harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
ds^{2}=-4dx^{+}dx^{-}-\bigg[\sum_{a=1}^{4}x_{a}^{2}+(1+{\frac{\tilde{m}}{{\tilde{r}}^{4}}})\sum_{b=5}^{7}x_{b}^{2}+(1-3{\frac{\tilde{m}}{{\tilde{r}}^{4}}})x_{8}^{2}\bigg](d{\frac{\mux^{+}}{R^{2}}})^{2}+\sum_{i=1}^{8}dx_{i}^{2}.
\omega_{\muAB}=\frac{1}{2}[E_{A}^{\nu}(\partial_{\mu}e_{\nuB}-\partial_{\nu}e_{\muB})-E_{B}^{\nu}(\partial_{\mu}e_{\nuA}-\partial_{\nu}e_{\muA})-E_{A}^{\alpha}E_{B}^{\beta}(\partial_{\betaC}-\partial_{\beta}e_{\alphaC})e_{\mu}^{C}].
m_{\tilde{f}_{1/2,i}}^{2}=\frac{1}{2}\left(m_{\tilde{f}_{L,i}}^{2}+m_{\tilde{f}_{R,i}}^{2}\right)\mp\frac{1}{2}\sqrt{\left(m_{\tilde{f}_{L,i}}^{2}-m_{\tilde{f}_{R,i}}^{2}\right)^{2}+4\left|m_{\tilde{f}_{L/R}}^{2}\right|^{2}}\quad,
\intd{\bfk}\left[\,\Psi_{{\bfk}}^{(+)}({\bfr_{1}})\,\Psi_{{\bfk}}^{(+)\,\dagger}({\bfr_{2}})\,\tau_{3}-\Psi_{{\bfk}}^{(-)}({\bfr_{1}})\,\Psi_{{\bfk}}^{(-)\,\dagger}({\bfr_{2}})\,\tau_{3}\,\right]={\bfI}\,\,\delta({\bfr_{1}-r_{2}})
{\calA}_{\mu}^{\prime\prime}+\frac{\ddot{{\calA}}_{\mu}}{{\calL}^{2}}-\frac{2}{m_{H}^{2}~M^{2}}\Box{\calA}_{\mu}+(2H+F){\calA}_{\mu}^{\prime}+\biggl[2\frac{\nu}{\alpha}\frac{{P^{\prime}}^{2}}{{\calL}^{2}}-\alphaf^{2}\biggr]{\calA}_{\mu}=0.
E_{2}^{m_{2}}\;E_{1}^{m_{1}}\;E_{3}^{m_{3}}\;\cdots\;E_{M}^{m_{M}}\;=\;\omega^{m_{1}m_{2}\cdotsm_{M}}\;\cdot\;E_{1}^{m_{1}}\;E_{2}^{m_{2}}\cdotsE_{M}^{m_{M}},
g_{L,R}=\left(\begin{array}{cc}{1}&{0}\\{X_{L,R}^{-}}&{1}\\\end{array}\right)\left(\begin{array}{cc}{:e^{-\frac{u_{L,R}}{k-2}}:}&{0}\\{0}&{:e^{\frac{u_{L,R}}{k-2}}:}\\\end{array}\right)\left(\begin{array}{cc}{1}&{X_{L,R}^{+}}\\{0}&{1}\\\end{array}\right)
Q(\phi)[\phi^{\prime\prime}+\frac1s\phi^{\prime}]=2{\phi^{\prime}}^{2}(P(\epsilon,\phi)-\frac12\omega\kappae^{2\phi})-\frac{\lambda^{2}}{2s}e^{2\rho}(P(\epsilon,\phi)-\frac12\kappae^{2\phi})
{\frac{d{F}}{dt}}={\frac{\partial{F}}{\partialt}}+\left\{F,\H_{0}\right\}^{*}+\mu^{A}\left\{F,\\psi_{A}\right\}^{*}-\{F,\phi_{\alpha}\}C^{\alpha\beta}\frac{\partial\phi_{\beta}}{\partialt},
E_{M}\,=\,\frac{l}{4}\sum_{n=1}^{\infty}\left[\left(1\,+\,\frac{r^{2}k_{n1}^{2}}{l^{2}}\right)\omega_{n1}^{2}A_{n1}^{2}\,+\,\left(1\,+\,\frac{r^{2}k_{n2}^{2}}{l^{2}}\right)\omega_{n2}^{2}A_{n2}^{2}\right]
\begin{array}{ccc}{}&{\Gamma_{A}}&{H_{A}^{^{\prime}}}\\{1}&{\mathbb{Z}_{3}}&{SU(3)}\\{2}&{U(1)}&{U(2)}\\{3}&{U(1)\timesU(1)}&{U(1)\timesU(1)}\\{4}&{U(2)}&{U(1)}\\{5}&{SU(3)}&{\mathbb{Z}_{3}}\\\end{array}
T_{DD}(L,m,d,z)=\frac{g}{2L}\frac{1}{(2\pi)^{d-1}}\sum_{n=-\infty}^{\infty}\intd^{d-1}p\frac{1}{(\vec{p}^{\,2}+(\frac{n\pi}{L})^{2}+m^{2})}-gf_{2}(d,L,m,z)
\begin{array}{lll}{\frac{\deltaW}{\deltaA_{\mu}^{j}}}&{=}&{\langle\frac{\delta(S_{0}+S_{fix}+S_{ghost})}{\deltaA_{\mu}^{j}}\rangle+Y_{\nu}^{i}\langle\frac{\deltaj^{\nu,i}}{\deltaA_{\mu}^{j}}\rangle}\\{}&{=}&{D_{\nu}^{jk}[A]F^{\nu\mu,k}[A]-\langlej^{\mu,j}\rangle+Y_{\nu}^{k}\langle\frac{\deltaj^{\nu,k}}{\deltaA_{\mu}^{j}}\rangle.}\\\end{array}
M_{0}^{2}=\frac{\intd^{p-4}z\sqrt{g_{\mathrm{brane}}}e^{4A}}{\intd^{p-4}z\sqrt{g_{\mathrm{brane}}}e^{2A}}M_{H}^{2}\sime^{2A_{\mathrm{SM}}}M_{D}^{2}\
