc}\hat{a}^{a}\hat{a}^{b}\hat{a}^{c}+3\hat{a}_{a}^{\dagger}V_{\;\;bc}^{a}\hat{a}^{b}\hat{a}^{c}+3\hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}V_{\quadc}^{ab}\hat{a}^{c}+\hat{a}_{a}^{\dagger}\hat{a}_{b}^{\dagger}\hat{a}_{c}^{\dagger}V^{abc}\right)
V/N_{f}=(Q_{0}^{2}+\bar{Q}_{0}^{2})\bigl[Q_{0}\bar{Q}_{0}\bigr]^{-2\gamma}+\frac{g^{2}}{4\gamma}(Q_{0}^{2}-\bar{Q}_{0}^{2})^{2}-m_{Q}^{2}Q_{0}^{2}+m_{\bar{Q}}^{2}\bar{Q}_{0}^{2}
\begin{array}{lcl}{V_{a}^{\mu}}&{=}&{(W_{+}^{\mu},W_{-}^{\mu},Z^{\mu},A^{\mu})}\\{B_{a}}&{=}&{(B_{+},B_{-},B_{Z},B_{A})}\\\end{array}\quad\begin{array}{lcl}{c_{a}}&{=}&{(c_{+},c_{-},c_{Z},c_{A})}\\{\bar{c}_{a}}&{=}&{(\bar{c}_{+},\bar{c}_{-},\bar{c}_{Z},\bar{c}_{A})}\\\end{array}
\Theta_{\alpha,m}(\rho,\bar{\rho},[\omega]):=\sum_{\lambda\inM_{\alpha}}q_{\rho}^{\frac{1}{2m}\bigl(\lambda_{+}^{[\omega]},\lambda_{+}^{[\omega]}\bigr)}\bar{q}_{\rho}^{-\frac{1}{2m}\bigl(\lambda_{-}^{[\omega]},\lambda_{-}^{[\omega]}\bigr)}\,.
\hat{\Gamma}^{(n)}[\vec{\varphi}_{1},\vec{\varphi}_{2}]=\sum_{i_{1},\,i_{2},\,...,\,i_{n}=1}^{2}\hat{\calF}_{i_{1},\,i_{2},\,...,\,i_{n}}[\vec{\varphi}_{1},\vec{\varphi}_{2}],
U_{I}(t)=I-\frac{i}{\hbar}\int_{0}^{t}dt^{\prime}H_{I}(t^{\prime})+\left(-\frac{i}{\hbar}\right)^{2}\int_{0}^{t}dt^{\prime}\int_{0}^{t^{\prime}}dt^{\prime\prime}H_{I}(t^{\prime})H_{I}(t^{\prime\prime})+\cdots
\frac{\nu_{3}}{\pi^{d}}\!\!\int\!\!\int\!\!d^{d}k_{1}~d^{d}k_{2}~(k_{1}k_{2})~P_{k_{1},m_{1}}^{\nu_{1}}P_{k_{1}-k_{2},m_{2}}^{\nu_{2}}P_{k_{2},m_{3}}^{\nu_{3}+1}+\!\nu_{1}m_{1}^{2}~J_{\nu_{1}+1~\nu_{2}\nu_{3}}^{(d)}\!-\left(\!\frac{d}{2}-\!\nu_{1}\!\right)\!J_{\nu_{1}\nu_{2}\nu_{3}}^{(d)}\equiv0.
u_{r}\equivU_{(0,\,r-1)}^{2}+\Lambda_{1}^{4}\Omega_{(2,\,r-1)}+\Lambda_{r}^{4}\Omega_{(1,\,r-2)}-M_{0}\Omega_{(1,\,r-1)}\quad\mathrm{and}\quad\Lambda_{(1,\,r)}^{2}\equiv2(\prod_{i=1}^{r}\Lambda_{i}^{4})^{1/2}.