e^{iS_{eff}[g^{+},g^{-}]}=Ne^{i(S_{grav}[g^{+}]-S_{grav}[g^{-}])}\int{\calD}\phi^{+}{\calD}\phi^{-}e^{i(S_{matter}[g^{+},\phi^{+}]-S_{matter}[g^{-},\phi^{-}])},
\intdx^{4}\nabla_{\nu}\Theta^{\mu\nu}\theta(\varepsilon_{2},\rho,\varepsilon_{1})\theta(\tau_{2},\tau,\tau_{1})=\Phi_{1}(\varepsilon_{2})-\Phi_{1}(\varepsilon_{1})+\Phi_{2}(\varepsilon_{2})-\Phi_{2}(\varepsilon_{1})
E_{C}(R,T,m)=-4\piT^{2}R\mathop{{\sum}^{\prime}}_{n=0}^{n=\infty}\frac{n}{g(2\pinT)-1}\left(1-\frac{2mM_{0}^{2}}{R(4\pi^{2}n^{2}T^{2}m^{2}-M_{0}^{4})}\right),
S=\frac{2}{{\alpha^{\prime}}^{3}}\int\sqrt{-h}d^{3}\xi+\frac{4!4!}{{\alpha^{\prime}}^{3}}\intB_{MNP}(X)\partial_{I}X^{M}\partial_{J}X^{N}\partial_{K}X^{P}\epsilon^{IJK}d^{3}\xi\,,
L_{cl}^{\alpha}=e_{\underline{\alpha}}^{\alpha}(X,\theta)d\theta^{\underline{\alpha}}\,\qquadL_{cl}^{A}=e_{M}^{A}(X,\theta=0)dX^{M}+e_{\underline{\alpha}}^{A}(X,\theta)d\theta^{\underline{\alpha}}\,
{\int}d{\bfx}\,\left\{{\calF}^{ba}({\bfr},{\bfx}_{0})+X_{i}\partial_{i}{\calF}^{ba}({\bfr},{\bfx}_{0})+\frac{1}{2}\,X_{i}X_{j}\,\partial_{i}\partial_{j}{\calF}^{ba}({\bfr},{\bfx}_{0})\;+\;\;\cdots\right\}{j}_{0}^{a}({\bfx})
\tilde{S}=\int_{0}^{t}{dt^{\prime}}\left\{\frac{M_{o}}{2}(\dot{x}^{2}(t^{\prime})-\dot{y}^{2}(t^{\prime}))+(\dot{x}(t^{\prime})-\dot{y}(t^{\prime}))\int_{0}^{t}{dt"}\;\Gamma_{I}(t^{\prime}-t")(\dot{x}(t")+\dot{y}(t"))\right\}
S_{l-e}=\intd^{2}x\sqrt{-G}e^{-2\Phi}\big\{R+4(\nabla\Phi)^{2}+\alpha^{2}-{\frac{1}{2}}e^{\epsilon\Phi}F^{2}-{\frac{1}{2}}(\nablaT)^{2}+T^{2}\big\}.
\left<{\calO}\left[\vec{\xi}{\,}\right]\right>_{\vec{\xi}}=\frac{\int_{\vec{\xi}(0)=\vec{\xi}(1)}^{}{\calD}\vec{\xi}e^{-S}{\calO}\left[\vec{\xi}{\,}\right]}{\int_{\vec{\xi}(0)=\vec{\xi}(1)}^{}{\calD}\vec{\xi}e^{-S}}.
v_{\mu}^{(1)}=-\frac{1}{2\pi\kappa}H_{\mu}^{(2)}+G_{\mu}^{(1)}\;\;\;,\;\;\;v_{\mu}^{(2)}=-\frac{1}{2\pi\kappa}H_{\mu}^{(1)}+G_{\mu}^{(2)}\;\;,
\Gamma_{j\bar{\jmath}}\simc_{j_{0},j}c_{j_{0},\bar{\jmath}}+d_{j_{0},j}d_{j_{0},\bar{\jmath}}+\frac{1}{2}c_{j_{0},j}d_{j_{0},\bar{\jmath}}+\frac{1}{2}d_{j_{0},j}c_{j_{0},\bar{\jmath}},
\delta{\calL}_{\mathrm{anomaly}}\=\\frac{\theta_{i}}{32\pi^{2}}\\sum_{a}\,b_{a}^{\primei}\,F^{a}\widetilde{F}^{a}\-\\frac{\theta_{i}}{768\pi^{2}}\b_{grav}^{\primei}\,R\widetilde{R}\,
{\calL}_{\mathrm{{spinor}}}=\frac{e}{2}ie_{a}^{~\mu}\left(\overline{{\psi}}\gamma^{a}\partial_{\mu}\psi-\partial_{\mu}\overline{{\psi}}\gamma^{a}\psi+~\frac{i}{2}\overline{{\psi}}\omega_{\mubc}\epsilon^{abc}\psi\right),
\sigma^{1}=\left(\begin{array}{}{c]{cc}0}&{1}\\{1}&{0}\\\end{array}\right),\quad\sigma^{2}=\left(\begin{array}{}{c]{cc}0}&{-i}\\{i}&{0}\\\end{array}\right),\quad\sigma^{3}=\left(\begin{array}{}{c]{cc}1}&{0}\\{0}&{-1}\\\end{array}\right)
V(t)~=~\left\{\begin{array}{cc}{{\frac{1}{2}}\,w(t)^{2}}&{\mathrm{for~the~ordinary~(rational~or~trigonometric)~case}}\\{{\frac{1}{2}}\,\wp(t)}&{\mathrm{for~the~elliptic~case}}\\\end{array}\right\}~,
\left.\mathrm{Trln}{\calG}^{-1}\right\vert^{div}=\frac{iV_{4}}{(4\pir^{2})^{2}\epsilon}\left[\frac{106}{15}-\frac{257}{10}(\Lambdar^{2})+36(\Lambdar^{2})^{2}-\frac{189}{8}(\Lambdar^{2})^{3}\right].