{\hat{\Pi}}=D{\hat{X}}^{\hat{\mu}}D{\hat{X}}^{\hat{\nu}}{\hat{g}}_{{\hat{\mu}}{\hat{\nu}}}=\partial{\hat{X}}^{\hat{\mu}}\partial{\hat{X}}^{\hat{\nu}}\left({\hat{g}}_{{\hat{\mu}}{\hat{\nu}}}+|{\hat{k}}|^{-2}{\hat{k}}_{\hat{\mu}}{\hat{k}}_{\hat{\nu}}\right)\,,
H=\intd^{\nu}x\left[{\frac{\Pi^{2}(\vec{x})}{2}}+{\frac{1}{2}}\left(\vec{\nabla}\phi(\vec{x})\right)^{2}+{\frac{1}{2}}m^{2}{\phi^{2}(\vec{x})}+{\frac{\lambda}{4!}}{\phi}^{4}(\vec{x})\right]\,.
S_{1}=\int_{M_{5}}[\frac{1}{2}P^{ij}\partial_{5}B_{ij}-\frac{1}{16}P^{ij}P_{ij}-2{}^{*}H^{0i}{}^{*}H_{0i}+\rho_{j}\partial_{i}P^{ij}]d^{5}x,
\varepsilon^{i}\rightarrow{\varepsilon^{i}}=\varepsilon^{i}+\delta\varepsilon^{i}={{W_{5}(p,q,r)}^{i}}_{j}\varepsilon^{j}=\varepsilon^{i}+(p\varepsilon^{1}+q\varepsilon^{2}+r\varepsilon^{3})\frac{k^{i}}{\omega}
\rho^{(n)}(L):=\frac{\sum_{\alpha_{i},\beta_{j}}{N^{(L)}}_{\alpha_{1}...\alpha_{n}\beta_{1}...\beta_{n}}^{\dagger}N_{\beta_{1}...\beta_{n}\alpha_{1}...\alpha_{n}}^{(L)}}{\langle{W^{(L)}}^{n}\rangle}\;=
\left\langle{\mathcalL}\right\rangle=-\frac{1}{4}\left\langle\mathcal{F^{A}}_{\mu\nu}\mathcalF^{\mathcalA\mu\nu}\right\rangle=-\frac{1}{4}f_{\mu\nu}f^{\mu\nu}+\left(D_{\mu}\varphi^{*}\right)\left(D^{\mu}\varphi\right)-\frac{8g^{2}}{25}\left|\varphi\right|^{4}
\phi_{+}(\tilde{\tau})=L\int_{0}^{\tilde{\tau}}\frac{dx}{r_{+}^{2}(x)}=\sqrt{\frac{L^{2}}{m^{2}+L^{2}}}F[\psi_{+}(r_{+}(\tilde{\tau})),\;\sqrt{\frac{m^{2}}{m^{2}+L^{2}}}\;].
\frac{(-1)^{p}}{\prod_{i=0}^{q}(w_{i}-v)^{\beta-1}\prod_{i=1}^{q-1}(\xi_{i}-v)}\frac{d^{p}}{dv^{p}}\left[\prod_{i=0}^{q}(w_{i}-v)^{\beta-1}\prod_{i=1}^{q-1}(\xi_{i}-v)\right].
\kappa={\frac{1}{2r_{H}}}\;{\frac{r_{H}}{\sqrt{r_{H}^{2}+a^{2}}}}\;\left(1-8\pi\;\rho_{H}\;r_{H}^{2}\right)={\frac{1}{2(2M-a)}}\left(1-8\pi\;\rho_{H}\;r_{H}^{2}\right).
G^{(4)}{}_{\mu\nu\rho\sigma}=4\left[\partial_{[\mu}C^{(3)}{}_{\nu\rho\sigma]}-3\left(\partial_{[\mu}B_{\nu\rho}\right)C^{(1)}{}_{\sigma]}+{\textstyle\frac{3}{8}}mB_{[\mu\nu}B_{\rho\sigma]}\right]\,.
\frac{1}{\calN}\int[d\chi][d\bar{\chi}]\,\chi_{+}(x)\chi_{+}^{\dagger}(y)e^{\intd^{2}x^{\prime}\,\bar{\chi}i\gamma_{\mu}\partial_{\mu}\chi}\int[d\phi]\,e^{\intd^{2}x^{\prime}\,[n-i\varepsilon_{\mu\nu}\partial_{\mu}j_{\nu}]\phi(x^{\prime})}e^{-\intd^{2}x^{\prime}\,\frac{1}{2}\phi\hat{O}\phi},
\frac{d}{d\tau}\,=\,\sqrt{\dot{x}^{2}}\,\frac{d}{ds\,}{;}\qquad\frac{d^{2}x_{\mu}}{ds^{2}\,}\,=\,\frac{\dot{x}^{2}\ddot{x}_{\mu}\,-\,(\dot{x}\ddot{x})\,\dot{x}_{\mu}}{(\dot{x}^{2})^{2}}\,{;}\qquadk_{1}\,\frac{\partialk_{1}}{\partial\ddot{x}_{\mu}}\,=\,-\,\frac{1}{\dot{x}^{2}}\,\frac{d^{2}x_{\mu}}{ds^{2}\,}{;}
+\frac{1}{3!}g_{0}\left[\left(\frac{\deltaW}{\deltaj(x)}\right)^{3}+3\frac{\deltaW}{\deltaj(x)}\,\frac{\delta^{2}W}{\deltaj(x)^{2}}+\frac{\delta^{3}W}{\deltaj(x)^{3}}\right]-j(x)=0\,.
\left\langleU_{\lambda}\left(w\right)\right\rangle_{w^{\prime}}=\prod_{X_{i^{\prime}}^{\prime}\in{\itG}}\prod_{\left(n_{i^{\prime}}^{*}\right)_{i^{\prime}=0}^{k}}\left\langleU_{\lambda}\left(w\right)\right\rangle_{w_{n_{i^{\prime}}^{*}}^{\prime}}^{\left({\itG}_{1}\right)_{X_{i^{\prime}}^{\prime}}}
\begin{array}{rll}{A_{+}}&{=}&{s_{0}(x^{+},x^{-})A(x^{+})s_{0}^{-1}(x^{+},x^{-})\,}\\{A_{-}}&{=}&{\tilde{s}_{0}(x^{+},x^{-})B(x^{-})\tilde{s}_{0}^{-1}(x^{+},x^{-})\,}\\\end{array}
A_{\mub}^{a}(\gamma(0))\rightarrowA_{\mub}^{\primea}(\gamma(0))=A_{\mub}^{a}(\gamma(0)-\epsilon\left(L(\gamma)A_{\mub}^{a}(\gamma(0))\right)=A_{\mub}^{a}(\gamma(0))+\epsilon\dot{\gamma}_{\mu}(0)U(\gamma)_{b}^{a}
\Psi=\mathrm{Tr}\int\left(\bar{c}\,\,d^{\ast}\!\!\,A+\bar{b}\wedge\,\,d^{\ast}\,\!B+\bar{a}\,\,d^{\ast}\!\,\Omega+\bar{\theta}\,\,d^{\ast}\,\!b-\,d^{\ast}\,\!\bar{b}\wedge\Xi\right)
\vec{\chi}(x)=\left(\begin{array}{c}{\vec{\chi}_{\mathrm{vec}}(x)}\\{\vec{\chi}_{\mathrm{gr}}(x)}\\\end{array}\right)=\left(\begin{array}{c}{A(x)\,\vec{n}}\\{B(x)\,\vec{p}}\\\end{array}\right)\,\qquad\partial_{i}\partial_{i}\vec{\chi}(x)=0\,
|{\calB},m,I\rangle=\frac{1}{2}\left(|+,m,I\rangle_{\scriptscriptstyle\mathrm{NS-NS}}-|-,m,I\rangle_{\scriptscriptstyle\mathrm{NS-NS}}+|+,m,I\rangle_{\scriptscriptstyle\mathrm{R-R}}+|-,m,I\rangle_{\scriptscriptstyle\mathrm{R-R}}\right),
Z_{F}[C]=\sum_{n=1}^{\infty}\frac{1}{n!}\int\prod_{i=1}^{n}{\calD}{\bfX}_{i}exp\{-\sum_{i=1}^{n}[\intdt(M\sqrt{\dot{\bfX}_{i}^{2}}+i{\bfC}\cdot\dot{\bfX}_{i})-i\Phi[\frac{\dot{\bfX}_{i}}{|\dot{\bfX}_{i}|}]-i\frac{\alpha-\frac{1}{2}}{2}\Theta_{ii}]+i(\alpha-\frac{1}{2})\sum_{i<j}\Theta_{ij}\}\;.