L_{0}=\frac{M}{2}{\dot{q}}^{2}-V(q)+\sum_{\alpha=1}^{N}\{\frac{m_{\alpha}}{2}{\dot{x}_{\alpha}}^{2}-\frac{m_{\alpha}}{2}\omega_{\alpha}^{2}x_{\alpha}^{2}+{\epsilon}_{\alpha}x_{\alpha}q\},
\widehat{D}_{0}\pi_{a}+i\varsigma{\frac{\partialE_{a}}{\partial\phi_{i}}}\tau_{i}-i\overline{{p}}^{\prime}{\frac{\partial\overline{{J}}^{a}}{\partial\overline{{\phi}}_{i}}}\overline{{\tau}}_{i}+iE_{a}\beta-i\overline{{J}}^{a}\overline{{\gamma}}=0
T_{ij}=-\frac{gM}{m}\gamma^{3}\varepsilon_{ijk}S_{k}\left[\frac{3(\vec{a}\vec{n})^{2}}{(1-(\vec{n}\vec{v}))^{2}}-\frac{\omega^{2}(\vec{n}\vec{v})}{1-(\vec{n}\vec{v})}\right].
\begin{array}{llll}{\theta^{1}=e^{\sigma_{1}},}&{\theta^{2}=e^{\sigma_{2}},}&{\bar{\theta}^{1}=e^{\bar{\sigma}_{1}},}&{\bar{\theta}^{2}=e^{\bar{\sigma}_{2}},\nonumber}\\{p_{1}=e^{-\sigma_{1}},}&{p_{2}=e^{-\sigma_{2}},}&{\bar{p}_{1}=e^{-\bar{\sigma}_{1}},}&{\bar{p}_{2}=e^{-\bar{\sigma}_{2}},}\\\end{array}
\Psi_{N_{r},M}(z,\bar{z})=C_{N_{r},|M|}(\lambda|z|)^{|M|}\mathrm{e}^{-\frac{\lambda^{2}|z|^{2}}{2}}L_{N_{r}}^{|M|}(\lambda^{2}|z|^{2})\mathrm{e}^{iM\mathrm{arg}z}/{\sqrt2\pi}.
=\frac1{m!}\epsilon^{\mu_{1}\cdot\cdot\cdot\mu_{k}\mu_{k+1}\cdot\cdot\cdot\mu_{n}}\epsilon_{A_{1}\cdot\cdot\cdotA_{m}}J(\frac\phiv)\frac{\partialv^{A_{1}}}{\partialx^{\mu_{k+1}}}\cdot\cdot\cdot\frac{\partialv^{A_{1}}}{\partialx^{\mu_{n}}},
S_{R^{4}}=\intd^{10}x\sqrt{-g}\left[2\zeta(3)e^{-2\phi}(t_{8}t_{8}+\frac{1}{8}\epsilon_{10}\epsilon_{10})R^{4}+\frac{2\pi^{2}}{3}(t_{8}t_{8}+(-)^{\mu}\frac{1}{8}\epsilon_{10}\epsilon_{10})R^{4}\right]
GL(-i\Omega_{m},\stackrel{\rightharpoonup}{p})=G\frac{d_{Q}(R)}{2}\int\frac{d^{3}l}{(2\pi)^{3}}T\sum_{n}\frac{-\mathrm{tr}[\Gamma^{-}(m_{U}-\bar{\notl}_{U})\Gamma^{+}(m_{D}-\bar{\notl}_{D}-\bar{\not\!{p}})]}{[(\omega_{n}+i\mu_{U})^{2}+{\stackrel{\rightharpoonup}{l}}^{2}+m_{U}^{2}][(\omega_{n}+\Omega_{m}+i\mu_{D})^{2}+{(\stackrel{\rightharpoonup}{l}+\stackrel{\rightharpoonup}{p})}^{2}+m_{D}^{2}]}
C_{ik~pr}^{~jn~ms}\theta_{m}^{~p}t_{s}^{~r}\otimest_{n}^{~l}+D_{ir~k}^{~pj~n}\otimes\theta_{p}^{~r}t_{n}^{~l}+C_{pn~bd}^{~rl~ms}t_{i}^{~p}S(t_{r}^{~j})t_{k}^{~n}\otimes\theta_{m}^{~b}t_{s}^{~d}=
M_{\mathrm{D}2j}=\sqrt{2}\frac{\prod_{i=1}^{2j}R_{i}}{g_{s}\left(\alpha^{\prime}\right)^{\frac{2j+1}{2}}}\left(\prod_{i=1}^{l}\left[1+\left(2\pi\alpha^{\prime}B_{i}\right)^{2}\right]^{\frac{1}{2}}\right);
{\calG}_{a}\Gamma^{(0)}=\left({{\frac\delta{\delta\bar{c}_{a}}}}+\partial^{\mu}{\frac\delta{\deltaA_{a}^{*\mu}}}+\rho^{ai}\frac\delta{\delta{\phi}_{i}^{*}}\right)\Gamma^{(0)}=-\frac12\chi\,b^{a}-\eta^{ai}{\phi}_{i}~~.
e_{0}=\left(\begin{matrix}{\omega_{2n}^{b}}&{0}\\{0}&{\omega_{2n}^{b}}\\\end{matrix}\right),\\e_{1}=\left(\begin{matrix}{\omega_{2k}^{c}}&{0}\\{0}&{\omega_{2k}^{-c}}\\\end{matrix}\right),\\e_{2}=\left(\begin{matrix}{0}&{i^{c}}\\{i^{c}}&{0}\\\end{matrix}\right),
F=\vec{e}_{L}\cdot\vec{f}=\left(\begin{array}{cccc}{0}&{-f_{3}}&{f_{2}}&{f_{1}}\\{f_{3}}&{0}&{-f_{1}}&{f_{2}}\\{-f_{2}}&{-f_{1}}&{0}&{f_{3}}\\{-f_{1}}&{-f_{2}}&{-f_{3}}&{0}\\\end{array}\right)=\vec{e}\cdot\vec{f}
A(p)|_{p^{2}=4m^{2}}=\frac{1}{2|\stackrel{\rightharpoonup}{p}|}\left.f(-\sqrt{{\stackrel{\rightharpoonup}{l}}^{2}+m^{2}},p^{0},\mu)\right|_{|\stackrel{\rightharpoonup}{l}|=|\stackrel{\rightharpoonup}{p}|/2}.