=e^{-\frac{i}{2R}\frac{\partial}{\partialt_{0}}}\sum_{n}\frac{1}{(\frac{1}{2\sqrt{\alpha^{\prime}}}(2n+1)i-t_{0})^{2}}=e^{-\frac{i}{2R}\frac{\partial}{\partialt_{0}}}\frac{\partial}{\partialt_{0}}\sum_{n}\frac{1}{\frac{1}{2\sqrt{\alpha^{\prime}}}(2n+1)i-t_{0}}.
{\calV}({\bfp})={\frac{m}{\sqrt{2}}}\left(\begin{matrix}{D^{J}\left(\alpha({\bfp})\Theta_{[1/2]}\right)\xi_{\sigma}^{*}}\\{D^{J}\left(\alpha^{-1\,\dagger}({\bfp})\Theta_{[1/2]}\right)(-1)^{2J}\xi_{\sigma}^{*}}\\\end{matrix}\right)\quad,
\left(1+\kappa_{0}e^{-\theta_{\infty}}\left[{\frac{\sqrt{||Z||^{2}}}{\vec{r}^{2}}}+{\frac{1}{(\vec{r}-\vec{r}_{0})^{2}}}\right]\right)\left(1+\kappa_{0}e^{\theta_{\infty}}\left[{\frac{\sqrt{||Z||^{2}}}{\vec{r}^{2}}}+{\frac{1}{(\vec{r}-\vec{r}_{0})^{2}}}\right]\right)
{\frac{\Lambda}{\Lambda_{0}}}={\frac{1+c_{N/2+1}\left(\Lambdar^{2}\right)^{N/2+1}(G/\pi\epsilonr^{2})/N(N-1)}{1-c_{N/2+2}\left(\Lambdar^{2}\right)^{N/2+1}(G/\pi\epsilonr^{2})/2.}}
{\frac{\partial\Sigma}{\partialp^{-}}}=N_{c}{\frac{\partial}{\partialp^{-}}}\int{\frac{d^{4}K}{(2\pi)^{4}}}(ig\gamma^{+}){\frac{-i}{\gamma^{\alpha}(p+K_{\|})_{\alpha}+m}}(ig\gamma^{+})D^{--}(K).
S^{reg}[\varphi,{\bar{\chi}},\chi,\eta,g_{\mu\nu},M]\;=\;-\frac{1}{2}\left[\langle\varphi|{\Box^{\prime}}|\varphi\rangle+\langle{\bar{\chi}}|({\Box^{\prime}}+M^{2}\sqrt{-g})|\chi\rangle+\langle\eta|({\Box^{\prime}}+2M^{2}\sqrt{-g})|\eta\rangle\right]
F_{p+1}[\mathbf{z}\cdot\mathbf{u}]=\sum_{i=1}^{N}\left\{\sum_{r=1}^{p}(u_{i})_{r+1}\frac{\partial}{\partial(u_{i})_{r}}+z_{i}(u_{i})_{1}\right\}F_{p}[\mathbf{z}\cdot\mathbf{u}].