K={\kappa}^{-2}{\hat{K}({M},{\bar{M}})}+Z_{{\bar{I}}J}{\Pi}^{\bar{I}}{\Pi}^{J}+(\frac{1}{2}H_{IJ}({\Pi},{\bar{\Pi}}){\Pi}^{I}{\Pi}^{J}+c.c)+higher\;order\;terms\;in\;\Pi,
F_{\mathrm{div}}^{C}(\beta,D)=-{\frac{\Gamma\left(1-\fracD2\right)}{(4\pi)^{D/2}}}\,\,{\frac{\pi^{2}m^{D-4}}{3\kappa\beta^{2}}}\int_{\Sigma}\left[m^{2}-\left({\frac{1}{60}}{\frac{4\pi^{2}}{\kappa^{2}\beta^{2}}}{\calP}+\left(\frac16-\xi\right)R\right)\right].
e^{-1}{\calL}_{B}=-\,\frac{1}{2}R-\frac{1}{2}\partial_{M}\phi\,\partial^{M}\phi-\frac{e^{-2\phi}}{12}\;G_{MNP}G^{MNP}-\frac{e^{-\phi}}{4}\;F_{MN}F^{MN}-{2g^{2}e^{\phi}},
\Pi=\langle0|T\left\{\mathrm{Tr}\,\lambda^{2}(x_{L},\theta),\mathrm{Tr}\,\lambda^{2}(x_{L}^{\prime},\theta^{\prime})\right\}|0\rangle=\langle0|\mathrm{Tr}\,\lambda^{2}|0\rangle^{2}\,.
\frac{\delta}{\deltaF(x,\theta,{\overline{\theta}})}\intd^{4}x^{\prime}d^{4}\theta^{\prime}\,F(x^{\prime},\theta^{\prime},{\overline{\theta}}^{\prime})G(x^{\prime},\theta^{\prime},{\overline{\theta}}^{\prime})=G(x,\theta,{\overline{\theta}})\,.
L^{\prime\prime}=L-\frac{1}{2}\chi^{\alpha}\gamma_{\alpha\beta}\chi^{\beta}-\bar{c}_{\alpha}s(q^{r})\chi_{;r}^{\alpha}+\dot{\bar{c}}_{\alpha}s(q^{r})\frac{\partial\chi^{\alpha}}{\partial\dot{q}^{r}}-\frac{d}{dt}\left(\bar{c}_{\alpha}s(q^{r})\frac{\partial\chi^{\alpha}}{\partial\dot{q}^{r}}\right),
\hat{X}^{\Lambda}={\frac{1}{\sqrt2}}\left(\begin{matrix}{1-TU}\\{-(T+U)}\\{-(1+TU)}\\{T-U}\\\end{matrix}\right),\qquad\hat{F}_{\Lambda}=\left(\begin{matrix}{S\hat{X}^{0}}\\{S\hat{X}^{1}}\\{-S\hat{X}^{2}}\\{-S\hat{X}^{3}}\\\end{matrix}\right)=S\eta_{\Lambda\Sigma}\hat{X}^{\Sigma}\,
\alpha_{i}~=~\left[\begin{array}{cc}{0}&{\sigma_{i}}\\{\sigma_{i}}&{0}\\\end{array}\right],~~~\beta~=~\left[\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right],~~~\Sigma_{i}~=~\left[\begin{array}{cc}{\sigma_{i}}&{0}\\{0}&{\sigma_{i}}\\\end{array}\right].
\begin{aligned}{V_{A}(\chi)}&{{}=\frac13\epsilon^{-2}e^{-2\chi}\left(X_{+}(\chi)\right)^{2}\left(3\alpha+2X_{+}(\chi)\right),}\\{V_{B}(\chi)}&{{}=\frac13\epsilon^{-2}e^{-2\chi}\left(X_{-}(\chi)\right)^{2}\left(3\alpha+2X_{-}(\chi)\right),}\\\end{aligned}
V(\eta_{3})=-2i(2\pi)^{-4}\intd^{4}\etaV(\eta)f^{-1}(\eta)\times(m_{q}^{2}+\eta^{2}-M_{B}^{2}/9)^{-1}(m_{q}^{2}+(\eta_{3}+\eta)^{2}-M_{B}^{2}/9)^{-1}
\frac{1}{A(\rho)}\frac{\partialA}{\partial\rho}\underset{(\rho\,\,\mathrm{large})}{\rightarrow}\mathrm{constant}_{1}\,,\quad\frac{1}{B(\rho)}\frac{\partialA}{\partial\rho}\underset{(\rho\,\,\mathrm{large})}{\rightarrow}0\,,\quad\frac{1}{B(\rho)}\frac{\partialB}{\partial\rho}\underset{(\rho\,\,\mathrm{large})}{\rightarrow}\mathrm{constant}_{2}\,,
\nu(\alpha)=\left(2b^{2Q\left(1/b-b\right)}C_{0}\right)^{-1/3}\kappa^{\alpha-\frac{Q}{3}}\left(\frac{\gamma\left(\frac{\alpha}{b}-\frac{Q}{2b}\right)}{\gamma\left(-\alphab+\frac{Qb}{2}\right)}\right)^{1/2},
(\alpha_{opt}^{2})^{od}=\sqrt{\frac{2\mid\beta\mid^{2}coth\mid\beta\mid^{2}+2\mid\beta\mid^{2}cos2\theta_{1}+1}{2\mid\beta\mid^{2}coth\mid\beta\mid^{2}-2\mid\beta\mid^{2}cos2\theta_{1}+1}}(\alpha_{opt}^{2})^{o}.