\frac{1}{((x_{\mu(1)}-x_{\mu(2)})^{2}+|z_{i(1)}-z_{i(2)}|^{2})^{4}}\left(tr(f_{\mu\nu(1)}f_{\nu\lambda(1)})\tr(f_{\mu\sigma(2)}f_{\sigma\lambda(2)})+\cdots\right).
\{g_{_E}(x,z)\}=\left(\begin{array}{cc}{g_{\mu\nu}(x,z)+A_{\mu}^{\,\,i}(x,z)\,A_{\nu}^{\,\,j}(x,z)\,g_{0\,ij}(x,z)}&{A_{\mu}^{\,\,i}(x,z)\,g_{0\,il}(x,z)}\\{g_{0\,ki}(x,z)\,A_{\nu}^{i}(x,z)}&{g_{0\,kl}(x,z)}\\\end{array}\right)
\frac{\partial}{\partialx^{\nu}}\left(\sum_{\kappa=0}^{3}A_{\kappa\mu}^{\kappa}\right)=\frac{1}{2}\sum_{\kappa,\lambda=0}^{3}g^{\kappa\lambda}\frac{\partial^{2}g_{\kappa\lambda}}{\partialx^{\mu}\partialx^{\nu}}-\frac{1}{2}\sum_{\kappa,\lambda,\sigma,\tau=0}^{3}g^{\kappa\sigma}g^{\lambda\tau}\frac{\partialg_{\kappa\lambda}}{\partialx^{\mu}}\frac{\partialg_{\sigma\tau}}{\partialx^{\nu}}.
{\tilde{H}}_{coll}^{\beta,\pm}=\sum_{n>0}(1-\beta)n\alpha_{n}^{\dagger}\alpha_{n}\mp\sqrt{\beta}\sum_{n,n^{\prime}\geq0}(\alpha_{n}^{\dagger}\alpha_{n^{\prime}}^{\dagger}\alpha_{n+n^{\prime}}+\alpha_{n+n^{\prime}}^{\dagger}\alpha_{n}\alpha_{n^{\prime}}).
S(D,a,p)=\int\mathrm{d}^{D}x\;{\frac{\sqrt{-g}}{2\kappa_{D}^{2}}}\Big[-R-{\textstyle{\frac{4}{D-2}}}(\partial\phi)^{2}-{\frac{g_{s}^{2k}}{2(d+1)!}}\left({\frac{e^{\phi}}{g_{s}}}\right)^{a}F_{d+1}^{2}\Big]\,,
\timesM_{E_{D}\alpha/\sqrt{M^{2}c^{4}-E_{D}^{2}},\sqrt{(2l_{D}+1)^{2}-4(\alpha^{2}+q^{2})}/2}\left(\frac2{\hbarc}\sqrt{M^{2}c^{4}-E_{D}^{2}}r_{a}\right).
[(hr^{3}\partial_{r})^{2}+\omega^{2}r^{6}{\mathcalH}-\frac{8hr^{4}(R^{2}+2r_{K}^{2})^{2}}{(3r^{2}+(R^{2}+2r_{K}^{2}))^{2}}(1+\frac{r_{0}^{2}}{(R^{2}+2r_{K}^{2})})]\tilde{\lambda}=0,
\int_{SO(4)}dg\\overline{{D_{\alpha\beta}^{(\Lambda)}(g)}}D_{\alpha^{\prime}\beta^{\prime}}^{(\Lambda^{\prime})}(g)={\frac{1}{\mathrm{dim}_{\Lambda}}}\delta^{\Lambda\Lambda^{\prime}}\\delta_{\alpha\alpha^{\prime}}\\delta_{\beta\beta^{\prime}}.
\begin{aligned}{}&{{}\alpha_{n}-\alpha_{-n+d}^{\prime}=a_{n},}&{}&{{}\qquad\beta_{n}+\beta_{-n+2d}^{\prime}=b_{n}}\\{}&{{}\gamma_{n}-\gamma_{-n-d}^{\prime}=c_{n},}&{}&{{}\qquad\delta_{n}+\delta_{-n-2d}^{\prime}=d_{n}}\\\end{aligned}
\int_{\itbd}\frac{\sqrt{\left|{g}\right|}}{16\piG}\left[\nabla^{\mu}\left(\xi_{m}^{\rho}{\nabla}_{\rho}\xi_{n}^{\nu}-\xi_{n}^{\rho}{\nabla}_{\rho}\xi_{m}^{\nu}\right)-(\mu\leftrightarrow\nu)\right]dS_{\mu\nu}.
f_{D=2}^{\mathrm{TM}}=-{\frac{i}{8\pi^{2}a^{3}}}\int_{-\infty}^{\infty}dx\,x\sum_{m=-\infty}^{\infty}\left(1-{\frac{m^{2}}{x^{2}}}\right)\left({\frac{J_{m}(x)}{J_{m}^{\prime}(x)}}+{\frac{H_{m}^{(1)}(x)}{H_{m}^{(1)\prime}(x)}}\right),
R_{\sigma}^{2}={\frac{1+s_{3}r_{3}}{2}}+{\frac{q_{3}^{2}}{q_{3}^{2}+M^{2}}}{\frac{1-s_{3}r_{3}}{2}},\quadR_{\pi}^{2}={\frac{1-s_{3}r_{3}}{2}}+{\frac{q_{3}^{2}}{q_{3}^{2}+M^{2}}}{\frac{1+s_{3}r_{3}}{2}}
\epsilon{\frac{\partial}{\partial\epsilon}}\Psi_{1}[Y]={\frac{1}{2!}}\ointd\tau_{1}\ointd\tau_{2}\epsilon\partial_{\epsilon}G_{ab}(t_{1},t_{2}){\frac{\partial}{\partialY^{a}(t_{1})}}{\frac{\partial}{\partialY^{b}(t_{2})}}\Psi_{1}[Y].
\begin{aligned}{F_{0,1,0}(d_{1},0;d_{1},0)}&{{}={\frac{1}{d_{1}}}\int_{{pt}_{S^{1}}}{\frac{e(B_{5}^{m})e(B_{1}^{m})}{e(B_{2}^{m})}}}\\{}&{{}={\frac{1}{d_{1}^{2}(d_{1}-1)!}}{\frac{1}{(\lambda_{1}-\lambda_{2})^{d_{1}-1}}}\prod_{k=1}^{d_{1}-1}\left((\lambda_{2}-2\lambda_{1})d_{1}+{k}(\lambda_{1}-\lambda_{2})\right).}\\\end{aligned}
{\calL}=\frac{g^{2}}{32\pi^{2}}G_{\mu\nu}^{a}G^{\mu\nu,a}-\frac{1}{8\pi}\epsilon^{\mu\nu\rho}(A_{\mu}^{a}\partial_{\nu}A_{\rho}^{a}+\frac{1}{3}\epsilon^{abc}A_{\mu}^{a}A_{\nu}^{b}A_{\rho}^{c})
B_{\mu\nu}^{c}=ie^{2}\{\epsilon_{\mu\nu\lambda}\partial_{\lambda}\phi-2\pi\int_{\Sigma_{C}}\delta(x-y(\sigma))d\sigma_{\mu\nu}+2\pi\oint\psi_{\mu}(\tau)\psi_{\nu}(\tau)\delta(x-y(\tau))d\tau\}.