{\bfB}^{\mathrm{lin.}}=U{\bfB}^{\mathrm{circ.}}=B^{(0)}\left[\left(\begin{matrix}{1}\\{0}\\{0}\\\end{matrix}\right)e^{+i\phi}+\left(\begin{matrix}{0}\\{1}\\{0}\\\end{matrix}\right)e^{-i\phi}\right]\.
{\frac{\partial}{\partial\tau}}\langlep_{1}p_{2}p_{3}p_{4}\rangle\sim\intd^{4}x_{L0}d^{4}\theta_{0}\langle\mathrm{tr}\;W^{2}(0)(q^{+})^{p_{1}}(1)\ldots(q^{+})^{p_{4}}(4)\rangle\;.
\left(\begin{array}{c}{p_{R}}\\{p_{L}}\\\end{array}\right)=\left(\begin{array}{cc}{p_{R}^{i},}&{p_{R}^{I}}\\{p_{L}^{j},}&{p_{L}^{J}}\\\end{array}\right),\quadi,j=1,\ldots,d,\quadI,J=1,\ldots,16.
\pi(x^{\mu})=\int_{(\infty)}\frac{d^{3}\vec{k}}{(2\pi)^{3}2\omega(\vec{k}\,)}\left(-i\omega(\vec{k})\,\right)\left[a(\vec{k}\,)e^{-ik\cdotx}-a^{*}(\vec{k}\,)e^{ik\cdotx}\right]\.
2\pi(1-\alpha)\left(\sum_{p=1}^{k_{d}-1}\lambda_{p+1}\delta_{ii_{1}i_{2}...i_{2p-1}i_{2p}}^{jj_{1}j_{2}...j_{2p-1}j_{2p}}R_{\\\j_{1}j_{2}}^{i_{1}i_{2}}...R_{\\\j_{2p-1}j_{2p}}^{i_{2p-1}i_{2p}}+\lambda_{1}\delta_{i}^{j}\right)=\mu\delta_{i}^{j}~~~
|n_{1}^{\taup},n_{2}^{\taup}>\equiv\{\prod_{\tau=1}^{2^{m}-1}\prod_{p=1}^{2^{m}}(a_{1}^{\taup\dagger})^{n_{1}^{\taup}}(a_{2}^{\taup\dagger})^{n_{2}^{\taup}}\}|0>
R_{a}^{~b}\equivd\omega_{a}^{~b}+\omega_{a}^{~c}\omega_{c}^{~b}=u_{a}^{~\underline{c}}R_{\underline{c}}^{~\underline{d}}u_{\underline{d}}^{~b}-\hat{\Omega}_{a}^{~i}\hat{\Omega}_{i}^{~b},
\Pi_{\mu\nu}^{\Sigma}(p_{b})={\frac{\alpha}{\beta}}\sum_{n}\int{\frac{d^{2}k}{(2\pi)^{2}}}\mathrm{tr}\left(\gamma_{\mu}{\frac{1}{p_{b}\!\!\!\!\!/+k_{f}\!\!\!\!\!/+\Sigma(k_{f}+p_{b})}}\gamma_{\nu}{\frac{1}{k_{f}\!\!\!\!\!/+\Sigma(k_{f})}}\right).
|\Phi_{v}^{\nu=1/k}\rangle={\calN}\varepsilon^{\alpha_{1}\dots\alpha_{N}}({\bfA}{_{\alpha_{1}}^{+}})\Big({\bfA}_{\alpha_{2}}^{+}\Big)^{2}\dots\Big({\bfA}_{\alpha_{N}}^{+}\Big)^{N}\{\bfI}\;|v\rangle
{\calL}=\intdx\frac{dt}{2\piR^{\prime}}\frac{1}{2g}[\mathrm{~tr}\dot{Y}^{i}\dot{Y}^{i}+\mathrm{~tr~}\dot{A}^{1}\dot{A}^{1}+\frac{1}{2}\mathrm{~tr~}[Y^{i},Y^{j}]^{2}-\mathrm{~tr~}[\partial_{1}Y^{i}-i[A^{1},Y^{i}]]^{2},
{\calA}_{open}=2\,V\,(8\pi^{2}\alpha^{\prime})^{-1/2}\int_{0}^{\infty}{\frac{ds}{2s}}\,s^{-1/2}\left(\mathrm{Z}_{\mathrm{NS}+}^{I}(r)+\mathrm{Z}_{\mathrm{NS}-}^{I}(r)-\mathrm{Z}_{\mathrm{R}}^{I}(r)\right)
S_{WZ}^{NKK}=\mu_{6}\int_{R^{6+1}}\mathrm{Tr}\left(i_{\hat{k}}{\hat{N}}^{(8)}+l_{p}^{2}i_{\hat{k}}{\hat{\tilde{C}}}\wedge{\hat{\calF}}+l_{p}^{4}{\hat{C}}_{{\hat{\mu}}{\hat{\nu}}{\hat{\rho}}}D{\hat{X}}^{\hat{\mu}}D{\hat{X}}^{\hat{\nu}}D{\hat{X}}^{\hat{\rho}}\wedge{\hat{\calF}}\wedge{\hat{\calF}}+\dots\right)\,.
{\frac{1}{N}}\sum_{i}g(\psi_{i}^{\prime})^{2}=\intd^{m}\psi\intd^{m}\psi^{\prime}[g(\psi^{\prime})^{2}f(\psi,\psi^{\prime})]=\intd^{m}\psi[n_{b}v_{b}^{2}+2{\frac{U_{\psi}}{N}}],
{\textstyle\frac{1}{18}}(dxxx)^{2}\,g_{AE}\,g_{BF}\,g_{CG}\,\frac{\partial}{\partialx^{D}}C^{EFG}=-6g_{DF}\,E_{ABCE}^{F}\,x^{E}=6g_{E}\,E_{ABCD}^{E}\.