ds_{\mathrm{rel}}^{2}=\left(2-\frac{\zeta}{r^{2}}\right)\left\{\frac{dr^{2}}{1-\zeta/r^{2}}+\frac{r^{2}}{4}(\sigma_{1}^{2}+\sigma_{2}^{2})+\frac{r^{4}(r^{2}-\zeta)}{(2r^{2}-\zeta)^{2}}\sigma_{3}^{2}\right\}\,,
\widehatD_{\mu}\widehat\Sigma_{\nu\rho}=D_{\mu}\Sigma_{\nu\rho}-\frac{1}{4}\theta^{\zeta\tau}\bigg(\left\{\omega_{\zeta},(D_{\tau}+\partial_{\tau})D_{\mu}\Sigma_{\nu\rho}\right\}+2\left\{R_{\mu\zeta},D_{\tau}\Sigma_{\nu\rho}\right\}\bigg)+{\calO}\left(\theta^{2}\right),
\left\{\nabla_{\bfr^{\prime}}^{2}+\frac{2M}{\hbar^{2}}\left[E-V({\bfr^{\prime}})\right]\right\}{\mathcalG}_{D}({\bfr^{\prime\prime}},{\bfr^{\prime}};E)=\delta({\bfr^{\prime\prime}}-{\bfr^{\prime}})\;,
+H_{1}^{-1}(dz_{1}^{2}+dz_{2}^{2})+H_{2}^{-1}(dz_{3}^{2}+dz_{4}^{2})+H_{3}^{-1}(dz_{5}^{2}+dz_{6}^{2})+{\sum_{i=1}^{4}}dx_{i}^{2}],
P=\frac{2^{8}}{3^{4}5^{2}7^{2}11}\frac{2\kappa^{2}N^{2}T_{0}^{2}}{\Omega_{8}}\left(\omega_{1}^{12}R_{1}^{4}+\omega_{2}^{12}R_{2}^{4}+\omega_{3}^{12}R_{3}^{4}\right)
S^{\mp,a,p}(x,x^{\prime})=\int_{-\infty}^{+\infty}\,dz\,\int_{-\infty}^{+\infty}\,\frac{dp_{3}}{2\pi}\,e^{-ip_{3}y^{3}}S_{Q}^{\mp,a,p}\;,\quady_{\mu}=x_{\mu}-x_{\mu}^{\prime}\;,
\frac{1}{g_{\Lambda}}-\frac{1}{4}\,\mathrm{Log}\left(\frac{\Lambda^{2}}{M^{2}}\right)+\frac{1}{6}\frac{\Lambda^{2}}{M^{2}}e^{-4/g_{\Lambda}}+O\left(\frac{\Lambda^{4}}{M^{4}}e^{-8/g_{\Lambda}}\right)
\begin{array}{lcl}{p~\partial_{\sigma}V}&{=}&{{\displaystyle2\sum_{j=2}^{D-1}\partial_{\tau}X^{j}\partial_{\sigma}X^{j}}}\\{}&{}&{}\\{p~\partial_{\tau}V}&{=}&{{\displaystyle\frac{\alpha}{\tau^{2}}~(X^{2}-Y^{2})+\sum_{j=2}^{D-1}\left\{(\partial_{\tau}X^{j})^{2}+(\partial_{\sigma}X^{j})^{2}\right\}}}\\\end{array}
\Phi_{+}^{(2)}=\sum_{i}z_{i}b_{i+}^{\dagger}(\underline{{P}})|0\rangle+\sum_{ijs}\intd\underline{{l}}f_{ijs}(\underline{{l}})b_{i,s}^{\dagger}(\underline{{P}}-\underline{{l}})a_{j}^{\dagger}(\underline{{l}})|0\rangle
\Theta\equiv\begin{array}{c}{lim}\\{\varepsilon\rightarrow0}\\\end{array}\begin{array}{c}{lim}\\{p_{i}^{2}\rightarrowm_{i}^{2}-i\varepsilon}\\\end{array}\Pi_{i}(p_{i}^{2}-m_{i}^{2}+i\varepsilon)\frac{\phi_{i}(p_{i})}{\sqrt{Z_{i}}}F.T.[\frac{\delta^{n}}{\deltaJ^{n}(x)}]
F_{p}\left(\phi_{0},\phi_{1},\cdots,\phi_{p}\right)=\sum_{m_{0}\cdotsm_{p}}\alpha_{m_{0}\cdotsm_{p}}\,\,a_{m_{0}}\left(\phi_{0}\right)a_{m_{1}}\left(\phi_{1}\right)\cdotsa_{m_{p}}\left(\phi_{p}\right)
g_{20}(q_{1},q_{2})={\frac{g}{4}}({\frac{q_{1}q_{2}}{q_{1}^{2}+q_{2}^{2}+q_{1}q_{2}}})[\hat{f}^{2}(q_{1})C(g,q_{1})+\hat{f}^{2}(q_{2})C(g,q_{2})+2\phi_{0}].