{\omega}_{\gamma_{i}}\sim{\frac{\rho^{3}R^{d_{i}}(\rhod^{4-d_{i}}x_{0}+x_{0}d\rho\,d^{3-d_{i}}x_{0})}{(\rho^{2}+x_{0}^{2})^{4}}},\quad{\Omega}_{\gamma_{i}}\sim{\frac{R^{d_{i}}(3\rho^{2}+x_{0}^{2})x_{0}d^{3-d_{i}}x_{0}}{(\rho^{2}+x_{0}^{2})^{3}}},\quad\mathrm{for}\,\,x_{0}\rightarrow\infty,
{\mu}_{IJ}B^{I}C^{J},\;{\mu}_{IJ}=m_{3/2}H_{IJ}-{\bar{F}}^{\bar{j}}{\bar{\partial}}_{\bar{j}}H_{IJ}+{\tilde{\mu}}_{IJ},\;H_{IJ}=\frac{1}{(T+{\bar{T}})(U+{\bar{U}})},
\begin{array}{rcl}{{\calH}}&{=}&{\frac{1}{2}\pi^{2}+\frac{1}{2}\left(\vec{\nabla}\phi\right)^{2}+\frac{1}{2}m^{2}\phi^{2}+\frac{1}{4}\lambda\phi^{4}}\\{}&{}&{}\\{}&{=}&{{\calH}_{0}\+\{\calH}_{\mathrm{int}}\\,\\{\calH}_{\mathrm{int}}=-{\calL}_{\mathrm{int}}=\frac{1}{4}\lambda\phi^{4}\.}\\\end{array}
Z=\sqrt{\pi}\int{\calD}G{\calD}\bar{A}{\calD}\Omega{\calD}\Lambda\,,exp\bigg(\int\{G_{a}^{\mu\nu}[-\alphaG_{\mu\nu}^{a}+F_{\mu\nu}^{a}(\bar{A})-2{\calD}_{\mu}^{(\bar{A})}\Lambda_{\nu}^{a}]-{M^{-1}}_{ab}^{\mu\nu}\Omega_{\mu}^{a}\Omega_{\nu}^{b}\}dx\bigg),
\gamma^{\mu}=\left(\begin{matrix}{{\widetilde{\gamma}}^{\mu}}&{0}\\{0}&{{\widehat{\gamma}}^{\mu}}\\\end{matrix}\right)=\left(\begin{matrix}{{\widetilde{\gamma}}^{\mu}}&{0}\\{0}&{-{\widetilde{\gamma}}^{\mu}}\\\end{matrix}\right)\qquad\psi(x)=\left(\begin{matrix}{\psi_{1}(x)}\\{\psi_{2}(x)}\\\end{matrix}\right)\.
W=\left(\zeta+a_{0}\frac{x_{1}^{4-n}z^{8-n}}{\zeta}\right)+x^{2}+z^{5}-y^{2}z+2(c_{0})^{1/2}y+c_{5}z^{4}+c_{4}z^{3}+c_{3}z^{2}+c_{2}z+c_{1}+\cdots.
\begin{array}{lclcll}{X^{\alpha\dot{\alpha}}}&{=}&{x^{\alpha\dot{\alpha}}+2{i}\theta_{i}^{\alpha}\bar{\theta}^{\dot{\alpha}i}}&{;}&{\theta_{i}^{\alpha}}&{\mathrm{chiral}}\\{\bar{X}^{\alpha\dot{\alpha}}}&{=}&{x^{\alpha\dot{\alpha}}-2{i}\theta_{i}^{\alpha}\bar{\theta}^{\dot{\alpha}i}}&{;}&{\bar{\theta}^{\dot{\alpha}i}}&{\mathrm{anti-chiral}}\\\end{array}
\sum_{m=0}^{\infty}\alpha_{m,-d}\cdotz^{m}=\sum_{m=0}^{\infty}c_{m}z^{m}+\sum_{m=1}^{\infty}\sum_{l=1}^{m}(\frac{d-l}{d})\cdot\alpha_{l,-d}\cdotc_{m-l}\cdotz^{m}
Z[T/e^{2};x_{1},\ldots,x_{K}]\;=\;\int[dA][dg]~\left<A\right|e^{-H/T}\left|A^{g}\right>\mathrm{Tr}~g^{R_{1}}(x_{1})\ldots\mathrm{Tr}~g^{R_{K}}(x_{K})~~,
F_{I_{1}\cdotsI_{n}}^{J_{1}\cdotsJ_{m}}\equiv{\frac{\delta^{n+m}F}{\delta\Phi^{I_{1}}\cdots\delta\Phi^{I_{n}}\delta\Phi_{J_{1}}\cdots\cdots\delta\Phi_{J_{m}}}}\equiv{\frac{\overrightarrow\delta^{n}}{\delta\Phi^{I_{1}}\cdots\delta\Phi^{I_{n}}}}F{\frac{\overleftarrow\delta^{m}}{\delta\Phi_{J_{1}}\cdots\cdots\delta\Phi_{J_{m}}}}
\begin{array}{rcl}{s\omega(s)}&{=}&{\displaystyle{\frac{1}{2(1-c)}}\Bigl[2s^{3}\omega(s)+s^{3}\omega(s)(\phi_{-}(s)-1)}\\{}&{}&{\qquad\qquad\displaystyle+2g(\omega(s)-\omega_{0}-s\omega_{1})+2gs\omega_{1}+2s(\phi_{-}(s)+1)\Bigr]\,}\\\end{array}
d\,L_{g}(A|_{h})=a^{M}(h)\frac{\partialx^{N}(gh)}{\partialx^{M}(h)}\frac{\partial}{\partialx^{N}(gh)}=a^{M}(h)L_{M}^{N}(gh,h)\frac{\partial}{\partialx^{N}(gh)}\,,
G^{\pm}(z)V_{s}^{\mp}(w)={\frac{{\frac{1}{2}}k^{2}W_{s}(w)}{(z-w)^{5/3}}}+{\frac{{\frac{1}{2}}\partialW_{s}(w)\pm\widetildeW_{s}(w)}{(z-w)^{2/3}}}+\ldots,
v_{f}^{i}(\sigma,t)=\sum_{n=1}^{\infty}W(n+\frac{\epsilon}{\tau})\Biggl[\frac{c_{n}^{i}}{\sqrt{n}}\,\dot{x}_{n}^{+}+\frac{\tilde{c}_{n}^{i}}{\sqrt{n}}\,\dot{x}_{n}^{-}+h.c.\Biggr]\,.