\left(\begin{array}{l}{u_{\nu}\left(\eta\right)}\\{v_{\nu}\left(\eta\right)}\\\end{array}\right)=\frac{\sqrt{2M}e^{\frac{\eta}{2}}}{\Gamma\left(\frac{1}{2}-i\nu\right)}\left(\frac{M}{2}\right)^{-i\nu}\left(\begin{array}{l}{K_{\frac{1}{2}-i\nu}\left(Me^{\eta}\right)}\\{K_{\frac{1}{2}+i\nu}\left(Me^{\eta}\right)}\\\end{array}\right),
S_{0}\;=\;\frac{1}{2}\int\,d^{2}\xi\,g^{\frac{1}{2}}\,[{\bar{\chi}}(\xi)\,\gamma^{a}{\calP}_{L}\,\partial_{a}\chi(\xi)\,-\,\partial_{a}{\bar{\chi}}(\xi)\,\gamma^{a}{\calP}_{L}\,\chi(\xi)]\;.
=\;e^{\frac{2}{N}\frac{\pi}{\pi+gN}\mathrm{K}_{0}(m_{d}|x-y|)}\left(\frac{1}{(x-y)^{2}}\right)^{\frac{N-1}{N}}\left(\frac{e^{2\gamma}}{4}\right)^{-\frac{N-1}{N}}\;.
W_{\mathrm{T}}^{(\xi=1)}=\frac{1}{4}\intd^{4}x\,\mathrm{tr}\,F_{\mu\nu}\tilde{F}^{\mu\nu}-\frac{1}{24}\epsilon^{\mu\nu\rho\sigma}\bar{Q}_{\mu}\bar{Q}_{\nu}\bar{Q}_{\rho}\bar{Q}_{\sigma}\intd^{4}x\,\mathrm{tr}\,\phi^{2},
{\tilde{H}}_{quark}={\psi^{\dagger}}({\bfr})\left[\,\betam-i\alpha_{i}\left(\,\partial_{i}-igA_{{\sfGI}\,i}^{a}({\bfr}){\textstyle\frac{\lambda^{\alpha}}{2}}\,\right)\,\right]\psi({\bfr})\,.
\nabla_{\mu}\Phi_{\alpha_{1}\ldots\alpha_{m}\dot{\alpha}_{1}\ldots\dot{\alpha}_{n}}=\frac{i}2mn(\sigma_{\mu})_{\alpha_{1}\dot{\alpha}_{1}}\Phi_{\alpha_{2}\ldots\alpha_{m}\dot{\alpha}_{2}\ldots\dot{\alpha}_{n}}-\frac{i}2(\sigma_{\mu})^{\beta\dot{\beta}}\Phi_{\beta\alpha_{1}\ldots\alpha_{m}\dot{\beta}\dot{\alpha}_{1}\ldots\dot{\alpha}_{n}},
e^{\textstyle-J\alpha_{-}\Phi(\sigma,\tau)}={\tilde{c}}_{J}a(\varpi)\sum_{M=-J}^{J}(-1)^{J-M}e^{ih(J-M)}\xi_{M}^{(J)}(x_{+})\,{\overline{\xi}_{-M}^{(J)}}(x_{-})\bigr/a(\varpi)
B_{\beta\alpha}^{ba(n)}=\Sigma_{m=0}^{n}\{\tilde{\theta}_{\beta}^{b(n-m)},\tilde{\theta}_{\alpha}^{a(m)}\}_{(q,p)}+\Sigma_{m=0}^{n}\{\tilde{\theta}_{\beta}^{b(n-m)},\tilde{\theta}_{\alpha}^{a(m+2)}\}_{(\phi)}~~;~~n\ge2