\Pi_{i_{1}\ldotsi_{p-2k-1}}=-\frac1{\triangle}\left(\partial^{i}\bar{\gamma}_{ii_{1}\cdotsi_{p-2k-1}}^{(2)}+\frac1{p-2k-1}\partial_{\left[i_{1}\right.}^{}\bar{\gamma}_{\left.i_{2}\cdotsi_{p-2k-1}\right]}^{(2)}\right).
Y^{[i]}=\int_{w_{-}}^{w_{+}}dwe^{-3A(w)}\int_{w_{-}}^{w}dw^{\prime}e^{3A(w^{\prime})}\int_{w_{-}}^{w^{\prime}}dw^{\prime\prime}e^{-A(w^{\prime\prime})}F_{(T)}^{[i]}(w^{\prime\prime}),
R(t)=\Big|e^{{\frac{-\gamma_{1}}{2}}t}e^{-im_{1}t}{\frac{b}{\sqrt{2}}}+\int_{0}^{-\infty_{II}}dE{\frac{<\pi\pi|H_{w}|E,cp=+1^{\enspace-}>}{<\pi\pi|H_{w}|K_{1}^{\enspace-}>}}{\frac{b(E)}{\sqrt{2}}}e^{-iEt}\Big|^{2},
i\Pi_{odd}^{\mu\nu}(p)=-\frac{mg^{2}}{2\pi^{\frac{3}{2}}}p^{\alpha}\epsilon^{\mu\nu\alpha}\intdx[\frac{\sqrt{\pi}}{M}-\sqrt{\frac{2\tilde{p}}{M}}~{\mathit{K}}_{\frac{1}{2}}(\tilde{p}~M)]~~~,
F_{\mathrm{M2}}=-\frac{2^{7/2}\pi^{2}}{3^{4}}N^{3/2}V_{2}T^{3}\left[1+\frac{9}{8\pi^{2}}\sum_{i=1}^{4}\omega_{i}^{2}-\frac{27}{128\pi^{4}}\left(\sum_{i=1}^{4}\omega_{i}^{2}\right)^{2}+\frac{81}{64\pi^{4}}\sum_{i=1}^{4}\omega_{i}^{4}+\ldots\right]
G_{\mu\nu}^{11}=e^{-\frac{2}{3}\phi}G_{\mu\nu}+e^{\frac{4}{3}\phi}(-\partial_{\mu}B+e^{-\phi}C_{\mu})(-\partial_{\nu}B+e^{-\phi}C_{\nu})=e^{-\frac{2}{3}\phi}G_{\mu\nu}^{\prime}.
S_{\mathrm{graviton}}^{\mathrm{red}}[g_{\mu\nu}]=\frac1{8\piG_{4}}\intd^{4}x\,\sqrtg\left(R_{\mu\nu}\frac{F_{\mathrm{red}}(\Box)}{l^{2}\,\Box^{2}}\,R^{\mu\nu}-\frac13\,R\frac{F_{\mathrm{red}}(\Box)}{l^{2}\,\Box^{2}}\,R\right)\.
\alpha\frac{\partial}{\partial\alpha}\left(\left.V_{1}\right|_{\varphi_{0}}+\left.\frac{\partialV_{0}}{\partial\varphi}\right|_{\varphi_{0}}\varphi_{1}\right)=C_{1}\left.\frac{\partialV_{0}}{\partial\varphi}\right|_{\varphi_{0}}
I=\intd^{4}xd^{4}yJ^{\mua}(x)\check{G}_{\mu}^{ab}(x-y)\eta^{b}(y)+\frac12\intd^{4}xd^{4}yA^{\mua}(x)\check{\Gamma}_{\mu\nu}^{ab}(x-y)A^{\nub}(y)+\cdots
\langlex_{a},{\bfn}_{i}|m_{1},m_{2},m_{3},m_{4}\rangle=\sqrt{\frac{p!}{m_{1}!m_{2}!m_{3}!m_{4}!}}\Psi_{1}^{m_{1}}\Psi_{2}^{m_{2}}\Psi_{3}^{m_{3}}\Psi_{4}^{m_{4}},
W(x)=\sqrt{\frac{\pi\lambda}{2}}\frac{i^{l+2}p(x)}{N_{1}(\lambda,l)}\left\{H_{l+2}^{(1)}(\lambda\,p(x))\sum_{n=0}^{\infty}A_{n}(x)\lambda^{-2n}-\frac{p(x)}{\lambda}H_{l+3}^{(1)}(\lambda\,p(x))\sum_{n=0}^{\infty}B_{n}(x)\lambda^{-2n}\right\}\,,
\begin{array}{ll}{p\lambda_{\Lambda}=0,\qquad}&{\Lambda\lambda_{\Lambda}=0,}\\{p\lambda_{A}=0,\qquad}&{p\lambda_{B}=0,}\\{\lambda_{\omega}=0,\qquad}&{\lambda_{\Phi}=0,}\\{\lambda_{\omega_{1}}=0,\qquad}&{\lambda_{\omega_{2}}=0,}\\\end{array}
\begin{array}{lll}{A\alpha_{1}=\alpha_{6},\qquad}&{A\alpha_{2}=\alpha_{2},\qquad}&{A\alpha_{3}=\alpha_{5},}\\{A\alpha_{4}=\alpha_{4},}&{A\alpha_{5}=\alpha_{3},}&{A\alpha_{6}=\alpha_{1},}\\{A\alpha_{7}=\alpha_{0},}&{A\alpha_{0}=\alpha_{7}.}&{}\\\end{array}
\left(\tilde{R}_{MN}-\frac{1}{2}G_{MN}\tilde{R}\right)+\frac{\Lambda}{M_{5}^{3}}G_{MN}+\lambda\left[\left(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\right)+\frac{v_{b}}{M_{4}^{2}}g_{\mu\nu}\right]\delta_{M}^{\mu}\delta_{N}^{\nu}\delta(y)=0
\left[-\omega^{2}\gamma^{tt}(r)f(r)-{\frac{l(l+n)}{r^{2}}}{\frac{f(r)}{h(r)}}+{\frac{\sqrt{-\gamma^{tt}f(r)}}{U(r)}}{\frac{d}{dr}}{\frac{U(r)}{\sqrt{-\gamma^{tt}f(r)}}}{\frac{d}{dr}}\right]\phi_{\omega}^{l}(r)=0.
\langle\bar{\psi}\psi\rangle_{T\mu}~\simeq~-~\frac{2\verteH\vert}{\pi^{2}}T\sum_{l}\int_{0}^{\sqrt{\verteH\vert}}dp_{3}\frac{m(T,\mu)}{[(2l+1)\piT+i\mu]^{2}+p_{3}^{2}+m^{2}(T,\mu)}_{,}
\mathrm{ind}_{0}^{D=4}(\mathrm{SO}(2N))=\frac{2}{2^{N-1}N!}~\frac{1}{E^{N}}~\oint\prod_{i=1}^{N}\frac{dx_{i}}{2\pii}~\prod_{i<j}^{N}\frac{(x_{i}^{2}-x_{j}^{2})^{2}}{\Big[(x_{i}-x_{j})^{2}-E^{2}\Big]\Big[(x_{i}+x_{j})^{2}-E^{2}\Big]}
(D_{\mu})_{i}{}^{j}=D_{\mu}^{G}\delta_{i}{}^{j}\pm\frac14\varepsilon^{abc}(S_{\mu}^{-}-P_{\mu})_{ab}(\tau^{c})_{i}{}^{j}+e^{\varphi}Q_{\mu}^{a+}(\tau^{a})_{i}{}^{j}
\mathrm{Inh}_{1}^{2}=A_{1}^{\bar{z}}\partial_{z}A_{1}^{z}-A_{1}^{z}\partial_{z}A_{1}^{\bar{z}}\quad\mathrm{and}\quad\mathrm{Inh}_{2}^{2}=-A_{1}^{\bar{z}}\partial_{\bar{z}}A_{1}^{z}+A_{1}^{z}\partial_{\bar{z}}A_{1}^{\bar{z}}
I({\hat{k}};M_{1},M_{2})\;=\;\int\frac{d^{2}{\hat{p}}}{(2\pi)^{2}}\,\frac{M_{2}}{[({\hat{k}}-{\hat{p}})^{2}+M_{1}^{2}]({\hat{p}}^{2}+M_{2}^{2})}
\begin{aligned}{\big(\rho(t)\,v\big)^{2}}&{{}\ll1\,,}&{}&{{}\quad\quad\big(\dot{\rho}(t)\big)^{2}}&{\ll1\,,}\\{\big(\rho(t)\,\dot{\Theta}(t)\big)^{2}}&{{}\ll1\,,}&{}&{{}\quad\quad\big(\dot{\rho}(t)\big)^{2}}&{\ll1\,,}\\{\big(\rho(t)\,\dot{\phi}(t)\big)^{2}}&{{}\ll1\,,}&{}&{{}\quad\quad(\rho^{2}/|\tau|^{2})^{2}}&{\ll1\,.}\\\end{aligned}
V_{i}\cdot{\calN}_{\overline{{\alphaV}}}+f_{i}({\overline{{\alphaV}}},{\bfn})+{\frac{\alpha^{*}t^{*}}{2t_{i}}}{\vec{Q}}^{2}({\vec{P}}_{\overline{{\alphaV}}})=\sum_{j}k_{ij}\alpha_{j}+s_{i}-V_{i}\cdot({\overline{{\alphaV}}}-W({\overline{{\alphaV}}}))~(\mathrm{mod}~1)~.
L({\bfx},{\bfv},t)=-mc^{2}(1-\frac{1}{c^{2}}|{\bfv}|^{2})^{1/2}+\frac{e}{c}\sum_{i=1}^{3}A_{i}({\bfx},t)v^{i}+eA_{0}({\bfx},t),
I_{\mathrm{IIB}}=\frac{1}{2}\intd^{6}x{\sqrt{-{\hat{g}}}}\biggl[-\hat{R}+\frac{1}{8}\mathrm{Tr}\bigl(\partial_{\hat{\mu}}{\hat{\calM}}\partial^{\hat{\mu}}{\hat{\calM}}^{-1}\bigr)+\frac{3}{8}(d\hat{B})_{\hat{\mu}\hat{\nu}\hat{\rho}}^{i}{\hat{\calM}}_{ij}^{-1}(d\hat{B})^{\hat{\mu}\hat{\nu}\hat{\rho}j}\biggr]\,.
\hat{G}[\vec{r}_{1},\vec{r}_{2};E]=\hat{G}_{V}[\vec{r}_{1},\vec{r}_{2};E]-\frac{\hat{G}_{V}[\vec{r}_{1},\vec{0};E]\hat{G}_{V}[\vec{0},\vec{r}_{2};E]}{\frac{1}{\hat{v}(E)}+\hat{G}_{V}[\vec{0},\vec{0};E]}.
(\Psi\star\Phi)[\{l_{2k+1}\};\{r_{2k+1}\}]=\int\;\prod_{k=0}^{\infty}ds_{2k+1}\;\Psi[\{l_{2k+1}\};\{s_{2k+1}\}]\;\Phi[\{s_{2k+1}\};\{r_{2k+1}\}]\,.
\rho_{0}(q,q^{\prime})=\intdxdx^{\prime}e^{i(qx-q^{\prime}x^{\prime})}\rho(x,x^{\prime})=\intdx^{+}dx^{-}e^{i(q^{-}x^{+}+q^{+}x^{-})}\rho(x,x^{\prime})
\tilde{G}_{\mu\nu}=\left(\begin{array}{ll}{-g+\xi^{i}\left(\omega\right)g_{ij}\xi^{j}\left(\omega\right)}&{\xi^{k}\left(\omega\right)g_{kj}}\\{\xi^{k}\left(\omega\right)g_{ki}}&{g_{ij}}\\\end{array}\right)